Subgroups — C₆₇×#34286 #30713537

Order 17152.

2062 conjugacy classes of subgroups.

The subgroup lattice has 922 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#541×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2373	C₂	1	2	yes	maximal
3	C₆₇×#2373	C₂	1	2	yes	maximal
4	C₆₇×#2371	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2370	C₂	1	1	yes	maximal, characteristic, complemented
6	C₆₇×#2370	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#2370	C₂	1	2	yes	maximal, complemented
8	C₆₇×#2365	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#2365	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#2362	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#2362	C₂	1	1	yes	maximal, characteristic
12	C₆₇×#2360	C₂	1	1	yes	maximal, characteristic
13	C₆₇×#2360	C₂	1	1	yes	maximal, characteristic
14	C₆₇×#2336	C₂	1	2	yes	maximal, complemented
15	C₆₇×#2334	C₂	1	2	yes	maximal
16	C₆₇×#2320	C₂	1	1	yes	maximal, characteristic, complemented
17	C₄×#125×C₆₇	C₂	1	2	yes	maximal, complemented
18	C₄×#126×C₆₇	C₂	1	2	yes	maximal
19	C₆₇×#2298	C₂	1	1	yes	maximal, characteristic
20	C₆₇×#2295	C₂	1	1	yes	maximal, characteristic, complemented
21	C₆₇×#2276	C₂	1	1	yes	maximal, characteristic, complemented
22	C₆₇×#2274	C₂	1	1	yes	maximal, characteristic
23	C₄²×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
24	#541×C₆₇	C₄	1	16	yes	complemented
25	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
26	C₂×#126×C₆₇	C₂²	1	2	yes	—
27	C₂×#126×C₆₇	C₂²	1	2	yes	—
28	C₂×#125×C₆₇	C₂²	1	2	yes	—
29	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×#125×C₆₇	C₂²	1	2	yes	—
31	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
33	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
35	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
36	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
37	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
38	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
39	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
40	C₂×#117×C₆₇	C₂²	1	1	yes	characteristic
41	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
42	#391×C₆₇	C₂²	1	1	yes	characteristic
43	#391×C₆₇	C₂²	1	1	yes	characteristic
44	#391×C₆₇	C₂²	1	1	yes	characteristic
45	#391×C₆₇	C₂²	1	1	yes	characteristic
46	#389×C₆₇	C₂²	1	1	yes	characteristic
47	#389×C₆₇	C₂²	1	1	yes	characteristic
48	#389×C₆₇	C₂²	1	2	yes	—
49	#389×C₆₇	C₂²	1	2	yes	—
50	#389×C₆₇	C₂²	1	1	yes	characteristic
51	#389×C₆₇	C₂²	1	1	yes	characteristic
52	#389×C₆₇	C₂²	1	2	yes	—
53	#389×C₆₇	C₂²	1	2	yes	—
54	#389×C₆₇	C₂²	1	2	yes	—
55	#389×C₆₇	C₂²	1	2	yes	—
56	#388×C₆₇	C₂²	1	1	yes	characteristic
57	#388×C₆₇	C₂²	1	1	yes	characteristic
58	#388×C₆₇	C₂²	1	1	yes	characteristic
59	#388×C₆₇	C₂²	1	1	yes	characteristic
60	#388×C₆₇	C₂²	1	2	yes	—
61	#388×C₆₇	C₂²	1	2	yes	—
62	#388×C₆₇	C₂²	1	2	yes	—
63	#388×C₆₇	C₂²	1	2	yes	—
64	#387×C₆₇	C₂²	1	2	yes	—
65	#387×C₆₇	C₂²	1	2	yes	—
66	#387×C₆₇	C₂²	1	1	yes	characteristic
67	#387×C₆₇	C₂²	1	1	yes	characteristic
68	#387×C₆₇	C₂²	1	1	yes	characteristic
69	#387×C₆₇	C₂²	1	1	yes	characteristic
70	#387×C₆₇	C₂²	1	2	yes	—
71	#387×C₆₇	C₂²	1	2	yes	—
72	#387×C₆₇	C₂²	1	2	yes	—
73	#387×C₆₇	C₂²	1	2	yes	—
74	#387×C₆₇	C₂²	1	2	yes	—
75	#387×C₆₇	C₂²	1	2	yes	—
76	#387×C₆₇	C₂²	1	2	yes	—
77	#387×C₆₇	C₂²	1	2	yes	—
78	#387×C₆₇	C₂²	1	2	yes	—
79	#387×C₆₇	C₂²	1	2	yes	—
80	#387×C₆₇	C₂²	1	2	yes	—
81	#387×C₆₇	C₂²	1	2	yes	—
82	#387×C₆₇	C₂²	1	2	yes	—
83	#387×C₆₇	C₂²	1	2	yes	—
84	#385×C₆₇	C₂²	1	2	yes	—
85	#385×C₆₇	C₂²	1	2	yes	—
86	#383×C₆₇	C₂²	1	2	yes	—
87	#382×C₆₇	C₂²	1	2	yes	—
88	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
89	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
90	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
91	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
92	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
93	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
94	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
95	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
96	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
97	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
98	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
99	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
100	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
101	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
102	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
103	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
104	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
105	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
106	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
107	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
108	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
109	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
110	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
111	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
112	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
113	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
114	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
115	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
116	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
117	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
118	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
119	#376×C₆₇	C₂²	1	1	yes	characteristic
120	#376×C₆₇	C₂²	1	2	yes	—
121	#376×C₆₇	C₂²	1	2	yes	—
122	C₄³×C₆₇	C₂²	1	1	yes	characteristic
123	C₄³×C₆₇	C₂²	1	2	yes	—
124	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
125	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
126	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
127	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
128	C₄⋊Q₈×C₆₇	C₂×C₄	1	8	yes	complemented
129	#126×C₆₇	C₂×C₄	1	16	yes	—
130	#126×C₆₇	C₂×C₄	1	16	yes	—
131	#125×C₆₇	C₂×C₄	1	8	yes	complemented
132	#125×C₆₇	C₂×C₄	1	16	yes	complemented
133	#125×C₆₇	C₂×C₄	1	16	yes	complemented
134	C₂²⋊Q₈×C₆₇	C₂×C₄	1	8	yes	—
135	C₂²⋊Q₈×C₆₇	C₂×C₄	1	8	yes	—
136	C₄×Q₈×C₆₇	—	2	4	no	—
137	C₄×Q₈×C₆₇	—	2	4	no	—
138	C₄×Q₈×C₆₇	—	2	4	no	—
139	C₄×Q₈×C₆₇	—	2	4	no	—
140	C₄×Q₈×C₆₇	C₂×C₄	1	8	yes	complemented
141	C₄×Q₈×C₆₇	C₂×C₄	1	8	yes	complemented
142	#117×C₆₇	—	2	4	no	—
143	#117×C₆₇	C₂×C₄	1	8	yes	—
144	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
145	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
146	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
147	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
148	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
149	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
150	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
151	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
152	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
153	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
154	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
155	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
156	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
157	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
158	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
159	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
160	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
161	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
162	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
163	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
164	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
165	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
166	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
167	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
168	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
169	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
170	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
171	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
172	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
173	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
174	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
175	C₂×C₂²⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
176	C₂×C₂²⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
177	C₂×C₂²⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
178	C₂×C₂²⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
179	C₂×C₂²⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
180	C₂×C₂²⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
181	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
182	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
183	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
184	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
185	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
186	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
187	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
188	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
189	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
190	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
191	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
192	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
193	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
194	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
195	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
196	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
197	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
198	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
199	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
200	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
201	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
202	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
203	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
204	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
205	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
206	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
207	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
208	#95×C₆₇	C₂³	1	1	yes	characteristic
209	#95×C₆₇	C₂³	1	2	yes	—
210	#95×C₆₇	C₂³	1	2	yes	—
211	#95×C₆₇	C₂³	1	1	yes	characteristic
212	#95×C₆₇	C₂³	1	1	yes	characteristic
213	#95×C₆₇	C₂³	1	2	yes	—
214	#95×C₆₇	C₂³	1	1	yes	characteristic
215	#95×C₆₇	C₂³	1	2	yes	—
216	#95×C₆₇	C₂³	1	2	yes	—
217	#95×C₆₇	C₂³	1	2	yes	—
218	#95×C₆₇	C₂³	1	2	yes	—
219	#95×C₆₇	C₂³	1	2	yes	—
220	#95×C₆₇	C₂³	1	1	yes	characteristic
221	#95×C₆₇	C₂³	1	2	yes	—
222	#95×C₆₇	C₂³	1	2	yes	—
223	#95×C₆₇	C₂³	1	1	yes	characteristic
224	#95×C₆₇	C₂³	1	2	yes	—
225	#95×C₆₇	C₂³	1	2	yes	—
226	#95×C₆₇	C₂³	1	2	yes	—
227	#95×C₆₇	C₂³	1	1	yes	characteristic
228	#95×C₆₇	C₂³	1	1	yes	characteristic
229	#95×C₆₇	C₂³	1	2	yes	—
230	#95×C₆₇	C₂³	1	1	yes	characteristic
231	#95×C₆₇	C₂³	1	1	yes	characteristic
232	#95×C₆₇	C₂³	1	2	yes	—
233	#95×C₆₇	C₂³	1	2	yes	—
234	#95×C₆₇	C₂³	1	2	yes	—
235	#95×C₆₇	C₂³	1	2	yes	—
236	#95×C₆₇	C₂³	1	2	yes	—
237	#95×C₆₇	C₂³	1	1	yes	characteristic
238	#95×C₆₇	C₂³	1	1	yes	characteristic
239	#95×C₆₇	C₂³	1	2	yes	—
240	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
241	C₂×Q₈×C₆₇	—	2	2	no	—
242	C₂×Q₈×C₆₇	—	2	2	no	—
243	C₂×Q₈×C₆₇	—	2	2	no	—
244	C₂×Q₈×C₆₇	—	2	2	no	—
245	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
246	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
247	C₂²×C₄×C₆₇	—	2	2	no	—
248	C₂²×C₄×C₆₇	—	2	2	no	—
249	C₂²×C₄×C₆₇	—	2	4	no	—
250	C₂²×C₄×C₆₇	—	4	2	no	—
251	C₂²×C₄×C₆₇	—	4	2	no	—
252	C₂²×C₄×C₆₇	C₂²×C₄	1	4	yes	—
253	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
254	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
255	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
256	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
257	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
258	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
259	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
260	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
261	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
262	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
263	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
264	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
265	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
266	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
267	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
268	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
269	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
270	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
271	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
272	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
273	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
274	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
275	C₄⋊C₄×C₆₇	—	2	4	no	—
276	C₄⋊C₄×C₆₇	—	2	2	no	—
277	C₄⋊C₄×C₆₇	—	2	2	no	—
278	C₄⋊C₄×C₆₇	—	2	4	no	—
279	C₄⋊C₄×C₆₇	—	2	4	no	—
280	C₄⋊C₄×C₆₇	—	2	4	no	—
281	C₄⋊C₄×C₆₇	—	2	4	no	—
282	C₄⋊C₄×C₆₇	—	2	2	no	—
283	C₄⋊C₄×C₆₇	—	2	4	no	—
284	C₄⋊C₄×C₆₇	—	2	2	no	—
285	C₄⋊C₄×C₆₇	—	4	1	no	—
286	C₄⋊C₄×C₆₇	—	4	1	no	—
287	C₄⋊C₄×C₆₇	—	4	2	no	—
288	C₄⋊C₄×C₆₇	—	4	2	no	—
289	C₄⋊C₄×C₆₇	—	4	2	no	—
290	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
291	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
292	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
293	C₄⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
294	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
295	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
296	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
297	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
298	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
299	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
300	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
301	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
302	C₄⋊C₄×C₆₇	#41	1	4	yes	—
303	C₄⋊C₄×C₆₇	#41	1	4	yes	—
304	C₂²⋊C₄×C₆₇	—	2	4	no	—
305	C₂²⋊C₄×C₆₇	—	2	2	no	—
306	C₂²⋊C₄×C₆₇	—	2	2	no	—
307	C₂²⋊C₄×C₆₇	—	4	1	no	—
308	C₂²⋊C₄×C₆₇	—	4	1	no	—
309	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
310	C₂²⋊C₄×C₆₇	C₂²×C₄	1	4	yes	—
311	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
312	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
313	C₄²×C₆₇	—	2	4	no	—
314	C₄²×C₆₇	—	2	4	no	—
315	C₄²×C₆₇	—	2	4	no	—
316	C₄²×C₆₇	—	2	2	no	—
317	C₄²×C₆₇	—	2	2	no	—
318	C₄²×C₆₇	—	2	4	no	—
319	C₄²×C₆₇	—	2	4	no	—
320	C₄²×C₆₇	—	2	2	no	—
321	C₄²×C₆₇	—	2	2	no	—
322	C₄²×C₆₇	—	2	4	no	—
323	C₄²×C₆₇	—	2	2	no	—
324	C₄²×C₆₇	—	2	2	no	—
325	C₄²×C₆₇	—	4	2	no	—
326	C₄²×C₆₇	—	4	1	no	—
327	C₄²×C₆₇	—	4	2	no	—
328	C₄²×C₆₇	—	4	2	no	—
329	C₄²×C₆₇	—	4	2	no	—
330	C₄²×C₆₇	—	4	2	no	—
331	C₄²×C₆₇	—	4	1	no	—
332	C₄²×C₆₇	—	4	2	no	—
333	C₄²×C₆₇	—	4	2	no	—
334	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
335	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
336	C₄²×C₆₇	C₂²×C₄	1	8	yes	—
337	C₄²×C₆₇	C₂²×C₄	1	8	yes	—
338	C₂³×C₆₇	—	2	1	no	—
339	C₂³×C₆₇	—	2	1	no	—
340	C₂³×C₆₇	—	2	1	no	—
341	C₂³×C₆₇	—	2	1	no	—
342	C₂³×C₆₇	—	2	1	no	—
343	C₂³×C₆₇	—	2	1	no	—
344	C₂³×C₆₇	C₂³×C₄	1	2	yes	—
345	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
346	Q₈×C₆₇	—	2	8	no	—
347	Q₈×C₆₇	—	2	8	no	—
348	C₂×C₄×C₆₇	—	2	1	no	—
349	C₂×C₄×C₆₇	—	2	1	no	—
350	C₂×C₄×C₆₇	—	2	1	no	—
351	C₂×C₄×C₆₇	—	2	2	no	—
352	C₂×C₄×C₆₇	—	2	2	no	—
353	C₂×C₄×C₆₇	—	2	2	no	—
354	C₂×C₄×C₆₇	—	2	2	no	—
355	C₂×C₄×C₆₇	—	2	1	no	—
356	C₂×C₄×C₆₇	—	2	1	no	—
357	C₂×C₄×C₆₇	—	2	2	no	—
358	C₂×C₄×C₆₇	—	2	1	no	—
359	C₂×C₄×C₆₇	—	2	2	no	—
360	C₂×C₄×C₆₇	—	2	1	no	—
361	C₂×C₄×C₆₇	—	2	1	no	—
362	C₂×C₄×C₆₇	—	2	1	no	—
363	C₂×C₄×C₆₇	—	2	2	no	—
364	C₂×C₄×C₆₇	—	2	1	no	—
365	C₂×C₄×C₆₇	—	2	1	no	—
366	C₂×C₄×C₆₇	—	2	2	no	—
367	C₂×C₄×C₆₇	—	2	1	no	—
368	C₂×C₄×C₆₇	—	2	2	no	—
369	C₂×C₄×C₆₇	—	2	1	no	—
370	C₂×C₄×C₆₇	—	2	1	no	—
371	C₂×C₄×C₆₇	—	2	2	no	—
372	C₂×C₄×C₆₇	—	2	2	no	—
373	C₂×C₄×C₆₇	—	2	2	no	—
374	C₂×C₄×C₆₇	—	2	8	no	—
375	C₂×C₄×C₆₇	—	2	1	no	—
376	C₂×C₄×C₆₇	—	2	1	no	—
377	C₂×C₄×C₆₇	—	2	2	no	—
378	C₂×C₄×C₆₇	—	2	1	no	—
379	C₂×C₄×C₆₇	—	2	1	no	—
380	C₂×C₄×C₆₇	—	2	1	no	—
381	C₂×C₄×C₆₇	—	2	1	no	—
382	C₂×C₄×C₆₇	—	2	2	no	—
383	C₂×C₄×C₆₇	—	2	1	no	—
384	C₂×C₄×C₆₇	—	2	1	no	—
385	C₂×C₄×C₆₇	—	2	1	no	—
386	C₂×C₄×C₆₇	—	2	1	no	—
387	C₂×C₄×C₆₇	—	2	2	no	—
388	C₂×C₄×C₆₇	—	2	1	no	—
389	C₂×C₄×C₆₇	—	2	1	no	—
390	C₂×C₄×C₆₇	—	2	1	no	—
391	C₂×C₄×C₆₇	—	2	2	no	—
392	C₂×C₄×C₆₇	—	2	1	no	—
393	C₂×C₄×C₆₇	—	2	1	no	—
394	C₂×C₄×C₆₇	—	2	1	no	—
395	C₂×C₄×C₆₇	—	2	2	no	—
396	C₂×C₄×C₆₇	—	2	2	no	—
397	C₂×C₄×C₆₇	—	2	1	no	—
398	C₂×C₄×C₆₇	—	2	1	no	—
399	C₂×C₄×C₆₇	—	8	4	no	—
400	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
401	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
402	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
403	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
404	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
405	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
406	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
407	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
408	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
409	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
410	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
411	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
412	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
413	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
414	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
415	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
416	C₂²×C₆₇	—	2	2	no	—
417	C₂²×C₆₇	—	2	2	no	—
418	C₂²×C₆₇	—	2	2	no	—
419	C₂²×C₆₇	—	4	1	no	—
420	C₂²×C₆₇	—	4	1	no	—
421	C₂²×C₆₇	—	4	1	no	—
422	C₂²×C₆₇	—	4	1	no	—
423	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
424	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
425	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
426	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
427	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
428	C₂²×C₆₇	#585	1	1	yes	characteristic
429	C₂²×C₆₇	#585	1	1	yes	characteristic
430	C₄×C₆₇	—	2	2	no	—
431	C₄×C₆₇	—	2	4	no	—
432	C₄×C₆₇	—	2	4	no	—
433	C₄×C₆₇	—	2	2	no	—
434	C₄×C₆₇	—	2	2	no	—
435	C₄×C₆₇	—	2	4	no	—
436	C₄×C₆₇	—	2	2	no	—
437	C₄×C₆₇	—	2	2	no	—
438	C₄×C₆₇	—	2	4	no	—
439	C₄×C₆₇	—	2	4	no	—
440	C₄×C₆₇	—	2	2	no	—
441	C₄×C₆₇	—	2	4	no	—
442	C₄×C₆₇	—	2	2	no	—
443	C₄×C₆₇	—	4	1	no	—
444	C₄×C₆₇	—	4	1	no	—
445	C₄×C₆₇	—	4	1	no	—
446	C₄×C₆₇	—	4	2	no	—
447	C₄×C₆₇	—	4	1	no	—
448	C₄×C₆₇	—	4	1	no	—
449	C₄×C₆₇	—	4	2	no	—
450	C₄×C₆₇	—	4	1	no	—
451	C₄×C₆₇	C₄×#41	1	4	yes	—
452	#34286	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
453	C₂×C₆₇	—	4	2	no	—
454	C₂×C₆₇	C₂×C₄×#41	1	1	yes	characteristic
455	C₂×C₆₇	C₄×2₋¹⁺⁴	1	1	yes	characteristic
456	C₂×C₆₇	C₄×2₋¹⁺⁴	1	1	yes	characteristic
457	C₂×C₆₇	#3515	1	1	yes	characteristic
458	C₂×C₆₇	#3516	1	1	yes	characteristic
459	C₂×C₆₇	#3521	1	1	yes	characteristic
460	C₂×C₆₇	#3523	1	1	yes	characteristic
461	C₂×#541	C₂×C₆₇	1	1	yes	characteristic
462	#2373	C₂×C₆₇	1	2	yes	—
463	#2373	C₂×C₆₇	1	2	yes	—
464	#2371	C₂×C₆₇	1	2	yes	complemented
465	#2370	C₂×C₆₇	1	1	yes	characteristic, complemented
466	#2370	C₂×C₆₇	1	1	yes	characteristic, complemented
467	#2370	C₂×C₆₇	1	2	yes	complemented
468	#2365	C₂×C₆₇	1	1	yes	characteristic, complemented
469	#2365	C₂×C₆₇	1	1	yes	characteristic, complemented
470	#2362	C₂×C₆₇	1	1	yes	characteristic
471	#2362	C₂×C₆₇	1	1	yes	characteristic
472	#2360	C₂×C₆₇	1	1	yes	characteristic
473	#2360	C₂×C₆₇	1	1	yes	characteristic
474	#2336	C₂×C₆₇	1	2	yes	complemented
475	#2334	C₂×C₆₇	1	2	yes	—
476	#2320	C₂×C₆₇	1	1	yes	characteristic, complemented
477	C₄×#125	C₂×C₆₇	1	2	yes	complemented
478	C₄×#126	C₂×C₆₇	1	2	yes	—
479	#2298	C₂×C₆₇	1	1	yes	characteristic
480	#2295	C₂×C₆₇	1	1	yes	characteristic, complemented
481	#2276	C₂×C₆₇	1	1	yes	characteristic, complemented
482	#2274	C₂×C₆₇	1	1	yes	characteristic
483	C₄²×Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
484	C₆₇	#34286	1	1	yes	characteristic, direct product factor, Sylow
485	#541	C₄×C₆₇	1	16	yes	complemented
486	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
487	C₂×#126	C₂²×C₆₇	1	2	yes	—
488	C₂×#126	C₂²×C₆₇	1	2	yes	—
489	C₂×#125	C₂²×C₆₇	1	2	yes	—
490	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
491	C₂×#125	C₂²×C₆₇	1	2	yes	—
492	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
493	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
494	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
495	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
496	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
497	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
498	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
499	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
500	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
501	C₂×#117	C₂²×C₆₇	1	1	yes	characteristic
502	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
503	#391	C₂²×C₆₇	1	1	yes	characteristic
504	#391	C₂²×C₆₇	1	1	yes	characteristic
505	#391	C₂²×C₆₇	1	1	yes	characteristic
506	#391	C₂²×C₆₇	1	1	yes	characteristic
507	#389	C₂²×C₆₇	1	2	yes	—
508	#389	C₂²×C₆₇	1	2	yes	—
509	#389	C₂²×C₆₇	1	1	yes	characteristic
510	#389	C₂²×C₆₇	1	1	yes	characteristic
511	#389	C₂²×C₆₇	1	2	yes	—
512	#389	C₂²×C₆₇	1	2	yes	—
513	#389	C₂²×C₆₇	1	2	yes	—
514	#389	C₂²×C₆₇	1	2	yes	—
515	#389	C₂²×C₆₇	1	1	yes	characteristic
516	#389	C₂²×C₆₇	1	1	yes	characteristic
517	#388	C₂²×C₆₇	1	1	yes	characteristic
518	#388	C₂²×C₆₇	1	2	yes	—
519	#388	C₂²×C₆₇	1	2	yes	—
520	#388	C₂²×C₆₇	1	1	yes	characteristic
521	#388	C₂²×C₆₇	1	2	yes	—
522	#388	C₂²×C₆₇	1	2	yes	—
523	#388	C₂²×C₆₇	1	1	yes	characteristic
524	#388	C₂²×C₆₇	1	1	yes	characteristic
525	#387	C₂²×C₆₇	1	2	yes	—
526	#387	C₂²×C₆₇	1	2	yes	—
527	#387	C₂²×C₆₇	1	2	yes	—
528	#387	C₂²×C₆₇	1	1	yes	characteristic
529	#387	C₂²×C₆₇	1	2	yes	—
530	#387	C₂²×C₆₇	1	2	yes	—
531	#387	C₂²×C₆₇	1	2	yes	—
532	#387	C₂²×C₆₇	1	2	yes	—
533	#387	C₂²×C₆₇	1	2	yes	—
534	#387	C₂²×C₆₇	1	2	yes	—
535	#387	C₂²×C₆₇	1	2	yes	—
536	#387	C₂²×C₆₇	1	1	yes	characteristic
537	#387	C₂²×C₆₇	1	2	yes	—
538	#387	C₂²×C₆₇	1	2	yes	—
539	#387	C₂²×C₆₇	1	2	yes	—
540	#387	C₂²×C₆₇	1	2	yes	—
541	#387	C₂²×C₆₇	1	2	yes	—
542	#387	C₂²×C₆₇	1	2	yes	—
543	#387	C₂²×C₆₇	1	1	yes	characteristic
544	#387	C₂²×C₆₇	1	1	yes	characteristic
545	#385	C₂²×C₆₇	1	2	yes	—
546	#385	C₂²×C₆₇	1	2	yes	—
547	#383	C₂²×C₆₇	1	2	yes	—
548	#382	C₂²×C₆₇	1	2	yes	—
549	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
550	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
551	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
552	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
553	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
554	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
555	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
556	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
557	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
558	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
559	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
560	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
561	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
562	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
563	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
564	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
565	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
566	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
567	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
568	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
569	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
570	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
571	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
572	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
573	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
574	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
575	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
576	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
577	C₄×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
578	C₄×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
579	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
580	#376	C₂²×C₆₇	1	1	yes	characteristic
581	#376	C₂²×C₆₇	1	2	yes	—
582	#376	C₂²×C₆₇	1	2	yes	—
583	C₄³	C₂²×C₆₇	1	2	yes	—
584	C₄³	C₂²×C₆₇	1	1	yes	characteristic
585	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
586	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
587	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
588	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
589	C₄⋊Q₈	C₂×C₄×C₆₇	1	8	yes	complemented
590	#126	C₂×C₄×C₆₇	1	16	yes	—
591	#126	C₂×C₄×C₆₇	1	16	yes	—
592	#125	C₂×C₄×C₆₇	1	8	yes	complemented
593	#125	C₂×C₄×C₆₇	1	16	yes	complemented
594	#125	C₂×C₄×C₆₇	1	16	yes	complemented
595	C₂²⋊Q₈	C₂×C₄×C₆₇	1	8	yes	—
596	C₂²⋊Q₈	C₂×C₄×C₆₇	1	8	yes	—
597	C₄×Q₈	—	2	4	no	—
598	C₄×Q₈	—	2	4	no	—
599	C₄×Q₈	—	2	4	no	—
600	C₄×Q₈	—	2	4	no	—
601	C₄×Q₈	C₂×C₄×C₆₇	1	8	yes	complemented
602	C₄×Q₈	C₂×C₄×C₆₇	1	8	yes	complemented
603	#117	—	2	4	no	—
604	#117	C₂×C₄×C₆₇	1	8	yes	—
605	C₂×C₄⋊C₄	—	2	4	no	—
606	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
607	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
608	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
609	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
610	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
611	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
612	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
613	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
614	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
615	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
616	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
617	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
618	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
619	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
620	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
621	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
622	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
623	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
624	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
625	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
626	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
627	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
628	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
629	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
630	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
631	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
632	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
633	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
634	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
635	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
636	C₂×C₂²⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
637	C₂×C₂²⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
638	C₂×C₂²⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
639	C₂×C₂²⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
640	C₂×C₂²⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
641	C₂×C₂²⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
642	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
643	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
644	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
645	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
646	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
647	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
648	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
649	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
650	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
651	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
652	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
653	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
654	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
655	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
656	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
657	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
658	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
659	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
660	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
661	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
662	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
663	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
664	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
665	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
666	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
667	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
668	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
669	#95	C₂³×C₆₇	1	1	yes	characteristic
670	#95	C₂³×C₆₇	1	2	yes	—
671	#95	C₂³×C₆₇	1	2	yes	—
672	#95	C₂³×C₆₇	1	1	yes	characteristic
673	#95	C₂³×C₆₇	1	1	yes	characteristic
674	#95	C₂³×C₆₇	1	2	yes	—
675	#95	C₂³×C₆₇	1	2	yes	—
676	#95	C₂³×C₆₇	1	1	yes	characteristic
677	#95	C₂³×C₆₇	1	2	yes	—
678	#95	C₂³×C₆₇	1	2	yes	—
679	#95	C₂³×C₆₇	1	2	yes	—
680	#95	C₂³×C₆₇	1	2	yes	—
681	#95	C₂³×C₆₇	1	1	yes	characteristic
682	#95	C₂³×C₆₇	1	2	yes	—
683	#95	C₂³×C₆₇	1	2	yes	—
684	#95	C₂³×C₆₇	1	1	yes	characteristic
685	#95	C₂³×C₆₇	1	2	yes	—
686	#95	C₂³×C₆₇	1	2	yes	—
687	#95	C₂³×C₆₇	1	2	yes	—
688	#95	C₂³×C₆₇	1	1	yes	characteristic
689	#95	C₂³×C₆₇	1	1	yes	characteristic
690	#95	C₂³×C₆₇	1	2	yes	—
691	#95	C₂³×C₆₇	1	1	yes	characteristic
692	#95	C₂³×C₆₇	1	1	yes	characteristic
693	#95	C₂³×C₆₇	1	2	yes	—
694	#95	C₂³×C₆₇	1	2	yes	—
695	#95	C₂³×C₆₇	1	2	yes	—
696	#95	C₂³×C₆₇	1	2	yes	—
697	#95	C₂³×C₆₇	1	2	yes	—
698	#95	C₂³×C₆₇	1	1	yes	characteristic
699	#95	C₂³×C₆₇	1	1	yes	characteristic
700	#95	C₂³×C₆₇	1	2	yes	—
701	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
702	C₂×Q₈	—	2	2	no	—
703	C₂×Q₈	—	2	2	no	—
704	C₂×Q₈	—	2	2	no	—
705	C₂×Q₈	—	2	2	no	—
706	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
707	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
708	C₂²×C₄	—	2	2	no	—
709	C₂²×C₄	—	2	2	no	—
710	C₂²×C₄	—	2	4	no	—
711	C₂²×C₄	—	4	2	no	—
712	C₂²×C₄	—	4	2	no	—
713	C₂²×C₄	C₂²×C₄×C₆₇	1	4	yes	—
714	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
715	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
716	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
717	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
718	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
719	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
720	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
721	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
722	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
723	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
724	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
725	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
726	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
727	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
728	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
729	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
730	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
731	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
732	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
733	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
734	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
735	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
736	C₄⋊C₄	—	2	4	no	—
737	C₄⋊C₄	—	2	2	no	—
738	C₄⋊C₄	—	2	2	no	—
739	C₄⋊C₄	—	2	4	no	—
740	C₄⋊C₄	—	2	4	no	—
741	C₄⋊C₄	—	2	4	no	—
742	C₄⋊C₄	—	2	4	no	—
743	C₄⋊C₄	—	2	2	no	—
744	C₄⋊C₄	—	2	4	no	—
745	C₄⋊C₄	—	2	2	no	—
746	C₄⋊C₄	—	4	1	no	—
747	C₄⋊C₄	—	4	1	no	—
748	C₄⋊C₄	—	4	2	no	—
749	C₄⋊C₄	—	4	2	no	—
750	C₄⋊C₄	—	4	2	no	—
751	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
752	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
753	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
754	C₄⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
755	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
756	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
757	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
758	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
759	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
760	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
761	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
762	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
763	C₄⋊C₄	#41×C₆₇	1	4	yes	—
764	C₄⋊C₄	#41×C₆₇	1	4	yes	—
765	C₂²⋊C₄	—	2	4	no	—
766	C₂²⋊C₄	—	2	2	no	—
767	C₂²⋊C₄	—	2	2	no	—
768	C₂²⋊C₄	—	4	1	no	—
769	C₂²⋊C₄	—	4	1	no	—
770	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
771	C₂²⋊C₄	C₂²×C₄×C₆₇	1	4	yes	—
772	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
773	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
774	C₄²	—	2	4	no	—
775	C₄²	—	2	4	no	—
776	C₄²	—	2	4	no	—
777	C₄²	—	2	2	no	—
778	C₄²	—	2	2	no	—
779	C₄²	—	2	4	no	—
780	C₄²	—	2	4	no	—
781	C₄²	—	2	2	no	—
782	C₄²	—	2	2	no	—
783	C₄²	—	2	4	no	—
784	C₄²	—	2	2	no	—
785	C₄²	—	2	2	no	—
786	C₄²	—	4	2	no	—
787	C₄²	—	4	1	no	—
788	C₄²	—	4	2	no	—
789	C₄²	—	4	2	no	—
790	C₄²	—	4	2	no	—
791	C₄²	—	4	2	no	—
792	C₄²	—	4	1	no	—
793	C₄²	—	4	2	no	—
794	C₄²	—	4	2	no	—
795	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
796	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
797	C₄²	C₂²×C₄×C₆₇	1	8	yes	—
798	C₄²	C₂²×C₄×C₆₇	1	8	yes	—
799	C₂³	—	2	1	no	—
800	C₂³	—	2	1	no	—
801	C₂³	—	2	1	no	—
802	C₂³	—	2	1	no	—
803	C₂³	—	2	1	no	—
804	C₂³	—	2	1	no	—
805	C₂³	C₂³×C₄×C₆₇	1	2	yes	—
806	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
807	Q₈	—	2	8	no	—
808	Q₈	—	2	8	no	—
809	C₂×C₄	—	2	1	no	—
810	C₂×C₄	—	2	1	no	—
811	C₂×C₄	—	2	1	no	—
812	C₂×C₄	—	2	2	no	—
813	C₂×C₄	—	2	2	no	—
814	C₂×C₄	—	2	2	no	—
815	C₂×C₄	—	2	2	no	—
816	C₂×C₄	—	2	1	no	—
817	C₂×C₄	—	2	1	no	—
818	C₂×C₄	—	2	1	no	—
819	C₂×C₄	—	2	2	no	—
820	C₂×C₄	—	2	1	no	—
821	C₂×C₄	—	2	2	no	—
822	C₂×C₄	—	2	1	no	—
823	C₂×C₄	—	2	2	no	—
824	C₂×C₄	—	2	1	no	—
825	C₂×C₄	—	2	1	no	—
826	C₂×C₄	—	2	1	no	—
827	C₂×C₄	—	2	2	no	—
828	C₂×C₄	—	2	1	no	—
829	C₂×C₄	—	2	2	no	—
830	C₂×C₄	—	2	1	no	—
831	C₂×C₄	—	2	1	no	—
832	C₂×C₄	—	2	2	no	—
833	C₂×C₄	—	2	2	no	—
834	C₂×C₄	—	2	2	no	—
835	C₂×C₄	—	2	8	no	—
836	C₂×C₄	—	2	1	no	—
837	C₂×C₄	—	2	1	no	—
838	C₂×C₄	—	2	2	no	—
839	C₂×C₄	—	2	1	no	—
840	C₂×C₄	—	2	1	no	—
841	C₂×C₄	—	2	1	no	—
842	C₂×C₄	—	2	1	no	—
843	C₂×C₄	—	2	2	no	—
844	C₂×C₄	—	2	1	no	—
845	C₂×C₄	—	2	1	no	—
846	C₂×C₄	—	2	1	no	—
847	C₂×C₄	—	2	1	no	—
848	C₂×C₄	—	2	1	no	—
849	C₂×C₄	—	2	2	no	—
850	C₂×C₄	—	2	1	no	—
851	C₂×C₄	—	2	1	no	—
852	C₂×C₄	—	2	2	no	—
853	C₂×C₄	—	2	1	no	—
854	C₂×C₄	—	2	1	no	—
855	C₂×C₄	—	2	1	no	—
856	C₂×C₄	—	2	2	no	—
857	C₂×C₄	—	2	2	no	—
858	C₂×C₄	—	2	1	no	—
859	C₂×C₄	—	2	1	no	—
860	C₂×C₄	—	8	4	no	—
861	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
862	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
863	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
864	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
865	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
866	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
867	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
868	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
869	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
870	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
871	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
872	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
873	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
874	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
875	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
876	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
877	C₂²	—	2	2	no	—
878	C₂²	—	2	2	no	—
879	C₂²	—	2	2	no	—
880	C₂²	—	4	1	no	—
881	C₂²	—	4	1	no	—
882	C₂²	—	4	1	no	—
883	C₂²	—	4	1	no	—
884	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
885	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
886	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
887	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
888	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
889	C₂²	#585×C₆₇	1	1	yes	characteristic
890	C₂²	#585×C₆₇	1	1	yes	characteristic
891	C₄	—	2	4	no	—
892	C₄	—	2	2	no	—
893	C₄	—	2	2	no	—
894	C₄	—	2	4	no	—
895	C₄	—	2	4	no	—
896	C₄	—	2	2	no	—
897	C₄	—	2	2	no	—
898	C₄	—	2	4	no	—
899	C₄	—	2	2	no	—
900	C₄	—	2	2	no	—
901	C₄	—	2	4	no	—
902	C₄	—	2	4	no	—
903	C₄	—	2	2	no	—
904	C₄	—	4	2	no	—
905	C₄	—	4	1	no	—
906	C₄	—	4	1	no	—
907	C₄	—	4	1	no	—
908	C₄	—	4	1	no	—
909	C₄	—	4	1	no	—
910	C₄	—	4	1	no	—
911	C₄	—	4	2	no	—
912	C₄	C₄×#41×C₆₇	1	4	yes	—
913	C₂	—	4	2	no	—
914	C₂	C₂×C₄×#41×C₆₇	1	1	yes	characteristic
915	C₂	C₄×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
916	C₂	C₄×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
917	C₂	C₆₇×#3515	1	1	yes	characteristic
918	C₂	C₆₇×#3516	1	1	yes	characteristic
919	C₂	C₆₇×#3521	1	1	yes	characteristic
920	C₂	C₆₇×#3523	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂⁴×C₆₇	C₂⁴
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
4. C₆₇×#2371	C₂	semidirect
5. C₆₇×#2370	C₂	semidirect
6. C₆₇×#2370	C₂	semidirect
7. C₆₇×#2370	C₂	semidirect
8. C₆₇×#2365	C₂	semidirect
9. C₆₇×#2365	C₂	semidirect
14. C₆₇×#2336	C₂	semidirect
16. C₆₇×#2320	C₂	semidirect
17. C₄×#125×C₆₇	C₂	semidirect
20. C₆₇×#2295	C₂	semidirect
21. C₆₇×#2276	C₂	semidirect
23. C₄²×Q₈×C₆₇	C₂	semidirect
24. #541×C₆₇	C₄	semidirect
128. C₄⋊Q₈×C₆₇	C₂×C₄	semidirect
131. #125×C₆₇	C₂×C₄	semidirect
132. #125×C₆₇	C₂×C₄	semidirect
133. #125×C₆₇	C₂×C₄	semidirect
140. C₄×Q₈×C₆₇	C₂×C₄	semidirect
141. C₄×Q₈×C₆₇	C₂×C₄	semidirect
452. #34286	C₆₇	direct (N × H)
464. #2371	C₂×C₆₇	semidirect
465. #2370	C₂×C₆₇	semidirect
466. #2370	C₂×C₆₇	semidirect
467. #2370	C₂×C₆₇	semidirect
468. #2365	C₂×C₆₇	semidirect
469. #2365	C₂×C₆₇	semidirect
474. #2336	C₂×C₆₇	semidirect
476. #2320	C₂×C₆₇	semidirect
477. C₄×#125	C₂×C₆₇	semidirect
480. #2295	C₂×C₆₇	semidirect
481. #2276	C₂×C₆₇	semidirect
483. C₄²×Q₈	C₂×C₆₇	semidirect
484. C₆₇	#34286	direct (N × H)
485. #541	C₄×C₆₇	semidirect
589. C₄⋊Q₈	C₂×C₄×C₆₇	semidirect
592. #125	C₂×C₄×C₆₇	semidirect
593. #125	C₂×C₄×C₆₇	semidirect
594. #125	C₂×C₄×C₆₇	semidirect
601. C₄×Q₈	C₂×C₄×C₆₇	semidirect
602. C₄×Q₈	C₂×C₄×C₆₇	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#541×C₆₇	2
C₆₇×#2373	2
C₆₇×#2371	2
C₆₇×#2370	2
C₆₇×#2365	2
C₆₇×#2362	2
C₆₇×#2360	2
C₆₇×#2336	2
C₆₇×#2334	2
C₆₇×#2320	2
C₄×#125×C₆₇	2
C₄×#126×C₆₇	2
C₆₇×#2298	2
C₆₇×#2295	2
C₆₇×#2276	2
C₆₇×#2274	2
C₄²×Q₈×C₆₇	2
#34286	67

Subgroups — C67×#34286 #30713537

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#34286 #30713537