Subgroups — C₆₇×#59663 #30738914

Order 17152.

1766 conjugacy classes of subgroups.

The subgroup lattice has 938 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₆₇×#2802	C₂	1	2	yes	maximal, complemented
2	C₆₇×#2755	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2588	C₂	1	2	yes	maximal, complemented
4	C₆₇×#2584	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2584	C₂	1	2	yes	maximal, complemented
6	C₆₇×#2582	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₇×#2549	C₂	1	2	yes	maximal
8	C₆₇×#2509	C₂	1	1	yes	maximal, characteristic, complemented
9	C₆₇×#2506	C₂	1	1	yes	maximal, characteristic, complemented
10	C₆₇×#2505	C₂	1	2	yes	maximal
11	C₆₇×#2473	C₂	1	2	yes	maximal
12	C₆₇×#2453	C₂	1	1	yes	maximal, characteristic, complemented
13	C₆₇×#2451	C₂	1	1	yes	maximal, characteristic, complemented
14	C₆₇×#2442	C₂	1	1	yes	maximal, characteristic
15	C₆₇×#2440	C₂	1	1	yes	maximal, characteristic
16	C₆₇×#2365	C₂	1	1	yes	maximal, characteristic, complemented
17	C₆₇×#2360	C₂	1	1	yes	maximal, characteristic
18	C₆₇×#2360	C₂	1	1	yes	maximal, characteristic
19	C₆₇×#2360	C₂	1	2	yes	maximal
20	C₆₇×#2351	C₂	1	1	yes	maximal, characteristic, complemented
21	C₄×C₂²⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
22	C₄×C₂²⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
23	C₂×#389×C₆₇	C₂	1	1	yes	maximal, characteristic
24	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
25	C₂×#125×C₆₇	C₂²	1	2	yes	—
26	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
28	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
29	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
30	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
31	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
33	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
35	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
36	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
37	C₂²×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
38	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
39	#400×C₆₇	C₂²	1	2	yes	—
40	#400×C₆₇	C₂²	1	2	yes	—
41	#400×C₆₇	C₂²	1	2	yes	—
42	#400×C₆₇	C₂²	1	2	yes	—
43	#400×C₆₇	C₂²	1	2	yes	—
44	#398×C₆₇	C₂²	1	2	yes	—
45	#398×C₆₇	C₂²	1	2	yes	—
46	#398×C₆₇	C₂²	1	2	yes	—
47	#398×C₆₇	C₂²	1	2	yes	—
48	#398×C₆₇	C₂²	1	2	yes	—
49	#398×C₆₇	C₂²	1	2	yes	—
50	#398×C₆₇	C₂²	1	2	yes	—
51	#398×C₆₇	C₂²	1	2	yes	—
52	#395×C₆₇	C₂²	1	2	yes	—
53	#395×C₆₇	C₂²	1	2	yes	—
54	#395×C₆₇	C₂²	1	2	yes	—
55	#391×C₆₇	C₂²	1	2	yes	—
56	#391×C₆₇	C₂²	1	2	yes	—
57	#391×C₆₇	C₂²	1	2	yes	—
58	#391×C₆₇	C₂²	1	1	yes	characteristic
59	#391×C₆₇	C₂²	1	1	yes	characteristic
60	#391×C₆₇	C₂²	1	1	yes	characteristic
61	#391×C₆₇	C₂²	1	1	yes	characteristic
62	#389×C₆₇	C₂²	1	2	yes	—
63	#389×C₆₇	C₂²	1	2	yes	—
64	#389×C₆₇	C₂²	1	2	yes	—
65	#389×C₆₇	C₂²	1	1	yes	characteristic
66	#389×C₆₇	C₂²	1	1	yes	characteristic
67	#389×C₆₇	C₂²	1	1	yes	characteristic
68	#389×C₆₇	C₂²	1	1	yes	characteristic
69	#389×C₆₇	C₂²	1	1	yes	characteristic
70	#389×C₆₇	C₂²	1	2	yes	—
71	#389×C₆₇	C₂²	1	1	yes	characteristic
72	#389×C₆₇	C₂²	1	1	yes	characteristic
73	#389×C₆₇	C₂²	1	2	yes	—
74	#389×C₆₇	C₂²	1	2	yes	—
75	#389×C₆₇	C₂²	1	2	yes	—
76	#389×C₆₇	C₂²	1	2	yes	—
77	#389×C₆₇	C₂²	1	2	yes	—
78	#389×C₆₇	C₂²	1	1	yes	characteristic
79	#389×C₆₇	C₂²	1	1	yes	characteristic
80	#389×C₆₇	C₂²	1	1	yes	characteristic
81	#389×C₆₇	C₂²	1	1	yes	characteristic
82	#389×C₆₇	C₂²	1	1	yes	characteristic
83	#389×C₆₇	C₂²	1	2	yes	—
84	#389×C₆₇	C₂²	1	2	yes	—
85	#387×C₆₇	C₂²	1	1	yes	characteristic
86	#387×C₆₇	C₂²	1	2	yes	—
87	#387×C₆₇	C₂²	1	2	yes	—
88	#387×C₆₇	C₂²	1	2	yes	—
89	#387×C₆₇	C₂²	1	2	yes	—
90	#387×C₆₇	C₂²	1	2	yes	—
91	#387×C₆₇	C₂²	1	2	yes	—
92	#387×C₆₇	C₂²	1	1	yes	characteristic
93	#387×C₆₇	C₂²	1	2	yes	—
94	#387×C₆₇	C₂²	1	2	yes	—
95	#387×C₆₇	C₂²	1	2	yes	—
96	#385×C₆₇	C₂²	1	2	yes	—
97	#385×C₆₇	C₂²	1	2	yes	—
98	#385×C₆₇	C₂²	1	2	yes	—
99	#385×C₆₇	C₂²	1	2	yes	—
100	#385×C₆₇	C₂²	1	2	yes	—
101	#385×C₆₇	C₂²	1	1	yes	characteristic
102	#385×C₆₇	C₂²	1	1	yes	characteristic
103	#384×C₆₇	C₂²	1	1	yes	characteristic
104	#382×C₆₇	C₂²	1	2	yes	—
105	#382×C₆₇	C₂²	1	1	yes	characteristic
106	#380×C₆₇	C₂²	1	1	yes	characteristic
107	#380×C₆₇	C₂²	1	2	yes	—
108	#380×C₆₇	C₂²	1	1	yes	characteristic
109	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
110	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
111	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
112	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
113	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
114	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
115	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
116	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
117	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
118	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
119	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
120	C₄×C₂²⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
121	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
122	C₂×#95×C₆₇	C₂²	1	2	yes	—
123	C₂²×Q₈×C₆₇	C₂³	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	1	yes	characteristic
128	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
129	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
130	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
131	C₄⋊Q₈×C₆₇	—	2	4	no	—
132	#125×C₆₇	—	4	2	no	—
133	#125×C₆₇	—	4	4	no	—
134	C₂²⋊Q₈×C₆₇	—	2	4	no	—
135	C₂²⋊Q₈×C₆₇	—	2	8	no	—
136	C₂²⋊Q₈×C₆₇	Q₈	1	8	yes	complemented
137	C₄×Q₈×C₆₇	—	2	4	no	—
138	C₄×Q₈×C₆₇	—	4	2	no	—
139	C₄×Q₈×C₆₇	—	4	4	no	—
140	C₄×Q₈×C₆₇	—	4	2	no	—
141	C₄×Q₈×C₆₇	—	4	2	no	—
142	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
143	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
144	C₂×C₄⋊C₄×C₆₇	—	4	4	no	—
145	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
146	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
147	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
148	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
149	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
150	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
151	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
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	1	yes	characteristic
156	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
157	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
158	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
159	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
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	2	yes	—
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	1	yes	characteristic
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	1	yes	characteristic
172	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
173	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
174	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
175	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
176	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
177	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
178	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
179	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
180	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
181	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
182	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
183	C₂×C₂²⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
184	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
185	C₂×C₂²⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
186	C₂×C₂²⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
187	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
188	C₂×C₂²⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
189	C₂×C₄²×C₆₇	—	4	2	no	—
190	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
191	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
192	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
193	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
194	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
195	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
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	1	yes	characteristic
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	#95×C₆₇	C₂³	1	2	yes	—
205	#95×C₆₇	C₂³	1	2	yes	—
206	#95×C₆₇	C₂³	1	2	yes	—
207	#95×C₆₇	C₂³	1	2	yes	—
208	#95×C₆₇	C₂³	1	2	yes	—
209	#95×C₆₇	C₂³	1	1	yes	characteristic
210	#95×C₆₇	C₂³	1	1	yes	characteristic
211	#95×C₆₇	C₂³	1	2	yes	—
212	#95×C₆₇	C₂³	1	2	yes	—
213	#95×C₆₇	C₂³	1	1	yes	characteristic
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	1	yes	characteristic
219	#95×C₆₇	C₂³	1	1	yes	characteristic
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	1	yes	characteristic
225	#95×C₆₇	C₂³	1	2	yes	—
226	#95×C₆₇	C₂³	1	2	yes	—
227	#95×C₆₇	C₂³	1	2	yes	—
228	#95×C₆₇	C₂³	1	2	yes	—
229	#95×C₆₇	C₂³	1	1	yes	characteristic
230	#95×C₆₇	C₂³	1	2	yes	—
231	#95×C₆₇	C₂³	1	2	yes	—
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	C₂⁴×C₆₇	C₂⁴	1	1	yes	characteristic
238	C₂×Q₈×C₆₇	—	2	2	no	—
239	C₂×Q₈×C₆₇	—	2	2	no	—
240	C₂×Q₈×C₆₇	—	2	4	no	—
241	C₂×Q₈×C₆₇	—	2	4	no	—
242	C₂×Q₈×C₆₇	—	2	2	no	—
243	C₂×Q₈×C₆₇	—	2	2	no	—
244	C₂×Q₈×C₆₇	—	4	1	no	—
245	C₂×Q₈×C₆₇	—	4	2	no	—
246	C₂×Q₈×C₆₇	C₂×Q₈	1	4	yes	—
247	C₂²×C₄×C₆₇	—	2	2	no	—
248	C₂²×C₄×C₆₇	—	2	2	no	—
249	C₂²×C₄×C₆₇	—	2	2	no	—
250	C₂²×C₄×C₆₇	—	2	4	no	—
251	C₂²×C₄×C₆₇	—	2	2	no	—
252	C₂²×C₄×C₆₇	—	2	4	no	—
253	C₂²×C₄×C₆₇	—	2	4	no	—
254	C₂²×C₄×C₆₇	—	2	2	no	—
255	C₂²×C₄×C₆₇	—	2	2	no	—
256	C₂²×C₄×C₆₇	—	4	1	no	—
257	C₂²×C₄×C₆₇	—	4	1	no	—
258	C₂²×C₄×C₆₇	—	4	2	no	—
259	C₂²×C₄×C₆₇	—	4	2	no	—
260	C₂²×C₄×C₆₇	—	4	2	no	—
261	C₂²×C₄×C₆₇	C₂×Q₈	1	4	yes	—
262	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
263	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
264	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
265	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
266	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
267	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
268	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
269	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
270	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
271	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
272	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
273	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
274	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
275	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
276	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
277	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
278	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
279	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
280	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
281	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
282	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
283	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
284	C₄⋊C₄×C₆₇	—	2	4	no	—
285	C₄⋊C₄×C₆₇	—	2	2	no	—
286	C₄⋊C₄×C₆₇	—	2	4	no	—
287	C₄⋊C₄×C₆₇	—	2	4	no	—
288	C₄⋊C₄×C₆₇	—	2	2	no	—
289	C₄⋊C₄×C₆₇	—	2	2	no	—
290	C₄⋊C₄×C₆₇	—	2	4	no	—
291	C₄⋊C₄×C₆₇	—	2	4	no	—
292	C₄⋊C₄×C₆₇	—	2	2	no	—
293	C₄⋊C₄×C₆₇	—	2	4	no	—
294	C₄⋊C₄×C₆₇	—	2	2	no	—
295	C₄⋊C₄×C₆₇	—	2	4	no	—
296	C₄⋊C₄×C₆₇	—	2	2	no	—
297	C₄⋊C₄×C₆₇	—	2	4	no	—
298	C₄⋊C₄×C₆₇	—	2	2	no	—
299	C₄⋊C₄×C₆₇	—	2	2	no	—
300	C₄⋊C₄×C₆₇	—	2	2	no	—
301	C₄⋊C₄×C₆₇	—	2	2	no	—
302	C₄⋊C₄×C₆₇	—	2	4	no	—
303	C₄⋊C₄×C₆₇	—	2	2	no	—
304	C₄⋊C₄×C₆₇	—	4	2	no	—
305	C₄⋊C₄×C₆₇	—	4	2	no	—
306	C₄⋊C₄×C₆₇	—	4	2	no	—
307	C₄⋊C₄×C₆₇	—	4	1	no	—
308	C₄⋊C₄×C₆₇	—	4	1	no	—
309	C₄⋊C₄×C₆₇	—	4	2	no	—
310	C₄⋊C₄×C₆₇	—	4	2	no	—
311	C₄⋊C₄×C₆₇	—	4	1	no	—
312	C₄⋊C₄×C₆₇	—	4	2	no	—
313	C₄⋊C₄×C₆₇	—	4	2	no	—
314	C₄⋊C₄×C₆₇	—	4	2	no	—
315	C₄⋊C₄×C₆₇	—	4	2	no	—
316	C₄⋊C₄×C₆₇	—	4	2	no	—
317	C₄⋊C₄×C₆₇	—	4	2	no	—
318	C₄⋊C₄×C₆₇	—	4	2	no	—
319	C₄⋊C₄×C₆₇	C₂×Q₈	1	4	yes	—
320	C₄⋊C₄×C₆₇	C₂×Q₈	1	8	yes	—
321	C₄⋊C₄×C₆₇	#41	1	4	yes	—
322	C₂²⋊C₄×C₆₇	—	2	2	no	—
323	C₂²⋊C₄×C₆₇	—	2	4	no	—
324	C₂²⋊C₄×C₆₇	—	2	2	no	—
325	C₂²⋊C₄×C₆₇	—	2	2	no	—
326	C₂²⋊C₄×C₆₇	C₂×Q₈	1	8	yes	—
327	C₂²⋊C₄×C₆₇	#41	1	4	yes	—
328	C₄²×C₆₇	—	2	4	no	—
329	C₄²×C₆₇	—	2	2	no	—
330	C₄²×C₆₇	—	2	2	no	—
331	C₄²×C₆₇	—	2	2	no	—
332	C₄²×C₆₇	—	2	2	no	—
333	C₄²×C₆₇	—	2	2	no	—
334	C₄²×C₆₇	—	2	4	no	—
335	C₄²×C₆₇	—	2	2	no	—
336	C₄²×C₆₇	—	4	1	no	—
337	C₄²×C₆₇	—	4	1	no	—
338	C₄²×C₆₇	—	4	2	no	—
339	C₄²×C₆₇	—	4	2	no	—
340	C₄²×C₆₇	—	4	2	no	—
341	C₄²×C₆₇	—	4	2	no	—
342	C₂³×C₆₇	—	2	1	no	—
343	C₂³×C₆₇	—	2	1	no	—
344	C₂³×C₆₇	—	2	1	no	—
345	C₂³×C₆₇	—	2	1	no	—
346	C₂³×C₆₇	C₂²×Q₈	1	2	yes	—
347	C₂³×C₆₇	C₂×#41	1	2	yes	—
348	C₂³×C₆₇	2₋¹⁺⁴	1	2	yes	—
349	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
350	Q₈×C₆₇	—	2	8	no	—
351	Q₈×C₆₇	—	4	8	no	—
352	Q₈×C₆₇	—	4	4	no	—
353	C₂×C₄×C₆₇	—	2	2	no	—
354	C₂×C₄×C₆₇	—	2	1	no	—
355	C₂×C₄×C₆₇	—	2	1	no	—
356	C₂×C₄×C₆₇	—	2	2	no	—
357	C₂×C₄×C₆₇	—	2	1	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	1	no	—
364	C₂×C₄×C₆₇	—	2	2	no	—
365	C₂×C₄×C₆₇	—	2	2	no	—
366	C₂×C₄×C₆₇	—	2	2	no	—
367	C₂×C₄×C₆₇	—	2	1	no	—
368	C₂×C₄×C₆₇	—	2	1	no	—
369	C₂×C₄×C₆₇	—	2	8	no	—
370	C₂×C₄×C₆₇	—	2	1	no	—
371	C₂×C₄×C₆₇	—	2	1	no	—
372	C₂×C₄×C₆₇	—	2	1	no	—
373	C₂×C₄×C₆₇	—	2	1	no	—
374	C₂×C₄×C₆₇	—	2	2	no	—
375	C₂×C₄×C₆₇	—	2	2	no	—
376	C₂×C₄×C₆₇	—	2	1	no	—
377	C₂×C₄×C₆₇	—	2	2	no	—
378	C₂×C₄×C₆₇	—	2	2	no	—
379	C₂×C₄×C₆₇	—	2	1	no	—
380	C₂×C₄×C₆₇	—	2	2	no	—
381	C₂×C₄×C₆₇	—	2	1	no	—
382	C₂×C₄×C₆₇	—	2	1	no	—
383	C₂×C₄×C₆₇	—	2	1	no	—
384	C₂×C₄×C₆₇	—	2	1	no	—
385	C₂×C₄×C₆₇	—	2	2	no	—
386	C₂×C₄×C₆₇	—	2	1	no	—
387	C₂×C₄×C₆₇	—	2	1	no	—
388	C₂×C₄×C₆₇	—	2	2	no	—
389	C₂×C₄×C₆₇	—	2	2	no	—
390	C₂×C₄×C₆₇	—	2	2	no	—
391	C₂×C₄×C₆₇	—	2	1	no	—
392	C₂×C₄×C₆₇	—	2	1	no	—
393	C₂×C₄×C₆₇	—	2	2	no	—
394	C₂×C₄×C₆₇	—	2	2	no	—
395	C₂×C₄×C₆₇	—	2	2	no	—
396	C₂×C₄×C₆₇	—	2	1	no	—
397	C₂×C₄×C₆₇	—	2	1	no	—
398	C₂×C₄×C₆₇	—	2	1	no	—
399	C₂×C₄×C₆₇	—	2	1	no	—
400	C₂×C₄×C₆₇	—	2	1	no	—
401	C₂×C₄×C₆₇	—	2	2	no	—
402	C₂×C₄×C₆₇	—	2	1	no	—
403	C₂×C₄×C₆₇	—	2	1	no	—
404	C₂×C₄×C₆₇	—	2	1	no	—
405	C₂×C₄×C₆₇	—	2	1	no	—
406	C₂×C₄×C₆₇	—	2	2	no	—
407	C₂×C₄×C₆₇	—	2	2	no	—
408	C₂×C₄×C₆₇	—	2	1	no	—
409	C₂×C₄×C₆₇	—	4	8	no	—
410	C₂×C₄×C₆₇	—	4	4	no	—
411	C₂×C₄×C₆₇	—	4	4	no	—
412	C₂×C₄×C₆₇	—	8	4	no	—
413	C₂×C₄×C₆₇	C₂²×Q₈	1	4	yes	—
414	C₂×C₄×C₆₇	C₂²×Q₈	1	4	yes	—
415	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
416	C₂×C₄×C₆₇	C₂²×Q₈	1	2	yes	—
417	C₂×C₄×C₆₇	C₂×#41	1	4	yes	—
418	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
419	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
420	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
421	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
422	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
423	C₂×C₄×C₆₇	2₋¹⁺⁴	1	2	yes	—
424	C₂²×C₆₇	—	2	2	no	—
425	C₂²×C₆₇	—	2	2	no	—
426	C₂²×C₆₇	—	2	2	no	—
427	C₂²×C₆₇	—	2	2	no	—
428	C₂²×C₆₇	—	2	2	no	—
429	C₂²×C₆₇	—	2	2	no	—
430	C₂²×C₆₇	#557	1	4	yes	—
431	C₂²×C₆₇	C₂³×Q₈	1	1	yes	characteristic
432	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
433	C₂²×C₆₇	C₂²×#41	1	1	yes	characteristic
434	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
435	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
436	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
437	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	1	yes	characteristic
438	C₄×C₆₇	—	2	2	no	—
439	C₄×C₆₇	—	2	4	no	—
440	C₄×C₆₇	—	2	4	no	—
441	C₄×C₆₇	—	2	2	no	—
442	C₄×C₆₇	—	2	4	no	—
443	C₄×C₆₇	—	2	2	no	—
444	C₄×C₆₇	—	2	2	no	—
445	C₄×C₆₇	—	2	2	no	—
446	C₄×C₆₇	—	4	1	no	—
447	C₄×C₆₇	—	4	2	no	—
448	C₄×C₆₇	—	4	2	no	—
449	C₄×C₆₇	—	4	2	no	—
450	C₄×C₆₇	—	4	1	no	—
451	C₄×C₆₇	—	4	1	no	—
452	C₄×C₆₇	—	4	1	no	—
453	C₄×C₆₇	—	4	1	no	—
454	C₄×C₆₇	—	4	1	no	—
455	C₄×C₆₇	—	4	1	no	—
456	C₄×C₆₇	—	4	1	no	—
457	C₄×C₆₇	—	4	2	no	—
458	C₄×C₆₇	—	4	2	no	—
459	C₄×C₆₇	#554	1	4	yes	—
460	#59663	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
461	C₂×C₆₇	—	2	4	no	—
462	C₂×C₆₇	C₂×#554	1	1	yes	characteristic
463	C₂×C₆₇	C₂×#557	1	1	yes	characteristic
464	C₂×C₆₇	#3481	1	1	yes	characteristic
465	C₂×C₆₇	#3503	1	1	yes	characteristic
466	C₂×C₆₇	#3520	1	1	yes	characteristic
467	C₂×C₆₇	#3523	1	1	yes	characteristic
468	C₂×C₆₇	#3536	1	1	yes	characteristic
469	#2802	C₂×C₆₇	1	2	yes	complemented
470	#2755	C₂×C₆₇	1	2	yes	complemented
471	#2588	C₂×C₆₇	1	2	yes	complemented
472	#2584	C₂×C₆₇	1	1	yes	characteristic, complemented
473	#2584	C₂×C₆₇	1	2	yes	complemented
474	#2582	C₂×C₆₇	1	1	yes	characteristic, complemented
475	#2549	C₂×C₆₇	1	2	yes	—
476	#2509	C₂×C₆₇	1	1	yes	characteristic, complemented
477	#2506	C₂×C₆₇	1	1	yes	characteristic, complemented
478	#2505	C₂×C₆₇	1	2	yes	—
479	#2473	C₂×C₆₇	1	2	yes	—
480	#2453	C₂×C₆₇	1	1	yes	characteristic, complemented
481	#2451	C₂×C₆₇	1	1	yes	characteristic, complemented
482	#2442	C₂×C₆₇	1	1	yes	characteristic
483	#2440	C₂×C₆₇	1	1	yes	characteristic
484	#2365	C₂×C₆₇	1	1	yes	characteristic, complemented
485	#2360	C₂×C₆₇	1	1	yes	characteristic
486	#2360	C₂×C₆₇	1	1	yes	characteristic
487	#2360	C₂×C₆₇	1	2	yes	—
488	#2351	C₂×C₆₇	1	1	yes	characteristic, complemented
489	C₄×C₂²⋊Q₈	C₂×C₆₇	1	1	yes	characteristic
490	C₄×C₂²⋊Q₈	C₂×C₆₇	1	1	yes	characteristic
491	C₂×#389	C₂×C₆₇	1	1	yes	characteristic
492	C₆₇	#59663	1	1	yes	characteristic, direct product factor, Sylow
493	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
494	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
495	C₂×#125	C₂²×C₆₇	1	2	yes	—
496	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
497	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
498	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
499	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
500	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
501	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
502	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
503	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
504	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
505	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
506	C₂²×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
507	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
508	#400	C₂²×C₆₇	1	2	yes	—
509	#400	C₂²×C₆₇	1	2	yes	—
510	#400	C₂²×C₆₇	1	2	yes	—
511	#400	C₂²×C₆₇	1	2	yes	—
512	#400	C₂²×C₆₇	1	2	yes	—
513	#398	C₂²×C₆₇	1	2	yes	—
514	#398	C₂²×C₆₇	1	2	yes	—
515	#398	C₂²×C₆₇	1	2	yes	—
516	#398	C₂²×C₆₇	1	2	yes	—
517	#398	C₂²×C₆₇	1	2	yes	—
518	#398	C₂²×C₆₇	1	2	yes	—
519	#398	C₂²×C₆₇	1	2	yes	—
520	#398	C₂²×C₆₇	1	2	yes	—
521	#395	C₂²×C₆₇	1	2	yes	—
522	#395	C₂²×C₆₇	1	2	yes	—
523	#395	C₂²×C₆₇	1	2	yes	—
524	#391	C₂²×C₆₇	1	2	yes	—
525	#391	C₂²×C₆₇	1	1	yes	characteristic
526	#391	C₂²×C₆₇	1	1	yes	characteristic
527	#391	C₂²×C₆₇	1	1	yes	characteristic
528	#391	C₂²×C₆₇	1	1	yes	characteristic
529	#391	C₂²×C₆₇	1	2	yes	—
530	#391	C₂²×C₆₇	1	2	yes	—
531	#389	C₂²×C₆₇	1	2	yes	—
532	#389	C₂²×C₆₇	1	1	yes	characteristic
533	#389	C₂²×C₆₇	1	1	yes	characteristic
534	#389	C₂²×C₆₇	1	2	yes	—
535	#389	C₂²×C₆₇	1	1	yes	characteristic
536	#389	C₂²×C₆₇	1	2	yes	—
537	#389	C₂²×C₆₇	1	1	yes	characteristic
538	#389	C₂²×C₆₇	1	2	yes	—
539	#389	C₂²×C₆₇	1	2	yes	—
540	#389	C₂²×C₆₇	1	2	yes	—
541	#389	C₂²×C₆₇	1	1	yes	characteristic
542	#389	C₂²×C₆₇	1	2	yes	—
543	#389	C₂²×C₆₇	1	2	yes	—
544	#389	C₂²×C₆₇	1	1	yes	characteristic
545	#389	C₂²×C₆₇	1	2	yes	—
546	#389	C₂²×C₆₇	1	1	yes	characteristic
547	#389	C₂²×C₆₇	1	1	yes	characteristic
548	#389	C₂²×C₆₇	1	2	yes	—
549	#389	C₂²×C₆₇	1	1	yes	characteristic
550	#389	C₂²×C₆₇	1	1	yes	characteristic
551	#389	C₂²×C₆₇	1	1	yes	characteristic
552	#389	C₂²×C₆₇	1	1	yes	characteristic
553	#389	C₂²×C₆₇	1	2	yes	—
554	#387	C₂²×C₆₇	1	2	yes	—
555	#387	C₂²×C₆₇	1	2	yes	—
556	#387	C₂²×C₆₇	1	2	yes	—
557	#387	C₂²×C₆₇	1	2	yes	—
558	#387	C₂²×C₆₇	1	2	yes	—
559	#387	C₂²×C₆₇	1	2	yes	—
560	#387	C₂²×C₆₇	1	2	yes	—
561	#387	C₂²×C₆₇	1	2	yes	—
562	#387	C₂²×C₆₇	1	1	yes	characteristic
563	#387	C₂²×C₆₇	1	2	yes	—
564	#387	C₂²×C₆₇	1	1	yes	characteristic
565	#385	C₂²×C₆₇	1	1	yes	characteristic
566	#385	C₂²×C₆₇	1	2	yes	—
567	#385	C₂²×C₆₇	1	2	yes	—
568	#385	C₂²×C₆₇	1	2	yes	—
569	#385	C₂²×C₆₇	1	2	yes	—
570	#385	C₂²×C₆₇	1	1	yes	characteristic
571	#385	C₂²×C₆₇	1	2	yes	—
572	#384	C₂²×C₆₇	1	1	yes	characteristic
573	#382	C₂²×C₆₇	1	1	yes	characteristic
574	#382	C₂²×C₆₇	1	2	yes	—
575	#380	C₂²×C₆₇	1	1	yes	characteristic
576	#380	C₂²×C₆₇	1	2	yes	—
577	#380	C₂²×C₆₇	1	1	yes	characteristic
578	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
579	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
580	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
581	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
582	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
583	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
584	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
585	C₄×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
586	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
587	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
588	C₄×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
589	C₄×C₂²⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
590	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
591	C₂×#95	C₂²×C₆₇	1	2	yes	—
592	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
593	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
594	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
595	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
596	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
597	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
598	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
599	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
600	C₄⋊Q₈	—	2	4	no	—
601	#125	—	4	2	no	—
602	#125	—	4	4	no	—
603	C₂²⋊Q₈	—	2	8	no	—
604	C₂²⋊Q₈	—	2	4	no	—
605	C₂²⋊Q₈	Q₈×C₆₇	1	8	yes	complemented
606	C₄×Q₈	—	2	4	no	—
607	C₄×Q₈	—	4	2	no	—
608	C₄×Q₈	—	4	4	no	—
609	C₄×Q₈	—	4	2	no	—
610	C₄×Q₈	—	4	2	no	—
611	C₂×C₄⋊C₄	—	2	4	no	—
612	C₂×C₄⋊C₄	—	4	2	no	—
613	C₂×C₄⋊C₄	—	4	4	no	—
614	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
615	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
616	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
617	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
618	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
619	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
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	2	yes	—
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	1	yes	characteristic
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	1	yes	characteristic
630	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
631	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
632	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
633	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
634	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
635	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
636	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
637	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
638	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
639	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
640	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
641	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
642	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
643	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
644	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
645	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
646	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
647	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
648	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
649	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
650	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
651	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
652	C₂×C₂²⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
653	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
654	C₂×C₂²⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
655	C₂×C₂²⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
656	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
657	C₂×C₂²⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
658	C₂×C₄²	—	4	2	no	—
659	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
660	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
661	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
662	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
663	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
664	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
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	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
670	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
671	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
672	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
673	#95	C₂³×C₆₇	1	2	yes	—
674	#95	C₂³×C₆₇	1	2	yes	—
675	#95	C₂³×C₆₇	1	2	yes	—
676	#95	C₂³×C₆₇	1	2	yes	—
677	#95	C₂³×C₆₇	1	2	yes	—
678	#95	C₂³×C₆₇	1	1	yes	characteristic
679	#95	C₂³×C₆₇	1	1	yes	characteristic
680	#95	C₂³×C₆₇	1	2	yes	—
681	#95	C₂³×C₆₇	1	2	yes	—
682	#95	C₂³×C₆₇	1	1	yes	characteristic
683	#95	C₂³×C₆₇	1	1	yes	characteristic
684	#95	C₂³×C₆₇	1	2	yes	—
685	#95	C₂³×C₆₇	1	2	yes	—
686	#95	C₂³×C₆₇	1	2	yes	—
687	#95	C₂³×C₆₇	1	1	yes	characteristic
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	2	yes	—
692	#95	C₂³×C₆₇	1	1	yes	characteristic
693	#95	C₂³×C₆₇	1	1	yes	characteristic
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	2	yes	—
700	#95	C₂³×C₆₇	1	2	yes	—
701	#95	C₂³×C₆₇	1	2	yes	—
702	#95	C₂³×C₆₇	1	2	yes	—
703	#95	C₂³×C₆₇	1	2	yes	—
704	#95	C₂³×C₆₇	1	2	yes	—
705	#95	C₂³×C₆₇	1	2	yes	—
706	C₂⁴	C₂⁴×C₆₇	1	1	yes	characteristic
707	C₂×Q₈	—	2	2	no	—
708	C₂×Q₈	—	2	2	no	—
709	C₂×Q₈	—	2	4	no	—
710	C₂×Q₈	—	2	4	no	—
711	C₂×Q₈	—	2	2	no	—
712	C₂×Q₈	—	2	2	no	—
713	C₂×Q₈	—	4	1	no	—
714	C₂×Q₈	—	4	2	no	—
715	C₂×Q₈	C₂×Q₈×C₆₇	1	4	yes	—
716	C₂²×C₄	—	2	2	no	—
717	C₂²×C₄	—	2	2	no	—
718	C₂²×C₄	—	2	4	no	—
719	C₂²×C₄	—	2	2	no	—
720	C₂²×C₄	—	2	2	no	—
721	C₂²×C₄	—	2	4	no	—
722	C₂²×C₄	—	2	4	no	—
723	C₂²×C₄	—	2	2	no	—
724	C₂²×C₄	—	2	2	no	—
725	C₂²×C₄	—	4	1	no	—
726	C₂²×C₄	—	4	1	no	—
727	C₂²×C₄	—	4	2	no	—
728	C₂²×C₄	—	4	2	no	—
729	C₂²×C₄	—	4	2	no	—
730	C₂²×C₄	C₂×Q₈×C₆₇	1	4	yes	—
731	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
732	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
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	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
737	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
738	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
739	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
740	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
741	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
742	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
743	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
744	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
745	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
746	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
747	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
748	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
749	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
750	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
751	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
752	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
753	C₄⋊C₄	—	2	4	no	—
754	C₄⋊C₄	—	2	2	no	—
755	C₄⋊C₄	—	2	4	no	—
756	C₄⋊C₄	—	2	4	no	—
757	C₄⋊C₄	—	2	2	no	—
758	C₄⋊C₄	—	2	2	no	—
759	C₄⋊C₄	—	2	4	no	—
760	C₄⋊C₄	—	2	4	no	—
761	C₄⋊C₄	—	2	2	no	—
762	C₄⋊C₄	—	2	4	no	—
763	C₄⋊C₄	—	2	2	no	—
764	C₄⋊C₄	—	2	4	no	—
765	C₄⋊C₄	—	2	2	no	—
766	C₄⋊C₄	—	2	4	no	—
767	C₄⋊C₄	—	2	2	no	—
768	C₄⋊C₄	—	2	2	no	—
769	C₄⋊C₄	—	2	2	no	—
770	C₄⋊C₄	—	2	2	no	—
771	C₄⋊C₄	—	2	4	no	—
772	C₄⋊C₄	—	2	2	no	—
773	C₄⋊C₄	—	4	2	no	—
774	C₄⋊C₄	—	4	2	no	—
775	C₄⋊C₄	—	4	2	no	—
776	C₄⋊C₄	—	4	1	no	—
777	C₄⋊C₄	—	4	1	no	—
778	C₄⋊C₄	—	4	2	no	—
779	C₄⋊C₄	—	4	2	no	—
780	C₄⋊C₄	—	4	1	no	—
781	C₄⋊C₄	—	4	2	no	—
782	C₄⋊C₄	—	4	2	no	—
783	C₄⋊C₄	—	4	2	no	—
784	C₄⋊C₄	—	4	2	no	—
785	C₄⋊C₄	—	4	2	no	—
786	C₄⋊C₄	—	4	2	no	—
787	C₄⋊C₄	—	4	2	no	—
788	C₄⋊C₄	C₂×Q₈×C₆₇	1	4	yes	—
789	C₄⋊C₄	C₂×Q₈×C₆₇	1	8	yes	—
790	C₄⋊C₄	#41×C₆₇	1	4	yes	—
791	C₂²⋊C₄	—	2	2	no	—
792	C₂²⋊C₄	—	2	4	no	—
793	C₂²⋊C₄	—	2	2	no	—
794	C₂²⋊C₄	—	2	2	no	—
795	C₂²⋊C₄	C₂×Q₈×C₆₇	1	8	yes	—
796	C₂²⋊C₄	#41×C₆₇	1	4	yes	—
797	C₄²	—	2	4	no	—
798	C₄²	—	2	2	no	—
799	C₄²	—	2	2	no	—
800	C₄²	—	2	2	no	—
801	C₄²	—	2	2	no	—
802	C₄²	—	2	2	no	—
803	C₄²	—	2	4	no	—
804	C₄²	—	2	2	no	—
805	C₄²	—	4	1	no	—
806	C₄²	—	4	1	no	—
807	C₄²	—	4	2	no	—
808	C₄²	—	4	2	no	—
809	C₄²	—	4	2	no	—
810	C₄²	—	4	2	no	—
811	C₂³	—	2	1	no	—
812	C₂³	—	2	1	no	—
813	C₂³	—	2	1	no	—
814	C₂³	—	2	1	no	—
815	C₂³	C₂²×Q₈×C₆₇	1	2	yes	—
816	C₂³	C₂×#41×C₆₇	1	2	yes	—
817	C₂³	2₋¹⁺⁴×C₆₇	1	2	yes	—
818	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
819	Q₈	—	2	8	no	—
820	Q₈	—	4	8	no	—
821	Q₈	—	4	4	no	—
822	C₂×C₄	—	2	2	no	—
823	C₂×C₄	—	2	1	no	—
824	C₂×C₄	—	2	1	no	—
825	C₂×C₄	—	2	2	no	—
826	C₂×C₄	—	2	1	no	—
827	C₂×C₄	—	2	1	no	—
828	C₂×C₄	—	2	2	no	—
829	C₂×C₄	—	2	1	no	—
830	C₂×C₄	—	2	1	no	—
831	C₂×C₄	—	2	1	no	—
832	C₂×C₄	—	2	1	no	—
833	C₂×C₄	—	2	2	no	—
834	C₂×C₄	—	2	2	no	—
835	C₂×C₄	—	2	2	no	—
836	C₂×C₄	—	2	1	no	—
837	C₂×C₄	—	2	1	no	—
838	C₂×C₄	—	2	8	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	2	no	—
845	C₂×C₄	—	2	1	no	—
846	C₂×C₄	—	2	2	no	—
847	C₂×C₄	—	2	2	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	1	no	—
853	C₂×C₄	—	2	1	no	—
854	C₂×C₄	—	2	2	no	—
855	C₂×C₄	—	2	1	no	—
856	C₂×C₄	—	2	1	no	—
857	C₂×C₄	—	2	2	no	—
858	C₂×C₄	—	2	2	no	—
859	C₂×C₄	—	2	2	no	—
860	C₂×C₄	—	2	1	no	—
861	C₂×C₄	—	2	1	no	—
862	C₂×C₄	—	2	2	no	—
863	C₂×C₄	—	2	2	no	—
864	C₂×C₄	—	2	2	no	—
865	C₂×C₄	—	2	1	no	—
866	C₂×C₄	—	2	1	no	—
867	C₂×C₄	—	2	1	no	—
868	C₂×C₄	—	2	1	no	—
869	C₂×C₄	—	2	1	no	—
870	C₂×C₄	—	2	1	no	—
871	C₂×C₄	—	2	1	no	—
872	C₂×C₄	—	2	2	no	—
873	C₂×C₄	—	2	1	no	—
874	C₂×C₄	—	2	2	no	—
875	C₂×C₄	—	2	2	no	—
876	C₂×C₄	—	2	1	no	—
877	C₂×C₄	—	2	1	no	—
878	C₂×C₄	—	4	4	no	—
879	C₂×C₄	—	4	8	no	—
880	C₂×C₄	—	4	4	no	—
881	C₂×C₄	—	8	4	no	—
882	C₂×C₄	C₂²×Q₈×C₆₇	1	4	yes	—
883	C₂×C₄	C₂²×Q₈×C₆₇	1	4	yes	—
884	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
885	C₂×C₄	C₂²×Q₈×C₆₇	1	2	yes	—
886	C₂×C₄	C₂×#41×C₆₇	1	4	yes	—
887	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
888	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
889	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
890	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
891	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
892	C₂×C₄	2₋¹⁺⁴×C₆₇	1	2	yes	—
893	C₂²	—	2	2	no	—
894	C₂²	—	2	2	no	—
895	C₂²	—	2	2	no	—
896	C₂²	—	2	2	no	—
897	C₂²	—	2	2	no	—
898	C₂²	—	2	2	no	—
899	C₂²	#557×C₆₇	1	4	yes	—
900	C₂²	C₂³×Q₈×C₆₇	1	1	yes	characteristic
901	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
902	C₂²	C₂²×#41×C₆₇	1	1	yes	characteristic
903	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
904	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
905	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
906	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
907	C₄	—	2	2	no	—
908	C₄	—	2	2	no	—
909	C₄	—	2	4	no	—
910	C₄	—	2	4	no	—
911	C₄	—	2	2	no	—
912	C₄	—	2	4	no	—
913	C₄	—	2	2	no	—
914	C₄	—	2	2	no	—
915	C₄	—	4	1	no	—
916	C₄	—	4	1	no	—
917	C₄	—	4	1	no	—
918	C₄	—	4	2	no	—
919	C₄	—	4	2	no	—
920	C₄	—	4	1	no	—
921	C₄	—	4	1	no	—
922	C₄	—	4	2	no	—
923	C₄	—	4	2	no	—
924	C₄	—	4	1	no	—
925	C₄	—	4	1	no	—
926	C₄	—	4	1	no	—
927	C₄	—	4	2	no	—
928	C₄	#554×C₆₇	1	4	yes	—
929	C₂	—	2	4	no	—
930	C₂	C₂×#554×C₆₇	1	1	yes	characteristic
931	C₂	C₂×#557×C₆₇	1	1	yes	characteristic
932	C₂	C₆₇×#3481	1	1	yes	characteristic
933	C₂	C₆₇×#3503	1	1	yes	characteristic
934	C₂	C₆₇×#3520	1	1	yes	characteristic
935	C₂	C₆₇×#3523	1	1	yes	characteristic
936	C₂	C₆₇×#3536	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. C₆₇×#2802	C₂	semidirect
2. C₆₇×#2755	C₂	semidirect
3. C₆₇×#2588	C₂	semidirect
4. C₆₇×#2584	C₂	semidirect
5. C₆₇×#2584	C₂	semidirect
6. C₆₇×#2582	C₂	semidirect
8. C₆₇×#2509	C₂	semidirect
9. C₆₇×#2506	C₂	semidirect
12. C₆₇×#2453	C₂	semidirect
13. C₆₇×#2451	C₂	semidirect
16. C₆₇×#2365	C₂	semidirect
20. C₆₇×#2351	C₂	semidirect
136. C₂²⋊Q₈×C₆₇	Q₈	semidirect
460. #59663	C₆₇	direct (N × H)
469. #2802	C₂×C₆₇	semidirect
470. #2755	C₂×C₆₇	semidirect
471. #2588	C₂×C₆₇	semidirect
472. #2584	C₂×C₆₇	semidirect
473. #2584	C₂×C₆₇	semidirect
474. #2582	C₂×C₆₇	semidirect
476. #2509	C₂×C₆₇	semidirect
477. #2506	C₂×C₆₇	semidirect
480. #2453	C₂×C₆₇	semidirect
481. #2451	C₂×C₆₇	semidirect
484. #2365	C₂×C₆₇	semidirect
488. #2351	C₂×C₆₇	semidirect
492. C₆₇	#59663	direct (N × H)
605. C₂²⋊Q₈	Q₈×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₆₇×#2802	2
C₆₇×#2755	2
C₆₇×#2588	2
C₆₇×#2584	2
C₆₇×#2582	2
C₆₇×#2549	2
C₆₇×#2509	2
C₆₇×#2506	2
C₆₇×#2505	2
C₆₇×#2473	2
C₆₇×#2453	2
C₆₇×#2451	2
C₆₇×#2442	2
C₆₇×#2440	2
C₆₇×#2365	2
C₆₇×#2360	2
C₆₇×#2351	2
C₄×C₂²⋊Q₈×C₆₇	2
C₂×#389×C₆₇	2
#59663	67

Subgroups — C67×#59663 #30738914

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#59663 #30738914