Subgroups — C₆₇×#34521 #30713772

Order 17152.

2008 conjugacy classes of subgroups.

The subgroup lattice has 810 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#563×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₆₇×#2383	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2381	C₂	1	1	yes	maximal, characteristic, complemented
4	C₆₇×#2373	C₂	1	2	yes	maximal, complemented
5	C₆₇×#2372	C₂	1	1	yes	maximal, characteristic
6	C₆₇×#2371	C₂	1	2	yes	maximal, complemented
7	C₆₇×#2369	C₂	1	1	yes	maximal, characteristic, complemented
8	C₆₇×#2368	C₂	1	2	yes	maximal, complemented
9	C₆₇×#2362	C₂	1	2	yes	maximal, complemented
10	C₆₇×#2359	C₂	1	2	yes	maximal
11	C₆₇×#2335	C₂	1	2	yes	maximal
12	C₆₇×#2334	C₂	1	2	yes	maximal, complemented
13	C₆₇×#2330	C₂	1	2	yes	maximal, complemented
14	C₆₇×#2329	C₂	1	2	yes	maximal, complemented
15	C₆₇×#2327	C₂	1	1	yes	maximal, characteristic, complemented
16	C₆₇×#2326	C₂	1	1	yes	maximal, characteristic
17	C₆₇×#2323	C₂	1	2	yes	maximal, complemented
18	C₄×C₄⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
19	C₄×#126×C₆₇	C₂	1	2	yes	maximal, complemented
20	#563×C₆₇	C₄	1	16	yes	complemented
21	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
22	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₂×#126×C₆₇	C₂²	1	2	yes	—
24	C₂×#126×C₆₇	C₂²	1	2	yes	—
25	C₂×#125×C₆₇	C₂²	1	2	yes	—
26	C₂×#124×C₆₇	C₂²	1	1	yes	characteristic
27	C₂×#123×C₆₇	C₂²	1	2	yes	—
28	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
29	C₂×C₂²⋊Q₈×C₆₇	C₂²	1	2	yes	—
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×D₄×C₆₇	C₂²	1	1	yes	characteristic
34	C₂×#117×C₆₇	C₂²	1	2	yes	—
35	#391×C₆₇	C₂²	1	1	yes	characteristic
36	#391×C₆₇	C₂²	1	1	yes	characteristic
37	#391×C₆₇	C₂²	1	2	yes	—
38	#391×C₆₇	C₂²	1	1	yes	characteristic
39	#391×C₆₇	C₂²	1	1	yes	characteristic
40	#391×C₆₇	C₂²	1	1	yes	characteristic
41	#391×C₆₇	C₂²	1	2	yes	—
42	#390×C₆₇	C₂²	1	1	yes	characteristic
43	#389×C₆₇	C₂²	1	2	yes	—
44	#389×C₆₇	C₂²	1	2	yes	—
45	#389×C₆₇	C₂²	1	2	yes	—
46	#389×C₆₇	C₂²	1	2	yes	—
47	#389×C₆₇	C₂²	1	2	yes	—
48	#389×C₆₇	C₂²	1	2	yes	—
49	#389×C₆₇	C₂²	1	2	yes	—
50	#389×C₆₇	C₂²	1	2	yes	—
51	#388×C₆₇	C₂²	1	2	yes	—
52	#388×C₆₇	C₂²	1	2	yes	—
53	#388×C₆₇	C₂²	1	2	yes	—
54	#388×C₆₇	C₂²	1	2	yes	—
55	#388×C₆₇	C₂²	1	2	yes	—
56	#388×C₆₇	C₂²	1	2	yes	—
57	#388×C₆₇	C₂²	1	2	yes	—
58	#388×C₆₇	C₂²	1	2	yes	—
59	#388×C₆₇	C₂²	1	2	yes	—
60	#388×C₆₇	C₂²	1	2	yes	—
61	#387×C₆₇	C₂²	1	2	yes	—
62	#387×C₆₇	C₂²	1	2	yes	—
63	#387×C₆₇	C₂²	1	2	yes	—
64	#387×C₆₇	C₂²	1	2	yes	—
65	#387×C₆₇	C₂²	1	2	yes	—
66	#387×C₆₇	C₂²	1	2	yes	—
67	#387×C₆₇	C₂²	1	2	yes	—
68	#387×C₆₇	C₂²	1	2	yes	—
69	#387×C₆₇	C₂²	1	2	yes	—
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	#386×C₆₇	C₂²	1	2	yes	—
78	#385×C₆₇	C₂²	1	2	yes	—
79	#385×C₆₇	C₂²	1	2	yes	—
80	#385×C₆₇	C₂²	1	2	yes	—
81	#385×C₆₇	C₂²	1	2	yes	—
82	#385×C₆₇	C₂²	1	2	yes	—
83	#384×C₆₇	C₂²	1	1	yes	characteristic
84	#384×C₆₇	C₂²	1	1	yes	characteristic
85	#383×C₆₇	C₂²	1	2	yes	—
86	#382×C₆₇	C₂²	1	1	yes	characteristic
87	#380×C₆₇	C₂²	1	2	yes	—
88	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
89	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
90	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
91	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
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	2	yes	—
95	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
96	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
97	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
98	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
99	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
100	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
101	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
102	C₄×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
103	#376×C₆₇	C₂²	1	1	yes	characteristic
104	#376×C₆₇	C₂²	1	1	yes	characteristic
105	#376×C₆₇	C₂²	1	2	yes	—
106	C₂×#95×C₆₇	C₂²	1	2	yes	—
107	C₄³×C₆₇	C₂²	1	1	yes	characteristic
108	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
109	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
110	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
111	C₂²×D₄×C₆₇	C₂³	1	1	yes	characteristic
112	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
113	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
114	C₂³×C₄×C₆₇	C₂³	1	2	yes	—
115	C₄⋊Q₈×C₆₇	C₂×C₄	1	8	yes	complemented
116	C₄⋊Q₈×C₆₇	C₂×C₄	1	8	yes	complemented
117	#126×C₆₇	C₂×C₄	1	16	yes	complemented
118	#126×C₆₇	C₂×C₄	1	16	yes	complemented
119	#125×C₆₇	C₂×C₄	1	16	yes	complemented
120	#124×C₆₇	C₂×C₄	1	8	yes	—
121	#123×C₆₇	C₂×C₄	1	16	yes	—
122	C₂²⋊Q₈×C₆₇	C₂×C₄	1	16	yes	complemented
123	C₂²⋊Q₈×C₆₇	C₂×C₄	1	16	yes	complemented
124	C₄×Q₈×C₆₇	—	4	2	no	—
125	C₄×Q₈×C₆₇	—	4	2	no	—
126	C₄×Q₈×C₆₇	—	4	2	no	—
127	C₄×D₄×C₆₇	—	4	2	no	—
128	#117×C₆₇	—	4	4	no	—
129	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
130	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
131	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
132	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
133	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
134	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
135	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
136	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
137	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
138	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
139	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
140	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
141	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
142	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
143	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
144	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
145	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
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	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	2	yes	—
153	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
154	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
155	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
156	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
157	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
158	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
159	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
160	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
161	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
162	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
163	C₂×C₂²⋊C₄×C₆₇	C₂³	1	2	yes	—
164	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
165	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
166	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
167	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
168	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
169	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
170	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
171	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
172	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
173	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
174	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
175	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
176	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
177	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
178	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
179	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
180	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
181	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
182	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
183	#95×C₆₇	C₂³	1	2	yes	—
184	#95×C₆₇	C₂³	1	2	yes	—
185	#95×C₆₇	C₂³	1	2	yes	—
186	#95×C₆₇	C₂³	1	2	yes	—
187	#95×C₆₇	C₂³	1	2	yes	—
188	#95×C₆₇	C₂³	1	2	yes	—
189	#95×C₆₇	C₂³	1	2	yes	—
190	#95×C₆₇	C₂³	1	2	yes	—
191	#95×C₆₇	C₂³	1	2	yes	—
192	#95×C₆₇	C₂³	1	2	yes	—
193	#95×C₆₇	C₂³	1	2	yes	—
194	#95×C₆₇	C₂³	1	2	yes	—
195	#95×C₆₇	C₂³	1	2	yes	—
196	#95×C₆₇	C₂³	1	2	yes	—
197	#95×C₆₇	C₂³	1	2	yes	—
198	#95×C₆₇	C₂³	1	2	yes	—
199	#95×C₆₇	C₂³	1	2	yes	—
200	#95×C₆₇	C₂³	1	2	yes	—
201	#95×C₆₇	C₂³	1	2	yes	—
202	#95×C₆₇	C₂³	1	2	yes	—
203	#95×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	C₂⁴×C₆₇	C₂⁴	1	2	yes	—
210	C₂×Q₈×C₆₇	—	4	1	no	—
211	C₂×Q₈×C₆₇	—	4	1	no	—
212	C₂×Q₈×C₆₇	—	4	1	no	—
213	C₂×Q₈×C₆₇	—	4	1	no	—
214	C₂×Q₈×C₆₇	—	4	1	no	—
215	C₂×Q₈×C₆₇	—	4	1	no	—
216	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
217	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
218	C₂×Q₈×C₆₇	C₂²×C₄	1	4	yes	—
219	C₂×D₄×C₆₇	—	4	1	no	—
220	C₂×D₄×C₆₇	—	4	1	no	—
221	C₂×D₄×C₆₇	C₂²×C₄	1	4	yes	—
222	C₂²×C₄×C₆₇	—	2	4	no	—
223	C₂²×C₄×C₆₇	—	4	2	no	—
224	C₂²×C₄×C₆₇	—	4	2	no	—
225	C₂²×C₄×C₆₇	—	4	2	no	—
226	C₂²×C₄×C₆₇	—	4	2	no	—
227	C₂²×C₄×C₆₇	—	4	2	no	—
228	C₂²×C₄×C₆₇	—	4	2	no	—
229	C₂²×C₄×C₆₇	—	4	2	no	—
230	C₂²×C₄×C₆₇	C₂²×C₄	1	8	yes	—
231	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
232	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
233	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
234	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
235	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
236	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
237	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
238	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
239	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
240	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
241	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
242	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
243	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
244	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
245	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
246	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
247	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
248	C₂²×C₄×C₆₇	C₂⁴	1	2	yes	—
249	C₄⋊C₄×C₆₇	—	2	4	no	—
250	C₄⋊C₄×C₆₇	—	2	4	no	—
251	C₄⋊C₄×C₆₇	—	4	2	no	—
252	C₄⋊C₄×C₆₇	—	4	2	no	—
253	C₄⋊C₄×C₆₇	—	4	2	no	—
254	C₄⋊C₄×C₆₇	—	4	2	no	—
255	C₄⋊C₄×C₆₇	—	4	2	no	—
256	C₄⋊C₄×C₆₇	—	4	2	no	—
257	C₄⋊C₄×C₆₇	—	4	2	no	—
258	C₄⋊C₄×C₆₇	—	4	1	no	—
259	C₄⋊C₄×C₆₇	—	4	2	no	—
260	C₄⋊C₄×C₆₇	—	4	1	no	—
261	C₄⋊C₄×C₆₇	—	4	1	no	—
262	C₄⋊C₄×C₆₇	—	4	1	no	—
263	C₄⋊C₄×C₆₇	—	4	1	no	—
264	C₄⋊C₄×C₆₇	—	4	1	no	—
265	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
266	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
267	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
268	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
269	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
270	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
271	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
272	C₄⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
273	C₂²⋊C₄×C₆₇	—	2	4	no	—
274	C₂²⋊C₄×C₆₇	—	2	4	no	—
275	C₂²⋊C₄×C₆₇	—	4	2	no	—
276	C₂²⋊C₄×C₆₇	—	4	2	no	—
277	C₂²⋊C₄×C₆₇	—	4	2	no	—
278	C₂²⋊C₄×C₆₇	—	4	2	no	—
279	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
280	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
281	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
282	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
283	C₂²⋊C₄×C₆₇	C₂²×C₄	1	8	yes	—
284	C₄²×C₆₇	—	2	4	no	—
285	C₄²×C₆₇	—	2	2	no	—
286	C₄²×C₆₇	—	2	2	no	—
287	C₄²×C₆₇	—	2	2	no	—
288	C₄²×C₆₇	—	2	4	no	—
289	C₄²×C₆₇	—	2	4	no	—
290	C₄²×C₆₇	—	4	2	no	—
291	C₄²×C₆₇	—	4	2	no	—
292	C₄²×C₆₇	—	4	1	no	—
293	C₄²×C₆₇	—	4	2	no	—
294	C₄²×C₆₇	—	4	2	no	—
295	C₄²×C₆₇	—	4	2	no	—
296	C₄²×C₆₇	—	4	1	no	—
297	C₄²×C₆₇	—	4	1	no	—
298	C₄²×C₆₇	—	4	2	no	—
299	C₄²×C₆₇	—	4	2	no	—
300	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
301	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
302	C₄²×C₆₇	C₂²×C₄	1	4	yes	—
303	C₂³×C₆₇	—	2	2	no	—
304	C₂³×C₆₇	—	2	2	no	—
305	C₂³×C₆₇	—	2	2	no	—
306	C₂³×C₆₇	—	2	2	no	—
307	C₂³×C₆₇	—	2	2	no	—
308	C₂³×C₆₇	—	2	2	no	—
309	C₂³×C₆₇	C₂³×C₄	1	4	yes	—
310	C₂³×C₆₇	C₂⁵	1	1	yes	characteristic
311	Q₈×C₆₇	—	4	4	no	—
312	Q₈×C₆₇	—	4	4	no	—
313	Q₈×C₆₇	—	4	4	no	—
314	D₄×C₆₇	—	4	4	no	—
315	C₂×C₄×C₆₇	—	2	2	no	—
316	C₂×C₄×C₆₇	—	2	1	no	—
317	C₂×C₄×C₆₇	—	2	1	no	—
318	C₂×C₄×C₆₇	—	2	1	no	—
319	C₂×C₄×C₆₇	—	2	2	no	—
320	C₂×C₄×C₆₇	—	2	2	no	—
321	C₂×C₄×C₆₇	—	2	2	no	—
322	C₂×C₄×C₆₇	—	2	2	no	—
323	C₂×C₄×C₆₇	—	2	2	no	—
324	C₂×C₄×C₆₇	—	2	2	no	—
325	C₂×C₄×C₆₇	—	2	2	no	—
326	C₂×C₄×C₆₇	—	2	2	no	—
327	C₂×C₄×C₆₇	—	2	2	no	—
328	C₂×C₄×C₆₇	—	2	2	no	—
329	C₂×C₄×C₆₇	—	2	2	no	—
330	C₂×C₄×C₆₇	—	2	1	no	—
331	C₂×C₄×C₆₇	—	2	1	no	—
332	C₂×C₄×C₆₇	—	2	1	no	—
333	C₂×C₄×C₆₇	—	2	2	no	—
334	C₂×C₄×C₆₇	—	2	1	no	—
335	C₂×C₄×C₆₇	—	2	2	no	—
336	C₂×C₄×C₆₇	—	2	1	no	—
337	C₂×C₄×C₆₇	—	2	1	no	—
338	C₂×C₄×C₆₇	—	2	2	no	—
339	C₂×C₄×C₆₇	—	2	2	no	—
340	C₂×C₄×C₆₇	—	2	2	no	—
341	C₂×C₄×C₆₇	—	2	2	no	—
342	C₂×C₄×C₆₇	—	2	2	no	—
343	C₂×C₄×C₆₇	—	2	2	no	—
344	C₂×C₄×C₆₇	—	2	2	no	—
345	C₂×C₄×C₆₇	—	2	2	no	—
346	C₂×C₄×C₆₇	—	2	2	no	—
347	C₂×C₄×C₆₇	—	2	2	no	—
348	C₂×C₄×C₆₇	—	2	2	no	—
349	C₂×C₄×C₆₇	—	2	2	no	—
350	C₂×C₄×C₆₇	—	2	2	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	—	4	8	no	—
356	C₂×C₄×C₆₇	—	8	4	no	—
357	C₂×C₄×C₆₇	—	8	4	no	—
358	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
359	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
360	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
361	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
362	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
363	C₂×C₄×C₆₇	C₂³×C₄	1	2	yes	—
364	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
365	C₂×C₄×C₆₇	C₂³×C₄	1	4	yes	—
366	C₂×C₄×C₆₇	H(5,𝔽₂)	1	2	yes	—
367	C₂×C₄×C₆₇	2₋¹⁺⁴	1	4	yes	—
368	C₂²×C₆₇	—	2	4	no	—
369	C₂²×C₆₇	—	2	4	no	—
370	C₂²×C₆₇	—	2	4	no	—
371	C₂²×C₆₇	—	4	2	no	—
372	C₂²×C₆₇	—	4	2	no	—
373	C₂²×C₆₇	—	4	2	no	—
374	C₂²×C₆₇	—	4	2	no	—
375	C₂²×C₆₇	C₂⁴×C₄	1	1	yes	characteristic
376	C₂²×C₆₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic
377	C₂²×C₆₇	C₂×2₋¹⁺⁴	1	2	yes	—
378	C₂²×C₆₇	#585	1	1	yes	characteristic
379	C₂²×C₆₇	#585	1	2	yes	—
380	C₄×C₆₇	—	2	4	no	—
381	C₄×C₆₇	—	2	4	no	—
382	C₄×C₆₇	—	2	2	no	—
383	C₄×C₆₇	—	2	2	no	—
384	C₄×C₆₇	—	2	2	no	—
385	C₄×C₆₇	—	2	4	no	—
386	C₄×C₆₇	—	2	4	no	—
387	C₄×C₆₇	—	2	4	no	—
388	C₄×C₆₇	—	2	2	no	—
389	C₄×C₆₇	—	2	4	no	—
390	C₄×C₆₇	—	4	2	no	—
391	C₄×C₆₇	—	4	1	no	—
392	C₄×C₆₇	—	4	2	no	—
393	C₄×C₆₇	—	4	1	no	—
394	C₄×C₆₇	—	4	2	no	—
395	C₄×C₆₇	—	4	1	no	—
396	C₄×C₆₇	—	4	2	no	—
397	C₄×C₆₇	—	4	2	no	—
398	#34521	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
399	C₂×C₆₇	—	4	4	no	—
400	C₂×C₆₇	C₄×H(5,𝔽₂)	1	1	yes	characteristic
401	C₂×C₆₇	C₄×2₋¹⁺⁴	1	2	yes	—
402	C₂×C₆₇	#3544	1	1	yes	characteristic
403	C₂×C₆₇	#3551	1	1	yes	characteristic
404	C₂×C₆₇	#3564	1	2	yes	—
405	C₂×#563	C₂×C₆₇	1	1	yes	characteristic
406	#2383	C₂×C₆₇	1	2	yes	complemented
407	#2381	C₂×C₆₇	1	1	yes	characteristic, complemented
408	#2373	C₂×C₆₇	1	2	yes	complemented
409	#2372	C₂×C₆₇	1	1	yes	characteristic
410	#2371	C₂×C₆₇	1	2	yes	complemented
411	#2369	C₂×C₆₇	1	1	yes	characteristic, complemented
412	#2368	C₂×C₆₇	1	2	yes	complemented
413	#2362	C₂×C₆₇	1	2	yes	complemented
414	#2359	C₂×C₆₇	1	2	yes	—
415	#2335	C₂×C₆₇	1	2	yes	—
416	#2334	C₂×C₆₇	1	2	yes	complemented
417	#2330	C₂×C₆₇	1	2	yes	complemented
418	#2329	C₂×C₆₇	1	2	yes	complemented
419	#2327	C₂×C₆₇	1	1	yes	characteristic, complemented
420	#2326	C₂×C₆₇	1	1	yes	characteristic
421	#2323	C₂×C₆₇	1	2	yes	complemented
422	C₄×C₄⋊Q₈	C₂×C₆₇	1	1	yes	characteristic, complemented
423	C₄×#126	C₂×C₆₇	1	2	yes	complemented
424	C₆₇	#34521	1	1	yes	characteristic, direct product factor, Sylow
425	#563	C₄×C₆₇	1	16	yes	complemented
426	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
427	C₂×C₄⋊Q₈	C₂²×C₆₇	1	1	yes	characteristic
428	C₂×#126	C₂²×C₆₇	1	2	yes	—
429	C₂×#126	C₂²×C₆₇	1	2	yes	—
430	C₂×#125	C₂²×C₆₇	1	2	yes	—
431	C₂×#124	C₂²×C₆₇	1	1	yes	characteristic
432	C₂×#123	C₂²×C₆₇	1	2	yes	—
433	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
434	C₂×C₂²⋊Q₈	C₂²×C₆₇	1	2	yes	—
435	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
436	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
437	C₂×C₄×Q₈	C₂²×C₆₇	1	1	yes	characteristic
438	C₂×C₄×D₄	C₂²×C₆₇	1	1	yes	characteristic
439	C₂×#117	C₂²×C₆₇	1	2	yes	—
440	#391	C₂²×C₆₇	1	1	yes	characteristic
441	#391	C₂²×C₆₇	1	1	yes	characteristic
442	#391	C₂²×C₆₇	1	2	yes	—
443	#391	C₂²×C₆₇	1	1	yes	characteristic
444	#391	C₂²×C₆₇	1	1	yes	characteristic
445	#391	C₂²×C₆₇	1	1	yes	characteristic
446	#391	C₂²×C₆₇	1	2	yes	—
447	#390	C₂²×C₆₇	1	1	yes	characteristic
448	#389	C₂²×C₆₇	1	2	yes	—
449	#389	C₂²×C₆₇	1	2	yes	—
450	#389	C₂²×C₆₇	1	2	yes	—
451	#389	C₂²×C₆₇	1	2	yes	—
452	#389	C₂²×C₆₇	1	2	yes	—
453	#389	C₂²×C₆₇	1	2	yes	—
454	#389	C₂²×C₆₇	1	2	yes	—
455	#389	C₂²×C₆₇	1	2	yes	—
456	#388	C₂²×C₆₇	1	2	yes	—
457	#388	C₂²×C₆₇	1	2	yes	—
458	#388	C₂²×C₆₇	1	2	yes	—
459	#388	C₂²×C₆₇	1	2	yes	—
460	#388	C₂²×C₆₇	1	2	yes	—
461	#388	C₂²×C₆₇	1	2	yes	—
462	#388	C₂²×C₆₇	1	2	yes	—
463	#388	C₂²×C₆₇	1	2	yes	—
464	#388	C₂²×C₆₇	1	2	yes	—
465	#388	C₂²×C₆₇	1	2	yes	—
466	#387	C₂²×C₆₇	1	2	yes	—
467	#387	C₂²×C₆₇	1	2	yes	—
468	#387	C₂²×C₆₇	1	2	yes	—
469	#387	C₂²×C₆₇	1	2	yes	—
470	#387	C₂²×C₆₇	1	2	yes	—
471	#387	C₂²×C₆₇	1	2	yes	—
472	#387	C₂²×C₆₇	1	2	yes	—
473	#387	C₂²×C₆₇	1	2	yes	—
474	#387	C₂²×C₆₇	1	2	yes	—
475	#387	C₂²×C₆₇	1	2	yes	—
476	#387	C₂²×C₆₇	1	2	yes	—
477	#387	C₂²×C₆₇	1	2	yes	—
478	#387	C₂²×C₆₇	1	2	yes	—
479	#387	C₂²×C₆₇	1	2	yes	—
480	#387	C₂²×C₆₇	1	2	yes	—
481	#387	C₂²×C₆₇	1	2	yes	—
482	#386	C₂²×C₆₇	1	2	yes	—
483	#385	C₂²×C₆₇	1	2	yes	—
484	#385	C₂²×C₆₇	1	2	yes	—
485	#385	C₂²×C₆₇	1	2	yes	—
486	#385	C₂²×C₆₇	1	2	yes	—
487	#385	C₂²×C₆₇	1	2	yes	—
488	#384	C₂²×C₆₇	1	1	yes	characteristic
489	#384	C₂²×C₆₇	1	1	yes	characteristic
490	#383	C₂²×C₆₇	1	2	yes	—
491	#382	C₂²×C₆₇	1	1	yes	characteristic
492	#380	C₂²×C₆₇	1	2	yes	—
493	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
494	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
495	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
496	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
497	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
498	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
499	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
500	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
501	C₄×C₄⋊C₄	C₂²×C₆₇	1	2	yes	—
502	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
503	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
504	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
505	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
506	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
507	C₄×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
508	#376	C₂²×C₆₇	1	1	yes	characteristic
509	#376	C₂²×C₆₇	1	1	yes	characteristic
510	#376	C₂²×C₆₇	1	2	yes	—
511	C₂×#95	C₂²×C₆₇	1	2	yes	—
512	C₄³	C₂²×C₆₇	1	1	yes	characteristic
513	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
514	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
515	C₂²×Q₈	C₂³×C₆₇	1	1	yes	characteristic
516	C₂²×D₄	C₂³×C₆₇	1	1	yes	characteristic
517	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
518	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
519	C₂³×C₄	C₂³×C₆₇	1	2	yes	—
520	C₄⋊Q₈	C₂×C₄×C₆₇	1	8	yes	complemented
521	C₄⋊Q₈	C₂×C₄×C₆₇	1	8	yes	complemented
522	#126	C₂×C₄×C₆₇	1	16	yes	complemented
523	#126	C₂×C₄×C₆₇	1	16	yes	complemented
524	#125	C₂×C₄×C₆₇	1	16	yes	complemented
525	#124	C₂×C₄×C₆₇	1	8	yes	—
526	#123	C₂×C₄×C₆₇	1	16	yes	—
527	C₂²⋊Q₈	C₂×C₄×C₆₇	1	16	yes	complemented
528	C₂²⋊Q₈	C₂×C₄×C₆₇	1	16	yes	complemented
529	C₄×Q₈	—	4	2	no	—
530	C₄×Q₈	—	4	2	no	—
531	C₄×Q₈	—	4	2	no	—
532	C₄×D₄	—	4	2	no	—
533	#117	—	4	4	no	—
534	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
535	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
536	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
537	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
538	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
539	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
540	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
541	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
542	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
543	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
544	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
545	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
546	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
547	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
548	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
549	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
550	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
551	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
552	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
553	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
554	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
555	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
556	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
557	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
558	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
559	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
560	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
561	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
562	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
563	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
564	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
565	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
566	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
567	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
568	C₂×C₂²⋊C₄	C₂³×C₆₇	1	2	yes	—
569	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
570	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
571	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
572	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
573	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
574	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
575	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
576	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
577	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
578	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
579	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
580	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
581	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
582	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
583	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
584	C₂×C₄²	C₂³×C₆₇	1	2	yes	—
585	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
586	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
587	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
588	#95	C₂³×C₆₇	1	2	yes	—
589	#95	C₂³×C₆₇	1	2	yes	—
590	#95	C₂³×C₆₇	1	2	yes	—
591	#95	C₂³×C₆₇	1	2	yes	—
592	#95	C₂³×C₆₇	1	2	yes	—
593	#95	C₂³×C₆₇	1	2	yes	—
594	#95	C₂³×C₆₇	1	2	yes	—
595	#95	C₂³×C₆₇	1	2	yes	—
596	#95	C₂³×C₆₇	1	2	yes	—
597	#95	C₂³×C₆₇	1	2	yes	—
598	#95	C₂³×C₆₇	1	2	yes	—
599	#95	C₂³×C₆₇	1	2	yes	—
600	#95	C₂³×C₆₇	1	2	yes	—
601	#95	C₂³×C₆₇	1	2	yes	—
602	#95	C₂³×C₆₇	1	2	yes	—
603	#95	C₂³×C₆₇	1	2	yes	—
604	#95	C₂³×C₆₇	1	2	yes	—
605	#95	C₂³×C₆₇	1	2	yes	—
606	#95	C₂³×C₆₇	1	2	yes	—
607	#95	C₂³×C₆₇	1	2	yes	—
608	#95	C₂³×C₆₇	1	2	yes	—
609	#95	C₂³×C₆₇	1	2	yes	—
610	#95	C₂³×C₆₇	1	2	yes	—
611	#95	C₂³×C₆₇	1	2	yes	—
612	#95	C₂³×C₆₇	1	2	yes	—
613	#95	C₂³×C₆₇	1	2	yes	—
614	C₂⁴	C₂⁴×C₆₇	1	2	yes	—
615	C₂×Q₈	—	4	1	no	—
616	C₂×Q₈	—	4	1	no	—
617	C₂×Q₈	—	4	1	no	—
618	C₂×Q₈	—	4	1	no	—
619	C₂×Q₈	—	4	1	no	—
620	C₂×Q₈	—	4	1	no	—
621	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
622	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
623	C₂×Q₈	C₂²×C₄×C₆₇	1	4	yes	—
624	C₂×D₄	—	4	1	no	—
625	C₂×D₄	—	4	1	no	—
626	C₂×D₄	C₂²×C₄×C₆₇	1	4	yes	—
627	C₂²×C₄	—	2	4	no	—
628	C₂²×C₄	—	4	2	no	—
629	C₂²×C₄	—	4	2	no	—
630	C₂²×C₄	—	4	2	no	—
631	C₂²×C₄	—	4	2	no	—
632	C₂²×C₄	—	4	2	no	—
633	C₂²×C₄	—	4	2	no	—
634	C₂²×C₄	—	4	2	no	—
635	C₂²×C₄	C₂²×C₄×C₆₇	1	8	yes	—
636	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
637	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
638	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
639	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
640	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
641	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
642	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
643	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
644	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
645	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
646	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
647	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
648	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
649	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
650	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
651	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
652	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
653	C₂²×C₄	C₂⁴×C₆₇	1	2	yes	—
654	C₄⋊C₄	—	2	4	no	—
655	C₄⋊C₄	—	2	4	no	—
656	C₄⋊C₄	—	4	2	no	—
657	C₄⋊C₄	—	4	1	no	—
658	C₄⋊C₄	—	4	2	no	—
659	C₄⋊C₄	—	4	1	no	—
660	C₄⋊C₄	—	4	1	no	—
661	C₄⋊C₄	—	4	1	no	—
662	C₄⋊C₄	—	4	1	no	—
663	C₄⋊C₄	—	4	1	no	—
664	C₄⋊C₄	—	4	2	no	—
665	C₄⋊C₄	—	4	2	no	—
666	C₄⋊C₄	—	4	2	no	—
667	C₄⋊C₄	—	4	2	no	—
668	C₄⋊C₄	—	4	2	no	—
669	C₄⋊C₄	—	4	2	no	—
670	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
671	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
672	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
673	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
674	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
675	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
676	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
677	C₄⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
678	C₂²⋊C₄	—	2	4	no	—
679	C₂²⋊C₄	—	2	4	no	—
680	C₂²⋊C₄	—	4	2	no	—
681	C₂²⋊C₄	—	4	2	no	—
682	C₂²⋊C₄	—	4	2	no	—
683	C₂²⋊C₄	—	4	2	no	—
684	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
685	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
686	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
687	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
688	C₂²⋊C₄	C₂²×C₄×C₆₇	1	8	yes	—
689	C₄²	—	2	4	no	—
690	C₄²	—	2	2	no	—
691	C₄²	—	2	2	no	—
692	C₄²	—	2	2	no	—
693	C₄²	—	2	4	no	—
694	C₄²	—	2	4	no	—
695	C₄²	—	4	2	no	—
696	C₄²	—	4	2	no	—
697	C₄²	—	4	2	no	—
698	C₄²	—	4	2	no	—
699	C₄²	—	4	1	no	—
700	C₄²	—	4	2	no	—
701	C₄²	—	4	2	no	—
702	C₄²	—	4	2	no	—
703	C₄²	—	4	1	no	—
704	C₄²	—	4	1	no	—
705	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
706	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
707	C₄²	C₂²×C₄×C₆₇	1	4	yes	—
708	C₂³	—	2	2	no	—
709	C₂³	—	2	2	no	—
710	C₂³	—	2	2	no	—
711	C₂³	—	2	2	no	—
712	C₂³	—	2	2	no	—
713	C₂³	—	2	2	no	—
714	C₂³	C₂³×C₄×C₆₇	1	4	yes	—
715	C₂³	C₂⁵×C₆₇	1	1	yes	characteristic
716	Q₈	—	4	4	no	—
717	Q₈	—	4	4	no	—
718	Q₈	—	4	4	no	—
719	D₄	—	4	4	no	—
720	C₂×C₄	—	2	2	no	—
721	C₂×C₄	—	2	1	no	—
722	C₂×C₄	—	2	1	no	—
723	C₂×C₄	—	2	1	no	—
724	C₂×C₄	—	2	2	no	—
725	C₂×C₄	—	2	2	no	—
726	C₂×C₄	—	2	2	no	—
727	C₂×C₄	—	2	2	no	—
728	C₂×C₄	—	2	2	no	—
729	C₂×C₄	—	2	2	no	—
730	C₂×C₄	—	2	2	no	—
731	C₂×C₄	—	2	2	no	—
732	C₂×C₄	—	2	2	no	—
733	C₂×C₄	—	2	2	no	—
734	C₂×C₄	—	2	2	no	—
735	C₂×C₄	—	2	1	no	—
736	C₂×C₄	—	2	1	no	—
737	C₂×C₄	—	2	1	no	—
738	C₂×C₄	—	2	2	no	—
739	C₂×C₄	—	2	1	no	—
740	C₂×C₄	—	2	2	no	—
741	C₂×C₄	—	2	1	no	—
742	C₂×C₄	—	2	1	no	—
743	C₂×C₄	—	2	2	no	—
744	C₂×C₄	—	2	2	no	—
745	C₂×C₄	—	2	2	no	—
746	C₂×C₄	—	2	2	no	—
747	C₂×C₄	—	2	2	no	—
748	C₂×C₄	—	2	2	no	—
749	C₂×C₄	—	2	2	no	—
750	C₂×C₄	—	2	2	no	—
751	C₂×C₄	—	2	2	no	—
752	C₂×C₄	—	2	2	no	—
753	C₂×C₄	—	2	2	no	—
754	C₂×C₄	—	2	2	no	—
755	C₂×C₄	—	2	2	no	—
756	C₂×C₄	—	2	2	no	—
757	C₂×C₄	—	2	2	no	—
758	C₂×C₄	—	2	2	no	—
759	C₂×C₄	—	2	2	no	—
760	C₂×C₄	—	4	8	no	—
761	C₂×C₄	—	8	4	no	—
762	C₂×C₄	—	8	4	no	—
763	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
764	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
765	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
766	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
767	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
768	C₂×C₄	C₂³×C₄×C₆₇	1	2	yes	—
769	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
770	C₂×C₄	C₂³×C₄×C₆₇	1	4	yes	—
771	C₂×C₄	H(5,𝔽₂)×C₆₇	1	2	yes	—
772	C₂×C₄	2₋¹⁺⁴×C₆₇	1	4	yes	—
773	C₂²	—	2	4	no	—
774	C₂²	—	2	4	no	—
775	C₂²	—	2	4	no	—
776	C₂²	—	4	2	no	—
777	C₂²	—	4	2	no	—
778	C₂²	—	4	2	no	—
779	C₂²	—	4	2	no	—
780	C₂²	C₂⁴×C₄×C₆₇	1	1	yes	characteristic
781	C₂²	C₂×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
782	C₂²	C₂×2₋¹⁺⁴×C₆₇	1	2	yes	—
783	C₂²	#585×C₆₇	1	2	yes	—
784	C₂²	#585×C₆₇	1	1	yes	characteristic
785	C₄	—	2	2	no	—
786	C₄	—	2	4	no	—
787	C₄	—	2	4	no	—
788	C₄	—	2	4	no	—
789	C₄	—	2	2	no	—
790	C₄	—	2	2	no	—
791	C₄	—	2	2	no	—
792	C₄	—	2	4	no	—
793	C₄	—	2	4	no	—
794	C₄	—	2	4	no	—
795	C₄	—	4	2	no	—
796	C₄	—	4	1	no	—
797	C₄	—	4	2	no	—
798	C₄	—	4	1	no	—
799	C₄	—	4	2	no	—
800	C₄	—	4	1	no	—
801	C₄	—	4	2	no	—
802	C₄	—	4	2	no	—
803	C₂	—	4	4	no	—
804	C₂	C₄×H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
805	C₂	C₄×2₋¹⁺⁴×C₆₇	1	2	yes	—
806	C₂	C₆₇×#3544	1	1	yes	characteristic
807	C₂	C₆₇×#3551	1	1	yes	characteristic
808	C₂	C₆₇×#3564	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂⁵
commutator	C₂²	C₂⁴×C₄×C₆₇
Frattini, cosocle	C₂³	C₂⁵×C₆₇
prime radical	C₂²×D₄×C₆₇	C₂³
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
2. C₆₇×#2383	C₂	semidirect
3. C₆₇×#2381	C₂	semidirect
4. C₆₇×#2373	C₂	semidirect
6. C₆₇×#2371	C₂	semidirect
7. C₆₇×#2369	C₂	semidirect
8. C₆₇×#2368	C₂	semidirect
9. C₆₇×#2362	C₂	semidirect
12. C₆₇×#2334	C₂	semidirect
13. C₆₇×#2330	C₂	semidirect
14. C₆₇×#2329	C₂	semidirect
15. C₆₇×#2327	C₂	semidirect
17. C₆₇×#2323	C₂	semidirect
18. C₄×C₄⋊Q₈×C₆₇	C₂	semidirect
19. C₄×#126×C₆₇	C₂	semidirect
20. #563×C₆₇	C₄	semidirect
115. C₄⋊Q₈×C₆₇	C₂×C₄	semidirect
116. C₄⋊Q₈×C₆₇	C₂×C₄	semidirect
117. #126×C₆₇	C₂×C₄	semidirect
118. #126×C₆₇	C₂×C₄	semidirect
119. #125×C₆₇	C₂×C₄	semidirect
122. C₂²⋊Q₈×C₆₇	C₂×C₄	semidirect
123. C₂²⋊Q₈×C₆₇	C₂×C₄	semidirect
398. #34521	C₆₇	direct (N × H)
406. #2383	C₂×C₆₇	semidirect
407. #2381	C₂×C₆₇	semidirect
408. #2373	C₂×C₆₇	semidirect
410. #2371	C₂×C₆₇	semidirect
411. #2369	C₂×C₆₇	semidirect
412. #2368	C₂×C₆₇	semidirect
413. #2362	C₂×C₆₇	semidirect
416. #2334	C₂×C₆₇	semidirect
417. #2330	C₂×C₆₇	semidirect
418. #2329	C₂×C₆₇	semidirect
419. #2327	C₂×C₆₇	semidirect
421. #2323	C₂×C₆₇	semidirect
422. C₄×C₄⋊Q₈	C₂×C₆₇	semidirect
423. C₄×#126	C₂×C₆₇	semidirect
424. C₆₇	#34521	direct (N × H)
425. #563	C₄×C₆₇	semidirect
520. C₄⋊Q₈	C₂×C₄×C₆₇	semidirect
521. C₄⋊Q₈	C₂×C₄×C₆₇	semidirect
522. #126	C₂×C₄×C₆₇	semidirect
523. #126	C₂×C₄×C₆₇	semidirect
524. #125	C₂×C₄×C₆₇	semidirect
527. C₂²⋊Q₈	C₂×C₄×C₆₇	semidirect
528. C₂²⋊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×#563×C₆₇	2
C₆₇×#2383	2
C₆₇×#2381	2
C₆₇×#2373	2
C₆₇×#2372	2
C₆₇×#2371	2
C₆₇×#2369	2
C₆₇×#2368	2
C₆₇×#2362	2
C₆₇×#2359	2
C₆₇×#2335	2
C₆₇×#2334	2
C₆₇×#2330	2
C₆₇×#2329	2
C₆₇×#2327	2
C₆₇×#2326	2
C₆₇×#2323	2
C₄×C₄⋊Q₈×C₆₇	2
C₄×#126×C₆₇	2
#34521	67

Subgroups — C67×#34521 #30713772

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#34521 #30713772