Subgroups — C₆₇×#32960 #30712211

Order 17152.

1288 conjugacy classes of subgroups.

The subgroup lattice has 530 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#479×C₆₇	C₂	1	2	yes	maximal, complemented
3	C₆₇×#2851	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1865	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1863	C₂	1	2	yes	maximal, complemented
6	C₆₇×#1853	C₂	1	1	yes	maximal, characteristic
7	C₆₇×#1852	C₂	1	1	yes	maximal, characteristic
8	C₆₇×#1805	C₂	1	2	yes	maximal, complemented
9	C₆₇×#1797	C₂	1	1	yes	maximal, characteristic
10	C₆₇×#1787	C₂	1	1	yes	maximal, characteristic
11	C₆₇×#1787	C₂	1	1	yes	maximal, characteristic
12	C₂³×Q₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#125×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₆₇	—	2	8	no	—
15	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
16	C₂×C₄×Q₈×C₆₇	—	2	1	no	—
17	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
18	C₂×C₄×Q₈×C₆₇	C₂²	1	2	yes	—
19	C₂²×C₄⋊C₄×C₆₇	—	2	1	no	—
20	C₂²×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
21	C₂²×C₄²×C₆₇	—	2	1	no	—
22	C₂²×C₄²×C₆₇	C₂²	1	1	yes	characteristic
23	#479×C₆₇	—	2	8	no	—
24	C₂×#107×C₆₇	C₂²	1	2	yes	—
25	C₂×#106×C₆₇	C₂²	1	2	yes	—
26	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
27	C₂×#103×C₆₇	C₂²	1	2	yes	—
28	C₂×#103×C₆₇	C₂²	1	2	yes	—
29	C₂×#103×C₆₇	C₂²	1	2	yes	—
30	C₂×#103×C₆₇	C₂²	1	2	yes	—
31	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
32	C₂×C₂²⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
33	#389×C₆₇	C₂²	1	1	yes	characteristic
34	#389×C₆₇	C₂²	1	1	yes	characteristic
35	#387×C₆₇	—	2	1	no	—
36	#387×C₆₇	—	2	1	no	—
37	#385×C₆₇	—	2	1	no	—
38	#385×C₆₇	—	2	1	no	—
39	#382×C₆₇	C₂²	1	1	yes	characteristic
40	#380×C₆₇	C₂²	1	1	yes	characteristic
41	#380×C₆₇	C₂²	1	1	yes	characteristic
42	#340×C₆₇	C₂²	1	2	yes	—
43	#340×C₆₇	C₂²	1	2	yes	—
44	#336×C₆₇	C₂²	1	2	yes	—
45	C₂²×Q₈×C₆₇	—	2	4	no	—
46	C₂²×Q₈×C₆₇	—	2	4	no	—
47	C₂²×Q₈×C₆₇	—	2	1	no	—
48	C₂²×Q₈×C₆₇	—	2	2	no	—
49	C₂²×Q₈×C₆₇	Q₈	1	4	yes	—
50	C₂²×Q₈×C₆₇	C₂³	1	2	yes	—
51	C₂³×C₄×C₆₇	—	2	1	no	—
52	C₂³×C₄×C₆₇	—	2	1	no	—
53	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
54	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
55	C₂³×C₄×C₆₇	C₂³	1	1	yes	characteristic
56	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
57	C₂²×C₈×C₆₇	C₂³	1	2	yes	—
58	#125×C₆₇	—	2	4	no	—
59	C₄×Q₈×C₆₇	—	2	8	no	—
60	C₄×Q₈×C₆₇	—	2	8	no	—
61	C₄×Q₈×C₆₇	—	2	4	no	—
62	C₄×Q₈×C₆₇	—	2	4	no	—
63	C₄×Q₈×C₆₇	—	2	4	no	—
64	C₄×Q₈×C₆₇	D₄	1	8	yes	—
65	C₂×C₄⋊C₄×C₆₇	—	2	8	no	—
66	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
67	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
68	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
69	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
70	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
71	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
72	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
73	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
74	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
75	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
76	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
77	C₂×C₄⋊C₄×C₆₇	C₂³	1	1	yes	characteristic
78	C₂×C₄⋊C₄×C₆₇	C₂³	1	2	yes	—
79	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
80	C₂×C₂²⋊C₄×C₆₇	—	2	1	no	—
81	C₂×C₄²×C₆₇	—	2	8	no	—
82	C₂×C₄²×C₆₇	—	2	2	no	—
83	C₂×C₄²×C₆₇	—	2	2	no	—
84	C₂×C₄²×C₆₇	—	2	4	no	—
85	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
86	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
87	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
88	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
89	#107×C₆₇	—	2	4	no	—
90	#106×C₆₇	—	2	4	no	—
91	C₄⋊C₈×C₆₇	—	2	4	no	—
92	#103×C₆₇	—	2	4	no	—
93	#103×C₆₇	—	2	4	no	—
94	#103×C₆₇	—	2	4	no	—
95	#103×C₆₇	—	2	4	no	—
96	C₂²⋊C₈×C₆₇	—	2	2	no	—
97	C₂²⋊C₈×C₆₇	—	2	2	no	—
98	#95×C₆₇	—	2	1	no	—
99	#95×C₆₇	—	2	1	no	—
100	#95×C₆₇	—	2	1	no	—
101	#95×C₆₇	—	2	1	no	—
102	#95×C₆₇	—	2	1	no	—
103	#95×C₆₇	—	2	1	no	—
104	C₂⁴×C₆₇	C₂×D₄	1	1	yes	characteristic
105	C₂×Q₈×C₆₇	—	2	4	no	—
106	C₂×Q₈×C₆₇	—	2	16	no	—
107	C₂×Q₈×C₆₇	—	2	4	no	—
108	C₂×Q₈×C₆₇	—	2	16	no	—
109	C₂×Q₈×C₆₇	—	2	4	no	—
110	C₂×Q₈×C₆₇	—	2	2	no	—
111	C₂×Q₈×C₆₇	—	2	2	no	—
112	C₂×Q₈×C₆₇	—	2	2	no	—
113	C₂×Q₈×C₆₇	C₂×D₄	1	4	yes	—
114	C₂×Q₈×C₆₇	C₂×Q₈	1	4	yes	—
115	C₂×Q₈×C₆₇	#41	1	4	yes	—
116	C₂²×C₄×C₆₇	—	2	4	no	—
117	C₂²×C₄×C₆₇	—	2	4	no	—
118	C₂²×C₄×C₆₇	—	2	4	no	—
119	C₂²×C₄×C₆₇	—	2	2	no	—
120	C₂²×C₄×C₆₇	—	2	2	no	—
121	C₂²×C₄×C₆₇	—	2	2	no	—
122	C₂²×C₄×C₆₇	—	2	2	no	—
123	C₂²×C₄×C₆₇	—	2	2	no	—
124	C₂²×C₄×C₆₇	—	2	2	no	—
125	C₂²×C₄×C₆₇	—	2	4	no	—
126	C₂²×C₄×C₆₇	—	2	2	no	—
127	C₂²×C₄×C₆₇	—	2	2	no	—
128	C₂²×C₄×C₆₇	—	2	2	no	—
129	C₂²×C₄×C₆₇	—	2	1	no	—
130	C₂²×C₄×C₆₇	—	2	1	no	—
131	C₂²×C₄×C₆₇	—	2	1	no	—
132	C₂²×C₄×C₆₇	—	2	1	no	—
133	C₂²×C₄×C₆₇	—	4	1	no	—
134	C₂²×C₄×C₆₇	—	4	2	no	—
135	C₂²×C₄×C₆₇	—	4	2	no	—
136	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
137	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
138	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
139	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
140	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
141	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
142	C₂²×C₄×C₆₇	#41	1	2	yes	—
143	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
144	C₂×C₈×C₆₇	—	2	2	no	—
145	C₂×C₈×C₆₇	—	2	2	no	—
146	C₂×C₈×C₆₇	—	2	2	no	—
147	C₂×C₈×C₆₇	—	2	2	no	—
148	C₂×C₈×C₆₇	—	2	2	no	—
149	C₂×C₈×C₆₇	—	2	2	no	—
150	C₄⋊C₄×C₆₇	—	2	8	no	—
151	C₄⋊C₄×C₆₇	—	2	4	no	—
152	C₄⋊C₄×C₆₇	—	2	8	no	—
153	C₄⋊C₄×C₆₇	—	2	2	no	—
154	C₄⋊C₄×C₆₇	—	2	2	no	—
155	C₄⋊C₄×C₆₇	—	2	2	no	—
156	C₄⋊C₄×C₆₇	—	2	2	no	—
157	C₄⋊C₄×C₆₇	—	2	2	no	—
158	C₄⋊C₄×C₆₇	—	4	2	no	—
159	C₄⋊C₄×C₆₇	—	4	2	no	—
160	C₄⋊C₄×C₆₇	—	4	2	no	—
161	C₄⋊C₄×C₆₇	—	4	2	no	—
162	C₄⋊C₄×C₆₇	C₂×D₄	1	4	yes	—
163	C₂²⋊C₄×C₆₇	—	4	2	no	—
164	C₂²⋊C₄×C₆₇	—	4	2	no	—
165	C₄²×C₆₇	—	2	8	no	—
166	C₄²×C₆₇	—	2	8	no	—
167	C₄²×C₆₇	—	2	2	no	—
168	C₄²×C₆₇	—	2	2	no	—
169	C₄²×C₆₇	—	2	2	no	—
170	C₄²×C₆₇	C₂×D₄	1	4	yes	—
171	C₂³×C₆₇	—	2	1	no	—
172	C₂³×C₆₇	—	2	1	no	—
173	C₂³×C₆₇	—	2	1	no	—
174	C₂³×C₆₇	—	2	1	no	—
175	C₂³×C₆₇	C₂²⋊Q₈	1	2	yes	—
176	C₂³×C₆₇	#135	1	2	yes	—
177	C₂³×C₆₇	#137	1	2	yes	—
178	C₂³×C₆₇	C₂²×D₄	1	1	yes	characteristic
179	Q₈×C₆₇	—	2	16	no	—
180	Q₈×C₆₇	—	2	4	no	—
181	Q₈×C₆₇	—	2	4	no	—
182	Q₈×C₆₇	—	2	4	no	—
183	Q₈×C₆₇	C₂²⋊Q₈	1	8	yes	—
184	C₂×C₄×C₆₇	—	2	8	no	—
185	C₂×C₄×C₆₇	—	2	16	no	—
186	C₂×C₄×C₆₇	—	2	1	no	—
187	C₂×C₄×C₆₇	—	2	4	no	—
188	C₂×C₄×C₆₇	—	2	1	no	—
189	C₂×C₄×C₆₇	—	2	4	no	—
190	C₂×C₄×C₆₇	—	2	4	no	—
191	C₂×C₄×C₆₇	—	2	1	no	—
192	C₂×C₄×C₆₇	—	2	1	no	—
193	C₂×C₄×C₆₇	—	2	1	no	—
194	C₂×C₄×C₆₇	—	2	1	no	—
195	C₂×C₄×C₆₇	—	2	1	no	—
196	C₂×C₄×C₆₇	—	2	1	no	—
197	C₂×C₄×C₆₇	—	2	4	no	—
198	C₂×C₄×C₆₇	—	2	4	no	—
199	C₂×C₄×C₆₇	—	2	1	no	—
200	C₂×C₄×C₆₇	—	2	4	no	—
201	C₂×C₄×C₆₇	—	2	1	no	—
202	C₂×C₄×C₆₇	—	2	1	no	—
203	C₂×C₄×C₆₇	—	4	4	no	—
204	C₂×C₄×C₆₇	—	4	8	no	—
205	C₂×C₄×C₆₇	—	4	2	no	—
206	C₂×C₄×C₆₇	—	4	2	no	—
207	C₂×C₄×C₆₇	—	4	1	no	—
208	C₂×C₄×C₆₇	—	4	1	no	—
209	C₂×C₄×C₆₇	—	4	1	no	—
210	C₂×C₄×C₆₇	—	4	1	no	—
211	C₂×C₄×C₆₇	—	4	1	no	—
212	C₂×C₄×C₆₇	—	4	1	no	—
213	C₂×C₄×C₆₇	—	4	1	no	—
214	C₂×C₄×C₆₇	—	4	1	no	—
215	C₂×C₄×C₆₇	—	4	1	no	—
216	C₂×C₄×C₆₇	—	4	1	no	—
217	C₂×C₄×C₆₇	—	4	1	no	—
218	C₂×C₄×C₆₇	—	4	1	no	—
219	C₂×C₄×C₆₇	#120	1	2	yes	—
220	C₂×C₄×C₆₇	#120	1	2	yes	—
221	C₂×C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
222	C₂×C₄×C₆₇	#137	1	2	yes	—
223	C₂×C₄×C₆₇	C₂²×D₄	1	2	yes	—
224	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
225	C₂×C₄×C₆₇	C₂×#41	1	2	yes	—
226	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
227	C₈×C₆₇	—	4	2	no	—
228	C₈×C₆₇	—	4	2	no	—
229	C₂²×C₆₇	—	2	2	no	—
230	C₂²×C₆₇	—	2	2	no	—
231	C₂²×C₆₇	—	2	2	no	—
232	C₂²×C₆₇	—	2	2	no	—
233	C₂²×C₆₇	—	2	2	no	—
234	C₂²×C₆₇	—	2	2	no	—
235	C₂²×C₆₇	#479	1	4	yes	complemented
236	C₂²×C₆₇	C₂×#120	1	1	yes	characteristic
237	C₂²×C₆₇	C₂×C₂²⋊Q₈	1	1	yes	characteristic
238	C₂²×C₆₇	#525	1	1	yes	characteristic
239	C₂²×C₆₇	C₂×#135	1	1	yes	characteristic
240	C₂²×C₆₇	C₂×#137	1	1	yes	characteristic
241	C₂²×C₆₇	C₂×#137	1	1	yes	characteristic
242	C₂²×C₆₇	#575	1	1	yes	characteristic
243	C₄×C₆₇	—	2	8	no	—
244	C₄×C₆₇	—	2	2	no	—
245	C₄×C₆₇	—	2	2	no	—
246	C₄×C₆₇	—	2	2	no	—
247	C₄×C₆₇	—	2	2	no	—
248	C₄×C₆₇	—	4	1	no	—
249	C₄×C₆₇	—	4	4	no	—
250	C₄×C₆₇	—	8	1	no	—
251	C₄×C₆₇	—	8	1	no	—
252	C₄×C₆₇	—	8	1	no	—
253	C₄×C₆₇	—	8	1	no	—
254	C₄×C₆₇	C₂×C₂²⋊Q₈	1	2	yes	—
255	C₄×C₆₇	#525	1	2	yes	—
256	#32960	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
257	C₂×C₆₇	—	2	4	no	—
258	C₂×C₆₇	#2848	1	1	yes	characteristic
259	C₂×C₆₇	#3002	1	1	yes	characteristic
260	C₂×C₆₇	#3004	1	1	yes	characteristic
261	C₂×C₆₇	C₂×#479	1	1	yes	characteristic
262	C₂×C₆₇	#3078	1	1	yes	characteristic
263	C₂×C₆₇	#3108	1	1	yes	characteristic
264	C₂×C₆₇	#3117	1	1	yes	characteristic
265	C₂²×C₄×Q₈	C₂×C₆₇	1	1	yes	characteristic
266	C₂×#479	C₂×C₆₇	1	2	yes	complemented
267	#2851	C₂×C₆₇	1	1	yes	characteristic
268	#1865	C₂×C₆₇	1	2	yes	complemented
269	#1863	C₂×C₆₇	1	2	yes	complemented
270	#1853	C₂×C₆₇	1	1	yes	characteristic
271	#1852	C₂×C₆₇	1	1	yes	characteristic
272	#1805	C₂×C₆₇	1	2	yes	complemented
273	#1797	C₂×C₆₇	1	1	yes	characteristic
274	#1787	C₂×C₆₇	1	1	yes	characteristic
275	#1787	C₂×C₆₇	1	1	yes	characteristic
276	C₆₇	#32960	1	1	yes	characteristic, direct product factor, Sylow
277	C₂³×Q₈	C₂²×C₆₇	1	1	yes	characteristic
278	C₂×#125	C₂²×C₆₇	1	1	yes	characteristic
279	C₂×C₄×Q₈	—	2	8	no	—
280	C₂×C₄×Q₈	—	2	1	no	—
281	C₂×C₄×Q₈	—	2	1	no	—
282	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
283	C₂×C₄×Q₈	C₂²×C₆₇	1	2	yes	—
284	C₂²×C₄⋊C₄	—	2	1	no	—
285	C₂²×C₄⋊C₄	C₂²×C₆₇	1	1	yes	characteristic
286	C₂²×C₄²	—	2	1	no	—
287	C₂²×C₄²	C₂²×C₆₇	1	1	yes	characteristic
288	#479	—	2	8	no	—
289	C₂×#107	C₂²×C₆₇	1	2	yes	—
290	C₂×#106	C₂²×C₆₇	1	2	yes	—
291	C₂×C₄⋊C₈	C₂²×C₆₇	1	2	yes	—
292	C₂×#103	C₂²×C₆₇	1	2	yes	—
293	C₂×#103	C₂²×C₆₇	1	2	yes	—
294	C₂×#103	C₂²×C₆₇	1	2	yes	—
295	C₂×#103	C₂²×C₆₇	1	2	yes	—
296	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
297	C₂×C₂²⋊C₈	C₂²×C₆₇	1	1	yes	characteristic
298	#389	C₂²×C₆₇	1	1	yes	characteristic
299	#389	C₂²×C₆₇	1	1	yes	characteristic
300	#387	—	2	1	no	—
301	#387	—	2	1	no	—
302	#385	—	2	1	no	—
303	#385	—	2	1	no	—
304	#382	C₂²×C₆₇	1	1	yes	characteristic
305	#380	C₂²×C₆₇	1	1	yes	characteristic
306	#380	C₂²×C₆₇	1	1	yes	characteristic
307	#340	C₂²×C₆₇	1	2	yes	—
308	#340	C₂²×C₆₇	1	2	yes	—
309	#336	C₂²×C₆₇	1	2	yes	—
310	C₂²×Q₈	—	2	4	no	—
311	C₂²×Q₈	—	2	4	no	—
312	C₂²×Q₈	—	2	1	no	—
313	C₂²×Q₈	—	2	2	no	—
314	C₂²×Q₈	Q₈×C₆₇	1	4	yes	—
315	C₂²×Q₈	C₂³×C₆₇	1	2	yes	—
316	C₂³×C₄	—	2	1	no	—
317	C₂³×C₄	—	2	1	no	—
318	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
319	C₂³×C₄	D₄×C₆₇	1	1	yes	characteristic
320	C₂³×C₄	C₂³×C₆₇	1	1	yes	characteristic
321	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
322	C₂²×C₈	C₂³×C₆₇	1	2	yes	—
323	#125	—	2	4	no	—
324	C₄×Q₈	—	2	8	no	—
325	C₄×Q₈	—	2	8	no	—
326	C₄×Q₈	—	2	4	no	—
327	C₄×Q₈	—	2	4	no	—
328	C₄×Q₈	—	2	4	no	—
329	C₄×Q₈	D₄×C₆₇	1	8	yes	—
330	C₂×C₄⋊C₄	—	2	8	no	—
331	C₂×C₄⋊C₄	—	2	2	no	—
332	C₂×C₄⋊C₄	—	2	2	no	—
333	C₂×C₄⋊C₄	—	2	1	no	—
334	C₂×C₄⋊C₄	—	2	4	no	—
335	C₂×C₄⋊C₄	—	2	1	no	—
336	C₂×C₄⋊C₄	—	2	1	no	—
337	C₂×C₄⋊C₄	—	2	1	no	—
338	C₂×C₄⋊C₄	—	2	1	no	—
339	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
340	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
341	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
342	C₂×C₄⋊C₄	C₂³×C₆₇	1	1	yes	characteristic
343	C₂×C₄⋊C₄	C₂³×C₆₇	1	2	yes	—
344	C₂×C₂²⋊C₄	—	2	1	no	—
345	C₂×C₂²⋊C₄	—	2	1	no	—
346	C₂×C₄²	—	2	8	no	—
347	C₂×C₄²	—	2	2	no	—
348	C₂×C₄²	—	2	2	no	—
349	C₂×C₄²	—	2	4	no	—
350	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
351	C₂×C₄²	D₄×C₆₇	1	1	yes	characteristic
352	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
353	C₂×C₄²	C₂³×C₆₇	1	1	yes	characteristic
354	#107	—	2	4	no	—
355	#106	—	2	4	no	—
356	C₄⋊C₈	—	2	4	no	—
357	#103	—	2	4	no	—
358	#103	—	2	4	no	—
359	#103	—	2	4	no	—
360	#103	—	2	4	no	—
361	C₂²⋊C₈	—	2	2	no	—
362	C₂²⋊C₈	—	2	2	no	—
363	#95	—	2	1	no	—
364	#95	—	2	1	no	—
365	#95	—	2	1	no	—
366	#95	—	2	1	no	—
367	#95	—	2	1	no	—
368	#95	—	2	1	no	—
369	C₂⁴	C₂×D₄×C₆₇	1	1	yes	characteristic
370	C₂×Q₈	—	2	4	no	—
371	C₂×Q₈	—	2	16	no	—
372	C₂×Q₈	—	2	4	no	—
373	C₂×Q₈	—	2	16	no	—
374	C₂×Q₈	—	2	4	no	—
375	C₂×Q₈	—	2	2	no	—
376	C₂×Q₈	—	2	2	no	—
377	C₂×Q₈	—	2	2	no	—
378	C₂×Q₈	C₂×D₄×C₆₇	1	4	yes	—
379	C₂×Q₈	C₂×Q₈×C₆₇	1	4	yes	—
380	C₂×Q₈	#41×C₆₇	1	4	yes	—
381	C₂²×C₄	—	2	4	no	—
382	C₂²×C₄	—	2	4	no	—
383	C₂²×C₄	—	2	4	no	—
384	C₂²×C₄	—	2	2	no	—
385	C₂²×C₄	—	2	2	no	—
386	C₂²×C₄	—	2	2	no	—
387	C₂²×C₄	—	2	2	no	—
388	C₂²×C₄	—	2	2	no	—
389	C₂²×C₄	—	2	2	no	—
390	C₂²×C₄	—	2	4	no	—
391	C₂²×C₄	—	2	2	no	—
392	C₂²×C₄	—	2	2	no	—
393	C₂²×C₄	—	2	2	no	—
394	C₂²×C₄	—	2	1	no	—
395	C₂²×C₄	—	2	1	no	—
396	C₂²×C₄	—	2	1	no	—
397	C₂²×C₄	—	2	1	no	—
398	C₂²×C₄	—	4	1	no	—
399	C₂²×C₄	—	4	2	no	—
400	C₂²×C₄	—	4	2	no	—
401	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
402	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
403	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
404	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
405	C₂²×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
406	C₂²×C₄	C₂×Q₈×C₆₇	1	2	yes	—
407	C₂²×C₄	#41×C₆₇	1	2	yes	—
408	C₂²×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
409	C₂×C₈	—	2	2	no	—
410	C₂×C₈	—	2	2	no	—
411	C₂×C₈	—	2	2	no	—
412	C₂×C₈	—	2	2	no	—
413	C₂×C₈	—	2	2	no	—
414	C₂×C₈	—	2	2	no	—
415	C₄⋊C₄	—	2	8	no	—
416	C₄⋊C₄	—	2	4	no	—
417	C₄⋊C₄	—	2	8	no	—
418	C₄⋊C₄	—	2	2	no	—
419	C₄⋊C₄	—	2	2	no	—
420	C₄⋊C₄	—	2	2	no	—
421	C₄⋊C₄	—	2	2	no	—
422	C₄⋊C₄	—	2	2	no	—
423	C₄⋊C₄	—	4	2	no	—
424	C₄⋊C₄	—	4	2	no	—
425	C₄⋊C₄	—	4	2	no	—
426	C₄⋊C₄	—	4	2	no	—
427	C₄⋊C₄	C₂×D₄×C₆₇	1	4	yes	—
428	C₂²⋊C₄	—	4	2	no	—
429	C₂²⋊C₄	—	4	2	no	—
430	C₄²	—	2	8	no	—
431	C₄²	—	2	8	no	—
432	C₄²	—	2	2	no	—
433	C₄²	—	2	2	no	—
434	C₄²	—	2	2	no	—
435	C₄²	C₂×D₄×C₆₇	1	4	yes	—
436	C₂³	—	2	1	no	—
437	C₂³	—	2	1	no	—
438	C₂³	—	2	1	no	—
439	C₂³	—	2	1	no	—
440	C₂³	C₂²⋊Q₈×C₆₇	1	2	yes	—
441	C₂³	#135×C₆₇	1	2	yes	—
442	C₂³	#137×C₆₇	1	2	yes	—
443	C₂³	C₂²×D₄×C₆₇	1	1	yes	characteristic
444	Q₈	—	2	16	no	—
445	Q₈	—	2	4	no	—
446	Q₈	—	2	4	no	—
447	Q₈	—	2	4	no	—
448	Q₈	C₂²⋊Q₈×C₆₇	1	8	yes	—
449	C₂×C₄	—	2	16	no	—
450	C₂×C₄	—	2	8	no	—
451	C₂×C₄	—	2	1	no	—
452	C₂×C₄	—	2	4	no	—
453	C₂×C₄	—	2	1	no	—
454	C₂×C₄	—	2	4	no	—
455	C₂×C₄	—	2	4	no	—
456	C₂×C₄	—	2	1	no	—
457	C₂×C₄	—	2	1	no	—
458	C₂×C₄	—	2	1	no	—
459	C₂×C₄	—	2	1	no	—
460	C₂×C₄	—	2	1	no	—
461	C₂×C₄	—	2	1	no	—
462	C₂×C₄	—	2	4	no	—
463	C₂×C₄	—	2	4	no	—
464	C₂×C₄	—	2	1	no	—
465	C₂×C₄	—	2	4	no	—
466	C₂×C₄	—	2	1	no	—
467	C₂×C₄	—	2	1	no	—
468	C₂×C₄	—	4	4	no	—
469	C₂×C₄	—	4	8	no	—
470	C₂×C₄	—	4	2	no	—
471	C₂×C₄	—	4	2	no	—
472	C₂×C₄	—	4	1	no	—
473	C₂×C₄	—	4	1	no	—
474	C₂×C₄	—	4	1	no	—
475	C₂×C₄	—	4	1	no	—
476	C₂×C₄	—	4	1	no	—
477	C₂×C₄	—	4	1	no	—
478	C₂×C₄	—	4	1	no	—
479	C₂×C₄	—	4	1	no	—
480	C₂×C₄	—	4	1	no	—
481	C₂×C₄	—	4	1	no	—
482	C₂×C₄	—	4	1	no	—
483	C₂×C₄	—	4	1	no	—
484	C₂×C₄	#120×C₆₇	1	2	yes	—
485	C₂×C₄	#120×C₆₇	1	2	yes	—
486	C₂×C₄	C₂²⋊Q₈×C₆₇	1	2	yes	—
487	C₂×C₄	#137×C₆₇	1	2	yes	—
488	C₂×C₄	C₂²×D₄×C₆₇	1	2	yes	—
489	C₂×C₄	C₂²×Q₈×C₆₇	1	1	yes	characteristic
490	C₂×C₄	C₂×#41×C₆₇	1	2	yes	—
491	C₂×C₄	C₂×#41×C₆₇	1	1	yes	characteristic
492	C₈	—	4	2	no	—
493	C₈	—	4	2	no	—
494	C₂²	—	2	2	no	—
495	C₂²	—	2	2	no	—
496	C₂²	—	2	2	no	—
497	C₂²	—	2	2	no	—
498	C₂²	—	2	2	no	—
499	C₂²	—	2	2	no	—
500	C₂²	#479×C₆₇	1	4	yes	complemented
501	C₂²	C₂×#120×C₆₇	1	1	yes	characteristic
502	C₂²	C₂×C₂²⋊Q₈×C₆₇	1	1	yes	characteristic
503	C₂²	#525×C₆₇	1	1	yes	characteristic
504	C₂²	C₂×#135×C₆₇	1	1	yes	characteristic
505	C₂²	C₂×#137×C₆₇	1	1	yes	characteristic
506	C₂²	C₂×#137×C₆₇	1	1	yes	characteristic
507	C₂²	#575×C₆₇	1	1	yes	characteristic
508	C₄	—	2	8	no	—
509	C₄	—	2	2	no	—
510	C₄	—	2	2	no	—
511	C₄	—	2	2	no	—
512	C₄	—	2	2	no	—
513	C₄	—	4	1	no	—
514	C₄	—	4	4	no	—
515	C₄	—	8	1	no	—
516	C₄	—	8	1	no	—
517	C₄	—	8	1	no	—
518	C₄	—	8	1	no	—
519	C₄	C₂×C₂²⋊Q₈×C₆₇	1	2	yes	—
520	C₄	#525×C₆₇	1	2	yes	—
521	C₂	—	2	4	no	—
522	C₂	C₆₇×#2848	1	1	yes	characteristic
523	C₂	C₆₇×#3002	1	1	yes	characteristic
524	C₂	C₆₇×#3004	1	1	yes	characteristic
525	C₂	C₂×#479×C₆₇	1	1	yes	characteristic
526	C₂	C₆₇×#3078	1	1	yes	characteristic
527	C₂	C₆₇×#3108	1	1	yes	characteristic
528	C₂	C₆₇×#3117	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₆₇
prime radical	C₂⁴×C₆₇	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₆₇ ◁ C₂²×C₄²×C₆₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#479×C₆₇	C₂	semidirect
4. C₆₇×#1865	C₂	semidirect
5. C₆₇×#1863	C₂	semidirect
8. C₆₇×#1805	C₂	semidirect
235. C₂²×C₆₇	#479	semidirect
256. #32960	C₆₇	direct (N × H)
266. C₂×#479	C₂×C₆₇	semidirect
268. #1865	C₂×C₆₇	semidirect
269. #1863	C₂×C₆₇	semidirect
272. #1805	C₂×C₆₇	semidirect
276. C₆₇	#32960	direct (N × H)
500. C₂²	#479×C₆₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×Q₈×C₆₇	2
C₂×#479×C₆₇	2
C₆₇×#2851	2
C₆₇×#1865	2
C₆₇×#1863	2
C₆₇×#1853	2
C₆₇×#1852	2
C₆₇×#1805	2
C₆₇×#1797	2
C₆₇×#1787	2
#32960	67

Subgroups — C67×#32960 #30712211

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#32960 #30712211