Subgroups — #30748529

Order 17152.

568 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22560326	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
3	C₂×Q₈⋊(C₆₇⋊C₈)	C₂	1	2	yes	maximal
4	#22558050	C₂	1	2	yes	maximal
5	C₄²×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
7	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
9	#16863151	—	2	1	no	—
10	#16863151	—	2	1	no	—
11	#16863151	—	2	1	no	—
12	Q₈⋊(C₆₇⋊C₈)	—	2	4	no	—
13	C₂×C₄×Q₈×C₆₇	C₄	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₆₇	C₄	1	1	yes	characteristic
15	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
17	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
18	C₄×C₄⋊C₄×C₆₇	—	2	1	no	—
19	C₄×C₄⋊C₄×C₆₇	C₄	1	1	yes	characteristic
20	C₄×C₄⋊C₄×C₆₇	C₄	1	1	yes	characteristic
21	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
22	C₄³×C₆₇	—	2	1	no	—
23	C₄³×C₆₇	C₂²	1	1	yes	characteristic
24	C₂²×C₆₇⋊C₈	—	2	2	no	—
25	C₂²×C₆₇⋊C₈	—	2	2	no	—
26	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
27	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
28	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
29	C₄×C₆₇⋊C₈	—	2	2	no	—
30	C₄×C₆₇⋊C₈	—	2	2	no	—
31	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₄×Q₈×C₆₇	—	2	2	no	—
33	C₄×Q₈×C₆₇	—	2	2	no	—
34	C₄×Q₈×C₆₇	—	2	2	no	—
35	C₄×Q₈×C₆₇	—	2	1	no	—
36	C₄×Q₈×C₆₇	—	2	1	no	—
37	C₄×Q₈×C₆₇	C₈	1	4	yes	complemented
38	C₄×Q₈×C₆₇	D₄	1	2	yes	—
39	C₄×Q₈×C₆₇	Q₈	1	2	yes	—
40	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
41	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
42	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
43	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
44	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
45	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
46	C₂×C₄²×C₆₇	—	2	1	no	—
47	C₂×C₄²×C₆₇	—	2	1	no	—
48	C₂×C₄²×C₆₇	—	2	1	no	—
49	C₂×C₄²×C₆₇	—	2	1	no	—
50	C₂×C₄²×C₆₇	—	2	1	no	—
51	C₂×C₄²×C₆₇	—	2	1	no	—
52	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
56	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
57	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
58	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
59	#95×C₆₇	—	2	1	no	—
60	#95×C₆₇	—	2	1	no	—
61	#95×C₆₇	—	2	1	no	—
62	C₂×C₆₇⋊C₈	—	2	4	no	—
63	C₂×C₆₇⋊C₈	—	4	2	no	—
64	C₂×C₆₇⋊C₈	—	4	2	no	—
65	C₂×C₆₇⋊C₈	—	4	2	no	—
66	C₂×C₆₇⋊C₈	—	4	2	no	—
67	C₂×C₆₇⋊C₈	—	4	2	no	—
68	C₂×Q₈×C₆₇	—	2	1	no	—
69	C₂×Q₈×C₆₇	—	2	1	no	—
70	C₂×Q₈×C₆₇	—	2	1	no	—
71	C₂×Q₈×C₆₇	C₄⋊C₄	1	2	yes	—
72	C₂×Q₈×C₆₇	C₂×C₈	1	2	yes	—
73	C₂×Q₈×C₆₇	M₁₆	1	2	yes	—
74	C₂²×C₄×C₆₇	—	2	1	no	—
75	C₂²×C₄×C₆₇	—	2	1	no	—
76	C₂²×C₄×C₆₇	—	2	1	no	—
77	C₂²×C₄×C₆₇	—	2	1	no	—
78	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
79	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
81	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
85	C₄⋊C₄×C₆₇	—	2	4	no	—
86	C₄⋊C₄×C₆₇	—	2	4	no	—
87	C₄⋊C₄×C₆₇	—	2	1	no	—
88	C₄⋊C₄×C₆₇	—	2	2	no	—
89	C₄⋊C₄×C₆₇	—	2	2	no	—
90	C₄⋊C₄×C₆₇	—	2	1	no	—
91	C₄⋊C₄×C₆₇	—	2	1	no	—
92	C₄⋊C₄×C₆₇	C₄⋊C₄	1	2	yes	—
93	C₄⋊C₄×C₆₇	C₂×C₈	1	2	yes	—
94	C₄⋊C₄×C₆₇	M₁₆	1	2	yes	—
95	C₄²×C₆₇	—	2	4	no	—
96	C₄²×C₆₇	—	2	4	no	—
97	C₄²×C₆₇	—	2	2	no	—
98	C₄²×C₆₇	—	2	2	no	—
99	C₄²×C₆₇	—	2	2	no	—
100	C₄²×C₆₇	—	2	1	no	—
101	C₄²×C₆₇	—	2	1	no	—
102	C₄²×C₆₇	—	2	2	no	—
103	C₄²×C₆₇	—	2	2	no	—
104	C₄²×C₆₇	—	4	1	no	—
105	C₄²×C₆₇	—	4	2	no	—
106	C₄²×C₆₇	—	4	2	no	—
107	C₄²×C₆₇	—	4	2	no	—
108	C₄²×C₆₇	C₂×C₈	1	2	yes	—
109	C₄²×C₆₇	M₁₆	1	2	yes	—
110	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
112	C₄²×C₆₇	#41	1	1	yes	characteristic
113	C₄²×C₆₇	#41	1	1	yes	characteristic
114	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
115	Q₈×C₆₇	—	2	2	no	—
116	Q₈×C₆₇	—	2	2	no	—
117	Q₈×C₆₇	—	2	2	no	—
118	Q₈×C₆₇	C₄⋊C₈	1	4	yes	complemented
119	C₂×C₄×C₆₇	—	2	2	no	—
120	C₂×C₄×C₆₇	—	2	2	no	—
121	C₂×C₄×C₆₇	—	2	1	no	—
122	C₂×C₄×C₆₇	—	2	2	no	—
123	C₂×C₄×C₆₇	—	2	1	no	—
124	C₂×C₄×C₆₇	—	2	1	no	—
125	C₂×C₄×C₆₇	—	2	1	no	—
126	C₂×C₄×C₆₇	—	2	1	no	—
127	C₂×C₄×C₆₇	—	2	1	no	—
128	C₂×C₄×C₆₇	—	2	1	no	—
129	C₂×C₄×C₆₇	—	2	1	no	—
130	C₂×C₄×C₆₇	—	2	2	no	—
131	C₂×C₄×C₆₇	—	2	2	no	—
132	C₂×C₄×C₆₇	—	2	1	no	—
133	C₂×C₄×C₆₇	—	2	2	no	—
134	C₂×C₄×C₆₇	—	2	1	no	—
135	C₂×C₄×C₆₇	—	2	1	no	—
136	C₂×C₄×C₆₇	—	4	1	no	—
137	C₂×C₄×C₆₇	—	4	1	no	—
138	C₂×C₄×C₆₇	—	4	1	no	—
139	C₂×C₄×C₆₇	—	4	1	no	—
140	C₂×C₄×C₆₇	—	4	1	no	—
141	C₂×C₄×C₆₇	—	4	1	no	—
142	C₂×C₄×C₆₇	C₂²⋊C₈	1	2	yes	—
143	C₂×C₄×C₆₇	C₂²⋊C₈	1	2	yes	—
144	C₂×C₄×C₆₇	#104	1	2	yes	—
145	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
146	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
147	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
148	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
149	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
150	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
151	C₂×C₄×C₆₇	C₂²×C₈	1	1	yes	characteristic
152	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
153	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
154	C₂×C₄×C₆₇	C₂×M₁₆	1	1	yes	characteristic
155	C₂×C₄×C₆₇	#135	1	2	yes	—
156	C₂×C₄×C₆₇	#137	1	2	yes	—
157	C₆₇⋊C₈	—	4	4	no	—
158	C₆₇⋊C₈	—	8	2	no	—
159	C₂²×C₆₇	#380	1	1	yes	characteristic
160	C₂²×C₆₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
161	C₂²×C₆₇	#407	1	1	yes	characteristic
162	C₂²×C₆₇	#416	1	1	yes	characteristic
163	C₂²×C₆₇	#419	1	1	yes	characteristic
164	C₂²×C₆₇	C₂×#104	1	1	yes	characteristic
165	C₂²×C₆₇	#421	1	1	yes	characteristic
166	C₄×C₆₇	—	2	4	no	—
167	C₄×C₆₇	—	2	1	no	—
168	C₄×C₆₇	—	2	1	no	—
169	C₄×C₆₇	—	2	2	no	—
170	C₄×C₆₇	—	2	2	no	—
171	C₄×C₆₇	—	2	2	no	—
172	C₄×C₆₇	—	4	2	no	—
173	C₄×C₆₇	—	4	2	no	—
174	C₄×C₆₇	—	4	1	no	—
175	C₄×C₆₇	—	4	2	no	—
176	C₄×C₆₇	C₂×C₄⋊C₈	1	1	yes	characteristic
177	C₄×C₆₇	#423	1	1	yes	characteristic
178	C₄×C₆₇	#431	1	1	yes	characteristic
179	C₄×C₆₇	#432	1	1	yes	characteristic
180	C₄×C₆₇	#464	1	2	yes	—
181	C₄×C₆₇	#479	1	2	yes	—
182	#8566	—	67	1	no	maximal, Sylow
183	C₂×C₆₇	#1476	1	1	yes	characteristic
184	C₂×C₆₇	#1479	1	1	yes	characteristic
185	C₂×C₆₇	#1488	1	1	yes	characteristic
186	C₂×C₆₇	#1491	1	1	yes	characteristic
187	C₂×C₆₇	#1797	1	1	yes	characteristic
188	C₂×C₆₇	#1805	1	1	yes	characteristic
189	C₂×C₆₇	#1806	1	1	yes	characteristic
190	C₄²×Q₈	D₆₇	1	1	yes	characteristic
191	#1831	—	67	1	no	—
192	C₄×C₄⋊C₈	—	67	1	no	—
193	C₂×Q₈⋊C₈	—	67	2	no	—
194	#1279	—	67	2	no	—
195	C₆₇	#8566	1	1	yes	characteristic, complemented, Sylow
196	C₂×C₄×Q₈	Dic₆₇	1	1	yes	characteristic
197	C₂×C₄×Q₈	Dic₆₇	1	1	yes	characteristic
198	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
199	C₂×C₄⋊C₈	—	67	2	no	—
200	C₂×C₄⋊C₈	—	134	1	no	—
201	C₂×C₄×C₈	—	67	2	no	—
202	C₄×C₄⋊C₄	—	2	1	no	—
203	C₄×C₄⋊C₄	—	2	1	no	—
204	C₄×C₄⋊C₄	—	2	1	no	—
205	C₄×C₄⋊C₄	Dic₆₇	1	1	yes	characteristic
206	C₄×C₄⋊C₄	Dic₆₇	1	1	yes	characteristic
207	C₄×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
208	C₄³	—	2	1	no	—
209	C₄³	C₂×D₆₇	1	1	yes	characteristic
210	#336	—	134	1	no	—
211	#336	—	134	1	no	—
212	#336	—	134	1	no	—
213	Q₈⋊C₈	—	134	4	no	—
214	C₂²×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
215	C₂²×C₈	—	134	2	no	—
216	C₂²×C₈	—	134	2	no	—
217	C₄×Q₈	—	2	2	no	—
218	C₄×Q₈	—	2	2	no	—
219	C₄×Q₈	—	2	2	no	—
220	C₄×Q₈	—	2	1	no	—
221	C₄×Q₈	—	2	1	no	—
222	C₄×Q₈	C₆₇⋊C₈	1	4	yes	complemented
223	C₄×Q₈	Dic₁₃₄	1	2	yes	—
224	C₄×Q₈	D₂₆₈	1	2	yes	—
225	C₂×C₄⋊C₄	—	2	1	no	—
226	C₂×C₄⋊C₄	—	2	1	no	—
227	C₂×C₄⋊C₄	—	2	1	no	—
228	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
229	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
230	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
231	C₂×C₄²	—	2	1	no	—
232	C₂×C₄²	—	2	1	no	—
233	C₂×C₄²	—	2	1	no	—
234	C₂×C₄²	—	2	1	no	—
235	C₂×C₄²	—	2	1	no	—
236	C₂×C₄²	—	2	1	no	—
237	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
238	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
239	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
240	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
241	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
242	C₂×C₄²	C₂²⋊D₆₇	1	1	yes	characteristic
243	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
244	C₄⋊C₈	—	134	4	no	—
245	C₄⋊C₈	—	134	2	no	—
246	C₄⋊C₈	—	134	2	no	—
247	C₄×C₈	—	134	2	no	—
248	C₄×C₈	—	134	2	no	—
249	#95	—	2	1	no	—
250	#95	—	2	1	no	—
251	#95	—	2	1	no	—
252	C₂×Q₈	—	2	1	no	—
253	C₂×Q₈	—	2	1	no	—
254	C₂×Q₈	—	2	1	no	—
255	C₂×Q₈	C₄⋊Dic₆₇	1	2	yes	—
256	C₂×Q₈	C₂×C₆₇⋊C₈	1	2	yes	—
257	C₂×Q₈	#11761612	1	2	yes	—
258	C₂²×C₄	—	2	1	no	—
259	C₂²×C₄	—	2	1	no	—
260	C₂²×C₄	—	2	1	no	—
261	C₂²×C₄	—	2	1	no	—
262	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
263	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
264	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
265	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
266	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
267	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
268	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
269	C₂×C₈	—	268	2	no	—
270	C₂×C₈	—	268	2	no	—
271	C₂×C₈	—	268	2	no	—
272	C₂×C₈	—	268	2	no	—
273	C₂×C₈	—	268	2	no	—
274	C₂×C₈	—	134	4	no	—
275	C₄⋊C₄	—	2	4	no	—
276	C₄⋊C₄	—	2	4	no	—
277	C₄⋊C₄	—	2	1	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	2	1	no	—
281	C₄⋊C₄	—	2	1	no	—
282	C₄⋊C₄	C₄⋊Dic₆₇	1	2	yes	—
283	C₄⋊C₄	C₂×C₆₇⋊C₈	1	2	yes	—
284	C₄⋊C₄	#11761612	1	2	yes	—
285	C₄²	—	2	4	no	—
286	C₄²	—	2	4	no	—
287	C₄²	—	2	2	no	—
288	C₄²	—	2	2	no	—
289	C₄²	—	2	2	no	—
290	C₄²	—	2	1	no	—
291	C₄²	—	2	1	no	—
292	C₄²	—	2	2	no	—
293	C₄²	—	2	2	no	—
294	C₄²	—	4	1	no	—
295	C₄²	—	4	2	no	—
296	C₄²	—	4	2	no	—
297	C₄²	—	4	2	no	—
298	C₄²	C₂×C₆₇⋊C₈	1	2	yes	—
299	C₄²	#11761612	1	2	yes	—
300	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
301	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
302	C₄²	#11761628	1	1	yes	characteristic
303	C₄²	#11761628	1	1	yes	characteristic
304	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
305	Q₈	—	2	2	no	—
306	Q₈	—	2	2	no	—
307	Q₈	—	2	2	no	—
308	Q₈	C₄⋊(C₆₇⋊C₈)	1	4	yes	complemented
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	1	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	1	no	—
323	C₂×C₄	—	2	2	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	4	1	no	—
327	C₂×C₄	—	4	1	no	—
328	C₂×C₄	—	4	1	no	—
329	C₂×C₄	—	4	1	no	—
330	C₂×C₄	—	4	1	no	—
331	C₂×C₄	—	4	1	no	—
332	C₂×C₄	C₂²⋊(C₆₇⋊C₈)	1	2	yes	—
333	C₂×C₄	C₂²⋊(C₆₇⋊C₈)	1	2	yes	—
334	C₂×C₄	#14523970	1	2	yes	—
335	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
336	C₂×C₄	#14524002	1	1	yes	characteristic
337	C₂×C₄	#14524017	1	1	yes	characteristic
338	C₂×C₄	#14524017	1	1	yes	characteristic
339	C₂×C₄	#14524022	1	1	yes	characteristic
340	C₂×C₄	#14524022	1	1	yes	characteristic
341	C₂×C₄	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
342	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
343	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
344	C₂×C₄	C₂×#11761612	1	1	yes	characteristic
345	C₂×C₄	#14524063	1	2	yes	—
346	C₂×C₄	#14524074	1	2	yes	—
347	C₈	—	268	4	no	—
348	C₈	—	536	2	no	—
349	C₂²	#16863256	1	1	yes	characteristic
350	C₂²	C₂×C₂²⋊(C₆₇⋊C₈)	1	1	yes	characteristic
351	C₂²	#16863360	1	1	yes	characteristic
352	C₂²	#16863392	1	1	yes	characteristic
353	C₂²	#16863405	1	1	yes	characteristic
354	C₂²	C₂×#14523970	1	1	yes	characteristic
355	C₂²	#16863413	1	1	yes	characteristic
356	C₄	—	2	2	no	—
357	C₄	—	2	2	no	—
358	C₄	—	2	4	no	—
359	C₄	—	2	2	no	—
360	C₄	—	2	1	no	—
361	C₄	—	2	1	no	—
362	C₄	—	4	2	no	—
363	C₄	—	4	1	no	—
364	C₄	—	4	2	no	—
365	C₄	—	4	2	no	—
366	C₄	C₂×C₄⋊(C₆₇⋊C₈)	1	1	yes	characteristic
367	C₄	#16863420	1	1	yes	characteristic
368	C₄	#16863449	1	1	yes	characteristic
369	C₄	#16863453	1	1	yes	characteristic
370	C₄	#16863618	1	2	yes	—
371	C₄	#16863703	1	2	yes	—
372	C₂	#22558591	1	1	yes	characteristic
373	C₂	#22558604	1	1	yes	characteristic
374	C₂	#22558643	1	1	yes	characteristic
375	C₂	#22558656	1	1	yes	characteristic
376	C₂	#22560189	1	1	yes	characteristic
377	C₂	#22560224	1	1	yes	characteristic
378	C₂	#22560229	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊D₆₇
commutator	C₂×C₄×C₆₇	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₄²×Q₈×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₄²×Q₈	D₆₇
hypocenter	C₆₇	#8566
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ C₄²×Q₈

Complements

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

N	H ≅ G/N	type
37. C₄×Q₈×C₆₇	C₈	semidirect
118. Q₈×C₆₇	C₄⋊C₈	semidirect
195. C₆₇	#8566	semidirect
222. C₄×Q₈	C₆₇⋊C₈	semidirect
308. Q₈	C₄⋊(C₆₇⋊C₈)	semidirect

Maximal subgroups

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

subgroup	index
#22560326	2
C₄×C₄⋊(C₆₇⋊C₈)	2
C₂×Q₈⋊(C₆₇⋊C₈)	2
#22558050	2
C₄²×Q₈×C₆₇	2
#8566	67