Subgroups — #30875723

Order 17152.

784 conjugacy classes of subgroups.

The subgroup lattice has 370 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22561146	C₂	1	2	yes	maximal
2	#22560768	C₂	1	4	yes	maximal
3	#22560361	C₂	1	1	yes	maximal, characteristic
4	#22560326	C₂	1	2	yes	maximal
5	C₄⋊C₄×C₆₇⋊C₈	C₂	1	2	yes	maximal
6	C₄×C₄⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
7	C₄×C₄⋊(C₆₇⋊C₈)	C₂	1	2	yes	maximal
8	C₄²×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
9	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
11	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
12	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
13	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
14	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
15	C₂×C₄×C₆₇⋊C₈	C₂²	1	4	yes	—
16	#16863151	C₂²	1	2	yes	—
17	#16863151	C₂²	1	4	yes	—
18	#16863151	C₂²	1	4	yes	—
19	#16863151	C₂²	1	4	yes	—
20	C₂×C₄×Q₈×C₆₇	C₄	1	1	yes	characteristic
21	C₂×C₄×Q₈×C₆₇	C₄	1	1	yes	characteristic
22	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
23	C₄×C₄⋊C₄×C₆₇	C₄	1	1	yes	characteristic
24	C₄×C₄⋊C₄×C₆₇	C₄	1	2	yes	—
25	C₄×C₄⋊C₄×C₆₇	C₄	1	2	yes	—
26	C₄×C₄⋊C₄×C₆₇	C₄	1	1	yes	characteristic
27	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
28	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
29	C₄³×C₆₇	C₂²	1	2	yes	—
30	C₄³×C₆₇	C₂²	1	1	yes	characteristic
31	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
32	C₂²×C₆₇⋊C₈	C₂³	1	4	yes	—
33	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
34	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
35	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
36	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
37	C₄⋊(C₆₇⋊C₈)	—	4	2	no	—
38	C₄⋊(C₆₇⋊C₈)	Q₈	1	4	yes	complemented
39	C₄×C₆₇⋊C₈	—	2	4	no	—
40	C₄×C₆₇⋊C₈	—	4	4	no	—
41	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
42	C₄×Q₈×C₆₇	—	2	4	no	—
43	C₄×Q₈×C₆₇	—	2	2	no	—
44	C₄×Q₈×C₆₇	—	2	2	no	—
45	C₄×Q₈×C₆₇	C₈	1	8	yes	complemented
46	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
47	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
49	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
50	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
54	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
56	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
57	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
58	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
59	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
60	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
61	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
62	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
63	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
64	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
65	#95×C₆₇	C₂×C₄	1	2	yes	—
66	#95×C₆₇	C₂×C₄	1	2	yes	—
67	#95×C₆₇	C₂×C₄	1	2	yes	—
68	C₂×C₆₇⋊C₈	—	2	2	no	—
69	C₂×C₆₇⋊C₈	—	2	2	no	—
70	C₂×C₆₇⋊C₈	—	2	2	no	—
71	C₂×C₆₇⋊C₈	—	2	4	no	—
72	C₂×C₆₇⋊C₈	—	2	2	no	—
73	C₂×C₆₇⋊C₈	—	2	4	no	—
74	C₂×C₆₇⋊C₈	—	2	4	no	—
75	C₂×C₆₇⋊C₈	—	2	2	no	—
76	C₂×C₆₇⋊C₈	C₂×Q₈	1	4	yes	—
77	C₂×Q₈×C₆₇	—	2	2	no	—
78	C₂×Q₈×C₆₇	—	2	2	no	—
79	C₂×Q₈×C₆₇	C₂×C₈	1	4	yes	—
80	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
82	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
83	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
84	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
85	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
86	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
87	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
88	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
89	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
90	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
91	C₄⋊C₄×C₆₇	—	2	4	no	—
92	C₄⋊C₄×C₆₇	—	2	2	no	—
93	C₄⋊C₄×C₆₇	—	2	4	no	—
94	C₄⋊C₄×C₆₇	C₂×C₈	1	8	yes	—
95	C₄⋊C₄×C₆₇	C₂×C₈	1	4	yes	—
96	C₄⋊C₄×C₆₇	M₁₆	1	4	yes	—
97	C₄²×C₆₇	—	2	4	no	—
98	C₄²×C₆₇	—	2	2	no	—
99	C₄²×C₆₇	—	2	2	no	—
100	C₄²×C₆₇	—	2	4	no	—
101	C₄²×C₆₇	—	2	2	no	—
102	C₄²×C₆₇	—	2	2	no	—
103	C₄²×C₆₇	—	2	2	no	—
104	C₄²×C₆₇	—	2	1	no	—
105	C₄²×C₆₇	—	2	2	no	—
106	C₄²×C₆₇	—	2	1	no	—
107	C₄²×C₆₇	—	4	1	no	—
108	C₄²×C₆₇	—	4	2	no	—
109	C₄²×C₆₇	C₂×C₈	1	8	yes	—
110	C₄²×C₆₇	C₂×C₈	1	4	yes	—
111	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
112	C₄²×C₆₇	#41	1	2	yes	—
113	C₄²×C₆₇	#41	1	2	yes	—
114	C₄²×C₆₇	#41	1	2	yes	—
115	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
116	Q₈×C₆₇	—	2	8	no	—
117	C₂×C₄×C₆₇	—	2	2	no	—
118	C₂×C₄×C₆₇	—	2	2	no	—
119	C₂×C₄×C₆₇	—	2	1	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	2	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	2	no	—
129	C₂×C₄×C₆₇	—	2	2	no	—
130	C₂×C₄×C₆₇	—	2	1	no	—
131	C₂×C₄×C₆₇	—	2	1	no	—
132	C₂×C₄×C₆₇	—	2	2	no	—
133	C₂×C₄×C₆₇	—	2	1	no	—
134	C₂×C₄×C₆₇	—	2	2	no	—
135	C₂×C₄×C₆₇	#117	1	2	yes	—
136	C₂×C₄×C₆₇	#117	1	2	yes	—
137	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
138	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
139	C₂×C₄×C₆₇	C₂²×C₈	1	4	yes	—
140	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
141	C₂×C₄×C₆₇	C₂²×C₈	1	4	yes	—
142	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
143	C₂×C₄×C₆₇	C₂²×C₈	1	2	yes	—
144	C₂×C₄×C₆₇	C₂×M₁₆	1	4	yes	—
145	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
146	C₂×C₄×C₆₇	#131	1	2	yes	—
147	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
148	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
149	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
150	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
151	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
152	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
153	C₆₇⋊C₈	—	2	4	no	—
154	C₆₇⋊C₈	—	4	2	no	—
155	C₆₇⋊C₈	—	4	4	no	—
156	C₂²×C₆₇	C₂×#117	1	1	yes	characteristic
157	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
158	C₂²×C₆₇	#520	1	1	yes	characteristic
159	C₂²×C₆₇	C₂³×C₈	1	1	yes	characteristic
160	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
161	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
162	C₂²×C₆₇	#568	1	1	yes	characteristic
163	C₄×C₆₇	—	2	4	no	—
164	C₄×C₆₇	—	2	2	no	—
165	C₄×C₆₇	—	2	2	no	—
166	C₄×C₆₇	—	2	4	no	—
167	C₄×C₆₇	—	2	2	no	—
168	C₄×C₆₇	—	2	4	no	—
169	C₄×C₆₇	—	4	2	no	—
170	C₄×C₆₇	—	4	1	no	—
171	C₄×C₆₇	#431	1	4	yes	—
172	C₄×C₆₇	C₈×Q₈	1	4	yes	—
173	C₄×C₆₇	#554	1	1	yes	characteristic
174	C₄×C₆₇	#556	1	1	yes	characteristic
175	C₄×C₆₇	#557	1	1	yes	characteristic
176	C₄×C₆₇	#559	1	1	yes	characteristic
177	#22821	—	67	1	no	maximal, Sylow
178	C₂×C₆₇	#2355	1	1	yes	characteristic
179	C₂×C₆₇	C₂×#431	1	1	yes	characteristic
180	C₂×C₆₇	#2925	1	1	yes	characteristic
181	C₂×C₆₇	C₂×C₈×Q₈	1	1	yes	characteristic
182	C₂×C₆₇	Q₈×M₁₆	1	1	yes	characteristic
183	C₂×C₆₇	#2988	1	1	yes	characteristic
184	C₂×C₆₇	#2990	1	1	yes	characteristic
185	C₄²×Q₈	D₆₇	1	1	yes	characteristic
186	#1984	—	67	2	no	—
187	#1916	—	67	4	no	—
188	#1842	—	67	1	no	—
189	#1831	—	67	2	no	—
190	C₈×C₄⋊C₄	—	67	2	no	—
191	C₄×C₄⋊C₈	—	67	1	no	—
192	C₄×C₄⋊C₈	—	67	2	no	—
193	C₆₇	#22821	1	1	yes	characteristic, complemented, Sylow
194	C₂×C₄×Q₈	Dic₆₇	1	1	yes	characteristic
195	C₂×C₄×Q₈	Dic₆₇	1	1	yes	characteristic
196	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
197	C₂×C₄⋊C₈	—	67	2	no	—
198	C₂×C₄⋊C₈	—	67	1	no	—
199	C₂×C₄⋊C₈	—	67	1	no	—
200	C₂×C₄⋊C₈	—	67	2	no	—
201	C₂×C₄⋊C₈	—	67	2	no	—
202	C₂×C₄×C₈	—	67	4	no	—
203	C₂×C₄×C₈	—	67	2	no	—
204	C₄×C₄⋊C₄	Dic₆₇	1	1	yes	characteristic
205	C₄×C₄⋊C₄	Dic₆₇	1	2	yes	—
206	C₄×C₄⋊C₄	Dic₆₇	1	2	yes	—
207	C₄×C₄⋊C₄	Dic₆₇	1	1	yes	characteristic
208	C₄×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
209	C₄×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
210	C₄³	C₂×D₆₇	1	2	yes	—
211	C₄³	C₂×D₆₇	1	1	yes	characteristic
212	#336	—	67	4	no	—
213	#336	—	67	4	no	—
214	#336	—	67	2	no	—
215	#336	—	67	4	no	—
216	C₂²×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
217	C₂²×C₈	—	67	2	no	—
218	C₂²×C₈	—	67	2	no	—
219	C₂²×C₈	—	67	4	no	—
220	C₄×Q₈	—	2	4	no	—
221	C₄×Q₈	—	2	2	no	—
222	C₄×Q₈	—	2	2	no	—
223	C₄×Q₈	C₆₇⋊C₈	1	8	yes	complemented
224	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
225	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
226	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
227	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
228	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
229	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
230	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
231	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
232	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
233	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
234	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
235	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
236	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
237	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
238	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
239	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
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	C₄⋊C₈	—	268	2	no	—
244	C₄⋊C₈	—	67	4	no	—
245	C₄⋊C₈	—	134	4	no	—
246	C₄⋊C₈	—	134	2	no	—
247	C₄⋊C₈	—	134	4	no	—
248	C₄×C₈	—	268	4	no	—
249	C₄×C₈	—	134	4	no	—
250	#95	C₂×Dic₆₇	1	2	yes	—
251	#95	C₂×Dic₆₇	1	2	yes	—
252	#95	C₂×Dic₆₇	1	2	yes	—
253	C₂×Q₈	—	2	2	no	—
254	C₂×Q₈	—	2	2	no	—
255	C₂×Q₈	C₂×C₆₇⋊C₈	1	4	yes	—
256	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
257	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
258	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
259	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
260	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
261	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
262	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
263	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
264	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
265	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
266	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
267	C₂×C₈	—	67	4	no	—
268	C₂×C₈	—	134	2	no	—
269	C₂×C₈	—	134	4	no	—
270	C₂×C₈	—	134	2	no	—
271	C₂×C₈	—	134	4	no	—
272	C₂×C₈	—	134	4	no	—
273	C₂×C₈	—	134	2	no	—
274	C₂×C₈	—	134	2	no	—
275	C₂×C₈	—	134	2	no	—
276	C₄⋊C₄	—	2	4	no	—
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	2	4	no	—
279	C₄⋊C₄	C₂×C₆₇⋊C₈	1	8	yes	—
280	C₄⋊C₄	C₂×C₆₇⋊C₈	1	4	yes	—
281	C₄⋊C₄	#11761612	1	4	yes	—
282	C₄²	—	2	4	no	—
283	C₄²	—	2	2	no	—
284	C₄²	—	2	4	no	—
285	C₄²	—	2	2	no	—
286	C₄²	—	2	2	no	—
287	C₄²	—	2	2	no	—
288	C₄²	—	2	2	no	—
289	C₄²	—	2	1	no	—
290	C₄²	—	2	2	no	—
291	C₄²	—	2	1	no	—
292	C₄²	—	4	1	no	—
293	C₄²	—	4	2	no	—
294	C₄²	C₂×C₆₇⋊C₈	1	8	yes	—
295	C₄²	C₂×C₆₇⋊C₈	1	4	yes	—
296	C₄²	Q₈×D₆₇	1	2	yes	—
297	C₄²	#11761628	1	2	yes	—
298	C₄²	#11761628	1	2	yes	—
299	C₄²	#11761629	1	2	yes	—
300	C₂³	C₂³×Dic₆₇	1	1	yes	characteristic
301	Q₈	—	2	8	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	1	no	—
307	C₂×C₄	—	2	2	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	#14524002	1	2	yes	—
321	C₂×C₄	#14524002	1	2	yes	—
322	C₂×C₄	Q₈×Dic₆₇	1	2	yes	—
323	C₂×C₄	Q₈×Dic₆₇	1	2	yes	—
324	C₂×C₄	C₂²×C₆₇⋊C₈	1	4	yes	—
325	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
326	C₂×C₄	C₂²×C₆₇⋊C₈	1	4	yes	—
327	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
328	C₂×C₄	C₂²×C₆₇⋊C₈	1	2	yes	—
329	C₂×C₄	C₂×#11761612	1	4	yes	—
330	C₂×C₄	C₂×#11761612	1	2	yes	—
331	C₂×C₄	#14524050	1	2	yes	—
332	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
333	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
334	C₂×C₄	C₂×#11761628	1	1	yes	characteristic
335	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
336	C₂×C₄	#14524089	1	1	yes	characteristic
337	C₂×C₄	#14524090	1	1	yes	characteristic
338	C₈	—	268	2	no	—
339	C₈	—	268	4	no	—
340	C₈	—	134	4	no	—
341	C₂²	C₂×#14524002	1	1	yes	characteristic
342	C₂²	C₂×Q₈×Dic₆₇	1	1	yes	characteristic
343	C₂²	#16863879	1	1	yes	characteristic
344	C₂²	C₂³×C₆₇⋊C₈	1	1	yes	characteristic
345	C₂²	C₂²×#11761612	1	1	yes	characteristic
346	C₂²	C₂×#14524050	1	1	yes	characteristic
347	C₂²	#16864163	1	1	yes	characteristic
348	C₄	—	2	4	no	—
349	C₄	—	2	2	no	—
350	C₄	—	2	4	no	—
351	C₄	—	2	4	no	—
352	C₄	—	2	2	no	—
353	C₄	—	2	2	no	—
354	C₄	—	4	1	no	—
355	C₄	—	4	2	no	—
356	C₄	#16863449	1	4	yes	—
357	C₄	Q₈×C₆₇⋊C₈	1	4	yes	—
358	C₄	#16864106	1	1	yes	characteristic
359	C₄	#16864119	1	1	yes	characteristic
360	C₄	#16864124	1	1	yes	characteristic
361	C₄	#16864131	1	1	yes	characteristic
362	C₂	#22563166	1	1	yes	characteristic
363	C₂	C₂×#16863449	1	1	yes	characteristic
364	C₂	#22568875	1	1	yes	characteristic
365	C₂	C₂×Q₈×C₆₇⋊C₈	1	1	yes	characteristic
366	C₂	Q₈×#11761612	1	1	yes	characteristic
367	C₂	#22569320	1	1	yes	characteristic
368	C₂	#22569334	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₆₇
commutator	C₂²×C₆₇	C₂³×C₈
Fitting	C₄²×Q₈×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂³×C₄
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₄²×Q₈	D₆₇
hypocenter	C₆₇	#22821
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×Q₈

Complements

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

N	H ≅ G/N	type
38. C₄⋊(C₆₇⋊C₈)	Q₈	semidirect
45. C₄×Q₈×C₆₇	C₈	semidirect
193. C₆₇	#22821	semidirect
223. C₄×Q₈	C₆₇⋊C₈	semidirect

Maximal subgroups

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

subgroup	index
#22561146	2
#22560768	2
#22560361	2
#22560326	2
C₄⋊C₄×C₆₇⋊C₈	2
C₄×C₄⋊(C₆₇⋊C₈)	2
C₄²×Q₈×C₆₇	2
#22821	67