Subgroups — #30874643

Order 17152.

832 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16863527	C₂	1	2	yes	maximal
2	#22560933	C₂	1	4	yes	maximal
3	#22560928	C₂	1	2	yes	maximal
4	#22560361	C₂	1	1	yes	maximal, characteristic
5	#22560361	C₂	1	2	yes	maximal
6	#22560062	C₂	1	2	yes	maximal
7	C₄×C₄⋊(C₆₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
8	C₄²×Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
9	#16863527	—	2	8	no	—
10	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
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	4	yes	—
14	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	4	yes	—
15	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
16	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
17	C₂×#14523954	C₂²	1	4	yes	—
18	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
19	#16863151	C₂²	1	2	yes	—
20	#16863151	C₂²	1	4	yes	—
21	C₂×C₄×Q₈×C₆₇	C₄	1	1	yes	characteristic
22	C₂×C₄×Q₈×C₆₇	C₄	1	1	yes	characteristic
23	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
24	C₄×C₄⋊C₄×C₆₇	C₄	1	2	yes	—
25	C₄×C₄⋊C₄×C₆₇	C₄	1	1	yes	characteristic
26	C₄×C₄⋊C₄×C₆₇	C₄	1	2	yes	—
27	C₄×C₄⋊C₄×C₆₇	C₄	1	1	yes	characteristic
28	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
29	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
30	C₄³×C₆₇	C₂²	1	1	yes	characteristic
31	C₄³×C₆₇	C₂²	1	2	yes	—
32	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
33	C₂²×C₆₇⋊C₈	C₂³	1	4	yes	—
34	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
35	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
36	C₄⋊(C₆₇⋊C₈)	—	2	8	no	—
37	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
38	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
39	C₄⋊(C₆₇⋊C₈)	—	4	4	no	—
40	C₄⋊(C₆₇⋊C₈)	—	4	2	no	—
41	C₄⋊(C₆₇⋊C₈)	Q₈	1	4	yes	complemented
42	#14523954	—	2	8	no	—
43	C₄×C₆₇⋊C₈	—	2	4	no	—
44	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
45	C₄×Q₈×C₆₇	—	2	4	no	—
46	C₄×Q₈×C₆₇	—	2	4	no	—
47	C₄×Q₈×C₆₇	D₄	1	4	yes	—
48	C₄×Q₈×C₆₇	Q₈	1	4	yes	—
49	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
51	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
52	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
53	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
55	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
56	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
57	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
58	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
59	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
60	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
61	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
63	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
64	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
65	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
66	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
67	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
68	#95×C₆₇	C₂×C₄	1	2	yes	—
69	#95×C₆₇	C₂×C₄	1	2	yes	—
70	#95×C₆₇	C₂×C₄	1	2	yes	—
71	C₂×C₆₇⋊C₈	—	2	2	no	—
72	C₂×C₆₇⋊C₈	—	2	2	no	—
73	C₂×C₆₇⋊C₈	—	2	2	no	—
74	C₂×C₆₇⋊C₈	—	2	4	no	—
75	C₂×C₆₇⋊C₈	—	2	2	no	—
76	C₂×C₆₇⋊C₈	—	2	4	no	—
77	C₂×C₆₇⋊C₈	—	2	2	no	—
78	C₂×C₆₇⋊C₈	—	2	4	no	—
79	C₂×C₆₇⋊C₈	C₂×Q₈	1	4	yes	—
80	C₂×Q₈×C₆₇	—	2	2	no	—
81	C₂×Q₈×C₆₇	—	2	2	no	—
82	C₂×Q₈×C₆₇	C₄⋊C₄	1	4	yes	—
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	2	yes	—
90	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
94	C₄⋊C₄×C₆₇	—	2	4	no	—
95	C₄⋊C₄×C₆₇	—	2	4	no	—
96	C₄⋊C₄×C₆₇	—	2	2	no	—
97	C₄⋊C₄×C₆₇	C₄⋊C₄	1	8	yes	—
98	C₄⋊C₄×C₆₇	C₄⋊C₄	1	4	yes	—
99	C₄⋊C₄×C₆₇	M₁₆	1	4	yes	—
100	C₄²×C₆₇	—	2	4	no	—
101	C₄²×C₆₇	—	2	2	no	—
102	C₄²×C₆₇	—	2	4	no	—
103	C₄²×C₆₇	—	2	4	no	—
104	C₄²×C₆₇	—	2	2	no	—
105	C₄²×C₆₇	—	2	1	no	—
106	C₄²×C₆₇	—	2	2	no	—
107	C₄²×C₆₇	—	2	1	no	—
108	C₄²×C₆₇	—	2	2	no	—
109	C₄²×C₆₇	—	4	1	no	—
110	C₄²×C₆₇	—	4	2	no	—
111	C₄²×C₆₇	M₁₆	1	4	yes	—
112	C₄²×C₆₇	C₂×D₄	1	2	yes	—
113	C₄²×C₆₇	C₂×D₄	1	4	yes	—
114	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
115	C₄²×C₆₇	C₂×Q₈	1	4	yes	—
116	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
117	C₄²×C₆₇	#41	1	2	yes	—
118	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
119	Q₈×C₆₇	—	2	8	no	—
120	C₂×C₄×C₆₇	—	2	1	no	—
121	C₂×C₄×C₆₇	—	2	2	no	—
122	C₂×C₄×C₆₇	—	2	1	no	—
123	C₂×C₄×C₆₇	—	2	2	no	—
124	C₂×C₄×C₆₇	—	2	2	no	—
125	C₂×C₄×C₆₇	—	2	2	no	—
126	C₂×C₄×C₆₇	—	2	1	no	—
127	C₂×C₄×C₆₇	—	2	2	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	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	—	2	1	no	—
136	C₂×C₄×C₆₇	—	2	1	no	—
137	C₂×C₄×C₆₇	—	2	2	no	—
138	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
139	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
140	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
141	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
142	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
143	C₂×C₄×C₆₇	C₄×Q₈	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	C₂×M₁₆	1	2	yes	—
147	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
148	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
149	C₂×C₄×C₆₇	#131	1	2	yes	—
150	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
151	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
152	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
153	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
154	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
155	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
156	C₆₇⋊C₈	—	2	4	no	—
157	C₆₇⋊C₈	—	4	2	no	—
158	C₆₇⋊C₈	—	4	4	no	—
159	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
160	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
161	C₂²×C₆₇	#520	1	1	yes	characteristic
162	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
163	C₂²×C₆₇	C₂²×M₁₆	1	1	yes	characteristic
164	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
165	C₂²×C₆₇	#568	1	1	yes	characteristic
166	C₄×C₆₇	—	2	4	no	—
167	C₄×C₆₇	—	2	2	no	—
168	C₄×C₆₇	—	2	4	no	—
169	C₄×C₆₇	—	2	2	no	—
170	C₄×C₆₇	—	2	2	no	—
171	C₄×C₆₇	—	2	4	no	—
172	C₄×C₆₇	—	4	2	no	—
173	C₄×C₆₇	—	4	1	no	—
174	C₄×C₆₇	#423	1	4	yes	—
175	C₄×C₆₇	#446	1	4	yes	complemented
176	C₄×C₆₇	D₄×Q₈	1	1	yes	characteristic
177	C₄×C₆₇	#550	1	1	yes	characteristic
178	C₄×C₆₇	#557	1	1	yes	characteristic
179	C₄×C₆₇	Q₈²	1	1	yes	characteristic
180	#22722	—	67	1	no	maximal, Sylow
181	C₂×C₆₇	Q₈×C₄⋊C₄	1	1	yes	characteristic
182	C₂×C₆₇	C₂×#423	1	1	yes	characteristic
183	C₂×C₆₇	#2906	1	1	yes	characteristic
184	C₂×C₆₇	C₂×#446	1	1	yes	characteristic
185	C₂×C₆₇	Q₈×M₁₆	1	1	yes	characteristic
186	C₂×C₆₇	#2989	1	1	yes	characteristic
187	C₂×C₆₇	#2990	1	1	yes	characteristic
188	C₂×#446	—	67	2	no	—
189	C₄²×Q₈	D₆₇	1	1	yes	characteristic
190	#1940	—	67	4	no	—
191	#1939	—	67	2	no	—
192	#1842	—	67	1	no	—
193	#1842	—	67	2	no	—
194	#1767	—	67	2	no	—
195	C₄×C₄⋊C₈	—	67	1	no	—
196	C₆₇	#22722	1	1	yes	characteristic, complemented, Sylow
197	C₂×C₄×Q₈	Dic₆₇	1	1	yes	characteristic
198	C₂×C₄×Q₈	Dic₆₇	1	1	yes	characteristic
199	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
200	#446	—	134	8	no	—
201	C₂×C₄⋊C₈	—	67	4	no	—
202	C₂×C₄⋊C₈	—	67	2	no	—
203	C₂×C₄⋊C₈	—	67	1	no	—
204	C₂×C₄⋊C₈	—	67	1	no	—
205	C₂×C₄⋊C₈	—	67	2	no	—
206	C₂×C₄⋊C₈	—	67	2	no	—
207	C₂×C₄⋊C₈	—	67	4	no	—
208	C₂×#97	—	67	4	no	—
209	C₂×C₄×C₈	—	67	2	no	—
210	C₄×C₄⋊C₄	Dic₆₇	1	2	yes	—
211	C₄×C₄⋊C₄	Dic₆₇	1	1	yes	characteristic
212	C₄×C₄⋊C₄	Dic₆₇	1	2	yes	—
213	C₄×C₄⋊C₄	Dic₆₇	1	1	yes	characteristic
214	C₄×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
215	C₄×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
216	C₄³	C₂×D₆₇	1	1	yes	characteristic
217	C₄³	C₂×D₆₇	1	2	yes	—
218	#336	—	67	4	no	—
219	#336	—	67	2	no	—
220	C₂²×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
221	C₂²×C₈	—	67	2	no	—
222	C₂²×C₈	—	67	2	no	—
223	C₂²×C₈	—	67	4	no	—
224	C₄×Q₈	—	2	4	no	—
225	C₄×Q₈	—	2	4	no	—
226	C₄×Q₈	Dic₁₃₄	1	4	yes	—
227	C₄×Q₈	D₂₆₈	1	4	yes	—
228	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
229	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
230	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
231	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
232	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
233	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
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	C₂×Dic₆₇	1	2	yes	—
239	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
240	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
241	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
242	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
243	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
244	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
245	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
246	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
247	C₄⋊C₈	—	268	4	no	—
248	C₄⋊C₈	—	268	2	no	—
249	C₄⋊C₈	—	67	4	no	—
250	C₄⋊C₈	—	134	4	no	—
251	C₄⋊C₈	—	134	8	no	—
252	C₄⋊C₈	—	134	2	no	—
253	C₄⋊C₈	—	134	4	no	—
254	#97	—	134	8	no	—
255	C₄×C₈	—	134	4	no	—
256	#95	C₂×Dic₆₇	1	2	yes	—
257	#95	C₂×Dic₆₇	1	2	yes	—
258	#95	C₂×Dic₆₇	1	2	yes	—
259	C₂×Q₈	—	2	2	no	—
260	C₂×Q₈	—	2	2	no	—
261	C₂×Q₈	C₄⋊Dic₆₇	1	4	yes	—
262	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
263	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
264	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
265	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
266	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
267	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
268	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
269	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
270	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
271	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
272	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
273	C₂×C₈	—	67	4	no	—
274	C₂×C₈	—	134	2	no	—
275	C₂×C₈	—	134	4	no	—
276	C₂×C₈	—	134	2	no	—
277	C₂×C₈	—	134	4	no	—
278	C₂×C₈	—	134	2	no	—
279	C₂×C₈	—	134	4	no	—
280	C₂×C₈	—	134	2	no	—
281	C₂×C₈	—	134	2	no	—
282	C₄⋊C₄	—	2	4	no	—
283	C₄⋊C₄	—	2	4	no	—
284	C₄⋊C₄	—	2	2	no	—
285	C₄⋊C₄	C₄⋊Dic₆₇	1	8	yes	—
286	C₄⋊C₄	C₄⋊Dic₆₇	1	4	yes	—
287	C₄⋊C₄	#11761612	1	4	yes	—
288	C₄²	—	2	4	no	—
289	C₄²	—	2	2	no	—
290	C₄²	—	2	4	no	—
291	C₄²	—	2	4	no	—
292	C₄²	—	2	2	no	—
293	C₄²	—	2	1	no	—
294	C₄²	—	2	2	no	—
295	C₄²	—	2	1	no	—
296	C₄²	—	2	2	no	—
297	C₄²	—	4	1	no	—
298	C₄²	—	4	2	no	—
299	C₄²	#11761612	1	4	yes	—
300	C₄²	C₂×D₂₆₈	1	2	yes	—
301	C₄²	C₂×D₂₆₈	1	4	yes	—
302	C₄²	Q₈×D₆₇	1	2	yes	—
303	C₄²	C₂×Dic₁₃₄	1	4	yes	—
304	C₄²	C₂×Dic₁₃₄	1	2	yes	—
305	C₄²	#11761629	1	2	yes	—
306	C₂³	C₂³×Dic₆₇	1	1	yes	characteristic
307	Q₈	—	2	8	no	—
308	C₂×C₄	—	2	1	no	—
309	C₂×C₄	—	2	2	no	—
310	C₂×C₄	—	2	1	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	1	no	—
317	C₂×C₄	—	2	1	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	2	1	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	2	1	no	—
324	C₂×C₄	—	2	1	no	—
325	C₂×C₄	—	2	2	no	—
326	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
327	C₂×C₄	C₂×C₄⋊Dic₆₇	1	4	yes	—
328	C₂×C₄	C₂×C₄⋊Dic₆₇	1	4	yes	—
329	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
330	C₂×C₄	Q₈×Dic₆₇	1	2	yes	—
331	C₂×C₄	Q₈×Dic₆₇	1	2	yes	—
332	C₂×C₄	C₂×#11761612	1	4	yes	—
333	C₂×C₄	C₂×#11761612	1	2	yes	—
334	C₂×C₄	C₂×#11761612	1	2	yes	—
335	C₂×C₄	C₂×#11761612	1	2	yes	—
336	C₂×C₄	C₂×#11761612	1	2	yes	—
337	C₂×C₄	#14524050	1	2	yes	—
338	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
339	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
340	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
341	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
342	C₂×C₄	#14524089	1	1	yes	characteristic
343	C₂×C₄	#14524090	1	1	yes	characteristic
344	C₈	—	268	2	no	—
345	C₈	—	268	4	no	—
346	C₈	—	134	4	no	—
347	C₂²	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
348	C₂²	C₂×Q₈×Dic₆₇	1	1	yes	characteristic
349	C₂²	#16863879	1	1	yes	characteristic
350	C₂²	C₂²×#11761612	1	1	yes	characteristic
351	C₂²	C₂²×#11761612	1	1	yes	characteristic
352	C₂²	C₂×#14524050	1	1	yes	characteristic
353	C₂²	#16864163	1	1	yes	characteristic
354	C₄	—	2	4	no	—
355	C₄	—	2	2	no	—
356	C₄	—	2	4	no	—
357	C₄	—	2	2	no	—
358	C₄	—	2	4	no	—
359	C₄	—	2	2	no	—
360	C₄	—	4	2	no	—
361	C₄	—	4	1	no	—
362	C₄	#16863420	1	4	yes	—
363	C₄	#16863527	1	4	yes	complemented
364	C₄	Q₈×D₂₆₈	1	1	yes	characteristic
365	C₄	#16864068	1	1	yes	characteristic
366	C₄	#16864125	1	1	yes	characteristic
367	C₄	Q₈×Dic₁₃₄	1	1	yes	characteristic
368	C₂	Q₈×C₄⋊Dic₆₇	1	1	yes	characteristic
369	C₂	C₂×#16863420	1	1	yes	characteristic
370	C₂	#22568777	1	1	yes	characteristic
371	C₂	C₂×#16863527	1	1	yes	characteristic
372	C₂	Q₈×#11761612	1	1	yes	characteristic
373	C₂	#22569326	1	1	yes	characteristic
374	C₂	#22569334	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₆₇
commutator, socle, prime radical	C₂³×C₆₇	C₂³×C₄
Fitting	C₄²×Q₈×C₆₇	C₂
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	C₄²×Q₈	D₆₇
hypocenter	C₆₇	#22722
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×Q₈

Complements

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

N	H ≅ G/N	type
41. C₄⋊(C₆₇⋊C₈)	Q₈	semidirect
175. C₄×C₆₇	#446	semidirect
196. C₆₇	#22722	semidirect
363. C₄	#16863527	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16863527	2
#22560933	2
#22560928	2
#22560361	2
#22560062	2
C₄×C₄⋊(C₆₇⋊C₈)	2
C₄²×Q₈×C₆₇	2
#22722	67