Subgroups — #30874651

Order 17152.

732 conjugacy classes of subgroups.

The subgroup lattice has 394 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	2	yes	maximal
3	#22560933	C₂	1	2	yes	maximal
4	#22560928	C₂	1	2	yes	maximal
5	#22560363	C₂	1	2	yes	maximal
6	#22560361	C₂	1	1	yes	maximal, characteristic
7	#22560066	C₂	1	2	yes	maximal
8	#22560062	C₂	1	1	yes	maximal, characteristic
9	C₆₇×#2295	C₂	1	1	yes	maximal, characteristic
10	#16863527	—	2	8	no	—
11	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
12	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
13	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
14	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
15	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
16	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
17	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
18	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
19	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
20	C₂×C₄⋊(C₆₇⋊C₈)	C₂²	1	1	yes	characteristic
21	C₂×#14523954	C₂²	1	2	yes	—
22	C₂×#14523954	C₂²	1	2	yes	—
23	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
24	#16863151	C₂²	1	2	yes	—
25	#16863151	C₂²	1	2	yes	—
26	#16863151	C₂²	1	2	yes	—
27	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
28	#391×C₆₇	C₄	1	1	yes	characteristic
29	#391×C₆₇	C₄	1	1	yes	characteristic
30	#389×C₆₇	C₄	1	1	yes	characteristic
31	#389×C₆₇	C₄	1	1	yes	characteristic
32	#387×C₆₇	C₄	1	2	yes	—
33	#387×C₆₇	C₄	1	2	yes	—
34	C₄×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
35	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
36	#376×C₆₇	C₂²	1	2	yes	—
37	C₄³×C₆₇	C₂²	1	1	yes	characteristic
38	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
39	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
40	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
41	C₂²×C₆₇⋊C₈	C₂³	1	2	yes	—
42	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
43	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
44	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
45	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
46	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
47	C₄⋊(C₆₇⋊C₈)	—	4	1	no	—
48	C₄⋊(C₆₇⋊C₈)	—	4	1	no	—
49	C₄⋊(C₆₇⋊C₈)	—	4	2	no	—
50	C₄⋊(C₆₇⋊C₈)	—	4	2	no	—
51	C₄⋊(C₆₇⋊C₈)	Q₈	1	4	yes	complemented
52	#14523954	—	2	4	no	—
53	#14523954	—	2	4	no	—
54	C₄×C₆₇⋊C₈	—	2	4	no	—
55	C₂²×Q₈×C₆₇	C₂×C₄	1	1	yes	characteristic
56	C₄×Q₈×C₆₇	D₄	1	4	yes	—
57	C₄×Q₈×C₆₇	Q₈	1	4	yes	—
58	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
59	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
60	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
61	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
62	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
63	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
65	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
66	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
69	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
70	C₂×C₄²×C₆₇	C₂³	1	2	yes	—
71	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
72	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
73	#95×C₆₇	C₂×C₄	1	2	yes	—
74	#95×C₆₇	C₂×C₄	1	2	yes	—
75	#95×C₆₇	C₂×C₄	1	2	yes	—
76	#95×C₆₇	C₂×C₄	1	2	yes	—
77	#95×C₆₇	C₂×C₄	1	2	yes	—
78	#95×C₆₇	C₂×C₄	1	2	yes	—
79	#95×C₆₇	C₂×C₄	1	2	yes	—
80	C₂×C₆₇⋊C₈	—	2	2	no	—
81	C₂×C₆₇⋊C₈	—	2	2	no	—
82	C₂×C₆₇⋊C₈	—	2	2	no	—
83	C₂×C₆₇⋊C₈	—	2	2	no	—
84	C₂×C₆₇⋊C₈	—	2	2	no	—
85	C₂×C₆₇⋊C₈	—	2	2	no	—
86	C₂×C₆₇⋊C₈	—	2	2	no	—
87	C₂×C₆₇⋊C₈	—	2	2	no	—
88	C₂×C₆₇⋊C₈	—	2	2	no	—
89	C₂×C₆₇⋊C₈	—	2	2	no	—
90	C₂×C₆₇⋊C₈	—	2	2	no	—
91	C₂×C₆₇⋊C₈	C₂×Q₈	1	4	yes	—
92	C₂×Q₈×C₆₇	—	4	1	no	—
93	C₂×Q₈×C₆₇	—	4	1	no	—
94	C₂×Q₈×C₆₇	C₄⋊C₄	1	4	yes	—
95	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
96	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
97	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
98	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
99	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
100	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
101	C₂²×C₄×C₆₇	C₂²×C₄	1	2	yes	—
102	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
103	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
105	C₂²×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
106	C₄⋊C₄×C₆₇	—	2	4	no	—
107	C₄⋊C₄×C₆₇	—	4	1	no	—
108	C₄⋊C₄×C₆₇	—	4	1	no	—
109	C₄⋊C₄×C₆₇	—	4	2	no	—
110	C₄⋊C₄×C₆₇	C₄⋊C₄	1	8	yes	—
111	C₄⋊C₄×C₆₇	C₄⋊C₄	1	4	yes	—
112	C₄²×C₆₇	—	2	2	no	—
113	C₄²×C₆₇	—	2	1	no	—
114	C₄²×C₆₇	—	2	2	no	—
115	C₄²×C₆₇	—	2	1	no	—
116	C₄²×C₆₇	—	2	2	no	—
117	C₄²×C₆₇	—	4	1	no	—
118	C₄²×C₆₇	—	4	1	no	—
119	C₄²×C₆₇	—	4	1	no	—
120	C₄²×C₆₇	C₂×D₄	1	4	yes	—
121	C₄²×C₆₇	C₂×D₄	1	2	yes	—
122	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
123	C₄²×C₆₇	C₂×Q₈	1	4	yes	—
124	C₄²×C₆₇	C₂×Q₈	1	2	yes	—
125	C₄²×C₆₇	#41	1	2	yes	—
126	C₂³×C₆₇	C₂³×C₄	1	1	yes	characteristic
127	Q₈×C₆₇	—	4	4	no	—
128	C₂×C₄×C₆₇	—	2	1	no	—
129	C₂×C₄×C₆₇	—	2	1	no	—
130	C₂×C₄×C₆₇	—	2	1	no	—
131	C₂×C₄×C₆₇	—	2	2	no	—
132	C₂×C₄×C₆₇	—	2	1	no	—
133	C₂×C₄×C₆₇	—	2	1	no	—
134	C₂×C₄×C₆₇	—	2	1	no	—
135	C₂×C₄×C₆₇	—	2	1	no	—
136	C₂×C₄×C₆₇	—	2	1	no	—
137	C₂×C₄×C₆₇	—	2	1	no	—
138	C₂×C₄×C₆₇	—	2	1	no	—
139	C₂×C₄×C₆₇	—	2	2	no	—
140	C₂×C₄×C₆₇	—	2	2	no	—
141	C₂×C₄×C₆₇	—	2	2	no	—
142	C₂×C₄×C₆₇	—	2	2	no	—
143	C₂×C₄×C₆₇	—	2	1	no	—
144	C₂×C₄×C₆₇	—	2	2	no	—
145	C₂×C₄×C₆₇	—	2	2	no	—
146	C₂×C₄×C₆₇	—	2	1	no	—
147	C₂×C₄×C₆₇	—	2	1	no	—
148	C₂×C₄×C₆₇	—	2	2	no	—
149	C₂×C₄×C₆₇	—	2	2	no	—
150	C₂×C₄×C₆₇	—	2	2	no	—
151	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
152	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
153	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	4	yes	—
154	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
155	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
156	C₂×C₄×C₆₇	C₄×Q₈	1	2	yes	—
157	C₂×C₄×C₆₇	#131	1	2	yes	—
158	C₂×C₄×C₆₇	C₂²×D₄	1	1	yes	characteristic
159	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
160	C₂×C₄×C₆₇	C₂²×Q₈	1	1	yes	characteristic
161	C₂×C₄×C₆₇	C₂×#41	1	1	yes	characteristic
162	C₂×C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
163	C₂×C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
164	C₆₇⋊C₈	—	2	4	no	—
165	C₆₇⋊C₈	—	4	2	no	—
166	C₆₇⋊C₈	—	4	2	no	—
167	C₆₇⋊C₈	—	4	2	no	—
168	C₂²×C₆₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
169	C₂²×C₆₇	C₂×C₄×Q₈	1	1	yes	characteristic
170	C₂²×C₆₇	#520	1	1	yes	characteristic
171	C₂²×C₆₇	C₂×#131	1	1	yes	characteristic
172	C₂²×C₆₇	#568	1	1	yes	characteristic
173	C₂²×C₆₇	#568	1	1	yes	characteristic
174	C₂²×C₆₇	#568	1	1	yes	characteristic
175	C₄×C₆₇	—	2	2	no	—
176	C₄×C₆₇	—	2	4	no	—
177	C₄×C₆₇	—	2	2	no	—
178	C₄×C₆₇	—	2	4	no	—
179	C₄×C₆₇	—	2	2	no	—
180	C₄×C₆₇	—	2	2	no	—
181	C₄×C₆₇	—	2	2	no	—
182	C₄×C₆₇	—	4	2	no	—
183	C₄×C₆₇	—	4	1	no	—
184	C₄×C₆₇	—	4	2	no	—
185	C₄×C₆₇	D₄×Q₈	1	1	yes	characteristic
186	C₄×C₆₇	#550	1	1	yes	characteristic
187	C₄×C₆₇	#557	1	1	yes	characteristic
188	C₄×C₆₇	Q₈²	1	1	yes	characteristic
189	#22723	—	67	1	no	maximal, Sylow
190	C₂×C₆₇	Q₈×C₄⋊C₄	1	1	yes	characteristic
191	C₂×C₆₇	#2905	1	1	yes	characteristic
192	C₂×C₆₇	#2906	1	1	yes	characteristic
193	C₂×C₆₇	#2961	1	1	yes	characteristic
194	C₂×C₆₇	#2962	1	1	yes	characteristic
195	C₂×C₆₇	#2992	1	1	yes	characteristic
196	C₂×C₆₇	#2995	1	1	yes	characteristic
197	C₂×#446	—	67	2	no	—
198	#2295	D₆₇	1	1	yes	characteristic
199	#1940	—	67	2	no	—
200	#1940	—	67	2	no	—
201	#1939	—	67	2	no	—
202	#1843	—	67	2	no	—
203	#1842	—	67	1	no	—
204	#1768	—	67	2	no	—
205	#1767	—	67	1	no	—
206	C₆₇	#22723	1	1	yes	characteristic, complemented, Sylow
207	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
208	#446	—	134	8	no	—
209	C₂×C₄⋊C₈	—	67	1	no	—
210	C₂×C₄⋊C₈	—	67	1	no	—
211	C₂×C₄⋊C₈	—	67	2	no	—
212	C₂×C₄⋊C₈	—	67	2	no	—
213	C₂×C₄⋊C₈	—	67	1	no	—
214	C₂×C₄⋊C₈	—	67	2	no	—
215	C₂×C₄⋊C₈	—	67	2	no	—
216	C₂×C₄⋊C₈	—	67	1	no	—
217	C₂×C₄⋊C₈	—	67	2	no	—
218	C₂×C₄⋊C₈	—	67	2	no	—
219	C₂×#97	—	67	2	no	—
220	C₂×#97	—	67	2	no	—
221	C₂×C₄×C₈	—	67	2	no	—
222	#391	Dic₆₇	1	1	yes	characteristic
223	#391	Dic₆₇	1	1	yes	characteristic
224	#389	Dic₆₇	1	1	yes	characteristic
225	#389	Dic₆₇	1	1	yes	characteristic
226	#387	Dic₆₇	1	2	yes	—
227	#387	Dic₆₇	1	2	yes	—
228	C₄×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
229	C₄×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
230	#376	C₂×D₆₇	1	2	yes	—
231	C₄³	C₂×D₆₇	1	1	yes	characteristic
232	#336	—	67	2	no	—
233	#336	—	67	2	no	—
234	#336	—	67	2	no	—
235	C₂²×Q₈	C₂×Dic₆₇	1	1	yes	characteristic
236	C₂²×C₈	—	67	2	no	—
237	C₂²×C₈	—	67	2	no	—
238	C₂²×C₈	—	67	2	no	—
239	C₂²×C₈	—	67	2	no	—
240	C₄×Q₈	Dic₁₃₄	1	4	yes	—
241	C₄×Q₈	D₂₆₈	1	4	yes	—
242	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
243	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
244	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
245	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
246	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
247	C₂×C₄⋊C₄	C₂×Dic₆₇	1	1	yes	characteristic
248	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
249	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
250	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
251	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
252	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
253	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
254	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
255	C₂×C₄²	C₂²×D₆₇	1	2	yes	—
256	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
257	C₄⋊C₈	—	268	1	no	—
258	C₄⋊C₈	—	268	1	no	—
259	C₄⋊C₈	—	268	2	no	—
260	C₄⋊C₈	—	268	2	no	—
261	C₄⋊C₈	—	67	4	no	—
262	C₄⋊C₈	—	134	4	no	—
263	C₄⋊C₈	—	134	4	no	—
264	C₄⋊C₈	—	134	4	no	—
265	C₄⋊C₈	—	134	4	no	—
266	C₄⋊C₈	—	134	2	no	—
267	#97	—	134	4	no	—
268	#97	—	134	4	no	—
269	C₄×C₈	—	134	4	no	—
270	#95	C₂×Dic₆₇	1	2	yes	—
271	#95	C₂×Dic₆₇	1	2	yes	—
272	#95	C₂×Dic₆₇	1	2	yes	—
273	#95	C₂×Dic₆₇	1	2	yes	—
274	#95	C₂×Dic₆₇	1	2	yes	—
275	#95	C₂×Dic₆₇	1	2	yes	—
276	#95	C₂×Dic₆₇	1	2	yes	—
277	C₂×Q₈	—	4	1	no	—
278	C₂×Q₈	—	4	1	no	—
279	C₂×Q₈	C₄⋊Dic₆₇	1	4	yes	—
280	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
281	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
282	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
283	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
284	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
285	C₂²×C₄	C₂²×Dic₆₇	1	2	yes	—
286	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
287	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
288	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
289	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
290	C₂²×C₄	C₂³×D₆₇	1	1	yes	characteristic
291	C₂×C₈	—	67	4	no	—
292	C₂×C₈	—	134	2	no	—
293	C₂×C₈	—	134	2	no	—
294	C₂×C₈	—	134	2	no	—
295	C₂×C₈	—	134	2	no	—
296	C₂×C₈	—	134	2	no	—
297	C₂×C₈	—	134	2	no	—
298	C₂×C₈	—	134	2	no	—
299	C₂×C₈	—	134	2	no	—
300	C₂×C₈	—	134	2	no	—
301	C₂×C₈	—	134	2	no	—
302	C₂×C₈	—	134	2	no	—
303	C₄⋊C₄	—	2	4	no	—
304	C₄⋊C₄	—	4	1	no	—
305	C₄⋊C₄	—	4	1	no	—
306	C₄⋊C₄	—	4	2	no	—
307	C₄⋊C₄	C₄⋊Dic₆₇	1	8	yes	—
308	C₄⋊C₄	C₄⋊Dic₆₇	1	4	yes	—
309	C₄²	—	2	2	no	—
310	C₄²	—	2	1	no	—
311	C₄²	—	2	2	no	—
312	C₄²	—	2	1	no	—
313	C₄²	—	2	2	no	—
314	C₄²	—	4	1	no	—
315	C₄²	—	4	1	no	—
316	C₄²	—	4	1	no	—
317	C₄²	C₂×D₂₆₈	1	4	yes	—
318	C₄²	C₂×D₂₆₈	1	2	yes	—
319	C₄²	Q₈×D₆₇	1	2	yes	—
320	C₄²	C₂×Dic₁₃₄	1	2	yes	—
321	C₄²	C₂×Dic₁₃₄	1	4	yes	—
322	C₄²	#11761629	1	2	yes	—
323	C₂³	C₂³×Dic₆₇	1	1	yes	characteristic
324	Q₈	—	4	4	no	—
325	C₂×C₄	—	2	1	no	—
326	C₂×C₄	—	2	1	no	—
327	C₂×C₄	—	2	1	no	—
328	C₂×C₄	—	2	2	no	—
329	C₂×C₄	—	2	1	no	—
330	C₂×C₄	—	2	1	no	—
331	C₂×C₄	—	2	1	no	—
332	C₂×C₄	—	2	1	no	—
333	C₂×C₄	—	2	1	no	—
334	C₂×C₄	—	2	1	no	—
335	C₂×C₄	—	2	1	no	—
336	C₂×C₄	—	2	2	no	—
337	C₂×C₄	—	2	2	no	—
338	C₂×C₄	—	2	2	no	—
339	C₂×C₄	—	2	2	no	—
340	C₂×C₄	—	2	2	no	—
341	C₂×C₄	—	2	1	no	—
342	C₂×C₄	—	2	2	no	—
343	C₂×C₄	—	2	1	no	—
344	C₂×C₄	—	2	1	no	—
345	C₂×C₄	—	2	2	no	—
346	C₂×C₄	—	2	2	no	—
347	C₂×C₄	—	2	2	no	—
348	C₂×C₄	C₂×C₄⋊Dic₆₇	1	4	yes	—
349	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
350	C₂×C₄	C₂×C₄⋊Dic₆₇	1	4	yes	—
351	C₂×C₄	C₂×C₄⋊Dic₆₇	1	2	yes	—
352	C₂×C₄	Q₈×Dic₆₇	1	2	yes	—
353	C₂×C₄	Q₈×Dic₆₇	1	2	yes	—
354	C₂×C₄	#14524050	1	2	yes	—
355	C₂×C₄	C₂²×D₂₆₈	1	1	yes	characteristic
356	C₂×C₄	C₂×Q₈×D₆₇	1	1	yes	characteristic
357	C₂×C₄	C₂²×Dic₁₃₄	1	1	yes	characteristic
358	C₂×C₄	C₂×#11761629	1	1	yes	characteristic
359	C₂×C₄	#14524089	1	1	yes	characteristic
360	C₂×C₄	#14524090	1	1	yes	characteristic
361	C₈	—	268	2	no	—
362	C₈	—	268	2	no	—
363	C₈	—	268	2	no	—
364	C₈	—	134	4	no	—
365	C₂²	C₂²×C₄⋊Dic₆₇	1	1	yes	characteristic
366	C₂²	C₂×Q₈×Dic₆₇	1	1	yes	characteristic
367	C₂²	#16863879	1	1	yes	characteristic
368	C₂²	C₂×#14524050	1	1	yes	characteristic
369	C₂²	#16864163	1	1	yes	characteristic
370	C₂²	#16864163	1	1	yes	characteristic
371	C₂²	#16864163	1	1	yes	characteristic
372	C₄	—	2	2	no	—
373	C₄	—	2	4	no	—
374	C₄	—	2	2	no	—
375	C₄	—	2	4	no	—
376	C₄	—	2	2	no	—
377	C₄	—	2	2	no	—
378	C₄	—	2	2	no	—
379	C₄	—	4	2	no	—
380	C₄	—	4	1	no	—
381	C₄	—	4	2	no	—
382	C₄	Q₈×D₂₆₈	1	1	yes	characteristic
383	C₄	#16864068	1	1	yes	characteristic
384	C₄	#16864125	1	1	yes	characteristic
385	C₄	Q₈×Dic₁₃₄	1	1	yes	characteristic
386	C₂	Q₈×C₄⋊Dic₆₇	1	1	yes	characteristic
387	C₂	#22568773	1	1	yes	characteristic
388	C₂	#22568777	1	1	yes	characteristic
389	C₂	#22569150	1	1	yes	characteristic
390	C₂	#22569158	1	1	yes	characteristic
391	C₂	#22569348	1	1	yes	characteristic
392	C₂	#22569379	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₆₇
commutator, socle, prime radical	C₂³×C₆₇	C₂³×C₄
Fitting	C₆₇×#2295	C₂
cosocle	C₂²×C₄×C₆₇	C₂⁴
hypercenter	#2295	D₆₇
hypocenter	C₆₇	#22723
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂²×C₄ ◁ #2295

Complements

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

N	H ≅ G/N	type
51. C₄⋊(C₆₇⋊C₈)	Q₈	semidirect
206. C₆₇	#22723	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
#22560363	2
#22560361	2
#22560066	2
#22560062	2
C₆₇×#2295	2
#22723	67