Subgroups — #22601285

Order 9088.

334 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17986727	C₂	1	1	yes	maximal, characteristic, complemented
2	#17986409	C₂	1	1	yes	maximal, characteristic, complemented
3	#17986405	C₂	1	1	yes	maximal, characteristic, complemented
4	#17986398	C₂	1	1	yes	maximal, characteristic, complemented
5	#17986303	C₂	1	1	yes	maximal, characteristic, complemented
6	#17986303	C₂	1	1	yes	maximal, characteristic, complemented
7	#17986299	C₂	1	1	yes	maximal, characteristic
8	#17986299	C₂	1	1	yes	maximal, characteristic
9	#17986289	C₂	1	1	yes	maximal, characteristic
10	#17986281	C₂	1	1	yes	maximal, characteristic, complemented
11	#17986130	C₂	1	1	yes	maximal, characteristic, complemented
12	#17986099	C₂	1	1	yes	maximal, characteristic
13	#17986099	C₂	1	1	yes	maximal, characteristic
14	C₄×#11765853	C₂	1	1	yes	maximal, characteristic
15	#554×C₇₁	C₂	1	1	yes	maximal, characteristic
16	C₂×#11765853	C₂²	1	1	yes	characteristic
17	#14556128	C₂²	1	1	yes	characteristic
18	#14556126	—	2	1	no	—
19	#14556120	C₂²	1	1	yes	characteristic
20	#14556120	C₂²	1	1	yes	characteristic
21	#14556118	—	2	1	no	—
22	#14556118	—	2	1	no	—
23	C₄×Dic₁₄₂	C₂²	1	1	yes	characteristic
24	C₄×Dic₁₄₂	C₂²	1	1	yes	characteristic
25	C₄×Dic₁₄₂	C₂²	1	1	yes	characteristic
26	#14556093	—	2	1	no	—
27	#14556061	C₂²	1	1	yes	characteristic
28	#14556061	C₂²	1	1	yes	characteristic
29	#14556061	C₂²	1	1	yes	characteristic
30	#14556061	C₂²	1	1	yes	characteristic
31	#14556059	C₂²	1	1	yes	characteristic
32	#14556059	C₂²	1	1	yes	characteristic
33	#14556059	C₂²	1	1	yes	characteristic
34	#14556059	C₂²	1	1	yes	characteristic
35	#14556059	C₂²	1	1	yes	characteristic
36	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
37	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
38	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
39	#14556051	C₂²	1	1	yes	characteristic
40	#14556051	C₂²	1	1	yes	characteristic
41	#14556051	C₂²	1	1	yes	characteristic
42	#14556049	C₂²	1	1	yes	characteristic
43	#14556049	C₂²	1	1	yes	characteristic
44	#14556049	C₂²	1	1	yes	characteristic
45	#14556039	C₂²	1	1	yes	characteristic
46	#14556039	C₂²	1	1	yes	characteristic
47	C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
48	C₄⋊Q₈×C₇₁	C₂²	1	1	yes	characteristic
49	#125×C₇₁	C₂²	1	1	yes	characteristic
50	#125×C₇₁	C₂²	1	1	yes	characteristic
51	C₂²⋊Q₈×C₇₁	—	2	1	no	—
52	C₂²⋊Q₈×C₇₁	—	2	1	no	—
53	C₂²⋊Q₈×C₇₁	—	2	1	no	—
54	C₄×Q₈×C₇₁	C₂²	1	1	yes	characteristic
55	C₄×D₄×C₇₁	C₂²	1	1	yes	characteristic
56	C₄×D₄×C₇₁	C₂²	1	1	yes	characteristic
57	C₄×D₄×C₇₁	C₂²	1	1	yes	characteristic
58	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
59	C₂×Dic₁₄₂	—	2	1	no	—
60	C₂×Dic₁₄₂	C₂³	1	1	yes	characteristic
61	#11765853	Q₈	1	4	yes	complemented
62	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
63	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
64	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
65	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
66	C₄⋊Dic₇₁	—	2	1	no	—
67	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
68	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
69	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
70	Dic₇₁⋊C₄	—	2	1	no	—
71	Dic₇₁⋊C₄	—	2	1	no	—
72	Dic₇₁⋊C₄	—	2	1	no	—
73	Dic₇₁⋊C₄	—	2	1	no	—
74	Dic₇₁⋊C₄	—	2	1	no	—
75	Dic₇₁⋊C₄	—	2	1	no	—
76	Dic₇₁⋊C₄	—	2	1	no	—
77	C₄×Dic₇₁	—	2	1	no	—
78	C₄×Dic₇₁	—	2	1	no	—
79	C₄×Dic₇₁	—	2	1	no	—
80	C₂×Q₈×C₇₁	—	2	1	no	—
81	C₂×Q₈×C₇₁	D₄	1	1	yes	characteristic
82	C₂×D₄×C₇₁	C₂³	1	1	yes	characteristic
83	C₂²×C₄×C₇₁	—	2	1	no	—
84	C₂²×C₄×C₇₁	—	2	1	no	—
85	C₂²×C₄×C₇₁	—	2	1	no	—
86	C₄⋊C₄×C₇₁	—	2	1	no	—
87	C₄⋊C₄×C₇₁	—	2	1	no	—
88	C₄⋊C₄×C₇₁	—	2	1	no	—
89	C₄⋊C₄×C₇₁	—	2	1	no	—
90	C₄⋊C₄×C₇₁	—	2	1	no	—
91	C₄⋊C₄×C₇₁	D₄	1	1	yes	characteristic
92	C₄⋊C₄×C₇₁	D₄	1	1	yes	characteristic
93	C₄⋊C₄×C₇₁	D₄	1	1	yes	characteristic
94	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
95	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
96	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
97	C₂²⋊C₄×C₇₁	—	2	1	no	—
98	C₂²⋊C₄×C₇₁	—	2	1	no	—
99	C₂²⋊C₄×C₇₁	—	2	1	no	—
100	C₄²×C₇₁	C₂³	1	1	yes	characteristic
101	C₄²×C₇₁	C₂³	1	1	yes	characteristic
102	C₄²×C₇₁	C₂³	1	1	yes	characteristic
103	C₂³×C₇₁	—	2	1	no	—
104	Q₈×C₇₁	—	4	1	no	—
105	Q₈×C₇₁	—	8	1	no	—
106	D₄×C₇₁	—	2	1	no	—
107	D₄×C₇₁	C₂×Q₈	1	2	yes	—
108	C₂×C₄×C₇₁	—	2	1	no	—
109	C₂×C₄×C₇₁	—	2	1	no	—
110	C₂×C₄×C₇₁	—	2	1	no	—
111	C₂×C₄×C₇₁	—	4	2	no	—
112	C₂×C₄×C₇₁	—	8	1	no	—
113	C₂×C₄×C₇₁	—	8	1	no	—
114	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
121	C₂×Dic₇₁	—	2	1	no	—
122	C₂×Dic₇₁	—	2	1	no	—
123	C₂×Dic₇₁	—	2	1	no	—
124	C₂×Dic₇₁	—	2	1	no	—
125	Dic₁₄₂	—	2	1	no	—
126	Dic₁₄₂	—	4	2	no	—
127	Dic₁₄₂	C₂×Q₈	1	2	yes	—
128	C₇₁⋊C₈	—	2	1	no	—
129	C₇₁⋊C₈	—	2	1	no	—
130	C₇₁⋊C₈	—	2	1	no	—
131	C₇₁⋊C₈	C₂×Q₈	1	2	yes	—
132	C₂²×C₇₁	—	2	2	no	—
133	C₂²×C₇₁	—	4	1	no	—
134	C₂²×C₇₁	—	4	1	no	—
135	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
136	C₄×C₇₁	—	2	1	no	—
137	C₄×C₇₁	—	2	1	no	—
138	C₄×C₇₁	—	2	1	no	—
139	C₄×C₇₁	—	2	1	no	—
140	C₄×C₇₁	—	2	1	no	—
141	C₄×C₇₁	—	2	1	no	—
142	C₄×C₇₁	—	4	1	no	—
143	C₄×C₇₁	—	4	1	no	—
144	C₄×C₇₁	—	4	1	no	—
145	C₄×C₇₁	C₂²×Q₈	1	1	yes	characteristic
146	C₄×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
147	Dic₇₁	—	2	2	no	—
148	Dic₇₁	—	4	1	no	—
149	Dic₇₁	—	4	1	no	—
150	Dic₇₁	—	4	1	no	—
151	C₂×C₇₁	—	4	2	no	—
152	C₂×C₇₁	D₄×Q₈	1	1	yes	characteristic
153	C₂×C₇₁	#575	1	1	yes	characteristic
154	C₂×C₇₁	#577	1	1	yes	characteristic
155	#3388	—	71	1	no	maximal, Sylow
156	C₇₁	#3388	1	1	yes	characteristic, complemented, Sylow
157	#557	—	71	1	no	—
158	#554	D₇₁	1	1	yes	characteristic
159	#501	—	71	1	no	—
160	#501	—	71	1	no	—
161	#499	—	71	1	no	—
162	#479	—	71	1	no	—
163	#479	—	71	1	no	—
164	#478	—	71	1	no	—
165	#478	—	71	1	no	—
166	#476	—	71	1	no	—
167	#475	—	71	1	no	—
168	#446	—	71	1	no	—
169	#440	—	71	1	no	—
170	#440	—	71	1	no	—
171	C₄×SD₁₆	—	71	1	no	—
172	C₂×SD₁₆	—	71	1	no	—
173	C₄⋊Q₈	—	71	1	no	—
174	C₄⋊Q₈	—	142	1	no	—
175	C₄⋊Q₈	C₂×D₇₁	1	1	yes	characteristic
176	#125	—	71	1	no	—
177	#125	—	71	1	no	—
178	#125	—	142	1	no	—
179	#125	—	142	1	no	—
180	#125	C₂×D₇₁	1	1	yes	characteristic
181	#125	C₂×D₇₁	1	1	yes	characteristic
182	C₂²⋊Q₈	—	2	1	no	—
183	C₂²⋊Q₈	—	2	1	no	—
184	C₂²⋊Q₈	—	2	1	no	—
185	C₄×Q₈	—	71	1	no	—
186	C₄×Q₈	—	71	1	no	—
187	C₄×Q₈	—	71	1	no	—
188	C₄×Q₈	—	142	1	no	—
189	C₄×Q₈	C₂×D₇₁	1	1	yes	characteristic
190	C₄×D₄	C₂×D₇₁	1	1	yes	characteristic
191	C₄×D₄	C₂×D₇₁	1	1	yes	characteristic
192	C₄×D₄	C₂×D₇₁	1	1	yes	characteristic
193	C₂×C₄⋊C₄	—	2	1	no	—
194	#107	—	71	1	no	—
195	#107	—	71	1	no	—
196	#107	—	71	1	no	—
197	#107	—	71	1	no	—
198	#106	—	71	1	no	—
199	#106	—	71	1	no	—
200	#106	—	71	1	no	—
201	#106	—	71	1	no	—
202	#106	—	71	1	no	—
203	C₄⋊C₈	—	71	1	no	—
204	C₄⋊C₈	—	71	1	no	—
205	C₄⋊C₈	—	71	1	no	—
206	#103	—	71	1	no	—
207	#103	—	71	1	no	—
208	#103	—	71	1	no	—
209	#102	—	71	1	no	—
210	#102	—	71	1	no	—
211	#102	—	71	1	no	—
212	#97	—	71	1	no	—
213	#97	—	71	1	no	—
214	C₄×C₈	—	71	1	no	—
215	C₂×Q₈	—	2	1	no	—
216	C₂×Q₈	—	71	1	no	—
217	C₂×Q₈	—	142	1	no	—
218	C₂×Q₈	C₂²⋊D₇₁	1	1	yes	characteristic
219	C₂×D₄	C₂²×D₇₁	1	1	yes	characteristic
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	1	no	—
222	C₂²×C₄	—	2	1	no	—
223	SD₁₆	—	71	4	no	—
224	C₂×C₈	—	71	1	no	—
225	C₂×C₈	—	71	1	no	—
226	C₂×C₈	—	71	1	no	—
227	C₂×C₈	—	71	1	no	—
228	C₄⋊C₄	—	2	1	no	—
229	C₄⋊C₄	—	2	1	no	—
230	C₄⋊C₄	—	2	1	no	—
231	C₄⋊C₄	—	2	1	no	—
232	C₄⋊C₄	—	2	1	no	—
233	C₄⋊C₄	—	71	1	no	—
234	C₄⋊C₄	—	71	1	no	—
235	C₄⋊C₄	—	71	1	no	—
236	C₄⋊C₄	—	142	1	no	—
237	C₄⋊C₄	—	142	1	no	—
238	C₄⋊C₄	—	142	1	no	—
239	C₄⋊C₄	—	142	1	no	—
240	C₄⋊C₄	—	142	1	no	—
241	C₄⋊C₄	—	142	1	no	—
242	C₄⋊C₄	—	142	1	no	—
243	C₄⋊C₄	—	142	1	no	—
244	C₄⋊C₄	C₂²⋊D₇₁	1	1	yes	characteristic
245	C₄⋊C₄	C₂²⋊D₇₁	1	1	yes	characteristic
246	C₄⋊C₄	C₂²⋊D₇₁	1	1	yes	characteristic
247	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
248	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
249	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
250	C₂²⋊C₄	—	2	1	no	—
251	C₂²⋊C₄	—	2	1	no	—
252	C₂²⋊C₄	—	2	1	no	—
253	C₄²	—	142	1	no	—
254	C₄²	—	142	1	no	—
255	C₄²	—	142	1	no	—
256	C₄²	C₂²×D₇₁	1	1	yes	characteristic
257	C₄²	C₂²×D₇₁	1	1	yes	characteristic
258	C₄²	C₂²×D₇₁	1	1	yes	characteristic
259	C₂³	—	2	1	no	—
260	Q₈	—	4	1	no	—
261	Q₈	—	8	1	no	—
262	Q₈	—	284	2	no	—
263	Q₈	—	71	2	no	—
264	Q₈	—	142	1	no	—
265	D₄	—	2	1	no	—
266	D₄	Q₈×D₇₁	1	2	yes	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	2	1	no	—
270	C₂×C₄	—	4	2	no	—
271	C₂×C₄	—	8	1	no	—
272	C₂×C₄	—	8	1	no	—
273	C₂×C₄	—	142	1	no	—
274	C₂×C₄	—	142	1	no	—
275	C₂×C₄	—	142	1	no	—
276	C₂×C₄	—	142	1	no	—
277	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
278	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
279	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
280	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
281	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
282	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
283	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
284	C₈	—	71	2	no	—
285	C₈	—	142	1	no	—
286	C₈	—	142	1	no	—
287	C₈	—	142	1	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	4	1	no	—
290	C₂²	—	4	1	no	—
291	C₂²	C₂²×C₂²⋊D₇₁	1	1	yes	characteristic
292	C₄	—	2	1	no	—
293	C₄	—	2	1	no	—
294	C₄	—	2	1	no	—
295	C₄	—	2	1	no	—
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	4	1	no	—
299	C₄	—	4	1	no	—
300	C₄	—	4	1	no	—
301	C₄	—	284	1	no	—
302	C₄	—	284	1	no	—
303	C₄	—	284	1	no	—
304	C₄	—	142	2	no	—
305	C₄	C₂×Q₈×D₇₁	1	1	yes	characteristic
306	C₄	#14556176	1	1	yes	characteristic
307	C₂	—	4	2	no	—
308	C₂	Q₈×C₂²⋊D₇₁	1	1	yes	characteristic
309	C₂	#17986797	1	1	yes	characteristic
310	C₂	#17986810	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₇₁
commutator	C₂×C₄×C₇₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₁
Fitting	#554×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₄
hypercenter	#554	D₇₁
hypocenter	C₇₁	#3388
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ #554

Complements

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

N	H ≅ G/N	type
1. #17986727	C₂	semidirect
2. #17986409	C₂	semidirect
3. #17986405	C₂	semidirect
4. #17986398	C₂	semidirect
5. #17986303	C₂	semidirect
6. #17986303	C₂	semidirect
10. #17986281	C₂	semidirect
11. #17986130	C₂	semidirect
61. #11765853	Q₈	semidirect
156. C₇₁	#3388	semidirect