Subgroups — #22598125

Order 9088.

388 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17986555	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#11765864	C₂	1	1	yes	maximal, characteristic
3	#17986307	C₂	1	1	yes	maximal, characteristic, complemented
4	#17986303	C₂	1	1	yes	maximal, characteristic, complemented
5	#17986299	C₂	1	1	yes	maximal, characteristic, complemented
6	#17986293	C₂	1	1	yes	maximal, characteristic, complemented
7	#17986287	C₂	1	1	yes	maximal, characteristic, complemented
8	#17986283	C₂	1	1	yes	maximal, characteristic, complemented
9	#17986277	C₂	1	1	yes	maximal, characteristic, complemented
10	#17986036	C₂	1	1	yes	maximal, characteristic, complemented
11	#17986034	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄⋊C₈×D₇₁	C₂	1	1	yes	maximal, characteristic, complemented
13	#17985992	C₂	1	1	yes	maximal, characteristic
14	#17985992	C₂	1	1	yes	maximal, characteristic
15	#478×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11765864	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₇₁	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₇₁	C₂²	1	1	yes	characteristic
19	#14556127	C₂²	1	1	yes	characteristic
20	#14556127	C₂²	1	1	yes	characteristic
21	#14556120	C₂²	1	1	yes	characteristic
22	#14556106	—	2	1	no	—
23	#14556106	—	2	1	no	—
24	C₄×Dic₁₄₂	C₂²	1	1	yes	characteristic
25	#14556093	—	2	1	no	—
26	#14556093	—	2	1	no	—
27	#14556093	C₂²	1	1	yes	characteristic
28	D₄×Dic₇₁	C₂²	1	1	yes	characteristic
29	C₄×C₂²⋊D₇₁	—	2	1	no	—
30	(C₂²⋊D₇₁)⋊C₄	—	2	1	no	—
31	#14556086	C₂²	1	1	yes	characteristic
32	#14556086	C₂²	1	1	yes	characteristic
33	#14556086	C₂²	1	1	yes	characteristic
34	#14556084	—	2	1	no	—
35	C₂×C₄×Dic₇₁	—	2	1	no	—
36	C₄²×D₇₁	C₂²	1	1	yes	characteristic
37	#14556062	C₂²	1	1	yes	characteristic
38	#14556061	C₂²	1	1	yes	characteristic
39	#14556061	C₂²	1	1	yes	characteristic
40	#14556060	C₂²	1	1	yes	characteristic
41	#14556059	C₂²	1	1	yes	characteristic
42	#14556059	C₂²	1	1	yes	characteristic
43	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
44	Dic₇₁⋊C₈	C₂²	1	1	yes	characteristic
45	Dic₇₁⋊C₈	C₂²	1	1	yes	characteristic
46	#14556053	C₂²	1	1	yes	characteristic
47	#14556053	C₂²	1	1	yes	characteristic
48	#14556051	C₂²	1	1	yes	characteristic
49	#14556051	C₂²	1	1	yes	characteristic
50	#14556049	C₂²	1	1	yes	characteristic
51	#14556049	C₂²	1	1	yes	characteristic
52	#125×C₇₁	C₂²	1	1	yes	characteristic
53	C₄×D₄×C₇₁	C₂²	1	1	yes	characteristic
54	#107×C₇₁	C₂²	1	1	yes	characteristic
55	#106×C₇₁	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
57	#102×C₇₁	C₂²	1	1	yes	characteristic
58	#102×C₇₁	C₂²	1	1	yes	characteristic
59	#11765864	—	2	2	no	—
60	#11765864	Q₈	1	4	yes	complemented
61	C₂×Dic₁₄₂	—	2	1	no	—
62	C₂×Dic₁₄₂	—	2	1	no	—
63	C₂×Dic₁₄₂	C₂³	1	1	yes	characteristic
64	C₂×C₂²⋊D₇₁	—	2	1	no	—
65	C₂²×Dic₇₁	—	2	1	no	—
66	C₂²×Dic₇₁	—	2	1	no	—
67	C₂×C₄×D₇₁	D₄	1	1	yes	characteristic
68	C₂×C₄×D₇₁	D₄	1	1	yes	characteristic
69	C₂×C₄×D₇₁	C₂³	1	1	yes	characteristic
70	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
71	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
72	C₈×D₇₁	—	2	2	no	—
73	C₈×D₇₁	—	2	2	no	—
74	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
77	Dic₇₁⋊C₄	—	2	1	no	—
78	Dic₇₁⋊C₄	—	2	1	no	—
79	Dic₇₁⋊C₄	—	2	1	no	—
80	Dic₇₁⋊C₄	—	2	1	no	—
81	Dic₇₁⋊C₄	—	2	1	no	—
82	Dic₇₁⋊C₄	—	2	1	no	—
83	C₂²⋊Dic₇₁	—	2	1	no	—
84	(C₂×D₇₁)⋊C₄	—	2	1	no	—
85	(C₂×D₇₁)⋊C₄	—	2	1	no	—
86	(C₂×D₇₁)⋊C₄	—	2	1	no	—
87	C₄×Dic₇₁	—	2	1	no	—
88	C₄×Dic₇₁	—	2	1	no	—
89	C₄×Dic₇₁	—	2	1	no	—
90	C₄×Dic₇₁	—	2	1	no	—
91	C₄×Dic₇₁	—	2	1	no	—
92	C₄×Dic₇₁	D₄	1	1	yes	characteristic
93	C₄×Dic₇₁	D₄	1	1	yes	characteristic
94	C₄×Dic₇₁	C₂³	1	1	yes	characteristic
95	C₂×D₄×C₇₁	C₂³	1	1	yes	characteristic
96	C₂²×C₄×C₇₁	—	2	1	no	—
97	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
98	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
99	C₄⋊C₄×C₇₁	—	2	1	no	—
100	C₄⋊C₄×C₇₁	—	2	1	no	—
101	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
104	C₂²⋊C₄×C₇₁	—	2	1	no	—
105	C₄²×C₇₁	C₂³	1	1	yes	characteristic
106	C₂³×C₇₁	—	2	1	no	—
107	C₂²×D₇₁	C₂×D₄	1	1	yes	characteristic
108	D₄×C₇₁	—	2	1	no	—
109	D₄×C₇₁	C₂×Q₈	1	2	yes	—
110	C₂×C₄×C₇₁	—	2	1	no	—
111	C₂×C₄×C₇₁	—	2	1	no	—
112	C₂×C₄×C₇₁	—	2	1	no	—
113	C₂×C₄×C₇₁	—	4	2	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	C₂⁴	1	1	yes	characteristic
117	C₂²⋊D₇₁	—	2	4	no	—
118	C₂×Dic₇₁	—	2	1	no	—
119	C₂×Dic₇₁	—	2	4	no	—
120	C₂×Dic₇₁	—	2	1	no	—
121	C₂×Dic₇₁	—	2	1	no	—
122	C₂×Dic₇₁	—	2	1	no	—
123	C₂×Dic₇₁	—	4	2	no	—
124	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
126	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
127	C₄×D₇₁	—	2	2	no	—
128	C₄×D₇₁	—	4	1	no	—
129	C₄×D₇₁	C₂×Q₈	1	2	yes	—
130	C₄×D₇₁	#41	1	2	yes	—
131	Dic₁₄₂	—	2	1	no	—
132	Dic₁₄₂	—	4	2	no	—
133	Dic₁₄₂	—	4	2	no	—
134	Dic₁₄₂	C₂×Q₈	1	2	yes	—
135	C₈×C₇₁	—	2	1	no	—
136	C₈×C₇₁	—	2	1	no	—
137	C₇₁⋊C₈	—	2	1	no	—
138	C₇₁⋊C₈	—	2	1	no	—
139	C₂²×C₇₁	—	2	2	no	—
140	C₂²×C₇₁	—	4	1	no	—
141	C₂²×C₇₁	—	4	1	no	—
142	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
143	C₂×D₇₁	—	2	1	no	—
144	C₂×D₇₁	—	2	1	no	—
145	C₂×D₇₁	C₂²⋊Q₈	1	2	yes	—
146	C₄×C₇₁	—	2	1	no	—
147	C₄×C₇₁	—	2	1	no	—
148	C₄×C₇₁	—	4	1	no	—
149	C₄×C₇₁	—	4	1	no	—
150	C₄×C₇₁	—	4	1	no	—
151	C₄×C₇₁	C₂²×Q₈	1	1	yes	characteristic
152	C₄×C₇₁	C₂×#41	1	1	yes	characteristic
153	Dic₇₁	—	2	2	no	—
154	Dic₇₁	—	2	1	no	—
155	Dic₇₁	—	4	1	no	—
156	Dic₇₁	—	4	1	no	—
157	Dic₇₁	—	4	1	no	—
158	Dic₇₁	C₂²⋊Q₈	1	2	yes	—
159	Dic₇₁	#135	1	2	yes	—
160	C₂×C₇₁	—	4	2	no	—
161	C₂×C₇₁	C₂×C₂²⋊Q₈	1	1	yes	characteristic
162	C₂×C₇₁	C₂×#135	1	1	yes	characteristic
163	C₂×C₇₁	#575	1	1	yes	characteristic
164	D₇₁	—	2	2	no	—
165	#3076	—	71	1	no	maximal, Sylow
166	C₇₁	#3076	1	1	yes	characteristic, complemented, Sylow
167	#533	—	71	1	no	—
168	C₄×#41	—	71	1	no	—
169	#479	—	71	1	no	—
170	#479	—	71	1	no	—
171	#478	—	71	1	no	—
172	#478	D₇₁	1	1	yes	characteristic, complemented
173	#477	—	71	1	no	—
174	#476	—	71	1	no	—
175	#475	—	71	1	no	—
176	#474	—	71	1	no	—
177	#427	—	71	1	no	—
178	#427	—	71	1	no	—
179	C₂×C₄⋊C₈	—	71	1	no	—
180	#416	—	71	1	no	—
181	#416	—	71	1	no	—
182	C₂×#41	—	71	1	no	—
183	C₂²×C₈	—	71	1	no	—
184	C₂²×C₈	—	71	1	no	—
185	C₄⋊Q₈	—	71	1	no	—
186	C₄⋊Q₈	—	71	1	no	—
187	#125	—	71	1	no	—
188	#125	C₂×D₇₁	1	1	yes	characteristic
189	C₂²⋊Q₈	—	142	1	no	—
190	C₂²⋊Q₈	—	142	1	no	—
191	C₄×Q₈	—	71	1	no	—
192	C₄×Q₈	—	71	1	no	—
193	C₄×Q₈	—	142	1	no	—
194	C₄×Q₈	—	142	1	no	—
195	C₄×D₄	—	71	1	no	—
196	C₄×D₄	—	142	1	no	—
197	C₄×D₄	—	142	1	no	—
198	C₄×D₄	C₂×D₇₁	1	1	yes	characteristic
199	#117	—	71	1	no	—
200	#117	—	71	1	no	—
201	#117	—	71	1	no	—
202	#117	—	142	1	no	—
203	C₂×C₄²	—	71	1	no	—
204	C₂×C₄²	—	142	1	no	—
205	#107	—	71	1	no	—
206	#107	—	71	1	no	—
207	#107	—	71	1	no	—
208	#107	C₂×D₇₁	1	1	yes	characteristic
209	#106	—	71	1	no	—
210	#106	—	71	1	no	—
211	#106	—	71	1	no	—
212	#106	C₂×D₇₁	1	1	yes	characteristic
213	C₄⋊C₈	—	71	1	no	—
214	C₄⋊C₈	—	71	1	no	—
215	C₄⋊C₈	—	71	1	no	—
216	C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
217	#103	—	71	1	no	—
218	#103	—	71	1	no	—
219	#103	—	71	1	no	—
220	#103	—	71	1	no	—
221	#102	—	71	1	no	—
222	#102	—	71	1	no	—
223	#102	C₂×D₇₁	1	1	yes	characteristic
224	#102	C₂×D₇₁	1	1	yes	characteristic
225	#41	—	71	4	no	—
226	#41	—	142	2	no	—
227	C₂×Q₈	—	71	1	no	—
228	C₂×Q₈	—	142	1	no	—
229	C₂×Q₈	—	142	1	no	—
230	C₂×D₄	—	142	1	no	—
231	C₂×D₄	C₂²×D₇₁	1	1	yes	characteristic
232	C₂²×C₄	—	2	1	no	—
233	C₂²×C₄	—	71	1	no	—
234	C₂²×C₄	—	71	1	no	—
235	C₂²×C₄	—	71	1	no	—
236	C₂²×C₄	—	142	1	no	—
237	C₂²×C₄	—	142	1	no	—
238	C₂×C₈	—	71	1	no	—
239	C₂×C₈	—	71	1	no	—
240	C₂×C₈	—	142	2	no	—
241	C₂×C₈	—	142	2	no	—
242	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
243	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	71	1	no	—
247	C₄⋊C₄	—	71	1	no	—
248	C₄⋊C₄	—	71	1	no	—
249	C₄⋊C₄	—	142	1	no	—
250	C₄⋊C₄	—	142	1	no	—
251	C₄⋊C₄	—	142	1	no	—
252	C₄⋊C₄	—	142	1	no	—
253	C₄⋊C₄	—	142	1	no	—
254	C₄⋊C₄	—	142	1	no	—
255	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
256	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
257	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
258	C₂²⋊C₄	—	2	1	no	—
259	C₂²⋊C₄	—	142	1	no	—
260	C₂²⋊C₄	—	142	1	no	—
261	C₂²⋊C₄	—	142	1	no	—
262	C₂²⋊C₄	—	142	1	no	—
263	C₄²	—	71	1	no	—
264	C₄²	—	71	1	no	—
265	C₄²	—	71	1	no	—
266	C₄²	—	142	1	no	—
267	C₄²	—	142	1	no	—
268	C₄²	—	142	1	no	—
269	C₄²	—	142	1	no	—
270	C₄²	—	142	1	no	—
271	C₄²	C₂²×D₇₁	1	1	yes	characteristic
272	C₂³	—	2	1	no	—
273	C₂³	—	71	1	no	—
274	Q₈	—	284	2	no	—
275	Q₈	—	284	2	no	—
276	Q₈	—	71	2	no	—
277	Q₈	—	142	1	no	—
278	D₄	—	2	1	no	—
279	D₄	—	142	4	no	—
280	D₄	Q₈×D₇₁	1	2	yes	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	284	1	no	—
286	C₂×C₄	—	284	2	no	—
287	C₂×C₄	—	71	1	no	—
288	C₂×C₄	—	71	1	no	—
289	C₂×C₄	—	71	2	no	—
290	C₂×C₄	—	71	2	no	—
291	C₂×C₄	—	71	1	no	—
292	C₂×C₄	—	142	2	no	—
293	C₂×C₄	—	142	1	no	—
294	C₂×C₄	—	142	1	no	—
295	C₂×C₄	—	142	1	no	—
296	C₂×C₄	—	142	4	no	—
297	C₂×C₄	—	142	1	no	—
298	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
299	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
300	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
301	C₈	—	2	1	no	—
302	C₈	—	2	1	no	—
303	C₈	—	142	1	no	—
304	C₈	—	142	1	no	—
305	C₂²	—	2	2	no	—
306	C₂²	—	4	1	no	—
307	C₂²	—	4	1	no	—
308	C₂²	—	71	2	no	—
309	C₂²	—	142	1	no	—
310	C₂²	—	142	1	no	—
311	C₂²	C₂×D₄×D₇₁	1	1	yes	characteristic
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	4	1	no	—
315	C₄	—	4	1	no	—
316	C₄	—	4	1	no	—
317	C₄	—	284	1	no	—
318	C₄	—	284	1	no	—
319	C₄	—	284	1	no	—
320	C₄	—	71	2	no	—
321	C₄	—	71	2	no	—
322	C₄	—	142	2	no	—
323	C₄	—	142	1	no	—
324	C₄	C₂×Q₈×D₇₁	1	1	yes	characteristic
325	C₄	#41×D₇₁	1	1	yes	characteristic
326	C₂	—	4	2	no	—
327	C₂	—	142	2	no	—
328	C₂	C₂²⋊Q₈×D₇₁	1	1	yes	characteristic
329	C₂	#135×D₇₁	1	1	yes	characteristic
330	C₂	#17986795	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₁
Fitting	#478×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₄
hypercenter	#478	D₇₁
hypocenter	C₇₁	#3076
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #478

Complements

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

N	H ≅ G/N	type
1. #17986555	C₂	semidirect
3. #17986307	C₂	semidirect
4. #17986303	C₂	semidirect
5. #17986299	C₂	semidirect
6. #17986293	C₂	semidirect
7. #17986287	C₂	semidirect
8. #17986283	C₂	semidirect
9. #17986277	C₂	semidirect
10. #17986036	C₂	semidirect
11. #17986034	C₂	semidirect
12. C₄⋊C₈×D₇₁	C₂	semidirect
15. #478×C₇₁	C₂	semidirect
60. #11765864	Q₈	semidirect
166. C₇₁	#3076	semidirect
172. #478	D₇₁	semidirect