Subgroups — #22600919

Order 9088.

374 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17986720	C₂	1	1	yes	maximal, characteristic, complemented
2	#17986649	C₂	1	1	yes	maximal, characteristic
3	#17986403	C₂	1	1	yes	maximal, characteristic, complemented
4	#17986333	C₂	1	1	yes	maximal, characteristic, complemented
5	#17986332	C₂	1	1	yes	maximal, characteristic, complemented
6	#17986322	C₂	1	1	yes	maximal, characteristic, complemented
7	#17986320	C₂	1	1	yes	maximal, characteristic, complemented
8	#17986282	C₂	1	1	yes	maximal, characteristic, complemented
9	#17986277	C₂	1	1	yes	maximal, characteristic, complemented
10	#17986242	C₂	1	1	yes	maximal, characteristic, complemented
11	SD₁₆×Dic₇₁	C₂	1	1	yes	maximal, characteristic, complemented
12	#17986075	C₂	1	1	yes	maximal, characteristic, complemented
13	#17986041	C₂	1	1	yes	maximal, characteristic, complemented
14	#17986040	C₂	1	1	yes	maximal, characteristic, complemented
15	#468×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11765864	—	2	1	no	—
17	C₂×#11765848	C₂²	1	1	yes	characteristic
18	#14556127	C₂²	1	1	yes	characteristic
19	#14556127	C₂²	1	1	yes	characteristic
20	#14556123	—	2	1	no	—
21	#14556121	—	2	1	no	—
22	#14556114	—	2	1	no	—
23	#14556110	—	2	1	no	—
24	#14556109	—	2	1	no	—
25	#14556105	—	2	1	no	—
26	#14556104	C₂²	1	1	yes	characteristic
27	#14556099	C₂²	1	1	yes	characteristic
28	Q₈×Dic₇₁	C₂²	1	1	yes	characteristic
29	#14556093	—	2	1	no	—
30	D₄×Dic₇₁	C₂²	1	1	yes	characteristic
31	C₄×C₂²⋊D₇₁	C₂²	1	1	yes	characteristic
32	#14556087	C₂²	1	1	yes	characteristic
33	#14556086	C₂²	1	1	yes	characteristic
34	C₂×C₄⋊Dic₇₁	C₂²	1	1	yes	characteristic
35	C₄⋊C₄×D₇₁	C₂²	1	1	yes	characteristic
36	#14556062	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	#14556061	C₂²	1	1	yes	characteristic
41	#14556060	C₂²	1	1	yes	characteristic
42	#14556060	C₂²	1	1	yes	characteristic
43	#14556059	C₂²	1	1	yes	characteristic
44	Dic₇₁⋊C₈	C₂²	1	1	yes	characteristic
45	#14556052	C₂²	1	1	yes	characteristic
46	#14556052	C₂²	1	1	yes	characteristic
47	#14556049	C₂²	1	1	yes	characteristic
48	#14556049	C₂²	1	1	yes	characteristic
49	C₂²⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
50	(C₂×D₇₁)⋊C₈	C₂²	1	1	yes	characteristic
51	C₈×Dic₇₁	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₇₁	C₂²	1	1	yes	characteristic
53	C₂×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
54	C₂²⋊Q₈×C₇₁	C₂²	1	1	yes	characteristic
55	#121×C₇₁	C₂²	1	1	yes	characteristic
56	#107×C₇₁	C₂²	1	1	yes	characteristic
57	#103×C₇₁	C₂²	1	1	yes	characteristic
58	#102×C₇₁	C₂²	1	1	yes	characteristic
59	#11765864	—	4	4	no	—
60	C₂×Dic₁₄₂	—	2	1	no	—
61	C₂×Dic₁₄₂	—	2	1	no	—
62	C₂×C₂²⋊D₇₁	—	2	1	no	—
63	C₂×C₂²⋊D₇₁	D₄	1	1	yes	characteristic
64	C₂²×Dic₇₁	—	2	1	no	—
65	C₂²×Dic₇₁	—	2	1	no	—
66	C₂×C₄×D₇₁	—	2	1	no	—
67	C₂×C₄×D₇₁	C₂³	1	1	yes	characteristic
68	#11765848	—	2	2	no	—
69	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
70	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
71	C₄⋊Dic₇₁	—	2	1	no	—
72	C₄⋊Dic₇₁	—	2	1	no	—
73	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
82	C₂²⋊Dic₇₁	—	2	1	no	—
83	C₂²⋊Dic₇₁	—	2	1	no	—
84	C₂²⋊Dic₇₁	—	2	1	no	—
85	C₂²⋊Dic₇₁	D₄	1	1	yes	characteristic
86	(C₂×D₇₁)⋊C₄	—	2	1	no	—
87	(C₂×D₇₁)⋊C₄	—	2	1	no	—
88	(C₂×D₇₁)⋊C₄	D₄	1	1	yes	characteristic
89	C₄×Dic₇₁	—	2	1	no	—
90	C₄×Dic₇₁	—	2	1	no	—
91	C₄×Dic₇₁	—	2	1	no	—
92	C₄×Dic₇₁	C₂³	1	1	yes	characteristic
93	C₂×Q₈×C₇₁	C₂³	1	1	yes	characteristic
94	C₂×D₄×C₇₁	—	2	1	no	—
95	C₂×D₄×C₇₁	C₂³	1	1	yes	characteristic
96	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
97	SD₁₆×C₇₁	—	2	2	no	—
98	M₁₆×C₇₁	—	2	2	no	—
99	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
100	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
101	C₄⋊C₄×C₇₁	—	2	1	no	—
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	C₂²⋊C₄×C₇₁	—	2	1	no	—
106	C₂³×C₇₁	—	2	1	no	—
107	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
108	C₂²×D₇₁	C₂×D₄	1	1	yes	characteristic
109	Q₈×C₇₁	—	2	1	no	—
110	Q₈×C₇₁	—	2	1	no	—
111	D₄×C₇₁	—	2	1	no	—
112	D₄×C₇₁	—	2	1	no	—
113	D₄×C₇₁	—	4	2	no	—
114	C₂×C₄×C₇₁	—	2	1	no	—
115	C₂×C₄×C₇₁	—	2	1	no	—
116	C₂×C₄×C₇₁	—	2	1	no	—
117	C₂×C₄×C₇₁	—	2	1	no	—
118	C₂×C₄×C₇₁	—	2	1	no	—
119	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
121	C₂²⋊D₇₁	—	2	2	no	—
122	C₂²⋊D₇₁	—	4	2	no	—
123	C₂×Dic₇₁	—	2	1	no	—
124	C₂×Dic₇₁	—	2	1	no	—
125	C₂×Dic₇₁	—	2	1	no	—
126	C₂×Dic₇₁	—	2	1	no	—
127	C₂×Dic₇₁	—	4	2	no	—
128	C₂×Dic₇₁	—	4	2	no	—
129	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
131	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
132	C₄×D₇₁	—	2	1	no	—
133	C₄×D₇₁	—	2	1	no	—
134	C₄×D₇₁	—	4	2	no	—
135	Dic₁₄₂	—	4	2	no	—
136	Dic₁₄₂	—	4	2	no	—
137	C₈×C₇₁	—	2	1	no	—
138	C₈×C₇₁	#41	1	2	yes	complemented
139	C₇₁⋊C₈	—	2	1	no	—
140	C₇₁⋊C₈	—	2	1	no	—
141	C₂²×C₇₁	—	2	1	no	—
142	C₂²×C₇₁	—	2	1	no	—
143	C₂²×C₇₁	—	2	1	no	—
144	C₂²×C₇₁	—	4	1	no	—
145	C₂²×C₇₁	—	4	1	no	—
146	C₂²×C₇₁	—	4	1	no	—
147	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
148	C₂×D₇₁	—	2	1	no	—
149	C₂×D₇₁	—	2	1	no	—
150	C₂×D₇₁	—	2	1	no	—
151	C₄×C₇₁	—	2	1	no	—
152	C₄×C₇₁	—	4	1	no	—
153	C₄×C₇₁	—	4	1	no	—
154	C₄×C₇₁	—	4	1	no	—
155	C₄×C₇₁	C₂×#41	1	1	yes	characteristic
156	C₄×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
157	Dic₇₁	—	2	1	no	—
158	Dic₇₁	—	2	1	no	—
159	Dic₇₁	—	4	1	no	—
160	Dic₇₁	—	4	1	no	—
161	Dic₇₁	—	4	1	no	—
162	Dic₇₁	—	4	1	no	—
163	Dic₇₁	AGL(1,ℤ/8)	1	2	yes	—
164	C₂×C₇₁	—	4	1	no	—
165	C₂×C₇₁	—	8	1	no	—
166	C₂×C₇₁	#547	1	1	yes	characteristic
167	C₂×C₇₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
168	C₂×C₇₁	#578	1	1	yes	characteristic
169	D₇₁	—	4	1	no	—
170	#3353	—	71	1	no	maximal, Sylow
171	C₇₁	#3353	1	1	yes	characteristic, complemented, Sylow
172	#556	—	71	1	no	—
173	#547	—	71	1	no	—
174	#500	—	71	1	no	—
175	#484	—	71	1	no	—
176	#484	—	71	1	no	—
177	#482	—	71	1	no	—
178	#482	—	71	1	no	—
179	#475	—	71	1	no	—
180	#474	—	71	1	no	—
181	#468	—	71	1	no	—
182	#468	D₇₁	1	1	yes	characteristic, complemented
183	C₄×SD₁₆	—	71	1	no	—
184	#436	—	71	1	no	—
185	#428	—	71	1	no	—
186	#428	—	71	1	no	—
187	C₂×#41	—	142	1	no	—
188	C₂×SD₁₆	C₂×D₇₁	1	1	yes	characteristic
189	C₂×M₁₆	—	71	1	no	—
190	C₂×M₁₆	C₂×D₇₁	1	1	yes	characteristic
191	C₄⋊Q₈	—	71	1	no	—
192	C₄⋊Q₈	—	71	1	no	—
193	#126	—	142	1	no	—
194	#126	—	142	1	no	—
195	#124	—	142	1	no	—
196	#123	—	142	1	no	—
197	#123	—	142	1	no	—
198	C₂²⋊Q₈	—	71	1	no	—
199	C₂²⋊Q₈	—	142	1	no	—
200	C₂²⋊Q₈	C₂×D₇₁	1	1	yes	characteristic
201	#121	—	71	1	no	—
202	#121	C₂×D₇₁	1	1	yes	characteristic
203	C₄×Q₈	—	71	1	no	—
204	C₄×Q₈	—	142	1	no	—
205	C₄×D₄	—	71	1	no	—
206	C₄×D₄	—	71	1	no	—
207	#117	—	71	1	no	—
208	#117	—	71	1	no	—
209	C₂×C₄⋊C₄	—	71	1	no	—
210	C₂×C₄⋊C₄	—	71	1	no	—
211	#107	—	71	1	no	—
212	#107	—	71	1	no	—
213	#107	—	71	1	no	—
214	#107	—	71	1	no	—
215	#107	—	71	1	no	—
216	#107	C₂×D₇₁	1	1	yes	characteristic
217	#106	—	71	1	no	—
218	#106	—	71	1	no	—
219	#106	—	71	1	no	—
220	C₄⋊C₈	—	71	1	no	—
221	#103	—	71	1	no	—
222	#103	—	71	1	no	—
223	#103	C₂×D₇₁	1	1	yes	characteristic
224	#102	—	71	1	no	—
225	#102	—	71	1	no	—
226	#102	C₂×D₇₁	1	1	yes	characteristic
227	C₂²⋊C₈	—	71	1	no	—
228	C₂²⋊C₈	—	71	1	no	—
229	C₄×C₈	—	71	1	no	—
230	#41	—	284	4	no	—
231	C₂×Q₈	—	142	1	no	—
232	C₂×Q₈	—	142	1	no	—
233	C₂×Q₈	C₂²×D₇₁	1	1	yes	characteristic
234	C₂×D₄	—	2	1	no	—
235	C₂×D₄	—	71	1	no	—
236	C₂×D₄	—	142	1	no	—
237	C₂×D₄	C₂²×D₇₁	1	1	yes	characteristic
238	C₂²×C₄	—	71	1	no	—
239	C₂²×C₄	—	142	1	no	—
240	C₂²×C₄	—	142	1	no	—
241	C₂²×C₄	—	142	1	no	—
242	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
243	SD₁₆	—	2	2	no	—
244	M₁₆	—	2	2	no	—
245	M₁₆	—	142	2	no	—
246	C₂×C₈	—	71	1	no	—
247	C₂×C₈	—	71	1	no	—
248	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
249	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
250	C₄⋊C₄	—	2	1	no	—
251	C₄⋊C₄	—	71	1	no	—
252	C₄⋊C₄	—	71	1	no	—
253	C₄⋊C₄	—	71	1	no	—
254	C₄⋊C₄	—	71	1	no	—
255	C₄⋊C₄	—	71	1	no	—
256	C₄⋊C₄	—	142	1	no	—
257	C₄⋊C₄	—	142	1	no	—
258	C₄⋊C₄	—	142	1	no	—
259	C₄⋊C₄	—	142	1	no	—
260	C₄⋊C₄	—	142	1	no	—
261	C₄⋊C₄	—	142	1	no	—
262	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
263	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
264	C₂²⋊C₄	—	2	1	no	—
265	C₂²⋊C₄	—	2	1	no	—
266	C₂²⋊C₄	—	71	1	no	—
267	C₂²⋊C₄	—	71	1	no	—
268	C₂²⋊C₄	—	142	1	no	—
269	C₂²⋊C₄	—	142	1	no	—
270	C₂²⋊C₄	—	142	1	no	—
271	C₂²⋊C₄	—	142	1	no	—
272	C₂²⋊C₄	—	142	1	no	—
273	C₄²	—	71	1	no	—
274	C₄²	—	142	1	no	—
275	C₄²	—	142	1	no	—
276	C₄²	—	142	1	no	—
277	C₂³	—	2	1	no	—
278	C₂³	—	71	1	no	—
279	C₂³	D₄×D₇₁	1	1	yes	characteristic
280	Q₈	—	2	1	no	—
281	Q₈	—	2	1	no	—
282	Q₈	—	284	2	no	—
283	Q₈	—	284	2	no	—
284	D₄	—	2	1	no	—
285	D₄	—	2	1	no	—
286	D₄	—	4	2	no	—
287	D₄	—	284	2	no	—
288	D₄	—	142	2	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	284	2	no	—
295	C₂×C₄	—	284	2	no	—
296	C₂×C₄	—	284	2	no	—
297	C₂×C₄	—	71	1	no	—
298	C₂×C₄	—	71	1	no	—
299	C₂×C₄	—	71	1	no	—
300	C₂×C₄	—	142	1	no	—
301	C₂×C₄	—	142	1	no	—
302	C₂×C₄	—	142	1	no	—
303	C₂×C₄	—	142	1	no	—
304	C₂×C₄	—	142	1	no	—
305	C₂×C₄	—	142	1	no	—
306	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
307	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	142	1	no	—
310	C₈	—	142	1	no	—
311	C₈	#11765864	1	2	yes	complemented
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	4	1	no	—
316	C₂²	—	4	1	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	142	1	no	—
319	C₂²	—	142	1	no	—
320	C₂²	—	142	1	no	—
321	C₂²	C₂×D₄×D₇₁	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	4	1	no	—
324	C₄	—	4	1	no	—
325	C₄	—	4	1	no	—
326	C₄	—	284	1	no	—
327	C₄	—	284	1	no	—
328	C₄	—	284	1	no	—
329	C₄	—	284	1	no	—
330	C₄	—	71	2	no	—
331	C₄	—	142	1	no	—
332	C₄	—	142	1	no	—
333	C₄	C₂×#11765864	1	1	yes	characteristic
334	C₄	#14556175	1	1	yes	characteristic
335	C₂	—	4	1	no	—
336	C₂	—	8	1	no	—
337	C₂	—	284	1	no	—
338	C₂	#17986649	1	1	yes	characteristic
339	C₂	AGL(1,ℤ/8)×D₇₁	1	1	yes	characteristic
340	C₂	#17986815	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₁
Fitting	#468×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₄
hypercenter	#468	D₇₁
hypocenter	C₇₁	#3353
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #468

Complements

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

N	H ≅ G/N	type
1. #17986720	C₂	semidirect
3. #17986403	C₂	semidirect
4. #17986333	C₂	semidirect
5. #17986332	C₂	semidirect
6. #17986322	C₂	semidirect
7. #17986320	C₂	semidirect
8. #17986282	C₂	semidirect
9. #17986277	C₂	semidirect
10. #17986242	C₂	semidirect
11. SD₁₆×Dic₇₁	C₂	semidirect
12. #17986075	C₂	semidirect
13. #17986041	C₂	semidirect
14. #17986040	C₂	semidirect
15. #468×C₇₁	C₂	semidirect
138. C₈×C₇₁	#41	semidirect
171. C₇₁	#3353	semidirect
182. #468	D₇₁	semidirect
311. C₈	#11765864	semidirect