Subgroups — #22600781

Order 9088.

376 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	#17986688	C₂	1	1	yes	maximal, characteristic, complemented
2	#17986650	C₂	1	1	yes	maximal, characteristic
3	#17986406	C₂	1	1	yes	maximal, characteristic, complemented
4	#17986334	C₂	1	1	yes	maximal, characteristic, complemented
5	#17986331	C₂	1	1	yes	maximal, characteristic, complemented
6	#17986323	C₂	1	1	yes	maximal, characteristic, complemented
7	#17986314	C₂	1	1	yes	maximal, characteristic, complemented
8	#17986298	C₂	1	1	yes	maximal, characteristic, complemented
9	#17986293	C₂	1	1	yes	maximal, characteristic, complemented
10	#17986245	C₂	1	1	yes	maximal, characteristic, complemented
11	#17986229	C₂	1	1	yes	maximal, characteristic, complemented
12	#17986098	C₂	1	1	yes	maximal, characteristic, complemented
13	#17986068	C₂	1	1	yes	maximal, characteristic, complemented
14	#107×D₇₁	C₂	1	1	yes	maximal, characteristic, complemented
15	#428×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11765865	—	2	1	no	—
17	C₂×#11765854	C₂²	1	1	yes	characteristic
18	C₂×C₈×D₇₁	C₂²	1	1	yes	characteristic
19	#14556127	C₂²	1	1	yes	characteristic
20	#14556121	—	2	1	no	—
21	#14556120	—	2	1	no	—
22	#14556119	C₂²	1	1	yes	characteristic
23	#14556113	—	2	1	no	—
24	#14556111	—	2	1	no	—
25	#14556110	—	2	1	no	—
26	#14556108	C₂²	1	1	yes	characteristic
27	#14556107	C₂²	1	1	yes	characteristic
28	#14556106	—	2	1	no	—
29	#14556103	C₂²	1	1	yes	characteristic
30	#14556102	C₂²	1	1	yes	characteristic
31	#14556093	C₂²	1	1	yes	characteristic
32	#14556090	C₂²	1	1	yes	characteristic
33	C₄×C₂²⋊D₇₁	C₂²	1	1	yes	characteristic
34	#14556085	—	2	1	no	—
35	C₄⋊C₄×D₇₁	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×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	#14556061	C₂²	1	1	yes	characteristic
41	#14556059	C₂²	1	1	yes	characteristic
42	Dic₇₁⋊C₈	C₂²	1	1	yes	characteristic
43	#14556053	C₂²	1	1	yes	characteristic
44	#14556051	C₂²	1	1	yes	characteristic
45	#14556051	C₂²	1	1	yes	characteristic
46	#14556050	C₂²	1	1	yes	characteristic
47	#14556048	C₂²	1	1	yes	characteristic
48	#14556048	C₂²	1	1	yes	characteristic
49	C₂²⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
50	(C₂×D₇₁)⋊C₈	C₂²	1	1	yes	characteristic
51	#14556038	C₂²	1	1	yes	characteristic
52	C₂×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
53	#117×C₇₁	C₂²	1	1	yes	characteristic
54	C₂×C₄⋊C₄×C₇₁	C₂²	1	1	yes	characteristic
55	#107×C₇₁	C₂²	1	1	yes	characteristic
56	#107×C₇₁	C₂²	1	1	yes	characteristic
57	#106×C₇₁	C₂²	1	1	yes	characteristic
58	#106×C₇₁	C₂²	1	1	yes	characteristic
59	#11765865	—	4	4	no	—
60	C₂×Dic₁₄₂	—	2	1	no	—
61	C₂×Dic₁₄₂	C₂³	1	1	yes	characteristic
62	C₂×D₂₈₄	—	2	1	no	—
63	C₂×D₂₈₄	C₂³	1	1	yes	characteristic
64	C₂×C₂²⋊D₇₁	—	2	1	no	—
65	C₂×C₂²⋊D₇₁	D₄	1	1	yes	characteristic
66	C₂×C₄×D₇₁	—	2	1	no	—
67	C₂×C₄×D₇₁	—	2	1	no	—
68	C₂×C₄×D₇₁	—	2	1	no	—
69	C₂×C₄×D₇₁	C₂³	1	1	yes	characteristic
70	#11765854	—	2	2	no	—
71	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
72	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
73	C₈×D₇₁	—	2	2	no	—
74	C₄⋊Dic₇₁	—	2	1	no	—
75	C₄⋊Dic₇₁	—	2	1	no	—
76	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
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	Dic₇₁⋊C₄	—	2	1	no	—
84	Dic₇₁⋊C₄	D₄	1	1	yes	characteristic
85	C₂²⋊Dic₇₁	—	2	1	no	—
86	C₂²⋊Dic₇₁	D₄	1	1	yes	characteristic
87	(C₂×D₇₁)⋊C₄	—	2	1	no	—
88	(C₂×D₇₁)⋊C₄	—	2	1	no	—
89	(C₂×D₇₁)⋊C₄	—	2	1	no	—
90	(C₂×D₇₁)⋊C₄	—	2	1	no	—
91	(C₂×D₇₁)⋊C₄	D₄	1	1	yes	characteristic
92	C₄×Dic₇₁	—	2	1	no	—
93	C₄×Dic₇₁	C₂³	1	1	yes	characteristic
94	C₂²×C₄×C₇₁	—	2	1	no	—
95	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
96	M₁₆×C₇₁	—	2	2	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	C₂³	1	1	yes	characteristic
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	—	2	1	no	—
106	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
107	C₂²×D₇₁	—	2	1	no	—
108	C₂²×D₇₁	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₇₁	—	2	1	no	—
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	—	2	1	no	—
114	C₂×C₄×C₇₁	—	2	1	no	—
115	C₂×C₄×C₇₁	—	4	2	no	—
116	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
118	C₂²⋊D₇₁	—	2	2	no	—
119	C₂²⋊D₇₁	—	4	2	no	—
120	C₂×Dic₇₁	—	2	1	no	—
121	C₂×Dic₇₁	—	2	1	no	—
122	C₂×Dic₇₁	—	2	1	no	—
123	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
124	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
126	D₂₈₄	—	2	1	no	—
127	D₂₈₄	—	2	1	no	—
128	D₂₈₄	—	4	2	no	—
129	C₄×D₇₁	—	2	1	no	—
130	C₄×D₇₁	—	2	1	no	—
131	C₄×D₇₁	—	4	2	no	—
132	C₄×D₇₁	—	4	2	no	—
133	C₄×D₇₁	—	4	2	no	—
134	Dic₁₄₂	—	2	1	no	—
135	Dic₁₄₂	—	2	1	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	C₂²×D₄	1	1	yes	characteristic
145	C₂×D₇₁	—	2	1	no	—
146	C₂×D₇₁	—	2	1	no	—
147	C₂×D₇₁	—	4	1	no	—
148	C₂×D₇₁	—	4	1	no	—
149	C₂×D₇₁	—	4	1	no	—
150	C₂×D₇₁	AGL(1,ℤ/8)	1	2	yes	—
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	—	4	1	no	—
156	C₄×C₇₁	C₂×#41	1	1	yes	characteristic
157	C₄×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
158	Dic₇₁	—	2	1	no	—
159	Dic₇₁	—	2	1	no	—
160	Dic₇₁	—	2	1	no	—
161	Dic₇₁	—	4	1	no	—
162	Dic₇₁	—	4	1	no	—
163	Dic₇₁	—	4	1	no	—
164	C₂×C₇₁	—	4	1	no	—
165	C₂×C₇₁	#547	1	1	yes	characteristic
166	C₂×C₇₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
167	C₂×C₇₁	#578	1	1	yes	characteristic
168	D₇₁	—	2	2	no	—
169	D₇₁	—	8	1	no	—
170	#3343	—	71	1	no	maximal, Sylow
171	C₇₁	#3343	1	1	yes	characteristic, complemented, Sylow
172	#552	—	71	1	no	—
173	#547	—	71	1	no	—
174	#501	—	71	1	no	—
175	#484	—	71	1	no	—
176	#483	—	71	1	no	—
177	#482	—	71	1	no	—
178	#480	—	71	1	no	—
179	#478	—	71	1	no	—
180	#477	—	71	1	no	—
181	#468	—	71	1	no	—
182	#465	—	71	1	no	—
183	#440	—	71	1	no	—
184	#435	—	71	1	no	—
185	#428	D₇₁	1	1	yes	characteristic, complemented
186	C₂×#107	—	71	1	no	—
187	C₂×#41	—	142	1	no	—
188	C₂×SD₁₆	—	71	1	no	—
189	C₂×M₁₆	C₂×D₇₁	1	1	yes	characteristic
190	C₂²×C₈	—	71	1	no	—
191	C₄⋊Q₈	—	71	1	no	—
192	#126	—	142	1	no	—
193	#125	—	71	1	no	—
194	#125	—	142	1	no	—
195	#123	—	142	1	no	—
196	#123	—	142	1	no	—
197	#123	—	142	1	no	—
198	C₂²⋊Q₈	—	71	1	no	—
199	C₂²⋊Q₈	—	71	1	no	—
200	C₂²⋊Q₈	—	142	1	no	—
201	#121	—	71	1	no	—
202	#121	—	71	1	no	—
203	C₄×Q₈	—	71	1	no	—
204	C₄×D₄	—	71	1	no	—
205	C₄×D₄	—	71	1	no	—
206	#117	—	142	1	no	—
207	#117	C₂×D₇₁	1	1	yes	characteristic
208	C₂×C₄⋊C₄	—	71	1	no	—
209	C₂×C₄⋊C₄	—	71	1	no	—
210	C₂×C₄⋊C₄	C₂×D₇₁	1	1	yes	characteristic
211	#107	—	71	1	no	—
212	#107	—	71	1	no	—
213	#107	—	71	1	no	—
214	#107	—	71	1	no	—
215	#107	C₂×D₇₁	1	1	yes	characteristic
216	#107	C₂×D₇₁	1	1	yes	characteristic
217	#106	—	71	1	no	—
218	#106	C₂×D₇₁	1	1	yes	characteristic
219	#106	C₂×D₇₁	1	1	yes	characteristic
220	C₄⋊C₈	—	71	1	no	—
221	#103	—	71	1	no	—
222	#103	—	71	1	no	—
223	#103	—	71	1	no	—
224	#102	—	71	1	no	—
225	#102	—	71	1	no	—
226	#102	—	71	1	no	—
227	C₂²⋊C₈	—	71	1	no	—
228	C₂²⋊C₈	—	71	1	no	—
229	#97	—	71	1	no	—
230	#41	—	284	4	no	—
231	C₂×Q₈	—	71	1	no	—
232	C₂×Q₈	—	142	1	no	—
233	C₂×D₄	—	71	1	no	—
234	C₂×D₄	—	71	1	no	—
235	C₂×D₄	—	142	1	no	—
236	C₂×D₄	—	142	1	no	—
237	C₂²×C₄	—	2	1	no	—
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	—	142	2	no	—
244	M₁₆	—	2	2	no	—
245	C₂×C₈	—	71	1	no	—
246	C₂×C₈	—	71	1	no	—
247	C₂×C₈	—	142	2	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	—	142	1	no	—
255	C₄⋊C₄	—	142	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	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
265	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
266	C₂²⋊C₄	—	2	1	no	—
267	C₂²⋊C₄	—	71	1	no	—
268	C₂²⋊C₄	—	71	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	C₂²⋊C₄	—	142	1	no	—
274	C₄²	—	2	1	no	—
275	C₄²	—	71	1	no	—
276	C₄²	—	142	1	no	—
277	C₂³	—	71	1	no	—
278	C₂³	—	142	1	no	—
279	C₂³	D₄×D₇₁	1	1	yes	characteristic
280	Q₈	—	284	2	no	—
281	Q₈	—	142	1	no	—
282	Q₈	—	142	1	no	—
283	D₄	—	284	2	no	—
284	D₄	—	284	2	no	—
285	D₄	—	142	2	no	—
286	D₄	—	142	1	no	—
287	D₄	—	142	1	no	—
288	C₂×C₄	—	2	1	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	—	4	2	no	—
295	C₂×C₄	—	284	2	no	—
296	C₂×C₄	—	284	2	no	—
297	C₂×C₄	—	284	2	no	—
298	C₂×C₄	—	71	1	no	—
299	C₂×C₄	—	71	1	no	—
300	C₂×C₄	—	71	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	#11765865	1	2	yes	complemented
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	2	1	no	—
315	C₂²	—	284	1	no	—
316	C₂²	—	284	1	no	—
317	C₂²	—	284	1	no	—
318	C₂²	—	71	2	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	—	4	1	no	—
327	C₄	—	284	1	no	—
328	C₄	—	284	1	no	—
329	C₄	—	284	1	no	—
330	C₄	—	142	1	no	—
331	C₄	—	142	1	no	—
332	C₄	—	142	1	no	—
333	C₄	C₂×#11765865	1	1	yes	characteristic
334	C₄	#14556176	1	1	yes	characteristic
335	C₂	—	4	1	no	—
336	C₂	—	568	1	no	—
337	C₂	—	142	2	no	—
338	C₂	#17986650	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	#428×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₄
hypercenter	#428	D₇₁
hypocenter	C₇₁	#3343
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #428

Complements

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

N	H ≅ G/N	type
1. #17986688	C₂	semidirect
3. #17986406	C₂	semidirect
4. #17986334	C₂	semidirect
5. #17986331	C₂	semidirect
6. #17986323	C₂	semidirect
7. #17986314	C₂	semidirect
8. #17986298	C₂	semidirect
9. #17986293	C₂	semidirect
10. #17986245	C₂	semidirect
11. #17986229	C₂	semidirect
12. #17986098	C₂	semidirect
13. #17986068	C₂	semidirect
14. #107×D₇₁	C₂	semidirect
15. #428×C₇₁	C₂	semidirect
138. C₈×C₇₁	#41	semidirect
171. C₇₁	#3343	semidirect
185. #428	D₇₁	semidirect
311. C₈	#11765865	semidirect