Subgroups — #22598567

Order 9088.

370 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17986614	C₂	1	1	yes	maximal, characteristic
2	#17986475	C₂	1	1	yes	maximal, characteristic, complemented
3	#17986342	C₂	1	1	yes	maximal, characteristic, complemented
4	#17986322	C₂	1	1	yes	maximal, characteristic
5	#17986293	C₂	1	1	yes	maximal, characteristic, complemented
6	#17986282	C₂	1	1	yes	maximal, characteristic, complemented
7	#17986220	C₂	1	1	yes	maximal, characteristic, complemented
8	#17986219	C₂	1	1	yes	maximal, characteristic, complemented
9	#17986156	C₂	1	1	yes	maximal, characteristic, complemented
10	#17986136	C₂	1	1	yes	maximal, characteristic
11	#17986107	C₂	1	1	yes	maximal, characteristic, complemented
12	C₄×#11765855	C₂	1	1	yes	maximal, characteristic, complemented
13	C₄×#11765853	C₂	1	1	yes	maximal, characteristic, complemented
14	#17986061	C₂	1	1	yes	maximal, characteristic, complemented
15	#440×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×Q₈×D₇₁	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₈₄	C₂²	1	1	yes	characteristic
18	C₂×#11765855	C₂²	1	1	yes	characteristic
19	C₂×#11765853	C₂²	1	1	yes	characteristic
20	#14556127	C₂²	1	1	yes	characteristic
21	#14556121	—	2	1	no	—
22	#14556115	C₂²	1	1	yes	characteristic
23	#14556114	—	2	1	no	—
24	#14556111	—	2	1	no	—
25	#14556108	C₂²	1	1	yes	characteristic
26	#14556107	—	2	1	no	—
27	#14556099	C₂²	1	1	yes	characteristic
28	Q₈×Dic₇₁	C₂²	1	1	yes	characteristic
29	C₄×Dic₁₄₂	—	2	1	no	—
30	C₄×Dic₁₄₂	C₂²	1	1	yes	characteristic
31	C₄×Dic₁₄₂	C₂²	1	1	yes	characteristic
32	#14556093	—	2	1	no	—
33	#14556093	C₂²	1	1	yes	characteristic
34	#14556086	—	2	1	no	—
35	#14556086	C₂²	1	1	yes	characteristic
36	#14556085	—	2	1	no	—
37	#14556085	C₂²	1	1	yes	characteristic
38	#14556061	C₂²	1	1	yes	characteristic
39	#14556060	C₂²	1	1	yes	characteristic
40	#14556059	C₂²	1	1	yes	characteristic
41	Dic₇₁⋊C₈	C₂²	1	1	yes	characteristic
42	Dic₇₁⋊C₈	C₂²	1	1	yes	characteristic
43	#14556053	C₂²	1	1	yes	characteristic
44	#14556053	C₂²	1	1	yes	characteristic
45	#14556052	C₂²	1	1	yes	characteristic
46	#14556051	C₂²	1	1	yes	characteristic
47	#14556051	C₂²	1	1	yes	characteristic
48	#14556049	C₂²	1	1	yes	characteristic
49	(C₂×D₇₁)⋊C₈	C₂²	1	1	yes	characteristic
50	(C₂×D₇₁)⋊C₈	C₂²	1	1	yes	characteristic
51	C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₇₁	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₇₁	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₇₁	C₂²	1	1	yes	characteristic
55	#107×C₇₁	C₂²	1	1	yes	characteristic
56	#103×C₇₁	C₂²	1	1	yes	characteristic
57	#102×C₇₁	C₂²	1	1	yes	characteristic
58	#97×C₇₁	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₄₂	—	2	1	no	—
60	C₂×Dic₁₄₂	—	2	1	no	—
61	C₂×Dic₁₄₂	C₂³	1	1	yes	characteristic
62	C₂×Dic₁₄₂	C₂³	1	1	yes	characteristic
63	Q₈×D₇₁	—	2	2	no	—
64	Q₈×D₇₁	—	2	2	no	—
65	C₂×C₂²⋊D₇₁	—	2	1	no	—
66	C₂×C₄×D₇₁	—	2	1	no	—
67	C₂×C₄×D₇₁	C₂³	1	1	yes	characteristic
68	Dic₂₈₄	—	2	2	no	—
69	#11765855	—	2	2	no	—
70	#11765853	—	2	2	no	—
71	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
72	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₇₁	—	2	1	no	—
74	C₄⋊Dic₇₁	—	2	1	no	—
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	Dic₇₁⋊C₄	D₄	1	1	yes	characteristic
83	C₂²⋊Dic₇₁	—	2	1	no	—
84	C₂²⋊Dic₇₁	—	2	1	no	—
85	C₂²⋊Dic₇₁	—	2	1	no	—
86	(C₂×D₇₁)⋊C₄	—	2	1	no	—
87	(C₂×D₇₁)⋊C₄	—	2	1	no	—
88	(C₂×D₇₁)⋊C₄	D₄	1	1	yes	characteristic
89	(C₂×D₇₁)⋊C₄	D₄	1	1	yes	characteristic
90	C₄×Dic₇₁	—	2	1	no	—
91	C₄×Dic₇₁	—	2	1	no	—
92	C₄×Dic₇₁	—	2	1	no	—
93	C₄×Dic₇₁	—	2	1	no	—
94	C₄×Dic₇₁	—	2	1	no	—
95	C₄×Dic₇₁	C₂³	1	1	yes	characteristic
96	C₂×Q₈×C₇₁	C₂³	1	1	yes	characteristic
97	C₂×D₄×C₇₁	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₇₁	—	2	1	no	—
99	SD₁₆×C₇₁	—	2	2	no	—
100	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
101	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
102	C₄⋊C₄×C₇₁	—	2	1	no	—
103	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
105	C₂²⋊C₄×C₇₁	—	2	1	no	—
106	C₄²×C₇₁	—	2	1	no	—
107	C₄²×C₇₁	C₂³	1	1	yes	characteristic
108	C₂³×C₇₁	—	2	1	no	—
109	C₂²×D₇₁	C₂×D₄	1	1	yes	characteristic
110	Q₈×C₇₁	—	2	1	no	—
111	Q₈×C₇₁	#41	1	2	yes	—
112	D₄×C₇₁	—	2	1	no	—
113	D₄×C₇₁	—	2	1	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	—	8	1	no	—
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	C₂²⋊D₇₁	—	8	1	no	—
122	C₂×Dic₇₁	—	2	1	no	—
123	C₂×Dic₇₁	—	2	1	no	—
124	C₂×Dic₇₁	—	2	1	no	—
125	C₂×Dic₇₁	—	2	1	no	—
126	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
127	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
128	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
129	C₄×D₇₁	—	2	4	no	—
130	C₄×D₇₁	—	2	1	no	—
131	C₄×D₇₁	—	2	1	no	—
132	Dic₁₄₂	—	2	4	no	—
133	Dic₁₄₂	—	2	1	no	—
134	Dic₁₄₂	—	2	1	no	—
135	Dic₁₄₂	—	2	1	no	—
136	Dic₁₄₂	—	8	1	no	—
137	Dic₁₄₂	#41	1	2	yes	—
138	C₈×C₇₁	—	2	1	no	—
139	C₈×C₇₁	—	2	1	no	—
140	C₇₁⋊C₈	—	2	1	no	—
141	C₇₁⋊C₈	—	2	1	no	—
142	C₂²×C₇₁	—	4	1	no	—
143	C₂²×C₇₁	—	4	1	no	—
144	C₂²×C₇₁	—	4	1	no	—
145	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
146	C₂×D₇₁	—	2	1	no	—
147	C₂×D₇₁	—	2	1	no	—
148	C₂×D₇₁	—	2	1	no	—
149	C₄×C₇₁	—	2	2	no	—
150	C₄×C₇₁	—	2	1	no	—
151	C₄×C₇₁	—	2	1	no	—
152	C₄×C₇₁	—	4	1	no	—
153	C₄×C₇₁	—	4	1	no	—
154	C₄×C₇₁	C₂×#41	1	1	yes	characteristic
155	C₄×C₇₁	C₂×#41	1	1	yes	characteristic
156	Dic₇₁	—	2	1	no	—
157	Dic₇₁	—	2	2	no	—
158	Dic₇₁	—	2	1	no	—
159	Dic₇₁	—	2	1	no	—
160	Dic₇₁	—	4	1	no	—
161	Dic₇₁	—	4	1	no	—
162	Dic₇₁	—	4	1	no	—
163	C₂×C₇₁	—	8	1	no	—
164	C₂×C₇₁	#525	1	1	yes	characteristic
165	C₂×C₇₁	#577	1	1	yes	characteristic
166	C₂×C₇₁	#578	1	1	yes	characteristic
167	D₇₁	—	4	1	no	—
168	#3121	—	71	1	no	maximal, Sylow
169	C₇₁	#3121	1	1	yes	characteristic, complemented, Sylow
170	#542	—	71	1	no	—
171	#519	—	71	1	no	—
172	#485	—	71	1	no	—
173	#482	—	71	1	no	—
174	#477	—	71	1	no	—
175	#475	—	71	1	no	—
176	#464	—	71	1	no	—
177	#464	—	71	1	no	—
178	#451	—	71	1	no	—
179	#448	—	71	1	no	—
180	#441	—	71	1	no	—
181	#440	D₇₁	1	1	yes	characteristic, complemented
182	C₄×Q₁₆	—	71	1	no	—
183	C₄×SD₁₆	—	71	1	no	—
184	#433	—	71	1	no	—
185	C₂²×Q₈	—	71	1	no	—
186	C₂×Q₁₆	—	71	1	no	—
187	C₂×Q₁₆	—	71	1	no	—
188	C₂×SD₁₆	—	71	1	no	—
189	C₂×SD₁₆	C₂×D₇₁	1	1	yes	characteristic
190	C₄⋊Q₈	—	71	1	no	—
191	#126	—	142	1	no	—
192	#124	—	71	1	no	—
193	#124	—	142	1	no	—
194	#123	—	142	1	no	—
195	C₂²⋊Q₈	—	71	1	no	—
196	C₂²⋊Q₈	—	142	1	no	—
197	#121	—	71	1	no	—
198	C₄×Q₈	—	71	1	no	—
199	C₄×Q₈	—	71	1	no	—
200	C₄×Q₈	—	71	1	no	—
201	C₄×Q₈	—	71	1	no	—
202	C₄×Q₈	—	142	1	no	—
203	C₄×Q₈	—	142	1	no	—
204	C₄×Q₈	C₂×D₇₁	1	1	yes	characteristic
205	C₄×D₄	C₂×D₇₁	1	1	yes	characteristic
206	#117	—	71	1	no	—
207	#117	—	71	1	no	—
208	#117	—	142	1	no	—
209	#117	—	142	1	no	—
210	#107	—	71	1	no	—
211	#107	C₂×D₇₁	1	1	yes	characteristic
212	#106	—	71	1	no	—
213	#106	—	71	1	no	—
214	C₄⋊C₈	—	71	1	no	—
215	C₄⋊C₈	—	71	1	no	—
216	#103	—	71	1	no	—
217	#103	—	71	1	no	—
218	#103	—	71	1	no	—
219	#103	—	71	1	no	—
220	#103	—	71	1	no	—
221	#103	C₂×D₇₁	1	1	yes	characteristic
222	#102	—	71	1	no	—
223	#102	C₂×D₇₁	1	1	yes	characteristic
224	C₂²⋊C₈	—	71	1	no	—
225	C₂²⋊C₈	—	71	1	no	—
226	#97	C₂×D₇₁	1	1	yes	characteristic
227	C₄×C₈	—	71	1	no	—
228	C₂×Q₈	—	71	1	no	—
229	C₂×Q₈	—	71	1	no	—
230	C₂×Q₈	—	142	1	no	—
231	C₂×Q₈	—	142	1	no	—
232	C₂×Q₈	—	142	2	no	—
233	C₂×Q₈	—	142	2	no	—
234	C₂×Q₈	C₂²×D₇₁	1	1	yes	characteristic
235	C₂×D₄	—	142	1	no	—
236	C₂×D₄	C₂²×D₇₁	1	1	yes	characteristic
237	C₂²×C₄	—	2	1	no	—
238	C₂²×C₄	—	71	1	no	—
239	C₂²×C₄	—	142	1	no	—
240	Q₁₆	—	142	2	no	—
241	Q₁₆	—	142	2	no	—
242	SD₁₆	—	2	2	no	—
243	SD₁₆	—	142	2	no	—
244	C₂×C₈	—	71	1	no	—
245	C₂×C₈	—	71	1	no	—
246	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
247	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	71	1	no	—
250	C₄⋊C₄	—	71	1	no	—
251	C₄⋊C₄	—	71	1	no	—
252	C₄⋊C₄	—	71	1	no	—
253	C₄⋊C₄	—	142	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	C₂²×D₇₁	1	1	yes	characteristic
260	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
261	C₂²⋊C₄	—	2	1	no	—
262	C₂²⋊C₄	—	71	1	no	—
263	C₂²⋊C₄	—	71	1	no	—
264	C₂²⋊C₄	—	142	1	no	—
265	C₂²⋊C₄	—	142	1	no	—
266	C₂²⋊C₄	—	142	1	no	—
267	C₂²⋊C₄	—	142	1	no	—
268	C₂²⋊C₄	—	142	1	no	—
269	C₄²	—	2	1	no	—
270	C₄²	—	71	1	no	—
271	C₄²	—	142	1	no	—
272	C₄²	—	142	1	no	—
273	C₄²	—	142	1	no	—
274	C₄²	—	142	1	no	—
275	C₄²	—	142	1	no	—
276	C₄²	C₂²×D₇₁	1	1	yes	characteristic
277	C₂³	—	2	1	no	—
278	C₂³	—	71	1	no	—
279	Q₈	—	2	1	no	—
280	Q₈	—	568	1	no	—
281	Q₈	—	71	2	no	—
282	Q₈	—	142	4	no	—
283	Q₈	—	142	1	no	—
284	Q₈	—	142	1	no	—
285	Q₈	—	142	1	no	—
286	Q₈	#11765865	1	2	yes	—
287	D₄	—	2	1	no	—
288	D₄	—	2	1	no	—
289	D₄	—	568	1	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	8	1	no	—
294	C₂×C₄	—	71	1	no	—
295	C₂×C₄	—	71	1	no	—
296	C₂×C₄	—	71	1	no	—
297	C₂×C₄	—	142	4	no	—
298	C₂×C₄	—	142	1	no	—
299	C₂×C₄	—	142	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	D₄×D₇₁	1	1	yes	characteristic
305	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
306	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	142	1	no	—
310	C₈	—	142	1	no	—
311	C₂²	—	4	1	no	—
312	C₂²	—	4	1	no	—
313	C₂²	—	4	1	no	—
314	C₂²	—	142	1	no	—
315	C₂²	—	142	1	no	—
316	C₂²	—	142	1	no	—
317	C₂²	C₂×D₄×D₇₁	1	1	yes	characteristic
318	C₄	—	2	2	no	—
319	C₄	—	2	1	no	—
320	C₄	—	2	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	4	1	no	—
323	C₄	—	284	1	no	—
324	C₄	—	284	1	no	—
325	C₄	—	284	1	no	—
326	C₄	—	142	1	no	—
327	C₄	—	142	2	no	—
328	C₄	—	142	1	no	—
329	C₄	—	142	1	no	—
330	C₄	#41×D₇₁	1	1	yes	characteristic
331	C₄	C₂×#11765865	1	1	yes	characteristic
332	C₂	—	8	1	no	—
333	C₂	—	284	1	no	—
334	C₂	#17986508	1	1	yes	characteristic
335	C₂	#17986809	1	1	yes	characteristic
336	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	#440×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₄
hypercenter	#440	D₇₁
hypocenter	C₇₁	#3121
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #440

Complements

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

N	H ≅ G/N	type
2. #17986475	C₂	semidirect
3. #17986342	C₂	semidirect
5. #17986293	C₂	semidirect
6. #17986282	C₂	semidirect
7. #17986220	C₂	semidirect
8. #17986219	C₂	semidirect
9. #17986156	C₂	semidirect
11. #17986107	C₂	semidirect
12. C₄×#11765855	C₂	semidirect
13. C₄×#11765853	C₂	semidirect
14. #17986061	C₂	semidirect
15. #440×C₇₁	C₂	semidirect
169. C₇₁	#3121	semidirect
181. #440	D₇₁	semidirect