Subgroups — #22601149

Order 9088.

350 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17986731	C₂	1	1	yes	maximal, characteristic
2	Q₈×Dic₁₄₂	C₂	1	1	yes	maximal, characteristic, complemented
3	#17986360	C₂	1	1	yes	maximal, characteristic, complemented
4	#17986358	C₂	1	1	yes	maximal, characteristic, complemented
5	#17986344	C₂	1	1	yes	maximal, characteristic
6	#17986292	C₂	1	1	yes	maximal, characteristic, complemented
7	#17986280	C₂	1	1	yes	maximal, characteristic, complemented
8	#17986277	C₂	1	1	yes	maximal, characteristic, complemented
9	#17986223	C₂	1	1	yes	maximal, characteristic, complemented
10	#17986197	C₂	1	1	yes	maximal, characteristic
11	#17986128	C₂	1	1	yes	maximal, characteristic, complemented
12	#17986100	C₂	1	1	yes	maximal, characteristic, complemented
13	#17986097	C₂	1	1	yes	maximal, characteristic, complemented
14	Q₁₆×Dic₇₁	C₂	1	1	yes	maximal, characteristic, complemented
15	#464×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11765852	C₂²	1	1	yes	characteristic
17	#14556128	—	2	1	no	—
18	#14556128	C₂²	1	1	yes	characteristic
19	#14556127	—	2	1	no	—
20	#14556127	—	2	1	no	—
21	#14556127	C₂²	1	1	yes	characteristic
22	#14556127	C₂²	1	1	yes	characteristic
23	#14556124	C₂²	1	1	yes	characteristic
24	#14556117	C₂²	1	1	yes	characteristic
25	#14556114	—	2	1	no	—
26	#14556112	—	2	1	no	—
27	#14556110	—	2	1	no	—
28	Q₈×Dic₇₁	C₂²	1	1	yes	characteristic
29	Q₈×Dic₇₁	C₂²	1	1	yes	characteristic
30	Q₈×Dic₇₁	C₂²	1	1	yes	characteristic
31	C₄×Dic₁₄₂	—	2	1	no	—
32	C₄×Dic₁₄₂	C₂²	1	1	yes	characteristic
33	C₄×D₂₈₄	C₂²	1	1	yes	characteristic
34	D₄×Dic₇₁	C₂²	1	1	yes	characteristic
35	(C₂²⋊D₇₁)⋊C₄	—	2	1	no	—
36	#14556062	C₂²	1	1	yes	characteristic
37	#14556062	C₂²	1	1	yes	characteristic
38	#14556061	C₂²	1	1	yes	characteristic
39	Dic₇₁⋊C₈	C₂²	1	1	yes	characteristic
40	Dic₇₁⋊C₈	C₂²	1	1	yes	characteristic
41	#14556052	C₂²	1	1	yes	characteristic
42	#14556052	C₂²	1	1	yes	characteristic
43	#14556052	C₂²	1	1	yes	characteristic
44	#14556050	C₂²	1	1	yes	characteristic
45	#14556050	C₂²	1	1	yes	characteristic
46	#14556049	C₂²	1	1	yes	characteristic
47	#14556049	C₂²	1	1	yes	characteristic
48	#14556048	C₂²	1	1	yes	characteristic
49	#14556039	C₂²	1	1	yes	characteristic
50	#14556038	C₂²	1	1	yes	characteristic
51	C₈×Dic₇₁	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₇₁	C₂²	1	1	yes	characteristic
53	C₂×SD₁₆×C₇₁	C₂²	1	1	yes	characteristic
54	#124×C₇₁	C₂²	1	1	yes	characteristic
55	C₄×Q₈×C₇₁	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
57	#103×C₇₁	C₂²	1	1	yes	characteristic
58	#102×C₇₁	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₄₂	—	2	1	no	—
60	C₂×Dic₁₄₂	—	2	1	no	—
61	C₂×Dic₁₄₂	—	2	1	no	—
62	C₂×Dic₁₄₂	D₄	1	1	yes	characteristic
63	C₂×D₂₈₄	C₂³	1	1	yes	characteristic
64	C₂×C₂²⋊D₇₁	—	2	1	no	—
65	C₂×C₂²⋊D₇₁	—	2	1	no	—
66	C₂²×Dic₇₁	—	2	1	no	—
67	C₂×C₄×D₇₁	—	2	1	no	—
68	#11765852	—	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	—	2	1	no	—
74	C₄⋊Dic₇₁	—	2	1	no	—
75	C₄⋊Dic₇₁	D₄	1	1	yes	characteristic
76	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
78	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
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	(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	C₂³	1	1	yes	characteristic
91	C₄×Dic₇₁	C₂³	1	1	yes	characteristic
92	C₂×Q₈×C₇₁	C₂³	1	1	yes	characteristic
93	C₂×Q₈×C₇₁	C₂³	1	1	yes	characteristic
94	C₂×D₄×C₇₁	C₂³	1	1	yes	characteristic
95	Q₁₆×C₇₁	—	2	2	no	—
96	SD₁₆×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	C₂²⋊C₄×C₇₁	—	2	1	no	—
102	C₂²⋊C₄×C₇₁	—	2	1	no	—
103	C₄²×C₇₁	—	2	1	no	—
104	C₄²×C₇₁	C₂³	1	1	yes	characteristic
105	C₂³×C₇₁	—	2	1	no	—
106	C₂²×D₇₁	—	2	1	no	—
107	Q₈×C₇₁	—	2	1	no	—
108	Q₈×C₇₁	—	2	1	no	—
109	Q₈×C₇₁	—	2	1	no	—
110	Q₈×C₇₁	#41	1	2	yes	—
111	D₄×C₇₁	—	2	1	no	—
112	D₄×C₇₁	—	2	1	no	—
113	C₂×C₄×C₇₁	—	2	1	no	—
114	C₂×C₄×C₇₁	—	2	1	no	—
115	C₂×C₄×C₇₁	—	2	1	no	—
116	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
117	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
119	C₂²⋊D₇₁	—	4	2	no	—
120	C₂²⋊D₇₁	—	8	1	no	—
121	C₂×Dic₇₁	—	2	1	no	—
122	C₂×Dic₇₁	—	2	1	no	—
123	C₂×Dic₇₁	—	2	1	no	—
124	C₂×Dic₇₁	—	8	1	no	—
125	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
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	D₂₈₄	—	2	1	no	—
130	D₂₈₄	—	2	1	no	—
131	C₄×D₇₁	—	8	1	no	—
132	Dic₁₄₂	—	2	2	no	—
133	Dic₁₄₂	—	8	1	no	—
134	Dic₁₄₂	—	8	1	no	—
135	Dic₁₄₂	—	8	1	no	—
136	C₈×C₇₁	—	2	1	no	—
137	C₈×C₇₁	—	2	1	no	—
138	C₇₁⋊C₈	—	2	1	no	—
139	C₇₁⋊C₈	—	2	1	no	—
140	C₂²×C₇₁	—	4	1	no	—
141	C₂²×C₇₁	—	4	1	no	—
142	C₂²×C₇₁	—	4	1	no	—
143	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
144	C₂×D₇₁	—	4	1	no	—
145	C₂×D₇₁	—	4	1	no	—
146	C₂×D₇₁	—	4	1	no	—
147	C₄×C₇₁	—	2	2	no	—
148	C₄×C₇₁	—	2	1	no	—
149	C₄×C₇₁	—	2	1	no	—
150	C₄×C₇₁	—	4	1	no	—
151	C₄×C₇₁	—	4	1	no	—
152	C₄×C₇₁	C₂×#41	1	1	yes	characteristic
153	C₄×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
154	Dic₇₁	—	2	1	no	—
155	Dic₇₁	—	2	1	no	—
156	Dic₇₁	—	2	1	no	—
157	Dic₇₁	—	4	1	no	—
158	Dic₇₁	—	4	1	no	—
159	Dic₇₁	—	4	1	no	—
160	Dic₇₁	#137	1	2	yes	—
161	C₂×C₇₁	—	8	1	no	—
162	C₂×C₇₁	#548	1	1	yes	characteristic
163	C₂×C₇₁	C₂×#137	1	1	yes	characteristic
164	C₂×C₇₁	#576	1	1	yes	characteristic
165	D₇₁	—	8	1	no	—
166	#3375	—	71	1	no	maximal, Sylow
167	C₇₁	#3375	1	1	yes	characteristic, complemented, Sylow
168	#559	—	71	1	no	—
169	Q₈²	—	71	1	no	—
170	#489	—	71	1	no	—
171	#489	—	71	1	no	—
172	#486	—	71	1	no	—
173	#477	—	71	1	no	—
174	#474	—	71	1	no	—
175	#474	—	71	1	no	—
176	#464	—	71	1	no	—
177	#464	D₇₁	1	1	yes	characteristic, complemented
178	#460	—	71	1	no	—
179	#446	—	71	1	no	—
180	#440	—	71	1	no	—
181	#440	—	71	1	no	—
182	C₄×Q₁₆	—	71	1	no	—
183	C₂×Q₁₆	C₂×D₇₁	1	1	yes	characteristic
184	C₂×SD₁₆	—	71	1	no	—
185	C₂×SD₁₆	C₂×D₇₁	1	1	yes	characteristic
186	C₄⋊Q₈	—	71	1	no	—
187	C₄⋊Q₈	—	71	1	no	—
188	C₄⋊Q₈	—	71	1	no	—
189	C₄⋊Q₈	—	142	1	no	—
190	C₄⋊Q₈	—	142	1	no	—
191	C₄⋊Q₈	—	142	1	no	—
192	#127	—	71	1	no	—
193	#124	—	71	1	no	—
194	#124	—	142	1	no	—
195	#124	C₂×D₇₁	1	1	yes	characteristic
196	#123	—	142	1	no	—
197	#123	—	142	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	C₂×D₇₁	1	1	yes	characteristic
204	C₄×D₄	—	71	1	no	—
205	C₄×D₄	—	71	1	no	—
206	C₄×D₄	—	142	1	no	—
207	#107	—	71	1	no	—
208	#107	—	71	1	no	—
209	#107	—	71	1	no	—
210	C₄⋊C₈	—	71	1	no	—
211	C₄⋊C₈	—	71	1	no	—
212	C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
213	#103	—	71	1	no	—
214	#103	—	71	1	no	—
215	#103	—	71	1	no	—
216	#103	C₂×D₇₁	1	1	yes	characteristic
217	#102	—	71	1	no	—
218	#102	—	71	1	no	—
219	#102	—	71	1	no	—
220	#102	—	71	1	no	—
221	#102	—	71	1	no	—
222	#102	C₂×D₇₁	1	1	yes	characteristic
223	#97	—	71	1	no	—
224	#97	—	71	1	no	—
225	C₄×C₈	—	71	1	no	—
226	C₂×Q₈	—	71	1	no	—
227	C₂×Q₈	—	142	1	no	—
228	C₂×Q₈	—	142	1	no	—
229	C₂×Q₈	—	142	1	no	—
230	C₂×Q₈	C₂²×D₇₁	1	1	yes	characteristic
231	C₂×Q₈	C₂²×D₇₁	1	1	yes	characteristic
232	C₂×D₄	—	71	1	no	—
233	C₂×D₄	—	142	1	no	—
234	C₂×D₄	—	142	1	no	—
235	C₂×D₄	C₂²×D₇₁	1	1	yes	characteristic
236	C₂²×C₄	—	142	1	no	—
237	C₂²×C₄	—	142	1	no	—
238	Q₁₆	—	2	2	no	—
239	SD₁₆	—	2	2	no	—
240	SD₁₆	—	142	2	no	—
241	C₂×C₈	—	71	1	no	—
242	C₂×C₈	—	71	1	no	—
243	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
244	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
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	—	71	1	no	—
250	C₄⋊C₄	—	71	1	no	—
251	C₄⋊C₄	—	71	1	no	—
252	C₄⋊C₄	—	142	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	C₂²×D₇₁	1	1	yes	characteristic
259	C₂²⋊C₄	—	2	1	no	—
260	C₂²⋊C₄	—	2	1	no	—
261	C₂²⋊C₄	—	142	1	no	—
262	C₂²⋊C₄	—	142	1	no	—
263	C₂²⋊C₄	—	142	1	no	—
264	C₂²⋊C₄	—	142	1	no	—
265	C₄²	—	2	1	no	—
266	C₄²	—	71	1	no	—
267	C₄²	—	71	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	—	142	1	no	—
274	Q₈	—	2	1	no	—
275	Q₈	—	2	1	no	—
276	Q₈	—	2	1	no	—
277	Q₈	—	568	1	no	—
278	Q₈	—	568	1	no	—
279	Q₈	—	568	1	no	—
280	Q₈	—	142	2	no	—
281	Q₈	#11765864	1	2	yes	—
282	D₄	—	2	1	no	—
283	D₄	—	2	1	no	—
284	D₄	—	284	2	no	—
285	D₄	—	568	1	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	—	568	1	no	—
292	C₂×C₄	—	568	1	no	—
293	C₂×C₄	—	71	1	no	—
294	C₂×C₄	—	71	1	no	—
295	C₂×C₄	—	71	1	no	—
296	C₂×C₄	—	71	1	no	—
297	C₂×C₄	—	142	1	no	—
298	C₂×C₄	—	142	1	no	—
299	C₂×C₄	—	142	1	no	—
300	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
301	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
302	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
303	C₈	—	2	1	no	—
304	C₈	—	2	1	no	—
305	C₈	—	142	1	no	—
306	C₈	—	142	1	no	—
307	C₂²	—	4	1	no	—
308	C₂²	—	4	1	no	—
309	C₂²	—	4	1	no	—
310	C₂²	—	284	1	no	—
311	C₂²	—	284	1	no	—
312	C₂²	—	284	1	no	—
313	C₂²	C₂×D₄×D₇₁	1	1	yes	characteristic
314	C₄	—	2	2	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	1	no	—
317	C₄	—	4	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	284	1	no	—
320	C₄	—	284	1	no	—
321	C₄	—	284	1	no	—
322	C₄	—	71	2	no	—
323	C₄	—	142	1	no	—
324	C₄	—	142	1	no	—
325	C₄	—	142	1	no	—
326	C₄	C₂×#11765864	1	1	yes	characteristic
327	C₄	#14556175	1	1	yes	characteristic
328	C₂	—	8	1	no	—
329	C₂	—	568	1	no	—
330	C₂	#17986655	1	1	yes	characteristic
331	C₂	#137×D₇₁	1	1	yes	characteristic
332	C₂	#17986801	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₁
Fitting	#464×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₄
hypercenter	#464	D₇₁
hypocenter	C₇₁	#3375
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #464

Complements

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

N	H ≅ G/N	type
2. Q₈×Dic₁₄₂	C₂	semidirect
3. #17986360	C₂	semidirect
4. #17986358	C₂	semidirect
6. #17986292	C₂	semidirect
7. #17986280	C₂	semidirect
8. #17986277	C₂	semidirect
9. #17986223	C₂	semidirect
11. #17986128	C₂	semidirect
12. #17986100	C₂	semidirect
13. #17986097	C₂	semidirect
14. Q₁₆×Dic₇₁	C₂	semidirect
15. #464×C₇₁	C₂	semidirect
167. C₇₁	#3375	semidirect
177. #464	D₇₁	semidirect