Subgroups — #22601251

Order 9088.

340 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×Dic₁₄₂	C₂	1	1	yes	maximal, characteristic, complemented
2	#17986708	C₂	1	1	yes	maximal, characteristic
3	#17986408	C₂	1	1	yes	maximal, characteristic, complemented
4	#17986407	C₂	1	1	yes	maximal, characteristic, complemented
5	#17986403	C₂	1	1	yes	maximal, characteristic, complemented
6	#17986306	C₂	1	1	yes	maximal, characteristic, complemented
7	#17986304	C₂	1	1	yes	maximal, characteristic, complemented
8	#17986292	C₂	1	1	yes	maximal, characteristic, complemented
9	#17986287	C₂	1	1	yes	maximal, characteristic
10	#17986279	C₂	1	1	yes	maximal, characteristic
11	#17986277	C₂	1	1	yes	maximal, characteristic
12	#17986127	C₂	1	1	yes	maximal, characteristic, complemented
13	#17986100	C₂	1	1	yes	maximal, characteristic
14	SD₁₆×Dic₇₁	C₂	1	1	yes	maximal, characteristic
15	#440×C₇₁	C₂	1	1	yes	maximal, characteristic
16	#14556128	—	2	1	no	—
17	#14556128	C₂²	1	1	yes	characteristic
18	#14556127	—	2	1	no	—
19	#14556127	—	2	1	no	—
20	#14556127	C₂²	1	1	yes	characteristic
21	#14556127	C₂²	1	1	yes	characteristic
22	#14556127	C₂²	1	1	yes	characteristic
23	#14556120	C₂²	1	1	yes	characteristic
24	#14556119	C₂²	1	1	yes	characteristic
25	#14556107	—	2	1	no	—
26	#14556105	—	2	1	no	—
27	#14556105	—	2	1	no	—
28	Q₈×Dic₇₁	C₂²	1	1	yes	characteristic
29	Q₈×Dic₇₁	C₂²	1	1	yes	characteristic
30	C₄×Dic₁₄₂	—	2	1	no	—
31	C₄×Dic₁₄₂	C₂²	1	1	yes	characteristic
32	D₄×Dic₇₁	C₂²	1	1	yes	characteristic
33	D₄×Dic₇₁	C₂²	1	1	yes	characteristic
34	C₂×Dic₇₁⋊C₄	—	2	1	no	—
35	#14556062	C₂²	1	1	yes	characteristic
36	#14556062	C₂²	1	1	yes	characteristic
37	#14556061	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	#14556060	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	#14556052	C₂²	1	1	yes	characteristic
47	#14556052	C₂²	1	1	yes	characteristic
48	#14556049	C₂²	1	1	yes	characteristic
49	#14556049	C₂²	1	1	yes	characteristic
50	#14556038	C₂²	1	1	yes	characteristic
51	C₈×Dic₇₁	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	—	2	1	no	—
62	C₂×Dic₁₄₂	—	2	1	no	—
63	C₂×Dic₁₄₂	D₄	1	1	yes	characteristic
64	C₂²×Dic₇₁	—	2	1	no	—
65	C₂²×Dic₇₁	—	2	1	no	—
66	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
67	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
68	C₄⋊Dic₇₁	—	2	1	no	—
69	C₄⋊Dic₇₁	—	2	1	no	—
70	C₄⋊Dic₇₁	—	2	1	no	—
71	C₄⋊Dic₇₁	—	2	1	no	—
72	C₄⋊Dic₇₁	D₄	1	1	yes	characteristic
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	—	2	1	no	—
82	Dic₇₁⋊C₄	—	2	1	no	—
83	Dic₇₁⋊C₄	D₄	1	1	yes	characteristic
84	Dic₇₁⋊C₄	D₄	1	1	yes	characteristic
85	C₂²⋊Dic₇₁	—	2	1	no	—
86	C₂²⋊Dic₇₁	—	2	1	no	—
87	C₄×Dic₇₁	—	2	1	no	—
88	C₄×Dic₇₁	—	2	1	no	—
89	C₄×Dic₇₁	C₂³	1	1	yes	characteristic
90	C₄×Dic₇₁	C₂³	1	1	yes	characteristic
91	C₂×Q₈×C₇₁	C₂³	1	1	yes	characteristic
92	C₂×D₄×C₇₁	C₂³	1	1	yes	characteristic
93	C₂²×C₄×C₇₁	—	2	1	no	—
94	SD₁₆×C₇₁	Q₈	1	4	yes	complemented
95	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
96	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
97	C₄⋊C₄×C₇₁	—	2	1	no	—
98	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
99	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
100	C₂²⋊C₄×C₇₁	—	2	1	no	—
101	C₄²×C₇₁	—	2	1	no	—
102	C₄²×C₇₁	C₂³	1	1	yes	characteristic
103	C₂³×C₇₁	—	2	1	no	—
104	Q₈×C₇₁	—	2	1	no	—
105	Q₈×C₇₁	C₂×Q₈	1	2	yes	—
106	D₄×C₇₁	—	2	1	no	—
107	D₄×C₇₁	C₂×Q₈	1	2	yes	—
108	C₂×C₄×C₇₁	—	2	1	no	—
109	C₂×C₄×C₇₁	—	2	1	no	—
110	C₂×C₄×C₇₁	—	2	1	no	—
111	C₂×C₄×C₇₁	—	8	1	no	—
112	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
115	C₂×Dic₇₁	—	2	1	no	—
116	C₂×Dic₇₁	—	2	1	no	—
117	C₂×Dic₇₁	—	2	1	no	—
118	C₂×Dic₇₁	—	2	1	no	—
119	C₂×Dic₇₁	—	4	2	no	—
120	C₂×Dic₇₁	—	8	1	no	—
121	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
122	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
123	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
124	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
125	Dic₁₄₂	—	2	2	no	—
126	Dic₁₄₂	—	8	1	no	—
127	Dic₁₄₂	—	8	1	no	—
128	Dic₁₄₂	—	8	1	no	—
129	Dic₁₄₂	—	8	1	no	—
130	C₈×C₇₁	—	2	1	no	—
131	C₈×C₇₁	C₂×Q₈	1	2	yes	—
132	C₇₁⋊C₈	—	2	1	no	—
133	C₇₁⋊C₈	—	2	1	no	—
134	C₂²×C₇₁	—	2	2	no	—
135	C₂²×C₇₁	—	4	1	no	—
136	C₂²×C₇₁	—	4	1	no	—
137	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
138	C₄×C₇₁	—	2	2	no	—
139	C₄×C₇₁	—	2	1	no	—
140	C₄×C₇₁	—	2	1	no	—
141	C₄×C₇₁	—	4	1	no	—
142	C₄×C₇₁	—	4	1	no	—
143	C₄×C₇₁	C₂²×Q₈	1	1	yes	characteristic
144	C₄×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
145	Dic₇₁	—	2	1	no	—
146	Dic₇₁	—	2	1	no	—
147	Dic₇₁	—	2	1	no	—
148	Dic₇₁	—	4	1	no	—
149	Dic₇₁	—	4	1	no	—
150	Dic₇₁	—	4	1	no	—
151	Dic₇₁	—	4	1	no	—
152	Dic₇₁	#137	1	2	yes	—
153	C₂×C₇₁	—	4	2	no	—
154	C₂×C₇₁	D₄×Q₈	1	1	yes	characteristic
155	C₂×C₇₁	C₂×#137	1	1	yes	characteristic
156	C₂×C₇₁	#576	1	1	yes	characteristic
157	#3385	—	71	1	no	maximal, Sylow
158	C₇₁	#3385	1	1	yes	characteristic, complemented, Sylow
159	Q₈²	—	71	1	no	—
160	#554	—	71	1	no	—
161	#501	—	71	1	no	—
162	#501	—	71	1	no	—
163	#500	—	71	1	no	—
164	#479	—	71	1	no	—
165	#479	—	71	1	no	—
166	#477	—	71	1	no	—
167	#476	—	71	1	no	—
168	#474	—	71	1	no	—
169	#474	—	71	1	no	—
170	#446	—	71	1	no	—
171	#440	—	71	1	no	—
172	#440	D₇₁	1	1	yes	characteristic
173	C₄×SD₁₆	—	71	1	no	—
174	C₂×SD₁₆	C₂×D₇₁	1	1	yes	characteristic
175	C₄⋊Q₈	—	71	1	no	—
176	C₄⋊Q₈	—	71	1	no	—
177	C₄⋊Q₈	—	71	1	no	—
178	C₄⋊Q₈	—	71	1	no	—
179	C₄⋊Q₈	—	142	1	no	—
180	C₄⋊Q₈	—	142	1	no	—
181	C₄⋊Q₈	—	142	1	no	—
182	#125	—	71	1	no	—
183	#125	—	71	1	no	—
184	C₂²⋊Q₈	—	142	1	no	—
185	C₂²⋊Q₈	—	142	1	no	—
186	C₂²⋊Q₈	—	142	1	no	—
187	C₄×Q₈	—	71	1	no	—
188	C₄×Q₈	—	71	1	no	—
189	C₄×Q₈	—	71	1	no	—
190	C₄×Q₈	—	142	1	no	—
191	C₄×Q₈	C₂×D₇₁	1	1	yes	characteristic
192	C₄×D₄	—	71	1	no	—
193	C₄×D₄	—	71	1	no	—
194	C₄×D₄	C₂×D₇₁	1	1	yes	characteristic
195	C₂×C₄⋊C₄	—	142	1	no	—
196	#107	—	71	1	no	—
197	#107	—	71	1	no	—
198	#107	—	71	1	no	—
199	#107	—	71	1	no	—
200	#107	—	71	1	no	—
201	#107	C₂×D₇₁	1	1	yes	characteristic
202	#106	—	71	1	no	—
203	#106	—	71	1	no	—
204	#106	—	71	1	no	—
205	C₄⋊C₈	—	71	1	no	—
206	C₄⋊C₈	—	71	1	no	—
207	C₄⋊C₈	—	71	1	no	—
208	#103	—	71	1	no	—
209	#103	—	71	1	no	—
210	#103	C₂×D₇₁	1	1	yes	characteristic
211	#102	—	71	1	no	—
212	#102	—	71	1	no	—
213	#102	C₂×D₇₁	1	1	yes	characteristic
214	#97	—	71	1	no	—
215	#97	C₂×D₇₁	1	1	yes	characteristic
216	C₄×C₈	—	71	1	no	—
217	C₂×Q₈	—	71	1	no	—
218	C₂×Q₈	—	142	1	no	—
219	C₂×Q₈	—	142	1	no	—
220	C₂×Q₈	—	142	1	no	—
221	C₂×Q₈	—	142	1	no	—
222	C₂×Q₈	C₂²×D₇₁	1	1	yes	characteristic
223	C₂×D₄	C₂²×D₇₁	1	1	yes	characteristic
224	C₂²×C₄	—	2	1	no	—
225	C₂²×C₄	—	142	1	no	—
226	C₂²×C₄	—	142	1	no	—
227	SD₁₆	Dic₁₄₂	1	4	yes	complemented
228	C₂×C₈	—	71	1	no	—
229	C₂×C₈	—	71	1	no	—
230	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
231	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
232	C₄⋊C₄	—	2	1	no	—
233	C₄⋊C₄	—	71	1	no	—
234	C₄⋊C₄	—	71	1	no	—
235	C₄⋊C₄	—	71	1	no	—
236	C₄⋊C₄	—	71	1	no	—
237	C₄⋊C₄	—	71	1	no	—
238	C₄⋊C₄	—	71	1	no	—
239	C₄⋊C₄	—	71	1	no	—
240	C₄⋊C₄	—	142	1	no	—
241	C₄⋊C₄	—	142	1	no	—
242	C₄⋊C₄	—	142	1	no	—
243	C₄⋊C₄	—	142	1	no	—
244	C₄⋊C₄	—	142	1	no	—
245	C₄⋊C₄	—	142	1	no	—
246	C₄⋊C₄	—	142	1	no	—
247	C₄⋊C₄	—	142	1	no	—
248	C₄⋊C₄	—	142	1	no	—
249	C₄⋊C₄	—	142	1	no	—
250	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
251	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
252	C₂²⋊C₄	—	2	1	no	—
253	C₂²⋊C₄	—	142	1	no	—
254	C₂²⋊C₄	—	142	1	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	71	1	no	—
257	C₄²	—	71	1	no	—
258	C₄²	—	142	1	no	—
259	C₄²	—	142	1	no	—
260	C₄²	C₂²×D₇₁	1	1	yes	characteristic
261	C₂³	—	2	1	no	—
262	Q₈	—	2	1	no	—
263	Q₈	—	568	1	no	—
264	Q₈	—	568	1	no	—
265	Q₈	—	568	1	no	—
266	Q₈	—	568	1	no	—
267	Q₈	—	142	2	no	—
268	Q₈	C₂×Dic₁₄₂	1	2	yes	—
269	D₄	—	2	1	no	—
270	D₄	C₂×Dic₁₄₂	1	2	yes	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	8	1	no	—
275	C₂×C₄	—	284	2	no	—
276	C₂×C₄	—	568	1	no	—
277	C₂×C₄	—	71	1	no	—
278	C₂×C₄	—	71	1	no	—
279	C₂×C₄	—	71	1	no	—
280	C₂×C₄	—	71	1	no	—
281	C₂×C₄	—	142	1	no	—
282	C₂×C₄	—	142	1	no	—
283	C₂×C₄	—	142	1	no	—
284	C₂×C₄	—	142	1	no	—
285	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
286	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
287	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
288	C₈	—	2	1	no	—
289	C₈	—	142	1	no	—
290	C₈	—	142	1	no	—
291	C₈	C₂×Dic₁₄₂	1	2	yes	—
292	C₂²	—	2	2	no	—
293	C₂²	—	4	1	no	—
294	C₂²	—	4	1	no	—
295	C₂²	C₂×D₄×D₇₁	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	4	1	no	—
300	C₄	—	4	1	no	—
301	C₄	—	284	1	no	—
302	C₄	—	284	1	no	—
303	C₄	—	284	1	no	—
304	C₄	—	284	1	no	—
305	C₄	—	71	2	no	—
306	C₄	—	142	1	no	—
307	C₄	—	142	1	no	—
308	C₄	—	142	1	no	—
309	C₄	C₂²×Dic₁₄₂	1	1	yes	characteristic
310	C₄	#14556175	1	1	yes	characteristic
311	C₂	—	4	2	no	—
312	C₂	D₄×Dic₁₄₂	1	1	yes	characteristic
313	C₂	#137×D₇₁	1	1	yes	characteristic
314	C₂	#17986803	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	#3385
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
1. Q₈×Dic₁₄₂	C₂	semidirect
3. #17986408	C₂	semidirect
4. #17986407	C₂	semidirect
5. #17986403	C₂	semidirect
6. #17986306	C₂	semidirect
7. #17986304	C₂	semidirect
8. #17986292	C₂	semidirect
12. #17986127	C₂	semidirect
94. SD₁₆×C₇₁	Q₈	semidirect
158. C₇₁	#3385	semidirect
227. SD₁₆	Dic₁₄₂	semidirect