Subgroups — #22598639

Order 9088.

358 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17986607	C₂	1	1	yes	maximal, characteristic
2	#17986475	C₂	1	1	yes	maximal, characteristic
3	#17986343	C₂	1	1	yes	maximal, characteristic
4	#17986338	C₂	1	1	yes	maximal, characteristic
5	#17986309	C₂	1	1	yes	maximal, characteristic, complemented
6	#17986304	C₂	1	1	yes	maximal, characteristic, complemented
7	#17986208	C₂	1	1	yes	maximal, characteristic, complemented
8	#17986207	C₂	1	1	yes	maximal, characteristic, complemented
9	#17986156	C₂	1	1	yes	maximal, characteristic
10	#17986153	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×#11765855	C₂	1	1	yes	maximal, characteristic, complemented
14	#17986061	C₂	1	1	yes	maximal, characteristic
15	#441×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×#11765855	C₂²	1	1	yes	characteristic
20	#14556126	C₂²	1	1	yes	characteristic
21	#14556123	—	2	1	no	—
22	#14556122	—	2	1	no	—
23	#14556120	—	2	1	no	—
24	#14556119	C₂²	1	1	yes	characteristic
25	#14556118	—	2	1	no	—
26	#14556108	C₂²	1	1	yes	characteristic
27	#14556104	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	#14556061	C₂²	1	1	yes	characteristic
40	#14556060	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	#14556052	C₂²	1	1	yes	characteristic
47	#14556051	C₂²	1	1	yes	characteristic
48	#14556051	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×Q₁₆×C₇₁	C₂²	1	1	yes	characteristic
53	C₄×Q₈×C₇₁	C₂²	1	1	yes	characteristic
54	C₄×Q₈×C₇₁	C₂²	1	1	yes	characteristic
55	#106×C₇₁	C₂²	1	1	yes	characteristic
56	#103×C₇₁	C₂²	1	1	yes	characteristic
57	#103×C₇₁	C₂²	1	1	yes	characteristic
58	#97×C₇₁	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₄₂	—	2	1	no	—
60	C₂×Dic₁₄₂	C₂³	1	1	yes	characteristic
61	C₂×Dic₁₄₂	C₂³	1	1	yes	characteristic
62	Q₈×D₇₁	—	2	2	no	—
63	Q₈×D₇₁	—	2	2	no	—
64	C₂×C₄×D₇₁	—	2	1	no	—
65	C₂×C₄×D₇₁	C₂³	1	1	yes	characteristic
66	Dic₂₈₄	—	2	2	no	—
67	#11765855	—	2	2	no	—
68	#11765855	—	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	C₂³	1	1	yes	characteristic
75	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
76	Dic₇₁⋊C₄	—	2	1	no	—
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	(C₂×D₇₁)⋊C₄	—	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×Q₈×C₇₁	C₂³	1	1	yes	characteristic
98	Q₁₆×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	—	2	1	no	—
103	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
104	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
105	C₄²×C₇₁	—	2	1	no	—
106	C₄²×C₇₁	—	2	1	no	—
107	C₄²×C₇₁	C₂³	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	Q₈×C₇₁	—	2	1	no	—
112	Q₈×C₇₁	#41	1	2	yes	—
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	—	2	1	no	—
117	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
118	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
120	C₂×Dic₇₁	—	2	1	no	—
121	C₂×Dic₇₁	—	2	1	no	—
122	C₂×Dic₇₁	—	2	1	no	—
123	C₂×Dic₇₁	—	2	1	no	—
124	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
125	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
126	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
127	C₄×D₇₁	—	2	4	no	—
128	C₄×D₇₁	—	2	1	no	—
129	C₄×D₇₁	—	2	1	no	—
130	Dic₁₄₂	—	2	4	no	—
131	Dic₁₄₂	—	2	1	no	—
132	Dic₁₄₂	—	2	1	no	—
133	Dic₁₄₂	—	2	1	no	—
134	Dic₁₄₂	#41	1	2	yes	—
135	C₈×C₇₁	—	2	1	no	—
136	C₈×C₇₁	—	2	1	no	—
137	C₇₁⋊C₈	—	2	1	no	—
138	C₇₁⋊C₈	—	2	1	no	—
139	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
140	C₂×D₇₁	—	2	1	no	—
141	C₂×D₇₁	—	2	1	no	—
142	C₂×D₇₁	—	2	1	no	—
143	C₄×C₇₁	—	2	2	no	—
144	C₄×C₇₁	—	2	1	no	—
145	C₄×C₇₁	—	2	1	no	—
146	C₄×C₇₁	—	4	1	no	—
147	C₄×C₇₁	—	4	1	no	—
148	C₄×C₇₁	—	4	1	no	—
149	C₄×C₇₁	C₂×#41	1	1	yes	characteristic
150	C₄×C₇₁	C₂×#41	1	1	yes	characteristic
151	Dic₇₁	—	2	1	no	—
152	Dic₇₁	—	2	2	no	—
153	Dic₇₁	—	2	1	no	—
154	Dic₇₁	—	2	1	no	—
155	Dic₇₁	—	4	1	no	—
156	Dic₇₁	—	4	1	no	—
157	Dic₇₁	—	4	1	no	—
158	C₂×C₇₁	#525	1	1	yes	characteristic
159	C₂×C₇₁	#578	1	1	yes	characteristic
160	C₂×C₇₁	#578	1	1	yes	characteristic
161	D₇₁	—	4	1	no	—
162	#3127	—	71	1	no	maximal, Sylow
163	C₇₁	#3127	1	1	yes	characteristic, complemented, Sylow
164	#541	—	71	1	no	—
165	#519	—	71	1	no	—
166	#485	—	71	1	no	—
167	#485	—	71	1	no	—
168	#479	—	71	1	no	—
169	#479	—	71	1	no	—
170	#462	—	71	1	no	—
171	#462	—	71	1	no	—
172	#451	—	71	1	no	—
173	#451	—	71	1	no	—
174	#441	—	71	1	no	—
175	#441	D₇₁	1	1	yes	characteristic, complemented
176	C₄×Q₁₆	—	71	1	no	—
177	C₄×Q₁₆	—	71	1	no	—
178	#433	—	71	1	no	—
179	C₂²×Q₈	—	71	1	no	—
180	C₂×Q₁₆	—	71	1	no	—
181	C₂×Q₁₆	—	71	1	no	—
182	C₂×Q₁₆	—	71	1	no	—
183	C₂×Q₁₆	C₂×D₇₁	1	1	yes	characteristic
184	C₄⋊Q₈	—	71	1	no	—
185	#126	—	142	1	no	—
186	#126	—	142	1	no	—
187	#125	—	71	1	no	—
188	#125	—	142	1	no	—
189	#125	—	142	1	no	—
190	C₂²⋊Q₈	—	71	1	no	—
191	C₂²⋊Q₈	—	71	1	no	—
192	C₄×Q₈	—	71	1	no	—
193	C₄×Q₈	—	71	1	no	—
194	C₄×Q₈	—	71	1	no	—
195	C₄×Q₈	—	71	1	no	—
196	C₄×Q₈	—	142	1	no	—
197	C₄×Q₈	—	142	1	no	—
198	C₄×Q₈	C₂×D₇₁	1	1	yes	characteristic
199	C₄×Q₈	C₂×D₇₁	1	1	yes	characteristic
200	#117	—	71	1	no	—
201	#117	—	71	1	no	—
202	#117	—	142	1	no	—
203	#117	—	142	1	no	—
204	#107	—	71	1	no	—
205	#107	—	71	1	no	—
206	#106	—	71	1	no	—
207	#106	C₂×D₇₁	1	1	yes	characteristic
208	C₄⋊C₈	—	71	1	no	—
209	C₄⋊C₈	—	71	1	no	—
210	#103	—	71	1	no	—
211	#103	—	71	1	no	—
212	#103	—	71	1	no	—
213	#103	—	71	1	no	—
214	#103	—	71	1	no	—
215	#103	—	71	1	no	—
216	#103	C₂×D₇₁	1	1	yes	characteristic
217	#103	C₂×D₇₁	1	1	yes	characteristic
218	C₂²⋊C₈	—	71	1	no	—
219	C₂²⋊C₈	—	71	1	no	—
220	#97	C₂×D₇₁	1	1	yes	characteristic
221	C₄×C₈	—	71	1	no	—
222	C₂×Q₈	—	71	1	no	—
223	C₂×Q₈	—	71	1	no	—
224	C₂×Q₈	—	142	1	no	—
225	C₂×Q₈	—	142	2	no	—
226	C₂×Q₈	—	142	2	no	—
227	C₂×Q₈	C₂²×D₇₁	1	1	yes	characteristic
228	C₂×Q₈	C₂²×D₇₁	1	1	yes	characteristic
229	C₂²×C₄	—	71	1	no	—
230	C₂²×C₄	—	142	1	no	—
231	Q₁₆	—	2	2	no	—
232	Q₁₆	—	142	2	no	—
233	Q₁₆	—	142	2	no	—
234	Q₁₆	—	142	2	no	—
235	C₂×C₈	—	71	1	no	—
236	C₂×C₈	—	71	1	no	—
237	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
238	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	—	2	1	no	—
241	C₄⋊C₄	—	71	1	no	—
242	C₄⋊C₄	—	71	1	no	—
243	C₄⋊C₄	—	71	1	no	—
244	C₄⋊C₄	—	71	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	—	142	1	no	—
251	C₄⋊C₄	—	142	1	no	—
252	C₄⋊C₄	—	142	1	no	—
253	C₄⋊C₄	—	142	1	no	—
254	C₄⋊C₄	—	142	1	no	—
255	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
256	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
257	C₂²⋊C₄	—	71	1	no	—
258	C₂²⋊C₄	—	71	1	no	—
259	C₂²⋊C₄	—	142	1	no	—
260	C₂²⋊C₄	—	142	1	no	—
261	C₂²⋊C₄	—	142	1	no	—
262	C₄²	—	2	1	no	—
263	C₄²	—	2	1	no	—
264	C₄²	—	71	1	no	—
265	C₄²	—	142	1	no	—
266	C₄²	—	142	1	no	—
267	C₄²	—	142	1	no	—
268	C₄²	—	142	1	no	—
269	C₄²	—	142	1	no	—
270	C₄²	C₂²×D₇₁	1	1	yes	characteristic
271	C₂³	—	71	1	no	—
272	Q₈	—	2	1	no	—
273	Q₈	—	2	1	no	—
274	Q₈	—	2	1	no	—
275	Q₈	—	71	2	no	—
276	Q₈	—	142	4	no	—
277	Q₈	—	142	1	no	—
278	Q₈	—	142	1	no	—
279	Q₈	—	142	1	no	—
280	Q₈	#11765865	1	2	yes	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	71	1	no	—
286	C₂×C₄	—	71	1	no	—
287	C₂×C₄	—	71	1	no	—
288	C₂×C₄	—	142	4	no	—
289	C₂×C₄	—	142	1	no	—
290	C₂×C₄	—	142	1	no	—
291	C₂×C₄	—	142	1	no	—
292	C₂×C₄	—	142	1	no	—
293	C₂×C₄	—	142	1	no	—
294	C₂×C₄	—	142	1	no	—
295	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
296	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
297	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
298	C₈	—	2	1	no	—
299	C₈	—	2	1	no	—
300	C₈	—	142	1	no	—
301	C₈	—	142	1	no	—
302	C₂²	—	142	1	no	—
303	C₂²	—	142	1	no	—
304	C₂²	—	142	1	no	—
305	C₂²	C₂×D₄×D₇₁	1	1	yes	characteristic
306	C₄	—	2	2	no	—
307	C₄	—	2	1	no	—
308	C₄	—	2	1	no	—
309	C₄	—	4	1	no	—
310	C₄	—	4	1	no	—
311	C₄	—	4	1	no	—
312	C₄	—	284	1	no	—
313	C₄	—	284	1	no	—
314	C₄	—	284	1	no	—
315	C₄	—	142	1	no	—
316	C₄	—	142	2	no	—
317	C₄	—	142	1	no	—
318	C₄	—	142	1	no	—
319	C₄	#41×D₇₁	1	1	yes	characteristic
320	C₄	C₂×#11765865	1	1	yes	characteristic
321	C₂	—	284	1	no	—
322	C₂	#17986508	1	1	yes	characteristic
323	C₂	#17986815	1	1	yes	characteristic
324	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	#441×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₄
hypercenter	#441	D₇₁
hypocenter	C₇₁	#3127
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #441

Complements

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

N	H ≅ G/N	type
5. #17986309	C₂	semidirect
6. #17986304	C₂	semidirect
7. #17986208	C₂	semidirect
8. #17986207	C₂	semidirect
11. #17986107	C₂	semidirect
12. C₄×#11765855	C₂	semidirect
13. C₄×#11765855	C₂	semidirect
15. #441×C₇₁	C₂	semidirect
163. C₇₁	#3127	semidirect
175. #441	D₇₁	semidirect