Subgroups — #22601077

Order 9088.

338 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17986729	C₂	1	1	yes	maximal, characteristic, complemented
2	#17986718	C₂	1	1	yes	maximal, characteristic
3	#17986357	C₂	1	1	yes	maximal, characteristic
4	#17986348	C₂	1	1	yes	maximal, characteristic, complemented
5	#17986287	C₂	1	1	yes	maximal, characteristic
6	#17986282	C₂	1	1	yes	maximal, characteristic, complemented
7	#17986281	C₂	1	1	yes	maximal, characteristic, complemented
8	#17986220	C₂	1	1	yes	maximal, characteristic
9	#17986218	C₂	1	1	yes	maximal, characteristic
10	#17986208	C₂	1	1	yes	maximal, characteristic, complemented
11	#17986127	C₂	1	1	yes	maximal, characteristic, complemented
12	#17986107	C₂	1	1	yes	maximal, characteristic, complemented
13	#17986103	C₂	1	1	yes	maximal, characteristic, complemented
14	#17986090	C₂	1	1	yes	maximal, characteristic
15	#489×C₇₁	C₂	1	1	yes	maximal, characteristic
16	C₂×Dic₂₈₄	C₂²	1	1	yes	characteristic
17	C₂×#11765855	C₂²	1	1	yes	characteristic
18	C₂×#11765853	C₂²	1	1	yes	characteristic
19	#14556127	—	2	1	no	—
20	#14556127	C₂²	1	1	yes	characteristic
21	#14556127	C₂²	1	1	yes	characteristic
22	#14556126	—	2	1	no	—
23	#14556126	—	2	1	no	—
24	#14556126	C₂²	1	1	yes	characteristic
25	#14556121	—	2	1	no	—
26	#14556121	—	2	1	no	—
27	#14556115	C₂²	1	1	yes	characteristic
28	#14556105	—	2	1	no	—
29	Q₈×Dic₇₁	C₂²	1	1	yes	characteristic
30	C₄×Dic₁₄₂	C₂²	1	1	yes	characteristic
31	C₄×Dic₁₄₂	C₂²	1	1	yes	characteristic
32	C₄×Dic₁₄₂	C₂²	1	1	yes	characteristic
33	#14556093	—	2	1	no	—
34	#14556093	C₂²	1	1	yes	characteristic
35	#14556093	C₂²	1	1	yes	characteristic
36	D₄×Dic₇₁	C₂²	1	1	yes	characteristic
37	#14556084	—	2	1	no	—
38	#14556060	C₂²	1	1	yes	characteristic
39	#14556059	C₂²	1	1	yes	characteristic
40	#14556059	C₂²	1	1	yes	characteristic
41	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
42	Dic₇₁⋊C₈	C₂²	1	1	yes	characteristic
43	Dic₇₁⋊C₈	C₂²	1	1	yes	characteristic
44	#14556053	C₂²	1	1	yes	characteristic
45	#14556053	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	#14556049	C₂²	1	1	yes	characteristic
50	#14556038	C₂²	1	1	yes	characteristic
51	C₈×Dic₇₁	C₂²	1	1	yes	characteristic
52	C₄⋊Q₈×C₇₁	C₂²	1	1	yes	characteristic
53	#124×C₇₁	C₂²	1	1	yes	characteristic
54	#103×C₇₁	C₂²	1	1	yes	characteristic
55	#103×C₇₁	C₂²	1	1	yes	characteristic
56	#102×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	D₄	1	1	yes	characteristic
62	C₂×Dic₁₄₂	C₂³	1	1	yes	characteristic
63	C₂×Dic₁₄₂	C₂³	1	1	yes	characteristic
64	C₂²×Dic₇₁	—	2	1	no	—
65	Dic₂₈₄	—	2	2	no	—
66	#11765855	—	2	2	no	—
67	#11765853	—	2	2	no	—
68	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
69	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
70	C₄⋊Dic₇₁	—	2	1	no	—
71	C₄⋊Dic₇₁	D₄	1	1	yes	characteristic
72	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
74	Dic₇₁⋊C₄	—	2	1	no	—
75	Dic₇₁⋊C₄	—	2	1	no	—
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	D₄	1	1	yes	characteristic
83	Dic₇₁⋊C₄	D₄	1	1	yes	characteristic
84	C₂²⋊Dic₇₁	—	2	1	no	—
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	—	2	1	no	—
90	C₄×Dic₇₁	—	2	1	no	—
91	C₄×Dic₇₁	—	2	1	no	—
92	C₄×Dic₇₁	C₂³	1	1	yes	characteristic
93	C₄×Dic₇₁	C₂³	1	1	yes	characteristic
94	C₂×Q₈×C₇₁	—	2	1	no	—
95	C₂×Q₈×C₇₁	C₂³	1	1	yes	characteristic
96	C₂×D₄×C₇₁	C₂³	1	1	yes	characteristic
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	C₂²⋊C₄×C₇₁	—	2	1	no	—
103	C₂²⋊C₄×C₇₁	—	2	1	no	—
104	C₄²×C₇₁	C₂³	1	1	yes	characteristic
105	C₂³×C₇₁	—	2	1	no	—
106	Q₈×C₇₁	—	2	1	no	—
107	Q₈×C₇₁	—	2	1	no	—
108	Q₈×C₇₁	—	8	1	no	—
109	D₄×C₇₁	—	2	1	no	—
110	D₄×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	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
116	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
117	C₂×Dic₇₁	—	2	1	no	—
118	C₂×Dic₇₁	—	2	1	no	—
119	C₂×Dic₇₁	—	2	1	no	—
120	C₂×Dic₇₁	—	2	1	no	—
121	C₂×Dic₇₁	—	4	2	no	—
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	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
126	Dic₁₄₂	—	2	1	no	—
127	Dic₁₄₂	—	2	2	no	—
128	Dic₁₄₂	—	2	1	no	—
129	Dic₁₄₂	—	2	1	no	—
130	Dic₁₄₂	—	8	1	no	—
131	Dic₁₄₂	—	8	1	no	—
132	Dic₁₄₂	#41	1	2	yes	—
133	C₈×C₇₁	—	2	1	no	—
134	C₈×C₇₁	—	2	1	no	—
135	C₇₁⋊C₈	—	2	1	no	—
136	C₇₁⋊C₈	—	2	1	no	—
137	C₂²×C₇₁	—	4	1	no	—
138	C₂²×C₇₁	—	4	1	no	—
139	C₂²×C₇₁	—	4	1	no	—
140	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
141	C₄×C₇₁	—	2	1	no	—
142	C₄×C₇₁	—	2	1	no	—
143	C₄×C₇₁	—	4	1	no	—
144	C₄×C₇₁	—	4	1	no	—
145	C₄×C₇₁	—	4	1	no	—
146	C₄×C₇₁	C₂×#41	1	1	yes	characteristic
147	C₄×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
148	Dic₇₁	—	2	2	no	—
149	Dic₇₁	—	2	1	no	—
150	Dic₇₁	—	2	1	no	—
151	Dic₇₁	—	2	1	no	—
152	Dic₇₁	—	4	1	no	—
153	Dic₇₁	—	4	1	no	—
154	Dic₇₁	—	4	1	no	—
155	Dic₇₁	#137	1	2	yes	—
156	C₂×C₇₁	—	8	1	no	—
157	C₂×C₇₁	#548	1	1	yes	characteristic
158	C₂×C₇₁	C₂×#137	1	1	yes	characteristic
159	C₂×C₇₁	#577	1	1	yes	characteristic
160	#3369	—	71	1	no	maximal, Sylow
161	C₇₁	#3369	1	1	yes	characteristic, complemented, Sylow
162	Q₈²	—	71	1	no	—
163	#556	—	71	1	no	—
164	#489	—	71	1	no	—
165	#489	D₇₁	1	1	yes	characteristic
166	#487	—	71	1	no	—
167	#476	—	71	1	no	—
168	#475	—	71	1	no	—
169	#475	—	71	1	no	—
170	#464	—	71	1	no	—
171	#464	—	71	1	no	—
172	#462	—	71	1	no	—
173	#446	—	71	1	no	—
174	#441	—	71	1	no	—
175	#441	—	71	1	no	—
176	C₄×SD₁₆	—	71	1	no	—
177	C₂×Q₁₆	—	71	1	no	—
178	C₂×Q₁₆	—	71	1	no	—
179	C₂×SD₁₆	—	71	1	no	—
180	C₄⋊Q₈	—	71	1	no	—
181	C₄⋊Q₈	—	71	1	no	—
182	C₄⋊Q₈	—	71	1	no	—
183	C₄⋊Q₈	—	142	1	no	—
184	C₄⋊Q₈	—	142	1	no	—
185	C₄⋊Q₈	—	142	1	no	—
186	C₄⋊Q₈	C₂×D₇₁	1	1	yes	characteristic
187	#126	—	142	1	no	—
188	#126	—	142	1	no	—
189	#124	—	71	1	no	—
190	#124	C₂×D₇₁	1	1	yes	characteristic
191	C₂²⋊Q₈	—	142	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	—	71	1	no	—
197	C₄×Q₈	—	71	1	no	—
198	C₄×Q₈	—	142	1	no	—
199	C₄×D₄	—	71	1	no	—
200	#117	—	142	1	no	—
201	#106	—	71	1	no	—
202	#106	—	71	1	no	—
203	#106	—	71	1	no	—
204	C₄⋊C₈	—	71	1	no	—
205	C₄⋊C₈	—	71	1	no	—
206	C₄⋊C₈	—	71	1	no	—
207	#103	—	71	1	no	—
208	#103	—	71	1	no	—
209	#103	—	71	1	no	—
210	#103	—	71	1	no	—
211	#103	—	71	1	no	—
212	#103	C₂×D₇₁	1	1	yes	characteristic
213	#103	C₂×D₇₁	1	1	yes	characteristic
214	#102	—	71	1	no	—
215	#102	C₂×D₇₁	1	1	yes	characteristic
216	#102	C₂×D₇₁	1	1	yes	characteristic
217	#97	—	71	1	no	—
218	#97	C₂×D₇₁	1	1	yes	characteristic
219	C₄×C₈	—	71	1	no	—
220	C₂×Q₈	—	2	1	no	—
221	C₂×Q₈	—	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	1	no	—
226	C₂×Q₈	C₂²×D₇₁	1	1	yes	characteristic
227	C₂×D₄	C₂²×D₇₁	1	1	yes	characteristic
228	C₂²×C₄	—	142	1	no	—
229	Q₁₆	—	142	2	no	—
230	Q₁₆	—	142	2	no	—
231	SD₁₆	—	142	2	no	—
232	C₂×C₈	—	71	1	no	—
233	C₂×C₈	—	71	1	no	—
234	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
235	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
236	C₄⋊C₄	—	2	1	no	—
237	C₄⋊C₄	—	71	1	no	—
238	C₄⋊C₄	—	71	1	no	—
239	C₄⋊C₄	—	71	1	no	—
240	C₄⋊C₄	—	71	1	no	—
241	C₄⋊C₄	—	71	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	—	142	1	no	—
251	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
252	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
253	C₂²⋊C₄	—	2	1	no	—
254	C₂²⋊C₄	—	2	1	no	—
255	C₂²⋊C₄	—	142	1	no	—
256	C₂²⋊C₄	—	142	1	no	—
257	C₂²⋊C₄	—	142	1	no	—
258	C₄²	—	71	1	no	—
259	C₄²	—	71	1	no	—
260	C₄²	—	142	1	no	—
261	C₄²	—	142	1	no	—
262	C₄²	—	142	1	no	—
263	C₄²	—	142	1	no	—
264	C₄²	—	142	1	no	—
265	C₄²	C₂²×D₇₁	1	1	yes	characteristic
266	C₂³	—	2	1	no	—
267	Q₈	—	2	1	no	—
268	Q₈	—	2	1	no	—
269	Q₈	—	8	1	no	—
270	Q₈	—	568	1	no	—
271	Q₈	—	568	1	no	—
272	Q₈	—	71	2	no	—
273	Q₈	—	142	2	no	—
274	Q₈	—	142	1	no	—
275	Q₈	—	142	1	no	—
276	Q₈	—	142	1	no	—
277	D₄	—	2	1	no	—
278	D₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	284	2	no	—
283	C₂×C₄	—	71	1	no	—
284	C₂×C₄	—	71	1	no	—
285	C₂×C₄	—	71	1	no	—
286	C₂×C₄	—	71	1	no	—
287	C₂×C₄	—	142	1	no	—
288	C₂×C₄	—	142	1	no	—
289	C₂×C₄	—	142	1	no	—
290	C₂×C₄	—	142	1	no	—
291	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
292	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
293	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
294	C₈	—	2	1	no	—
295	C₈	—	2	1	no	—
296	C₈	—	142	1	no	—
297	C₈	—	142	1	no	—
298	C₂²	—	4	1	no	—
299	C₂²	—	4	1	no	—
300	C₂²	—	4	1	no	—
301	C₂²	C₂×D₄×D₇₁	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	4	1	no	—
305	C₄	—	4	1	no	—
306	C₄	—	4	1	no	—
307	C₄	—	284	1	no	—
308	C₄	—	284	1	no	—
309	C₄	—	284	1	no	—
310	C₄	—	71	2	no	—
311	C₄	—	142	2	no	—
312	C₄	—	142	1	no	—
313	C₄	—	142	1	no	—
314	C₄	—	142	1	no	—
315	C₄	#41×D₇₁	1	1	yes	characteristic
316	C₄	#14556175	1	1	yes	characteristic
317	C₂	—	8	1	no	—
318	C₂	#17986657	1	1	yes	characteristic
319	C₂	#137×D₇₁	1	1	yes	characteristic
320	C₂	#17986809	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₁
Fitting	#489×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₄
hypercenter	#489	D₇₁
hypocenter	C₇₁	#3369
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #489

Complements

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

N	H ≅ G/N	type
1. #17986729	C₂	semidirect
4. #17986348	C₂	semidirect
6. #17986282	C₂	semidirect
7. #17986281	C₂	semidirect
10. #17986208	C₂	semidirect
11. #17986127	C₂	semidirect
12. #17986107	C₂	semidirect
13. #17986103	C₂	semidirect
161. C₇₁	#3369	semidirect