Subgroups — #22601092

Order 9088.

358 conjugacy classes of subgroups.

The subgroup lattice has 342 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, complemented
2	#17986710	C₂	1	1	yes	maximal, characteristic, complemented
3	#17986362	C₂	1	1	yes	maximal, characteristic, complemented
4	#17986355	C₂	1	1	yes	maximal, characteristic, complemented
5	#17986304	C₂	1	1	yes	maximal, characteristic, complemented
6	#17986301	C₂	1	1	yes	maximal, characteristic, complemented
7	#17986276	C₂	1	1	yes	maximal, characteristic, complemented
8	#17986217	C₂	1	1	yes	maximal, characteristic, complemented
9	#17986211	C₂	1	1	yes	maximal, characteristic, complemented
10	#17986193	C₂	1	1	yes	maximal, characteristic
11	#17986127	C₂	1	1	yes	maximal, characteristic, complemented
12	#17986112	C₂	1	1	yes	maximal, characteristic, complemented
13	#17986102	C₂	1	1	yes	maximal, characteristic, complemented
14	#17986079	C₂	1	1	yes	maximal, characteristic, complemented
15	#489×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11765852	C₂²	1	1	yes	characteristic
17	C₂×D₅₆₈	C₂²	1	1	yes	characteristic
18	C₂×#11765850	C₂²	1	1	yes	characteristic
19	#14556124	C₂²	1	1	yes	characteristic
20	#14556119	C₂²	1	1	yes	characteristic
21	#14556119	C₂²	1	1	yes	characteristic
22	#14556117	C₂²	1	1	yes	characteristic
23	#14556114	—	2	1	no	—
24	#14556112	—	2	1	no	—
25	#14556110	—	2	1	no	—
26	#14556108	—	2	1	no	—
27	#14556104	—	2	1	no	—
28	#14556104	—	2	1	no	—
29	Q₈×Dic₇₁	C₂²	1	1	yes	characteristic
30	C₄×Dic₁₄₂	C₂²	1	1	yes	characteristic
31	C₄×D₂₈₄	C₂²	1	1	yes	characteristic
32	C₄×D₂₈₄	C₂²	1	1	yes	characteristic
33	D₄×Dic₇₁	C₂²	1	1	yes	characteristic
34	#14556090	C₂²	1	1	yes	characteristic
35	#14556090	C₂²	1	1	yes	characteristic
36	(C₂²⋊D₇₁)⋊C₄	—	2	1	no	—
37	C₄⋊C₄×D₇₁	—	2	1	no	—
38	#14556061	C₂²	1	1	yes	characteristic
39	#14556061	C₂²	1	1	yes	characteristic
40	#14556060	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	#14556052	C₂²	1	1	yes	characteristic
45	#14556050	C₂²	1	1	yes	characteristic
46	#14556050	C₂²	1	1	yes	characteristic
47	#14556049	C₂²	1	1	yes	characteristic
48	#14556048	C₂²	1	1	yes	characteristic
49	#14556048	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	D₄	1	1	yes	characteristic
60	C₂×D₂₈₄	C₂³	1	1	yes	characteristic
61	C₂×D₂₈₄	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₇₁	—	2	1	no	—
63	C₂×C₂²⋊D₇₁	—	2	1	no	—
64	C₂²×Dic₇₁	—	2	1	no	—
65	C₂×C₄×D₇₁	—	2	1	no	—
66	C₂×C₄×D₇₁	—	2	1	no	—
67	C₂×C₄×D₇₁	—	2	1	no	—
68	C₂×C₄×D₇₁	—	2	1	no	—
69	#11765852	—	2	2	no	—
70	D₅₆₈	—	2	2	no	—
71	#11765850	—	2	2	no	—
72	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
73	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₇₁	—	2	1	no	—
75	C₄⋊Dic₇₁	—	2	1	no	—
76	C₄⋊Dic₇₁	—	2	1	no	—
77	C₄⋊Dic₇₁	D₄	1	1	yes	characteristic
78	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
79	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
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	(C₂×D₇₁)⋊C₄	—	2	1	no	—
86	(C₂×D₇₁)⋊C₄	—	2	1	no	—
87	(C₂×D₇₁)⋊C₄	—	2	1	no	—
88	(C₂×D₇₁)⋊C₄	—	2	1	no	—
89	(C₂×D₇₁)⋊C₄	—	2	1	no	—
90	(C₂×D₇₁)⋊C₄	—	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	C₂²×D₇₁	—	2	1	no	—
107	C₂²×D₇₁	—	2	1	no	—
108	Q₈×C₇₁	—	2	1	no	—
109	Q₈×C₇₁	—	2	1	no	—
110	Q₈×C₇₁	—	8	1	no	—
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	—	8	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	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
128	D₂₈₄	—	2	1	no	—
129	D₂₈₄	—	2	1	no	—
130	D₂₈₄	—	2	1	no	—
131	D₂₈₄	#41	1	2	yes	—
132	C₄×D₇₁	—	4	2	no	—
133	C₄×D₇₁	—	8	1	no	—
134	C₄×D₇₁	—	8	1	no	—
135	C₄×D₇₁	—	8	1	no	—
136	Dic₁₄₂	—	4	1	no	—
137	C₈×C₇₁	—	2	1	no	—
138	C₈×C₇₁	—	2	1	no	—
139	C₇₁⋊C₈	—	2	1	no	—
140	C₇₁⋊C₈	—	2	1	no	—
141	C₂²×C₇₁	—	4	1	no	—
142	C₂²×C₇₁	—	4	1	no	—
143	C₂²×C₇₁	—	4	1	no	—
144	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
145	C₂×D₇₁	—	2	2	no	—
146	C₂×D₇₁	—	4	1	no	—
147	C₂×D₇₁	—	4	1	no	—
148	C₂×D₇₁	—	4	1	no	—
149	C₂×D₇₁	—	4	1	no	—
150	C₂×D₇₁	—	4	1	no	—
151	C₄×C₇₁	—	2	1	no	—
152	C₄×C₇₁	—	2	1	no	—
153	C₄×C₇₁	—	4	1	no	—
154	C₄×C₇₁	—	4	1	no	—
155	C₄×C₇₁	—	4	1	no	—
156	C₄×C₇₁	C₂×#41	1	1	yes	characteristic
157	C₄×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
158	Dic₇₁	—	2	1	no	—
159	Dic₇₁	—	2	1	no	—
160	Dic₇₁	—	2	1	no	—
161	Dic₇₁	—	2	1	no	—
162	Dic₇₁	—	4	1	no	—
163	Dic₇₁	—	4	1	no	—
164	C₂×C₇₁	—	8	1	no	—
165	C₂×C₇₁	#548	1	1	yes	characteristic
166	C₂×C₇₁	#575	1	1	yes	characteristic
167	C₂×C₇₁	#576	1	1	yes	characteristic
168	D₇₁	—	4	2	no	—
169	D₇₁	—	8	1	no	—
170	#3370	—	71	1	no	maximal, Sylow
171	C₇₁	#3370	1	1	yes	characteristic, complemented, Sylow
172	#559	—	71	1	no	—
173	#554	—	71	1	no	—
174	#490	—	71	1	no	—
175	#489	D₇₁	1	1	yes	characteristic, complemented
176	#488	—	71	1	no	—
177	#479	—	71	1	no	—
178	#478	—	71	1	no	—
179	#474	—	71	1	no	—
180	#463	—	71	1	no	—
181	#463	—	71	1	no	—
182	#459	—	71	1	no	—
183	#446	—	71	1	no	—
184	#442	—	71	1	no	—
185	#440	—	71	1	no	—
186	C₄×D₈	—	71	1	no	—
187	C₂×SD₁₆	—	71	1	no	—
188	C₂×D₈	—	71	1	no	—
189	C₂×D₈	—	71	1	no	—
190	C₄⋊Q₈	C₂×D₇₁	1	1	yes	characteristic
191	#127	—	71	1	no	—
192	#125	—	71	1	no	—
193	#125	—	71	1	no	—
194	#124	—	71	1	no	—
195	#124	—	142	1	no	—
196	#124	C₂×D₇₁	1	1	yes	characteristic
197	#123	—	142	1	no	—
198	#123	—	142	1	no	—
199	C₂²⋊Q₈	—	142	1	no	—
200	C₂²⋊Q₈	—	142	1	no	—
201	C₂²⋊Q₈	—	142	1	no	—
202	C₄×Q₈	—	71	1	no	—
203	C₄×Q₈	—	71	1	no	—
204	C₄×D₄	—	71	1	no	—
205	C₄×D₄	—	71	1	no	—
206	C₄×D₄	—	71	1	no	—
207	C₄×D₄	—	71	1	no	—
208	C₄×D₄	—	71	1	no	—
209	C₄×D₄	—	142	1	no	—
210	C₂×C₄⋊C₄	—	142	1	no	—
211	#107	—	71	1	no	—
212	#107	—	71	1	no	—
213	#106	—	71	1	no	—
214	C₄⋊C₈	—	71	1	no	—
215	C₄⋊C₈	—	71	1	no	—
216	C₄⋊C₈	—	71	1	no	—
217	#103	—	71	1	no	—
218	#103	C₂×D₇₁	1	1	yes	characteristic
219	#103	C₂×D₇₁	1	1	yes	characteristic
220	#102	—	71	1	no	—
221	#102	—	71	1	no	—
222	#102	—	71	1	no	—
223	#102	—	71	1	no	—
224	#102	—	71	1	no	—
225	#102	C₂×D₇₁	1	1	yes	characteristic
226	#102	C₂×D₇₁	1	1	yes	characteristic
227	#97	—	71	1	no	—
228	#97	C₂×D₇₁	1	1	yes	characteristic
229	C₄×C₈	—	71	1	no	—
230	C₂×Q₈	—	2	1	no	—
231	C₂×Q₈	—	71	1	no	—
232	C₂×Q₈	C₂²×D₇₁	1	1	yes	characteristic
233	C₂×D₄	—	71	1	no	—
234	C₂×D₄	—	71	1	no	—
235	C₂×D₄	—	142	1	no	—
236	C₂×D₄	—	142	1	no	—
237	C₂×D₄	C₂²×D₇₁	1	1	yes	characteristic
238	C₂²×C₄	—	142	1	no	—
239	C₂²×C₄	—	142	1	no	—
240	C₂²×C₄	—	142	1	no	—
241	C₂²×C₄	—	142	1	no	—
242	C₂²×C₄	—	142	1	no	—
243	SD₁₆	—	142	2	no	—
244	D₈	—	142	2	no	—
245	D₈	—	142	2	no	—
246	C₂×C₈	—	71	1	no	—
247	C₂×C₈	—	71	1	no	—
248	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
249	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
250	C₄⋊C₄	—	2	1	no	—
251	C₄⋊C₄	—	71	1	no	—
252	C₄⋊C₄	—	71	1	no	—
253	C₄⋊C₄	—	71	1	no	—
254	C₄⋊C₄	—	71	1	no	—
255	C₄⋊C₄	—	71	1	no	—
256	C₄⋊C₄	—	142	1	no	—
257	C₄⋊C₄	—	142	1	no	—
258	C₄⋊C₄	—	142	1	no	—
259	C₄⋊C₄	—	142	1	no	—
260	C₄⋊C₄	—	142	1	no	—
261	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
262	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
263	C₂²⋊C₄	—	2	1	no	—
264	C₂²⋊C₄	—	2	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	C₂²⋊C₄	—	142	1	no	—
270	C₂²⋊C₄	—	142	1	no	—
271	C₂²⋊C₄	—	142	1	no	—
272	C₄²	—	71	1	no	—
273	C₄²	—	71	1	no	—
274	C₄²	—	142	1	no	—
275	C₄²	C₂²×D₇₁	1	1	yes	characteristic
276	C₂³	—	2	1	no	—
277	C₂³	—	142	1	no	—
278	C₂³	—	142	1	no	—
279	Q₈	—	2	1	no	—
280	Q₈	—	2	1	no	—
281	Q₈	—	8	1	no	—
282	Q₈	—	284	1	no	—
283	D₄	—	2	1	no	—
284	D₄	—	2	1	no	—
285	D₄	—	284	2	no	—
286	D₄	—	568	1	no	—
287	D₄	—	71	2	no	—
288	D₄	—	142	1	no	—
289	D₄	—	142	1	no	—
290	D₄	—	142	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	284	2	no	—
295	C₂×C₄	—	568	1	no	—
296	C₂×C₄	—	568	1	no	—
297	C₂×C₄	—	568	1	no	—
298	C₂×C₄	—	568	1	no	—
299	C₂×C₄	—	71	1	no	—
300	C₂×C₄	—	71	1	no	—
301	C₂×C₄	—	71	1	no	—
302	C₂×C₄	—	71	1	no	—
303	C₂×C₄	—	142	1	no	—
304	C₂×C₄	—	142	1	no	—
305	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
306	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
307	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
308	C₈	—	2	1	no	—
309	C₈	—	2	1	no	—
310	C₈	—	142	1	no	—
311	C₈	—	142	1	no	—
312	C₂²	—	4	1	no	—
313	C₂²	—	4	1	no	—
314	C₂²	—	4	1	no	—
315	C₂²	—	284	1	no	—
316	C₂²	—	284	1	no	—
317	C₂²	—	284	1	no	—
318	C₂²	—	284	1	no	—
319	C₂²	—	284	1	no	—
320	C₂²	—	142	2	no	—
321	C₂²	C₂×D₄×D₇₁	1	1	yes	characteristic
322	C₄	—	2	1	no	—
323	C₄	—	2	1	no	—
324	C₄	—	4	1	no	—
325	C₄	—	4	1	no	—
326	C₄	—	4	1	no	—
327	C₄	—	284	1	no	—
328	C₄	—	284	1	no	—
329	C₄	—	142	1	no	—
330	C₄	—	142	1	no	—
331	C₄	—	142	1	no	—
332	C₄	—	142	1	no	—
333	C₄	#41×D₇₁	1	1	yes	characteristic
334	C₄	#14556175	1	1	yes	characteristic
335	C₂	—	8	1	no	—
336	C₂	—	284	2	no	—
337	C₂	—	568	1	no	—
338	C₂	#17986657	1	1	yes	characteristic
339	C₂	#17986798	1	1	yes	characteristic
340	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	#489×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₄
hypercenter	#489	D₇₁
hypocenter	C₇₁	#3370
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. #17986731	C₂	semidirect
2. #17986710	C₂	semidirect
3. #17986362	C₂	semidirect
4. #17986355	C₂	semidirect
5. #17986304	C₂	semidirect
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15. #489×C₇₁	C₂	semidirect
171. C₇₁	#3370	semidirect
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