Subgroups — #22598909

Order 9088.

380 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17986534	C₂	1	1	yes	maximal, characteristic, complemented
2	#17986528	C₂	1	1	yes	maximal, characteristic, complemented
3	#17986398	C₂	1	1	yes	maximal, characteristic, complemented
4	#17986381	C₂	1	1	yes	maximal, characteristic, complemented
5	#17986354	C₂	1	1	yes	maximal, characteristic, complemented
6	#17986239	C₂	1	1	yes	maximal, characteristic, complemented
7	#17986234	C₂	1	1	yes	maximal, characteristic
8	#17986231	C₂	1	1	yes	maximal, characteristic, complemented
9	#17986227	C₂	1	1	yes	maximal, characteristic, complemented
10	#17986091	C₂	1	1	yes	maximal, characteristic, complemented
11	#17986090	C₂	1	1	yes	maximal, characteristic, complemented
12	#17986084	C₂	1	1	yes	maximal, characteristic, complemented
13	#17986079	C₂	1	1	yes	maximal, characteristic, complemented
14	C₄×C₈×D₇₁	C₂	1	1	yes	maximal, characteristic, complemented
15	#488×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11765855	C₂²	1	1	yes	characteristic
17	C₂×#11765853	C₂²	1	1	yes	characteristic
18	C₂×#11765852	C₂²	1	1	yes	characteristic
19	C₂×#11765850	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₇₁	C₂²	1	1	yes	characteristic
21	C₂×C₈×D₇₁	C₂²	1	1	yes	characteristic
22	#14556123	—	2	1	no	—
23	#14556121	—	2	1	no	—
24	#14556120	C₂²	1	1	yes	characteristic
25	#14556116	C₂²	1	1	yes	characteristic
26	#14556113	—	2	1	no	—
27	#14556110	—	2	1	no	—
28	#14556108	C₂²	1	1	yes	characteristic
29	#14556104	C₂²	1	1	yes	characteristic
30	#14556103	C₂²	1	1	yes	characteristic
31	#14556099	C₂²	1	1	yes	characteristic
32	Q₈×Dic₇₁	C₂²	1	1	yes	characteristic
33	#14556093	C₂²	1	1	yes	characteristic
34	D₄×Dic₇₁	C₂²	1	1	yes	characteristic
35	#14556090	—	2	1	no	—
36	#14556090	C₂²	1	1	yes	characteristic
37	(C₂²⋊D₇₁)⋊C₄	—	2	1	no	—
38	#14556086	—	2	1	no	—
39	#14556084	—	2	1	no	—
40	C₄²×D₇₁	C₂²	1	1	yes	characteristic
41	#14556061	C₂²	1	1	yes	characteristic
42	#14556061	C₂²	1	1	yes	characteristic
43	#14556059	C₂²	1	1	yes	characteristic
44	#14556059	C₂²	1	1	yes	characteristic
45	#14556053	C₂²	1	1	yes	characteristic
46	#14556053	C₂²	1	1	yes	characteristic
47	#14556050	C₂²	1	1	yes	characteristic
48	#14556050	C₂²	1	1	yes	characteristic
49	C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
50	C₈×Dic₇₁	C₂²	1	1	yes	characteristic
51	C₈×Dic₇₁	C₂²	1	1	yes	characteristic
52	#125×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	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₄₂	C₂³	1	1	yes	characteristic
60	C₂×D₂₈₄	—	2	1	no	—
61	C₂×D₂₈₄	C₂³	1	1	yes	characteristic
62	C₂×C₂²⋊D₇₁	—	2	1	no	—
63	C₂²×Dic₇₁	—	2	1	no	—
64	C₂×C₄×D₇₁	—	2	1	no	—
65	C₂×C₄×D₇₁	D₄	1	1	yes	characteristic
66	C₂×C₄×D₇₁	D₄	1	1	yes	characteristic
67	C₂×C₄×D₇₁	C₂³	1	1	yes	characteristic
68	#11765855	—	2	2	no	—
69	#11765853	—	2	2	no	—
70	#11765852	—	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	C₈×D₇₁	—	2	2	no	—
75	C₈×D₇₁	—	2	2	no	—
76	C₄⋊Dic₇₁	—	2	1	no	—
77	C₄⋊Dic₇₁	—	2	1	no	—
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	—	2	1	no	—
83	Dic₇₁⋊C₄	—	2	1	no	—
84	C₂²⋊Dic₇₁	—	2	1	no	—
85	C₂²⋊Dic₇₁	—	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	(C₂×D₇₁)⋊C₄	—	2	1	no	—
92	C₄×Dic₇₁	—	2	1	no	—
93	C₄×Dic₇₁	—	2	1	no	—
94	C₄×Dic₇₁	D₄	1	1	yes	characteristic
95	C₄×Dic₇₁	D₄	1	1	yes	characteristic
96	C₄×Dic₇₁	C₂³	1	1	yes	characteristic
97	C₂×Q₈×C₇₁	C₂³	1	1	yes	characteristic
98	C₂×D₄×C₇₁	C₂³	1	1	yes	characteristic
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	C₂²⋊C₄×C₇₁	—	2	1	no	—
106	C₂²⋊C₄×C₇₁	—	2	1	no	—
107	C₄²×C₇₁	C₂³	1	1	yes	characteristic
108	C₂³×C₇₁	—	2	1	no	—
109	C₂²×D₇₁	—	2	1	no	—
110	C₂²×D₇₁	C₂×D₄	1	1	yes	characteristic
111	Q₈×C₇₁	—	2	1	no	—
112	Q₈×C₇₁	—	2	1	no	—
113	D₄×C₇₁	—	2	1	no	—
114	D₄×C₇₁	—	2	1	no	—
115	C₂×C₄×C₇₁	—	2	1	no	—
116	C₂×C₄×C₇₁	—	2	1	no	—
117	C₂×C₄×C₇₁	—	2	1	no	—
118	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
121	C₂²⋊D₇₁	—	4	2	no	—
122	C₂×Dic₇₁	—	2	1	no	—
123	C₂×Dic₇₁	—	2	1	no	—
124	C₂×Dic₇₁	—	2	1	no	—
125	C₂×Dic₇₁	—	4	2	no	—
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	D₂₈₄	—	4	2	no	—
132	C₄×D₇₁	—	2	2	no	—
133	C₄×D₇₁	—	2	1	no	—
134	C₄×D₇₁	—	2	1	no	—
135	C₄×D₇₁	—	2	2	no	—
136	C₄×D₇₁	—	4	2	no	—
137	Dic₁₄₂	—	2	1	no	—
138	Dic₁₄₂	—	2	1	no	—
139	C₈×C₇₁	—	2	1	no	—
140	C₈×C₇₁	—	2	1	no	—
141	C₇₁⋊C₈	#41	1	2	yes	—
142	C₇₁⋊C₈	#41	1	2	yes	—
143	C₂²×C₇₁	—	4	1	no	—
144	C₂²×C₇₁	—	4	1	no	—
145	C₂²×C₇₁	—	4	1	no	—
146	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
147	C₂×D₇₁	—	2	1	no	—
148	C₂×D₇₁	—	2	1	no	—
149	C₂×D₇₁	—	4	1	no	—
150	C₂×D₇₁	—	4	1	no	—
151	C₂×D₇₁	—	4	1	no	—
152	C₂×D₇₁	#135	1	2	yes	—
153	C₄×C₇₁	—	2	1	no	—
154	C₄×C₇₁	—	2	1	no	—
155	C₄×C₇₁	—	4	1	no	—
156	C₄×C₇₁	—	4	1	no	—
157	C₄×C₇₁	—	4	1	no	—
158	C₄×C₇₁	C₂×#41	1	1	yes	characteristic
159	C₄×C₇₁	C₂×#41	1	1	yes	characteristic
160	Dic₇₁	—	2	1	no	—
161	Dic₇₁	—	2	1	no	—
162	Dic₇₁	—	4	1	no	—
163	Dic₇₁	—	4	1	no	—
164	Dic₇₁	—	4	1	no	—
165	Dic₇₁	#135	1	2	yes	—
166	C₂×C₇₁	—	8	1	no	—
167	C₂×C₇₁	#532	1	1	yes	characteristic
168	C₂×C₇₁	C₂×#135	1	1	yes	characteristic
169	C₂×C₇₁	C₂×#135	1	1	yes	characteristic
170	D₇₁	—	2	2	no	—
171	D₇₁	—	8	1	no	—
172	#3168	—	71	1	no	maximal, Sylow
173	C₇₁	#3168	1	1	yes	characteristic, complemented, Sylow
174	#529	—	71	1	no	—
175	#529	—	71	1	no	—
176	#499	—	71	1	no	—
177	#495	—	71	1	no	—
178	#488	—	71	1	no	—
179	#488	D₇₁	1	1	yes	characteristic, complemented
180	#467	—	71	1	no	—
181	#466	—	71	1	no	—
182	#465	—	71	1	no	—
183	#465	—	71	1	no	—
184	C₄×Q₁₆	—	71	1	no	—
185	C₄×SD₁₆	—	71	1	no	—
186	C₄×SD₁₆	—	71	1	no	—
187	C₄×D₈	—	71	1	no	—
188	C₂×C₄×C₈	—	71	1	no	—
189	C₂×Q₁₆	—	71	1	no	—
190	C₂×SD₁₆	—	71	1	no	—
191	C₂×SD₁₆	—	71	1	no	—
192	C₂×D₈	—	71	1	no	—
193	C₂²×C₈	—	71	1	no	—
194	C₂²×C₈	—	71	1	no	—
195	#126	—	142	1	no	—
196	#126	—	142	1	no	—
197	#125	—	71	1	no	—
198	#125	C₂×D₇₁	1	1	yes	characteristic
199	#124	—	71	1	no	—
200	#124	C₂×D₇₁	1	1	yes	characteristic
201	#123	—	142	1	no	—
202	#123	—	142	1	no	—
203	C₂²⋊Q₈	—	71	1	no	—
204	C₂²⋊Q₈	—	71	1	no	—
205	#121	—	71	1	no	—
206	#121	—	71	1	no	—
207	C₄×Q₈	—	71	1	no	—
208	C₄×Q₈	—	71	1	no	—
209	C₄×D₄	—	71	1	no	—
210	C₄×D₄	—	71	1	no	—
211	C₄×D₄	—	142	1	no	—
212	C₄×D₄	—	142	1	no	—
213	#117	—	142	1	no	—
214	#117	—	142	1	no	—
215	C₂×C₄²	—	71	1	no	—
216	#107	—	71	1	no	—
217	#107	—	71	1	no	—
218	#106	—	71	1	no	—
219	#106	—	71	1	no	—
220	#103	—	71	1	no	—
221	#103	—	71	1	no	—
222	#103	C₂×D₇₁	1	1	yes	characteristic
223	#103	C₂×D₇₁	1	1	yes	characteristic
224	#102	—	71	1	no	—
225	#102	—	71	1	no	—
226	#102	C₂×D₇₁	1	1	yes	characteristic
227	#102	C₂×D₇₁	1	1	yes	characteristic
228	C₄×C₈	—	71	1	no	—
229	C₄×C₈	—	71	1	no	—
230	C₄×C₈	—	71	1	no	—
231	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
232	C₂×Q₈	—	71	1	no	—
233	C₂×Q₈	C₂²×D₇₁	1	1	yes	characteristic
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	—	71	1	no	—
239	C₂²×C₄	—	71	1	no	—
240	C₂²×C₄	—	71	1	no	—
241	C₂²×C₄	—	142	1	no	—
242	C₂²×C₄	—	142	1	no	—
243	Q₁₆	—	142	2	no	—
244	SD₁₆	—	142	2	no	—
245	SD₁₆	—	142	2	no	—
246	D₈	—	142	2	no	—
247	C₂×C₈	—	71	1	no	—
248	C₂×C₈	—	71	1	no	—
249	C₂×C₈	—	142	2	no	—
250	C₂×C₈	—	142	2	no	—
251	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
252	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
253	C₄⋊C₄	—	2	1	no	—
254	C₄⋊C₄	—	2	1	no	—
255	C₄⋊C₄	—	71	1	no	—
256	C₄⋊C₄	—	71	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	—	142	1	no	—
262	C₄⋊C₄	—	142	1	no	—
263	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
264	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
265	C₂²⋊C₄	—	2	1	no	—
266	C₂²⋊C₄	—	2	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	C₂²⋊C₄	—	142	1	no	—
273	C₂²⋊C₄	—	142	1	no	—
274	C₂²⋊C₄	—	142	1	no	—
275	C₄²	—	71	1	no	—
276	C₄²	—	71	1	no	—
277	C₄²	—	71	1	no	—
278	C₄²	—	142	1	no	—
279	C₄²	—	142	1	no	—
280	C₄²	C₂²×D₇₁	1	1	yes	characteristic
281	C₂³	—	2	1	no	—
282	C₂³	—	71	1	no	—
283	C₂³	—	142	1	no	—
284	Q₈	—	2	1	no	—
285	Q₈	—	2	1	no	—
286	Q₈	—	142	1	no	—
287	Q₈	—	142	1	no	—
288	D₄	—	2	1	no	—
289	D₄	—	2	1	no	—
290	D₄	—	284	2	no	—
291	D₄	—	284	2	no	—
292	D₄	—	142	1	no	—
293	D₄	—	142	1	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	284	2	no	—
298	C₂×C₄	—	284	2	no	—
299	C₂×C₄	—	71	1	no	—
300	C₂×C₄	—	71	1	no	—
301	C₂×C₄	—	71	1	no	—
302	C₂×C₄	—	142	2	no	—
303	C₂×C₄	—	142	2	no	—
304	C₂×C₄	—	142	1	no	—
305	C₂×C₄	—	142	1	no	—
306	C₂×C₄	—	142	1	no	—
307	C₂×C₄	—	142	1	no	—
308	C₂×C₄	—	142	1	no	—
309	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
310	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
311	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
312	C₈	—	2	1	no	—
313	C₈	—	2	1	no	—
314	C₈	—	71	2	no	—
315	C₈	—	71	2	no	—
316	C₂²	—	4	1	no	—
317	C₂²	—	4	1	no	—
318	C₂²	—	4	1	no	—
319	C₂²	—	284	1	no	—
320	C₂²	—	284	1	no	—
321	C₂²	—	284	1	no	—
322	C₂²	—	71	2	no	—
323	C₂²	—	142	1	no	—
324	C₂²	—	142	1	no	—
325	C₂²	C₂×D₄×D₇₁	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	4	1	no	—
329	C₄	—	4	1	no	—
330	C₄	—	4	1	no	—
331	C₄	—	284	1	no	—
332	C₄	—	284	1	no	—
333	C₄	—	284	1	no	—
334	C₄	—	71	2	no	—
335	C₄	—	142	1	no	—
336	C₄	—	142	1	no	—
337	C₄	#41×D₇₁	1	1	yes	characteristic
338	C₄	#41×D₇₁	1	1	yes	characteristic
339	C₂	—	8	1	no	—
340	C₂	—	568	1	no	—
341	C₂	—	142	2	no	—
342	C₂	#17986548	1	1	yes	characteristic
343	C₂	#135×D₇₁	1	1	yes	characteristic
344	C₂	#135×D₇₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₁
Fitting	#488×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₄
hypercenter	#488	D₇₁
hypocenter	C₇₁	#3168
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #488

Complements

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

N	H ≅ G/N	type
1. #17986534	C₂	semidirect
2. #17986528	C₂	semidirect
3. #17986398	C₂	semidirect
4. #17986381	C₂	semidirect
5. #17986354	C₂	semidirect
6. #17986239	C₂	semidirect
8. #17986231	C₂	semidirect
9. #17986227	C₂	semidirect
10. #17986091	C₂	semidirect
11. #17986090	C₂	semidirect
12. #17986084	C₂	semidirect
13. #17986079	C₂	semidirect
14. C₄×C₈×D₇₁	C₂	semidirect
15. #488×C₇₁	C₂	semidirect
173. C₇₁	#3168	semidirect
179. #488	D₇₁	semidirect