Subgroups — #22598607

Order 9088.

384 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17986623	C₂	1	1	yes	maximal, characteristic, complemented
2	#17986478	C₂	1	1	yes	maximal, characteristic, complemented
3	#17986336	C₂	1	1	yes	maximal, characteristic, complemented
4	#17986314	C₂	1	1	yes	maximal, characteristic, complemented
5	#17986291	C₂	1	1	yes	maximal, characteristic, complemented
6	#17986222	C₂	1	1	yes	maximal, characteristic, complemented
7	#17986214	C₂	1	1	yes	maximal, characteristic, complemented
8	#17986162	C₂	1	1	yes	maximal, characteristic, complemented
9	#17986145	C₂	1	1	yes	maximal, characteristic
10	#17986098	C₂	1	1	yes	maximal, characteristic, complemented
11	#17986096	C₂	1	1	yes	maximal, characteristic, complemented
12	#17986093	C₂	1	1	yes	maximal, characteristic, complemented
13	#17986079	C₂	1	1	yes	maximal, characteristic, complemented
14	#17986059	C₂	1	1	yes	maximal, characteristic, complemented
15	#474×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11765864	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₈₄	C₂²	1	1	yes	characteristic
18	C₂×#11765854	C₂²	1	1	yes	characteristic
19	C₂×#11765853	C₂²	1	1	yes	characteristic
20	C₂×#11765850	C₂²	1	1	yes	characteristic
21	#14556123	—	2	1	no	—
22	#14556117	—	2	1	no	—
23	#14556116	C₂²	1	1	yes	characteristic
24	#14556113	—	2	1	no	—
25	#14556108	C₂²	1	1	yes	characteristic
26	#14556104	—	2	1	no	—
27	#14556103	C₂²	1	1	yes	characteristic
28	C₄×Dic₁₄₂	C₂²	1	1	yes	characteristic
29	#14556093	C₂²	1	1	yes	characteristic
30	#14556093	C₂²	1	1	yes	characteristic
31	D₄×Dic₇₁	C₂²	1	1	yes	characteristic
32	#14556090	C₂²	1	1	yes	characteristic
33	C₄×C₂²⋊D₇₁	—	2	1	no	—
34	(C₂²⋊D₇₁)⋊C₄	—	2	1	no	—
35	#14556085	C₂²	1	1	yes	characteristic
36	#14556084	—	2	1	no	—
37	C₂×Dic₇₁⋊C₄	—	2	1	no	—
38	C₄⋊C₄×D₇₁	C₂²	1	1	yes	characteristic
39	#14556061	C₂²	1	1	yes	characteristic
40	#14556061	C₂²	1	1	yes	characteristic
41	C₄⋊(C₇₁⋊C₈)	C₂²	1	1	yes	characteristic
42	#14556053	C₂²	1	1	yes	characteristic
43	#14556051	C₂²	1	1	yes	characteristic
44	#14556051	C₂²	1	1	yes	characteristic
45	#14556051	C₂²	1	1	yes	characteristic
46	#14556050	C₂²	1	1	yes	characteristic
47	#14556048	C₂²	1	1	yes	characteristic
48	(C₂×D₇₁)⋊C₈	C₂²	1	1	yes	characteristic
49	(C₂×D₇₁)⋊C₈	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	C₄×D₄×C₇₁	C₂²	1	1	yes	characteristic
54	#107×C₇₁	C₂²	1	1	yes	characteristic
55	#107×C₇₁	C₂²	1	1	yes	characteristic
56	C₄⋊C₈×C₇₁	C₂²	1	1	yes	characteristic
57	#102×C₇₁	C₂²	1	1	yes	characteristic
58	#102×C₇₁	C₂²	1	1	yes	characteristic
59	#11765864	—	2	2	no	—
60	#11765864	—	2	2	no	—
61	C₂×Dic₁₄₂	C₂³	1	1	yes	characteristic
62	C₂×Dic₁₄₂	C₂³	1	1	yes	characteristic
63	C₂×D₂₈₄	—	2	1	no	—
64	C₂×D₂₈₄	C₂³	1	1	yes	characteristic
65	C₂×C₂²⋊D₇₁	—	2	1	no	—
66	C₂²×Dic₇₁	—	2	1	no	—
67	C₂²×Dic₇₁	—	2	1	no	—
68	C₂×C₄×D₇₁	—	2	1	no	—
69	C₂×C₄×D₇₁	—	2	1	no	—
70	C₂×C₄×D₇₁	C₂³	1	1	yes	characteristic
71	Dic₂₈₄	—	2	2	no	—
72	#11765854	—	2	2	no	—
73	#11765853	—	2	2	no	—
74	#11765850	—	2	2	no	—
75	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
76	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
77	C₄⋊Dic₇₁	—	2	1	no	—
78	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
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	(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	D₄	1	1	yes	characteristic
92	(C₂×D₇₁)⋊C₄	D₄	1	1	yes	characteristic
93	C₄×Dic₇₁	—	2	1	no	—
94	C₄×Dic₇₁	—	2	1	no	—
95	C₄×Dic₇₁	—	2	1	no	—
96	C₄×Dic₇₁	C₂³	1	1	yes	characteristic
97	C₂×Q₈×C₇₁	—	2	1	no	—
98	C₂×D₄×C₇₁	C₂³	1	1	yes	characteristic
99	C₂²×C₄×C₇₁	—	2	1	no	—
100	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
101	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
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	C₄⋊C₄×C₇₁	C₂³	1	1	yes	characteristic
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	—	8	1	no	—
112	D₄×C₇₁	—	2	1	no	—
113	D₄×C₇₁	#41	1	2	yes	—
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	—	8	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	—	2	4	no	—
122	C₂×Dic₇₁	—	2	4	no	—
123	C₂×Dic₇₁	—	2	1	no	—
124	C₂×Dic₇₁	—	2	1	no	—
125	C₂×Dic₇₁	—	2	1	no	—
126	C₂×Dic₇₁	—	8	1	no	—
127	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
128	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
130	D₂₈₄	—	2	1	no	—
131	D₂₈₄	—	2	1	no	—
132	D₂₈₄	—	8	1	no	—
133	C₄×D₇₁	—	2	1	no	—
134	C₄×D₇₁	—	2	1	no	—
135	C₄×D₇₁	—	8	1	no	—
136	C₄×D₇₁	—	8	1	no	—
137	Dic₁₄₂	—	2	1	no	—
138	Dic₁₄₂	—	2	1	no	—
139	Dic₁₄₂	—	2	1	no	—
140	Dic₁₄₂	#41	1	2	yes	—
141	C₈×C₇₁	—	2	1	no	—
142	C₈×C₇₁	—	2	1	no	—
143	C₇₁⋊C₈	—	2	1	no	—
144	C₇₁⋊C₈	—	2	1	no	—
145	C₂²×C₇₁	—	2	2	no	—
146	C₂²×C₇₁	—	4	1	no	—
147	C₂²×C₇₁	—	4	1	no	—
148	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
149	C₂×D₇₁	—	2	1	no	—
150	C₂×D₇₁	—	2	1	no	—
151	C₂×D₇₁	—	2	1	no	—
152	C₂×D₇₁	—	4	1	no	—
153	C₂×D₇₁	—	4	1	no	—
154	C₂×D₇₁	—	4	1	no	—
155	C₄×C₇₁	—	2	1	no	—
156	C₄×C₇₁	—	2	1	no	—
157	C₄×C₇₁	—	4	1	no	—
158	C₄×C₇₁	—	4	1	no	—
159	C₄×C₇₁	—	4	1	no	—
160	C₄×C₇₁	C₂×#41	1	1	yes	characteristic
161	C₄×C₇₁	C₂×#41	1	1	yes	characteristic
162	Dic₇₁	—	2	2	no	—
163	Dic₇₁	—	2	1	no	—
164	Dic₇₁	—	2	1	no	—
165	Dic₇₁	—	2	1	no	—
166	Dic₇₁	—	4	1	no	—
167	Dic₇₁	—	4	1	no	—
168	C₂×C₇₁	—	4	2	no	—
169	C₂×C₇₁	#525	1	1	yes	characteristic
170	C₂×C₇₁	#576	1	1	yes	characteristic
171	C₂×C₇₁	#578	1	1	yes	characteristic
172	D₇₁	—	4	1	no	—
173	D₇₁	—	8	1	no	—
174	#3124	—	71	1	no	maximal, Sylow
175	C₇₁	#3124	1	1	yes	characteristic, complemented, Sylow
176	#542	—	71	1	no	—
177	#520	—	71	1	no	—
178	#484	—	71	1	no	—
179	#480	—	71	1	no	—
180	#477	—	71	1	no	—
181	#474	D₇₁	1	1	yes	characteristic, complemented
182	#464	—	71	1	no	—
183	#463	—	71	1	no	—
184	#452	—	71	1	no	—
185	#449	—	71	1	no	—
186	#440	—	71	1	no	—
187	#440	—	71	1	no	—
188	C₄×Q₁₆	—	71	1	no	—
189	C₄×D₈	—	71	1	no	—
190	#433	—	71	1	no	—
191	C₂×#41	—	71	1	no	—
192	C₂×Q₁₆	—	71	1	no	—
193	C₂×SD₁₆	—	71	1	no	—
194	C₂×SD₁₆	—	71	1	no	—
195	C₂×D₈	—	71	1	no	—
196	C₄⋊Q₈	C₂×D₇₁	1	1	yes	characteristic
197	#126	—	142	1	no	—
198	#124	—	71	1	no	—
199	#124	—	142	1	no	—
200	#123	—	142	1	no	—
201	C₂²⋊Q₈	—	71	1	no	—
202	C₂²⋊Q₈	—	142	1	no	—
203	#121	—	71	1	no	—
204	C₄×Q₈	—	71	1	no	—
205	C₄×Q₈	—	71	1	no	—
206	C₄×Q₈	—	71	1	no	—
207	C₄×D₄	—	71	1	no	—
208	C₄×D₄	—	71	1	no	—
209	C₄×D₄	—	142	1	no	—
210	C₄×D₄	—	142	1	no	—
211	C₄×D₄	C₂×D₇₁	1	1	yes	characteristic
212	#117	—	71	1	no	—
213	#117	—	142	1	no	—
214	C₂×C₄⋊C₄	—	71	1	no	—
215	C₂×C₄⋊C₄	—	142	1	no	—
216	#107	—	71	1	no	—
217	#107	—	71	1	no	—
218	#107	C₂×D₇₁	1	1	yes	characteristic
219	#107	C₂×D₇₁	1	1	yes	characteristic
220	C₄⋊C₈	—	71	1	no	—
221	C₄⋊C₈	C₂×D₇₁	1	1	yes	characteristic
222	#103	—	71	1	no	—
223	#103	—	71	1	no	—
224	#103	—	71	1	no	—
225	#103	—	71	1	no	—
226	#102	—	71	1	no	—
227	#102	—	71	1	no	—
228	#102	C₂×D₇₁	1	1	yes	characteristic
229	#102	C₂×D₇₁	1	1	yes	characteristic
230	C₂²⋊C₈	—	71	1	no	—
231	C₂²⋊C₈	—	71	1	no	—
232	#97	—	71	1	no	—
233	C₄×C₈	—	71	1	no	—
234	#41	—	142	2	no	—
235	#41	—	142	2	no	—
236	C₂×Q₈	—	2	1	no	—
237	C₂×Q₈	—	71	1	no	—
238	C₂×Q₈	—	71	1	no	—
239	C₂×D₄	—	71	1	no	—
240	C₂×D₄	—	142	1	no	—
241	C₂×D₄	—	142	1	no	—
242	C₂×D₄	C₂²×D₇₁	1	1	yes	characteristic
243	C₂²×C₄	—	2	1	no	—
244	C₂²×C₄	—	71	1	no	—
245	C₂²×C₄	—	142	1	no	—
246	C₂²×C₄	—	142	1	no	—
247	C₂²×C₄	—	142	1	no	—
248	C₂²×C₄	—	142	1	no	—
249	Q₁₆	—	142	2	no	—
250	SD₁₆	—	142	2	no	—
251	SD₁₆	—	142	2	no	—
252	D₈	—	142	2	no	—
253	C₂×C₈	—	71	1	no	—
254	C₂×C₈	—	71	1	no	—
255	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
256	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
257	C₄⋊C₄	—	2	1	no	—
258	C₄⋊C₄	—	71	1	no	—
259	C₄⋊C₄	—	71	1	no	—
260	C₄⋊C₄	—	71	1	no	—
261	C₄⋊C₄	—	142	1	no	—
262	C₄⋊C₄	—	142	1	no	—
263	C₄⋊C₄	—	142	1	no	—
264	C₄⋊C₄	—	142	1	no	—
265	C₄⋊C₄	—	142	1	no	—
266	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
267	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
268	C₄⋊C₄	C₂²×D₇₁	1	1	yes	characteristic
269	C₂²⋊C₄	—	2	1	no	—
270	C₂²⋊C₄	—	71	1	no	—
271	C₂²⋊C₄	—	71	1	no	—
272	C₂²⋊C₄	—	142	1	no	—
273	C₂²⋊C₄	—	142	1	no	—
274	C₂²⋊C₄	—	142	1	no	—
275	C₂²⋊C₄	—	142	1	no	—
276	C₂²⋊C₄	—	142	1	no	—
277	C₂²⋊C₄	—	142	1	no	—
278	C₄²	—	71	1	no	—
279	C₄²	—	142	1	no	—
280	C₄²	—	142	1	no	—
281	C₄²	—	142	1	no	—
282	C₄²	C₂²×D₇₁	1	1	yes	characteristic
283	C₂³	—	2	1	no	—
284	C₂³	—	71	1	no	—
285	C₂³	—	142	1	no	—
286	Q₈	—	8	1	no	—
287	Q₈	—	71	2	no	—
288	Q₈	—	142	1	no	—
289	Q₈	—	142	1	no	—
290	Q₈	—	142	1	no	—
291	D₄	—	2	1	no	—
292	D₄	—	568	1	no	—
293	D₄	—	142	4	no	—
294	D₄	—	142	1	no	—
295	D₄	—	142	1	no	—
296	D₄	#11765866	1	2	yes	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	8	1	no	—
301	C₂×C₄	—	568	1	no	—
302	C₂×C₄	—	568	1	no	—
303	C₂×C₄	—	568	1	no	—
304	C₂×C₄	—	71	1	no	—
305	C₂×C₄	—	71	1	no	—
306	C₂×C₄	—	71	1	no	—
307	C₂×C₄	—	142	1	no	—
308	C₂×C₄	—	142	1	no	—
309	C₂×C₄	—	142	4	no	—
310	C₂×C₄	—	142	1	no	—
311	C₂×C₄	—	142	1	no	—
312	C₂×C₄	—	142	1	no	—
313	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
314	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
315	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
316	C₈	—	2	1	no	—
317	C₈	—	2	1	no	—
318	C₈	—	142	1	no	—
319	C₈	—	142	1	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	4	1	no	—
322	C₂²	—	4	1	no	—
323	C₂²	—	284	1	no	—
324	C₂²	—	284	1	no	—
325	C₂²	—	284	1	no	—
326	C₂²	—	142	1	no	—
327	C₂²	—	142	1	no	—
328	C₂²	—	142	1	no	—
329	C₂²	C₂×D₄×D₇₁	1	1	yes	characteristic
330	C₄	—	2	1	no	—
331	C₄	—	2	1	no	—
332	C₄	—	4	1	no	—
333	C₄	—	4	1	no	—
334	C₄	—	4	1	no	—
335	C₄	—	284	1	no	—
336	C₄	—	284	1	no	—
337	C₄	—	142	1	no	—
338	C₄	—	142	2	no	—
339	C₄	—	142	1	no	—
340	C₄	—	142	1	no	—
341	C₄	#41×D₇₁	1	1	yes	characteristic
342	C₄	C₂×#11765866	1	1	yes	characteristic
343	C₂	—	4	2	no	—
344	C₂	—	284	1	no	—
345	C₂	—	568	1	no	—
346	C₂	#17986512	1	1	yes	characteristic
347	C₂	#17986800	1	1	yes	characteristic
348	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	#474×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₄
hypercenter	#474	D₇₁
hypocenter	C₇₁	#3124
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #474

Complements

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

N	H ≅ G/N	type
1. #17986623	C₂	semidirect
2. #17986478	C₂	semidirect
3. #17986336	C₂	semidirect
4. #17986314	C₂	semidirect
5. #17986291	C₂	semidirect
6. #17986222	C₂	semidirect
7. #17986214	C₂	semidirect
8. #17986162	C₂	semidirect
10. #17986098	C₂	semidirect
11. #17986096	C₂	semidirect
12. #17986093	C₂	semidirect
13. #17986079	C₂	semidirect
14. #17986059	C₂	semidirect
15. #474×C₇₁	C₂	semidirect
175. C₇₁	#3124	semidirect
181. #474	D₇₁	semidirect