Subgroups — #22598950

Order 9088.

362 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	#17986622	C₂	1	1	yes	maximal, characteristic, complemented
2	#17986616	C₂	1	1	yes	maximal, characteristic, complemented
3	#17986396	C₂	1	1	yes	maximal, characteristic, complemented
4	#17986380	C₂	1	1	yes	maximal, characteristic, complemented
5	#17986360	C₂	1	1	yes	maximal, characteristic, complemented
6	#17986357	C₂	1	1	yes	maximal, characteristic, complemented
7	#17986255	C₂	1	1	yes	maximal, characteristic, complemented
8	#17986250	C₂	1	1	yes	maximal, characteristic
9	#17986228	C₂	1	1	yes	maximal, characteristic, complemented
10	#17986108	C₂	1	1	yes	maximal, characteristic, complemented
11	#17986103	C₂	1	1	yes	maximal, characteristic, complemented
12	SD₁₆×Dic₇₁	C₂	1	1	yes	maximal, characteristic, complemented
13	#17986086	C₂	1	1	yes	maximal, characteristic, complemented
14	#17985954	C₂	1	1	yes	maximal, characteristic, complemented
15	#465×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11765855	C₂²	1	1	yes	characteristic
17	C₂×#11765854	C₂²	1	1	yes	characteristic
18	C₂×#11765850	C₂²	1	1	yes	characteristic
19	C₂×#11765849	C₂²	1	1	yes	characteristic
20	#14556127	C₂²	1	1	yes	characteristic
21	#14556127	C₂²	1	1	yes	characteristic
22	#14556123	—	2	1	no	—
23	#14556121	—	2	1	no	—
24	#14556117	C₂²	1	1	yes	characteristic
25	#14556115	—	2	1	no	—
26	#14556115	C₂²	1	1	yes	characteristic
27	#14556114	—	2	1	no	—
28	#14556110	—	2	1	no	—
29	#14556109	—	2	1	no	—
30	#14556108	—	2	1	no	—
31	#14556107	C₂²	1	1	yes	characteristic
32	#14556105	—	2	1	no	—
33	#14556102	C₂²	1	1	yes	characteristic
34	Q₈×Dic₇₁	C₂²	1	1	yes	characteristic
35	#14556093	C₂²	1	1	yes	characteristic
36	D₄×Dic₇₁	C₂²	1	1	yes	characteristic
37	#14556090	C₂²	1	1	yes	characteristic
38	#14556087	C₂²	1	1	yes	characteristic
39	#14556060	C₂²	1	1	yes	characteristic
40	#14556059	C₂²	1	1	yes	characteristic
41	#14556059	C₂²	1	1	yes	characteristic
42	#14556053	C₂²	1	1	yes	characteristic
43	#14556052	C₂²	1	1	yes	characteristic
44	#14556051	C₂²	1	1	yes	characteristic
45	#14556050	C₂²	1	1	yes	characteristic
46	#14556049	C₂²	1	1	yes	characteristic
47	#14556048	C₂²	1	1	yes	characteristic
48	#14556038	C₂²	1	1	yes	characteristic
49	#14556038	C₂²	1	1	yes	characteristic
50	C₈×Dic₇₁	C₂²	1	1	yes	characteristic
51	C₈×Dic₇₁	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₇₁	C₂²	1	1	yes	characteristic
53	C₂²×C₈×C₇₁	C₂²	1	1	yes	characteristic
54	C₂²⋊Q₈×C₇₁	C₂²	1	1	yes	characteristic
55	#121×C₇₁	C₂²	1	1	yes	characteristic
56	#106×C₇₁	C₂²	1	1	yes	characteristic
57	#103×C₇₁	C₂²	1	1	yes	characteristic
58	#102×C₇₁	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₄₂	—	2	1	no	—
60	C₂×Dic₁₄₂	—	2	1	no	—
61	C₂×Dic₁₄₂	C₂³	1	1	yes	characteristic
62	C₂×D₂₈₄	C₂³	1	1	yes	characteristic
63	C₂×C₂²⋊D₇₁	—	2	1	no	—
64	C₂²×Dic₇₁	—	2	1	no	—
65	C₂×C₄×D₇₁	—	2	1	no	—
66	#11765855	—	2	2	no	—
67	#11765854	—	2	2	no	—
68	#11765850	—	2	2	no	—
69	#11765849	—	2	2	no	—
70	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
71	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
72	C₄⋊Dic₇₁	—	2	1	no	—
73	C₄⋊Dic₇₁	D₄	1	1	yes	characteristic
74	C₄⋊Dic₇₁	D₄	1	1	yes	characteristic
75	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
76	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
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	C₂²⋊Dic₇₁	—	2	1	no	—
82	C₂²⋊Dic₇₁	—	2	1	no	—
83	C₂²⋊Dic₇₁	—	2	1	no	—
84	C₂²⋊Dic₇₁	—	2	1	no	—
85	C₂²⋊Dic₇₁	D₄	1	1	yes	characteristic
86	C₂²⋊Dic₇₁	D₄	1	1	yes	characteristic
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	—	2	1	no	—
93	C₄×Dic₇₁	C₂³	1	1	yes	characteristic
94	C₄×Dic₇₁	C₂³	1	1	yes	characteristic
95	C₂×Q₈×C₇₁	C₂³	1	1	yes	characteristic
96	C₂×D₄×C₇₁	—	2	1	no	—
97	C₂×D₄×C₇₁	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
99	SD₁₆×C₇₁	—	2	2	no	—
100	C₂×C₈×C₇₁	—	2	2	no	—
101	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
102	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
103	C₄⋊C₄×C₇₁	—	2	1	no	—
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	C₂²⋊C₄×C₇₁	—	2	1	no	—
108	C₂³×C₇₁	—	2	1	no	—
109	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
110	C₂²×D₇₁	—	2	1	no	—
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	D₄×C₇₁	—	8	1	no	—
116	C₂×C₄×C₇₁	—	2	1	no	—
117	C₂×C₄×C₇₁	—	2	1	no	—
118	C₂×C₄×C₇₁	—	2	1	no	—
119	C₂×C₄×C₇₁	—	2	1	no	—
120	C₂×C₄×C₇₁	—	2	1	no	—
121	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
122	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
123	C₂²⋊D₇₁	—	8	1	no	—
124	C₂×Dic₇₁	—	2	1	no	—
125	C₂×Dic₇₁	—	2	1	no	—
126	C₂×Dic₇₁	—	2	1	no	—
127	C₂×Dic₇₁	—	8	1	no	—
128	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
129	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
130	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
131	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
132	D₂₈₄	—	2	1	no	—
133	D₂₈₄	—	2	1	no	—
134	C₄×D₇₁	—	8	1	no	—
135	Dic₁₄₂	—	2	1	no	—
136	Dic₁₄₂	—	2	1	no	—
137	Dic₁₄₂	—	8	1	no	—
138	Dic₁₄₂	—	8	1	no	—
139	C₈×C₇₁	—	2	1	no	—
140	C₈×C₇₁	#41	1	2	yes	—
141	C₇₁⋊C₈	—	2	1	no	—
142	C₇₁⋊C₈	#41	1	2	yes	—
143	C₂²×C₇₁	—	2	1	no	—
144	C₂²×C₇₁	—	2	1	no	—
145	C₂²×C₇₁	—	2	1	no	—
146	C₂²×C₇₁	—	4	1	no	—
147	C₂²×C₇₁	—	4	1	no	—
148	C₂²×C₇₁	—	4	1	no	—
149	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
150	C₂×D₇₁	—	4	1	no	—
151	C₂×D₇₁	—	4	1	no	—
152	C₂×D₇₁	—	4	1	no	—
153	C₄×C₇₁	—	2	1	no	—
154	C₄×C₇₁	—	4	1	no	—
155	C₄×C₇₁	—	4	1	no	—
156	C₄×C₇₁	—	4	1	no	—
157	C₄×C₇₁	C₂×#41	1	1	yes	characteristic
158	C₄×C₇₁	C₂×#41	1	1	yes	characteristic
159	Dic₇₁	—	2	1	no	—
160	Dic₇₁	—	2	1	no	—
161	Dic₇₁	—	2	1	no	—
162	Dic₇₁	—	2	1	no	—
163	Dic₇₁	—	4	1	no	—
164	Dic₇₁	—	4	1	no	—
165	Dic₇₁	—	4	1	no	—
166	C₂×C₇₁	—	4	1	no	—
167	C₂×C₇₁	—	8	1	no	—
168	C₂×C₇₁	#532	1	1	yes	characteristic
169	C₂×C₇₁	#577	1	1	yes	characteristic
170	C₂×C₇₁	#577	1	1	yes	characteristic
171	D₇₁	—	8	1	no	—
172	#3173	—	71	1	no	maximal, Sylow
173	C₇₁	#3173	1	1	yes	characteristic, complemented, Sylow
174	#542	—	71	1	no	—
175	#542	—	71	1	no	—
176	#498	—	71	1	no	—
177	#495	—	71	1	no	—
178	#489	—	71	1	no	—
179	#489	—	71	1	no	—
180	#470	—	71	1	no	—
181	#469	—	71	1	no	—
182	#465	—	71	1	no	—
183	#465	D₇₁	1	1	yes	characteristic, complemented
184	#442	—	71	1	no	—
185	#441	—	71	1	no	—
186	C₄×SD₁₆	—	71	1	no	—
187	C₄×SD₁₆	—	71	1	no	—
188	#405	—	71	1	no	—
189	C₂×Q₁₆	—	71	1	no	—
190	C₂×SD₁₆	—	71	1	no	—
191	C₂×SD₁₆	C₂×D₇₁	1	1	yes	characteristic
192	C₂×D₈	—	71	1	no	—
193	C₂×M₁₆	—	71	1	no	—
194	C₂²×C₈	C₂×D₇₁	1	1	yes	characteristic
195	C₄⋊Q₈	—	71	1	no	—
196	C₄⋊Q₈	—	71	1	no	—
197	#126	—	142	1	no	—
198	#126	—	142	1	no	—
199	#124	—	71	1	no	—
200	#124	—	71	1	no	—
201	#124	—	142	1	no	—
202	#124	—	142	1	no	—
203	#123	—	142	1	no	—
204	#123	—	142	1	no	—
205	C₂²⋊Q₈	—	71	1	no	—
206	C₂²⋊Q₈	—	142	1	no	—
207	C₂²⋊Q₈	—	142	1	no	—
208	C₂²⋊Q₈	C₂×D₇₁	1	1	yes	characteristic
209	#121	—	71	1	no	—
210	#121	C₂×D₇₁	1	1	yes	characteristic
211	C₄×Q₈	—	71	1	no	—
212	C₄×Q₈	—	71	1	no	—
213	C₄×D₄	—	71	1	no	—
214	C₄×D₄	—	71	1	no	—
215	#117	—	71	1	no	—
216	#106	—	71	1	no	—
217	#106	—	71	1	no	—
218	#106	—	71	1	no	—
219	#106	C₂×D₇₁	1	1	yes	characteristic
220	#103	—	71	1	no	—
221	#103	—	71	1	no	—
222	#103	—	71	1	no	—
223	#103	C₂×D₇₁	1	1	yes	characteristic
224	#102	—	71	1	no	—
225	#102	—	71	1	no	—
226	#102	—	71	1	no	—
227	#102	C₂×D₇₁	1	1	yes	characteristic
228	#97	—	71	1	no	—
229	#97	—	71	1	no	—
230	C₄×C₈	—	71	1	no	—
231	C₄×C₈	—	71	1	no	—
232	C₂×Q₈	—	71	1	no	—
233	C₂×Q₈	—	142	1	no	—
234	C₂×Q₈	—	142	1	no	—
235	C₂×Q₈	C₂²×D₇₁	1	1	yes	characteristic
236	C₂×D₄	—	2	1	no	—
237	C₂×D₄	—	71	1	no	—
238	C₂×D₄	—	142	1	no	—
239	C₂×D₄	C₂²×D₇₁	1	1	yes	characteristic
240	C₂²×C₄	—	142	1	no	—
241	C₂²×C₄	—	142	1	no	—
242	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
243	Q₁₆	—	142	2	no	—
244	SD₁₆	—	2	2	no	—
245	SD₁₆	—	142	2	no	—
246	D₈	—	142	2	no	—
247	M₁₆	—	142	2	no	—
248	C₂×C₈	—	2	2	no	—
249	C₂×C₈	—	71	1	no	—
250	C₂×C₈	—	71	1	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	—	71	1	no	—
255	C₄⋊C₄	—	71	1	no	—
256	C₄⋊C₄	—	71	1	no	—
257	C₄⋊C₄	—	71	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	—	71	1	no	—
268	C₂²⋊C₄	—	71	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	C₂²⋊C₄	—	142	1	no	—
276	C₂²⋊C₄	—	142	1	no	—
277	C₄²	—	71	1	no	—
278	C₄²	—	71	1	no	—
279	C₄²	—	142	1	no	—
280	C₄²	—	142	1	no	—
281	C₂³	—	2	1	no	—
282	C₂³	—	142	1	no	—
283	C₂³	D₄×D₇₁	1	1	yes	characteristic
284	Q₈	—	2	1	no	—
285	Q₈	—	2	1	no	—
286	Q₈	—	568	1	no	—
287	Q₈	—	568	1	no	—
288	Q₈	—	142	1	no	—
289	Q₈	—	142	1	no	—
290	D₄	—	2	1	no	—
291	D₄	—	2	1	no	—
292	D₄	—	8	1	no	—
293	D₄	—	568	1	no	—
294	D₄	—	142	1	no	—
295	D₄	—	142	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	1	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	1	no	—
301	C₂×C₄	—	568	1	no	—
302	C₂×C₄	—	568	1	no	—
303	C₂×C₄	—	71	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	1	no	—
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	—	71	2	no	—
314	C₈	—	142	1	no	—
315	C₈	#11765864	1	2	yes	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	4	1	no	—
320	C₂²	—	4	1	no	—
321	C₂²	—	4	1	no	—
322	C₂²	—	284	1	no	—
323	C₂²	—	284	1	no	—
324	C₂²	—	284	1	no	—
325	C₂²	C₂×D₄×D₇₁	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	4	1	no	—
328	C₄	—	4	1	no	—
329	C₄	—	4	1	no	—
330	C₄	—	284	1	no	—
331	C₄	—	284	1	no	—
332	C₄	—	284	1	no	—
333	C₄	—	142	1	no	—
334	C₄	—	142	1	no	—
335	C₄	—	142	1	no	—
336	C₄	—	142	1	no	—
337	C₄	#41×D₇₁	1	1	yes	characteristic
338	C₄	C₂×#11765864	1	1	yes	characteristic
339	C₂	—	4	1	no	—
340	C₂	—	8	1	no	—
341	C₂	—	568	1	no	—
342	C₂	#17986544	1	1	yes	characteristic
343	C₂	#17986805	1	1	yes	characteristic
344	C₂	#17986806	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₁
Fitting	#465×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₄
hypercenter	#465	D₇₁
hypocenter	C₇₁	#3173
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #465

Complements

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

N	H ≅ G/N	type
1. #17986622	C₂	semidirect
2. #17986616	C₂	semidirect
3. #17986396	C₂	semidirect
4. #17986380	C₂	semidirect
5. #17986360	C₂	semidirect
6. #17986357	C₂	semidirect
7. #17986255	C₂	semidirect
9. #17986228	C₂	semidirect
10. #17986108	C₂	semidirect
11. #17986103	C₂	semidirect
12. SD₁₆×Dic₇₁	C₂	semidirect
13. #17986086	C₂	semidirect
14. #17985954	C₂	semidirect
15. #465×C₇₁	C₂	semidirect
173. C₇₁	#3173	semidirect
183. #465	D₇₁	semidirect