Subgroups — #22599005

Order 9088.

350 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17986618	C₂	1	1	yes	maximal, characteristic
2	#17986610	C₂	1	1	yes	maximal, characteristic, complemented
3	#17986408	C₂	1	1	yes	maximal, characteristic, complemented
4	#17986353	C₂	1	1	yes	maximal, characteristic, complemented
5	#17986349	C₂	1	1	yes	maximal, characteristic, complemented
6	#17986254	C₂	1	1	yes	maximal, characteristic, complemented
7	#17986242	C₂	1	1	yes	maximal, characteristic
8	#17986237	C₂	1	1	yes	maximal, characteristic, complemented
9	#17986225	C₂	1	1	yes	maximal, characteristic
10	#17986106	C₂	1	1	yes	maximal, characteristic, complemented
11	#17986100	C₂	1	1	yes	maximal, characteristic, complemented
12	Q₁₆×Dic₇₁	C₂	1	1	yes	maximal, characteristic, complemented
13	SD₁₆×Dic₇₁	C₂	1	1	yes	maximal, characteristic, complemented
14	#17985956	C₂	1	1	yes	maximal, characteristic, complemented
15	#497×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#11765848	C₂²	1	1	yes	characteristic
17	C₂×C₈×D₇₁	C₂²	1	1	yes	characteristic
18	#14556128	C₂²	1	1	yes	characteristic
19	#14556123	—	2	1	no	—
20	#14556123	—	2	1	no	—
21	#14556121	—	2	1	no	—
22	#14556120	C₂²	1	1	yes	characteristic
23	#14556118	—	2	1	no	—
24	#14556118	—	2	1	no	—
25	#14556114	—	2	1	no	—
26	#14556112	—	2	1	no	—
27	#14556105	—	2	1	no	—
28	#14556104	C₂²	1	1	yes	characteristic
29	#14556104	C₂²	1	1	yes	characteristic
30	#14556104	C₂²	1	1	yes	characteristic
31	#14556099	C₂²	1	1	yes	characteristic
32	Q₈×Dic₇₁	C₂²	1	1	yes	characteristic
33	Q₈×Dic₇₁	C₂²	1	1	yes	characteristic
34	Q₈×Dic₇₁	C₂²	1	1	yes	characteristic
35	D₄×Dic₇₁	C₂²	1	1	yes	characteristic
36	#14556085	C₂²	1	1	yes	characteristic
37	#14556062	C₂²	1	1	yes	characteristic
38	#14556062	C₂²	1	1	yes	characteristic
39	#14556060	C₂²	1	1	yes	characteristic
40	#14556060	C₂²	1	1	yes	characteristic
41	#14556052	C₂²	1	1	yes	characteristic
42	#14556052	C₂²	1	1	yes	characteristic
43	#14556052	C₂²	1	1	yes	characteristic
44	#14556052	C₂²	1	1	yes	characteristic
45	#14556052	C₂²	1	1	yes	characteristic
46	#14556052	C₂²	1	1	yes	characteristic
47	#14556049	C₂²	1	1	yes	characteristic
48	#14556049	C₂²	1	1	yes	characteristic
49	#14556038	C₂²	1	1	yes	characteristic
50	C₄×C₇₁⋊C₈	C₂²	1	1	yes	characteristic
51	C₈×Dic₇₁	C₂²	1	1	yes	characteristic
52	C₂×Q₁₆×C₇₁	C₂²	1	1	yes	characteristic
53	C₂×Q₁₆×C₇₁	C₂²	1	1	yes	characteristic
54	C₂×SD₁₆×C₇₁	C₂²	1	1	yes	characteristic
55	C₂×SD₁₆×C₇₁	C₂²	1	1	yes	characteristic
56	C₄⋊Q₈×C₇₁	C₂²	1	1	yes	characteristic
57	#124×C₇₁	C₂²	1	1	yes	characteristic
58	#97×C₇₁	C₂²	1	1	yes	characteristic
59	C₂×Dic₁₄₂	—	2	1	no	—
60	C₂×C₂²⋊D₇₁	—	2	1	no	—
61	C₂²×Dic₇₁	—	2	1	no	—
62	C₂×C₄×D₇₁	C₂³	1	1	yes	characteristic
63	#11765848	—	2	2	no	—
64	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
65	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
66	C₈×D₇₁	—	2	2	no	—
67	C₄⋊Dic₇₁	—	2	1	no	—
68	C₄⋊Dic₇₁	—	2	1	no	—
69	C₄⋊Dic₇₁	—	2	1	no	—
70	C₄⋊Dic₇₁	—	2	1	no	—
71	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
72	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
73	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
74	C₄⋊Dic₇₁	C₂³	1	1	yes	characteristic
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	D₄	1	1	yes	characteristic
81	Dic₇₁⋊C₄	D₄	1	1	yes	characteristic
82	C₂²⋊Dic₇₁	—	2	1	no	—
83	C₂²⋊Dic₇₁	—	2	1	no	—
84	(C₂×D₇₁)⋊C₄	—	2	1	no	—
85	(C₂×D₇₁)⋊C₄	—	2	1	no	—
86	(C₂×D₇₁)⋊C₄	—	2	1	no	—
87	(C₂×D₇₁)⋊C₄	D₄	1	1	yes	characteristic
88	(C₂×D₇₁)⋊C₄	D₄	1	1	yes	characteristic
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	C₂×Q₈×C₇₁	C₂³	1	1	yes	characteristic
94	C₂×Q₈×C₇₁	C₂³	1	1	yes	characteristic
95	C₂×Q₈×C₇₁	C₂³	1	1	yes	characteristic
96	C₂×D₄×C₇₁	C₂³	1	1	yes	characteristic
97	Q₁₆×C₇₁	—	2	2	no	—
98	Q₁₆×C₇₁	—	2	2	no	—
99	SD₁₆×C₇₁	—	2	2	no	—
100	SD₁₆×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	—	2	1	no	—
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	C₂×D₄	1	1	yes	characteristic
110	Q₈×C₇₁	—	2	1	no	—
111	Q₈×C₇₁	—	2	1	no	—
112	Q₈×C₇₁	—	2	1	no	—
113	Q₈×C₇₁	—	2	1	no	—
114	Q₈×C₇₁	—	2	1	no	—
115	Q₈×C₇₁	—	2	1	no	—
116	D₄×C₇₁	—	2	1	no	—
117	D₄×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	C₂×D₄	1	1	yes	characteristic
123	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
124	C₂²⋊D₇₁	—	8	1	no	—
125	C₂×Dic₇₁	—	2	1	no	—
126	C₂×Dic₇₁	—	2	1	no	—
127	C₂×Dic₇₁	—	2	1	no	—
128	C₂×Dic₇₁	—	2	1	no	—
129	C₂×Dic₇₁	—	8	1	no	—
130	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
131	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
132	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
133	C₄×D₇₁	—	2	1	no	—
134	C₄×D₇₁	—	2	1	no	—
135	Dic₁₄₂	—	8	1	no	—
136	C₈×C₇₁	#41	1	2	yes	—
137	C₈×C₇₁	#41	1	2	yes	—
138	C₇₁⋊C₈	—	2	1	no	—
139	C₇₁⋊C₈	—	2	1	no	—
140	C₂²×C₇₁	—	4	1	no	—
141	C₂²×C₇₁	—	4	1	no	—
142	C₂²×C₇₁	—	4	1	no	—
143	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
144	C₂×D₇₁	—	2	1	no	—
145	C₂×D₇₁	—	2	1	no	—
146	C₂×D₇₁	—	2	1	no	—
147	C₄×C₇₁	—	2	1	no	—
148	C₄×C₇₁	—	2	1	no	—
149	C₄×C₇₁	—	4	1	no	—
150	C₄×C₇₁	—	4	1	no	—
151	C₄×C₇₁	—	4	1	no	—
152	C₄×C₇₁	C₂×#41	1	1	yes	characteristic
153	C₄×C₇₁	C₂×#41	1	1	yes	characteristic
154	Dic₇₁	—	2	1	no	—
155	Dic₇₁	—	2	1	no	—
156	Dic₇₁	—	2	1	no	—
157	Dic₇₁	—	4	1	no	—
158	Dic₇₁	—	4	1	no	—
159	Dic₇₁	—	4	1	no	—
160	Dic₇₁	—	4	1	no	—
161	C₂×C₇₁	—	8	1	no	—
162	C₂×C₇₁	#532	1	1	yes	characteristic
163	C₂×C₇₁	#577	1	1	yes	characteristic
164	C₂×C₇₁	#578	1	1	yes	characteristic
165	D₇₁	—	4	1	no	—
166	#3178	—	71	1	no	maximal, Sylow
167	C₇₁	#3178	1	1	yes	characteristic, complemented, Sylow
168	#542	—	71	1	no	—
169	#541	—	71	1	no	—
170	#501	—	71	1	no	—
171	#497	D₇₁	1	1	yes	characteristic, complemented
172	#488	—	71	1	no	—
173	#487	—	71	1	no	—
174	#470	—	71	1	no	—
175	#468	—	71	1	no	—
176	#467	—	71	1	no	—
177	#465	—	71	1	no	—
178	#441	—	71	1	no	—
179	#440	—	71	1	no	—
180	C₄×Q₁₆	—	71	1	no	—
181	C₄×SD₁₆	—	71	1	no	—
182	#405	—	71	1	no	—
183	C₂×Q₁₆	C₂×D₇₁	1	1	yes	characteristic
184	C₂×Q₁₆	C₂×D₇₁	1	1	yes	characteristic
185	C₂×SD₁₆	C₂×D₇₁	1	1	yes	characteristic
186	C₂×SD₁₆	C₂×D₇₁	1	1	yes	characteristic
187	C₂×M₁₆	—	71	1	no	—
188	C₂²×C₈	—	71	1	no	—
189	C₄⋊Q₈	—	71	1	no	—
190	C₄⋊Q₈	C₂×D₇₁	1	1	yes	characteristic
191	#126	—	142	1	no	—
192	#126	—	142	1	no	—
193	#126	—	142	1	no	—
194	#125	—	71	1	no	—
195	#125	—	142	1	no	—
196	#125	—	142	1	no	—
197	#124	—	142	1	no	—
198	#124	C₂×D₇₁	1	1	yes	characteristic
199	#123	—	142	1	no	—
200	C₂²⋊Q₈	—	71	1	no	—
201	C₂²⋊Q₈	—	71	1	no	—
202	C₂²⋊Q₈	—	71	1	no	—
203	C₂²⋊Q₈	—	142	1	no	—
204	#121	—	71	1	no	—
205	C₄×Q₈	—	71	1	no	—
206	C₄×Q₈	—	71	1	no	—
207	C₄×Q₈	—	71	1	no	—
208	C₄×D₄	—	71	1	no	—
209	#117	—	71	1	no	—
210	#107	—	71	1	no	—
211	#107	—	71	1	no	—
212	#106	—	71	1	no	—
213	#106	—	71	1	no	—
214	#103	—	71	1	no	—
215	#103	—	71	1	no	—
216	#103	—	71	1	no	—
217	#103	—	71	1	no	—
218	#103	—	71	1	no	—
219	#103	—	71	1	no	—
220	#102	—	71	1	no	—
221	#102	—	71	1	no	—
222	#97	—	71	1	no	—
223	#97	C₂×D₇₁	1	1	yes	characteristic
224	C₄×C₈	—	71	1	no	—
225	C₄×C₈	—	71	1	no	—
226	C₂×Q₈	—	142	1	no	—
227	C₂×Q₈	C₂²×D₇₁	1	1	yes	characteristic
228	C₂×Q₈	C₂²×D₇₁	1	1	yes	characteristic
229	C₂×Q₈	C₂²×D₇₁	1	1	yes	characteristic
230	C₂×D₄	—	142	1	no	—
231	C₂×D₄	C₂²×D₇₁	1	1	yes	characteristic
232	C₂²×C₄	—	71	1	no	—
233	C₂²×C₄	—	142	1	no	—
234	Q₁₆	—	2	2	no	—
235	Q₁₆	—	2	2	no	—
236	SD₁₆	—	2	2	no	—
237	SD₁₆	—	2	2	no	—
238	M₁₆	—	142	2	no	—
239	C₂×C₈	—	71	1	no	—
240	C₂×C₈	—	71	1	no	—
241	C₂×C₈	—	142	2	no	—
242	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
243	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
244	C₄⋊C₄	—	2	1	no	—
245	C₄⋊C₄	—	2	1	no	—
246	C₄⋊C₄	—	71	1	no	—
247	C₄⋊C₄	—	71	1	no	—
248	C₄⋊C₄	—	71	1	no	—
249	C₄⋊C₄	—	71	1	no	—
250	C₄⋊C₄	—	71	1	no	—
251	C₄⋊C₄	—	71	1	no	—
252	C₄⋊C₄	—	142	1	no	—
253	C₄⋊C₄	—	142	1	no	—
254	C₄⋊C₄	—	142	1	no	—
255	C₄⋊C₄	—	142	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	C₂²⋊C₄	—	2	1	no	—
262	C₂²⋊C₄	—	2	1	no	—
263	C₂²⋊C₄	—	71	1	no	—
264	C₂²⋊C₄	—	71	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	C₄²	—	71	1	no	—
271	C₄²	—	142	1	no	—
272	C₄²	—	142	1	no	—
273	C₄²	—	142	1	no	—
274	C₄²	C₂²×D₇₁	1	1	yes	characteristic
275	C₂³	—	2	1	no	—
276	C₂³	—	71	1	no	—
277	Q₈	—	2	1	no	—
278	Q₈	—	2	1	no	—
279	Q₈	—	2	1	no	—
280	Q₈	—	2	1	no	—
281	Q₈	—	2	1	no	—
282	Q₈	—	2	1	no	—
283	Q₈	—	568	1	no	—
284	D₄	—	2	1	no	—
285	D₄	—	2	1	no	—
286	D₄	—	568	1	no	—
287	C₂×C₄	—	2	1	no	—
288	C₂×C₄	—	2	1	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	568	1	no	—
291	C₂×C₄	—	71	1	no	—
292	C₂×C₄	—	71	1	no	—
293	C₂×C₄	—	71	1	no	—
294	C₂×C₄	—	142	1	no	—
295	C₂×C₄	—	142	1	no	—
296	C₂×C₄	—	142	1	no	—
297	C₂×C₄	—	142	1	no	—
298	C₂×C₄	—	142	1	no	—
299	C₂×C₄	—	142	1	no	—
300	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
301	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
302	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
303	C₈	—	142	1	no	—
304	C₈	—	142	1	no	—
305	C₈	#11765864	1	2	yes	—
306	C₈	#11765864	1	2	yes	—
307	C₂²	—	4	1	no	—
308	C₂²	—	4	1	no	—
309	C₂²	—	4	1	no	—
310	C₂²	—	142	1	no	—
311	C₂²	—	142	1	no	—
312	C₂²	—	142	1	no	—
313	C₂²	C₂×D₄×D₇₁	1	1	yes	characteristic
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	4	1	no	—
317	C₄	—	4	1	no	—
318	C₄	—	4	1	no	—
319	C₄	—	284	1	no	—
320	C₄	—	284	1	no	—
321	C₄	—	284	1	no	—
322	C₄	—	284	1	no	—
323	C₄	—	142	1	no	—
324	C₄	—	142	1	no	—
325	C₄	—	142	1	no	—
326	C₄	C₂×#11765864	1	1	yes	characteristic
327	C₄	C₂×#11765864	1	1	yes	characteristic
328	C₂	—	8	1	no	—
329	C₂	—	284	1	no	—
330	C₂	#17986547	1	1	yes	characteristic
331	C₂	#17986808	1	1	yes	characteristic
332	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	#497×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₄
hypercenter	#497	D₇₁
hypocenter	C₇₁	#3178
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #497

Complements

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

N	H ≅ G/N	type
2. #17986610	C₂	semidirect
3. #17986408	C₂	semidirect
4. #17986353	C₂	semidirect
5. #17986349	C₂	semidirect
6. #17986254	C₂	semidirect
8. #17986237	C₂	semidirect
10. #17986106	C₂	semidirect
11. #17986100	C₂	semidirect
12. Q₁₆×Dic₇₁	C₂	semidirect
13. SD₁₆×Dic₇₁	C₂	semidirect
14. #17985956	C₂	semidirect
15. #497×C₇₁	C₂	semidirect
167. C₇₁	#3178	semidirect
171. #497	D₇₁	semidirect