Subgroups — #22599706

Order 9088.

348 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#17986551	C₂	1	1	yes	maximal, characteristic, complemented
2	#17986530	C₂	1	1	yes	maximal, characteristic
3	#17986305	C₂	1	1	yes	maximal, characteristic, complemented
4	#17986304	C₂	1	1	yes	maximal, characteristic, complemented
5	#17986293	C₂	1	1	yes	maximal, characteristic, complemented
6	#17986287	C₂	1	1	yes	maximal, characteristic, complemented
7	#17986257	C₂	1	1	yes	maximal, characteristic, complemented
8	#17986255	C₂	1	1	yes	maximal, characteristic, complemented
9	#17986243	C₂	1	1	yes	maximal, characteristic
10	#17986220	C₂	1	1	yes	maximal, characteristic, complemented
11	#17986219	C₂	1	1	yes	maximal, characteristic, complemented
12	#17986208	C₂	1	1	yes	maximal, characteristic, complemented
13	#17986202	C₂	1	1	yes	maximal, characteristic, complemented
14	#17986022	C₂	1	1	yes	maximal, characteristic, complemented
15	#468×C₇₁	C₂	1	1	yes	maximal, characteristic
16	C₂×Dic₂₈₄	C₂²	1	1	yes	characteristic
17	C₂×#11765855	C₂²	1	1	yes	characteristic
18	C₂×#11765853	C₂²	1	1	yes	characteristic
19	C₂×#11765849	C₂²	1	1	yes	characteristic
20	#14556127	C₂²	1	1	yes	characteristic
21	#14556126	C₂²	1	1	yes	characteristic
22	#14556121	—	2	1	no	—
23	#14556119	C₂²	1	1	yes	characteristic
24	#14556118	—	2	1	no	—
25	#14556115	C₂²	1	1	yes	characteristic
26	#14556109	—	2	1	no	—
27	#14556107	C₂²	1	1	yes	characteristic
28	#14556107	C₂²	1	1	yes	characteristic
29	#14556105	—	2	1	no	—
30	Q₈×Dic₇₁	C₂²	1	1	yes	characteristic
31	#14556093	—	2	1	no	—
32	#14556093	C₂²	1	1	yes	characteristic
33	#14556093	C₂²	1	1	yes	characteristic
34	D₄×Dic₇₁	—	2	1	no	—
35	D₄×Dic₇₁	C₂²	1	1	yes	characteristic
36	#14556084	—	2	1	no	—
37	#14556084	—	2	1	no	—
38	C₂×C₄×Dic₇₁	C₂²	1	1	yes	characteristic
39	#14556061	C₂²	1	1	yes	characteristic
40	#14556060	C₂²	1	1	yes	characteristic
41	#14556059	C₂²	1	1	yes	characteristic
42	Dic₇₁⋊C₈	C₂²	1	1	yes	characteristic
43	Dic₇₁⋊C₈	C₂²	1	1	yes	characteristic
44	Dic₇₁⋊C₈	C₂²	1	1	yes	characteristic
45	Dic₇₁⋊C₈	C₂²	1	1	yes	characteristic
46	#14556053	C₂²	1	1	yes	characteristic
47	#14556053	C₂²	1	1	yes	characteristic
48	#14556052	C₂²	1	1	yes	characteristic
49	#14556051	C₂²	1	1	yes	characteristic
50	#14556051	C₂²	1	1	yes	characteristic
51	#14556049	C₂²	1	1	yes	characteristic
52	C₂×SD₁₆×C₇₁	C₂²	1	1	yes	characteristic
53	C₂×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
54	C₂²⋊Q₈×C₇₁	C₂²	1	1	yes	characteristic
55	#121×C₇₁	C₂²	1	1	yes	characteristic
56	#107×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	C₂³	1	1	yes	characteristic
61	C₂×Dic₁₄₂	C₂³	1	1	yes	characteristic
62	C₂²×Dic₇₁	—	2	1	no	—
63	C₂²×Dic₇₁	D₄	1	1	yes	characteristic
64	C₂²×Dic₇₁	D₄	1	1	yes	characteristic
65	Dic₂₈₄	—	4	1	no	—
66	#11765855	—	4	1	no	—
67	#11765853	—	4	1	no	—
68	#11765849	—	4	1	no	—
69	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
70	C₂×C₇₁⋊C₈	C₂³	1	1	yes	characteristic
71	C₄⋊Dic₇₁	—	2	1	no	—
72	C₄⋊Dic₇₁	—	2	1	no	—
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	—	2	1	no	—
81	Dic₇₁⋊C₄	—	2	1	no	—
82	Dic₇₁⋊C₄	—	2	1	no	—
83	C₂²⋊Dic₇₁	—	2	1	no	—
84	C₂²⋊Dic₇₁	—	2	1	no	—
85	C₂²⋊Dic₇₁	—	2	1	no	—
86	C₂²⋊Dic₇₁	—	2	1	no	—
87	C₂²⋊Dic₇₁	—	2	1	no	—
88	C₄×Dic₇₁	—	2	1	no	—
89	C₄×Dic₇₁	—	2	1	no	—
90	C₄×Dic₇₁	—	2	1	no	—
91	C₄×Dic₇₁	D₄	1	1	yes	characteristic
92	C₄×Dic₇₁	D₄	1	1	yes	characteristic
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	—	4	1	no	—
100	M₁₆×C₇₁	—	4	1	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	Q₈×C₇₁	—	2	1	no	—
111	Q₈×C₇₁	—	2	1	no	—
112	D₄×C₇₁	—	2	1	no	—
113	D₄×C₇₁	—	2	1	no	—
114	D₄×C₇₁	—	4	2	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	—	2	1	no	—
119	C₂×C₄×C₇₁	—	2	1	no	—
120	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
122	C₂×Dic₇₁	—	2	2	no	—
123	C₂×Dic₇₁	—	2	1	no	—
124	C₂×Dic₇₁	—	2	1	no	—
125	C₂×Dic₇₁	—	2	1	no	—
126	C₂×Dic₇₁	—	2	1	no	—
127	C₂×Dic₇₁	—	4	1	no	—
128	C₂×Dic₇₁	—	4	2	no	—
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	C₂×Dic₇₁	C₂×D₄	1	1	yes	characteristic
133	Dic₁₄₂	—	2	1	no	—
134	Dic₁₄₂	—	2	1	no	—
135	Dic₁₄₂	—	2	1	no	—
136	Dic₁₄₂	—	2	1	no	—
137	Dic₁₄₂	—	4	2	no	—
138	C₈×C₇₁	—	2	1	no	—
139	C₈×C₇₁	—	2	1	no	—
140	C₇₁⋊C₈	—	2	1	no	—
141	C₇₁⋊C₈	—	2	1	no	—
142	C₂²×C₇₁	—	2	1	no	—
143	C₂²×C₇₁	—	2	1	no	—
144	C₂²×C₇₁	—	2	1	no	—
145	C₂²×C₇₁	—	4	1	no	—
146	C₂²×C₇₁	—	4	1	no	—
147	C₂²×C₇₁	—	4	1	no	—
148	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
149	C₄×C₇₁	—	2	1	no	—
150	C₄×C₇₁	—	4	1	no	—
151	C₄×C₇₁	—	4	1	no	—
152	C₄×C₇₁	—	4	1	no	—
153	C₄×C₇₁	H(5,𝔽₂)	1	1	yes	characteristic
154	C₄×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
155	Dic₇₁	—	2	1	no	—
156	Dic₇₁	—	2	1	no	—
157	Dic₇₁	—	2	1	no	—
158	Dic₇₁	—	4	1	no	—
159	Dic₇₁	—	4	1	no	—
160	Dic₇₁	—	4	1	no	—
161	Dic₇₁	—	4	1	no	—
162	Dic₇₁	#137	1	2	yes	—
163	C₂×C₇₁	—	4	1	no	—
164	C₂×C₇₁	—	8	1	no	—
165	C₂×C₇₁	#537	1	1	yes	characteristic
166	C₂×C₇₁	C₂×#137	1	1	yes	characteristic
167	C₂×C₇₁	#575	1	1	yes	characteristic
168	#3252	—	71	1	no	maximal, Sylow
169	C₇₁	#3252	1	1	yes	characteristic, complemented, Sylow
170	#533	—	71	1	no	—
171	#529	—	71	1	no	—
172	#479	—	71	1	no	—
173	#479	—	71	1	no	—
174	#477	—	71	1	no	—
175	#476	—	71	1	no	—
176	#470	—	71	1	no	—
177	#470	—	71	1	no	—
178	#468	—	71	1	no	—
179	#468	D₇₁	1	1	yes	characteristic
180	#464	—	71	1	no	—
181	#464	—	71	1	no	—
182	#462	—	71	1	no	—
183	#461	—	71	1	no	—
184	#423	—	71	1	no	—
185	C₂×Q₁₆	—	71	1	no	—
186	C₂×Q₁₆	—	71	1	no	—
187	C₂×SD₁₆	—	71	1	no	—
188	C₂×SD₁₆	C₂×D₇₁	1	1	yes	characteristic
189	C₂×M₁₆	—	71	1	no	—
190	C₂×M₁₆	C₂×D₇₁	1	1	yes	characteristic
191	C₄⋊Q₈	—	71	1	no	—
192	C₄⋊Q₈	—	71	1	no	—
193	#126	—	142	1	no	—
194	#125	—	71	1	no	—
195	#125	—	142	1	no	—
196	#124	—	71	1	no	—
197	#123	—	142	1	no	—
198	C₂²⋊Q₈	—	71	1	no	—
199	C₂²⋊Q₈	—	71	1	no	—
200	C₂²⋊Q₈	—	142	1	no	—
201	C₂²⋊Q₈	C₂×D₇₁	1	1	yes	characteristic
202	#121	C₂×D₇₁	1	1	yes	characteristic
203	C₄×Q₈	—	71	1	no	—
204	C₄×Q₈	—	71	1	no	—
205	C₄×Q₈	—	71	1	no	—
206	C₄×Q₈	—	142	1	no	—
207	C₄×D₄	—	71	1	no	—
208	C₄×D₄	—	142	1	no	—
209	#117	—	142	1	no	—
210	#117	—	142	1	no	—
211	C₂×C₄²	—	71	1	no	—
212	#107	—	71	1	no	—
213	#107	C₂×D₇₁	1	1	yes	characteristic
214	#106	—	71	1	no	—
215	#106	—	71	1	no	—
216	C₄⋊C₈	—	71	1	no	—
217	C₄⋊C₈	—	71	1	no	—
218	C₄⋊C₈	—	71	1	no	—
219	C₄⋊C₈	—	71	1	no	—
220	#103	—	71	1	no	—
221	#103	—	71	1	no	—
222	#103	—	71	1	no	—
223	#103	—	71	1	no	—
224	#103	—	71	1	no	—
225	#103	C₂×D₇₁	1	1	yes	characteristic
226	#102	—	71	1	no	—
227	#102	C₂×D₇₁	1	1	yes	characteristic
228	C₂×Q₈	—	71	1	no	—
229	C₂×Q₈	—	71	1	no	—
230	C₂×Q₈	—	142	1	no	—
231	C₂×Q₈	C₂²×D₇₁	1	1	yes	characteristic
232	C₂×D₄	—	2	1	no	—
233	C₂×D₄	C₂²×D₇₁	1	1	yes	characteristic
234	C₂²×C₄	—	71	1	no	—
235	C₂²×C₄	—	71	1	no	—
236	C₂²×C₄	—	142	1	no	—
237	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
238	Q₁₆	—	284	1	no	—
239	Q₁₆	—	284	1	no	—
240	SD₁₆	—	4	1	no	—
241	SD₁₆	—	284	1	no	—
242	M₁₆	—	4	1	no	—
243	M₁₆	—	284	1	no	—
244	C₂×C₈	—	71	1	no	—
245	C₂×C₈	—	71	1	no	—
246	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
247	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
248	C₄⋊C₄	—	2	1	no	—
249	C₄⋊C₄	—	71	1	no	—
250	C₄⋊C₄	—	71	1	no	—
251	C₄⋊C₄	—	142	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	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	C₄²	—	71	1	no	—
271	C₄²	—	71	1	no	—
272	C₄²	—	71	1	no	—
273	C₄²	—	71	1	no	—
274	C₄²	—	142	1	no	—
275	C₄²	—	142	1	no	—
276	C₄²	—	142	1	no	—
277	C₂³	—	2	1	no	—
278	C₂³	D₄×D₇₁	1	1	yes	characteristic
279	Q₈	—	2	1	no	—
280	Q₈	—	2	1	no	—
281	Q₈	—	284	2	no	—
282	Q₈	—	142	1	no	—
283	Q₈	—	142	1	no	—
284	Q₈	—	142	1	no	—
285	Q₈	—	142	1	no	—
286	D₄	—	2	1	no	—
287	D₄	—	2	1	no	—
288	D₄	—	4	2	no	—
289	C₂×C₄	—	2	1	no	—
290	C₂×C₄	—	2	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	—	284	1	no	—
296	C₂×C₄	—	71	1	no	—
297	C₂×C₄	—	71	1	no	—
298	C₂×C₄	—	71	1	no	—
299	C₂×C₄	—	71	1	no	—
300	C₂×C₄	—	142	1	no	—
301	C₂×C₄	—	142	1	no	—
302	C₂×C₄	—	142	1	no	—
303	C₂×C₄	—	142	1	no	—
304	C₂×C₄	—	142	2	no	—
305	C₂×C₄	D₄×D₇₁	1	1	yes	characteristic
306	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
307	C₈	—	2	1	no	—
308	C₈	—	2	1	no	—
309	C₈	—	142	1	no	—
310	C₈	—	142	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	2	1	no	—
313	C₂²	—	2	1	no	—
314	C₂²	—	4	1	no	—
315	C₂²	—	4	1	no	—
316	C₂²	—	4	1	no	—
317	C₂²	C₂×D₄×D₇₁	1	1	yes	characteristic
318	C₄	—	2	1	no	—
319	C₄	—	4	1	no	—
320	C₄	—	4	1	no	—
321	C₄	—	4	1	no	—
322	C₄	—	284	1	no	—
323	C₄	—	284	1	no	—
324	C₄	—	284	1	no	—
325	C₄	—	284	1	no	—
326	C₄	—	71	2	no	—
327	C₄	—	142	1	no	—
328	C₄	—	142	1	no	—
329	C₄	—	142	1	no	—
330	C₄	#14556173	1	1	yes	characteristic
331	C₄	#14556175	1	1	yes	characteristic
332	C₂	—	4	1	no	—
333	C₂	—	8	1	no	—
334	C₂	#17986576	1	1	yes	characteristic
335	C₂	#137×D₇₁	1	1	yes	characteristic
336	C₂	#17986795	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₇₁
commutator	C₂×C₄×C₇₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₁
Fitting	#468×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₄
hypercenter	#468	D₇₁
hypocenter	C₇₁	#3252
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #468

Complements

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

N	H ≅ G/N	type
1. #17986551	C₂	semidirect
3. #17986305	C₂	semidirect
4. #17986304	C₂	semidirect
5. #17986293	C₂	semidirect
6. #17986287	C₂	semidirect
7. #17986257	C₂	semidirect
8. #17986255	C₂	semidirect
10. #17986220	C₂	semidirect
11. #17986219	C₂	semidirect
12. #17986208	C₂	semidirect
13. #17986202	C₂	semidirect
14. #17986022	C₂	semidirect
169. C₇₁	#3252	semidirect