Subgroups — #36708867

Order 22784.

710 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19152030	C₂	1	1	yes	maximal, characteristic, complemented
2	#26029833	C₂	1	2	yes	maximal, complemented
3	#26022348	C₂	1	4	yes	maximal, complemented
4	#26022343	C₂	1	2	yes	maximal, complemented
5	#26022338	C₂	1	2	yes	maximal, complemented
6	#26022036	C₂	1	1	yes	maximal, characteristic, complemented
7	#26021960	C₂	1	2	yes	maximal, complemented
8	C₂³×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#15668034	C₂²	1	2	yes	—
10	#19152109	—	2	2	no	—
11	#19152030	—	2	4	no	—
12	C₂×Dic₇₁₂	C₂²	1	1	yes	characteristic, complemented
13	C₂×#15667957	C₂²	1	2	yes	complemented
14	C₂×D₁₄₂₄	C₂²	1	1	yes	characteristic, complemented
15	#19151839	C₂²	1	4	yes	—
16	#19151834	C₂²	1	4	yes	—
17	#19151827	—	2	4	no	—
18	#19151636	C₂²	1	2	yes	—
19	#19151595	—	2	2	no	—
20	#19151426	C₂²	1	2	yes	complemented
21	#19151424	C₂²	1	2	yes	complemented
22	#19151403	C₂²	1	4	yes	complemented
23	#19151400	C₂²	1	4	yes	complemented
24	C₂³×C₈×C₈₉	C₂²	1	1	yes	characteristic
25	C₂²×C₁₆×C₈₉	—	2	4	no	—
26	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
27	C₂²×C₁₆×C₈₉	C₂²	1	2	yes	—
28	C₂²×C₁₆×C₈₉	C₂²	1	2	yes	—
29	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
30	C₂×#13083574	—	2	2	no	—
31	#15668034	—	2	4	no	—
32	#15668034	—	2	4	no	—
33	C₂×Dic₃₅₆	C₂³	1	2	yes	—
34	C₂×#13083563	—	2	2	no	—
35	C₂×D₇₁₂	C₂³	1	2	yes	—
36	#15667995	—	4	2	no	—
37	#15667991	—	2	4	no	—
38	#15667986	—	2	4	no	—
39	#15667981	—	4	2	no	—
40	C₄×C₂²⋊D₈₉	—	4	4	no	—
41	#15667971	—	2	2	no	—
42	Dic₇₁₂	—	2	2	no	—
43	#15667957	—	2	4	no	—
44	D₁₄₂₄	—	2	2	no	—
45	#15667946	C₂³	1	4	yes	—
46	#15667944	—	2	2	no	—
47	#15667937	—	2	4	no	—
48	#15667934	—	2	4	no	—
49	C₂³×C₄×C₈₉	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₈₉	—	2	2	no	—
51	C₂²×C₈×C₈₉	—	2	2	no	—
52	C₂²×C₈×C₈₉	D₄	1	2	yes	—
53	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
54	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
55	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
56	C₂×C₁₆×C₈₉	—	2	2	no	—
57	C₂×C₁₆×C₈₉	—	2	4	no	—
58	C₂×C₁₆×C₈₉	—	2	4	no	—
59	C₂×C₁₆×C₈₉	D₄	1	4	yes	complemented
60	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
61	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
62	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
63	#13083574	—	4	4	no	—
64	#13083574	—	4	4	no	—
65	C₂×Dic₁₇₈	—	2	2	no	—
66	C₂×D₃₅₆	—	2	2	no	—
67	C₂×C₂²⋊D₈₉	—	4	4	no	—
68	C₂×C₂²⋊D₈₉	—	4	2	no	—
69	C₂×C₄×D₈₉	—	4	2	no	—
70	Dic₃₅₆	—	2	2	no	—
71	Dic₃₅₆	—	2	2	no	—
72	#13083563	—	4	4	no	—
73	D₇₁₂	—	2	2	no	—
74	D₇₁₂	—	2	2	no	—
75	C₄⋊Dic₈₉	—	2	4	no	—
76	#13083553	—	4	4	no	—
77	C₂²⋊Dic₈₉	—	4	2	no	—
78	C₂²⋊Dic₈₉	—	4	4	no	—
79	#13083551	—	4	4	no	—
80	C₄×Dic₈₉	—	4	2	no	—
81	C₂⁴×C₈₉	D₈	1	1	yes	characteristic
82	C₂²×C₄×C₈₉	—	2	2	no	—
83	C₂²×C₄×C₈₉	—	2	2	no	—
84	C₂²×C₄×C₈₉	D₈	1	2	yes	—
85	C₂²×C₄×C₈₉	D₈	1	1	yes	characteristic
86	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₈₉	—	2	2	no	—
89	C₂×C₈×C₈₉	—	2	4	no	—
90	C₂×C₈×C₈₉	—	2	2	no	—
91	C₂×C₈×C₈₉	—	2	2	no	—
92	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
95	C₂×C₈×C₈₉	#41	1	2	yes	—
96	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
97	C₁₆×C₈₉	—	2	2	no	—
98	C₁₆×C₈₉	C₂×D₄	1	2	yes	—
99	C₁₆×C₈₉	C₂×D₄	1	2	yes	—
100	C₂³×C₈₉	—	2	2	no	—
101	C₂³×C₈₉	—	2	4	no	—
102	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
103	C₂³×C₈₉	C₂×D₈	1	2	yes	—
104	C₂²×D₈₉	—	4	2	no	—
105	C₂×C₄×C₈₉	—	2	2	no	—
106	C₂×C₄×C₈₉	—	2	2	no	—
107	C₂×C₄×C₈₉	—	2	2	no	—
108	C₂×C₄×C₈₉	—	2	4	no	—
109	C₂×C₄×C₈₉	#121	1	2	yes	—
110	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
111	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
112	C₂×C₄×C₈₉	#135	1	2	yes	—
113	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
114	C₂²⋊D₈₉	—	8	4	no	—
115	C₂²⋊D₈₉	—	8	8	no	—
116	C₂×Dic₈₉	—	4	4	no	—
117	C₂×Dic₈₉	—	4	2	no	—
118	D₃₅₆	—	4	2	no	—
119	D₃₅₆	—	4	2	no	—
120	C₄×D₈₉	—	8	4	no	—
121	Dic₁₇₈	—	4	2	no	—
122	Dic₁₇₈	—	4	2	no	—
123	C₈×C₈₉	—	2	2	no	—
124	C₈×C₈₉	#121	1	2	yes	—
125	C₈×C₈₉	C₂²×D₄	1	1	yes	characteristic
126	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
127	C₂²×C₈₉	—	2	8	no	—
128	C₂²×C₈₉	—	2	2	no	—
129	C₂²×C₈₉	—	2	2	no	—
130	C₂²×C₈₉	—	2	2	no	—
131	C₂²×C₈₉	#466	1	2	yes	—
132	C₂²×C₈₉	#508	1	4	yes	complemented
133	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
134	C₂×D₈₉	—	8	4	no	—
135	C₂×D₈₉	—	8	2	no	—
136	C₄×C₈₉	—	2	2	no	—
137	C₄×C₈₉	#466	1	2	yes	—
138	C₄×C₈₉	C₂×#121	1	1	yes	characteristic
139	C₄×C₈₉	C₂×#135	1	1	yes	characteristic
140	Dic₈₉	—	8	4	no	—
141	Dic₈₉	—	8	2	no	—
142	#33600	—	89	1	no	maximal, Sylow
143	C₂×C₈₉	—	2	2	no	—
144	C₂×C₈₉	—	2	4	no	—
145	C₂×C₈₉	C₂×#466	1	1	yes	characteristic
146	C₂×C₈₉	C₂×#508	1	2	yes	—
147	D₈₉	—	16	2	no	—
148	C₂×#508	—	89	1	no	—
149	C₂³×C₁₆	D₈₉	1	1	yes	characteristic, complemented
150	#3050	—	89	2	no	—
151	#2217	—	89	4	no	—
152	#2216	—	89	2	no	—
153	#2215	—	89	2	no	—
154	#2158	—	89	1	no	—
155	#2138	—	89	2	no	—
156	C₈₉	#33600	1	1	yes	characteristic, complemented, Sylow
157	C₂×#135	—	89	2	no	—
158	C₂³×C₈	C₂×D₈₉	1	1	yes	characteristic
159	#525	—	178	2	no	—
160	#508	—	178	4	no	—
161	C₂×Q₃₂	—	89	1	no	—
162	C₂×SD₃₂	—	89	2	no	—
163	C₂×D₁₆	—	89	1	no	—
164	C₂²×C₁₆	—	2	4	no	—
165	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
166	C₂²×C₁₆	C₂×D₈₉	1	2	yes	—
167	C₂²×C₁₆	C₂×D₈₉	1	2	yes	—
168	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
169	#467	—	89	4	no	—
170	#466	—	89	4	no	—
171	#465	—	178	4	no	—
172	#427	—	89	2	no	—
173	#416	—	178	2	no	—
174	#367	—	89	2	no	—
175	#366	—	89	2	no	—
176	#358	—	89	4	no	—
177	#357	—	89	4	no	—
178	C₂×#41	—	178	2	no	—
179	C₂³×C₄	D₃₅₆	1	1	yes	characteristic
180	#135	—	178	4	no	—
181	#135	—	178	4	no	—
182	C₂×Q₁₆	—	89	2	no	—
183	C₂×SD₁₆	—	178	2	no	—
184	C₂×D₈	—	89	2	no	—
185	C₂²×C₈	—	2	2	no	—
186	C₂²×C₈	—	2	2	no	—
187	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
188	C₂²×C₈	D₃₅₆	1	2	yes	—
189	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
190	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
191	#123	—	356	2	no	—
192	C₂²⋊Q₈	—	178	4	no	—
193	#121	—	178	4	no	—
194	#120	—	356	2	no	—
195	C₄×D₄	—	356	4	no	—
196	#117	—	178	2	no	—
197	Q₃₂	—	178	2	no	—
198	SD₃₂	—	178	4	no	—
199	D₁₆	—	178	2	no	—
200	C₂×C₁₆	—	2	2	no	—
201	C₂×C₁₆	—	2	4	no	—
202	C₂×C₁₆	—	2	4	no	—
203	C₂×C₁₆	C₂²⋊D₈₉	1	4	yes	complemented
204	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
205	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
206	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
207	#107	—	89	4	no	—
208	#106	—	178	2	no	—
209	#103	—	178	4	no	—
210	#102	—	178	4	no	—
211	C₂⁴	D₇₁₂	1	1	yes	characteristic
212	#41	—	356	4	no	—
213	#41	—	356	4	no	—
214	C₂×Q₈	—	178	2	no	—
215	C₂×D₄	—	356	4	no	—
216	C₂×D₄	—	356	2	no	—
217	C₂×D₄	—	178	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	356	2	no	—
221	C₂²×C₄	D₇₁₂	1	1	yes	characteristic
222	C₂²×C₄	D₇₁₂	1	2	yes	—
223	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
224	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
225	Q₁₆	—	178	2	no	—
226	Q₁₆	—	178	2	no	—
227	SD₁₆	—	356	4	no	—
228	D₈	—	178	2	no	—
229	D₈	—	178	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	4	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	C₂×C₂²⋊D₈₉	1	2	yes	—
235	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
236	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
237	C₂×C₈	#13083574	1	2	yes	—
238	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
239	C₄⋊C₄	—	356	4	no	—
240	C₄⋊C₄	—	178	4	no	—
241	C₂²⋊C₄	—	356	2	no	—
242	C₂²⋊C₄	—	356	4	no	—
243	C₂²⋊C₄	—	356	4	no	—
244	C₄²	—	356	2	no	—
245	C₁₆	—	2	2	no	—
246	C₁₆	C₂×C₂²⋊D₈₉	1	2	yes	—
247	C₁₆	C₂×C₂²⋊D₈₉	1	2	yes	—
248	C₂³	—	2	4	no	—
249	C₂³	—	2	2	no	—
250	C₂³	—	356	2	no	—
251	C₂³	C₂×D₇₁₂	1	2	yes	—
252	C₂³	C₂×D₇₁₂	1	1	yes	characteristic
253	Q₈	—	356	2	no	—
254	Q₈	—	356	2	no	—
255	D₄	—	356	2	no	—
256	D₄	—	356	2	no	—
257	D₄	—	712	8	no	—
258	D₄	—	712	4	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	356	4	no	—
264	C₂×C₄	—	356	2	no	—
265	C₂×C₄	—	712	4	no	—
266	C₂×C₄	#15667986	1	2	yes	—
267	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
268	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
269	C₂×C₄	#15668034	1	2	yes	—
270	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
271	C₈	—	2	2	no	—
272	C₈	#15667986	1	2	yes	—
273	C₈	C₂²×C₂²⋊D₈₉	1	1	yes	characteristic
274	C₈	C₂×#13083574	1	1	yes	characteristic
275	C₂²	—	2	8	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	712	4	no	—
280	C₂²	—	712	2	no	—
281	C₂²	#19151834	1	2	yes	—
282	C₂²	#19152030	1	4	yes	complemented
283	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
284	C₄	—	2	2	no	—
285	C₄	—	712	4	no	—
286	C₄	—	712	2	no	—
287	C₄	#19151834	1	2	yes	—
288	C₄	C₂×#15667986	1	1	yes	characteristic
289	C₄	C₂×#15668034	1	1	yes	characteristic
290	C₂	—	2	4	no	—
291	C₂	—	2	2	no	—
292	C₂	—	1424	2	no	—
293	C₂	C₂×#19151834	1	1	yes	characteristic
294	C₂	C₂×#19152030	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₇₁₂
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂³×C₁₆×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₈
hypercenter	C₂³×C₁₆	D₈₉
hypocenter	C₈₉	#33600
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#19152030	C₂	semidirect
2. #26029833	C₂	semidirect
3. #26022348	C₂	semidirect
4. #26022343	C₂	semidirect
5. #26022338	C₂	semidirect
6. #26022036	C₂	semidirect
7. #26021960	C₂	semidirect
8. C₂³×C₁₆×C₈₉	C₂	semidirect
12. C₂×Dic₇₁₂	C₂²	semidirect
13. C₂×#15667957	C₂²	semidirect
14. C₂×D₁₄₂₄	C₂²	semidirect
20. #19151426	C₂²	semidirect
21. #19151424	C₂²	semidirect
22. #19151403	C₂²	semidirect
23. #19151400	C₂²	semidirect
59. C₂×C₁₆×C₈₉	D₄	semidirect
132. C₂²×C₈₉	#508	semidirect
149. C₂³×C₁₆	D₈₉	semidirect
156. C₈₉	#33600	semidirect
203. C₂×C₁₆	C₂²⋊D₈₉	semidirect
282. C₂²	#19152030	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#19152030	2
#26029833	2
#26022348	2
#26022343	2
#26022338	2
#26022036	2
#26021960	2
C₂³×C₁₆×C₈₉	2
#33600	89