Subgroups — #36711807

Order 22784.

470 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26032621	C₂	1	2	yes	maximal
2	#26022531	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022461	C₂	1	2	yes	maximal, complemented
4	#26022450	C₂	1	2	yes	maximal, complemented
5	#26022417	C₂	1	2	yes	maximal, complemented
6	#26022343	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022343	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×Dic₇₁₂	C₂	1	1	yes	maximal, characteristic, complemented
9	#26022036	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#19151424	C₂	1	1	yes	maximal, characteristic, complemented
11	D₄×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic
12	#19152288	—	2	2	no	—
13	C₂×Dic₇₁₂	C₂²	1	1	yes	characteristic
14	#19152000	C₂²	1	2	yes	—
15	#19151940	—	2	2	no	—
16	#19151929	—	2	2	no	—
17	#19151906	—	2	2	no	—
18	#19151839	C₂²	1	2	yes	—
19	#19151839	C₂²	1	2	yes	—
20	C₄×Dic₃₅₆	C₂²	1	2	yes	—
21	#19151636	C₂²	1	2	yes	—
22	C₂×#15667946	C₂²	1	2	yes	—
23	#19151426	C₂²	1	2	yes	—
24	#19151426	C₂²	1	2	yes	—
25	#19151424	C₂²	1	2	yes	—
26	#19151424	C₂²	1	1	yes	characteristic
27	#19151424	C₂²	1	1	yes	characteristic
28	#19151424	C₂²	1	1	yes	characteristic
29	#19151403	C₂²	1	2	yes	—
30	#19151403	C₂²	1	2	yes	—
31	#19151403	C₂²	1	2	yes	—
32	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
33	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
34	C₈×D₄×C₈₉	C₂²	1	1	yes	characteristic
35	C₄⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
36	C₂²⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
37	C₂²⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
38	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
39	C₂×Dic₃₅₆	C₂³	1	2	yes	—
40	#15668005	—	4	2	no	—
41	#15668004	—	2	2	no	—
42	#15668002	—	4	2	no	—
43	#15667993	—	4	2	no	—
44	#15667991	—	2	2	no	—
45	#15667991	—	2	2	no	—
46	C₄×Dic₁₇₈	—	2	2	no	—
47	#15667971	—	2	2	no	—
48	#15667968	—	4	2	no	—
49	C₂×C₄⋊Dic₈₉	—	2	2	no	—
50	Dic₇₁₂	—	2	2	no	—
51	#15667946	—	2	2	no	—
52	#15667946	C₂³	1	2	yes	—
53	#15667946	C₂³	1	2	yes	—
54	#15667946	C₂³	1	2	yes	—
55	#15667944	—	2	2	no	—
56	#15667944	—	2	2	no	—
57	#15667937	—	2	2	no	—
58	#15667937	—	2	2	no	—
59	#15667937	—	2	2	no	—
60	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
61	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
62	C₄×D₄×C₈₉	D₄	1	1	yes	characteristic
63	C₂×C₁₆×C₈₉	—	2	2	no	—
64	C₂×C₁₆×C₈₉	—	2	2	no	—
65	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
66	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
67	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
68	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
69	C₄⋊C₈×C₈₉	D₄	1	1	yes	characteristic
70	C₂²⋊C₈×C₈₉	D₄	1	1	yes	characteristic
71	C₂²⋊C₈×C₈₉	D₄	1	1	yes	characteristic
72	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
73	C₂×Dic₁₇₈	—	2	2	no	—
74	C₂²×Dic₈₉	—	4	2	no	—
75	Dic₃₅₆	—	2	2	no	—
76	Dic₃₅₆	—	2	2	no	—
77	C₄⋊Dic₈₉	—	2	2	no	—
78	C₄⋊Dic₈₉	—	2	2	no	—
79	C₄⋊Dic₈₉	—	2	2	no	—
80	C₄⋊Dic₈₉	—	4	2	no	—
81	#13083553	—	4	2	no	—
82	#13083553	—	4	2	no	—
83	#13083553	—	4	2	no	—
84	#13083553	—	4	2	no	—
85	#13083553	—	4	2	no	—
86	C₂²⋊Dic₈₉	—	4	2	no	—
87	C₂²⋊Dic₈₉	—	4	2	no	—
88	C₂²⋊Dic₈₉	—	4	2	no	—
89	C₄×Dic₈₉	—	4	2	no	—
90	C₄×Dic₈₉	—	4	2	no	—
91	C₂×D₄×C₈₉	D₈	1	1	yes	characteristic
92	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
93	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₈₉	—	2	1	no	—
95	C₂×C₈×C₈₉	—	2	1	no	—
96	C₂×C₈×C₈₉	—	2	1	no	—
97	C₂×C₈×C₈₉	—	2	1	no	—
98	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
102	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
103	C₂²⋊C₄×C₈₉	D₈	1	1	yes	characteristic
104	C₂²⋊C₄×C₈₉	D₈	1	1	yes	characteristic
105	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
106	C₁₆×C₈₉	—	2	1	no	—
107	C₁₆×C₈₉	—	2	1	no	—
108	C₁₆×C₈₉	—	2	1	no	—
109	C₁₆×C₈₉	#41	1	2	yes	—
110	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
111	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
112	D₄×C₈₉	—	2	2	no	—
113	C₂×C₄×C₈₉	—	2	1	no	—
114	C₂×C₄×C₈₉	—	2	1	no	—
115	C₂×C₄×C₈₉	—	2	1	no	—
116	C₂×C₄×C₈₉	—	2	1	no	—
117	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
118	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
119	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
120	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
121	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
122	C₂×Dic₈₉	—	4	2	no	—
123	C₂×Dic₈₉	—	4	2	no	—
124	C₂×Dic₈₉	—	4	2	no	—
125	C₂×Dic₈₉	—	4	2	no	—
126	C₂×Dic₈₉	—	8	4	no	—
127	Dic₁₇₈	—	4	2	no	—
128	Dic₁₇₈	—	4	2	no	—
129	C₈×C₈₉	—	2	1	no	—
130	C₈×C₈₉	—	2	1	no	—
131	C₈×C₈₉	—	2	1	no	—
132	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
133	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
134	C₂²×C₈₉	—	2	1	no	—
135	C₂²×C₈₉	—	2	1	no	—
136	C₂²×C₈₉	—	2	1	no	—
137	C₂²×C₈₉	—	2	1	no	—
138	C₂²×C₈₉	—	2	1	no	—
139	C₂²×C₈₉	#508	1	2	yes	—
140	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
141	C₄×C₈₉	—	2	1	no	—
142	C₄×C₈₉	—	2	1	no	—
143	C₄×C₈₉	—	2	1	no	—
144	C₄×C₈₉	—	2	1	no	—
145	C₄×C₈₉	#547	1	1	yes	characteristic
146	C₄×C₈₉	#578	1	1	yes	characteristic
147	Dic₈₉	—	8	2	no	—
148	Dic₈₉	—	8	2	no	—
149	Dic₈₉	—	8	2	no	—
150	Dic₈₉	—	8	2	no	—
151	#33903	—	89	1	no	maximal, Sylow
152	C₂×C₈₉	—	2	2	no	—
153	C₂×C₈₉	—	4	1	no	—
154	C₂×C₈₉	#3334	1	1	yes	characteristic
155	C₂×C₈₉	C₂×#508	1	1	yes	characteristic
156	C₂×C₈₉	#3417	1	1	yes	characteristic
157	#3336	—	89	2	no	—
158	#2253	—	89	1	no	—
159	#2237	—	89	2	no	—
160	#2235	—	89	2	no	—
161	#2229	—	89	2	no	—
162	#2216	—	89	1	no	—
163	#2216	—	89	1	no	—
164	C₄×Q₃₂	—	89	1	no	—
165	D₄×C₁₆	D₈₉	1	1	yes	characteristic
166	#2158	—	89	1	no	—
167	C₂×#366	—	89	1	no	—
168	C₈₉	#33903	1	1	yes	characteristic, complemented, Sylow
169	#552	—	178	2	no	—
170	C₂×Q₃₂	—	89	1	no	—
171	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
172	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
173	#499	—	89	2	no	—
174	#485	—	178	2	no	—
175	#483	—	178	2	no	—
176	#479	—	178	2	no	—
177	#467	—	89	2	no	—
178	#467	—	89	2	no	—
179	C₄×Q₁₆	—	89	2	no	—
180	C₈×D₄	C₂×D₈₉	1	1	yes	characteristic
181	#427	—	89	2	no	—
182	C₂×#107	—	89	2	no	—
183	#367	—	89	2	no	—
184	#367	—	89	2	no	—
185	#366	—	89	1	no	—
186	#366	—	89	1	no	—
187	#366	—	89	2	no	—
188	#366	—	89	1	no	—
189	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
190	#358	—	89	2	no	—
191	#358	—	89	2	no	—
192	#358	—	89	2	no	—
193	C₂²⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
194	C₂²⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
195	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
196	C₂×Q₁₆	—	89	2	no	—
197	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
198	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
199	#126	—	356	2	no	—
200	#125	—	356	2	no	—
201	#125	—	178	2	no	—
202	#123	—	356	2	no	—
203	C₂²⋊Q₈	—	178	2	no	—
204	C₂²⋊Q₈	—	178	2	no	—
205	C₄×Q₈	—	178	2	no	—
206	C₄×D₄	D₃₅₆	1	1	yes	characteristic
207	#117	—	356	2	no	—
208	#117	—	178	2	no	—
209	C₂×C₄⋊C₄	—	178	2	no	—
210	Q₃₂	—	178	2	no	—
211	C₂×C₁₆	—	2	2	no	—
212	C₂×C₁₆	—	2	2	no	—
213	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
214	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
215	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
216	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
217	#107	—	89	2	no	—
218	#107	—	89	2	no	—
219	#107	—	89	2	no	—
220	#107	—	178	2	no	—
221	#106	—	178	2	no	—
222	#106	—	178	2	no	—
223	C₄⋊C₈	D₃₅₆	1	1	yes	characteristic
224	#103	—	178	2	no	—
225	#103	—	178	2	no	—
226	#103	—	178	2	no	—
227	C₂²⋊C₈	D₃₅₆	1	1	yes	characteristic
228	C₂²⋊C₈	D₃₅₆	1	1	yes	characteristic
229	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
230	C₂×Q₈	—	178	2	no	—
231	C₂×D₄	D₇₁₂	1	1	yes	characteristic
232	C₂²×C₄	—	356	2	no	—
233	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
234	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
235	Q₁₆	—	178	2	no	—
236	Q₁₆	—	178	2	no	—
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
242	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
243	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
244	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
245	C₄⋊C₄	—	356	2	no	—
246	C₄⋊C₄	—	356	2	no	—
247	C₄⋊C₄	—	356	2	no	—
248	C₄⋊C₄	—	356	2	no	—
249	C₄⋊C₄	—	356	2	no	—
250	C₄⋊C₄	—	356	2	no	—
251	C₄⋊C₄	—	178	2	no	—
252	C₄⋊C₄	—	178	2	no	—
253	C₄⋊C₄	—	178	2	no	—
254	C₄⋊C₄	D₇₁₂	1	1	yes	characteristic
255	C₂²⋊C₄	—	356	2	no	—
256	C₂²⋊C₄	—	356	2	no	—
257	C₂²⋊C₄	—	356	2	no	—
258	C₂²⋊C₄	D₇₁₂	1	1	yes	characteristic
259	C₂²⋊C₄	D₇₁₂	1	1	yes	characteristic
260	C₄²	—	356	2	no	—
261	C₄²	—	356	2	no	—
262	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
263	C₁₆	—	2	1	no	—
264	C₁₆	—	2	1	no	—
265	C₁₆	—	2	1	no	—
266	C₁₆	#13083573	1	2	yes	—
267	C₂³	C₂×D₇₁₂	1	1	yes	characteristic
268	C₂³	C₂×D₇₁₂	1	1	yes	characteristic
269	Q₈	—	356	2	no	—
270	Q₈	—	356	2	no	—
271	D₄	—	2	2	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	356	2	no	—
277	C₂×C₄	—	356	2	no	—
278	C₂×C₄	—	356	2	no	—
279	C₂×C₄	—	356	2	no	—
280	C₂×C₄	—	712	4	no	—
281	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
282	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
283	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
284	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
285	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
286	C₈	—	2	1	no	—
287	C₈	—	2	1	no	—
288	C₈	—	2	1	no	—
289	C₈	C₂×#13083573	1	1	yes	characteristic
290	C₈	#15668059	1	1	yes	characteristic
291	C₂²	—	2	1	no	—
292	C₂²	—	2	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	2	1	no	—
296	C₂²	#19152030	1	2	yes	—
297	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	712	2	no	—
303	C₄	—	712	2	no	—
304	C₄	—	712	2	no	—
305	C₄	—	712	2	no	—
306	C₄	#19152252	1	1	yes	characteristic
307	C₄	#19152415	1	1	yes	characteristic
308	C₂	—	2	2	no	—
309	C₂	—	4	1	no	—
310	C₂	#26032600	1	1	yes	characteristic
311	C₂	C₂×#19152030	1	1	yes	characteristic
312	C₂	#26033405	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₇₁₂
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	D₄×C₁₆×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂×D₄×C₈₉	D₈
hypercenter	D₄×C₁₆	D₈₉
hypocenter	C₈₉	#33903
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ D₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. #26022531	C₂	semidirect
3. #26022461	C₂	semidirect
4. #26022450	C₂	semidirect
5. #26022417	C₂	semidirect
6. #26022343	C₂	semidirect
7. #26022343	C₂	semidirect
8. C₄×Dic₇₁₂	C₂	semidirect
9. #26022036	C₂	semidirect
10. C₂×#19151424	C₂	semidirect
168. C₈₉	#33903	semidirect

Maximal subgroups

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

subgroup	index
#26032621	2
#26022531	2
#26022461	2
#26022450	2
#26022417	2
#26022343	2
C₄×Dic₇₁₂	2
#26022036	2
C₂×#19151424	2
D₄×C₁₆×C₈₉	2
#33903	89