Subgroups — #36712023

Order 22784.

502 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	#26032683	C₂	1	1	yes	maximal, characteristic, complemented
2	#26032600	C₂	1	1	yes	maximal, characteristic
3	#26022527	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022455	C₂	1	2	yes	maximal, complemented
5	#26022444	C₂	1	2	yes	maximal, complemented
6	#26022395	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022390	C₂	1	1	yes	maximal, characteristic, complemented
8	#26022365	C₂	1	2	yes	maximal, complemented
9	C₄×#15667957	C₂	1	1	yes	maximal, characteristic, complemented
10	#26022040	C₂	1	2	yes	maximal, complemented
11	C₈₉×#2169	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152318	—	2	1	no	—
13	#19152252	—	2	1	no	—
14	C₂×#15667957	C₂²	1	1	yes	characteristic
15	#19152003	C₂²	1	1	yes	characteristic
16	#19152003	C₂²	1	1	yes	characteristic
17	#19151940	—	2	2	no	—
18	#19151912	—	2	2	no	—
19	#19151884	—	2	1	no	—
20	#19151879	—	2	1	no	—
21	#19151839	C₂²	1	2	yes	—
22	#19151834	C₂²	1	2	yes	—
23	C₄×Dic₃₅₆	C₂²	1	1	yes	characteristic
24	C₄×D₇₁₂	C₂²	1	1	yes	characteristic
25	#19151636	C₂²	1	2	yes	—
26	C₂×#15667946	C₂²	1	2	yes	—
27	#19151426	C₂²	1	2	yes	—
28	#19151426	C₂²	1	1	yes	characteristic
29	#19151424	C₂²	1	2	yes	—
30	#19151424	C₂²	1	2	yes	—
31	#19151424	C₂²	1	2	yes	—
32	#19151403	C₂²	1	1	yes	characteristic
33	#19151403	C₂²	1	2	yes	—
34	#19151400	C₂²	1	1	yes	characteristic
35	#19151400	C₂²	1	2	yes	—
36	C₂×M₃₂×C₈₉	C₂²	1	2	yes	—
37	C₈×D₄×C₈₉	C₂²	1	1	yes	characteristic
38	C₄⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
39	C₂²⋊C₁₆×C₈₉	C₂²	1	2	yes	—
40	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
41	C₂×#13083573	—	4	1	no	—
42	C₂×Dic₃₅₆	C₂³	1	1	yes	characteristic
43	C₂×D₇₁₂	C₂³	1	1	yes	characteristic
44	#15668012	—	2	1	no	—
45	#15668012	—	2	1	no	—
46	#15668005	—	4	2	no	—
47	#15667999	—	4	1	no	—
48	#15667996	—	4	2	no	—
49	#15667992	—	4	1	no	—
50	#15667991	—	2	2	no	—
51	#15667986	—	2	2	no	—
52	C₄×Dic₁₇₈	—	2	1	no	—
53	#15667977	—	4	1	no	—
54	C₄×D₃₅₆	—	2	1	no	—
55	#15667971	—	2	2	no	—
56	C₂×C₄⋊Dic₈₉	—	2	2	no	—
57	#15667957	—	2	2	no	—
58	#15667946	—	2	2	no	—
59	#15667946	—	2	1	no	—
60	#15667946	—	2	1	no	—
61	#15667946	C₂³	1	1	yes	characteristic
62	#15667946	C₂³	1	2	yes	—
63	#15667946	C₂³	1	1	yes	characteristic
64	#15667946	C₂³	1	2	yes	—
65	#15667944	—	2	2	no	—
66	#15667937	—	2	2	no	—
67	#15667937	—	2	1	no	—
68	#15667934	—	2	2	no	—
69	#15667934	—	2	1	no	—
70	C₂²×C₈×C₈₉	C₂³	1	2	yes	—
71	C₄×D₄×C₈₉	D₄	1	1	yes	characteristic
72	M₃₂×C₈₉	—	2	4	no	—
73	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
75	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
76	C₄⋊C₈×C₈₉	D₄	1	1	yes	characteristic
77	C₂²⋊C₈×C₈₉	D₄	1	2	yes	—
78	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
79	#13083573	—	8	4	no	—
80	C₂×Dic₁₇₈	—	2	1	no	—
81	C₂×Dic₁₇₈	—	4	1	no	—
82	C₂×Dic₁₇₈	—	4	1	no	—
83	C₂×D₃₅₆	—	2	1	no	—
84	C₂×C₂²⋊D₈₉	—	4	2	no	—
85	C₂²×Dic₈₉	—	4	2	no	—
86	C₂×C₄×D₈₉	—	4	1	no	—
87	Dic₃₅₆	—	2	1	no	—
88	Dic₃₅₆	—	2	1	no	—
89	D₇₁₂	—	2	1	no	—
90	D₇₁₂	—	2	1	no	—
91	C₄⋊Dic₈₉	—	2	1	no	—
92	C₄⋊Dic₈₉	—	2	2	no	—
93	C₄⋊Dic₈₉	—	2	1	no	—
94	C₄⋊Dic₈₉	—	2	2	no	—
95	C₄⋊Dic₈₉	—	4	2	no	—
96	C₄⋊Dic₈₉	—	4	1	no	—
97	C₄⋊Dic₈₉	—	4	1	no	—
98	#13083553	—	4	2	no	—
99	#13083553	—	4	1	no	—
100	C₂²⋊Dic₈₉	—	4	2	no	—
101	C₂²⋊Dic₈₉	—	4	2	no	—
102	#13083551	—	4	1	no	—
103	#13083551	—	4	2	no	—
104	C₄×Dic₈₉	—	4	2	no	—
105	C₄×Dic₈₉	—	4	1	no	—
106	C₂×D₄×C₈₉	D₈	1	1	yes	characteristic
107	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
108	C₂×C₈×C₈₉	—	2	2	no	—
109	C₂×C₈×C₈₉	—	2	2	no	—
110	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
111	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
112	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
113	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
114	C₂²⋊C₄×C₈₉	D₈	1	2	yes	—
115	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
116	C₁₆×C₈₉	—	2	2	no	—
117	C₁₆×C₈₉	—	2	1	no	—
118	C₁₆×C₈₉	#41	1	2	yes	complemented
119	C₂³×C₈₉	C₂×D₈	1	2	yes	—
120	C₂²×D₈₉	—	4	1	no	—
121	D₄×C₈₉	—	2	2	no	—
122	C₂×C₄×C₈₉	—	2	2	no	—
123	C₂×C₄×C₈₉	—	2	2	no	—
124	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
126	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
128	C₂²⋊D₈₉	—	8	4	no	—
129	C₂×Dic₈₉	—	4	1	no	—
130	C₂×Dic₈₉	—	4	1	no	—
131	C₂×Dic₈₉	—	4	2	no	—
132	C₂×Dic₈₉	—	4	1	no	—
133	C₂×Dic₈₉	—	4	2	no	—
134	C₂×Dic₈₉	—	8	4	no	—
135	D₃₅₆	—	4	1	no	—
136	D₃₅₆	—	4	1	no	—
137	C₄×D₈₉	—	8	2	no	—
138	Dic₁₇₈	—	4	1	no	—
139	Dic₁₇₈	—	4	1	no	—
140	Dic₁₇₈	—	8	2	no	—
141	Dic₁₇₈	—	8	2	no	—
142	C₈×C₈₉	—	2	2	no	—
143	C₈×C₈₉	—	2	1	no	—
144	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
145	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
146	C₂²×C₈₉	—	2	2	no	—
147	C₂²×C₈₉	—	2	2	no	—
148	C₂²×C₈₉	—	2	2	no	—
149	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
150	C₂×D₈₉	—	8	1	no	—
151	C₂×D₈₉	—	8	1	no	—
152	C₂×D₈₉	—	8	1	no	—
153	C₄×C₈₉	—	2	2	no	—
154	C₄×C₈₉	—	2	1	no	—
155	C₄×C₈₉	#509	1	2	yes	—
156	C₄×C₈₉	#547	1	1	yes	characteristic
157	C₄×C₈₉	#578	1	1	yes	characteristic
158	Dic₈₉	—	8	1	no	—
159	Dic₈₉	—	8	2	no	—
160	Dic₈₉	—	8	2	no	—
161	Dic₈₉	—	8	1	no	—
162	Dic₈₉	—	8	1	no	—
163	#33920	—	89	1	no	maximal, Sylow
164	C₂×C₈₉	—	4	2	no	—
165	C₂×C₈₉	#3334	1	1	yes	characteristic
166	C₂×C₈₉	C₂×#509	1	1	yes	characteristic
167	C₂×C₈₉	#3417	1	1	yes	characteristic
168	D₈₉	—	16	1	no	—
169	#3342	—	89	1	no	—
170	#3334	—	89	1	no	—
171	#2252	—	89	1	no	—
172	#2236	—	89	2	no	—
173	#2234	—	89	2	no	—
174	#2225	—	89	1	no	—
175	#2224	—	89	1	no	—
176	#2220	—	89	2	no	—
177	C₄×SD₃₂	—	89	1	no	—
178	#2169	D₈₉	1	1	yes	characteristic, complemented
179	#2159	—	89	2	no	—
180	C₈₉	#33920	1	1	yes	characteristic, complemented, Sylow
181	#556	—	178	1	no	—
182	#547	—	178	1	no	—
183	C₂×SD₃₂	—	89	1	no	—
184	C₂×M₃₂	C₂×D₈₉	1	2	yes	—
185	#500	—	89	1	no	—
186	#500	—	89	1	no	—
187	#485	—	178	2	no	—
188	#480	—	178	2	no	—
189	#475	—	178	1	no	—
190	#474	—	178	1	no	—
191	#467	—	89	2	no	—
192	#466	—	89	2	no	—
193	C₄×Q₁₆	—	89	1	no	—
194	C₄×D₈	—	89	1	no	—
195	C₈×D₄	C₂×D₈₉	1	1	yes	characteristic
196	#427	—	89	2	no	—
197	C₂×#107	—	89	2	no	—
198	#367	—	89	1	no	—
199	#367	—	89	2	no	—
200	#366	—	89	2	no	—
201	#366	—	89	2	no	—
202	#366	—	89	2	no	—
203	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
204	#358	—	89	1	no	—
205	#358	—	89	2	no	—
206	#357	—	89	1	no	—
207	#357	—	89	2	no	—
208	C₂²⋊C₁₆	C₂×D₈₉	1	2	yes	—
209	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
210	C₂×#41	—	356	1	no	—
211	C₂×Q₁₆	—	89	1	no	—
212	C₂×D₈	—	89	1	no	—
213	C₂²×C₈	C₂²×D₈₉	1	2	yes	—
214	C₄⋊Q₈	—	178	1	no	—
215	C₄⋊Q₈	—	178	1	no	—
216	#126	—	356	2	no	—
217	#124	—	356	1	no	—
218	#123	—	356	2	no	—
219	C₂²⋊Q₈	—	356	1	no	—
220	C₂²⋊Q₈	—	178	2	no	—
221	#121	—	178	2	no	—
222	C₄×Q₈	—	356	1	no	—
223	C₄×Q₈	—	178	1	no	—
224	C₄×D₄	—	178	1	no	—
225	C₄×D₄	D₃₅₆	1	1	yes	characteristic
226	#117	—	178	2	no	—
227	C₂×C₄⋊C₄	—	178	2	no	—
228	SD₃₂	—	178	2	no	—
229	M₃₂	—	2	4	no	—
230	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
231	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
232	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
233	#107	—	89	2	no	—
234	#107	—	89	1	no	—
235	#107	—	89	1	no	—
236	#107	—	89	2	no	—
237	#107	—	178	2	no	—
238	#107	—	178	1	no	—
239	#107	—	178	1	no	—
240	#106	—	178	2	no	—
241	C₄⋊C₈	D₃₅₆	1	1	yes	characteristic
242	#103	—	178	1	no	—
243	#103	—	178	2	no	—
244	#102	—	178	1	no	—
245	#102	—	178	2	no	—
246	C₂²⋊C₈	D₃₅₆	1	2	yes	—
247	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
248	#41	—	712	4	no	—
249	C₂×Q₈	—	356	1	no	—
250	C₂×Q₈	—	356	1	no	—
251	C₂×Q₈	—	178	1	no	—
252	C₂×D₄	—	356	2	no	—
253	C₂×D₄	—	178	1	no	—
254	C₂×D₄	D₇₁₂	1	1	yes	characteristic
255	C₂²×C₄	—	356	2	no	—
256	C₂²×C₄	—	356	1	no	—
257	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
258	Q₁₆	—	178	1	no	—
259	Q₁₆	—	178	1	no	—
260	D₈	—	178	1	no	—
261	D₈	—	178	1	no	—
262	C₂×C₈	—	2	2	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
265	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
266	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
267	C₄⋊C₄	—	356	2	no	—
268	C₄⋊C₄	—	356	2	no	—
269	C₄⋊C₄	—	356	1	no	—
270	C₄⋊C₄	—	356	1	no	—
271	C₄⋊C₄	—	356	1	no	—
272	C₄⋊C₄	—	178	1	no	—
273	C₄⋊C₄	—	178	2	no	—
274	C₄⋊C₄	—	178	1	no	—
275	C₄⋊C₄	—	178	2	no	—
276	C₄⋊C₄	D₇₁₂	1	1	yes	characteristic
277	C₂²⋊C₄	—	356	2	no	—
278	C₂²⋊C₄	—	356	2	no	—
279	C₂²⋊C₄	—	356	1	no	—
280	C₂²⋊C₄	—	356	2	no	—
281	C₂²⋊C₄	D₇₁₂	1	2	yes	—
282	C₄²	—	356	2	no	—
283	C₄²	—	356	1	no	—
284	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
285	C₁₆	—	2	2	no	—
286	C₁₆	—	2	1	no	—
287	C₁₆	#13083573	1	2	yes	complemented
288	C₂³	—	356	1	no	—
289	C₂³	C₂×D₇₁₂	1	2	yes	—
290	Q₈	—	356	1	no	—
291	Q₈	—	356	1	no	—
292	Q₈	—	712	2	no	—
293	Q₈	—	712	2	no	—
294	D₄	—	2	2	no	—
295	D₄	—	356	1	no	—
296	D₄	—	356	1	no	—
297	D₄	—	712	4	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	356	1	no	—
301	C₂×C₄	—	356	1	no	—
302	C₂×C₄	—	356	2	no	—
303	C₂×C₄	—	356	1	no	—
304	C₂×C₄	—	356	2	no	—
305	C₂×C₄	—	712	4	no	—
306	C₂×C₄	—	712	2	no	—
307	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
308	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
309	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
310	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
311	C₈	—	2	1	no	—
312	C₈	—	2	2	no	—
313	C₈	C₂×#13083573	1	1	yes	characteristic
314	C₈	#15668059	1	1	yes	characteristic
315	C₂²	—	2	2	no	—
316	C₂²	—	2	2	no	—
317	C₂²	—	2	2	no	—
318	C₂²	—	712	1	no	—
319	C₂²	—	712	1	no	—
320	C₂²	—	712	1	no	—
321	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
322	C₄	—	2	2	no	—
323	C₄	—	2	1	no	—
324	C₄	—	712	1	no	—
325	C₄	—	712	1	no	—
326	C₄	—	712	2	no	—
327	C₄	—	712	2	no	—
328	C₄	—	712	1	no	—
329	C₄	#19152036	1	2	yes	—
330	C₄	#19152252	1	1	yes	characteristic
331	C₄	#19152415	1	1	yes	characteristic
332	C₂	—	4	2	no	—
333	C₂	—	1424	1	no	—
334	C₂	#26032600	1	1	yes	characteristic
335	C₂	C₂×#19152036	1	1	yes	characteristic
336	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	C₈₉×#2169	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	#26032600	C₂
hypercenter	#2169	D₈₉
hypocenter	C₈₉	#33920
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×D₄ ◁ C₈×D₄ ◁ #2169

Complements

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

N	H ≅ G/N	type
1. #26032683	C₂	semidirect
3. #26022527	C₂	semidirect
4. #26022455	C₂	semidirect
5. #26022444	C₂	semidirect
6. #26022395	C₂	semidirect
7. #26022390	C₂	semidirect
8. #26022365	C₂	semidirect
9. C₄×#15667957	C₂	semidirect
10. #26022040	C₂	semidirect
11. C₈₉×#2169	C₂	semidirect
118. C₁₆×C₈₉	#41	semidirect
178. #2169	D₈₉	semidirect
180. C₈₉	#33920	semidirect
287. C₁₆	#13083573	semidirect

Maximal subgroups

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

subgroup	index
#26032683	2
#26032600	2
#26022527	2
#26022455	2
#26022444	2
#26022395	2
#26022390	2
#26022365	2
C₄×#15667957	2
#26022040	2
C₈₉×#2169	2
#33920	89