Subgroups — #36709605

Order 22784.

556 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030911	C₂	1	1	yes	maximal, characteristic, complemented
2	#26029105	C₂	1	1	yes	maximal, characteristic
3	#26022461	C₂	1	2	yes	maximal, complemented
4	#26022410	C₂	1	2	yes	maximal, complemented
5	#26022298	C₂	1	2	yes	maximal, complemented
6	#26022183	C₂	1	2	yes	maximal
7	#26022118	C₂	1	2	yes	maximal, complemented
8	C₄×#15667957	C₂	1	2	yes	maximal, complemented
9	C₈₉×#2164	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₇₁₂	C₂²	1	1	yes	characteristic
11	#19152210	—	2	1	no	—
12	#19152072	—	2	1	no	—
13	C₂×#15667957	C₂²	1	2	yes	—
14	C₂×#15667957	C₂²	1	2	yes	—
15	#19152000	C₂²	1	1	yes	characteristic, complemented
16	#19151978	C₂²	1	1	yes	characteristic, complemented
17	#19151967	—	2	1	no	—
18	#19151856	—	2	1	no	—
19	#19151839	C₂²	1	2	yes	—
20	#19151711	—	2	1	no	—
21	#19151711	—	2	1	no	—
22	#19151706	—	2	1	no	—
23	C₄×Dic₃₅₆	C₂²	1	2	yes	complemented
24	C₄×D₇₁₂	C₂²	1	2	yes	—
25	C₄×D₇₁₂	C₂²	1	2	yes	—
26	#19151636	C₂²	1	1	yes	characteristic
27	#19151604	—	2	1	no	—
28	#19151426	C₂²	1	2	yes	complemented
29	#19151424	C₂²	1	2	yes	complemented
30	#19151403	C₂²	1	2	yes	complemented
31	#19151403	C₂²	1	2	yes	complemented
32	#19151400	C₂²	1	2	yes	—
33	#19151400	C₂²	1	2	yes	—
34	#405×C₈₉	C₂²	1	1	yes	characteristic
35	C₄⋊C₁₆×C₈₉	C₂²	1	2	yes	complemented
36	C₂²⋊C₁₆×C₈₉	C₂²	1	2	yes	—
37	#346×C₈₉	C₂²	1	1	yes	characteristic, complemented
38	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic, complemented
39	C₂²×D₃₅₆	—	2	1	no	—
40	C₂×Dic₃₅₆	—	2	1	no	—
41	C₂×Dic₃₅₆	C₂³	1	2	yes	—
42	C₂×D₇₁₂	—	2	4	no	—
43	C₂×D₇₁₂	—	2	1	no	—
44	C₂×D₇₁₂	C₂³	1	2	yes	—
45	#15668012	—	2	1	no	—
46	#15668005	—	4	2	no	—
47	#15668004	—	2	1	no	—
48	#15668002	—	4	2	no	—
49	#15667991	—	2	2	no	—
50	C₄×Dic₁₇₈	—	2	2	no	—
51	C₄×D₃₅₆	—	2	2	no	—
52	C₄×D₃₅₆	—	2	2	no	—
53	#15667974	—	4	2	no	—
54	#15667971	—	2	1	no	—
55	C₂²⋊C₄×D₈₉	—	4	2	no	—
56	#15667957	—	2	4	no	—
57	#15667957	—	2	4	no	—
58	#15667946	C₂³	1	2	yes	—
59	#15667946	C₂³	1	2	yes	—
60	#15667944	—	2	1	no	—
61	#15667944	—	2	1	no	—
62	#15667937	—	2	2	no	—
63	#15667937	—	2	2	no	—
64	#15667934	—	2	2	no	—
65	#15667934	—	2	2	no	—
66	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
67	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
68	#117×C₈₉	D₄	1	1	yes	characteristic
69	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
70	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
71	#97×C₈₉	D₄	1	1	yes	characteristic
72	#97×C₈₉	D₄	1	1	yes	characteristic
73	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
75	C₂³×D₈₉	—	4	1	no	—
76	C₂×Dic₁₇₈	—	2	2	no	—
77	C₂×D₃₅₆	—	2	4	no	—
78	C₂×D₃₅₆	—	2	2	no	—
79	C₂×D₃₅₆	—	4	2	no	—
80	C₂×D₃₅₆	—	4	1	no	—
81	C₂×C₄×D₈₉	—	4	2	no	—
82	C₂×C₄×D₈₉	—	4	2	no	—
83	Dic₃₅₆	—	2	2	no	—
84	Dic₃₅₆	—	2	2	no	—
85	Dic₃₅₆	—	4	2	no	—
86	D₇₁₂	—	2	4	no	—
87	D₇₁₂	—	2	2	no	—
88	D₇₁₂	#41	1	4	yes	—
89	C₄⋊Dic₈₉	—	2	2	no	—
90	C₄⋊Dic₈₉	—	2	2	no	—
91	C₄⋊Dic₈₉	—	4	2	no	—
92	#13083553	—	4	2	no	—
93	#13083553	—	4	2	no	—
94	#13083553	—	4	2	no	—
95	C₂²⋊Dic₈₉	—	4	1	no	—
96	C₂²⋊Dic₈₉	—	4	2	no	—
97	#13083551	—	4	2	no	—
98	#13083551	—	4	2	no	—
99	C₄×Dic₈₉	—	4	1	no	—
100	C₄×Dic₈₉	—	4	2	no	—
101	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
102	M₁₆×C₈₉	—	4	1	no	—
103	C₂×C₈×C₈₉	—	2	2	no	—
104	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
108	C₄⋊C₄×C₈₉	D₈	1	2	yes	—
109	C₂²⋊C₄×C₈₉	D₈	1	2	yes	—
110	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
111	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
112	C₁₆×C₈₉	—	2	2	no	—
113	C₁₆×C₈₉	—	2	2	no	—
114	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
115	C₂²×D₈₉	—	4	2	no	—
116	C₂²×D₈₉	—	4	4	no	—
117	C₂²×D₈₉	—	16	2	no	—
118	C₂×C₄×C₈₉	—	2	1	no	—
119	C₂×C₄×C₈₉	—	2	1	no	—
120	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
121	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
122	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
123	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
124	C₂×Dic₈₉	—	4	2	no	—
125	C₂×Dic₈₉	—	4	2	no	—
126	C₂×Dic₈₉	—	4	2	no	—
127	D₃₅₆	—	2	4	no	—
128	D₃₅₆	—	4	4	no	—
129	D₃₅₆	—	4	2	no	—
130	C₄×D₈₉	—	16	2	no	—
131	C₄×D₈₉	—	16	2	no	—
132	Dic₁₇₈	—	4	2	no	—
133	Dic₁₇₈	—	4	2	no	—
134	C₈×C₈₉	—	2	1	no	—
135	C₈×C₈₉	—	2	1	no	—
136	C₈×C₈₉	—	2	1	no	—
137	C₈×C₈₉	C₂×#41	1	2	yes	—
138	C₂²×C₈₉	—	2	1	no	—
139	C₂²×C₈₉	—	2	2	no	—
140	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
141	C₂×D₈₉	—	4	4	no	—
142	C₂×D₈₉	—	8	2	no	—
143	C₂×D₈₉	—	8	2	no	—
144	C₂×D₈₉	—	16	4	no	—
145	C₄×C₈₉	—	2	1	no	—
146	C₄×C₈₉	—	2	2	no	—
147	C₄×C₈₉	—	2	2	no	—
148	C₄×C₈₉	#525	1	1	yes	characteristic
149	C₄×C₈₉	#575	1	1	yes	characteristic
150	Dic₈₉	—	8	2	no	—
151	Dic₈₉	—	8	2	no	—
152	Dic₈₉	—	8	2	no	—
153	#33683	—	89	1	no	maximal, Sylow
154	C₂×C₈₉	—	4	1	no	—
155	C₂×C₈₉	#3100	1	1	yes	characteristic
156	C₂×C₈₉	#3416	1	2	yes	—
157	D₈₉	—	8	4	no	—
158	#3177	—	89	1	no	—
159	#2948	—	89	1	no	—
160	#2237	—	89	2	no	—
161	#2228	—	89	2	no	—
162	#2208	—	89	2	no	—
163	#2187	—	89	2	no	—
164	#2175	—	89	2	no	—
165	C₄×SD₃₂	—	89	2	no	—
166	#2164	D₈₉	1	1	yes	characteristic, complemented
167	C₈₉	#33683	1	1	yes	characteristic, complemented, Sylow
168	C₂²×D₈	—	89	1	no	—
169	#541	—	178	1	no	—
170	#518	—	178	1	no	—
171	C₂×SD₃₂	—	89	2	no	—
172	C₂×SD₃₂	—	89	2	no	—
173	#499	—	89	1	no	—
174	#494	—	89	1	no	—
175	#491	—	178	1	no	—
176	#470	—	178	1	no	—
177	#467	—	89	2	no	—
178	#442	—	178	1	no	—
179	#442	—	178	1	no	—
180	#441	—	178	1	no	—
181	C₄×Q₁₆	—	89	2	no	—
182	C₄×D₈	—	89	2	no	—
183	C₄×D₈	—	89	2	no	—
184	#427	—	89	1	no	—
185	#418	—	178	1	no	—
186	#405	C₂×D₈₉	1	1	yes	characteristic
187	#367	—	89	2	no	—
188	#366	—	89	2	no	—
189	C₄⋊C₁₆	C₂×D₈₉	1	2	yes	complemented
190	#358	—	89	2	no	—
191	#358	—	89	2	no	—
192	#357	—	89	2	no	—
193	#357	—	89	2	no	—
194	C₂²⋊C₁₆	C₂×D₈₉	1	2	yes	—
195	#346	C₂×D₈₉	1	1	yes	characteristic, complemented
196	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic, complemented
197	C₂²×D₄	—	178	1	no	—
198	C₂×Q₁₆	—	89	2	no	—
199	C₂×Q₁₆	—	178	1	no	—
200	C₂×D₈	—	89	2	no	—
201	C₂×D₈	—	178	4	no	—
202	C₂×D₈	—	178	1	no	—
203	C₂×M₁₆	D₃₅₆	1	1	yes	characteristic
204	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
205	C₄⋊Q₈	—	178	1	no	—
206	#126	—	356	2	no	—
207	#125	—	356	2	no	—
208	#125	—	178	1	no	—
209	C₂²⋊Q₈	—	178	2	no	—
210	C₄×Q₈	—	178	2	no	—
211	C₄×D₄	—	356	2	no	—
212	C₄×D₄	—	178	2	no	—
213	C₄×D₄	—	178	2	no	—
214	#117	—	178	1	no	—
215	#117	D₃₅₆	1	1	yes	characteristic
216	C₂×C₂²⋊C₄	—	356	2	no	—
217	SD₃₂	—	178	4	no	—
218	SD₃₂	—	178	4	no	—
219	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
220	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
221	#107	—	89	2	no	—
222	#107	—	89	2	no	—
223	#106	—	178	1	no	—
224	#106	—	178	1	no	—
225	#103	—	178	2	no	—
226	#103	—	178	2	no	—
227	#102	—	178	2	no	—
228	#102	—	178	2	no	—
229	#97	D₃₅₆	1	1	yes	characteristic
230	#97	D₃₅₆	1	1	yes	characteristic
231	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
232	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
233	C₂⁴	—	356	1	no	—
234	C₂×Q₈	—	178	2	no	—
235	C₂×D₄	—	356	2	no	—
236	C₂×D₄	—	356	1	no	—
237	C₂×D₄	—	178	4	no	—
238	C₂×D₄	—	178	2	no	—
239	C₂²×C₄	—	356	2	no	—
240	C₂²×C₄	—	356	2	no	—
241	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
242	Q₁₆	—	356	2	no	—
243	Q₁₆	—	178	2	no	—
244	Q₁₆	—	178	2	no	—
245	D₈	—	89	4	no	—
246	D₈	—	178	4	no	—
247	D₈	—	178	2	no	—
248	M₁₆	—	4	1	no	—
249	C₂×C₈	—	2	2	no	—
250	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
251	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
252	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
253	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
254	C₄⋊C₄	—	356	2	no	—
255	C₄⋊C₄	—	356	2	no	—
256	C₄⋊C₄	—	356	2	no	—
257	C₄⋊C₄	—	356	2	no	—
258	C₄⋊C₄	—	178	2	no	—
259	C₄⋊C₄	—	178	2	no	—
260	C₄⋊C₄	D₇₁₂	1	2	yes	—
261	C₂²⋊C₄	—	356	1	no	—
262	C₂²⋊C₄	—	356	2	no	—
263	C₂²⋊C₄	—	356	2	no	—
264	C₂²⋊C₄	—	356	2	no	—
265	C₂²⋊C₄	D₇₁₂	1	2	yes	—
266	C₄²	—	356	1	no	—
267	C₄²	—	356	2	no	—
268	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
269	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
270	C₁₆	—	2	2	no	—
271	C₁₆	—	2	2	no	—
272	C₂³	—	356	4	no	—
273	C₂³	—	356	2	no	—
274	C₂³	—	1424	2	no	—
275	C₂³	C₂×D₇₁₂	1	1	yes	characteristic
276	Q₈	—	356	2	no	—
277	Q₈	—	356	2	no	—
278	D₄	—	356	4	no	—
279	D₄	—	356	2	no	—
280	D₄	—	178	4	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	356	2	no	—
284	C₂×C₄	—	356	2	no	—
285	C₂×C₄	—	356	2	no	—
286	C₂×C₄	—	1424	2	no	—
287	C₂×C₄	—	1424	2	no	—
288	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
289	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
290	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
291	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
292	C₈	—	2	1	no	—
293	C₈	—	2	1	no	—
294	C₈	—	2	1	no	—
295	C₈	#41×D₈₉	1	2	yes	—
296	C₂²	—	2	1	no	—
297	C₂²	—	2	2	no	—
298	C₂²	—	356	4	no	—
299	C₂²	—	1424	4	no	—
300	C₂²	—	712	2	no	—
301	C₂²	—	712	2	no	—
302	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
303	C₄	—	2	1	no	—
304	C₄	—	2	2	no	—
305	C₄	—	2	2	no	—
306	C₄	—	712	2	no	—
307	C₄	—	712	2	no	—
308	C₄	—	712	2	no	—
309	C₄	#19152110	1	1	yes	characteristic
310	C₄	#19152398	1	1	yes	characteristic
311	C₂	—	4	1	no	—
312	C₂	—	712	4	no	—
313	C₂	#26030253	1	1	yes	characteristic
314	C₂	#26033400	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×D₇₁₂
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2164	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂²×D₇₁₂	C₂²
hypercenter	#2164	D₈₉
hypocenter	C₈₉	#33683
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #2164

Complements

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

N	H ≅ G/N	type
1. #26030911	C₂	semidirect
3. #26022461	C₂	semidirect
4. #26022410	C₂	semidirect
5. #26022298	C₂	semidirect
7. #26022118	C₂	semidirect
8. C₄×#15667957	C₂	semidirect
9. C₈₉×#2164	C₂	semidirect
15. #19152000	C₂²	semidirect
16. #19151978	C₂²	semidirect
23. C₄×Dic₃₅₆	C₂²	semidirect
28. #19151426	C₂²	semidirect
29. #19151424	C₂²	semidirect
30. #19151403	C₂²	semidirect
31. #19151403	C₂²	semidirect
35. C₄⋊C₁₆×C₈₉	C₂²	semidirect
37. #346×C₈₉	C₂²	semidirect
38. C₄×C₁₆×C₈₉	C₂²	semidirect
166. #2164	D₈₉	semidirect
167. C₈₉	#33683	semidirect
189. C₄⋊C₁₆	C₂×D₈₉	semidirect
195. #346	C₂×D₈₉	semidirect
196. C₄×C₁₆	C₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26030911	2
#26029105	2
#26022461	2
#26022410	2
#26022298	2
#26022183	2
#26022118	2
C₄×#15667957	2
C₈₉×#2164	2
#33683	89