Subgroups — #36709599

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	#26029105	C₂	1	1	yes	maximal, characteristic
2	#26022457	C₂	1	2	yes	maximal, complemented
3	#26022406	C₂	1	2	yes	maximal, complemented
4	#26022294	C₂	1	2	yes	maximal, complemented
5	#26022180	C₂	1	2	yes	maximal
6	#26022114	C₂	1	2	yes	maximal, complemented
7	C₄×#15667955	C₂	1	2	yes	maximal, complemented
8	#26022059	C₂	1	1	yes	maximal, characteristic, complemented
9	C₈₉×#3177	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×D₇₁₂	C₂²	1	1	yes	characteristic
11	#19152072	—	2	1	no	—
12	C₂×#15667955	C₂²	1	2	yes	—
13	C₂×#15667955	C₂²	1	2	yes	—
14	#19151711	—	2	1	no	—
15	#19151711	—	2	1	no	—
16	C₄×D₇₁₂	C₂²	1	2	yes	—
17	C₄×D₇₁₂	C₂²	1	2	yes	—
18	#19151636	C₂²	1	1	yes	characteristic
19	#19151604	—	2	1	no	—
20	#19151425	C₂²	1	2	yes	complemented
21	#19151423	C₂²	1	2	yes	complemented
22	C₄⋊#13083549	C₂²	1	2	yes	complemented
23	#19151402	C₂²	1	2	yes	complemented
24	#19151402	C₂²	1	2	yes	complemented
25	#19151398	C₂²	1	2	yes	—
26	#19151398	C₂²	1	2	yes	—
27	C₂²⋊#13083549	C₂²	1	2	yes	—
28	#19151374	C₂²	1	1	yes	characteristic, complemented
29	C₄×#13083549	C₂²	1	1	yes	characteristic, complemented
30	#541×C₈₉	—	2	1	no	—
31	#499×C₈₉	C₂²	1	1	yes	characteristic, complemented
32	#494×C₈₉	C₂²	1	1	yes	characteristic, complemented
33	#491×C₈₉	—	2	1	no	—
34	#470×C₈₉	—	2	1	no	—
35	#467×C₈₉	C₂²	1	2	yes	—
36	#441×C₈₉	—	2	1	no	—
37	C₄×Q₁₆×C₈₉	C₂²	1	2	yes	complemented
38	#405×C₈₉	C₂²	1	1	yes	characteristic
39	C₂²×D₃₅₆	—	2	1	no	—
40	C₂×D₇₁₂	—	2	4	no	—
41	C₂×D₇₁₂	—	2	1	no	—
42	C₂×D₇₁₂	C₂³	1	2	yes	—
43	C₄×D₃₅₆	—	2	2	no	—
44	C₄×D₃₅₆	—	2	2	no	—
45	#15667974	—	4	2	no	—
46	#15667971	—	2	1	no	—
47	C₂²⋊C₄×D₈₉	—	4	2	no	—
48	#15667955	—	2	4	no	—
49	#15667955	—	2	4	no	—
50	C₂×#13083549	C₂³	1	2	yes	—
51	C₂×#13083549	C₂³	1	2	yes	—
52	#15667946	C₂³	1	2	yes	—
53	#15667944	—	2	1	no	—
54	#15667934	—	2	2	no	—
55	#15667934	—	2	2	no	—
56	C₂×Q₁₆×C₈₉	—	2	1	no	—
57	C₂×Q₁₆×C₈₉	C₂³	1	2	yes	—
58	C₂×M₁₆×C₈₉	D₄	1	1	yes	characteristic
59	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
60	C₄⋊Q₈×C₈₉	—	2	1	no	—
61	#126×C₈₉	—	4	2	no	—
62	#125×C₈₉	—	2	1	no	—
63	#125×C₈₉	—	4	2	no	—
64	C₂²⋊Q₈×C₈₉	—	2	2	no	—
65	C₄×Q₈×C₈₉	—	2	2	no	—
66	#117×C₈₉	D₄	1	1	yes	characteristic
67	#107×C₈₉	C₂³	1	2	yes	—
68	#106×C₈₉	—	2	1	no	—
69	#103×C₈₉	—	2	2	no	—
70	#103×C₈₉	—	2	2	no	—
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×D₃₅₆	—	2	4	no	—
77	C₂×D₃₅₆	—	2	2	no	—
78	C₂×D₃₅₆	—	4	2	no	—
79	C₂×D₃₅₆	—	4	1	no	—
80	C₂×C₄×D₈₉	—	4	2	no	—
81	C₂×C₄×D₈₉	—	4	2	no	—
82	D₇₁₂	—	2	4	no	—
83	D₇₁₂	—	2	2	no	—
84	D₇₁₂	#41	1	4	yes	—
85	C₄⋊Dic₈₉	—	2	2	no	—
86	C₂²⋊Dic₈₉	—	4	1	no	—
87	#13083551	—	4	2	no	—
88	#13083551	—	4	2	no	—
89	C₄×Dic₈₉	—	4	1	no	—
90	#13083549	—	2	2	no	—
91	#13083549	—	2	2	no	—
92	C₂×Q₈×C₈₉	—	2	2	no	—
93	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
94	Q₁₆×C₈₉	—	2	2	no	—
95	Q₁₆×C₈₉	—	2	2	no	—
96	Q₁₆×C₈₉	—	4	2	no	—
97	M₁₆×C₈₉	—	4	1	no	—
98	C₂×C₈×C₈₉	—	2	2	no	—
99	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
103	C₄⋊C₄×C₈₉	—	2	2	no	—
104	C₄⋊C₄×C₈₉	—	4	2	no	—
105	C₄⋊C₄×C₈₉	—	4	2	no	—
106	C₄⋊C₄×C₈₉	—	4	2	no	—
107	C₄⋊C₄×C₈₉	—	4	2	no	—
108	C₄⋊C₄×C₈₉	D₈	1	2	yes	—
109	C₂²⋊C₄×C₈₉	—	4	2	no	—
110	C₂²⋊C₄×C₈₉	D₈	1	2	yes	—
111	C₄²×C₈₉	—	4	2	no	—
112	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
113	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
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	Q₈×C₈₉	—	4	2	no	—
119	Q₈×C₈₉	—	4	2	no	—
120	C₂×C₄×C₈₉	—	2	1	no	—
121	C₂×C₄×C₈₉	—	2	1	no	—
122	C₂×C₄×C₈₉	—	4	2	no	—
123	C₂×C₄×C₈₉	—	4	2	no	—
124	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
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	C₂×Dic₈₉	—	4	2	no	—
129	D₃₅₆	—	2	4	no	—
130	D₃₅₆	—	4	4	no	—
131	D₃₅₆	—	4	2	no	—
132	C₄×D₈₉	—	16	2	no	—
133	C₄×D₈₉	—	16	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	—	8	2	no	—
149	C₄×C₈₉	—	8	2	no	—
150	C₄×C₈₉	#525	1	1	yes	characteristic
151	C₄×C₈₉	#575	1	1	yes	characteristic
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	D₈₉	1	1	yes	characteristic, complemented
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	—	89	1	no	—
167	C₈₉	#33683	1	1	yes	characteristic, complemented, Sylow
168	C₂²×D₈	—	89	1	no	—
169	#541	—	2	1	no	—
170	#518	—	178	1	no	—
171	C₂×SD₃₂	—	89	2	no	—
172	C₂×SD₃₂	—	89	2	no	—
173	#499	C₂×D₈₉	1	1	yes	characteristic, complemented
174	#494	C₂×D₈₉	1	1	yes	characteristic, complemented
175	#491	—	2	1	no	—
176	#470	—	2	1	no	—
177	#467	C₂×D₈₉	1	2	yes	—
178	#442	—	178	1	no	—
179	#442	—	178	1	no	—
180	#441	—	2	1	no	—
181	C₄×Q₁₆	C₂×D₈₉	1	2	yes	complemented
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	—	89	2	no	—
190	#358	—	89	2	no	—
191	#358	—	89	2	no	—
192	#357	—	89	2	no	—
193	#357	—	89	2	no	—
194	C₂²⋊C₁₆	—	89	2	no	—
195	#346	—	89	1	no	—
196	C₄×C₁₆	—	89	1	no	—
197	C₂²×D₄	—	178	1	no	—
198	C₂×Q₁₆	—	2	1	no	—
199	C₂×Q₁₆	C₂²×D₈₉	1	2	yes	—
200	C₂×D₈	—	89	2	no	—
201	C₂×D₈	—	178	4	no	—
202	C₂×D₈	—	178	1	no	—
203	C₂×M₁₆	C₂²⋊D₈₉	1	1	yes	characteristic
204	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
205	C₄⋊Q₈	—	2	1	no	—
206	#126	—	4	2	no	—
207	#125	—	2	1	no	—
208	#125	—	4	2	no	—
209	C₂²⋊Q₈	—	2	2	no	—
210	C₄×Q₈	—	2	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	C₂²⋊D₈₉	1	1	yes	characteristic
216	C₂×C₂²⋊C₄	—	356	2	no	—
217	SD₃₂	—	178	4	no	—
218	SD₃₂	—	178	4	no	—
219	C₂×C₁₆	—	89	2	no	—
220	C₂×C₁₆	—	89	2	no	—
221	#107	—	89	2	no	—
222	#107	C₂²×D₈₉	1	2	yes	—
223	#106	—	2	1	no	—
224	#106	—	178	1	no	—
225	#103	—	2	2	no	—
226	#103	—	2	2	no	—
227	#102	—	178	2	no	—
228	#102	—	178	2	no	—
229	#97	C₂²⋊D₈₉	1	1	yes	characteristic
230	#97	C₂²⋊D₈₉	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	—	2	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×C₂²⋊D₈₉	1	1	yes	characteristic
242	Q₁₆	—	2	2	no	—
243	Q₁₆	—	2	2	no	—
244	Q₁₆	—	4	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×C₂²⋊D₈₉	1	1	yes	characteristic
251	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
252	C₂×C₈	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
253	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	—	4	2	no	—
259	C₄⋊C₄	—	178	2	no	—
260	C₄⋊C₄	#13083559	1	2	yes	—
261	C₂²⋊C₄	—	4	2	no	—
262	C₂²⋊C₄	—	356	1	no	—
263	C₂²⋊C₄	—	356	2	no	—
264	C₂²⋊C₄	—	356	2	no	—
265	C₂²⋊C₄	#13083559	1	2	yes	—
266	C₄²	—	4	2	no	—
267	C₄²	—	356	1	no	—
268	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
269	C₄²	C₂×C₂²⋊D₈₉	1	1	yes	characteristic
270	C₁₆	—	178	2	no	—
271	C₁₆	—	178	2	no	—
272	C₂³	—	356	4	no	—
273	C₂³	—	356	2	no	—
274	C₂³	—	1424	2	no	—
275	C₂³	C₂×#13083559	1	1	yes	characteristic
276	Q₈	—	4	2	no	—
277	Q₈	—	4	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	—	4	2	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	—	356	2	no	—
286	C₂×C₄	—	1424	2	no	—
287	C₂×C₄	—	1424	2	no	—
288	C₂×C₄	C₂×#13083559	1	2	yes	—
289	C₂×C₄	C₂×#13083559	1	1	yes	characteristic
290	C₂×C₄	C₂×#13083559	1	2	yes	—
291	C₂×C₄	C₂²×C₂²⋊D₈₉	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×#13083559	1	1	yes	characteristic
303	C₄	—	2	1	no	—
304	C₄	—	2	2	no	—
305	C₄	—	2	2	no	—
306	C₄	—	8	2	no	—
307	C₄	—	8	2	no	—
308	C₄	—	712	2	no	—
309	C₄	#19152105	1	1	yes	characteristic
310	C₄	#19152396	1	1	yes	characteristic
311	C₂	—	4	1	no	—
312	C₂	—	712	4	no	—
313	C₂	#26030248	1	1	yes	characteristic
314	C₂	#26033396	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂²×#13083559
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#3177	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂²×D₇₁₂	C₂²
hypercenter	#3177	D₈₉
hypocenter	C₈₉	#33683
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ #117 ◁ #405 ◁ #3177

Complements

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

N	H ≅ G/N	type
2. #26022457	C₂	semidirect
3. #26022406	C₂	semidirect
4. #26022294	C₂	semidirect
6. #26022114	C₂	semidirect
7. C₄×#15667955	C₂	semidirect
8. #26022059	C₂	semidirect
9. C₈₉×#3177	C₂	semidirect
20. #19151425	C₂²	semidirect
21. #19151423	C₂²	semidirect
22. C₄⋊#13083549	C₂²	semidirect
23. #19151402	C₂²	semidirect
24. #19151402	C₂²	semidirect
28. #19151374	C₂²	semidirect
29. C₄×#13083549	C₂²	semidirect
31. #499×C₈₉	C₂²	semidirect
32. #494×C₈₉	C₂²	semidirect
37. C₄×Q₁₆×C₈₉	C₂²	semidirect
158. #3177	D₈₉	semidirect
167. C₈₉	#33683	semidirect
173. #499	C₂×D₈₉	semidirect
174. #494	C₂×D₈₉	semidirect
181. C₄×Q₁₆	C₂×D₈₉	semidirect

Maximal subgroups

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

subgroup	index
#26029105	2
#26022457	2
#26022406	2
#26022294	2
#26022180	2
#26022114	2
C₄×#15667955	2
#26022059	2
C₈₉×#3177	2
#33683	89