Subgroups — #36711808

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	#26032622	C₂	1	2	yes	maximal
2	#26022531	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022458	C₂	1	2	yes	maximal, complemented
4	#26022448	C₂	1	2	yes	maximal, complemented
5	#26022416	C₂	1	2	yes	maximal, complemented
6	#26022340	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022340	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₆×D₃₅₆	C₂	1	1	yes	maximal, characteristic
9	#26022033	C₂	1	1	yes	maximal, characteristic, complemented
10	#366×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×Q₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152289	—	2	2	no	—
13	C₂×C₁₆×D₈₉	C₂²	1	1	yes	characteristic
14	C₂×C₁₆×D₈₉	C₂²	1	1	yes	characteristic
15	#19152000	C₂²	1	2	yes	—
16	#19151937	—	2	2	no	—
17	#19151927	—	2	2	no	—
18	#19151905	—	2	2	no	—
19	#19151836	C₂²	1	2	yes	—
20	#19151836	C₂²	1	2	yes	—
21	C₈×D₃₅₆	C₂²	1	1	yes	characteristic
22	#19151635	C₂²	1	2	yes	—
23	#107×D₈₉	C₂²	1	2	yes	—
24	#19151426	C₂²	1	2	yes	—
25	#19151425	C₂²	1	2	yes	—
26	#19151424	C₂²	1	1	yes	characteristic
27	#19151424	C₂²	1	1	yes	characteristic
28	#19151423	C₂²	1	2	yes	—
29	C₄⋊#13083549	C₂²	1	1	yes	characteristic
30	#19151402	C₂²	1	2	yes	—
31	#19151402	C₂²	1	2	yes	—
32	#19151377	C₂²	1	1	yes	characteristic
33	#19151377	C₂²	1	1	yes	characteristic
34	C₂×Q₃₂×C₈₉	C₂²	1	1	yes	characteristic
35	C₄×Q₁₆×C₈₉	C₂²	1	2	yes	—
36	#366×C₈₉	C₂²	1	1	yes	characteristic
37	#358×C₈₉	C₂²	1	2	yes	—
38	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
39	C₂×C₈×D₈₉	C₂³	1	1	yes	characteristic
40	C₂×C₈×D₈₉	C₂³	1	1	yes	characteristic
41	#15668006	—	4	2	no	—
42	#15668004	—	2	2	no	—
43	#15668002	—	4	2	no	—
44	#15667997	—	4	2	no	—
45	#15667988	—	2	2	no	—
46	#15667988	—	2	2	no	—
47	C₄×D₃₅₆	D₄	1	1	yes	characteristic
48	#15667970	—	2	2	no	—
49	#15667969	—	4	2	no	—
50	C₄⋊C₄×D₈₉	—	2	2	no	—
51	C₂×#13083549	C₂³	1	1	yes	characteristic
52	C₂×#13083549	C₂³	1	1	yes	characteristic
53	C₁₆×D₈₉	—	2	2	no	—
54	C₁₆×D₈₉	—	2	2	no	—
55	#15667946	C₂³	1	2	yes	—
56	#15667946	C₂³	1	2	yes	—
57	#15667945	—	2	2	no	—
58	#15667944	—	2	2	no	—
59	#15667943	—	2	2	no	—
60	C₄⋊#10623494	D₄	1	1	yes	characteristic
61	#15667936	—	2	2	no	—
62	#15667936	—	2	2	no	—
63	#15667924	D₄	1	1	yes	characteristic
64	#15667924	D₄	1	1	yes	characteristic
65	C₂×Q₁₆×C₈₉	C₂³	1	2	yes	—
66	C₄×Q₈×C₈₉	—	2	2	no	—
67	Q₃₂×C₈₉	—	2	2	no	—
68	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
69	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
70	#107×C₈₉	C₂³	1	2	yes	—
71	#103×C₈₉	—	2	2	no	—
72	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
73	C₂×D₃₅₆	D₈	1	1	yes	characteristic
74	C₂×C₄×D₈₉	—	4	2	no	—
75	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
76	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
77	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
78	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
79	C₈×D₈₉	—	2	1	no	—
80	C₈×D₈₉	—	2	1	no	—
81	C₈×D₈₉	—	2	1	no	—
82	C₈×D₈₉	—	2	1	no	—
83	C₄⋊Dic₈₉	—	2	2	no	—
84	C₄⋊Dic₈₉	—	2	2	no	—
85	C₄⋊Dic₈₉	D₈	1	1	yes	characteristic
86	#13083553	—	4	2	no	—
87	#13083553	—	4	2	no	—
88	#13083553	—	4	2	no	—
89	#13083553	—	4	2	no	—
90	#13083553	—	4	2	no	—
91	#13083551	—	4	2	no	—
92	#13083551	—	4	2	no	—
93	#13083551	—	4	2	no	—
94	#13083551	D₈	1	1	yes	characteristic
95	#13083551	D₈	1	1	yes	characteristic
96	C₄×Dic₈₉	—	4	2	no	—
97	#13083549	—	2	1	no	—
98	#13083549	—	2	1	no	—
99	C₂×Q₈×C₈₉	—	2	2	no	—
100	Q₁₆×C₈₉	—	2	2	no	—
101	Q₁₆×C₈₉	—	2	2	no	—
102	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
103	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
104	C₄⋊C₄×C₈₉	—	2	2	no	—
105	C₄⋊C₄×C₈₉	—	4	2	no	—
106	C₄²×C₈₉	—	4	2	no	—
107	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
108	C₁₆×C₈₉	—	2	1	no	—
109	C₁₆×C₈₉	#41	1	2	yes	—
110	C₂²×D₈₉	C₂×D₈	1	1	yes	characteristic
111	C₂²×D₈₉	C₂×D₈	1	1	yes	characteristic
112	Q₈×C₈₉	—	4	2	no	—
113	Q₈×C₈₉	—	4	2	no	—
114	C₂×C₄×C₈₉	—	4	2	no	—
115	C₂×C₄×C₈₉	—	4	2	no	—
116	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
117	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
118	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
119	C₂×Dic₈₉	—	4	2	no	—
120	C₂×Dic₈₉	—	4	2	no	—
121	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
122	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
123	D₃₅₆	—	2	2	no	—
124	C₄×D₈₉	—	2	1	no	—
125	C₄×D₈₉	—	2	1	no	—
126	C₄×D₈₉	—	2	1	no	—
127	C₄×D₈₉	—	2	1	no	—
128	C₄×D₈₉	—	8	4	no	—
129	C₈×C₈₉	—	2	1	no	—
130	C₈×C₈₉	C₂×#41	1	1	yes	characteristic
131	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
132	#10623494	—	2	1	no	—
133	#10623494	—	2	1	no	—
134	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
135	C₂×D₈₉	—	2	1	no	—
136	C₂×D₈₉	—	2	1	no	—
137	C₂×D₈₉	—	2	1	no	—
138	C₂×D₈₉	—	2	1	no	—
139	C₂×D₈₉	—	2	1	no	—
140	C₂×D₈₉	#508	1	2	yes	—
141	C₄×C₈₉	—	2	1	no	—
142	C₄×C₈₉	—	2	1	no	—
143	C₄×C₈₉	—	8	2	no	—
144	C₄×C₈₉	—	8	2	no	—
145	C₄×C₈₉	#547	1	1	yes	characteristic
146	C₄×C₈₉	#578	1	1	yes	characteristic
147	Dic₈₉	—	2	1	no	—
148	Dic₈₉	—	2	1	no	—
149	Dic₈₉	—	8	2	no	—
150	Dic₈₉	—	8	2	no	—
151	#33903	—	89	1	no	maximal, Sylow
152	C₂×C₈₉	#3334	1	1	yes	characteristic
153	C₂×C₈₉	C₂×#508	1	1	yes	characteristic
154	C₂×C₈₉	#3417	1	1	yes	characteristic
155	D₈₉	—	2	2	no	—
156	D₈₉	—	4	1	no	—
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	D₈₉	1	1	yes	characteristic, complemented
165	D₄×C₁₆	—	89	1	no	—
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	C₂×D₈₉	1	1	yes	characteristic
171	C₂²×C₁₆	—	89	1	no	—
172	C₂²×C₁₆	—	89	1	no	—
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	C₂×D₈₉	1	2	yes	—
180	C₈×D₄	—	89	1	no	—
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	C₂×D₈₉	1	1	yes	characteristic
189	C₄⋊C₁₆	—	89	1	no	—
190	#358	—	89	2	no	—
191	#358	—	89	2	no	—
192	#358	C₂×D₈₉	1	2	yes	—
193	C₂²⋊C₁₆	—	89	1	no	—
194	C₂²⋊C₁₆	—	89	1	no	—
195	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
196	C₂×Q₁₆	C₂²×D₈₉	1	2	yes	—
197	C₂²×C₈	—	89	1	no	—
198	C₂²×C₈	—	89	1	no	—
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	—	2	2	no	—
206	C₄×D₄	—	89	1	no	—
207	#117	—	356	2	no	—
208	#117	—	178	2	no	—
209	C₂×C₄⋊C₄	—	178	2	no	—
210	Q₃₂	—	2	2	no	—
211	C₂×C₁₆	—	89	1	no	—
212	C₂×C₁₆	—	89	1	no	—
213	C₂×C₁₆	—	178	2	no	—
214	C₂×C₁₆	—	178	2	no	—
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	—	178	2	no	—
220	#107	C₂²×D₈₉	1	2	yes	—
221	#106	—	178	2	no	—
222	#106	—	178	2	no	—
223	C₄⋊C₈	—	89	1	no	—
224	#103	—	2	2	no	—
225	#103	—	178	2	no	—
226	#103	—	178	2	no	—
227	C₂²⋊C₈	—	89	1	no	—
228	C₂²⋊C₈	—	89	1	no	—
229	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
230	C₂×Q₈	—	2	2	no	—
231	C₂×D₄	—	89	1	no	—
232	C₂²×C₄	—	89	1	no	—
233	C₂²×C₄	—	89	1	no	—
234	C₂²×C₄	—	356	2	no	—
235	Q₁₆	—	2	2	no	—
236	Q₁₆	—	2	2	no	—
237	C₂×C₈	—	89	1	no	—
238	C₂×C₈	—	89	1	no	—
239	C₂×C₈	—	178	1	no	—
240	C₂×C₈	—	178	1	no	—
241	C₂×C₈	—	178	1	no	—
242	C₂×C₈	—	178	1	no	—
243	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
244	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	4	2	no	—
247	C₄⋊C₄	—	89	1	no	—
248	C₄⋊C₄	—	356	2	no	—
249	C₄⋊C₄	—	356	2	no	—
250	C₄⋊C₄	—	356	2	no	—
251	C₄⋊C₄	—	356	2	no	—
252	C₄⋊C₄	—	356	2	no	—
253	C₄⋊C₄	—	178	2	no	—
254	C₄⋊C₄	—	178	2	no	—
255	C₂²⋊C₄	—	89	1	no	—
256	C₂²⋊C₄	—	89	1	no	—
257	C₂²⋊C₄	—	356	2	no	—
258	C₂²⋊C₄	—	356	2	no	—
259	C₂²⋊C₄	—	356	2	no	—
260	C₄²	—	4	2	no	—
261	C₄²	—	356	2	no	—
262	C₄²	D₄×D₈₉	1	1	yes	characteristic
263	C₁₆	—	2	1	no	—
264	C₁₆	—	178	1	no	—
265	C₁₆	—	178	1	no	—
266	C₁₆	#13083574	1	2	yes	—
267	C₂³	—	89	1	no	—
268	C₂³	—	89	1	no	—
269	Q₈	—	4	2	no	—
270	Q₈	—	4	2	no	—
271	D₄	—	178	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	89	1	no	—
275	C₂×C₄	—	89	1	no	—
276	C₂×C₄	—	356	2	no	—
277	C₂×C₄	—	356	2	no	—
278	C₂×C₄	—	178	1	no	—
279	C₂×C₄	—	178	1	no	—
280	C₂×C₄	—	178	1	no	—
281	C₂×C₄	—	178	1	no	—
282	C₂×C₄	—	712	4	no	—
283	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
284	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
285	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
286	C₈	—	2	1	no	—
287	C₈	—	178	1	no	—
288	C₈	—	178	1	no	—
289	C₈	C₂×#13083574	1	1	yes	characteristic
290	C₈	#15668059	1	1	yes	characteristic
291	C₂²	—	89	2	no	—
292	C₂²	—	178	1	no	—
293	C₂²	—	178	1	no	—
294	C₂²	—	178	1	no	—
295	C₂²	—	178	1	no	—
296	C₂²	—	178	1	no	—
297	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	8	2	no	—
301	C₄	—	8	2	no	—
302	C₄	—	178	1	no	—
303	C₄	—	178	1	no	—
304	C₄	—	712	2	no	—
305	C₄	—	712	2	no	—
306	C₄	#19152253	1	1	yes	characteristic
307	C₄	#19152412	1	1	yes	characteristic
308	C₂	—	356	1	no	—
309	C₂	—	178	2	no	—
310	C₂	#26032601	1	1	yes	characteristic
311	C₂	#508×D₈₉	1	1	yes	characteristic
312	C₂	#26033401	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₈₉
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₄×Q₃₂×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂×D₃₅₆	D₈
hypercenter	C₄×Q₃₂	D₈₉
hypocenter	C₈₉	#33903
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₃₂

Complements

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

N	H ≅ G/N	type
2. #26022531	C₂	semidirect
3. #26022458	C₂	semidirect
4. #26022448	C₂	semidirect
5. #26022416	C₂	semidirect
6. #26022340	C₂	semidirect
7. #26022340	C₂	semidirect
9. #26022033	C₂	semidirect
10. #366×D₈₉	C₂	semidirect
11. C₄×Q₃₂×C₈₉	C₂	semidirect
164. C₄×Q₃₂	D₈₉	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
#26032622	2
#26022531	2
#26022458	2
#26022448	2
#26022416	2
#26022340	2
C₁₆×D₃₅₆	2
#26022033	2
#366×D₈₉	2
C₄×Q₃₂×C₈₉	2
#33903	89