Subgroups — #36711827

Order 22784.

484 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26032641	C₂	1	2	yes	maximal
2	#26022526	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022453	C₂	1	2	yes	maximal, complemented
4	#26022452	C₂	1	2	yes	maximal, complemented
5	#26022402	C₂	1	2	yes	maximal, complemented
6	#26022340	C₂	1	1	yes	maximal, characteristic, complemented
7	Q₃₂×Dic₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₆×C₂²⋊D₈₉	C₂	1	1	yes	maximal, characteristic
9	C₂×#19151424	C₂	1	1	yes	maximal, characteristic, complemented
10	#366×D₈₉	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#2216	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152306	—	2	2	no	—
13	C₂×C₁₆×D₈₉	C₂²	1	1	yes	characteristic
14	#19152002	C₂²	1	2	yes	—
15	#19151932	—	2	2	no	—
16	#19151931	—	2	2	no	—
17	#19151891	—	2	2	no	—
18	#19151836	C₂²	1	2	yes	—
19	Q₁₆×Dic₈₉	C₂²	1	2	yes	—
20	C₈×C₂²⋊D₈₉	C₂²	1	1	yes	characteristic
21	C₂×#15667946	C₂²	1	2	yes	—
22	#107×D₈₉	C₂²	1	2	yes	—
23	#19151424	C₂²	1	2	yes	—
24	#19151424	C₂²	1	1	yes	characteristic
25	#19151424	C₂²	1	1	yes	characteristic
26	#19151423	C₂²	1	2	yes	—
27	#19151423	C₂²	1	2	yes	—
28	#19151416	C₂²	1	1	yes	characteristic
29	#19151402	C₂²	1	2	yes	—
30	#19151402	C₂²	1	2	yes	—
31	C₂²⋊#13083549	C₂²	1	1	yes	characteristic
32	#19151377	C₂²	1	1	yes	characteristic
33	C₁₆×Dic₈₉	C₂²	1	1	yes	characteristic
34	C₂×Q₃₂×C₈₉	C₂²	1	1	yes	characteristic
35	C₂²×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
36	#467×C₈₉	C₂²	1	2	yes	—
37	#366×C₈₉	C₂²	1	1	yes	characteristic
38	#358×C₈₉	C₂²	1	2	yes	—
39	C₂×C₈×D₈₉	C₂³	1	1	yes	characteristic
40	#15668011	—	2	2	no	—
41	#15668002	—	4	2	no	—
42	#15667992	—	4	2	no	—
43	#15667989	—	4	2	no	—
44	#15667988	—	2	2	no	—
45	Q₈×Dic₈₉	—	2	2	no	—
46	C₄×C₂²⋊D₈₉	D₄	1	1	yes	characteristic
47	#15667972	—	4	2	no	—
48	C₂×C₄⋊Dic₈₉	—	2	2	no	—
49	C₄⋊C₄×D₈₉	—	2	2	no	—
50	C₂×#13083549	C₂³	1	1	yes	characteristic
51	C₂×#13083549	C₂³	1	1	yes	characteristic
52	C₁₆×D₈₉	—	2	2	no	—
53	#15667946	—	2	2	no	—
54	#15667946	C₂³	1	2	yes	—
55	#15667946	C₂³	1	2	yes	—
56	#15667945	—	2	2	no	—
57	#15667945	—	2	2	no	—
58	#15667941	D₄	1	1	yes	characteristic
59	#15667936	—	2	2	no	—
60	#15667936	—	2	2	no	—
61	C₂²⋊#10623494	D₄	1	1	yes	characteristic
62	#15667924	D₄	1	1	yes	characteristic
63	C₈×Dic₈₉	C₂³	1	1	yes	characteristic
64	C₂×Q₁₆×C₈₉	C₂³	1	2	yes	—
65	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
66	C₂²⋊Q₈×C₈₉	—	2	2	no	—
67	Q₃₂×C₈₉	—	2	2	no	—
68	C₂×C₁₆×C₈₉	—	2	2	no	—
69	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
70	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
71	#107×C₈₉	C₂³	1	2	yes	—
72	#103×C₈₉	—	2	2	no	—
73	C₂×Dic₁₇₈	—	4	2	no	—
74	C₂×C₂²⋊D₈₉	D₈	1	1	yes	characteristic
75	C₂²×Dic₈₉	—	4	2	no	—
76	C₂×C₄×D₈₉	—	4	2	no	—
77	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
78	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
79	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
80	C₈×D₈₉	—	2	1	no	—
81	C₈×D₈₉	—	2	1	no	—
82	C₄⋊Dic₈₉	—	2	2	no	—
83	C₄⋊Dic₈₉	—	2	2	no	—
84	C₄⋊Dic₈₉	—	4	2	no	—
85	C₄⋊Dic₈₉	—	4	2	no	—
86	#13083553	—	4	2	no	—
87	#13083553	—	4	2	no	—
88	#13083553	—	4	2	no	—
89	#13083553	D₈	1	1	yes	characteristic
90	C₂²⋊Dic₈₉	D₈	1	1	yes	characteristic
91	#13083551	—	4	2	no	—
92	#13083551	D₈	1	1	yes	characteristic
93	C₄×Dic₈₉	—	4	2	no	—
94	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
95	#13083549	—	2	1	no	—
96	#13083549	—	2	1	no	—
97	C₂×Q₈×C₈₉	—	2	2	no	—
98	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
99	Q₁₆×C₈₉	—	2	2	no	—
100	Q₁₆×C₈₉	—	2	2	no	—
101	C₂×C₈×C₈₉	—	2	1	no	—
102	C₂×C₈×C₈₉	—	2	1	no	—
103	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
105	C₄⋊C₄×C₈₉	—	2	2	no	—
106	C₄⋊C₄×C₈₉	—	4	2	no	—
107	C₂²⋊C₄×C₈₉	—	4	2	no	—
108	C₁₆×C₈₉	—	2	1	no	—
109	C₁₆×C₈₉	#41	1	2	yes	—
110	C₂³×C₈₉	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	—	2	1	no	—
115	C₂×C₄×C₈₉	—	2	1	no	—
116	C₂×C₄×C₈₉	—	4	2	no	—
117	C₂×C₄×C₈₉	—	4	2	no	—
118	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
119	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
120	C₂²⋊D₈₉	Q₃₂	1	4	yes	complemented
121	C₂×Dic₈₉	—	4	2	no	—
122	C₂×Dic₈₉	—	4	2	no	—
123	C₂×Dic₈₉	—	16	2	no	—
124	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
125	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
126	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
127	C₄×D₈₉	—	2	1	no	—
128	C₄×D₈₉	—	2	1	no	—
129	C₄×D₈₉	—	16	2	no	—
130	Dic₁₇₈	—	16	2	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	#10623494	—	2	1	no	—
135	#10623494	—	2	1	no	—
136	C₂²×C₈₉	—	2	1	no	—
137	C₂²×C₈₉	—	2	1	no	—
138	C₂²×C₈₉	C₂×Q₃₂	1	2	yes	—
139	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
140	C₂×D₈₉	—	2	1	no	—
141	C₂×D₈₉	—	2	1	no	—
142	C₂×D₈₉	C₂×Q₃₂	1	2	yes	—
143	C₄×C₈₉	—	2	1	no	—
144	C₄×C₈₉	—	8	2	no	—
145	C₄×C₈₉	—	8	2	no	—
146	C₄×C₈₉	#547	1	1	yes	characteristic
147	C₄×C₈₉	#578	1	1	yes	characteristic
148	Dic₈₉	—	2	1	no	—
149	Dic₈₉	—	2	1	no	—
150	Dic₈₉	—	8	2	no	—
151	Dic₈₉	—	8	2	no	—
152	Dic₈₉	C₂×Q₃₂	1	2	yes	—
153	#33905	—	89	1	no	maximal, Sylow
154	C₂×C₈₉	—	2	2	no	—
155	C₂×C₈₉	#3334	1	1	yes	characteristic
156	C₂×C₈₉	C₂²×Q₃₂	1	1	yes	characteristic
157	C₂×C₈₉	#3415	1	1	yes	characteristic
158	D₈₉	—	2	2	no	—
159	#3338	—	89	2	no	—
160	#2252	—	89	1	no	—
161	#2236	—	89	2	no	—
162	#2236	—	89	2	no	—
163	#2227	—	89	2	no	—
164	#2216	—	89	1	no	—
165	#2216	D₈₉	1	1	yes	characteristic, complemented
166	C₄×Q₃₂	—	89	1	no	—
167	D₄×C₁₆	—	89	1	no	—
168	C₂×#366	—	89	1	no	—
169	C₂×#366	—	89	1	no	—
170	C₈₉	#33905	1	1	yes	characteristic, complemented, Sylow
171	#554	—	178	2	no	—
172	C₂×Q₃₂	C₂×D₈₉	1	1	yes	characteristic
173	C₂²×C₁₆	—	89	1	no	—
174	C₂²×C₁₆	C₂×D₈₉	1	1	yes	characteristic
175	#500	—	89	2	no	—
176	#484	—	178	2	no	—
177	#484	—	178	2	no	—
178	#477	—	178	2	no	—
179	#467	—	89	2	no	—
180	#467	C₂×D₈₉	1	2	yes	—
181	C₄×Q₁₆	—	89	2	no	—
182	C₈×D₄	—	89	1	no	—
183	C₂×#107	—	89	2	no	—
184	C₂×#107	—	89	2	no	—
185	#366	—	89	1	no	—
186	#366	—	89	1	no	—
187	#366	—	89	2	no	—
188	#366	—	89	2	no	—
189	#366	—	89	2	no	—
190	#366	C₂×D₈₉	1	1	yes	characteristic
191	C₄⋊C₁₆	—	89	1	no	—
192	#358	—	89	2	no	—
193	#358	—	89	2	no	—
194	#358	C₂×D₈₉	1	2	yes	—
195	C₂²⋊C₁₆	—	89	1	no	—
196	C₂²⋊C₁₆	—	89	1	no	—
197	C₄×C₁₆	—	89	1	no	—
198	C₂×Q₁₆	C₂²×D₈₉	1	2	yes	—
199	C₂²×C₈	—	89	1	no	—
200	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
201	C₄⋊Q₈	—	178	2	no	—
202	#125	—	356	2	no	—
203	C₂²⋊Q₈	—	2	2	no	—
204	C₂²⋊Q₈	—	356	2	no	—
205	C₂²⋊Q₈	—	356	2	no	—
206	C₂²⋊Q₈	—	178	2	no	—
207	C₄×Q₈	—	178	2	no	—
208	C₄×D₄	—	89	1	no	—
209	C₄×D₄	—	356	2	no	—
210	C₂×C₄⋊C₄	—	178	2	no	—
211	C₂×C₄⋊C₄	—	178	2	no	—
212	Q₃₂	—	2	2	no	—
213	C₂×C₁₆	—	2	2	no	—
214	C₂×C₁₆	—	89	1	no	—
215	C₂×C₁₆	—	89	1	no	—
216	C₂×C₁₆	—	178	2	no	—
217	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
218	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
219	#107	—	89	2	no	—
220	#107	—	89	2	no	—
221	#107	—	178	2	no	—
222	#107	—	178	2	no	—
223	#107	—	178	2	no	—
224	#107	C₂²×D₈₉	1	2	yes	—
225	C₄⋊C₈	—	89	1	no	—
226	#103	—	2	2	no	—
227	#103	—	178	2	no	—
228	#103	—	178	2	no	—
229	C₂²⋊C₈	—	89	1	no	—
230	C₂²⋊C₈	—	89	1	no	—
231	C₄×C₈	—	89	1	no	—
232	C₂×Q₈	—	2	2	no	—
233	C₂×Q₈	—	356	2	no	—
234	C₂×D₄	—	89	1	no	—
235	C₂²×C₄	—	89	1	no	—
236	C₂²×C₄	—	356	2	no	—
237	C₂²×C₄	—	356	2	no	—
238	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
239	Q₁₆	—	2	2	no	—
240	Q₁₆	—	2	2	no	—
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	—	2	1	no	—
243	C₂×C₈	—	89	1	no	—
244	C₂×C₈	—	89	1	no	—
245	C₂×C₈	—	178	1	no	—
246	C₂×C₈	—	178	1	no	—
247	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
248	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	4	2	no	—
251	C₄⋊C₄	—	89	1	no	—
252	C₄⋊C₄	—	356	2	no	—
253	C₄⋊C₄	—	356	2	no	—
254	C₄⋊C₄	—	356	2	no	—
255	C₄⋊C₄	—	356	2	no	—
256	C₄⋊C₄	—	356	2	no	—
257	C₄⋊C₄	—	178	2	no	—
258	C₄⋊C₄	—	178	2	no	—
259	C₂²⋊C₄	—	4	2	no	—
260	C₂²⋊C₄	—	89	1	no	—
261	C₂²⋊C₄	—	89	1	no	—
262	C₂²⋊C₄	—	356	2	no	—
263	C₄²	—	89	1	no	—
264	C₄²	—	356	2	no	—
265	C₁₆	—	2	1	no	—
266	C₁₆	—	178	1	no	—
267	C₁₆	—	178	1	no	—
268	C₁₆	#13083573	1	2	yes	—
269	C₂³	—	89	1	no	—
270	C₂³	D₈×D₈₉	1	1	yes	characteristic
271	Q₈	—	4	2	no	—
272	Q₈	—	4	2	no	—
273	Q₈	—	1424	2	no	—
274	D₄	—	89	4	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	89	1	no	—
280	C₂×C₄	—	89	1	no	—
281	C₂×C₄	—	89	1	no	—
282	C₂×C₄	—	356	2	no	—
283	C₂×C₄	—	356	2	no	—
284	C₂×C₄	—	1424	2	no	—
285	C₂×C₄	—	1424	2	no	—
286	C₂×C₄	—	178	1	no	—
287	C₂×C₄	—	178	1	no	—
288	C₂×C₄	D₈×D₈₉	1	1	yes	characteristic
289	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
290	C₈	—	2	1	no	—
291	C₈	—	178	1	no	—
292	C₈	—	178	1	no	—
293	C₈	C₂×#13083573	1	1	yes	characteristic
294	C₈	#15668059	1	1	yes	characteristic
295	C₂²	—	2	1	no	—
296	C₂²	—	2	1	no	—
297	C₂²	—	89	2	no	—
298	C₂²	—	178	1	no	—
299	C₂²	—	178	1	no	—
300	C₂²	Q₃₂×D₈₉	1	2	yes	—
301	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
302	C₄	—	2	1	no	—
303	C₄	—	8	2	no	—
304	C₄	—	8	2	no	—
305	C₄	—	89	2	no	—
306	C₄	—	178	1	no	—
307	C₄	—	178	1	no	—
308	C₄	—	712	2	no	—
309	C₄	—	712	2	no	—
310	C₄	#19152248	1	1	yes	characteristic
311	C₄	#19152412	1	1	yes	characteristic
312	C₂	—	2	2	no	—
313	C₂	—	178	2	no	—
314	C₂	#26032596	1	1	yes	characteristic
315	C₂	C₂×Q₃₂×D₈₉	1	1	yes	characteristic
316	C₂	#26033390	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₈×D₈₉
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2216	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂×C₂²⋊D₈₉	D₈
hypercenter	#2216	D₈₉
hypocenter	C₈₉	#33905
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #2216

Complements

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

N	H ≅ G/N	type
2. #26022526	C₂	semidirect
3. #26022453	C₂	semidirect
4. #26022452	C₂	semidirect
5. #26022402	C₂	semidirect
6. #26022340	C₂	semidirect
7. Q₃₂×Dic₈₉	C₂	semidirect
9. C₂×#19151424	C₂	semidirect
10. #366×D₈₉	C₂	semidirect
11. C₈₉×#2216	C₂	semidirect
120. C₂²⋊D₈₉	Q₃₂	semidirect
165. #2216	D₈₉	semidirect
170. C₈₉	#33905	semidirect

Maximal subgroups

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

subgroup	index
#26032641	2
#26022526	2
#26022453	2
#26022452	2
#26022402	2
#26022340	2
Q₃₂×Dic₈₉	2
C₁₆×C₂²⋊D₈₉	2
C₂×#19151424	2
#366×D₈₉	2
C₈₉×#2216	2
#33905	89