Subgroups — #36711809

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	#26032623	C₂	1	2	yes	maximal
2	#26022529	C₂	1	1	yes	maximal, characteristic, complemented
3	#26022459	C₂	1	2	yes	maximal, complemented
4	#26022447	C₂	1	2	yes	maximal, complemented
5	#26022418	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	#26022033	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×#19151424	C₂	1	1	yes	maximal, characteristic, complemented
11	C₈₉×#2216	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152290	—	2	2	no	—
13	C₂×C₁₆×D₈₉	C₂²	1	1	yes	characteristic
14	#19151999	C₂²	1	2	yes	—
15	#19151938	—	2	2	no	—
16	#19151926	—	2	2	no	—
17	#19151907	—	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	#19151635	C₂²	1	2	yes	—
22	C₂×#15667946	C₂²	1	2	yes	—
23	#19151425	C₂²	1	2	yes	—
24	#19151425	C₂²	1	2	yes	—
25	#19151424	C₂²	1	2	yes	—
26	#19151424	C₂²	1	1	yes	characteristic
27	#19151424	C₂²	1	1	yes	characteristic
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	#15668007	—	4	2	no	—
41	#15668003	—	2	2	no	—
42	#15668003	—	4	2	no	—
43	#15667996	—	4	2	no	—
44	#15667988	—	2	2	no	—
45	Q₈×Dic₈₉	—	2	2	no	—
46	C₄×C₂²⋊D₈₉	D₄	1	1	yes	characteristic
47	#15667970	—	2	2	no	—
48	#15667968	—	4	2	no	—
49	C₂×C₄⋊Dic₈₉	—	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	#15667943	—	2	2	no	—
57	#15667943	—	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×C₂²⋊D₈₉	D₈	1	1	yes	characteristic
74	C₂²×Dic₈₉	—	4	2	no	—
75	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
76	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
77	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
78	C₈×D₈₉	—	2	1	no	—
79	C₈×D₈₉	—	2	1	no	—
80	C₄⋊Dic₈₉	—	2	2	no	—
81	C₄⋊Dic₈₉	—	2	2	no	—
82	C₄⋊Dic₈₉	—	4	2	no	—
83	C₄⋊Dic₈₉	—	4	2	no	—
84	C₄⋊Dic₈₉	—	4	2	no	—
85	#13083553	—	4	2	no	—
86	#13083553	—	4	2	no	—
87	#13083553	D₈	1	1	yes	characteristic
88	C₂²⋊Dic₈₉	D₈	1	1	yes	characteristic
89	#13083551	—	4	2	no	—
90	#13083551	—	4	2	no	—
91	#13083551	D₈	1	1	yes	characteristic
92	C₄×Dic₈₉	—	4	2	no	—
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	—	2	2	no	—
121	C₂×Dic₈₉	—	4	2	no	—
122	C₂×Dic₈₉	—	4	2	no	—
123	C₂×Dic₈₉	—	8	4	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	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	—	2	1	no	—
135	C₂²×C₈₉	—	2	1	no	—
136	C₂²×C₈₉	#508	1	2	yes	—
137	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
138	C₂×D₈₉	—	2	1	no	—
139	C₂×D₈₉	—	2	1	no	—
140	C₂×D₈₉	—	2	1	no	—
141	C₄×C₈₉	—	2	1	no	—
142	C₄×C₈₉	—	8	2	no	—
143	C₄×C₈₉	—	8	2	no	—
144	C₄×C₈₉	#547	1	1	yes	characteristic
145	C₄×C₈₉	#578	1	1	yes	characteristic
146	Dic₈₉	—	2	1	no	—
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	—	2	2	no	—
153	C₂×C₈₉	#3334	1	1	yes	characteristic
154	C₂×C₈₉	C₂×#508	1	1	yes	characteristic
155	C₂×C₈₉	#3417	1	1	yes	characteristic
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	D₈₉	1	1	yes	characteristic, complemented
164	C₄×Q₃₂	—	89	1	no	—
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	C₂×D₈₉	1	1	yes	characteristic
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	C₂×D₈₉	1	2	yes	—
179	C₄×Q₁₆	—	89	2	no	—
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	—	89	1	no	—
196	C₂×Q₁₆	C₂²×D₈₉	1	2	yes	—
197	C₂²×C₈	—	89	1	no	—
198	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
199	#126	—	356	2	no	—
200	#125	—	356	2	no	—
201	#125	—	178	2	no	—
202	#123	—	356	2	no	—
203	C₂²⋊Q₈	—	2	2	no	—
204	C₂²⋊Q₈	—	178	2	no	—
205	C₄×Q₈	—	178	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	—	2	2	no	—
212	C₂×C₁₆	—	89	1	no	—
213	C₂×C₁₆	—	89	1	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	—	89	1	no	—
230	C₂×Q₈	—	2	2	no	—
231	C₂×D₄	—	89	1	no	—
232	C₂²×C₄	—	89	1	no	—
233	C₂²×C₄	—	356	2	no	—
234	C₂²×C₄	D₄×D₈₉	1	1	yes	characteristic
235	Q₁₆	—	2	2	no	—
236	Q₁₆	—	2	2	no	—
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	89	1	no	—
240	C₂×C₈	—	89	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	—	4	2	no	—
256	C₂²⋊C₄	—	89	1	no	—
257	C₂²⋊C₄	—	89	1	no	—
258	C₂²⋊C₄	—	356	2	no	—
259	C₂²⋊C₄	—	356	2	no	—
260	C₄²	—	89	1	no	—
261	C₄²	—	356	2	no	—
262	C₄²	—	356	2	no	—
263	C₁₆	—	2	1	no	—
264	C₁₆	—	178	1	no	—
265	C₁₆	—	178	1	no	—
266	C₁₆	#13083573	1	2	yes	—
267	C₂³	—	89	1	no	—
268	C₂³	D₈×D₈₉	1	1	yes	characteristic
269	Q₈	—	4	2	no	—
270	Q₈	—	4	2	no	—
271	D₄	—	178	2	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	89	1	no	—
277	C₂×C₄	—	89	1	no	—
278	C₂×C₄	—	89	1	no	—
279	C₂×C₄	—	356	2	no	—
280	C₂×C₄	—	356	2	no	—
281	C₂×C₄	—	178	1	no	—
282	C₂×C₄	—	178	1	no	—
283	C₂×C₄	—	712	4	no	—
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×#13083573	1	1	yes	characteristic
290	C₈	#15668059	1	1	yes	characteristic
291	C₂²	—	2	1	no	—
292	C₂²	—	2	1	no	—
293	C₂²	—	178	1	no	—
294	C₂²	—	178	1	no	—
295	C₂²	—	178	1	no	—
296	C₂²	#19152031	1	2	yes	—
297	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	8	2	no	—
300	C₄	—	8	2	no	—
301	C₄	—	178	1	no	—
302	C₄	—	178	1	no	—
303	C₄	—	178	1	no	—
304	C₄	—	712	2	no	—
305	C₄	—	712	2	no	—
306	C₄	#19152248	1	1	yes	characteristic
307	C₄	#19152412	1	1	yes	characteristic
308	C₂	—	2	2	no	—
309	C₂	—	356	1	no	—
310	C₂	#26032596	1	1	yes	characteristic
311	C₂	C₂×#19152031	1	1	yes	characteristic
312	C₂	#26033402	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	#33903
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. #26022529	C₂	semidirect
3. #26022459	C₂	semidirect
4. #26022447	C₂	semidirect
5. #26022418	C₂	semidirect
6. #26022340	C₂	semidirect
7. Q₃₂×Dic₈₉	C₂	semidirect
9. #26022033	C₂	semidirect
10. C₂×#19151424	C₂	semidirect
11. C₈₉×#2216	C₂	semidirect
163. #2216	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
#26032623	2
#26022529	2
#26022459	2
#26022447	2
#26022418	2
#26022340	2
Q₃₂×Dic₈₉	2
C₁₆×C₂²⋊D₈₉	2
#26022033	2
C₂×#19151424	2
C₈₉×#2216	2
#33903	89