Subgroups — #36709954

Order 22784.

482 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030765	C₂	1	1	yes	maximal, characteristic
2	#499×D₈₉	C₂	1	1	yes	maximal, characteristic
3	#26022537	C₂	1	2	yes	maximal, complemented
4	#26022536	C₂	1	2	yes	maximal, complemented
5	#26022531	C₂	1	1	yes	maximal, characteristic, complemented
6	#26022528	C₂	1	1	yes	maximal, characteristic, complemented
7	#26022528	C₂	1	1	yes	maximal, characteristic, complemented
8	#26022039	C₂	1	2	yes	maximal
9	#26022037	C₂	1	2	yes	maximal
10	C₄×#15667951	C₂	1	1	yes	maximal, characteristic
11	C₈₉×#2253	C₂	1	1	yes	maximal, characteristic, complemented
12	#19152154	—	2	1	no	—
13	#125×D₈₉	—	2	1	no	—
14	C₂×#15667951	C₂²	1	1	yes	characteristic
15	C₂×#15667951	C₂²	1	1	yes	characteristic
16	#19152002	C₂²	1	2	yes	—
17	#19152000	C₂²	1	1	yes	characteristic
18	#19152000	C₂²	1	1	yes	characteristic
19	#19151999	C₂²	1	2	yes	—
20	#19151998	—	2	1	no	—
21	#19151997	—	2	1	no	—
22	#19151997	—	2	1	no	—
23	#19151995	—	2	1	no	—
24	#19151635	C₂²	1	2	yes	—
25	#19151633	—	2	1	no	—
26	#107×D₈₉	C₂²	1	2	yes	—
27	#106×D₈₉	—	2	1	no	—
28	C₄×C₈×D₈₉	C₂²	1	1	yes	characteristic
29	#19151426	C₂²	1	2	yes	—
30	#19151425	C₂²	1	2	yes	—
31	#19151425	C₂²	1	2	yes	—
32	#19151424	C₂²	1	2	yes	—
33	#19151423	C₂²	1	2	yes	—
34	#19151423	C₂²	1	2	yes	—
35	#19151373	C₂²	1	1	yes	characteristic
36	#19151373	C₂²	1	1	yes	characteristic
37	C₄×#13083549	C₂²	1	1	yes	characteristic
38	#499×C₈₉	C₂²	1	1	yes	characteristic
39	#499×C₈₉	C₂²	1	1	yes	characteristic
40	#367×C₈₉	C₂²	1	2	yes	—
41	#366×C₈₉	C₂²	1	2	yes	—
42	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
43	C₂×C₈×D₈₉	D₄	1	1	yes	characteristic
44	C₂×C₈×D₈₉	C₂³	1	1	yes	characteristic
45	#15668011	—	2	2	no	—
46	#15668004	—	2	1	no	—
47	#15668004	—	2	1	no	—
48	#15668003	—	2	2	no	—
49	#15668002	—	4	2	no	—
50	#15667990	—	4	2	no	—
51	#15667977	—	4	2	no	—
52	#15667970	—	2	2	no	—
53	C₄⋊C₄×D₈₉	—	2	2	no	—
54	C₄⋊C₄×D₈₉	—	4	2	no	—
55	C₄²×D₈₉	D₄	1	1	yes	characteristic
56	#15667951	Q₈	1	4	yes	complemented
57	#15667951	Q₈	1	4	yes	complemented
58	C₂×#13083549	C₂³	1	1	yes	characteristic
59	C₂×#13083549	C₂³	1	1	yes	characteristic
60	#15667946	C₂³	1	2	yes	—
61	#15667946	C₂³	1	2	yes	—
62	#15667945	—	2	2	no	—
63	#15667945	—	2	2	no	—
64	#15667944	—	2	1	no	—
65	#15667944	—	2	1	no	—
66	#15667943	—	2	2	no	—
67	#15667943	—	2	2	no	—
68	C₄×#10623494	D₄	1	1	yes	characteristic
69	C₈×Dic₈₉	D₄	1	1	yes	characteristic
70	C₈×Dic₈₉	C₂³	1	1	yes	characteristic
71	#125×C₈₉	—	2	1	no	—
72	#125×C₈₉	—	2	1	no	—
73	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
74	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
75	#107×C₈₉	C₂³	1	2	yes	—
76	#107×C₈₉	C₂³	1	2	yes	—
77	#106×C₈₉	—	2	1	no	—
78	#106×C₈₉	—	2	1	no	—
79	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
80	C₂×Dic₁₇₈	—	4	2	no	—
81	C₂×C₄×D₈₉	—	4	2	no	—
82	C₂×C₄×D₈₉	D₈	1	2	yes	—
83	C₂×C₄×D₈₉	C₂×D₄	1	1	yes	characteristic
84	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
85	C₂×#10623494	C₂×D₄	1	1	yes	characteristic
86	C₈×D₈₉	Q₁₆	1	4	yes	—
87	C₈×D₈₉	C₂×Q₈	1	2	yes	—
88	C₈×D₈₉	C₂×Q₈	1	2	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	#13083553	—	4	2	no	—
96	#13083553	—	4	2	no	—
97	#13083551	—	4	2	no	—
98	C₄×Dic₈₉	—	4	2	no	—
99	C₄×Dic₈₉	D₈	1	2	yes	—
100	C₄×Dic₈₉	C₂×D₄	1	1	yes	characteristic
101	#13083549	C₂×Q₈	1	2	yes	—
102	#13083549	C₂×Q₈	1	2	yes	—
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	—	2	2	no	—
107	C₄⋊C₄×C₈₉	—	4	2	no	—
108	C₄⋊C₄×C₈₉	—	4	2	no	—
109	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
110	C₁₆×C₈₉	C₂×Q₈	1	2	yes	—
111	C₁₆×C₈₉	C₂×Q₈	1	2	yes	—
112	C₂²×D₈₉	C₂×D₈	1	1	yes	characteristic
113	C₂×C₄×C₈₉	—	4	2	no	—
114	C₂×C₄×C₈₉	—	4	2	no	—
115	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
116	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
117	C₂×Dic₈₉	—	4	2	no	—
118	C₂×Dic₈₉	—	4	2	no	—
119	C₂×Dic₈₉	C₂×D₈	1	2	yes	—
120	C₂×Dic₈₉	C₂×D₈	1	1	yes	characteristic
121	C₄×D₈₉	—	4	2	no	—
122	C₄×D₈₉	—	8	4	no	—
123	C₄×D₈₉	C₄⋊Q₈	1	2	yes	—
124	C₄×D₈₉	C₂×Q₁₆	1	2	yes	—
125	Dic₁₇₈	—	8	4	no	—
126	C₈×C₈₉	C₂×Q₁₆	1	2	yes	—
127	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
128	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
129	#10623494	C₄⋊Q₈	1	2	yes	—
130	#10623494	C₂×Q₁₆	1	2	yes	—
131	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
132	C₂×D₈₉	—	2	2	no	—
133	C₂×D₈₉	#500	1	2	yes	—
134	C₄×C₈₉	—	2	2	no	—
135	C₄×C₈₉	—	8	2	no	—
136	C₄×C₈₉	—	8	2	no	—
137	C₄×C₈₉	C₂×C₄⋊Q₈	1	1	yes	characteristic
138	C₄×C₈₉	C₂²×Q₁₆	1	1	yes	characteristic
139	Dic₈₉	—	2	2	no	—
140	Dic₈₉	—	8	2	no	—
141	Dic₈₉	—	8	2	no	—
142	Dic₈₉	#500	1	2	yes	—
143	#33732	—	89	1	no	maximal, Sylow
144	C₂×C₈₉	C₂×#500	1	1	yes	characteristic
145	C₂×C₈₉	#3414	1	2	yes	—
146	D₈₉	—	2	2	no	—
147	#3160	—	89	1	no	—
148	C₂×#499	—	89	1	no	—
149	#2255	—	89	2	no	—
150	#2255	—	89	2	no	—
151	#2253	—	89	1	no	—
152	#2253	—	89	1	no	—
153	#2253	—	89	1	no	—
154	#2253	D₈₉	1	1	yes	characteristic, complemented
155	#2159	—	89	2	no	—
156	#2159	—	89	2	no	—
157	C₄×M₃₂	—	89	1	no	—
158	C₈₉	#33732	1	1	yes	characteristic, complemented, Sylow
159	#533	—	178	1	no	—
160	C₂×#125	—	178	1	no	—
161	C₂×M₃₂	—	89	1	no	—
162	C₂×M₃₂	—	89	1	no	—
163	#500	—	89	2	no	—
164	#499	—	89	2	no	—
165	#499	—	89	1	no	—
166	#499	—	89	1	no	—
167	#499	—	178	1	no	—
168	#499	—	178	1	no	—
169	#499	—	178	1	no	—
170	#499	C₂×D₈₉	1	1	yes	characteristic
171	#499	C₂×D₈₉	1	1	yes	characteristic
172	#498	—	178	1	no	—
173	#427	—	89	2	no	—
174	#427	—	178	1	no	—
175	C₂×#107	—	89	2	no	—
176	C₂×#106	—	178	1	no	—
177	C₂×C₄×C₈	—	89	1	no	—
178	#367	—	89	2	no	—
179	#367	—	89	2	no	—
180	#367	—	89	2	no	—
181	#367	C₂×D₈₉	1	2	yes	—
182	#366	—	89	2	no	—
183	#366	—	89	2	no	—
184	#366	—	89	2	no	—
185	#366	C₂×D₈₉	1	2	yes	—
186	#346	—	89	1	no	—
187	#346	—	89	1	no	—
188	C₄×C₁₆	—	89	1	no	—
189	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
190	C₂²×C₈	—	89	1	no	—
191	C₂²×C₈	—	89	1	no	—
192	C₄⋊Q₈	—	178	2	no	—
193	#125	—	2	1	no	—
194	#125	—	2	1	no	—
195	#125	—	356	2	no	—
196	#125	—	178	2	no	—
197	#125	—	178	1	no	—
198	#125	—	178	1	no	—
199	C₂²⋊Q₈	—	356	2	no	—
200	C₄×Q₈	—	356	2	no	—
201	#117	—	178	2	no	—
202	C₂×C₄⋊C₄	—	356	2	no	—
203	C₂×C₄⋊C₄	—	178	2	no	—
204	C₂×C₄²	—	89	1	no	—
205	M₃₂	—	89	4	no	—
206	M₃₂	—	89	4	no	—
207	C₂×C₁₆	—	89	1	no	—
208	C₂×C₁₆	—	89	1	no	—
209	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
210	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
211	#107	—	89	2	no	—
212	#107	—	89	2	no	—
213	#107	—	178	2	no	—
214	#107	—	178	2	no	—
215	#107	C₂²×D₈₉	1	2	yes	—
216	#107	C₂²×D₈₉	1	2	yes	—
217	#106	—	2	1	no	—
218	#106	—	2	1	no	—
219	#106	—	178	2	no	—
220	#106	—	178	2	no	—
221	#106	—	178	1	no	—
222	#106	—	178	1	no	—
223	C₄×C₈	—	89	1	no	—
224	C₄×C₈	—	89	1	no	—
225	C₄×C₈	—	89	1	no	—
226	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
227	C₂×Q₈	—	356	2	no	—
228	C₂²×C₄	—	89	1	no	—
229	C₂²×C₄	—	89	2	no	—
230	C₂²×C₄	—	356	2	no	—
231	C₂×C₈	—	89	1	no	—
232	C₂×C₈	—	89	1	no	—
233	C₂×C₈	—	89	4	no	—
234	C₂×C₈	—	89	2	no	—
235	C₂×C₈	—	89	2	no	—
236	C₂×C₈	D₄×D₈₉	1	1	yes	characteristic
237	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	4	2	no	—
241	C₄⋊C₄	—	4	2	no	—
242	C₄⋊C₄	—	356	2	no	—
243	C₄⋊C₄	—	356	2	no	—
244	C₄⋊C₄	—	356	2	no	—
245	C₄⋊C₄	—	356	2	no	—
246	C₄⋊C₄	—	356	2	no	—
247	C₄⋊C₄	—	356	2	no	—
248	C₄⋊C₄	—	178	2	no	—
249	C₄⋊C₄	—	178	2	no	—
250	C₂²⋊C₄	—	356	2	no	—
251	C₄²	—	89	2	no	—
252	C₄²	—	89	1	no	—
253	C₄²	—	356	2	no	—
254	C₄²	D₄×D₈₉	1	1	yes	characteristic
255	C₁₆	—	89	2	no	—
256	C₁₆	—	89	2	no	—
257	C₁₆	Q₈×D₈₉	1	2	yes	—
258	C₁₆	Q₈×D₈₉	1	2	yes	—
259	C₂³	—	89	1	no	—
260	Q₈	—	712	4	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	89	1	no	—
264	C₂×C₄	—	89	2	no	—
265	C₂×C₄	—	89	2	no	—
266	C₂×C₄	—	89	2	no	—
267	C₂×C₄	—	356	2	no	—
268	C₂×C₄	—	356	2	no	—
269	C₂×C₄	—	356	2	no	—
270	C₂×C₄	—	712	4	no	—
271	C₂×C₄	D₈×D₈₉	1	2	yes	—
272	C₂×C₄	C₂×D₄×D₈₉	1	1	yes	characteristic
273	C₈	—	89	2	no	—
274	C₈	—	89	2	no	—
275	C₈	Q₁₆×D₈₉	1	2	yes	—
276	C₈	C₂×Q₈×D₈₉	1	1	yes	characteristic
277	C₈	C₂×Q₈×D₈₉	1	1	yes	characteristic
278	C₂²	—	89	2	no	—
279	C₂²	—	178	2	no	—
280	C₂²	C₂×D₈×D₈₉	1	1	yes	characteristic
281	C₄	—	2	2	no	—
282	C₄	—	8	2	no	—
283	C₄	—	8	2	no	—
284	C₄	—	89	2	no	—
285	C₄	—	178	2	no	—
286	C₄	—	712	2	no	—
287	C₄	—	712	2	no	—
288	C₄	C₄⋊Q₈×D₈₉	1	1	yes	characteristic
289	C₄	C₂×Q₁₆×D₈₉	1	1	yes	characteristic
290	C₂	—	178	2	no	—
291	C₂	#500×D₈₉	1	1	yes	characteristic
292	C₂	#26033385	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₈₉
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2253	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂²×D₈₉	C₂×D₈
hypercenter	#2253	D₈₉
hypocenter	C₈₉	#33732
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2253

Complements

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

N	H ≅ G/N	type
3. #26022537	C₂	semidirect
4. #26022536	C₂	semidirect
5. #26022531	C₂	semidirect
6. #26022528	C₂	semidirect
7. #26022528	C₂	semidirect
11. C₈₉×#2253	C₂	semidirect
56. #15667951	Q₈	semidirect
57. #15667951	Q₈	semidirect
154. #2253	D₈₉	semidirect
158. C₈₉	#33732	semidirect

Maximal subgroups

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

subgroup	index
#26030765	2
#499×D₈₉	2
#26022537	2
#26022536	2
#26022531	2
#26022528	2
#26022039	2
#26022037	2
C₄×#15667951	2
C₈₉×#2253	2
#33732	89