Subgroups — #36712398

Order 22784.

472 conjugacy classes of subgroups.

The subgroup lattice has 262 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26033120	C₂	1	2	yes	maximal, complemented
2	#26022538	C₂	1	2	yes	maximal, complemented
3	#26022531	C₂	1	1	yes	maximal, characteristic, complemented
4	#26022410	C₂	1	2	yes	maximal, complemented
5	#26022410	C₂	1	2	yes	maximal, complemented
6	#26022390	C₂	1	2	yes	maximal, complemented
7	#26022129	C₂	1	2	yes	maximal
8	C₄×D₁₄₂₄	C₂	1	1	yes	maximal, characteristic
9	C₈₉×#2183	C₂	1	1	yes	maximal, characteristic, complemented
10	#19152308	—	2	2	no	—
11	C₂×D₁₄₂₄	C₂²	1	1	yes	characteristic
12	#19152003	C₂²	1	2	yes	—
13	#19152000	C₂²	1	2	yes	—
14	#19152000	C₂²	1	2	yes	—
15	#19151899	—	2	2	no	—
16	#19151899	—	2	2	no	—
17	#19151889	—	2	2	no	—
18	C₄×D₇₁₂	C₂²	1	2	yes	—
19	C₄×D₇₁₂	C₂²	1	2	yes	—
20	C₄×D₇₁₂	C₂²	1	2	yes	—
21	#19151426	C₂²	1	2	yes	—
22	#19151426	C₂²	1	2	yes	—
23	#19151424	C₂²	1	1	yes	characteristic
24	#19151424	C₂²	1	2	yes	—
25	#19151424	C₂²	1	1	yes	characteristic
26	#19151424	C₂²	1	1	yes	characteristic
27	#19151400	C₂²	1	2	yes	—
28	#19151400	C₂²	1	2	yes	—
29	#19151400	C₂²	1	2	yes	—
30	C₈×Q₈×C₈₉	C₂²	1	1	yes	characteristic
31	C₄⋊C₁₆×C₈₉	C₂²	1	1	yes	characteristic
32	C₄⋊C₁₆×C₈₉	C₂²	1	2	yes	—
33	#346×C₈₉	C₂²	1	2	yes	—
34	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
35	C₂×D₇₁₂	C₂³	1	2	yes	—
36	#15668012	—	2	2	no	—
37	#15668004	—	2	2	no	—
38	#15668004	—	2	2	no	—
39	#15667990	—	4	2	no	—
40	#15667990	—	4	2	no	—
41	#15667988	—	4	2	no	—
42	C₄×D₃₅₆	—	2	2	no	—
43	C₄×D₃₅₆	—	2	2	no	—
44	C₄×D₃₅₆	—	2	2	no	—
45	C₄⋊C₄×D₈₉	—	4	2	no	—
46	D₁₄₂₄	Q₈	1	4	yes	complemented
47	#15667946	—	2	2	no	—
48	#15667946	C₂³	1	2	yes	—
49	#15667946	C₂³	1	2	yes	—
50	#15667946	C₂³	1	2	yes	—
51	#15667944	—	2	2	no	—
52	#15667944	—	2	2	no	—
53	#15667934	—	2	2	no	—
54	#15667934	—	2	2	no	—
55	#15667934	—	2	2	no	—
56	C₄×Q₈×C₈₉	D₄	1	1	yes	characteristic
57	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
58	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
59	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
60	C₄⋊C₈×C₈₉	D₄	1	2	yes	—
61	C₄⋊C₈×C₈₉	D₄	1	1	yes	characteristic
62	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
63	C₄×C₈×C₈₉	C₂³	1	2	yes	—
64	C₂×Dic₁₇₈	—	4	2	no	—
65	C₂×D₃₅₆	—	2	2	no	—
66	C₂×C₄×D₈₉	—	4	2	no	—
67	C₂×C₄×D₈₉	—	4	2	no	—
68	C₂×C₄×D₈₉	—	4	2	no	—
69	D₇₁₂	—	2	2	no	—
70	D₇₁₂	C₂×Q₈	1	4	yes	—
71	C₄⋊Dic₈₉	—	2	2	no	—
72	C₄⋊Dic₈₉	—	2	2	no	—
73	C₄⋊Dic₈₉	—	2	2	no	—
74	C₄⋊Dic₈₉	—	4	2	no	—
75	#13083553	—	4	2	no	—
76	#13083553	—	4	2	no	—
77	#13083553	—	4	2	no	—
78	#13083553	—	4	2	no	—
79	#13083551	—	4	2	no	—
80	#13083551	—	4	2	no	—
81	#13083551	—	4	2	no	—
82	C₂×Q₈×C₈₉	D₈	1	1	yes	characteristic
83	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
86	C₄⋊C₄×C₈₉	D₈	1	1	yes	characteristic
87	C₄⋊C₄×C₈₉	D₈	1	2	yes	—
88	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₈₉	C₂×D₄	1	2	yes	—
90	C₁₆×C₈₉	—	2	1	no	—
91	C₁₆×C₈₉	—	2	2	no	—
92	C₁₆×C₈₉	C₂×Q₈	1	2	yes	—
93	C₂²×D₈₉	—	4	2	no	—
94	Q₈×C₈₉	—	4	1	no	—
95	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
96	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
97	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
98	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
99	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
100	C₂×Dic₈₉	—	4	2	no	—
101	C₂×Dic₈₉	—	4	2	no	—
102	C₂×Dic₈₉	—	4	2	no	—
103	D₃₅₆	—	2	4	no	—
104	D₃₅₆	—	4	2	no	—
105	C₄×D₈₉	—	8	4	no	—
106	C₄×D₈₉	—	16	2	no	—
107	C₄×D₈₉	—	16	2	no	—
108	Dic₁₇₈	—	16	2	no	—
109	C₈×C₈₉	—	2	1	no	—
110	C₈×C₈₉	—	2	2	no	—
111	C₈×C₈₉	C₂²×Q₈	1	1	yes	characteristic
112	C₈×C₈₉	2₋¹⁺⁴	1	1	yes	characteristic
113	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
114	C₂×D₈₉	—	4	4	no	—
115	C₂×D₈₉	—	8	2	no	—
116	C₂×D₈₉	—	8	2	no	—
117	C₄×C₈₉	—	2	2	no	—
118	C₄×C₈₉	—	2	1	no	—
119	C₄×C₈₉	—	2	2	no	—
120	C₄×C₈₉	—	2	1	no	—
121	C₄×C₈₉	D₄×Q₈	1	1	yes	characteristic
122	C₄×C₈₉	#578	1	1	yes	characteristic
123	Dic₈₉	—	8	2	no	—
124	Dic₈₉	—	8	2	no	—
125	Dic₈₉	—	8	2	no	—
126	#33956	—	89	1	no	maximal, Sylow
127	C₂×C₈₉	Q₈×D₈	1	1	yes	characteristic
128	C₂×C₈₉	#3414	1	1	yes	characteristic
129	C₂×C₈₉	#3415	1	1	yes	characteristic
130	D₈₉	—	8	4	no	—
131	#3380	—	89	2	no	—
132	#2255	—	89	2	no	—
133	#2253	—	89	1	no	—
134	#2228	—	89	2	no	—
135	#2228	—	89	2	no	—
136	#2224	—	89	2	no	—
137	#2183	D₈₉	1	1	yes	characteristic, complemented
138	#2177	—	89	2	no	—
139	C₄×D₁₆	—	89	1	no	—
140	C₈₉	#33956	1	1	yes	characteristic, complemented, Sylow
141	#554	—	178	2	no	—
142	C₂×D₁₆	—	89	1	no	—
143	#500	—	89	2	no	—
144	#499	—	89	2	no	—
145	#499	—	89	2	no	—
146	#478	—	178	2	no	—
147	#478	—	178	2	no	—
148	#476	—	178	2	no	—
149	C₈×Q₈	C₂×D₈₉	1	1	yes	characteristic
150	C₄×D₈	—	89	2	no	—
151	C₄×D₈	—	89	2	no	—
152	C₄×D₈	—	89	2	no	—
153	#367	—	89	2	no	—
154	#367	—	89	2	no	—
155	#366	—	89	1	no	—
156	#366	—	89	1	no	—
157	#366	—	89	2	no	—
158	#366	—	89	1	no	—
159	C₄⋊C₁₆	C₂×D₈₉	1	2	yes	—
160	C₄⋊C₁₆	C₂×D₈₉	1	1	yes	characteristic
161	#357	—	89	2	no	—
162	#357	—	89	2	no	—
163	#357	—	89	2	no	—
164	#346	C₂×D₈₉	1	2	yes	—
165	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
166	C₂×D₈	—	89	2	no	—
167	C₄⋊Q₈	—	178	2	no	—
168	#125	—	178	2	no	—
169	#125	—	178	2	no	—
170	C₂²⋊Q₈	—	356	2	no	—
171	C₂²⋊Q₈	—	356	2	no	—
172	C₂²⋊Q₈	—	356	2	no	—
173	C₄×Q₈	D₃₅₆	1	1	yes	characteristic
174	C₄×D₄	—	178	2	no	—
175	C₄×D₄	—	178	2	no	—
176	C₄×D₄	—	178	2	no	—
177	C₂×C₄⋊C₄	—	356	2	no	—
178	D₁₆	—	89	4	no	—
179	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
180	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
181	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
182	#107	—	89	2	no	—
183	#107	—	89	2	no	—
184	#107	—	89	2	no	—
185	#107	—	178	2	no	—
186	#106	—	178	2	no	—
187	#106	—	178	2	no	—
188	C₄⋊C₈	D₃₅₆	1	2	yes	—
189	C₄⋊C₈	D₃₅₆	1	1	yes	characteristic
190	#102	—	178	2	no	—
191	#102	—	178	2	no	—
192	#102	—	178	2	no	—
193	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
194	C₄×C₈	C₂²×D₈₉	1	2	yes	—
195	C₂×Q₈	—	356	2	no	—
196	C₂×Q₈	D₇₁₂	1	1	yes	characteristic
197	C₂×D₄	—	178	2	no	—
198	C₂²×C₄	—	356	2	no	—
199	C₂²×C₄	—	356	2	no	—
200	C₂²×C₄	—	356	2	no	—
201	D₈	—	89	4	no	—
202	D₈	—	178	2	no	—
203	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
204	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
205	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
206	C₄⋊C₄	—	356	2	no	—
207	C₄⋊C₄	—	356	2	no	—
208	C₄⋊C₄	—	356	2	no	—
209	C₄⋊C₄	—	356	2	no	—
210	C₄⋊C₄	—	356	2	no	—
211	C₄⋊C₄	—	178	2	no	—
212	C₄⋊C₄	—	178	2	no	—
213	C₄⋊C₄	—	178	2	no	—
214	C₄⋊C₄	D₇₁₂	1	2	yes	—
215	C₄⋊C₄	D₇₁₂	1	1	yes	characteristic
216	C₂²⋊C₄	—	356	2	no	—
217	C₂²⋊C₄	—	356	2	no	—
218	C₂²⋊C₄	—	356	2	no	—
219	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
220	C₄²	C₂×D₃₅₆	1	2	yes	—
221	C₁₆	—	2	1	no	—
222	C₁₆	—	2	2	no	—
223	C₁₆	Q₈×D₈₉	1	2	yes	—
224	C₂³	—	356	2	no	—
225	Q₈	—	4	1	no	—
226	Q₈	—	1424	2	no	—
227	D₄	—	356	2	no	—
228	D₄	—	178	4	no	—
229	C₂×C₄	—	356	2	no	—
230	C₂×C₄	—	356	2	no	—
231	C₂×C₄	—	356	2	no	—
232	C₂×C₄	—	1424	2	no	—
233	C₂×C₄	—	1424	2	no	—
234	C₂×C₄	—	712	4	no	—
235	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
236	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
237	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
238	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
239	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
240	C₈	—	2	1	no	—
241	C₈	—	2	2	no	—
242	C₈	C₂×Q₈×D₈₉	1	1	yes	characteristic
243	C₈	#15668059	1	1	yes	characteristic
244	C₂²	—	356	4	no	—
245	C₂²	—	712	2	no	—
246	C₂²	—	712	2	no	—
247	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
248	C₄	—	2	2	no	—
249	C₄	—	2	1	no	—
250	C₄	—	2	1	no	—
251	C₄	—	2	2	no	—
252	C₄	—	712	2	no	—
253	C₄	—	712	2	no	—
254	C₄	—	712	2	no	—
255	C₄	Q₈×D₃₅₆	1	1	yes	characteristic
256	C₄	#19152415	1	1	yes	characteristic
257	C₂	—	712	4	no	—
258	C₂	Q₈×D₇₁₂	1	1	yes	characteristic
259	C₂	#26033388	1	1	yes	characteristic
260	C₂	#26033393	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×D₇₁₂
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₈₉×#2183	C₂
socle	C₂²×C₈₉	C₂²×D₈
prime radical	C₂×D₁₄₂₄	C₂²
hypercenter	#2183	D₈₉
hypocenter	C₈₉	#33956
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈ ◁ #2183

Complements

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

N	H ≅ G/N	type
1. #26033120	C₂	semidirect
2. #26022538	C₂	semidirect
3. #26022531	C₂	semidirect
4. #26022410	C₂	semidirect
5. #26022410	C₂	semidirect
6. #26022390	C₂	semidirect
9. C₈₉×#2183	C₂	semidirect
46. D₁₄₂₄	Q₈	semidirect
137. #2183	D₈₉	semidirect
140. C₈₉	#33956	semidirect

Maximal subgroups

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

subgroup	index
#26033120	2
#26022538	2
#26022531	2
#26022410	2
#26022390	2
#26022129	2
C₄×D₁₄₂₄	2
C₈₉×#2183	2
#33956	89