Subgroups — #36542090

Order 22784.

814 conjugacy classes of subgroups.

The subgroup lattice has 252 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26030811	C₂	1	4	yes	maximal, complemented
2	#26030766	C₂	1	1	yes	maximal, characteristic, complemented
3	#26030680	C₂	1	1	yes	maximal, characteristic
4	#26019853	C₂	1	2	yes	maximal, complemented
5	#26019849	C₂	1	2	yes	maximal, complemented
6	#26019844	C₂	1	2	yes	maximal, complemented
7	#26019839	C₂	1	2	yes	maximal, complemented
8	C₂×C₈²×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#15668034	—	2	2	no	—
10	#19152155	C₂²	1	2	yes	—
11	#19152148	C₂²	1	2	yes	—
12	#19152003	C₂²	1	2	yes	—
13	#19152000	—	2	2	no	—
14	#19151996	C₂²	1	4	yes	—
15	#19151996	C₂²	1	2	yes	—
16	#19151982	—	2	2	no	—
17	#19151978	C₂²	1	1	yes	characteristic
18	#19151975	C₂²	1	1	yes	characteristic
19	#19151971	C₂²	1	4	yes	—
20	#19151971	C₂²	1	2	yes	—
21	#19151949	C₂²	1	4	yes	—
22	#19151945	C₂²	1	4	yes	—
23	#19151827	—	2	8	no	—
24	C₄×#13083563	—	2	8	no	—
25	#19151636	—	2	2	no	—
26	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₈₉	C₂²	1	2	yes	—
28	C₈²×C₈₉	C₂²	1	2	yes	—
29	C₈²×C₈₉	C₂²	1	1	yes	characteristic
30	C₈²×C₈₉	C₂²	1	1	yes	characteristic
31	C₂×#13083574	—	2	2	no	—
32	#15668034	—	4	8	no	—
33	C₂×Dic₃₅₆	—	2	2	no	—
34	C₂×#13083563	—	2	8	no	—
35	C₂×#13083563	—	2	4	no	—
36	C₂×D₇₁₂	—	2	2	no	—
37	#15668012	C₂³	1	2	yes	—
38	#15668012	C₂³	1	4	yes	—
39	#15668009	C₂³	1	2	yes	—
40	#15668004	—	2	2	no	—
41	#15668000	—	2	2	no	—
42	#15667991	—	2	4	no	—
43	#15667986	—	2	4	no	—
44	C₄×Dic₁₇₈	—	2	4	no	—
45	C₄×D₃₅₆	—	2	4	no	—
46	#15667971	—	2	2	no	—
47	#15667946	—	2	4	no	—
48	#15667944	—	2	4	no	—
49	#15667944	—	2	8	no	—
50	#15667937	—	2	8	no	—
51	#15667934	—	2	8	no	—
52	C₂²×C₈×C₈₉	D₄	1	4	yes	—
53	C₂²×C₈×C₈₉	D₄	1	2	yes	—
54	C₂×C₄²×C₈₉	C₂³	1	1	yes	characteristic
55	C₄×C₈×C₈₉	D₄	1	4	yes	—
56	C₄×C₈×C₈₉	D₄	1	2	yes	—
57	C₄×C₈×C₈₉	C₂³	1	2	yes	—
58	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
59	C₄×C₈×C₈₉	C₂³	1	2	yes	—
60	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
61	#13083574	—	4	8	no	—
62	C₂×Dic₁₇₈	—	2	4	no	—
63	C₂×Dic₁₇₈	—	2	2	no	—
64	C₂×D₃₅₆	—	2	4	no	—
65	C₂×D₃₅₆	—	2	2	no	—
66	C₂×C₂²⋊D₈₉	—	4	2	no	—
67	C₂×C₄×D₈₉	—	4	2	no	—
68	Dic₃₅₆	—	4	4	no	—
69	#13083563	—	4	8	no	—
70	#13083563	—	4	16	no	—
71	D₇₁₂	—	4	4	no	—
72	C₄⋊Dic₈₉	—	2	4	no	—
73	C₄⋊Dic₈₉	—	2	4	no	—
74	C₄⋊Dic₈₉	—	2	4	no	—
75	#13083553	—	4	4	no	—
76	C₂²⋊Dic₈₉	—	4	2	no	—
77	#13083551	—	4	4	no	—
78	C₄×Dic₈₉	—	4	2	no	—
79	C₂²×C₄×C₈₉	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₈₉	—	2	8	no	—
82	C₂×C₈×C₈₉	SD₁₆	1	8	yes	complemented
83	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
84	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₈₉	C₂×D₄	1	2	yes	—
86	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
87	C₄²×C₈₉	C₂×D₄	1	2	yes	—
88	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
89	C₄²×C₈₉	C₂⁴	1	1	yes	characteristic
90	C₂³×C₈₉	#127	1	1	yes	characteristic
91	C₂²×D₈₉	—	4	2	no	—
92	C₂×C₄×C₈₉	—	2	4	no	—
93	C₂×C₄×C₈₉	#127	1	2	yes	—
94	C₂×C₄×C₈₉	#127	1	1	yes	characteristic
95	C₂×C₄×C₈₉	C₂×SD₁₆	1	4	yes	—
96	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
97	C₂×C₄×C₈₉	C₂²×D₄	1	2	yes	—
98	C₂²⋊D₈₉	—	8	4	no	—
99	C₂×Dic₈₉	—	4	4	no	—
100	C₂×Dic₈₉	—	4	2	no	—
101	D₃₅₆	—	4	4	no	—
102	D₃₅₆	—	4	8	no	—
103	C₄×D₈₉	—	8	4	no	—
104	Dic₁₇₈	—	4	4	no	—
105	Dic₁₇₈	—	4	8	no	—
106	C₈×C₈₉	—	2	4	no	—
107	C₈×C₈₉	C₂×SD₁₆	1	4	yes	—
108	C₈×C₈₉	C₂×SD₁₆	1	4	yes	—
109	C₈×C₈₉	#135	1	8	yes	complemented
110	C₂²×C₈₉	—	2	2	no	—
111	C₂²×C₈₉	#492	1	2	yes	—
112	C₂²×C₈₉	C₂×#127	1	1	yes	characteristic
113	C₂×D₈₉	—	8	2	no	—
114	C₂×D₈₉	—	8	4	no	—
115	C₄×C₈₉	—	2	2	no	—
116	C₄×C₈₉	#492	1	2	yes	—
117	C₄×C₈₉	C₂²×SD₁₆	1	2	yes	—
118	C₄×C₈₉	C₂×#135	1	4	yes	—
119	Dic₈₉	—	8	2	no	—
120	Dic₈₉	—	8	4	no	—
121	#19596	—	89	1	no	maximal, Sylow
122	C₂×C₈₉	—	2	2	no	—
123	C₂×C₈₉	C₂×#492	1	1	yes	characteristic
124	C₂×C₈₉	#3147	1	2	yes	—
125	D₈₉	—	16	2	no	—
126	#3167	—	89	4	no	—
127	#3160	—	89	1	no	—
128	#3147	—	89	1	no	—
129	#1708	—	89	2	no	—
130	#1707	—	89	2	no	—
131	#1706	—	89	2	no	—
132	#1705	—	89	2	no	—
133	C₂×C₈²	D₈₉	1	1	yes	characteristic, complemented
134	C₈₉	#19596	1	1	yes	characteristic, complemented, Sylow
135	C₂×#135	—	178	2	no	—
136	#533	—	89	2	no	—
137	#532	—	89	2	no	—
138	#500	—	89	2	no	—
139	#499	—	178	2	no	—
140	#498	—	89	2	no	—
141	#498	—	89	4	no	—
142	#495	—	178	2	no	—
143	#494	—	89	1	no	—
144	#493	—	89	1	no	—
145	#492	—	89	2	no	—
146	#492	—	89	4	no	—
147	#487	—	89	4	no	—
148	#486	—	89	4	no	—
149	#465	—	178	8	no	—
150	C₄×SD₁₆	—	178	8	no	—
151	#427	—	178	2	no	—
152	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
153	C₂×C₄×C₈	C₂×D₈₉	1	2	yes	—
154	C₈²	C₂×D₈₉	1	1	yes	characteristic
155	C₈²	C₂×D₈₉	1	2	yes	—
156	C₈²	C₂×D₈₉	1	1	yes	characteristic
157	C₂×#41	—	178	2	no	—
158	#135	—	356	8	no	—
159	C₂×Q₁₆	—	178	2	no	—
160	C₂×SD₁₆	—	178	4	no	—
161	C₂×SD₁₆	—	178	8	no	—
162	C₂×D₈	—	178	2	no	—
163	C₂²×C₈	D₃₅₆	1	4	yes	—
164	C₂²×C₈	D₃₅₆	1	2	yes	—
165	C₄⋊Q₈	—	89	2	no	—
166	C₄⋊Q₈	—	89	4	no	—
167	#127	—	89	2	no	—
168	#125	—	178	2	no	—
169	#124	—	178	2	no	—
170	C₂²⋊Q₈	—	178	4	no	—
171	#121	—	178	4	no	—
172	C₄×Q₈	—	178	4	no	—
173	C₄×D₄	—	178	4	no	—
174	#117	—	178	2	no	—
175	C₂×C₄²	C₂²×D₈₉	1	1	yes	characteristic
176	#107	—	178	4	no	—
177	#106	—	178	4	no	—
178	#106	—	178	8	no	—
179	#103	—	178	8	no	—
180	#102	—	178	8	no	—
181	C₄×C₈	D₃₅₆	1	4	yes	—
182	C₄×C₈	D₃₅₆	1	2	yes	—
183	C₄×C₈	C₂²×D₈₉	1	2	yes	—
184	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
185	C₄×C₈	C₂²×D₈₉	1	2	yes	—
186	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
187	#41	—	356	8	no	—
188	C₂×Q₈	—	178	4	no	—
189	C₂×Q₈	—	178	2	no	—
190	C₂×D₄	—	356	2	no	—
191	C₂×D₄	—	178	4	no	—
192	C₂×D₄	—	178	2	no	—
193	C₂²×C₄	—	356	2	no	—
194	C₂²×C₄	C₂×D₃₅₆	1	2	yes	—
195	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
196	Q₁₆	—	356	4	no	—
197	SD₁₆	—	356	8	no	—
198	SD₁₆	—	356	16	no	—
199	D₈	—	356	4	no	—
200	C₂×C₈	—	2	8	no	—
201	C₂×C₈	#13083563	1	8	yes	complemented
202	C₂×C₈	C₂×D₃₅₆	1	4	yes	—
203	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
204	C₂×C₈	C₂×D₃₅₆	1	2	yes	—
205	C₂×C₈	C₂×D₃₅₆	1	4	yes	—
206	C₄⋊C₄	—	356	4	no	—
207	C₄⋊C₄	—	178	4	no	—
208	C₄⋊C₄	—	178	4	no	—
209	C₄⋊C₄	—	178	4	no	—
210	C₂²⋊C₄	—	356	2	no	—
211	C₂²⋊C₄	—	356	4	no	—
212	C₄²	—	356	2	no	—
213	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
214	C₄²	C₂×D₃₅₆	1	2	yes	—
215	C₄²	C₂³×D₈₉	1	1	yes	characteristic
216	C₂³	—	356	2	no	—
217	C₂³	#15668009	1	1	yes	characteristic
218	Q₈	—	356	4	no	—
219	Q₈	—	356	8	no	—
220	D₄	—	356	4	no	—
221	D₄	—	356	8	no	—
222	D₄	—	712	4	no	—
223	C₂×C₄	—	2	4	no	—
224	C₂×C₄	—	356	4	no	—
225	C₂×C₄	—	356	2	no	—
226	C₂×C₄	—	712	4	no	—
227	C₂×C₄	#15668009	1	2	yes	—
228	C₂×C₄	#15668009	1	1	yes	characteristic
229	C₂×C₄	C₂×#13083563	1	4	yes	—
230	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
231	C₂×C₄	C₂²×D₃₅₆	1	2	yes	—
232	C₈	—	2	4	no	—
233	C₈	C₂×#13083563	1	4	yes	—
234	C₈	C₂×#13083563	1	4	yes	—
235	C₈	#15668034	1	8	yes	complemented
236	C₂²	—	2	2	no	—
237	C₂²	—	712	2	no	—
238	C₂²	—	712	4	no	—
239	C₂²	#19151971	1	2	yes	—
240	C₂²	C₂×#15668009	1	1	yes	characteristic
241	C₄	—	2	2	no	—
242	C₄	—	712	2	no	—
243	C₄	—	712	4	no	—
244	C₄	#19151971	1	2	yes	—
245	C₄	C₂²×#13083563	1	2	yes	—
246	C₄	C₂×#15668034	1	4	yes	—
247	C₂	—	2	2	no	—
248	C₂	—	1424	2	no	—
249	C₂	C₂×#19151971	1	1	yes	characteristic
250	C₂	#26030680	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	#15668009
commutator, cosocle	C₄²×C₈₉	C₂⁴
Frattini	C₄²	C₂³×D₈₉
Fitting	C₂×C₈²×C₈₉	C₂
socle	C₂²×C₈₉	C₂×#127
prime radical	#26030680	C₂
hypercenter	C₂×C₈²	D₈₉
hypocenter	C₈₉	#19596
LCS	G ▷ C₄²×C₈₉ ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
1. #26030811	C₂	semidirect
2. #26030766	C₂	semidirect
4. #26019853	C₂	semidirect
5. #26019849	C₂	semidirect
6. #26019844	C₂	semidirect
7. #26019839	C₂	semidirect
8. C₂×C₈²×C₈₉	C₂	semidirect
82. C₂×C₈×C₈₉	SD₁₆	semidirect
109. C₈×C₈₉	#135	semidirect
133. C₂×C₈²	D₈₉	semidirect
134. C₈₉	#19596	semidirect
201. C₂×C₈	#13083563	semidirect
235. C₈	#15668034	semidirect

Maximal subgroups

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

subgroup	index
#26030811	2
#26030766	2
#26030680	2
#26019853	2
#26019849	2
#26019844	2
#26019839	2
C₂×C₈²×C₈₉	2
#19596	89