Subgroups — #36709834

Order 22784.

834 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19152030	C₂	1	4	yes	maximal, complemented
2	#26030680	C₂	1	2	yes	maximal, complemented
3	#26022513	C₂	1	4	yes	maximal, complemented
4	#26022508	C₂	1	2	yes	maximal, complemented
5	#26022505	C₂	1	1	yes	maximal, characteristic, complemented
6	#26022502	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₄×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#15668034	—	2	2	no	—
9	C₂×#15668034	C₂²	1	4	yes	—
10	#19152148	—	2	2	no	—
11	#19152030	—	2	16	no	—
12	C₂×Dic₇₁₂	C₂²	1	4	yes	—
13	C₂×#15667957	C₂²	1	8	yes	—
14	C₂×D₁₄₂₄	C₂²	1	4	yes	—
15	#19151982	—	2	2	no	—
16	#19151982	C₂²	1	4	yes	—
17	#19151978	C₂²	1	2	yes	—
18	#19151975	C₂²	1	2	yes	—
19	#19151971	—	2	2	no	—
20	C₂²×C₁₆×C₈₉	C₂²	1	2	yes	—
21	C₂×C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
22	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
23	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
24	C₄×C₁₆×C₈₉	C₂²	1	2	yes	—
25	C₂×#13083574	—	2	4	no	—
26	#15668034	—	2	16	no	—
27	#15668034	—	4	8	no	—
28	C₂×Dic₃₅₆	—	2	2	no	—
29	C₂×Dic₃₅₆	C₂³	1	4	yes	—
30	C₂×#13083563	—	2	4	no	—
31	C₂×#13083563	—	2	4	no	—
32	C₂×D₇₁₂	—	2	2	no	—
33	C₂×D₇₁₂	C₂³	1	4	yes	—
34	#15668012	—	2	2	no	—
35	#15668009	—	2	2	no	—
36	#15668000	—	2	4	no	—
37	#15667986	—	4	4	no	—
38	C₄×D₃₅₆	—	4	4	no	—
39	Dic₇₁₂	—	2	8	no	—
40	#15667957	—	2	16	no	—
41	D₁₄₂₄	—	2	8	no	—
42	C₂²×C₈×C₈₉	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₈₉	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₈₉	D₄	1	1	yes	characteristic
45	C₂×C₁₆×C₈₉	D₄	1	8	yes	complemented
46	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
47	C₂×C₁₆×C₈₉	C₂³	1	2	yes	—
48	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
49	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
50	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
51	C₄×C₈×C₈₉	C₂³	1	1	yes	characteristic
52	#13083574	—	4	8	no	—
53	#13083574	—	4	8	no	—
54	C₂×Dic₁₇₈	—	2	4	no	—
55	C₂×D₃₅₆	—	2	4	no	—
56	C₂×D₃₅₆	—	4	4	no	—
57	C₂×C₂²⋊D₈₉	—	4	4	no	—
58	C₂×C₄×D₈₉	—	4	4	no	—
59	Dic₃₅₆	—	2	8	no	—
60	Dic₃₅₆	—	4	4	no	—
61	#13083563	—	4	8	no	—
62	#13083563	—	4	8	no	—
63	D₇₁₂	—	2	8	no	—
64	D₇₁₂	—	4	4	no	—
65	C₄⋊Dic₈₉	—	4	4	no	—
66	#13083551	—	4	8	no	—
67	C₂²×C₄×C₈₉	D₈	1	2	yes	—
68	C₂²×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₈₉	D₈	1	4	yes	—
70	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₈₉	C₂×D₄	1	4	yes	—
72	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₈₉	C₂⁴	1	1	yes	characteristic
75	C₄²×C₈₉	D₈	1	2	yes	—
76	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₈₉	C₂×D₄	1	1	yes	characteristic
78	C₁₆×C₈₉	C₂×D₄	1	4	yes	—
79	C₁₆×C₈₉	C₂×D₄	1	4	yes	—
80	C₂³×C₈₉	C₂×D₈	1	1	yes	characteristic
81	C₂²×D₈₉	—	4	4	no	—
82	C₂×C₄×C₈₉	—	2	4	no	—
83	C₂×C₄×C₈₉	#127	1	2	yes	—
84	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
85	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
86	C₂×C₄×C₈₉	C₂×D₈	1	2	yes	—
87	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
88	C₂×C₄×C₈₉	C₂²×D₄	1	1	yes	characteristic
89	C₂²⋊D₈₉	—	8	8	no	—
90	C₂×Dic₈₉	—	4	4	no	—
91	D₃₅₆	—	4	4	no	—
92	D₃₅₆	—	4	4	no	—
93	D₃₅₆	—	8	8	no	—
94	C₄×D₈₉	—	8	8	no	—
95	Dic₁₇₈	—	4	4	no	—
96	Dic₁₇₈	—	4	4	no	—
97	C₈×C₈₉	#127	1	2	yes	—
98	C₈×C₈₉	C₂×D₈	1	2	yes	—
99	C₈×C₈₉	C₂×D₈	1	2	yes	—
100	C₈×C₈₉	C₂²×D₄	1	2	yes	—
101	C₂²×C₈₉	—	2	2	no	—
102	C₂²×C₈₉	#493	1	2	yes	—
103	C₂²×C₈₉	C₂²×D₈	1	1	yes	characteristic
104	C₂×D₈₉	—	8	4	no	—
105	C₂×D₈₉	—	8	8	no	—
106	C₄×C₈₉	—	2	2	no	—
107	C₄×C₈₉	#493	1	2	yes	—
108	C₄×C₈₉	#508	1	4	yes	complemented
109	C₄×C₈₉	C₂×#127	1	1	yes	characteristic
110	C₄×C₈₉	C₂²×D₈	1	1	yes	characteristic
111	Dic₈₉	—	8	4	no	—
112	#33713	—	89	1	no	maximal, Sylow
113	C₂×C₈₉	—	2	2	no	—
114	C₂×C₈₉	C₂×#493	1	1	yes	characteristic
115	C₂×C₈₉	C₂×#508	1	2	yes	—
116	D₈₉	—	16	4	no	—
117	C₂×#508	—	89	4	no	—
118	#3147	—	89	2	no	—
119	#2249	—	89	4	no	—
120	#2248	—	89	2	no	—
121	#2247	—	89	1	no	—
122	#2246	—	89	1	no	—
123	C₂×C₄×C₁₆	D₈₉	1	1	yes	characteristic, complemented
124	C₈₉	#33713	1	1	yes	characteristic, complemented, Sylow
125	C₂×#135	—	89	4	no	—
126	C₂×#135	—	178	2	no	—
127	#532	—	178	2	no	—
128	#508	—	178	16	no	—
129	C₂×Q₃₂	—	89	4	no	—
130	C₂×SD₃₂	—	89	8	no	—
131	C₂×D₁₆	—	89	4	no	—
132	C₂²×C₁₆	C₂×D₈₉	1	2	yes	—
133	#495	—	89	4	no	—
134	#495	—	178	2	no	—
135	#494	—	89	2	no	—
136	#493	—	89	2	no	—
137	#492	—	178	2	no	—
138	C₂×C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
139	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
140	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
141	C₄×C₁₆	C₂×D₈₉	1	2	yes	—
142	C₂×#41	—	178	4	no	—
143	#135	—	356	8	no	—
144	#135	—	178	16	no	—
145	C₂×Q₁₆	—	89	4	no	—
146	C₂×Q₁₆	—	178	2	no	—
147	C₂×SD₁₆	—	178	4	no	—
148	C₂×SD₁₆	—	178	4	no	—
149	C₂×D₈	—	89	4	no	—
150	C₂×D₈	—	178	2	no	—
151	C₂²×C₈	D₃₅₆	1	1	yes	characteristic
152	C₂²×C₈	C₂²×D₈₉	1	1	yes	characteristic
153	C₄⋊Q₈	—	178	2	no	—
154	#127	—	178	2	no	—
155	#124	—	178	4	no	—
156	#121	—	356	4	no	—
157	C₄×D₄	—	356	4	no	—
158	C₂×C₄²	D₃₅₆	1	1	yes	characteristic
159	Q₃₂	—	178	8	no	—
160	SD₃₂	—	178	16	no	—
161	D₁₆	—	178	8	no	—
162	C₂×C₁₆	D₃₅₆	1	8	yes	complemented
163	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
164	C₂×C₁₆	C₂²×D₈₉	1	2	yes	—
165	C₄×C₈	D₃₅₆	1	1	yes	characteristic
166	C₄×C₈	D₃₅₆	1	1	yes	characteristic
167	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
168	C₄×C₈	C₂²×D₈₉	1	1	yes	characteristic
169	#41	—	356	8	no	—
170	#41	—	356	8	no	—
171	C₂×Q₈	—	178	4	no	—
172	C₂×D₄	—	356	4	no	—
173	C₂×D₄	—	356	4	no	—
174	C₂×D₄	—	178	4	no	—
175	C₂²×C₄	—	356	4	no	—
176	C₂²×C₄	D₇₁₂	1	2	yes	—
177	C₂²×C₄	C₂×D₃₅₆	1	1	yes	characteristic
178	Q₁₆	—	356	4	no	—
179	Q₁₆	—	178	8	no	—
180	SD₁₆	—	356	8	no	—
181	SD₁₆	—	356	8	no	—
182	D₈	—	356	4	no	—
183	D₈	—	178	8	no	—
184	C₂×C₈	D₇₁₂	1	4	yes	—
185	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
186	C₂×C₈	C₂×D₃₅₆	1	4	yes	—
187	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
188	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
189	C₂×C₈	C₂³×D₈₉	1	1	yes	characteristic
190	C₄⋊C₄	—	356	4	no	—
191	C₂²⋊C₄	—	356	8	no	—
192	C₄²	D₇₁₂	1	2	yes	—
193	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
194	C₄²	C₂×D₃₅₆	1	1	yes	characteristic
195	C₁₆	C₂×D₃₅₆	1	4	yes	—
196	C₁₆	C₂×D₃₅₆	1	4	yes	—
197	C₂³	—	356	4	no	—
198	C₂³	C₂×D₇₁₂	1	1	yes	characteristic
199	Q₈	—	356	4	no	—
200	Q₈	—	356	4	no	—
201	D₄	—	356	4	no	—
202	D₄	—	356	4	no	—
203	D₄	—	712	8	no	—
204	D₄	—	712	8	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	356	4	no	—
207	C₂×C₄	—	712	8	no	—
208	C₂×C₄	#15668009	1	2	yes	—
209	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
210	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
211	C₂×C₄	C₂×D₇₁₂	1	2	yes	—
212	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
213	C₂×C₄	C₂²×D₃₅₆	1	1	yes	characteristic
214	C₈	#15668009	1	2	yes	—
215	C₈	C₂×D₇₁₂	1	2	yes	—
216	C₈	C₂×D₇₁₂	1	2	yes	—
217	C₈	C₂²×D₃₅₆	1	2	yes	—
218	C₂²	—	2	2	no	—
219	C₂²	—	712	4	no	—
220	C₂²	—	712	8	no	—
221	C₂²	#19151975	1	2	yes	—
222	C₂²	C₂²×D₇₁₂	1	1	yes	characteristic
223	C₄	—	2	2	no	—
224	C₄	—	712	4	no	—
225	C₄	#19151975	1	2	yes	—
226	C₄	#19152030	1	4	yes	complemented
227	C₄	C₂×#15668009	1	1	yes	characteristic
228	C₄	C₂²×D₇₁₂	1	1	yes	characteristic
229	C₂	—	2	2	no	—
230	C₂	—	1424	4	no	—
231	C₂	C₂×#19151975	1	1	yes	characteristic
232	C₂	C₂×#19152030	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×D₇₁₂
commutator, cosocle	C₂×C₈×C₈₉	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₉
Fitting	C₂×C₄×C₁₆×C₈₉	C₂
socle	C₂²×C₈₉	C₂²×D₈
hypercenter	C₂×C₄×C₁₆	D₈₉
hypocenter	C₈₉	#33713
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#19152030	C₂	semidirect
2. #26030680	C₂	semidirect
3. #26022513	C₂	semidirect
4. #26022508	C₂	semidirect
5. #26022505	C₂	semidirect
6. #26022502	C₂	semidirect
7. C₂×C₄×C₁₆×C₈₉	C₂	semidirect
45. C₂×C₁₆×C₈₉	D₄	semidirect
108. C₄×C₈₉	#508	semidirect
123. C₂×C₄×C₁₆	D₈₉	semidirect
124. C₈₉	#33713	semidirect
162. C₂×C₁₆	D₃₅₆	semidirect
226. C₄	#19152030	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19152030	2
#26030680	2
#26022513	2
#26022508	2
#26022505	2
#26022502	2
C₂×C₄×C₁₆×C₈₉	2
#33713	89