Subgroups — #36475482

Order 22784.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26022561	C₂	1	2	yes	maximal, complemented
2	Dic₂₈₄₈	C₂	1	1	yes	maximal, characteristic, complemented
3	#26018468	C₂	1	2	yes	maximal, complemented
4	D₅₆₉₆	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×C₆₄×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
6	#19152030	—	2	2	no	—
7	Dic₁₄₂₄	C₂²	1	2	yes	—
8	#19151437	—	2	2	no	—
9	D₂₈₄₈	C₂²	1	2	yes	—
10	C₂×C₃₂×C₈₉	C₂²	1	1	yes	characteristic
11	C₆₄×C₈₉	C₂²	1	1	yes	characteristic
12	C₆₄×C₈₉	C₂²	1	1	yes	characteristic
13	#15668034	—	4	2	no	—
14	Dic₇₁₂	—	2	2	no	—
15	#15667957	—	4	2	no	—
16	D₁₄₂₄	—	2	2	no	—
17	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
18	C₃₂×C₈₉	D₄	1	1	yes	characteristic
19	C₃₂×C₈₉	C₂³	1	1	yes	characteristic
20	#13083574	—	8	2	no	—
21	Dic₃₅₆	—	4	2	no	—
22	#13083563	—	8	2	no	—
23	D₇₁₂	—	4	2	no	—
24	C₂×C₈×C₈₉	D₈	1	1	yes	characteristic
25	C₁₆×C₈₉	D₈	1	1	yes	characteristic
26	C₁₆×C₈₉	C₂×D₄	1	1	yes	characteristic
27	C₂×C₄×C₈₉	D₁₆	1	1	yes	characteristic
28	C₂²⋊D₈₉	—	16	2	no	—
29	D₃₅₆	—	8	2	no	—
30	C₄×D₈₉	—	16	2	no	—
31	Dic₁₇₈	—	8	2	no	—
32	C₈×C₈₉	D₁₆	1	1	yes	characteristic
33	C₈×C₈₉	C₂×D₈	1	1	yes	characteristic
34	C₂²×C₈₉	D₃₂	1	1	yes	characteristic
35	C₂×D₈₉	—	16	2	no	—
36	C₄×C₈₉	D₃₂	1	1	yes	characteristic
37	C₄×C₈₉	C₂×D₁₆	1	1	yes	characteristic
38	Dic₈₉	—	16	2	no	—
39	#13741	—	89	1	no	maximal, Sylow
40	C₂×C₈₉	—	2	1	no	—
41	C₂×C₈₉	C₂×D₃₂	1	1	yes	characteristic
42	D₈₉	—	32	2	no	—
43	#2262	—	89	2	no	—
44	Q₁₂₈	—	89	1	no	—
45	SD₁₂₈	—	89	2	no	—
46	D₆₄	—	89	1	no	—
47	C₂×C₆₄	D₈₉	1	1	yes	characteristic, complemented
48	C₈₉	#13741	1	1	yes	characteristic, complemented, Sylow
49	#508	—	178	2	no	—
50	Q₆₄	—	89	2	no	—
51	SD₆₄	—	178	2	no	—
52	D₃₂	—	89	2	no	—
53	C₂×C₃₂	C₂×D₈₉	1	1	yes	characteristic
54	C₆₄	C₂×D₈₉	1	1	yes	characteristic
55	C₆₄	C₂×D₈₉	1	1	yes	characteristic
56	#135	—	356	2	no	—
57	Q₃₂	—	178	2	no	—
58	SD₃₂	—	356	2	no	—
59	D₁₆	—	178	2	no	—
60	C₂×C₁₆	D₃₅₆	1	1	yes	characteristic
61	C₃₂	D₃₅₆	1	1	yes	characteristic
62	C₃₂	C₂²×D₈₉	1	1	yes	characteristic
63	#41	—	712	2	no	—
64	Q₁₆	—	356	2	no	—
65	SD₁₆	—	712	2	no	—
66	D₈	—	356	2	no	—
67	C₂×C₈	D₇₁₂	1	1	yes	characteristic
68	C₁₆	D₇₁₂	1	1	yes	characteristic
69	C₁₆	C₂×D₃₅₆	1	1	yes	characteristic
70	Q₈	—	712	2	no	—
71	D₄	—	1424	2	no	—
72	D₄	—	712	2	no	—
73	C₂×C₄	—	1424	2	no	—
74	C₂×C₄	D₁₄₂₄	1	1	yes	characteristic
75	C₈	D₁₄₂₄	1	1	yes	characteristic
76	C₈	C₂×D₇₁₂	1	1	yes	characteristic
77	C₂²	—	1424	2	no	—
78	C₂²	D₂₈₄₈	1	1	yes	characteristic
79	C₄	—	1424	2	no	—
80	C₄	D₂₈₄₈	1	1	yes	characteristic
81	C₄	C₂×D₁₄₂₄	1	1	yes	characteristic
82	C₂	—	2	1	no	—
83	C₂	—	2848	2	no	—
84	C₂	C₂×D₂₈₄₈	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₂₈₄₈
commutator, cosocle	C₃₂×C₈₉	C₂³
Frattini	C₃₂	C₂²×D₈₉
Fitting	C₂×C₆₄×C₈₉	C₂
socle	C₂×C₈₉	C₂×D₃₂
hypercenter	C₂×C₆₄	D₈₉
hypocenter	C₈₉	#13741
LCS	G ▷ C₃₂×C₈₉ ▷ C₁₆×C₈₉ ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×C₃₂ ◁ C₂×C₆₄

Complements

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

N	H ≅ G/N	type
1. #26022561	C₂	semidirect
2. Dic₂₈₄₈	C₂	semidirect
3. #26018468	C₂	semidirect
4. D₅₆₉₆	C₂	semidirect
5. C₂×C₆₄×C₈₉	C₂	semidirect
47. C₂×C₆₄	D₈₉	semidirect
48. C₈₉	#13741	semidirect

Maximal subgroups

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

subgroup	index
#26022561	2
Dic₂₈₄₈	2
#26018468	2
D₅₆₉₆	2
C₂×C₆₄×C₈₉	2
#13741	89