Subgroups — #36475448

Order 22784.

158 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#26022527	C₂	1	2	yes	maximal
2	#26018452	C₂	1	4	yes	maximal
3	C₄×C₃₂×C₈₉	C₂	1	1	yes	maximal, characteristic
4	#19152003	—	2	2	no	—
5	#19151424	—	2	2	no	—
6	#19151424	C₂²	1	4	yes	—
7	C₂×C₃₂×C₈₉	C₂²	1	2	yes	—
8	C₄×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
9	#15668012	—	4	2	no	—
10	#15667946	—	2	4	no	—
11	#15667946	—	4	2	no	—
12	C₂×C₁₆×C₈₉	D₄	1	1	yes	characteristic
13	C₂×C₁₆×C₈₉	C₂³	1	1	yes	characteristic
14	C₄×C₈×C₈₉	D₄	1	1	yes	characteristic
15	C₃₂×C₈₉	Q₈	1	4	yes	complemented
16	C₂×Dic₁₇₈	—	8	2	no	—
17	C₄⋊Dic₈₉	—	4	4	no	—
18	C₄⋊Dic₈₉	—	8	2	no	—
19	C₂×C₈×C₈₉	D₈	1	1	yes	characteristic
20	C₂×C₈×C₈₉	C₂×D₄	1	1	yes	characteristic
21	C₄²×C₈₉	D₈	1	1	yes	characteristic
22	C₁₆×C₈₉	Q₁₆	1	2	yes	—
23	C₁₆×C₈₉	C₂×Q₈	1	2	yes	—
24	C₂×C₄×C₈₉	D₁₆	1	1	yes	characteristic
25	C₂×C₄×C₈₉	D₁₆	1	1	yes	characteristic
26	C₂×C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
27	C₂×Dic₈₉	—	8	4	no	—
28	Dic₁₇₈	—	16	4	no	—
29	C₈×C₈₉	Q₃₂	1	2	yes	—
30	C₈×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
31	C₈×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
32	C₂²×C₈₉	C₂×D₁₆	1	1	yes	characteristic
33	C₄×C₈₉	D₃₂	1	2	yes	—
34	C₄×C₈₉	Q₆₄	1	2	yes	—
35	C₄×C₈₉	#500	1	1	yes	characteristic
36	C₄×C₈₉	C₂×Q₃₂	1	1	yes	characteristic
37	Dic₈₉	—	16	4	no	—
38	#13731	—	89	1	no	maximal, Sylow
39	C₂×C₈₉	#2252	1	1	yes	characteristic
40	C₂×C₈₉	C₂×D₃₂	1	1	yes	characteristic
41	C₂×C₈₉	C₂×Q₆₄	1	1	yes	characteristic
42	#2252	—	89	2	no	—
43	#1423	—	89	4	no	—
44	C₄×C₃₂	D₈₉	1	1	yes	characteristic
45	C₈₉	#13731	1	1	yes	characteristic, complemented, Sylow
46	#500	—	178	2	no	—
47	C₂×C₃₂	C₂×D₈₉	1	2	yes	—
48	#366	—	89	4	no	—
49	#366	—	178	2	no	—
50	C₄×C₁₆	C₂×D₈₉	1	1	yes	characteristic
51	C₄⋊Q₈	—	356	2	no	—
52	C₂×C₁₆	D₃₅₆	1	1	yes	characteristic
53	C₂×C₁₆	C₂²×D₈₉	1	1	yes	characteristic
54	#107	—	356	2	no	—
55	#107	—	178	4	no	—
56	C₄×C₈	D₃₅₆	1	1	yes	characteristic
57	C₃₂	Dic₁₇₈	1	4	yes	complemented
58	C₂×Q₈	—	712	2	no	—
59	C₂×C₈	D₇₁₂	1	1	yes	characteristic
60	C₂×C₈	C₂×D₃₅₆	1	1	yes	characteristic
61	C₄⋊C₄	—	356	4	no	—
62	C₄⋊C₄	—	712	2	no	—
63	C₄²	D₇₁₂	1	1	yes	characteristic
64	C₁₆	Dic₃₅₆	1	2	yes	—
65	C₁₆	C₂×Dic₁₇₈	1	2	yes	—
66	Q₈	—	1424	4	no	—
67	C₂×C₄	—	712	4	no	—
68	C₂×C₄	D₁₄₂₄	1	1	yes	characteristic
69	C₂×C₄	D₁₄₂₄	1	1	yes	characteristic
70	C₂×C₄	C₂×D₇₁₂	1	1	yes	characteristic
71	C₈	Dic₇₁₂	1	2	yes	—
72	C₈	#15668012	1	1	yes	characteristic
73	C₈	C₂×Dic₃₅₆	1	1	yes	characteristic
74	C₂²	C₂×D₁₄₂₄	1	1	yes	characteristic
75	C₄	—	1424	4	no	—
76	C₄	D₂₈₄₈	1	2	yes	—
77	C₄	Dic₁₄₂₄	1	2	yes	—
78	C₄	#19152003	1	1	yes	characteristic
79	C₄	C₂×Dic₇₁₂	1	1	yes	characteristic
80	C₂	#26022527	1	1	yes	characteristic
81	C₂	C₂×D₂₈₄₈	1	1	yes	characteristic
82	C₂	C₂×Dic₁₄₂₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₄₂₄
commutator, cosocle	C₂×C₁₆×C₈₉	C₂³
Frattini	C₂×C₁₆	C₂²×D₈₉
Fitting	C₄×C₃₂×C₈₉	C₂
socle, prime radical	C₂²×C₈₉	C₂×D₁₆
hypercenter	C₄×C₃₂	D₈₉
hypocenter	C₈₉	#13731
LCS	G ▷ C₂×C₁₆×C₈₉ ▷ C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ C₄×C₃₂

Complements

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

N	H ≅ G/N	type
15. C₃₂×C₈₉	Q₈	semidirect
45. C₈₉	#13731	semidirect
57. C₃₂	Dic₁₇₈	semidirect

Maximal subgroups

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

subgroup	index
#26022527	2
#26018452	2
C₄×C₃₂×C₈₉	2
#13731	89