Subgroups — #26022527

Order 11392.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center	C₂²	C₂×D₇₁₂
commutator, cosocle	C₂×C₈×C₈₉	C₂³
Frattini	C₂×C₈	C₂²×D₈₉
Fitting	C₄×C₁₆×C₈₉	C₂
socle, prime radical	C₂²×C₈₉	C₂×D₈
hypercenter	C₄×C₁₆	D₈₉
hypocenter	C₈₉	#2252
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

Complements

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

N	H ≅ G/N	type
15. C₁₆×C₈₉	Q₈	semidirect
33. C₈₉	#2252	semidirect
48. C₁₆	Dic₁₇₈	semidirect

Maximal subgroups

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

subgroup	index
#19152003	2
#19151424	2
C₄×C₁₆×C₈₉	2
#2252	89