Subgroups — #26022533

Order 11392.

128 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#19152002	C₂	1	2	yes	maximal
2	#19151426	C₂	1	1	yes	maximal, characteristic
3	#19151425	C₂	1	2	yes	maximal
4	C₁₆×Dic₈₉	C₂	1	1	yes	maximal, characteristic
5	#367×C₈₉	C₂	1	1	yes	maximal, characteristic
6	#15668011	—	2	2	no	—
7	C₂×#13083549	C₂²	1	1	yes	characteristic
8	#15667946	C₂²	1	2	yes	—
9	#15667945	—	2	2	no	—
10	C₈×Dic₈₉	C₂²	1	1	yes	characteristic
11	C₂×C₁₆×C₈₉	C₂²	1	1	yes	characteristic
12	#107×C₈₉	C₂²	1	2	yes	—
13	C₂×Dic₁₇₈	—	4	2	no	—
14	C₂×#10623494	D₄	1	1	yes	characteristic
15	C₄⋊Dic₈₉	—	2	2	no	—
16	#13083553	—	4	2	no	—
17	C₄×Dic₈₉	D₄	1	1	yes	characteristic
18	#13083549	Q₈	1	2	yes	complemented
19	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
20	C₄⋊C₄×C₈₉	—	2	2	no	—
21	C₁₆×C₈₉	Q₈	1	2	yes	complemented
22	C₂×C₄×C₈₉	—	4	2	no	—
23	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
24	C₂×Dic₈₉	—	4	2	no	—
25	C₂×Dic₈₉	D₈	1	2	yes	—
26	Dic₁₇₈	—	8	4	no	—
27	C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
28	C₈×C₈₉	C₂×Q₈	1	1	yes	characteristic
29	#10623494	Q₁₆	1	2	yes	—
30	C₂²×C₈₉	C₂×D₈	1	1	yes	characteristic
31	C₄×C₈₉	—	8	2	no	—
32	C₄×C₈₉	C₄⋊Q₈	1	1	yes	characteristic
33	C₄×C₈₉	C₂×Q₁₆	1	1	yes	characteristic
34	Dic₈₉	—	8	2	no	—
35	Dic₈₉	SD₃₂	1	4	yes	—
36	C₂×C₈₉	#500	1	1	yes	characteristic
37	C₂×C₈₉	C₂×SD₃₂	1	2	yes	—
38	#2254	—	89	1	no	maximal, Sylow
39	C₈₉	#2254	1	1	yes	characteristic, complemented, Sylow
40	#500	—	89	2	no	—
41	#367	—	89	1	no	—
42	#367	—	89	2	no	—
43	#367	D₈₉	1	1	yes	characteristic
44	C₄×C₁₆	—	89	1	no	—
45	C₄⋊Q₈	—	178	2	no	—
46	C₂×C₁₆	—	89	1	no	—
47	C₂×C₁₆	C₂×D₈₉	1	1	yes	characteristic
48	#107	—	89	2	no	—
49	#107	—	178	2	no	—
50	#107	C₂×D₈₉	1	2	yes	—
51	C₄×C₈	—	89	1	no	—
52	C₂×Q₈	—	356	2	no	—
53	C₂×C₈	—	89	1	no	—
54	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
55	C₄⋊C₄	—	2	2	no	—
56	C₄⋊C₄	—	356	2	no	—
57	C₄⋊C₄	—	178	2	no	—
58	C₄²	—	89	1	no	—
59	C₁₆	—	89	2	no	—
60	C₁₆	Dic₁₇₈	1	2	yes	complemented
61	Q₈	—	712	4	no	—
62	C₂×C₄	—	4	2	no	—
63	C₂×C₄	—	89	2	no	—
64	C₂×C₄	—	356	2	no	—
65	C₂×C₄	D₄×D₈₉	1	1	yes	characteristic
66	C₈	—	89	2	no	—
67	C₈	Q₈×D₈₉	1	1	yes	characteristic
68	C₈	C₂×Dic₁₇₈	1	1	yes	characteristic
69	C₂²	D₈×D₈₉	1	1	yes	characteristic
70	C₄	—	8	2	no	—
71	C₄	—	89	4	no	—
72	C₄	—	712	2	no	—
73	C₄	#15668011	1	1	yes	characteristic
74	C₄	Q₁₆×D₈₉	1	1	yes	characteristic
75	C₂	#19152002	1	1	yes	characteristic
76	C₂	SD₃₂×D₈₉	1	2	yes	—

	subgroup	G/H
center	C₂²	D₈×D₈₉
commutator, cosocle	C₂×C₈×C₈₉	C₂³
Frattini	C₂×C₈	C₂²×D₈₉
Fitting	#367×C₈₉	C₂
socle, prime radical	C₂²×C₈₉	C₂×D₈
hypercenter	#367	D₈₉
hypocenter	C₈₉	#2254
LCS	G ▷ C₂×C₈×C₈₉ ▷ C₄×C₈₉ ▷ C₂×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ #367

Complements

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

N	H ≅ G/N	type
18. #13083549	Q₈	semidirect
21. C₁₆×C₈₉	Q₈	semidirect
39. C₈₉	#2254	semidirect
60. C₁₆	Dic₁₇₈	semidirect

Maximal subgroups

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

subgroup	index
#19152002	2
#19151426	2
#19151425	2
C₁₆×Dic₈₉	2
#367×C₈₉	2
#2254	89