Subgroups — #26022502

Order 11392.

216 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₄₂₄	C₂	1	4	yes	maximal, complemented
2	#19151975	C₂	1	2	yes	maximal, complemented
3	C₄×C₁₆×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×D₇₁₂	—	2	2	no	—
5	C₂×D₇₁₂	C₂²	1	4	yes	—
6	#15668009	—	2	2	no	—
7	D₁₄₂₄	—	2	8	no	—
8	C₂×C₁₆×C₈₉	C₂²	1	2	yes	—
9	C₄×C₈×C₈₉	C₂²	1	1	yes	characteristic
10	C₂×D₃₅₆	—	2	4	no	—
11	C₂×D₃₅₆	—	4	4	no	—
12	D₇₁₂	—	2	8	no	—
13	D₇₁₂	—	4	4	no	—
14	C₂×C₈×C₈₉	D₄	1	1	yes	characteristic
15	C₂×C₈×C₈₉	C₂³	1	1	yes	characteristic
16	C₄²×C₈₉	D₄	1	1	yes	characteristic
17	C₁₆×C₈₉	D₄	1	4	yes	complemented
18	C₂²×D₈₉	—	4	4	no	—
19	C₂×C₄×C₈₉	D₈	1	2	yes	—
20	C₂×C₄×C₈₉	C₂×D₄	1	1	yes	characteristic
21	D₃₅₆	—	4	4	no	—
22	D₃₅₆	—	4	4	no	—
23	D₃₅₆	—	8	8	no	—
24	C₈×C₈₉	D₈	1	2	yes	—
25	C₈×C₈₉	C₂×D₄	1	2	yes	—
26	C₂²×C₈₉	C₂×D₈	1	1	yes	characteristic
27	C₂×D₈₉	—	8	4	no	—
28	C₂×D₈₉	—	8	8	no	—
29	C₄×C₈₉	D₁₆	1	4	yes	complemented
30	C₄×C₈₉	#127	1	1	yes	characteristic
31	C₄×C₈₉	C₂×D₈	1	1	yes	characteristic
32	C₂×C₈₉	#493	1	1	yes	characteristic
33	C₂×C₈₉	C₂×D₁₆	1	2	yes	—
34	D₈₉	—	16	4	no	—
35	#2246	—	89	1	no	maximal, Sylow
36	C₈₉	#2246	1	1	yes	characteristic, complemented, Sylow
37	C₂×D₁₆	—	89	4	no	—
38	#493	—	89	2	no	—
39	C₄×C₁₆	D₈₉	1	1	yes	characteristic, complemented
40	C₂×D₈	—	89	4	no	—
41	C₂×D₈	—	178	2	no	—
42	#127	—	178	2	no	—
43	D₁₆	—	178	8	no	—
44	C₂×C₁₆	C₂×D₈₉	1	2	yes	—
45	C₄×C₈	C₂×D₈₉	1	1	yes	characteristic
46	C₂×D₄	—	356	4	no	—
47	C₂×D₄	—	178	4	no	—
48	D₈	—	356	4	no	—
49	D₈	—	178	8	no	—
50	C₂×C₈	D₃₅₆	1	1	yes	characteristic
51	C₂×C₈	C₂²×D₈₉	1	1	yes	characteristic
52	C₄²	D₃₅₆	1	1	yes	characteristic
53	C₁₆	D₃₅₆	1	4	yes	complemented
54	C₂³	—	356	4	no	—
55	D₄	—	356	4	no	—
56	D₄	—	356	4	no	—
57	D₄	—	712	8	no	—
58	C₂×C₄	D₇₁₂	1	2	yes	—
59	C₂×C₄	C₂×D₃₅₆	1	1	yes	characteristic
60	C₈	D₇₁₂	1	2	yes	—
61	C₈	C₂×D₃₅₆	1	2	yes	—
62	C₂²	—	712	4	no	—
63	C₂²	—	712	8	no	—
64	C₂²	C₂×D₇₁₂	1	1	yes	characteristic
65	C₄	D₁₄₂₄	1	4	yes	complemented
66	C₄	#15668009	1	1	yes	characteristic
67	C₄	C₂×D₇₁₂	1	1	yes	characteristic
68	C₂	—	1424	4	no	—
69	C₂	#19151975	1	1	yes	characteristic
70	C₂	C₂×D₁₄₂₄	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₇₁₂
commutator, cosocle	C₂×C₈×C₈₉	C₂³
Frattini	C₂×C₈	C₂²×D₈₉
Fitting	C₄×C₁₆×C₈₉	C₂
socle	C₂²×C₈₉	C₂×D₈
hypercenter	C₄×C₁₆	D₈₉
hypocenter	C₈₉	#2246
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
1. C₂×D₁₄₂₄	C₂	semidirect
2. #19151975	C₂	semidirect
3. C₄×C₁₆×C₈₉	C₂	semidirect
17. C₁₆×C₈₉	D₄	semidirect
29. C₄×C₈₉	D₁₆	semidirect
36. C₈₉	#2246	semidirect
39. C₄×C₁₆	D₈₉	semidirect
53. C₁₆	D₃₅₆	semidirect
65. C₄	D₁₄₂₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₄₂₄	2
#19151975	2
C₄×C₁₆×C₈₉	2
#2246	89