Subgroups — C₂²×#15668009 #26033524

Order 11392.

3000 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₃₅₆	C₂	1	6	yes	maximal, complemented
2	C₂×#15668009	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₄²×C₈₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁴×D₈₉	C₂²	1	4	yes	—
5	C₂²×D₃₅₆	—	2	48	no	—
6	C₂²×D₃₅₆	C₂²	1	72	yes	complemented
7	#15668009	C₂²	1	64	yes	direct product factor
8	C₂³×C₄×C₈₉	C₂²	1	3	yes	—
9	C₂×C₄²×C₈₉	C₂²	1	12	yes	complemented
10	C₂³×D₈₉	—	2	48	no	—
11	C₂³×D₈₉	C₂³	1	24	yes	—
12	C₂×D₃₅₆	—	2	288	no	—
13	C₂×D₃₅₆	C₂³	1	96	yes	complemented
14	C₂⁴×C₈₉	C₂³	1	1	yes	characteristic
15	C₂²×C₄×C₈₉	D₄	1	24	yes	complemented
16	C₂²×C₄×C₈₉	C₂³	1	18	yes	—
17	C₄²×C₈₉	C₂³	1	16	yes	complemented
18	C₂³×C₈₉	C₂×D₄	1	12	yes	—
19	C₂³×C₈₉	C₂⁴	1	3	yes	—
20	C₂²×D₈₉	—	2	144	no	—
21	C₂²×D₈₉	—	4	64	no	—
22	C₂²×D₈₉	C₂⁴	1	16	yes	—
23	C₂×C₄×C₈₉	C₂×D₄	1	72	yes	complemented
24	C₂×C₄×C₈₉	C₂⁴	1	12	yes	—
25	D₃₅₆	—	2	192	no	—
26	C₂²×C₈₉	#127	1	16	yes	complemented
27	C₂²×C₈₉	C₂²×D₄	1	18	yes	—
28	C₂²×C₈₉	C₂⁵	1	1	yes	characteristic
29	C₂×D₈₉	—	2	48	no	—
30	C₂×D₈₉	—	4	96	no	—
31	C₄×C₈₉	C₂²×D₄	1	24	yes	complemented
32	C₂×C₈₉	C₂×#127	1	12	yes	complemented
33	C₂×C₈₉	C₂³×D₄	1	3	yes	—
34	D₈₉	—	4	16	no	—
35	C₂²×#127	—	89	1	no	maximal, Sylow
36	C₈₉	C₂²×#127	1	1	yes	characteristic, complemented, Sylow
37	C₂³×D₄	—	89	6	no	—
38	C₂×#127	—	89	24	no	—
39	C₂²×C₄²	D₈₉	1	1	yes	characteristic, complemented
40	C₂⁵	—	89	4	no	—
41	C₂²×D₄	—	89	72	no	—
42	C₂²×D₄	—	178	48	no	—
43	C₂³×C₄	C₂×D₈₉	1	3	yes	—
44	#127	—	89	64	no	—
45	C₂×C₄²	C₂×D₈₉	1	12	yes	complemented
46	C₂⁴	—	89	24	no	—
47	C₂⁴	—	178	48	no	—
48	C₂⁴	C₂²×D₈₉	1	1	yes	characteristic
49	C₂×D₄	—	89	96	no	—
50	C₂×D₄	—	178	288	no	—
51	C₂²×C₄	D₃₅₆	1	24	yes	complemented
52	C₂²×C₄	C₂²×D₈₉	1	18	yes	—
53	C₄²	C₂²×D₈₉	1	16	yes	complemented
54	C₂³	—	89	16	no	—
55	C₂³	—	356	64	no	—
56	C₂³	—	178	144	no	—
57	C₂³	C₂×D₃₅₆	1	12	yes	—
58	C₂³	C₂³×D₈₉	1	3	yes	—
59	D₄	—	178	192	no	—
60	C₂×C₄	C₂×D₃₅₆	1	72	yes	complemented
61	C₂×C₄	C₂³×D₈₉	1	12	yes	—
62	C₂²	—	356	96	no	—
63	C₂²	—	178	48	no	—
64	C₂²	#15668009	1	16	yes	direct product factor
65	C₂²	C₂²×D₃₅₆	1	18	yes	—
66	C₂²	C₂⁴×D₈₉	1	1	yes	characteristic
67	C₄	C₂²×D₃₅₆	1	24	yes	complemented
68	C₂	—	356	16	no	—
69	C₂	C₂×#15668009	1	12	yes	direct product factor
70	C₂	C₂³×D₃₅₆	1	3	yes	—

	subgroup	G/H
center	C₂⁴	C₂²×D₈₉
commutator, cosocle	C₂²×C₈₉	C₂⁵
Frattini	C₂²	C₂⁴×D₈₉
Fitting	C₂²×C₄²×C₈₉	C₂
socle	C₂⁴×C₈₉	C₂³
hypercenter	C₂²×C₄²	D₈₉
hypocenter	C₈₉	C₂²×#127
LCS	G ▷ C₂²×C₈₉ ▷ C₈₉
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
1. C₂³×D₃₅₆	C₂	semidirect
2. C₂×#15668009	C₂	direct (N × H)
3. C₂²×C₄²×C₈₉	C₂	semidirect
6. C₂²×D₃₅₆	C₂²	semidirect
7. #15668009	C₂²	direct (N × H)
9. C₂×C₄²×C₈₉	C₂²	semidirect
13. C₂×D₃₅₆	C₂³	semidirect
15. C₂²×C₄×C₈₉	D₄	semidirect
17. C₄²×C₈₉	C₂³	semidirect
23. C₂×C₄×C₈₉	C₂×D₄	semidirect
26. C₂²×C₈₉	#127	semidirect
31. C₄×C₈₉	C₂²×D₄	semidirect
32. C₂×C₈₉	C₂×#127	semidirect
36. C₈₉	C₂²×#127	semidirect
39. C₂²×C₄²	D₈₉	semidirect
45. C₂×C₄²	C₂×D₈₉	semidirect
51. C₂²×C₄	D₃₅₆	semidirect
53. C₄²	C₂²×D₈₉	semidirect
60. C₂×C₄	C₂×D₃₅₆	semidirect
64. C₂²	#15668009	direct (N × H)
67. C₄	C₂²×D₃₅₆	semidirect
69. C₂	C₂×#15668009	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×D₃₅₆	2
C₂×#15668009	2
C₂²×C₄²×C₈₉	2
C₂²×#127	89

Subgroups — C22×#15668009 #26033524

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#15668009 #26033524