Subgroups — C₂²×#11745236 #16832764

Order 3712.

3000 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₁₁₆	C₂	1	6	yes	maximal, complemented
2	C₂×#11745236	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₄²×C₂₉	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁴×D₂₉	C₂²	1	4	yes	—
5	C₂³×C₄×C₂₉	C₂²	1	3	yes	—
6	C₂²×D₁₁₆	—	2	48	no	—
7	C₂²×D₁₁₆	C₂²	1	72	yes	complemented
8	C₂×C₄²×C₂₉	C₂²	1	12	yes	complemented
9	#11745236	C₂²	1	64	yes	direct product factor
10	C₂⁴×C₂₉	C₂³	1	1	yes	characteristic
11	C₂³×D₂₉	—	2	48	no	—
12	C₂³×D₂₉	C₂³	1	24	yes	—
13	C₂²×C₄×C₂₉	D₄	1	24	yes	complemented
14	C₂²×C₄×C₂₉	C₂³	1	18	yes	—
15	C₂×D₁₁₆	—	2	288	no	—
16	C₂×D₁₁₆	C₂³	1	96	yes	complemented
17	C₄²×C₂₉	C₂³	1	16	yes	complemented
18	C₂³×C₂₉	C₂×D₄	1	12	yes	—
19	C₂³×C₂₉	C₂⁴	1	3	yes	—
20	C₂²×D₂₉	—	2	144	no	—
21	C₂²×D₂₉	—	4	64	no	—
22	C₂²×D₂₉	C₂⁴	1	16	yes	—
23	C₂×C₄×C₂₉	C₂×D₄	1	72	yes	complemented
24	C₂×C₄×C₂₉	C₂⁴	1	12	yes	—
25	D₁₁₆	—	2	192	no	—
26	C₂²×#127	—	29	1	no	maximal, Sylow
27	C₂²×C₂₉	#127	1	16	yes	complemented
28	C₂²×C₂₉	C₂²×D₄	1	18	yes	—
29	C₂²×C₂₉	C₂⁵	1	1	yes	characteristic
30	C₂×D₂₉	—	2	48	no	—
31	C₂×D₂₉	—	4	96	no	—
32	C₄×C₂₉	C₂²×D₄	1	24	yes	complemented
33	C₂³×D₄	—	29	6	no	—
34	C₂×#127	—	29	24	no	—
35	C₂²×C₄²	D₂₉	1	1	yes	characteristic, complemented
36	C₂×C₂₉	C₂×#127	1	12	yes	complemented
37	C₂×C₂₉	C₂³×D₄	1	3	yes	—
38	D₂₉	—	4	16	no	—
39	C₂⁵	—	29	4	no	—
40	C₂²×D₄	—	29	72	no	—
41	C₂²×D₄	—	58	48	no	—
42	C₂³×C₄	C₂×D₂₉	1	3	yes	—
43	#127	—	29	64	no	—
44	C₂×C₄²	C₂×D₂₉	1	12	yes	complemented
45	C₂₉	C₂²×#127	1	1	yes	characteristic, complemented, Sylow
46	C₂⁴	—	29	24	no	—
47	C₂⁴	—	58	48	no	—
48	C₂⁴	C₂²×D₂₉	1	1	yes	characteristic
49	C₂×D₄	—	29	96	no	—
50	C₂×D₄	—	58	288	no	—
51	C₂²×C₄	D₁₁₆	1	24	yes	complemented
52	C₂²×C₄	C₂²×D₂₉	1	18	yes	—
53	C₄²	C₂²×D₂₉	1	16	yes	complemented
54	C₂³	—	29	16	no	—
55	C₂³	—	58	144	no	—
56	C₂³	—	116	64	no	—
57	C₂³	C₂×D₁₁₆	1	12	yes	—
58	C₂³	C₂³×D₂₉	1	3	yes	—
59	D₄	—	58	192	no	—
60	C₂×C₄	C₂×D₁₁₆	1	72	yes	complemented
61	C₂×C₄	C₂³×D₂₉	1	12	yes	—
62	C₂²	—	58	48	no	—
63	C₂²	—	116	96	no	—
64	C₂²	#11745236	1	16	yes	direct product factor
65	C₂²	C₂²×D₁₁₆	1	18	yes	—
66	C₂²	C₂⁴×D₂₉	1	1	yes	characteristic
67	C₄	C₂²×D₁₁₆	1	24	yes	complemented
68	C₂	—	116	16	no	—
69	C₂	C₂×#11745236	1	12	yes	direct product factor
70	C₂	C₂³×D₁₁₆	1	3	yes	—

	subgroup	G/H
center	C₂⁴	C₂²×D₂₉
commutator, cosocle	C₂²×C₂₉	C₂⁵
Frattini	C₂²	C₂⁴×D₂₉
Fitting	C₂²×C₄²×C₂₉	C₂
socle	C₂⁴×C₂₉	C₂³
hypercenter	C₂²×C₄²	D₂₉
hypocenter	C₂₉	C₂²×#127
LCS	G ▷ C₂²×C₂₉ ▷ C₂₉
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
1. C₂³×D₁₁₆	C₂	semidirect
2. C₂×#11745236	C₂	direct (N × H)
3. C₂²×C₄²×C₂₉	C₂	semidirect
7. C₂²×D₁₁₆	C₂²	semidirect
8. C₂×C₄²×C₂₉	C₂²	semidirect
9. #11745236	C₂²	direct (N × H)
13. C₂²×C₄×C₂₉	D₄	semidirect
16. C₂×D₁₁₆	C₂³	semidirect
17. C₄²×C₂₉	C₂³	semidirect
23. C₂×C₄×C₂₉	C₂×D₄	semidirect
27. C₂²×C₂₉	#127	semidirect
32. C₄×C₂₉	C₂²×D₄	semidirect
35. C₂²×C₄²	D₂₉	semidirect
36. C₂×C₂₉	C₂×#127	semidirect
44. C₂×C₄²	C₂×D₂₉	semidirect
45. C₂₉	C₂²×#127	semidirect
51. C₂²×C₄	D₁₁₆	semidirect
53. C₄²	C₂²×D₂₉	semidirect
60. C₂×C₄	C₂×D₁₁₆	semidirect
64. C₂²	#11745236	direct (N × H)
67. C₄	C₂²×D₁₁₆	semidirect
69. C₂	C₂×#11745236	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×D₁₁₆	2
C₂×#11745236	2
C₂²×C₄²×C₂₉	2
C₂²×#127	29

Subgroups — C22×#11745236 #16832764

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11745236 #16832764