Subgroups — C₂²×#11758451 #16858983

Order 3968.

3000 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×D₁₂₄	C₂	1	6	yes	maximal, complemented
2	C₂×#11758451	C₂	1	24	yes	maximal, direct product factor
3	C₂²×C₄²×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁴×D₃₁	C₂²	1	4	yes	—
5	C₂³×C₄×C₃₁	C₂²	1	3	yes	—
6	C₂²×D₁₂₄	—	2	48	no	—
7	C₂²×D₁₂₄	C₂²	1	72	yes	complemented
8	C₂×C₄²×C₃₁	C₂²	1	12	yes	complemented
9	#11758451	C₂²	1	64	yes	direct product factor
10	C₂⁴×C₃₁	C₂³	1	1	yes	characteristic
11	C₂³×D₃₁	—	2	48	no	—
12	C₂³×D₃₁	C₂³	1	24	yes	—
13	C₂²×C₄×C₃₁	D₄	1	24	yes	complemented
14	C₂²×C₄×C₃₁	C₂³	1	18	yes	—
15	C₂×D₁₂₄	—	2	288	no	—
16	C₂×D₁₂₄	C₂³	1	96	yes	complemented
17	C₄²×C₃₁	C₂³	1	16	yes	complemented
18	C₂³×C₃₁	C₂×D₄	1	12	yes	—
19	C₂³×C₃₁	C₂⁴	1	3	yes	—
20	C₂²×D₃₁	—	2	144	no	—
21	C₂²×D₃₁	—	4	64	no	—
22	C₂²×D₃₁	C₂⁴	1	16	yes	—
23	C₂×C₄×C₃₁	C₂×D₄	1	72	yes	complemented
24	C₂×C₄×C₃₁	C₂⁴	1	12	yes	—
25	D₁₂₄	—	2	192	no	—
26	C₂²×#127	—	31	1	no	maximal, Sylow
27	C₂²×C₃₁	#127	1	16	yes	complemented
28	C₂²×C₃₁	C₂²×D₄	1	18	yes	—
29	C₂²×C₃₁	C₂⁵	1	1	yes	characteristic
30	C₂×D₃₁	—	2	48	no	—
31	C₂×D₃₁	—	4	96	no	—
32	C₄×C₃₁	C₂²×D₄	1	24	yes	complemented
33	C₂³×D₄	—	31	6	no	—
34	C₂×#127	—	31	24	no	—
35	C₂²×C₄²	D₃₁	1	1	yes	characteristic, complemented
36	C₂×C₃₁	C₂×#127	1	12	yes	complemented
37	C₂×C₃₁	C₂³×D₄	1	3	yes	—
38	D₃₁	—	4	16	no	—
39	C₂⁵	—	31	4	no	—
40	C₂²×D₄	—	31	72	no	—
41	C₂²×D₄	—	62	48	no	—
42	C₂³×C₄	C₂×D₃₁	1	3	yes	—
43	#127	—	31	64	no	—
44	C₂×C₄²	C₂×D₃₁	1	12	yes	complemented
45	C₃₁	C₂²×#127	1	1	yes	characteristic, complemented, Sylow
46	C₂⁴	—	31	24	no	—
47	C₂⁴	—	62	48	no	—
48	C₂⁴	C₂²×D₃₁	1	1	yes	characteristic
49	C₂×D₄	—	31	96	no	—
50	C₂×D₄	—	62	288	no	—
51	C₂²×C₄	D₁₂₄	1	24	yes	complemented
52	C₂²×C₄	C₂²×D₃₁	1	18	yes	—
53	C₄²	C₂²×D₃₁	1	16	yes	complemented
54	C₂³	—	31	16	no	—
55	C₂³	—	62	144	no	—
56	C₂³	—	124	64	no	—
57	C₂³	C₂×D₁₂₄	1	12	yes	—
58	C₂³	C₂³×D₃₁	1	3	yes	—
59	D₄	—	62	192	no	—
60	C₂×C₄	C₂×D₁₂₄	1	72	yes	complemented
61	C₂×C₄	C₂³×D₃₁	1	12	yes	—
62	C₂²	—	62	48	no	—
63	C₂²	—	124	96	no	—
64	C₂²	#11758451	1	16	yes	direct product factor
65	C₂²	C₂²×D₁₂₄	1	18	yes	—
66	C₂²	C₂⁴×D₃₁	1	1	yes	characteristic
67	C₄	C₂²×D₁₂₄	1	24	yes	complemented
68	C₂	—	124	16	no	—
69	C₂	C₂×#11758451	1	12	yes	direct product factor
70	C₂	C₂³×D₁₂₄	1	3	yes	—

	subgroup	G/H
center	C₂⁴	C₂²×D₃₁
commutator, cosocle	C₂²×C₃₁	C₂⁵
Frattini	C₂²	C₂⁴×D₃₁
Fitting	C₂²×C₄²×C₃₁	C₂
socle	C₂⁴×C₃₁	C₂³
hypercenter	C₂²×C₄²	D₃₁
hypocenter	C₃₁	C₂²×#127
LCS	G ▷ C₂²×C₃₁ ▷ C₃₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄²

Complements

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

N	H ≅ G/N	type
1. C₂³×D₁₂₄	C₂	semidirect
2. C₂×#11758451	C₂	direct (N × H)
3. C₂²×C₄²×C₃₁	C₂	semidirect
7. C₂²×D₁₂₄	C₂²	semidirect
8. C₂×C₄²×C₃₁	C₂²	semidirect
9. #11758451	C₂²	direct (N × H)
13. C₂²×C₄×C₃₁	D₄	semidirect
16. C₂×D₁₂₄	C₂³	semidirect
17. C₄²×C₃₁	C₂³	semidirect
23. C₂×C₄×C₃₁	C₂×D₄	semidirect
27. C₂²×C₃₁	#127	semidirect
32. C₄×C₃₁	C₂²×D₄	semidirect
35. C₂²×C₄²	D₃₁	semidirect
36. C₂×C₃₁	C₂×#127	semidirect
44. C₂×C₄²	C₂×D₃₁	semidirect
45. C₃₁	C₂²×#127	semidirect
51. C₂²×C₄	D₁₂₄	semidirect
53. C₄²	C₂²×D₃₁	semidirect
60. C₂×C₄	C₂×D₁₂₄	semidirect
64. C₂²	#11758451	direct (N × H)
67. C₄	C₂²×D₁₂₄	semidirect
69. C₂	C₂×#11758451	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×D₁₂₄	2
C₂×#11758451	2
C₂²×C₄²×C₃₁	2
C₂²×#127	31

Subgroups — C22×#11758451 #16858983

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#11758451 #16858983