Subgroups — C₂×C₃²×#127 #10596331

Order 576.

1494 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃²×D₄	C₂	1	6	yes	maximal, complemented
2	C₃²×#127	C₂	1	8	yes	maximal, direct product factor
3	C₂×C₃²×C₄²	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃×#127	C₃	1	4	yes	maximal, direct product factor
5	C₂⁴×C₃²	C₂²	1	4	yes	—
6	C₂×C₃²×D₄	—	2	24	no	—
7	C₂×C₃²×D₄	C₂²	1	24	yes	complemented
8	C₂²×C₃²×C₄	C₂²	1	3	yes	—
9	C₃²×C₄²	C₂²	1	4	yes	complemented
10	C₂²×C₃×D₄	C₂×C₃	1	24	yes	complemented
11	C₃×#127	C₂×C₃	1	32	yes	direct product factor
12	C₂×C₃×C₄²	C₂×C₃	1	4	yes	complemented
13	C₂³×C₃²	—	2	24	no	—
14	C₂³×C₃²	C₂³	1	8	yes	—
15	C₂³×C₃²	C₂³	1	1	yes	characteristic
16	C₃²×D₄	—	2	48	no	—
17	C₂×C₃²×C₄	D₄	1	12	yes	complemented
18	C₂×C₃²×C₄	C₂³	1	6	yes	—
19	C₂×#127	C₃²	1	1	yes	characteristic, direct product factor, Sylow
20	C₂⁴×C₃	C₂²×C₃	1	16	yes	—
21	C₂×C₃×D₄	—	2	96	no	—
22	C₂×C₃×D₄	C₂²×C₃	1	96	yes	complemented
23	C₂²×C₃×C₄	C₂²×C₃	1	12	yes	—
24	C₃×C₄²	C₂²×C₃	1	16	yes	complemented
25	C₂²×C₃²	—	2	24	no	—
26	C₂²×C₃²	—	4	16	no	—
27	C₂²×C₃²	C₂×D₄	1	6	yes	—
28	C₂²×C₃²	C₂⁴	1	1	yes	characteristic
29	C₃²×C₄	C₂×D₄	1	12	yes	complemented
30	C₂²×D₄	C₂×C₃²	1	6	yes	complemented
31	#127	C₂×C₃²	1	8	yes	direct product factor
32	C₂×C₄²	C₂×C₃²	1	1	yes	characteristic, complemented
33	C₂³×C₃	—	2	96	no	—
34	C₂³×C₃	C₂³×C₃	1	32	yes	—
35	C₂³×C₃	C₂³×C₃	1	4	yes	—
36	C₃×D₄	—	2	192	no	—
37	C₂×C₃×C₄	C₃×D₄	1	48	yes	complemented
38	C₂×C₃×C₄	C₂³×C₃	1	24	yes	—
39	C₂×C₃²	—	4	8	no	—
40	C₂×C₃²	#127	1	4	yes	direct product factor
41	C₂×C₃²	C₂²×D₄	1	3	yes	—
42	C₂⁴	C₂²×C₃²	1	4	yes	—
43	C₂×D₄	—	2	24	no	—
44	C₂×D₄	C₂²×C₃²	1	24	yes	complemented
45	C₂²×C₄	C₂²×C₃²	1	3	yes	—
46	C₄²	C₂²×C₃²	1	4	yes	complemented
47	C₂²×C₃	—	2	96	no	—
48	C₂²×C₃	—	4	64	no	—
49	C₂²×C₃	C₂×C₃×D₄	1	24	yes	—
50	C₂²×C₃	C₂⁴×C₃	1	4	yes	—
51	C₃×C₄	C₂×C₃×D₄	1	48	yes	complemented
52	C₃²	C₂×#127	1	1	yes	characteristic, direct product factor, Sylow
53	C₂³	—	2	24	no	—
54	C₂³	C₂³×C₃²	1	8	yes	—
55	C₂³	C₂³×C₃²	1	1	yes	characteristic
56	D₄	—	2	48	no	—
57	C₂×C₄	C₃²×D₄	1	12	yes	complemented
58	C₂×C₄	C₂³×C₃²	1	6	yes	—
59	C₂×C₃	—	4	32	no	—
60	C₂×C₃	C₃×#127	1	16	yes	direct product factor
61	C₂×C₃	C₂²×C₃×D₄	1	12	yes	—
62	C₂²	—	2	24	no	—
63	C₂²	—	4	16	no	—
64	C₂²	C₂×C₃²×D₄	1	6	yes	—
65	C₂²	C₂⁴×C₃²	1	1	yes	characteristic
66	C₄	C₂×C₃²×D₄	1	12	yes	complemented
67	C₃	C₂×C₃×#127	1	4	yes	direct product factor
68	C₂	—	4	8	no	—
69	C₂	C₃²×#127	1	4	yes	direct product factor
70	C₂	C₂²×C₃²×D₄	1	3	yes	—

	subgroup	G/H
center, socle	C₂³×C₃²	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁴×C₃²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₃² ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₃²×D₄	C₂	semidirect
2. C₃²×#127	C₂	direct (N × H)
3. C₂×C₃²×C₄²	C₂	semidirect
4. C₂×C₃×#127	C₃	direct (N × H)
7. C₂×C₃²×D₄	C₂²	semidirect
9. C₃²×C₄²	C₂²	semidirect
10. C₂²×C₃×D₄	C₂×C₃	semidirect
11. C₃×#127	C₂×C₃	direct (N × H)
12. C₂×C₃×C₄²	C₂×C₃	semidirect
17. C₂×C₃²×C₄	D₄	semidirect
19. C₂×#127	C₃²	direct (N × H)
22. C₂×C₃×D₄	C₂²×C₃	semidirect
24. C₃×C₄²	C₂²×C₃	semidirect
29. C₃²×C₄	C₂×D₄	semidirect
30. C₂²×D₄	C₂×C₃²	semidirect
31. #127	C₂×C₃²	direct (N × H)
32. C₂×C₄²	C₂×C₃²	semidirect
37. C₂×C₃×C₄	C₃×D₄	semidirect
40. C₂×C₃²	#127	direct (N × H)
44. C₂×D₄	C₂²×C₃²	semidirect
46. C₄²	C₂²×C₃²	semidirect
51. C₃×C₄	C₂×C₃×D₄	semidirect
52. C₃²	C₂×#127	direct (N × H)
57. C₂×C₄	C₃²×D₄	semidirect
60. C₂×C₃	C₃×#127	direct (N × H)
66. C₄	C₂×C₃²×D₄	semidirect
67. C₃	C₂×C₃×#127	direct (N × H)
69. C₂	C₃²×#127	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃²×D₄	2
C₃²×#127	2
C₂×C₃²×C₄²	2
C₂×C₃×#127	3

Subgroups — C2×C32×#127 #10596331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃²×#127 #10596331