Subgroups — C₂²×C₂₃×#127 #15671059

Order 2944.

3000 conjugacy classes of subgroups.

Conjugacy classes of subgroups

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

	subgroup	G/H
center, socle	C₂⁴×C₂₃	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁵×C₂₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₂₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×D₄×C₂₃	C₂	semidirect
2. C₂×C₂₃×#127	C₂	direct (N × H)
3. C₂²×C₄²×C₂₃	C₂	semidirect
6. C₂²×D₄×C₂₃	C₂²	semidirect
8. C₂₃×#127	C₂²	direct (N × H)
9. C₂×C₄²×C₂₃	C₂²	semidirect
14. C₂×D₄×C₂₃	C₂³	semidirect
15. C₂²×C₄×C₂₃	D₄	semidirect
17. C₄²×C₂₃	C₂³	semidirect
24. C₂×C₄×C₂₃	C₂×D₄	semidirect
26. C₂²×#127	C₂₃	direct (N × H)
29. C₂²×C₂₃	#127	direct (N × H)
32. C₄×C₂₃	C₂²×D₄	semidirect
33. C₂³×D₄	C₂×C₂₃	semidirect
34. C₂×#127	C₂×C₂₃	direct (N × H)
35. C₂²×C₄²	C₂×C₂₃	semidirect
37. C₂×C₂₃	C₂×#127	direct (N × H)
41. C₂²×D₄	C₂²×C₂₃	semidirect
43. #127	C₂²×C₂₃	direct (N × H)
44. C₂×C₄²	C₂²×C₂₃	semidirect
45. C₂₃	C₂²×#127	direct (N × H)
50. C₂×D₄	C₂³×C₂₃	semidirect
51. C₂²×C₄	D₄×C₂₃	semidirect
53. C₄²	C₂³×C₂₃	semidirect
60. C₂×C₄	C₂×D₄×C₂₃	semidirect
64. C₂²	C₂₃×#127	direct (N × H)
67. C₄	C₂²×D₄×C₂₃	semidirect
69. C₂	C₂×C₂₃×#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
C₂³×D₄×C₂₃	2
C₂×C₂₃×#127	2
C₂²×C₄²×C₂₃	2
C₂²×#127	23

Subgroups — C22×C23×#127 #15671059

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₂₃×#127 #15671059