Subgroups — C₂²×#120 #3431

Order 128.

2272 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁶	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×D₄	C₂	1	3	yes	maximal, complemented
3	C₂×#120	C₂	1	24	yes	maximal, direct product factor
4	C₂²×C₂²⋊C₄	C₂	1	3	yes	maximal, complemented
5	C₂⁵	—	2	24	no	—
6	C₂⁵	C₂²	1	3	yes	—
7	C₂⁵	C₂²	1	12	yes	complemented
8	C₂⁵	C₂²	1	1	yes	characteristic, complemented
9	C₂²×D₄	—	2	24	no	—
10	C₂²×D₄	C₂²	1	36	yes	complemented
11	C₂³×C₄	C₂²	1	3	yes	complemented
12	#120	C₂²	1	64	yes	direct product factor
13	C₂×C₂²⋊C₄	C₂²	1	36	yes	complemented
14	C₂⁴	—	2	128	no	—
15	C₂⁴	—	2	24	no	—
16	C₂⁴	—	2	144	no	—
17	C₂⁴	—	2	12	no	—
18	C₂⁴	D₄	1	24	yes	complemented
19	C₂⁴	C₂³	1	18	yes	—
20	C₂⁴	C₂³	1	1	yes	characteristic
21	C₂⁴	C₂³	1	16	yes	complemented
22	C₂⁴	C₂³	1	6	yes	complemented
23	C₂×D₄	—	2	144	no	—
24	C₂×D₄	C₂³	1	48	yes	complemented
25	C₂²×C₄	—	2	12	no	—
26	C₂²×C₄	C₂³	1	18	yes	complemented
27	C₂²⋊C₄	C₂³	1	48	yes	complemented
28	C₂³	—	2	384	no	—
29	C₂³	—	2	96	no	—
30	C₂³	—	2	72	no	—
31	C₂³	—	2	96	no	—
32	C₂³	—	2	36	no	—
33	C₂³	—	4	16	no	—
34	C₂³	C₂×D₄	1	12	yes	—
35	C₂³	C₂×D₄	1	72	yes	complemented
36	C₂³	C₂⁴	1	12	yes	—
37	C₂³	C₂⁴	1	3	yes	—
38	C₂³	C₂⁴	1	4	yes	complemented
39	D₄	—	2	96	no	—
40	C₂×C₄	—	2	36	no	—
41	C₂×C₄	C₂⁴	1	12	yes	complemented
42	C₂²	—	2	128	no	—
43	C₂²	—	2	144	no	—
44	C₂²	—	2	24	no	—
45	C₂²	—	2	12	no	—
46	C₂²	—	4	24	no	—
47	C₂²	#120	1	16	yes	direct product factor
48	C₂²	C₂²×D₄	1	18	yes	—
49	C₂²	C₂²×D₄	1	24	yes	complemented
50	C₂²	C₂⁵	1	1	yes	characteristic
51	C₄	—	2	12	no	—
52	C₂	—	2	24	no	—
53	C₂	—	4	4	no	—
54	C₂	C₂×#120	1	12	yes	direct product factor
55	C₂	C₂³×D₄	1	3	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂⁶	C₂	semidirect
2. C₂³×D₄	C₂	semidirect
3. C₂×#120	C₂	direct (N × H)
4. C₂²×C₂²⋊C₄	C₂	semidirect
7. C₂⁵	C₂²	semidirect
8. C₂⁵	C₂²	semidirect
10. C₂²×D₄	C₂²	semidirect
11. C₂³×C₄	C₂²	semidirect
12. #120	C₂²	direct (N × H)
13. C₂×C₂²⋊C₄	C₂²	semidirect
18. C₂⁴	D₄	semidirect
21. C₂⁴	C₂³	semidirect
22. C₂⁴	C₂³	semidirect
24. C₂×D₄	C₂³	semidirect
26. C₂²×C₄	C₂³	semidirect
27. C₂²⋊C₄	C₂³	semidirect
35. C₂³	C₂×D₄	semidirect
38. C₂³	C₂⁴	semidirect
41. C₂×C₄	C₂⁴	semidirect
47. C₂²	#120	direct (N × H)
49. C₂²	C₂²×D₄	semidirect
54. C₂	C₂×#120	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁶	2
C₂³×D₄	2
C₂×#120	2
C₂²×C₂²⋊C₄	2

Subgroups — C22×#120 #3431

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#120 #3431