Subgroups — C₂³×#120 #62655

Order 256.

21431 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁷	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂⁴×D₄	C₂	1	3	yes	maximal, complemented
3	C₂²×#120	C₂	1	56	yes	maximal, direct product factor
4	C₂³×C₂²⋊C₄	C₂	1	3	yes	maximal, complemented
5	C₂⁶	—	2	48	no	—
6	C₂⁶	C₂²	1	3	yes	—
7	C₂⁶	C₂²	1	28	yes	complemented
8	C₂⁶	C₂²	1	1	yes	characteristic, complemented
9	C₂³×D₄	—	2	48	no	—
10	C₂³×D₄	C₂²	1	84	yes	complemented
11	C₂⁴×C₄	C₂²	1	3	yes	complemented
12	C₂×#120	C₂²	1	448	yes	direct product factor
13	C₂²×C₂²⋊C₄	C₂²	1	84	yes	complemented
14	C₂⁵	—	2	512	no	—
15	C₂⁵	—	2	48	no	—
16	C₂⁵	—	2	672	no	—
17	C₂⁵	—	2	24	no	—
18	C₂⁵	D₄	1	48	yes	complemented
19	C₂⁵	C₂³	1	42	yes	—
20	C₂⁵	C₂³	1	1	yes	characteristic
21	C₂⁵	C₂³	1	112	yes	complemented
22	C₂⁵	C₂³	1	14	yes	complemented
23	C₂²×D₄	—	2	672	no	—
24	C₂²×D₄	C₂³	1	336	yes	complemented
25	C₂³×C₄	—	2	24	no	—
26	C₂³×C₄	C₂³	1	42	yes	complemented
27	#120	C₂³	1	512	yes	direct product factor
28	C₂×C₂²⋊C₄	C₂³	1	336	yes	complemented
29	C₂⁴	—	2	3584	no	—
30	C₂⁴	—	2	384	no	—
31	C₂⁴	—	2	336	no	—
32	C₂⁴	—	2	1344	no	—
33	C₂⁴	—	2	168	no	—
34	C₂⁴	—	4	64	no	—
35	C₂⁴	C₂×D₄	1	24	yes	—
36	C₂⁴	C₂×D₄	1	336	yes	complemented
37	C₂⁴	C₂⁴	1	84	yes	—
38	C₂⁴	C₂⁴	1	7	yes	—
39	C₂⁴	C₂⁴	1	64	yes	complemented
40	C₂⁴	C₂⁴	1	28	yes	complemented
41	C₂×D₄	—	2	1344	no	—
42	C₂×D₄	C₂⁴	1	192	yes	complemented
43	C₂²×C₄	—	2	168	no	—
44	C₂²×C₄	C₂⁴	1	84	yes	complemented
45	C₂²⋊C₄	C₂⁴	1	192	yes	complemented
46	C₂³	—	2	3584	no	—
47	C₂³	—	2	1344	no	—
48	C₂³	—	2	336	no	—
49	C₂³	—	2	384	no	—
50	C₂³	—	2	168	no	—
51	C₂³	—	4	224	no	—
52	C₂³	#120	1	64	yes	direct product factor
53	C₂³	C₂²×D₄	1	84	yes	—
54	C₂³	C₂²×D₄	1	336	yes	complemented
55	C₂³	C₂⁵	1	24	yes	—
56	C₂³	C₂⁵	1	7	yes	—
57	C₂³	C₂⁵	1	8	yes	complemented
58	D₄	—	2	384	no	—
59	C₂×C₄	—	2	168	no	—
60	C₂×C₄	C₂⁵	1	24	yes	complemented
61	C₂²	—	2	512	no	—
62	C₂²	—	2	48	no	—
63	C₂²	—	2	672	no	—
64	C₂²	—	2	24	no	—
65	C₂²	—	4	112	no	—
66	C₂²	C₂×#120	1	112	yes	direct product factor
67	C₂²	C₂³×D₄	1	42	yes	—
68	C₂²	C₂³×D₄	1	48	yes	complemented
69	C₂²	C₂⁶	1	1	yes	characteristic
70	C₄	—	2	24	no	—
71	C₂	—	2	48	no	—
72	C₂	—	4	8	no	—
73	C₂	C₂²×#120	1	28	yes	direct product factor
74	C₂	C₂⁴×D₄	1	3	yes	—

	subgroup	G/H
center, socle	C₂⁵	C₂³
commutator, Frattini, cosocle	C₂²	C₂⁶
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂² ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂⁷	C₂	semidirect
2. C₂⁴×D₄	C₂	semidirect
3. C₂²×#120	C₂	direct (N × H)
4. C₂³×C₂²⋊C₄	C₂	semidirect
7. C₂⁶	C₂²	semidirect
8. C₂⁶	C₂²	semidirect
10. C₂³×D₄	C₂²	semidirect
11. C₂⁴×C₄	C₂²	semidirect
12. C₂×#120	C₂²	direct (N × H)
13. C₂²×C₂²⋊C₄	C₂²	semidirect
18. C₂⁵	D₄	semidirect
21. C₂⁵	C₂³	semidirect
22. C₂⁵	C₂³	semidirect
24. C₂²×D₄	C₂³	semidirect
26. C₂³×C₄	C₂³	semidirect
27. #120	C₂³	direct (N × H)
28. C₂×C₂²⋊C₄	C₂³	semidirect
36. C₂⁴	C₂×D₄	semidirect
39. C₂⁴	C₂⁴	semidirect
40. C₂⁴	C₂⁴	semidirect
42. C₂×D₄	C₂⁴	semidirect
44. C₂²×C₄	C₂⁴	semidirect
45. C₂²⋊C₄	C₂⁴	semidirect
52. C₂³	#120	direct (N × H)
54. C₂³	C₂²×D₄	semidirect
57. C₂³	C₂⁵	semidirect
60. C₂×C₄	C₂⁵	semidirect
66. C₂²	C₂×#120	direct (N × H)
68. C₂²	C₂³×D₄	semidirect
73. C₂	C₂²×#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
C₂⁷	2
C₂⁴×D₄	2
C₂²×#120	2
C₂³×C₂²⋊C₄	2

Subgroups — C23×#120 #62655

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#120 #62655