Subgroups — C₂×#2846 #36610

Order 256.

16445 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	C₂²×#120	C₂	1	7	yes	maximal, complemented
3	#2846	C₂	1	16	yes	maximal, direct product factor
4	C₂×#379	C₂	1	7	yes	maximal, complemented
5	C₂⁶	—	2	56	no	—
6	C₂⁶	C₂²	1	7	yes	—
7	C₂⁶	C₂²	1	8	yes	complemented
8	C₂³×D₄	C₂²	1	7	yes	complemented
9	C₂×#120	—	2	56	no	—
10	C₂×#120	C₂²	1	56	yes	complemented
11	C₂²×C₂²⋊C₄	C₂²	1	21	yes	complemented
12	#379	C₂²	1	56	yes	complemented
13	C₂⁵	—	2	896	no	—
14	C₂⁵	—	2	168	no	—
15	C₂⁵	—	2	224	no	—
16	C₂⁵	D₄	1	56	yes	complemented
17	C₂⁵	C₂³	1	28	yes	—
18	C₂⁵	C₂³	1	7	yes	—
19	C₂⁵	C₂³	1	1	yes	characteristic, complemented
20	C₂²×D₄	—	2	84	no	—
21	C₂²×D₄	C₂³	1	28	yes	complemented
22	C₂³×C₄	C₂³	1	7	yes	complemented
23	#120	—	2	224	no	—
24	C₂×C₂²⋊C₄	—	2	84	no	—
25	C₂×C₂²⋊C₄	C₂³	1	84	yes	complemented
26	C₂⁴	—	2	2048	no	—
27	C₂⁴	—	2	896	no	—
28	C₂⁴	—	2	1792	no	—
29	C₂⁴	—	2	672	no	—
30	C₂⁴	—	2	336	no	—
31	C₂⁴	—	2	56	no	—
32	C₂⁴	—	2	14	no	—
33	C₂⁴	C₂×D₄	1	84	yes	—
34	C₂⁴	C₂×D₄	1	112	yes	complemented
35	C₂⁴	C₂⁴	1	14	yes	—
36	C₂⁴	C₂⁴	1	1	yes	characteristic
37	C₂⁴	C₂⁴	1	2	yes	complemented
38	C₂×D₄	—	2	168	no	—
39	C₂×D₄	—	4	112	no	—
40	C₂²×C₄	—	2	42	no	—
41	C₂²×C₄	C₂⁴	1	14	yes	complemented
42	C₂²⋊C₄	—	2	168	no	—
43	C₂³	—	2	2048	no	—
44	C₂³	—	2	1792	no	—
45	C₂³	—	2	896	no	—
46	C₂³	—	2	336	no	—
47	C₂³	—	2	672	no	—
48	C₂³	—	2	56	no	—
49	C₂³	—	2	14	no	—
50	C₂³	—	4	28	no	—
51	C₂³	#120	1	112	yes	complemented
52	C₂³	C₂²×D₄	1	84	yes	—
53	C₂³	C₂²×D₄	1	14	yes	—
54	C₂³	C₂⁵	1	1	yes	characteristic
55	D₄	—	4	112	no	—
56	C₂×C₄	—	2	42	no	—
57	C₂×C₄	—	4	28	no	—
58	C₂²	—	2	896	no	—
59	C₂²	—	2	224	no	—
60	C₂²	—	2	168	no	—
61	C₂²	—	4	14	no	—
62	C₂²	—	8	8	no	—
63	C₂²	C₂×#120	1	28	yes	—
64	C₂²	C₂×#120	1	56	yes	complemented
65	C₂²	C₂³×D₄	1	7	yes	—
66	C₄	—	4	14	no	—
67	C₂	—	2	56	no	—
68	C₂	—	8	2	no	—
69	C₂	#2846	1	8	yes	direct product factor
70	C₂	C₂²×#120	1	7	yes	—

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

Complements

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

N	H ≅ G/N	type
1. C₂⁷	C₂	semidirect
2. C₂²×#120	C₂	semidirect
3. #2846	C₂	direct (N × H)
4. C₂×#379	C₂	semidirect
7. C₂⁶	C₂²	semidirect
8. C₂³×D₄	C₂²	semidirect
10. C₂×#120	C₂²	semidirect
11. C₂²×C₂²⋊C₄	C₂²	semidirect
12. #379	C₂²	semidirect
16. C₂⁵	D₄	semidirect
19. C₂⁵	C₂³	semidirect
21. C₂²×D₄	C₂³	semidirect
22. C₂³×C₄	C₂³	semidirect
25. C₂×C₂²⋊C₄	C₂³	semidirect
34. C₂⁴	C₂×D₄	semidirect
37. C₂⁴	C₂⁴	semidirect
41. C₂²×C₄	C₂⁴	semidirect
51. C₂³	#120	semidirect
64. C₂²	C₂×#120	semidirect
69. C₂	#2846	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
C₂²×#120	2
#2846	2
C₂×#379	2

Subgroups — C2×#2846 #36610

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#2846 #36610