Subgroups — C₂²×#2867 #7633348

Order 512.

33858 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×#127	C₂	1	14	yes	maximal, complemented
2	C₂×#2867	C₂	1	48	yes	maximal, direct product factor
3	C₂²×C₄³	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂⁴×D₄	C₂²	1	28	yes	—
5	C₂²×#127	—	2	224	no	—
6	C₂²×#127	C₂²	1	336	yes	complemented
7	C₂³×C₄²	C₂²	1	7	yes	—
8	#2867	C₂²	1	256	yes	direct product factor
9	C₂×C₄³	C₂²	1	24	yes	complemented
10	C₂⁶	C₂³	1	8	yes	—
11	C₂³×D₄	—	2	672	no	—
12	C₂³×D₄	C₂³	1	336	yes	—
13	C₂⁴×C₄	C₂³	1	7	yes	—
14	C₂×#127	—	2	2688	no	—
15	C₂×#127	C₂³	1	896	yes	complemented
16	C₂²×C₄²	D₄	1	112	yes	complemented
17	C₂²×C₄²	C₂³	1	84	yes	—
18	C₄³	C₂³	1	64	yes	complemented
19	C₂⁵	—	2	224	no	—
20	C₂⁵	C₂⁴	1	48	yes	—
21	C₂⁵	C₂⁴	1	1	yes	characteristic
22	C₂²×D₄	—	2	4032	no	—
23	C₂²×D₄	—	4	1792	no	—
24	C₂²×D₄	C₂⁴	1	448	yes	—
25	C₂³×C₄	C₂×D₄	1	168	yes	—
26	C₂³×C₄	C₂⁴	1	42	yes	—
27	#127	—	2	3584	no	—
28	C₂×C₄²	C₂×D₄	1	672	yes	complemented
29	C₂×C₄²	C₂⁴	1	112	yes	—
30	C₂⁴	—	2	672	no	—
31	C₂⁴	—	4	896	no	—
32	C₂⁴	C₂²×D₄	1	28	yes	—
33	C₂⁴	C₂⁵	1	32	yes	—
34	C₂⁴	C₂⁵	1	3	yes	—
35	C₂×D₄	—	2	2688	no	—
36	C₂×D₄	—	4	5376	no	—
37	C₂²×C₄	#127	1	448	yes	complemented
38	C₂²×C₄	C₂²×D₄	1	504	yes	—
39	C₂²×C₄	C₂⁵	1	28	yes	—
40	C₄²	C₂²×D₄	1	448	yes	complemented
41	C₂³	—	2	224	no	—
42	C₂³	—	4	1344	no	—
43	C₂³	—	8	512	no	—
44	C₂³	C₂×#127	1	112	yes	—
45	C₂³	C₂³×D₄	1	42	yes	—
46	C₂³	C₂⁶	1	1	yes	characteristic
47	D₄	—	4	1792	no	—
48	C₂×C₄	C₂×#127	1	672	yes	complemented
49	C₂×C₄	C₂³×D₄	1	168	yes	—
50	C₂²	—	4	224	no	—
51	C₂²	—	8	384	no	—
52	C₂²	#2867	1	64	yes	direct product factor
53	C₂²	C₂²×#127	1	84	yes	—
54	C₂²	C₂⁴×D₄	1	7	yes	—
55	C₄	C₂²×#127	1	112	yes	complemented
56	C₂	—	8	32	no	—
57	C₂	C₂×#2867	1	24	yes	direct product factor
58	C₂	C₂³×#127	1	7	yes	—

	subgroup	G/H
center, socle	C₂⁵	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂³ ▷ C₁

Complements

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

N	H ≅ G/N	type
1. C₂³×#127	C₂	semidirect
2. C₂×#2867	C₂	direct (N × H)
3. C₂²×C₄³	C₂	semidirect
6. C₂²×#127	C₂²	semidirect
8. #2867	C₂²	direct (N × H)
9. C₂×C₄³	C₂²	semidirect
15. C₂×#127	C₂³	semidirect
16. C₂²×C₄²	D₄	semidirect
18. C₄³	C₂³	semidirect
28. C₂×C₄²	C₂×D₄	semidirect
37. C₂²×C₄	#127	semidirect
40. C₄²	C₂²×D₄	semidirect
48. C₂×C₄	C₂×#127	semidirect
52. C₂²	#2867	direct (N × H)
55. C₄	C₂²×#127	semidirect
57. C₂	C₂×#2867	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×#127	2
C₂×#2867	2
C₂²×C₄³	2

Subgroups — C22×#2867 #7633348

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#2867 #7633348