Subgroups — C₈×H(5,𝔽₂) #60303

Order 256.

1211 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×H(5,𝔽₂)	C₂	1	1	yes	maximal, characteristic
2	#2988	C₂	1	6	yes	maximal, complemented
3	#2972	C₂	1	9	yes	maximal, complemented
4	C₈×#41	C₂	1	6	yes	maximal, complemented
5	C₂×C₈×D₄	C₂	1	9	yes	maximal, complemented
6	C₂×H(5,𝔽₂)	C₄	1	1	yes	characteristic
7	C₂³×C₈	C₂²	1	6	yes	complemented
8	#520	C₄	1	6	yes	—
9	#518	C₄	1	9	yes	—
10	C₄×#41	C₂²	1	6	yes	—
11	C₂×C₄×D₄	C₂²	1	9	yes	—
12	C₈×Q₈	C₂²	1	2	yes	complemented
13	C₈×Q₈	C₂²	1	6	yes	complemented
14	C₈×D₄	C₂²	1	18	yes	complemented
15	C₈×D₄	C₂²	1	36	yes	complemented
16	C₈×D₄	C₂²	1	18	yes	complemented
17	#431	C₂²	1	18	yes	complemented
18	C₂×C₄⋊C₈	C₂²	1	9	yes	complemented
19	C₂×C₂²⋊C₈	C₂²	1	18	yes	complemented
20	C₂×C₄×C₈	C₂²	1	9	yes	complemented
21	H(5,𝔽₂)	C₈	1	16	yes	direct product factor
22	C₂×#41	C₂×C₄	1	6	yes	—
23	C₂²×D₄	C₂×C₄	1	9	yes	—
24	C₂³×C₄	C₂³	1	6	yes	—
25	C₂²×C₈	—	4	12	no	—
26	C₂²×C₈	C₂³	1	9	yes	—
27	C₂²×C₈	C₂³	1	18	yes	—
28	C₂²×C₈	C₂³	1	9	yes	—
29	C₄×Q₈	C₂×C₄	1	6	yes	—
30	C₄×Q₈	C₂³	1	2	yes	—
31	C₄×D₄	C₂×C₄	1	36	yes	—
32	C₄×D₄	C₂×C₄	1	18	yes	—
33	C₄×D₄	C₂³	1	18	yes	—
34	#117	C₂×C₄	1	18	yes	—
35	C₂×C₄⋊C₄	C₂×C₄	1	9	yes	—
36	C₂×C₂²⋊C₄	C₂×C₄	1	18	yes	—
37	C₂×C₄²	C₂³	1	9	yes	—
38	C₄⋊C₈	C₂³	1	18	yes	—
39	C₄⋊C₈	C₂³	1	6	yes	—
40	C₂²⋊C₈	C₂³	1	36	yes	—
41	C₄×C₈	C₂³	1	18	yes	—
42	C₄×C₈	C₂³	1	6	yes	—
43	C₂⁴	C₂²×C₄	1	6	yes	—
44	#41	C₂×C₈	1	48	yes	complemented
45	C₂×Q₈	C₂²×C₄	1	2	yes	—
46	C₂×D₄	C₂×C₈	1	72	yes	complemented
47	C₂×D₄	C₂²×C₄	1	18	yes	—
48	C₂²×C₄	—	4	6	no	—
49	C₂²×C₄	—	4	6	no	—
50	C₂²×C₄	C₂²×C₄	1	9	yes	—
51	C₂²×C₄	C₂²×C₄	1	18	yes	—
52	C₂²×C₄	C₂²×C₄	1	9	yes	—
53	C₂²×C₄	C₂⁴	1	9	yes	—
54	C₂×C₈	—	2	18	no	—
55	C₂×C₈	—	4	18	no	—
56	C₂×C₈	—	4	9	no	—
57	C₂×C₈	C₂⁴	1	6	yes	—
58	C₂×C₈	C₂⁴	1	9	yes	—
59	C₂×C₈	C₂⁴	1	1	yes	characteristic
60	C₄⋊C₄	C₂²×C₄	1	18	yes	—
61	C₄⋊C₄	C₂²×C₄	1	6	yes	—
62	C₂²⋊C₄	C₂²×C₄	1	36	yes	—
63	C₄²	C₂²×C₄	1	18	yes	—
64	C₄²	C₂⁴	1	6	yes	—
65	C₂³	—	4	6	no	—
66	C₂³	—	4	6	no	—
67	C₂³	C₂²×C₈	1	24	yes	complemented
68	C₂³	C₂³×C₄	1	9	yes	—
69	Q₈	C₂²×C₈	1	8	yes	complemented
70	D₄	C₂²×C₈	1	72	yes	complemented
71	C₂×C₄	—	2	9	no	—
72	C₂×C₄	—	2	9	no	—
73	C₂×C₄	—	4	18	no	—
74	C₂×C₄	—	4	9	no	—
75	C₂×C₄	C₂²×C₈	1	36	yes	complemented
76	C₂×C₄	C₂³×C₄	1	6	yes	—
77	C₂×C₄	C₂³×C₄	1	9	yes	—
78	C₂×C₄	C₂³×C₄	1	6	yes	—
79	C₂×C₄	C₂⁵	1	1	yes	characteristic
80	C₈	—	2	9	no	—
81	C₈	—	2	6	no	—
82	C₈	H(5,𝔽₂)	1	2	yes	direct product factor
83	C₂²	—	2	9	no	—
84	C₂²	—	2	9	no	—
85	C₂²	—	4	24	no	—
86	C₂²	C₂³×C₈	1	18	yes	—
87	C₂²	C₂⁴×C₄	1	1	yes	characteristic
88	C₄	—	2	9	no	—
89	C₄	—	2	6	no	—
90	C₄	C₂³×C₈	1	12	yes	—
91	C₄	C₂×H(5,𝔽₂)	1	1	yes	characteristic
92	C₄	#585	1	1	yes	characteristic
93	C₂	—	2	18	no	—
94	C₂	C₄×H(5,𝔽₂)	1	1	yes	characteristic
95	C₂	C₂⁴×C₈	1	1	yes	characteristic
96	C₂	#3573	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂⁴
commutator	C₂	C₂⁴×C₈
Frattini, cosocle	C₂×C₄	C₂⁵
socle	C₂²	C₂⁴×C₄
prime radical	C₂×H(5,𝔽₂)	C₄
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

Complements

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

N	H ≅ G/N	type
2. #2988	C₂	semidirect
3. #2972	C₂	semidirect
4. C₈×#41	C₂	semidirect
5. C₂×C₈×D₄	C₂	semidirect
7. C₂³×C₈	C₂²	semidirect
12. C₈×Q₈	C₂²	semidirect
13. C₈×Q₈	C₂²	semidirect
14. C₈×D₄	C₂²	semidirect
15. C₈×D₄	C₂²	semidirect
16. C₈×D₄	C₂²	semidirect
17. #431	C₂²	semidirect
18. C₂×C₄⋊C₈	C₂²	semidirect
19. C₂×C₂²⋊C₈	C₂²	semidirect
20. C₂×C₄×C₈	C₂²	semidirect
21. H(5,𝔽₂)	C₈	direct (N × H)
44. #41	C₂×C₈	semidirect
46. C₂×D₄	C₂×C₈	semidirect
67. C₂³	C₂²×C₈	semidirect
69. Q₈	C₂²×C₈	semidirect
70. D₄	C₂²×C₈	semidirect
75. C₂×C₄	C₂²×C₈	semidirect
82. C₈	H(5,𝔽₂)	direct (N × H)

Maximal subgroups

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

subgroup	index
C₄×H(5,𝔽₂)	2
#2988	2
#2972	2
C₈×#41	2
C₂×C₈×D₄	2

Subgroups — C8×H(5,𝔽2) #60303

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈×H(5,𝔽₂) #60303