Subgroups — H(5,𝔽₂)×D₃₁₇ #35204052

Order 20288.

898 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23778826	C₂	1	6	yes	maximal, complemented
2	#23778825	C₂	1	9	yes	maximal, complemented
3	#41×D₃₁₇	C₂	1	6	yes	maximal, complemented
4	C₂×D₄×D₃₁₇	C₂	1	9	yes	maximal, complemented
5	H(5,𝔽₂)×C₃₁₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂³×D₃₁₇	C₂²	1	6	yes	complemented
7	#19099575	C₂²	1	6	yes	complemented
8	#19099574	C₂²	1	18	yes	complemented
9	#19099573	C₂²	1	18	yes	complemented
10	Q₈×D₃₁₇	C₂²	1	2	yes	complemented
11	C₂×D₁₂₆₈	C₂²	1	9	yes	complemented
12	C₂×C₂²⋊D₃₁₇	C₂²	1	18	yes	complemented
13	D₄×D₃₁₇	C₂²	1	18	yes	complemented
14	D₄×D₃₁₇	C₂²	1	36	yes	complemented
15	C₂×C₄×D₃₁₇	C₂²	1	9	yes	complemented
16	#41×C₃₁₇	C₂²	1	6	yes	complemented
17	C₂×D₄×C₃₁₇	C₂²	1	9	yes	complemented
18	C₂²×D₃₁₇	—	4	12	no	—
19	C₂²×D₃₁₇	C₂³	1	9	yes	—
20	C₂²×D₃₁₇	C₂³	1	18	yes	complemented
21	Dic₆₃₄	C₂³	1	6	yes	complemented
22	D₁₂₆₈	C₂³	1	18	yes	complemented
23	C₂²⋊D₃₁₇	C₂³	1	36	yes	complemented
24	C₂×Dic₃₁₇	C₂³	1	9	yes	complemented
25	C₄×D₃₁₇	C₂³	1	6	yes	—
26	C₄×D₃₁₇	C₂³	1	18	yes	complemented
27	C₂³×C₃₁₇	C₂³	1	6	yes	complemented
28	Q₈×C₃₁₇	C₂³	1	2	yes	complemented
29	D₄×C₃₁₇	C₂³	1	18	yes	complemented
30	C₂×C₄×C₃₁₇	C₂³	1	9	yes	complemented
31	C₂²×C₃₁₇	—	4	6	no	—
32	C₂²×C₃₁₇	C₂⁴	1	9	yes	—
33	C₂×D₃₁₇	—	2	18	no	—
34	C₂×D₃₁₇	—	4	18	no	—
35	C₂×D₃₁₇	C₂⁴	1	9	yes	—
36	C₂×D₃₁₇	C₂⁴	1	1	yes	characteristic
37	C₄×C₃₁₇	C₂⁴	1	6	yes	—
38	Dic₃₁₇	C₂⁴	1	6	yes	—
39	C₂×C₃₁₇	—	2	9	no	—
40	C₂×C₃₁₇	C₂⁵	1	1	yes	characteristic
41	D₃₁₇	—	2	9	no	—
42	D₃₁₇	H(5,𝔽₂)	1	2	yes	direct product factor
43	C₃₁₇	C₂×H(5,𝔽₂)	1	1	yes	characteristic, complemented, Sylow
44	C₂×H(5,𝔽₂)	—	317	1	no	maximal, Sylow
45	H(5,𝔽₂)	—	317	6	no	—
46	H(5,𝔽₂)	—	317	9	no	—
47	H(5,𝔽₂)	D₃₁₇	1	1	yes	characteristic, direct product factor
48	C₂×#41	—	317	6	no	—
49	C₂²×D₄	—	317	9	no	—
50	C₂⁴	—	317	6	no	—
51	#41	—	317	6	no	—
52	#41	—	317	18	no	—
53	#41	—	317	18	no	—
54	#41	C₂×D₃₁₇	1	6	yes	complemented
55	C₂×Q₈	—	317	2	no	—
56	C₂×D₄	—	317	18	no	—
57	C₂×D₄	—	317	18	no	—
58	C₂×D₄	—	317	36	no	—
59	C₂×D₄	—	317	9	no	—
60	C₂×D₄	C₂×D₃₁₇	1	9	yes	complemented
61	C₂²×C₄	—	317	9	no	—
62	C₂³	—	317	9	no	—
63	C₂³	—	317	18	no	—
64	C₂³	—	1268	12	no	—
65	C₂³	C₂²×D₃₁₇	1	6	yes	complemented
66	Q₈	—	317	6	no	—
67	Q₈	C₂²×D₃₁₇	1	2	yes	complemented
68	D₄	—	317	36	no	—
69	D₄	—	317	18	no	—
70	D₄	C₂²×D₃₁₇	1	18	yes	complemented
71	C₂×C₄	—	317	9	no	—
72	C₂×C₄	—	317	6	no	—
73	C₂×C₄	—	317	18	no	—
74	C₂×C₄	C₂²×D₃₁₇	1	9	yes	complemented
75	C₂²	—	4	6	no	—
76	C₂²	—	317	1	no	—
77	C₂²	—	317	9	no	—
78	C₂²	—	634	18	no	—
79	C₂²	—	1268	18	no	—
80	C₂²	C₂³×D₃₁₇	1	9	yes	—
81	C₄	—	317	6	no	—
82	C₄	C₂³×D₃₁₇	1	6	yes	—
83	C₂	—	2	9	no	—
84	C₂	—	317	2	no	—
85	C₂	—	634	9	no	—
86	C₂	C₂⁴×D₃₁₇	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂	C₂⁴×D₃₁₇
commutator, socle, cosocle	C₂×C₃₁₇	C₂⁵
Fitting	H(5,𝔽₂)×C₃₁₇	C₂
hypercenter	H(5,𝔽₂)	D₃₁₇
hypocenter	C₃₁₇	C₂×H(5,𝔽₂)
LCS	G ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂ ◁ H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. #23778826	C₂	semidirect
2. #23778825	C₂	semidirect
3. #41×D₃₁₇	C₂	semidirect
4. C₂×D₄×D₃₁₇	C₂	semidirect
5. H(5,𝔽₂)×C₃₁₇	C₂	semidirect
6. C₂³×D₃₁₇	C₂²	semidirect
7. #19099575	C₂²	semidirect
8. #19099574	C₂²	semidirect
9. #19099573	C₂²	semidirect
10. Q₈×D₃₁₇	C₂²	semidirect
11. C₂×D₁₂₆₈	C₂²	semidirect
12. C₂×C₂²⋊D₃₁₇	C₂²	semidirect
13. D₄×D₃₁₇	C₂²	semidirect
14. D₄×D₃₁₇	C₂²	semidirect
15. C₂×C₄×D₃₁₇	C₂²	semidirect
16. #41×C₃₁₇	C₂²	semidirect
17. C₂×D₄×C₃₁₇	C₂²	semidirect
20. C₂²×D₃₁₇	C₂³	semidirect
21. Dic₆₃₄	C₂³	semidirect
22. D₁₂₆₈	C₂³	semidirect
23. C₂²⋊D₃₁₇	C₂³	semidirect
24. C₂×Dic₃₁₇	C₂³	semidirect
26. C₄×D₃₁₇	C₂³	semidirect
27. C₂³×C₃₁₇	C₂³	semidirect
28. Q₈×C₃₁₇	C₂³	semidirect
29. D₄×C₃₁₇	C₂³	semidirect
30. C₂×C₄×C₃₁₇	C₂³	semidirect
42. D₃₁₇	H(5,𝔽₂)	direct (N × H)
43. C₃₁₇	C₂×H(5,𝔽₂)	semidirect
47. H(5,𝔽₂)	D₃₁₇	direct (N × H)
54. #41	C₂×D₃₁₇	semidirect
60. C₂×D₄	C₂×D₃₁₇	semidirect
65. C₂³	C₂²×D₃₁₇	semidirect
67. Q₈	C₂²×D₃₁₇	semidirect
70. D₄	C₂²×D₃₁₇	semidirect
74. C₂×C₄	C₂²×D₃₁₇	semidirect

Maximal subgroups

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

subgroup	index
#23778826	2
#23778825	2
#41×D₃₁₇	2
C₂×D₄×D₃₁₇	2
H(5,𝔽₂)×C₃₁₇	2
C₂×H(5,𝔽₂)	317

Subgroups — H(5,𝔽2)×D317 #35204052

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — H(5,𝔽₂)×D₃₁₇ #35204052