Subgroups — C₂×#23778855 #35204280

Order 20288.

90 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23778855	C₂	1	4	yes	maximal, direct product factor
2	C₂×#19099577	C₂	1	2	yes	maximal, complemented
3	C₂×C₈×D₃₁₇	C₂	1	1	yes	maximal, characteristic
4	#19099577	C₂²	1	4	yes	complemented
5	C₂×C₄×D₃₁₇	C₄	1	1	yes	characteristic
6	C₂×#14579261	C₂²	1	1	yes	characteristic
7	C₈×D₃₁₇	C₄	1	2	yes	—
8	C₈×D₃₁₇	C₂²	1	2	yes	—
9	C₂×C₈×C₃₁₇	C₄	1	1	yes	characteristic
10	C₂²×D₃₁₇	C₈	1	1	yes	characteristic
11	C₂×Dic₃₁₇	C₈	1	1	yes	characteristic
12	C₄×D₃₁₇	C₈	1	2	yes	—
13	C₄×D₃₁₇	C₂×C₄	1	2	yes	—
14	#14579261	C₂×C₄	1	1	yes	characteristic
15	#14579261	C₂³	1	1	yes	characteristic
16	C₂×C₄×C₃₁₇	C₂×C₄	1	1	yes	characteristic
17	C₈×C₃₁₇	C₂×C₄	1	2	yes	—
18	C₂²×C₃₁₇	C₂×C₈	1	1	yes	characteristic
19	C₂×D₃₁₇	—	2	2	no	—
20	C₂×D₃₁₇	C₂×C₈	1	2	yes	—
21	C₄×C₃₁₇	C₂×C₈	1	1	yes	characteristic
22	C₄×C₃₁₇	C₂²×C₄	1	1	yes	characteristic
23	Dic₃₁₇	C₂×C₈	1	2	yes	—
24	C₂×C₃₁₇	M₃₂	1	2	yes	complemented
25	C₂×C₃₁₇	C₂²×C₈	1	1	yes	characteristic
26	D₃₁₇	—	2	2	no	—
27	C₃₁₇	C₂×M₃₂	1	1	yes	characteristic, complemented, Sylow
28	C₂×M₃₂	—	317	1	no	maximal, Sylow
29	C₂²×C₈	—	317	1	no	—
30	M₃₂	—	317	4	no	—
31	C₂×C₁₆	—	317	2	no	—
32	C₂²×C₄	—	317	1	no	—
33	C₂×C₈	—	317	1	no	—
34	C₂×C₈	—	317	2	no	—
35	C₂×C₈	—	317	2	no	—
36	C₂×C₈	#11929639	1	1	yes	characteristic
37	C₁₆	—	317	4	no	—
38	C₂³	—	317	1	no	—
39	C₂×C₄	—	317	1	no	—
40	C₂×C₄	—	317	2	no	—
41	C₂×C₄	—	317	2	no	—
42	C₂×C₄	C₂×#11929639	1	1	yes	characteristic
43	C₈	—	317	1	no	—
44	C₈	—	317	1	no	—
45	C₈	C₂×#11929639	1	2	yes	—
46	C₂²	—	317	2	no	—
47	C₂²	—	634	2	no	—
48	C₂²	D₃₁₇⋊C₈	1	1	yes	characteristic
49	C₄	—	317	2	no	—
50	C₄	D₃₁₇⋊C₈	1	1	yes	characteristic
51	C₄	C₂²×#11929639	1	1	yes	characteristic
52	C₂	—	634	2	no	—
53	C₂	#23778855	1	2	yes	direct product factor
54	C₂	C₂×D₃₁₇⋊C₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×#11929639
commutator	C₂×C₃₁₇	C₂²×C₈
Frattini	C₄	C₂²×#11929639
Fitting	C₂×C₈×C₃₁₇	C₄
socle	C₂²×C₃₁₇	C₂×C₈
prime radical	C₂²×D₃₁₇	C₈
cosocle	#14579261	C₂³
hypercenter	C₂×C₈	#11929639
hypocenter	C₃₁₇	C₂×M₃₂
LCS	G ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
1. #23778855	C₂	direct (N × H)
2. C₂×#19099577	C₂	semidirect
4. #19099577	C₂²	semidirect
24. C₂×C₃₁₇	M₃₂	semidirect
27. C₃₁₇	C₂×M₃₂	semidirect
53. C₂	#23778855	direct (N × H)

Maximal subgroups

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

subgroup	index
#23778855	2
C₂×#19099577	2
C₂×C₈×D₃₁₇	2
C₂×M₃₂	317

Subgroups — C2×#23778855 #35204280

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23778855 #35204280