Subgroups — #35204144

Order 20288.

50 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19099577	C₂	1	2	yes	maximal
2	C₈×Dic₃₁₇	C₂	1	1	yes	maximal, characteristic
3	#19099577	—	2	2	no	—
4	C₂×#14579261	C₂²	1	1	yes	characteristic
5	C₄×Dic₃₁₇	C₄	1	1	yes	characteristic
6	C₂×C₈×C₃₁₇	C₄	1	1	yes	characteristic
7	C₂×Dic₃₁₇	C₈	1	1	yes	characteristic
8	C₂×Dic₃₁₇	C₈	1	1	yes	characteristic
9	#14579261	D₄	1	1	yes	characteristic
10	#14579261	Q₈	1	1	yes	characteristic
11	C₂×C₄×C₃₁₇	C₂×C₄	1	1	yes	characteristic
12	C₈×C₃₁₇	—	2	1	no	—
13	C₂²×C₃₁₇	C₂×C₈	1	1	yes	characteristic
14	C₄×C₃₁₇	C₄⋊C₄	1	1	yes	characteristic
15	C₄×C₃₁₇	M₁₆	1	1	yes	characteristic
16	Dic₃₁₇	—	2	1	no	—
17	Dic₃₁₇	C₁₆	1	2	yes	complemented
18	C₂×C₃₁₇	C₄⋊C₈	1	1	yes	characteristic
19	C₂×C₃₁₇	C₂×C₁₆	1	1	yes	characteristic
20	C₂×C₃₁₇	M₃₂	1	1	yes	characteristic
21	C₃₁₇	C₄⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
22	C₄⋊C₁₆	—	317	1	no	maximal, Sylow
23	C₂×C₁₆	—	317	2	no	—
24	C₄×C₈	—	317	1	no	—
25	C₂×C₈	—	317	1	no	—
26	C₂×C₈	#11929639	1	1	yes	characteristic
27	C₄²	—	317	1	no	—
28	C₁₆	—	634	2	no	—
29	C₂×C₄	—	317	1	no	—
30	C₂×C₄	—	317	1	no	—
31	C₂×C₄	C₂×#11929639	1	1	yes	characteristic
32	C₈	—	2	1	no	—
33	C₈	—	317	1	no	—
34	C₈	—	317	1	no	—
35	C₂²	D₃₁₇⋊C₈	1	1	yes	characteristic
36	C₄	—	317	2	no	—
37	C₄	—	634	1	no	—
38	C₄	C₄⋊#11929639	1	1	yes	characteristic
39	C₄	#19099583	1	1	yes	characteristic
40	C₂	#23778847	1	1	yes	characteristic
41	C₂	D₃₁₇⋊C₁₆	1	1	yes	characteristic
42	C₂	#23778855	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂×#11929639
commutator	C₂×C₃₁₇	C₂×C₁₆
Fitting	C₂×C₈×C₃₁₇	C₄
socle, prime radical	C₂²×C₃₁₇	C₂×C₈
cosocle	C₂×#14579261	C₂²
hypercenter	C₂×C₈	#11929639
hypocenter	C₃₁₇	C₄⋊C₁₆
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
17. Dic₃₁₇	C₁₆	semidirect
21. C₃₁₇	C₄⋊C₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19099577	2
C₈×Dic₃₁₇	2
C₄⋊C₁₆	317