Subgroups — #35204092

Order 20288.

86 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₃₁₇⋊M₁₆	C₂	1	2	yes	maximal
2	C₂×#19099557	C₂	1	1	yes	maximal, characteristic
3	#19099584	—	2	2	no	—
4	#19099583	—	2	2	no	—
5	D₃₁₇⋊C₈	C₄	1	4	yes	—
6	C₂×C₄×D₃₁₇	C₂²	1	1	yes	characteristic
7	#19099557	—	2	2	no	—
8	C₂×#14579261	C₄	1	1	yes	characteristic
9	C₂×C₈×C₃₁₇	C₄	1	1	yes	characteristic
10	#14579268	—	2	4	no	—
11	C₂²×D₃₁₇	Q₈	1	1	yes	characteristic
12	C₂×Dic₃₁₇	D₄	1	1	yes	characteristic
13	C₄×D₃₁₇	C₂×C₄	1	2	yes	—
14	C₄×D₃₁₇	D₄	1	2	yes	—
15	#14579261	—	2	1	no	—
16	C₂×C₄×C₃₁₇	C₂×C₄	1	1	yes	characteristic
17	C₈×C₃₁₇	—	2	1	no	—
18	C₂²×C₃₁₇	C₄⋊C₄	1	1	yes	characteristic
19	C₂×D₃₁₇	—	4	1	no	—
20	C₂×D₃₁₇	C₄⋊C₄	1	2	yes	—
21	C₄×C₃₁₇	C₄²	1	1	yes	characteristic
22	C₄×C₃₁₇	C₂²⋊C₄	1	1	yes	characteristic
23	Dic₃₁₇	C₂²⋊C₄	1	2	yes	—
24	C₂×C₃₁₇	—	2	1	no	—
25	C₂×C₃₁₇	#95	1	1	yes	characteristic
26	D₃₁₇	—	2	2	no	—
27	C₃₁₇	#338	1	1	yes	characteristic, complemented, Sylow
28	#338	—	317	1	no	maximal, Sylow
29	C₂×M₁₆	—	317	2	no	—
30	C₂×M₁₆	—	317	1	no	—
31	C₂²×C₄	—	317	1	no	—
32	M₁₆	—	634	2	no	—
33	M₁₆	—	634	2	no	—
34	M₁₆	—	634	2	no	—
35	C₂×C₈	—	317	4	no	—
36	C₂×C₈	—	317	1	no	—
37	C₂×C₈	#11929639	1	1	yes	characteristic
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	—	2	1	no	—
44	C₈	—	634	4	no	—
45	C₈	—	634	1	no	—
46	C₂²	—	317	2	no	—
47	C₂²	—	1268	1	no	—
48	C₂²	C₄⋊#11929639	1	1	yes	characteristic
49	C₄	—	317	2	no	—
50	C₄	C₄×#11929639	1	1	yes	characteristic
51	C₄	C₂²⋊#11929639	1	1	yes	characteristic
52	C₂	—	2	1	no	—
53	C₂	—	634	2	no	—
54	C₂	(C₂×#11929639)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄×#11929639
commutator	C₄×C₃₁₇	C₄²
Frattini	C₂×C₄	C₂×#11929639
Fitting	C₂×C₈×C₃₁₇	C₄
socle	C₂×C₃₁₇	#95
prime radical	C₂²×D₃₁₇	Q₈
cosocle	C₂×C₄×D₃₁₇	C₂²
hypercenter	C₂×C₈	#11929639
hypocenter	C₃₁₇	#338
LCS	G ▷ C₄×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
27. C₃₁₇	#338	semidirect

Maximal subgroups

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

subgroup	index
D₃₁₇⋊M₁₆	2
C₂×#19099557	2
#338	317