Subgroups — #35203008

Order 20288.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23778786	C₂	1	1	yes	maximal, characteristic
2	C₂×#19099549	C₂	1	1	yes	maximal, characteristic, complemented
3	M₃₂×C₃₁₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#19099574	C₄	1	1	yes	characteristic
5	#19099558	C₄	1	1	yes	characteristic
6	#19099557	—	2	1	no	—
7	C₈×D₃₁₇	—	2	1	no	—
8	#19099549	—	2	1	no	—
9	C₂×C₈×C₃₁₇	C₂²	1	1	yes	characteristic
10	C₁₆×C₃₁₇	—	2	1	no	—
11	Dic₆₃₄	C₈	1	1	yes	characteristic
12	D₁₂₆₈	C₈	1	1	yes	characteristic
13	C₂²⋊D₃₁₇	—	2	1	no	—
14	C₄×D₃₁₇	—	2	1	no	—
15	#14579261	—	2	1	no	—
16	C₂×C₄×C₃₁₇	C₂×C₄	1	1	yes	characteristic
17	C₈×C₃₁₇	D₄	1	1	yes	characteristic
18	C₈×C₃₁₇	D₄	1	1	yes	characteristic
19	C₂²×C₃₁₇	M₁₆	1	1	yes	characteristic
20	C₂×D₃₁₇	—	2	1	no	—
21	C₄×C₃₁₇	C₂²⋊C₄	1	1	yes	characteristic
22	C₄×C₃₁₇	C₂×C₈	1	1	yes	characteristic
23	Dic₃₁₇	—	2	1	no	—
24	C₂×C₃₁₇	—	2	1	no	—
25	C₂×C₃₁₇	C₂²⋊C₈	1	1	yes	characteristic
26	D₃₁₇	—	4	1	no	—
27	C₃₁₇	#350	1	1	yes	characteristic, complemented, Sylow
28	#350	—	317	1	no	maximal, Sylow
29	#131	—	317	1	no	—
30	M₃₂	D₃₁₇	1	1	yes	characteristic, complemented
31	C₂×C₁₆	—	317	1	no	—
32	#41	—	317	1	no	—
33	M₁₆	—	317	1	no	—
34	M₁₆	—	634	1	no	—
35	C₂×C₈	—	634	1	no	—
36	C₂×C₈	C₂×D₃₁₇	1	1	yes	characteristic
37	C₁₆	—	2	1	no	—
38	C₁₆	—	634	1	no	—
39	Q₈	—	317	1	no	—
40	D₄	—	317	1	no	—
41	D₄	—	634	1	no	—
42	C₂×C₄	—	634	1	no	—
43	C₂×C₄	C₄×D₃₁₇	1	1	yes	characteristic
44	C₈	—	634	1	no	—
45	C₈	C₂²⋊D₃₁₇	1	1	yes	characteristic
46	C₈	D₁₂₆₈	1	1	yes	characteristic
47	C₂²	—	634	1	no	—
48	C₂²	#19099557	1	1	yes	characteristic
49	C₄	—	634	1	no	—
50	C₄	#19099551	1	1	yes	characteristic
51	C₄	C₈×D₃₁₇	1	1	yes	characteristic
52	C₂	—	2	1	no	—
53	C₂	—	1268	1	no	—
54	C₂	#23778692	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²⋊D₃₁₇
commutator	C₄×C₃₁₇	C₂×C₈
Frattini	C₂×C₈	C₂×D₃₁₇
Fitting	M₃₂×C₃₁₇	C₂
socle	C₂×C₃₁₇	C₂²⋊C₈
prime radical	#19099574	C₄
cosocle	C₂×C₈×C₃₁₇	C₂²
hypercenter	M₃₂	D₃₁₇
hypocenter	C₃₁₇	#350
LCS	G ▷ C₄×C₃₁₇ ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ M₃₂

Complements

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

N	H ≅ G/N	type
2. C₂×#19099549	C₂	semidirect
3. M₃₂×C₃₁₇	C₂	semidirect
27. C₃₁₇	#350	semidirect
30. M₃₂	D₃₁₇	semidirect

Maximal subgroups

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

subgroup	index
#23778786	2
C₂×#19099549	2
M₃₂×C₃₁₇	2
#350	317