Subgroups — #35203272

Order 20288.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19099558	C₂	1	2	yes	maximal
2	#23778716	C₂	1	4	yes	maximal, complemented
3	C₂×M₁₆×C₃₁₇	C₂	1	1	yes	maximal, characteristic
4	#19099558	—	2	2	no	—
5	#19099558	C₂²	1	4	yes	—
6	C₂×#14579261	—	2	2	no	—
7	C₂²×C₄×C₃₁₇	C₂²	1	1	yes	characteristic
8	M₁₆×C₃₁₇	C₄	1	4	yes	—
9	C₂×C₈×C₃₁₇	C₂²	1	2	yes	—
10	#14579261	—	2	4	no	—
11	C₂³×C₃₁₇	Q₈	1	1	yes	characteristic
12	C₂×C₄×C₃₁₇	C₂×C₄	1	2	yes	—
13	C₂×C₄×C₃₁₇	D₄	1	2	yes	—
14	C₂×C₄×C₃₁₇	Q₈	1	1	yes	characteristic
15	C₂×C₄×C₃₁₇	C₂³	1	1	yes	characteristic
16	C₈×C₃₁₇	C₂×C₄	1	4	yes	—
17	C₂²×C₃₁₇	—	4	1	no	—
18	C₂²×C₃₁₇	C₄⋊C₄	1	2	yes	—
19	C₂²×C₃₁₇	C₂×Q₈	1	1	yes	characteristic
20	C₄×C₃₁₇	C₄⋊C₄	1	2	yes	—
21	C₄×C₃₁₇	C₂²×C₄	1	1	yes	characteristic
22	C₄×C₃₁₇	C₂×D₄	1	1	yes	characteristic
23	C₂×C₃₁₇	—	2	2	no	—
24	C₂×C₃₁₇	—	2	1	no	—
25	C₂×C₃₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
26	C₃₁₇	#430	1	1	yes	characteristic, complemented, Sylow
27	#430	—	317	1	no	maximal, Sylow
28	C₂×M₁₆	—	317	2	no	—
29	C₂×M₁₆	D₃₁₇	1	1	yes	characteristic
30	#108	—	317	4	no	—
31	C₂²×C₄	C₂×D₃₁₇	1	1	yes	characteristic
32	M₁₆	—	317	4	no	—
33	M₁₆	—	634	2	no	—
34	M₁₆	Dic₃₁₇	1	4	yes	—
35	C₂×C₈	—	634	2	no	—
36	C₂×C₈	C₂×D₃₁₇	1	2	yes	—
37	C₂³	Dic₆₃₄	1	1	yes	characteristic
38	C₂×C₄	C₂×Dic₃₁₇	1	2	yes	—
39	C₂×C₄	D₁₂₆₈	1	2	yes	—
40	C₂×C₄	Dic₆₃₄	1	1	yes	characteristic
41	C₂×C₄	C₂²×D₃₁₇	1	1	yes	characteristic
42	C₈	—	634	4	no	—
43	C₈	C₂×Dic₃₁₇	1	4	yes	—
44	C₂²	—	4	1	no	—
45	C₂²	C₄⋊Dic₃₁₇	1	2	yes	—
46	C₂²	C₂×Dic₆₃₄	1	1	yes	characteristic
47	C₄	C₄⋊Dic₃₁₇	1	2	yes	—
48	C₄	C₂²×Dic₃₁₇	1	1	yes	characteristic
49	C₄	C₂×D₁₂₆₈	1	1	yes	characteristic
50	C₂	—	2	2	no	—
51	C₂	—	2	1	no	—
52	C₂	C₂×C₄⋊Dic₃₁₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₂×D₁₂₆₈
commutator	C₄×C₃₁₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃₁₇
Fitting	C₂×M₁₆×C₃₁₇	C₂
socle	C₂×C₃₁₇	C₂×C₄⋊C₄
prime radical	C₂³×C₃₁₇	Q₈
cosocle	C₂×C₄×C₃₁₇	C₂³
hypercenter	C₂×M₁₆	D₃₁₇
hypocenter	C₃₁₇	#430
LCS	G ▷ C₄×C₃₁₇ ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×M₁₆

Complements

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

N	H ≅ G/N	type
2. #23778716	C₂	semidirect
26. C₃₁₇	#430	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19099558	2
#23778716	2
C₂×M₁₆×C₃₁₇	2
#430	317