Subgroups — #35203249

Order 20288.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19099558	C₂	1	1	yes	maximal, characteristic
2	#23778739	C₂	1	1	yes	maximal, characteristic
3	#23778709	C₂	1	4	yes	maximal, complemented
4	C₂²×C₈×C₃₁₇	C₂	1	1	yes	maximal, characteristic
5	#19099558	—	2	2	no	—
6	C₂×#14579261	C₂²	1	2	yes	—
7	C₄⋊Dic₃₁₇	C₄	1	2	yes	—
8	C₂²⋊Dic₃₁₇	C₄	1	2	yes	—
9	C₄×Dic₃₁₇	C₂²	1	2	yes	—
10	C₂²×C₄×C₃₁₇	C₂²	1	1	yes	characteristic
11	C₂×C₈×C₃₁₇	—	2	2	no	—
12	C₂×C₈×C₃₁₇	C₂²	1	2	yes	—
13	C₂×Dic₃₁₇	C₂×C₄	1	4	yes	—
14	#14579261	—	2	2	no	—
15	C₂³×C₃₁₇	C₂×C₄	1	1	yes	characteristic
16	C₂×C₄×C₃₁₇	C₂×C₄	1	1	yes	characteristic
17	C₂×C₄×C₃₁₇	D₄	1	2	yes	—
18	C₂×C₄×C₃₁₇	Q₈	1	2	yes	—
19	C₂×C₄×C₃₁₇	C₂³	1	1	yes	characteristic
20	C₈×C₃₁₇	—	2	2	no	—
21	C₂²×C₃₁₇	—	2	2	no	—
22	C₂²×C₃₁₇	C₄⋊C₄	1	2	yes	—
23	C₂²×C₃₁₇	C₂²×C₄	1	1	yes	characteristic
24	C₄×C₃₁₇	C₄⋊C₄	1	2	yes	—
25	C₄×C₃₁₇	C₂×D₄	1	1	yes	characteristic
26	C₄×C₃₁₇	C₂×Q₈	1	1	yes	characteristic
27	Dic₃₁₇	—	2	4	no	—
28	C₂×C₃₁₇	—	2	2	no	—
29	C₂×C₃₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₃₁₇	#131	1	2	yes	—
31	C₃₁₇	#424	1	1	yes	characteristic, complemented, Sylow
32	#424	—	317	1	no	maximal, Sylow
33	C₂×M₁₆	—	317	1	no	—
34	C₂²×C₈	D₃₁₇	1	1	yes	characteristic
35	#117	—	317	1	no	—
36	C₄⋊C₈	—	317	4	no	—
37	C₂²×C₄	C₂×D₃₁₇	1	1	yes	characteristic
38	M₁₆	—	634	2	no	—
39	C₂×C₈	—	2	2	no	—
40	C₂×C₈	—	317	2	no	—
41	C₂×C₈	C₂×D₃₁₇	1	2	yes	—
42	C₄⋊C₄	—	317	2	no	—
43	C₂²⋊C₄	—	317	2	no	—
44	C₄²	—	317	2	no	—
45	C₂³	C₄×D₃₁₇	1	1	yes	characteristic
46	C₂×C₄	—	317	4	no	—
47	C₂×C₄	C₄×D₃₁₇	1	1	yes	characteristic
48	C₂×C₄	C₂²⋊D₃₁₇	1	2	yes	—
49	C₂×C₄	Dic₆₃₄	1	2	yes	—
50	C₂×C₄	C₂²×D₃₁₇	1	1	yes	characteristic
51	C₈	—	2	2	no	—
52	C₈	—	634	2	no	—
53	C₂²	—	2	2	no	—
54	C₂²	#19099553	1	2	yes	—
55	C₂²	C₂×C₄×D₃₁₇	1	1	yes	characteristic
56	C₄	—	634	4	no	—
57	C₄	#19099553	1	2	yes	—
58	C₄	C₂×C₂²⋊D₃₁₇	1	1	yes	characteristic
59	C₄	C₂×Dic₆₃₄	1	1	yes	characteristic
60	C₂	—	2	2	no	—
61	C₂	C₂×#19099553	1	1	yes	characteristic
62	C₂	#23778786	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂²×D₃₁₇
commutator, socle	C₂²×C₃₁₇	C₂²×C₄
Fitting	C₂²×C₈×C₃₁₇	C₂
prime radical	C₂³×C₃₁₇	C₂×C₄
cosocle	C₂×C₄×C₃₁₇	C₂³
hypercenter	C₂²×C₈	D₃₁₇
hypocenter	C₃₁₇	#424
LCS	G ▷ C₂²×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
3. #23778709	C₂	semidirect
31. C₃₁₇	#424	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
#23778739	2
#23778709	2
C₂²×C₈×C₃₁₇	2
#424	317