Subgroups — #35203413

Order 20288.

160 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23778761	C₂	1	3	yes	maximal, complemented
2	#23778695	C₂	1	3	yes	maximal, complemented
3	H(5,𝔽₂)×C₃₁₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×Dic₃₁₇	C₂²	1	1	yes	characteristic, complemented
5	#19099553	—	2	3	no	—
6	C₂²⋊Dic₃₁₇	—	2	3	no	—
7	C₂²⋊Dic₃₁₇	C₂²	1	3	yes	complemented
8	#41×C₃₁₇	—	2	3	no	—
9	C₂×D₄×C₃₁₇	—	2	3	no	—
10	C₂×D₄×C₃₁₇	C₂²	1	3	yes	—
11	C₂×Dic₃₁₇	—	2	3	no	—
12	C₂×Dic₃₁₇	—	2	3	no	—
13	C₂³×C₃₁₇	—	2	1	no	—
14	C₂³×C₃₁₇	D₄	1	3	yes	—
15	C₂³×C₃₁₇	C₂³	1	1	yes	characteristic
16	Q₈×C₃₁₇	—	2	1	no	—
17	D₄×C₃₁₇	—	2	3	no	—
18	D₄×C₃₁₇	—	2	6	no	—
19	C₂×C₄×C₃₁₇	—	2	3	no	—
20	C₂×C₄×C₃₁₇	D₄	1	3	yes	—
21	C₂²×C₃₁₇	—	2	3	no	—
22	C₂²×C₃₁₇	—	4	1	no	—
23	C₂²×C₃₁₇	—	4	3	no	—
24	C₂²×C₃₁₇	—	8	1	no	—
25	C₂²×C₃₁₇	C₂×D₄	1	3	yes	—
26	C₄×C₃₁₇	—	2	3	no	—
27	Dic₃₁₇	—	4	3	no	—
28	Dic₃₁₇	—	4	1	no	—
29	C₂×C₃₁₇	—	2	3	no	—
30	C₂×C₃₁₇	—	4	3	no	—
31	C₂×C₃₁₇	#120	1	1	yes	characteristic
32	C₃₁₇	#458	1	1	yes	characteristic, complemented, Sylow
33	#458	—	317	1	no	maximal, Sylow
34	H(5,𝔽₂)	D₃₁₇	1	1	yes	characteristic
35	#123	—	317	3	no	—
36	#99	—	317	3	no	—
37	#41	—	2	3	no	—
38	C₂×D₄	—	2	3	no	—
39	C₂×D₄	C₂×D₃₁₇	1	3	yes	—
40	C₂²×C₄	—	317	1	no	—
41	C₄⋊C₄	—	634	3	no	—
42	C₂²⋊C₄	—	317	3	no	—
43	C₂²⋊C₄	—	634	3	no	—
44	C₂³	—	2	1	no	—
45	C₂³	C₂²⋊D₃₁₇	1	3	yes	—
46	C₂³	C₂²×D₃₁₇	1	1	yes	characteristic
47	Q₈	—	2	1	no	—
48	D₄	—	2	3	no	—
49	D₄	—	2	6	no	—
50	C₂×C₄	—	2	3	no	—
51	C₂×C₄	—	634	3	no	—
52	C₂×C₄	—	634	3	no	—
53	C₂×C₄	C₂²⋊D₃₁₇	1	3	yes	—
54	C₂²	—	2	3	no	—
55	C₂²	—	4	1	no	—
56	C₂²	—	4	3	no	—
57	C₂²	—	8	1	no	—
58	C₂²	C₂×C₂²⋊D₃₁₇	1	3	yes	—
59	C₄	—	2	3	no	—
60	C₄	—	1268	3	no	—
61	C₄	—	1268	1	no	—
62	C₂	—	2	3	no	—
63	C₂	—	4	3	no	—
64	C₂	#23778749	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#23778749
commutator, cosocle	C₂³×C₃₁₇	C₂³
Frattini	C₂³	C₂²×D₃₁₇
Fitting, prime radical	H(5,𝔽₂)×C₃₁₇	C₂
socle	C₂×C₃₁₇	#120
hypercenter	H(5,𝔽₂)	D₃₁₇
hypocenter	C₃₁₇	#458
LCS	G ▷ C₂³×C₃₁₇ ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ H(5,𝔽₂)

Complements

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

N	H ≅ G/N	type
1. #23778761	C₂	semidirect
2. #23778695	C₂	semidirect
4. C₂²×Dic₃₁₇	C₂²	semidirect
7. C₂²⋊Dic₃₁₇	C₂²	semidirect
32. C₃₁₇	#458	semidirect

Maximal subgroups

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

subgroup	index
#23778761	2
#23778695	2
H(5,𝔽₂)×C₃₁₇	2
#458	317