Subgroups — #35203701

Order 20288.

266 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	Q₈×Dic₃₁₇	C₂	1	8	yes	maximal, complemented
2	#23778739	C₂	1	6	yes	maximal
3	C₂²×Q₈×C₃₁₇	C₂	1	1	yes	maximal, characteristic
4	C₄⋊Dic₃₁₇	C₂²	1	12	yes	—
5	C₂²⋊Dic₃₁₇	C₂²	1	4	yes	—
6	C₄×Dic₃₁₇	C₂²	1	12	yes	—
7	C₂×Q₈×C₃₁₇	C₄	1	8	yes	complemented
8	C₂×Q₈×C₃₁₇	C₂²	1	4	yes	—
9	C₂²×C₄×C₃₁₇	C₂²	1	3	yes	—
10	C₂×Dic₃₁₇	C₂³	1	8	yes	—
11	C₂³×C₃₁₇	C₂³	1	1	yes	characteristic
12	Q₈×C₃₁₇	C₂×C₄	1	16	yes	—
13	C₂×C₄×C₃₁₇	C₂×C₄	1	12	yes	—
14	C₂×C₄×C₃₁₇	C₂³	1	6	yes	—
15	C₂²×C₃₁₇	—	2	2	no	—
16	C₂²×C₃₁₇	C₂²×C₄	1	2	yes	—
17	C₂²×C₃₁₇	C₂⁴	1	1	yes	characteristic
18	C₄×C₃₁₇	C₂²×C₄	1	12	yes	—
19	Dic₃₁₇	—	2	8	no	—
20	C₂×C₃₁₇	—	2	2	no	—
21	C₂×C₃₁₇	C₂³×C₄	1	1	yes	characteristic
22	C₂×C₃₁₇	2₋¹⁺⁴	1	2	yes	—
23	C₃₁₇	#519	1	1	yes	characteristic, complemented, Sylow
24	#519	—	317	1	no	maximal, Sylow
25	C₂²×Q₈	D₃₁₇	1	1	yes	characteristic
26	C₄×Q₈	—	317	8	no	—
27	#117	—	317	6	no	—
28	C₂×Q₈	Dic₃₁₇	1	8	yes	complemented
29	C₂×Q₈	C₂×D₃₁₇	1	4	yes	—
30	C₂²×C₄	C₂×D₃₁₇	1	3	yes	—
31	C₄⋊C₄	—	317	12	no	—
32	C₂²⋊C₄	—	317	4	no	—
33	C₄²	—	317	12	no	—
34	C₂³	C₂²×D₃₁₇	1	1	yes	characteristic
35	Q₈	C₂×Dic₃₁₇	1	16	yes	—
36	C₂×C₄	—	317	8	no	—
37	C₂×C₄	C₂×Dic₃₁₇	1	12	yes	—
38	C₂×C₄	C₂²×D₃₁₇	1	6	yes	—
39	C₂²	—	2	2	no	—
40	C₂²	C₂²×Dic₃₁₇	1	2	yes	—
41	C₂²	C₂³×D₃₁₇	1	1	yes	characteristic
42	C₄	—	634	8	no	—
43	C₄	C₂²×Dic₃₁₇	1	12	yes	—
44	C₂	—	2	2	no	—
45	C₂	C₂³×Dic₃₁₇	1	1	yes	characteristic
46	C₂	#23778828	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₁₇
commutator	C₂×C₃₁₇	C₂³×C₄
Fitting	C₂²×Q₈×C₃₁₇	C₂
socle, cosocle	C₂²×C₃₁₇	C₂⁴
prime radical	C₂³×C₃₁₇	C₂³
hypercenter	C₂²×Q₈	D₃₁₇
hypocenter	C₃₁₇	#519
LCS	G ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
1. Q₈×Dic₃₁₇	C₂	semidirect
7. C₂×Q₈×C₃₁₇	C₄	semidirect
23. C₃₁₇	#519	semidirect
28. C₂×Q₈	Dic₃₁₇	semidirect

Maximal subgroups

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

subgroup	index
Q₈×Dic₃₁₇	2
#23778739	2
C₂²×Q₈×C₃₁₇	2
#519	317