Subgroups — #35204142

Order 20288.

54 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23778855	C₂	1	1	yes	maximal, characteristic
2	#23778850	C₂	1	1	yes	maximal, characteristic
3	Q₁₆×D₃₁₇	C₂	1	1	yes	maximal, characteristic
4	#19099584	—	2	1	no	—
5	#19099577	—	2	1	no	—
6	Q₈×D₃₁₇	—	2	1	no	—
7	Dic₁₂₆₈	C₄	1	1	yes	characteristic
8	#19099564	—	2	1	no	—
9	C₈×D₃₁₇	C₂²	1	1	yes	characteristic
10	Q₁₆×C₃₁₇	C₄	1	1	yes	characteristic
11	#14579268	—	4	1	no	—
12	Dic₆₃₄	—	2	1	no	—
13	Dic₆₃₄	—	4	1	no	—
14	C₄×D₃₁₇	—	4	1	no	—
15	C₄×D₃₁₇	D₄	1	1	yes	characteristic
16	#14579261	D₄	1	1	yes	characteristic
17	Q₈×C₃₁₇	—	2	1	no	—
18	C₈×C₃₁₇	C₂×C₄	1	1	yes	characteristic
19	C₂×D₃₁₇	SD₁₆	1	1	yes	characteristic
20	C₄×C₃₁₇	—	4	1	no	—
21	C₄×C₃₁₇	C₂²⋊C₄	1	1	yes	characteristic
22	Dic₃₁₇	—	4	1	no	—
23	Dic₃₁₇	D₈	1	1	yes	characteristic
24	C₂×C₃₁₇	#102	1	1	yes	characteristic
25	D₃₁₇	—	2	1	no	—
26	C₃₁₇	#362	1	1	yes	characteristic, complemented, Sylow
27	#362	—	317	1	no	maximal, Sylow
28	C₂×Q₁₆	—	317	1	no	—
29	M₃₂	—	317	1	no	—
30	#108	—	317	1	no	—
31	C₂×Q₈	—	634	1	no	—
32	Q₁₆	—	317	1	no	—
33	Q₁₆	—	634	1	no	—
34	Q₁₆	#11929639	1	1	yes	characteristic
35	M₁₆	—	634	1	no	—
36	C₂×C₈	—	317	1	no	—
37	C₁₆	—	634	1	no	—
38	Q₈	—	2	1	no	—
39	Q₈	—	634	1	no	—
40	Q₈	—	1268	1	no	—
41	C₂×C₄	—	317	1	no	—
42	C₂×C₄	—	1268	1	no	—
43	C₈	—	317	1	no	—
44	C₈	—	1268	1	no	—
45	C₈	C₂×#11929639	1	1	yes	characteristic
46	C₂²	—	317	1	no	—
47	C₄	—	4	1	no	—
48	C₄	—	317	1	no	—
49	C₄	—	1268	1	no	—
50	C₄	C₂²⋊#11929639	1	1	yes	characteristic
51	C₂	—	634	1	no	—
52	C₂	#23778842	1	1	yes	characteristic

	subgroup	G/H
center	C₂	#23778842
commutator	C₈×C₃₁₇	C₂×C₄
Frattini	C₈	C₂×#11929639
Fitting	Q₁₆×C₃₁₇	C₄
socle	C₂×C₃₁₇	#102
prime radical	C₂×D₃₁₇	SD₁₆
cosocle	C₈×D₃₁₇	C₂²
hypercenter	Q₁₆	#11929639
hypocenter	C₃₁₇	#362
LCS	G ▷ C₈×C₃₁₇ ▷ C₄×C₃₁₇ ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₁₆

Complements

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

N	H ≅ G/N	type
26. C₃₁₇	#362	semidirect

Maximal subgroups

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

subgroup	index
#23778855	2
#23778850	2
Q₁₆×D₃₁₇	2
#362	317