Subgroups — #35204097

Order 20288.

114 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#11929639	C₂	1	2	yes	maximal
2	C₂×C₈×D₃₁₇	C₂	1	1	yes	maximal, characteristic
3	C₂²×#11929639	—	2	2	no	—
4	C₄⋊#11929639	—	2	2	no	—
5	C₄⋊#11929639	C₄	1	4	yes	complemented
6	C₂×C₄×D₃₁₇	C₂²	1	1	yes	characteristic
7	C₂×#14579261	C₄	1	1	yes	characteristic, complemented
8	C₈×D₃₁₇	—	2	2	no	—
9	C₂×C₈×C₃₁₇	C₄	1	1	yes	characteristic, complemented
10	C₂×#11929639	—	2	4	no	—
11	C₂×#11929639	—	4	2	no	—
12	C₂×#11929639	—	4	2	no	—
13	C₂²×D₃₁₇	D₄	1	1	yes	characteristic
14	C₂×Dic₃₁₇	Q₈	1	1	yes	characteristic
15	C₄×D₃₁₇	C₂×C₄	1	2	yes	—
16	C₄×D₃₁₇	D₄	1	2	yes	—
17	#14579261	—	2	1	no	—
18	C₂×C₄×C₃₁₇	C₂×C₄	1	1	yes	characteristic
19	C₈×C₃₁₇	—	2	1	no	—
20	C₂²×C₃₁₇	C₄⋊C₄	1	1	yes	characteristic
21	C₂×D₃₁₇	C₂²⋊C₄	1	2	yes	—
22	C₂×D₃₁₇	D₈	1	1	yes	characteristic
23	C₂×D₃₁₇	SD₁₆	1	2	yes	—
24	C₂×D₃₁₇	Q₁₆	1	1	yes	characteristic
25	#11929639	—	4	4	no	—
26	C₄×C₃₁₇	C₄²	1	1	yes	characteristic
27	C₄×C₃₁₇	C₂²⋊C₄	1	1	yes	characteristic
28	Dic₃₁₇	C₄⋊C₄	1	2	yes	—
29	C₂×C₃₁₇	#95	1	1	yes	characteristic
30	C₂×C₃₁₇	#106	1	1	yes	characteristic
31	C₂×C₃₁₇	#107	1	1	yes	characteristic
32	D₃₁₇	#102	1	2	yes	—
33	D₃₁₇	#103	1	2	yes	—
34	C₃₁₇	#340	1	1	yes	characteristic, complemented, Sylow
35	#340	—	317	1	no	maximal, Sylow
36	C₂²×C₈	—	317	1	no	—
37	C₂×C₄⋊C₄	—	317	2	no	—
38	C₂²×C₄	—	317	1	no	—
39	C₂²×C₄	—	634	2	no	—
40	C₂×C₈	—	317	1	no	—
41	C₂×C₈	—	634	2	no	—
42	C₂×C₈	#11929639	1	1	yes	characteristic, complemented
43	C₄⋊C₄	—	317	4	no	—
44	C₄⋊C₄	—	634	2	no	—
45	C₂³	—	317	1	no	—
46	C₂×C₄	—	317	1	no	—
47	C₂×C₄	—	317	2	no	—
48	C₂×C₄	—	317	2	no	—
49	C₂×C₄	—	634	4	no	—
50	C₂×C₄	—	1268	2	no	—
51	C₂×C₄	—	1268	2	no	—
52	C₂×C₄	C₂×#11929639	1	1	yes	characteristic
53	C₈	—	2	1	no	—
54	C₈	—	634	1	no	—
55	C₂²	—	317	2	no	—
56	C₂²	—	317	2	no	—
57	C₂²	—	317	1	no	—
58	C₂²	—	317	1	no	—
59	C₂²	C₄⋊#11929639	1	1	yes	characteristic
60	C₄	—	317	2	no	—
61	C₄	—	1268	4	no	—
62	C₄	C₄×#11929639	1	1	yes	characteristic
63	C₄	C₂²⋊#11929639	1	1	yes	characteristic
64	C₂	—	317	2	no	—
65	C₂	—	317	2	no	—
66	C₂	(C₂×#11929639)⋊C₄	1	1	yes	characteristic
67	C₂	#23778848	1	1	yes	characteristic
68	C₂	#23778849	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₄⋊#11929639
commutator	C₄×C₃₁₇	C₄²
Frattini	C₂×C₄	C₂×#11929639
Fitting	C₂×C₈×C₃₁₇	C₄
socle	C₂²×C₃₁₇	C₄⋊C₄
prime radical	C₂²×D₃₁₇	D₄
cosocle	C₂×C₄×D₃₁₇	C₂²
hypercenter	C₂×C₈	#11929639
hypocenter	C₃₁₇	#340
LCS	G ▷ C₄×C₃₁₇ ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
5. C₄⋊#11929639	C₄	semidirect
7. C₂×#14579261	C₄	semidirect
9. C₂×C₈×C₃₁₇	C₄	semidirect
34. C₃₁₇	#340	semidirect
42. C₂×C₈	#11929639	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊#11929639	2
C₂×C₈×D₃₁₇	2
#340	317