Subgroups — C₂×#23778848 #35204231

Order 20288.

130 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	#23778848	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₈×D₃₁₇	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11929639	—	2	2	no	—
5	C₄⋊#11929639	—	2	2	no	—
6	C₄⋊#11929639	C₂²	1	4	yes	—
7	C₂×C₄×D₃₁₇	C₂²	1	1	yes	characteristic
8	C₂×#14579261	C₄	1	1	yes	characteristic, complemented
9	C₈×D₃₁₇	C₄	1	2	yes	complemented
10	C₈×D₃₁₇	C₂²	1	2	yes	—
11	C₂×C₈×C₃₁₇	C₄	1	1	yes	characteristic, complemented
12	C₂×#11929639	—	2	4	no	—
13	C₂×#11929639	—	4	4	no	—
14	C₂²×D₃₁₇	D₄	1	1	yes	characteristic
15	C₂×Dic₃₁₇	Q₈	1	1	yes	characteristic
16	C₄×D₃₁₇	C₂×C₄	1	1	yes	characteristic
17	C₄×D₃₁₇	D₄	1	1	yes	characteristic
18	C₄×D₃₁₇	Q₈	1	1	yes	characteristic
19	C₄×D₃₁₇	C₂³	1	1	yes	characteristic
20	#14579261	C₂×C₄	1	2	yes	complemented
21	C₂×C₄×C₃₁₇	C₂×C₄	1	1	yes	characteristic
22	C₈×C₃₁₇	C₂×C₄	1	2	yes	complemented
23	C₂²×C₃₁₇	C₄⋊C₄	1	1	yes	characteristic
24	C₂×D₃₁₇	C₄⋊C₄	1	1	yes	characteristic
25	C₂×D₃₁₇	SD₁₆	1	4	yes	—
26	C₂×D₃₁₇	C₂×D₄	1	1	yes	characteristic
27	#11929639	—	4	4	no	—
28	C₄×C₃₁₇	C₄⋊C₄	1	1	yes	characteristic
29	C₄×C₃₁₇	C₂²×C₄	1	1	yes	characteristic
30	Dic₃₁₇	C₄⋊C₄	1	1	yes	characteristic
31	Dic₃₁₇	C₂×Q₈	1	1	yes	characteristic
32	C₂×C₃₁₇	#106	1	2	yes	complemented
33	C₂×C₃₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
34	D₃₁₇	#106	1	2	yes	complemented
35	D₃₁₇	C₂×SD₁₆	1	2	yes	—
36	C₃₁₇	C₂×#106	1	1	yes	characteristic, complemented, Sylow
37	C₂×#106	—	317	1	no	maximal, Sylow
38	C₂²×C₈	—	317	1	no	—
39	C₂×C₄⋊C₄	—	317	2	no	—
40	#106	—	317	4	no	—
41	C₂²×C₄	—	317	1	no	—
42	C₂²×C₄	—	634	2	no	—
43	C₂×C₈	—	317	1	no	—
44	C₂×C₈	—	317	2	no	—
45	C₂×C₈	—	317	2	no	—
46	C₂×C₈	#11929639	1	1	yes	characteristic, complemented
47	C₄⋊C₄	—	317	4	no	—
48	C₄⋊C₄	—	634	2	no	—
49	C₂³	—	317	1	no	—
50	C₂×C₄	—	317	1	no	—
51	C₂×C₄	—	317	1	no	—
52	C₂×C₄	—	317	1	no	—
53	C₂×C₄	—	317	1	no	—
54	C₂×C₄	—	317	1	no	—
55	C₂×C₄	—	634	4	no	—
56	C₂×C₄	—	1268	4	no	—
57	C₂×C₄	C₂×#11929639	1	1	yes	characteristic
58	C₈	—	317	2	no	—
59	C₈	C₂×#11929639	1	2	yes	complemented
60	C₂²	—	317	4	no	—
61	C₂²	—	317	1	no	—
62	C₂²	—	317	1	no	—
63	C₂²	C₄⋊#11929639	1	1	yes	characteristic
64	C₄	—	317	1	no	—
65	C₄	—	317	1	no	—
66	C₄	—	1268	4	no	—
67	C₄	C₄⋊#11929639	1	1	yes	characteristic
68	C₄	C₂²×#11929639	1	1	yes	characteristic
69	C₂	—	317	2	no	—
70	C₂	—	317	2	no	—
71	C₂	#23778848	1	2	yes	direct product factor
72	C₂	C₂×C₄⋊#11929639	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₄⋊#11929639
commutator	C₄×C₃₁₇	C₂²×C₄
Frattini	C₄	C₂²×#11929639
Fitting	C₂×C₈×C₃₁₇	C₄
socle	C₂²×C₃₁₇	C₄⋊C₄
prime radical	C₂²×D₃₁₇	D₄
cosocle	C₄×D₃₁₇	C₂³
hypercenter	C₂×C₈	#11929639
hypocenter	C₃₁₇	C₂×#106
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
2. #23778848	C₂	direct (N × H)
8. C₂×#14579261	C₄	semidirect
9. C₈×D₃₁₇	C₄	semidirect
11. C₂×C₈×C₃₁₇	C₄	semidirect
20. #14579261	C₂×C₄	semidirect
22. C₈×C₃₁₇	C₂×C₄	semidirect
32. C₂×C₃₁₇	#106	semidirect
34. D₃₁₇	#106	semidirect
36. C₃₁₇	C₂×#106	semidirect
46. C₂×C₈	#11929639	semidirect
59. C₈	C₂×#11929639	semidirect
71. C₂	#23778848	direct (N × H)

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
#23778848	2
C₂×C₈×D₃₁₇	2
C₂×#106	317

Subgroups — C2×#23778848 #35204231

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23778848 #35204231