Subgroups — #35203336

Order 20288.

132 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₂₅₃₆	C₂	1	1	yes	maximal, characteristic
2	C₄×D₁₂₆₈	C₂	1	2	yes	maximal, complemented
3	#23778712	C₂	1	1	yes	maximal, characteristic, complemented
4	#23778702	C₂	1	2	yes	maximal, complemented
5	#97×C₃₁₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₂₆₈	C₂²	1	2	yes	—
7	C₂×C₄×D₃₁₇	—	2	2	no	—
8	D₂₅₃₆	C₄	1	4	yes	complemented
9	C₄⋊Dic₃₁₇	C₂²	1	2	yes	—
10	#19099551	—	2	2	no	—
11	C₂×C₈×C₃₁₇	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₃₁₇	C₂²	1	1	yes	characteristic
13	C₄²×C₃₁₇	C₂²	1	1	yes	characteristic
14	C₂²×D₃₁₇	—	2	2	no	—
15	D₁₂₆₈	—	2	2	no	—
16	D₁₂₆₈	C₂×C₄	1	4	yes	complemented
17	C₂×Dic₃₁₇	—	2	2	no	—
18	C₄×D₃₁₇	—	4	4	no	—
19	C₂×C₄×C₃₁₇	D₄	1	2	yes	—
20	C₂×C₄×C₃₁₇	C₂³	1	1	yes	characteristic
21	C₈×C₃₁₇	—	2	1	no	—
22	C₈×C₃₁₇	C₂×C₄	1	2	yes	complemented
23	C₂²×C₃₁₇	C₂×D₄	1	1	yes	characteristic
24	C₂×D₃₁₇	—	2	4	no	—
25	C₂×D₃₁₇	—	4	2	no	—
26	C₂×D₃₁₇	—	4	2	no	—
27	C₄×C₃₁₇	—	2	2	no	—
28	C₄×C₃₁₇	C₂²×C₄	1	1	yes	characteristic
29	C₄×C₃₁₇	#41	1	1	yes	characteristic
30	Dic₃₁₇	—	4	2	no	—
31	C₂×C₃₁₇	C₄×D₄	1	1	yes	characteristic
32	C₂×C₃₁₇	AGL(1,ℤ/8)	1	2	yes	—
33	D₃₁₇	—	4	4	no	—
34	C₃₁₇	#442	1	1	yes	characteristic, complemented, Sylow
35	#442	—	317	1	no	maximal, Sylow
36	C₂×D₈	—	317	1	no	—
37	C₄×D₄	—	317	2	no	—
38	#106	—	317	1	no	—
39	#102	—	317	2	no	—
40	#97	D₃₁₇	1	1	yes	characteristic, complemented
41	C₂×D₄	—	317	2	no	—
42	C₂²×C₄	—	634	2	no	—
43	D₈	—	317	4	no	—
44	C₂×C₈	C₂×D₃₁₇	1	1	yes	characteristic
45	C₂×C₈	C₂×D₃₁₇	1	1	yes	characteristic
46	C₄⋊C₄	—	317	2	no	—
47	C₂²⋊C₄	—	634	2	no	—
48	C₄²	C₂×D₃₁₇	1	1	yes	characteristic
49	C₂³	—	634	2	no	—
50	D₄	—	317	4	no	—
51	D₄	—	634	2	no	—
52	C₂×C₄	—	634	2	no	—
53	C₂×C₄	—	1268	4	no	—
54	C₂×C₄	D₁₂₆₈	1	2	yes	—
55	C₂×C₄	C₂²×D₃₁₇	1	1	yes	characteristic
56	C₈	—	2	1	no	—
57	C₈	C₄×D₃₁₇	1	2	yes	complemented
58	C₂²	—	634	4	no	—
59	C₂²	—	1268	2	no	—
60	C₂²	—	1268	2	no	—
61	C₂²	C₂×D₁₂₆₈	1	1	yes	characteristic
62	C₄	—	2	2	no	—
63	C₄	—	1268	2	no	—
64	C₄	C₂×C₄×D₃₁₇	1	1	yes	characteristic
65	C₄	#19099574	1	1	yes	characteristic
66	C₂	—	1268	4	no	—
67	C₂	C₄×D₁₂₆₈	1	1	yes	characteristic
68	C₂	#23778808	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₁₂₆₈
commutator	C₄×C₃₁₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃₁₇
Fitting	#97×C₃₁₇	C₂
socle	C₂²×C₃₁₇	C₂×D₄
prime radical	C₂×D₂₅₃₆	C₂
cosocle	C₂×C₄×C₃₁₇	C₂³
hypercenter	#97	D₃₁₇
hypocenter	C₃₁₇	#442
LCS	G ▷ C₄×C₃₁₇ ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #97

Complements

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

N	H ≅ G/N	type
2. C₄×D₁₂₆₈	C₂	semidirect
3. #23778712	C₂	semidirect
4. #23778702	C₂	semidirect
5. #97×C₃₁₇	C₂	semidirect
8. D₂₅₃₆	C₄	semidirect
16. D₁₂₆₈	C₂×C₄	semidirect
22. C₈×C₃₁₇	C₂×C₄	semidirect
34. C₃₁₇	#442	semidirect
40. #97	D₃₁₇	semidirect
57. C₈	C₄×D₃₁₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₂₅₃₆	2
C₄×D₁₂₆₈	2
#23778712	2
#23778702	2
#97×C₃₁₇	2
#442	317