Subgroups — #35202983

Order 20288.

110 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19099558	C₂	1	1	yes	maximal, characteristic
2	#23778739	C₂	1	1	yes	maximal, characteristic
3	C₂×C₄²×C₃₁₇	C₂	1	1	yes	maximal, characteristic
4	#19099558	—	2	1	no	—
5	#19099558	C₄	1	2	yes	complemented
6	C₂×#14579261	—	2	1	no	—
7	C₄⋊Dic₃₁₇	C₄	1	2	yes	complemented
8	C₂²⋊Dic₃₁₇	—	2	1	no	—
9	C₄×Dic₃₁₇	—	2	1	no	—
10	C₂²×C₄×C₃₁₇	—	2	1	no	—
11	C₂²×C₄×C₃₁₇	C₂²	1	1	yes	characteristic
12	C₄²×C₃₁₇	—	2	1	no	—
13	C₄²×C₃₁₇	C₄	1	2	yes	complemented
14	C₂×Dic₃₁₇	—	2	2	no	—
15	#14579261	—	2	2	no	—
16	C₂³×C₃₁₇	D₄	1	1	yes	characteristic
17	C₂×C₄×C₃₁₇	—	2	4	no	—
18	C₂×C₄×C₃₁₇	—	2	2	no	—
19	C₂×C₄×C₃₁₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×C₃₁₇	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×C₃₁₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×C₃₁₇	D₄	1	1	yes	characteristic
23	C₂×C₄×C₃₁₇	D₄	1	1	yes	characteristic
24	C₂×C₄×C₃₁₇	Q₈	1	1	yes	characteristic
25	C₂²×C₃₁₇	—	2	2	no	—
26	C₂²×C₃₁₇	C₂²⋊C₄	1	1	yes	characteristic
27	C₂²×C₃₁₇	C₂²⋊C₄	1	1	yes	characteristic
28	C₂²×C₃₁₇	C₄⋊C₄	1	1	yes	characteristic
29	C₄×C₃₁₇	—	2	4	no	—
30	C₄×C₃₁₇	C₄²	1	1	yes	characteristic
31	C₄×C₃₁₇	C₂²⋊C₄	1	1	yes	characteristic
32	C₄×C₃₁₇	C₄⋊C₄	1	1	yes	characteristic
33	C₄×C₃₁₇	C₄⋊C₄	1	1	yes	characteristic
34	Dic₃₁₇	—	4	2	no	—
35	C₂×C₃₁₇	—	2	1	no	—
36	C₂×C₃₁₇	—	2	1	no	—
37	C₂×C₃₁₇	#95	1	1	yes	characteristic
38	C₂×C₃₁₇	#104	1	2	yes	—
39	C₃₁₇	#339	1	1	yes	characteristic, complemented, Sylow
40	#339	—	317	1	no	maximal, Sylow
41	C₂×M₁₆	—	317	1	no	—
42	#117	—	317	1	no	—
43	C₂×C₄²	D₃₁₇	1	1	yes	characteristic
44	C₂²×C₄	—	2	1	no	—
45	C₂²×C₄	C₂×D₃₁₇	1	1	yes	characteristic
46	M₁₆	—	317	2	no	—
47	M₁₆	—	634	1	no	—
48	C₂×C₈	—	634	1	no	—
49	C₄⋊C₄	—	317	2	no	—
50	C₂²⋊C₄	—	634	1	no	—
51	C₄²	—	2	1	no	—
52	C₄²	—	634	1	no	—
53	C₄²	Dic₃₁₇	1	2	yes	complemented
54	C₂³	C₂²⋊D₃₁₇	1	1	yes	characteristic
55	C₂×C₄	—	2	4	no	—
56	C₂×C₄	—	2	2	no	—
57	C₂×C₄	—	634	2	no	—
58	C₂×C₄	C₄×D₃₁₇	1	1	yes	characteristic
59	C₂×C₄	C₄×D₃₁₇	1	1	yes	characteristic
60	C₂×C₄	C₂×Dic₃₁₇	1	1	yes	characteristic
61	C₂×C₄	C₂²⋊D₃₁₇	1	1	yes	characteristic
62	C₂×C₄	D₁₂₆₈	1	1	yes	characteristic
63	C₂×C₄	Dic₆₃₄	1	1	yes	characteristic
64	C₈	—	634	2	no	—
65	C₂²	—	2	2	no	—
66	C₂²	#19099551	1	1	yes	characteristic
67	C₂²	C₂²⋊Dic₃₁₇	1	1	yes	characteristic
68	C₂²	#19099553	1	1	yes	characteristic
69	C₄	—	2	4	no	—
70	C₄	—	1268	2	no	—
71	C₄	C₄×Dic₃₁₇	1	1	yes	characteristic
72	C₄	#19099551	1	1	yes	characteristic
73	C₄	#19099553	1	1	yes	characteristic
74	C₄	C₄⋊Dic₃₁₇	1	1	yes	characteristic
75	C₂	—	2	1	no	—
76	C₂	—	2	1	no	—
77	C₂	#23778687	1	1	yes	characteristic
78	C₂	#23778708	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	D₁₂₆₈
commutator	C₄×C₃₁₇	C₄²
Frattini	C₂²×C₄	C₂×D₃₁₇
Fitting	C₂×C₄²×C₃₁₇	C₂
socle	C₂²×C₃₁₇	C₂²⋊C₄
prime radical	C₂³×C₃₁₇	D₄
cosocle	C₂²×C₄×C₃₁₇	C₂²
hypercenter	C₂×C₄²	D₃₁₇
hypocenter	C₃₁₇	#339
LCS	G ▷ C₄×C₃₁₇ ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
5. #19099558	C₄	semidirect
7. C₄⋊Dic₃₁₇	C₄	semidirect
13. C₄²×C₃₁₇	C₄	semidirect
39. C₃₁₇	#339	semidirect
53. C₄²	Dic₃₁₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19099558	2
#23778739	2
C₂×C₄²×C₃₁₇	2
#339	317