Subgroups — #35203178

Order 20288.

130 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	#23778690	C₂	1	2	yes	maximal, complemented
4	C₈×Dic₃₁₇	C₂	1	2	yes	maximal, complemented
5	C₂²×C₈×C₃₁₇	C₂	1	1	yes	maximal, characteristic
6	#19099558	C₄	1	4	yes	complemented
7	C₂×#14579261	C₂²	1	2	yes	—
8	C₄⋊Dic₃₁₇	C₄	1	2	yes	—
9	C₂²⋊Dic₃₁₇	C₄	1	2	yes	—
10	C₄×Dic₃₁₇	C₂²	1	2	yes	—
11	C₂²×C₄×C₃₁₇	C₂²	1	1	yes	characteristic
12	C₂×C₈×C₃₁₇	C₄	1	4	yes	complemented
13	C₂×C₈×C₃₁₇	C₂²	1	1	yes	characteristic
14	C₂×C₈×C₃₁₇	C₂²	1	1	yes	characteristic
15	C₂×Dic₃₁₇	C₂×C₄	1	4	yes	—
16	#14579261	C₂×C₄	1	4	yes	—
17	C₂³×C₃₁₇	C₂×C₄	1	1	yes	characteristic
18	C₂×C₄×C₃₁₇	C₂×C₄	1	1	yes	characteristic
19	C₂×C₄×C₃₁₇	C₂×C₄	1	2	yes	—
20	C₂×C₄×C₃₁₇	C₂×C₄	1	2	yes	—
21	C₂×C₄×C₃₁₇	C₂³	1	1	yes	characteristic
22	C₈×C₃₁₇	C₂×C₄	1	2	yes	—
23	C₈×C₃₁₇	C₂×C₄	1	2	yes	—
24	C₂²×C₃₁₇	—	2	2	no	—
25	C₂²×C₃₁₇	C₄²	1	2	yes	—
26	C₂²×C₃₁₇	C₂²×C₄	1	1	yes	characteristic
27	C₄×C₃₁₇	C₄²	1	2	yes	—
28	C₄×C₃₁₇	C₂²×C₄	1	1	yes	characteristic
29	C₄×C₃₁₇	C₂²×C₄	1	1	yes	characteristic
30	Dic₃₁₇	—	2	4	no	—
31	C₂×C₃₁₇	—	2	2	no	—
32	C₂×C₃₁₇	C₂×C₄²	1	1	yes	characteristic
33	C₂×C₃₁₇	#131	1	2	yes	—
34	C₃₁₇	#405	1	1	yes	characteristic, complemented, Sylow
35	#405	—	317	1	no	maximal, Sylow
36	C₂×M₁₆	—	317	1	no	—
37	C₂²×C₈	D₃₁₇	1	1	yes	characteristic
38	#117	—	317	1	no	—
39	#97	—	317	2	no	—
40	C₄×C₈	—	317	2	no	—
41	C₂²×C₄	C₂×D₃₁₇	1	1	yes	characteristic
42	M₁₆	—	317	4	no	—
43	C₂×C₈	—	317	2	no	—
44	C₂×C₈	Dic₃₁₇	1	4	yes	complemented
45	C₂×C₈	C₂×D₃₁₇	1	1	yes	characteristic
46	C₂×C₈	C₂×D₃₁₇	1	1	yes	characteristic
47	C₄⋊C₄	—	317	2	no	—
48	C₂²⋊C₄	—	317	2	no	—
49	C₄²	—	317	2	no	—
50	C₂³	C₄×D₃₁₇	1	1	yes	characteristic
51	C₂×C₄	—	317	4	no	—
52	C₂×C₄	C₄×D₃₁₇	1	1	yes	characteristic
53	C₂×C₄	C₄×D₃₁₇	1	2	yes	—
54	C₂×C₄	C₂×Dic₃₁₇	1	2	yes	—
55	C₂×C₄	C₂²×D₃₁₇	1	1	yes	characteristic
56	C₈	—	317	4	no	—
57	C₈	C₂×Dic₃₁₇	1	2	yes	—
58	C₈	C₂×Dic₃₁₇	1	2	yes	—
59	C₂²	—	2	2	no	—
60	C₂²	C₄×Dic₃₁₇	1	2	yes	—
61	C₂²	C₂×C₄×D₃₁₇	1	1	yes	characteristic
62	C₄	—	634	4	no	—
63	C₄	C₄×Dic₃₁₇	1	2	yes	—
64	C₄	C₂×C₄×D₃₁₇	1	1	yes	characteristic
65	C₄	C₂²×Dic₃₁₇	1	1	yes	characteristic
66	C₂	—	2	2	no	—
67	C₂	C₂×C₄×Dic₃₁₇	1	1	yes	characteristic
68	C₂	#23778786	1	2	yes	—

	subgroup	G/H
center	C₂×C₈	C₂×D₃₁₇
commutator	C₂×C₃₁₇	C₂×C₄²
Frattini	C₂×C₄	C₂²×D₃₁₇
Fitting	C₂²×C₈×C₃₁₇	C₂
socle	C₂²×C₃₁₇	C₂²×C₄
prime radical	C₂³×C₃₁₇	C₂×C₄
cosocle	C₂×C₄×C₃₁₇	C₂³
hypercenter	C₂²×C₈	D₃₁₇
hypocenter	C₃₁₇	#405
LCS	G ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈

Complements

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

N	H ≅ G/N	type
3. #23778690	C₂	semidirect
4. C₈×Dic₃₁₇	C₂	semidirect
6. #19099558	C₄	semidirect
12. C₂×C₈×C₃₁₇	C₄	semidirect
34. C₃₁₇	#405	semidirect
44. C₂×C₈	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
#23778690	2
C₈×Dic₃₁₇	2
C₂²×C₈×C₃₁₇	2
#405	317