Subgroups — #35203345

Order 20288.

104 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23778787	C₂	1	2	yes	maximal, complemented
2	#23778716	C₂	1	1	yes	maximal, characteristic, complemented
3	#23778707	C₂	1	2	yes	maximal, complemented
4	#23778690	C₂	1	1	yes	maximal, characteristic, complemented
5	#135×C₃₁₇	C₂	1	1	yes	maximal, characteristic
6	#19099558	—	2	2	no	—
7	#19099558	C₂²	1	2	yes	—
8	C₂×#14579261	—	2	2	no	—
9	C₂×#14579261	C₂²	1	1	yes	characteristic
10	C₄×Dic₃₁₇	C₂²	1	1	yes	characteristic
11	#41×C₃₁₇	C₂²	1	2	yes	—
12	Q₁₆×C₃₁₇	C₄	1	1	yes	characteristic, complemented
13	SD₁₆×C₃₁₇	C₄	1	2	yes	complemented
14	D₈×C₃₁₇	C₄	1	1	yes	characteristic, complemented
15	C₂×C₈×C₃₁₇	C₂²	1	1	yes	characteristic
16	C₂×Dic₃₁₇	—	2	1	no	—
17	#14579261	—	2	2	no	—
18	#14579261	D₄	1	2	yes	—
19	Q₈×C₃₁₇	C₂×C₄	1	2	yes	—
20	D₄×C₃₁₇	—	2	2	no	—
21	D₄×C₃₁₇	C₂×C₄	1	2	yes	—
22	C₂×C₄×C₃₁₇	—	2	2	no	—
23	C₂×C₄×C₃₁₇	C₂³	1	1	yes	characteristic
24	C₈×C₃₁₇	C₂×C₄	1	1	yes	characteristic
25	C₈×C₃₁₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₃₁₇	—	2	2	no	—
27	C₂²×C₃₁₇	#41	1	1	yes	characteristic
28	C₄×C₃₁₇	—	2	2	no	—
29	C₄×C₃₁₇	C₂²×C₄	1	1	yes	characteristic
30	C₄×C₃₁₇	C₂×D₄	1	1	yes	characteristic
31	Dic₃₁₇	—	4	1	no	—
32	C₂×C₃₁₇	—	2	1	no	—
33	C₂×C₃₁₇	—	4	2	no	—
34	C₂×C₃₁₇	C₄×D₄	1	1	yes	characteristic
35	C₃₁₇	#444	1	1	yes	characteristic, complemented, Sylow
36	#444	—	317	1	no	maximal, Sylow
37	#135	D₃₁₇	1	1	yes	characteristic
38	#131	—	317	2	no	—
39	#108	—	317	1	no	—
40	#104	—	317	2	no	—
41	#97	—	317	1	no	—
42	#41	C₂×D₃₁₇	1	2	yes	—
43	Q₁₆	Dic₃₁₇	1	1	yes	characteristic, complemented
44	SD₁₆	Dic₃₁₇	1	2	yes	complemented
45	D₈	Dic₃₁₇	1	1	yes	characteristic, complemented
46	M₁₆	—	317	2	no	—
47	M₁₆	—	634	2	no	—
48	C₂×C₈	—	317	1	no	—
49	C₂×C₈	—	634	2	no	—
50	C₂×C₈	C₂×D₃₁₇	1	1	yes	characteristic
51	C₄²	—	317	1	no	—
52	Q₈	C₂×Dic₃₁₇	1	2	yes	—
53	D₄	—	2	2	no	—
54	D₄	C₂×Dic₃₁₇	1	2	yes	—
55	C₂×C₄	—	2	2	no	—
56	C₂×C₄	—	634	1	no	—
57	C₂×C₄	C₂²×D₃₁₇	1	1	yes	characteristic
58	C₈	—	317	2	no	—
59	C₈	—	634	2	no	—
60	C₈	C₂×Dic₃₁₇	1	1	yes	characteristic
61	C₈	C₂×Dic₃₁₇	1	1	yes	characteristic
62	C₂²	—	2	2	no	—
63	C₂²	#19099573	1	1	yes	characteristic
64	C₄	—	2	2	no	—
65	C₄	—	1268	1	no	—
66	C₄	C₂²×Dic₃₁₇	1	1	yes	characteristic
67	C₄	D₄×D₃₁₇	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	4	2	no	—
70	C₂	D₄×Dic₃₁₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×D₃₁₇
commutator	C₄×C₃₁₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃₁₇
Fitting, prime radical	#135×C₃₁₇	C₂
socle	C₂×C₃₁₇	C₄×D₄
cosocle	C₂×C₄×C₃₁₇	C₂³
hypercenter	#135	D₃₁₇
hypocenter	C₃₁₇	#444
LCS	G ▷ C₄×C₃₁₇ ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #135

Complements

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

N	H ≅ G/N	type
1. #23778787	C₂	semidirect
2. #23778716	C₂	semidirect
3. #23778707	C₂	semidirect
4. #23778690	C₂	semidirect
12. Q₁₆×C₃₁₇	C₄	semidirect
13. SD₁₆×C₃₁₇	C₄	semidirect
14. D₈×C₃₁₇	C₄	semidirect
35. C₃₁₇	#444	semidirect
43. Q₁₆	Dic₃₁₇	semidirect
44. SD₁₆	Dic₃₁₇	semidirect
45. D₈	Dic₃₁₇	semidirect

Maximal subgroups

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

subgroup	index
#23778787	2
#23778716	2
#23778707	2
#23778690	2
#135×C₃₁₇	2
#444	317