Subgroups — #108×D₃₁₇ #35203267

Order 20288.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	M₁₆×D₃₁₇	C₂	1	2	yes	maximal
2	C₂×C₈×D₃₁₇	C₂	1	1	yes	maximal, characteristic
3	#23778716	C₂	1	1	yes	maximal, characteristic, complemented
4	#23778715	C₂	1	2	yes	maximal, complemented
5	#108×C₃₁₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×D₃₁₇	C₂²	1	1	yes	characteristic
7	#19099558	C₂²	1	2	yes	—
8	#19099557	—	2	2	no	—
9	C₂×#14579261	C₂²	1	1	yes	characteristic
10	C₈×D₃₁₇	—	2	2	no	—
11	C₈×D₃₁₇	C₄	1	2	yes	—
12	C₈×D₃₁₇	C₄	1	2	yes	—
13	M₁₆×C₃₁₇	C₂²	1	2	yes	—
14	C₂×C₈×C₃₁₇	C₂²	1	1	yes	characteristic
15	C₂²×D₃₁₇	Q₈	1	1	yes	characteristic
16	C₂×Dic₃₁₇	Q₈	1	1	yes	characteristic
17	C₄×D₃₁₇	C₂×C₄	1	1	yes	characteristic
18	C₄×D₃₁₇	C₂×C₄	1	1	yes	characteristic
19	C₄×D₃₁₇	D₄	1	1	yes	characteristic
20	C₄×D₃₁₇	D₄	1	1	yes	characteristic
21	#14579261	—	2	2	no	—
22	#14579261	C₂×C₄	1	2	yes	—
23	C₂×C₄×C₃₁₇	C₂³	1	1	yes	characteristic
24	C₈×C₃₁₇	—	2	2	no	—
25	C₈×C₃₁₇	C₂×C₄	1	2	yes	—
26	C₂²×C₃₁₇	C₂×Q₈	1	1	yes	characteristic
27	C₂×D₃₁₇	—	2	2	no	—
28	C₂×D₃₁₇	C₄⋊C₄	1	1	yes	characteristic
29	C₂×D₃₁₇	C₄⋊C₄	1	1	yes	characteristic
30	C₄×C₃₁₇	C₂²×C₄	1	1	yes	characteristic
31	C₄×C₃₁₇	C₂×D₄	1	1	yes	characteristic
32	Dic₃₁₇	C₄⋊C₄	1	1	yes	characteristic
33	Dic₃₁₇	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₃₁₇	—	2	1	no	—
35	C₂×C₃₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
36	D₃₁₇	—	2	1	no	—
37	D₃₁₇	#108	1	2	yes	direct product factor
38	C₃₁₇	C₂×#108	1	1	yes	characteristic, complemented, Sylow
39	C₂×#108	—	317	1	no	maximal, Sylow
40	C₂×M₁₆	—	317	2	no	—
41	C₂²×C₈	—	317	1	no	—
42	#108	—	317	2	no	—
43	#108	—	317	1	no	—
44	#108	D₃₁₇	1	1	yes	characteristic, direct product factor
45	C₂²×C₄	—	317	1	no	—
46	M₁₆	—	317	2	no	—
47	M₁₆	—	634	2	no	—
48	M₁₆	C₂×D₃₁₇	1	2	yes	—
49	C₂×C₈	—	317	1	no	—
50	C₂×C₈	—	317	2	no	—
51	C₂×C₈	—	317	2	no	—
52	C₂×C₈	—	634	2	no	—
53	C₂×C₈	C₂×D₃₁₇	1	1	yes	characteristic
54	C₂³	—	317	1	no	—
55	C₂×C₄	—	317	1	no	—
56	C₂×C₄	—	317	1	no	—
57	C₂×C₄	—	317	1	no	—
58	C₂×C₄	—	317	1	no	—
59	C₂×C₄	—	317	1	no	—
60	C₂×C₄	C₂²×D₃₁₇	1	1	yes	characteristic
61	C₈	—	2	2	no	—
62	C₈	—	317	2	no	—
63	C₈	—	634	2	no	—
64	C₈	C₄×D₃₁₇	1	2	yes	—
65	C₂²	—	317	1	no	—
66	C₂²	—	317	1	no	—
67	C₂²	—	634	2	no	—
68	C₂²	Q₈×D₃₁₇	1	1	yes	characteristic
69	C₄	—	317	1	no	—
70	C₄	—	317	1	no	—
71	C₄	C₂×C₄×D₃₁₇	1	1	yes	characteristic
72	C₄	D₄×D₃₁₇	1	1	yes	characteristic
73	C₂	—	2	1	no	—
74	C₂	—	317	2	no	—
75	C₂	—	634	1	no	—
76	C₂	C₄⋊C₄×D₃₁₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄	D₄×D₃₁₇
commutator	C₄×C₃₁₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃₁₇
Fitting	#108×C₃₁₇	C₂
socle	C₂×C₃₁₇	C₂×C₄⋊C₄
prime radical	C₂²×D₃₁₇	Q₈
cosocle	C₂×C₄×C₃₁₇	C₂³
hypercenter	#108	D₃₁₇
hypocenter	C₃₁₇	C₂×#108
LCS	G ▷ C₄×C₃₁₇ ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ #108

Complements

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

N	H ≅ G/N	type
3. #23778716	C₂	semidirect
4. #23778715	C₂	semidirect
5. #108×C₃₁₇	C₂	semidirect
37. D₃₁₇	#108	direct (N × H)
38. C₃₁₇	C₂×#108	semidirect
44. #108	D₃₁₇	direct (N × H)

Maximal subgroups

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

subgroup	index
M₁₆×D₃₁₇	2
C₂×C₈×D₃₁₇	2
#23778716	2
#23778715	2
#108×C₃₁₇	2
C₂×#108	317

Subgroups — #108×D317 #35203267

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #108×D₃₁₇ #35203267