Subgroups — D₃₁₇⋊#108 #35204236

Order 20288.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	D₃₁₇⋊M₁₆	C₂	1	2	yes	maximal
2	#23778851	C₂	1	2	yes	maximal, complemented
3	#23778850	C₂	1	2	yes	maximal, complemented
4	C₂×C₈×D₃₁₇	C₂	1	1	yes	maximal, characteristic
5	#19099584	C₂²	1	4	yes	—
6	#19099583	—	2	2	no	—
7	D₃₁₇⋊C₈	—	2	2	no	—
8	C₂×C₄×D₃₁₇	C₂²	1	1	yes	characteristic
9	C₂×#14579261	C₄	1	1	yes	characteristic
10	C₈×D₃₁₇	C₄	1	1	yes	characteristic
11	C₈×D₃₁₇	C₄	1	1	yes	characteristic
12	C₈×D₃₁₇	C₂²	1	1	yes	characteristic
13	C₈×D₃₁₇	C₂²	1	1	yes	characteristic
14	C₂×C₈×C₃₁₇	C₄	1	1	yes	characteristic
15	#14579268	—	2	4	no	—
16	C₂²×D₃₁₇	Q₈	1	1	yes	characteristic
17	C₂×Dic₃₁₇	D₄	1	1	yes	characteristic
18	C₄×D₃₁₇	C₂×C₄	1	1	yes	characteristic
19	C₄×D₃₁₇	D₄	1	1	yes	characteristic
20	C₄×D₃₁₇	Q₈	1	1	yes	characteristic
21	C₄×D₃₁₇	C₂³	1	1	yes	characteristic
22	#14579261	C₂×C₄	1	1	yes	characteristic
23	#14579261	C₂×C₄	1	1	yes	characteristic
24	C₂×C₄×C₃₁₇	C₂×C₄	1	1	yes	characteristic
25	C₈×C₃₁₇	C₂×C₄	1	1	yes	characteristic
26	C₈×C₃₁₇	C₂×C₄	1	1	yes	characteristic
27	C₂²×C₃₁₇	C₄⋊C₄	1	1	yes	characteristic
28	C₂×D₃₁₇	—	2	2	no	—
29	C₂×D₃₁₇	C₄⋊C₄	1	1	yes	characteristic
30	C₂×D₃₁₇	C₂×Q₈	1	1	yes	characteristic
31	C₄×C₃₁₇	C₄⋊C₄	1	1	yes	characteristic
32	C₄×C₃₁₇	C₂²×C₄	1	1	yes	characteristic
33	Dic₃₁₇	C₄⋊C₄	1	1	yes	characteristic
34	Dic₃₁₇	C₂×D₄	1	1	yes	characteristic
35	C₂×C₃₁₇	—	2	1	no	—
36	C₂×C₃₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
37	D₃₁₇	—	2	1	no	—
38	D₃₁₇	#108	1	2	yes	complemented
39	C₃₁₇	C₂×#108	1	1	yes	characteristic, complemented, Sylow
40	C₂×#108	—	317	1	no	maximal, Sylow
41	C₂×M₁₆	—	317	2	no	—
42	C₂²×C₈	—	317	1	no	—
43	#108	—	317	2	no	—
44	#108	—	317	2	no	—
45	C₂²×C₄	—	317	1	no	—
46	M₁₆	—	317	4	no	—
47	M₁₆	—	634	2	no	—
48	C₂×C₈	—	317	1	no	—
49	C₂×C₈	—	317	1	no	—
50	C₂×C₈	—	317	1	no	—
51	C₂×C₈	—	317	1	no	—
52	C₂×C₈	—	317	1	no	—
53	C₂×C₈	—	634	2	no	—
54	C₂×C₈	#11929639	1	1	yes	characteristic
55	C₂³	—	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	—	317	1	no	—
61	C₂×C₄	C₂×#11929639	1	1	yes	characteristic
62	C₈	—	317	1	no	—
63	C₈	—	317	1	no	—
64	C₈	—	634	4	no	—
65	C₈	C₂×#11929639	1	1	yes	characteristic
66	C₈	C₂×#11929639	1	1	yes	characteristic
67	C₂²	—	317	1	no	—
68	C₂²	—	317	1	no	—
69	C₂²	—	634	2	no	—
70	C₂²	C₄⋊#11929639	1	1	yes	characteristic
71	C₄	—	317	1	no	—
72	C₄	—	317	1	no	—
73	C₄	C₄⋊#11929639	1	1	yes	characteristic
74	C₄	C₂²×#11929639	1	1	yes	characteristic
75	C₂	—	2	1	no	—
76	C₂	—	317	2	no	—
77	C₂	—	634	1	no	—
78	C₂	C₂×C₄⋊#11929639	1	1	yes	characteristic

	subgroup	G/H
center	C₄	C₄⋊#11929639
commutator	C₄×C₃₁₇	C₂²×C₄
Frattini	C₄	C₂²×#11929639
Fitting	C₂×C₈×C₃₁₇	C₄
socle	C₂×C₃₁₇	C₂×C₄⋊C₄
prime radical	C₂²×D₃₁₇	Q₈
cosocle	C₄×D₃₁₇	C₂³
hypercenter	C₂×C₈	#11929639
hypocenter	C₃₁₇	C₂×#108
LCS	G ▷ C₄×C₃₁₇ ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈

Complements

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

N	H ≅ G/N	type
2. #23778851	C₂	semidirect
3. #23778850	C₂	semidirect
38. D₃₁₇	#108	semidirect
39. C₃₁₇	C₂×#108	semidirect

Maximal subgroups

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

subgroup	index
D₃₁₇⋊M₁₆	2
#23778851	2
#23778850	2
C₂×C₈×D₃₁₇	2
C₂×#108	317

Subgroups — D317⋊#108 #35204236

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₃₁₇⋊#108 #35204236