Subgroups — #35203264

Order 20288.

118 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19099557	C₂	1	1	yes	maximal, characteristic
2	#23778738	C₂	1	1	yes	maximal, characteristic
3	C₄⋊C₄×D₃₁₇	C₂	1	1	yes	maximal, characteristic
4	#23778713	C₂	1	1	yes	maximal, characteristic, complemented
5	#23778712	C₂	1	1	yes	maximal, characteristic, complemented
6	#23778711	C₂	1	1	yes	maximal, characteristic, complemented
7	#107×C₃₁₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₄×D₃₁₇	—	2	1	no	—
9	C₂×C₄×D₃₁₇	C₂²	1	1	yes	characteristic
10	#19099557	C₄	1	4	yes	complemented
11	C₂×#14579261	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₃₁₇	C₂²	1	1	yes	characteristic
13	C₄⋊Dic₃₁₇	C₂²	1	1	yes	characteristic
14	#19099553	—	2	1	no	—
15	#19099551	—	2	1	no	—
16	C₄×Dic₃₁₇	—	2	1	no	—
17	C₂×C₈×C₃₁₇	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₃₁₇	C₂²	1	1	yes	characteristic
19	C₄⋊C₄×C₃₁₇	C₂²	1	1	yes	characteristic
20	C₂²×D₃₁₇	D₄	1	1	yes	characteristic
21	C₂×Dic₃₁₇	—	2	1	no	—
22	C₂×Dic₃₁₇	—	2	1	no	—
23	C₂×Dic₃₁₇	D₄	1	1	yes	characteristic
24	C₄×D₃₁₇	—	4	2	no	—
25	C₄×D₃₁₇	C₂×C₄	1	2	yes	—
26	C₄×D₃₁₇	Q₈	1	2	yes	—
27	#14579261	C₂×C₄	1	2	yes	complemented
28	C₂×C₄×C₃₁₇	—	2	1	no	—
29	C₂×C₄×C₃₁₇	—	2	1	no	—
30	C₂×C₄×C₃₁₇	C₂³	1	1	yes	characteristic
31	C₈×C₃₁₇	C₂×C₄	1	2	yes	complemented
32	C₂²×C₃₁₇	C₂×D₄	1	1	yes	characteristic
33	C₂×D₃₁₇	—	2	1	no	—
34	C₂×D₃₁₇	—	2	1	no	—
35	C₂×D₃₁₇	C₄⋊C₄	1	2	yes	—
36	C₄×C₃₁₇	—	4	1	no	—
37	C₄×C₃₁₇	—	4	1	no	—
38	C₄×C₃₁₇	C₂²×C₄	1	1	yes	characteristic
39	C₄×C₃₁₇	C₂×Q₈	1	1	yes	characteristic
40	Dic₃₁₇	—	4	1	no	—
41	Dic₃₁₇	—	4	1	no	—
42	Dic₃₁₇	C₄⋊C₄	1	2	yes	—
43	C₂×C₃₁₇	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₃₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
45	C₂×C₃₁₇	#137	1	1	yes	characteristic
46	D₃₁₇	—	2	2	no	—
47	C₃₁₇	#428	1	1	yes	characteristic, complemented, Sylow
48	#428	—	317	1	no	maximal, Sylow
49	C₂×M₁₆	—	317	1	no	—
50	#117	—	317	1	no	—
51	C₂×C₄⋊C₄	—	317	1	no	—
52	#107	—	317	1	no	—
53	#107	D₃₁₇	1	1	yes	characteristic, complemented
54	#106	—	317	1	no	—
55	#106	—	317	1	no	—
56	C₂²×C₄	—	317	1	no	—
57	C₂²×C₄	—	634	1	no	—
58	M₁₆	—	317	4	no	—
59	C₂×C₈	—	317	1	no	—
60	C₂×C₈	C₂×D₃₁₇	1	1	yes	characteristic
61	C₄⋊C₄	—	317	1	no	—
62	C₄⋊C₄	—	317	1	no	—
63	C₄⋊C₄	—	634	1	no	—
64	C₄⋊C₄	C₂×D₃₁₇	1	1	yes	characteristic
65	C₄⋊C₄	C₂×D₃₁₇	1	1	yes	characteristic
66	C₂²⋊C₄	—	634	1	no	—
67	C₄²	—	634	1	no	—
68	C₂³	—	317	1	no	—
69	C₂×C₄	—	2	1	no	—
70	C₂×C₄	—	2	1	no	—
71	C₂×C₄	—	317	1	no	—
72	C₂×C₄	—	317	2	no	—
73	C₂×C₄	—	317	2	no	—
74	C₂×C₄	—	634	1	no	—
75	C₂×C₄	—	634	1	no	—
76	C₂×C₄	—	1268	2	no	—
77	C₂×C₄	C₂²×D₃₁₇	1	1	yes	characteristic
78	C₈	—	317	2	no	—
79	C₈	C₄×D₃₁₇	1	2	yes	complemented
80	C₂²	—	317	2	no	—
81	C₂²	—	634	1	no	—
82	C₂²	—	634	1	no	—
83	C₂²	D₄×D₃₁₇	1	1	yes	characteristic
84	C₄	—	4	1	no	—
85	C₄	—	4	1	no	—
86	C₄	—	317	2	no	—
87	C₄	—	1268	1	no	—
88	C₄	—	1268	1	no	—
89	C₄	C₂×C₄×D₃₁₇	1	1	yes	characteristic
90	C₄	Q₈×D₃₁₇	1	1	yes	characteristic
91	C₂	—	634	2	no	—
92	C₂	C₄⋊C₄×D₃₁₇	1	1	yes	characteristic
93	C₂	#23778804	1	1	yes	characteristic
94	C₂	#23778812	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	D₄×D₃₁₇
commutator	C₄×C₃₁₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃₁₇
Fitting	#107×C₃₁₇	C₂
socle	C₂²×C₃₁₇	C₂×D₄
prime radical	C₂²×D₃₁₇	D₄
cosocle	C₂×C₄×C₃₁₇	C₂³
hypercenter	#107	D₃₁₇
hypocenter	C₃₁₇	#428
LCS	G ▷ C₄×C₃₁₇ ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ #107

Complements

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

N	H ≅ G/N	type
4. #23778713	C₂	semidirect
5. #23778712	C₂	semidirect
6. #23778711	C₂	semidirect
7. #107×C₃₁₇	C₂	semidirect
10. #19099557	C₄	semidirect
27. #14579261	C₂×C₄	semidirect
31. C₈×C₃₁₇	C₂×C₄	semidirect
47. C₃₁₇	#428	semidirect
53. #107	D₃₁₇	semidirect
79. C₈	C₄×D₃₁₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19099557	2
#23778738	2
C₄⋊C₄×D₃₁₇	2
#23778713	2
#23778712	2
#23778711	2
#107×C₃₁₇	2
#428	317