Subgroups — #35203524

Order 20288.

102 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23778772	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×D₁₂₆₈	C₂	1	1	yes	maximal, characteristic
3	#23778714	C₂	1	1	yes	maximal, characteristic, complemented
4	#23778712	C₂	1	1	yes	maximal, characteristic, complemented
5	#23778702	C₂	1	1	yes	maximal, characteristic
6	#23778702	C₂	1	1	yes	maximal, characteristic
7	C₄⋊C₈×C₃₁₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×D₁₂₆₈	C₂²	1	1	yes	characteristic
9	C₂×C₄×D₃₁₇	—	2	1	no	—
10	C₄⋊Dic₃₁₇	C₂²	1	1	yes	characteristic
11	C₄⋊Dic₃₁₇	C₂²	1	1	yes	characteristic
12	C₄⋊Dic₃₁₇	C₂²	1	1	yes	characteristic
13	#19099553	—	2	1	no	—
14	#19099553	—	2	1	no	—
15	#19099551	—	2	1	no	—
16	C₂×C₈×C₃₁₇	C₂²	1	1	yes	characteristic
17	C₂×C₈×C₃₁₇	C₂²	1	1	yes	characteristic
18	C₄²×C₃₁₇	C₂²	1	1	yes	characteristic
19	C₂²×D₃₁₇	—	2	1	no	—
20	D₁₂₆₈	—	2	1	no	—
21	D₁₂₆₈	Q₈	1	2	yes	—
22	C₂×Dic₃₁₇	—	2	1	no	—
23	C₂×Dic₃₁₇	—	2	1	no	—
24	C₂×Dic₃₁₇	—	2	1	no	—
25	C₄×D₃₁₇	—	4	2	no	—
26	C₂×C₄×C₃₁₇	D₄	1	1	yes	characteristic
27	C₂×C₄×C₃₁₇	D₄	1	1	yes	characteristic
28	C₂×C₄×C₃₁₇	C₂³	1	1	yes	characteristic
29	C₈×C₃₁₇	—	2	1	no	—
30	C₈×C₃₁₇	—	2	1	no	—
31	C₂²×C₃₁₇	C₂×D₄	1	1	yes	characteristic
32	C₂×D₃₁₇	—	2	2	no	—
33	C₂×D₃₁₇	—	4	1	no	—
34	C₂×D₃₁₇	—	4	1	no	—
35	C₄×C₃₁₇	—	2	1	no	—
36	C₄×C₃₁₇	—	2	1	no	—
37	C₄×C₃₁₇	C₂×Q₈	1	1	yes	characteristic
38	C₄×C₃₁₇	#41	1	1	yes	characteristic
39	Dic₃₁₇	—	4	1	no	—
40	Dic₃₁₇	—	4	1	no	—
41	Dic₃₁₇	—	4	1	no	—
42	C₂×C₃₁₇	C₂²⋊Q₈	1	1	yes	characteristic
43	C₂×C₃₁₇	#135	1	1	yes	characteristic
44	C₂×C₃₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
45	D₃₁₇	—	4	2	no	—
46	C₃₁₇	#478	1	1	yes	characteristic, complemented, Sylow
47	#478	—	317	1	no	maximal, Sylow
48	#125	—	317	1	no	—
49	C₄×D₄	—	317	1	no	—
50	#107	—	317	1	no	—
51	#106	—	317	1	no	—
52	C₄⋊C₈	D₃₁₇	1	1	yes	characteristic, complemented
53	#102	—	317	1	no	—
54	#102	—	317	1	no	—
55	C₂×D₄	—	317	1	no	—
56	C₂²×C₄	—	634	1	no	—
57	C₂×C₈	C₂×D₃₁₇	1	1	yes	characteristic
58	C₂×C₈	C₂×D₃₁₇	1	1	yes	characteristic
59	C₄⋊C₄	—	317	1	no	—
60	C₄⋊C₄	—	317	1	no	—
61	C₄⋊C₄	—	317	1	no	—
62	C₄⋊C₄	—	634	1	no	—
63	C₄⋊C₄	—	634	1	no	—
64	C₂²⋊C₄	—	634	1	no	—
65	C₄²	C₂×D₃₁₇	1	1	yes	characteristic
66	C₂³	—	634	1	no	—
67	D₄	—	317	2	no	—
68	D₄	—	634	1	no	—
69	C₂×C₄	—	634	1	no	—
70	C₂×C₄	—	634	1	no	—
71	C₂×C₄	—	634	1	no	—
72	C₂×C₄	—	1268	2	no	—
73	C₂×C₄	D₁₂₆₈	1	1	yes	characteristic
74	C₂×C₄	D₁₂₆₈	1	1	yes	characteristic
75	C₂×C₄	C₂²×D₃₁₇	1	1	yes	characteristic
76	C₈	—	2	1	no	—
77	C₈	—	2	1	no	—
78	C₂²	—	634	2	no	—
79	C₂²	—	1268	1	no	—
80	C₂²	—	1268	1	no	—
81	C₂²	C₂×D₁₂₆₈	1	1	yes	characteristic
82	C₄	—	2	1	no	—
83	C₄	—	2	1	no	—
84	C₄	—	1268	1	no	—
85	C₄	—	1268	1	no	—
86	C₄	—	1268	1	no	—
87	C₄	Q₈×D₃₁₇	1	1	yes	characteristic
88	C₄	#19099573	1	1	yes	characteristic
89	C₂	—	1268	2	no	—
90	C₂	#23778760	1	1	yes	characteristic
91	C₂	#23778802	1	1	yes	characteristic
92	C₂	#23778808	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×D₁₂₆₈
commutator, cosocle	C₂×C₄×C₃₁₇	C₂³
Frattini	C₂×C₄	C₂²×D₃₁₇
Fitting	C₄⋊C₈×C₃₁₇	C₂
socle	C₂²×C₃₁₇	C₂×D₄
prime radical	C₂×D₁₂₆₈	C₂²
hypercenter	C₄⋊C₈	D₃₁₇
hypocenter	C₃₁₇	#478
LCS	G ▷ C₂×C₄×C₃₁₇ ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊C₈

Complements

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

N	H ≅ G/N	type
1. #23778772	C₂	semidirect
3. #23778714	C₂	semidirect
4. #23778712	C₂	semidirect
7. C₄⋊C₈×C₃₁₇	C₂	semidirect
46. C₃₁₇	#478	semidirect
52. C₄⋊C₈	D₃₁₇	semidirect

Maximal subgroups

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

subgroup	index
#23778772	2
C₄×D₁₂₆₈	2
#23778714	2
#23778712	2
#23778702	2
C₄⋊C₈×C₃₁₇	2
#478	317