Subgroups — #35203419

Order 20288.

140 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₂₅₃₆	C₂	1	2	yes	maximal, complemented
2	#23778777	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄×D₁₂₆₈	C₂	1	1	yes	maximal, characteristic, complemented
4	#23778702	C₂	1	2	yes	maximal, complemented
5	C₄⋊C₈×C₃₁₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₁₂₆₈	—	2	1	no	—
7	C₂×D₁₂₆₈	—	2	1	no	—
8	C₂×D₁₂₆₈	C₂²	1	2	yes	—
9	C₂×D₁₂₆₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₃₁₇	—	2	1	no	—
11	D₂₅₃₆	—	2	4	no	—
12	C₄⋊Dic₃₁₇	C₂²	1	1	yes	characteristic
13	#19099551	—	2	1	no	—
14	C₂×C₈×C₃₁₇	C₂²	1	2	yes	—
15	C₄²×C₃₁₇	C₂²	1	1	yes	characteristic
16	C₂²×D₃₁₇	—	2	1	no	—
17	C₂²×D₃₁₇	—	2	2	no	—
18	D₁₂₆₈	—	2	2	no	—
19	D₁₂₆₈	—	2	1	no	—
20	D₁₂₆₈	—	2	2	no	—
21	D₁₂₆₈	—	4	2	no	—
22	D₁₂₆₈	—	4	2	no	—
23	D₁₂₆₈	D₄	1	2	yes	complemented
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	2	no	—
30	C₂²×C₃₁₇	C₂×D₄	1	1	yes	characteristic
31	C₂×D₃₁₇	—	2	2	no	—
32	C₂×D₃₁₇	—	4	1	no	—
33	C₂×D₃₁₇	—	4	1	no	—
34	C₂×D₃₁₇	—	4	2	no	—
35	C₂×D₃₁₇	—	4	2	no	—
36	C₂×D₃₁₇	—	4	2	no	—
37	C₄×C₃₁₇	—	2	1	no	—
38	C₄×C₃₁₇	D₈	1	2	yes	complemented
39	C₄×C₃₁₇	C₂×D₄	1	1	yes	characteristic
40	C₄×C₃₁₇	#41	1	1	yes	characteristic
41	Dic₃₁₇	—	4	1	no	—
42	C₂×C₃₁₇	#121	1	1	yes	characteristic
43	C₂×C₃₁₇	C₂×D₈	1	1	yes	characteristic
44	C₂×C₃₁₇	AGL(1,ℤ/8)	1	1	yes	characteristic
45	D₃₁₇	—	4	2	no	—
46	D₃₁₇	—	8	2	no	—
47	C₃₁₇	#459	1	1	yes	characteristic, complemented, Sylow
48	#459	—	317	1	no	maximal, Sylow
49	C₂×D₈	—	317	2	no	—
50	#127	—	317	1	no	—
51	C₄×D₄	—	317	1	no	—
52	C₄⋊C₈	D₃₁₇	1	1	yes	characteristic, complemented
53	#102	—	317	2	no	—
54	C₂×D₄	—	317	1	no	—
55	C₂×D₄	—	317	2	no	—
56	C₂×D₄	—	634	1	no	—
57	C₂×D₄	—	634	1	no	—
58	C₂²×C₄	—	634	1	no	—
59	D₈	—	634	4	no	—
60	C₂×C₈	C₂×D₃₁₇	1	2	yes	—
61	C₄⋊C₄	—	317	1	no	—
62	C₂²⋊C₄	—	634	1	no	—
63	C₄²	C₂×D₃₁₇	1	1	yes	characteristic
64	C₂³	—	634	1	no	—
65	C₂³	—	634	2	no	—
66	D₄	—	317	2	no	—
67	D₄	—	634	2	no	—
68	D₄	—	634	1	no	—
69	D₄	—	634	2	no	—
70	D₄	—	1268	2	no	—
71	D₄	—	1268	2	no	—
72	C₂×C₄	—	634	1	no	—
73	C₂×C₄	—	1268	2	no	—
74	C₂×C₄	D₁₂₆₈	1	1	yes	characteristic
75	C₂×C₄	D₁₂₆₈	1	1	yes	characteristic
76	C₂×C₄	C₂²×D₃₁₇	1	1	yes	characteristic
77	C₈	—	2	2	no	—
78	C₂²	—	634	2	no	—
79	C₂²	—	1268	2	no	—
80	C₂²	—	1268	2	no	—
81	C₂²	—	1268	2	no	—
82	C₂²	—	1268	1	no	—
83	C₂²	—	1268	1	no	—
84	C₂²	C₂×D₁₂₆₈	1	1	yes	characteristic
85	C₄	—	2	1	no	—
86	C₄	—	1268	1	no	—
87	C₄	D₂₅₃₆	1	2	yes	complemented
88	C₄	D₄×D₃₁₇	1	1	yes	characteristic
89	C₄	#19099575	1	1	yes	characteristic
90	C₂	—	2536	2	no	—
91	C₂	—	1268	2	no	—
92	C₂	#23778755	1	1	yes	characteristic
93	C₂	C₂×D₂₅₃₆	1	1	yes	characteristic
94	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
hypercenter	C₄⋊C₈	D₃₁₇
hypocenter	C₃₁₇	#459
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. C₂×D₂₅₃₆	C₂	semidirect
2. #23778777	C₂	semidirect
3. C₄×D₁₂₆₈	C₂	semidirect
4. #23778702	C₂	semidirect
5. C₄⋊C₈×C₃₁₇	C₂	semidirect
23. D₁₂₆₈	D₄	semidirect
38. C₄×C₃₁₇	D₈	semidirect
47. C₃₁₇	#459	semidirect
52. C₄⋊C₈	D₃₁₇	semidirect
87. C₄	D₂₅₃₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₂₅₃₆	2
#23778777	2
C₄×D₁₂₆₈	2
#23778702	2
C₄⋊C₈×C₃₁₇	2
#459	317