Subgroups — #35203697

Order 20288.

338 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₁₂₆₈	C₂	1	1	yes	maximal, characteristic
2	C₄×D₁₂₆₈	C₂	1	4	yes	maximal, complemented
3	#23778742	C₂	1	4	yes	maximal, complemented
4	#23778739	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²⋊C₄×D₃₁₇	C₂	1	4	yes	maximal, complemented
6	#117×C₃₁₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₃₁₇	C₂²	1	2	yes	—
8	C₂×D₁₂₆₈	C₄	1	8	yes	complemented
9	C₂×D₁₂₆₈	C₂²	1	2	yes	—
10	C₂×D₁₂₆₈	C₂²	1	2	yes	—
11	C₂×C₄×D₃₁₇	C₂²	1	8	yes	complemented
12	C₄⋊Dic₃₁₇	C₂²	1	2	yes	complemented
13	C₂²⋊Dic₃₁₇	C₂²	1	2	yes	—
14	#19099551	C₂²	1	8	yes	complemented
15	C₄×Dic₃₁₇	C₂²	1	2	yes	complemented
16	C₂²×C₄×C₃₁₇	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₃₁₇	C₂²	1	2	yes	complemented
18	C₂²⋊C₄×C₃₁₇	C₂²	1	2	yes	—
19	C₄²×C₃₁₇	C₂²	1	2	yes	complemented
20	C₂²×D₃₁₇	—	4	4	no	—
21	C₂²×D₃₁₇	C₂×C₄	1	8	yes	complemented
22	C₂²×D₃₁₇	C₂³	1	4	yes	—
23	D₁₂₆₈	C₂×C₄	1	8	yes	complemented
24	D₁₂₆₈	C₂×C₄	1	8	yes	complemented
25	C₂×Dic₃₁₇	C₂³	1	4	yes	—
26	C₄×D₃₁₇	—	4	8	no	—
27	C₂³×C₃₁₇	C₂³	1	1	yes	characteristic
28	C₂×C₄×C₃₁₇	C₂×C₄	1	4	yes	complemented
29	C₂×C₄×C₃₁₇	C₂³	1	4	yes	—
30	C₂×C₄×C₃₁₇	C₂³	1	1	yes	characteristic
31	C₂×C₄×C₃₁₇	C₂³	1	1	yes	characteristic
32	C₂²×C₃₁₇	—	2	2	no	—
33	C₂²×C₃₁₇	C₂²×C₄	1	2	yes	—
34	C₂²×C₃₁₇	C₂⁴	1	1	yes	characteristic
35	C₂×D₃₁₇	—	2	8	no	—
36	C₂×D₃₁₇	—	4	8	no	—
37	C₂×D₃₁₇	C₂²×C₄	1	8	yes	—
38	C₄×C₃₁₇	—	2	4	no	—
39	C₄×C₃₁₇	C₂²×C₄	1	2	yes	—
40	C₄×C₃₁₇	C₂²×C₄	1	2	yes	—
41	Dic₃₁₇	—	2	4	no	—
42	C₂×C₃₁₇	—	2	2	no	—
43	C₂×C₃₁₇	C₂³×C₄	1	1	yes	characteristic
44	C₂×C₃₁₇	H(5,𝔽₂)	1	2	yes	—
45	D₃₁₇	—	2	8	no	—
46	C₃₁₇	#518	1	1	yes	characteristic, complemented, Sylow
47	#518	—	317	1	no	maximal, Sylow
48	C₂²×D₄	—	317	1	no	—
49	C₄×D₄	—	317	4	no	—
50	C₄×D₄	—	317	4	no	—
51	#117	—	317	1	no	—
52	#117	D₃₁₇	1	1	yes	characteristic, complemented
53	C₂×C₂²⋊C₄	—	317	4	no	—
54	C₂⁴	—	317	2	no	—
55	C₂×D₄	—	317	8	no	—
56	C₂×D₄	—	317	2	no	—
57	C₂×D₄	—	317	2	no	—
58	C₂²×C₄	—	317	8	no	—
59	C₂²×C₄	C₂×D₃₁₇	1	1	yes	characteristic
60	C₄⋊C₄	—	317	2	no	—
61	C₄⋊C₄	C₂×D₃₁₇	1	2	yes	complemented
62	C₂²⋊C₄	—	317	2	no	—
63	C₂²⋊C₄	—	317	8	no	—
64	C₂²⋊C₄	C₂×D₃₁₇	1	2	yes	—
65	C₄²	—	317	2	no	—
66	C₄²	C₂×D₃₁₇	1	2	yes	complemented
67	C₂³	—	317	4	no	—
68	C₂³	—	317	8	no	—
69	C₂³	—	1268	4	no	—
70	C₂³	C₂²×D₃₁₇	1	1	yes	characteristic
71	D₄	—	317	8	no	—
72	D₄	—	317	8	no	—
73	C₂×C₄	—	317	4	no	—
74	C₂×C₄	—	1268	8	no	—
75	C₂×C₄	C₄×D₃₁₇	1	4	yes	complemented
76	C₂×C₄	C₂²×D₃₁₇	1	4	yes	—
77	C₂×C₄	C₂²×D₃₁₇	1	1	yes	characteristic
78	C₂×C₄	C₂²×D₃₁₇	1	1	yes	characteristic
79	C₂²	—	2	2	no	—
80	C₂²	—	317	8	no	—
81	C₂²	—	634	8	no	—
82	C₂²	—	1268	8	no	—
83	C₂²	C₂×C₄×D₃₁₇	1	2	yes	—
84	C₂²	C₂³×D₃₁₇	1	1	yes	characteristic
85	C₄	—	2	4	no	—
86	C₄	—	634	4	no	—
87	C₄	C₂×C₄×D₃₁₇	1	2	yes	—
88	C₄	C₂×C₄×D₃₁₇	1	2	yes	—
89	C₂	—	2	2	no	—
90	C₂	—	634	8	no	—
91	C₂	C₂²×C₄×D₃₁₇	1	1	yes	characteristic
92	C₂	#23778826	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₃₁₇
commutator	C₂×C₃₁₇	C₂³×C₄
Fitting	#117×C₃₁₇	C₂
socle, cosocle	C₂²×C₃₁₇	C₂⁴
prime radical	C₂²×D₁₂₆₈	C₂
hypercenter	#117	D₃₁₇
hypocenter	C₃₁₇	#518
LCS	G ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂² ◁ #117

Complements

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

N	H ≅ G/N	type
2. C₄×D₁₂₆₈	C₂	semidirect
3. #23778742	C₂	semidirect
4. #23778739	C₂	semidirect
5. C₂²⋊C₄×D₃₁₇	C₂	semidirect
6. #117×C₃₁₇	C₂	semidirect
8. C₂×D₁₂₆₈	C₄	semidirect
11. C₂×C₄×D₃₁₇	C₂²	semidirect
12. C₄⋊Dic₃₁₇	C₂²	semidirect
14. #19099551	C₂²	semidirect
15. C₄×Dic₃₁₇	C₂²	semidirect
17. C₄⋊C₄×C₃₁₇	C₂²	semidirect
19. C₄²×C₃₁₇	C₂²	semidirect
21. C₂²×D₃₁₇	C₂×C₄	semidirect
23. D₁₂₆₈	C₂×C₄	semidirect
24. D₁₂₆₈	C₂×C₄	semidirect
28. C₂×C₄×C₃₁₇	C₂×C₄	semidirect
46. C₃₁₇	#518	semidirect
52. #117	D₃₁₇	semidirect
61. C₄⋊C₄	C₂×D₃₁₇	semidirect
66. C₄²	C₂×D₃₁₇	semidirect
75. C₂×C₄	C₄×D₃₁₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×D₁₂₆₈	2
C₄×D₁₂₆₈	2
#23778742	2
#23778739	2
C₂²⋊C₄×D₃₁₇	2
#117×C₃₁₇	2
#518	317