Subgroups — #35203227

Order 20288.

190 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	C₂×#19099558	C₂	1	1	yes	maximal, characteristic, complemented
3	#23778700	C₂	1	4	yes	maximal, complemented
4	#117×C₃₁₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×D₃₁₇	—	2	1	no	—
6	C₂×D₁₂₆₈	—	2	1	no	—
7	C₂×D₁₂₆₈	—	2	2	no	—
8	C₂×D₁₂₆₈	C₄	1	4	yes	complemented
9	C₂×D₁₂₆₈	C₂²	1	2	yes	—
10	#19099558	—	2	2	no	—
11	C₂×#14579261	C₂²	1	2	yes	complemented
12	C₂²×C₄×C₃₁₇	C₂²	1	1	yes	characteristic
13	C₄⋊C₄×C₃₁₇	C₂²	1	2	yes	complemented
14	C₂²⋊C₄×C₃₁₇	—	2	1	no	—
15	C₄²×C₃₁₇	—	2	1	no	—
16	C₂²×D₃₁₇	—	2	4	no	—
17	C₂²×D₃₁₇	—	2	2	no	—
18	C₂²×D₃₁₇	—	4	4	no	—
19	D₁₂₆₈	—	2	4	no	—
20	D₁₂₆₈	—	2	2	no	—
21	D₁₂₆₈	C₂×C₄	1	4	yes	—
22	#14579261	—	2	2	no	—
23	C₂³×C₃₁₇	D₄	1	1	yes	characteristic
24	C₂×C₄×C₃₁₇	—	2	2	no	—
25	C₂×C₄×C₃₁₇	C₂×C₄	1	2	yes	—
26	C₂×C₄×C₃₁₇	D₄	1	1	yes	characteristic
27	C₂×C₄×C₃₁₇	D₄	1	2	yes	—
28	C₂×C₄×C₃₁₇	C₂³	1	1	yes	characteristic
29	C₂²×C₃₁₇	—	2	2	no	—
30	C₂²×C₃₁₇	C₂²⋊C₄	1	2	yes	—
31	C₂²×C₃₁₇	C₂×D₄	1	1	yes	characteristic
32	C₂×D₃₁₇	—	2	4	no	—
33	C₂×D₃₁₇	—	4	8	no	—
34	C₂×D₃₁₇	—	4	4	no	—
35	C₄×C₃₁₇	—	4	2	no	—
36	C₄×C₃₁₇	C₂²⋊C₄	1	2	yes	—
37	C₄×C₃₁₇	C₂²×C₄	1	1	yes	characteristic
38	C₄×C₃₁₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₃₁₇	—	2	2	no	—
40	C₂×C₃₁₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₃₁₇	AGL(1,ℤ/8)	1	2	yes	—
42	D₃₁₇	—	4	4	no	—
43	C₃₁₇	#418	1	1	yes	characteristic, complemented, Sylow
44	#418	—	317	1	no	maximal, Sylow
45	C₂²×D₄	—	317	1	no	—
46	C₂×M₁₆	—	317	1	no	—
47	#117	D₃₁₇	1	1	yes	characteristic, complemented
48	#102	—	317	4	no	—
49	C₂⁴	—	634	1	no	—
50	C₂×D₄	—	317	4	no	—
51	C₂×D₄	—	317	2	no	—
52	C₂×D₄	—	634	1	no	—
53	C₂×D₄	—	634	2	no	—
54	C₂²×C₄	C₂×D₃₁₇	1	1	yes	characteristic
55	M₁₆	—	634	2	no	—
56	C₂×C₈	—	317	2	no	—
57	C₄⋊C₄	C₂×D₃₁₇	1	2	yes	complemented
58	C₂²⋊C₄	—	2	1	no	—
59	C₄²	—	2	1	no	—
60	C₂³	—	634	4	no	—
61	C₂³	—	634	2	no	—
62	C₂³	—	1268	4	no	—
63	C₂³	C₂²⋊D₃₁₇	1	1	yes	characteristic
64	D₄	—	317	4	no	—
65	D₄	—	634	4	no	—
66	D₄	—	634	2	no	—
67	C₂×C₄	—	2	2	no	—
68	C₂×C₄	C₄×D₃₁₇	1	2	yes	—
69	C₂×C₄	C₂²⋊D₃₁₇	1	1	yes	characteristic
70	C₂×C₄	D₁₂₆₈	1	2	yes	—
71	C₂×C₄	C₂²×D₃₁₇	1	1	yes	characteristic
72	C₈	—	634	2	no	—
73	C₂²	—	2	2	no	—
74	C₂²	—	634	4	no	—
75	C₂²	—	1268	8	no	—
76	C₂²	—	1268	4	no	—
77	C₂²	#19099551	1	2	yes	—
78	C₂²	C₂×C₂²⋊D₃₁₇	1	1	yes	characteristic
79	C₄	—	4	2	no	—
80	C₄	#19099551	1	2	yes	—
81	C₄	C₂×C₄×D₃₁₇	1	1	yes	characteristic
82	C₄	C₂×D₁₂₆₈	1	1	yes	characteristic
83	C₂	—	2	2	no	—
84	C₂	—	1268	4	no	—
85	C₂	C₂×#19099551	1	1	yes	characteristic
86	C₂	#23778805	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊D₃₁₇
commutator	C₄×C₃₁₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×D₃₁₇
Fitting	#117×C₃₁₇	C₂
socle	C₂²×C₃₁₇	C₂×D₄
prime radical	C₂²×D₁₂₆₈	C₂
cosocle	C₂×C₄×C₃₁₇	C₂³
hypercenter	#117	D₃₁₇
hypocenter	C₃₁₇	#418
LCS	G ▷ C₄×C₃₁₇ ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #117

Complements

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

N	H ≅ G/N	type
2. C₂×#19099558	C₂	semidirect
3. #23778700	C₂	semidirect
4. #117×C₃₁₇	C₂	semidirect
8. C₂×D₁₂₆₈	C₄	semidirect
11. C₂×#14579261	C₂²	semidirect
13. C₄⋊C₄×C₃₁₇	C₂²	semidirect
43. C₃₁₇	#418	semidirect
47. #117	D₃₁₇	semidirect
57. C₄⋊C₄	C₂×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
C₂×#19099558	2
#23778700	2
#117×C₃₁₇	2
#418	317