Subgroups — C₂²×#19099551 #35203670

Order 20288.

674 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×D₃₁₇	C₂	1	1	yes	maximal, characteristic
2	C₂³×Dic₃₁₇	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#19099551	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₄×C₃₁₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×D₃₁₇	—	2	4	no	—
6	C₂³×D₃₁₇	—	2	4	no	—
7	C₂³×D₃₁₇	C₄	1	8	yes	complemented
8	C₂³×D₃₁₇	C₂²	1	6	yes	—
9	C₂²×Dic₃₁₇	—	2	4	no	—
10	C₂²×Dic₃₁₇	C₂²	1	6	yes	complemented
11	#19099551	C₂²	1	16	yes	direct product factor
12	C₂⁴×C₃₁₇	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₃₁₇	—	2	4	no	—
14	C₂²×C₄×C₃₁₇	C₂²	1	6	yes	complemented
15	C₂²×D₃₁₇	—	2	32	no	—
16	C₂²×D₃₁₇	—	2	12	no	—
17	C₂²×D₃₁₇	—	2	12	no	—
18	C₂²×D₃₁₇	C₂×C₄	1	24	yes	complemented
19	C₂²×D₃₁₇	C₂³	1	4	yes	—
20	C₂×Dic₃₁₇	—	2	12	no	—
21	C₂×Dic₃₁₇	C₂³	1	4	yes	complemented
22	C₂³×C₃₁₇	C₂×C₄	1	4	yes	—
23	C₂³×C₃₁₇	D₄	1	4	yes	—
24	C₂³×C₃₁₇	D₄	1	4	yes	—
25	C₂³×C₃₁₇	C₂³	1	3	yes	—
26	C₂×C₄×C₃₁₇	—	2	12	no	—
27	C₂×C₄×C₃₁₇	C₂³	1	4	yes	complemented
28	C₂²×C₃₁₇	C₂²⋊C₄	1	16	yes	complemented
29	C₂²×C₃₁₇	C₂²×C₄	1	6	yes	—
30	C₂²×C₃₁₇	C₂×D₄	1	6	yes	—
31	C₂²×C₃₁₇	C₂×D₄	1	6	yes	—
32	C₂²×C₃₁₇	C₂⁴	1	1	yes	characteristic
33	C₂×D₃₁₇	—	2	48	no	—
34	C₂×D₃₁₇	—	2	4	no	—
35	C₂×D₃₁₇	—	2	4	no	—
36	C₂×D₃₁₇	C₂²×C₄	1	8	yes	complemented
37	C₄×C₃₁₇	—	2	4	no	—
38	Dic₃₁₇	—	2	4	no	—
39	C₂×C₃₁₇	C₂×C₂²⋊C₄	1	12	yes	complemented
40	C₂×C₃₁₇	C₂³×C₄	1	1	yes	characteristic
41	C₂×C₃₁₇	C₂²×D₄	1	1	yes	characteristic
42	C₂×C₃₁₇	C₂²×D₄	1	1	yes	characteristic
43	D₃₁₇	—	2	8	no	—
44	C₃₁₇	C₂²×C₂²⋊C₄	1	1	yes	characteristic, complemented, Sylow
45	C₂²×C₂²⋊C₄	—	317	1	no	maximal, Sylow
46	C₂⁵	—	317	1	no	—
47	C₂³×C₄	—	317	1	no	—
48	C₂³×C₄	D₃₁₇	1	1	yes	characteristic, complemented
49	C₂×C₂²⋊C₄	—	317	12	no	—
50	C₂⁴	—	317	8	no	—
51	C₂⁴	—	317	6	no	—
52	C₂⁴	—	634	4	no	—
53	C₂⁴	—	634	4	no	—
54	C₂⁴	C₂×D₃₁₇	1	1	yes	characteristic
55	C₂²×C₄	—	2	4	no	—
56	C₂²×C₄	—	317	6	no	—
57	C₂²×C₄	—	634	4	no	—
58	C₂²×C₄	C₂×D₃₁₇	1	6	yes	complemented
59	C₂²⋊C₄	—	317	16	no	—
60	C₂³	—	317	24	no	—
61	C₂³	—	317	4	no	—
62	C₂³	—	634	32	no	—
63	C₂³	—	634	12	no	—
64	C₂³	—	634	12	no	—
65	C₂³	C₄×D₃₁₇	1	4	yes	—
66	C₂³	C₂²⋊D₃₁₇	1	4	yes	—
67	C₂³	D₁₂₆₈	1	4	yes	—
68	C₂³	C₂²×D₃₁₇	1	3	yes	—
69	C₂×C₄	—	2	12	no	—
70	C₂×C₄	—	317	4	no	—
71	C₂×C₄	—	634	12	no	—
72	C₂×C₄	C₂²×D₃₁₇	1	4	yes	complemented
73	C₂²	—	317	8	no	—
74	C₂²	—	634	4	no	—
75	C₂²	—	634	4	no	—
76	C₂²	—	634	48	no	—
77	C₂²	#19099551	1	16	yes	direct product factor
78	C₂²	C₂×C₄×D₃₁₇	1	6	yes	—
79	C₂²	C₂×C₂²⋊D₃₁₇	1	6	yes	—
80	C₂²	C₂×D₁₂₆₈	1	6	yes	—
81	C₂²	C₂³×D₃₁₇	1	1	yes	characteristic
82	C₄	—	2	4	no	—
83	C₄	—	634	4	no	—
84	C₂	—	634	8	no	—
85	C₂	C₂×#19099551	1	12	yes	direct product factor
86	C₂	C₂²×C₄×D₃₁₇	1	1	yes	characteristic
87	C₂	C₂²×C₂²⋊D₃₁₇	1	1	yes	characteristic
88	C₂	C₂²×D₁₂₆₈	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
2. C₂³×Dic₃₁₇	C₂	semidirect
3. C₂×#19099551	C₂	direct (N × H)
4. C₂³×C₄×C₃₁₇	C₂	semidirect
7. C₂³×D₃₁₇	C₄	semidirect
10. C₂²×Dic₃₁₇	C₂²	semidirect
11. #19099551	C₂²	direct (N × H)
14. C₂²×C₄×C₃₁₇	C₂²	semidirect
18. C₂²×D₃₁₇	C₂×C₄	semidirect
21. C₂×Dic₃₁₇	C₂³	semidirect
27. C₂×C₄×C₃₁₇	C₂³	semidirect
28. C₂²×C₃₁₇	C₂²⋊C₄	semidirect
36. C₂×D₃₁₇	C₂²×C₄	semidirect
39. C₂×C₃₁₇	C₂×C₂²⋊C₄	semidirect
44. C₃₁₇	C₂²×C₂²⋊C₄	semidirect
48. C₂³×C₄	D₃₁₇	semidirect
58. C₂²×C₄	C₂×D₃₁₇	semidirect
72. C₂×C₄	C₂²×D₃₁₇	semidirect
77. C₂²	#19099551	direct (N × H)
85. C₂	C₂×#19099551	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×D₃₁₇	2
C₂³×Dic₃₁₇	2
C₂×#19099551	2
C₂³×C₄×C₃₁₇	2
C₂²×C₂²⋊C₄	317

Subgroups — C22×#19099551 #35203670

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#19099551 #35203670