Subgroups — C₂²×#19099559 #35203992

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, complemented
2	C₂×#19099559	C₂	1	12	yes	maximal, direct product factor
3	C₂²×#14579261	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×D₄×C₃₁₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂³×D₃₁₇	—	2	1	no	—
6	C₂×D₁₂₆₈	—	2	3	no	—
7	C₂×D₁₂₆₈	C₂²	1	6	yes	complemented
8	#19099559	C₂²	1	16	yes	direct product factor
9	C₂×#14579261	C₂²	1	6	yes	complemented
10	C₂⁴×C₃₁₇	—	2	1	no	—
11	C₂×D₄×C₃₁₇	—	2	3	no	—
12	C₂×D₄×C₃₁₇	C₂²	1	6	yes	complemented
13	C₂²×C₄×C₃₁₇	C₂²	1	1	yes	characteristic
14	C₂²×D₃₁₇	—	2	6	no	—
15	C₂²×D₃₁₇	—	4	4	no	—
16	D₁₂₆₈	—	2	6	no	—
17	D₁₂₆₈	C₂³	1	4	yes	complemented
18	#14579261	C₂³	1	4	yes	complemented
19	C₂³×C₃₁₇	—	2	6	no	—
20	C₂³×C₃₁₇	—	4	4	no	—
21	C₂³×C₃₁₇	D₄	1	1	yes	characteristic
22	D₄×C₃₁₇	—	2	6	no	—
23	D₄×C₃₁₇	C₂³	1	4	yes	complemented
24	C₂×C₄×C₃₁₇	D₄	1	3	yes	—
25	C₂×C₄×C₃₁₇	C₂³	1	3	yes	—
26	C₂²×C₃₁₇	—	2	4	no	—
27	C₂²×C₃₁₇	—	4	12	no	—
28	C₂²×C₃₁₇	D₈	1	4	yes	complemented
29	C₂²×C₃₁₇	C₂×D₄	1	3	yes	—
30	C₂×D₃₁₇	—	2	4	no	—
31	C₂×D₃₁₇	—	4	12	no	—
32	C₄×C₃₁₇	C₂×D₄	1	3	yes	—
33	C₄×C₃₁₇	C₂⁴	1	1	yes	characteristic
34	C₂×C₃₁₇	—	4	4	no	—
35	C₂×C₃₁₇	C₂×D₈	1	6	yes	complemented
36	C₂×C₃₁₇	C₂²×D₄	1	1	yes	characteristic
37	D₃₁₇	—	4	4	no	—
38	C₃₁₇	C₂²×D₈	1	1	yes	characteristic, complemented, Sylow
39	C₂²×D₈	—	317	1	no	maximal, Sylow
40	C₂²×D₄	—	317	1	no	—
41	C₂²×D₄	D₃₁₇	1	1	yes	characteristic, complemented
42	C₂×D₈	—	317	12	no	—
43	C₂²×C₈	—	317	1	no	—
44	C₂⁴	—	2	1	no	—
45	C₂⁴	—	634	1	no	—
46	C₂×D₄	—	2	3	no	—
47	C₂×D₄	—	317	6	no	—
48	C₂×D₄	—	634	3	no	—
49	C₂×D₄	C₂×D₃₁₇	1	6	yes	complemented
50	C₂²×C₄	C₂×D₃₁₇	1	1	yes	characteristic
51	D₈	—	317	16	no	—
52	C₂×C₈	—	317	6	no	—
53	C₂³	—	2	6	no	—
54	C₂³	—	4	4	no	—
55	C₂³	—	634	6	no	—
56	C₂³	—	1268	4	no	—
57	C₂³	C₂²⋊D₃₁₇	1	1	yes	characteristic
58	D₄	—	2	6	no	—
59	D₄	—	317	4	no	—
60	D₄	—	634	6	no	—
61	D₄	C₂²×D₃₁₇	1	4	yes	complemented
62	C₂×C₄	C₂²⋊D₃₁₇	1	3	yes	—
63	C₂×C₄	C₂²×D₃₁₇	1	3	yes	—
64	C₈	—	317	4	no	—
65	C₂²	—	2	4	no	—
66	C₂²	—	4	12	no	—
67	C₂²	—	634	4	no	—
68	C₂²	—	1268	12	no	—
69	C₂²	#19099559	1	4	yes	direct product factor
70	C₂²	C₂×C₂²⋊D₃₁₇	1	3	yes	—
71	C₄	C₂×C₂²⋊D₃₁₇	1	3	yes	—
72	C₄	C₂³×D₃₁₇	1	1	yes	characteristic
73	C₂	—	4	4	no	—
74	C₂	—	1268	4	no	—
75	C₂	C₂×#19099559	1	6	yes	direct product factor
76	C₂	C₂²×C₂²⋊D₃₁₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂²⋊D₃₁₇
commutator, cosocle	C₄×C₃₁₇	C₂⁴
Frattini	C₄	C₂³×D₃₁₇
Fitting	C₂²×D₄×C₃₁₇	C₂
socle	C₂³×C₃₁₇	D₄
hypercenter	C₂²×D₄	D₃₁₇
hypocenter	C₃₁₇	C₂²×D₈
LCS	G ▷ C₄×C₃₁₇ ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×D₄

Complements

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

N	H ≅ G/N	type
1. C₂²×D₁₂₆₈	C₂	semidirect
2. C₂×#19099559	C₂	direct (N × H)
3. C₂²×#14579261	C₂	semidirect
4. C₂²×D₄×C₃₁₇	C₂	semidirect
7. C₂×D₁₂₆₈	C₂²	semidirect
8. #19099559	C₂²	direct (N × H)
9. C₂×#14579261	C₂²	semidirect
12. C₂×D₄×C₃₁₇	C₂²	semidirect
17. D₁₂₆₈	C₂³	semidirect
18. #14579261	C₂³	semidirect
23. D₄×C₃₁₇	C₂³	semidirect
28. C₂²×C₃₁₇	D₈	semidirect
35. C₂×C₃₁₇	C₂×D₈	semidirect
38. C₃₁₇	C₂²×D₈	semidirect
41. C₂²×D₄	D₃₁₇	semidirect
49. C₂×D₄	C₂×D₃₁₇	semidirect
61. D₄	C₂²×D₃₁₇	semidirect
69. C₂²	#19099559	direct (N × H)
75. C₂	C₂×#19099559	direct (N × H)

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×#19099559	2
C₂²×#14579261	2
C₂²×D₄×C₃₁₇	2
C₂²×D₈	317

Subgroups — C22×#19099559 #35203992

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#19099559 #35203992