Subgroups — C₂²×#19099561 #35203995

Order 20288.

298 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×#19099561	C₂	1	12	yes	maximal, direct product factor
3	C₂²×#14579261	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×Q₈×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	—
8	#19099561	C₂²	1	16	yes	direct product factor
9	C₂×#14579261	C₂²	1	6	yes	complemented
10	C₂×Q₈×C₃₁₇	—	2	3	no	—
11	C₂×Q₈×C₃₁₇	C₂²	1	6	yes	complemented
12	C₂²×C₄×C₃₁₇	—	2	1	no	—
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	—
18	#14579261	C₂³	1	4	yes	complemented
19	C₂³×C₃₁₇	D₄	1	1	yes	characteristic
20	Q₈×C₃₁₇	—	2	6	no	—
21	Q₈×C₃₁₇	C₂³	1	4	yes	complemented
22	C₂×C₄×C₃₁₇	—	2	6	no	—
23	C₂×C₄×C₃₁₇	D₄	1	3	yes	—
24	C₂×C₄×C₃₁₇	C₂³	1	3	yes	—
25	C₂²×C₃₁₇	SD₁₆	1	4	yes	complemented
26	C₂²×C₃₁₇	C₂×D₄	1	3	yes	—
27	C₂×D₃₁₇	—	2	4	no	—
28	C₂×D₃₁₇	—	4	12	no	—
29	C₄×C₃₁₇	—	2	4	no	—
30	C₄×C₃₁₇	C₂×D₄	1	3	yes	—
31	C₄×C₃₁₇	C₂⁴	1	1	yes	characteristic
32	C₂×C₃₁₇	C₂×SD₁₆	1	6	yes	complemented
33	C₂×C₃₁₇	C₂²×D₄	1	1	yes	characteristic
34	D₃₁₇	—	4	4	no	—
35	C₃₁₇	C₂²×SD₁₆	1	1	yes	characteristic, complemented, Sylow
36	C₂²×SD₁₆	—	317	1	no	maximal, Sylow
37	C₂²×Q₈	D₃₁₇	1	1	yes	characteristic, complemented
38	C₂²×D₄	—	317	1	no	—
39	C₂×SD₁₆	—	317	12	no	—
40	C₂²×C₈	—	317	1	no	—
41	C₂⁴	—	634	1	no	—
42	C₂×Q₈	—	2	3	no	—
43	C₂×Q₈	C₂×D₃₁₇	1	6	yes	complemented
44	C₂×D₄	—	317	6	no	—
45	C₂×D₄	—	634	3	no	—
46	C₂²×C₄	—	2	1	no	—
47	C₂²×C₄	C₂×D₃₁₇	1	1	yes	characteristic
48	SD₁₆	—	317	16	no	—
49	C₂×C₈	—	317	6	no	—
50	C₂³	—	634	6	no	—
51	C₂³	—	1268	4	no	—
52	C₂³	C₂²⋊D₃₁₇	1	1	yes	characteristic
53	Q₈	—	2	6	no	—
54	Q₈	C₂²×D₃₁₇	1	4	yes	complemented
55	D₄	—	317	4	no	—
56	D₄	—	634	6	no	—
57	C₂×C₄	—	2	6	no	—
58	C₂×C₄	C₂²⋊D₃₁₇	1	3	yes	—
59	C₂×C₄	C₂²×D₃₁₇	1	3	yes	—
60	C₈	—	317	4	no	—
61	C₂²	—	634	4	no	—
62	C₂²	—	1268	12	no	—
63	C₂²	#19099561	1	4	yes	direct product factor
64	C₂²	C₂×C₂²⋊D₃₁₇	1	3	yes	—
65	C₄	—	2	4	no	—
66	C₄	C₂×C₂²⋊D₃₁₇	1	3	yes	—
67	C₄	C₂³×D₃₁₇	1	1	yes	characteristic
68	C₂	—	1268	4	no	—
69	C₂	C₂×#19099561	1	6	yes	direct product factor
70	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×Q₈×C₃₁₇	C₂
socle	C₂³×C₃₁₇	D₄
prime radical	C₂²×D₁₂₆₈	C₂
hypercenter	C₂²×Q₈	D₃₁₇
hypocenter	C₃₁₇	C₂²×SD₁₆
LCS	G ▷ C₄×C₃₁₇ ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₈

Complements

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

N	H ≅ G/N	type
2. C₂×#19099561	C₂	direct (N × H)
3. C₂²×#14579261	C₂	semidirect
4. C₂²×Q₈×C₃₁₇	C₂	semidirect
8. #19099561	C₂²	direct (N × H)
9. C₂×#14579261	C₂²	semidirect
11. C₂×Q₈×C₃₁₇	C₂²	semidirect
18. #14579261	C₂³	semidirect
21. Q₈×C₃₁₇	C₂³	semidirect
25. C₂²×C₃₁₇	SD₁₆	semidirect
32. C₂×C₃₁₇	C₂×SD₁₆	semidirect
35. C₃₁₇	C₂²×SD₁₆	semidirect
37. C₂²×Q₈	D₃₁₇	semidirect
43. C₂×Q₈	C₂×D₃₁₇	semidirect
54. Q₈	C₂²×D₃₁₇	semidirect
63. C₂²	#19099561	direct (N × H)
69. C₂	C₂×#19099561	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×#19099561	2
C₂²×#14579261	2
C₂²×Q₈×C₃₁₇	2
C₂²×SD₁₆	317

Subgroups — C22×#19099561 #35203995

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#19099561 #35203995