Subgroups — C₂²×#19099553 #35203673

Order 20288.

418 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Dic₃₁₇	C₂	1	1	yes	maximal, characteristic
2	C₂³×Dic₃₁₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#19099553	C₂	1	12	yes	maximal, direct product factor
4	C₂³×C₄×C₃₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₃₁₇	—	2	4	no	—
6	C₂²×Dic₃₁₇	C₄	1	8	yes	complemented
7	C₂²×Dic₃₁₇	C₂²	1	6	yes	—
8	C₂²×Dic₃₁₇	C₂²	1	6	yes	—
9	#19099553	C₂²	1	16	yes	direct product factor
10	C₂⁴×C₃₁₇	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₃₁₇	—	2	4	no	—
12	C₂²×C₄×C₃₁₇	C₂²	1	6	yes	—
13	C₂×Dic₃₁₇	—	2	12	no	—
14	C₂×Dic₃₁₇	C₂×C₄	1	24	yes	complemented
15	C₂×Dic₃₁₇	C₂³	1	4	yes	—
16	C₂×Dic₃₁₇	C₂³	1	4	yes	—
17	C₂³×C₃₁₇	C₂×C₄	1	4	yes	—
18	C₂³×C₃₁₇	D₄	1	4	yes	—
19	C₂³×C₃₁₇	Q₈	1	4	yes	—
20	C₂³×C₃₁₇	C₂³	1	3	yes	—
21	C₂×C₄×C₃₁₇	—	2	12	no	—
22	C₂×C₄×C₃₁₇	C₂³	1	4	yes	—
23	C₂²×C₃₁₇	C₄⋊C₄	1	16	yes	complemented
24	C₂²×C₃₁₇	C₂²×C₄	1	6	yes	—
25	C₂²×C₃₁₇	C₂×D₄	1	6	yes	—
26	C₂²×C₃₁₇	C₂×Q₈	1	6	yes	—
27	C₂²×C₃₁₇	C₂⁴	1	1	yes	characteristic
28	C₄×C₃₁₇	—	2	4	no	—
29	Dic₃₁₇	—	2	4	no	—
30	Dic₃₁₇	C₂²×C₄	1	8	yes	complemented
31	C₂×C₃₁₇	C₂×C₄⋊C₄	1	12	yes	complemented
32	C₂×C₃₁₇	C₂³×C₄	1	1	yes	characteristic
33	C₂×C₃₁₇	C₂²×D₄	1	1	yes	characteristic
34	C₂×C₃₁₇	C₂²×Q₈	1	1	yes	characteristic
35	C₃₁₇	C₂²×C₄⋊C₄	1	1	yes	characteristic, complemented, Sylow
36	C₂²×C₄⋊C₄	—	317	1	no	maximal, Sylow
37	C₂³×C₄	—	317	1	no	—
38	C₂³×C₄	—	317	1	no	—
39	C₂³×C₄	D₃₁₇	1	1	yes	characteristic
40	C₂×C₄⋊C₄	—	317	12	no	—
41	C₂⁴	C₂×D₃₁₇	1	1	yes	characteristic
42	C₂²×C₄	—	2	4	no	—
43	C₂²×C₄	—	317	8	no	—
44	C₂²×C₄	—	317	6	no	—
45	C₂²×C₄	—	317	6	no	—
46	C₂²×C₄	—	634	4	no	—
47	C₂²×C₄	C₂×D₃₁₇	1	6	yes	—
48	C₄⋊C₄	—	317	16	no	—
49	C₂³	C₄×D₃₁₇	1	4	yes	—
50	C₂³	C₂²⋊D₃₁₇	1	4	yes	—
51	C₂³	Dic₆₃₄	1	4	yes	—
52	C₂³	C₂²×D₃₁₇	1	3	yes	—
53	C₂×C₄	—	2	12	no	—
54	C₂×C₄	—	317	24	no	—
55	C₂×C₄	—	317	4	no	—
56	C₂×C₄	—	317	4	no	—
57	C₂×C₄	—	634	12	no	—
58	C₂×C₄	C₂²×D₃₁₇	1	4	yes	—
59	C₂²	#19099553	1	16	yes	direct product factor
60	C₂²	C₂×C₄×D₃₁₇	1	6	yes	—
61	C₂²	C₂×C₂²⋊D₃₁₇	1	6	yes	—
62	C₂²	C₂×Dic₆₃₄	1	6	yes	—
63	C₂²	C₂³×D₃₁₇	1	1	yes	characteristic
64	C₄	—	2	4	no	—
65	C₄	—	317	8	no	—
66	C₄	—	634	4	no	—
67	C₂	C₂×#19099553	1	12	yes	direct product factor
68	C₂	C₂²×C₄×D₃₁₇	1	1	yes	characteristic
69	C₂	C₂²×C₂²⋊D₃₁₇	1	1	yes	characteristic
70	C₂	C₂²×Dic₆₃₄	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, prime radical	C₂⁴×C₃₁₇	C₂²
cosocle	C₂²×C₃₁₇	C₂⁴
hypercenter	C₂³×C₄	D₃₁₇
hypocenter	C₃₁₇	C₂²×C₄⋊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
3. C₂×#19099553	C₂	direct (N × H)
6. C₂²×Dic₃₁₇	C₄	semidirect
9. #19099553	C₂²	direct (N × H)
14. C₂×Dic₃₁₇	C₂×C₄	semidirect
23. C₂²×C₃₁₇	C₄⋊C₄	semidirect
30. Dic₃₁₇	C₂²×C₄	semidirect
31. C₂×C₃₁₇	C₂×C₄⋊C₄	semidirect
35. C₃₁₇	C₂²×C₄⋊C₄	semidirect
59. C₂²	#19099553	direct (N × H)
67. C₂	C₂×#19099553	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×Dic₃₁₇	2
C₂×#19099553	2
C₂³×C₄×C₃₁₇	2
C₂²×C₄⋊C₄	317

Subgroups — C22×#19099553 #35203673

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×#19099553 #35203673