Subgroups — C₂×#23778772 #35203745

Order 20288.

226 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23778772	C₂	1	8	yes	maximal, direct product factor
2	C₂×C₄⋊Dic₃₁₇	C₂	1	2	yes	maximal
3	C₂×#19099553	C₂	1	4	yes	maximal
4	C₂×C₄²×C₃₁₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₃₁₇	C₂²	1	4	yes	—
6	C₄⋊Dic₃₁₇	C₂²	1	8	yes	—
7	#19099553	C₂²	1	16	yes	—
8	C₂²×C₄×C₃₁₇	C₂²	1	2	yes	—
9	C₂²×C₄×C₃₁₇	C₂²	1	1	yes	characteristic
10	C₄²×C₃₁₇	C₂²	1	4	yes	—
11	C₂×Dic₃₁₇	—	2	8	no	—
12	C₂×Dic₃₁₇	C₂³	1	8	yes	—
13	C₂³×C₃₁₇	C₂³	1	1	yes	characteristic
14	C₂×C₄×C₃₁₇	—	2	4	no	—
15	C₂×C₄×C₃₁₇	Q₈	1	4	yes	—
16	C₂×C₄×C₃₁₇	C₂³	1	4	yes	—
17	C₂×C₄×C₃₁₇	C₂³	1	2	yes	—
18	C₂²×C₃₁₇	C₂×Q₈	1	2	yes	—
19	C₂²×C₃₁₇	#41	1	4	yes	—
20	C₂²×C₃₁₇	C₂⁴	1	1	yes	characteristic
21	C₄×C₃₁₇	—	2	4	no	—
22	C₄×C₃₁₇	C₂×Q₈	1	4	yes	—
23	Dic₃₁₇	—	2	8	no	—
24	C₂×C₃₁₇	#125	1	4	yes	complemented
25	C₂×C₃₁₇	C₂²×Q₈	1	1	yes	characteristic
26	C₂×C₃₁₇	C₂×#41	1	2	yes	—
27	C₃₁₇	C₂×#125	1	1	yes	characteristic, complemented, Sylow
28	C₂×#125	—	317	1	no	maximal, Sylow
29	#125	—	317	8	no	—
30	C₂×C₄⋊C₄	—	317	4	no	—
31	C₂×C₄⋊C₄	—	317	2	no	—
32	C₂×C₄²	D₃₁₇	1	1	yes	characteristic
33	C₂²×C₄	—	317	4	no	—
34	C₂²×C₄	C₂×D₃₁₇	1	2	yes	—
35	C₂²×C₄	C₂×D₃₁₇	1	1	yes	characteristic
36	C₄⋊C₄	—	317	16	no	—
37	C₄⋊C₄	—	317	8	no	—
38	C₄²	C₂×D₃₁₇	1	4	yes	—
39	C₂³	C₂²×D₃₁₇	1	1	yes	characteristic
40	C₂×C₄	—	2	4	no	—
41	C₂×C₄	—	317	8	no	—
42	C₂×C₄	—	634	8	no	—
43	C₂×C₄	Dic₆₃₄	1	4	yes	—
44	C₂×C₄	C₂²×D₃₁₇	1	4	yes	—
45	C₂×C₄	C₂²×D₃₁₇	1	2	yes	—
46	C₂²	C₂×Dic₆₃₄	1	2	yes	—
47	C₂²	#19099574	1	4	yes	—
48	C₂²	C₂³×D₃₁₇	1	1	yes	characteristic
49	C₄	—	2	4	no	—
50	C₄	—	634	8	no	—
51	C₄	C₂×Dic₆₃₄	1	4	yes	—
52	C₂	#23778772	1	4	yes	direct product factor
53	C₂	C₂²×Dic₆₃₄	1	1	yes	characteristic
54	C₂	C₂×#19099574	1	2	yes	—

	subgroup	G/H
center	C₂³	C₂²×D₃₁₇
commutator, cosocle	C₂²×C₃₁₇	C₂⁴
Frattini	C₂²	C₂³×D₃₁₇
Fitting	C₂×C₄²×C₃₁₇	C₂
socle, prime radical	C₂³×C₃₁₇	C₂³
hypercenter	C₂×C₄²	D₃₁₇
hypocenter	C₃₁₇	C₂×#125
LCS	G ▷ C₂²×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄²

Complements

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

N	H ≅ G/N	type
1. #23778772	C₂	direct (N × H)
24. C₂×C₃₁₇	#125	semidirect
27. C₃₁₇	C₂×#125	semidirect
52. C₂	#23778772	direct (N × H)

Maximal subgroups

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

subgroup	index
#23778772	2
C₂×C₄⋊Dic₃₁₇	2
C₂×#19099553	2
C₂×C₄²×C₃₁₇	2
C₂×#125	317

Subgroups — C2×#23778772 #35203745

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23778772 #35203745