Subgroups — C₂×#23778787 #35203986

Order 20288.

266 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23778787	C₂	1	8	yes	maximal, direct product factor
2	C₂×#19099558	C₂	1	3	yes	maximal, complemented
3	C₂²×#14579261	C₂	1	3	yes	maximal, complemented
4	C₂×#41×C₃₁₇	C₂	1	1	yes	maximal, characteristic
5	#19099558	C₂²	1	12	yes	complemented
6	C₂×#14579261	C₂²	1	1	yes	characteristic
7	C₂×#14579261	C₂²	1	3	yes	—
8	C₂×#14579261	C₂²	1	12	yes	complemented
9	#41×C₃₁₇	C₄	1	4	yes	—
10	#41×C₃₁₇	C₂²	1	4	yes	—
11	C₂×Q₈×C₃₁₇	C₄	1	1	yes	characteristic
12	C₂×D₄×C₃₁₇	C₄	1	3	yes	—
13	C₂²×C₄×C₃₁₇	C₂²	1	3	yes	—
14	#14579261	C₂³	1	2	yes	—
15	#14579261	C₂³	1	6	yes	—
16	C₂³×C₃₁₇	C₂×C₄	1	3	yes	—
17	Q₈×C₃₁₇	C₂×C₄	1	4	yes	—
18	D₄×C₃₁₇	C₂×C₄	1	12	yes	—
19	C₂×C₄×C₃₁₇	C₂×C₄	1	3	yes	—
20	C₂×C₄×C₃₁₇	C₂×C₄	1	6	yes	—
21	C₂×C₄×C₃₁₇	C₂³	1	6	yes	—
22	C₂×C₄×C₃₁₇	C₂³	1	1	yes	characteristic
23	C₂²×C₃₁₇	—	2	6	no	—
24	C₂²×C₃₁₇	C₂²×C₄	1	6	yes	—
25	C₂²×C₃₁₇	C₂²×C₄	1	1	yes	characteristic
26	C₄×C₃₁₇	C₂²×C₄	1	1	yes	characteristic
27	C₄×C₃₁₇	C₂²×C₄	1	6	yes	—
28	C₄×C₃₁₇	C₂⁴	1	1	yes	characteristic
29	C₂×C₃₁₇	—	2	6	no	—
30	C₂×C₃₁₇	#131	1	2	yes	complemented
31	C₂×C₃₁₇	C₂³×C₄	1	1	yes	characteristic
32	C₃₁₇	C₂×#131	1	1	yes	characteristic, complemented, Sylow
33	C₂×#131	—	317	1	no	maximal, Sylow
34	C₂×#41	D₃₁₇	1	1	yes	characteristic
35	#131	—	317	8	no	—
36	C₂×M₁₆	—	317	3	no	—
37	C₂²×C₈	—	317	3	no	—
38	#41	Dic₃₁₇	1	4	yes	—
39	#41	C₂×D₃₁₇	1	4	yes	—
40	C₂×Q₈	Dic₃₁₇	1	1	yes	characteristic
41	C₂×D₄	Dic₃₁₇	1	3	yes	—
42	C₂²×C₄	C₂×D₃₁₇	1	3	yes	—
43	M₁₆	—	317	12	no	—
44	C₂×C₈	—	317	12	no	—
45	C₂×C₈	—	317	1	no	—
46	C₂×C₈	—	317	3	no	—
47	C₂³	C₂×Dic₃₁₇	1	3	yes	—
48	Q₈	C₂×Dic₃₁₇	1	4	yes	—
49	D₄	C₂×Dic₃₁₇	1	12	yes	—
50	C₂×C₄	C₂×Dic₃₁₇	1	3	yes	—
51	C₂×C₄	C₂×Dic₃₁₇	1	6	yes	—
52	C₂×C₄	C₂²×D₃₁₇	1	6	yes	—
53	C₂×C₄	C₂²×D₃₁₇	1	1	yes	characteristic
54	C₈	—	317	2	no	—
55	C₈	—	317	6	no	—
56	C₂²	—	2	6	no	—
57	C₂²	C₂²×Dic₃₁₇	1	6	yes	—
58	C₂²	C₂²×Dic₃₁₇	1	1	yes	characteristic
59	C₄	C₂²×Dic₃₁₇	1	1	yes	characteristic
60	C₄	C₂²×Dic₃₁₇	1	6	yes	—
61	C₄	C₂³×D₃₁₇	1	1	yes	characteristic
62	C₂	—	2	6	no	—
63	C₂	#23778787	1	2	yes	direct product factor
64	C₂	C₂³×Dic₃₁₇	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂²×D₃₁₇
commutator	C₂×C₃₁₇	C₂³×C₄
Frattini	C₄	C₂³×D₃₁₇
Fitting, prime radical	C₂×#41×C₃₁₇	C₂
socle	C₂²×C₃₁₇	C₂²×C₄
cosocle	C₄×C₃₁₇	C₂⁴
hypercenter	C₂×#41	D₃₁₇
hypocenter	C₃₁₇	C₂×#131
LCS	G ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#41

Complements

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

N	H ≅ G/N	type
1. #23778787	C₂	direct (N × H)
2. C₂×#19099558	C₂	semidirect
3. C₂²×#14579261	C₂	semidirect
5. #19099558	C₂²	semidirect
8. C₂×#14579261	C₂²	semidirect
30. C₂×C₃₁₇	#131	semidirect
32. C₃₁₇	C₂×#131	semidirect
63. C₂	#23778787	direct (N × H)

Maximal subgroups

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

subgroup	index
#23778787	2
C₂×#19099558	2
C₂²×#14579261	2
C₂×#41×C₃₁₇	2
C₂×#131	317

Subgroups — C2×#23778787 #35203986

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23778787 #35203986