Subgroups — C₂×#23778868 #35204300

Order 20288.

266 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#23778868	C₂	1	8	yes	maximal, direct product factor
2	C₂×#19099584	C₂	1	3	yes	maximal, complemented
3	C₂×D₃₁₇⋊C₈	C₂	1	3	yes	maximal, complemented
4	C₂×#19099575	C₂	1	1	yes	maximal, characteristic
5	#19099584	C₂²	1	12	yes	complemented
6	D₃₁₇⋊C₈	C₂²	1	12	yes	complemented
7	C₂×#14579268	C₂²	1	1	yes	characteristic
8	C₂×#14579268	C₂²	1	3	yes	—
9	#19099575	C₄	1	4	yes	—
10	#19099575	C₂²	1	4	yes	—
11	C₂×D₁₂₆₈	C₄	1	3	yes	—
12	C₂×C₄×D₃₁₇	C₂²	1	3	yes	—
13	C₂×Q₈×C₃₁₇	C₄	1	1	yes	characteristic
14	#14579268	C₂³	1	2	yes	—
15	#14579268	C₂³	1	6	yes	—
16	C₂²×D₃₁₇	C₂×C₄	1	3	yes	—
17	D₁₂₆₈	C₂×C₄	1	12	yes	—
18	C₂×Dic₃₁₇	C₂³	1	1	yes	characteristic
19	C₄×D₃₁₇	C₂×C₄	1	6	yes	—
20	C₄×D₃₁₇	C₂³	1	6	yes	—
21	Q₈×C₃₁₇	C₂×C₄	1	4	yes	—
22	C₂×C₄×C₃₁₇	C₂×C₄	1	3	yes	—
23	C₂²×C₃₁₇	C₂²×C₄	1	1	yes	characteristic
24	C₂×D₃₁₇	—	2	6	no	—
25	C₂×D₃₁₇	C₂²×C₄	1	6	yes	—
26	C₄×C₃₁₇	C₂²×C₄	1	6	yes	—
27	Dic₃₁₇	C₂²×C₄	1	1	yes	characteristic
28	Dic₃₁₇	C₂⁴	1	1	yes	characteristic
29	C₂×C₃₁₇	#131	1	2	yes	complemented
30	C₂×C₃₁₇	C₂³×C₄	1	1	yes	characteristic
31	D₃₁₇	—	2	6	no	—
32	C₃₁₇	C₂×#131	1	1	yes	characteristic, complemented, Sylow
33	C₂×#131	—	317	1	no	maximal, Sylow
34	C₂×#41	—	317	1	no	—
35	#131	—	317	8	no	—
36	C₂×M₁₆	—	317	3	no	—
37	C₂²×C₈	—	317	3	no	—
38	#41	—	317	4	no	—
39	#41	—	317	4	no	—
40	C₂×Q₈	#11929639	1	1	yes	characteristic
41	C₂×D₄	—	317	3	no	—
42	C₂²×C₄	—	317	3	no	—
43	M₁₆	—	317	12	no	—
44	C₂×C₈	—	317	1	no	—
45	C₂×C₈	—	317	3	no	—
46	C₂×C₈	—	317	12	no	—
47	C₂³	—	317	3	no	—
48	Q₈	C₂×#11929639	1	4	yes	—
49	D₄	—	317	12	no	—
50	C₂×C₄	—	317	1	no	—
51	C₂×C₄	—	317	6	no	—
52	C₂×C₄	—	317	6	no	—
53	C₂×C₄	C₂×#11929639	1	3	yes	—
54	C₈	—	317	2	no	—
55	C₈	—	317	6	no	—
56	C₂²	—	317	6	no	—
57	C₂²	—	634	6	no	—
58	C₂²	C₂²×#11929639	1	1	yes	characteristic
59	C₄	—	317	1	no	—
60	C₄	—	317	1	no	—
61	C₄	C₂²×#11929639	1	6	yes	—
62	C₂	—	634	6	no	—
63	C₂	#23778868	1	2	yes	direct product factor
64	C₂	C₂³×#11929639	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×#11929639
commutator	C₂×C₃₁₇	C₂³×C₄
Frattini	C₂	C₂³×#11929639
Fitting	C₂×Q₈×C₃₁₇	C₄
socle	C₂²×C₃₁₇	C₂²×C₄
prime radical	C₂×#19099575	C₂
cosocle	Dic₃₁₇	C₂⁴
hypercenter	C₂×Q₈	#11929639
hypocenter	C₃₁₇	C₂×#131
LCS	G ▷ C₂×C₃₁₇ ▷ C₃₁₇
UCS	C₁ ◁ C₂² ◁ C₂×Q₈

Complements

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

N	H ≅ G/N	type
1. #23778868	C₂	direct (N × H)
2. C₂×#19099584	C₂	semidirect
3. C₂×D₃₁₇⋊C₈	C₂	semidirect
5. #19099584	C₂²	semidirect
6. D₃₁₇⋊C₈	C₂²	semidirect
29. C₂×C₃₁₇	#131	semidirect
32. C₃₁₇	C₂×#131	semidirect
63. C₂	#23778868	direct (N × H)

Maximal subgroups

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

subgroup	index
#23778868	2
C₂×#19099584	2
C₂×D₃₁₇⋊C₈	2
C₂×#19099575	2
C₂×#131	317

Subgroups — C2×#23778868 #35204300

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#23778868 #35204300