Subgroups — C₂³×#14579261 #35203980

Order 20288.

338 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14579261	C₂	1	14	yes	maximal, direct product factor
2	C₂³×C₄×C₃₁₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#14579261	C₂²	1	28	yes	direct product factor
4	C₂⁴×C₃₁₇	C₄	1	1	yes	characteristic
5	C₂²×C₄×C₃₁₇	C₄	1	7	yes	—
6	C₂²×C₄×C₃₁₇	C₂²	1	7	yes	—
7	#14579261	C₂³	1	8	yes	direct product factor
8	C₂³×C₃₁₇	C₈	1	8	yes	complemented
9	C₂³×C₃₁₇	C₂×C₄	1	7	yes	—
10	C₂×C₄×C₃₁₇	C₂×C₄	1	21	yes	—
11	C₂×C₄×C₃₁₇	C₂³	1	7	yes	—
12	C₂²×C₃₁₇	C₂×C₈	1	28	yes	complemented
13	C₂²×C₃₁₇	C₂²×C₄	1	7	yes	—
14	C₄×C₃₁₇	C₂²×C₄	1	7	yes	—
15	C₄×C₃₁₇	C₂⁴	1	1	yes	characteristic
16	C₂×C₃₁₇	C₂²×C₈	1	14	yes	complemented
17	C₂×C₃₁₇	C₂³×C₄	1	1	yes	characteristic
18	C₃₁₇	C₂³×C₈	1	1	yes	characteristic, complemented, Sylow
19	C₂³×C₈	—	317	1	no	maximal, Sylow
20	C₂³×C₄	D₃₁₇	1	1	yes	characteristic
21	C₂²×C₈	—	317	14	no	—
22	C₂⁴	Dic₃₁₇	1	1	yes	characteristic
23	C₂²×C₄	Dic₃₁₇	1	7	yes	—
24	C₂²×C₄	C₂×D₃₁₇	1	7	yes	—
25	C₂×C₈	—	317	28	no	—
26	C₂³	#14579261	1	8	yes	direct product factor
27	C₂³	C₂×Dic₃₁₇	1	7	yes	—
28	C₂×C₄	C₂×Dic₃₁₇	1	21	yes	—
29	C₂×C₄	C₂²×D₃₁₇	1	7	yes	—
30	C₈	—	317	8	no	—
31	C₂²	C₂×#14579261	1	28	yes	direct product factor
32	C₂²	C₂²×Dic₃₁₇	1	7	yes	—
33	C₄	C₂²×Dic₃₁₇	1	7	yes	—
34	C₄	C₂³×D₃₁₇	1	1	yes	characteristic
35	C₂	C₂²×#14579261	1	14	yes	direct product factor
36	C₂	C₂³×Dic₃₁₇	1	1	yes	characteristic

	subgroup	G/H
center, hypercenter	C₂³×C₄	D₃₁₇
commutator, hypocenter	C₃₁₇	C₂³×C₈
Frattini	C₄	C₂³×D₃₁₇
Fitting	C₂³×C₄×C₃₁₇	C₂
socle, prime radical	C₂⁴×C₃₁₇	C₄
cosocle	C₄×C₃₁₇	C₂⁴
LCS	G ▷ C₃₁₇
UCS	C₁ ◁ C₂³×C₄

Complements

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

N	H ≅ G/N	type
1. C₂²×#14579261	C₂	direct (N × H)
3. C₂×#14579261	C₂²	direct (N × H)
7. #14579261	C₂³	direct (N × H)
8. C₂³×C₃₁₇	C₈	semidirect
12. C₂²×C₃₁₇	C₂×C₈	semidirect
16. C₂×C₃₁₇	C₂²×C₈	semidirect
18. C₃₁₇	C₂³×C₈	semidirect
26. C₂³	#14579261	direct (N × H)
31. C₂²	C₂×#14579261	direct (N × H)
35. C₂	C₂²×#14579261	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×#14579261	2
C₂³×C₄×C₃₁₇	2
C₂³×C₈	317

Subgroups — C23×#14579261 #35203980

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×#14579261 #35203980