Subgroups — #117×C₃₁₇ #23778658

Order 10144.

76 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₄×C₃₁₇	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₄×C₃₁₇	C₂	1	2	yes	maximal, complemented
3	C₂²⋊C₄×C₃₁₇	C₂	1	2	yes	maximal
4	C₄²×C₃₁₇	C₂	1	2	yes	maximal, complemented
5	C₂³×C₃₁₇	C₂²	1	1	yes	characteristic
6	C₂×C₄×C₃₁₇	C₄	1	4	yes	complemented
7	C₂×C₄×C₃₁₇	C₂²	1	4	yes	—
8	C₂×C₄×C₃₁₇	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₃₁₇	C₂²	1	1	yes	characteristic
10	C₂²×C₃₁₇	—	2	2	no	—
11	C₂²×C₃₁₇	C₂×C₄	1	2	yes	—
12	C₂²×C₃₁₇	C₂³	1	1	yes	characteristic
13	C₄×C₃₁₇	—	2	4	no	—
14	C₄×C₃₁₇	C₂×C₄	1	2	yes	—
15	C₄×C₃₁₇	C₂×C₄	1	2	yes	—
16	C₂×C₃₁₇	—	2	2	no	—
17	C₂×C₃₁₇	C₂²×C₄	1	1	yes	characteristic
18	C₂×C₃₁₇	#41	1	2	yes	—
19	C₃₁₇	#117	1	1	yes	characteristic, direct product factor, Sylow
20	#117	C₃₁₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
21	C₂²×C₄	C₂×C₃₁₇	1	1	yes	characteristic
22	C₄⋊C₄	C₂×C₃₁₇	1	2	yes	complemented
23	C₂²⋊C₄	C₂×C₃₁₇	1	2	yes	—
24	C₄²	C₂×C₃₁₇	1	2	yes	complemented
25	C₂³	C₂²×C₃₁₇	1	1	yes	characteristic
26	C₂×C₄	C₄×C₃₁₇	1	4	yes	complemented
27	C₂×C₄	C₂²×C₃₁₇	1	4	yes	—
28	C₂×C₄	C₂²×C₃₁₇	1	1	yes	characteristic
29	C₂×C₄	C₂²×C₃₁₇	1	1	yes	characteristic
30	C₂²	—	2	2	no	—
31	C₂²	C₂×C₄×C₃₁₇	1	2	yes	—
32	C₂²	C₂³×C₃₁₇	1	1	yes	characteristic
33	C₄	—	2	4	no	—
34	C₄	C₂×C₄×C₃₁₇	1	2	yes	—
35	C₄	C₂×C₄×C₃₁₇	1	2	yes	—
36	C₂	—	2	2	no	—
37	C₂	C₂²×C₄×C₃₁₇	1	1	yes	characteristic
38	C₂	#41×C₃₁₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₃₁₇	C₂²
commutator	C₂	C₂²×C₄×C₃₁₇
Frattini, cosocle	C₂²	C₂³×C₃₁₇
socle	C₂²×C₃₁₇	C₂³
prime radical	C₂³×C₃₁₇	C₂²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₃₁₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₄⋊C₄×C₃₁₇	C₂	semidirect
4. C₄²×C₃₁₇	C₂	semidirect
6. C₂×C₄×C₃₁₇	C₄	semidirect
19. C₃₁₇	#117	direct (N × H)
20. #117	C₃₁₇	direct (N × H)
22. C₄⋊C₄	C₂×C₃₁₇	semidirect
24. C₄²	C₂×C₃₁₇	semidirect
26. C₂×C₄	C₄×C₃₁₇	semidirect

Maximal subgroups

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

subgroup	index
C₂²×C₄×C₃₁₇	2
C₄⋊C₄×C₃₁₇	2
C₂²⋊C₄×C₃₁₇	2
C₄²×C₃₁₇	2
#117	317

Subgroups — #117×C317 #23778658

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #117×C₃₁₇ #23778658