Subgroups — #1423×C₁₃₇ #31776434

Order 17536.

56 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₂×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#366×C₁₃₇	C₂	1	2	yes	maximal
3	C₂×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
4	#107×C₁₃₇	—	2	2	no	—
5	C₃₂×C₁₃₇	C₄	1	2	yes	complemented
6	C₂×C₈×C₁₃₇	D₄	1	1	yes	characteristic
7	C₄⋊C₄×C₁₃₇	—	4	2	no	—
8	C₁₆×C₁₃₇	C₂×C₄	1	1	yes	characteristic
9	C₁₆×C₁₃₇	Q₈	1	1	yes	characteristic
10	C₂×C₄×C₁₃₇	—	8	2	no	—
11	C₂×C₄×C₁₃₇	D₈	1	1	yes	characteristic
12	C₈×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
13	C₈×C₁₃₇	Q₁₆	1	1	yes	characteristic
14	C₂²×C₁₃₇	D₁₆	1	1	yes	characteristic
15	C₄×C₁₃₇	—	16	2	no	—
16	C₄×C₁₃₇	#107	1	1	yes	characteristic
17	C₄×C₁₃₇	Q₃₂	1	1	yes	characteristic
18	C₂×C₁₃₇	#366	1	1	yes	characteristic
19	C₂×C₁₃₇	D₃₂	1	1	yes	characteristic
20	C₂×C₁₃₇	Q₆₄	1	1	yes	characteristic
21	C₁₃₇	#1423	1	1	yes	characteristic, direct product factor, Sylow
22	#1423	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
23	C₂×C₃₂	C₂×C₁₃₇	1	1	yes	characteristic
24	#366	C₂×C₁₃₇	1	2	yes	—
25	C₂×C₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
26	#107	—	2	2	no	—
27	C₃₂	C₄×C₁₃₇	1	2	yes	complemented
28	C₂×C₈	D₄×C₁₃₇	1	1	yes	characteristic
29	C₄⋊C₄	—	4	2	no	—
30	C₁₆	C₂×C₄×C₁₃₇	1	1	yes	characteristic
31	C₁₆	Q₈×C₁₃₇	1	1	yes	characteristic
32	C₂×C₄	—	8	2	no	—
33	C₂×C₄	D₈×C₁₃₇	1	1	yes	characteristic
34	C₈	C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
35	C₈	Q₁₆×C₁₃₇	1	1	yes	characteristic
36	C₂²	D₁₆×C₁₃₇	1	1	yes	characteristic
37	C₄	—	16	2	no	—
38	C₄	#107×C₁₃₇	1	1	yes	characteristic
39	C₄	Q₃₂×C₁₃₇	1	1	yes	characteristic
40	C₂	#366×C₁₃₇	1	1	yes	characteristic
41	C₂	D₃₂×C₁₃₇	1	1	yes	characteristic
42	C₂	Q₆₄×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₁₃₇	D₁₆
commutator	C₁₆	C₂×C₄×C₁₃₇
Frattini, cosocle	C₂×C₁₆	C₂²×C₁₃₇
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₂×C₄×C₁₃₇ ◁ C₂×C₈×C₁₃₇ ◁ C₂×C₁₆×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
5. C₃₂×C₁₃₇	C₄	semidirect
21. C₁₃₇	#1423	direct (N × H)
22. #1423	C₁₃₇	direct (N × H)
27. C₃₂	C₄×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₂×C₁₃₇	2
#366×C₁₃₇	2
#1423	137

Subgroups — #1423×C137 #31776434

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1423×C₁₃₇ #31776434