Subgroups — #1422×C₁₃₇ #31776433

Order 17536.

58 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	M₆₄×C₁₃₇	C₂	1	2	yes	maximal
2	C₄×C₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₆×C₁₃₇	C₄	1	1	yes	characteristic
4	C₂×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
5	C₄×C₈×C₁₃₇	C₄	1	1	yes	characteristic
6	C₃₂×C₁₃₇	—	2	2	no	—
7	C₂×C₈×C₁₃₇	C₈	1	1	yes	characteristic
8	C₂×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
9	C₄²×C₁₃₇	C₈	1	1	yes	characteristic
10	C₁₆×C₁₃₇	—	2	1	no	—
11	C₁₆×C₁₃₇	D₄	1	1	yes	characteristic
12	C₁₆×C₁₃₇	Q₈	1	1	yes	characteristic
13	C₂×C₄×C₁₃₇	—	2	1	no	—
14	C₂×C₄×C₁₃₇	C₂×C₈	1	1	yes	characteristic
15	C₈×C₁₃₇	C₁₆	1	2	yes	—
16	C₈×C₁₃₇	C₄⋊C₄	1	1	yes	characteristic
17	C₈×C₁₃₇	M₁₆	1	1	yes	characteristic
18	C₂²×C₁₃₇	M₃₂	1	1	yes	characteristic
19	C₄×C₁₃₇	—	2	2	no	—
20	C₄×C₁₃₇	C₄⋊C₈	1	1	yes	characteristic
21	C₄×C₁₃₇	C₂×C₁₆	1	1	yes	characteristic
22	C₂×C₁₃₇	—	2	1	no	—
23	C₂×C₁₃₇	C₄⋊C₁₆	1	1	yes	characteristic
24	C₁₃₇	#1422	1	1	yes	characteristic, direct product factor, Sylow
25	#1422	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
26	M₆₄	C₂×C₁₃₇	1	2	yes	—
27	C₄×C₁₆	C₂×C₁₃₇	1	1	yes	characteristic
28	C₂×C₁₆	C₄×C₁₃₇	1	1	yes	characteristic
29	C₂×C₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
30	C₄×C₈	C₄×C₁₃₇	1	1	yes	characteristic
31	C₃₂	—	2	2	no	—
32	C₂×C₈	C₈×C₁₃₇	1	1	yes	characteristic
33	C₂×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
34	C₄²	C₈×C₁₃₇	1	1	yes	characteristic
35	C₁₆	—	2	1	no	—
36	C₁₆	D₄×C₁₃₇	1	1	yes	characteristic
37	C₁₆	Q₈×C₁₃₇	1	1	yes	characteristic
38	C₂×C₄	—	2	1	no	—
39	C₂×C₄	C₂×C₈×C₁₃₇	1	1	yes	characteristic
40	C₈	C₁₆×C₁₃₇	1	2	yes	—
41	C₈	C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
42	C₈	M₁₆×C₁₃₇	1	1	yes	characteristic
43	C₂²	M₃₂×C₁₃₇	1	1	yes	characteristic
44	C₄	—	2	2	no	—
45	C₄	C₄⋊C₈×C₁₃₇	1	1	yes	characteristic
46	C₄	C₂×C₁₆×C₁₃₇	1	1	yes	characteristic
47	C₂	—	2	1	no	—
48	C₂	C₄⋊C₁₆×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆×C₁₃₇	D₄
commutator	C₄	C₂×C₁₆×C₁₃₇
Frattini, cosocle	C₂×C₁₆	C₂²×C₁₃₇
socle	C₂×C₁₃₇	C₄⋊C₁₆
prime radical	C₂²×C₁₃₇	M₃₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆×C₁₃₇ ◁ C₂×C₁₆×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
24. C₁₃₇	#1422	direct (N × H)
25. #1422	C₁₃₇	direct (N × H)

Maximal subgroups

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

subgroup	index
M₆₄×C₁₃₇	2
C₄×C₁₆×C₁₃₇	2
#1422	137

Subgroups — #1422×C137 #31776433

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1422×C₁₃₇ #31776433