Subgroups — #1355×C₁₃₇ #31776366

Order 17536.

88 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#490×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#347×C₁₃₇	C₂	1	2	yes	maximal, complemented
3	#127×C₁₃₇	C₄	1	1	yes	characteristic
4	#125×C₁₃₇	C₄	1	1	yes	characteristic
5	M₃₂×C₁₃₇	—	2	2	no	—
6	#102×C₁₃₇	—	2	2	no	—
7	#97×C₁₃₇	C₂²	1	1	yes	characteristic
8	C₂×D₄×C₁₃₇	—	2	1	no	—
9	C₂×D₄×C₁₃₇	—	4	1	no	—
10	C₂×C₈×C₁₃₇	D₄	1	2	yes	—
11	C₄⋊C₄×C₁₃₇	—	2	1	no	—
12	C₄⋊C₄×C₁₃₇	—	4	1	no	—
13	C₄²×C₁₃₇	C₂×C₄	1	1	yes	characteristic
14	C₁₆×C₁₃₇	—	2	4	no	—
15	C₂³×C₁₃₇	—	4	1	no	—
16	D₄×C₁₃₇	—	4	2	no	—
17	D₄×C₁₃₇	—	8	2	no	—
18	C₂×C₄×C₁₃₇	—	2	1	no	—
19	C₂×C₄×C₁₃₇	—	4	1	no	—
20	C₂×C₄×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
21	C₈×C₁₃₇	—	2	2	no	—
22	C₂²×C₁₃₇	—	8	2	no	—
23	C₂²×C₁₃₇	—	8	1	no	—
24	C₂²×C₁₃₇	#100	1	1	yes	characteristic
25	C₄×C₁₃₇	—	4	1	no	—
26	C₄×C₁₃₇	—	8	1	no	—
27	C₄×C₁₃₇	#104	1	2	yes	—
28	C₂×C₁₃₇	—	2	1	no	—
29	C₂×C₁₃₇	—	16	1	no	—
30	C₂×C₁₃₇	#329	1	1	yes	characteristic
31	C₁₃₇	#1355	1	1	yes	characteristic, direct product factor, Sylow
32	#1355	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
33	#490	C₂×C₁₃₇	1	1	yes	characteristic
34	#347	C₂×C₁₃₇	1	2	yes	complemented
35	#127	C₄×C₁₃₇	1	1	yes	characteristic
36	#125	C₄×C₁₃₇	1	1	yes	characteristic
37	M₃₂	—	2	2	no	—
38	#102	—	2	2	no	—
39	#97	C₂²×C₁₃₇	1	1	yes	characteristic
40	C₂×D₄	—	2	1	no	—
41	C₂×D₄	—	4	1	no	—
42	C₂×C₈	D₄×C₁₃₇	1	2	yes	—
43	C₄⋊C₄	—	2	1	no	—
44	C₄⋊C₄	—	4	1	no	—
45	C₄²	C₂×C₄×C₁₃₇	1	1	yes	characteristic
46	C₁₆	—	2	4	no	—
47	C₂³	—	4	1	no	—
48	D₄	—	4	2	no	—
49	D₄	—	8	2	no	—
50	C₂×C₄	—	2	1	no	—
51	C₂×C₄	—	4	1	no	—
52	C₂×C₄	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
53	C₈	—	2	2	no	—
54	C₂²	—	8	2	no	—
55	C₂²	—	8	1	no	—
56	C₂²	#100×C₁₃₇	1	1	yes	characteristic
57	C₄	—	4	1	no	—
58	C₄	—	8	1	no	—
59	C₄	#104×C₁₃₇	1	2	yes	—
60	C₂	—	2	1	no	—
61	C₂	—	16	1	no	—
62	C₂	#329×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₃₇	#329
commutator	C₄²	C₂×C₄×C₁₃₇
Frattini, cosocle	#97	C₂²×C₁₃₇
prime radical	#127×C₁₃₇	C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃₇ ◁ C₂×C₄×C₁₃₇ ◁ #97×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
2. #347×C₁₃₇	C₂	semidirect
31. C₁₃₇	#1355	direct (N × H)
32. #1355	C₁₃₇	direct (N × H)
34. #347	C₂×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#490×C₁₃₇	2
#347×C₁₃₇	2
#1355	137

Subgroups — #1355×C137 #31776366

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1355×C₁₃₇ #31776366