Subgroups — C₁₃₇×#7332 #46856718

Order 35072.

106 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1397×C₁₃₇	C₂	1	2	yes	maximal
2	#1311×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₂×C₁₃₇	C₄	1	4	yes	—
4	C₄×C₁₆×C₁₃₇	C₄	1	1	yes	characteristic
5	C₄×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
6	C₈²×C₁₃₇	C₄	1	1	yes	characteristic
7	C₂×C₁₆×C₁₃₇	C₈	1	2	yes	—
8	C₂×C₁₆×C₁₃₇	C₂×C₄	1	2	yes	—
9	C₄×C₈×C₁₃₇	C₈	1	2	yes	—
10	C₄×C₈×C₁₃₇	C₂×C₄	1	1	yes	characteristic
11	C₃₂×C₁₃₇	—	2	4	no	—
12	C₂×C₈×C₁₃₇	—	2	2	no	—
13	C₂×C₈×C₁₃₇	C₄²	1	1	yes	characteristic
14	C₂×C₈×C₁₃₇	C₂×C₈	1	1	yes	characteristic
15	C₄²×C₁₃₇	C₂×C₈	1	1	yes	characteristic
16	C₁₆×C₁₃₇	C₁₆	1	4	yes	—
17	C₁₆×C₁₃₇	M₁₆	1	4	yes	—
18	C₂×C₄×C₁₃₇	C₄×C₈	1	1	yes	characteristic
19	C₂×C₄×C₁₃₇	M₃₂	1	2	yes	—
20	C₈×C₁₃₇	—	4	2	no	—
21	C₈×C₁₃₇	#97	1	2	yes	—
22	C₈×C₁₃₇	C₂×C₁₆	1	2	yes	—
23	C₂²×C₁₃₇	#346	1	1	yes	characteristic
24	C₄×C₁₃₇	—	2	2	no	—
25	C₄×C₁₃₇	#322	1	1	yes	characteristic
26	C₄×C₁₃₇	C₄×C₁₆	1	1	yes	characteristic
27	C₂×C₁₃₇	#1312	1	1	yes	characteristic
28	C₂×C₁₃₇	#1398	1	2	yes	—
29	#7332	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₁₃₇	#7332	1	1	yes	characteristic, direct product factor, Sylow
31	#1397	C₂×C₁₃₇	1	2	yes	—
32	#1311	C₂×C₁₃₇	1	1	yes	characteristic
33	C₂×C₃₂	C₄×C₁₃₇	1	4	yes	—
34	C₄×C₁₆	C₄×C₁₃₇	1	1	yes	characteristic
35	C₄×C₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
36	C₈²	C₄×C₁₃₇	1	1	yes	characteristic
37	C₂×C₁₆	C₈×C₁₃₇	1	2	yes	—
38	C₂×C₁₆	C₂×C₄×C₁₃₇	1	2	yes	—
39	C₄×C₈	C₈×C₁₃₇	1	2	yes	—
40	C₄×C₈	C₂×C₄×C₁₃₇	1	1	yes	characteristic
41	C₃₂	—	2	4	no	—
42	C₂×C₈	—	2	2	no	—
43	C₂×C₈	C₄²×C₁₃₇	1	1	yes	characteristic
44	C₂×C₈	C₂×C₈×C₁₃₇	1	1	yes	characteristic
45	C₄²	C₂×C₈×C₁₃₇	1	1	yes	characteristic
46	C₁₆	C₁₆×C₁₃₇	1	4	yes	—
47	C₁₆	M₁₆×C₁₃₇	1	4	yes	—
48	C₂×C₄	C₄×C₈×C₁₃₇	1	1	yes	characteristic
49	C₂×C₄	M₃₂×C₁₃₇	1	2	yes	—
50	C₈	—	4	2	no	—
51	C₈	#97×C₁₃₇	1	2	yes	—
52	C₈	C₂×C₁₆×C₁₃₇	1	2	yes	—
53	C₂²	#346×C₁₃₇	1	1	yes	characteristic
54	C₄	—	2	2	no	—
55	C₄	#322×C₁₃₇	1	1	yes	characteristic
56	C₄	C₄×C₁₆×C₁₃₇	1	1	yes	characteristic
57	C₂	#1312×C₁₃₇	1	1	yes	characteristic
58	C₂	#1398×C₁₃₇	1	2	yes	—

	subgroup	G/H
center	C₂×C₈×C₁₃₇	C₄²
commutator	C₄	C₄×C₁₆×C₁₃₇
Frattini, cosocle	C₄×C₁₆	C₂²×C₁₃₇
socle, prime radical	C₂²×C₁₃₇	#346
LCS	G ▷ C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₈×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
29. #7332	C₁₃₇	direct (N × H)
30. C₁₃₇	#7332	direct (N × H)

Maximal subgroups

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

subgroup	index
#1397×C₁₃₇	2
#1311×C₁₃₇	2
#7332	137

Subgroups — C137×#7332 #46856718

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#7332 #46856718