Subgroups — #1417×C₁₃₇ #31776428

Order 17536.

72 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#508×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃₂×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#368×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#135×C₁₃₇	—	2	1	no	—
5	Q₃₂×C₁₃₇	C₄	1	1	yes	characteristic
6	SD₃₂×C₁₃₇	—	2	1	no	—
7	D₁₆×C₁₃₇	C₄	1	1	yes	characteristic
8	C₂×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
9	#108×C₁₃₇	—	2	1	no	—
10	C₃₂×C₁₃₇	—	2	1	no	—
11	#41×C₁₃₇	—	4	1	no	—
12	Q₁₆×C₁₃₇	—	2	1	no	—
13	SD₁₆×C₁₃₇	—	4	1	no	—
14	D₈×C₁₃₇	—	2	1	no	—
15	M₁₆×C₁₃₇	—	4	1	no	—
16	C₂×C₈×C₁₃₇	D₄	1	1	yes	characteristic
17	C₁₆×C₁₃₇	C₂×C₄	1	1	yes	characteristic
18	C₁₆×C₁₃₇	D₄	1	1	yes	characteristic
19	Q₈×C₁₃₇	—	4	1	no	—
20	D₄×C₁₃₇	—	4	1	no	—
21	D₄×C₁₃₇	—	8	1	no	—
22	C₂×C₄×C₁₃₇	—	8	1	no	—
23	C₂×C₄×C₁₃₇	D₈	1	1	yes	characteristic
24	C₈×C₁₃₇	—	8	1	no	—
25	C₈×C₁₃₇	C₂²⋊C₄	1	1	yes	characteristic
26	C₈×C₁₃₇	SD₁₆	1	1	yes	characteristic
27	C₂²×C₁₃₇	—	8	1	no	—
28	C₂²×C₁₃₇	SD₃₂	1	1	yes	characteristic
29	C₄×C₁₃₇	—	8	1	no	—
30	C₄×C₁₃₇	#102	1	1	yes	characteristic
31	C₄×C₁₃₇	D₁₆	1	1	yes	characteristic
32	C₂×C₁₃₇	—	2	1	no	—
33	C₂×C₁₃₇	—	16	1	no	—
34	C₂×C₁₃₇	#357	1	1	yes	characteristic
35	C₁₃₇	#1417	1	1	yes	characteristic, direct product factor, Sylow
36	#1417	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
37	#508	C₂×C₁₃₇	1	1	yes	characteristic
38	C₂×C₃₂	C₂×C₁₃₇	1	1	yes	characteristic, complemented
39	#368	C₂×C₁₃₇	1	1	yes	characteristic, complemented
40	#135	—	2	1	no	—
41	Q₃₂	C₄×C₁₃₇	1	1	yes	characteristic
42	SD₃₂	—	2	1	no	—
43	D₁₆	C₄×C₁₃₇	1	1	yes	characteristic
44	C₂×C₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
45	#108	—	2	1	no	—
46	C₃₂	—	2	1	no	—
47	#41	—	4	1	no	—
48	Q₁₆	—	2	1	no	—
49	SD₁₆	—	4	1	no	—
50	D₈	—	2	1	no	—
51	M₁₆	—	4	1	no	—
52	C₂×C₈	D₄×C₁₃₇	1	1	yes	characteristic
53	C₁₆	C₂×C₄×C₁₃₇	1	1	yes	characteristic
54	C₁₆	D₄×C₁₃₇	1	1	yes	characteristic
55	Q₈	—	4	1	no	—
56	D₄	—	4	1	no	—
57	D₄	—	8	1	no	—
58	C₂×C₄	—	8	1	no	—
59	C₂×C₄	D₈×C₁₃₇	1	1	yes	characteristic
60	C₈	—	8	1	no	—
61	C₈	C₂²⋊C₄×C₁₃₇	1	1	yes	characteristic
62	C₈	SD₁₆×C₁₃₇	1	1	yes	characteristic
63	C₂²	—	8	1	no	—
64	C₂²	SD₃₂×C₁₃₇	1	1	yes	characteristic
65	C₄	—	8	1	no	—
66	C₄	#102×C₁₃₇	1	1	yes	characteristic
67	C₄	D₁₆×C₁₃₇	1	1	yes	characteristic
68	C₂	—	2	1	no	—
69	C₂	—	16	1	no	—
70	C₂	#357×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₃₇	D₁₆
commutator	C₁₆	C₂×C₄×C₁₃₇
Frattini, cosocle	C₂×C₁₆	C₂²×C₁₃₇
socle	C₂×C₁₃₇	#357
prime radical	#508×C₁₃₇	C₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×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
2. C₂×C₃₂×C₁₃₇	C₂	semidirect
3. #368×C₁₃₇	C₂	semidirect
35. C₁₃₇	#1417	direct (N × H)
36. #1417	C₁₃₇	direct (N × H)
38. C₂×C₃₂	C₂×C₁₃₇	semidirect
39. #368	C₂×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
#508×C₁₃₇	2
C₂×C₃₂×C₁₃₇	2
#368×C₁₃₇	2
#1417	137

Subgroups — #1417×C137 #31776428

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1417×C₁₃₇ #31776428