Subgroups — #472×C₁₃₇ #22577007

Order 8768.

108 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	AGL(1,ℤ/8)×C₁₃₇	C₂	1	2	yes	maximal, complemented
2	C₂×D₈×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#131×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#108×C₁₃₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#100×C₁₃₇	C₂	1	2	yes	maximal, complemented
6	#41×C₁₃₇	C₂²	1	1	yes	characteristic
7	C₂×D₄×C₁₃₇	C₂²	1	2	yes	—
8	SD₁₆×C₁₃₇	—	2	2	no	—
9	D₈×C₁₃₇	—	2	2	no	—
10	D₈×C₁₃₇	—	2	1	no	—
11	D₈×C₁₃₇	—	2	1	no	—
12	M₁₆×C₁₃₇	—	2	1	no	—
13	M₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
14	M₁₆×C₁₃₇	C₂²	1	2	yes	—
15	C₂×C₈×C₁₃₇	—	2	1	no	—
16	C₂×C₈×C₁₃₇	C₂²	1	1	yes	characteristic
17	C₂³×C₁₃₇	—	2	2	no	—
18	Q₈×C₁₃₇	D₄	1	1	yes	characteristic
19	D₄×C₁₃₇	—	2	1	no	—
20	D₄×C₁₃₇	—	2	2	no	—
21	D₄×C₁₃₇	—	2	2	no	—
22	D₄×C₁₃₇	D₄	1	1	yes	characteristic
23	C₂×C₄×C₁₃₇	—	2	1	no	—
24	C₂×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
25	C₈×C₁₃₇	—	2	1	no	—
26	C₈×C₁₃₇	—	2	2	no	—
27	C₈×C₁₃₇	D₄	1	1	yes	characteristic
28	C₈×C₁₃₇	D₄	1	1	yes	characteristic
29	C₂²×C₁₃₇	—	2	1	no	—
30	C₂²×C₁₃₇	—	4	2	no	—
31	C₂²×C₁₃₇	—	8	2	no	—
32	C₂²×C₁₃₇	#41	1	1	yes	characteristic
33	C₄×C₁₃₇	—	2	1	no	—
34	C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
35	C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
36	C₂×C₁₃₇	—	2	1	no	—
37	C₂×C₁₃₇	—	4	1	no	—
38	C₂×C₁₃₇	—	8	2	no	—
39	C₂×C₁₃₇	#121	1	1	yes	characteristic
40	C₁₃₇	#472	1	1	yes	characteristic, direct product factor, Sylow
41	#472	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
42	AGL(1,ℤ/8)	C₂×C₁₃₇	1	2	yes	complemented
43	C₂×D₈	C₂×C₁₃₇	1	1	yes	characteristic, complemented
44	#131	C₂×C₁₃₇	1	1	yes	characteristic, complemented
45	#108	C₂×C₁₃₇	1	1	yes	characteristic, complemented
46	#100	C₂×C₁₃₇	1	2	yes	complemented
47	#41	C₂²×C₁₃₇	1	1	yes	characteristic
48	C₂×D₄	C₂²×C₁₃₇	1	2	yes	—
49	SD₁₆	—	2	2	no	—
50	D₈	—	2	2	no	—
51	D₈	—	2	1	no	—
52	D₈	—	2	1	no	—
53	M₁₆	—	2	1	no	—
54	M₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
55	M₁₆	C₂²×C₁₃₇	1	2	yes	—
56	C₂×C₈	—	2	1	no	—
57	C₂×C₈	C₂²×C₁₃₇	1	1	yes	characteristic
58	C₂³	—	2	2	no	—
59	Q₈	D₄×C₁₃₇	1	1	yes	characteristic
60	D₄	—	2	1	no	—
61	D₄	—	2	2	no	—
62	D₄	—	2	2	no	—
63	D₄	D₄×C₁₃₇	1	1	yes	characteristic
64	C₂×C₄	—	2	1	no	—
65	C₂×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
66	C₈	—	2	1	no	—
67	C₈	—	2	2	no	—
68	C₈	D₄×C₁₃₇	1	1	yes	characteristic
69	C₈	D₄×C₁₃₇	1	1	yes	characteristic
70	C₂²	—	2	1	no	—
71	C₂²	—	4	2	no	—
72	C₂²	—	8	2	no	—
73	C₂²	#41×C₁₃₇	1	1	yes	characteristic
74	C₄	—	2	1	no	—
75	C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
76	C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
77	C₂	—	2	1	no	—
78	C₂	—	4	1	no	—
79	C₂	—	8	2	no	—
80	C₂	#121×C₁₃₇	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₃₇	#121
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₁₃₇
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃₇ ◁ C₂×C₄×C₁₃₇ ◁ G

Complements

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

N	H ≅ G/N	type
1. AGL(1,ℤ/8)×C₁₃₇	C₂	semidirect
2. C₂×D₈×C₁₃₇	C₂	semidirect
3. #131×C₁₃₇	C₂	semidirect
4. #108×C₁₃₇	C₂	semidirect
5. #100×C₁₃₇	C₂	semidirect
40. C₁₃₇	#472	direct (N × H)
41. #472	C₁₃₇	direct (N × H)
42. AGL(1,ℤ/8)	C₂×C₁₃₇	semidirect
43. C₂×D₈	C₂×C₁₃₇	semidirect
44. #131	C₂×C₁₃₇	semidirect
45. #108	C₂×C₁₃₇	semidirect
46. #100	C₂×C₁₃₇	semidirect

Maximal subgroups

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

subgroup	index
AGL(1,ℤ/8)×C₁₃₇	2
C₂×D₈×C₁₃₇	2
#131×C₁₃₇	2
#108×C₁₃₇	2
#100×C₁₃₇	2
#472	137

Subgroups — #472×C137 #22577007

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #472×C₁₃₇ #22577007