Group #46858195 — subgroups

Subgroups — C₁₃₇×#8809 #46858195

Order 35072.

402 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 222 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#1842×C₁₃₇	C₂	1	2	yes	maximal
2	#1744×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	C₂×#334×C₁₃₇	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₈×C₁₃₇	—	2	4	no	—
5	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	4	yes	—
6	C₂×C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
7	C₄³×C₁₃₇	C₂²	1	1	yes	characteristic
8	#336×C₁₃₇	C₄	1	4	yes	—
9	#334×C₁₃₇	—	2	8	no	—
10	C₂²×C₈×C₁₃₇	—	2	4	no	—
11	C₂²×C₈×C₁₃₇	C₂×C₄	1	4	yes	—
12	C₂×C₄²×C₁₃₇	C₂×C₄	1	2	yes	—
13	C₂×C₄²×C₁₃₇	D₄	1	2	yes	—
14	C₂×C₄²×C₁₃₇	Q₈	1	2	yes	—
15	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
16	C₄⋊C₈×C₁₃₇	—	2	8	no	—
17	C₄⋊C₈×C₁₃₇	—	4	8	no	—
18	C₄×C₈×C₁₃₇	D₄	1	4	yes	—
19	C₄×C₈×C₁₃₇	Q₈	1	4	yes	—
20	C₂²×C₄×C₁₃₇	C₄⋊C₄	1	4	yes	—
21	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
23	C₂²×C₄×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
24	C₂×C₈×C₁₃₇	—	2	8	no	—
25	C₂×C₈×C₁₃₇	—	4	4	no	—
26	C₂×C₈×C₁₃₇	—	4	8	no	—
27	C₂×C₈×C₁₃₇	C₄⋊C₄	1	8	yes	—
28	C₄²×C₁₃₇	—	2	4	no	—
29	C₄²×C₁₃₇	—	4	2	no	—
30	C₄²×C₁₃₇	SD₁₆	1	8	yes	—
31	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
32	C₄²×C₁₃₇	C₂×Q₈	1	2	yes	—
33	C₂³×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₁₃₇	—	2	4	no	—
35	C₂×C₄×C₁₃₇	—	2	4	no	—
36	C₂×C₄×C₁₃₇	—	2	2	no	—
37	C₂×C₄×C₁₃₇	#106	1	8	yes	—
38	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
39	C₂×C₄×C₁₃₇	#127	1	1	yes	characteristic
40	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	2	yes	—
41	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
42	C₂×C₄×C₁₃₇	#131	1	4	yes	—
43	C₂×C₄×C₁₃₇	C₂×SD₁₆	1	4	yes	—
44	C₈×C₁₃₇	—	2	8	no	—
45	C₈×C₁₃₇	—	8	4	no	—
46	C₂²×C₁₃₇	#384	1	1	yes	characteristic
47	C₂²×C₁₃₇	#424	1	2	yes	—
48	C₂²×C₁₃₇	C₂×#106	1	2	yes	—
49	C₂²×C₁₃₇	#430	1	2	yes	—
50	C₄×C₁₃₇	—	2	8	no	—
51	C₄×C₁₃₇	—	2	2	no	—
52	C₄×C₁₃₇	#424	1	4	yes	—
53	C₄×C₁₃₇	#492	1	2	yes	—
54	C₄×C₁₃₇	#498	1	2	yes	—
55	C₂×C₁₃₇	#1570	1	4	yes	—
56	C₂×C₁₃₇	#1843	1	1	yes	characteristic
57	C₂×C₁₃₇	#1844	1	1	yes	characteristic
58	C₂×C₁₃₇	#1849	1	1	yes	characteristic
59	#8809	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₁₃₇	#8809	1	1	yes	characteristic, direct product factor, Sylow
61	#1842	C₂×C₁₃₇	1	2	yes	—
62	#1744	C₂×C₁₃₇	1	1	yes	characteristic
63	C₂×#334	C₂×C₁₃₇	1	4	yes	—
64	C₂×C₄⋊C₈	—	2	4	no	—
65	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	4	yes	—
66	C₂×C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
67	C₄³	C₂²×C₁₃₇	1	1	yes	characteristic
68	#336	C₄×C₁₃₇	1	4	yes	—
69	#334	—	2	8	no	—
70	C₂²×C₈	—	2	4	no	—
71	C₂²×C₈	C₂×C₄×C₁₃₇	1	4	yes	—
72	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
73	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
74	C₂×C₄²	Q₈×C₁₃₇	1	2	yes	—
75	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
76	C₄⋊C₈	—	2	8	no	—
77	C₄⋊C₈	—	4	8	no	—
78	C₄×C₈	D₄×C₁₃₇	1	4	yes	—
79	C₄×C₈	Q₈×C₁₃₇	1	4	yes	—
80	C₂²×C₄	C₄⋊C₄×C₁₃₇	1	4	yes	—
81	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
82	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
83	C₂²×C₄	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
84	C₂×C₈	—	2	8	no	—
85	C₂×C₈	—	4	4	no	—
86	C₂×C₈	—	4	8	no	—
87	C₂×C₈	C₄⋊C₄×C₁₃₇	1	8	yes	—
88	C₄²	—	2	4	no	—
89	C₄²	—	4	2	no	—
90	C₄²	SD₁₆×C₁₃₇	1	8	yes	—
91	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
92	C₄²	C₂×Q₈×C₁₃₇	1	2	yes	—
93	C₂³	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	2	2	no	—
97	C₂×C₄	#106×C₁₃₇	1	8	yes	—
98	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
99	C₂×C₄	#127×C₁₃₇	1	1	yes	characteristic
100	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	2	yes	—
101	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
102	C₂×C₄	#131×C₁₃₇	1	4	yes	—
103	C₂×C₄	C₂×SD₁₆×C₁₃₇	1	4	yes	—
104	C₈	—	2	8	no	—
105	C₈	—	8	4	no	—
106	C₂²	#384×C₁₃₇	1	1	yes	characteristic
107	C₂²	#424×C₁₃₇	1	2	yes	—
108	C₂²	C₂×#106×C₁₃₇	1	2	yes	—
109	C₂²	#430×C₁₃₇	1	2	yes	—
110	C₄	—	2	8	no	—
111	C₄	—	2	2	no	—
112	C₄	#424×C₁₃₇	1	4	yes	—
113	C₄	#492×C₁₃₇	1	2	yes	—
114	C₄	#498×C₁₃₇	1	2	yes	—
115	C₂	#1570×C₁₃₇	1	4	yes	—
116	C₂	#1843×C₁₃₇	1	1	yes	characteristic
117	C₂	#1844×C₁₃₇	1	1	yes	characteristic
118	C₂	#1849×C₁₃₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#1842×C₁₃₇

C₂

yes

maximal

#1744×C₁₃₇

C₂

yes

maximal, characteristic

C₂×#334×C₁₃₇

C₂

yes

maximal

C₂×C₄⋊C₈×C₁₃₇

—

C₂×C₄⋊C₈×C₁₃₇

C₂²

yes

—

C₂×C₄×C₈×C₁₃₇

C₂²

yes

—

C₄³×C₁₃₇

C₂²

yes

characteristic

#336×C₁₃₇

C₄

yes

—

#334×C₁₃₇

—

C₂²×C₈×C₁₃₇

—

C₂²×C₈×C₁₃₇

C₂×C₄

yes

—

C₂×C₄²×C₁₃₇

C₂×C₄

yes

—

C₂×C₄²×C₁₃₇

D₄

yes

—

C₂×C₄²×C₁₃₇

Q₈

yes

—

C₂×C₄²×C₁₃₇

C₂³

yes

characteristic

C₄⋊C₈×C₁₃₇

—

C₄⋊C₈×C₁₃₇

—

C₄×C₈×C₁₃₇

D₄

yes

—

C₄×C₈×C₁₃₇

Q₈

yes

—

C₂²×C₄×C₁₃₇

C₄⋊C₄

yes

—

C₂²×C₄×C₁₃₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₁₃₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₁₃₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

C₄⋊C₄

yes

—

C₄²×C₁₃₇

—

C₄²×C₁₃₇

—

C₄²×C₁₃₇

SD₁₆

yes

—

C₄²×C₁₃₇

C₂×D₄

yes

—

C₄²×C₁₃₇

C₂×Q₈

yes

—

C₂³×C₁₃₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

#106

yes

—

C₂×C₄×C₁₃₇

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₁₃₇

#127

yes

characteristic

C₂×C₄×C₁₃₇

C₄⋊Q₈

yes

—

C₂×C₄×C₁₃₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₁₃₇

#131

yes

—

C₂×C₄×C₁₃₇

C₂×SD₁₆

yes

—

C₈×C₁₃₇

—

C₈×C₁₃₇

—

C₂²×C₁₃₇

#384

yes

characteristic

C₂²×C₁₃₇

#424

yes

—

C₂²×C₁₃₇

C₂×#106

yes

—

C₂²×C₁₃₇

#430

yes

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

#424

yes

—

C₄×C₁₃₇

#492

yes

—

C₄×C₁₃₇

#498

yes

—

C₂×C₁₃₇

#1570

yes

—

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

C₁₃₇

yes

maximal, characteristic, direct product factor, Sylow

C₁₃₇

#8809

yes

characteristic, direct product factor, Sylow

#1842

C₂×C₁₃₇

yes

—

#1744

C₂×C₁₃₇

yes

characteristic

C₂×#334

C₂×C₁₃₇

yes

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂²×C₁₃₇

yes

—

C₂×C₄×C₈

C₂²×C₁₃₇

yes

—

C₄³

C₂²×C₁₃₇

yes

characteristic

#336

C₄×C₁₃₇

yes

—

#334

—

C₂²×C₈

—

C₂²×C₈

C₂×C₄×C₁₃₇

yes

—

C₂×C₄²

C₂×C₄×C₁₃₇

yes

—

C₂×C₄²

D₄×C₁₃₇

yes

—

C₂×C₄²

Q₈×C₁₃₇

yes

—

C₂×C₄²

C₂³×C₁₃₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

D₄×C₁₃₇

yes

—

C₄×C₈

Q₈×C₁₃₇

yes

—

C₂²×C₄

C₄⋊C₄×C₁₃₇

yes

—

C₂²×C₄

C₂²×C₄×C₁₃₇

yes

characteristic

C₂²×C₄

C₂×D₄×C₁₃₇

yes

characteristic

C₂²×C₄

C₂×Q₈×C₁₃₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊C₄×C₁₃₇

yes

—

C₄²

—

C₄²

—

C₄²

SD₁₆×C₁₃₇

yes

—

C₄²

C₂×D₄×C₁₃₇

yes

—

C₄²

C₂×Q₈×C₁₃₇

yes

—

C₂³

C₂×C₄⋊C₄×C₁₃₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#106×C₁₃₇

yes

—

C₂×C₄

C₂×C₄⋊C₄×C₁₃₇

yes

—

C₂×C₄

#127×C₁₃₇

yes

characteristic

100

C₂×C₄

C₄⋊Q₈×C₁₃₇

yes

—

101

C₂×C₄

C₄⋊Q₈×C₁₃₇

yes

characteristic

102

C₂×C₄

#131×C₁₃₇

yes

—

103

C₂×C₄

C₂×SD₁₆×C₁₃₇

yes

—

104

C₈

—

105

C₈

—

106

C₂²

#384×C₁₃₇

yes

characteristic

107

C₂²

#424×C₁₃₇

yes

—

108

C₂²

C₂×#106×C₁₃₇

yes

—

109

C₂²

#430×C₁₃₇

yes

—

110

C₄

—

111

C₄

—

112

C₄

#424×C₁₃₇

yes

—

113

C₄

#492×C₁₃₇

yes

—

114

C₄

#498×C₁₃₇

yes

—

115

C₂

#1570×C₁₃₇

yes

—

116

C₂

#1843×C₁₃₇

yes

characteristic

117

C₂

#1844×C₁₃₇

yes

characteristic

118

C₂

#1849×C₁₃₇

yes

characteristic

	subgroup	G/H
center	C₂²×C₄×C₁₃₇	C₂×D₄
commutator	C₂²×C₄	C₂²×C₄×C₁₃₇
Frattini, cosocle	C₂×C₄²	C₂³×C₁₃₇
socle, prime radical	C₂³×C₁₃₇	C₂×C₄⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃₇ ◁ C₄³×C₁₃₇ ◁ G

subgroup

G/H

center

C₂²×C₄×C₁₃₇

C₂×D₄

commutator

C₂²×C₄

C₂²×C₄×C₁₃₇

Frattini, cosocle

C₂×C₄²

C₂³×C₁₃₇

socle, prime radical

C₂³×C₁₃₇

C₂×C₄⋊C₄

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₄×C₁₃₇ ◁ C₄³×C₁₃₇ ◁ G

N	H ≅ G/N	type
59. #8809	C₁₃₇	direct (N × H)
60. C₁₃₇	#8809	direct (N × H)

H ≅ G/N

type

59. #8809

C₁₃₇

direct (N × H)

60. C₁₃₇

#8809

direct (N × H)

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

subgroup	index
#1842×C₁₃₇	2
#1744×C₁₃₇	2
C₂×#334×C₁₃₇	2
#8809	137

subgroup

index

#1842×C₁₃₇

#1744×C₁₃₇

C₂×#334×C₁₃₇

#8809

137

Subgroups — C₁₃₇×#8809 #46858195

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C137×#8809 #46858195

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#8809 #46858195