Group #46858196 — subgroups

Subgroups — C₁₃₇×#8810 #46858196

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×#335×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	#335×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	D₈	1	4	yes	—
31	C₄²×C₁₃₇	Q₁₆	1	4	yes	—
32	C₄²×C₁₃₇	C₂×D₄	1	2	yes	—
33	C₄²×C₁₃₇	C₂×Q₈	1	2	yes	—
34	C₂³×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₁₃₇	—	2	4	no	—
36	C₂×C₄×C₁₃₇	—	2	4	no	—
37	C₂×C₄×C₁₃₇	—	2	2	no	—
38	C₂×C₄×C₁₃₇	#107	1	8	yes	—
39	C₂×C₄×C₁₃₇	C₂×C₄⋊C₄	1	2	yes	—
40	C₂×C₄×C₁₃₇	#127	1	1	yes	characteristic
41	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	2	yes	—
42	C₂×C₄×C₁₃₇	C₄⋊Q₈	1	1	yes	characteristic
43	C₂×C₄×C₁₃₇	#131	1	4	yes	—
44	C₂×C₄×C₁₃₇	C₂×D₈	1	2	yes	—
45	C₂×C₄×C₁₃₇	C₂×Q₁₆	1	2	yes	—
46	C₈×C₁₃₇	—	2	8	no	—
47	C₈×C₁₃₇	—	8	4	no	—
48	C₂²×C₁₃₇	#384	1	1	yes	characteristic
49	C₂²×C₁₃₇	#424	1	2	yes	—
50	C₂²×C₁₃₇	C₂×#107	1	2	yes	—
51	C₂²×C₁₃₇	#430	1	2	yes	—
52	C₄×C₁₃₇	—	2	8	no	—
53	C₄×C₁₃₇	—	2	2	no	—
54	C₄×C₁₃₇	#424	1	4	yes	—
55	C₄×C₁₃₇	#493	1	1	yes	characteristic
56	C₄×C₁₃₇	#494	1	1	yes	characteristic
57	C₄×C₁₃₇	#500	1	2	yes	—
58	C₂×C₁₃₇	#1571	1	4	yes	—
59	C₂×C₁₃₇	#1843	1	1	yes	characteristic
60	C₂×C₁₃₇	#1845	1	1	yes	characteristic
61	C₂×C₁₃₇	#1849	1	1	yes	characteristic
62	#8810	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₁₃₇	#8810	1	1	yes	characteristic, direct product factor, Sylow
64	#1842	C₂×C₁₃₇	1	2	yes	—
65	#1744	C₂×C₁₃₇	1	1	yes	characteristic
66	C₂×#335	C₂×C₁₃₇	1	4	yes	—
67	C₂×C₄⋊C₈	—	2	4	no	—
68	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	4	yes	—
69	C₂×C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
70	C₄³	C₂²×C₁₃₇	1	1	yes	characteristic
71	#336	C₄×C₁₃₇	1	4	yes	—
72	#335	—	2	8	no	—
73	C₂²×C₈	—	2	4	no	—
74	C₂²×C₈	C₂×C₄×C₁₃₇	1	4	yes	—
75	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
76	C₂×C₄²	D₄×C₁₃₇	1	2	yes	—
77	C₂×C₄²	Q₈×C₁₃₇	1	2	yes	—
78	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
79	C₄⋊C₈	—	2	8	no	—
80	C₄⋊C₈	—	4	8	no	—
81	C₄×C₈	D₄×C₁₃₇	1	4	yes	—
82	C₄×C₈	Q₈×C₁₃₇	1	4	yes	—
83	C₂²×C₄	C₄⋊C₄×C₁₃₇	1	4	yes	—
84	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
85	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
86	C₂²×C₄	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
87	C₂×C₈	—	2	8	no	—
88	C₂×C₈	—	4	4	no	—
89	C₂×C₈	—	4	8	no	—
90	C₂×C₈	C₄⋊C₄×C₁₃₇	1	8	yes	—
91	C₄²	—	2	4	no	—
92	C₄²	—	4	2	no	—
93	C₄²	D₈×C₁₃₇	1	4	yes	—
94	C₄²	Q₁₆×C₁₃₇	1	4	yes	—
95	C₄²	C₂×D₄×C₁₃₇	1	2	yes	—
96	C₄²	C₂×Q₈×C₁₃₇	1	2	yes	—
97	C₂³	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
98	C₂×C₄	—	2	4	no	—
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	#107×C₁₃₇	1	8	yes	—
102	C₂×C₄	C₂×C₄⋊C₄×C₁₃₇	1	2	yes	—
103	C₂×C₄	#127×C₁₃₇	1	1	yes	characteristic
104	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	2	yes	—
105	C₂×C₄	C₄⋊Q₈×C₁₃₇	1	1	yes	characteristic
106	C₂×C₄	#131×C₁₃₇	1	4	yes	—
107	C₂×C₄	C₂×D₈×C₁₃₇	1	2	yes	—
108	C₂×C₄	C₂×Q₁₆×C₁₃₇	1	2	yes	—
109	C₈	—	2	8	no	—
110	C₈	—	8	4	no	—
111	C₂²	#384×C₁₃₇	1	1	yes	characteristic
112	C₂²	#424×C₁₃₇	1	2	yes	—
113	C₂²	C₂×#107×C₁₃₇	1	2	yes	—
114	C₂²	#430×C₁₃₇	1	2	yes	—
115	C₄	—	2	8	no	—
116	C₄	—	2	2	no	—
117	C₄	#424×C₁₃₇	1	4	yes	—
118	C₄	#493×C₁₃₇	1	1	yes	characteristic
119	C₄	#494×C₁₃₇	1	1	yes	characteristic
120	C₄	#500×C₁₃₇	1	2	yes	—
121	C₂	#1571×C₁₃₇	1	4	yes	—
122	C₂	#1843×C₁₃₇	1	1	yes	characteristic
123	C₂	#1845×C₁₃₇	1	1	yes	characteristic
124	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×#335×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

—

#335×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

D₈

yes

—

C₄²×C₁₃₇

Q₁₆

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

#107

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×D₈

yes

—

C₂×C₄×C₁₃₇

C₂×Q₁₆

yes

—

C₈×C₁₃₇

—

C₈×C₁₃₇

—

C₂²×C₁₃₇

#384

yes

characteristic

C₂²×C₁₃₇

#424

yes

—

C₂²×C₁₃₇

C₂×#107

yes

—

C₂²×C₁₃₇

#430

yes

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

#424

yes

—

C₄×C₁₃₇

yes

C₄×C₁₃₇

yes

C₄×C₁₃₇

yes

—

C₂×C₁₃₇

#1571

yes

—

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

C₁₃₇

yes

maximal, characteristic, direct product factor, Sylow

C₁₃₇

#8810

yes

characteristic, direct product factor, Sylow

#1842

C₂×C₁₃₇

yes

—

#1744

C₂×C₁₃₇

yes

characteristic

C₂×#335

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

—

#335

—

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

D₈×C₁₃₇

yes

—

C₄²

Q₁₆×C₁₃₇

yes

—

C₄²

C₂×D₄×C₁₃₇

yes

—

C₄²

C₂×Q₈×C₁₃₇

yes

—

C₂³

C₂×C₄⋊C₄×C₁₃₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

#107×C₁₃₇

yes

—

102

C₂×C₄

C₂×C₄⋊C₄×C₁₃₇

yes

—

103

C₂×C₄

#127×C₁₃₇

yes

characteristic

104

C₂×C₄

C₄⋊Q₈×C₁₃₇

yes

—

105

C₂×C₄

C₄⋊Q₈×C₁₃₇

yes

characteristic

106

C₂×C₄

#131×C₁₃₇

yes

—

107

C₂×C₄

C₂×D₈×C₁₃₇

yes

—

108

C₂×C₄

C₂×Q₁₆×C₁₃₇

yes

—

109

C₈

—

110

C₈

—

111

C₂²

#384×C₁₃₇

yes

characteristic

112

C₂²

#424×C₁₃₇

yes

—

113

C₂²

C₂×#107×C₁₃₇

yes

—

114

C₂²

#430×C₁₃₇

yes

—

115

C₄

—

116

C₄

—

117

C₄

#424×C₁₃₇

yes

—

118

C₄

#493×C₁₃₇

yes

characteristic

119

C₄

#494×C₁₃₇

yes

characteristic

120

C₄

#500×C₁₃₇

yes

—

121

C₂

#1571×C₁₃₇

yes

—

122

C₂

#1843×C₁₃₇

yes

characteristic

123

C₂

#1845×C₁₃₇

yes

characteristic

124

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
62. #8810	C₁₃₇	direct (N × H)
63. C₁₃₇	#8810	direct (N × H)

H ≅ G/N

type

62. #8810

C₁₃₇

direct (N × H)

63. C₁₃₇

#8810

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×#335×C₁₃₇	2
#8810	137

subgroup

index

#1842×C₁₃₇

#1744×C₁₃₇

C₂×#335×C₁₃₇

#8810

137

Subgroups — C₁₃₇×#8810 #46858196

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C137×#8810 #46858196

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#8810 #46858196