Group #46858173 — subgroups

Subgroups — C₁₃₇×#8787 #46858173

Order 35072.

362 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 362 conjugacy classes of subgroups — too many to draw. Only its 190 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	#1277×C₁₃₇	C₂	1	4	yes	maximal
4	C₂×C₄⋊C₈×C₁₃₇	—	2	2	no	—
5	C₂×C₄⋊C₈×C₁₃₇	—	2	2	no	—
6	C₂×C₄⋊C₈×C₁₃₇	C₂²	1	4	yes	—
7	C₂×C₄×C₈×C₁₃₇	C₂²	1	2	yes	—
8	C₄³×C₁₃₇	C₂²	1	1	yes	characteristic
9	#336×C₁₃₇	C₄	1	4	yes	—
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	1	yes	characteristic
14	C₂×C₄²×C₁₃₇	D₄	1	1	yes	characteristic
15	C₂×C₄²×C₁₃₇	Q₈	1	2	yes	—
16	C₂×C₄²×C₁₃₇	C₂³	1	1	yes	characteristic
17	C₄⋊C₈×C₁₃₇	—	2	8	no	—
18	C₄⋊C₈×C₁₃₇	—	4	4	no	—
19	C₄⋊C₈×C₁₃₇	—	4	4	no	—
20	C₄×C₈×C₁₃₇	—	2	4	no	—
21	C₂²×C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
22	C₂²×C₄×C₁₃₇	C₄⋊C₄	1	2	yes	—
23	C₂²×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₁₃₇	C₂×D₄	1	1	yes	characteristic
25	C₂²×C₄×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
26	C₂×C₈×C₁₃₇	—	2	4	no	—
27	C₂×C₈×C₁₃₇	—	2	8	no	—
28	C₂×C₈×C₁₃₇	—	4	4	no	—
29	C₂×C₈×C₁₃₇	—	4	4	no	—
30	C₂×C₈×C₁₃₇	—	4	4	no	—
31	C₄²×C₁₃₇	—	2	4	no	—
32	C₄²×C₁₃₇	—	4	2	no	—
33	C₄²×C₁₃₇	D₈	1	2	yes	—
34	C₄²×C₁₃₇	SD₁₆	1	4	yes	—
35	C₄²×C₁₃₇	Q₁₆	1	2	yes	—
36	C₄²×C₁₃₇	#41	1	4	yes	—
37	C₂³×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₃₇	—	2	2	no	—
39	C₂×C₄×C₁₃₇	—	2	2	no	—
40	C₂×C₄×C₁₃₇	—	2	2	no	—
41	C₂×C₄×C₁₃₇	—	2	2	no	—
42	C₂×C₄×C₁₃₇	—	2	2	no	—
43	C₂×C₄×C₁₃₇	#106	1	4	yes	—
44	C₂×C₄×C₁₃₇	#107	1	4	yes	—
45	C₂×C₄×C₁₃₇	#117	1	2	yes	—
46	C₂×C₄×C₁₃₇	#124	1	2	yes	—
47	C₂×C₄×C₁₃₇	#125	1	2	yes	—
48	C₂×C₄×C₁₃₇	#131	1	4	yes	—
49	C₂×C₄×C₁₃₇	C₂×D₈	1	1	yes	characteristic
50	C₂×C₄×C₁₃₇	C₂×SD₁₆	1	2	yes	—
51	C₂×C₄×C₁₃₇	C₂×Q₁₆	1	1	yes	characteristic
52	C₈×C₁₃₇	—	4	4	no	—
53	C₈×C₁₃₇	—	8	4	no	—
54	C₂²×C₁₃₇	#382	1	1	yes	characteristic
55	C₂²×C₁₃₇	#424	1	2	yes	—
56	C₂²×C₁₃₇	C₂×#106	1	1	yes	characteristic
57	C₂²×C₁₃₇	C₂×#107	1	1	yes	characteristic
58	C₂²×C₁₃₇	#430	1	2	yes	—
59	C₄×C₁₃₇	—	2	4	no	—
60	C₄×C₁₃₇	—	2	4	no	—
61	C₄×C₁₃₇	—	2	2	no	—
62	C₄×C₁₃₇	#433	1	4	yes	—
63	C₄×C₁₃₇	#486	1	2	yes	—
64	C₄×C₁₃₇	#487	1	2	yes	—
65	C₂×C₁₃₇	#1570	1	2	yes	—
66	C₂×C₁₃₇	#1571	1	2	yes	—
67	C₂×C₁₃₇	#1832	1	1	yes	characteristic
68	C₂×C₁₃₇	#1834	1	1	yes	characteristic
69	C₂×C₁₃₇	#1838	1	1	yes	characteristic
70	#8787	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₁₃₇	#8787	1	1	yes	characteristic, direct product factor, Sylow
72	#1842	C₂×C₁₃₇	1	2	yes	—
73	#1744	C₂×C₁₃₇	1	1	yes	characteristic
74	#1277	C₂×C₁₃₇	1	4	yes	—
75	C₂×C₄⋊C₈	—	2	2	no	—
76	C₂×C₄⋊C₈	—	2	2	no	—
77	C₂×C₄⋊C₈	C₂²×C₁₃₇	1	4	yes	—
78	C₂×C₄×C₈	C₂²×C₁₃₇	1	2	yes	—
79	C₄³	C₂²×C₁₃₇	1	1	yes	characteristic
80	#336	C₄×C₁₃₇	1	4	yes	—
81	C₂²×C₈	—	2	4	no	—
82	C₂²×C₈	C₂×C₄×C₁₃₇	1	4	yes	—
83	C₂×C₄²	C₂×C₄×C₁₃₇	1	2	yes	—
84	C₂×C₄²	D₄×C₁₃₇	1	1	yes	characteristic
85	C₂×C₄²	D₄×C₁₃₇	1	1	yes	characteristic
86	C₂×C₄²	Q₈×C₁₃₇	1	2	yes	—
87	C₂×C₄²	C₂³×C₁₃₇	1	1	yes	characteristic
88	C₄⋊C₈	—	2	8	no	—
89	C₄⋊C₈	—	4	4	no	—
90	C₄⋊C₈	—	4	4	no	—
91	C₄×C₈	—	2	4	no	—
92	C₂²×C₄	C₄⋊C₄×C₁₃₇	1	2	yes	—
93	C₂²×C₄	C₄⋊C₄×C₁₃₇	1	2	yes	—
94	C₂²×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
95	C₂²×C₄	C₂×D₄×C₁₃₇	1	1	yes	characteristic
96	C₂²×C₄	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
97	C₂×C₈	—	2	4	no	—
98	C₂×C₈	—	2	8	no	—
99	C₂×C₈	—	4	4	no	—
100	C₂×C₈	—	4	4	no	—
101	C₂×C₈	—	4	4	no	—
102	C₄²	—	2	4	no	—
103	C₄²	—	4	2	no	—
104	C₄²	D₈×C₁₃₇	1	2	yes	—
105	C₄²	SD₁₆×C₁₃₇	1	4	yes	—
106	C₄²	Q₁₆×C₁₃₇	1	2	yes	—
107	C₄²	#41×C₁₃₇	1	4	yes	—
108	C₂³	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	2	no	—
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	#106×C₁₃₇	1	4	yes	—
115	C₂×C₄	#107×C₁₃₇	1	4	yes	—
116	C₂×C₄	#117×C₁₃₇	1	2	yes	—
117	C₂×C₄	#124×C₁₃₇	1	2	yes	—
118	C₂×C₄	#125×C₁₃₇	1	2	yes	—
119	C₂×C₄	#131×C₁₃₇	1	4	yes	—
120	C₂×C₄	C₂×D₈×C₁₃₇	1	1	yes	characteristic
121	C₂×C₄	C₂×SD₁₆×C₁₃₇	1	2	yes	—
122	C₂×C₄	C₂×Q₁₆×C₁₃₇	1	1	yes	characteristic
123	C₈	—	4	4	no	—
124	C₈	—	8	4	no	—
125	C₂²	#382×C₁₃₇	1	1	yes	characteristic
126	C₂²	#424×C₁₃₇	1	2	yes	—
127	C₂²	C₂×#106×C₁₃₇	1	1	yes	characteristic
128	C₂²	C₂×#107×C₁₃₇	1	1	yes	characteristic
129	C₂²	#430×C₁₃₇	1	2	yes	—
130	C₄	—	2	4	no	—
131	C₄	—	2	4	no	—
132	C₄	—	2	2	no	—
133	C₄	#433×C₁₃₇	1	4	yes	—
134	C₄	#486×C₁₃₇	1	2	yes	—
135	C₄	#487×C₁₃₇	1	2	yes	—
136	C₂	#1570×C₁₃₇	1	2	yes	—
137	C₂	#1571×C₁₃₇	1	2	yes	—
138	C₂	#1832×C₁₃₇	1	1	yes	characteristic
139	C₂	#1834×C₁₃₇	1	1	yes	characteristic
140	C₂	#1838×C₁₃₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#1842×C₁₃₇

C₂

yes

maximal

#1744×C₁₃₇

C₂

yes

maximal, characteristic

#1277×C₁₃₇

C₂

yes

maximal

C₂×C₄⋊C₈×C₁₃₇

—

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

—

C₂²×C₈×C₁₃₇

—

C₂²×C₈×C₁₃₇

C₂×C₄

yes

—

C₂×C₄²×C₁₃₇

C₂×C₄

yes

—

C₂×C₄²×C₁₃₇

D₄

yes

characteristic

C₂×C₄²×C₁₃₇

D₄

yes

characteristic

C₂×C₄²×C₁₃₇

Q₈

yes

—

C₂×C₄²×C₁₃₇

C₂³

yes

characteristic

C₄⋊C₈×C₁₃₇

—

C₄⋊C₈×C₁₃₇

—

C₄⋊C₈×C₁₃₇

—

C₄×C₈×C₁₃₇

—

C₂²×C₄×C₁₃₇

C₄⋊C₄

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

—

C₂×C₈×C₁₃₇

—

C₄²×C₁₃₇

—

C₄²×C₁₃₇

—

C₄²×C₁₃₇

D₈

yes

—

C₄²×C₁₃₇

SD₁₆

yes

—

C₄²×C₁₃₇

Q₁₆

yes

—

C₄²×C₁₃₇

#41

yes

—

C₂³×C₁₃₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

#106

yes

—

C₂×C₄×C₁₃₇

#107

yes

—

C₂×C₄×C₁₃₇

#117

yes

—

C₂×C₄×C₁₃₇

#124

yes

—

C₂×C₄×C₁₃₇

#125

yes

—

C₂×C₄×C₁₃₇

#131

yes

—

C₂×C₄×C₁₃₇

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃₇

C₂×SD₁₆

yes

—

C₂×C₄×C₁₃₇

C₂×Q₁₆

yes

characteristic

C₈×C₁₃₇

—

C₈×C₁₃₇

—

C₂²×C₁₃₇

#382

yes

characteristic

C₂²×C₁₃₇

#424

yes

—

C₂²×C₁₃₇

yes

C₂²×C₁₃₇

yes

C₂²×C₁₃₇

yes

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

#433

yes

—

C₄×C₁₃₇

#486

yes

—

C₄×C₁₃₇

#487

yes

—

C₂×C₁₃₇

#1570

yes

—

C₂×C₁₃₇

#1571

yes

—

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

C₁₃₇

yes

maximal, characteristic, direct product factor, Sylow

C₁₃₇

#8787

yes

characteristic, direct product factor, Sylow

#1842

C₂×C₁₃₇

yes

—

#1744

C₂×C₁₃₇

yes

characteristic

#1277

C₂×C₁₃₇

yes

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂²×C₁₃₇

yes

—

C₂×C₄×C₈

C₂²×C₁₃₇

yes

—

C₄³

C₂²×C₁₃₇

yes

characteristic

#336

C₄×C₁₃₇

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×C₄×C₁₃₇

yes

—

C₂×C₄²

C₂×C₄×C₁₃₇

yes

—

C₂×C₄²

D₄×C₁₃₇

yes

characteristic

C₂×C₄²

D₄×C₁₃₇

yes

characteristic

C₂×C₄²

Q₈×C₁₃₇

yes

—

C₂×C₄²

C₂³×C₁₃₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₂²×C₄

C₄⋊C₄×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

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₄²

—

103

C₄²

—

104

C₄²

D₈×C₁₃₇

yes

—

105

C₄²

SD₁₆×C₁₃₇

yes

—

106

C₄²

Q₁₆×C₁₃₇

yes

—

107

C₄²

#41×C₁₃₇

yes

—

108

C₂³

C₂×C₄⋊C₄×C₁₃₇

yes

characteristic

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

#106×C₁₃₇

yes

—

115

C₂×C₄

#107×C₁₃₇

yes

—

116

C₂×C₄

#117×C₁₃₇

yes

—

117

C₂×C₄

#124×C₁₃₇

yes

—

118

C₂×C₄

#125×C₁₃₇

yes

—

119

C₂×C₄

#131×C₁₃₇

yes

—

120

C₂×C₄

C₂×D₈×C₁₃₇

yes

characteristic

121

C₂×C₄

C₂×SD₁₆×C₁₃₇

yes

—

122

C₂×C₄

C₂×Q₁₆×C₁₃₇

yes

characteristic

123

C₈

—

124

C₈

—

125

C₂²

#382×C₁₃₇

yes

characteristic

126

C₂²

#424×C₁₃₇

yes

—

127

C₂²

C₂×#106×C₁₃₇

yes

characteristic

128

C₂²

C₂×#107×C₁₃₇

yes

characteristic

129

C₂²

#430×C₁₃₇

yes

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

#433×C₁₃₇

yes

—

134

C₄

#486×C₁₃₇

yes

—

135

C₄

#487×C₁₃₇

yes

—

136

C₂

#1570×C₁₃₇

yes

—

137

C₂

#1571×C₁₃₇

yes

—

138

C₂

#1832×C₁₃₇

yes

characteristic

139

C₂

#1834×C₁₃₇

yes

characteristic

140

C₂

#1838×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
70. #8787	C₁₃₇	direct (N × H)
71. C₁₃₇	#8787	direct (N × H)

H ≅ G/N

type

70. #8787

C₁₃₇

direct (N × H)

71. C₁₃₇

#8787

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
#1277×C₁₃₇	2
#8787	137

subgroup

index

#1842×C₁₃₇

#1744×C₁₃₇

#1277×C₁₃₇

#8787

137

Subgroups — C₁₃₇×#8787 #46858173

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C137×#8787 #46858173

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#8787 #46858173