Group #46861844 — subgroups

Subgroups — C₁₃₇×#12458 #46861844

Order 35072.

192 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 192 conjugacy classes of subgroups — too many to draw. Only its 86 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	#2252×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₁₆×C₁₃₇	C₂	1	1	yes	maximal, characteristic
3	#1676×C₁₃₇	C₂	1	2	yes	maximal
4	#1371×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	#1365×C₁₃₇	C₂	1	2	yes	maximal
6	#500×C₁₃₇	C₂²	1	2	yes	—
7	#477×C₁₃₇	—	2	2	no	—
8	C₈×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
9	#366×C₁₃₇	—	2	1	no	—
10	#366×C₁₃₇	—	2	1	no	—
11	C₄⋊C₁₆×C₁₃₇	—	2	1	no	—
12	C₄⋊C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
13	C₄×C₁₆×C₁₃₇	—	2	1	no	—
14	C₄×C₁₆×C₁₃₇	C₂²	1	1	yes	characteristic
15	#335×C₁₃₇	C₂²	1	2	yes	—
16	#333×C₁₃₇	—	2	2	no	—
17	C₄⋊Q₈×C₁₃₇	—	2	2	no	—
18	C₄×Q₈×C₁₃₇	D₄	1	1	yes	characteristic
19	C₂×C₁₆×C₁₃₇	—	2	1	no	—
20	C₂×C₁₆×C₁₃₇	D₄	1	1	yes	characteristic
21	C₂×C₁₆×C₁₃₇	D₄	1	1	yes	characteristic
22	#107×C₁₃₇	—	2	2	no	—
23	#107×C₁₃₇	—	2	2	no	—
24	#107×C₁₃₇	—	4	2	no	—
25	C₄⋊C₈×C₁₃₇	—	2	1	no	—
26	C₄⋊C₈×C₁₃₇	—	2	2	no	—
27	C₄⋊C₈×C₁₃₇	D₄	1	1	yes	characteristic
28	#103×C₁₃₇	—	4	2	no	—
29	C₄×C₈×C₁₃₇	—	2	1	no	—
30	C₄×C₈×C₁₃₇	C₂³	1	1	yes	characteristic
31	C₂×Q₈×C₁₃₇	—	4	2	no	—
32	C₂×Q₈×C₁₃₇	D₈	1	1	yes	characteristic
33	C₂×C₈×C₁₃₇	—	2	1	no	—
34	C₂×C₈×C₁₃₇	—	4	2	no	—
35	C₂×C₈×C₁₃₇	C₂×D₄	1	1	yes	characteristic
36	C₂×C₈×C₁₃₇	#41	1	1	yes	characteristic
37	C₄⋊C₄×C₁₃₇	—	2	1	no	—
38	C₄⋊C₄×C₁₃₇	—	4	2	no	—
39	C₄⋊C₄×C₁₃₇	—	4	2	no	—
40	C₄⋊C₄×C₁₃₇	D₈	1	1	yes	characteristic
41	C₄²×C₁₃₇	—	2	1	no	—
42	C₄²×C₁₃₇	C₂×D₄	1	1	yes	characteristic
43	C₁₆×C₁₃₇	—	2	1	no	—
44	C₁₆×C₁₃₇	—	4	1	no	—
45	C₁₆×C₁₃₇	Q₁₆	1	2	yes	—
46	Q₈×C₁₃₇	—	2	1	no	—
47	Q₈×C₁₃₇	—	16	2	no	—
48	Q₈×C₁₃₇	Q₃₂	1	2	yes	—
49	C₂×C₄×C₁₃₇	—	2	1	no	—
50	C₂×C₄×C₁₃₇	—	2	1	no	—
51	C₂×C₄×C₁₃₇	—	8	2	no	—
52	C₂×C₄×C₁₃₇	#121	1	1	yes	characteristic
53	C₂×C₄×C₁₃₇	C₂×D₈	1	1	yes	characteristic
54	C₂×C₄×C₁₃₇	#135	1	1	yes	characteristic
55	C₈×C₁₃₇	—	2	1	no	—
56	C₈×C₁₃₇	—	4	1	no	—
57	C₈×C₁₃₇	—	8	2	no	—
58	C₈×C₁₃₇	C₂×Q₁₆	1	1	yes	characteristic
59	C₈×C₁₃₇	#137	1	1	yes	characteristic
60	C₂²×C₁₃₇	#466	1	1	yes	characteristic
61	C₄×C₁₃₇	—	2	2	no	—
62	C₄×C₁₃₇	—	2	1	no	—
63	C₄×C₁₃₇	—	4	1	no	—
64	C₄×C₁₃₇	—	16	2	no	—
65	C₄×C₁₃₇	#462	1	1	yes	characteristic
66	C₄×C₁₃₇	#473	1	1	yes	characteristic
67	C₄×C₁₃₇	C₂×Q₃₂	1	1	yes	characteristic
68	C₄×C₁₃₇	#508	1	1	yes	characteristic
69	C₂×C₁₃₇	#1674	1	1	yes	characteristic
70	C₂×C₁₃₇	#2216	1	1	yes	characteristic
71	C₂×C₁₃₇	#2221	1	1	yes	characteristic
72	#12458	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₁₃₇	#12458	1	1	yes	characteristic, direct product factor, Sylow
74	#2252	C₂×C₁₃₇	1	1	yes	characteristic
75	Q₈×C₁₆	C₂×C₁₃₇	1	1	yes	characteristic
76	#1676	C₂×C₁₃₇	1	2	yes	—
77	#1371	C₂×C₁₃₇	1	1	yes	characteristic
78	#1365	C₂×C₁₃₇	1	2	yes	—
79	#500	C₂²×C₁₃₇	1	2	yes	—
80	#477	—	2	2	no	—
81	C₈×Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
82	#366	—	2	1	no	—
83	#366	—	2	1	no	—
84	C₄⋊C₁₆	—	2	1	no	—
85	C₄⋊C₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
86	C₄×C₁₆	—	2	1	no	—
87	C₄×C₁₆	C₂²×C₁₃₇	1	1	yes	characteristic
88	#335	C₂²×C₁₃₇	1	2	yes	—
89	#333	—	2	2	no	—
90	C₄⋊Q₈	—	2	2	no	—
91	C₄×Q₈	D₄×C₁₃₇	1	1	yes	characteristic
92	C₂×C₁₆	—	2	1	no	—
93	C₂×C₁₆	D₄×C₁₃₇	1	1	yes	characteristic
94	C₂×C₁₆	D₄×C₁₃₇	1	1	yes	characteristic
95	#107	—	2	2	no	—
96	#107	—	2	2	no	—
97	#107	—	4	2	no	—
98	C₄⋊C₈	—	2	1	no	—
99	C₄⋊C₈	—	2	2	no	—
100	C₄⋊C₈	D₄×C₁₃₇	1	1	yes	characteristic
101	#103	—	4	2	no	—
102	C₄×C₈	—	2	1	no	—
103	C₄×C₈	C₂³×C₁₃₇	1	1	yes	characteristic
104	C₂×Q₈	—	4	2	no	—
105	C₂×Q₈	D₈×C₁₃₇	1	1	yes	characteristic
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	4	2	no	—
108	C₂×C₈	C₂×D₄×C₁₃₇	1	1	yes	characteristic
109	C₂×C₈	#41×C₁₃₇	1	1	yes	characteristic
110	C₄⋊C₄	—	2	1	no	—
111	C₄⋊C₄	—	4	2	no	—
112	C₄⋊C₄	—	4	2	no	—
113	C₄⋊C₄	D₈×C₁₃₇	1	1	yes	characteristic
114	C₄²	—	2	1	no	—
115	C₄²	C₂×D₄×C₁₃₇	1	1	yes	characteristic
116	C₁₆	—	2	1	no	—
117	C₁₆	—	4	1	no	—
118	C₁₆	Q₁₆×C₁₃₇	1	2	yes	—
119	Q₈	—	2	1	no	—
120	Q₈	—	16	2	no	—
121	Q₈	Q₃₂×C₁₃₇	1	2	yes	—
122	C₂×C₄	—	2	1	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	8	2	no	—
125	C₂×C₄	#121×C₁₃₇	1	1	yes	characteristic
126	C₂×C₄	C₂×D₈×C₁₃₇	1	1	yes	characteristic
127	C₂×C₄	#135×C₁₃₇	1	1	yes	characteristic
128	C₈	—	2	1	no	—
129	C₈	—	4	1	no	—
130	C₈	—	8	2	no	—
131	C₈	C₂×Q₁₆×C₁₃₇	1	1	yes	characteristic
132	C₈	#137×C₁₃₇	1	1	yes	characteristic
133	C₂²	#466×C₁₃₇	1	1	yes	characteristic
134	C₄	—	2	2	no	—
135	C₄	—	2	1	no	—
136	C₄	—	4	1	no	—
137	C₄	—	16	2	no	—
138	C₄	#462×C₁₃₇	1	1	yes	characteristic
139	C₄	#473×C₁₃₇	1	1	yes	characteristic
140	C₄	C₂×Q₃₂×C₁₃₇	1	1	yes	characteristic
141	C₄	#508×C₁₃₇	1	1	yes	characteristic
142	C₂	#1674×C₁₃₇	1	1	yes	characteristic
143	C₂	#2216×C₁₃₇	1	1	yes	characteristic
144	C₂	#2221×C₁₃₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#2252×C₁₃₇

C₂

yes

maximal, characteristic

Q₈×C₁₆×C₁₃₇

C₂

yes

maximal, characteristic

#1676×C₁₃₇

C₂

yes

maximal

#1371×C₁₃₇

C₂

yes

maximal, characteristic

#1365×C₁₃₇

C₂

yes

maximal

#500×C₁₃₇

C₂²

yes

—

#477×C₁₃₇

—

C₈×Q₈×C₁₃₇

C₂²

yes

characteristic

#366×C₁₃₇

—

#366×C₁₃₇

—

C₄⋊C₁₆×C₁₃₇

—

C₄⋊C₁₆×C₁₃₇

C₂²

yes

characteristic

C₄×C₁₆×C₁₃₇

—

C₄×C₁₆×C₁₃₇

C₂²

yes

characteristic

#335×C₁₃₇

C₂²

yes

—

#333×C₁₃₇

—

C₄⋊Q₈×C₁₃₇

—

C₄×Q₈×C₁₃₇

D₄

yes

characteristic

C₂×C₁₆×C₁₃₇

—

C₂×C₁₆×C₁₃₇

D₄

yes

characteristic

C₂×C₁₆×C₁₃₇

D₄

yes

characteristic

#107×C₁₃₇

—

#107×C₁₃₇

—

#107×C₁₃₇

—

C₄⋊C₈×C₁₃₇

—

C₄⋊C₈×C₁₃₇

—

C₄⋊C₈×C₁₃₇

D₄

yes

characteristic

#103×C₁₃₇

—

C₄×C₈×C₁₃₇

—

C₄×C₈×C₁₃₇

C₂³

yes

characteristic

C₂×Q₈×C₁₃₇

—

C₂×Q₈×C₁₃₇

D₈

yes

characteristic

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

—

C₂×C₈×C₁₃₇

C₂×D₄

yes

characteristic

C₂×C₈×C₁₃₇

#41

yes

characteristic

C₄⋊C₄×C₁₃₇

—

C₄⋊C₄×C₁₃₇

—

C₄⋊C₄×C₁₃₇

—

C₄⋊C₄×C₁₃₇

D₈

yes

characteristic

C₄²×C₁₃₇

—

C₄²×C₁₃₇

C₂×D₄

yes

characteristic

C₁₆×C₁₃₇

—

C₁₆×C₁₃₇

—

C₁₆×C₁₃₇

Q₁₆

yes

—

Q₈×C₁₃₇

—

Q₈×C₁₃₇

—

Q₈×C₁₃₇

Q₃₂

yes

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

#121

yes

characteristic

C₂×C₄×C₁₃₇

C₂×D₈

yes

characteristic

C₂×C₄×C₁₃₇

#135

yes

characteristic

C₈×C₁₃₇

—

C₈×C₁₃₇

—

C₈×C₁₃₇

—

C₈×C₁₃₇

C₂×Q₁₆

yes

C₈×C₁₃₇

yes

C₂²×C₁₃₇

yes

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

yes

C₄×C₁₃₇

yes

C₄×C₁₃₇

C₂×Q₃₂

yes

C₄×C₁₃₇

yes

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

C₁₃₇

yes

maximal, characteristic, direct product factor, Sylow

C₁₃₇

#12458

yes

characteristic, direct product factor, Sylow

#2252

C₂×C₁₃₇

yes

characteristic

Q₈×C₁₆

C₂×C₁₃₇

yes

characteristic

#1676

C₂×C₁₃₇

yes

—

#1371

C₂×C₁₃₇

yes

characteristic

#1365

C₂×C₁₃₇

yes

—

#500

C₂²×C₁₃₇

yes

—

#477

—

C₈×Q₈

C₂²×C₁₃₇

yes

characteristic

#366

—

#366

—

C₄⋊C₁₆

—

C₄⋊C₁₆

C₂²×C₁₃₇

yes

characteristic

C₄×C₁₆

—

C₄×C₁₆

C₂²×C₁₃₇

yes

characteristic

#335

C₂²×C₁₃₇

yes

—

#333

—

C₄⋊Q₈

—

C₄×Q₈

D₄×C₁₃₇

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

D₄×C₁₃₇

yes

characteristic

C₂×C₁₆

D₄×C₁₃₇

yes

—

—

—

C₄⋊C₈

—

C₄⋊C₈

—

100

C₄⋊C₈

D₄×C₁₃₇

yes

characteristic

101

#103

—

102

C₄×C₈

—

103

C₄×C₈

C₂³×C₁₃₇

yes

characteristic

104

C₂×Q₈

—

105

C₂×Q₈

D₈×C₁₃₇

yes

characteristic

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

C₂×D₄×C₁₃₇

yes

characteristic

109

C₂×C₈

#41×C₁₃₇

yes

characteristic

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

D₈×C₁₃₇

yes

characteristic

114

C₄²

—

115

C₄²

C₂×D₄×C₁₃₇

yes

characteristic

116

C₁₆

—

117

C₁₆

—

118

C₁₆

Q₁₆×C₁₃₇

yes

—

119

Q₈

—

120

Q₈

—

121

Q₈

Q₃₂×C₁₃₇

yes

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

#121×C₁₃₇

yes

characteristic

126

C₂×C₄

C₂×D₈×C₁₃₇

yes

characteristic

127

C₂×C₄

#135×C₁₃₇

yes

characteristic

128

C₈

—

129

C₈

—

130

C₈

—

131

C₈

C₂×Q₁₆×C₁₃₇

yes

characteristic

132

C₈

#137×C₁₃₇

yes

characteristic

133

C₂²

#466×C₁₃₇

yes

characteristic

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

#462×C₁₃₇

yes

characteristic

139

C₄

#473×C₁₃₇

yes

characteristic

140

C₄

C₂×Q₃₂×C₁₃₇

yes

characteristic

141

C₄

#508×C₁₃₇

yes

characteristic

142

C₂

#1674×C₁₃₇

yes

characteristic

143

C₂

#2216×C₁₃₇

yes

characteristic

144

C₂

#2221×C₁₃₇

yes

characteristic

	subgroup	G/H
center, socle, prime radical	C₂²×C₁₃₇	#466
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₁₃₇
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₄²×C₁₃₇ ◁ C₈×Q₈×C₁₃₇ ◁ G

subgroup

G/H

center, socle, prime radical

C₂²×C₁₃₇

#466

commutator, Frattini, cosocle

C₄×C₈

C₂³×C₁₃₇

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₃₇ ◁ C₄²×C₁₃₇ ◁ C₈×Q₈×C₁₃₇ ◁ G

N	H ≅ G/N	type
72. #12458	C₁₃₇	direct (N × H)
73. C₁₃₇	#12458	direct (N × H)

H ≅ G/N

type

72. #12458

C₁₃₇

direct (N × H)

73. C₁₃₇

#12458

direct (N × H)

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

subgroup	index
#2252×C₁₃₇	2
Q₈×C₁₆×C₁₃₇	2
#1676×C₁₃₇	2
#1371×C₁₃₇	2
#1365×C₁₃₇	2
#12458	137

subgroup

index

#2252×C₁₃₇

Q₈×C₁₆×C₁₃₇

#1676×C₁₃₇

#1371×C₁₃₇

#1365×C₁₃₇

#12458

137

Subgroups — C₁₃₇×#12458 #46861844

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C137×#12458 #46861844

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃₇×#12458 #46861844