Group #31776868 — subgroups

Subgroups — #1857×C₁₃₇ #31776868

Order 17536.

272 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 112 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	#519×C₁₃₇	C₂	1	1	yes	maximal, characteristic
2	#428×C₁₃₇	C₂	1	2	yes	maximal
3	C₂×#101×C₁₃₇	C₂	1	1	yes	maximal, characteristic
4	#391×C₁₃₇	C₂	1	1	yes	maximal, characteristic
5	#339×C₁₃₇	C₂	1	2	yes	maximal
6	C₂²×Q₈×C₁₃₇	C₂²	1	1	yes	characteristic
7	C₂×M₁₆×C₁₃₇	C₂²	1	2	yes	—
8	C₄×Q₈×C₁₃₇	—	2	2	no	—
9	C₄×Q₈×C₁₃₇	—	2	2	no	—
10	#117×C₁₃₇	—	2	2	no	—
11	#117×C₁₃₇	C₂²	1	2	yes	—
12	C₂×C₄⋊C₄×C₁₃₇	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₁₃₇	C₂²	1	1	yes	characteristic
14	#107×C₁₃₇	—	2	2	no	—
15	#106×C₁₃₇	—	2	2	no	—
16	#101×C₁₃₇	C₄	1	4	yes	complemented
17	#95×C₁₃₇	—	2	2	no	—
18	C₂×Q₈×C₁₃₇	—	2	2	no	—
19	C₂×Q₈×C₁₃₇	—	2	2	no	—
20	C₂×Q₈×C₁₃₇	C₂×C₄	1	2	yes	—
21	C₂×Q₈×C₁₃₇	Q₈	1	2	yes	—
22	C₂²×C₄×C₁₃₇	—	2	1	no	—
23	C₂²×C₄×C₁₃₇	—	2	1	no	—
24	C₂²×C₄×C₁₃₇	D₄	1	2	yes	—
25	C₂²×C₄×C₁₃₇	C₂³	1	1	yes	characteristic
26	M₁₆×C₁₃₇	—	2	2	no	—
27	M₁₆×C₁₃₇	C₂×C₄	1	4	yes	—
28	C₂×C₈×C₁₃₇	—	2	2	no	—
29	C₄⋊C₄×C₁₃₇	—	2	2	no	—
30	C₄⋊C₄×C₁₃₇	—	2	2	no	—
31	C₄⋊C₄×C₁₃₇	—	2	2	no	—
32	C₄⋊C₄×C₁₃₇	D₄	1	2	yes	—
33	C₄⋊C₄×C₁₃₇	D₄	1	2	yes	—
34	C₂²⋊C₄×C₁₃₇	—	2	2	no	—
35	C₄²×C₁₃₇	—	2	4	no	—
36	C₄²×C₁₃₇	—	2	2	no	—
37	C₄²×C₁₃₇	—	2	2	no	—
38	C₂³×C₁₃₇	C₂×D₄	1	1	yes	characteristic
39	Q₈×C₁₃₇	—	2	4	no	—
40	Q₈×C₁₃₇	—	2	4	no	—
41	C₂×C₄×C₁₃₇	—	2	2	no	—
42	C₂×C₄×C₁₃₇	—	2	2	no	—
43	C₂×C₄×C₁₃₇	—	2	2	no	—
44	C₂×C₄×C₁₃₇	—	2	2	no	—
45	C₂×C₄×C₁₃₇	—	2	2	no	—
46	C₂×C₄×C₁₃₇	—	4	1	no	—
47	C₂×C₄×C₁₃₇	—	4	1	no	—
48	C₂×C₄×C₁₃₇	—	4	2	no	—
49	C₂×C₄×C₁₃₇	—	4	1	no	—
50	C₂×C₄×C₁₃₇	C₄⋊C₄	1	4	yes	—
51	C₂×C₄×C₁₃₇	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₁₃₇	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₁₃₇	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₄×C₁₃₇	#41	1	2	yes	—
55	C₈×C₁₃₇	—	2	4	no	—
56	C₂²×C₁₃₇	—	2	1	no	—
57	C₂²×C₁₃₇	—	2	1	no	—
58	C₂²×C₁₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₁₃₇	#120	1	1	yes	characteristic
60	C₂²×C₁₃₇	#123	1	1	yes	characteristic
61	C₄×C₁₃₇	—	2	4	no	—
62	C₄×C₁₃₇	—	4	2	no	—
63	C₄×C₁₃₇	—	4	2	no	—
64	C₄×C₁₃₇	—	4	2	no	—
65	C₄×C₁₃₇	—	8	1	no	—
66	C₄×C₁₃₇	C₄×D₄	1	2	yes	—
67	C₄×C₁₃₇	C₂²⋊Q₈	1	2	yes	—
68	C₂×C₁₃₇	—	2	1	no	—
69	C₂×C₁₃₇	—	2	1	no	—
70	C₂×C₁₃₇	#385	1	1	yes	characteristic
71	C₂×C₁₃₇	#454	1	1	yes	characteristic
72	C₂×C₁₃₇	#456	1	1	yes	characteristic
73	C₁₃₇	#1857	1	1	yes	characteristic, direct product factor, Sylow
74	#1857	C₁₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	#519	C₂×C₁₃₇	1	1	yes	characteristic
76	#428	C₂×C₁₃₇	1	2	yes	—
77	C₂×#101	C₂×C₁₃₇	1	1	yes	characteristic
78	#391	C₂×C₁₃₇	1	1	yes	characteristic
79	#339	C₂×C₁₃₇	1	2	yes	—
80	C₂²×Q₈	C₂²×C₁₃₇	1	1	yes	characteristic
81	C₂×M₁₆	C₂²×C₁₃₇	1	2	yes	—
82	C₄×Q₈	—	2	2	no	—
83	C₄×Q₈	—	2	2	no	—
84	#117	—	2	2	no	—
85	#117	C₂²×C₁₃₇	1	2	yes	—
86	C₂×C₄⋊C₄	C₂²×C₁₃₇	1	1	yes	characteristic
87	C₂×C₄²	C₂²×C₁₃₇	1	1	yes	characteristic
88	#107	—	2	2	no	—
89	#106	—	2	2	no	—
90	#101	C₄×C₁₃₇	1	4	yes	complemented
91	#95	—	2	2	no	—
92	C₂×Q₈	—	2	2	no	—
93	C₂×Q₈	—	2	2	no	—
94	C₂×Q₈	C₂×C₄×C₁₃₇	1	2	yes	—
95	C₂×Q₈	Q₈×C₁₃₇	1	2	yes	—
96	C₂²×C₄	—	2	1	no	—
97	C₂²×C₄	—	2	1	no	—
98	C₂²×C₄	D₄×C₁₃₇	1	2	yes	—
99	C₂²×C₄	C₂³×C₁₃₇	1	1	yes	characteristic
100	M₁₆	—	2	2	no	—
101	M₁₆	C₂×C₄×C₁₃₇	1	4	yes	—
102	C₂×C₈	—	2	2	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	—	2	2	no	—
106	C₄⋊C₄	D₄×C₁₃₇	1	2	yes	—
107	C₄⋊C₄	D₄×C₁₃₇	1	2	yes	—
108	C₂²⋊C₄	—	2	2	no	—
109	C₄²	—	2	4	no	—
110	C₄²	—	2	2	no	—
111	C₄²	—	2	2	no	—
112	C₂³	C₂×D₄×C₁₃₇	1	1	yes	characteristic
113	Q₈	—	2	4	no	—
114	Q₈	—	2	4	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	4	1	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	4	1	no	—
124	C₂×C₄	C₄⋊C₄×C₁₃₇	1	4	yes	—
125	C₂×C₄	C₂²×C₄×C₁₃₇	1	1	yes	characteristic
126	C₂×C₄	C₂×D₄×C₁₃₇	1	2	yes	—
127	C₂×C₄	C₂×Q₈×C₁₃₇	1	1	yes	characteristic
128	C₂×C₄	#41×C₁₃₇	1	2	yes	—
129	C₈	—	2	4	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	1	no	—
132	C₂²	C₂×C₄⋊C₄×C₁₃₇	1	1	yes	characteristic
133	C₂²	#120×C₁₃₇	1	1	yes	characteristic
134	C₂²	#123×C₁₃₇	1	1	yes	characteristic
135	C₄	—	2	4	no	—
136	C₄	—	4	2	no	—
137	C₄	—	4	2	no	—
138	C₄	—	4	2	no	—
139	C₄	—	8	1	no	—
140	C₄	C₄×D₄×C₁₃₇	1	2	yes	—
141	C₄	C₂²⋊Q₈×C₁₃₇	1	2	yes	—
142	C₂	—	2	1	no	—
143	C₂	—	2	1	no	—
144	C₂	#385×C₁₃₇	1	1	yes	characteristic
145	C₂	#454×C₁₃₇	1	1	yes	characteristic
146	C₂	#456×C₁₃₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#519×C₁₃₇

C₂

yes

maximal, characteristic

#428×C₁₃₇

C₂

yes

maximal

C₂×#101×C₁₃₇

C₂

yes

maximal, characteristic

#391×C₁₃₇

C₂

yes

maximal, characteristic

#339×C₁₃₇

C₂

yes

maximal

C₂²×Q₈×C₁₃₇

C₂²

yes

characteristic

C₂×M₁₆×C₁₃₇

C₂²

yes

—

C₄×Q₈×C₁₃₇

—

C₄×Q₈×C₁₃₇

—

#117×C₁₃₇

—

#117×C₁₃₇

C₂²

yes

—

C₂×C₄⋊C₄×C₁₃₇

C₂²

yes

characteristic

C₂×C₄²×C₁₃₇

C₂²

yes

characteristic

#107×C₁₃₇

—

#106×C₁₃₇

—

#101×C₁₃₇

C₄

yes

complemented

#95×C₁₃₇

—

C₂×Q₈×C₁₃₇

—

C₂×Q₈×C₁₃₇

—

C₂×Q₈×C₁₃₇

C₂×C₄

yes

—

C₂×Q₈×C₁₃₇

Q₈

yes

—

C₂²×C₄×C₁₃₇

—

C₂²×C₄×C₁₃₇

—

C₂²×C₄×C₁₃₇

D₄

yes

—

C₂²×C₄×C₁₃₇

C₂³

yes

characteristic

M₁₆×C₁₃₇

—

M₁₆×C₁₃₇

C₂×C₄

yes

—

C₂×C₈×C₁₃₇

—

C₄⋊C₄×C₁₃₇

—

C₄⋊C₄×C₁₃₇

—

C₄⋊C₄×C₁₃₇

—

C₄⋊C₄×C₁₃₇

D₄

yes

—

C₄⋊C₄×C₁₃₇

D₄

yes

—

C₂²⋊C₄×C₁₃₇

—

C₄²×C₁₃₇

—

C₄²×C₁₃₇

—

C₄²×C₁₃₇

—

C₂³×C₁₃₇

C₂×D₄

yes

characteristic

Q₈×C₁₃₇

—

Q₈×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

—

C₂×C₄×C₁₃₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₃₇

C₂×D₄

yes

—

C₂×C₄×C₁₃₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₃₇

#41

yes

—

C₈×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

—

C₂²×C₁₃₇

C₂×C₄⋊C₄

yes

C₂²×C₁₃₇

yes

C₂²×C₁₃₇

yes

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

—

C₄×C₁₃₇

C₄×D₄

yes

—

C₄×C₁₃₇

C₂²⋊Q₈

yes

—

C₂×C₁₃₇

—

C₂×C₁₃₇

—

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

C₂×C₁₃₇

yes

C₁₃₇

yes

characteristic, direct product factor, Sylow

#1857

C₁₃₇

yes

maximal, characteristic, direct product factor, Sylow

#519

C₂×C₁₃₇

yes

characteristic

#428

C₂×C₁₃₇

yes

—

C₂×#101

C₂×C₁₃₇

yes

characteristic

#391

C₂×C₁₃₇

yes

characteristic

#339

C₂×C₁₃₇

yes

—

C₂²×Q₈

C₂²×C₁₃₇

yes

characteristic

C₂×M₁₆

C₂²×C₁₃₇

yes

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

#117

C₂²×C₁₃₇

yes

—

C₂×C₄⋊C₄

C₂²×C₁₃₇

yes

characteristic

C₂×C₄²

C₂²×C₁₃₇

yes

—

—

C₄×C₁₃₇

yes

complemented

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×C₄×C₁₃₇

yes

—

C₂×Q₈

Q₈×C₁₃₇

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₄×C₁₃₇

yes

—

C₂²×C₄

C₂³×C₁₃₇

yes

characteristic

100

M₁₆

—

101

M₁₆

C₂×C₄×C₁₃₇

yes

—

102

C₂×C₈

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

D₄×C₁₃₇

yes

—

107

C₄⋊C₄

D₄×C₁₃₇

yes

—

108

C₂²⋊C₄

—

109

C₄²

—

110

C₄²

—

111

C₄²

—

112

C₂³

C₂×D₄×C₁₃₇

yes

characteristic

113

Q₈

—

114

Q₈

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

C₄⋊C₄×C₁₃₇

yes

—

125

C₂×C₄

C₂²×C₄×C₁₃₇

yes

characteristic

126

C₂×C₄

C₂×D₄×C₁₃₇

yes

—

127

C₂×C₄

C₂×Q₈×C₁₃₇

yes

characteristic

128

C₂×C₄

#41×C₁₃₇

yes

—

129

C₈

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂×C₄⋊C₄×C₁₃₇

yes

characteristic

133

C₂²

#120×C₁₃₇

yes

characteristic

134

C₂²

#123×C₁₃₇

yes

characteristic

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

C₄×D₄×C₁₃₇

yes

—

141

C₄

C₂²⋊Q₈×C₁₃₇

yes

—

142

C₂

—

143

C₂

—

144

C₂

#385×C₁₃₇

yes

characteristic

145

C₂

#454×C₁₃₇

yes

characteristic

146

C₂

#456×C₁₃₇

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃₇	#120
commutator	C₂×C₄	C₂²×C₄×C₁₃₇
Frattini, cosocle	C₂²×C₄	C₂³×C₁₃₇
prime radical	C₂³×C₁₃₇	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃₇ ◁ C₂²×C₄×C₁₃₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₃₇

#120

commutator

C₂×C₄

C₂²×C₄×C₁₃₇

Frattini, cosocle

C₂²×C₄

C₂³×C₁₃₇

prime radical

C₂³×C₁₃₇

C₂×D₄

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₁₃₇ ◁ C₂²×C₄×C₁₃₇ ◁ G

N	H ≅ G/N	type
16. #101×C₁₃₇	C₄	semidirect
73. C₁₃₇	#1857	direct (N × H)
74. #1857	C₁₃₇	direct (N × H)
90. #101	C₄×C₁₃₇	semidirect

H ≅ G/N

type

16. #101×C₁₃₇

C₄

semidirect

73. C₁₃₇

#1857

direct (N × H)

74. #1857

C₁₃₇

direct (N × H)

90. #101

C₄×C₁₃₇

semidirect

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

subgroup	index
#519×C₁₃₇	2
#428×C₁₃₇	2
C₂×#101×C₁₃₇	2
#391×C₁₃₇	2
#339×C₁₃₇	2
#1857	137

subgroup

index

#519×C₁₃₇

#428×C₁₃₇

C₂×#101×C₁₃₇

#391×C₁₃₇

#339×C₁₃₇

#1857

137

Subgroups — #1857×C₁₃₇ #31776868

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1857×C137 #31776868

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1857×C₁₃₇ #31776868