Group #22630008 — subgroups

Subgroups — C₃₇×#8755 #22630008

Order 9472.

398 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 110 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	C₃₇×#2313	C₂	1	1	yes	maximal, characteristic
2	C₃₇×#1821	C₂	1	2	yes	maximal
3	C₃₇×#1306	C₂	1	4	yes	maximal
4	C₂²×C₄⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
5	C₃₇×#407	C₂²	1	4	yes	—
6	C₃₇×#387	—	2	4	no	—
7	C₃₇×#381	C₂²	1	2	yes	—
8	C₃₇×#380	C₄	1	2	yes	—
9	C₄×C₂²⋊C₄×C₃₇	C₄	1	2	yes	—
10	C₃₇×#343	—	2	4	no	—
11	C₂³×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
12	C₂³×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
13	C₂³×C₄×C₃₇	C₂³	1	1	yes	characteristic
14	C₂×M₁₆×C₃₇	—	2	4	no	—
15	C₂×C₄⋊C₄×C₃₇	—	2	4	no	—
16	C₂×C₄⋊C₄×C₃₇	—	2	2	no	—
17	C₂×C₄⋊C₄×C₃₇	D₄	1	4	yes	—
18	C₂×C₂²⋊C₄×C₃₇	C₂×C₄	1	4	yes	—
19	C₂×C₄²×C₃₇	—	2	2	no	—
20	C₂²⋊C₈×C₃₇	—	2	8	no	—
21	#95×C₃₇	—	2	2	no	—
22	#95×C₃₇	—	2	2	no	—
23	#95×C₃₇	—	2	4	no	—
24	C₂⁴×C₃₇	C₂²×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₃₇	—	2	2	no	—
26	C₂²×C₄×C₃₇	—	2	2	no	—
27	C₂²×C₄×C₃₇	—	2	4	no	—
28	C₂²×C₄×C₃₇	—	4	2	no	—
29	C₂²×C₄×C₃₇	—	8	1	no	—
30	C₂²×C₄×C₃₇	—	8	1	no	—
31	C₂²×C₄×C₃₇	C₂²⋊C₄	1	2	yes	—
32	C₂²×C₄×C₃₇	C₂²⋊C₄	1	2	yes	—
33	C₂²×C₄×C₃₇	C₂×D₄	1	2	yes	—
34	C₂²×C₄×C₃₇	#41	1	4	yes	—
35	M₁₆×C₃₇	—	4	8	no	—
36	C₂×C₈×C₃₇	—	4	4	no	—
37	C₄⋊C₄×C₃₇	—	2	8	no	—
38	C₄⋊C₄×C₃₇	—	8	2	no	—
39	C₂²⋊C₄×C₃₇	—	4	4	no	—
40	C₄²×C₃₇	—	8	2	no	—
41	C₂³×C₃₇	—	2	2	no	—
42	C₂³×C₃₇	—	4	1	no	—
43	C₂³×C₃₇	—	4	1	no	—
44	C₂³×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
45	C₂³×C₃₇	#117	1	2	yes	—
46	C₂×C₄×C₃₇	—	2	4	no	—
47	C₂×C₄×C₃₇	—	2	2	no	—
48	C₂×C₄×C₃₇	—	2	2	no	—
49	C₂×C₄×C₃₇	—	4	4	no	—
50	C₂×C₄×C₃₇	—	4	2	no	—
51	C₂×C₄×C₃₇	—	4	2	no	—
52	C₂×C₄×C₃₇	—	4	4	no	—
53	C₂×C₄×C₃₇	—	4	4	no	—
54	C₂×C₄×C₃₇	—	4	4	no	—
55	C₂×C₄×C₃₇	—	8	2	no	—
56	C₂×C₄×C₃₇	—	8	2	no	—
57	C₂×C₄×C₃₇	#123	1	4	yes	—
58	C₈×C₃₇	—	8	4	no	—
59	#8755	C₃₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
60	C₂²×C₃₇	—	2	1	no	—
61	C₂²×C₃₇	—	2	1	no	—
62	C₂²×C₃₇	—	2	2	no	—
63	C₂²×C₃₇	—	2	2	no	—
64	C₂²×C₃₇	—	2	2	no	—
65	C₂²×C₃₇	—	4	4	no	—
66	C₂²×C₃₇	#381	1	1	yes	characteristic
67	C₂²×C₃₇	#410	1	2	yes	—
68	C₄×C₃₇	—	2	4	no	—
69	C₄×C₃₇	—	4	2	no	—
70	C₄×C₃₇	—	4	2	no	—
71	C₄×C₃₇	—	8	4	no	—
72	#2313	C₂×C₃₇	1	1	yes	characteristic
73	#1821	C₂×C₃₇	1	2	yes	—
74	#1306	C₂×C₃₇	1	4	yes	—
75	C₂×C₃₇	—	2	2	no	—
76	C₂×C₃₇	—	4	2	no	—
77	C₂×C₃₇	#1820	1	1	yes	characteristic
78	C₂×C₃₇	#1829	1	1	yes	characteristic
79	C₂×C₃₇	#1830	1	1	yes	characteristic
80	C₂²×C₄⋊C₄	C₂²×C₃₇	1	1	yes	characteristic
81	#407	C₂²×C₃₇	1	4	yes	—
82	#387	—	2	4	no	—
83	#381	C₂²×C₃₇	1	2	yes	—
84	#380	C₄×C₃₇	1	2	yes	—
85	C₄×C₂²⋊C₄	C₄×C₃₇	1	2	yes	—
86	#343	—	2	4	no	—
87	C₃₇	#8755	1	1	yes	characteristic, direct product factor, Sylow
88	C₂³×C₄	C₂×C₄×C₃₇	1	1	yes	characteristic
89	C₂³×C₄	C₂×C₄×C₃₇	1	1	yes	characteristic
90	C₂³×C₄	C₂³×C₃₇	1	1	yes	characteristic
91	C₂×M₁₆	—	2	4	no	—
92	C₂×C₄⋊C₄	—	2	4	no	—
93	C₂×C₄⋊C₄	—	2	2	no	—
94	C₂×C₄⋊C₄	D₄×C₃₇	1	4	yes	—
95	C₂×C₂²⋊C₄	C₂×C₄×C₃₇	1	4	yes	—
96	C₂×C₄²	—	2	2	no	—
97	C₂²⋊C₈	—	2	8	no	—
98	#95	—	2	2	no	—
99	#95	—	2	2	no	—
100	#95	—	2	4	no	—
101	C₂⁴	C₂²×C₄×C₃₇	1	1	yes	characteristic
102	C₂²×C₄	—	2	2	no	—
103	C₂²×C₄	—	2	2	no	—
104	C₂²×C₄	—	2	4	no	—
105	C₂²×C₄	—	4	2	no	—
106	C₂²×C₄	—	8	1	no	—
107	C₂²×C₄	—	8	1	no	—
108	C₂²×C₄	C₂²⋊C₄×C₃₇	1	2	yes	—
109	C₂²×C₄	C₂²⋊C₄×C₃₇	1	2	yes	—
110	C₂²×C₄	C₂×D₄×C₃₇	1	2	yes	—
111	C₂²×C₄	#41×C₃₇	1	4	yes	—
112	M₁₆	—	4	8	no	—
113	C₂×C₈	—	4	4	no	—
114	C₄⋊C₄	—	2	8	no	—
115	C₄⋊C₄	—	8	2	no	—
116	C₂²⋊C₄	—	4	4	no	—
117	C₄²	—	8	2	no	—
118	C₂³	—	2	2	no	—
119	C₂³	—	4	1	no	—
120	C₂³	—	4	1	no	—
121	C₂³	C₂×C₂²⋊C₄×C₃₇	1	1	yes	characteristic
122	C₂³	#117×C₃₇	1	2	yes	—
123	C₂×C₄	—	2	4	no	—
124	C₂×C₄	—	2	2	no	—
125	C₂×C₄	—	2	2	no	—
126	C₂×C₄	—	4	4	no	—
127	C₂×C₄	—	4	2	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	4	4	no	—
130	C₂×C₄	—	4	4	no	—
131	C₂×C₄	—	4	4	no	—
132	C₂×C₄	—	8	2	no	—
133	C₂×C₄	—	8	2	no	—
134	C₂×C₄	#123×C₃₇	1	4	yes	—
135	C₈	—	8	4	no	—
136	C₂²	—	2	1	no	—
137	C₂²	—	2	1	no	—
138	C₂²	—	2	2	no	—
139	C₂²	—	2	2	no	—
140	C₂²	—	2	2	no	—
141	C₂²	—	4	4	no	—
142	C₂²	C₃₇×#381	1	1	yes	characteristic
143	C₂²	C₃₇×#410	1	2	yes	—
144	C₄	—	2	4	no	—
145	C₄	—	4	2	no	—
146	C₄	—	4	2	no	—
147	C₄	—	8	4	no	—
148	C₂	—	2	2	no	—
149	C₂	—	4	2	no	—
150	C₂	C₃₇×#1820	1	1	yes	characteristic
151	C₂	C₃₇×#1829	1	1	yes	characteristic
152	C₂	C₃₇×#1830	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₃₇×#2313

C₂

yes

maximal, characteristic

C₃₇×#1821

C₂

yes

maximal

C₃₇×#1306

C₂

yes

maximal

C₂²×C₄⋊C₄×C₃₇

C₂²

yes

characteristic

C₃₇×#407

C₂²

yes

—

C₃₇×#387

—

C₃₇×#381

C₂²

yes

—

C₃₇×#380

C₄

yes

—

C₄×C₂²⋊C₄×C₃₇

C₄

yes

—

C₃₇×#343

—

C₂³×C₄×C₃₇

C₂×C₄

yes

characteristic

C₂³×C₄×C₃₇

C₂×C₄

yes

characteristic

C₂³×C₄×C₃₇

C₂³

yes

characteristic

C₂×M₁₆×C₃₇

—

C₂×C₄⋊C₄×C₃₇

—

C₂×C₄⋊C₄×C₃₇

—

C₂×C₄⋊C₄×C₃₇

D₄

yes

—

C₂×C₂²⋊C₄×C₃₇

C₂×C₄

yes

—

C₂×C₄²×C₃₇

—

C₂²⋊C₈×C₃₇

—

#95×C₃₇

—

#95×C₃₇

—

#95×C₃₇

—

C₂⁴×C₃₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₃₇

—

C₂²×C₄×C₃₇

—

C₂²×C₄×C₃₇

—

C₂²×C₄×C₃₇

—

C₂²×C₄×C₃₇

—

C₂²×C₄×C₃₇

—

C₂²×C₄×C₃₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₃₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₃₇

C₂×D₄

yes

—

C₂²×C₄×C₃₇

#41

yes

—

M₁₆×C₃₇

—

C₂×C₈×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

—

C₂²⋊C₄×C₃₇

—

C₄²×C₃₇

—

C₂³×C₃₇

—

C₂³×C₃₇

—

C₂³×C₃₇

—

C₂³×C₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₂³×C₃₇

#117

yes

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

#123

yes

—

C₈×C₃₇

—

#8755

C₃₇

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

#381

yes

characteristic

C₂²×C₃₇

#410

yes

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

#2313

C₂×C₃₇

yes

characteristic

#1821

C₂×C₃₇

yes

—

#1306

C₂×C₃₇

yes

—

C₂×C₃₇

—

C₂×C₃₇

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂²×C₄⋊C₄

C₂²×C₃₇

yes

characteristic

#407

C₂²×C₃₇

yes

—

#387

—

#381

C₂²×C₃₇

yes

—

#380

C₄×C₃₇

yes

—

C₄×C₂²⋊C₄

C₄×C₃₇

yes

—

#343

—

C₃₇

#8755

yes

characteristic, direct product factor, Sylow

C₂³×C₄

C₂×C₄×C₃₇

yes

characteristic

C₂³×C₄

C₂×C₄×C₃₇

yes

characteristic

C₂³×C₄

C₂³×C₃₇

yes

characteristic

C₂×M₁₆

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

D₄×C₃₇

yes

—

C₂×C₂²⋊C₄

C₂×C₄×C₃₇

yes

—

C₂×C₄²

—

C₂²⋊C₈

—

#95

—

#95

—

100

#95

—

101

C₂⁴

C₂²×C₄×C₃₇

yes

characteristic

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

C₂²⋊C₄×C₃₇

yes

—

109

C₂²×C₄

C₂²⋊C₄×C₃₇

yes

—

110

C₂²×C₄

C₂×D₄×C₃₇

yes

—

111

C₂²×C₄

#41×C₃₇

yes

—

112

M₁₆

—

113

C₂×C₈

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₂²⋊C₄

—

117

C₄²

—

118

C₂³

—

119

C₂³

—

120

C₂³

—

121

C₂³

C₂×C₂²⋊C₄×C₃₇

yes

characteristic

122

C₂³

#117×C₃₇

yes

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

#123×C₃₇

yes

—

135

C₈

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

C₃₇×#381

yes

characteristic

143

C₂²

C₃₇×#410

yes

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₂

—

149

C₂

—

150

C₂

C₃₇×#1820

yes

characteristic

151

C₂

C₃₇×#1829

yes

characteristic

152

C₂

C₃₇×#1830

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₃₇	#381
commutator	C₂⁴	C₂²×C₄×C₃₇
Frattini, cosocle	C₂³×C₄	C₂³×C₃₇
prime radical	C₂⁴×C₃₇	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₇ ◁ C₂³×C₄×C₃₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₃₇

#381

commutator

C₂⁴

C₂²×C₄×C₃₇

Frattini, cosocle

C₂³×C₄

C₂³×C₃₇

prime radical

C₂⁴×C₃₇

C₂²×C₄

LCS

G ▷ C₂⁴ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₃₇ ◁ C₂³×C₄×C₃₇ ◁ G

N	H ≅ G/N	type
59. #8755	C₃₇	direct (N × H)
87. C₃₇	#8755	direct (N × H)

H ≅ G/N

type

59. #8755

C₃₇

direct (N × H)

87. C₃₇

#8755

direct (N × H)

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

subgroup	index
C₃₇×#2313	2
C₃₇×#1821	2
C₃₇×#1306	2
#8755	37

subgroup

index

C₃₇×#2313

C₃₇×#1821

C₃₇×#1306

#8755

Subgroups — C₃₇×#8755 #22630008

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C37×#8755 #22630008

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₇×#8755 #22630008