Group #17995737 — subgroups

Subgroups — #17995737

Order 4736.

290 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 93 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	C₂×#11924536	C₂	1	1	yes	maximal, characteristic
2	C₂×#11924460	C₂	1	2	yes	maximal
3	#14557295	C₂	1	2	yes	maximal
4	#14557288	C₂	1	1	yes	maximal, characteristic
5	C₃₇×#391	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₇₄	—	2	1	no	—
7	C₂²×#64436	C₂²	1	2	yes	—
8	#11924536	—	2	4	no	—
9	C₂×C₄⋊Dic₃₇	—	2	1	no	—
10	C₂×C₄⋊Dic₃₇	C₂²	1	2	yes	—
11	#11924460	—	2	4	no	—
12	C₂²×Q₈×C₃₇	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
15	#95×C₃₇	—	2	1	no	—
16	#95×C₃₇	—	2	1	no	—
17	C₂×Q₈×C₃₇	—	2	1	no	—
18	C₂×Q₈×C₃₇	—	2	2	no	—
19	C₂×Q₈×C₃₇	—	2	2	no	—
20	C₂×Q₈×C₃₇	Q₈	1	2	yes	complemented
21	C₂²×C₄×C₃₇	—	2	1	no	—
22	C₂²×C₄×C₃₇	—	2	1	no	—
23	C₂²×C₄×C₃₇	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₃₇	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
26	C₂²×Dic₃₇	—	2	2	no	—
27	C₂×Dic₇₄	—	2	4	no	—
28	C₂×Dic₇₄	—	4	2	no	—
29	C₂×Dic₇₄	—	4	2	no	—
30	C₄⋊C₄×C₃₇	—	2	2	no	—
31	C₄²×C₃₇	—	2	2	no	—
32	C₄⋊Dic₃₇	—	2	4	no	—
33	C₄⋊Dic₃₇	—	2	2	no	—
34	C₄⋊Dic₃₇	D₄	1	4	yes	—
35	C₂×#64436	—	2	2	no	—
36	C₂×#64436	—	2	2	no	—
37	C₂×#64436	—	2	2	no	—
38	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
39	Q₈×C₃₇	—	2	4	no	—
40	Q₈×C₃₇	—	4	2	no	—
41	C₂×C₄×C₃₇	—	2	1	no	—
42	C₂×C₄×C₃₇	—	2	1	no	—
43	C₂×C₄×C₃₇	—	2	1	no	—
44	C₂×C₄×C₃₇	—	2	1	no	—
45	C₂×C₄×C₃₇	—	2	2	no	—
46	C₂×C₄×C₃₇	—	4	1	no	—
47	C₂×C₄×C₃₇	—	4	1	no	—
48	C₂×C₄×C₃₇	—	4	1	no	—
49	C₂×C₄×C₃₇	—	4	1	no	—
50	C₂×C₄×C₃₇	—	4	1	no	—
51	C₂×C₄×C₃₇	SD₁₆	1	2	yes	—
52	C₂×C₄×C₃₇	Q₁₆	1	2	yes	—
53	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₃₇	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₄×C₃₇	#41	1	1	yes	characteristic
56	C₂×C₄×C₃₇	#41	1	2	yes	—
57	C₂×Dic₃₇	—	2	4	no	—
58	C₂×Dic₃₇	—	4	2	no	—
59	C₂×Dic₃₇	—	4	2	no	—
60	Dic₇₄	—	4	8	no	—
61	#64436	—	4	2	no	—
62	C₂²×C₃₇	#120	1	1	yes	characteristic
63	C₂²×C₃₇	C₂²⋊Q₈	1	1	yes	characteristic
64	C₂²×C₃₇	#124	1	1	yes	characteristic
65	C₂²×C₃₇	C₂×SD₁₆	1	1	yes	characteristic
66	C₂²×C₃₇	C₂×Q₁₆	1	1	yes	characteristic
67	C₂²×C₃₇	AGL(1,ℤ/8)	1	1	yes	characteristic
68	C₂²×C₃₇	#137	1	1	yes	characteristic
69	C₄×C₃₇	—	2	2	no	—
70	C₄×C₃₇	—	2	2	no	—
71	C₄×C₃₇	—	4	2	no	—
72	C₄×C₃₇	—	8	1	no	—
73	C₄×C₃₇	C₂²⋊Q₈	1	2	yes	—
74	C₄×C₃₇	#124	1	2	yes	—
75	Dic₃₇	—	4	4	no	—
76	#2024	—	37	1	no	maximal, Sylow
77	C₂×C₃₇	#393	1	1	yes	characteristic
78	C₂×C₃₇	#450	1	1	yes	characteristic
79	C₂×C₃₇	#451	1	1	yes	characteristic
80	C₂×C₃₇	#475	1	1	yes	characteristic
81	C₂×C₃₇	#477	1	1	yes	characteristic
82	C₂×C₃₇	#487	1	1	yes	characteristic
83	C₂×C₃₇	#489	1	1	yes	characteristic
84	C₂×C₄⋊Q₈	—	37	1	no	—
85	C₂×#103	—	37	2	no	—
86	#391	D₃₇	1	1	yes	characteristic
87	#340	—	37	2	no	—
88	#336	—	37	1	no	—
89	C₃₇	#2024	1	1	yes	characteristic, complemented, Sylow
90	C₂²×Q₈	—	74	1	no	—
91	C₂²×Q₈	C₂×D₃₇	1	1	yes	characteristic
92	C₂²×C₈	—	37	2	no	—
93	C₄⋊Q₈	—	74	4	no	—
94	C₂×C₄⋊C₄	—	37	2	no	—
95	C₂×C₄⋊C₄	—	74	1	no	—
96	C₂×C₄⋊C₄	C₂×D₃₇	1	1	yes	characteristic
97	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
98	#103	—	74	4	no	—
99	#95	—	2	1	no	—
100	#95	—	2	1	no	—
101	C₂×Q₈	—	2	1	no	—
102	C₂×Q₈	—	2	2	no	—
103	C₂×Q₈	—	2	2	no	—
104	C₂×Q₈	—	74	4	no	—
105	C₂×Q₈	—	148	2	no	—
106	C₂×Q₈	—	148	2	no	—
107	C₂×Q₈	Dic₇₄	1	2	yes	complemented
108	C₂²×C₄	—	2	1	no	—
109	C₂²×C₄	—	2	1	no	—
110	C₂²×C₄	—	74	2	no	—
111	C₂²×C₄	C₂²⋊D₃₇	1	1	yes	characteristic
112	C₂²×C₄	C₂²⋊D₃₇	1	1	yes	characteristic
113	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
114	C₂×C₈	—	74	2	no	—
115	C₂×C₈	—	74	2	no	—
116	C₂×C₈	—	74	2	no	—
117	C₄⋊C₄	—	2	2	no	—
118	C₄⋊C₄	—	37	4	no	—
119	C₄⋊C₄	—	74	4	no	—
120	C₄⋊C₄	—	74	2	no	—
121	C₄²	—	2	2	no	—
122	C₂³	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
123	Q₈	—	2	4	no	—
124	Q₈	—	4	2	no	—
125	Q₈	—	148	8	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	2	1	no	—
128	C₂×C₄	—	2	1	no	—
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	4	1	no	—
132	C₂×C₄	—	4	1	no	—
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	74	4	no	—
137	C₂×C₄	—	148	2	no	—
138	C₂×C₄	—	148	2	no	—
139	C₂×C₄	#10596941	1	2	yes	—
140	C₂×C₄	#10596942	1	2	yes	—
141	C₂×C₄	C₂×Dic₇₄	1	1	yes	characteristic
142	C₂×C₄	#10596962	1	1	yes	characteristic
143	C₂×C₄	D₄×D₃₇	1	2	yes	—
144	C₂×C₄	#10596964	1	2	yes	—
145	C₈	—	148	2	no	—
146	C₂²	#11924504	1	1	yes	characteristic
147	C₂²	#11924515	1	1	yes	characteristic
148	C₂²	#11924523	1	1	yes	characteristic
149	C₂²	C₂×#10596941	1	1	yes	characteristic
150	C₂²	C₂×#10596942	1	1	yes	characteristic
151	C₂²	#11924561	1	1	yes	characteristic
152	C₂²	#11924566	1	1	yes	characteristic
153	C₄	—	2	2	no	—
154	C₄	—	2	2	no	—
155	C₄	—	4	2	no	—
156	C₄	—	8	1	no	—
157	C₄	—	148	4	no	—
158	C₄	#11924513	1	2	yes	—
159	C₄	#11924522	1	2	yes	—
160	C₂	#14557445	1	1	yes	characteristic
161	C₂	#14557683	1	1	yes	characteristic
162	C₂	#14557688	1	1	yes	characteristic
163	C₂	#14557818	1	1	yes	characteristic
164	C₂	#14557828	1	1	yes	characteristic
165	C₂	#14557883	1	1	yes	characteristic
166	C₂	#14557892	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924536

C₂

yes

maximal, characteristic

C₂×#11924460

C₂

yes

maximal

#14557295

C₂

yes

maximal

#14557288

C₂

yes

maximal, characteristic

C₃₇×#391

C₂

yes

maximal, characteristic

C₂²×Dic₇₄

—

C₂²×#64436

C₂²

yes

—

#11924536

—

C₂×C₄⋊Dic₃₇

—

C₂×C₄⋊Dic₃₇

C₂²

yes

—

#11924460

—

C₂²×Q₈×C₃₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₃₇

C₂²

yes

characteristic

C₂×C₄²×C₃₇

C₂²

yes

characteristic

#95×C₃₇

—

#95×C₃₇

—

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

Q₈

yes

complemented

C₂²×C₄×C₃₇

—

C₂²×C₄×C₃₇

—

C₂²×C₄×C₃₇

D₄

yes

characteristic

C₂²×C₄×C₃₇

D₄

yes

characteristic

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

—

C₂×Dic₇₄

—

C₂×Dic₇₄

—

C₂×Dic₇₄

—

C₄⋊C₄×C₃₇

—

C₄²×C₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

D₄

yes

—

C₂×#64436

—

C₂×#64436

—

C₂×#64436

—

C₂³×C₃₇

C₂×D₄

yes

characteristic

Q₈×C₃₇

—

Q₈×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

SD₁₆

yes

—

C₂×C₄×C₃₇

Q₁₆

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×Q₈

yes

C₂×C₄×C₃₇

yes

C₂×C₄×C₃₇

yes

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

Dic₇₄

—

#64436

—

C₂²×C₃₇

#120

yes

characteristic

C₂²×C₃₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₃₇

#124

yes

characteristic

C₂²×C₃₇

C₂×SD₁₆

yes

characteristic

C₂²×C₃₇

C₂×Q₁₆

yes

C₂²×C₃₇

yes

C₂²×C₃₇

yes

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₂²⋊Q₈

yes

—

C₄×C₃₇

#124

yes

—

Dic₃₇

—

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₄⋊Q₈

—

C₂×#103

—

#391

D₃₇

yes

characteristic

#340

—

#336

—

C₃₇

#2024

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×Q₈

C₂×D₃₇

yes

characteristic

C₂²×C₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃₇

yes

characteristic

C₂×C₄²

C₂×D₃₇

yes

characteristic

#103

—

#95

—

100

#95

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

148

—

106

C₂×Q₈

—

148

—

107

C₂×Q₈

Dic₇₄

yes

complemented

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

C₂²⋊D₃₇

yes

characteristic

112

C₂²×C₄

C₂²⋊D₃₇

yes

characteristic

113

C₂²×C₄

C₂²×D₃₇

yes

characteristic

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄²

—

122

C₂³

C₂×C₂²⋊D₃₇

yes

characteristic

123

Q₈

—

124

Q₈

—

125

Q₈

—

148

—

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

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

148

—

138

C₂×C₄

—

148

—

139

C₂×C₄

#10596941

yes

—

140

C₂×C₄

#10596942

yes

—

141

C₂×C₄

C₂×Dic₇₄

yes

characteristic

142

C₂×C₄

#10596962

yes

characteristic

143

C₂×C₄

D₄×D₃₇

yes

—

144

C₂×C₄

#10596964

yes

—

145

C₈

—

148

—

146

C₂²

#11924504

yes

characteristic

147

C₂²

#11924515

yes

characteristic

148

C₂²

#11924523

yes

characteristic

149

C₂²

C₂×#10596941

yes

characteristic

150

C₂²

C₂×#10596942

yes

characteristic

151

C₂²

#11924561

yes

characteristic

152

C₂²

#11924566

yes

characteristic

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

148

—

158

C₄

#11924513

yes

—

159

C₄

#11924522

yes

—

160

C₂

#14557445

yes

characteristic

161

C₂

#14557683

yes

characteristic

162

C₂

#14557688

yes

characteristic

163

C₂

#14557818

yes

characteristic

164

C₂

#14557828

yes

characteristic

165

C₂

#14557883

yes

characteristic

166

C₂

#14557892

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₃₇
commutator, cosocle	C₂²×C₄×C₃₇	C₂³
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting	C₃₇×#391	C₂
socle, prime radical	C₂³×C₃₇	C₂×D₄
hypercenter	#391	D₃₇
hypocenter	C₃₇	#2024
LCS	G ▷ C₂²×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₃₇

commutator, cosocle

C₂²×C₄×C₃₇

C₂³

Frattini

C₂²×C₄

C₂²×D₃₇

Fitting

C₃₇×#391

C₂

socle, prime radical

C₂³×C₃₇

C₂×D₄

D₃₇

C₃₇

G ▷ C₂²×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #391

N	H ≅ G/N	type
20. C₂×Q₈×C₃₇	Q₈	semidirect
89. C₃₇	#2024	semidirect
107. C₂×Q₈	Dic₇₄	semidirect

H ≅ G/N

type

20. C₂×Q₈×C₃₇

Q₈

semidirect

89. C₃₇

#2024

semidirect

107. C₂×Q₈

Dic₇₄

semidirect

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

subgroup	index
C₂×#11924536	2
C₂×#11924460	2
#14557295	2
#14557288	2
C₃₇×#391	2
#2024	37

subgroup

index

C₃₇×#391

Subgroups — #17995737

Conjugacy classes of subgroups

Complements

Maximal subgroups