Group #37591057 — subgroups

Subgroups — #1440×C₁₉₃ #37591057

Order 24704.

736 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 736 conjugacy classes of subgroups — too many to draw. Only its 216 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	C₂²×C₂²⋊C₄×C₁₉₃	C₂	1	2	yes	maximal
2	C₂²×C₂²⋊C₄×C₁₉₃	C₂	1	1	yes	maximal, characteristic
3	C₂×#95×C₁₉₃	C₂	1	4	yes	maximal, complemented
4	C₂⁵×C₁₉₃	C₂²	1	1	yes	characteristic
5	C₂³×C₄×C₁₉₃	C₂²	1	2	yes	—
6	C₂³×C₄×C₁₉₃	C₂²	1	4	yes	—
7	C₂×C₂²⋊C₄×C₁₉₃	—	2	4	no	—
8	C₂×C₂²⋊C₄×C₁₉₃	—	2	4	no	—
9	C₂×C₂²⋊C₄×C₁₉₃	—	2	8	no	—
10	C₂×C₂²⋊C₄×C₁₉₃	C₄	1	4	yes	complemented
11	#95×C₁₉₃	—	2	8	no	—
12	C₂⁴×C₁₉₃	—	2	4	no	—
13	C₂⁴×C₁₉₃	—	2	4	no	—
14	C₂⁴×C₁₉₃	—	2	4	no	—
15	C₂⁴×C₁₉₃	C₂×C₄	1	2	yes	—
16	C₂⁴×C₁₉₃	D₄	1	4	yes	—
17	C₂⁴×C₁₉₃	C₂³	1	1	yes	characteristic
18	C₂²×C₄×C₁₉₃	—	2	4	no	—
19	C₂²×C₄×C₁₉₃	—	2	4	no	—
20	C₂²×C₄×C₁₉₃	—	2	2	no	—
21	C₂²×C₄×C₁₉₃	—	2	2	no	—
22	C₂²×C₄×C₁₉₃	—	2	4	no	—
23	C₂²×C₄×C₁₉₃	—	2	4	no	—
24	C₂²×C₄×C₁₉₃	—	2	4	no	—
25	C₂²×C₄×C₁₉₃	—	2	4	no	—
26	C₂²×C₄×C₁₉₃	—	2	4	no	—
27	C₂²×C₄×C₁₉₃	—	2	4	no	—
28	C₂²×C₄×C₁₉₃	C₂×C₄	1	4	yes	—
29	C₂²×C₄×C₁₉₃	D₄	1	8	yes	—
30	C₂²⋊C₄×C₁₉₃	—	2	16	no	—
31	C₂²⋊C₄×C₁₉₃	—	2	8	no	—
32	C₂³×C₁₉₃	—	2	4	no	—
33	C₂³×C₁₉₃	—	2	8	no	—
34	C₂³×C₁₉₃	—	2	4	no	—
35	C₂³×C₁₉₃	—	2	8	no	—
36	C₂³×C₁₉₃	—	2	8	no	—
37	C₂³×C₁₉₃	—	2	4	no	—
38	C₂³×C₁₉₃	—	4	4	no	—
39	C₂³×C₁₉₃	—	4	4	no	—
40	C₂³×C₁₉₃	—	4	4	no	—
41	C₂³×C₁₉₃	—	4	4	no	—
42	C₂³×C₁₉₃	C₂²⋊C₄	1	4	yes	—
43	C₂³×C₁₉₃	C₂²×C₄	1	1	yes	characteristic
44	C₂³×C₁₉₃	C₂×D₄	1	2	yes	—
45	C₂³×C₁₉₃	C₂×D₄	1	4	yes	—
46	C₂³×C₁₉₃	#41	1	4	yes	—
47	C₂³×C₁₉₃	#41	1	4	yes	—
48	C₂×C₄×C₁₉₃	—	2	4	no	—
49	C₂×C₄×C₁₉₃	—	2	8	no	—
50	C₂×C₄×C₁₉₃	—	2	8	no	—
51	C₂×C₄×C₁₉₃	—	2	8	no	—
52	C₂×C₄×C₁₉₃	—	2	8	no	—
53	C₂×C₄×C₁₉₃	—	4	2	no	—
54	C₂×C₄×C₁₉₃	—	4	4	no	—
55	C₂×C₄×C₁₉₃	—	4	4	no	—
56	C₂×C₄×C₁₉₃	—	4	4	no	—
57	C₂×C₄×C₁₉₃	—	4	2	no	—
58	C₂×C₄×C₁₉₃	—	4	4	no	—
59	C₂×C₄×C₁₉₃	—	4	4	no	—
60	C₂²×C₁₉₃	—	2	4	no	—
61	C₂²×C₁₉₃	—	2	8	no	—
62	C₂²×C₁₉₃	—	4	8	no	—
63	C₂²×C₁₉₃	—	4	8	no	—
64	C₂²×C₁₉₃	—	4	8	no	—
65	C₂²×C₁₉₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₁₉₃	#117	1	2	yes	—
67	C₂²×C₁₉₃	C₄×D₄	1	4	yes	—
68	C₂²×C₁₉₃	#120	1	2	yes	—
69	C₂²×C₁₉₃	#121	1	4	yes	—
70	C₂²×C₁₉₃	#121	1	4	yes	—
71	C₂²×C₁₉₃	#123	1	4	yes	—
72	C₂²×C₁₉₃	#123	1	2	yes	—
73	C₂²×C₁₉₃	#123	1	4	yes	—
74	C₂²×C₁₉₃	#124	1	4	yes	—
75	C₂²×C₁₉₃	#126	1	4	yes	—
76	C₄×C₁₉₃	—	4	4	no	—
77	C₄×C₁₉₃	—	4	8	no	—
78	C₂×C₁₉₃	—	4	4	no	—
79	C₂×C₁₉₃	#381	1	1	yes	characteristic
80	C₂×C₁₉₃	#386	1	2	yes	—
81	C₂×C₁₉₃	#388	1	4	yes	—
82	C₂×C₁₉₃	#394	1	4	yes	—
83	C₂×C₁₉₃	#397	1	4	yes	—
84	C₁₉₃	#1440	1	1	yes	characteristic, direct product factor, Sylow
85	#1440	C₁₉₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
86	C₂²×C₂²⋊C₄	C₂×C₁₉₃	1	2	yes	—
87	C₂²×C₂²⋊C₄	C₂×C₁₉₃	1	1	yes	characteristic
88	C₂×#95	C₂×C₁₉₃	1	4	yes	complemented
89	C₂⁵	C₂²×C₁₉₃	1	1	yes	characteristic
90	C₂³×C₄	C₂²×C₁₉₃	1	2	yes	—
91	C₂³×C₄	C₂²×C₁₉₃	1	4	yes	—
92	C₂×C₂²⋊C₄	—	2	4	no	—
93	C₂×C₂²⋊C₄	—	2	4	no	—
94	C₂×C₂²⋊C₄	—	2	8	no	—
95	C₂×C₂²⋊C₄	C₄×C₁₉₃	1	4	yes	complemented
96	#95	—	2	8	no	—
97	C₂⁴	—	2	4	no	—
98	C₂⁴	—	2	4	no	—
99	C₂⁴	—	2	4	no	—
100	C₂⁴	C₂×C₄×C₁₉₃	1	2	yes	—
101	C₂⁴	D₄×C₁₉₃	1	4	yes	—
102	C₂⁴	C₂³×C₁₉₃	1	1	yes	characteristic
103	C₂²×C₄	—	2	4	no	—
104	C₂²×C₄	—	2	4	no	—
105	C₂²×C₄	—	2	2	no	—
106	C₂²×C₄	—	2	2	no	—
107	C₂²×C₄	—	2	4	no	—
108	C₂²×C₄	—	2	4	no	—
109	C₂²×C₄	—	2	4	no	—
110	C₂²×C₄	—	2	4	no	—
111	C₂²×C₄	—	2	4	no	—
112	C₂²×C₄	—	2	4	no	—
113	C₂²×C₄	C₂×C₄×C₁₉₃	1	4	yes	—
114	C₂²×C₄	D₄×C₁₉₃	1	8	yes	—
115	C₂²⋊C₄	—	2	16	no	—
116	C₂²⋊C₄	—	2	8	no	—
117	C₂³	—	2	4	no	—
118	C₂³	—	2	8	no	—
119	C₂³	—	2	4	no	—
120	C₂³	—	2	8	no	—
121	C₂³	—	2	8	no	—
122	C₂³	—	2	4	no	—
123	C₂³	—	4	4	no	—
124	C₂³	—	4	4	no	—
125	C₂³	—	4	4	no	—
126	C₂³	—	4	4	no	—
127	C₂³	C₂²⋊C₄×C₁₉₃	1	4	yes	—
128	C₂³	C₂²×C₄×C₁₉₃	1	1	yes	characteristic
129	C₂³	C₂×D₄×C₁₉₃	1	2	yes	—
130	C₂³	C₂×D₄×C₁₉₃	1	4	yes	—
131	C₂³	#41×C₁₉₃	1	4	yes	—
132	C₂³	#41×C₁₉₃	1	4	yes	—
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	—	2	8	no	—
135	C₂×C₄	—	2	8	no	—
136	C₂×C₄	—	2	8	no	—
137	C₂×C₄	—	2	8	no	—
138	C₂×C₄	—	4	2	no	—
139	C₂×C₄	—	4	4	no	—
140	C₂×C₄	—	4	4	no	—
141	C₂×C₄	—	4	4	no	—
142	C₂×C₄	—	4	2	no	—
143	C₂×C₄	—	4	4	no	—
144	C₂×C₄	—	4	4	no	—
145	C₂²	—	2	4	no	—
146	C₂²	—	2	8	no	—
147	C₂²	—	4	8	no	—
148	C₂²	—	4	8	no	—
149	C₂²	—	4	8	no	—
150	C₂²	C₂×C₂²⋊C₄×C₁₉₃	1	1	yes	characteristic
151	C₂²	#117×C₁₉₃	1	2	yes	—
152	C₂²	C₄×D₄×C₁₉₃	1	4	yes	—
153	C₂²	#120×C₁₉₃	1	2	yes	—
154	C₂²	#121×C₁₉₃	1	4	yes	—
155	C₂²	#121×C₁₉₃	1	4	yes	—
156	C₂²	#123×C₁₉₃	1	4	yes	—
157	C₂²	#123×C₁₉₃	1	2	yes	—
158	C₂²	#123×C₁₉₃	1	4	yes	—
159	C₂²	#124×C₁₉₃	1	4	yes	—
160	C₂²	#126×C₁₉₃	1	4	yes	—
161	C₄	—	4	4	no	—
162	C₄	—	4	8	no	—
163	C₂	—	4	4	no	—
164	C₂	#381×C₁₉₃	1	1	yes	characteristic
165	C₂	#386×C₁₉₃	1	2	yes	—
166	C₂	#388×C₁₉₃	1	4	yes	—
167	C₂	#394×C₁₉₃	1	4	yes	—
168	C₂	#397×C₁₉₃	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₂²⋊C₄×C₁₉₃

C₂

yes

maximal

C₂²×C₂²⋊C₄×C₁₉₃

C₂

yes

maximal, characteristic

C₂×#95×C₁₉₃

C₂

yes

maximal, complemented

C₂⁵×C₁₉₃

C₂²

yes

characteristic

C₂³×C₄×C₁₉₃

C₂²

yes

—

C₂³×C₄×C₁₉₃

C₂²

yes

—

C₂×C₂²⋊C₄×C₁₉₃

—

C₂×C₂²⋊C₄×C₁₉₃

—

C₂×C₂²⋊C₄×C₁₉₃

—

C₂×C₂²⋊C₄×C₁₉₃

C₄

yes

complemented

#95×C₁₉₃

—

C₂⁴×C₁₉₃

—

C₂⁴×C₁₉₃

—

C₂⁴×C₁₉₃

—

C₂⁴×C₁₉₃

C₂×C₄

yes

—

C₂⁴×C₁₉₃

D₄

yes

—

C₂⁴×C₁₉₃

C₂³

yes

characteristic

C₂²×C₄×C₁₉₃

—

C₂²×C₄×C₁₉₃

—

C₂²×C₄×C₁₉₃

—

C₂²×C₄×C₁₉₃

—

C₂²×C₄×C₁₉₃

—

C₂²×C₄×C₁₉₃

—

C₂²×C₄×C₁₉₃

—

C₂²×C₄×C₁₉₃

—

C₂²×C₄×C₁₉₃

—

C₂²×C₄×C₁₉₃

—

C₂²×C₄×C₁₉₃

C₂×C₄

yes

—

C₂²×C₄×C₁₉₃

D₄

yes

—

C₂²⋊C₄×C₁₉₃

—

C₂²⋊C₄×C₁₉₃

—

C₂³×C₁₉₃

—

C₂³×C₁₉₃

—

C₂³×C₁₉₃

—

C₂³×C₁₉₃

—

C₂³×C₁₉₃

—

C₂³×C₁₉₃

—

C₂³×C₁₉₃

—

C₂³×C₁₉₃

—

C₂³×C₁₉₃

—

C₂³×C₁₉₃

—

C₂³×C₁₉₃

C₂²⋊C₄

yes

—

C₂³×C₁₉₃

C₂²×C₄

yes

characteristic

C₂³×C₁₉₃

C₂×D₄

yes

—

C₂³×C₁₉₃

C₂×D₄

yes

—

C₂³×C₁₉₃

#41

yes

—

C₂³×C₁₉₃

#41

yes

—

C₂×C₄×C₁₉₃

—

C₂×C₄×C₁₉₃

—

C₂×C₄×C₁₉₃

—

C₂×C₄×C₁₉₃

—

C₂×C₄×C₁₉₃

—

C₂×C₄×C₁₉₃

—

C₂×C₄×C₁₉₃

—

C₂×C₄×C₁₉₃

—

C₂×C₄×C₁₉₃

—

C₂×C₄×C₁₉₃

—

C₂×C₄×C₁₉₃

—

C₂×C₄×C₁₉₃

—

C₂²×C₁₉₃

—

C₂²×C₁₉₃

—

C₂²×C₁₉₃

—

C₂²×C₁₉₃

—

C₂²×C₁₉₃

—

C₂²×C₁₉₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₉₃

#117

yes

—

C₂²×C₁₉₃

C₄×D₄

yes

—

C₂²×C₁₉₃

#120

yes

—

C₂²×C₁₉₃

#121

yes

—

C₂²×C₁₉₃

#121

yes

—

C₂²×C₁₉₃

#123

yes

—

C₂²×C₁₉₃

#123

yes

—

C₂²×C₁₉₃

#123

yes

—

C₂²×C₁₉₃

#124

yes

—

C₂²×C₁₉₃

#126

yes

—

C₄×C₁₉₃

—

C₄×C₁₉₃

—

C₂×C₁₉₃

—

C₂×C₁₉₃

#381

yes

characteristic

C₂×C₁₉₃

#386

yes

—

C₂×C₁₉₃

#388

yes

—

C₂×C₁₉₃

#394

yes

—

C₂×C₁₉₃

#397

yes

—

C₁₉₃

#1440

yes

characteristic, direct product factor, Sylow

#1440

C₁₉₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₂²⋊C₄

C₂×C₁₉₃

yes

—

C₂²×C₂²⋊C₄

C₂×C₁₉₃

yes

characteristic

C₂×#95

C₂×C₁₉₃

yes

complemented

C₂⁵

C₂²×C₁₉₃

yes

characteristic

C₂³×C₄

C₂²×C₁₉₃

yes

—

C₂³×C₄

C₂²×C₁₉₃

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

C₄×C₁₉₃

yes

complemented

#95

—

C₂⁴

—

C₂⁴

—

C₂⁴

—

100

C₂⁴

C₂×C₄×C₁₉₃

yes

—

101

C₂⁴

D₄×C₁₉₃

yes

—

102

C₂⁴

C₂³×C₁₉₃

yes

characteristic

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

C₂×C₄×C₁₉₃

yes

—

114

C₂²×C₄

D₄×C₁₉₃

yes

—

115

C₂²⋊C₄

—

116

C₂²⋊C₄

—

117

C₂³

—

118

C₂³

—

119

C₂³

—

120

C₂³

—

121

C₂³

—

122

C₂³

—

123

C₂³

—

124

C₂³

—

125

C₂³

—

126

C₂³

—

127

C₂³

C₂²⋊C₄×C₁₉₃

yes

—

128

C₂³

C₂²×C₄×C₁₉₃

yes

characteristic

129

C₂³

C₂×D₄×C₁₉₃

yes

—

130

C₂³

C₂×D₄×C₁₉₃

yes

—

131

C₂³

#41×C₁₉₃

yes

—

132

C₂³

#41×C₁₉₃

yes

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

C₂×C₂²⋊C₄×C₁₉₃

yes

characteristic

151

C₂²

#117×C₁₉₃

yes

—

152

C₂²

C₄×D₄×C₁₉₃

yes

—

153

C₂²

#120×C₁₉₃

yes

—

154

C₂²

#121×C₁₉₃

yes

—

155

C₂²

#121×C₁₉₃

yes

—

156

C₂²

#123×C₁₉₃

yes

—

157

C₂²

#123×C₁₉₃

yes

—

158

C₂²

#123×C₁₉₃

yes

—

159

C₂²

#124×C₁₉₃

yes

—

160

C₂²

#126×C₁₉₃

yes

—

161

C₄

—

162

C₄

—

163

C₂

—

164

C₂

#381×C₁₉₃

yes

characteristic

165

C₂

#386×C₁₉₃

yes

—

166

C₂

#388×C₁₉₃

yes

—

167

C₂

#394×C₁₉₃

yes

—

168

C₂

#397×C₁₉₃

yes

—

	subgroup	G/H
center, socle	C₂⁴×C₁₉₃	C₂³
commutator	C₂³	C₂²×C₄×C₁₉₃
Frattini, cosocle	C₂⁴	C₂³×C₁₉₃
prime radical	C₂⁵×C₁₉₃	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₉₃ ◁ G

subgroup

G/H

center, socle

C₂⁴×C₁₉₃

C₂³

commutator

C₂³

C₂²×C₄×C₁₉₃

Frattini, cosocle

C₂⁴

C₂³×C₁₉₃

prime radical

C₂⁵×C₁₉₃

C₂²

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂⁴×C₁₉₃ ◁ G

N	H ≅ G/N	type
3. C₂×#95×C₁₉₃	C₂	semidirect
10. C₂×C₂²⋊C₄×C₁₉₃	C₄	semidirect
84. C₁₉₃	#1440	direct (N × H)
85. #1440	C₁₉₃	direct (N × H)
88. C₂×#95	C₂×C₁₉₃	semidirect
95. C₂×C₂²⋊C₄	C₄×C₁₉₃	semidirect

H ≅ G/N

type

3. C₂×#95×C₁₉₃

C₂

semidirect

10. C₂×C₂²⋊C₄×C₁₉₃

C₄

semidirect

84. C₁₉₃

#1440

direct (N × H)

85. #1440

C₁₉₃

direct (N × H)

88. C₂×#95

C₂×C₁₉₃

semidirect

95. C₂×C₂²⋊C₄

C₄×C₁₉₃

semidirect

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

subgroup	index
C₂²×C₂²⋊C₄×C₁₉₃	2
C₂×#95×C₁₉₃	2
#1440	193

subgroup

index

C₂²×C₂²⋊C₄×C₁₉₃

C₂×#95×C₁₉₃

#1440

193

Subgroups — #1440×C₁₉₃ #37591057

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1440×C193 #37591057

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1440×C₁₉₃ #37591057