Group #18004985 — subgroups

Subgroups — #18004985

Order 4736.

400 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 133 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×#11924524	C₂	1	1	yes	maximal, characteristic
2	#14557895	C₂	1	4	yes	maximal, complemented
3	#14557890	C₂	1	4	yes	maximal, complemented
4	#14557541	C₂	1	2	yes	maximal, complemented
5	#14557537	C₂	1	2	yes	maximal
6	C₄×#10596934	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₇×#533	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Dic₇₄	C₂²	1	1	yes	characteristic
9	C₂²×D₁₄₈	C₂²	1	1	yes	characteristic
10	C₂×#10596934	C₂²	1	2	yes	—
11	#11924524	—	2	2	no	—
12	#11924524	C₂²	1	2	yes	—
13	#11924524	C₂²	1	2	yes	—
14	C₂×#10596938	—	2	2	no	—
15	#11924459	C₂²	1	8	yes	complemented
16	#11924456	C₂²	1	8	yes	—
17	#11924447	C₂²	1	2	yes	complemented
18	C₄×#64436	C₂²	1	2	yes	complemented
19	C₄⋊Q₈×C₃₇	C₂²	1	2	yes	complemented
20	#125×C₃₇	C₂²	1	2	yes	complemented
21	C₂²⋊Q₈×C₃₇	—	2	2	no	—
22	C₄×Q₈×C₃₇	—	2	2	no	—
23	#117×C₃₇	C₂²	1	2	yes	—
24	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
25	C₂³×D₃₇	—	2	1	no	—
26	C₂×Q₈×C₃₇	—	2	2	no	—
27	C₂²×C₄×C₃₇	D₄	1	2	yes	—
28	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
29	C₂²×Dic₃₇	—	2	1	no	—
30	C₂×D₁₄₈	—	2	4	no	—
31	C₂×D₁₄₈	—	2	1	no	—
32	C₂×D₁₄₈	C₂³	1	2	yes	—
33	C₂×Dic₇₄	—	2	4	no	—
34	C₂×Dic₇₄	—	2	1	no	—
35	C₂×Dic₇₄	C₂³	1	2	yes	—
36	C₄⋊C₄×C₃₇	—	2	4	no	—
37	C₄⋊C₄×C₃₇	—	2	2	no	—
38	C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
39	C₂²⋊C₄×C₃₇	—	2	2	no	—
40	C₄²×C₃₇	—	2	2	no	—
41	C₄²×C₃₇	D₄	1	2	yes	—
42	C₄²×C₃₇	C₂³	1	1	yes	characteristic
43	C₄²×C₃₇	C₂³	1	1	yes	characteristic
44	#10596938	—	2	4	no	—
45	#10596938	—	2	4	no	—
46	#10596934	—	2	4	no	—
47	C₂×#64436	C₂³	1	4	yes	—
48	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
49	C₂²×D₃₇	—	2	2	no	—
50	C₂²×D₃₇	—	4	4	no	—
51	C₂²×D₃₇	—	4	2	no	—
52	Q₈×C₃₇	—	4	4	no	—
53	C₂×C₄×C₃₇	—	2	4	no	—
54	C₂×C₄×C₃₇	—	4	2	no	—
55	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
56	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₃₇	#41	1	4	yes	—
59	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
60	C₂×Dic₃₇	—	2	2	no	—
61	C₂×Dic₃₇	—	4	2	no	—
62	D₁₄₈	—	2	4	no	—
63	D₁₄₈	—	4	2	no	—
64	Dic₇₄	—	2	4	no	—
65	Dic₇₄	—	4	2	no	—
66	#64436	—	2	4	no	—
67	C₂²×C₃₇	—	2	2	no	—
68	C₂²×C₃₇	#124	1	2	yes	—
69	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
70	C₂×D₃₇	—	4	2	no	—
71	C₂×D₃₇	—	4	4	no	—
72	C₂×D₃₇	—	8	4	no	—
73	C₄×C₃₇	—	2	2	no	—
74	C₄×C₃₇	—	2	2	no	—
75	C₄×C₃₇	—	4	4	no	—
76	C₄×C₃₇	#124	1	2	yes	—
77	C₄×C₃₇	C₂×#41	1	2	yes	—
78	Dic₃₇	—	4	2	no	—
79	#3140	—	37	1	no	maximal, Sylow
80	C₂×C₃₇	—	2	2	no	—
81	C₂×C₃₇	C₂×#124	1	1	yes	characteristic
82	C₂×C₃₇	#575	1	2	yes	—
83	D₃₇	—	8	2	no	—
84	#533	D₃₇	1	1	yes	characteristic, complemented
85	C₂×#124	—	37	1	no	—
86	#489	—	37	4	no	—
87	#488	—	37	4	no	—
88	#419	—	37	2	no	—
89	#418	—	37	2	no	—
90	C₄×M₁₆	—	37	1	no	—
91	C₃₇	#3140	1	1	yes	characteristic, complemented, Sylow
92	C₂²×Q₈	—	37	1	no	—
93	C₂²×D₄	—	37	1	no	—
94	C₂×M₁₆	—	37	2	no	—
95	C₄⋊Q₈	C₂×D₃₇	1	2	yes	complemented
96	#125	C₂×D₃₇	1	2	yes	complemented
97	#124	—	37	2	no	—
98	#124	—	37	2	no	—
99	#124	—	74	2	no	—
100	C₂²⋊Q₈	—	2	2	no	—
101	C₄×Q₈	—	2	2	no	—
102	#117	C₂×D₃₇	1	2	yes	—
103	C₂×C₂²⋊C₄	—	74	2	no	—
104	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
105	#103	—	37	8	no	—
106	#102	—	37	8	no	—
107	#97	—	37	2	no	—
108	C₄×C₈	—	37	2	no	—
109	C₂⁴	—	74	1	no	—
110	C₂×Q₈	—	2	2	no	—
111	C₂×Q₈	—	37	2	no	—
112	C₂×Q₈	—	74	1	no	—
113	C₂×Q₈	—	74	4	no	—
114	C₂×D₄	—	37	2	no	—
115	C₂×D₄	—	74	4	no	—
116	C₂×D₄	—	74	1	no	—
117	C₂²×C₄	—	74	1	no	—
118	C₂²×C₄	C₂²⋊D₃₇	1	2	yes	—
119	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
120	M₁₆	—	74	4	no	—
121	C₂×C₈	—	37	4	no	—
122	C₄⋊C₄	—	2	4	no	—
123	C₄⋊C₄	—	2	2	no	—
124	C₄⋊C₄	C₂²×D₃₇	1	4	yes	—
125	C₂²⋊C₄	—	2	2	no	—
126	C₂²⋊C₄	—	74	4	no	—
127	C₂²⋊C₄	—	74	4	no	—
128	C₄²	—	2	2	no	—
129	C₄²	C₂²⋊D₃₇	1	2	yes	—
130	C₄²	C₂²×D₃₇	1	1	yes	characteristic
131	C₄²	C₂²×D₃₇	1	1	yes	characteristic
132	C₂³	—	74	2	no	—
133	C₂³	—	148	4	no	—
134	C₂³	—	148	2	no	—
135	C₂³	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
136	Q₈	—	4	4	no	—
137	Q₈	—	74	4	no	—
138	Q₈	—	148	2	no	—
139	D₄	—	74	4	no	—
140	D₄	—	148	2	no	—
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	4	2	no	—
143	C₂×C₄	—	74	2	no	—
144	C₂×C₄	—	148	2	no	—
145	C₂×C₄	#10596966	1	4	yes	—
146	C₂×C₄	C₂×C₂²⋊D₃₇	1	2	yes	—
147	C₂×C₄	C₂×C₂²⋊D₃₇	1	2	yes	—
148	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
149	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
150	C₈	—	74	4	no	—
151	C₂²	—	2	2	no	—
152	C₂²	—	296	4	no	—
153	C₂²	—	148	2	no	—
154	C₂²	—	148	4	no	—
155	C₂²	#11924525	1	2	yes	—
156	C₂²	C₂²×C₂²⋊D₃₇	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	—	2	2	no	—
159	C₄	—	4	4	no	—
160	C₄	—	148	2	no	—
161	C₄	#11924525	1	2	yes	—
162	C₄	C₂×#10596966	1	2	yes	—
163	C₂	—	2	2	no	—
164	C₂	—	296	2	no	—
165	C₂	C₂×#11924525	1	1	yes	characteristic
166	C₂	#14558331	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924524

C₂

yes

maximal, characteristic

#14557895

C₂

yes

maximal, complemented

#14557890

C₂

yes

maximal, complemented

#14557541

C₂

yes

maximal, complemented

#14557537

C₂

yes

maximal

C₄×#10596934

C₂

yes

maximal, characteristic, complemented

C₃₇×#533

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₇₄

C₂²

yes

characteristic

C₂²×D₁₄₈

C₂²

yes

characteristic

C₂×#10596934

C₂²

yes

—

#11924524

—

#11924524

C₂²

yes

—

#11924524

C₂²

yes

—

C₂×#10596938

—

#11924459

C₂²

yes

complemented

#11924456

C₂²

yes

—

#11924447

C₂²

yes

complemented

C₄×#64436

C₂²

yes

complemented

C₄⋊Q₈×C₃₇

C₂²

yes

complemented

#125×C₃₇

C₂²

yes

complemented

C₂²⋊Q₈×C₃₇

—

C₄×Q₈×C₃₇

—

#117×C₃₇

C₂²

yes

—

C₂×C₄²×C₃₇

C₂²

yes

characteristic

C₂³×D₃₇

—

C₂×Q₈×C₃₇

—

C₂²×C₄×C₃₇

D₄

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

—

C₂×D₁₄₈

—

C₂×D₁₄₈

—

C₂×D₁₄₈

C₂³

yes

—

C₂×Dic₇₄

—

C₂×Dic₇₄

—

C₂×Dic₇₄

C₂³

yes

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

C₂³

yes

—

C₂²⋊C₄×C₃₇

—

C₄²×C₃₇

—

C₄²×C₃₇

D₄

yes

—

C₄²×C₃₇

C₂³

yes

characteristic

C₄²×C₃₇

C₂³

yes

—

—

—

C₂³

yes

—

C₂³×C₃₇

C₂×D₄

yes

characteristic

C₂²×D₃₇

—

C₂²×D₃₇

—

C₂²×D₃₇

—

Q₈×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

#41

yes

—

C₂×C₄×C₃₇

C₂⁴

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

—

D₁₄₈

—

D₁₄₈

—

Dic₇₄

—

Dic₇₄

—

#64436

—

C₂²×C₃₇

—

C₂²×C₃₇

#124

yes

—

C₂²×C₃₇

C₂²×D₄

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

#124

yes

—

C₄×C₃₇

C₂×#41

yes

—

Dic₃₇

—

#3140

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

C₂×#124

yes

characteristic

C₂×C₃₇

#575

yes

—

D₃₇

—

#533

D₃₇

yes

characteristic, complemented

—

—

—

—

—

C₄×M₁₆

—

C₃₇

#3140

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×D₄

—

C₂×M₁₆

—

C₄⋊Q₈

C₂×D₃₇

yes

complemented

#125

C₂×D₃₇

yes

—

—

—

100

C₂²⋊Q₈

—

101

C₄×Q₈

—

102

#117

C₂×D₃₇

yes

—

103

C₂×C₂²⋊C₄

—

104

C₂×C₄²

C₂×D₃₇

yes

characteristic

105

#103

—

106

#102

—

107

#97

—

108

C₄×C₈

—

109

C₂⁴

—

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂×Q₈

—

113

C₂×Q₈

—

114

C₂×D₄

—

115

C₂×D₄

—

116

C₂×D₄

—

117

C₂²×C₄

—

118

C₂²×C₄

C₂²⋊D₃₇

yes

—

119

C₂²×C₄

C₂²×D₃₇

yes

characteristic

120

M₁₆

—

121

C₂×C₈

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

C₂²×D₃₇

yes

—

125

C₂²⋊C₄

—

126

C₂²⋊C₄

—

127

C₂²⋊C₄

—

128

C₄²

—

129

C₄²

C₂²⋊D₃₇

yes

—

130

C₄²

C₂²×D₃₇

yes

characteristic

131

C₄²

C₂²×D₃₇

yes

characteristic

132

C₂³

—

133

C₂³

—

148

—

134

C₂³

—

148

—

135

C₂³

C₂×C₂²⋊D₃₇

yes

characteristic

136

Q₈

—

137

Q₈

—

138

Q₈

—

148

—

139

D₄

—

140

D₄

—

148

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

148

—

145

C₂×C₄

#10596966

yes

—

146

C₂×C₄

C₂×C₂²⋊D₃₇

yes

—

147

C₂×C₄

C₂×C₂²⋊D₃₇

yes

—

148

C₂×C₄

C₂×C₂²⋊D₃₇

yes

characteristic

149

C₂×C₄

C₂³×D₃₇

yes

characteristic

150

C₈

—

151

C₂²

—

152

C₂²

—

296

—

153

C₂²

—

148

—

154

C₂²

—

148

—

155

C₂²

#11924525

yes

—

156

C₂²

C₂²×C₂²⋊D₃₇

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

148

—

161

C₄

#11924525

yes

—

162

C₄

C₂×#10596966

yes

—

163

C₂

—

164

C₂

—

296

—

165

C₂

C₂×#11924525

yes

characteristic

166

C₂

#14558331

yes

—

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₃₇
commutator, cosocle	C₂×C₄×C₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₃₇×#533	C₂
socle	C₂²×C₃₇	C₂²×D₄
prime radical	C₂²×D₁₄₈	C₂²
hypercenter	#533	D₃₇
hypocenter	C₃₇	#3140
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₃₇

commutator, cosocle

C₂×C₄×C₃₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₃₇

Fitting

C₃₇×#533

C₂

socle

C₂²×C₃₇

C₂²×D₄

prime radical

C₂²×D₁₄₈

C₂²

D₃₇

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #14557895	C₂	semidirect
3. #14557890	C₂	semidirect
4. #14557541	C₂	semidirect
6. C₄×#10596934	C₂	semidirect
7. C₃₇×#533	C₂	semidirect
15. #11924459	C₂²	semidirect
17. #11924447	C₂²	semidirect
18. C₄×#64436	C₂²	semidirect
19. C₄⋊Q₈×C₃₇	C₂²	semidirect
20. #125×C₃₇	C₂²	semidirect
84. #533	D₃₇	semidirect
91. C₃₇	#3140	semidirect
95. C₄⋊Q₈	C₂×D₃₇	semidirect
96. #125	C₂×D₃₇	semidirect

H ≅ G/N

type

2. #14557895

C₂

semidirect

3. #14557890

C₂

semidirect

4. #14557541

C₂

semidirect

6. C₄×#10596934

C₂

semidirect

7. C₃₇×#533

C₂

semidirect

15. #11924459

C₂²

semidirect

17. #11924447

C₂²

semidirect

18. C₄×#64436

C₂²

semidirect

19. C₄⋊Q₈×C₃₇

C₂²

semidirect

20. #125×C₃₇

C₂²

semidirect

84. #533

D₃₇

semidirect

91. C₃₇

#3140

semidirect

95. C₄⋊Q₈

C₂×D₃₇

semidirect

96. #125

C₂×D₃₇

semidirect

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

subgroup	index
C₂×#11924524	2
#14557895	2
#14557890	2
#14557541	2
#14557537	2
C₄×#10596934	2
C₃₇×#533	2
#3140	37

subgroup

index

C₃₇×#533

Subgroups — #18004985

Conjugacy classes of subgroups

Complements

Maximal subgroups