Group #18010775 — subgroups

Subgroups — #18010775

Order 4736.

292 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 88 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	D₃₇⋊(C₄⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₄×C₄⋊#3866	C₂	1	1	yes	maximal, characteristic
3	#14558406	C₂	1	4	yes	maximal
4	C₄⋊Q₈×D₃₇	C₂	1	1	yes	maximal, characteristic
5	C₂×D₃₇⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊#3866	C₂²	1	2	yes	—
7	C₂×C₄×#3866	—	2	1	no	—
8	#11924603	—	2	1	no	—
9	C₄⋊#64438	—	2	1	no	—
10	(C₂×#3866)⋊C₄	—	2	1	no	—
11	C₂×Q₈×D₃₇	—	2	1	no	—
12	#11924536	C₄	1	1	yes	characteristic, complemented
13	#11924535	—	2	1	no	—
14	#11924535	C₄	1	2	yes	complemented
15	#11924534	—	2	1	no	—
16	C₄⋊C₄×D₃₇	—	2	1	no	—
17	C₄⋊C₄×D₃₇	C₂²	1	2	yes	—
18	C₄²×D₃₇	C₂²	1	1	yes	characteristic
19	C₄⋊Q₈×C₃₇	C₄	1	1	yes	characteristic, complemented
20	C₂²×#3866	—	2	2	no	—
21	C₂×Q₈×C₃₇	—	2	1	no	—
22	Q₈×D₃₇	—	4	2	no	—
23	Q₈×D₃₇	—	4	2	no	—
24	C₂×C₄×D₃₇	—	2	2	no	—
25	C₂×C₄×D₃₇	D₄	1	1	yes	characteristic
26	C₂×C₄×D₃₇	D₄	1	1	yes	characteristic
27	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
28	C₂×Dic₇₄	—	2	1	no	—
29	C₂×Dic₇₄	—	2	2	no	—
30	C₂×#64438	—	2	2	no	—
31	C₄⋊#3866	—	2	2	no	—
32	C₄⋊#3866	Q₈	1	4	yes	complemented
33	C₄×#3866	—	4	2	no	—
34	D₃₇⋊C₈	—	2	4	no	—
35	C₄⋊C₄×C₃₇	—	2	1	no	—
36	C₄⋊C₄×C₃₇	C₂×C₄	1	2	yes	—
37	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
38	C₄⋊Dic₃₇	—	2	1	no	—
39	C₄⋊Dic₃₇	C₂×C₄	1	2	yes	—
40	#10596936	—	2	2	no	—
41	#10596936	—	2	2	no	—
42	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
43	C₄×Dic₃₇	D₄	1	1	yes	characteristic
44	C₄×Dic₃₇	D₄	1	1	yes	characteristic
45	C₂²×D₃₇	C₂×D₄	1	1	yes	characteristic
46	C₂×#3866	—	2	4	no	—
47	C₂×#3866	—	4	2	no	—
48	C₂×#3866	—	4	2	no	—
49	Q₈×C₃₇	—	4	2	no	—
50	C₂×C₄×C₃₇	—	2	2	no	—
51	C₂×C₄×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×C₄×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
54	C₂×Dic₃₇	—	2	2	no	—
55	C₂×Dic₃₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×Dic₃₇	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
58	C₄×D₃₇	—	2	2	no	—
59	C₄×D₃₇	—	4	2	no	—
60	C₄×D₃₇	—	4	2	no	—
61	C₄×D₃₇	D₈	1	2	yes	—
62	C₄×D₃₇	SD₁₆	1	2	yes	—
63	C₄×D₃₇	C₂×Q₈	1	2	yes	—
64	C₄×D₃₇	#41	1	2	yes	—
65	Dic₇₄	—	4	2	no	—
66	Dic₇₄	—	4	4	no	—
67	#64438	—	4	2	no	—
68	C₂²×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂×D₃₇	C₂²⋊Q₈	1	2	yes	—
70	C₂×D₃₇	C₂×D₈	1	1	yes	characteristic
71	C₂×D₃₇	C₂×SD₁₆	1	1	yes	characteristic
72	C₂×D₃₇	#137	1	2	yes	—
73	#3866	—	4	4	no	—
74	C₄×C₃₇	—	2	1	no	—
75	C₄×C₃₇	—	4	2	no	—
76	C₄×C₃₇	#102	1	2	yes	—
77	C₄×C₃₇	C₄×Q₈	1	2	yes	—
78	Dic₃₇	—	2	1	no	—
79	Dic₃₇	—	4	2	no	—
80	Dic₃₇	#102	1	2	yes	—
81	Dic₃₇	#124	1	1	yes	characteristic
82	Dic₃₇	C₄⋊Q₈	1	1	yes	characteristic
83	#1988	—	37	1	no	maximal, Sylow
84	C₂×C₃₇	#391	1	1	yes	characteristic
85	C₂×C₃₇	C₂×#102	1	1	yes	characteristic
86	C₂×C₃₇	#419	1	1	yes	characteristic
87	D₃₇	#474	1	2	yes	—
88	D₃₇	#475	1	2	yes	—
89	C₂×C₄⋊Q₈	—	37	1	no	—
90	C₂×C₄⋊C₈	—	37	1	no	—
91	C₄×C₄⋊C₄	—	37	1	no	—
92	#340	—	37	4	no	—
93	C₃₇	#1988	1	1	yes	characteristic, complemented, Sylow
94	C₂²×Q₈	—	74	1	no	—
95	C₂²×C₈	—	37	2	no	—
96	C₄⋊Q₈	—	37	2	no	—
97	C₄⋊Q₈	—	37	1	no	—
98	C₄⋊Q₈	—	74	1	no	—
99	C₄⋊Q₈	—	74	1	no	—
100	C₄⋊Q₈	#3866	1	1	yes	characteristic, complemented
101	C₂×C₄⋊C₄	—	37	2	no	—
102	C₂×C₄⋊C₄	—	37	2	no	—
103	C₂×C₄⋊C₄	—	74	1	no	—
104	C₂×C₄²	—	37	1	no	—
105	C₂×C₄²	—	74	1	no	—
106	C₄⋊C₈	—	74	1	no	—
107	C₄⋊C₈	—	74	1	no	—
108	#95	—	74	1	no	—
109	C₂×Q₈	—	2	1	no	—
110	C₂×Q₈	—	74	2	no	—
111	C₂×Q₈	—	74	1	no	—
112	C₂×Q₈	—	148	2	no	—
113	C₂×Q₈	—	148	2	no	—
114	C₂²×C₄	—	37	1	no	—
115	C₂²×C₄	—	37	1	no	—
116	C₂²×C₄	—	37	1	no	—
117	C₂²×C₄	—	74	2	no	—
118	C₂²×C₄	—	74	2	no	—
119	C₂×C₈	—	74	4	no	—
120	C₂×C₈	—	74	2	no	—
121	C₄⋊C₄	—	2	1	no	—
122	C₄⋊C₄	—	37	4	no	—
123	C₄⋊C₄	—	37	2	no	—
124	C₄⋊C₄	—	74	2	no	—
125	C₄⋊C₄	—	74	2	no	—
126	C₄⋊C₄	—	74	2	no	—
127	C₄⋊C₄	—	74	1	no	—
128	C₄⋊C₄	C₂×#3866	1	2	yes	—
129	C₄²	—	37	1	no	—
130	C₄²	—	37	1	no	—
131	C₄²	—	37	1	no	—
132	C₄²	—	148	2	no	—
133	C₄²	C₂×#3866	1	1	yes	characteristic
134	C₂³	—	37	1	no	—
135	Q₈	—	4	2	no	—
136	Q₈	—	148	4	no	—
137	Q₈	—	148	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	37	1	no	—
140	C₂×C₄	—	37	1	no	—
141	C₂×C₄	—	37	2	no	—
142	C₂×C₄	—	37	2	no	—
143	C₂×C₄	—	37	2	no	—
144	C₂×C₄	—	37	2	no	—
145	C₂×C₄	—	37	1	no	—
146	C₂×C₄	—	74	2	no	—
147	C₂×C₄	—	74	4	no	—
148	C₂×C₄	—	74	2	no	—
149	C₂×C₄	—	148	2	no	—
150	C₂×C₄	—	148	2	no	—
151	C₂×C₄	—	148	2	no	—
152	C₂×C₄	—	148	2	no	—
153	C₂×C₄	C₂²⋊#3866	1	1	yes	characteristic
154	C₂×C₄	C₂²⋊#3866	1	1	yes	characteristic
155	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
156	C₈	—	148	2	no	—
157	C₂²	—	37	2	no	—
158	C₂²	—	37	1	no	—
159	C₂²	—	37	1	no	—
160	C₂²	—	37	2	no	—
161	C₂²	C₂×C₂²⋊#3866	1	1	yes	characteristic
162	C₄	—	2	1	no	—
163	C₄	—	4	2	no	—
164	C₄	—	37	1	no	—
165	C₄	—	37	1	no	—
166	C₄	—	37	2	no	—
167	C₄	—	74	1	no	—
168	C₄	—	148	4	no	—
169	C₄	—	148	2	no	—
170	C₄	#11924598	1	2	yes	—
171	C₄	Q₈×#3866	1	2	yes	—
172	C₂	—	37	2	no	—
173	C₂	—	37	2	no	—
174	C₂	(C₂×#3866)⋊Q₈	1	1	yes	characteristic
175	C₂	C₂×#11924598	1	1	yes	characteristic
176	C₂	#14558526	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₃₇⋊(C₄⋊C₈)

C₂

yes

maximal, characteristic

C₄×C₄⋊#3866

C₂

yes

maximal, characteristic

#14558406

C₂

yes

maximal

C₄⋊Q₈×D₃₇

C₂

yes

maximal, characteristic

C₂×D₃₇⋊C₈

C₂²

yes

—

C₂×C₄⋊#3866

C₂²

yes

—

C₂×C₄×#3866

—

#11924603

—

C₄⋊#64438

—

(C₂×#3866)⋊C₄

—

C₂×Q₈×D₃₇

—

#11924536

C₄

yes

characteristic, complemented

#11924535

—

#11924535

C₄

yes

complemented

#11924534

—

C₄⋊C₄×D₃₇

—

C₄⋊C₄×D₃₇

C₂²

yes

—

C₄²×D₃₇

C₂²

yes

characteristic

C₄⋊Q₈×C₃₇

C₄

yes

characteristic, complemented

C₂²×#3866

—

C₂×Q₈×C₃₇

—

Q₈×D₃₇

—

Q₈×D₃₇

—

C₂×C₄×D₃₇

—

C₂×C₄×D₃₇

D₄

yes

characteristic

C₂×C₄×D₃₇

D₄

yes

characteristic

C₂×C₄×D₃₇

C₂³

yes

characteristic

C₂×Dic₇₄

—

C₂×Dic₇₄

—

C₂×#64438

—

C₄⋊#3866

—

C₄⋊#3866

Q₈

yes

complemented

C₄×#3866

—

D₃₇⋊C₈

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

C₂×C₄

yes

—

C₄²×C₃₇

C₂×C₄

yes

characteristic

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

C₂×C₄

yes

—

#10596936

—

#10596936

—

C₄×Dic₃₇

C₂×C₄

yes

characteristic

C₄×Dic₃₇

D₄

yes

characteristic

C₄×Dic₃₇

D₄

yes

characteristic

C₂²×D₃₇

C₂×D₄

yes

—

—

—

Q₈×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₃₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₃₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₃₇

C₂×D₄

yes

characteristic

C₄×D₃₇

—

C₄×D₃₇

—

C₄×D₃₇

—

C₄×D₃₇

D₈

yes

—

C₄×D₃₇

SD₁₆

yes

—

C₄×D₃₇

C₂×Q₈

yes

—

C₄×D₃₇

#41

yes

—

Dic₇₄

—

Dic₇₄

—

#64438

—

C₂²×C₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₃₇

C₂²⋊Q₈

yes

—

C₂×D₃₇

C₂×D₈

yes

characteristic

C₂×D₃₇

C₂×SD₁₆

yes

characteristic

C₂×D₃₇

#137

yes

—

#3866

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

#102

yes

—

C₄×C₃₇

C₄×Q₈

yes

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

#102

yes

—

Dic₃₇

#124

yes

characteristic

Dic₃₇

C₄⋊Q₈

yes

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

D₃₇

yes

—

D₃₇

#475

yes

—

C₂×C₄⋊Q₈

—

C₂×C₄⋊C₈

—

C₄×C₄⋊C₄

—

#340

—

C₃₇

#1988

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

100

C₄⋊Q₈

#3866

yes

characteristic, complemented

101

C₂×C₄⋊C₄

—

102

C₂×C₄⋊C₄

—

103

C₂×C₄⋊C₄

—

104

C₂×C₄²

—

105

C₂×C₄²

—

106

C₄⋊C₈

—

107

C₄⋊C₈

—

108

#95

—

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂×Q₈

—

148

—

113

C₂×Q₈

—

148

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

—

117

C₂²×C₄

—

118

C₂²×C₄

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

C₂×#3866

yes

—

129

C₄²

—

130

C₄²

—

131

C₄²

—

132

C₄²

—

148

—

133

C₄²

C₂×#3866

yes

characteristic

134

C₂³

—

135

Q₈

—

136

Q₈

—

148

—

137

Q₈

—

148

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

—

149

C₂×C₄

—

148

—

150

C₂×C₄

—

148

—

151

C₂×C₄

—

148

—

152

C₂×C₄

—

148

—

153

C₂×C₄

C₂²⋊#3866

yes

characteristic

154

C₂×C₄

C₂²⋊#3866

yes

characteristic

155

C₂×C₄

C₂²×#3866

yes

characteristic

156

C₈

—

148

—

157

C₂²

—

158

C₂²

—

159

C₂²

—

160

C₂²

—

161

C₂²

C₂×C₂²⋊#3866

yes

characteristic

162

C₄

—

163

C₄

—

164

C₄

—

165

C₄

—

166

C₄

—

167

C₄

—

168

C₄

—

148

—

169

C₄

—

148

—

170

C₄

#11924598

yes

—

171

C₄

Q₈×#3866

yes

—

172

C₂

—

173

C₂

—

174

C₂

(C₂×#3866)⋊Q₈

yes

characteristic

175

C₂

C₂×#11924598

yes

characteristic

176

C₂

#14558526

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#3866
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#3866
Fitting	C₄⋊Q₈×C₃₇	C₄
socle	C₂²×C₃₇	C₂×C₂²⋊C₄
prime radical	C₂²×D₃₇	C₂×D₄
cosocle	C₂×C₄×D₃₇	C₂³
hypercenter	C₄⋊Q₈	#3866
hypocenter	C₃₇	#1988
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊#3866

commutator

C₂×C₄×C₃₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×#3866

Fitting

C₄⋊Q₈×C₃₇

C₄

socle

C₂²×C₃₇

C₂×C₂²⋊C₄

prime radical

C₂²×D₃₇

C₂×D₄

cosocle

C₂×C₄×D₃₇

C₂³

C₄⋊Q₈

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄⋊Q₈

N	H ≅ G/N	type
12. #11924536	C₄	semidirect
14. #11924535	C₄	semidirect
19. C₄⋊Q₈×C₃₇	C₄	semidirect
32. C₄⋊#3866	Q₈	semidirect
93. C₃₇	#1988	semidirect
100. C₄⋊Q₈	#3866	semidirect

H ≅ G/N

type

12. #11924536

C₄

semidirect

14. #11924535

C₄

semidirect

19. C₄⋊Q₈×C₃₇

C₄

semidirect

32. C₄⋊#3866

Q₈

semidirect

93. C₃₇

#1988

semidirect

100. C₄⋊Q₈

#3866

semidirect

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

subgroup	index
D₃₇⋊(C₄⋊C₈)	2
C₄×C₄⋊#3866	2
#14558406	2
C₄⋊Q₈×D₃₇	2
#1988	37

subgroup

index

D₃₇⋊(C₄⋊C₈)

C₄×C₄⋊#3866

#14558406

C₄⋊Q₈×D₃₇

#1988

Subgroups — #18010775

Conjugacy classes of subgroups

Complements

Maximal subgroups