Group #17994455 — subgroups

Subgroups — #17994455

Order 4736.

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

normal subgroups only

This group has 260 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×C₄⋊#64436	C₂	1	1	yes	maximal, characteristic
2	C₂×#11924446	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×Dic₃₇	C₂	1	1	yes	maximal, characteristic
4	#14557379	C₂	1	1	yes	maximal, characteristic
5	#14557287	C₂	1	1	yes	maximal, characteristic
6	#14557287	C₂	1	1	yes	maximal, characteristic
7	C₂×C₄⋊C₈×C₃₇	C₂	1	1	yes	maximal, characteristic
8	C₂²×#64436	C₂²	1	1	yes	characteristic
9	C₂²×#64436	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
12	C₄⋊#64436	C₄	1	4	yes	complemented
13	#11924446	—	2	2	no	—
14	C₈×Dic₃₇	—	2	2	no	—
15	#11924443	C₄	1	2	yes	—
16	#11924443	C₄	1	2	yes	—
17	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
20	C₄⋊C₈×C₃₇	C₄	1	4	yes	complemented
21	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
24	C₂²×Dic₃₇	C₂×C₄	1	2	yes	—
25	C₂²×Dic₃₇	C₂×C₄	1	2	yes	—
26	C₂×C₈×C₃₇	—	2	1	no	—
27	C₂×C₈×C₃₇	—	2	1	no	—
28	C₂×C₈×C₃₇	—	2	1	no	—
29	C₂×C₈×C₃₇	—	2	1	no	—
30	C₂×C₈×C₃₇	C₂×C₄	1	2	yes	—
31	C₂×C₈×C₃₇	C₂×C₄	1	2	yes	—
32	C₄²×C₃₇	C₂×C₄	1	2	yes	—
33	C₄²×C₃₇	C₂×C₄	1	2	yes	—
34	C₄×Dic₃₇	—	2	1	no	—
35	C₄×Dic₃₇	—	2	1	no	—
36	C₄×Dic₃₇	D₄	1	2	yes	—
37	C₄×Dic₃₇	Q₈	1	2	yes	—
38	C₂×#64436	—	2	1	no	—
39	C₂×#64436	—	2	1	no	—
40	C₂×#64436	—	2	1	no	—
41	C₂×#64436	—	2	1	no	—
42	C₂×#64436	C₂×C₄	1	2	yes	—
43	C₂×#64436	C₂×C₄	1	2	yes	—
44	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₃₇	—	2	2	no	—
46	C₂×C₄×C₃₇	—	2	2	no	—
47	C₂×C₄×C₃₇	C₄²	1	2	yes	—
48	C₂×C₄×C₃₇	C₄²	1	2	yes	—
49	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₃₇	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₄×C₃₇	#41	1	1	yes	characteristic
54	C₂×C₄×C₃₇	#41	1	1	yes	characteristic
55	C₂×Dic₃₇	—	2	2	no	—
56	C₂×Dic₃₇	—	2	2	no	—
57	C₂×Dic₃₇	—	2	2	no	—
58	C₂×Dic₃₇	—	2	2	no	—
59	C₂×Dic₃₇	—	2	2	no	—
60	C₂×Dic₃₇	C₄⋊C₄	1	4	yes	—
61	C₈×C₃₇	—	2	2	no	—
62	C₈×C₃₇	—	2	2	no	—
63	#64436	—	2	2	no	—
64	#64436	—	2	2	no	—
65	C₂²×C₃₇	C₂×C₄²	1	1	yes	characteristic
66	C₂²×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₃₇	#117	1	1	yes	characteristic
68	C₂²×C₃₇	#131	1	2	yes	—
69	C₂²×C₃₇	#131	1	2	yes	—
70	C₄×C₃₇	—	2	2	no	—
71	C₄×C₃₇	—	2	2	no	—
72	C₄×C₃₇	C₄×D₄	1	1	yes	characteristic
73	C₄×C₃₇	C₄×D₄	1	1	yes	characteristic
74	C₄×C₃₇	C₄×Q₈	1	1	yes	characteristic
75	C₄×C₃₇	C₄×Q₈	1	1	yes	characteristic
76	Dic₃₇	—	2	4	no	—
77	Dic₃₇	—	4	2	no	—
78	#1768	—	37	1	no	maximal, Sylow
79	C₂×C₃₇	C₄×C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₃₇	#405	1	1	yes	characteristic
81	C₂×C₃₇	#405	1	1	yes	characteristic
82	C₂×C₃₇	#424	1	2	yes	—
83	C₂×C₃₇	#433	1	2	yes	—
84	C₂×C₄⋊C₈	—	37	1	no	—
85	C₂×C₄⋊C₈	D₃₇	1	1	yes	characteristic
86	C₂×#97	—	37	1	no	—
87	C₂×C₄×C₈	—	37	1	no	—
88	#376	—	37	1	no	—
89	#336	—	37	1	no	—
90	#336	—	37	1	no	—
91	C₃₇	#1768	1	1	yes	characteristic, complemented, Sylow
92	C₂²×C₈	—	37	1	no	—
93	C₂²×C₈	—	37	1	no	—
94	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic
95	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic
96	C₂×C₄²	—	37	1	no	—
97	C₂×C₄²	—	37	1	no	—
98	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
99	C₄⋊C₈	—	37	4	no	—
100	C₄⋊C₈	Dic₃₇	1	4	yes	complemented
101	#97	—	74	2	no	—
102	C₄×C₈	—	74	2	no	—
103	#95	—	37	2	no	—
104	#95	—	37	2	no	—
105	C₂²×C₄	—	37	2	no	—
106	C₂²×C₄	—	37	2	no	—
107	C₂²×C₄	C₄×D₃₇	1	1	yes	characteristic
108	C₂²×C₄	C₄×D₃₇	1	1	yes	characteristic
109	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
110	C₂×C₈	—	2	1	no	—
111	C₂×C₈	—	2	1	no	—
112	C₂×C₈	—	2	1	no	—
113	C₂×C₈	—	2	1	no	—
114	C₂×C₈	—	37	2	no	—
115	C₂×C₈	—	37	2	no	—
116	C₂×C₈	—	74	1	no	—
117	C₂×C₈	—	74	1	no	—
118	C₂×C₈	—	74	1	no	—
119	C₂×C₈	—	74	1	no	—
120	C₂×C₈	C₂×Dic₃₇	1	2	yes	—
121	C₂×C₈	C₂×Dic₃₇	1	2	yes	—
122	C₄²	—	37	2	no	—
123	C₄²	—	37	2	no	—
124	C₄²	—	74	1	no	—
125	C₄²	—	74	1	no	—
126	C₄²	C₄×D₃₇	1	2	yes	—
127	C₄²	C₂×Dic₃₇	1	2	yes	—
128	C₂³	C₂×C₄×D₃₇	1	1	yes	characteristic
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	37	4	no	—
132	C₂×C₄	—	74	2	no	—
133	C₂×C₄	—	74	2	no	—
134	C₂×C₄	—	74	2	no	—
135	C₂×C₄	—	74	2	no	—
136	C₂×C₄	—	74	2	no	—
137	C₂×C₄	C₄×Dic₃₇	1	2	yes	—
138	C₂×C₄	C₄×Dic₃₇	1	2	yes	—
139	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
140	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
141	C₂×C₄	#10596964	1	1	yes	characteristic
142	C₂×C₄	Q₈×D₃₇	1	1	yes	characteristic
143	C₂×C₄	#10596966	1	1	yes	characteristic
144	C₂×C₄	C₂²×Dic₃₇	1	1	yes	characteristic
145	C₈	—	2	2	no	—
146	C₈	—	2	2	no	—
147	C₈	—	74	2	no	—
148	C₈	—	74	2	no	—
149	C₂²	C₂×C₄×Dic₃₇	1	1	yes	characteristic
150	C₂²	C₄⋊C₄×D₃₇	1	1	yes	characteristic
151	C₂²	#11924494	1	1	yes	characteristic
152	C₂²	#11924542	1	2	yes	—
153	C₂²	#11924544	1	2	yes	—
154	C₄	—	2	2	no	—
155	C₄	—	2	2	no	—
156	C₄	—	74	4	no	—
157	C₄	—	148	2	no	—
158	C₄	#11924498	1	1	yes	characteristic
159	C₄	D₄×Dic₃₇	1	1	yes	characteristic
160	C₄	#11924501	1	1	yes	characteristic
161	C₄	Q₈×Dic₃₇	1	1	yes	characteristic
162	C₂	C₄⋊C₄×Dic₃₇	1	1	yes	characteristic
163	C₂	#14557488	1	1	yes	characteristic
164	C₂	#14557491	1	1	yes	characteristic
165	C₂	#14557558	1	2	yes	—
166	C₂	#14557593	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#64436

C₂

yes

maximal, characteristic

C₂×#11924446

C₂

yes

maximal, characteristic

C₂×C₈×Dic₃₇

C₂

yes

maximal, characteristic

#14557379

C₂

yes

maximal, characteristic

#14557287

C₂

yes

maximal, characteristic

#14557287

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₃₇

C₂

yes

maximal, characteristic

C₂²×#64436

C₂²

yes

characteristic

C₂²×#64436

C₂²

yes

characteristic

C₂×C₄×Dic₃₇

C₂²

yes

characteristic

C₂×C₄×Dic₃₇

C₂²

yes

characteristic

C₄⋊#64436

C₄

yes

complemented

#11924446

—

C₈×Dic₃₇

—

#11924443

C₄

yes

—

#11924443

C₄

yes

—

C₂²×C₈×C₃₇

C₂²

yes

characteristic

C₂²×C₈×C₃₇

C₂²

yes

characteristic

C₂×C₄²×C₃₇

C₂²

yes

characteristic

C₄⋊C₈×C₃₇

C₄

yes

complemented

C₂²×C₄×C₃₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

C₂×C₄

yes

—

C₂²×Dic₃₇

C₂×C₄

yes

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

C₂×C₄

yes

—

C₂×C₈×C₃₇

C₂×C₄

yes

—

C₄²×C₃₇

C₂×C₄

yes

—

C₄²×C₃₇

C₂×C₄

yes

—

C₄×Dic₃₇

—

C₄×Dic₃₇

—

C₄×Dic₃₇

D₄

yes

—

C₄×Dic₃₇

Q₈

yes

—

—

—

—

—

C₂×C₄

yes

—

C₂×#64436

C₂×C₄

yes

—

C₂³×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₄²

yes

—

C₂×C₄×C₃₇

C₄²

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×Q₈

yes

C₂×C₄×C₃₇

yes

C₂×C₄×C₃₇

yes

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

C₄⋊C₄

yes

—

C₈×C₃₇

—

C₈×C₃₇

—

#64436

—

#64436

—

C₂²×C₃₇

C₂×C₄²

yes

characteristic

C₂²×C₃₇

C₂×C₄⋊C₄

yes

C₂²×C₃₇

yes

C₂²×C₃₇

yes

—

C₂²×C₃₇

#131

yes

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₄×D₄

yes

characteristic

C₄×C₃₇

C₄×D₄

yes

characteristic

C₄×C₃₇

C₄×Q₈

yes

characteristic

C₄×C₃₇

C₄×Q₈

yes

characteristic

Dic₃₇

—

Dic₃₇

—

#1768

—

maximal, Sylow

C₂×C₃₇

C₄×C₄⋊C₄

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

—

C₂×C₃₇

#433

yes

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

D₃₇

yes

characteristic

C₂×#97

—

C₂×C₄×C₈

—

#376

—

#336

—

#336

—

C₃₇

#1768

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₃₇

yes

characteristic

C₂²×C₈

C₂×D₃₇

yes

characteristic

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₃₇

yes

characteristic

C₄⋊C₈

—

100

C₄⋊C₈

Dic₃₇

yes

complemented

101

#97

—

102

C₄×C₈

—

103

#95

—

104

#95

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

C₄×D₃₇

yes

characteristic

108

C₂²×C₄

C₄×D₃₇

yes

characteristic

109

C₂²×C₄

C₂²×D₃₇

yes

characteristic

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

C₂×Dic₃₇

yes

—

121

C₂×C₈

C₂×Dic₃₇

yes

—

122

C₄²

—

123

C₄²

—

124

C₄²

—

125

C₄²

—

126

C₄²

C₄×D₃₇

yes

—

127

C₄²

C₂×Dic₃₇

yes

—

128

C₂³

C₂×C₄×D₃₇

yes

characteristic

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

C₄×Dic₃₇

yes

—

138

C₂×C₄

C₄×Dic₃₇

yes

—

139

C₂×C₄

C₂×C₄×D₃₇

yes

characteristic

140

C₂×C₄

D₄×D₃₇

yes

characteristic

141

C₂×C₄

#10596964

yes

characteristic

142

C₂×C₄

Q₈×D₃₇

yes

characteristic

143

C₂×C₄

#10596966

yes

characteristic

144

C₂×C₄

C₂²×Dic₃₇

yes

characteristic

145

C₈

—

146

C₈

—

147

C₈

—

148

C₈

—

149

C₂²

C₂×C₄×Dic₃₇

yes

characteristic

150

C₂²

C₄⋊C₄×D₃₇

yes

characteristic

151

C₂²

#11924494

yes

characteristic

152

C₂²

#11924542

yes

—

153

C₂²

#11924544

yes

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

148

—

158

C₄

#11924498

yes

characteristic

159

C₄

D₄×Dic₃₇

yes

characteristic

160

C₄

#11924501

yes

characteristic

161

C₄

Q₈×Dic₃₇

yes

characteristic

162

C₂

C₄⋊C₄×Dic₃₇

yes

characteristic

163

C₂

#14557488

yes

characteristic

164

C₂

#14557491

yes

characteristic

165

C₂

#14557558

yes

—

166

C₂

#14557593

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₃₇
commutator	C₂²×C₃₇	C₂×C₄²
Fitting	C₂×C₄⋊C₈×C₃₇	C₂
socle, prime radical	C₂³×C₃₇	C₂²×C₄
cosocle	C₂²×C₄×C₃₇	C₂³
hypercenter	C₂×C₄⋊C₈	D₃₇
hypocenter	C₃₇	#1768
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₃₇

commutator

C₂²×C₃₇

C₂×C₄²

Fitting

C₂×C₄⋊C₈×C₃₇

C₂

socle, prime radical

C₂³×C₃₇

C₂²×C₄

cosocle

C₂²×C₄×C₃₇

C₂³

hypercenter

C₂×C₄⋊C₈

D₃₇

C₃₇

G ▷ C₂²×C₃₇ ▷ C₃₇

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
12. C₄⋊#64436	C₄	semidirect
20. C₄⋊C₈×C₃₇	C₄	semidirect
91. C₃₇	#1768	semidirect
100. C₄⋊C₈	Dic₃₇	semidirect

H ≅ G/N

type

12. C₄⋊#64436

C₄

semidirect

20. C₄⋊C₈×C₃₇

C₄

semidirect

91. C₃₇

#1768

semidirect

100. C₄⋊C₈

Dic₃₇

semidirect

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

subgroup	index
C₂×C₄⋊#64436	2
C₂×#11924446	2
C₂×C₈×Dic₃₇	2
#14557379	2
#14557287	2
C₂×C₄⋊C₈×C₃₇	2
#1768	37

subgroup

index

C₂×C₄⋊#64436

C₂×C₈×Dic₃₇

C₂×C₄⋊C₈×C₃₇

Subgroups — #17994455

Conjugacy classes of subgroups

Complements

Maximal subgroups