Group #17994666 — subgroups

Subgroups — #17994666

Order 4736.

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

normal subgroups only

This group has 292 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×#11924493	C₂	1	1	yes	maximal, characteristic
2	C₂×#11924448	C₂	1	2	yes	maximal
3	#14557288	C₂	1	1	yes	maximal, characteristic, complemented
4	#14557287	C₂	1	2	yes	maximal, complemented
5	C₃₇×#336	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₃₇	C₂²	1	1	yes	characteristic
7	C₂²×#64436	C₂²	1	2	yes	—
8	#11924493	—	2	4	no	—
9	C₂×#10596936	C₄	1	2	yes	—
10	C₂×#10596938	C₄	1	2	yes	—
11	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
12	#11924448	—	2	4	no	—
13	C₂²×C₈×C₃₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
15	C₂³×D₃₇	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₃₇	C₂×C₄	1	2	yes	—
17	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
18	C₂²×Dic₃₇	C₂×C₄	1	2	yes	—
19	C₂²×Dic₃₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×D₃₇	—	2	2	no	—
21	C₂×C₄×D₃₇	—	2	1	no	—
22	C₂×C₄×D₃₇	—	2	1	no	—
23	C₂×C₄×D₃₇	D₄	1	4	yes	—
24	C₂×C₈×C₃₇	—	2	2	no	—
25	C₂×C₈×C₃₇	—	2	2	no	—
26	C₂×C₈×C₃₇	—	2	2	no	—
27	C₄²×C₃₇	—	2	2	no	—
28	#10596938	—	2	4	no	—
29	#10596936	—	2	4	no	—
30	C₄×Dic₃₇	—	2	2	no	—
31	C₂×#64436	—	2	2	no	—
32	C₂×#64436	—	2	2	no	—
33	C₂×#64436	—	2	2	no	—
34	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
35	C₂²×D₃₇	—	2	1	no	—
36	C₂²×D₃₇	—	2	1	no	—
37	C₂²×D₃₇	—	2	2	no	—
38	C₂²×D₃₇	—	2	2	no	—
39	C₂²×D₃₇	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₃₇	—	2	2	no	—
41	C₂×C₄×C₃₇	—	2	2	no	—
42	C₂×C₄×C₃₇	—	2	2	no	—
43	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₃₇	#41	1	2	yes	—
45	C₂×C₄×C₃₇	#41	1	1	yes	characteristic
46	C₂×C₄×C₃₇	#41	1	1	yes	characteristic
47	C₂×Dic₃₇	—	2	1	no	—
48	C₂×Dic₃₇	—	2	1	no	—
49	C₂×Dic₃₇	—	2	2	no	—
50	C₂×Dic₃₇	—	2	2	no	—
51	C₂×Dic₃₇	—	2	2	no	—
52	C₂×Dic₃₇	C₂²⋊C₄	1	2	yes	—
53	C₄×D₃₇	—	2	4	no	—
54	C₄×D₃₇	—	2	4	no	—
55	C₈×C₃₇	—	4	2	no	—
56	#64436	—	4	2	no	—
57	C₂²×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₃₇	#117	1	1	yes	characteristic
59	C₂²×C₃₇	#117	1	1	yes	characteristic
60	C₂²×C₃₇	#131	1	2	yes	—
61	C₂²×C₃₇	#131	1	2	yes	—
62	C₂×D₃₇	—	2	2	no	—
63	C₂×D₃₇	—	2	4	no	—
64	C₂×D₃₇	—	4	1	no	—
65	C₂×D₃₇	—	4	2	no	—
66	C₂×D₃₇	—	4	1	no	—
67	C₄×C₃₇	—	4	2	no	—
68	C₄×C₃₇	#123	1	2	yes	—
69	C₄×C₃₇	#123	1	2	yes	—
70	Dic₃₇	—	2	2	no	—
71	Dic₃₇	—	4	2	no	—
72	#1818	—	37	1	no	maximal, Sylow
73	C₂×C₃₇	#381	1	1	yes	characteristic
74	C₂×C₃₇	#408	1	2	yes	—
75	C₂×C₃₇	#433	1	1	yes	characteristic
76	C₂×C₃₇	#433	1	2	yes	—
77	C₂×C₃₇	#433	1	1	yes	characteristic
78	D₃₇	—	4	2	no	—
79	C₂×#117	—	37	1	no	—
80	C₂×C₂²⋊C₈	—	37	2	no	—
81	#336	—	37	2	no	—
82	#336	—	37	1	no	—
83	#336	D₃₇	1	1	yes	characteristic, complemented
84	C₃₇	#1818	1	1	yes	characteristic, complemented, Sylow
85	C₂³×C₄	—	37	1	no	—
86	C₂²×C₈	—	37	2	no	—
87	C₂²×C₈	C₂×D₃₇	1	2	yes	—
88	#117	—	74	4	no	—
89	C₂×C₄⋊C₄	—	37	2	no	—
90	C₂×C₂²⋊C₄	—	37	2	no	—
91	C₂×C₄²	—	37	1	no	—
92	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
93	C₂²⋊C₈	—	74	4	no	—
94	C₂⁴	—	37	1	no	—
95	C₂²×C₄	—	37	1	no	—
96	C₂²×C₄	—	37	4	no	—
97	C₂²×C₄	—	37	2	no	—
98	C₂²×C₄	—	74	2	no	—
99	C₂²×C₄	—	74	1	no	—
100	C₂²×C₄	—	74	1	no	—
101	C₂²×C₄	C₄×D₃₇	1	2	yes	—
102	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
103	C₂×C₈	—	2	2	no	—
104	C₂×C₈	—	2	2	no	—
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	—	74	2	no	—
107	C₂×C₈	—	74	2	no	—
108	C₂×C₈	—	74	2	no	—
109	C₄⋊C₄	—	74	4	no	—
110	C₂²⋊C₄	—	74	4	no	—
111	C₄²	—	2	2	no	—
112	C₄²	—	74	2	no	—
113	C₂³	—	37	2	no	—
114	C₂³	—	74	1	no	—
115	C₂³	—	74	1	no	—
116	C₂³	—	74	2	no	—
117	C₂³	—	74	2	no	—
118	C₂³	C₂×C₄×D₃₇	1	1	yes	characteristic
119	C₂×C₄	—	2	2	no	—
120	C₂×C₄	—	2	2	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	37	2	no	—
123	C₂×C₄	—	74	1	no	—
124	C₂×C₄	—	74	1	no	—
125	C₂×C₄	—	74	4	no	—
126	C₂×C₄	—	74	4	no	—
127	C₂×C₄	—	74	2	no	—
128	C₂×C₄	—	74	2	no	—
129	C₂×C₄	—	74	2	no	—
130	C₂×C₄	#10596962	1	2	yes	—
131	C₂×C₄	D₄×D₃₇	1	2	yes	—
132	C₂×C₄	#10596964	1	1	yes	characteristic
133	C₂×C₄	#10596966	1	1	yes	characteristic
134	C₈	—	4	2	no	—
135	C₈	—	148	2	no	—
136	C₂²	—	74	4	no	—
137	C₂²	—	74	2	no	—
138	C₂²	—	148	1	no	—
139	C₂²	—	148	2	no	—
140	C₂²	—	148	1	no	—
141	C₂²	C₂²⋊C₄×D₃₇	1	1	yes	characteristic
142	C₂²	#11924493	1	1	yes	characteristic
143	C₂²	#11924494	1	1	yes	characteristic
144	C₂²	#11924542	1	2	yes	—
145	C₂²	#11924544	1	2	yes	—
146	C₄	—	4	2	no	—
147	C₄	—	74	2	no	—
148	C₄	—	148	2	no	—
149	C₄	#11924518	1	2	yes	—
150	C₄	#11924521	1	2	yes	—
151	C₂	—	148	2	no	—
152	C₂	#14557395	1	1	yes	characteristic
153	C₂	#14557498	1	2	yes	—
154	C₂	#14557592	1	1	yes	characteristic
155	C₂	#14557593	1	2	yes	—
156	C₂	#14557595	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924493

C₂

yes

maximal, characteristic

C₂×#11924448

C₂

yes

maximal

#14557288

C₂

yes

maximal, characteristic, complemented

#14557287

C₂

yes

maximal, complemented

C₃₇×#336

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₃₇

C₂²

yes

characteristic

C₂²×#64436

C₂²

yes

—

#11924493

—

C₂×#10596936

C₄

yes

—

C₂×#10596938

C₄

yes

—

C₂×C₄×Dic₃₇

C₂²

yes

characteristic

#11924448

—

C₂²×C₈×C₃₇

C₂²

yes

—

C₂×C₄²×C₃₇

C₂²

yes

characteristic

C₂³×D₃₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₇

C₂×C₄

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

C₂×C₄

yes

—

C₂²×Dic₃₇

C₂×C₄

yes

characteristic

C₂×C₄×D₃₇

—

C₂×C₄×D₃₇

—

C₂×C₄×D₃₇

—

C₂×C₄×D₃₇

D₄

yes

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

—

C₄²×C₃₇

—

#10596938

—

#10596936

—

C₄×Dic₃₇

—

C₂×#64436

—

C₂×#64436

—

C₂×#64436

—

C₂³×C₃₇

C₂²×C₄

yes

characteristic

C₂²×D₃₇

—

C₂²×D₃₇

—

C₂²×D₃₇

—

C₂²×D₃₇

—

C₂²×D₃₇

C₂²⋊C₄

yes

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

#41

yes

—

C₂×C₄×C₃₇

yes

C₂×C₄×C₃₇

yes

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

C₂²⋊C₄

yes

—

C₄×D₃₇

—

C₄×D₃₇

—

C₈×C₃₇

—

#64436

—

C₂²×C₃₇

C₂×C₂²⋊C₄

yes

C₂²×C₃₇

yes

C₂²×C₃₇

yes

C₂²×C₃₇

yes

—

C₂²×C₃₇

#131

yes

—

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

—

C₄×C₃₇

—

C₄×C₃₇

#123

yes

—

C₄×C₃₇

#123

yes

—

Dic₃₇

—

Dic₃₇

—

—

C₂×C₃₇

yes

C₂×C₃₇

yes

—

C₂×C₃₇

#433

yes

characteristic

C₂×C₃₇

#433

yes

—

C₂×C₃₇

#433

yes

characteristic

D₃₇

—

C₂×#117

—

C₂×C₂²⋊C₈

—

#336

—

#336

—

#336

D₃₇

yes

characteristic, complemented

C₃₇

#1818

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂²×C₈

—

C₂²×C₈

C₂×D₃₇

yes

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₃₇

yes

characteristic

C₂²⋊C₈

—

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

C₄×D₃₇

yes

—

102

C₂²×C₄

C₂²×D₃₇

yes

characteristic

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₄⋊C₄

—

110

C₂²⋊C₄

—

111

C₄²

—

112

C₄²

—

113

C₂³

—

114

C₂³

—

115

C₂³

—

116

C₂³

—

117

C₂³

—

118

C₂³

C₂×C₄×D₃₇

yes

characteristic

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

#10596962

yes

—

131

C₂×C₄

D₄×D₃₇

yes

—

132

C₂×C₄

#10596964

yes

characteristic

133

C₂×C₄

#10596966

yes

characteristic

134

C₈

—

135

C₈

—

148

—

136

C₂²

—

137

C₂²

—

138

C₂²

—

148

—

139

C₂²

—

148

—

140

C₂²

—

148

—

141

C₂²

C₂²⋊C₄×D₃₇

yes

characteristic

142

C₂²

#11924493

yes

characteristic

143

C₂²

#11924494

yes

characteristic

144

C₂²

#11924542

yes

—

145

C₂²

#11924544

yes

—

146

C₄

—

147

C₄

—

148

C₄

—

148

—

149

C₄

#11924518

yes

—

150

C₄

#11924521

yes

—

151

C₂

—

148

—

152

C₂

#14557395

yes

characteristic

153

C₂

#14557498

yes

—

154

C₂

#14557592

yes

characteristic

155

C₂

#14557593

yes

—

156

C₂

#14557595

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₃₇
commutator, socle	C₂³×C₃₇	C₂²×C₄
Fitting	C₃₇×#336	C₂
prime radical	C₂³×D₃₇	C₂×C₄
cosocle	C₂²×C₄×C₃₇	C₂³
hypercenter	#336	D₃₇
hypocenter	C₃₇	#1818
LCS	G ▷ C₂³×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₃₇

commutator, socle

C₂³×C₃₇

C₂²×C₄

Fitting

C₃₇×#336

C₂

prime radical

C₂³×D₃₇

C₂×C₄

cosocle

C₂²×C₄×C₃₇

C₂³

D₃₇

C₃₇

G ▷ C₂³×C₃₇ ▷ C₃₇

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
3. #14557288	C₂	semidirect
4. #14557287	C₂	semidirect
5. C₃₇×#336	C₂	semidirect
83. #336	D₃₇	semidirect
84. C₃₇	#1818	semidirect

H ≅ G/N

type

3. #14557288

C₂

semidirect

4. #14557287

C₂

semidirect

5. C₃₇×#336

C₂

semidirect

83. #336

D₃₇

semidirect

84. C₃₇

#1818

semidirect

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

subgroup	index
C₂×#11924493	2
C₂×#11924448	2
#14557288	2
#14557287	2
C₃₇×#336	2
#1818	37

subgroup

index

C₃₇×#336

Subgroups — #17994666

Conjugacy classes of subgroups

Complements

Maximal subgroups