Group #17993267 — subgroups

Subgroups — #17993267

Order 4736.

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

normal subgroups only

This group has 240 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×#11924536	C₂	1	1	yes	maximal, characteristic
2	#14557557	C₂	1	1	yes	maximal, characteristic
3	#14557278	C₂	1	4	yes	maximal, complemented
4	C₃₇×#431	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₇₄	—	2	1	no	—
6	C₂×#10596934	—	2	1	no	—
7	#11924536	—	2	1	no	—
8	#11924536	—	2	1	no	—
9	#11924536	C₄	1	2	yes	—
10	#11924536	C₂²	1	2	yes	—
11	C₂×C₄⋊Dic₃₇	—	2	1	no	—
12	C₂×C₄⋊Dic₃₇	C₄	1	2	yes	—
13	C₄⋊#64436	—	2	1	no	—
14	C₄⋊#64436	C₂²	1	2	yes	—
15	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₃₇	C₂²	1	2	yes	—
17	C₂²⋊C₈×C₃₇	—	2	1	no	—
18	C₄×C₈×C₃₇	—	2	1	no	—
19	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₃₇	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₃₇	D₄	1	1	yes	characteristic
22	C₂²×Dic₃₇	—	2	2	no	—
23	C₂×Dic₇₄	—	2	2	no	—
24	C₂×Dic₇₄	—	2	2	no	—
25	C₂×Dic₇₄	—	4	2	no	—
26	C₂×Dic₇₄	—	4	2	no	—
27	C₂×C₈×C₃₇	—	2	2	no	—
28	C₄²×C₃₇	C₂×C₄	1	1	yes	characteristic
29	C₄²×C₃₇	D₄	1	1	yes	characteristic
30	C₄²×C₃₇	D₄	1	1	yes	characteristic
31	C₄²×C₃₇	C₂³	1	1	yes	characteristic
32	C₄⋊Dic₃₇	—	2	2	no	—
33	C₄⋊Dic₃₇	—	2	2	no	—
34	C₄⋊Dic₃₇	—	2	2	no	—
35	C₄⋊Dic₃₇	C₂×C₄	1	4	yes	—
36	#10596934	—	4	2	no	—
37	C₂×#64436	—	2	2	no	—
38	C₂³×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
39	C₂×C₄×C₃₇	—	2	2	no	—
40	C₂×C₄×C₃₇	—	2	2	no	—
41	C₂×C₄×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₃₇	D₈	1	2	yes	—
45	C₂×C₄×C₃₇	SD₁₆	1	2	yes	—
46	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
49	C₂×Dic₃₇	—	2	4	no	—
50	C₂×Dic₃₇	—	4	4	no	—
51	Dic₇₄	—	4	4	no	—
52	Dic₇₄	—	4	2	no	—
53	Dic₇₄	—	4	2	no	—
54	C₈×C₃₇	—	4	2	no	—
55	#64436	—	4	2	no	—
56	C₂²×C₃₇	—	2	1	no	—
57	C₂²×C₃₇	—	2	1	no	—
58	C₂²×C₃₇	#102	1	2	yes	—
59	C₂²×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₄×C₃₇	—	2	1	no	—
61	C₄×C₃₇	—	2	1	no	—
62	C₄×C₃₇	#101	1	2	yes	—
63	C₄×C₃₇	#102	1	2	yes	—
64	C₄×C₃₇	C₂×D₈	1	1	yes	characteristic
65	C₄×C₃₇	C₂×SD₁₆	1	1	yes	characteristic
66	C₄×C₃₇	#137	1	2	yes	—
67	Dic₃₇	—	4	4	no	—
68	#1546	—	37	1	no	maximal, Sylow
69	C₂×C₃₇	—	2	2	no	—
70	C₂×C₃₇	C₂×#101	1	1	yes	characteristic
71	C₂×C₃₇	C₂×#102	1	1	yes	characteristic
72	C₂×C₃₇	#419	1	1	yes	characteristic
73	C₂×C₄⋊Q₈	—	37	1	no	—
74	#431	D₃₇	1	1	yes	characteristic
75	#423	—	37	1	no	—
76	#332	—	37	4	no	—
77	C₃₇	#1546	1	1	yes	characteristic, complemented, Sylow
78	C₂²×Q₈	—	74	1	no	—
79	C₂×M₁₆	—	74	1	no	—
80	C₄⋊Q₈	—	37	2	no	—
81	C₄⋊Q₈	—	37	2	no	—
82	C₄⋊Q₈	—	74	1	no	—
83	C₄⋊Q₈	—	74	1	no	—
84	C₂×C₄⋊C₄	—	37	2	no	—
85	C₂×C₄⋊C₄	—	74	1	no	—
86	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
87	C₄⋊C₈	—	37	2	no	—
88	C₄⋊C₈	—	74	1	no	—
89	C₄⋊C₈	C₂×D₃₇	1	2	yes	—
90	C₂²⋊C₈	—	2	1	no	—
91	C₄×C₈	—	2	1	no	—
92	C₂×Q₈	—	74	2	no	—
93	C₂×Q₈	—	74	2	no	—
94	C₂×Q₈	—	148	2	no	—
95	C₂×Q₈	—	148	2	no	—
96	C₂²×C₄	—	74	2	no	—
97	C₂²×C₄	C₄×D₃₇	1	1	yes	characteristic
98	C₂²×C₄	D₁₄₈	1	1	yes	characteristic
99	C₂²×C₄	C₂²⋊D₃₇	1	1	yes	characteristic
100	M₁₆	—	148	2	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	74	2	no	—
103	C₄⋊C₄	—	37	4	no	—
104	C₄⋊C₄	—	74	2	no	—
105	C₄⋊C₄	—	74	2	no	—
106	C₄⋊C₄	—	74	2	no	—
107	C₄²	C₄×D₃₇	1	1	yes	characteristic
108	C₄²	D₁₄₈	1	1	yes	characteristic
109	C₄²	C₂²⋊D₃₇	1	1	yes	characteristic
110	C₄²	C₂²×D₃₇	1	1	yes	characteristic
111	C₂³	#10596938	1	1	yes	characteristic
112	Q₈	—	148	2	no	—
113	Q₈	—	148	2	no	—
114	Q₈	—	148	4	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	74	4	no	—
118	C₂×C₄	—	148	4	no	—
119	C₂×C₄	#10596930	1	2	yes	—
120	C₂×C₄	D₂₉₆	1	2	yes	—
121	C₂×C₄	#10596938	1	1	yes	characteristic
122	C₂×C₄	#10596938	1	1	yes	characteristic
123	C₂×C₄	#10596938	1	1	yes	characteristic
124	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
125	C₂×C₄	C₂×D₁₄₈	1	1	yes	characteristic
126	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
127	C₈	—	4	2	no	—
128	C₈	—	148	2	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	#11924458	1	2	yes	—
132	C₂²	C₂×#10596938	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	148	4	no	—
136	C₄	#11924454	1	2	yes	—
137	C₄	#11924458	1	2	yes	—
138	C₄	C₂×D₂₉₆	1	1	yes	characteristic
139	C₄	C₂×#10596930	1	1	yes	characteristic
140	C₄	#11924566	1	2	yes	—
141	C₂	—	2	2	no	—
142	C₂	C₂×#11924454	1	1	yes	characteristic
143	C₂	C₂×#11924458	1	1	yes	characteristic
144	C₂	#14557541	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924536

C₂

yes

maximal, characteristic

#14557557

C₂

yes

maximal, characteristic

#14557278

C₂

yes

maximal, complemented

C₃₇×#431

C₂

yes

maximal, characteristic

C₂²×Dic₇₄

—

—

—

—

C₄

yes

—

#11924536

C₂²

yes

—

C₂×C₄⋊Dic₃₇

—

C₂×C₄⋊Dic₃₇

C₄

yes

—

C₄⋊#64436

—

C₄⋊#64436

C₂²

yes

—

C₂×C₄²×C₃₇

C₂²

yes

characteristic

C₄⋊C₈×C₃₇

C₂²

yes

—

C₂²⋊C₈×C₃₇

—

C₄×C₈×C₃₇

—

C₂²×C₄×C₃₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₃₇

D₄

yes

characteristic

C₂²×C₄×C₃₇

D₄

yes

characteristic

C₂²×Dic₃₇

—

C₂×Dic₇₄

—

C₂×Dic₇₄

—

C₂×Dic₇₄

—

C₂×Dic₇₄

—

C₂×C₈×C₃₇

—

C₄²×C₃₇

C₂×C₄

yes

characteristic

C₄²×C₃₇

D₄

yes

characteristic

C₄²×C₃₇

D₄

yes

characteristic

C₄²×C₃₇

C₂³

yes

characteristic

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

C₂×C₄

yes

—

#10596934

—

C₂×#64436

—

C₂³×C₃₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×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×C₄×C₃₇

D₈

yes

—

C₂×C₄×C₃₇

SD₁₆

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

—

Dic₇₄

—

Dic₇₄

—

Dic₇₄

—

C₈×C₃₇

—

#64436

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

#102

yes

—

C₂²×C₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

#101

yes

—

C₄×C₃₇

#102

yes

—

C₄×C₃₇

C₂×D₈

yes

characteristic

C₄×C₃₇

C₂×SD₁₆

yes

characteristic

C₄×C₃₇

#137

yes

—

Dic₃₇

—

#1546

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₄⋊Q₈

—

#431

D₃₇

yes

characteristic

#423

—

#332

—

C₃₇

#1546

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₃₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₃₇

yes

—

C₂²⋊C₈

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

148

—

C₂×Q₈

—

148

—

C₂²×C₄

—

C₂²×C₄

C₄×D₃₇

yes

characteristic

C₂²×C₄

D₁₄₈

yes

characteristic

C₂²×C₄

C₂²⋊D₃₇

yes

characteristic

100

M₁₆

—

148

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄²

C₄×D₃₇

yes

characteristic

108

C₄²

D₁₄₈

yes

characteristic

109

C₄²

C₂²⋊D₃₇

yes

characteristic

110

C₄²

C₂²×D₃₇

yes

characteristic

111

C₂³

#10596938

yes

characteristic

112

Q₈

—

148

—

113

Q₈

—

148

—

114

Q₈

—

148

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

148

—

119

C₂×C₄

#10596930

yes

—

120

C₂×C₄

D₂₉₆

yes

—

121

C₂×C₄

#10596938

yes

characteristic

122

C₂×C₄

#10596938

yes

characteristic

123

C₂×C₄

#10596938

yes

characteristic

124

C₂×C₄

C₂×C₄×D₃₇

yes

characteristic

125

C₂×C₄

C₂×D₁₄₈

yes

characteristic

126

C₂×C₄

C₂×C₂²⋊D₃₇

yes

characteristic

127

C₈

—

128

C₈

—

148

—

129

C₂²

—

130

C₂²

—

131

C₂²

#11924458

yes

—

132

C₂²

C₂×#10596938

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

148

—

136

C₄

#11924454

yes

—

137

C₄

#11924458

yes

—

138

C₄

C₂×D₂₉₆

yes

characteristic

139

C₄

C₂×#10596930

yes

characteristic

140

C₄

#11924566

yes

—

141

C₂

—

142

C₂

C₂×#11924454

yes

characteristic

143

C₂

C₂×#11924458

yes

characteristic

144

C₂

#14557541

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#10596938
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₄²	C₂²×D₃₇
Fitting	C₃₇×#431	C₂
socle	C₂²×C₃₇	C₂×C₂²⋊C₄
prime radical	C₂³×C₃₇	C₂²⋊C₄
cosocle	C₄²×C₃₇	C₂³
hypercenter	#431	D₃₇
hypocenter	C₃₇	#1546
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #431

subgroup

G/H

center

C₂²

C₂×#10596938

commutator

C₂×C₄×C₃₇

C₂²×C₄

Frattini

C₄²

C₂²×D₃₇

Fitting

C₃₇×#431

C₂

socle

C₂²×C₃₇

C₂×C₂²⋊C₄

prime radical

C₂³×C₃₇

C₂²⋊C₄

cosocle

C₄²×C₃₇

C₂³

D₃₇

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #431

N	H ≅ G/N	type
3. #14557278	C₂	semidirect
77. C₃₇	#1546	semidirect

H ≅ G/N

type

3. #14557278

C₂

semidirect

77. C₃₇

#1546

semidirect

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

subgroup	index
C₂×#11924536	2
#14557557	2
#14557278	2
C₃₇×#431	2
#1546	37

subgroup

index

C₃₇×#431

Subgroups — #17993267

Conjugacy classes of subgroups

Complements

Maximal subgroups