Group #17995445 — subgroups

Subgroups — #17995445

Order 4736.

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

normal subgroups only

This group has 312 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×#11924524	C₂	1	1	yes	maximal, characteristic
2	#14557557	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11924454	C₂	1	2	yes	maximal, complemented
4	C₂×#11924452	C₂	1	2	yes	maximal
5	C₄×M₁₆×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Dic₇₄	C₂²	1	1	yes	characteristic
7	C₂²×D₁₄₈	C₂²	1	1	yes	characteristic
8	C₂×#10596934	C₂²	1	2	yes	—
9	#11924524	—	2	2	no	—
10	#11924524	—	2	1	no	—
11	#11924524	—	2	1	no	—
12	C₂×#10596938	C₄	1	4	yes	—
13	C₄⋊#64436	—	2	2	no	—
14	#11924454	—	2	4	no	—
15	#11924452	—	2	4	no	—
16	C₂×M₁₆×C₃₇	C₂²	1	2	yes	—
17	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
18	#97×C₃₇	—	2	1	no	—
19	C₄×C₈×C₃₇	—	2	1	no	—
20	C₂³×D₃₇	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₃₇	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
23	C₂²×Dic₃₇	C₂×C₄	1	2	yes	—
24	C₂×D₁₄₈	—	2	1	no	—
25	C₂×D₁₄₈	—	2	1	no	—
26	C₂×D₁₄₈	—	2	2	no	—
27	C₂×D₁₄₈	—	2	2	no	—
28	C₂×Dic₇₄	—	2	1	no	—
29	C₂×Dic₇₄	—	2	1	no	—
30	C₂×Dic₇₄	—	2	2	no	—
31	C₂×Dic₇₄	—	2	2	no	—
32	M₁₆×C₃₇	—	2	2	no	—
33	M₁₆×C₃₇	D₄	1	4	yes	—
34	C₂×C₈×C₃₇	—	2	2	no	—
35	C₄²×C₃₇	D₄	1	2	yes	—
36	C₄²×C₃₇	D₄	1	1	yes	characteristic
37	C₄²×C₃₇	D₄	1	1	yes	characteristic
38	#10596938	—	2	4	no	—
39	#10596938	—	2	4	no	—
40	#10596934	—	2	2	no	—
41	#10596934	—	2	2	no	—
42	C₂×#64436	—	2	2	no	—
43	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
44	C₂²×D₃₇	—	2	2	no	—
45	C₂²×D₃₇	—	2	4	no	—
46	C₂²×D₃₇	—	4	4	no	—
47	C₂×C₄×C₃₇	—	4	2	no	—
48	C₂×C₄×C₃₇	C₂²⋊C₄	1	2	yes	—
49	C₂×C₄×C₃₇	C₂²⋊C₄	1	2	yes	—
50	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₃₇	#41	1	2	yes	—
54	C₂×Dic₃₇	—	2	2	no	—
55	C₂×Dic₃₇	—	2	4	no	—
56	D₁₄₈	—	4	2	no	—
57	D₁₄₈	—	4	1	no	—
58	D₁₄₈	—	4	1	no	—
59	Dic₇₄	—	4	2	no	—
60	Dic₇₄	—	4	1	no	—
61	Dic₇₄	—	4	1	no	—
62	C₈×C₃₇	—	2	4	no	—
63	#64436	—	4	2	no	—
64	C₂²×C₃₇	—	2	2	no	—
65	C₂²×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
66	C₂²×C₃₇	C₄×D₄	1	2	yes	—
67	C₂×D₃₇	—	4	2	no	—
68	C₂×D₃₇	—	4	4	no	—
69	C₂×D₃₇	—	8	4	no	—
70	C₄×C₃₇	—	2	2	no	—
71	C₄×C₃₇	—	2	2	no	—
72	C₄×C₃₇	#121	1	2	yes	—
73	C₄×C₃₇	#124	1	1	yes	characteristic
74	C₄×C₃₇	#127	1	1	yes	characteristic
75	Dic₃₇	—	4	2	no	—
76	#1967	—	37	1	no	maximal, Sylow
77	C₂×C₃₇	—	2	2	no	—
78	C₂×C₃₇	#390	1	1	yes	characteristic
79	C₂×C₃₇	#413	1	2	yes	—
80	D₃₇	—	8	2	no	—
81	C₂×#124	—	37	1	no	—
82	#423	—	37	1	no	—
83	C₂×#101	—	37	2	no	—
84	C₂×#100	—	37	2	no	—
85	C₄×M₁₆	D₃₇	1	1	yes	characteristic, complemented
86	C₃₇	#1967	1	1	yes	characteristic, complemented, Sylow
87	C₂²×Q₈	—	37	1	no	—
88	C₂²×D₄	—	37	1	no	—
89	C₂×M₁₆	—	37	2	no	—
90	C₂×M₁₆	C₂×D₃₇	1	2	yes	—
91	#124	—	74	2	no	—
92	#124	—	74	1	no	—
93	#124	—	74	1	no	—
94	C₂×C₂²⋊C₄	—	37	4	no	—
95	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
96	C₄⋊C₈	—	74	2	no	—
97	#101	—	74	4	no	—
98	#100	—	74	4	no	—
99	#97	—	2	1	no	—
100	C₄×C₈	—	2	1	no	—
101	C₂⁴	—	37	2	no	—
102	C₂×Q₈	—	74	2	no	—
103	C₂×Q₈	—	74	1	no	—
104	C₂×Q₈	—	74	2	no	—
105	C₂×Q₈	—	74	1	no	—
106	C₂×D₄	—	74	2	no	—
107	C₂×D₄	—	74	1	no	—
108	C₂×D₄	—	74	2	no	—
109	C₂×D₄	—	74	1	no	—
110	C₂²×C₄	—	37	2	no	—
111	C₂²×C₄	C₄×D₃₇	1	2	yes	—
112	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
113	M₁₆	—	2	2	no	—
114	M₁₆	—	74	2	no	—
115	M₁₆	—	74	2	no	—
116	M₁₆	D₁₄₈	1	4	yes	—
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	74	2	no	—
119	C₂²⋊C₄	—	74	4	no	—
120	C₂²⋊C₄	—	74	4	no	—
121	C₄²	D₁₄₈	1	2	yes	—
122	C₄²	C₂²⋊D₃₇	1	1	yes	characteristic
123	C₄²	C₂²⋊D₃₇	1	1	yes	characteristic
124	C₂³	—	74	4	no	—
125	C₂³	—	74	2	no	—
126	C₂³	—	148	4	no	—
127	C₂³	C₂×C₄×D₃₇	1	1	yes	characteristic
128	Q₈	—	148	2	no	—
129	Q₈	—	148	1	no	—
130	Q₈	—	148	1	no	—
131	D₄	—	148	2	no	—
132	D₄	—	148	1	no	—
133	D₄	—	148	1	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	—	74	2	no	—
136	C₂×C₄	—	74	4	no	—
137	C₂×C₄	#10596938	1	2	yes	—
138	C₂×C₄	#10596938	1	2	yes	—
139	C₂×C₄	C₂×D₁₄₈	1	2	yes	—
140	C₂×C₄	C₂×D₁₄₈	1	1	yes	characteristic
141	C₂×C₄	#10596962	1	2	yes	—
142	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
143	C₈	—	2	4	no	—
144	C₈	—	148	2	no	—
145	C₂²	—	2	2	no	—
146	C₂²	—	296	4	no	—
147	C₂²	—	148	4	no	—
148	C₂²	—	148	2	no	—
149	C₂²	C₂×#10596938	1	1	yes	characteristic
150	C₂²	C₄×D₁₄₈	1	2	yes	—
151	C₄	—	2	2	no	—
152	C₄	—	2	2	no	—
153	C₄	—	148	2	no	—
154	C₄	#11924510	1	2	yes	—
155	C₄	#11924524	1	1	yes	characteristic
156	C₄	#11924533	1	1	yes	characteristic
157	C₂	—	2	2	no	—
158	C₂	—	296	2	no	—
159	C₂	#14557435	1	1	yes	characteristic
160	C₂	#14557515	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924524

C₂

yes

maximal, characteristic

#14557557

C₂

yes

maximal, characteristic, complemented

C₂×#11924454

C₂

yes

maximal, complemented

C₂×#11924452

C₂

yes

maximal

C₄×M₁₆×C₃₇

C₂

yes

maximal, characteristic, complemented

C₂²×Dic₇₄

C₂²

yes

characteristic

C₂²×D₁₄₈

C₂²

yes

characteristic

C₂×#10596934

C₂²

yes

—

—

—

—

C₄

yes

—

C₄⋊#64436

—

#11924454

—

#11924452

—

C₂×M₁₆×C₃₇

C₂²

yes

—

C₂×C₄²×C₃₇

C₂²

yes

characteristic

#97×C₃₇

—

C₄×C₈×C₃₇

—

C₂³×D₃₇

C₂×C₄

yes

—

C₂²×C₄×C₃₇

C₂×C₄

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

C₂×C₄

yes

—

C₂×D₁₄₈

—

C₂×D₁₄₈

—

C₂×D₁₄₈

—

C₂×D₁₄₈

—

C₂×Dic₇₄

—

C₂×Dic₇₄

—

C₂×Dic₇₄

—

C₂×Dic₇₄

—

M₁₆×C₃₇

—

M₁₆×C₃₇

D₄

yes

—

C₂×C₈×C₃₇

—

C₄²×C₃₇

D₄

yes

—

C₄²×C₃₇

D₄

yes

characteristic

C₄²×C₃₇

D₄

yes

—

—

—

—

—

C₂³×C₃₇

C₂²×C₄

yes

characteristic

C₂²×D₃₇

—

C₂²×D₃₇

—

C₂²×D₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²⋊C₄

yes

—

C₂×C₄×C₃₇

C₂²⋊C₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

#41

yes

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

D₁₄₈

—

D₁₄₈

—

D₁₄₈

—

Dic₇₄

—

Dic₇₄

—

Dic₇₄

—

C₈×C₃₇

—

#64436

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₃₇

C₄×D₄

yes

—

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

#121

yes

—

C₄×C₃₇

yes

C₄×C₃₇

yes

Dic₃₇

—

#1967

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

#390

yes

characteristic

C₂×C₃₇

#413

yes

—

D₃₇

—

—

—

—

—

C₄×M₁₆

D₃₇

yes

characteristic, complemented

C₃₇

#1967

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×D₄

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₃₇

yes

—

#124

—

#124

—

#124

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂×D₃₇

yes

characteristic

C₄⋊C₈

—

#101

—

#100

—

#97

—

100

C₄×C₈

—

101

C₂⁴

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂²×C₄

—

111

C₂²×C₄

C₄×D₃₇

yes

—

112

C₂²×C₄

C₂²×D₃₇

yes

characteristic

113

M₁₆

—

114

M₁₆

—

115

M₁₆

—

116

M₁₆

D₁₄₈

yes

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

121

C₄²

D₁₄₈

yes

—

122

C₄²

C₂²⋊D₃₇

yes

characteristic

123

C₄²

C₂²⋊D₃₇

yes

characteristic

124

C₂³

—

125

C₂³

—

126

C₂³

—

148

—

127

C₂³

C₂×C₄×D₃₇

yes

characteristic

128

Q₈

—

148

—

129

Q₈

—

148

—

130

Q₈

—

148

—

131

D₄

—

148

—

132

D₄

—

148

—

133

D₄

—

148

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

#10596938

yes

—

138

C₂×C₄

#10596938

yes

—

139

C₂×C₄

C₂×D₁₄₈

yes

—

140

C₂×C₄

C₂×D₁₄₈

yes

characteristic

141

C₂×C₄

#10596962

yes

—

142

C₂×C₄

C₂×C₂²⋊D₃₇

yes

characteristic

143

C₈

—

144

C₈

—

148

—

145

C₂²

—

146

C₂²

—

296

—

147

C₂²

—

148

—

148

C₂²

—

148

—

149

C₂²

C₂×#10596938

yes

characteristic

150

C₂²

C₄×D₁₄₈

yes

—

151

C₄

—

152

C₄

—

153

C₄

—

148

—

154

C₄

#11924510

yes

—

155

C₄

#11924524

yes

characteristic

156

C₄

#11924533

yes

characteristic

157

C₂

—

158

C₂

—

296

—

159

C₂

#14557435

yes

characteristic

160

C₂

#14557515

yes

—

	subgroup	G/H
center	C₂²	C₂×#10596938
commutator	C₂³×C₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting	C₄×M₁₆×C₃₇	C₂
socle	C₂²×C₃₇	C₂×C₂²⋊C₄
prime radical	C₂²×D₁₄₈	C₂²
cosocle	C₂²×C₄×C₃₇	C₂³
hypercenter	C₄×M₁₆	D₃₇
hypocenter	C₃₇	#1967
LCS	G ▷ C₂³×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

subgroup

G/H

center

C₂²

C₂×#10596938

commutator

C₂³×C₃₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₇

Fitting

C₄×M₁₆×C₃₇

C₂

socle

C₂²×C₃₇

C₂×C₂²⋊C₄

prime radical

C₂²×D₁₄₈

C₂²

cosocle

C₂²×C₄×C₃₇

C₂³

hypercenter

C₄×M₁₆

D₃₇

hypocenter

C₃₇

#1967

LCS

G ▷ C₂³×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×M₁₆

N	H ≅ G/N	type
2. #14557557	C₂	semidirect
3. C₂×#11924454	C₂	semidirect
5. C₄×M₁₆×C₃₇	C₂	semidirect
85. C₄×M₁₆	D₃₇	semidirect
86. C₃₇	#1967	semidirect

H ≅ G/N

type

2. #14557557

C₂

semidirect

3. C₂×#11924454

C₂

semidirect

5. C₄×M₁₆×C₃₇

C₂

semidirect

85. C₄×M₁₆

D₃₇

semidirect

86. C₃₇

#1967

semidirect

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

subgroup	index
C₂×#11924524	2
#14557557	2
C₂×#11924454	2
C₂×#11924452	2
C₄×M₁₆×C₃₇	2
#1967	37

subgroup

index

C₄×M₁₆×C₃₇

Subgroups — #17995445

Conjugacy classes of subgroups

Complements

Maximal subgroups