Group #18003248 — subgroups

Subgroups — #18003248

Order 4736.

384 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 173 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	C₄×#10596962	C₂	1	1	yes	maximal, characteristic
2	#14557592	C₂	1	4	yes	maximal, complemented
3	#14557588	C₂	1	4	yes	maximal, complemented
4	#14557561	C₂	1	2	yes	maximal, complemented
5	#14557501	C₂	1	2	yes	maximal
6	C₂×#11924447	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×M₁₆×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10596962	C₂²	1	1	yes	characteristic
9	C₂²×#64436	C₂²	1	2	yes	—
10	C₄×Dic₇₄	C₄	1	2	yes	—
11	C₄×D₁₄₈	C₄	1	2	yes	—
12	C₄×C₂²⋊D₃₇	C₄	1	4	yes	—
13	#11924495	C₂²	1	1	yes	characteristic
14	#11924493	C₂²	1	2	yes	—
15	C₄²×D₃₇	C₂²	1	2	yes	—
16	#11924465	C₂²	1	8	yes	—
17	#11924448	C₂²	1	8	yes	—
18	#11924447	C₂²	1	2	yes	—
19	#11924447	C₂²	1	2	yes	—
20	C₂×M₁₆×C₃₇	C₂²	1	2	yes	—
21	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
22	#97×C₃₇	C₂²	1	2	yes	—
23	C₄×C₈×C₃₇	C₂²	1	2	yes	—
24	C₂²×C₄×C₃₇	C₂×C₄	1	2	yes	—
25	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
26	C₂×C₂²⋊D₃₇	C₂×C₄	1	2	yes	—
27	#10596962	—	2	4	no	—
28	C₂×D₁₄₈	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×D₃₇	C₂×C₄	1	4	yes	—
30	C₂×C₄×D₃₇	C₂³	1	2	yes	—
31	C₂×Dic₇₄	C₂×C₄	1	1	yes	characteristic
32	M₁₆×C₃₇	—	2	4	no	—
33	C₂×C₈×C₃₇	C₂³	1	4	yes	—
34	C₄²×C₃₇	C₂×C₄	1	2	yes	—
35	C₄²×C₃₇	C₂³	1	1	yes	characteristic
36	C₄²×C₃₇	C₂³	1	1	yes	characteristic
37	C₂²⋊Dic₃₇	C₂×C₄	1	2	yes	—
38	#10596938	C₂×C₄	1	4	yes	—
39	C₄⋊Dic₃₇	C₂×C₄	1	2	yes	—
40	#10596936	C₂×C₄	1	4	yes	—
41	C₄×Dic₃₇	C₂×C₄	1	4	yes	—
42	C₄×Dic₃₇	C₂³	1	2	yes	—
43	C₂×#64436	—	2	4	no	—
44	C₂×#64436	C₂³	1	4	yes	—
45	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
46	C₂²×D₃₇	C₂²×C₄	1	2	yes	—
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	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₃₇	#41	1	4	yes	—
52	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
53	C₂²⋊D₃₇	—	2	4	no	—
54	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
55	C₂×Dic₃₇	C₂²×C₄	1	4	yes	—
56	D₁₄₈	—	2	2	no	—
57	C₄×D₃₇	—	2	4	no	—
58	C₄×D₃₇	—	4	4	no	—
59	Dic₇₄	—	2	2	no	—
60	C₈×C₃₇	—	2	4	no	—
61	#64436	—	2	4	no	—
62	C₂²×C₃₇	—	2	2	no	—
63	C₂²×C₃₇	#117	1	2	yes	—
64	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
65	C₂×D₃₇	—	2	2	no	—
66	C₂×D₃₇	—	2	4	no	—
67	C₄×C₃₇	—	2	2	no	—
68	C₄×C₃₇	—	2	2	no	—
69	C₄×C₃₇	#117	1	2	yes	—
70	C₄×C₃₇	C₂×#41	1	2	yes	—
71	Dic₃₇	—	2	2	no	—
72	Dic₃₇	—	2	4	no	—
73	#2923	—	37	1	no	maximal, Sylow
74	C₂×C₃₇	—	2	2	no	—
75	C₂×C₃₇	C₂×#117	1	1	yes	characteristic
76	C₂×C₃₇	#568	1	2	yes	—
77	D₃₇	—	4	2	no	—
78	C₄×#41	—	37	1	no	—
79	#433	—	37	4	no	—
80	#432	—	37	4	no	—
81	#424	—	37	2	no	—
82	#408	—	37	2	no	—
83	C₄×M₁₆	D₃₇	1	1	yes	characteristic, complemented
84	C₂×#97	—	37	1	no	—
85	C₃₇	#2923	1	1	yes	characteristic, complemented, Sylow
86	C₂×#41	—	37	1	no	—
87	C₂×M₁₆	C₂×D₃₇	1	2	yes	—
88	C₂²×C₈	—	37	2	no	—
89	C₄×Q₈	—	37	2	no	—
90	C₄×D₄	—	37	4	no	—
91	C₄×D₄	—	37	2	no	—
92	#117	—	37	1	no	—
93	#117	—	37	2	no	—
94	C₂×C₄²	—	37	2	no	—
95	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
96	C₄⋊C₈	—	37	8	no	—
97	C₂²⋊C₈	—	37	8	no	—
98	#97	—	37	2	no	—
99	#97	—	37	2	no	—
100	#97	C₂×D₃₇	1	2	yes	—
101	C₄×C₈	C₂×D₃₇	1	2	yes	—
102	#41	—	74	4	no	—
103	C₂×Q₈	—	37	1	no	—
104	C₂×D₄	—	37	2	no	—
105	C₂×D₄	—	37	1	no	—
106	C₂²×C₄	—	37	2	no	—
107	C₂²×C₄	—	37	4	no	—
108	C₂²×C₄	C₄×D₃₇	1	2	yes	—
109	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
110	M₁₆	—	2	4	no	—
111	C₂×C₈	—	37	4	no	—
112	C₂×C₈	—	74	4	no	—
113	C₂×C₈	C₂²×D₃₇	1	4	yes	—
114	C₄⋊C₄	—	37	4	no	—
115	C₄⋊C₄	—	37	2	no	—
116	C₂²⋊C₄	—	37	2	no	—
117	C₂²⋊C₄	—	37	4	no	—
118	C₄²	—	37	4	no	—
119	C₄²	—	37	2	no	—
120	C₄²	C₄×D₃₇	1	2	yes	—
121	C₄²	C₂²×D₃₇	1	1	yes	characteristic
122	C₄²	C₂²×D₃₇	1	1	yes	characteristic
123	C₂³	—	37	2	no	—
124	C₂³	C₂×C₄×D₃₇	1	1	yes	characteristic
125	Q₈	—	74	2	no	—
126	D₄	—	74	4	no	—
127	D₄	—	74	2	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	37	4	no	—
130	C₂×C₄	—	37	2	no	—
131	C₂×C₄	—	74	4	no	—
132	C₂×C₄	—	148	4	no	—
133	C₂×C₄	C₂×C₄×D₃₇	1	2	yes	—
134	C₂×C₄	C₂×C₄×D₃₇	1	2	yes	—
135	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
136	C₂×C₄	#10596962	1	4	yes	—
137	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
138	C₈	—	2	4	no	—
139	C₈	—	74	4	no	—
140	C₂²	—	2	2	no	—
141	C₂²	—	74	2	no	—
142	C₂²	—	74	4	no	—
143	C₂²	#11924493	1	2	yes	—
144	C₂²	C₂²×C₄×D₃₇	1	1	yes	characteristic
145	C₄	—	2	2	no	—
146	C₄	—	2	2	no	—
147	C₄	—	74	2	no	—
148	C₄	—	74	4	no	—
149	C₄	#11924493	1	2	yes	—
150	C₄	C₂×#10596962	1	2	yes	—
151	C₂	—	2	2	no	—
152	C₂	—	148	2	no	—
153	C₂	C₂×#11924493	1	1	yes	characteristic
154	C₂	#14558299	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#10596962

C₂

yes

maximal, characteristic

#14557592

C₂

yes

maximal, complemented

#14557588

C₂

yes

maximal, complemented

#14557561

C₂

yes

maximal, complemented

#14557501

C₂

yes

maximal

C₂×#11924447

C₂

yes

maximal, characteristic, complemented

C₄×M₁₆×C₃₇

C₂

yes

maximal, characteristic, complemented

C₂×#10596962

C₂²

yes

characteristic

C₂²×#64436

C₂²

yes

—

C₄×Dic₇₄

C₄

yes

—

C₄×D₁₄₈

C₄

yes

—

C₄×C₂²⋊D₃₇

C₄

yes

—

#11924495

C₂²

yes

characteristic

#11924493

C₂²

yes

—

C₄²×D₃₇

C₂²

yes

—

#11924465

C₂²

yes

—

#11924448

C₂²

yes

—

#11924447

C₂²

yes

—

#11924447

C₂²

yes

—

C₂×M₁₆×C₃₇

C₂²

yes

—

C₂×C₄²×C₃₇

C₂²

yes

characteristic

#97×C₃₇

C₂²

yes

—

C₄×C₈×C₃₇

C₂²

yes

—

C₂²×C₄×C₃₇

C₂×C₄

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂×C₂²⋊D₃₇

C₂×C₄

yes

—

#10596962

—

C₂×D₁₄₈

C₂×C₄

yes

characteristic

C₂×C₄×D₃₇

C₂×C₄

yes

—

C₂×C₄×D₃₇

C₂³

yes

—

C₂×Dic₇₄

C₂×C₄

yes

characteristic

M₁₆×C₃₇

—

C₂×C₈×C₃₇

C₂³

yes

—

C₄²×C₃₇

C₂×C₄

yes

—

C₄²×C₃₇

C₂³

yes

characteristic

C₄²×C₃₇

C₂³

yes

characteristic

C₂²⋊Dic₃₇

C₂×C₄

yes

—

#10596938

C₂×C₄

yes

—

C₄⋊Dic₃₇

C₂×C₄

yes

—

#10596936

C₂×C₄

yes

—

C₄×Dic₃₇

C₂×C₄

yes

—

C₄×Dic₃₇

C₂³

yes

—

C₂×#64436

—

C₂×#64436

C₂³

yes

—

C₂³×C₃₇

C₂²×C₄

yes

characteristic

C₂²×D₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

#41

yes

—

C₂×C₄×C₃₇

C₂⁴

yes

characteristic

C₂²⋊D₃₇

—

C₂×Dic₃₇

C₂²×C₄

yes

—

C₂×Dic₃₇

C₂²×C₄

yes

—

D₁₄₈

—

C₄×D₃₇

—

C₄×D₃₇

—

Dic₇₄

—

C₈×C₃₇

—

#64436

—

C₂²×C₃₇

—

C₂²×C₃₇

#117

yes

—

C₂²×C₃₇

C₂³×C₄

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

#117

yes

—

C₄×C₃₇

C₂×#41

yes

—

Dic₃₇

—

Dic₃₇

—

#2923

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

C₂×#117

yes

characteristic

C₂×C₃₇

#568

yes

—

D₃₇

—

C₄×#41

—

—

—

—

—

C₄×M₁₆

D₃₇

yes

characteristic, complemented

C₂×#97

—

C₃₇

#2923

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×M₁₆

C₂×D₃₇

yes

—

C₂²×C₈

—

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

—

#117

—

#117

—

C₂×C₄²

—

C₂×C₄²

C₂×D₃₇

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

100

#97

C₂×D₃₇

yes

—

101

C₄×C₈

C₂×D₃₇

yes

—

102

#41

—

103

C₂×Q₈

—

104

C₂×D₄

—

105

C₂×D₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

C₄×D₃₇

yes

—

109

C₂²×C₄

C₂²×D₃₇

yes

characteristic

110

M₁₆

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

C₂²×D₃₇

yes

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₂²⋊C₄

—

117

C₂²⋊C₄

—

118

C₄²

—

119

C₄²

—

120

C₄²

C₄×D₃₇

yes

—

121

C₄²

C₂²×D₃₇

yes

characteristic

122

C₄²

C₂²×D₃₇

yes

characteristic

123

C₂³

—

124

C₂³

C₂×C₄×D₃₇

yes

characteristic

125

Q₈

—

126

D₄

—

127

D₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

148

—

133

C₂×C₄

C₂×C₄×D₃₇

yes

—

134

C₂×C₄

C₂×C₄×D₃₇

yes

—

135

C₂×C₄

C₂×C₄×D₃₇

yes

characteristic

136

C₂×C₄

#10596962

yes

—

137

C₂×C₄

C₂³×D₃₇

yes

characteristic

138

C₈

—

139

C₈

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

#11924493

yes

—

144

C₂²

C₂²×C₄×D₃₇

yes

characteristic

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

#11924493

yes

—

150

C₄

C₂×#10596962

yes

—

151

C₂

—

152

C₂

—

148

—

153

C₂

C₂×#11924493

yes

characteristic

154

C₂

#14558299

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃₇
commutator, socle	C₂²×C₃₇	C₂³×C₄
Fitting	C₄×M₁₆×C₃₇	C₂
prime radical	C₂×#10596962	C₂²
cosocle	C₂×C₄×C₃₇	C₂⁴
hypercenter	C₄×M₁₆	D₃₇
hypocenter	C₃₇	#2923
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₃₇

commutator, socle

C₂²×C₃₇

C₂³×C₄

Fitting

C₄×M₁₆×C₃₇

C₂

prime radical

C₂×#10596962

C₂²

cosocle

C₂×C₄×C₃₇

C₂⁴

hypercenter

C₄×M₁₆

D₃₇

C₃₇

G ▷ C₂²×C₃₇ ▷ C₃₇

C₁ ◁ C₂×C₄ ◁ C₄×M₁₆

N	H ≅ G/N	type
2. #14557592	C₂	semidirect
3. #14557588	C₂	semidirect
4. #14557561	C₂	semidirect
6. C₂×#11924447	C₂	semidirect
7. C₄×M₁₆×C₃₇	C₂	semidirect
83. C₄×M₁₆	D₃₇	semidirect
85. C₃₇	#2923	semidirect

H ≅ G/N

type

2. #14557592

C₂

semidirect

3. #14557588

C₂

semidirect

4. #14557561

C₂

semidirect

6. C₂×#11924447

C₂

semidirect

7. C₄×M₁₆×C₃₇

C₂

semidirect

83. C₄×M₁₆

D₃₇

semidirect

85. C₃₇

#2923

semidirect

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

subgroup	index
C₄×#10596962	2
#14557592	2
#14557588	2
#14557561	2
#14557501	2
C₂×#11924447	2
C₄×M₁₆×C₃₇	2
#2923	37

subgroup

index

C₄×M₁₆×C₃₇

Subgroups — #18003248

Conjugacy classes of subgroups

Complements

Maximal subgroups