Group #18003158 — subgroups

Subgroups — #18003158

Order 4736.

484 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 213 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×M₁₆×D₃₇	C₂	1	2	yes	maximal
2	C₂×#11924493	C₂	1	1	yes	maximal, characteristic
3	#14557558	C₂	1	8	yes	maximal, complemented
4	#14557557	C₂	1	1	yes	maximal, characteristic, complemented
5	#14557556	C₂	1	2	yes	maximal, complemented
6	C₃₇×#423	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₄×D₃₇	C₂²	1	1	yes	characteristic
8	C₂×#10596934	C₂²	1	2	yes	—
9	C₂×#10596929	C₂²	1	4	yes	—
10	M₁₆×D₃₇	—	2	8	no	—
11	C₂×C₈×D₃₇	C₂²	1	4	yes	—
12	#11924493	C₄	1	4	yes	—
13	#11924493	C₂²	1	4	yes	—
14	C₂×#10596936	C₄	1	2	yes	—
15	C₂×#10596938	C₄	1	2	yes	—
16	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
17	C₄⋊#64436	C₂²	1	4	yes	complemented
18	#11924465	C₂²	1	8	yes	complemented
19	C₂×M₁₆×C₃₇	C₂²	1	2	yes	—
20	C₂×C₄²×C₃₇	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₃₇	C₂²	1	4	yes	complemented
22	C₂³×D₃₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₃₇	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
25	C₂²×Dic₃₇	C₂×C₄	1	2	yes	—
26	C₂²×Dic₃₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₄×D₃₇	C₂×C₄	1	2	yes	—
28	C₂×C₄×D₃₇	D₄	1	4	yes	—
29	C₂×C₄×D₃₇	Q₈	1	4	yes	—
30	C₂×C₄×D₃₇	C₂³	1	2	yes	—
31	M₁₆×C₃₇	—	2	4	no	—
32	C₂×C₈×C₃₇	C₂³	1	4	yes	—
33	C₄²×C₃₇	C₂×C₄	1	2	yes	—
34	C₄²×C₃₇	C₂³	1	2	yes	—
35	#10596938	C₂×C₄	1	8	yes	—
36	#10596936	C₂×C₄	1	8	yes	—
37	C₄×Dic₃₇	C₂×C₄	1	2	yes	—
38	C₄×Dic₃₇	C₂³	1	2	yes	—
39	#10596934	—	2	4	no	—
40	C₂×#64436	C₂³	1	4	yes	—
41	#10596929	—	2	8	no	—
42	C₈×D₃₇	—	2	8	no	—
43	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
44	C₂²×D₃₇	—	4	2	no	—
45	C₂²×D₃₇	C₄⋊C₄	1	4	yes	—
46	C₂²×D₃₇	C₂²×C₄	1	2	yes	—
47	C₂×C₄×C₃₇	—	4	2	no	—
48	C₂×C₄×C₃₇	C₂²×C₄	1	4	yes	—
49	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₃₇	C₂×Q₈	1	2	yes	—
52	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
53	C₂×Dic₃₇	—	4	2	no	—
54	C₂×Dic₃₇	C₄⋊C₄	1	4	yes	—
55	C₂×Dic₃₇	C₂²×C₄	1	4	yes	—
56	C₂×Dic₃₇	C₂²×C₄	1	2	yes	—
57	C₄×D₃₇	C₄⋊C₄	1	8	yes	—
58	C₄×D₃₇	C₂×D₄	1	4	yes	—
59	C₄×D₃₇	C₂×Q₈	1	4	yes	—
60	C₈×C₃₇	—	2	4	no	—
61	#64436	—	2	4	no	—
62	C₂²×C₃₇	—	2	2	no	—
63	C₂²×C₃₇	C₂×C₄⋊C₄	1	2	yes	—
64	C₂²×C₃₇	C₂³×C₄	1	1	yes	characteristic
65	C₂×D₃₇	—	2	4	no	—
66	C₂×D₃₇	—	4	4	no	—
67	C₂×D₃₇	C₂×C₄⋊C₄	1	4	yes	—
68	C₄×C₃₇	—	2	4	no	—
69	C₄×C₃₇	C₂×C₄⋊C₄	1	2	yes	—
70	C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
71	C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
72	Dic₃₇	—	2	4	no	—
73	Dic₃₇	C₂×C₄⋊C₄	1	4	yes	—
74	#2905	—	37	1	no	maximal, Sylow
75	C₂×C₃₇	—	2	2	no	—
76	C₂×C₃₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₃₇	#568	1	2	yes	—
78	D₃₇	—	2	4	no	—
79	C₂²×M₁₆	—	37	2	no	—
80	C₂×#117	—	37	1	no	—
81	#424	—	37	8	no	—
82	#423	—	37	2	no	—
83	#423	—	37	1	no	—
84	#423	D₃₇	1	1	yes	characteristic, complemented
85	C₃₇	#2905	1	1	yes	characteristic, complemented, Sylow
86	C₂³×C₄	—	37	1	no	—
87	C₂×M₁₆	—	37	2	no	—
88	C₂×M₁₆	—	37	4	no	—
89	C₂×M₁₆	—	74	8	no	—
90	C₂×M₁₆	C₂×D₃₇	1	2	yes	—
91	C₂²×C₈	—	37	4	no	—
92	#117	—	37	4	no	—
93	#117	—	37	4	no	—
94	C₂×C₄⋊C₄	—	37	2	no	—
95	C₂×C₂²⋊C₄	—	37	2	no	—
96	C₂×C₄²	—	37	1	no	—
97	C₂×C₄²	C₂×D₃₇	1	1	yes	characteristic
98	C₄⋊C₈	—	37	8	no	—
99	C₄⋊C₈	—	37	4	no	—
100	C₄⋊C₈	C₂×D₃₇	1	4	yes	complemented
101	C₂⁴	—	37	1	no	—
102	C₂²×C₄	—	37	1	no	—
103	C₂²×C₄	—	37	2	no	—
104	C₂²×C₄	—	37	4	no	—
105	C₂²×C₄	—	37	4	no	—
106	C₂²×C₄	—	37	2	no	—
107	C₂²×C₄	—	37	2	no	—
108	C₂²×C₄	C₄×D₃₇	1	2	yes	—
109	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
110	M₁₆	—	2	4	no	—
111	M₁₆	—	74	8	no	—
112	M₁₆	—	74	4	no	—
113	C₂×C₈	—	37	4	no	—
114	C₂×C₈	—	74	8	no	—
115	C₂×C₈	C₂²×D₃₇	1	4	yes	—
116	C₄⋊C₄	—	37	8	no	—
117	C₂²⋊C₄	—	37	8	no	—
118	C₄²	—	37	2	no	—
119	C₄²	—	37	2	no	—
120	C₄²	C₄×D₃₇	1	2	yes	—
121	C₄²	C₂²×D₃₇	1	2	yes	—
122	C₂³	—	37	2	no	—
123	C₂³	—	37	4	no	—
124	C₂³	—	148	2	no	—
125	C₂³	C₂×C₄×D₃₇	1	1	yes	characteristic
126	C₂×C₄	—	4	2	no	—
127	C₂×C₄	—	37	4	no	—
128	C₂×C₄	—	37	2	no	—
129	C₂×C₄	—	37	8	no	—
130	C₂×C₄	—	37	4	no	—
131	C₂×C₄	—	37	4	no	—
132	C₂×C₄	—	37	4	no	—
133	C₂×C₄	—	148	2	no	—
134	C₂×C₄	C₂×C₄×D₃₇	1	4	yes	—
135	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
136	C₂×C₄	D₄×D₃₇	1	2	yes	—
137	C₂×C₄	Q₈×D₃₇	1	2	yes	—
138	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
139	C₈	—	2	4	no	—
140	C₈	—	74	4	no	—
141	C₂²	—	2	2	no	—
142	C₂²	—	37	4	no	—
143	C₂²	—	74	4	no	—
144	C₂²	—	148	4	no	—
145	C₂²	C₄⋊C₄×D₃₇	1	2	yes	—
146	C₂²	C₂²×C₄×D₃₇	1	1	yes	characteristic
147	C₄	—	2	4	no	—
148	C₄	—	37	4	no	—
149	C₄	—	74	4	no	—
150	C₄	C₄⋊C₄×D₃₇	1	2	yes	—
151	C₄	C₂×D₄×D₃₇	1	1	yes	characteristic
152	C₄	C₂×Q₈×D₃₇	1	1	yes	characteristic
153	C₂	—	2	2	no	—
154	C₂	—	74	4	no	—
155	C₂	C₂×C₄⋊C₄×D₃₇	1	1	yes	characteristic
156	C₂	#14558299	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×M₁₆×D₃₇

C₂

yes

maximal

C₂×#11924493

C₂

yes

maximal, characteristic

#14557558

C₂

yes

maximal, complemented

#14557557

C₂

yes

maximal, characteristic, complemented

#14557556

C₂

yes

maximal, complemented

C₃₇×#423

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₃₇

C₂²

yes

characteristic

C₂×#10596934

C₂²

yes

—

C₂×#10596929

C₂²

yes

—

M₁₆×D₃₇

—

C₂×C₈×D₃₇

C₂²

yes

—

#11924493

C₄

yes

—

#11924493

C₂²

yes

—

C₂×#10596936

C₄

yes

—

C₂×#10596938

C₄

yes

—

C₂×C₄×Dic₃₇

C₂²

yes

characteristic

C₄⋊#64436

C₂²

yes

complemented

#11924465

C₂²

yes

complemented

C₂×M₁₆×C₃₇

C₂²

yes

—

C₂×C₄²×C₃₇

C₂²

yes

characteristic

C₄⋊C₈×C₃₇

C₂²

yes

complemented

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

yes

—

C₂×C₄×D₃₇

D₄

yes

—

C₂×C₄×D₃₇

Q₈

yes

—

C₂×C₄×D₃₇

C₂³

yes

—

M₁₆×C₃₇

—

C₂×C₈×C₃₇

C₂³

yes

—

C₄²×C₃₇

C₂×C₄

yes

—

C₄²×C₃₇

C₂³

yes

—

#10596938

C₂×C₄

yes

—

#10596936

C₂×C₄

yes

—

C₄×Dic₃₇

C₂×C₄

yes

—

C₄×Dic₃₇

C₂³

yes

—

#10596934

—

C₂×#64436

C₂³

yes

—

#10596929

—

C₈×D₃₇

—

C₂³×C₃₇

C₂²×C₄

yes

characteristic

C₂²×D₃₇

—

C₂²×D₃₇

C₄⋊C₄

yes

—

C₂²×D₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×Q₈

yes

—

C₂×C₄×C₃₇

C₂⁴

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

C₄⋊C₄

yes

—

C₂×Dic₃₇

C₂²×C₄

yes

—

C₂×Dic₃₇

C₂²×C₄

yes

—

C₄×D₃₇

C₄⋊C₄

yes

—

C₄×D₃₇

C₂×D₄

yes

—

C₄×D₃₇

C₂×Q₈

yes

—

C₈×C₃₇

—

#64436

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂×C₄⋊C₄

yes

—

C₂²×C₃₇

C₂³×C₄

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

C₂×C₄⋊C₄

yes

—

C₄×C₃₇

—

C₄×C₃₇

C₂×C₄⋊C₄

yes

—

C₄×C₃₇

C₂²×D₄

yes

characteristic

C₄×C₃₇

C₂²×Q₈

yes

characteristic

Dic₃₇

—

Dic₃₇

C₂×C₄⋊C₄

yes

—

#2905

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

C₂²×C₄⋊C₄

yes

characteristic

C₂×C₃₇

#568

yes

—

D₃₇

—

C₂²×M₁₆

—

—

—

—

—

D₃₇

yes

characteristic, complemented

C₃₇

#2905

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₃₇

yes

—

C₂²×C₈

—

#117

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂×C₄²

C₂×D₃₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

100

C₄⋊C₈

C₂×D₃₇

yes

complemented

101

C₂⁴

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

C₄×D₃₇

yes

—

109

C₂²×C₄

C₂²×D₃₇

yes

characteristic

110

M₁₆

—

111

M₁₆

—

112

M₁₆

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

C₂²×D₃₇

yes

—

116

C₄⋊C₄

—

117

C₂²⋊C₄

—

118

C₄²

—

119

C₄²

—

120

C₄²

C₄×D₃₇

yes

—

121

C₄²

C₂²×D₃₇

yes

—

122

C₂³

—

123

C₂³

—

124

C₂³

—

148

—

125

C₂³

C₂×C₄×D₃₇

yes

characteristic

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

148

—

134

C₂×C₄

C₂×C₄×D₃₇

yes

—

135

C₂×C₄

C₂×C₄×D₃₇

yes

characteristic

136

C₂×C₄

D₄×D₃₇

yes

—

137

C₂×C₄

Q₈×D₃₇

yes

—

138

C₂×C₄

C₂³×D₃₇

yes

characteristic

139

C₈

—

140

C₈

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

—

148

—

145

C₂²

C₄⋊C₄×D₃₇

yes

—

146

C₂²

C₂²×C₄×D₃₇

yes

characteristic

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

C₄⋊C₄×D₃₇

yes

—

151

C₄

C₂×D₄×D₃₇

yes

characteristic

152

C₄

C₂×Q₈×D₃₇

yes

characteristic

153

C₂

—

154

C₂

—

155

C₂

C₂×C₄⋊C₄×D₃₇

yes

characteristic

156

C₂

#14558299

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₃₇
commutator, socle	C₂²×C₃₇	C₂³×C₄
Fitting	C₃₇×#423	C₂
prime radical	C₂³×D₃₇	C₂×C₄
cosocle	C₂×C₄×C₃₇	C₂⁴
hypercenter	#423	D₃₇
hypocenter	C₃₇	#2905
LCS	G ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂×C₄ ◁ #423

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₃₇

commutator, socle

C₂²×C₃₇

C₂³×C₄

Fitting

C₃₇×#423

C₂

prime radical

C₂³×D₃₇

C₂×C₄

cosocle

C₂×C₄×C₃₇

C₂⁴

D₃₇

C₃₇

G ▷ C₂²×C₃₇ ▷ C₃₇

C₁ ◁ C₂×C₄ ◁ #423

N	H ≅ G/N	type
3. #14557558	C₂	semidirect
4. #14557557	C₂	semidirect
5. #14557556	C₂	semidirect
6. C₃₇×#423	C₂	semidirect
17. C₄⋊#64436	C₂²	semidirect
18. #11924465	C₂²	semidirect
21. C₄⋊C₈×C₃₇	C₂²	semidirect
84. #423	D₃₇	semidirect
85. C₃₇	#2905	semidirect
100. C₄⋊C₈	C₂×D₃₇	semidirect

H ≅ G/N

type

3. #14557558

C₂

semidirect

4. #14557557

C₂

semidirect

5. #14557556

C₂

semidirect

6. C₃₇×#423

C₂

semidirect

17. C₄⋊#64436

C₂²

semidirect

18. #11924465

C₂²

semidirect

21. C₄⋊C₈×C₃₇

C₂²

semidirect

84. #423

D₃₇

semidirect

85. C₃₇

#2905

semidirect

100. C₄⋊C₈

C₂×D₃₇

semidirect

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

subgroup	index
C₂×M₁₆×D₃₇	2
C₂×#11924493	2
#14557558	2
#14557557	2
#14557556	2
C₃₇×#423	2
#2905	37

subgroup

index

C₂×M₁₆×D₃₇

C₃₇×#423

Subgroups — #18003158

Conjugacy classes of subgroups

Complements

Maximal subgroups