Group #18010739 — subgroups

Subgroups — #18010739

Order 4736.

292 conjugacy classes of subgroups; only the 76 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 76 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	#14558536	C₂	1	1	yes	maximal, characteristic
2	C₂×#11924597	C₂	1	4	yes	maximal
3	C₄×#10596957	C₂	1	1	yes	maximal, characteristic
4	C₂×#11924535	C₂	1	1	yes	maximal, characteristic
5	C₂×#10596957	C₂²	1	2	yes	—
6	C₂×#10596957	C₂²	1	2	yes	—
7	#11924603	—	2	1	no	—
8	C₄⋊#64438	—	2	1	no	—
9	#11924597	—	2	8	no	—
10	#11924591	—	2	1	no	—
11	C₄×#64438	—	2	1	no	—
12	C₂²×Dic₇₄	C₂²	1	2	yes	—
13	#11924535	—	2	1	no	—
14	#11924535	—	2	1	no	—
15	#11924535	—	2	1	no	—
16	#11924535	—	2	1	no	—
17	C₂×C₄⋊Dic₃₇	C₄	1	1	yes	characteristic
18	C₂×#10596936	C₄	1	2	yes	—
19	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
20	C₂×C₄⋊C₄×C₃₇	C₄	1	1	yes	characteristic
21	C₂²×C₄×C₃₇	C₂×C₄	1	2	yes	—
22	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×Dic₃₇	C₂×C₄	1	2	yes	—
24	C₂²×Dic₃₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
26	C₂×Dic₇₄	—	2	2	no	—
27	C₂×Dic₇₄	—	2	2	no	—
28	C₂×Dic₇₄	—	2	4	no	—
29	C₂×Dic₇₄	—	2	4	no	—
30	#10596957	—	2	2	no	—
31	#10596957	—	2	4	no	—
32	#10596957	D₄	1	4	yes	—
33	C₂×#64438	—	2	2	no	—
34	C₂×#64438	—	2	2	no	—
35	C₄⋊C₄×C₃₇	—	2	1	no	—
36	C₄⋊C₄×C₃₇	—	2	1	no	—
37	C₄⋊Dic₃₇	—	2	1	no	—
38	C₄⋊Dic₃₇	—	2	1	no	—
39	#10596936	—	2	2	no	—
40	#10596936	—	2	2	no	—
41	C₄×Dic₃₇	D₄	1	1	yes	characteristic
42	C₄×Dic₃₇	D₄	1	1	yes	characteristic
43	C₄×Dic₃₇	D₄	1	1	yes	characteristic
44	C₄×Dic₃₇	D₄	1	1	yes	characteristic
45	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₃₇	—	2	2	no	—
47	C₂×C₄×C₃₇	—	2	2	no	—
48	C₂×C₄×C₃₇	—	2	4	no	—
49	C₂×C₄×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×Dic₃₇	—	2	2	no	—
52	C₂×Dic₃₇	—	2	2	no	—
53	C₂×Dic₃₇	—	2	4	no	—
54	C₂×Dic₃₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×Dic₃₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
57	C₂×Dic₃₇	C₂×D₄	1	2	yes	—
58	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
59	C₂×Dic₃₇	#41	1	2	yes	—
60	Dic₇₄	—	4	4	no	—
61	Dic₇₄	—	4	2	no	—
62	Dic₇₄	—	4	2	no	—
63	#64438	—	2	4	no	—
64	#64438	—	4	2	no	—
65	C₂²×C₃₇	—	2	1	no	—
66	C₂²×C₃₇	—	2	1	no	—
67	C₂²×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₃₇	C₄×D₄	1	2	yes	—
69	C₄×C₃₇	—	2	1	no	—
70	C₄×C₃₇	—	4	2	no	—
71	C₄×C₃₇	#101	1	2	yes	complemented
72	Dic₃₇	—	2	1	no	—
73	Dic₃₇	—	4	2	no	—
74	Dic₃₇	#101	1	2	yes	complemented
75	Dic₃₇	#121	1	2	yes	—
76	Dic₃₇	#124	1	1	yes	characteristic
77	Dic₃₇	#127	1	1	yes	characteristic
78	#1966	—	37	1	no	maximal, Sylow
79	C₂×C₃₇	—	2	2	no	—
80	C₂×C₃₇	#390	1	1	yes	characteristic
81	C₂×C₃₇	C₂×#101	1	1	yes	characteristic
82	C₂×C₃₇	C₂×#101	1	1	yes	characteristic
83	C₂×C₄⋊Q₈	—	37	1	no	—
84	#423	—	37	1	no	—
85	C₂×#101	—	37	4	no	—
86	C₄×M₁₆	—	37	1	no	—
87	C₃₇	#1966	1	1	yes	characteristic, complemented, Sylow
88	C₂²×Q₈	—	37	2	no	—
89	C₂×M₁₆	—	37	2	no	—
90	C₂×M₁₆	—	37	2	no	—
91	C₄⋊Q₈	—	74	1	no	—
92	C₄⋊Q₈	—	74	1	no	—
93	C₄⋊Q₈	—	74	1	no	—
94	C₄⋊Q₈	—	74	1	no	—
95	C₂×C₄⋊C₄	—	37	2	no	—
96	C₂×C₄⋊C₄	—	37	1	no	—
97	C₂×C₄⋊C₄	#3866	1	1	yes	characteristic
98	C₂×C₄²	—	37	1	no	—
99	C₄⋊C₈	—	74	1	no	—
100	C₄⋊C₈	—	74	1	no	—
101	#101	—	74	8	no	—
102	#97	—	74	1	no	—
103	C₄×C₈	—	74	1	no	—
104	C₂×Q₈	—	74	4	no	—
105	C₂×Q₈	—	74	2	no	—
106	C₂×Q₈	—	74	2	no	—
107	C₂×Q₈	—	74	4	no	—
108	C₂²×C₄	—	37	1	no	—
109	C₂²×C₄	—	37	1	no	—
110	C₂²×C₄	—	37	2	no	—
111	C₂²×C₄	C₂×#3866	1	2	yes	—
112	C₂²×C₄	C₂×#3866	1	1	yes	characteristic
113	M₁₆	—	37	4	no	—
114	M₁₆	—	74	2	no	—
115	M₁₆	—	74	4	no	—
116	C₂×C₈	—	74	2	no	—
117	C₂×C₈	—	74	2	no	—
118	C₄⋊C₄	—	2	1	no	—
119	C₄⋊C₄	—	2	1	no	—
120	C₄⋊C₄	—	74	2	no	—
121	C₄⋊C₄	—	74	2	no	—
122	C₄⋊C₄	—	74	1	no	—
123	C₄⋊C₄	—	74	1	no	—
124	C₄²	—	37	1	no	—
125	C₄²	—	37	1	no	—
126	C₄²	—	37	1	no	—
127	C₄²	—	37	1	no	—
128	C₂³	C₂²×#3866	1	1	yes	characteristic
129	Q₈	—	148	4	no	—
130	Q₈	—	148	2	no	—
131	Q₈	—	148	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	4	no	—
135	C₂×C₄	—	37	1	no	—
136	C₂×C₄	—	37	1	no	—
137	C₂×C₄	—	37	1	no	—
138	C₂×C₄	—	37	2	no	—
139	C₂×C₄	—	37	2	no	—
140	C₂×C₄	—	37	1	no	—
141	C₂×C₄	—	74	2	no	—
142	C₂×C₄	—	74	2	no	—
143	C₂×C₄	—	74	4	no	—
144	C₂×C₄	C₂²⋊#3866	1	1	yes	characteristic
145	C₂×C₄	C₂²⋊#3866	1	1	yes	characteristic
146	C₈	—	74	4	no	—
147	C₈	—	148	2	no	—
148	C₂²	—	2	1	no	—
149	C₂²	—	2	1	no	—
150	C₂²	C₂×C₂²⋊#3866	1	1	yes	characteristic
151	C₂²	D₄×#3866	1	2	yes	—
152	C₄	—	2	1	no	—
153	C₄	—	4	2	no	—
154	C₄	—	37	2	no	—
155	C₄	—	37	1	no	—
156	C₄	—	37	1	no	—
157	C₄	—	37	2	no	—
158	C₄	—	74	1	no	—
159	C₄	—	148	2	no	—
160	C₄	#11924597	1	2	yes	complemented
161	C₂	—	2	2	no	—
162	C₂	#14558485	1	1	yes	characteristic
163	C₂	C₂×#11924597	1	1	yes	characteristic
164	C₂	D₃₇⋊#101	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14558536

C₂

yes

maximal, characteristic

C₂×#11924597

C₂

yes

maximal

C₄×#10596957

C₂

yes

maximal, characteristic

C₂×#11924535

C₂

yes

maximal, characteristic

C₂×#10596957

C₂²

yes

—

C₂×#10596957

C₂²

yes

—

—

—

—

—

—

C₂²×Dic₇₄

C₂²

yes

—

—

—

—

—

C₂×C₄⋊Dic₃₇

C₄

yes

characteristic

C₂×#10596936

C₄

yes

—

C₂×C₄×Dic₃₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₃₇

C₄

yes

characteristic

C₂²×C₄×C₃₇

C₂×C₄

yes

—

C₂²×C₄×C₃₇

C₂×C₄

yes

characteristic

C₂²×Dic₃₇

C₂×C₄

yes

—

C₂²×Dic₃₇

C₂×C₄

yes

characteristic

C₂²×Dic₃₇

C₂³

yes

characteristic

C₂×Dic₇₄

—

C₂×Dic₇₄

—

C₂×Dic₇₄

—

C₂×Dic₇₄

—

#10596957

—

#10596957

—

#10596957

D₄

yes

—

C₂×#64438

—

C₂×#64438

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

#10596936

—

#10596936

—

C₄×Dic₃₇

D₄

yes

characteristic

C₄×Dic₃₇

D₄

yes

characteristic

C₄×Dic₃₇

D₄

yes

characteristic

C₄×Dic₃₇

D₄

yes

characteristic

C₂³×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₃₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₃₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₃₇

C₂×D₄

yes

characteristic

C₂×Dic₃₇

C₂×D₄

yes

—

C₂×Dic₃₇

C₂×D₄

yes

characteristic

C₂×Dic₃₇

#41

yes

—

Dic₇₄

—

Dic₇₄

—

Dic₇₄

—

#64438

—

#64438

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₃₇

C₄×D₄

yes

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

#101

yes

complemented

Dic₃₇

—

Dic₃₇

—

Dic₃₇

#101

yes

complemented

Dic₃₇

#121

yes

—

Dic₃₇

yes

Dic₃₇

yes

—

C₂×C₃₇

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₄⋊Q₈

—

#423

—

C₂×#101

—

C₄×M₁₆

—

C₃₇

#1966

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×M₁₆

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

#3866

yes

characteristic

C₂×C₄²

—

C₄⋊C₈

—

100

C₄⋊C₈

—

101

#101

—

102

#97

—

103

C₄×C₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

C₂×#3866

yes

—

112

C₂²×C₄

C₂×#3866

yes

characteristic

113

M₁₆

—

114

M₁₆

—

115

M₁₆

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄²

—

125

C₄²

—

126

C₄²

—

127

C₄²

—

128

C₂³

C₂²×#3866

yes

characteristic

129

Q₈

—

148

—

130

Q₈

—

148

—

131

Q₈

—

148

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

C₂²⋊#3866

yes

characteristic

145

C₂×C₄

C₂²⋊#3866

yes

characteristic

146

C₈

—

147

C₈

—

148

—

148

C₂²

—

149

C₂²

—

150

C₂²

C₂×C₂²⋊#3866

yes

characteristic

151

C₂²

D₄×#3866

yes

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

148

—

160

C₄

#11924597

yes

complemented

161

C₂

—

162

C₂

#14558485

yes

characteristic

163

C₂

C₂×#11924597

yes

characteristic

164

C₂

D₃₇⋊#101

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#3866
commutator, prime radical	C₂³×C₃₇	C₂²×C₄
Frattini	C₂³	C₂²×#3866
Fitting	C₂×C₄⋊C₄×C₃₇	C₄
socle	C₂²×C₃₇	C₂×C₂²⋊C₄
cosocle	C₂²×Dic₃₇	C₂³
hypercenter	C₂×C₄⋊C₄	#3866
hypocenter	C₃₇	#1966
LCS	G ▷ C₂³×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

subgroup

G/H

center

C₂²

C₂×C₂²⋊#3866

commutator, prime radical

C₂³×C₃₇

C₂²×C₄

Frattini

C₂³

C₂²×#3866

Fitting

C₂×C₄⋊C₄×C₃₇

C₄

socle

C₂²×C₃₇

C₂×C₂²⋊C₄

cosocle

C₂²×Dic₃₇

C₂³

C₂×C₄⋊C₄

C₃₇

G ▷ C₂³×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
71. C₄×C₃₇	#101	semidirect
74. Dic₃₇	#101	semidirect
87. C₃₇	#1966	semidirect
160. C₄	#11924597	semidirect

H ≅ G/N

type

71. C₄×C₃₇

#101

semidirect

74. Dic₃₇

#101

semidirect

87. C₃₇

#1966

semidirect

160. C₄

#11924597

semidirect

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

subgroup	index
#14558536	2
C₂×#11924597	2
C₄×#10596957	2
C₂×#11924535	2
#1966	37

subgroup

index

Subgroups — #18010739

Conjugacy classes of subgroups

Complements

Maximal subgroups