Group #18010789 — subgroups

Subgroups — #18010789

Order 4736.

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

normal subgroups only

This group has 260 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	C₄×#10596957	C₂	1	1	yes	maximal, characteristic
3	#14558413	C₂	1	4	yes	maximal
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	#11924591	—	2	1	no	—
10	C₄×#64438	—	2	1	no	—
11	C₂²×Dic₇₄	C₄	1	2	yes	—
12	#11924535	—	2	1	no	—
13	#11924535	—	2	1	no	—
14	#11924535	—	2	1	no	—
15	#11924535	—	2	1	no	—
16	C₂×C₄⋊Dic₃₇	C₄	1	1	yes	characteristic
17	C₂×#10596936	C₂²	1	2	yes	—
18	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₃₇	C₄	1	1	yes	characteristic
20	C₂²×C₄×C₃₇	C₂×C₄	1	2	yes	—
21	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×Dic₃₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×Dic₃₇	C₂×C₄	1	2	yes	—
24	C₂²×Dic₃₇	C₂³	1	1	yes	characteristic
25	C₂×Dic₇₄	—	2	2	no	—
26	C₂×Dic₇₄	—	2	2	no	—
27	C₂×Dic₇₄	—	4	4	no	—
28	#10596957	—	2	2	no	—
29	#10596957	—	2	4	no	—
30	#10596957	Q₈	1	4	yes	complemented
31	C₂×#64438	—	2	2	no	—
32	C₂×#64438	—	2	2	no	—
33	C₄⋊C₄×C₃₇	—	2	1	no	—
34	C₄⋊C₄×C₃₇	—	2	1	no	—
35	C₄⋊Dic₃₇	—	2	1	no	—
36	C₄⋊Dic₃₇	—	2	1	no	—
37	#10596936	—	2	2	no	—
38	#10596936	—	2	2	no	—
39	C₄×Dic₃₇	D₄	1	1	yes	characteristic
40	C₄×Dic₃₇	D₄	1	1	yes	characteristic
41	C₄×Dic₃₇	D₄	1	1	yes	characteristic
42	C₄×Dic₃₇	D₄	1	1	yes	characteristic
43	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₃₇	—	2	2	no	—
45	C₂×C₄×C₃₇	—	2	2	no	—
46	C₂×C₄×C₃₇	—	4	2	no	—
47	C₂×C₄×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×Dic₃₇	—	2	2	no	—
50	C₂×Dic₃₇	—	2	2	no	—
51	C₂×Dic₃₇	—	4	2	no	—
52	C₂×Dic₃₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×Dic₃₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
55	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
56	C₂×Dic₃₇	C₂×Q₈	1	2	yes	—
57	C₂×Dic₃₇	#41	1	2	yes	—
58	Dic₇₄	—	4	4	no	—
59	Dic₇₄	—	4	4	no	—
60	#64438	—	2	4	no	—
61	#64438	—	4	2	no	—
62	C₂²×C₃₇	—	2	1	no	—
63	C₂²×C₃₇	—	2	1	no	—
64	C₂²×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₃₇	C₄×Q₈	1	2	yes	—
66	C₄×C₃₇	—	2	1	no	—
67	C₄×C₃₇	—	4	2	no	—
68	C₄×C₃₇	#101	1	2	yes	—
69	Dic₃₇	—	2	1	no	—
70	Dic₃₇	—	4	2	no	—
71	Dic₃₇	#100	1	2	yes	—
72	Dic₃₇	C₂²⋊Q₈	1	2	yes	—
73	Dic₃₇	#124	1	1	yes	characteristic
74	Dic₃₇	C₄⋊Q₈	1	1	yes	characteristic
75	#1997	—	37	1	no	maximal, Sylow
76	C₂×C₃₇	—	2	2	no	—
77	C₂×C₃₇	#391	1	1	yes	characteristic
78	C₂×C₃₇	C₂×#100	1	1	yes	characteristic
79	C₂×C₃₇	C₂×#101	1	1	yes	characteristic
80	C₂×C₄⋊Q₈	—	37	1	no	—
81	#423	—	37	1	no	—
82	C₄×M₁₆	—	37	1	no	—
83	#343	—	37	4	no	—
84	C₃₇	#1997	1	1	yes	characteristic, complemented, Sylow
85	C₂²×Q₈	—	37	2	no	—
86	C₂×M₁₆	—	37	2	no	—
87	C₂×M₁₆	—	37	2	no	—
88	C₄⋊Q₈	—	74	1	no	—
89	C₄⋊Q₈	—	74	1	no	—
90	C₄⋊Q₈	—	74	1	no	—
91	C₄⋊Q₈	—	74	1	no	—
92	C₂×C₄⋊C₄	—	37	2	no	—
93	C₂×C₄⋊C₄	—	37	1	no	—
94	C₂×C₄⋊C₄	#3866	1	1	yes	characteristic
95	C₂×C₄²	—	37	1	no	—
96	C₄⋊C₈	—	74	1	no	—
97	C₄⋊C₈	—	74	1	no	—
98	#97	—	74	1	no	—
99	C₄×C₈	—	74	1	no	—
100	C₂×Q₈	—	74	2	no	—
101	C₂×Q₈	—	74	2	no	—
102	C₂×Q₈	—	148	4	no	—
103	C₂²×C₄	—	37	1	no	—
104	C₂²×C₄	—	37	1	no	—
105	C₂²×C₄	—	37	2	no	—
106	C₂²×C₄	C₂×#3866	1	2	yes	—
107	C₂²×C₄	C₂×#3866	1	1	yes	characteristic
108	M₁₆	—	37	4	no	—
109	M₁₆	—	74	2	no	—
110	M₁₆	—	74	4	no	—
111	C₂×C₈	—	74	2	no	—
112	C₂×C₈	—	74	2	no	—
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	—	2	1	no	—
115	C₄⋊C₄	—	74	2	no	—
116	C₄⋊C₄	—	74	2	no	—
117	C₄⋊C₄	—	74	1	no	—
118	C₄⋊C₄	—	74	1	no	—
119	C₄²	—	37	1	no	—
120	C₄²	—	37	1	no	—
121	C₄²	—	37	1	no	—
122	C₄²	—	37	1	no	—
123	C₂³	C₂²×#3866	1	1	yes	characteristic
124	Q₈	—	148	4	no	—
125	Q₈	—	148	4	no	—
126	C₂×C₄	—	2	2	no	—
127	C₂×C₄	—	2	2	no	—
128	C₂×C₄	—	4	2	no	—
129	C₂×C₄	—	37	1	no	—
130	C₂×C₄	—	37	1	no	—
131	C₂×C₄	—	37	1	no	—
132	C₂×C₄	—	37	2	no	—
133	C₂×C₄	—	37	2	no	—
134	C₂×C₄	—	37	1	no	—
135	C₂×C₄	—	74	2	no	—
136	C₂×C₄	—	74	2	no	—
137	C₂×C₄	—	148	2	no	—
138	C₂×C₄	C₂²⋊#3866	1	1	yes	characteristic
139	C₂×C₄	C₂²⋊#3866	1	1	yes	characteristic
140	C₈	—	74	4	no	—
141	C₈	—	148	2	no	—
142	C₂²	—	2	1	no	—
143	C₂²	—	2	1	no	—
144	C₂²	C₂×C₂²⋊#3866	1	1	yes	characteristic
145	C₂²	Q₈×#3866	1	2	yes	—
146	C₄	—	2	1	no	—
147	C₄	—	4	2	no	—
148	C₄	—	37	2	no	—
149	C₄	—	37	1	no	—
150	C₄	—	37	1	no	—
151	C₄	—	37	2	no	—
152	C₄	—	74	1	no	—
153	C₄	—	148	2	no	—
154	C₄	#11924597	1	2	yes	—
155	C₂	—	2	2	no	—
156	C₂	(C₂×#3866)⋊Q₈	1	1	yes	characteristic
157	C₂	D₃₇⋊#100	1	1	yes	characteristic
158	C₂	C₂×#11924597	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14558536

C₂

yes

maximal, characteristic

C₄×#10596957

C₂

yes

maximal, characteristic

#14558413

C₂

yes

maximal

C₂×#11924535

C₂

yes

maximal, characteristic

C₂×#10596957

C₂²

yes

—

C₂×#10596957

C₂²

yes

—

—

—

—

—

C₂²×Dic₇₄

C₄

yes

—

—

—

—

—

C₂×C₄⋊Dic₃₇

C₄

yes

characteristic

C₂×#10596936

C₂²

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

characteristic

C₂²×Dic₃₇

C₂×C₄

yes

—

C₂²×Dic₃₇

C₂³

yes

characteristic

C₂×Dic₇₄

—

C₂×Dic₇₄

—

C₂×Dic₇₄

—

#10596957

—

#10596957

—

#10596957

Q₈

yes

complemented

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

characteristic

C₂×Dic₃₇

C₂×Q₈

yes

—

C₂×Dic₃₇

#41

yes

—

Dic₇₄

—

Dic₇₄

—

#64438

—

#64438

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₃₇

C₄×Q₈

yes

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

#101

yes

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

#100

yes

—

Dic₃₇

C₂²⋊Q₈

yes

—

Dic₃₇

#124

yes

characteristic

Dic₃₇

C₄⋊Q₈

yes

characteristic

#1997

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₄⋊Q₈

—

#423

—

C₄×M₁₆

—

#343

—

C₃₇

#1997

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

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×Q₈

—

148

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

C₂×#3866

yes

—

107

C₂²×C₄

C₂×#3866

yes

characteristic

108

M₁₆

—

109

M₁₆

—

110

M₁₆

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄²

—

120

C₄²

—

121

C₄²

—

122

C₄²

—

123

C₂³

C₂²×#3866

yes

characteristic

124

Q₈

—

148

—

125

Q₈

—

148

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

148

—

138

C₂×C₄

C₂²⋊#3866

yes

characteristic

139

C₂×C₄

C₂²⋊#3866

yes

characteristic

140

C₈

—

141

C₈

—

148

—

142

C₂²

—

143

C₂²

—

144

C₂²

C₂×C₂²⋊#3866

yes

characteristic

145

C₂²

Q₈×#3866

yes

—

146

C₄

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

148

—

154

C₄

#11924597

yes

—

155

C₂

—

156

C₂

(C₂×#3866)⋊Q₈

yes

characteristic

157

C₂

D₃₇⋊#100

yes

characteristic

158

C₂

C₂×#11924597

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	#1997
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
30. #10596957	Q₈	semidirect
84. C₃₇	#1997	semidirect

H ≅ G/N

type

30. #10596957

Q₈

semidirect

84. C₃₇

#1997

semidirect

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

subgroup	index
#14558536	2
C₄×#10596957	2
#14558413	2
C₂×#11924535	2
#1997	37

subgroup

index

Subgroups — #18010789

Conjugacy classes of subgroups

Complements

Maximal subgroups