Group #18010060 — subgroups

Subgroups — #18010060

Order 4736.

212 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 62 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	#14558554	C₂	1	1	yes	maximal, characteristic
2	C₄×#10596957	C₂	1	1	yes	maximal, characteristic
3	#14558390	C₂	1	4	yes	maximal, complemented
4	C₂×#11924527	C₂	1	1	yes	maximal, characteristic
5	C₂×#10596957	—	2	1	no	—
6	C₄⋊#64438	—	2	1	no	—
7	C₂²⋊#64438	—	2	1	no	—
8	#11924591	C₂²	1	2	yes	—
9	#11924591	C₂²	1	2	yes	—
10	C₄×#64438	—	2	1	no	—
11	#11924527	—	2	1	no	—
12	#11924527	—	2	1	no	—
13	#11924527	C₄	1	2	yes	—
14	#11924527	C₂²	1	2	yes	—
15	C₂×C₄⋊Dic₃₇	—	2	1	no	—
16	C₂×C₄⋊Dic₃₇	C₄	1	1	yes	characteristic
17	C₂×#10596936	—	2	1	no	—
18	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₃₇	C₄	1	1	yes	characteristic
20	C₂²×C₄×C₃₇	—	2	1	no	—
21	C₂²×C₄×C₃₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×Dic₃₇	—	2	1	no	—
23	C₂²×Dic₃₇	D₄	1	1	yes	characteristic
24	C₂²×Dic₃₇	D₄	1	1	yes	characteristic
25	#10596957	—	2	4	no	—
26	C₂×#64438	—	2	2	no	—
27	C₂×#64438	—	2	2	no	—
28	C₄⋊C₄×C₃₇	—	2	1	no	—
29	C₄⋊C₄×C₃₇	C₂×C₄	1	2	yes	—
30	C₄⋊Dic₃₇	—	2	1	no	—
31	C₄⋊Dic₃₇	—	2	2	no	—
32	C₄⋊Dic₃₇	—	2	2	no	—
33	C₄⋊Dic₃₇	C₂×C₄	1	2	yes	—
34	#10596936	—	2	2	no	—
35	#10596936	—	2	2	no	—
36	C₄×Dic₃₇	C₂×C₄	1	1	yes	characteristic
37	C₄×Dic₃₇	D₄	1	1	yes	characteristic
38	C₄×Dic₃₇	D₄	1	1	yes	characteristic
39	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
40	C₂³×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₃₇	—	2	2	no	—
42	C₂×C₄×C₃₇	—	2	2	no	—
43	C₂×C₄×C₃₇	—	4	2	no	—
44	C₂×C₄×C₃₇	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
46	C₂×Dic₃₇	—	2	2	no	—
47	C₂×Dic₃₇	—	2	2	no	—
48	C₂×Dic₃₇	—	4	1	no	—
49	C₂×Dic₃₇	—	4	2	no	—
50	C₂×Dic₃₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×Dic₃₇	C₂²⋊C₄	1	1	yes	characteristic
52	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
53	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
54	#64438	—	2	4	no	—
55	#64438	—	4	2	no	—
56	C₂²×C₃₇	—	2	1	no	—
57	C₂²×C₃₇	—	2	1	no	—
58	C₂²×C₃₇	#104	1	2	yes	—
59	C₂²×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₄×C₃₇	—	2	1	no	—
61	C₄×C₃₇	—	4	2	no	—
62	C₄×C₃₇	#101	1	2	yes	—
63	Dic₃₇	—	2	1	no	—
64	Dic₃₇	—	2	1	no	—
65	Dic₃₇	—	2	1	no	—
66	Dic₃₇	—	2	1	no	—
67	Dic₃₇	—	4	2	no	—
68	#1528	—	37	1	no	maximal, Sylow
69	C₂×C₃₇	—	2	2	no	—
70	C₂×C₃₇	C₂×#101	1	1	yes	characteristic
71	C₂×C₃₇	C₂×#104	1	1	yes	characteristic
72	C₂×C₃₇	#421	1	1	yes	characteristic
73	C₂×#125	—	37	1	no	—
74	#432	—	37	1	no	—
75	C₄×M₁₆	—	37	1	no	—
76	#330	—	37	4	no	—
77	C₃₇	#1528	1	1	yes	characteristic, complemented, Sylow
78	C₂×M₁₆	—	74	1	no	—
79	#125	—	37	2	no	—
80	#125	—	37	2	no	—
81	#125	—	74	1	no	—
82	#125	—	74	1	no	—
83	C₂×C₄⋊C₄	—	37	1	no	—
84	C₂×C₄⋊C₄	—	74	1	no	—
85	C₂×C₄⋊C₄	—	74	1	no	—
86	C₂×C₄⋊C₄	#3866	1	1	yes	characteristic
87	C₂×C₄²	—	37	1	no	—
88	C₄⋊C₈	—	74	1	no	—
89	C₂²⋊C₈	—	74	1	no	—
90	#97	—	37	2	no	—
91	#97	—	37	2	no	—
92	C₄×C₈	—	74	1	no	—
93	C₂²×C₄	—	2	1	no	—
94	C₂²×C₄	—	37	1	no	—
95	C₂²×C₄	—	37	1	no	—
96	C₂²×C₄	—	74	1	no	—
97	C₂²×C₄	C₂×#3866	1	1	yes	characteristic
98	M₁₆	—	74	4	no	—
99	C₂×C₈	—	74	2	no	—
100	C₂×C₈	—	74	2	no	—
101	C₄⋊C₄	—	2	1	no	—
102	C₄⋊C₄	—	37	2	no	—
103	C₄⋊C₄	—	74	2	no	—
104	C₄⋊C₄	—	74	2	no	—
105	C₄⋊C₄	—	74	2	no	—
106	C₄⋊C₄	—	74	2	no	—
107	C₄⋊C₄	—	74	1	no	—
108	C₄⋊C₄	C₂×#3866	1	2	yes	—
109	C₄²	—	37	1	no	—
110	C₄²	—	37	1	no	—
111	C₄²	—	37	1	no	—
112	C₄²	—	37	1	no	—
113	C₂³	C₂²⋊#3866	1	1	yes	characteristic
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	37	1	no	—
118	C₂×C₄	—	37	1	no	—
119	C₂×C₄	—	37	1	no	—
120	C₂×C₄	—	37	1	no	—
121	C₂×C₄	—	74	2	no	—
122	C₂×C₄	—	74	2	no	—
123	C₂×C₄	—	148	1	no	—
124	C₂×C₄	—	148	2	no	—
125	C₂×C₄	C₂²⋊#3866	1	1	yes	characteristic
126	C₂×C₄	C₂²×#3866	1	1	yes	characteristic
127	C₈	—	74	4	no	—
128	C₈	—	148	2	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	#11924601	1	2	yes	—
132	C₂²	C₂×C₂²⋊#3866	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	4	2	no	—
135	C₄	—	74	1	no	—
136	C₄	—	74	1	no	—
137	C₄	—	74	1	no	—
138	C₄	—	74	1	no	—
139	C₄	—	148	2	no	—
140	C₄	#11924597	1	2	yes	—
141	C₂	—	2	2	no	—
142	C₂	C₂×#11924597	1	1	yes	characteristic
143	C₂	C₂×#11924601	1	1	yes	characteristic
144	C₂	#14558531	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14558554

C₂

yes

maximal, characteristic

C₄×#10596957

C₂

yes

maximal, characteristic

#14558390

C₂

yes

maximal, complemented

C₂×#11924527

C₂

yes

maximal, characteristic

—

—

—

C₂²

yes

—

#11924591

C₂²

yes

—

—

—

—

C₄

yes

—

#11924527

C₂²

yes

—

C₂×C₄⋊Dic₃₇

—

C₂×C₄⋊Dic₃₇

C₄

yes

characteristic

C₂×#10596936

—

C₂×C₄×Dic₃₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₃₇

C₄

yes

characteristic

C₂²×C₄×C₃₇

—

C₂²×C₄×C₃₇

C₂×C₄

yes

characteristic

C₂²×Dic₃₇

—

C₂²×Dic₃₇

D₄

yes

characteristic

C₂²×Dic₃₇

D₄

yes

—

—

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

C₂×C₄

yes

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

C₂×C₄

yes

—

#10596936

—

#10596936

—

C₄×Dic₃₇

C₂×C₄

yes

characteristic

C₄×Dic₃₇

D₄

yes

characteristic

C₄×Dic₃₇

D₄

yes

characteristic

C₄×Dic₃₇

C₂³

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

—

C₂×Dic₃₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₃₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₃₇

C₂×D₄

yes

characteristic

C₂×Dic₃₇

C₂×D₄

yes

characteristic

#64438

—

#64438

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

#104

yes

—

C₂²×C₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

#101

yes

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

—

#1528

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

—

—

C₄×M₁₆

—

#330

—

C₃₇

#1528

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

—

—

—

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

#3866

yes

characteristic

C₂×C₄²

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×#3866

yes

characteristic

M₁₆

—

C₂×C₈

—

100

C₂×C₈

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

C₂×#3866

yes

—

109

C₄²

—

110

C₄²

—

111

C₄²

—

112

C₄²

—

113

C₂³

C₂²⋊#3866

yes

characteristic

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

148

—

124

C₂×C₄

—

148

—

125

C₂×C₄

C₂²⋊#3866

yes

characteristic

126

C₂×C₄

C₂²×#3866

yes

characteristic

127

C₈

—

128

C₈

—

148

—

129

C₂²

—

130

C₂²

—

131

C₂²

#11924601

yes

—

132

C₂²

C₂×C₂²⋊#3866

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

139

C₄

—

148

—

140

C₄

#11924597

yes

—

141

C₂

—

142

C₂

C₂×#11924597

yes

characteristic

143

C₂

C₂×#11924601

yes

characteristic

144

C₂

#14558531

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊#3866
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₂×C₄	C₂²×#3866
Fitting	C₂×C₄⋊C₄×C₃₇	C₄
socle	C₂²×C₃₇	C₂×C₂²⋊C₄
prime radical	C₂³×C₃₇	C₂²⋊C₄
cosocle	C₄×Dic₃₇	C₂³
hypercenter	C₂×C₄⋊C₄	#3866
hypocenter	C₃₇	#1528
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

subgroup

G/H

center

C₂²

C₂×C₂²⋊#3866

commutator

C₂×C₄×C₃₇

C₂²×C₄

Frattini

C₂×C₄

C₂²×#3866

Fitting

C₂×C₄⋊C₄×C₃₇

C₄

socle

C₂²×C₃₇

C₂×C₂²⋊C₄

prime radical

C₂³×C₃₇

C₂²⋊C₄

cosocle

C₄×Dic₃₇

C₂³

C₂×C₄⋊C₄

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
3. #14558390	C₂	semidirect
77. C₃₇	#1528	semidirect

H ≅ G/N

type

3. #14558390

C₂

semidirect

77. C₃₇

#1528

semidirect

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

subgroup	index
#14558554	2
C₄×#10596957	2
#14558390	2
C₂×#11924527	2
#1528	37

subgroup

index

Subgroups — #18010060

Conjugacy classes of subgroups

Complements

Maximal subgroups