Group #18005332 — subgroups

Subgroups — #18005332

Order 4736.

446 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 125 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	#14558081	C₂	1	2	yes	maximal, complemented
2	#14557931	C₂	1	1	yes	maximal, characteristic, complemented
3	#14557879	C₂	1	2	yes	maximal, complemented
4	#14557783	C₂	1	4	yes	maximal, complemented
5	#14557645	C₂	1	4	yes	maximal, complemented
6	C₄×#10596929	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₇×#493	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10596964	—	2	2	no	—
9	C₂×#10596929	C₂²	1	2	yes	—
10	#11924536	C₂²	1	2	yes	—
11	#11924523	—	2	2	no	—
12	#11924507	—	2	2	no	—
13	#11924507	C₂²	1	4	yes	—
14	D₄×Dic₃₇	—	2	2	no	—
15	D₄×Dic₃₇	C₂²	1	4	yes	—
16	C₄²×D₃₇	C₂²	1	1	yes	characteristic
17	#11924468	C₂²	1	4	yes	—
18	#11924457	C₂²	1	8	yes	—
19	#11924446	C₂²	1	2	yes	—
20	C₄×#64436	C₂²	1	1	yes	characteristic
21	C₂×D₈×C₃₇	C₂²	1	4	yes	—
22	#127×C₃₇	C₂²	1	2	yes	—
23	C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
24	C₂×D₄×C₃₇	—	2	2	no	—
25	C₂×D₄×C₃₇	—	2	2	no	—
26	C₂×D₄×C₃₇	C₂³	1	4	yes	—
27	C₂×C₂²⋊D₃₇	—	2	4	no	—
28	C₂²×Dic₃₇	—	2	4	no	—
29	#10596964	—	4	8	no	—
30	C₂×C₄×D₃₇	D₄	1	2	yes	—
31	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
32	C₂×Dic₇₄	—	2	2	no	—
33	D₈×C₃₇	—	2	8	no	—
34	C₂×C₈×C₃₇	C₂³	1	2	yes	—
35	C₄²×C₃₇	C₂³	1	1	yes	characteristic
36	C₂²⋊Dic₃₇	—	2	4	no	—
37	C₂²⋊Dic₃₇	—	2	4	no	—
38	C₄⋊Dic₃₇	—	2	2	no	—
39	C₄⋊Dic₃₇	C₂³	1	4	yes	—
40	C₄×Dic₃₇	D₄	1	2	yes	—
41	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
42	C₂×#64436	C₂³	1	2	yes	—
43	#10596929	—	2	4	no	—
44	C₂³×C₃₇	—	2	4	no	—
45	C₂²×D₃₇	C₂×D₄	1	1	yes	characteristic
46	D₄×C₃₇	—	2	8	no	—
47	D₄×C₃₇	—	4	4	no	—
48	D₄×C₃₇	—	4	4	no	—
49	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
51	C₂²⋊D₃₇	—	4	8	no	—
52	C₂×Dic₃₇	—	2	4	no	—
53	C₂×Dic₃₇	—	4	8	no	—
54	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
55	C₂×Dic₃₇	C₂×D₄	1	2	yes	—
56	C₄×D₃₇	—	2	4	no	—
57	C₄×D₃₇	—	2	2	no	—
58	Dic₇₄	—	4	4	no	—
59	C₈×C₃₇	#41	1	4	yes	complemented
60	#64436	—	2	2	no	—
61	C₂²×C₃₇	—	4	4	no	—
62	C₂²×C₃₇	—	4	4	no	—
63	C₂²×C₃₇	—	4	4	no	—
64	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
65	C₂×D₃₇	—	2	1	no	—
66	C₂×D₃₇	—	2	2	no	—
67	C₄×C₃₇	AGL(1,ℤ/8)	1	4	yes	—
68	C₄×C₃₇	C₂×#41	1	2	yes	—
69	Dic₃₇	—	2	1	no	—
70	Dic₃₇	—	2	2	no	—
71	Dic₃₇	—	4	4	no	—
72	#3184	—	37	1	no	maximal, Sylow
73	C₂×C₃₇	—	8	4	no	—
74	C₂×C₃₇	#532	1	1	yes	characteristic
75	C₂×C₃₇	C₂×AGL(1,ℤ/8)	1	2	yes	—
76	D₃₇	—	4	1	no	—
77	#532	—	37	2	no	—
78	#498	—	37	1	no	—
79	#493	D₃₇	1	1	yes	characteristic, complemented
80	#486	—	37	2	no	—
81	#469	—	37	4	no	—
82	#442	—	37	4	no	—
83	C₄×M₁₆	—	37	1	no	—
84	C₃₇	#3184	1	1	yes	characteristic, complemented, Sylow
85	C₂×#41	—	74	2	no	—
86	C₂×D₈	C₂×D₃₇	1	4	yes	—
87	C₂×M₁₆	—	37	2	no	—
88	C₄⋊Q₈	—	37	2	no	—
89	#127	C₂×D₃₇	1	2	yes	—
90	#124	—	74	2	no	—
91	#121	—	37	4	no	—
92	#121	—	74	2	no	—
93	C₄×D₄	—	37	4	no	—
94	C₄×D₄	—	74	2	no	—
95	C₂×C₄²	—	37	1	no	—
96	#106	—	37	4	no	—
97	#102	—	37	8	no	—
98	#97	—	37	2	no	—
99	C₄×C₈	—	37	1	no	—
100	C₄×C₈	C₂×D₃₇	1	1	yes	characteristic
101	#41	—	148	8	no	—
102	C₂×Q₈	—	74	2	no	—
103	C₂×D₄	—	2	2	no	—
104	C₂×D₄	—	2	2	no	—
105	C₂×D₄	—	74	4	no	—
106	C₂×D₄	C₂²×D₃₇	1	4	yes	—
107	C₂²×C₄	—	37	2	no	—
108	C₂²×C₄	—	37	1	no	—
109	C₂²×C₄	—	74	4	no	—
110	D₈	—	2	8	no	—
111	M₁₆	—	74	4	no	—
112	C₂×C₈	—	37	2	no	—
113	C₂×C₈	C₂²×D₃₇	1	2	yes	—
114	C₄⋊C₄	—	37	4	no	—
115	C₄⋊C₄	—	74	2	no	—
116	C₂²⋊C₄	—	74	4	no	—
117	C₂²⋊C₄	—	74	4	no	—
118	C₄²	—	37	2	no	—
119	C₄²	—	37	1	no	—
120	C₄²	C₂²×D₃₇	1	1	yes	characteristic
121	C₂³	—	2	4	no	—
122	C₂³	—	37	1	no	—
123	Q₈	—	148	4	no	—
124	D₄	—	2	8	no	—
125	D₄	—	4	4	no	—
126	D₄	—	4	4	no	—
127	D₄	—	148	8	no	—
128	C₂×C₄	—	37	1	no	—
129	C₂×C₄	—	37	2	no	—
130	C₂×C₄	—	74	4	no	—
131	C₂×C₄	—	74	2	no	—
132	C₂×C₄	—	74	4	no	—
133	C₂×C₄	—	148	8	no	—
134	C₂×C₄	D₄×D₃₇	1	2	yes	—
135	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
136	C₈	—	74	2	no	—
137	C₈	#10596964	1	4	yes	complemented
138	C₂²	—	4	4	no	—
139	C₂²	—	4	4	no	—
140	C₂²	—	4	4	no	—
141	C₂²	—	74	2	no	—
142	C₂²	—	74	1	no	—
143	C₂²	C₂×D₄×D₃₇	1	1	yes	characteristic
144	C₄	—	74	2	no	—
145	C₄	—	74	1	no	—
146	C₄	—	148	4	no	—
147	C₄	#11924560	1	4	yes	—
148	C₄	C₂×#10596964	1	2	yes	—
149	C₂	—	8	4	no	—
150	C₂	—	148	1	no	—
151	C₂	#14558081	1	1	yes	characteristic
152	C₂	C₂×#11924560	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14558081

C₂

yes

maximal, complemented

#14557931

C₂

yes

maximal, characteristic, complemented

#14557879

C₂

yes

maximal, complemented

#14557783

C₂

yes

maximal, complemented

#14557645

C₂

yes

maximal, complemented

C₄×#10596929

C₂

yes

maximal, characteristic, complemented

C₃₇×#493

C₂

yes

maximal, characteristic, complemented

C₂×#10596964

—

C₂×#10596929

C₂²

yes

—

#11924536

C₂²

yes

—

#11924523

—

#11924507

—

#11924507

C₂²

yes

—

D₄×Dic₃₇

—

D₄×Dic₃₇

C₂²

yes

—

C₄²×D₃₇

C₂²

yes

characteristic

#11924468

C₂²

yes

—

#11924457

C₂²

yes

—

#11924446

C₂²

yes

—

C₄×#64436

C₂²

yes

characteristic

C₂×D₈×C₃₇

C₂²

yes

—

#127×C₃₇

C₂²

yes

—

C₄×C₈×C₃₇

C₂²

yes

characteristic

C₂×D₄×C₃₇

—

C₂×D₄×C₃₇

—

C₂×D₄×C₃₇

C₂³

yes

—

C₂×C₂²⋊D₃₇

—

C₂²×Dic₃₇

—

#10596964

—

C₂×C₄×D₃₇

D₄

yes

—

C₂×C₄×D₃₇

C₂³

yes

characteristic

C₂×Dic₇₄

—

D₈×C₃₇

—

C₂×C₈×C₃₇

C₂³

yes

—

C₄²×C₃₇

C₂³

yes

characteristic

C₂²⋊Dic₃₇

—

C₂²⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

C₂³

yes

—

C₄×Dic₃₇

D₄

yes

—

C₄×Dic₃₇

C₂³

yes

characteristic

C₂×#64436

C₂³

yes

—

#10596929

—

C₂³×C₃₇

—

C₂²×D₃₇

C₂×D₄

yes

characteristic

D₄×C₃₇

—

D₄×C₃₇

—

D₄×C₃₇

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂⁴

yes

characteristic

C₂²⋊D₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

C₂×D₄

yes

characteristic

C₂×Dic₃₇

C₂×D₄

yes

—

C₄×D₃₇

—

C₄×D₃₇

—

Dic₇₄

—

C₈×C₃₇

#41

yes

complemented

#64436

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂²×D₄

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₄×C₃₇

AGL(1,ℤ/8)

yes

—

C₄×C₃₇

C₂×#41

yes

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

—

#3184

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

#532

yes

characteristic

C₂×C₃₇

C₂×AGL(1,ℤ/8)

yes

—

D₃₇

—

#532

—

#498

—

#493

D₃₇

yes

characteristic, complemented

#486

—

#469

—

#442

—

C₄×M₁₆

—

C₃₇

#3184

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×D₈

C₂×D₃₇

yes

—

C₂×M₁₆

—

C₄⋊Q₈

—

#127

C₂×D₃₇

yes

—

#124

—

#121

—

#121

—

C₄×D₄

—

C₄×D₄

—

C₂×C₄²

—

#106

—

#102

—

#97

—

C₄×C₈

—

100

C₄×C₈

C₂×D₃₇

yes

characteristic

101

#41

—

148

—

102

C₂×Q₈

—

103

C₂×D₄

—

104

C₂×D₄

—

105

C₂×D₄

—

106

C₂×D₄

C₂²×D₃₇

yes

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

D₈

—

111

M₁₆

—

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

C₂²×D₃₇

yes

characteristic

121

C₂³

—

122

C₂³

—

123

Q₈

—

148

—

124

D₄

—

125

D₄

—

126

D₄

—

127

D₄

—

148

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

148

—

134

C₂×C₄

D₄×D₃₇

yes

—

135

C₂×C₄

C₂³×D₃₇

yes

characteristic

136

C₈

—

137

C₈

#10596964

yes

complemented

138

C₂²

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

C₂×D₄×D₃₇

yes

characteristic

144

C₄

—

145

C₄

—

146

C₄

—

148

—

147

C₄

#11924560

yes

—

148

C₄

C₂×#10596964

yes

—

149

C₂

—

150

C₂

—

148

—

151

C₂

#14558081

yes

characteristic

152

C₂

C₂×#11924560

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₃₇
commutator, cosocle	C₂×C₄×C₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₃₇×#493	C₂
socle	C₂²×C₃₇	C₂²×D₄
hypercenter	#493	D₃₇
hypocenter	C₃₇	#3184
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

C₂×D₄×D₃₇

commutator, cosocle

C₂×C₄×C₃₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₃₇

Fitting

C₃₇×#493

C₂

socle

C₂²×C₃₇

C₂²×D₄

D₃₇

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #14558081	C₂	semidirect
2. #14557931	C₂	semidirect
3. #14557879	C₂	semidirect
4. #14557783	C₂	semidirect
5. #14557645	C₂	semidirect
6. C₄×#10596929	C₂	semidirect
7. C₃₇×#493	C₂	semidirect
59. C₈×C₃₇	#41	semidirect
79. #493	D₃₇	semidirect
84. C₃₇	#3184	semidirect
137. C₈	#10596964	semidirect

H ≅ G/N

type

1. #14558081

C₂

semidirect

2. #14557931

C₂

semidirect

3. #14557879

C₂

semidirect

4. #14557783

C₂

semidirect

5. #14557645

C₂

semidirect

6. C₄×#10596929

C₂

semidirect

7. C₃₇×#493

C₂

semidirect

59. C₈×C₃₇

#41

semidirect

79. #493

D₃₇

semidirect

84. C₃₇

#3184

semidirect

137. C₈

#10596964

semidirect

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

subgroup	index
#14558081	2
#14557931	2
#14557879	2
#14557783	2
#14557645	2
C₄×#10596929	2
C₃₇×#493	2
#3184	37

subgroup

index

C₃₇×#493

Subgroups — #18005332

Conjugacy classes of subgroups

Complements

Maximal subgroups