Group #18005109 — subgroups

Subgroups — #18005109

Order 4736.

356 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 356 conjugacy classes of subgroups — too many to draw. Only its 141 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	#14558089	C₂	1	2	yes	maximal
2	#14557937	C₂	1	2	yes	maximal, complemented
3	#14557935	C₂	1	1	yes	maximal, characteristic, complemented
4	#14557932	C₂	1	2	yes	maximal, complemented
5	#14557930	C₂	1	2	yes	maximal, complemented
6	#14557570	C₂	1	2	yes	maximal
7	#14557568	C₂	1	2	yes	maximal
8	C₄×C₈×D₃₇	C₂	1	1	yes	maximal, characteristic
9	C₃₇×#499	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₈×D₃₇	C₂²	1	1	yes	characteristic
11	C₂×C₈×D₃₇	C₂²	1	1	yes	characteristic
12	#11924535	C₂²	1	4	yes	—
13	#11924528	C₂²	1	2	yes	—
14	#11924514	—	2	4	no	—
15	#11924501	—	2	4	no	—
16	#11924494	C₂²	1	4	yes	—
17	C₄²×D₃₇	C₂²	1	1	yes	characteristic
18	#11924470	C₂²	1	2	yes	—
19	#11924469	C₂²	1	4	yes	—
20	#11924468	C₂²	1	2	yes	—
21	#11924467	C₂²	1	4	yes	—
22	C₄×#64436	C₂²	1	1	yes	characteristic
23	C₈×Dic₃₇	C₂²	1	1	yes	characteristic
24	C₈×Dic₃₇	C₂²	1	1	yes	characteristic
25	#125×C₃₇	C₂²	1	2	yes	—
26	#107×C₃₇	C₂²	1	2	yes	—
27	#106×C₃₇	C₂²	1	2	yes	—
28	C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic
29	C₂×C₄×D₃₇	D₄	1	2	yes	—
30	C₂×C₄×D₃₇	C₂³	1	1	yes	characteristic
31	C₂×Dic₇₄	—	2	4	no	—
32	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
33	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
34	C₄⋊C₄×C₃₇	—	2	4	no	—
35	C₄⋊C₄×C₃₇	C₂³	1	4	yes	—
36	C₄²×C₃₇	C₂³	1	1	yes	characteristic
37	#10596938	—	2	4	no	—
38	C₄⋊Dic₃₇	C₂³	1	4	yes	—
39	#10596936	—	2	4	no	—
40	#10596936	—	2	4	no	—
41	C₄×Dic₃₇	—	2	4	no	—
42	C₄×Dic₃₇	D₄	1	2	yes	—
43	C₄×Dic₃₇	C₂³	1	1	yes	characteristic
44	C₂×#64436	C₂³	1	1	yes	characteristic
45	C₂×#64436	C₂³	1	1	yes	characteristic
46	C₈×D₃₇	Q₈	1	4	yes	complemented
47	C₈×D₃₇	Q₈	1	4	yes	complemented
48	C₂²×D₃₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₃₇	—	2	4	no	—
50	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
52	C₂×Dic₃₇	—	2	4	no	—
53	C₂×Dic₃₇	C₂×D₄	1	1	yes	characteristic
54	C₂×Dic₃₇	C₂×D₄	1	2	yes	—
55	C₄×D₃₇	—	2	4	no	—
56	C₄×D₃₇	C₂×Q₈	1	2	yes	—
57	C₄×D₃₇	C₂×Q₈	1	2	yes	—
58	Dic₇₄	—	4	8	no	—
59	C₈×C₃₇	C₂×Q₈	1	2	yes	—
60	C₈×C₃₇	C₂×Q₈	1	2	yes	—
61	#64436	C₂×Q₈	1	2	yes	—
62	#64436	C₂×Q₈	1	2	yes	—
63	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
64	C₂×D₃₇	—	2	2	no	—
65	C₂×D₃₇	C₄⋊Q₈	1	2	yes	—
66	C₄×C₃₇	—	2	2	no	—
67	C₄×C₃₇	—	4	4	no	—
68	C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
69	C₄×C₃₇	C₂²×Q₈	1	1	yes	characteristic
70	Dic₃₇	—	4	4	no	—
71	Dic₃₇	C₄⋊Q₈	1	2	yes	—
72	Dic₃₇	#135	1	4	yes	—
73	#3160	—	37	1	no	maximal, Sylow
74	C₂×C₃₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
75	C₂×C₃₇	C₂×#135	1	2	yes	—
76	D₃₇	—	2	2	no	—
77	#533	—	37	2	no	—
78	#500	—	37	2	no	—
79	#499	—	37	2	no	—
80	#499	—	37	1	no	—
81	#499	D₃₇	1	1	yes	characteristic, complemented
82	#498	—	37	2	no	—
83	#427	—	37	2	no	—
84	#427	—	37	2	no	—
85	C₂×C₄×C₈	—	37	1	no	—
86	C₃₇	#3160	1	1	yes	characteristic, complemented, Sylow
87	C₂²×C₈	—	37	1	no	—
88	C₂²×C₈	—	37	1	no	—
89	C₄⋊Q₈	—	37	4	no	—
90	#125	—	37	2	no	—
91	#125	C₂×D₃₇	1	2	yes	—
92	C₂²⋊Q₈	—	74	4	no	—
93	C₄×Q₈	—	74	4	no	—
94	#117	—	37	4	no	—
95	C₂×C₄²	—	37	1	no	—
96	#107	—	37	2	no	—
97	#107	—	37	4	no	—
98	#107	C₂×D₃₇	1	2	yes	—
99	#106	—	37	2	no	—
100	#106	—	37	4	no	—
101	#106	C₂×D₃₇	1	2	yes	—
102	C₄×C₈	—	37	1	no	—
103	C₄×C₈	—	37	1	no	—
104	C₄×C₈	—	37	1	no	—
105	C₄×C₈	C₂×D₃₇	1	1	yes	characteristic
106	C₂×Q₈	—	74	4	no	—
107	C₂²×C₄	—	37	1	no	—
108	C₂²×C₄	—	37	2	no	—
109	C₂×C₈	—	37	1	no	—
110	C₂×C₈	—	37	1	no	—
111	C₂×C₈	—	37	4	no	—
112	C₂×C₈	—	37	4	no	—
113	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
114	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
115	C₄⋊C₄	—	2	4	no	—
116	C₄⋊C₄	—	37	4	no	—
117	C₄⋊C₄	—	74	4	no	—
118	C₄⋊C₄	—	74	4	no	—
119	C₄⋊C₄	C₂²×D₃₇	1	4	yes	—
120	C₂²⋊C₄	—	74	4	no	—
121	C₄²	—	37	2	no	—
122	C₄²	—	37	1	no	—
123	C₄²	—	74	4	no	—
124	C₄²	C₂²×D₃₇	1	1	yes	characteristic
125	C₂³	—	37	1	no	—
126	Q₈	—	148	8	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	—	37	1	no	—
129	C₂×C₄	—	37	2	no	—
130	C₂×C₄	—	37	2	no	—
131	C₂×C₄	—	37	2	no	—
132	C₂×C₄	—	74	4	no	—
133	C₂×C₄	—	74	4	no	—
134	C₂×C₄	D₄×D₃₇	1	2	yes	—
135	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
136	C₈	—	37	2	no	—
137	C₈	—	37	2	no	—
138	C₈	Q₈×D₃₇	1	2	yes	—
139	C₈	Q₈×D₃₇	1	2	yes	—
140	C₂²	—	37	2	no	—
141	C₂²	—	74	2	no	—
142	C₂²	C₂×D₄×D₃₇	1	1	yes	characteristic
143	C₄	—	2	2	no	—
144	C₄	—	4	4	no	—
145	C₄	—	37	4	no	—
146	C₄	—	37	2	no	—
147	C₄	—	148	4	no	—
148	C₄	C₂×Q₈×D₃₇	1	1	yes	characteristic
149	C₄	C₂×Q₈×D₃₇	1	1	yes	characteristic
150	C₂	—	74	2	no	—
151	C₂	C₄⋊Q₈×D₃₇	1	1	yes	characteristic
152	C₂	#135×D₃₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14558089

C₂

yes

maximal

#14557937

C₂

yes

maximal, complemented

#14557935

C₂

yes

maximal, characteristic, complemented

#14557932

C₂

yes

maximal, complemented

#14557930

C₂

yes

maximal, complemented

#14557570

C₂

yes

maximal

#14557568

C₂

yes

maximal

C₄×C₈×D₃₇

C₂

yes

maximal, characteristic

C₃₇×#499

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₃₇

C₂²

yes

characteristic

C₂×C₈×D₃₇

C₂²

yes

characteristic

#11924535

C₂²

yes

—

#11924528

C₂²

yes

—

#11924514

—

#11924501

—

#11924494

C₂²

yes

—

C₄²×D₃₇

C₂²

yes

characteristic

#11924470

C₂²

yes

—

#11924469

C₂²

yes

—

#11924468

C₂²

yes

—

#11924467

C₂²

yes

—

C₄×#64436

C₂²

yes

characteristic

C₈×Dic₃₇

C₂²

yes

characteristic

C₈×Dic₃₇

C₂²

yes

characteristic

#125×C₃₇

C₂²

yes

—

#107×C₃₇

C₂²

yes

—

#106×C₃₇

C₂²

yes

—

C₄×C₈×C₃₇

C₂²

yes

characteristic

C₂×C₄×D₃₇

D₄

yes

—

C₂×C₄×D₃₇

C₂³

yes

characteristic

C₂×Dic₇₄

—

C₂×C₈×C₃₇

C₂³

yes

characteristic

C₂×C₈×C₃₇

C₂³

yes

characteristic

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

C₂³

yes

—

C₄²×C₃₇

C₂³

yes

characteristic

#10596938

—

C₄⋊Dic₃₇

C₂³

yes

—

#10596936

—

#10596936

—

C₄×Dic₃₇

—

C₄×Dic₃₇

D₄

yes

—

C₄×Dic₃₇

C₂³

yes

characteristic

C₂×#64436

C₂³

yes

characteristic

C₂×#64436

C₂³

yes

characteristic

C₈×D₃₇

Q₈

yes

complemented

C₈×D₃₇

Q₈

yes

complemented

C₂²×D₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂⁴

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

C₂×D₄

yes

characteristic

C₂×Dic₃₇

C₂×D₄

yes

—

C₄×D₃₇

—

C₄×D₃₇

C₂×Q₈

yes

—

C₄×D₃₇

C₂×Q₈

yes

—

Dic₇₄

—

C₈×C₃₇

C₂×Q₈

yes

—

C₈×C₃₇

C₂×Q₈

yes

—

#64436

C₂×Q₈

yes

—

#64436

C₂×Q₈

yes

—

C₂²×C₃₇

C₂²×D₄

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

C₄⋊Q₈

yes

—

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₂²×Q₈

yes

characteristic

C₄×C₃₇

C₂²×Q₈

yes

characteristic

Dic₃₇

—

Dic₃₇

C₄⋊Q₈

yes

—

Dic₃₇

#135

yes

—

#3160

—

maximal, Sylow

C₂×C₃₇

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₃₇

C₂×#135

yes

—

D₃₇

—

—

—

—

—

D₃₇

yes

characteristic, complemented

#498

—

#427

—

#427

—

C₂×C₄×C₈

—

C₃₇

#3160

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

—

#125

—

#125

C₂×D₃₇

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

#107

—

#107

—

#107

C₂×D₃₇

yes

—

#106

—

100

#106

—

101

#106

C₂×D₃₇

yes

—

102

C₄×C₈

—

103

C₄×C₈

—

104

C₄×C₈

—

105

C₄×C₈

C₂×D₃₇

yes

characteristic

106

C₂×Q₈

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

C₂²×D₃₇

yes

characteristic

114

C₂×C₈

C₂²×D₃₇

yes

characteristic

115

C₄⋊C₄

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

C₂²×D₃₇

yes

—

120

C₂²⋊C₄

—

121

C₄²

—

122

C₄²

—

123

C₄²

—

124

C₄²

C₂²×D₃₇

yes

characteristic

125

C₂³

—

126

Q₈

—

148

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

D₄×D₃₇

yes

—

135

C₂×C₄

C₂³×D₃₇

yes

characteristic

136

C₈

—

137

C₈

—

138

C₈

Q₈×D₃₇

yes

—

139

C₈

Q₈×D₃₇

yes

—

140

C₂²

—

141

C₂²

—

142

C₂²

C₂×D₄×D₃₇

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

—

148

C₄

C₂×Q₈×D₃₇

yes

characteristic

149

C₄

C₂×Q₈×D₃₇

yes

characteristic

150

C₂

—

151

C₂

C₄⋊Q₈×D₃₇

yes

characteristic

152

C₂

#135×D₃₇

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₃₇
commutator, cosocle	C₂×C₄×C₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₃₇×#499	C₂
socle	C₂²×C₃₇	C₂²×D₄
prime radical	C₂²×D₃₇	C₂×D₄
hypercenter	#499	D₃₇
hypocenter	C₃₇	#3160
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #499

subgroup

G/H

center

C₂²

C₂×D₄×D₃₇

commutator, cosocle

C₂×C₄×C₃₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₃₇

Fitting

C₃₇×#499

C₂

socle

C₂²×C₃₇

C₂²×D₄

prime radical

C₂²×D₃₇

C₂×D₄

D₃₇

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #499

N	H ≅ G/N	type
2. #14557937	C₂	semidirect
3. #14557935	C₂	semidirect
4. #14557932	C₂	semidirect
5. #14557930	C₂	semidirect
9. C₃₇×#499	C₂	semidirect
46. C₈×D₃₇	Q₈	semidirect
47. C₈×D₃₇	Q₈	semidirect
81. #499	D₃₇	semidirect
86. C₃₇	#3160	semidirect

H ≅ G/N

type

2. #14557937

C₂

semidirect

3. #14557935

C₂

semidirect

4. #14557932

C₂

semidirect

5. #14557930

C₂

semidirect

9. C₃₇×#499

C₂

semidirect

46. C₈×D₃₇

Q₈

semidirect

47. C₈×D₃₇

Q₈

semidirect

81. #499

D₃₇

semidirect

86. C₃₇

#3160

semidirect

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

subgroup	index
#14558089	2
#14557937	2
#14557935	2
#14557932	2
#14557930	2
#14557570	2
#14557568	2
C₄×C₈×D₃₇	2
C₃₇×#499	2
#3160	37

subgroup

index

C₄×C₈×D₃₇

C₃₇×#499

#3160

Subgroups — #18005109

Conjugacy classes of subgroups

Complements

Maximal subgroups