Group #18003464 — subgroups

Subgroups — #18003464

Order 4736.

456 conjugacy classes of subgroups; only the 221 normal subgroups are shown.

normal subgroups only

This group has 456 conjugacy classes of subgroups — too many to draw. Only its 221 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	#14558011	C₂	1	2	yes	maximal
2	#14557640	C₂	1	4	yes	maximal, complemented
3	Q₁₆×Dic₃₇	C₂	1	4	yes	maximal, complemented
4	#14557571	C₂	1	1	yes	maximal, characteristic
5	#14557542	C₂	1	2	yes	maximal
6	#14557488	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₁₆×C₃₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#10596934	C₂²	1	1	yes	characteristic
9	Q₈×Dic₃₇	—	2	4	no	—
10	Q₈×Dic₃₇	C₂²	1	8	yes	—
11	#11924495	—	2	4	no	—
12	#11924495	C₂²	1	2	yes	—
13	#11924495	C₂²	1	1	yes	characteristic
14	#11924470	C₂²	1	2	yes	—
15	#11924468	C₂²	1	2	yes	—
16	#11924460	C₂²	1	8	yes	—
17	#11924446	C₂²	1	2	yes	—
18	C₈×Dic₃₇	C₂²	1	2	yes	—
19	C₂²×Q₈×C₃₇	C₂²	1	2	yes	—
20	C₂×Q₁₆×C₃₇	C₄	1	8	yes	complemented
21	C₂×Q₁₆×C₃₇	C₂²	1	2	yes	—
22	C₂×Q₁₆×C₃₇	C₂²	1	2	yes	—
23	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
24	C₂×Q₈×C₃₇	—	2	2	no	—
25	C₂×Q₈×C₃₇	—	2	4	no	—
26	C₂×Q₈×C₃₇	C₂×C₄	1	8	yes	—
27	C₂×Q₈×C₃₇	C₂³	1	4	yes	—
28	C₂²×C₄×C₃₇	—	2	2	no	—
29	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
30	Q₁₆×C₃₇	C₂×C₄	1	8	yes	—
31	Q₁₆×C₃₇	C₂×C₄	1	8	yes	—
32	C₂×C₈×C₃₇	C₂×C₄	1	4	yes	—
33	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₃₇	C₂³	1	1	yes	characteristic
35	C₂²⋊Dic₃₇	—	2	2	no	—
36	C₂²⋊Dic₃₇	D₄	1	2	yes	—
37	C₄⋊Dic₃₇	—	2	8	no	—
38	C₄⋊Dic₃₇	D₄	1	2	yes	—
39	C₄⋊Dic₃₇	C₂³	1	4	yes	—
40	C₄×Dic₃₇	—	2	2	no	—
41	C₄×Dic₃₇	—	2	8	no	—
42	C₄×Dic₃₇	C₂³	1	2	yes	—
43	#10596934	—	2	2	no	—
44	C₂×#64436	C₂³	1	2	yes	—
45	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
46	Q₈×C₃₇	—	2	8	no	—
47	Q₈×C₃₇	—	2	4	no	—
48	Q₈×C₃₇	C₂²×C₄	1	8	yes	—
49	C₂×C₄×C₃₇	—	2	8	no	—
50	C₂×C₄×C₃₇	—	2	4	no	—
51	C₂×C₄×C₃₇	C₂²×C₄	1	2	yes	—
52	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₃₇	#41	1	2	yes	—
54	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
55	C₂×Dic₃₇	—	2	4	no	—
56	C₂×Dic₃₇	C₂×D₄	1	4	yes	—
57	C₈×C₃₇	C₂²×C₄	1	2	yes	—
58	C₈×C₃₇	C₂²×C₄	1	2	yes	—
59	#64436	—	2	2	no	—
60	C₂²×C₃₇	—	2	2	no	—
61	C₂²×C₃₇	C₄×D₄	1	2	yes	—
62	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
63	C₄×C₃₇	—	2	8	no	—
64	C₄×C₃₇	C₄×D₄	1	2	yes	—
65	C₄×C₃₇	C₂³×C₄	1	1	yes	characteristic
66	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
67	Dic₃₇	—	2	4	no	—
68	Dic₃₇	—	4	4	no	—
69	#2950	—	37	1	no	maximal, Sylow
70	C₂×C₃₇	—	2	2	no	—
71	C₂×C₃₇	C₂×C₄×D₄	1	1	yes	characteristic
72	C₂×C₃₇	#578	1	2	yes	—
73	C₂²×Q₁₆	D₃₇	1	1	yes	characteristic
74	#519	—	37	2	no	—
75	#441	—	37	4	no	—
76	C₄×Q₁₆	—	37	4	no	—
77	#427	—	37	1	no	—
78	#419	—	37	2	no	—
79	#405	—	37	1	no	—
80	C₃₇	#2950	1	1	yes	characteristic, complemented, Sylow
81	C₂²×Q₈	C₂×D₃₇	1	2	yes	—
82	C₂×Q₁₆	Dic₃₇	1	8	yes	complemented
83	C₂×Q₁₆	C₂×D₃₇	1	2	yes	—
84	C₂×Q₁₆	C₂×D₃₇	1	2	yes	—
85	C₂×M₁₆	—	37	1	no	—
86	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic
87	C₄×Q₈	—	37	8	no	—
88	C₄×Q₈	—	74	4	no	—
89	#117	—	37	1	no	—
90	#117	—	37	2	no	—
91	#117	—	74	4	no	—
92	#107	—	37	2	no	—
93	#106	—	37	2	no	—
94	#103	—	37	8	no	—
95	#97	—	37	2	no	—
96	C₄×C₈	—	37	2	no	—
97	C₂×Q₈	—	2	2	no	—
98	C₂×Q₈	—	2	4	no	—
99	C₂×Q₈	C₂×Dic₃₇	1	8	yes	—
100	C₂×Q₈	C₂²×D₃₇	1	4	yes	—
101	C₂²×C₄	—	2	2	no	—
102	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
103	Q₁₆	C₂×Dic₃₇	1	8	yes	—
104	Q₁₆	C₂×Dic₃₇	1	8	yes	—
105	M₁₆	—	74	2	no	—
106	C₂×C₈	—	37	2	no	—
107	C₂×C₈	C₂×Dic₃₇	1	4	yes	—
108	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
109	C₂×C₈	C₂²×D₃₇	1	1	yes	characteristic
110	C₄⋊C₄	—	37	2	no	—
111	C₄⋊C₄	—	37	4	no	—
112	C₄⋊C₄	—	74	8	no	—
113	C₂²⋊C₄	—	37	2	no	—
114	C₂²⋊C₄	—	74	2	no	—
115	C₄²	—	37	2	no	—
116	C₄²	—	74	8	no	—
117	C₄²	—	74	2	no	—
118	C₂³	D₄×D₃₇	1	1	yes	characteristic
119	Q₈	—	2	8	no	—
120	Q₈	—	2	4	no	—
121	Q₈	C₂²×Dic₃₇	1	8	yes	—
122	C₂×C₄	—	2	8	no	—
123	C₂×C₄	—	2	4	no	—
124	C₂×C₄	—	37	4	no	—
125	C₂×C₄	—	74	4	no	—
126	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
127	C₂×C₄	#10596964	1	2	yes	—
128	C₂×C₄	C₂²×Dic₃₇	1	2	yes	—
129	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
130	C₈	—	74	2	no	—
131	C₈	C₂²×Dic₃₇	1	2	yes	—
132	C₈	C₂²×Dic₃₇	1	2	yes	—
133	C₂²	—	2	2	no	—
134	C₂²	D₄×Dic₃₇	1	2	yes	—
135	C₂²	C₂×D₄×D₃₇	1	1	yes	characteristic
136	C₄	—	2	8	no	—
137	C₄	—	74	4	no	—
138	C₄	—	148	4	no	—
139	C₄	D₄×Dic₃₇	1	2	yes	—
140	C₄	C₂³×Dic₃₇	1	1	yes	characteristic
141	C₄	C₂×#10596964	1	1	yes	characteristic
142	C₂	—	2	2	no	—
143	C₂	C₂×D₄×Dic₃₇	1	1	yes	characteristic
144	C₂	#14558346	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#14558011

C₂

yes

maximal

#14557640

C₂

yes

maximal, complemented

Q₁₆×Dic₃₇

C₂

yes

maximal, complemented

#14557571

C₂

yes

maximal, characteristic

#14557542

C₂

yes

maximal

#14557488

C₂

yes

maximal, characteristic

C₂²×Q₁₆×C₃₇

C₂

yes

maximal, characteristic

C₂×#10596934

C₂²

yes

characteristic

Q₈×Dic₃₇

—

Q₈×Dic₃₇

C₂²

yes

—

#11924495

—

#11924495

C₂²

yes

—

#11924495

C₂²

yes

characteristic

#11924470

C₂²

yes

—

#11924468

C₂²

yes

—

#11924460

C₂²

yes

—

#11924446

C₂²

yes

—

C₈×Dic₃₇

C₂²

yes

—

C₂²×Q₈×C₃₇

C₂²

yes

—

C₂×Q₁₆×C₃₇

C₄

yes

complemented

C₂×Q₁₆×C₃₇

C₂²

yes

—

C₂×Q₁₆×C₃₇

C₂²

yes

—

C₂²×C₈×C₃₇

C₂²

yes

characteristic

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

C₂×C₄

yes

—

C₂×Q₈×C₃₇

C₂³

yes

—

C₂²×C₄×C₃₇

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

Q₁₆×C₃₇

C₂×C₄

yes

—

Q₁₆×C₃₇

C₂×C₄

yes

—

C₂×C₈×C₃₇

C₂×C₄

yes

—

C₂×C₈×C₃₇

C₂³

yes

characteristic

C₂×C₈×C₃₇

C₂³

yes

characteristic

C₂²⋊Dic₃₇

—

C₂²⋊Dic₃₇

D₄

yes

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

D₄

yes

—

C₄⋊Dic₃₇

C₂³

yes

—

C₄×Dic₃₇

—

C₄×Dic₃₇

—

C₄×Dic₃₇

C₂³

yes

—

#10596934

—

C₂×#64436

C₂³

yes

—

C₂³×C₃₇

C₂×D₄

yes

characteristic

Q₈×C₃₇

—

Q₈×C₃₇

—

Q₈×C₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²×C₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

#41

yes

—

C₂×C₄×C₃₇

C₂⁴

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

C₂×D₄

yes

—

C₈×C₃₇

C₂²×C₄

yes

—

C₈×C₃₇

C₂²×C₄

yes

—

#64436

—

C₂²×C₃₇

—

C₂²×C₃₇

C₄×D₄

yes

—

C₂²×C₃₇

C₂²×D₄

yes

characteristic

C₄×C₃₇

—

C₄×C₃₇

C₄×D₄

yes

—

C₄×C₃₇

C₂³×C₄

yes

characteristic

C₄×C₃₇

C₂×#41

yes

characteristic

Dic₃₇

—

Dic₃₇

—

#2950

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

C₂×C₄×D₄

yes

characteristic

C₂×C₃₇

#578

yes

—

C₂²×Q₁₆

D₃₇

yes

characteristic

#519

—

#441

—

C₄×Q₁₆

—

#427

—

#419

—

#405

—

C₃₇

#2950

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₃₇

yes

—

C₂×Q₁₆

Dic₃₇

yes

complemented

C₂×Q₁₆

C₂×D₃₇

yes

—

C₂×Q₁₆

C₂×D₃₇

yes

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₃₇

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

—

—

—

—

—

—

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₃₇

yes

—

100

C₂×Q₈

C₂²×D₃₇

yes

—

101

C₂²×C₄

—

102

C₂²×C₄

C₂²×D₃₇

yes

characteristic

103

Q₁₆

C₂×Dic₃₇

yes

—

104

Q₁₆

C₂×Dic₃₇

yes

—

105

M₁₆

—

106

C₂×C₈

—

107

C₂×C₈

C₂×Dic₃₇

yes

—

108

C₂×C₈

C₂²×D₃₇

yes

characteristic

109

C₂×C₈

C₂²×D₃₇

yes

characteristic

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₂²⋊C₄

—

114

C₂²⋊C₄

—

115

C₄²

—

116

C₄²

—

117

C₄²

—

118

C₂³

D₄×D₃₇

yes

characteristic

119

Q₈

—

120

Q₈

—

121

Q₈

C₂²×Dic₃₇

yes

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

D₄×D₃₇

yes

characteristic

127

C₂×C₄

#10596964

yes

—

128

C₂×C₄

C₂²×Dic₃₇

yes

—

129

C₂×C₄

C₂³×D₃₇

yes

characteristic

130

C₈

—

131

C₈

C₂²×Dic₃₇

yes

—

132

C₈

C₂²×Dic₃₇

yes

—

133

C₂²

—

134

C₂²

D₄×Dic₃₇

yes

—

135

C₂²

C₂×D₄×D₃₇

yes

characteristic

136

C₄

—

137

C₄

—

138

C₄

—

148

—

139

C₄

D₄×Dic₃₇

yes

—

140

C₄

C₂³×Dic₃₇

yes

characteristic

141

C₄

C₂×#10596964

yes

characteristic

142

C₂

—

143

C₂

C₂×D₄×Dic₃₇

yes

characteristic

144

C₂

#14558346

yes

—

	subgroup	G/H
center	C₂²	C₂×D₄×D₃₇
commutator	C₄×C₃₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	C₂²×Q₁₆×C₃₇	C₂
socle	C₂²×C₃₇	C₂²×D₄
prime radical	C₂³×C₃₇	C₂×D₄
cosocle	C₂×C₄×C₃₇	C₂⁴
hypercenter	C₂²×Q₁₆	D₃₇
hypocenter	C₃₇	#2950
LCS	G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

subgroup

G/H

center

C₂²

C₂×D₄×D₃₇

commutator

C₄×C₃₇

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₃₇

Fitting

C₂²×Q₁₆×C₃₇

C₂

socle

C₂²×C₃₇

C₂²×D₄

prime radical

C₂³×C₃₇

C₂×D₄

cosocle

C₂×C₄×C₃₇

C₂⁴

hypercenter

C₂²×Q₁₆

D₃₇

hypocenter

C₃₇

#2950

LCS

G ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

N	H ≅ G/N	type
2. #14557640	C₂	semidirect
3. Q₁₆×Dic₃₇	C₂	semidirect
20. C₂×Q₁₆×C₃₇	C₄	semidirect
80. C₃₇	#2950	semidirect
82. C₂×Q₁₆	Dic₃₇	semidirect

H ≅ G/N

type

2. #14557640

C₂

semidirect

3. Q₁₆×Dic₃₇

C₂

semidirect

20. C₂×Q₁₆×C₃₇

C₄

semidirect

80. C₃₇

#2950

semidirect

82. C₂×Q₁₆

Dic₃₇

semidirect

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

subgroup	index
#14558011	2
#14557640	2
Q₁₆×Dic₃₇	2
#14557571	2
#14557542	2
#14557488	2
C₂²×Q₁₆×C₃₇	2
#2950	37

subgroup

index

Q₁₆×Dic₃₇

C₂²×Q₁₆×C₃₇

Subgroups — #18003464

Conjugacy classes of subgroups

Complements

Maximal subgroups