Group #17994641 — subgroups

Subgroups — #17994641

Order 4736.

260 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 81 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	C₂×#11924544	C₂	1	1	yes	maximal, characteristic
2	#14557569	C₂	1	2	yes	maximal, complemented
3	#14557506	C₂	1	2	yes	maximal
4	#14557290	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₇×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10596962	C₂²	1	1	yes	characteristic
7	#11924544	—	2	2	no	—
8	#11924544	—	2	2	no	—
9	C₂×#10596929	C₄	1	2	yes	complemented
10	C₂²×#64436	C₂²	1	1	yes	characteristic
11	C₂×C₈×D₃₇	C₄	1	2	yes	complemented
12	#11924495	C₂²	1	2	yes	—
13	#11924469	—	2	2	no	—
14	#11924467	—	2	2	no	—
15	#11924450	—	2	4	no	—
16	C₂×M₁₆×C₃₇	C₂²	1	1	yes	characteristic
17	#117×C₃₇	C₂²	1	2	yes	—
18	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
19	C₂×C₂²⋊D₃₇	D₄	1	2	yes	—
20	#10596962	—	2	2	no	—
21	#10596962	—	2	2	no	—
22	C₂×D₁₄₈	Q₈	1	1	yes	characteristic
23	C₂×C₄×D₃₇	C₂×C₄	1	2	yes	—
24	C₂×Dic₇₄	Q₈	1	1	yes	characteristic
25	M₁₆×C₃₇	—	2	2	no	—
26	C₂×C₈×C₃₇	C₂×C₄	1	2	yes	—
27	C₄⋊C₄×C₃₇	—	2	2	no	—
28	C₂²⋊C₄×C₃₇	—	2	2	no	—
29	C₄²×C₃₇	—	2	2	no	—
30	C₂²⋊Dic₃₇	—	2	2	no	—
31	C₄⋊Dic₃₇	—	2	2	no	—
32	C₄×Dic₃₇	—	2	2	no	—
33	C₂×#64436	C₂×C₄	1	1	yes	characteristic
34	C₂×#64436	C₂×C₄	1	1	yes	characteristic
35	C₂×#64436	D₄	1	2	yes	—
36	C₂×#64436	D₄	1	2	yes	—
37	#10596929	—	2	4	no	—
38	C₈×D₃₇	—	2	2	no	—
39	C₈×D₃₇	—	2	2	no	—
40	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
41	C₂²×D₃₇	C₄⋊C₄	1	2	yes	—
42	C₂×C₄×C₃₇	—	4	2	no	—
43	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₃₇	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₃₇	#41	1	2	yes	—
47	C₂²⋊D₃₇	—	2	4	no	—
48	C₂×Dic₃₇	—	4	2	no	—
49	C₂×Dic₃₇	C₄⋊C₄	1	2	yes	—
50	D₁₄₈	—	2	2	no	—
51	C₄×D₃₇	—	2	2	no	—
52	C₄×D₃₇	—	2	2	no	—
53	Dic₇₄	—	2	2	no	—
54	C₈×C₃₇	—	2	2	no	—
55	#64436	C₂²⋊C₄	1	1	yes	characteristic
56	#64436	C₂²⋊C₄	1	1	yes	characteristic
57	#64436	C₂²⋊C₄	1	2	yes	—
58	C₂²×C₃₇	—	4	1	no	—
59	C₂²×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
60	C₂²×C₃₇	#121	1	2	yes	—
61	C₂×D₃₇	—	2	2	no	—
62	C₂×D₃₇	—	4	2	no	—
63	C₄×C₃₇	—	8	2	no	—
64	C₄×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₄×C₃₇	#117	1	1	yes	characteristic
66	C₄×C₃₇	C₂²⋊Q₈	1	2	yes	—
67	Dic₃₇	—	2	2	no	—
68	Dic₃₇	—	8	2	no	—
69	#1812	—	37	1	no	maximal, Sylow
70	C₂×C₃₇	—	2	2	no	—
71	C₂×C₃₇	—	2	1	no	—
72	C₂×C₃₇	#380	1	1	yes	characteristic
73	D₃₇	—	4	2	no	—
74	C₂×#131	—	37	1	no	—
75	#427	—	37	2	no	—
76	#410	—	37	2	no	—
77	H(3,ℤ/4)	—	37	1	no	—
78	H(3,ℤ/4)	D₃₇	1	1	yes	characteristic, complemented
79	C₃₇	#1812	1	1	yes	characteristic, complemented, Sylow
80	C₂×#41	—	37	1	no	—
81	#131	—	74	2	no	—
82	#131	—	74	2	no	—
83	C₂×M₁₆	—	37	2	no	—
84	C₂×M₁₆	C₂×D₃₇	1	1	yes	characteristic
85	C₂²×C₈	—	37	1	no	—
86	C₂²×C₈	—	37	2	no	—
87	#117	—	37	2	no	—
88	#117	C₂×D₃₇	1	2	yes	—
89	#107	—	74	2	no	—
90	#106	—	74	2	no	—
91	#99	—	74	4	no	—
92	#41	—	74	2	no	—
93	#41	—	74	2	no	—
94	C₂×Q₈	—	37	1	no	—
95	C₂×D₄	—	37	2	no	—
96	C₂×D₄	—	37	1	no	—
97	C₂²×C₄	—	37	2	no	—
98	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
99	M₁₆	—	2	2	no	—
100	M₁₆	—	74	4	no	—
101	C₂×C₈	—	37	1	no	—
102	C₂×C₈	—	37	1	no	—
103	C₂×C₈	—	37	2	no	—
104	C₂×C₈	—	37	2	no	—
105	C₂×C₈	—	74	2	no	—
106	C₂×C₈	—	74	2	no	—
107	C₂×C₈	C₄×D₃₇	1	2	yes	—
108	C₄⋊C₄	—	2	2	no	—
109	C₄⋊C₄	—	74	2	no	—
110	C₂²⋊C₄	—	2	2	no	—
111	C₂²⋊C₄	—	74	2	no	—
112	C₄²	—	2	2	no	—
113	C₄²	—	74	2	no	—
114	C₂³	—	37	2	no	—
115	C₂³	D₄×D₃₇	1	1	yes	characteristic
116	Q₈	—	74	2	no	—
117	D₄	—	74	4	no	—
118	D₄	—	74	2	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	37	2	no	—
121	C₂×C₄	—	74	2	no	—
122	C₂×C₄	—	74	2	no	—
123	C₂×C₄	—	148	2	no	—
124	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
125	C₂×C₄	#10596962	1	2	yes	—
126	C₂×C₄	D₄×D₃₇	1	2	yes	—
127	C₂×C₄	Q₈×D₃₇	1	1	yes	characteristic
128	C₈	—	2	2	no	—
129	C₈	—	37	1	no	—
130	C₈	—	37	1	no	—
131	C₈	—	37	2	no	—
132	C₂²	—	4	1	no	—
133	C₂²	—	74	2	no	—
134	C₂²	—	148	2	no	—
135	C₂²	C₄⋊C₄×D₃₇	1	1	yes	characteristic
136	C₂²	#11924509	1	2	yes	—
137	C₄	—	8	2	no	—
138	C₄	—	296	2	no	—
139	C₄	—	74	2	no	—
140	C₄	C₂²⋊C₄×D₃₇	1	1	yes	characteristic
141	C₄	#11924493	1	1	yes	characteristic
142	C₄	#11924514	1	2	yes	—
143	C₂	—	2	2	no	—
144	C₂	—	2	1	no	—
145	C₂	—	148	2	no	—
146	C₂	#14557392	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924544

C₂

yes

maximal, characteristic

#14557569

C₂

yes

maximal, complemented

#14557506

C₂

yes

maximal

#14557290

C₂

yes

maximal, characteristic, complemented

C₃₇×H(3,ℤ/4)

C₂

yes

maximal, characteristic, complemented

C₂×#10596962

C₂²

yes

—

—

C₄

yes

complemented

C₂²×#64436

C₂²

yes

characteristic

C₂×C₈×D₃₇

C₄

yes

complemented

#11924495

C₂²

yes

—

#11924469

—

#11924467

—

#11924450

—

C₂×M₁₆×C₃₇

C₂²

yes

characteristic

#117×C₃₇

C₂²

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂×C₂²⋊D₃₇

D₄

yes

—

#10596962

—

#10596962

—

C₂×D₁₄₈

Q₈

yes

characteristic

C₂×C₄×D₃₇

C₂×C₄

yes

—

C₂×Dic₇₄

Q₈

yes

characteristic

M₁₆×C₃₇

—

C₂×C₈×C₃₇

C₂×C₄

yes

—

C₄⋊C₄×C₃₇

—

C₂²⋊C₄×C₃₇

—

C₄²×C₃₇

—

C₂²⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄×Dic₃₇

—

C₂×#64436

C₂×C₄

yes

characteristic

C₂×#64436

C₂×C₄

yes

characteristic

C₂×#64436

D₄

yes

—

C₂×#64436

D₄

yes

—

#10596929

—

C₈×D₃₇

—

C₈×D₃₇

—

C₂³×C₃₇

C₂×D₄

yes

characteristic

C₂²×D₃₇

C₄⋊C₄

yes

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₇

#41

yes

—

C₂²⋊D₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

C₄⋊C₄

yes

—

D₁₄₈

—

C₄×D₃₇

—

C₄×D₃₇

—

Dic₇₄

—

C₈×C₃₇

—

#64436

C₂²⋊C₄

yes

characteristic

#64436

C₂²⋊C₄

yes

characteristic

#64436

C₂²⋊C₄

yes

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃₇

#121

yes

—

C₂×D₃₇

—

C₂×D₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₃₇

#117

yes

characteristic

C₄×C₃₇

C₂²⋊Q₈

yes

—

Dic₃₇

—

Dic₃₇

—

#1812

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

—

C₂×C₃₇

#380

yes

characteristic

D₃₇

—

—

—

—

—

D₃₇

yes

characteristic, complemented

C₃₇

#1812

yes

characteristic, complemented, Sylow

C₂×#41

—

#131

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₃₇

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

#117

—

#117

C₂×D₃₇

yes

—

—

—

—

—

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₃₇

yes

characteristic

M₁₆

—

100

M₁₆

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

C₄×D₃₇

yes

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₂²⋊C₄

—

111

C₂²⋊C₄

—

112

C₄²

—

113

C₄²

—

114

C₂³

—

115

C₂³

D₄×D₃₇

yes

characteristic

116

Q₈

—

117

D₄

—

118

D₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

148

—

124

C₂×C₄

C₂×C₄×D₃₇

yes

characteristic

125

C₂×C₄

#10596962

yes

—

126

C₂×C₄

D₄×D₃₇

yes

—

127

C₂×C₄

Q₈×D₃₇

yes

characteristic

128

C₈

—

129

C₈

—

130

C₈

—

131

C₈

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

148

—

135

C₂²

C₄⋊C₄×D₃₇

yes

characteristic

136

C₂²

#11924509

yes

—

137

C₄

—

138

C₄

—

296

—

139

C₄

—

140

C₄

C₂²⋊C₄×D₃₇

yes

characteristic

141

C₄

#11924493

yes

characteristic

142

C₄

#11924514

yes

—

143

C₂

—

144

C₂

—

145

C₂

—

148

—

146

C₂

#14557392

yes

characteristic

	subgroup	G/H
center	C₄	#11924493
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting	C₃₇×H(3,ℤ/4)	C₂
socle	C₂×C₃₇	#380
prime radical	C₂×#10596962	C₂²
cosocle	C₂²×C₄×C₃₇	C₂³
hypercenter	H(3,ℤ/4)	D₃₇
hypocenter	C₃₇	#1812
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#11924493

commutator

C₂×C₄×C₃₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₇

Fitting

C₃₇×H(3,ℤ/4)

C₂

C₂×C₃₇

C₂²

C₂²×C₄×C₃₇

C₂³

D₃₇

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
2. #14557569	C₂	semidirect
4. #14557290	C₂	semidirect
5. C₃₇×H(3,ℤ/4)	C₂	semidirect
9. C₂×#10596929	C₄	semidirect
11. C₂×C₈×D₃₇	C₄	semidirect
78. H(3,ℤ/4)	D₃₇	semidirect
79. C₃₇	#1812	semidirect

H ≅ G/N

type

2. #14557569

C₂

semidirect

4. #14557290

C₂

semidirect

5. C₃₇×H(3,ℤ/4)

C₂

semidirect

9. C₂×#10596929

C₄

semidirect

11. C₂×C₈×D₃₇

C₄

semidirect

78. H(3,ℤ/4)

D₃₇

semidirect

79. C₃₇

#1812

semidirect

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

subgroup	index
C₂×#11924544	2
#14557569	2
#14557506	2
#14557290	2
C₃₇×H(3,ℤ/4)	2
#1812	37

subgroup

index

C₃₇×H(3,ℤ/4)

Subgroups — #17994641

Conjugacy classes of subgroups

Complements

Maximal subgroups