Group #17994591 — subgroups

Subgroups — #17994591

Order 4736.

272 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 87 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×#11924501	C₂	1	1	yes	maximal, characteristic
2	#14557556	C₂	1	2	yes	maximal
3	C₂×#11924454	C₂	1	2	yes	maximal
4	#14557302	C₂	1	1	yes	maximal, characteristic
5	C₃₇×#343	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₇₄	C₂²	1	1	yes	characteristic
7	C₂×#10596934	C₂²	1	2	yes	—
8	#11924501	—	2	2	no	—
9	#11924501	—	2	2	no	—
10	C₂×#10596936	C₄	1	2	yes	—
11	C₂×C₄×Dic₃₇	C₄	1	2	yes	—
12	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
13	#11924465	—	2	2	no	—
14	#11924465	—	2	2	no	—
15	#11924454	—	2	4	no	—
16	C₂×M₁₆×C₃₇	C₂²	1	2	yes	—
17	C₂×C₄⋊C₄×C₃₇	C₂²	1	1	yes	characteristic
18	C₂²×C₄×C₃₇	C₂×C₄	1	2	yes	—
19	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
20	C₂²×Dic₃₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×Dic₃₇	C₂×C₄	1	2	yes	—
22	C₂²×Dic₃₇	C₂×C₄	1	1	yes	characteristic
23	C₂×Dic₇₄	—	2	2	no	—
24	C₂×Dic₇₄	—	2	2	no	—
25	C₂×Dic₇₄	D₄	1	2	yes	—
26	C₂×Dic₇₄	Q₈	1	2	yes	—
27	M₁₆×C₃₇	—	2	2	no	—
28	M₁₆×C₃₇	—	2	2	no	—
29	C₂×C₈×C₃₇	—	2	2	no	—
30	C₄⋊C₄×C₃₇	—	2	2	no	—
31	#10596936	—	2	4	no	—
32	C₄×Dic₃₇	—	2	2	no	—
33	C₄×Dic₃₇	—	2	2	no	—
34	C₄×Dic₃₇	D₄	1	2	yes	—
35	C₄×Dic₃₇	D₄	1	2	yes	—
36	#10596934	—	2	2	no	—
37	#10596934	—	2	2	no	—
38	C₂×#64436	—	2	2	no	—
39	C₂³×C₃₇	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₃₇	—	2	2	no	—
41	C₂×C₄×C₃₇	—	4	2	no	—
42	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₃₇	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₄×C₃₇	#41	1	2	yes	—
46	C₂×Dic₃₇	—	2	2	no	—
47	C₂×Dic₃₇	—	2	2	no	—
48	C₂×Dic₃₇	—	2	2	no	—
49	C₂×Dic₃₇	—	4	1	no	—
50	C₂×Dic₃₇	C₂²⋊C₄	1	1	yes	characteristic
51	C₂×Dic₃₇	C₂²⋊C₄	1	2	yes	—
52	C₂×Dic₃₇	C₂²⋊C₄	1	1	yes	characteristic
53	C₂×Dic₃₇	C₄⋊C₄	1	4	yes	—
54	Dic₇₄	—	2	4	no	—
55	Dic₇₄	—	2	4	no	—
56	C₈×C₃₇	—	4	2	no	—
57	#64436	—	4	2	no	—
58	C₂²×C₃₇	—	2	1	no	—
59	C₂²×C₃₇	—	2	1	no	—
60	C₂²×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
61	C₂²×C₃₇	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂²×C₃₇	#117	1	1	yes	characteristic
63	C₄×C₃₇	—	4	2	no	—
64	C₄×C₃₇	#121	1	2	yes	—
65	C₄×C₃₇	C₂²⋊Q₈	1	2	yes	—
66	Dic₃₇	—	2	4	no	—
67	Dic₃₇	—	2	1	no	—
68	Dic₃₇	—	2	2	no	—
69	Dic₃₇	#101	1	2	yes	complemented
70	#1801	—	37	1	no	maximal, Sylow
71	C₂×C₃₇	—	2	1	no	—
72	C₂×C₃₇	—	2	1	no	—
73	C₂×C₃₇	#380	1	1	yes	characteristic
74	C₂×C₃₇	C₂×#101	1	1	yes	characteristic
75	C₂×C₃₇	#413	1	1	yes	characteristic
76	C₂×C₄×Q₈	—	37	1	no	—
77	#423	—	37	2	no	—
78	C₂×#101	—	37	2	no	—
79	#343	—	37	1	no	—
80	#343	D₃₇	1	1	yes	characteristic
81	C₃₇	#1801	1	1	yes	characteristic, complemented, Sylow
82	C₂²×Q₈	—	37	1	no	—
83	C₂×M₁₆	—	37	2	no	—
84	C₂×M₁₆	C₂×D₃₇	1	2	yes	—
85	C₄×Q₈	—	74	2	no	—
86	C₄×Q₈	—	74	2	no	—
87	C₂×C₄⋊C₄	—	37	2	no	—
88	C₂×C₄⋊C₄	C₂×D₃₇	1	1	yes	characteristic
89	C₂×C₄²	—	37	1	no	—
90	C₂×C₄²	—	37	2	no	—
91	C₄⋊C₈	—	74	2	no	—
92	C₄⋊C₈	—	74	2	no	—
93	#101	—	74	4	no	—
94	C₂×Q₈	—	37	2	no	—
95	C₂×Q₈	—	37	2	no	—
96	C₂×Q₈	—	74	2	no	—
97	C₂×Q₈	—	74	2	no	—
98	C₂²×C₄	—	37	1	no	—
99	C₂²×C₄	—	37	2	no	—
100	C₂²×C₄	—	37	1	no	—
101	C₂²×C₄	C₄×D₃₇	1	2	yes	—
102	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
103	M₁₆	—	2	2	no	—
104	M₁₆	—	2	2	no	—
105	M₁₆	—	74	2	no	—
106	M₁₆	—	74	2	no	—
107	C₂×C₈	—	2	2	no	—
108	C₂×C₈	—	74	2	no	—
109	C₄⋊C₄	—	2	2	no	—
110	C₄⋊C₄	—	74	4	no	—
111	C₄²	—	37	2	no	—
112	C₄²	—	37	2	no	—
113	C₄²	—	74	2	no	—
114	C₄²	—	74	2	no	—
115	C₂³	C₂×C₄×D₃₇	1	1	yes	characteristic
116	Q₈	—	74	4	no	—
117	Q₈	—	74	4	no	—
118	C₂×C₄	—	2	2	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	—	37	1	no	—
121	C₂×C₄	—	37	1	no	—
122	C₂×C₄	—	37	2	no	—
123	C₂×C₄	—	37	4	no	—
124	C₂×C₄	—	74	2	no	—
125	C₂×C₄	—	74	2	no	—
126	C₂×C₄	—	74	2	no	—
127	C₂×C₄	—	148	1	no	—
128	C₂×C₄	#10596962	1	2	yes	—
129	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
130	C₂×C₄	D₄×D₃₇	1	2	yes	—
131	C₂×C₄	Q₈×D₃₇	1	1	yes	characteristic
132	C₈	—	4	2	no	—
133	C₈	—	148	2	no	—
134	C₂²	—	2	1	no	—
135	C₂²	—	2	1	no	—
136	C₂²	C₂²⋊C₄×D₃₇	1	1	yes	characteristic
137	C₂²	C₄⋊C₄×D₃₇	1	1	yes	characteristic
138	C₂²	#11924493	1	1	yes	characteristic
139	C₄	—	4	2	no	—
140	C₄	—	37	2	no	—
141	C₄	—	74	1	no	—
142	C₄	—	74	2	no	—
143	C₄	—	74	4	no	—
144	C₄	#11924509	1	2	yes	—
145	C₄	#11924514	1	2	yes	—
146	C₂	—	2	1	no	—
147	C₂	—	2	1	no	—
148	C₂	#14557392	1	1	yes	characteristic
149	C₂	#101×D₃₇	1	1	yes	characteristic
150	C₂	#14557514	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924501

C₂

yes

maximal, characteristic

#14557556

C₂

yes

maximal

C₂×#11924454

C₂

yes

maximal

#14557302

C₂

yes

maximal, characteristic

C₃₇×#343

C₂

yes

maximal, characteristic

C₂²×Dic₇₄

C₂²

yes

characteristic

C₂×#10596934

C₂²

yes

—

#11924501

—

#11924501

—

C₂×#10596936

C₄

yes

—

C₂×C₄×Dic₃₇

C₄

yes

—

C₂×C₄×Dic₃₇

C₂²

yes

—

—

—

C₂×M₁₆×C₃₇

C₂²

yes

—

C₂×C₄⋊C₄×C₃₇

C₂²

yes

characteristic

C₂²×C₄×C₃₇

C₂×C₄

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

C₂×C₄

yes

characteristic

C₂²×Dic₃₇

C₂×C₄

yes

—

C₂²×Dic₃₇

C₂×C₄

yes

characteristic

C₂×Dic₇₄

—

C₂×Dic₇₄

—

C₂×Dic₇₄

D₄

yes

—

C₂×Dic₇₄

Q₈

yes

—

M₁₆×C₃₇

—

M₁₆×C₃₇

—

C₂×C₈×C₃₇

—

C₄⋊C₄×C₃₇

—

#10596936

—

C₄×Dic₃₇

—

C₄×Dic₃₇

—

C₄×Dic₃₇

D₄

yes

—

C₄×Dic₃₇

D₄

yes

—

#10596934

—

#10596934

—

C₂×#64436

—

C₂³×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₃₇

#41

yes

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₃₇

C₂²⋊C₄

yes

—

C₂×Dic₃₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₃₇

C₄⋊C₄

yes

—

Dic₇₄

—

Dic₇₄

—

C₈×C₃₇

—

#64436

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₃₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₃₇

#117

yes

characteristic

C₄×C₃₇

—

C₄×C₃₇

#121

yes

—

C₄×C₃₇

C₂²⋊Q₈

yes

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

—

Dic₃₇

yes

—

C₂×C₃₇

—

C₂×C₃₇

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂×C₄×Q₈

—

—

—

—

D₃₇

yes

characteristic

C₃₇

#1801

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₃₇

yes

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₃₇

yes

characteristic

C₂×C₄²

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

#101

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

C₄×D₃₇

yes

—

102

C₂²×C₄

C₂²×D₃₇

yes

characteristic

103

M₁₆

—

104

M₁₆

—

105

M₁₆

—

106

M₁₆

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄²

—

112

C₄²

—

113

C₄²

—

114

C₄²

—

115

C₂³

C₂×C₄×D₃₇

yes

characteristic

116

Q₈

—

117

Q₈

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

148

—

128

C₂×C₄

#10596962

yes

—

129

C₂×C₄

D₄×D₃₇

yes

characteristic

130

C₂×C₄

D₄×D₃₇

yes

—

131

C₂×C₄

Q₈×D₃₇

yes

characteristic

132

C₈

—

133

C₈

—

148

—

134

C₂²

—

135

C₂²

—

136

C₂²

C₂²⋊C₄×D₃₇

yes

characteristic

137

C₂²

C₄⋊C₄×D₃₇

yes

characteristic

138

C₂²

#11924493

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

#11924509

yes

—

145

C₄

#11924514

yes

—

146

C₂

—

147

C₂

—

148

C₂

#14557392

yes

characteristic

149

C₂

#101×D₃₇

yes

characteristic

150

C₂

#14557514

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₃₇
commutator, prime radical	C₂³×C₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting	C₃₇×#343	C₂
socle	C₂²×C₃₇	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₃₇	C₂³
hypercenter	#343	D₃₇
hypocenter	C₃₇	#1801
LCS	G ▷ C₂³×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₃₇

commutator, prime radical

C₂³×C₃₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₇

Fitting

C₃₇×#343

C₂

socle

C₂²×C₃₇

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₃₇

C₂³

D₃₇

C₃₇

G ▷ C₂³×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

N	H ≅ G/N	type
69. Dic₃₇	#101	semidirect
81. C₃₇	#1801	semidirect

H ≅ G/N

type

69. Dic₃₇

#101

semidirect

81. C₃₇

#1801

semidirect

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

subgroup	index
C₂×#11924501	2
#14557556	2
C₂×#11924454	2
#14557302	2
C₃₇×#343	2
#1801	37

subgroup

index

C₃₇×#343

Subgroups — #17994591

Conjugacy classes of subgroups

Complements

Maximal subgroups