Group #17995118 — subgroups

Subgroups — #17995118

Order 4736.

312 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 89 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	#14558104	C₂	1	1	yes	maximal, characteristic
2	#14557543	C₂	1	2	yes	maximal, complemented
3	#14557507	C₂	1	1	yes	maximal, characteristic
4	#14557290	C₂	1	2	yes	maximal, complemented
5	C₃₇×#568	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₇₄	C₂²	1	1	yes	characteristic
7	#11924534	C₄	1	2	yes	complemented
8	#11924523	C₄	1	2	yes	complemented
9	#11924517	—	2	2	no	—
10	#11924515	—	2	2	no	—
11	#11924495	C₂²	1	2	yes	—
12	#11924495	C₂²	1	1	yes	characteristic
13	#11924461	—	2	4	no	—
14	#11924451	—	2	2	no	—
15	C₂×#41×C₃₇	C₂²	1	1	yes	characteristic
16	#131×C₃₇	—	2	4	no	—
17	C₂×M₁₆×C₃₇	—	2	2	no	—
18	C₂×M₁₆×C₃₇	C₂²	1	2	yes	—
19	#41×C₃₇	—	2	2	no	—
20	#41×C₃₇	—	2	2	no	—
21	C₂×Q₈×C₃₇	C₂×C₄	1	1	yes	characteristic
22	C₂×D₄×C₃₇	C₂×C₄	1	1	yes	characteristic
23	C₂×D₄×C₃₇	D₄	1	2	yes	—
24	C₂²×C₄×C₃₇	D₄	1	2	yes	—
25	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
26	C₂²×Dic₃₇	—	2	1	no	—
27	C₂×Dic₇₄	—	2	2	no	—
28	C₂×Dic₇₄	—	2	2	no	—
29	C₂×Dic₇₄	—	2	1	no	—
30	C₂×Dic₇₄	C₂×C₄	1	2	yes	—
31	M₁₆×C₃₇	—	2	4	no	—
32	M₁₆×C₃₇	—	2	2	no	—
33	M₁₆×C₃₇	—	2	2	no	—
34	C₂×C₈×C₃₇	—	2	4	no	—
35	C₂×C₈×C₃₇	D₄	1	4	yes	—
36	C₂²⋊Dic₃₇	—	2	4	no	—
37	C₂²⋊Dic₃₇	—	2	1	no	—
38	C₂²⋊Dic₃₇	—	2	2	no	—
39	C₄⋊Dic₃₇	—	2	1	no	—
40	C₄⋊Dic₃₇	—	2	2	no	—
41	#10596936	—	2	4	no	—
42	C₄×Dic₃₇	—	2	2	no	—
43	C₄×Dic₃₇	C₂×C₄	1	2	yes	—
44	C₂³×C₃₇	C₂²⋊C₄	1	2	yes	—
45	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
46	Q₈×C₃₇	—	2	2	no	—
47	D₄×C₃₇	—	2	4	no	—
48	D₄×C₃₇	—	2	2	no	—
49	C₂×C₄×C₃₇	—	2	2	no	—
50	C₂×C₄×C₃₇	—	2	2	no	—
51	C₂×C₄×C₃₇	C₂²⋊C₄	1	2	yes	—
52	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₃₇	#41	1	2	yes	—
56	C₂×Dic₃₇	—	2	2	no	—
57	C₂×Dic₃₇	—	2	2	no	—
58	C₂×Dic₃₇	—	4	2	no	—
59	C₂×Dic₃₇	—	4	2	no	—
60	Dic₇₄	—	2	4	no	—
61	Dic₇₄	—	4	1	no	—
62	Dic₇₄	—	4	1	no	—
63	C₈×C₃₇	—	2	4	no	—
64	C₂²×C₃₇	—	2	2	no	—
65	C₂²×C₃₇	—	4	1	no	—
66	C₂²×C₃₇	—	4	2	no	—
67	C₂²×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
68	C₂²×C₃₇	#121	1	2	yes	—
69	C₄×C₃₇	—	2	2	no	—
70	C₄×C₃₇	C₄×D₄	1	2	yes	—
71	C₄×C₃₇	#120	1	1	yes	characteristic
72	C₄×C₃₇	#123	1	1	yes	characteristic
73	Dic₃₇	—	4	2	no	—
74	Dic₃₇	—	4	2	no	—
75	Dic₃₇	—	8	2	no	—
76	#1909	—	37	1	no	maximal, Sylow
77	C₂×C₃₇	—	2	1	no	—
78	C₂×C₃₇	—	2	2	no	—
79	C₂×C₃₇	—	4	2	no	—
80	C₂×C₃₇	#386	1	1	yes	characteristic
81	#568	D₃₇	1	1	yes	characteristic
82	#536	—	37	1	no	—
83	#419	—	37	2	no	—
84	#410	—	37	1	no	—
85	H(3,ℤ/4)	—	37	2	no	—
86	C₃₇	#1909	1	1	yes	characteristic, complemented, Sylow
87	C₂×#41	C₂×D₃₇	1	1	yes	characteristic
88	C₂²×Q₈	—	37	1	no	—
89	#131	—	2	4	no	—
90	C₂×M₁₆	—	2	2	no	—
91	C₂×M₁₆	C₂×D₃₇	1	2	yes	—
92	C₄⋊Q₈	—	37	2	no	—
93	#124	—	37	2	no	—
94	#123	—	74	2	no	—
95	C₂²⋊Q₈	—	74	2	no	—
96	#117	—	37	1	no	—
97	#117	—	37	2	no	—
98	#103	—	74	4	no	—
99	#99	—	74	2	no	—
100	#41	—	2	2	no	—
101	#41	—	2	2	no	—
102	C₂×Q₈	—	37	2	no	—
103	C₂×Q₈	—	74	1	no	—
104	C₂×Q₈	—	74	2	no	—
105	C₂×Q₈	—	74	2	no	—
106	C₂×Q₈	C₄×D₃₇	1	1	yes	characteristic
107	C₂×D₄	C₄×D₃₇	1	1	yes	characteristic
108	C₂×D₄	D₁₄₈	1	2	yes	—
109	C₂²×C₄	—	74	1	no	—
110	C₂²×C₄	C₂²⋊D₃₇	1	2	yes	—
111	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
112	M₁₆	—	2	4	no	—
113	M₁₆	—	2	2	no	—
114	M₁₆	—	2	2	no	—
115	C₂×C₈	—	2	4	no	—
116	C₂×C₈	C₂²⋊D₃₇	1	4	yes	—
117	C₄⋊C₄	—	74	4	no	—
118	C₄⋊C₄	—	74	1	no	—
119	C₄⋊C₄	—	74	2	no	—
120	C₂²⋊C₄	—	74	4	no	—
121	C₂²⋊C₄	—	74	1	no	—
122	C₂²⋊C₄	—	74	2	no	—
123	C₄²	—	37	2	no	—
124	C₄²	—	74	2	no	—
125	C₂³	#10596938	1	2	yes	—
126	C₂³	C₂×D₁₄₈	1	1	yes	characteristic
127	Q₈	—	2	2	no	—
128	Q₈	—	74	4	no	—
129	Q₈	—	148	1	no	—
130	Q₈	—	148	1	no	—
131	D₄	—	2	4	no	—
132	D₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	74	2	no	—
136	C₂×C₄	—	74	2	no	—
137	C₂×C₄	—	148	2	no	—
138	C₂×C₄	—	148	2	no	—
139	C₂×C₄	#10596938	1	2	yes	—
140	C₂×C₄	C₂×C₄×D₃₇	1	1	yes	characteristic
141	C₂×C₄	#10596962	1	2	yes	—
142	C₂×C₄	C₂×C₂²⋊D₃₇	1	2	yes	—
143	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
144	C₈	—	2	4	no	—
145	C₂²	—	2	2	no	—
146	C₂²	—	4	1	no	—
147	C₂²	—	4	2	no	—
148	C₂²	C₂×#10596938	1	1	yes	characteristic
149	C₂²	#11924510	1	2	yes	—
150	C₄	—	2	2	no	—
151	C₄	—	296	2	no	—
152	C₄	—	148	2	no	—
153	C₄	—	148	2	no	—
154	C₄	C₄×C₂²⋊D₃₇	1	2	yes	—
155	C₄	#11924505	1	1	yes	characteristic
156	C₄	#11924519	1	1	yes	characteristic
157	C₂	—	2	2	no	—
158	C₂	—	2	1	no	—
159	C₂	—	4	2	no	—
160	C₂	#14557412	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14558104

C₂

yes

maximal, characteristic

#14557543

C₂

yes

maximal, complemented

#14557507

C₂

yes

maximal, characteristic

#14557290

C₂

yes

maximal, complemented

C₃₇×#568

C₂

yes

maximal, characteristic

C₂²×Dic₇₄

C₂²

yes

characteristic

#11924534

C₄

yes

complemented

#11924523

C₄

yes

—

—

C₂²

yes

—

#11924495

C₂²

yes

characteristic

#11924461

—

#11924451

—

C₂×#41×C₃₇

C₂²

yes

characteristic

#131×C₃₇

—

C₂×M₁₆×C₃₇

—

C₂×M₁₆×C₃₇

C₂²

yes

—

#41×C₃₇

—

#41×C₃₇

—

C₂×Q₈×C₃₇

C₂×C₄

yes

characteristic

C₂×D₄×C₃₇

C₂×C₄

yes

characteristic

C₂×D₄×C₃₇

D₄

yes

—

C₂²×C₄×C₃₇

D₄

yes

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

—

C₂×Dic₇₄

—

C₂×Dic₇₄

—

C₂×Dic₇₄

—

C₂×Dic₇₄

C₂×C₄

yes

—

M₁₆×C₃₇

—

M₁₆×C₃₇

—

M₁₆×C₃₇

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

D₄

yes

—

C₂²⋊Dic₃₇

—

C₂²⋊Dic₃₇

—

C₂²⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

#10596936

—

C₄×Dic₃₇

—

C₄×Dic₃₇

C₂×C₄

yes

—

C₂³×C₃₇

C₂²⋊C₄

yes

—

C₂³×C₃₇

C₂×D₄

yes

characteristic

Q₈×C₃₇

—

D₄×C₃₇

—

D₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²⋊C₄

yes

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

#41

yes

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

Dic₇₄

—

Dic₇₄

—

Dic₇₄

—

C₈×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₃₇

#121

yes

—

C₄×C₃₇

—

C₄×C₃₇

C₄×D₄

yes

—

C₄×C₃₇

yes

C₄×C₃₇

yes

Dic₃₇

—

Dic₃₇

—

Dic₃₇

—

#1909

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

—

C₂×C₃₇

—

C₂×C₃₇

#386

yes

characteristic

#568

D₃₇

yes

—

—

—

—

C₃₇

#1909

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₃₇

yes

characteristic

C₂²×Q₈

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₃₇

yes

—

C₄⋊Q₈

—

#124

—

#123

—

C₂²⋊Q₈

—

—

—

—

—

100

#41

—

101

#41

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

C₄×D₃₇

yes

characteristic

107

C₂×D₄

C₄×D₃₇

yes

characteristic

108

C₂×D₄

D₁₄₈

yes

—

109

C₂²×C₄

—

110

C₂²×C₄

C₂²⋊D₃₇

yes

—

111

C₂²×C₄

C₂²×D₃₇

yes

characteristic

112

M₁₆

—

113

M₁₆

—

114

M₁₆

—

115

C₂×C₈

—

116

C₂×C₈

C₂²⋊D₃₇

yes

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₂²⋊C₄

—

121

C₂²⋊C₄

—

122

C₂²⋊C₄

—

123

C₄²

—

124

C₄²

—

125

C₂³

#10596938

yes

—

126

C₂³

C₂×D₁₄₈

yes

characteristic

127

Q₈

—

128

Q₈

—

129

Q₈

—

148

—

130

Q₈

—

148

—

131

D₄

—

132

D₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

148

—

138

C₂×C₄

—

148

—

139

C₂×C₄

#10596938

yes

—

140

C₂×C₄

C₂×C₄×D₃₇

yes

characteristic

141

C₂×C₄

#10596962

yes

—

142

C₂×C₄

C₂×C₂²⋊D₃₇

yes

—

143

C₂×C₄

C₂×C₂²⋊D₃₇

yes

characteristic

144

C₈

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

C₂×#10596938

yes

characteristic

149

C₂²

#11924510

yes

—

150

C₄

—

151

C₄

—

296

—

152

C₄

—

148

—

153

C₄

—

148

—

154

C₄

C₄×C₂²⋊D₃₇

yes

—

155

C₄

#11924505

yes

characteristic

156

C₄

#11924519

yes

characteristic

157

C₂

—

158

C₂

—

159

C₂

—

160

C₂

#14557412

yes

characteristic

	subgroup	G/H
center	C₂	#14557412
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting	C₃₇×#568	C₂
socle	C₂×C₃₇	#386
prime radical	C₂×#41×C₃₇	C₂²
cosocle	C₂²×C₄×C₃₇	C₂³
hypercenter	#568	D₃₇
hypocenter	C₃₇	#1909
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #568

subgroup

G/H

center

C₂

#14557412

commutator

C₂×C₄×C₃₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₇

Fitting

C₃₇×#568

C₂

socle

C₂×C₃₇

#386

prime radical

C₂×#41×C₃₇

C₂²

cosocle

C₂²×C₄×C₃₇

C₂³

D₃₇

C₃₇

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #568

N	H ≅ G/N	type
2. #14557543	C₂	semidirect
4. #14557290	C₂	semidirect
7. #11924534	C₄	semidirect
8. #11924523	C₄	semidirect
86. C₃₇	#1909	semidirect

H ≅ G/N

type

2. #14557543

C₂

semidirect

4. #14557290

C₂

semidirect

7. #11924534

C₄

semidirect

8. #11924523

C₄

semidirect

86. C₃₇

#1909

semidirect

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

subgroup	index
#14558104	2
#14557543	2
#14557507	2
#14557290	2
C₃₇×#568	2
#1909	37

subgroup

index

C₃₇×#568

Subgroups — #17995118

Conjugacy classes of subgroups

Complements

Maximal subgroups