Group #17995717 — subgroups

Subgroups — #17995717

Order 4736.

342 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 73 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×#11924566	C₂	1	2	yes	maximal
2	C₂²×Dic₁₄₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#11924454	C₂	1	2	yes	maximal, complemented
4	#14557298	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₇×#408	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₇₄	C₂²	1	2	yes	—
7	#11924566	—	2	4	no	—
8	#11924566	—	2	4	no	—
9	C₂×Dic₁₄₈	—	2	4	no	—
10	C₂×Dic₁₄₈	—	2	2	no	—
11	C₂×#10596942	—	2	2	no	—
12	C₂×#10596940	—	2	2	no	—
13	C₂×#10596934	C₂²	1	2	yes	—
14	#11924454	—	2	4	no	—
15	C₂×#41×C₃₇	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₃₇	C₂²	1	1	yes	characteristic
17	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
18	C₂²⋊C₈×C₃₇	—	2	2	no	—
19	#41×C₃₇	—	2	2	no	—
20	#41×C₃₇	—	2	2	no	—
21	C₂×Q₈×C₃₇	D₄	1	1	yes	characteristic
22	C₂×D₄×C₃₇	—	2	1	no	—
23	C₂×D₄×C₃₇	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₃₇	—	2	1	no	—
25	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
26	C₂²×Dic₃₇	—	2	2	no	—
27	C₂×Dic₇₄	—	2	2	no	—
28	C₂×Dic₇₄	—	2	2	no	—
29	C₂×Dic₇₄	—	2	2	no	—
30	C₂×Dic₇₄	—	2	2	no	—
31	C₂×Dic₇₄	—	2	2	no	—
32	C₂×Dic₇₄	D₄	1	4	yes	—
33	M₁₆×C₃₇	—	2	2	no	—
34	C₂×C₈×C₃₇	—	2	2	no	—
35	C₂×C₈×C₃₇	—	2	1	no	—
36	C₂×C₈×C₃₇	D₄	1	2	yes	—
37	#10596942	—	2	8	no	—
38	#10596940	—	2	8	no	—
39	#10596934	—	2	2	no	—
40	#10596934	D₄	1	4	yes	—
41	C₂×#64436	—	2	2	no	—
42	Dic₁₄₈	—	2	8	no	—
43	C₂³×C₃₇	—	2	1	no	—
44	C₂³×C₃₇	#41	1	1	yes	characteristic
45	Q₈×C₃₇	—	2	2	no	—
46	D₄×C₃₇	—	2	2	no	—
47	D₄×C₃₇	—	4	2	no	—
48	C₂×C₄×C₃₇	—	2	1	no	—
49	C₂×C₄×C₃₇	—	4	1	no	—
50	C₂×C₄×C₃₇	—	4	1	no	—
51	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
52	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
53	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
55	C₂×Dic₃₇	—	2	4	no	—
56	C₂×Dic₃₇	—	4	2	no	—
57	C₂×Dic₃₇	—	4	2	no	—
58	Dic₇₄	—	2	4	no	—
59	Dic₇₄	—	2	4	no	—
60	Dic₇₄	—	4	2	no	—
61	Dic₇₄	—	4	2	no	—
62	C₈×C₃₇	—	2	2	no	—
63	C₈×C₃₇	—	4	1	no	—
64	#64436	—	2	4	no	—
65	C₂²×C₃₇	—	2	2	no	—
66	C₂²×C₃₇	—	4	1	no	—
67	C₂²×C₃₇	—	4	2	no	—
68	C₂²×C₃₇	#121	1	2	yes	—
69	C₂²×C₃₇	#121	1	1	yes	characteristic
70	C₄×C₃₇	—	4	1	no	—
71	C₄×C₃₇	#120	1	1	yes	characteristic
72	C₄×C₃₇	#120	1	2	yes	—
73	C₄×C₃₇	#127	1	1	yes	characteristic
74	Dic₃₇	—	4	4	no	—
75	#2020	—	37	1	no	maximal, Sylow
76	C₂×C₃₇	—	2	2	no	—
77	C₂×C₃₇	—	8	1	no	—
78	C₂×C₃₇	#392	1	1	yes	characteristic
79	C₂×C₃₇	#473	1	2	yes	—
80	C₂×#137	—	37	2	no	—
81	C₂²×Q₁₆	—	37	1	no	—
82	C₂×#101	—	37	2	no	—
83	#408	D₃₇	1	1	yes	characteristic
84	#341	—	37	1	no	—
85	C₃₇	#2020	1	1	yes	characteristic, complemented, Sylow
86	C₂×#41	C₂×D₃₇	1	1	yes	characteristic
87	C₂²×Q₈	—	37	2	no	—
88	#137	—	74	4	no	—
89	#137	—	74	4	no	—
90	C₂×Q₁₆	—	74	4	no	—
91	C₂×Q₁₆	—	74	2	no	—
92	C₂×Q₁₆	—	74	2	no	—
93	C₂×SD₁₆	—	74	2	no	—
94	C₂×M₁₆	—	37	2	no	—
95	C₂×M₁₆	C₂×D₃₇	1	1	yes	characteristic
96	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic
97	#101	—	74	4	no	—
98	C₂²⋊C₈	—	2	2	no	—
99	#41	—	2	2	no	—
100	#41	—	2	2	no	—
101	C₂×Q₈	—	37	4	no	—
102	C₂×Q₈	—	74	2	no	—
103	C₂×Q₈	—	74	2	no	—
104	C₂×Q₈	—	74	2	no	—
105	C₂×Q₈	—	74	2	no	—
106	C₂×Q₈	—	74	2	no	—
107	C₂×Q₈	D₁₄₈	1	1	yes	characteristic
108	C₂×D₄	—	2	1	no	—
109	C₂×D₄	D₁₄₈	1	1	yes	characteristic
110	C₂²×C₄	—	2	1	no	—
111	C₂²×C₄	—	74	2	no	—
112	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
113	Q₁₆	—	74	8	no	—
114	Q₁₆	—	74	8	no	—
115	SD₁₆	—	74	8	no	—
116	M₁₆	—	2	2	no	—
117	M₁₆	—	37	4	no	—
118	M₁₆	—	74	2	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	2	1	no	—
121	C₂×C₈	—	74	2	no	—
122	C₂×C₈	C₂²⋊D₃₇	1	2	yes	—
123	C₂³	—	2	1	no	—
124	C₂³	#10596962	1	1	yes	characteristic
125	Q₈	—	2	2	no	—
126	Q₈	—	74	4	no	—
127	Q₈	—	74	4	no	—
128	Q₈	—	148	2	no	—
129	Q₈	—	148	2	no	—
130	D₄	—	2	2	no	—
131	D₄	—	4	2	no	—
132	C₂×C₄	—	2	1	no	—
133	C₂×C₄	—	4	1	no	—
134	C₂×C₄	—	4	1	no	—
135	C₂×C₄	—	74	4	no	—
136	C₂×C₄	—	148	2	no	—
137	C₂×C₄	—	148	2	no	—
138	C₂×C₄	C₂×D₁₄₈	1	1	yes	characteristic
139	C₂×C₄	D₄×D₃₇	1	2	yes	—
140	C₂×C₄	D₄×D₃₇	1	2	yes	—
141	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
142	C₈	—	2	2	no	—
143	C₈	—	4	1	no	—
144	C₈	—	74	4	no	—
145	C₂²	—	2	2	no	—
146	C₂²	—	4	1	no	—
147	C₂²	—	4	2	no	—
148	C₂²	#11924509	1	2	yes	—
149	C₂²	#11924510	1	1	yes	characteristic
150	C₄	—	4	1	no	—
151	C₄	—	148	4	no	—
152	C₄	#11924504	1	1	yes	characteristic
153	C₄	#11924506	1	2	yes	—
154	C₄	#11924532	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	—	8	1	no	—
157	C₂	#14557442	1	1	yes	characteristic
158	C₂	#14557808	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924566

C₂

yes

maximal

C₂²×Dic₁₄₈

C₂

yes

maximal, characteristic, complemented

C₂×#11924454

C₂

yes

maximal, complemented

#14557298

C₂

yes

maximal, characteristic, complemented

C₃₇×#408

C₂

yes

maximal, characteristic

C₂²×Dic₇₄

C₂²

yes

—

#11924566

—

#11924566

—

C₂×Dic₁₄₈

—

C₂×Dic₁₄₈

—

C₂×#10596942

—

C₂×#10596940

—

C₂×#10596934

C₂²

yes

—

#11924454

—

C₂×#41×C₃₇

C₂²

yes

characteristic

C₂×M₁₆×C₃₇

C₂²

yes

characteristic

C₂²×C₈×C₃₇

C₂²

yes

characteristic

C₂²⋊C₈×C₃₇

—

#41×C₃₇

—

#41×C₃₇

—

C₂×Q₈×C₃₇

D₄

yes

characteristic

C₂×D₄×C₃₇

—

C₂×D₄×C₃₇

D₄

yes

characteristic

C₂²×C₄×C₃₇

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

—

C₂×Dic₇₄

—

C₂×Dic₇₄

—

C₂×Dic₇₄

—

C₂×Dic₇₄

—

C₂×Dic₇₄

—

C₂×Dic₇₄

D₄

yes

—

M₁₆×C₃₇

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

D₄

yes

—

—

—

—

D₄

yes

—

C₂×#64436

—

Dic₁₄₈

—

C₂³×C₃₇

—

C₂³×C₃₇

#41

yes

characteristic

Q₈×C₃₇

—

D₄×C₃₇

—

D₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

Dic₇₄

—

Dic₇₄

—

Dic₇₄

—

Dic₇₄

—

C₈×C₃₇

—

C₈×C₃₇

—

#64436

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

#121

yes

—

C₂²×C₃₇

#121

yes

characteristic

C₄×C₃₇

—

C₄×C₃₇

#120

yes

characteristic

C₄×C₃₇

#120

yes

—

C₄×C₃₇

#127

yes

characteristic

Dic₃₇

—

#2020

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

—

C₂×C₃₇

#392

yes

characteristic

C₂×C₃₇

#473

yes

—

C₂×#137

—

C₂²×Q₁₆

—

C₂×#101

—

#408

D₃₇

yes

characteristic

#341

—

C₃₇

#2020

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₃₇

yes

characteristic

C₂²×Q₈

—

#137

—

#137

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₃₇

yes

characteristic

C₂²×C₈

C₂×D₃₇

yes

characteristic

#101

—

C₂²⋊C₈

—

#41

—

100

#41

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂×Q₈

—

104

C₂×Q₈

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

D₁₄₈

yes

characteristic

108

C₂×D₄

—

109

C₂×D₄

D₁₄₈

yes

characteristic

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

C₂²×D₃₇

yes

characteristic

113

Q₁₆

—

114

Q₁₆

—

115

SD₁₆

—

116

M₁₆

—

117

M₁₆

—

118

M₁₆

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

C₂²⋊D₃₇

yes

—

123

C₂³

—

124

C₂³

#10596962

yes

characteristic

125

Q₈

—

126

Q₈

—

127

Q₈

—

128

Q₈

—

148

—

129

Q₈

—

148

—

130

D₄

—

131

D₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

148

—

137

C₂×C₄

—

148

—

138

C₂×C₄

C₂×D₁₄₈

yes

characteristic

139

C₂×C₄

D₄×D₃₇

yes

—

140

C₂×C₄

D₄×D₃₇

yes

—

141

C₂×C₄

C₂×C₂²⋊D₃₇

yes

characteristic

142

C₈

—

143

C₈

—

144

C₈

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

#11924509

yes

—

149

C₂²

#11924510

yes

characteristic

150

C₄

—

151

C₄

—

148

—

152

C₄

#11924504

yes

characteristic

153

C₄

#11924506

yes

—

154

C₄

#11924532

yes

characteristic

155

C₂

—

156

C₂

—

157

C₂

#14557442

yes

characteristic

158

C₂

#14557808

yes

—

	subgroup	G/H
center	C₂²	#11924510
commutator, cosocle	C₂²×C₄×C₃₇	C₂³
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting	C₃₇×#408	C₂
socle	C₂²×C₃₇	#121
prime radical	C₂×#41×C₃₇	C₂²
hypercenter	#408	D₃₇
hypocenter	C₃₇	#2020
LCS	G ▷ C₂²×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #408

subgroup

G/H

center

C₂²

#11924510

commutator, cosocle

C₂²×C₄×C₃₇

C₂³

Frattini

C₂²×C₄

C₂²×D₃₇

Fitting

C₃₇×#408

C₂

socle

C₂²×C₃₇

#121

prime radical

C₂×#41×C₃₇

C₂²

D₃₇

C₃₇

G ▷ C₂²×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #408

N	H ≅ G/N	type
2. C₂²×Dic₁₄₈	C₂	semidirect
3. C₂×#11924454	C₂	semidirect
4. #14557298	C₂	semidirect
85. C₃₇	#2020	semidirect

H ≅ G/N

type

2. C₂²×Dic₁₄₈

C₂

semidirect

3. C₂×#11924454

C₂

semidirect

4. #14557298

C₂

semidirect

85. C₃₇

#2020

semidirect

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

subgroup	index
C₂×#11924566	2
C₂²×Dic₁₄₈	2
C₂×#11924454	2
#14557298	2
C₃₇×#408	2
#2020	37

subgroup

index

C₂²×Dic₁₄₈

C₃₇×#408

Subgroups — #17995717

Conjugacy classes of subgroups

Complements

Maximal subgroups