Group #17995715 — subgroups

Subgroups — #17995715

Order 4736.

342 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 75 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×#11924567	C₂	1	2	yes	maximal
2	C₂×#11924455	C₂	1	2	yes	maximal, complemented
3	#14557499	C₂	1	1	yes	maximal, characteristic
4	#14557298	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×Q₁₆×C₃₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10596962	C₂²	1	1	yes	characteristic
7	#11924567	—	2	4	no	—
8	#11924567	—	2	4	no	—
9	C₂×#10596942	—	2	2	no	—
10	C₂×#10596941	—	2	2	no	—
11	C₂×#10596934	C₂²	1	1	yes	characteristic
12	C₂×#10596934	C₂²	1	2	yes	—
13	#11924455	—	2	4	no	—
14	#11924448	—	2	2	no	—
15	C₂²×Q₈×C₃₇	C₂²	1	2	yes	—
16	C₂×Q₁₆×C₃₇	—	2	4	no	—
17	C₂×Q₁₆×C₃₇	—	2	2	no	—
18	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
19	C₂×Q₈×C₃₇	—	2	2	no	—
20	C₂×Q₈×C₃₇	—	2	2	no	—
21	C₂×Q₈×C₃₇	—	2	2	no	—
22	C₂×Q₈×C₃₇	—	2	2	no	—
23	C₂×Q₈×C₃₇	—	2	2	no	—
24	C₂×Q₈×C₃₇	D₄	1	4	yes	—
25	C₂²×C₄×C₃₇	—	2	2	no	—
26	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
27	C₂×C₂²⋊D₃₇	—	2	1	no	—
28	#10596962	—	2	2	no	—
29	#10596962	—	2	2	no	—
30	C₂×D₁₄₈	D₄	1	1	yes	characteristic
31	C₂×C₄×D₃₇	—	2	1	no	—
32	C₂×Dic₇₄	D₄	1	1	yes	characteristic
33	Q₁₆×C₃₇	—	2	8	no	—
34	C₂×C₈×C₃₇	—	2	2	no	—
35	C₂×C₈×C₃₇	D₄	1	2	yes	—
36	#10596942	—	2	8	no	—
37	#10596941	—	2	8	no	—
38	#10596934	—	2	2	no	—
39	#10596934	—	2	2	no	—
40	#10596934	D₄	1	4	yes	—
41	C₂×#64436	—	2	2	no	—
42	C₂×#64436	—	2	1	no	—
43	C₂³×C₃₇	#41	1	1	yes	characteristic
44	C₂²×D₃₇	—	2	1	no	—
45	Q₈×C₃₇	—	2	4	no	—
46	Q₈×C₃₇	—	2	4	no	—
47	Q₈×C₃₇	—	4	2	no	—
48	Q₈×C₃₇	—	4	2	no	—
49	C₂×C₄×C₃₇	—	2	4	no	—
50	C₂×C₄×C₃₇	—	4	2	no	—
51	C₂×C₄×C₃₇	—	4	2	no	—
52	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
55	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
56	C₂²⋊D₃₇	—	4	2	no	—
57	C₂×Dic₃₇	—	2	1	no	—
58	D₁₄₈	—	2	2	no	—
59	C₄×D₃₇	—	4	1	no	—
60	C₄×D₃₇	—	4	1	no	—
61	Dic₇₄	—	2	2	no	—
62	C₈×C₃₇	—	2	2	no	—
63	#64436	—	2	4	no	—
64	#64436	—	4	1	no	—
65	C₂²×C₃₇	—	2	2	no	—
66	C₂²×C₃₇	#121	1	1	yes	characteristic
67	C₂²×C₃₇	#121	1	2	yes	—
68	C₂×D₃₇	—	4	1	no	—
69	C₂×D₃₇	—	4	2	no	—
70	C₄×C₃₇	—	4	4	no	—
71	C₄×C₃₇	#120	1	2	yes	—
72	C₄×C₃₇	#120	1	1	yes	characteristic
73	C₄×C₃₇	#127	1	1	yes	characteristic
74	Dic₃₇	—	4	1	no	—
75	#2020	—	37	1	no	maximal, Sylow
76	C₂×C₃₇	—	2	2	no	—
77	C₂×C₃₇	#392	1	1	yes	characteristic
78	C₂×C₃₇	#473	1	2	yes	—
79	D₃₇	—	8	1	no	—
80	C₂×#137	—	37	2	no	—
81	C₂²×Q₁₆	D₃₇	1	1	yes	characteristic, complemented
82	C₂×#101	—	37	2	no	—
83	#408	—	37	1	no	—
84	#341	—	37	1	no	—
85	C₃₇	#2020	1	1	yes	characteristic, complemented, Sylow
86	C₂×#41	—	37	1	no	—
87	C₂²×Q₈	C₂×D₃₇	1	2	yes	—
88	#137	—	74	4	no	—
89	#137	—	74	4	no	—
90	C₂×Q₁₆	—	2	4	no	—
91	C₂×Q₁₆	—	2	2	no	—
92	C₂×Q₁₆	—	74	2	no	—
93	C₂×SD₁₆	—	74	2	no	—
94	C₂×M₁₆	—	37	2	no	—
95	C₂×M₁₆	—	37	1	no	—
96	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic
97	#101	—	74	4	no	—
98	C₂²⋊C₈	—	74	2	no	—
99	#41	—	74	2	no	—
100	#41	—	74	2	no	—
101	C₂×Q₈	—	2	2	no	—
102	C₂×Q₈	—	2	2	no	—
103	C₂×Q₈	—	2	2	no	—
104	C₂×Q₈	—	2	2	no	—
105	C₂×Q₈	—	2	2	no	—
106	C₂×Q₈	—	37	1	no	—
107	C₂×Q₈	C₂²⋊D₃₇	1	4	yes	—
108	C₂×D₄	—	37	1	no	—
109	C₂×D₄	—	74	1	no	—
110	C₂²×C₄	—	2	2	no	—
111	C₂²×C₄	—	74	1	no	—
112	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
113	Q₁₆	—	2	8	no	—
114	Q₁₆	—	74	8	no	—
115	SD₁₆	—	74	8	no	—
116	M₁₆	—	37	4	no	—
117	M₁₆	—	74	2	no	—
118	M₁₆	—	74	2	no	—
119	C₂×C₈	—	2	2	no	—
120	C₂×C₈	—	74	2	no	—
121	C₂×C₈	—	74	1	no	—
122	C₂×C₈	C₂²⋊D₃₇	1	2	yes	—
123	C₂³	—	74	1	no	—
124	C₂³	#10596964	1	1	yes	characteristic
125	Q₈	—	2	4	no	—
126	Q₈	—	2	4	no	—
127	Q₈	—	4	2	no	—
128	Q₈	—	4	2	no	—
129	Q₈	—	74	2	no	—
130	D₄	—	74	2	no	—
131	D₄	—	148	2	no	—
132	C₂×C₄	—	2	4	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	—	74	1	no	—
136	C₂×C₄	—	148	1	no	—
137	C₂×C₄	—	148	1	no	—
138	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
139	C₂×C₄	D₄×D₃₇	1	2	yes	—
140	C₂×C₄	C₂×C₂²⋊D₃₇	1	2	yes	—
141	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
142	C₈	—	2	2	no	—
143	C₈	—	74	4	no	—
144	C₈	—	148	1	no	—
145	C₂²	—	2	2	no	—
146	C₂²	—	148	1	no	—
147	C₂²	—	148	2	no	—
148	C₂²	#11924507	1	1	yes	characteristic
149	C₂²	#11924508	1	2	yes	—
150	C₄	—	4	4	no	—
151	C₄	—	148	1	no	—
152	C₄	#11924504	1	2	yes	—
153	C₄	#11924505	1	1	yes	characteristic
154	C₄	#11924532	1	1	yes	characteristic
155	C₂	—	2	2	no	—
156	C₂	—	296	1	no	—
157	C₂	#14557441	1	1	yes	characteristic
158	C₂	#14557805	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924567

C₂

yes

maximal

C₂×#11924455

C₂

yes

maximal, complemented

#14557499

C₂

yes

maximal, characteristic

#14557298

C₂

yes

maximal, characteristic, complemented

C₂²×Q₁₆×C₃₇

C₂

yes

maximal, characteristic, complemented

C₂×#10596962

C₂²

yes

—

—

—

—

C₂²

yes

characteristic

C₂×#10596934

C₂²

yes

—

#11924455

—

#11924448

—

C₂²×Q₈×C₃₇

C₂²

yes

—

C₂×Q₁₆×C₃₇

—

C₂×Q₁₆×C₃₇

—

C₂²×C₈×C₃₇

C₂²

yes

characteristic

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

D₄

yes

—

C₂²×C₄×C₃₇

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂×C₂²⋊D₃₇

—

#10596962

—

#10596962

—

C₂×D₁₄₈

D₄

yes

characteristic

C₂×C₄×D₃₇

—

C₂×Dic₇₄

D₄

yes

characteristic

Q₁₆×C₃₇

—

C₂×C₈×C₃₇

—

C₂×C₈×C₃₇

D₄

yes

—

—

—

—

—

D₄

yes

—

C₂×#64436

—

C₂×#64436

—

C₂³×C₃₇

#41

yes

characteristic

C₂²×D₃₇

—

Q₈×C₃₇

—

Q₈×C₃₇

—

Q₈×C₃₇

—

Q₈×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

C₂²⋊D₃₇

—

C₂×Dic₃₇

—

D₁₄₈

—

C₄×D₃₇

—

C₄×D₃₇

—

Dic₇₄

—

C₈×C₃₇

—

#64436

—

#64436

—

C₂²×C₃₇

—

C₂²×C₃₇

#121

yes

characteristic

C₂²×C₃₇

#121

yes

—

C₂×D₃₇

—

C₂×D₃₇

—

C₄×C₃₇

—

C₄×C₃₇

#120

yes

—

C₄×C₃₇

yes

C₄×C₃₇

yes

Dic₃₇

—

#2020

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

#392

yes

characteristic

C₂×C₃₇

#473

yes

—

D₃₇

—

C₂×#137

—

C₂²×Q₁₆

D₃₇

yes

characteristic, complemented

C₂×#101

—

#408

—

#341

—

C₃₇

#2020

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂²×Q₈

C₂×D₃₇

yes

—

#137

—

#137

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×SD₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

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

C₂²⋊D₃₇

yes

—

108

C₂×D₄

—

109

C₂×D₄

—

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

#10596964

yes

characteristic

125

Q₈

—

126

Q₈

—

127

Q₈

—

128

Q₈

—

129

Q₈

—

130

D₄

—

131

D₄

—

148

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

148

—

137

C₂×C₄

—

148

—

138

C₂×C₄

D₄×D₃₇

yes

characteristic

139

C₂×C₄

D₄×D₃₇

yes

—

140

C₂×C₄

C₂×C₂²⋊D₃₇

yes

—

141

C₂×C₄

C₂×C₂²⋊D₃₇

yes

characteristic

142

C₈

—

143

C₈

—

144

C₈

—

148

—

145

C₂²

—

146

C₂²

—

148

—

147

C₂²

—

148

—

148

C₂²

#11924507

yes

characteristic

149

C₂²

#11924508

yes

—

150

C₄

—

151

C₄

—

148

—

152

C₄

#11924504

yes

—

153

C₄

#11924505

yes

characteristic

154

C₄

#11924532

yes

characteristic

155

C₂

—

156

C₂

—

296

—

157

C₂

#14557441

yes

characteristic

158

C₂

#14557805

yes

—

	subgroup	G/H
center	C₂²	#11924507
commutator, cosocle	C₂²×C₄×C₃₇	C₂³
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting	C₂²×Q₁₆×C₃₇	C₂
socle	C₂²×C₃₇	#121
prime radical	C₂×#10596962	C₂²
hypercenter	C₂²×Q₁₆	D₃₇
hypocenter	C₃₇	#2020
LCS	G ▷ C₂²×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

subgroup

G/H

center

C₂²

#11924507

commutator, cosocle

C₂²×C₄×C₃₇

C₂³

Frattini

C₂²×C₄

C₂²×D₃₇

Fitting

C₂²×Q₁₆×C₃₇

C₂

C₂²×C₃₇

C₂²

C₂²×Q₁₆

D₃₇

hypocenter

C₃₇

#2020

LCS

G ▷ C₂²×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×Q₁₆

N	H ≅ G/N	type
2. C₂×#11924455	C₂	semidirect
4. #14557298	C₂	semidirect
5. C₂²×Q₁₆×C₃₇	C₂	semidirect
81. C₂²×Q₁₆	D₃₇	semidirect
85. C₃₇	#2020	semidirect

H ≅ G/N

type

2. C₂×#11924455

C₂

semidirect

4. #14557298

C₂

semidirect

5. C₂²×Q₁₆×C₃₇

C₂

semidirect

81. C₂²×Q₁₆

D₃₇

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×#11924567	2
C₂×#11924455	2
#14557499	2
#14557298	2
C₂²×Q₁₆×C₃₇	2
#2020	37

subgroup

index

C₂²×Q₁₆×C₃₇

Subgroups — #17995715

Conjugacy classes of subgroups

Complements

Maximal subgroups