Group #17995458 — subgroups

Subgroups — #17995458

Order 4736.

312 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 117 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×#11924470	C₂	1	1	yes	maximal, characteristic
2	C₂×#11924460	C₂	1	2	yes	maximal
3	C₂×C₈×Dic₃₇	C₂	1	1	yes	maximal, characteristic
4	#14557440	C₂	1	2	yes	maximal
5	C₂²×Q₁₆×C₃₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#64436	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₃₇	C₂²	1	2	yes	—
8	C₂×C₄×Dic₃₇	C₂²	1	1	yes	characteristic
9	#11924470	—	2	2	no	—
10	#11924460	—	2	4	no	—
11	C₈×Dic₃₇	—	2	2	no	—
12	#11924443	—	2	4	no	—
13	C₂²×Q₈×C₃₇	C₂²	1	2	yes	—
14	C₂×Q₁₆×C₃₇	—	2	2	no	—
15	C₂×Q₁₆×C₃₇	—	2	2	no	—
16	C₂×Q₁₆×C₃₇	C₄	1	4	yes	complemented
17	C₂²×C₈×C₃₇	C₂²	1	1	yes	characteristic
18	C₂×Q₈×C₃₇	—	2	2	no	—
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	C₂×C₄	1	4	yes	—
24	C₂²×C₄×C₃₇	—	2	2	no	—
25	C₂²×C₄×C₃₇	C₂³	1	1	yes	characteristic
26	C₂²×Dic₃₇	—	2	2	no	—
27	C₂²×Dic₃₇	D₄	1	2	yes	—
28	Q₁₆×C₃₇	—	2	8	no	—
29	C₂×C₈×C₃₇	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₃₇	D₄	1	2	yes	—
31	C₂×C₈×C₃₇	D₄	1	2	yes	—
32	C₄⋊Dic₃₇	—	2	2	no	—
33	C₄⋊Dic₃₇	—	2	2	no	—
34	C₄×Dic₃₇	—	2	1	no	—
35	C₄×Dic₃₇	—	2	1	no	—
36	C₂×#64436	—	2	1	no	—
37	C₂×#64436	—	2	1	no	—
38	C₂×#64436	D₄	1	2	yes	—
39	C₂³×C₃₇	C₂×D₄	1	1	yes	characteristic
40	Q₈×C₃₇	—	2	4	no	—
41	Q₈×C₃₇	—	2	4	no	—
42	Q₈×C₃₇	—	4	2	no	—
43	Q₈×C₃₇	—	4	2	no	—
44	C₂×C₄×C₃₇	—	2	4	no	—
45	C₂×C₄×C₃₇	—	4	2	no	—
46	C₂×C₄×C₃₇	—	4	2	no	—
47	C₂×C₄×C₃₇	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
49	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₃₇	#41	1	1	yes	characteristic
52	C₂×C₄×C₃₇	#41	1	1	yes	characteristic
53	C₂×Dic₃₇	—	2	2	no	—
54	C₂×Dic₃₇	—	2	2	no	—
55	C₂×Dic₃₇	—	4	2	no	—
56	C₂×Dic₃₇	—	4	4	no	—
57	C₂×Dic₃₇	Q₁₆	1	4	yes	complemented
58	C₈×C₃₇	C₂²⋊C₄	1	4	yes	—
59	#64436	—	2	2	no	—
60	C₂²×C₃₇	C₄×D₄	1	1	yes	characteristic
61	C₂²×C₃₇	#121	1	1	yes	characteristic
62	C₂²×C₃₇	#127	1	1	yes	characteristic
63	C₂²×C₃₇	C₂×Q₁₆	1	2	yes	—
64	C₂²×C₃₇	#135	1	2	yes	—
65	C₄×C₃₇	—	4	4	no	—
66	C₄×C₃₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₄×C₃₇	C₄×D₄	1	1	yes	characteristic
68	C₄×C₃₇	#121	1	1	yes	characteristic
69	C₄×C₃₇	#124	1	1	yes	characteristic
70	Dic₃₇	—	2	4	no	—
71	Dic₃₇	—	8	2	no	—
72	#1969	—	37	1	no	maximal, Sylow
73	C₂×C₃₇	#390	1	1	yes	characteristic
74	C₂×C₃₇	C₄×Q₁₆	1	2	yes	—
75	C₂×C₃₇	#467	1	2	yes	—
76	C₂×C₃₇	#494	1	1	yes	characteristic
77	C₂×C₃₇	#495	1	1	yes	characteristic
78	C₂²×Q₁₆	D₃₇	1	1	yes	characteristic
79	C₂×#107	—	37	1	no	—
80	C₂×#103	—	37	2	no	—
81	C₂×C₄×C₈	—	37	1	no	—
82	#391	—	37	2	no	—
83	C₃₇	#1969	1	1	yes	characteristic, complemented, Sylow
84	C₂²×Q₈	C₂×D₃₇	1	2	yes	—
85	C₂×Q₁₆	—	2	2	no	—
86	C₂×Q₁₆	—	2	2	no	—
87	C₂×Q₁₆	Dic₃₇	1	4	yes	complemented
88	C₂²×C₈	—	37	1	no	—
89	C₂²×C₈	C₂×D₃₇	1	1	yes	characteristic
90	C₂×C₄⋊C₄	—	37	2	no	—
91	C₂×C₄²	—	37	1	no	—
92	#107	—	74	2	no	—
93	#103	—	74	4	no	—
94	C₄×C₈	—	74	2	no	—
95	#95	—	74	4	no	—
96	C₂×Q₈	—	2	2	no	—
97	C₂×Q₈	—	2	2	no	—
98	C₂×Q₈	—	2	2	no	—
99	C₂×Q₈	—	2	2	no	—
100	C₂×Q₈	—	2	2	no	—
101	C₂×Q₈	C₂×Dic₃₇	1	4	yes	—
102	C₂²×C₄	—	2	2	no	—
103	C₂²×C₄	—	37	2	no	—
104	C₂²×C₄	—	74	2	no	—
105	C₂²×C₄	C₂²×D₃₇	1	1	yes	characteristic
106	Q₁₆	—	2	8	no	—
107	C₂×C₈	—	37	2	no	—
108	C₂×C₈	—	74	1	no	—
109	C₂×C₈	—	74	1	no	—
110	C₂×C₈	C₂×Dic₃₇	1	2	yes	—
111	C₂×C₈	C₂²⋊D₃₇	1	2	yes	—
112	C₂×C₈	C₂²⋊D₃₇	1	2	yes	—
113	C₄⋊C₄	—	74	2	no	—
114	C₄⋊C₄	—	74	2	no	—
115	C₄²	—	74	1	no	—
116	C₄²	—	74	1	no	—
117	C₂³	D₄×D₃₇	1	1	yes	characteristic
118	Q₈	—	2	4	no	—
119	Q₈	—	2	4	no	—
120	Q₈	—	4	2	no	—
121	Q₈	—	4	2	no	—
122	C₂×C₄	—	2	4	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	37	4	no	—
126	C₂×C₄	—	74	2	no	—
127	C₂×C₄	—	74	2	no	—
128	C₂×C₄	—	148	2	no	—
129	C₂×C₄	—	148	4	no	—
130	C₂×C₄	D₄×D₃₇	1	1	yes	characteristic
131	C₂×C₄	#10596964	1	1	yes	characteristic
132	C₂×C₄	#10596964	1	1	yes	characteristic
133	C₂×C₄	C₂²×Dic₃₇	1	1	yes	characteristic
134	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
135	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
136	C₈	—	74	2	no	—
137	C₈	C₂²⋊Dic₃₇	1	4	yes	—
138	C₂²	D₄×Dic₃₇	1	1	yes	characteristic
139	C₂²	#11924507	1	1	yes	characteristic
140	C₂²	#11924532	1	1	yes	characteristic
141	C₂²	Q₁₆×D₃₇	1	2	yes	—
142	C₂²	#11924559	1	2	yes	—
143	C₄	—	4	4	no	—
144	C₄	—	296	2	no	—
145	C₄	—	74	4	no	—
146	C₄	C₂×C₂²⋊Dic₃₇	1	1	yes	characteristic
147	C₄	D₄×Dic₃₇	1	1	yes	characteristic
148	C₄	#11924507	1	1	yes	characteristic
149	C₄	#11924523	1	1	yes	characteristic
150	C₂	#14557436	1	1	yes	characteristic
151	C₂	Q₁₆×Dic₃₇	1	2	yes	—
152	C₂	#14557771	1	2	yes	—
153	C₂	#14557911	1	1	yes	characteristic
154	C₂	#14557918	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#11924470

C₂

yes

maximal, characteristic

C₂×#11924460

C₂

yes

maximal

C₂×C₈×Dic₃₇

C₂

yes

maximal, characteristic

#14557440

C₂

yes

maximal

C₂²×Q₁₆×C₃₇

C₂

yes

maximal, characteristic

C₂²×#64436

C₂²

yes

characteristic

C₂×C₄⋊Dic₃₇

C₂²

yes

—

C₂×C₄×Dic₃₇

C₂²

yes

characteristic

#11924470

—

#11924460

—

C₈×Dic₃₇

—

#11924443

—

C₂²×Q₈×C₃₇

C₂²

yes

—

C₂×Q₁₆×C₃₇

—

C₂×Q₁₆×C₃₇

—

C₂×Q₁₆×C₃₇

C₄

yes

complemented

C₂²×C₈×C₃₇

C₂²

yes

characteristic

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

C₂×C₄

yes

—

C₂²×C₄×C₃₇

—

C₂²×C₄×C₃₇

C₂³

yes

characteristic

C₂²×Dic₃₇

—

C₂²×Dic₃₇

D₄

yes

—

Q₁₆×C₃₇

—

C₂×C₈×C₃₇

C₂×C₄

yes

—

C₂×C₈×C₃₇

D₄

yes

—

C₂×C₈×C₃₇

D₄

yes

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄×Dic₃₇

—

C₄×Dic₃₇

—

C₂×#64436

—

C₂×#64436

—

C₂×#64436

D₄

yes

—

C₂³×C₃₇

C₂×D₄

yes

characteristic

Q₈×C₃₇

—

Q₈×C₃₇

—

Q₈×C₃₇

—

Q₈×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂×C₄×C₃₇

C₂×D₄

yes

C₂×C₄×C₃₇

yes

C₂×C₄×C₃₇

yes

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

—

C₂×Dic₃₇

Q₁₆

yes

complemented

C₈×C₃₇

C₂²⋊C₄

yes

—

#64436

—

C₂²×C₃₇

C₄×D₄

yes

C₂²×C₃₇

yes

C₂²×C₃₇

yes

C₂²×C₃₇

C₂×Q₁₆

yes

—

C₂²×C₃₇

#135

yes

—

C₄×C₃₇

—

C₄×C₃₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₃₇

C₄×D₄

yes

C₄×C₃₇

yes

C₄×C₃₇

yes

Dic₃₇

—

Dic₃₇

—

—

C₂×C₃₇

yes

C₂×C₃₇

C₄×Q₁₆

yes

—

C₂×C₃₇

#467

yes

—

C₂×C₃₇

yes

C₂×C₃₇

yes

C₂²×Q₁₆

D₃₇

yes

characteristic

C₂×#107

—

C₂×#103

—

C₂×C₄×C₈

—

#391

—

C₃₇

#1969

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₃₇

yes

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

Dic₃₇

yes

complemented

C₂²×C₈

—

C₂²×C₈

C₂×D₃₇

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

—

#107

—

#103

—

C₄×C₈

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

C₂×Dic₃₇

yes

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

C₂²×D₃₇

yes

characteristic

106

Q₁₆

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

C₂×Dic₃₇

yes

—

111

C₂×C₈

C₂²⋊D₃₇

yes

—

112

C₂×C₈

C₂²⋊D₃₇

yes

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄²

—

116

C₄²

—

117

C₂³

D₄×D₃₇

yes

characteristic

118

Q₈

—

119

Q₈

—

120

Q₈

—

121

Q₈

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

148

—

129

C₂×C₄

—

148

—

130

C₂×C₄

D₄×D₃₇

yes

characteristic

131

C₂×C₄

#10596964

yes

characteristic

132

C₂×C₄

#10596964

yes

characteristic

133

C₂×C₄

C₂²×Dic₃₇

yes

characteristic

134

C₂×C₄

C₂×C₂²⋊D₃₇

yes

characteristic

135

C₂×C₄

C₂×C₂²⋊D₃₇

yes

characteristic

136

C₈

—

137

C₈

C₂²⋊Dic₃₇

yes

—

138

C₂²

D₄×Dic₃₇

yes

characteristic

139

C₂²

#11924507

yes

characteristic

140

C₂²

#11924532

yes

characteristic

141

C₂²

Q₁₆×D₃₇

yes

—

142

C₂²

#11924559

yes

—

143

C₄

—

144

C₄

—

296

—

145

C₄

—

146

C₄

C₂×C₂²⋊Dic₃₇

yes

characteristic

147

C₄

D₄×Dic₃₇

yes

characteristic

148

C₄

#11924507

yes

characteristic

149

C₄

#11924523

yes

characteristic

150

C₂

#14557436

yes

characteristic

151

C₂

Q₁₆×Dic₃₇

yes

—

152

C₂

#14557771

yes

—

153

C₂

#14557911

yes

characteristic

154

C₂

#14557918

yes

characteristic

	subgroup	G/H
center	C₂³	D₄×D₃₇
commutator	C₂×C₄×C₃₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₃₇
Fitting	C₂²×Q₁₆×C₃₇	C₂
socle, prime radical	C₂³×C₃₇	C₂×D₄
cosocle	C₂²×C₄×C₃₇	C₂³
hypercenter	C₂²×Q₁₆	D₃₇
hypocenter	C₃₇	#1969
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₁₆

subgroup

G/H

center

C₂³

D₄×D₃₇

commutator

C₂×C₄×C₃₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₃₇

Fitting

C₂²×Q₁₆×C₃₇

C₂

socle, prime radical

C₂³×C₃₇

C₂×D₄

cosocle

C₂²×C₄×C₃₇

C₂³

hypercenter

C₂²×Q₁₆

D₃₇

hypocenter

C₃₇

#1969

LCS

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×Q₁₆

N	H ≅ G/N	type
16. C₂×Q₁₆×C₃₇	C₄	semidirect
57. C₂×Dic₃₇	Q₁₆	semidirect
83. C₃₇	#1969	semidirect
87. C₂×Q₁₆	Dic₃₇	semidirect

H ≅ G/N

type

16. C₂×Q₁₆×C₃₇

C₄

semidirect

57. C₂×Dic₃₇

Q₁₆

semidirect

83. C₃₇

#1969

semidirect

87. C₂×Q₁₆

Dic₃₇

semidirect

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

subgroup	index
C₂×#11924470	2
C₂×#11924460	2
C₂×C₈×Dic₃₇	2
#14557440	2
C₂²×Q₁₆×C₃₇	2
#1969	37

subgroup

index

C₂×C₈×Dic₃₇

C₂²×Q₁₆×C₃₇

Subgroups — #17995458

Conjugacy classes of subgroups

Complements

Maximal subgroups