Group #18007238 — subgroups

Subgroups — #18007238

Order 4736.

370 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 141 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	#14558262	C₂	1	1	yes	maximal, characteristic, complemented
2	#14557885	C₂	1	3	yes	maximal, complemented
3	#14557815	C₂	1	3	yes	maximal, complemented
4	#14557738	C₂	1	3	yes	maximal
5	Q₈×#64436	C₂	1	1	yes	maximal, characteristic, complemented
6	C₄×#10596940	C₂	1	3	yes	maximal
7	D₄×Q₈×C₃₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#10596940	C₂²	1	3	yes	—
9	#11924536	C₂²	1	3	yes	—
10	#11924527	—	2	3	no	—
11	Q₈×Dic₃₇	—	2	1	no	—
12	C₄×Dic₇₄	C₂²	1	3	yes	—
13	#11924467	C₂²	1	3	yes	—
14	C₄⋊#64436	C₂²	1	3	yes	—
15	#11924459	C₂²	1	6	yes	—
16	#11924459	C₂²	1	3	yes	—
17	#11924457	C₂²	1	1	yes	characteristic
18	C₄×#64436	C₂²	1	3	yes	—
19	C₂²×Q₈×C₃₇	—	2	1	no	—
20	C₄⋊Q₈×C₃₇	C₂²	1	3	yes	—
21	C₂²⋊Q₈×C₃₇	—	2	3	no	—
22	C₄×Q₈×C₃₇	C₂²	1	1	yes	characteristic
23	C₄×D₄×C₃₇	C₂²	1	3	yes	—
24	C₂×Q₈×C₃₇	—	2	3	no	—
25	C₂×Q₈×C₃₇	—	4	4	no	—
26	C₂×Q₈×C₃₇	D₄	1	1	yes	characteristic
27	C₂×D₄×C₃₇	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₃₇	—	2	3	no	—
29	C₂×Dic₇₄	C₂³	1	3	yes	—
30	C₄⋊C₄×C₃₇	—	2	3	no	—
31	C₄⋊C₄×C₃₇	D₄	1	3	yes	—
32	C₄⋊C₄×C₃₇	C₂³	1	3	yes	—
33	C₂²⋊C₄×C₃₇	—	2	3	no	—
34	C₄²×C₃₇	C₂³	1	3	yes	—
35	#10596940	—	2	6	no	—
36	C₄⋊Dic₃₇	—	2	3	no	—
37	C₄⋊Dic₃₇	—	2	3	no	—
38	C₄⋊Dic₃₇	C₂³	1	1	yes	characteristic
39	#10596936	—	2	3	no	—
40	C₄×Dic₃₇	—	2	3	no	—
41	C₂×#64436	C₂³	1	3	yes	—
42	C₂×#64436	C₂³	1	1	yes	characteristic
43	C₂³×C₃₇	—	2	1	no	—
44	Q₈×C₃₇	—	4	6	no	—
45	Q₈×C₃₇	SD₁₆	1	4	yes	complemented
46	D₄×C₃₇	—	2	1	no	—
47	D₄×C₃₇	#41	1	2	yes	—
48	C₂×C₄×C₃₇	—	2	3	no	—
49	C₂×C₄×C₃₇	—	4	6	no	—
50	C₂×C₄×C₃₇	C₂×D₄	1	3	yes	—
51	C₂×C₄×C₃₇	C₂×D₄	1	3	yes	—
52	C₂×C₄×C₃₇	C₂⁴	1	1	yes	characteristic
53	C₂×Dic₃₇	—	2	1	no	—
54	C₂×Dic₃₇	—	2	3	no	—
55	Dic₇₄	—	2	3	no	—
56	Dic₇₄	—	2	3	no	—
57	#64436	—	2	3	no	—
58	#64436	—	2	1	no	—
59	C₂²×C₃₇	—	2	2	no	—
60	C₂²×C₃₇	—	4	1	no	—
61	C₂²×C₃₇	—	4	1	no	—
62	C₂²×C₃₇	C₂²×D₄	1	1	yes	characteristic
63	C₄×C₃₇	—	2	3	no	—
64	C₄×C₃₇	—	4	3	no	—
65	C₄×C₃₇	C₂×SD₁₆	1	6	yes	—
66	C₄×C₃₇	C₂×#41	1	1	yes	characteristic
67	C₄×C₃₇	2₋¹⁺⁴	1	1	yes	characteristic
68	Dic₃₇	—	4	3	no	—
69	Dic₃₇	—	4	1	no	—
70	#3359	—	37	1	no	maximal, Sylow
71	C₂×C₃₇	—	4	2	no	—
72	C₂×C₃₇	#548	1	1	yes	characteristic
73	C₂×C₃₇	C₂²×SD₁₆	1	1	yes	characteristic
74	C₂×C₃₇	#578	1	1	yes	characteristic
75	#557	—	37	1	no	—
76	D₄×Q₈	D₃₇	1	1	yes	characteristic
77	#487	—	37	3	no	—
78	#475	—	37	3	no	—
79	#461	—	37	3	no	—
80	C₈×Q₈	—	37	1	no	—
81	C₄×SD₁₆	—	37	3	no	—
82	C₃₇	#3359	1	1	yes	characteristic, complemented, Sylow
83	C₂²×Q₈	—	2	1	no	—
84	C₂×SD₁₆	—	37	3	no	—
85	C₄⋊Q₈	—	37	3	no	—
86	C₄⋊Q₈	C₂×D₃₇	1	3	yes	—
87	#125	—	74	3	no	—
88	C₂²⋊Q₈	—	2	3	no	—
89	C₄×Q₈	—	37	3	no	—
90	C₄×Q₈	—	74	1	no	—
91	C₄×Q₈	C₂×D₃₇	1	1	yes	characteristic
92	C₄×D₄	C₂×D₃₇	1	3	yes	—
93	#106	—	37	3	no	—
94	C₄⋊C₈	—	37	3	no	—
95	#103	—	37	3	no	—
96	#103	—	37	6	no	—
97	#102	—	37	1	no	—
98	C₄×C₈	—	37	3	no	—
99	C₂×Q₈	—	2	3	no	—
100	C₂×Q₈	—	4	4	no	—
101	C₂×Q₈	—	37	3	no	—
102	C₂×Q₈	C₂²⋊D₃₇	1	1	yes	characteristic
103	C₂×D₄	C₂²×D₃₇	1	1	yes	characteristic
104	C₂²×C₄	—	2	3	no	—
105	SD₁₆	—	74	6	no	—
106	C₂×C₈	—	37	1	no	—
107	C₂×C₈	—	37	3	no	—
108	C₄⋊C₄	—	2	3	no	—
109	C₄⋊C₄	—	37	1	no	—
110	C₄⋊C₄	—	74	3	no	—
111	C₄⋊C₄	—	74	3	no	—
112	C₄⋊C₄	—	74	3	no	—
113	C₄⋊C₄	C₂²⋊D₃₇	1	3	yes	—
114	C₄⋊C₄	C₂²×D₃₇	1	3	yes	—
115	C₂²⋊C₄	—	2	3	no	—
116	C₄²	—	74	3	no	—
117	C₄²	C₂²×D₃₇	1	3	yes	—
118	C₂³	—	2	1	no	—
119	Q₈	—	4	6	no	—
120	Q₈	—	74	3	no	—
121	Q₈	—	74	3	no	—
122	Q₈	#10596940	1	4	yes	complemented
123	D₄	—	2	1	no	—
124	D₄	#10596966	1	2	yes	—
125	C₂×C₄	—	2	3	no	—
126	C₂×C₄	—	4	6	no	—
127	C₂×C₄	—	74	1	no	—
128	C₂×C₄	—	74	3	no	—
129	C₂×C₄	C₂×C₂²⋊D₃₇	1	3	yes	—
130	C₂×C₄	C₂×C₂²⋊D₃₇	1	3	yes	—
131	C₂×C₄	C₂³×D₃₇	1	1	yes	characteristic
132	C₈	—	74	3	no	—
133	C₈	—	74	1	no	—
134	C₂²	—	2	2	no	—
135	C₂²	—	4	1	no	—
136	C₂²	—	4	1	no	—
137	C₂²	C₂²×C₂²⋊D₃₇	1	1	yes	characteristic
138	C₄	—	2	3	no	—
139	C₄	—	4	3	no	—
140	C₄	—	148	1	no	—
141	C₄	—	148	3	no	—
142	C₄	C₂×#10596940	1	6	yes	—
143	C₄	C₂×#10596966	1	1	yes	characteristic
144	C₄	#11924584	1	1	yes	characteristic
145	C₂	—	4	2	no	—
146	C₂	#14558194	1	1	yes	characteristic
147	C₂	C₂²×#10596940	1	1	yes	characteristic
148	C₂	#14558348	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14558262

C₂

yes

maximal, characteristic, complemented

#14557885

C₂

yes

maximal, complemented

#14557815

C₂

yes

maximal, complemented

#14557738

C₂

yes

maximal

Q₈×#64436

C₂

yes

maximal, characteristic, complemented

C₄×#10596940

C₂

yes

maximal

D₄×Q₈×C₃₇

C₂

yes

maximal, characteristic

C₂×#10596940

C₂²

yes

—

#11924536

C₂²

yes

—

#11924527

—

Q₈×Dic₃₇

—

C₄×Dic₇₄

C₂²

yes

—

#11924467

C₂²

yes

—

C₄⋊#64436

C₂²

yes

—

#11924459

C₂²

yes

—

#11924459

C₂²

yes

—

#11924457

C₂²

yes

characteristic

C₄×#64436

C₂²

yes

—

C₂²×Q₈×C₃₇

—

C₄⋊Q₈×C₃₇

C₂²

yes

—

C₂²⋊Q₈×C₃₇

—

C₄×Q₈×C₃₇

C₂²

yes

characteristic

C₄×D₄×C₃₇

C₂²

yes

—

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

—

C₂×Q₈×C₃₇

D₄

yes

characteristic

C₂×D₄×C₃₇

C₂³

yes

characteristic

C₂²×C₄×C₃₇

—

C₂×Dic₇₄

C₂³

yes

—

C₄⋊C₄×C₃₇

—

C₄⋊C₄×C₃₇

D₄

yes

—

C₄⋊C₄×C₃₇

C₂³

yes

—

C₂²⋊C₄×C₃₇

—

C₄²×C₃₇

C₂³

yes

—

#10596940

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

—

C₄⋊Dic₃₇

C₂³

yes

characteristic

#10596936

—

C₄×Dic₃₇

—

C₂×#64436

C₂³

yes

—

C₂×#64436

C₂³

yes

characteristic

C₂³×C₃₇

—

Q₈×C₃₇

—

Q₈×C₃₇

SD₁₆

yes

complemented

D₄×C₃₇

—

D₄×C₃₇

#41

yes

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂⁴

yes

characteristic

C₂×Dic₃₇

—

C₂×Dic₃₇

—

Dic₇₄

—

Dic₇₄

—

#64436

—

#64436

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂²×D₄

yes

characteristic

C₄×C₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₂×SD₁₆

yes

—

C₄×C₃₇

C₂×#41

yes

characteristic

C₄×C₃₇

2₋¹⁺⁴

yes

characteristic

Dic₃₇

—

Dic₃₇

—

#3359

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

#548

yes

characteristic

C₂×C₃₇

C₂²×SD₁₆

yes

C₂×C₃₇

yes

—

D₄×Q₈

D₃₇

yes

—

—

—

C₈×Q₈

—

C₄×SD₁₆

—

C₃₇

#3359

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×SD₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₃₇

yes

—

#125

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

C₂×D₃₇

yes

characteristic

C₄×D₄

C₂×D₃₇

yes

—

#106

—

C₄⋊C₈

—

#103

—

#103

—

#102

—

C₄×C₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×Q₈

C₂²⋊D₃₇

yes

characteristic

103

C₂×D₄

C₂²×D₃₇

yes

characteristic

104

C₂²×C₄

—

105

SD₁₆

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₄⋊C₄

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

C₂²⋊D₃₇

yes

—

114

C₄⋊C₄

C₂²×D₃₇

yes

—

115

C₂²⋊C₄

—

116

C₄²

—

117

C₄²

C₂²×D₃₇

yes

—

118

C₂³

—

119

Q₈

—

120

Q₈

—

121

Q₈

—

122

Q₈

#10596940

yes

complemented

123

D₄

—

124

D₄

#10596966

yes

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

C₂×C₂²⋊D₃₇

yes

—

130

C₂×C₄

C₂×C₂²⋊D₃₇

yes

—

131

C₂×C₄

C₂³×D₃₇

yes

characteristic

132

C₈

—

133

C₈

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

C₂²×C₂²⋊D₃₇

yes

characteristic

138

C₄

—

139

C₄

—

140

C₄

—

148

—

141

C₄

—

148

—

142

C₄

C₂×#10596940

yes

—

143

C₄

C₂×#10596966

yes

characteristic

144

C₄

#11924584

yes

characteristic

145

C₂

—

146

C₂

#14558194

yes

characteristic

147

C₂

C₂²×#10596940

yes

characteristic

148

C₂

#14558348

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₃₇
commutator, cosocle	C₂×C₄×C₃₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₃₇
Fitting	D₄×Q₈×C₃₇	C₂
socle	C₂²×C₃₇	C₂²×D₄
prime radical	C₂×D₄×C₃₇	C₂³
hypercenter	D₄×Q₈	D₃₇
hypocenter	C₃₇	#3359
LCS	G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ D₄×Q₈

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₃₇

commutator, cosocle

C₂×C₄×C₃₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₃₇

Fitting

D₄×Q₈×C₃₇

C₂

socle

C₂²×C₃₇

C₂²×D₄

prime radical

C₂×D₄×C₃₇

C₂³

hypercenter

D₄×Q₈

D₃₇

hypocenter

C₃₇

#3359

LCS

G ▷ C₂×C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ D₄×Q₈

N	H ≅ G/N	type
1. #14558262	C₂	semidirect
2. #14557885	C₂	semidirect
3. #14557815	C₂	semidirect
5. Q₈×#64436	C₂	semidirect
45. Q₈×C₃₇	SD₁₆	semidirect
82. C₃₇	#3359	semidirect
122. Q₈	#10596940	semidirect

H ≅ G/N

type

1. #14558262

C₂

semidirect

2. #14557885

C₂

semidirect

3. #14557815

C₂

semidirect

5. Q₈×#64436

C₂

semidirect

45. Q₈×C₃₇

SD₁₆

semidirect

82. C₃₇

#3359

semidirect

122. Q₈

#10596940

semidirect

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

subgroup	index
#14558262	2
#14557885	2
#14557815	2
#14557738	2
Q₈×#64436	2
C₄×#10596940	2
D₄×Q₈×C₃₇	2
#3359	37

subgroup

index

D₄×Q₈×C₃₇

Subgroups — #18007238

Conjugacy classes of subgroups

Complements

Maximal subgroups