Group #17993648 — subgroups

Subgroups — #17993648

Order 4736.

336 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 57 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	#14558168	C₂	1	1	yes	maximal, characteristic, complemented
2	#14557725	C₂	1	2	yes	maximal, complemented
3	#14557277	C₂	1	1	yes	maximal, characteristic, complemented
4	#14557262	C₂	1	2	yes	maximal, complemented
5	C₃₇×#493	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄×D₃₇	—	2	2	no	—
7	C₂×#10596939	—	2	2	no	—
8	#11924507	—	2	2	no	—
9	#11924504	—	2	2	no	—
10	C₄×D₁₄₈	C₂²	1	2	yes	—
11	C₄⋊#64436	C₂²	1	2	yes	complemented
12	#11924457	—	2	2	no	—
13	C₂×D₈×C₃₇	—	2	2	no	—
14	#127×C₃₇	C₂²	1	1	yes	characteristic
15	#127×C₃₇	C₂²	1	1	yes	characteristic, complemented
16	C₄×C₈×C₃₇	C₂²	1	1	yes	characteristic, complemented
17	C₂³×D₃₇	—	2	2	no	—
18	C₂×D₄×C₃₇	—	2	2	no	—
19	C₂×D₄×C₃₇	—	2	2	no	—
20	C₂×D₄×C₃₇	—	2	1	no	—
21	C₂×D₄×C₃₇	D₄	1	2	yes	—
22	C₂×C₂²⋊D₃₇	—	2	4	no	—
23	D₄×D₃₇	—	4	4	no	—
24	D₄×D₃₇	—	4	4	no	—
25	C₂×D₁₄₈	D₄	1	2	yes	—
26	C₂×C₄×D₃₇	—	2	2	no	—
27	D₈×C₃₇	—	4	4	no	—
28	C₂×C₈×C₃₇	—	2	1	no	—
29	C₄²×C₃₇	C₂³	1	1	yes	characteristic
30	C₂²⋊Dic₃₇	—	2	2	no	—
31	#10596939	—	4	4	no	—
32	#10596938	—	2	2	no	—
33	C₄⋊Dic₃₇	D₄	1	2	yes	—
34	C₂×#64436	—	2	2	no	—
35	C₂³×C₃₇	—	2	2	no	—
36	C₂³×C₃₇	—	4	1	no	—
37	C₂²×D₃₇	—	2	2	no	—
38	C₂²×D₃₇	—	4	8	no	—
39	C₂²×D₃₇	—	8	2	no	—
40	D₄×C₃₇	—	4	2	no	—
41	D₄×C₃₇	—	4	2	no	—
42	D₄×C₃₇	—	4	2	no	—
43	D₄×C₃₇	—	4	2	no	—
44	D₄×C₃₇	—	4	2	no	—
45	D₄×C₃₇	—	4	2	no	—
46	C₂×C₄×C₃₇	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₃₇	C₂×D₄	1	1	yes	characteristic
48	C₂²⋊D₃₇	—	4	8	no	—
49	C₂×Dic₃₇	—	2	2	no	—
50	D₁₄₈	—	2	2	no	—
51	D₁₄₈	D₈	1	4	yes	complemented
52	C₄×D₃₇	—	4	2	no	—
53	C₄×D₃₇	—	4	2	no	—
54	C₈×C₃₇	—	2	2	no	—
55	#64436	—	4	2	no	—
56	C₂²×C₃₇	—	4	4	no	—
57	C₂²×C₃₇	—	4	2	no	—
58	C₂²×C₃₇	—	8	2	no	—
59	C₂²×C₃₇	—	8	1	no	—
60	C₂²×C₃₇	#120	1	1	yes	characteristic
61	C₂×D₃₇	—	2	4	no	—
62	C₂×D₃₇	—	4	2	no	—
63	C₂×D₃₇	—	4	2	no	—
64	C₂×D₃₇	—	8	8	no	—
65	C₄×C₃₇	—	2	1	no	—
66	C₄×C₃₇	C₂×D₈	1	2	yes	—
67	C₄×C₃₇	AGL(1,ℤ/8)	1	2	yes	—
68	Dic₃₇	—	4	2	no	—
69	#1619	—	37	1	no	maximal, Sylow
70	C₂×C₃₇	—	8	2	no	—
71	C₂×C₃₇	—	16	1	no	—
72	C₂×C₃₇	#447	1	2	yes	—
73	C₂×C₃₇	#453	1	1	yes	characteristic
74	D₃₇	—	4	4	no	—
75	D₄²	—	37	1	no	—
76	#493	D₃₇	1	1	yes	characteristic, complemented
77	#459	—	37	2	no	—
78	#331	—	37	1	no	—
79	D₄⋊C₈	—	37	2	no	—
80	C₃₇	#1619	1	1	yes	characteristic, complemented, Sylow
81	C₂²×D₄	—	74	2	no	—
82	C₂×D₈	—	2	2	no	—
83	C₂×D₈	—	74	2	no	—
84	#127	C₂×D₃₇	1	1	yes	characteristic
85	#127	C₂×D₃₇	1	1	yes	characteristic, complemented
86	#121	—	74	2	no	—
87	#120	—	74	2	no	—
88	C₄×D₄	—	37	2	no	—
89	C₄⋊C₈	—	37	2	no	—
90	#102	—	74	2	no	—
91	C₄×C₈	C₂×D₃₇	1	1	yes	characteristic, complemented
92	C₂⁴	—	74	2	no	—
93	C₂×D₄	—	2	2	no	—
94	C₂×D₄	—	2	2	no	—
95	C₂×D₄	—	2	1	no	—
96	C₂×D₄	—	37	2	no	—
97	C₂×D₄	—	74	4	no	—
98	C₂×D₄	—	148	4	no	—
99	C₂×D₄	—	148	4	no	—
100	C₂×D₄	C₂²⋊D₃₇	1	2	yes	—
101	C₂²×C₄	—	74	2	no	—
102	D₈	—	4	4	no	—
103	D₈	—	148	4	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	74	2	no	—
106	C₄⋊C₄	—	37	2	no	—
107	C₂²⋊C₄	—	74	2	no	—
108	C₂²⋊C₄	—	74	2	no	—
109	C₄²	C₂²×D₃₇	1	1	yes	characteristic
110	C₂³	—	2	2	no	—
111	C₂³	—	4	1	no	—
112	C₂³	—	296	2	no	—
113	C₂³	—	74	2	no	—
114	C₂³	—	148	8	no	—
115	D₄	—	4	2	no	—
116	D₄	—	4	2	no	—
117	D₄	—	4	2	no	—
118	D₄	—	4	2	no	—
119	D₄	—	4	2	no	—
120	D₄	—	4	2	no	—
121	D₄	—	37	4	no	—
122	D₄	—	74	2	no	—
123	D₄	—	148	8	no	—
124	C₂×C₄	—	74	2	no	—
125	C₂×C₄	—	148	2	no	—
126	C₂×C₄	—	148	2	no	—
127	C₂×C₄	D₄×D₃₇	1	2	yes	—
128	C₂×C₄	C₂×C₂²⋊D₃₇	1	1	yes	characteristic
129	C₈	—	2	2	no	—
130	C₈	—	148	2	no	—
131	C₂²	—	4	2	no	—
132	C₂²	—	4	4	no	—
133	C₂²	—	8	1	no	—
134	C₂²	—	8	2	no	—
135	C₂²	—	296	8	no	—
136	C₂²	—	74	4	no	—
137	C₂²	—	148	2	no	—
138	C₂²	—	148	2	no	—
139	C₂²	#11924504	1	1	yes	characteristic
140	C₄	—	2	1	no	—
141	C₄	—	148	2	no	—
142	C₄	D₈×D₃₇	1	2	yes	—
143	C₄	#11924560	1	2	yes	—
144	C₂	—	8	2	no	—
145	C₂	—	16	1	no	—
146	C₂	—	148	4	no	—
147	C₂	#14557665	1	2	yes	—
148	C₂	#14557700	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14558168

C₂

yes

maximal, characteristic, complemented

#14557725

C₂

yes

maximal, complemented

#14557277

C₂

yes

maximal, characteristic, complemented

#14557262

C₂

yes

maximal, complemented

C₃₇×#493

C₂

yes

maximal, characteristic, complemented

C₂×D₄×D₃₇

—

C₂×#10596939

—

#11924507

—

#11924504

—

C₄×D₁₄₈

C₂²

yes

—

C₄⋊#64436

C₂²

yes

complemented

#11924457

—

C₂×D₈×C₃₇

—

#127×C₃₇

C₂²

yes

characteristic

#127×C₃₇

C₂²

yes

characteristic, complemented

C₄×C₈×C₃₇

C₂²

yes

characteristic, complemented

C₂³×D₃₇

—

C₂×D₄×C₃₇

—

C₂×D₄×C₃₇

—

C₂×D₄×C₃₇

—

C₂×D₄×C₃₇

D₄

yes

—

C₂×C₂²⋊D₃₇

—

D₄×D₃₇

—

D₄×D₃₇

—

C₂×D₁₄₈

D₄

yes

—

C₂×C₄×D₃₇

—

D₈×C₃₇

—

C₂×C₈×C₃₇

—

C₄²×C₃₇

C₂³

yes

characteristic

C₂²⋊Dic₃₇

—

#10596939

—

#10596938

—

C₄⋊Dic₃₇

D₄

yes

—

C₂×#64436

—

C₂³×C₃₇

—

C₂³×C₃₇

—

C₂²×D₃₇

—

C₂²×D₃₇

—

C₂²×D₃₇

—

D₄×C₃₇

—

D₄×C₃₇

—

D₄×C₃₇

—

D₄×C₃₇

—

D₄×C₃₇

—

D₄×C₃₇

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂×C₄×C₃₇

C₂×D₄

yes

characteristic

C₂²⋊D₃₇

—

C₂×Dic₃₇

—

D₁₄₈

—

D₁₄₈

D₈

yes

complemented

C₄×D₃₇

—

C₄×D₃₇

—

C₈×C₃₇

—

#64436

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

#120

yes

characteristic

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

—

C₂×D₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₂×D₈

yes

—

C₄×C₃₇

AGL(1,ℤ/8)

yes

—

Dic₃₇

—

#1619

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

—

C₂×C₃₇

#447

yes

—

C₂×C₃₇

#453

yes

characteristic

D₃₇

—

D₄²

—

#493

D₃₇

yes

characteristic, complemented

#459

—

#331

—

D₄⋊C₈

—

C₃₇

#1619

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂×D₈

—

C₂×D₈

—

#127

C₂×D₃₇

yes

characteristic

#127

C₂×D₃₇

yes

characteristic, complemented

#121

—

#120

—

C₄×D₄

—

C₄⋊C₈

—

#102

—

C₄×C₈

C₂×D₃₇

yes

characteristic, complemented

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

148

—

C₂×D₄

—

148

—

100

C₂×D₄

C₂²⋊D₃₇

yes

—

101

C₂²×C₄

—

102

D₈

—

103

D₈

—

148

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₄⋊C₄

—

107

C₂²⋊C₄

—

108

C₂²⋊C₄

—

109

C₄²

C₂²×D₃₇

yes

characteristic

110

C₂³

—

111

C₂³

—

112

C₂³

—

296

—

113

C₂³

—

114

C₂³

—

148

—

115

D₄

—

116

D₄

—

117

D₄

—

118

D₄

—

119

D₄

—

120

D₄

—

121

D₄

—

122

D₄

—

123

D₄

—

148

—

124

C₂×C₄

—

125

C₂×C₄

—

148

—

126

C₂×C₄

—

148

—

127

C₂×C₄

D₄×D₃₇

yes

—

128

C₂×C₄

C₂×C₂²⋊D₃₇

yes

characteristic

129

C₈

—

130

C₈

—

148

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

296

—

136

C₂²

—

137

C₂²

—

148

—

138

C₂²

—

148

—

139

C₂²

#11924504

yes

characteristic

140

C₄

—

141

C₄

—

148

—

142

C₄

D₈×D₃₇

yes

—

143

C₄

#11924560

yes

—

144

C₂

—

145

C₂

—

146

C₂

—

148

—

147

C₂

#14557665

yes

—

148

C₂

#14557700

yes

characteristic

	subgroup	G/H
center	C₂²	#11924504
commutator, cosocle	C₄²×C₃₇	C₂³
Frattini	C₄²	C₂²×D₃₇
Fitting	C₃₇×#493	C₂
socle	C₂²×C₃₇	#120
hypercenter	#493	D₃₇
hypocenter	C₃₇	#1619
LCS	G ▷ C₄²×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

#11924504

commutator, cosocle

C₄²×C₃₇

C₂³

Frattini

C₄²

C₂²×D₃₇

Fitting

C₃₇×#493

C₂

C₂²×C₃₇

D₃₇

C₃₇

G ▷ C₄²×C₃₇ ▷ C₂²×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #14558168	C₂	semidirect
2. #14557725	C₂	semidirect
3. #14557277	C₂	semidirect
4. #14557262	C₂	semidirect
5. C₃₇×#493	C₂	semidirect
11. C₄⋊#64436	C₂²	semidirect
15. #127×C₃₇	C₂²	semidirect
16. C₄×C₈×C₃₇	C₂²	semidirect
51. D₁₄₈	D₈	semidirect
76. #493	D₃₇	semidirect
80. C₃₇	#1619	semidirect
85. #127	C₂×D₃₇	semidirect
91. C₄×C₈	C₂×D₃₇	semidirect

H ≅ G/N

type

1. #14558168

C₂

semidirect

2. #14557725

C₂

semidirect

3. #14557277

C₂

semidirect

4. #14557262

C₂

semidirect

5. C₃₇×#493

C₂

semidirect

11. C₄⋊#64436

C₂²

semidirect

15. #127×C₃₇

C₂²

semidirect

16. C₄×C₈×C₃₇

C₂²

semidirect

51. D₁₄₈

D₈

semidirect

76. #493

D₃₇

semidirect

80. C₃₇

#1619

semidirect

85. #127

C₂×D₃₇

semidirect

91. C₄×C₈

C₂×D₃₇

semidirect

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

subgroup	index
#14558168	2
#14557725	2
#14557277	2
#14557262	2
C₃₇×#493	2
#1619	37

subgroup

index

C₃₇×#493

Subgroups — #17993648

Conjugacy classes of subgroups

Complements

Maximal subgroups