Group #18007743 — subgroups

Subgroups — #18007743

Order 4736.

360 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 129 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	#14558350	C₂	1	1	yes	maximal, characteristic, complemented
2	#14557973	C₂	1	3	yes	maximal, complemented
3	#14557969	C₂	1	3	yes	maximal, complemented
4	#14557963	C₂	1	3	yes	maximal, complemented
5	C₂×#11924480	C₂	1	3	yes	maximal, complemented
6	#14557950	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₇×#576	C₂	1	1	yes	maximal, characteristic, complemented
8	#11924583	—	2	1	no	—
9	#11924569	—	2	3	no	—
10	#11924558	C₂²	1	3	yes	complemented
11	C₂×Dic₁₄₈	C₂²	1	3	yes	complemented
12	#11924482	C₂²	1	3	yes	complemented
13	#11924480	C₂²	1	3	yes	complemented
14	#11924480	C₂²	1	6	yes	complemented
15	#11924479	C₂²	1	1	yes	characteristic, complemented
16	#11924477	C₂²	1	3	yes	complemented
17	#11924476	C₂²	1	3	yes	complemented
18	C₂×#10596925	C₂²	1	3	yes	complemented
19	H(5,𝔽₂)×C₃₇	—	2	1	no	—
20	AGL(1,ℤ/8)×C₃₇	—	2	3	no	—
21	#135×C₃₇	C₂²	1	3	yes	—
22	C₂×D₈×C₃₇	C₂²	1	3	yes	—
23	#131×C₃₇	C₂²	1	1	yes	characteristic
24	#41×C₃₇	—	2	3	no	—
25	#41×C₃₇	—	4	1	no	—
26	#41×C₃₇	D₄	1	1	yes	characteristic
27	C₂×D₄×C₃₇	—	2	3	no	—
28	C₂×D₄×C₃₇	—	4	3	no	—
29	Q₈×D₃₇	—	4	1	no	—
30	#10596964	—	4	3	no	—
31	#10596962	—	2	3	no	—
32	C₂×Dic₇₄	—	2	3	no	—
33	Q₁₆×C₃₇	C₂³	1	1	yes	characteristic
34	SD₁₆×C₃₇	—	2	3	no	—
35	D₈×C₃₇	—	2	3	no	—
36	D₈×C₃₇	C₂³	1	3	yes	—
37	M₁₆×C₃₇	D₄	1	3	yes	—
38	C₂×C₈×C₃₇	C₂³	1	3	yes	—
39	Dic₁₄₈	—	2	3	no	—
40	Dic₁₄₈	C₂³	1	3	yes	—
41	D₂₉₆	C₂³	1	1	yes	characteristic
42	#10596930	—	2	3	no	—
43	#10596925	C₂³	1	1	yes	characteristic
44	#10596925	C₂³	1	3	yes	—
45	C₂³×C₃₇	—	4	3	no	—
46	Q₈×C₃₇	—	2	1	no	—
47	Q₈×C₃₇	D₈	1	1	yes	characteristic
48	D₄×C₃₇	—	2	3	no	—
49	D₄×C₃₇	—	4	3	no	—
50	D₄×C₃₇	—	4	3	no	—
51	D₄×C₃₇	D₈	1	3	yes	—
52	C₂×C₄×C₃₇	—	4	3	no	—
53	C₂×C₄×C₃₇	C₂×D₄	1	3	yes	—
54	C₂²⋊D₃₇	—	4	3	no	—
55	C₂×Dic₃₇	—	4	3	no	—
56	D₁₄₈	—	2	1	no	—
57	C₄×D₃₇	—	4	3	no	—
58	Dic₇₄	—	2	3	no	—
59	Dic₇₄	—	4	3	no	—
60	C₈×C₃₇	C₂×D₄	1	3	yes	—
61	C₈×C₃₇	C₂⁴	1	1	yes	characteristic
62	C₂²×C₃₇	—	4	3	no	—
63	C₂²×C₃₇	—	16	3	no	—
64	C₂²×C₃₇	C₂×D₈	1	3	yes	—
65	C₂×D₃₇	—	4	1	no	—
66	C₄×C₃₇	—	4	1	no	—
67	C₄×C₃₇	C₂×D₈	1	3	yes	—
68	C₄×C₃₇	C₂²×D₄	1	1	yes	characteristic
69	Dic₃₇	—	4	3	no	—
70	#3416	—	37	1	no	maximal, Sylow
71	C₂×C₃₇	—	2	3	no	—
72	C₂×C₃₇	—	8	3	no	—
73	C₂×C₃₇	C₂²×D₈	1	1	yes	characteristic
74	D₃₇	—	8	1	no	—
75	#578	—	37	1	no	—
76	#576	D₃₇	1	1	yes	characteristic, complemented
77	#510	—	37	3	no	—
78	#509	—	37	3	no	—
79	#508	—	37	3	no	—
80	C₂×SD₃₂	—	37	3	no	—
81	#504	—	37	1	no	—
82	C₃₇	#3416	1	1	yes	characteristic, complemented, Sylow
83	2₋¹⁺⁴	—	74	1	no	—
84	H(5,𝔽₂)	—	2	1	no	—
85	#137	—	74	3	no	—
86	AGL(1,ℤ/8)	—	2	3	no	—
87	#135	—	37	3	no	—
88	#135	C₂×D₃₇	1	3	yes	—
89	C₂×Q₁₆	—	37	3	no	—
90	C₂×D₈	C₂×D₃₇	1	3	yes	—
91	#131	C₂×D₃₇	1	1	yes	characteristic
92	Q₃₂	—	37	3	no	—
93	SD₃₂	—	37	1	no	—
94	SD₃₂	—	37	3	no	—
95	SD₃₂	—	37	6	no	—
96	D₁₆	—	37	3	no	—
97	M₃₂	—	37	3	no	—
98	C₂×C₁₆	—	37	3	no	—
99	#41	—	2	3	no	—
100	#41	—	4	1	no	—
101	#41	—	74	3	no	—
102	#41	—	148	3	no	—
103	#41	C₂²⋊D₃₇	1	1	yes	characteristic
104	C₂×Q₈	—	74	3	no	—
105	C₂×Q₈	—	148	1	no	—
106	C₂×D₄	—	2	3	no	—
107	C₂×D₄	—	4	3	no	—
108	Q₁₆	—	37	3	no	—
109	Q₁₆	—	74	3	no	—
110	Q₁₆	C₂²×D₃₇	1	1	yes	characteristic
111	SD₁₆	—	2	3	no	—
112	SD₁₆	—	74	3	no	—
113	D₈	—	2	3	no	—
114	D₈	—	37	1	no	—
115	D₈	C₂²×D₃₇	1	3	yes	—
116	M₁₆	C₂²⋊D₃₇	1	3	yes	—
117	C₂×C₈	C₂²×D₃₇	1	3	yes	—
118	C₁₆	—	37	1	no	—
119	C₁₆	—	37	3	no	—
120	C₂³	—	4	3	no	—
121	Q₈	—	2	1	no	—
122	Q₈	—	74	3	no	—
123	Q₈	—	148	3	no	—
124	Q₈	#10596939	1	1	yes	characteristic
125	D₄	—	2	3	no	—
126	D₄	—	4	3	no	—
127	D₄	—	4	3	no	—
128	D₄	—	74	1	no	—
129	D₄	—	148	3	no	—
130	D₄	#10596939	1	3	yes	—
131	C₂×C₄	—	4	3	no	—
132	C₂×C₄	—	148	3	no	—
133	C₂×C₄	—	148	3	no	—
134	C₂×C₄	C₂×C₂²⋊D₃₇	1	3	yes	—
135	C₈	C₂×C₂²⋊D₃₇	1	3	yes	—
136	C₈	C₂³×D₃₇	1	1	yes	characteristic
137	C₂²	—	4	3	no	—
138	C₂²	—	16	3	no	—
139	C₂²	—	148	1	no	—
140	C₂²	C₂×#10596939	1	3	yes	—
141	C₄	—	4	1	no	—
142	C₄	—	148	3	no	—
143	C₄	C₂×#10596939	1	3	yes	—
144	C₄	C₂²×C₂²⋊D₃₇	1	1	yes	characteristic
145	C₂	—	2	3	no	—
146	C₂	—	8	3	no	—
147	C₂	—	296	1	no	—
148	C₂	C₂²×#10596939	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#14558350

C₂

yes

maximal, characteristic, complemented

#14557973

C₂

yes

maximal, complemented

#14557969

C₂

yes

maximal, complemented

#14557963

C₂

yes

maximal, complemented

C₂×#11924480

C₂

yes

maximal, complemented

#14557950

C₂

yes

maximal, characteristic, complemented

C₃₇×#576

C₂

yes

maximal, characteristic, complemented

#11924583

—

#11924569

—

#11924558

C₂²

yes

complemented

C₂×Dic₁₄₈

C₂²

yes

complemented

#11924482

C₂²

yes

complemented

#11924480

C₂²

yes

complemented

#11924480

C₂²

yes

complemented

#11924479

C₂²

yes

characteristic, complemented

#11924477

C₂²

yes

complemented

#11924476

C₂²

yes

complemented

C₂×#10596925

C₂²

yes

complemented

H(5,𝔽₂)×C₃₇

—

AGL(1,ℤ/8)×C₃₇

—

#135×C₃₇

C₂²

yes

—

C₂×D₈×C₃₇

C₂²

yes

—

#131×C₃₇

C₂²

yes

characteristic

#41×C₃₇

—

#41×C₃₇

—

#41×C₃₇

D₄

yes

characteristic

C₂×D₄×C₃₇

—

C₂×D₄×C₃₇

—

Q₈×D₃₇

—

#10596964

—

#10596962

—

C₂×Dic₇₄

—

Q₁₆×C₃₇

C₂³

yes

characteristic

SD₁₆×C₃₇

—

D₈×C₃₇

—

D₈×C₃₇

C₂³

yes

—

M₁₆×C₃₇

D₄

yes

—

C₂×C₈×C₃₇

C₂³

yes

—

Dic₁₄₈

—

Dic₁₄₈

C₂³

yes

—

D₂₉₆

C₂³

yes

characteristic

#10596930

—

#10596925

C₂³

yes

characteristic

#10596925

C₂³

yes

—

C₂³×C₃₇

—

Q₈×C₃₇

—

Q₈×C₃₇

D₈

yes

characteristic

D₄×C₃₇

—

D₄×C₃₇

—

D₄×C₃₇

—

D₄×C₃₇

D₈

yes

—

C₂×C₄×C₃₇

—

C₂×C₄×C₃₇

C₂×D₄

yes

—

C₂²⋊D₃₇

—

C₂×Dic₃₇

—

D₁₄₈

—

C₄×D₃₇

—

Dic₇₄

—

Dic₇₄

—

C₈×C₃₇

C₂×D₄

yes

—

C₈×C₃₇

C₂⁴

yes

characteristic

C₂²×C₃₇

—

C₂²×C₃₇

—

C₂²×C₃₇

C₂×D₈

yes

—

C₂×D₃₇

—

C₄×C₃₇

—

C₄×C₃₇

C₂×D₈

yes

—

C₄×C₃₇

C₂²×D₄

yes

characteristic

Dic₃₇

—

#3416

—

maximal, Sylow

C₂×C₃₇

—

C₂×C₃₇

—

C₂×C₃₇

C₂²×D₈

yes

characteristic

D₃₇

—

#578

—

#576

D₃₇

yes

characteristic, complemented

#510

—

#509

—

#508

—

C₂×SD₃₂

—

#504

—

C₃₇

#3416

yes

characteristic, complemented, Sylow

2₋¹⁺⁴

—

—

—

—

—

C₂×D₃₇

yes

—

C₂×Q₁₆

—

C₂×D₈

C₂×D₃₇

yes

—

#131

C₂×D₃₇

yes

characteristic

Q₃₂

—

SD₃₂

—

SD₃₂

—

SD₃₂

—

D₁₆

—

M₃₂

—

C₂×C₁₆

—

#41

—

100

#41

—

101

#41

—

102

#41

—

148

—

103

#41

C₂²⋊D₃₇

yes

characteristic

104

C₂×Q₈

—

105

C₂×Q₈

—

148

—

106

C₂×D₄

—

107

C₂×D₄

—

108

Q₁₆

—

109

Q₁₆

—

110

Q₁₆

C₂²×D₃₇

yes

characteristic

111

SD₁₆

—

112

SD₁₆

—

113

D₈

—

114

D₈

—

115

D₈

C₂²×D₃₇

yes

—

116

M₁₆

C₂²⋊D₃₇

yes

—

117

C₂×C₈

C₂²×D₃₇

yes

—

118

C₁₆

—

119

C₁₆

—

120

C₂³

—

121

Q₈

—

122

Q₈

—

123

Q₈

—

148

—

124

Q₈

#10596939

yes

characteristic

125

D₄

—

126

D₄

—

127

D₄

—

128

D₄

—

129

D₄

—

148

—

130

D₄

#10596939

yes

—

131

C₂×C₄

—

132

C₂×C₄

—

148

—

133

C₂×C₄

—

148

—

134

C₂×C₄

C₂×C₂²⋊D₃₇

yes

—

135

C₈

C₂×C₂²⋊D₃₇

yes

—

136

C₈

C₂³×D₃₇

yes

characteristic

137

C₂²

—

138

C₂²

—

139

C₂²

—

148

—

140

C₂²

C₂×#10596939

yes

—

141

C₄

—

142

C₄

—

148

—

143

C₄

C₂×#10596939

yes

—

144

C₄

C₂²×C₂²⋊D₃₇

yes

characteristic

145

C₂

—

146

C₂

—

147

C₂

—

296

—

148

C₂

C₂²×#10596939

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×#10596939
commutator, cosocle	C₈×C₃₇	C₂⁴
Frattini	C₈	C₂³×D₃₇
Fitting	C₃₇×#576	C₂
socle	C₂×C₃₇	C₂²×D₈
hypercenter	#576	D₃₇
hypocenter	C₃₇	#3416
LCS	G ▷ C₈×C₃₇ ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131 ◁ #576

subgroup

G/H

center

C₂

C₂²×#10596939

commutator, cosocle

C₈×C₃₇

C₂⁴

Frattini

C₈

C₂³×D₃₇

Fitting

C₃₇×#576

C₂

socle

C₂×C₃₇

C₂²×D₈

D₃₇

C₃₇

G ▷ C₈×C₃₇ ▷ C₄×C₃₇ ▷ C₂×C₃₇ ▷ C₃₇

UCS

C₁ ◁ C₂ ◁ #41 ◁ #131 ◁ #576

N	H ≅ G/N	type
1. #14558350	C₂	semidirect
2. #14557973	C₂	semidirect
3. #14557969	C₂	semidirect
4. #14557963	C₂	semidirect
5. C₂×#11924480	C₂	semidirect
6. #14557950	C₂	semidirect
7. C₃₇×#576	C₂	semidirect
10. #11924558	C₂²	semidirect
11. C₂×Dic₁₄₈	C₂²	semidirect
12. #11924482	C₂²	semidirect
13. #11924480	C₂²	semidirect
14. #11924480	C₂²	semidirect
15. #11924479	C₂²	semidirect
16. #11924477	C₂²	semidirect
17. #11924476	C₂²	semidirect
18. C₂×#10596925	C₂²	semidirect
76. #576	D₃₇	semidirect
82. C₃₇	#3416	semidirect

H ≅ G/N

type

1. #14558350

C₂

semidirect

2. #14557973

C₂

semidirect

3. #14557969

C₂

semidirect

4. #14557963

C₂

semidirect

5. C₂×#11924480

C₂

semidirect

6. #14557950

C₂

semidirect

7. C₃₇×#576

C₂

semidirect

10. #11924558

C₂²

semidirect

11. C₂×Dic₁₄₈

C₂²

semidirect

12. #11924482

C₂²

semidirect

13. #11924480

C₂²

semidirect

14. #11924480

C₂²

semidirect

15. #11924479

C₂²

semidirect

16. #11924477

C₂²

semidirect

17. #11924476

C₂²

semidirect

18. C₂×#10596925

C₂²

semidirect

76. #576

D₃₇

semidirect

82. C₃₇

#3416

semidirect

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

subgroup	index
#14558350	2
#14557973	2
#14557969	2
#14557963	2
C₂×#11924480	2
#14557950	2
C₃₇×#576	2
#3416	37

subgroup

index

C₃₇×#576

Subgroups — #18007743

Conjugacy classes of subgroups

Complements

Maximal subgroups