Group #20242440 — subgroups

Subgroups — #20242440

Order 6016.

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	#15689758	C₂	1	1	yes	maximal, characteristic, complemented
2	#15689315	C₂	1	2	yes	maximal, complemented
3	#15688867	C₂	1	1	yes	maximal, characteristic, complemented
4	#15688852	C₂	1	2	yes	maximal, complemented
5	C₄₇×#493	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₄×D₄₇	—	2	2	no	—
7	C₂×#10625334	—	2	2	no	—
8	#13093995	—	2	2	no	—
9	#13093992	—	2	2	no	—
10	C₄×D₁₈₈	C₂²	1	2	yes	—
11	C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	complemented
12	#13093945	—	2	2	no	—
13	C₂×D₈×C₄₇	—	2	2	no	—
14	#127×C₄₇	C₂²	1	1	yes	characteristic
15	#127×C₄₇	C₂²	1	1	yes	characteristic, complemented
16	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic, complemented
17	C₂³×D₄₇	—	2	2	no	—
18	C₂×D₄×C₄₇	—	2	2	no	—
19	C₂×D₄×C₄₇	—	2	1	no	—
20	C₂×D₄×C₄₇	—	2	2	no	—
21	C₂×D₄×C₄₇	D₄	1	2	yes	—
22	C₂×C₂²⋊D₄₇	—	2	4	no	—
23	D₄×D₄₇	—	4	4	no	—
24	D₄×D₄₇	—	4	4	no	—
25	C₂×D₁₈₈	D₄	1	2	yes	—
26	C₂×C₄×D₄₇	—	2	2	no	—
27	D₈×C₄₇	—	4	4	no	—
28	C₂×C₈×C₄₇	—	2	1	no	—
29	C₄²×C₄₇	C₂³	1	1	yes	characteristic
30	C₂²⋊Dic₄₇	—	2	2	no	—
31	#10625334	—	4	4	no	—
32	(C₂×D₄₇)⋊C₄	—	2	2	no	—
33	C₄⋊Dic₄₇	D₄	1	2	yes	—
34	C₂×C₄₇⋊C₈	—	2	2	no	—
35	C₂³×C₄₇	—	2	2	no	—
36	C₂³×C₄₇	—	4	1	no	—
37	C₂²×D₄₇	—	2	2	no	—
38	C₂²×D₄₇	—	4	8	no	—
39	C₂²×D₄₇	—	8	2	no	—
40	D₄×C₄₇	—	4	2	no	—
41	D₄×C₄₇	—	4	2	no	—
42	D₄×C₄₇	—	4	2	no	—
43	D₄×C₄₇	—	4	2	no	—
44	D₄×C₄₇	—	4	2	no	—
45	D₄×C₄₇	—	4	2	no	—
46	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
47	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
48	C₂²⋊D₄₇	—	4	8	no	—
49	C₂×Dic₄₇	—	2	2	no	—
50	D₁₈₈	—	2	2	no	—
51	D₁₈₈	D₈	1	4	yes	complemented
52	C₄×D₄₇	—	4	2	no	—
53	C₄×D₄₇	—	4	2	no	—
54	C₈×C₄₇	—	2	2	no	—
55	C₄₇⋊C₈	—	4	2	no	—
56	C₂²×C₄₇	—	4	4	no	—
57	C₂²×C₄₇	—	4	2	no	—
58	C₂²×C₄₇	—	8	2	no	—
59	C₂²×C₄₇	—	8	1	no	—
60	C₂²×C₄₇	#120	1	1	yes	characteristic
61	C₂×D₄₇	—	2	4	no	—
62	C₂×D₄₇	—	4	2	no	—
63	C₂×D₄₇	—	4	2	no	—
64	C₂×D₄₇	—	8	8	no	—
65	C₄×C₄₇	—	2	1	no	—
66	C₄×C₄₇	C₂×D₈	1	2	yes	—
67	C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
68	Dic₄₇	—	4	2	no	—
69	#1619	—	47	1	no	maximal, Sylow
70	C₂×C₄₇	—	8	2	no	—
71	C₂×C₄₇	—	16	1	no	—
72	C₂×C₄₇	#447	1	2	yes	—
73	C₂×C₄₇	#453	1	1	yes	characteristic
74	D₄₇	—	4	4	no	—
75	D₄²	—	47	1	no	—
76	#493	D₄₇	1	1	yes	characteristic, complemented
77	#459	—	47	2	no	—
78	#331	—	47	1	no	—
79	D₄⋊C₈	—	47	2	no	—
80	C₄₇	#1619	1	1	yes	characteristic, complemented, Sylow
81	C₂²×D₄	—	94	2	no	—
82	C₂×D₈	—	2	2	no	—
83	C₂×D₈	—	94	2	no	—
84	#127	C₂×D₄₇	1	1	yes	characteristic
85	#127	C₂×D₄₇	1	1	yes	characteristic, complemented
86	#121	—	94	2	no	—
87	#120	—	94	2	no	—
88	C₄×D₄	—	47	2	no	—
89	C₄⋊C₈	—	47	2	no	—
90	#102	—	94	2	no	—
91	C₄×C₈	C₂×D₄₇	1	1	yes	characteristic, complemented
92	C₂⁴	—	94	2	no	—
93	C₂×D₄	—	2	2	no	—
94	C₂×D₄	—	2	2	no	—
95	C₂×D₄	—	2	1	no	—
96	C₂×D₄	—	47	2	no	—
97	C₂×D₄	—	94	4	no	—
98	C₂×D₄	—	188	4	no	—
99	C₂×D₄	—	188	4	no	—
100	C₂×D₄	C₂²⋊D₄₇	1	2	yes	—
101	C₂²×C₄	—	94	2	no	—
102	D₈	—	4	4	no	—
103	D₈	—	188	4	no	—
104	C₂×C₈	—	2	1	no	—
105	C₂×C₈	—	94	2	no	—
106	C₄⋊C₄	—	47	2	no	—
107	C₂²⋊C₄	—	94	2	no	—
108	C₂²⋊C₄	—	94	2	no	—
109	C₄²	C₂²×D₄₇	1	1	yes	characteristic
110	C₂³	—	2	2	no	—
111	C₂³	—	4	1	no	—
112	C₂³	—	94	2	no	—
113	C₂³	—	376	2	no	—
114	C₂³	—	188	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	—	47	4	no	—
122	D₄	—	94	2	no	—
123	D₄	—	188	8	no	—
124	C₂×C₄	—	94	2	no	—
125	C₂×C₄	—	188	2	no	—
126	C₂×C₄	—	188	2	no	—
127	C₂×C₄	D₄×D₄₇	1	2	yes	—
128	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
129	C₈	—	2	2	no	—
130	C₈	—	188	2	no	—
131	C₂²	—	4	4	no	—
132	C₂²	—	4	2	no	—
133	C₂²	—	8	2	no	—
134	C₂²	—	8	1	no	—
135	C₂²	—	94	4	no	—
136	C₂²	—	376	8	no	—
137	C₂²	—	188	2	no	—
138	C₂²	—	188	2	no	—
139	C₂²	#13093992	1	1	yes	characteristic
140	C₄	—	2	1	no	—
141	C₄	—	188	2	no	—
142	C₄	D₈×D₄₇	1	2	yes	—
143	C₄	#13094048	1	2	yes	—
144	C₂	—	8	2	no	—
145	C₂	—	16	1	no	—
146	C₂	—	188	4	no	—
147	C₂	#15689255	1	2	yes	—
148	C₂	#15689290	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689758

C₂

yes

maximal, characteristic, complemented

#15689315

C₂

yes

maximal, complemented

#15688867

C₂

yes

maximal, characteristic, complemented

#15688852

C₂

yes

maximal, complemented

C₄₇×#493

C₂

yes

maximal, characteristic, complemented

C₂×D₄×D₄₇

—

C₂×#10625334

—

#13093995

—

#13093992

—

C₄×D₁₈₈

C₂²

yes

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

complemented

#13093945

—

C₂×D₈×C₄₇

—

#127×C₄₇

C₂²

yes

characteristic

#127×C₄₇

C₂²

yes

characteristic, complemented

C₄×C₈×C₄₇

C₂²

yes

characteristic, complemented

C₂³×D₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

D₄

yes

—

C₂×C₂²⋊D₄₇

—

D₄×D₄₇

—

D₄×D₄₇

—

C₂×D₁₈₈

D₄

yes

—

C₂×C₄×D₄₇

—

D₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

C₂³

yes

characteristic

C₂²⋊Dic₄₇

—

#10625334

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

D₄

yes

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²⋊D₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

D₈

yes

complemented

C₄×D₄₇

—

C₄×D₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#120

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×D₈

yes

—

C₄×C₄₇

AGL(1,ℤ/8)

yes

—

Dic₄₇

—

#1619

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#447

yes

—

C₂×C₄₇

#453

yes

characteristic

D₄₇

—

D₄²

—

#493

D₄₇

yes

characteristic, complemented

#459

—

#331

—

D₄⋊C₈

—

C₄₇

#1619

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂×D₈

—

C₂×D₈

—

#127

C₂×D₄₇

yes

characteristic

#127

C₂×D₄₇

yes

characteristic, complemented

#121

—

#120

—

C₄×D₄

—

C₄⋊C₈

—

#102

—

C₄×C₈

C₂×D₄₇

yes

characteristic, complemented

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

188

—

C₂×D₄

—

188

—

100

C₂×D₄

C₂²⋊D₄₇

yes

—

101

C₂²×C₄

—

102

D₈

—

103

D₈

—

188

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₄⋊C₄

—

107

C₂²⋊C₄

—

108

C₂²⋊C₄

—

109

C₄²

C₂²×D₄₇

yes

characteristic

110

C₂³

—

111

C₂³

—

112

C₂³

—

113

C₂³

—

376

—

114

C₂³

—

188

—

115

D₄

—

116

D₄

—

117

D₄

—

118

D₄

—

119

D₄

—

120

D₄

—

121

D₄

—

122

D₄

—

123

D₄

—

188

—

124

C₂×C₄

—

125

C₂×C₄

—

188

—

126

C₂×C₄

—

188

—

127

C₂×C₄

D₄×D₄₇

yes

—

128

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

129

C₈

—

130

C₈

—

188

—

131

C₂²

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

376

—

137

C₂²

—

188

—

138

C₂²

—

188

—

139

C₂²

#13093992

yes

characteristic

140

C₄

—

141

C₄

—

188

—

142

C₄

D₈×D₄₇

yes

—

143

C₄

#13094048

yes

—

144

C₂

—

145

C₂

—

146

C₂

—

188

—

147

C₂

#15689255

yes

—

148

C₂

#15689290

yes

characteristic

	subgroup	G/H
center	C₂²	#13093992
commutator	C₄²×C₄₇	C₂³
Frattini	C₄²	C₂²×D₄₇
Fitting	C₄₇×#493	C₂
socle	C₂²×C₄₇	#120
hypercenter	#493	D₄₇
hypocenter	C₄₇	#1619
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

#13093992

commutator

C₄²×C₄₇

C₂³

Frattini

C₄²

C₂²×D₄₇

Fitting

C₄₇×#493

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #15689758	C₂	semidirect
2. #15689315	C₂	semidirect
3. #15688867	C₂	semidirect
4. #15688852	C₂	semidirect
5. C₄₇×#493	C₂	semidirect
11. C₄⋊(C₄₇⋊C₈)	C₂²	semidirect
15. #127×C₄₇	C₂²	semidirect
16. C₄×C₈×C₄₇	C₂²	semidirect
51. D₁₈₈	D₈	semidirect
76. #493	D₄₇	semidirect
80. C₄₇	#1619	semidirect
85. #127	C₂×D₄₇	semidirect
91. C₄×C₈	C₂×D₄₇	semidirect

H ≅ G/N

type

1. #15689758

C₂

semidirect

2. #15689315

C₂

semidirect

3. #15688867

C₂

semidirect

4. #15688852

C₂

semidirect

5. C₄₇×#493

C₂

semidirect

11. C₄⋊(C₄₇⋊C₈)

C₂²

semidirect

15. #127×C₄₇

C₂²

semidirect

16. C₄×C₈×C₄₇

C₂²

semidirect

51. D₁₈₈

D₈

semidirect

76. #493

D₄₇

semidirect

80. C₄₇

#1619

semidirect

85. #127

C₂×D₄₇

semidirect

91. C₄×C₈

C₂×D₄₇

semidirect

Subgroups — #20242440

Conjugacy classes of subgroups

Complements