Group #20256527 — subgroups

Subgroups — #20256527

Order 6016.

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

normal subgroups only

This group has 370 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	#15689928	C₂	1	1	yes	maximal, characteristic, complemented
2	#15689559	C₂	1	3	yes	maximal, complemented
3	#15689558	C₂	1	3	yes	maximal, complemented
4	#15689553	C₂	1	3	yes	maximal, complemented
5	C₂×#13093965	C₂	1	3	yes	maximal, complemented
6	#15689540	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#576	C₂	1	1	yes	maximal, characteristic, complemented
8	#13094070	—	2	1	no	—
9	#13094052	—	2	3	no	—
10	#13094046	C₂²	1	3	yes	complemented
11	C₂×D₃₇₆	C₂²	1	3	yes	complemented
12	#13093970	C₂²	1	1	yes	characteristic, complemented
13	#13093968	C₂²	1	3	yes	complemented
14	#13093967	C₂²	1	3	yes	complemented
15	#13093965	C₂²	1	3	yes	complemented
16	#13093965	C₂²	1	6	yes	complemented
17	#13093964	C₂²	1	3	yes	complemented
18	C₂×C₄₇⋊C₁₆	C₂²	1	3	yes	complemented
19	H(5,𝔽₂)×C₄₇	—	2	1	no	—
20	AGL(1,ℤ/8)×C₄₇	—	2	3	no	—
21	#135×C₄₇	C₂²	1	3	yes	complemented
22	C₂×D₈×C₄₇	C₂²	1	3	yes	complemented
23	#131×C₄₇	C₂²	1	1	yes	characteristic
24	#41×C₄₇	—	2	3	no	—
25	#41×C₄₇	—	4	1	no	—
26	#41×C₄₇	D₄	1	1	yes	characteristic
27	C₂×D₄×C₄₇	—	2	3	no	—
28	C₂×D₄×C₄₇	—	4	3	no	—
29	#10625354	—	4	1	no	—
30	D₄×D₄₇	—	4	3	no	—
31	#10625350	—	2	3	no	—
32	C₂×D₁₈₈	—	2	3	no	—
33	Q₁₆×C₄₇	C₂³	1	1	yes	characteristic
34	SD₁₆×C₄₇	—	2	3	no	—
35	D₈×C₄₇	—	2	3	no	—
36	D₈×C₄₇	C₂³	1	3	yes	—
37	M₁₆×C₄₇	D₄	1	3	yes	—
38	C₂×C₈×C₄₇	C₂³	1	3	yes	—
39	Dic₁₈₈	C₂³	1	1	yes	characteristic
40	D₃₇₆	—	2	3	no	—
41	D₃₇₆	C₂³	1	3	yes	—
42	#10625325	—	2	3	no	—
43	C₄₇⋊C₁₆	C₂³	1	1	yes	characteristic
44	C₄₇⋊C₁₆	C₂³	1	3	yes	—
45	C₂³×C₄₇	—	4	3	no	—
46	C₂²×D₄₇	—	4	3	no	—
47	Q₈×C₄₇	—	2	1	no	—
48	Q₈×C₄₇	D₈	1	1	yes	characteristic
49	D₄×C₄₇	—	2	3	no	—
50	D₄×C₄₇	—	4	3	no	—
51	D₄×C₄₇	—	4	3	no	—
52	D₄×C₄₇	D₈	1	3	yes	—
53	C₂×C₄×C₄₇	—	4	3	no	—
54	C₂×C₄×C₄₇	C₂×D₄	1	3	yes	—
55	C₂²⋊D₄₇	—	4	3	no	—
56	D₁₈₈	—	2	3	no	—
57	D₁₈₈	—	4	3	no	—
58	C₄×D₄₇	—	4	3	no	—
59	Dic₉₄	—	2	1	no	—
60	C₈×C₄₇	C₂×D₄	1	3	yes	—
61	C₈×C₄₇	C₂⁴	1	1	yes	characteristic
62	C₂²×C₄₇	—	4	3	no	—
63	C₂²×C₄₇	—	16	3	no	—
64	C₂²×C₄₇	C₂×D₈	1	3	yes	—
65	C₂×D₄₇	—	4	3	no	—
66	C₂×D₄₇	—	16	3	no	—
67	C₄×C₄₇	—	4	1	no	—
68	C₄×C₄₇	C₂×D₈	1	3	yes	—
69	C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
70	Dic₄₇	—	4	1	no	—
71	#3415	—	47	1	no	maximal, Sylow
72	C₂×C₄₇	—	2	3	no	—
73	C₂×C₄₇	—	8	3	no	—
74	C₂×C₄₇	C₂²×D₈	1	1	yes	characteristic
75	D₄₇	—	8	3	no	—
76	#576	—	47	1	no	—
77	#576	D₄₇	1	1	yes	characteristic, complemented
78	#509	—	47	3	no	—
79	#509	—	47	3	no	—
80	#508	—	47	3	no	—
81	C₂×D₁₆	—	47	3	no	—
82	#504	—	47	1	no	—
83	C₄₇	#3415	1	1	yes	characteristic, complemented, Sylow
84	H(5,𝔽₂)	—	2	1	no	—
85	H(5,𝔽₂)	—	94	1	no	—
86	AGL(1,ℤ/8)	—	2	3	no	—
87	AGL(1,ℤ/8)	—	94	3	no	—
88	#135	—	47	3	no	—
89	#135	C₂×D₄₇	1	3	yes	complemented
90	C₂×D₈	—	47	3	no	—
91	C₂×D₈	C₂×D₄₇	1	3	yes	complemented
92	#131	C₂×D₄₇	1	1	yes	characteristic
93	Q₃₂	—	47	1	no	—
94	SD₃₂	—	47	3	no	—
95	SD₃₂	—	47	3	no	—
96	D₁₆	—	47	3	no	—
97	D₁₆	—	47	6	no	—
98	M₃₂	—	47	3	no	—
99	C₂×C₁₆	—	47	3	no	—
100	#41	—	2	3	no	—
101	#41	—	4	1	no	—
102	#41	—	94	3	no	—
103	#41	—	188	1	no	—
104	#41	C₂²⋊D₄₇	1	1	yes	characteristic
105	C₂×D₄	—	2	3	no	—
106	C₂×D₄	—	4	3	no	—
107	C₂×D₄	—	94	3	no	—
108	C₂×D₄	—	188	3	no	—
109	Q₁₆	—	47	1	no	—
110	Q₁₆	C₂²×D₄₇	1	1	yes	characteristic
111	SD₁₆	—	2	3	no	—
112	SD₁₆	—	94	3	no	—
113	D₈	—	2	3	no	—
114	D₈	—	47	3	no	—
115	D₈	—	94	3	no	—
116	D₈	C₂²×D₄₇	1	3	yes	—
117	M₁₆	C₂²⋊D₄₇	1	3	yes	—
118	C₂×C₈	C₂²×D₄₇	1	3	yes	—
119	C₁₆	—	47	1	no	—
120	C₁₆	—	47	3	no	—
121	C₂³	—	4	3	no	—
122	C₂³	—	188	3	no	—
123	Q₈	—	2	1	no	—
124	Q₈	—	94	1	no	—
125	Q₈	#10625334	1	1	yes	characteristic
126	D₄	—	2	3	no	—
127	D₄	—	4	3	no	—
128	D₄	—	4	3	no	—
129	D₄	—	94	3	no	—
130	D₄	—	188	3	no	—
131	D₄	—	188	3	no	—
132	D₄	#10625334	1	3	yes	—
133	C₂×C₄	—	4	3	no	—
134	C₂×C₄	—	188	3	no	—
135	C₂×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
136	C₈	C₂×C₂²⋊D₄₇	1	3	yes	—
137	C₈	C₂³×D₄₇	1	1	yes	characteristic
138	C₂²	—	4	3	no	—
139	C₂²	—	16	3	no	—
140	C₂²	—	188	3	no	—
141	C₂²	—	752	3	no	—
142	C₂²	C₂×#10625334	1	3	yes	—
143	C₄	—	4	1	no	—
144	C₄	—	188	1	no	—
145	C₄	C₂×#10625334	1	3	yes	—
146	C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
147	C₂	—	2	3	no	—
148	C₂	—	8	3	no	—
149	C₂	—	376	3	no	—
150	C₂	C₂²×#10625334	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689928

C₂

yes

maximal, characteristic, complemented

#15689559

C₂

yes

maximal, complemented

#15689558

C₂

yes

maximal, complemented

#15689553

C₂

yes

maximal, complemented

C₂×#13093965

C₂

yes

maximal, complemented

#15689540

C₂

yes

maximal, characteristic, complemented

C₄₇×#576

C₂

yes

maximal, characteristic, complemented

#13094070

—

#13094052

—

#13094046

C₂²

yes

complemented

C₂×D₃₇₆

C₂²

yes

complemented

#13093970

C₂²

yes

characteristic, complemented

#13093968

C₂²

yes

complemented

#13093967

C₂²

yes

complemented

#13093965

C₂²

yes

complemented

#13093965

C₂²

yes

complemented

#13093964

C₂²

yes

complemented

C₂×C₄₇⋊C₁₆

C₂²

yes

complemented

H(5,𝔽₂)×C₄₇

—

AGL(1,ℤ/8)×C₄₇

—

#135×C₄₇

C₂²

yes

complemented

C₂×D₈×C₄₇

C₂²

yes

complemented

#131×C₄₇

C₂²

yes

characteristic

#41×C₄₇

—

#41×C₄₇

—

#41×C₄₇

D₄

yes

characteristic

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

#10625354

—

D₄×D₄₇

—

#10625350

—

C₂×D₁₈₈

—

Q₁₆×C₄₇

C₂³

yes

characteristic

SD₁₆×C₄₇

—

D₈×C₄₇

—

D₈×C₄₇

C₂³

yes

—

M₁₆×C₄₇

D₄

yes

—

C₂×C₈×C₄₇

C₂³

yes

—

Dic₁₈₈

C₂³

yes

characteristic

D₃₇₆

—

D₃₇₆

C₂³

yes

—

#10625325

—

C₄₇⋊C₁₆

C₂³

yes

characteristic

C₄₇⋊C₁₆

C₂³

yes

—

C₂³×C₄₇

—

C₂²×D₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

D₈

yes

characteristic

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

D₈

yes

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂²⋊D₄₇

—

D₁₈₈

—

D₁₈₈

—

C₄×D₄₇

—

Dic₉₄

—

C₈×C₄₇

C₂×D₄

yes

—

C₈×C₄₇

C₂⁴

yes

characteristic

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×D₈

yes

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×D₈

yes

—

C₄×C₄₇

C₂²×D₄

yes

characteristic

Dic₄₇

—

#3415

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

C₂²×D₈

yes

characteristic

D₄₇

—

#576

—

#576

D₄₇

yes

characteristic, complemented

#509

—

#509

—

#508

—

C₂×D₁₆

—

#504

—

C₄₇

#3415

yes

characteristic, complemented, Sylow

—

—

—

—

—

C₂×D₄₇

yes

complemented

C₂×D₈

—

C₂×D₈

C₂×D₄₇

yes

complemented

#131

C₂×D₄₇

yes

characteristic

Q₃₂

—

SD₃₂

—

SD₃₂

—

D₁₆

—

D₁₆

—

M₃₂

—

C₂×C₁₆

—

100

#41

—

101

#41

—

102

#41

—

103

#41

—

188

—

104

#41

C₂²⋊D₄₇

yes

characteristic

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

188

—

109

Q₁₆

—

110

Q₁₆

C₂²×D₄₇

yes

characteristic

111

SD₁₆

—

112

SD₁₆

—

113

D₈

—

114

D₈

—

115

D₈

—

116

D₈

C₂²×D₄₇

yes

—

117

M₁₆

C₂²⋊D₄₇

yes

—

118

C₂×C₈

C₂²×D₄₇

yes

—

119

C₁₆

—

120

C₁₆

—

121

C₂³

—

122

C₂³

—

188

—

123

Q₈

—

124

Q₈

—

125

Q₈

#10625334

yes

characteristic

126

D₄

—

127

D₄

—

128

D₄

—

129

D₄

—

130

D₄

—

188

—

131

D₄

—

188

—

132

D₄

#10625334

yes

—

133

C₂×C₄

—

134

C₂×C₄

—

188

—

135

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

136

C₈

C₂×C₂²⋊D₄₇

yes

—

137

C₈

C₂³×D₄₇

yes

characteristic

138

C₂²

—

139

C₂²

—

140

C₂²

—

188

—

141

C₂²

—

752

—

142

C₂²

C₂×#10625334

yes

—

143

C₄

—

144

C₄

—

188

—

145

C₄

C₂×#10625334

yes

—

146

C₄

C₂²×C₂²⋊D₄₇

yes

characteristic

147

C₂

—

148

C₂

—

149

C₂

—

376

—

150

C₂

C₂²×#10625334

yes

characteristic

	subgroup	G/H
center	C₂	C₂²×#10625334
commutator	C₈×C₄₇	C₂⁴
Frattini	C₈	C₂³×D₄₇
Fitting	C₄₇×#576	C₂
socle	C₂×C₄₇	C₂²×D₈
hypercenter	#576	D₄₇
hypocenter	C₄₇	#3415
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131 ◁ #576

subgroup

G/H

center

C₂

C₂²×#10625334

commutator

C₈×C₄₇

C₂⁴

Frattini

C₈

C₂³×D₄₇

Fitting

C₄₇×#576

C₂

socle

C₂×C₄₇

C₂²×D₈

D₄₇

C₄₇

G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂ ◁ #41 ◁ #131 ◁ #576

N	H ≅ G/N	type
1. #15689928	C₂	semidirect
2. #15689559	C₂	semidirect
3. #15689558	C₂	semidirect
4. #15689553	C₂	semidirect
5. C₂×#13093965	C₂	semidirect
6. #15689540	C₂	semidirect
7. C₄₇×#576	C₂	semidirect
10. #13094046	C₂²	semidirect
11. C₂×D₃₇₆	C₂²	semidirect
12. #13093970	C₂²	semidirect
13. #13093968	C₂²	semidirect
14. #13093967	C₂²	semidirect
15. #13093965	C₂²	semidirect
16. #13093965	C₂²	semidirect
17. #13093964	C₂²	semidirect
18. C₂×C₄₇⋊C₁₆	C₂²	semidirect
21. #135×C₄₇	C₂²	semidirect
22. C₂×D₈×C₄₇	C₂²	semidirect
77. #576	D₄₇	semidirect
83. C₄₇	#3415	semidirect
89. #135	C₂×D₄₇	semidirect
91. C₂×D₈	C₂×D₄₇	semidirect

H ≅ G/N

type

1. #15689928

C₂

semidirect

2. #15689559

C₂

semidirect

3. #15689558

C₂

semidirect

4. #15689553

C₂

semidirect

5. C₂×#13093965

C₂

semidirect

6. #15689540

C₂

semidirect

7. C₄₇×#576

C₂

semidirect

10. #13094046

C₂²

semidirect

11. C₂×D₃₇₆

C₂²

semidirect

12. #13093970

C₂²

semidirect

13. #13093968

C₂²

semidirect

14. #13093967

C₂²

semidirect

15. #13093965

C₂²

semidirect

16. #13093965

C₂²

semidirect

17. #13093964

C₂²

semidirect

18. C₂×C₄₇⋊C₁₆

C₂²

semidirect

21. #135×C₄₇

C₂²

semidirect

22. C₂×D₈×C₄₇

C₂²

semidirect

77. #576

D₄₇

semidirect

83. C₄₇

#3415

semidirect

89. #135

C₂×D₄₇

semidirect

91. C₂×D₈

C₂×D₄₇

semidirect

Subgroups — #20256527

Conjugacy classes of subgroups

Complements