Group #20291122 — subgroups

Subgroups — #20291122

Order 6784.

142 conjugacy classes of subgroups; only the 44 normal subgroups are shown.

normal subgroups only

This group has 142 conjugacy classes of subgroups — too many to draw. Only its 44 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	#16811669	C₂	1	1	yes	maximal, characteristic
2	#16811663	C₂	1	1	yes	maximal, characteristic, complemented
3	#16811547	C₂	1	1	yes	maximal, characteristic, complemented
4	#16811542	C₂	1	1	yes	maximal, characteristic, complemented
5	#16811540	C₂	1	1	yes	maximal, characteristic
6	#16811513	C₂	1	1	yes	maximal, characteristic, complemented
7	#16811065	C₂	1	1	yes	maximal, characteristic
8	#13133305	—	2	1	no	—
9	Q₈×#6157	—	2	1	no	—
10	#13133292	C₂²	1	1	yes	characteristic
11	#13133287	C₂²	1	1	yes	characteristic
12	#13133285	C₂²	1	1	yes	characteristic
13	#13133282	—	2	1	no	—
14	#13133280	—	2	1	no	—
15	#13133271	C₂²	1	1	yes	characteristic
16	#13133240	C₂²	1	1	yes	characteristic
17	Q₁₆×D₅₃	C₂²	1	1	yes	characteristic
18	#13133163	C₄	1	1	yes	characteristic, complemented
19	#13133162	C₄	1	1	yes	characteristic, complemented
20	#13133160	C₄	1	1	yes	characteristic, complemented
21	#13133156	C₂²	1	1	yes	characteristic
22	Q₃₂×C₅₃	C₄	1	1	yes	characteristic, complemented
23	#11719949	—	2	1	no	—
24	Q₈×D₅₃	—	2	1	no	—
25	C₄⋊#6157	—	2	1	no	—
26	C₄⋊#6157	—	4	1	no	—
27	C₄×#6157	—	4	1	no	—
28	C₄×#6157	D₄	1	1	yes	characteristic
29	#11719936	—	2	1	no	—
30	#11719936	—	4	1	no	—
31	D₅₃⋊C₈	—	4	1	no	—
32	D₅₃⋊C₈	D₄	1	1	yes	characteristic
33	Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
34	Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
35	#11719925	—	2	1	no	—
36	#11719924	—	2	1	no	—
37	Dic₂₁₂	C₂×C₄	1	1	yes	characteristic
38	D₄₂₄	C₂×C₄	1	1	yes	characteristic
39	C₈×D₅₃	C₂³	1	1	yes	characteristic
40	#11719910	—	2	1	no	—
41	C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
42	#11719908	C₂×C₄	1	1	yes	characteristic
43	C₂×#6157	—	2	1	no	—
44	C₂×#6157	—	4	1	no	—
45	Q₈×C₅₃	—	2	1	no	—
46	Q₈×C₅₃	—	2	1	no	—
47	D₂₁₂	—	2	1	no	—
48	D₂₁₂	—	4	1	no	—
49	C₄×D₅₃	—	4	1	no	—
50	C₄×D₅₃	—	4	1	no	—
51	C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
52	Dic₁₀₆	—	2	1	no	—
53	Dic₁₀₆	—	4	1	no	—
54	#88324	—	2	1	no	—
55	#88324	—	4	1	no	—
56	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
57	#88322	#41	1	1	yes	characteristic
58	C₂×D₅₃	—	4	1	no	—
59	C₂×D₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
60	#6157	—	4	1	no	—
61	#6157	—	8	1	no	—
62	C₄×C₅₃	—	4	1	no	—
63	C₄×C₅₃	—	4	1	no	—
64	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
65	Dic₅₃	—	4	1	no	—
66	Dic₅₃	AGL(1,ℤ/8)	1	1	yes	characteristic
67	#2180	—	53	1	no	maximal, Sylow
68	C₂×C₅₃	#442	1	1	yes	characteristic
69	D₅₃	—	2	1	no	—
70	D₅₃	—	8	1	no	—
71	#510	—	53	1	no	—
72	#444	—	53	1	no	—
73	#441	—	53	1	no	—
74	#365	—	53	1	no	—
75	#362	—	53	1	no	—
76	#360	—	53	1	no	—
77	#347	—	53	1	no	—
78	C₅₃	#2180	1	1	yes	characteristic, complemented, Sylow
79	#135	—	53	1	no	—
80	C₂×Q₁₆	—	53	1	no	—
81	#131	—	106	1	no	—
82	C₄×Q₈	—	106	1	no	—
83	Q₃₂	—	53	1	no	—
84	Q₃₂	#6157	1	1	yes	characteristic, complemented
85	SD₃₂	—	53	1	no	—
86	SD₃₂	—	53	1	no	—
87	M₃₂	—	53	1	no	—
88	M₃₂	—	53	1	no	—
89	#108	—	53	1	no	—
90	#106	—	53	1	no	—
91	#104	—	106	1	no	—
92	#103	—	106	1	no	—
93	#97	—	53	1	no	—
94	#41	—	106	1	no	—
95	C₂×Q₈	—	106	1	no	—
96	Q₁₆	—	53	1	no	—
97	Q₁₆	—	106	1	no	—
98	Q₁₆	C₂×#6157	1	1	yes	characteristic
99	Q₁₆	C₂×#6157	1	1	yes	characteristic
100	SD₁₆	—	106	1	no	—
101	D₈	—	53	1	no	—
102	M₁₆	—	106	1	no	—
103	M₁₆	—	212	1	no	—
104	C₂×C₈	—	53	1	no	—
105	C₂×C₈	—	53	1	no	—
106	C₂×C₈	—	212	1	no	—
107	C₄⋊C₄	—	106	1	no	—
108	C₄⋊C₄	—	212	1	no	—
109	C₄²	—	53	1	no	—
110	C₄²	—	212	1	no	—
111	C₁₆	—	53	1	no	—
112	C₁₆	—	106	1	no	—
113	C₁₆	C₂×#6157	1	1	yes	characteristic
114	Q₈	—	2	1	no	—
115	Q₈	—	2	1	no	—
116	Q₈	—	106	1	no	—
117	Q₈	—	212	1	no	—
118	D₄	—	106	1	no	—
119	D₄	—	212	1	no	—
120	C₂×C₄	—	53	1	no	—
121	C₂×C₄	—	106	1	no	—
122	C₂×C₄	—	212	1	no	—
123	C₂×C₄	—	212	1	no	—
124	C₂×C₄	—	212	1	no	—
125	C₈	—	53	1	no	—
126	C₈	—	106	1	no	—
127	C₈	—	212	1	no	—
128	C₈	C₂²×#6157	1	1	yes	characteristic
129	C₂²	—	53	1	no	—
130	C₂²	—	212	1	no	—
131	C₄	—	4	1	no	—
132	C₄	—	4	1	no	—
133	C₄	—	53	1	no	—
134	C₄	—	424	1	no	—
135	C₄	—	212	1	no	—
136	C₄	—	212	1	no	—
137	C₄	D₄×#6157	1	1	yes	characteristic
138	C₂	—	106	1	no	—
139	C₂	—	424	1	no	—
140	C₂	#16811664	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811669

C₂

yes

maximal, characteristic

#16811663

C₂

yes

maximal, characteristic, complemented

#16811547

C₂

yes

maximal, characteristic, complemented

#16811542

C₂

yes

maximal, characteristic, complemented

#16811540

C₂

yes

maximal, characteristic

#16811513

C₂

yes

maximal, characteristic, complemented

#16811065

C₂

yes

maximal, characteristic

#13133305

—

Q₈×#6157

—

#13133292

C₂²

yes

characteristic

#13133287

C₂²

yes

characteristic

#13133285

C₂²

yes

—

—

C₂²

yes

characteristic

#13133240

C₂²

yes

characteristic

Q₁₆×D₅₃

C₂²

yes

characteristic

#13133163

C₄

yes

characteristic, complemented

#13133162

C₄

yes

characteristic, complemented

#13133160

C₄

yes

characteristic, complemented

#13133156

C₂²

yes

characteristic

Q₃₂×C₅₃

C₄

yes

characteristic, complemented

#11719949

—

Q₈×D₅₃

—

—

—

—

D₄

yes

characteristic

#11719936

—

#11719936

—

D₅₃⋊C₈

—

D₅₃⋊C₈

D₄

yes

characteristic

Q₁₆×C₅₃

C₂×C₄

yes

characteristic

Q₁₆×C₅₃

C₂×C₄

yes

characteristic

#11719925

—

#11719924

—

Dic₂₁₂

C₂×C₄

yes

characteristic

D₄₂₄

C₂×C₄

yes

characteristic

C₈×D₅₃

C₂³

yes

characteristic

#11719910

—

C₁₆×C₅₃

C₂×C₄

yes

characteristic

#11719908

C₂×C₄

yes

characteristic

C₂×#6157

—

C₂×#6157

—

Q₈×C₅₃

—

Q₈×C₅₃

—

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

C₂×D₄

yes

characteristic

Dic₁₀₆

—

Dic₁₀₆

—

#88324

—

#88324

—

C₈×C₅₃

C₂²×C₄

yes

yes

C₂×D₅₃

—

C₂×D₅₃

yes

—

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₄×D₄

yes

characteristic

Dic₅₃

—

Dic₅₃

yes

—

C₂×C₅₃

yes

D₅₃

—

D₅₃

—

—

—

—

—

—

—

—

C₅₃

#2180

yes

characteristic, complemented, Sylow

#135

—

C₂×Q₁₆

—

#131

—

106

—

C₄×Q₈

—

106

—

Q₃₂

—

Q₃₂

#6157

yes

characteristic, complemented

SD₃₂

—

SD₃₂

—

M₃₂

—

M₃₂

—

#108

—

#106

—

#104

—

106

—

#103

—

106

—

#97

—

#41

—

106

—

C₂×Q₈

—

106

—

Q₁₆

—

Q₁₆

—

106

—

Q₁₆

yes

Q₁₆

yes

100

SD₁₆

—

106

—

101

D₈

—

102

M₁₆

—

106

—

103

M₁₆

—

212

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

212

—

107

C₄⋊C₄

—

106

—

108

C₄⋊C₄

—

212

—

109

C₄²

—

110

C₄²

—

212

—

111

C₁₆

—

112

C₁₆

—

106

—

113

C₁₆

C₂×#6157

yes

characteristic

114

Q₈

—

115

Q₈

—

116

Q₈

—

106

—

117

Q₈

—

212

—

118

D₄

—

106

—

119

D₄

—

212

—

120

C₂×C₄

—

121

C₂×C₄

—

106

—

122

C₂×C₄

—

212

—

123

C₂×C₄

—

212

—

124

C₂×C₄

—

212

—

125

C₈

—

126

C₈

—

106

—

127

C₈

—

212

—

128

C₈

C₂²×#6157

yes

characteristic

129

C₂²

—

130

C₂²

—

212

—

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

424

—

135

C₄

—

212

—

136

C₄

—

212

—

137

C₄

D₄×#6157

yes

characteristic

138

C₂

—

106

—

139

C₂

—

424

—

140

C₂

#16811664

yes

characteristic

	subgroup	G/H
center	C₂	#16811664
commutator	C₈×C₅₃	C₂²×C₄
Frattini	C₈	C₂²×#6157
Fitting	Q₃₂×C₅₃	C₄
socle	C₂×C₅₃	#442
hypercenter	Q₃₂	#6157
hypocenter	C₅₃	#2180
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

subgroup

G/H

center

C₂

#16811664

commutator

C₈×C₅₃

C₂²×C₄

Frattini

C₈

C₂²×#6157

Fitting

Q₃₂×C₅₃

C₄

C₂×C₅₃

Q₃₂

C₅₃

G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ Q₃₂

N	H ≅ G/N	type
2. #16811663	C₂	semidirect
3. #16811547	C₂	semidirect
4. #16811542	C₂	semidirect
6. #16811513	C₂	semidirect
18. #13133163	C₄	semidirect
19. #13133162	C₄	semidirect
20. #13133160	C₄	semidirect
22. Q₃₂×C₅₃	C₄	semidirect
78. C₅₃	#2180	semidirect
84. Q₃₂	#6157	semidirect

H ≅ G/N

type

2. #16811663

C₂

semidirect

3. #16811547

C₂

semidirect

4. #16811542

C₂

semidirect

6. #16811513

C₂

semidirect

18. #13133163

C₄

semidirect

19. #13133162

C₄

semidirect

20. #13133160

C₄

semidirect

22. Q₃₂×C₅₃

C₄

semidirect

78. C₅₃

#2180

semidirect

84. Q₃₂

#6157

semidirect

Subgroups — #20291122

Conjugacy classes of subgroups

Complements