Group #20276593 — subgroups

Subgroups — #20276593

Order 6784.

146 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 53 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	#16811053	C₂	1	1	yes	maximal, characteristic
2	#16810740	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810739	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810452	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810431	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#359	C₂	1	1	yes	maximal, characteristic, complemented
8	#13133239	C₂²	1	1	yes	characteristic
9	#13133225	—	2	1	no	—
10	#13133224	—	2	1	no	—
11	#13133163	C₄	1	1	yes	characteristic, complemented
12	#13133161	C₄	1	1	yes	characteristic, complemented
13	#13133160	C₄	1	1	yes	characteristic, complemented
14	#13133158	C₄	1	1	yes	characteristic, complemented
15	C₂×#11719908	C₂²	1	1	yes	characteristic
16	#13133153	C₂²	1	1	yes	characteristic
17	#13133144	—	2	1	no	—
18	#13133143	—	2	1	no	—
19	C₈×Dic₅₃	C₂²	1	1	yes	characteristic
20	#135×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
22	#108×C₅₃	C₂²	1	1	yes	characteristic
23	#41×C₅₃	—	2	1	no	—
24	#11719945	—	2	1	no	—
25	Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
26	SD₁₆×C₅₃	—	2	1	no	—
27	D₈×C₅₃	C₂×C₄	1	1	yes	characteristic
28	M₁₆×C₅₃	—	2	1	no	—
29	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
30	C₄×Dic₅₃	D₄	1	1	yes	characteristic
31	#11719917	—	2	1	no	—
32	#11719917	—	4	1	no	—
33	C₂×#88322	—	4	1	no	—
34	C₂×#88322	D₄	1	1	yes	characteristic
35	Dic₂₁₂	C₂×C₄	1	1	yes	characteristic
36	D₄₂₄	C₂×C₄	1	1	yes	characteristic
37	#11719913	—	2	1	no	—
38	#11719912	—	4	1	no	—
39	C₈×D₅₃	—	4	1	no	—
40	C₁₆×C₅₃	—	2	1	no	—
41	#11719908	C₂×C₄	1	1	yes	characteristic
42	#11719908	C₂×C₄	1	1	yes	characteristic
43	Q₈×C₅₃	—	2	1	no	—
44	D₄×C₅₃	—	2	1	no	—
45	D₄×C₅₃	—	4	1	no	—
46	C₂×C₄×C₅₃	—	4	1	no	—
47	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
48	C₂²⋊D₅₃	—	4	1	no	—
49	C₂×Dic₅₃	—	2	1	no	—
50	D₂₁₂	—	2	1	no	—
51	C₄×D₅₃	—	4	1	no	—
52	Dic₁₀₆	—	2	1	no	—
53	C₈×C₅₃	—	4	1	no	—
54	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₈×C₅₃	#41	1	1	yes	characteristic
56	#88322	—	4	1	no	—
57	#88322	D₈	1	1	yes	characteristic
58	#88322	D₈	1	1	yes	characteristic
59	C₂²×C₅₃	—	4	1	no	—
60	C₂²×C₅₃	#135	1	1	yes	characteristic
61	C₂×D₅₃	—	4	1	no	—
62	C₄×C₅₃	—	4	1	no	—
63	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
64	C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
65	Dic₅₃	—	2	2	no	—
66	Dic₅₃	—	4	1	no	—
67	#2178	—	53	1	no	maximal, Sylow
68	C₂×C₅₃	—	2	1	no	—
69	C₂×C₅₃	—	8	1	no	—
70	C₂×C₅₃	C₄×D₈	1	1	yes	characteristic
71	D₅₃	—	8	1	no	—
72	#508	—	53	1	no	—
73	#443	—	53	1	no	—
74	#443	—	53	1	no	—
75	#368	—	53	1	no	—
76	#359	—	53	1	no	—
77	#359	D₅₃	1	1	yes	characteristic, complemented
78	C₄×C₁₆	—	53	1	no	—
79	C₅₃	#2178	1	1	yes	characteristic, complemented, Sylow
80	#135	—	53	1	no	—
81	#135	C₂×D₅₃	1	1	yes	characteristic
82	#131	—	106	1	no	—
83	#131	—	106	1	no	—
84	Q₃₂	—	53	1	no	—
85	SD₃₂	—	53	1	no	—
86	SD₃₂	—	53	1	no	—
87	D₁₆	—	53	1	no	—
88	C₂×C₁₆	—	53	1	no	—
89	C₂×C₁₆	C₂×D₅₃	1	1	yes	characteristic
90	#108	—	53	1	no	—
91	#108	C₂×D₅₃	1	1	yes	characteristic
92	#104	—	106	1	no	—
93	#104	—	106	1	no	—
94	C₄×C₈	—	53	1	no	—
95	#41	—	2	1	no	—
96	#41	—	106	1	no	—
97	Q₁₆	—	53	1	no	—
98	Q₁₆	C₄×D₅₃	1	1	yes	characteristic
99	SD₁₆	—	2	1	no	—
100	SD₁₆	—	106	1	no	—
101	D₈	—	53	1	no	—
102	D₈	C₄×D₅₃	1	1	yes	characteristic
103	M₁₆	—	2	1	no	—
104	M₁₆	—	106	1	no	—
105	M₁₆	—	212	1	no	—
106	M₁₆	—	212	1	no	—
107	C₂×C₈	—	53	1	no	—
108	C₂×C₈	—	212	1	no	—
109	C₂×C₈	—	212	1	no	—
110	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
111	C₄²	—	53	1	no	—
112	C₁₆	—	2	1	no	—
113	C₁₆	—	53	1	no	—
114	C₁₆	—	53	1	no	—
115	Q₈	—	2	1	no	—
116	Q₈	—	106	1	no	—
117	D₄	—	2	1	no	—
118	D₄	—	4	1	no	—
119	D₄	—	106	1	no	—
120	D₄	—	212	1	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	106	1	no	—
123	C₂×C₄	—	212	1	no	—
124	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
125	C₈	—	4	1	no	—
126	C₈	—	53	1	no	—
127	C₈	—	53	1	no	—
128	C₈	—	212	1	no	—
129	C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
130	C₈	#11719947	1	1	yes	characteristic
131	C₂²	—	4	1	no	—
132	C₂²	—	212	1	no	—
133	C₂²	#13133238	1	1	yes	characteristic
134	C₄	—	4	1	no	—
135	C₄	—	106	2	no	—
136	C₄	—	212	1	no	—
137	C₄	#13133177	1	1	yes	characteristic
138	C₄	D₈×D₅₃	1	1	yes	characteristic
139	C₂	—	2	1	no	—
140	C₂	—	8	1	no	—
141	C₂	—	424	1	no	—
142	C₂	#16810703	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811053

C₂

yes

maximal, characteristic

#16810740

C₂

yes

maximal, characteristic, complemented

#16810739

C₂

yes

maximal, characteristic, complemented

#16810452

C₂

yes

maximal, characteristic, complemented

#16810431

C₂

yes

maximal, characteristic, complemented

C₁₆×Dic₅₃

C₂

yes

maximal, characteristic, complemented

C₅₃×#359

C₂

yes

maximal, characteristic, complemented

#13133239

C₂²

yes

—

—

C₄

yes

characteristic, complemented

#13133161

C₄

yes

characteristic, complemented

#13133160

C₄

yes

characteristic, complemented

#13133158

C₄

yes

characteristic, complemented

C₂×#11719908

C₂²

yes

characteristic

#13133153

C₂²

yes

characteristic

#13133144

—

#13133143

—

C₈×Dic₅₃

C₂²

yes

characteristic

#135×C₅₃

C₂²

yes

characteristic

C₂×C₁₆×C₅₃

C₂²

yes

characteristic

#108×C₅₃

C₂²

yes

characteristic

#41×C₅₃

—

#11719945

—

Q₁₆×C₅₃

C₂×C₄

yes

characteristic

SD₁₆×C₅₃

—

D₈×C₅₃

C₂×C₄

yes

characteristic

M₁₆×C₅₃

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄×Dic₅₃

D₄

yes

—

—

—

D₄

yes

characteristic

Dic₂₁₂

C₂×C₄

yes

characteristic

D₄₂₄

C₂×C₄

yes

characteristic

#11719913

—

#11719912

—

C₈×D₅₃

—

C₁₆×C₅₃

—

#11719908

C₂×C₄

yes

characteristic

#11719908

C₂×C₄

yes

characteristic

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

C₄×D₅₃

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

C₂²×C₄

yes

C₈×C₅₃

yes

—

D₈

yes

characteristic

#88322

D₈

yes

characteristic

C₂²×C₅₃

—

C₂²×C₅₃

#135

yes

characteristic

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₄×D₄

yes

characteristic

C₄×C₅₃

C₂×D₈

yes

characteristic

Dic₅₃

—

Dic₅₃

—

#2178

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

C₄×D₈

yes

characteristic

D₅₃

—

—

—

—

—

—

D₅₃

yes

characteristic, complemented

C₄×C₁₆

—

C₅₃

#2178

yes

characteristic, complemented, Sylow

#135

—

#135

C₂×D₅₃

yes

characteristic

#131

—

106

—

#131

—

106

—

Q₃₂

—

SD₃₂

—

SD₃₂

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₅₃

yes

characteristic

#108

—

#108

C₂×D₅₃

yes

characteristic

#104

—

106

—

#104

—

106

—

C₄×C₈

—

#41

—

#41

—

106

—

Q₁₆

—

Q₁₆

C₄×D₅₃

yes

characteristic

SD₁₆

—

100

SD₁₆

—

106

—

101

D₈

—

102

D₈

C₄×D₅₃

yes

characteristic

103

M₁₆

—

104

M₁₆

—

106

—

105

M₁₆

—

212

—

106

M₁₆

—

212

—

107

C₂×C₈

—

108

C₂×C₈

—

212

—

109

C₂×C₈

—

212

—

110

C₂×C₈

C₂²×D₅₃

yes

characteristic

111

C₄²

—

112

C₁₆

—

113

C₁₆

—

114

C₁₆

—

115

Q₈

—

116

Q₈

—

106

—

117

D₄

—

118

D₄

—

119

D₄

—

106

—

120

D₄

—

212

—

121

C₂×C₄

—

122

C₂×C₄

—

106

—

123

C₂×C₄

—

212

—

124

C₂×C₄

D₄×D₅₃

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

212

—

129

C₈

C₂×C₄×D₅₃

yes

characteristic

130

C₈

#11719947

yes

characteristic

131

C₂²

—

132

C₂²

—

212

—

133

C₂²

#13133238

yes

characteristic

134

C₄

—

135

C₄

—

106

—

136

C₄

—

212

—

137

C₄

#13133177

yes

characteristic

138

C₄

D₈×D₅₃

yes

characteristic

139

C₂

—

140

C₂

—

141

C₂

—

424

—

142

C₂

#16810703

yes

characteristic

	subgroup	G/H
center	C₄	D₈×D₅₃
commutator	C₈×C₅₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₅₃×#359	C₂
socle	C₂×C₅₃	C₄×D₈
hypercenter	#359	D₅₃
hypocenter	C₅₃	#2178
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #359

subgroup

G/H

center

C₄

D₈×D₅₃

commutator

C₈×C₅₃

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₅₃×#359

C₂

socle

C₂×C₅₃

C₄×D₈

D₅₃

C₅₃

G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #359

N	H ≅ G/N	type
2. #16810740	C₂	semidirect
3. #16810739	C₂	semidirect
4. #16810452	C₂	semidirect
5. #16810431	C₂	semidirect
6. C₁₆×Dic₅₃	C₂	semidirect
7. C₅₃×#359	C₂	semidirect
11. #13133163	C₄	semidirect
12. #13133161	C₄	semidirect
13. #13133160	C₄	semidirect
14. #13133158	C₄	semidirect
77. #359	D₅₃	semidirect
79. C₅₃	#2178	semidirect

H ≅ G/N

type

2. #16810740

C₂

semidirect

3. #16810739

C₂

semidirect

4. #16810452

C₂

semidirect

5. #16810431

C₂

semidirect

6. C₁₆×Dic₅₃

C₂

semidirect

7. C₅₃×#359

C₂

semidirect

11. #13133163

C₄

semidirect

12. #13133161

C₄

semidirect

13. #13133160

C₄

semidirect

14. #13133158

C₄

semidirect

77. #359

D₅₃

semidirect

79. C₅₃

#2178

semidirect

Subgroups — #20276593

Conjugacy classes of subgroups

Complements