Group #20274383 — subgroups

Subgroups — #20274383

Order 6784.

242 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 242 conjugacy classes of subgroups — too many to draw. Only its 61 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	#16811000	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810999	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810999	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810997	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810971	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810345	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#486	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11719924	—	2	2	no	—
9	C₂×D₄₂₄	—	2	2	no	—
10	C₂×#11719922	—	2	2	no	—
11	C₂×#11719922	—	2	2	no	—
12	#13133214	C₂²	1	1	yes	characteristic
13	#13133214	C₂²	1	1	yes	characteristic
14	#13133137	—	2	2	no	—
15	C₄×#88322	C₂²	1	1	yes	characteristic
16	C₄×#88322	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
18	#127×C₅₃	C₂²	1	1	yes	characteristic
19	#102×C₅₃	—	2	2	no	—
20	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×Q₈×C₅₃	—	2	1	no	—
22	C₂×D₄×C₅₃	—	2	1	no	—
23	C₂×D₄×C₅₃	—	2	1	no	—
24	C₂×D₄×C₅₃	—	2	1	no	—
25	C₂×D₂₁₂	—	2	1	no	—
26	C₂×D₂₁₂	—	2	1	no	—
27	C₂×D₂₁₂	—	2	1	no	—
28	C₂×D₂₁₂	—	2	1	no	—
29	C₂×D₂₁₂	—	2	1	no	—
30	C₂×D₂₁₂	—	2	1	no	—
31	C₂×C₈×C₅₃	D₄	1	2	yes	—
32	C₄⋊C₄×C₅₃	—	2	1	no	—
33	C₄⋊C₄×C₅₃	—	2	1	no	—
34	C₄²×C₅₃	C₂³	1	1	yes	characteristic
35	#11719924	—	4	4	no	—
36	#11719922	—	4	4	no	—
37	#11719922	—	4	4	no	—
38	C₂×#88322	D₄	1	2	yes	—
39	C₂×#88322	D₄	1	2	yes	—
40	D₄₂₄	—	4	4	no	—
41	C₂³×C₅₃	—	4	1	no	—
42	C₂²×D₅₃	—	4	1	no	—
43	C₂²×D₅₃	—	4	1	no	—
44	Q₈×C₅₃	—	4	2	no	—
45	D₄×C₅₃	—	4	2	no	—
46	D₄×C₅₃	—	4	2	no	—
47	D₄×C₅₃	—	4	2	no	—
48	C₂×C₄×C₅₃	—	4	1	no	—
49	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
52	D₂₁₂	—	4	2	no	—
53	D₂₁₂	—	4	2	no	—
54	D₂₁₂	—	4	2	no	—
55	D₂₁₂	—	4	2	no	—
56	D₂₁₂	—	4	2	no	—
57	D₂₁₂	—	4	2	no	—
58	C₈×C₅₃	—	2	2	no	—
59	#88322	—	2	2	no	—
60	#88322	D₈	1	4	yes	complemented
61	C₂²×C₅₃	—	8	1	no	—
62	C₂²×C₅₃	—	8	1	no	—
63	C₂²×C₅₃	—	8	1	no	—
64	C₂²×C₅₃	#127	1	1	yes	characteristic
65	C₂×D₅₃	—	8	1	no	—
66	C₂×D₅₃	—	8	1	no	—
67	C₂×D₅₃	—	8	1	no	—
68	C₂×D₅₃	—	8	1	no	—
69	C₂×D₅₃	—	8	1	no	—
70	C₂×D₅₃	—	8	1	no	—
71	C₄×C₅₃	—	8	1	no	—
72	C₄×C₅₃	C₂×D₈	1	2	yes	—
73	C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
74	C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
75	#1721	—	53	1	no	maximal, Sylow
76	C₂×C₅₃	—	16	1	no	—
77	C₂×C₅₃	#493	1	1	yes	characteristic
78	C₂×C₅₃	#496	1	1	yes	characteristic
79	C₂×C₅₃	#496	1	1	yes	characteristic
80	D₅₃	—	16	1	no	—
81	D₅₃	—	16	1	no	—
82	#493	—	53	1	no	—
83	#493	—	53	1	no	—
84	#493	—	53	1	no	—
85	#492	—	53	1	no	—
86	#486	—	53	1	no	—
87	#486	D₅₃	1	1	yes	characteristic, complemented
88	#322	—	53	1	no	—
89	C₅₃	#1721	1	1	yes	characteristic, complemented, Sylow
90	C₂×SD₁₆	—	106	2	no	—
91	C₂×D₈	—	106	2	no	—
92	C₂×D₈	—	106	2	no	—
93	C₂×D₈	—	106	2	no	—
94	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
95	#127	—	53	1	no	—
96	#127	—	53	1	no	—
97	#127	C₂×D₅₃	1	1	yes	characteristic
98	#102	—	2	2	no	—
99	#102	—	106	2	no	—
100	C₄×C₈	—	53	1	no	—
101	C₄×C₈	—	53	1	no	—
102	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
103	C₂×Q₈	—	2	1	no	—
104	C₂×D₄	—	2	1	no	—
105	C₂×D₄	—	2	1	no	—
106	C₂×D₄	—	2	1	no	—
107	C₂×D₄	—	106	1	no	—
108	C₂×D₄	—	106	1	no	—
109	C₂×D₄	—	106	1	no	—
110	C₂×D₄	—	106	1	no	—
111	C₂×D₄	—	106	1	no	—
112	C₂×D₄	—	106	1	no	—
113	SD₁₆	—	212	4	no	—
114	D₈	—	212	4	no	—
115	D₈	—	212	4	no	—
116	D₈	—	212	4	no	—
117	C₂×C₈	—	53	2	no	—
118	C₂×C₈	—	53	2	no	—
119	C₂×C₈	C₂²⋊D₅₃	1	2	yes	—
120	C₄⋊C₄	—	2	1	no	—
121	C₄⋊C₄	—	2	1	no	—
122	C₄²	C₂²×D₅₃	1	1	yes	characteristic
123	C₂³	—	4	1	no	—
124	C₂³	—	212	1	no	—
125	C₂³	—	212	1	no	—
126	Q₈	—	4	2	no	—
127	D₄	—	4	2	no	—
128	D₄	—	4	2	no	—
129	D₄	—	4	2	no	—
130	D₄	—	212	2	no	—
131	D₄	—	212	2	no	—
132	D₄	—	212	2	no	—
133	D₄	—	212	2	no	—
134	D₄	—	212	2	no	—
135	D₄	—	212	2	no	—
136	C₂×C₄	—	4	1	no	—
137	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
138	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
139	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
140	C₈	—	2	2	no	—
141	C₈	—	53	4	no	—
142	C₈	—	106	2	no	—
143	C₂²	—	8	1	no	—
144	C₂²	—	8	1	no	—
145	C₂²	—	8	1	no	—
146	C₂²	—	424	1	no	—
147	C₂²	—	424	1	no	—
148	C₂²	—	424	1	no	—
149	C₂²	—	424	1	no	—
150	C₂²	—	424	1	no	—
151	C₂²	—	424	1	no	—
152	C₂²	#13133213	1	1	yes	characteristic
153	C₄	—	8	1	no	—
154	C₄	D₈×D₅₃	1	2	yes	—
155	C₄	#13133242	1	2	yes	—
156	C₄	#13133244	1	2	yes	—
157	C₂	—	16	1	no	—
158	C₂	—	848	1	no	—
159	C₂	—	848	1	no	—
160	C₂	#16810998	1	1	yes	characteristic
161	C₂	#16811009	1	1	yes	characteristic
162	C₂	#16811013	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811000

C₂

yes

maximal, characteristic, complemented

#16810999

C₂

yes

maximal, characteristic, complemented

#16810999

C₂

yes

maximal, characteristic, complemented

#16810997

C₂

yes

maximal, characteristic, complemented

#16810971

C₂

yes

maximal, characteristic, complemented

#16810345

C₂

yes

maximal, characteristic, complemented

C₅₃×#486

C₂

yes

maximal, characteristic, complemented

C₂×#11719924

—

C₂×D₄₂₄

—

C₂×#11719922

—

C₂×#11719922

—

#13133214

C₂²

yes

characteristic

#13133214

C₂²

yes

characteristic

#13133137

—

C₄×#88322

C₂²

yes

characteristic

C₄×#88322

C₂²

yes

characteristic

C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

#127×C₅₃

C₂²

yes

characteristic

#102×C₅₃

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×C₈×C₅₃

D₄

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂³

yes

—

—

—

D₄

yes

—

C₂×#88322

D₄

yes

—

D₄₂₄

—

C₂³×C₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

C₈×C₅₃

—

#88322

—

#88322

D₈

yes

complemented

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#127

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×D₈

yes

—

C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₄×C₅₃

AGL(1,ℤ/8)

yes

—

#1721

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

D₅₃

—

—

—

—

—

—

D₅₃

yes

characteristic, complemented

#322

—

C₅₃

#1721

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

106

—

C₂×D₈

—

106

—

C₂×D₈

—

106

—

C₂×D₈

—

106

—

C₄⋊Q₈

C₂×D₅₃

yes

—

—

C₂×D₅₃

yes

characteristic

#102

—

#102

—

106

—

100

C₄×C₈

—

101

C₄×C₈

—

102

C₄×C₈

C₂×D₅₃

yes

characteristic

103

C₂×Q₈

—

104

C₂×D₄

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

106

—

108

C₂×D₄

—

106

—

109

C₂×D₄

—

106

—

110

C₂×D₄

—

106

—

111

C₂×D₄

—

106

—

112

C₂×D₄

—

106

—

113

SD₁₆

—

212

—

114

D₈

—

212

—

115

D₈

—

212

—

116

D₈

—

212

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

C₂²⋊D₅₃

yes

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄²

C₂²×D₅₃

yes

characteristic

123

C₂³

—

124

C₂³

—

212

—

125

C₂³

—

212

—

126

Q₈

—

127

D₄

—

128

D₄

—

129

D₄

—

130

D₄

—

212

—

131

D₄

—

212

—

132

D₄

—

212

—

133

D₄

—

212

—

134

D₄

—

212

—

135

D₄

—

212

—

136

C₂×C₄

—

137

C₂×C₄

D₄×D₅₃

yes

characteristic

138

C₂×C₄

D₄×D₅₃

yes

characteristic

139

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

140

C₈

—

141

C₈

—

142

C₈

—

106

—

143

C₂²

—

144

C₂²

—

145

C₂²

—

146

C₂²

—

424

—

147

C₂²

—

424

—

148

C₂²

—

424

—

149

C₂²

—

424

—

150

C₂²

—

424

—

151

C₂²

—

424

—

152

C₂²

#13133213

yes

characteristic

153

C₄

—

154

C₄

D₈×D₅₃

yes

—

155

C₄

#13133242

yes

—

156

C₄

#13133244

yes

—

157

C₂

—

158

C₂

—

848

—

159

C₂

—

848

—

160

C₂

#16810998

yes

characteristic

161

C₂

#16811009

yes

characteristic

162

C₂

#16811013

yes

characteristic

	subgroup	G/H
center	C₂²	#13133213
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#486	C₂
socle	C₂²×C₅₃	#127
hypercenter	#486	D₅₃
hypocenter	C₅₃	#1721
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #486

subgroup

G/H

center

C₂²

#13133213

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#486

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #486

N	H ≅ G/N	type
1. #16811000	C₂	semidirect
2. #16810999	C₂	semidirect
3. #16810999	C₂	semidirect
4. #16810997	C₂	semidirect
5. #16810971	C₂	semidirect
6. #16810345	C₂	semidirect
7. C₅₃×#486	C₂	semidirect
60. #88322	D₈	semidirect
87. #486	D₅₃	semidirect
89. C₅₃	#1721	semidirect

H ≅ G/N

type

1. #16811000

C₂

semidirect

2. #16810999

C₂

semidirect

3. #16810999

C₂

semidirect

4. #16810997

C₂

semidirect

5. #16810971

C₂

semidirect

6. #16810345

C₂

semidirect

7. C₅₃×#486

C₂

semidirect

60. #88322

D₈

semidirect

87. #486

D₅₃

semidirect

89. C₅₃

#1721

semidirect

Subgroups — #20274383

Conjugacy classes of subgroups

Complements