Group #20274357 — subgroups

Subgroups — #20274357

Order 6784.

208 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 208 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	#16811000	C₂	1	1	yes	maximal, characteristic
2	#16810997	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810971	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810971	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×#487	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11719924	—	2	2	no	—
9	C₂×D₄₂₄	—	2	2	no	—
10	#13133214	C₂²	1	1	yes	characteristic
11	#13133214	C₂²	1	1	yes	characteristic
12	#13133137	—	2	2	no	—
13	#13133137	—	2	2	no	—
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	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
19	#103×C₅₃	—	2	2	no	—
20	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×Q₈×C₅₃	—	2	1	no	—
22	C₂×Q₈×C₅₃	—	2	1	no	—
23	C₂×D₂₁₂	—	2	1	no	—
24	C₂×D₂₁₂	—	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×C₈×C₅₃	D₄	1	2	yes	—
30	C₄⋊C₄×C₅₃	—	2	1	no	—
31	C₄⋊C₄×C₅₃	—	2	1	no	—
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	C₂×#88322	D₄	1	2	yes	—
37	C₂×#88322	D₄	1	2	yes	—
38	D₄₂₄	—	4	4	no	—
39	C₂²×D₅₃	—	4	1	no	—
40	C₂²×D₅₃	—	4	1	no	—
41	Q₈×C₅₃	—	4	2	no	—
42	Q₈×C₅₃	—	4	2	no	—
43	C₂×C₄×C₅₃	—	4	1	no	—
44	C₂×C₄×C₅₃	—	4	1	no	—
45	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
48	D₂₁₂	—	4	2	no	—
49	D₂₁₂	—	4	2	no	—
50	D₂₁₂	—	4	2	no	—
51	D₂₁₂	—	4	2	no	—
52	D₂₁₂	—	4	2	no	—
53	D₂₁₂	—	4	2	no	—
54	C₈×C₅₃	—	2	2	no	—
55	#88322	—	2	2	no	—
56	#88322	—	2	2	no	—
57	C₂²×C₅₃	#127	1	1	yes	characteristic
58	C₂×D₅₃	—	8	1	no	—
59	C₂×D₅₃	—	8	1	no	—
60	C₂×D₅₃	—	8	1	no	—
61	C₂×D₅₃	—	8	1	no	—
62	C₂×D₅₃	—	8	1	no	—
63	C₂×D₅₃	—	8	1	no	—
64	C₄×C₅₃	—	8	1	no	—
65	C₄×C₅₃	—	8	1	no	—
66	C₄×C₅₃	#135	1	2	yes	—
67	C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
68	C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
69	#1717	—	53	1	no	maximal, Sylow
70	C₂×C₅₃	#495	1	1	yes	characteristic
71	C₂×C₅₃	#496	1	1	yes	characteristic
72	C₂×C₅₃	#496	1	1	yes	characteristic
73	D₅₃	—	16	1	no	—
74	D₅₃	—	16	1	no	—
75	#493	—	53	1	no	—
76	#492	—	53	1	no	—
77	#487	D₅₃	1	1	yes	characteristic, complemented
78	#486	—	53	1	no	—
79	#486	—	53	1	no	—
80	#486	—	53	1	no	—
81	#322	—	53	1	no	—
82	C₅₃	#1717	1	1	yes	characteristic, complemented, Sylow
83	C₂×SD₁₆	—	106	2	no	—
84	C₂×D₈	—	106	2	no	—
85	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
86	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
87	#127	—	53	1	no	—
88	#127	—	53	1	no	—
89	#103	—	2	2	no	—
90	#102	—	106	2	no	—
91	#102	—	106	2	no	—
92	#102	—	106	2	no	—
93	C₄×C₈	—	53	1	no	—
94	C₄×C₈	—	53	1	no	—
95	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
96	C₂×Q₈	—	2	1	no	—
97	C₂×Q₈	—	2	1	no	—
98	C₂×D₄	—	106	1	no	—
99	C₂×D₄	—	106	1	no	—
100	C₂×D₄	—	106	1	no	—
101	C₂×D₄	—	106	1	no	—
102	C₂×D₄	—	106	1	no	—
103	C₂×D₄	—	106	1	no	—
104	SD₁₆	—	212	4	no	—
105	D₈	—	212	4	no	—
106	C₂×C₈	—	53	2	no	—
107	C₂×C₈	—	53	2	no	—
108	C₂×C₈	C₂²⋊D₅₃	1	2	yes	—
109	C₄⋊C₄	—	2	1	no	—
110	C₄⋊C₄	—	2	1	no	—
111	C₄⋊C₄	—	2	1	no	—
112	C₄⋊C₄	—	2	1	no	—
113	C₄²	C₂²×D₅₃	1	1	yes	characteristic
114	C₂³	—	212	1	no	—
115	C₂³	—	212	1	no	—
116	Q₈	—	4	2	no	—
117	Q₈	—	4	2	no	—
118	D₄	—	212	2	no	—
119	D₄	—	212	2	no	—
120	D₄	—	212	2	no	—
121	D₄	—	212	2	no	—
122	D₄	—	212	2	no	—
123	D₄	—	212	2	no	—
124	C₂×C₄	—	4	1	no	—
125	C₂×C₄	—	4	1	no	—
126	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
127	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
128	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
129	C₈	—	2	2	no	—
130	C₈	—	106	2	no	—
131	C₈	—	106	2	no	—
132	C₂²	—	424	1	no	—
133	C₂²	—	424	1	no	—
134	C₂²	—	424	1	no	—
135	C₂²	—	424	1	no	—
136	C₂²	—	424	1	no	—
137	C₂²	—	424	1	no	—
138	C₂²	#13133213	1	1	yes	characteristic
139	C₄	—	8	1	no	—
140	C₄	—	8	1	no	—
141	C₄	#13133240	1	2	yes	—
142	C₄	#13133242	1	2	yes	—
143	C₄	#13133244	1	2	yes	—
144	C₂	—	848	1	no	—
145	C₂	—	848	1	no	—
146	C₂	#16811008	1	1	yes	characteristic
147	C₂	#16811009	1	1	yes	characteristic
148	C₂	#16811013	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811000

C₂

yes

maximal, characteristic

#16810997

C₂

yes

maximal, characteristic, complemented

#16810971

C₂

yes

maximal, characteristic, complemented

#16810971

C₂

yes

maximal, characteristic, complemented

#16810971

C₂

yes

maximal, characteristic, complemented

#16810345

C₂

yes

maximal, characteristic, complemented

C₅₃×#487

C₂

yes

maximal, characteristic, complemented

C₂×#11719924

—

C₂×D₄₂₄

—

#13133214

C₂²

yes

characteristic

#13133214

C₂²

yes

—

—

—

C₂²

yes

characteristic

C₄×#88322

C₂²

yes

characteristic

C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

#103×C₅₃

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×C₈×C₅₃

D₄

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂³

yes

characteristic

#11719924

—

C₂×#88322

D₄

yes

—

C₂×#88322

D₄

yes

—

D₄₂₄

—

C₂²×D₅₃

—

C₂²×D₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×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

—

C₂²×C₅₃

#127

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#135

yes

—

C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₄×C₅₃

AGL(1,ℤ/8)

yes

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

D₅₃

—

#493

—

#492

—

#487

D₅₃

yes

characteristic, complemented

—

—

—

—

C₅₃

#1717

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

106

—

C₂×D₈

—

106

—

C₄⋊Q₈

C₂×D₅₃

yes

characteristic

C₄⋊Q₈

C₂×D₅₃

yes

—

—

—

—

106

—

#102

—

106

—

#102

—

106

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

106

—

C₂×D₄

—

106

—

100

C₂×D₄

—

106

—

101

C₂×D₄

—

106

—

102

C₂×D₄

—

106

—

103

C₂×D₄

—

106

—

104

SD₁₆

—

212

—

105

D₈

—

212

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

C₂²⋊D₅₃

yes

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄²

C₂²×D₅₃

yes

characteristic

114

C₂³

—

212

—

115

C₂³

—

212

—

116

Q₈

—

117

Q₈

—

118

D₄

—

212

—

119

D₄

—

212

—

120

D₄

—

212

—

121

D₄

—

212

—

122

D₄

—

212

—

123

D₄

—

212

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

D₄×D₅₃

yes

characteristic

127

C₂×C₄

D₄×D₅₃

yes

characteristic

128

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

129

C₈

—

130

C₈

—

106

—

131

C₈

—

106

—

132

C₂²

—

424

—

133

C₂²

—

424

—

134

C₂²

—

424

—

135

C₂²

—

424

—

136

C₂²

—

424

—

137

C₂²

—

424

—

138

C₂²

#13133213

yes

characteristic

139

C₄

—

140

C₄

—

141

C₄

#13133240

yes

—

142

C₄

#13133242

yes

—

143

C₄

#13133244

yes

—

144

C₂

—

848

—

145

C₂

—

848

—

146

C₂

#16811008

yes

characteristic

147

C₂

#16811009

yes

characteristic

148

C₂

#16811013

yes

characteristic

	subgroup	G/H
center	C₂²	#13133213
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#487	C₂
socle	C₂²×C₅₃	#127
hypercenter	#487	D₅₃
hypocenter	C₅₃	#1717
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #487

subgroup

G/H

center

C₂²

#13133213

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#487

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #487

N	H ≅ G/N	type
2. #16810997	C₂	semidirect
3. #16810971	C₂	semidirect
4. #16810971	C₂	semidirect
5. #16810971	C₂	semidirect
6. #16810345	C₂	semidirect
7. C₅₃×#487	C₂	semidirect
77. #487	D₅₃	semidirect
82. C₅₃	#1717	semidirect

H ≅ G/N

type

2. #16810997

C₂

semidirect

3. #16810971

C₂

semidirect

4. #16810971

C₂

semidirect

5. #16810971

C₂

semidirect

6. #16810345

C₂

semidirect

7. C₅₃×#487

C₂

semidirect

77. #487

D₅₃

semidirect

82. C₅₃

#1717

semidirect

Subgroups — #20274357

Conjugacy classes of subgroups

Complements