Group #27283597 — subgroups

Subgroups — #27283597

Order 13568.

558 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 127 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	#20274853	C₂	1	1	yes	maximal, characteristic, complemented
2	#20273383	C₂	1	4	yes	maximal, complemented
3	#20273007	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₅₃×#493	C₂	1	1	yes	maximal, characteristic
5	#16810676	C₂²	1	2	yes	—
6	#16810647	C₂²	1	2	yes	—
7	#16810388	—	2	2	no	—
8	#16810367	—	2	8	no	—
9	#16810345	—	2	2	no	—
10	C₂²×D₈×C₅₃	—	2	2	no	—
11	C₂×#127×C₅₃	C₂²	1	2	yes	—
12	C₅₃×#493	—	2	2	no	—
13	C₅₃×#493	C₄	1	4	yes	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×#11719917	—	2	2	no	—
16	C₄⋊#88322	—	2	4	no	—
17	C₄⋊#88322	D₄	1	4	yes	—
18	C₂²⋊#88322	—	2	2	no	—
19	C₄×#88322	—	2	2	no	—
20	C₂²×D₄×C₅₃	—	2	2	no	—
21	C₂²×D₄×C₅₃	—	2	4	no	—
22	C₂×D₈×C₅₃	—	2	8	no	—
23	C₂×D₈×C₅₃	—	4	8	no	—
24	C₂²×C₈×C₅₃	D₄	1	2	yes	—
25	#127×C₅₃	—	2	2	no	—
26	#127×C₅₃	—	2	4	no	—
27	#127×C₅₃	C₂×C₄	1	4	yes	—
28	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
29	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₅₃	D₄	1	2	yes	—
31	C₂⁴×C₅₃	—	4	2	no	—
32	C₂×D₄×C₅₃	—	2	4	no	—
33	C₂×D₄×C₅₃	—	2	8	no	—
34	C₂×D₄×C₅₃	—	4	4	no	—
35	C₂×D₄×C₅₃	—	4	2	no	—
36	C₂×D₄×C₅₃	—	4	8	no	—
37	C₂×D₄×C₅₃	—	8	8	no	—
38	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₅₃	#41	1	2	yes	—
40	D₈×C₅₃	—	4	16	no	—
41	C₂×C₈×C₅₃	—	2	4	no	—
42	C₂×C₈×C₅₃	C₂²⋊C₄	1	4	yes	—
43	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
44	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₄²×C₅₃	C₂×D₄	1	2	yes	—
46	#11719917	—	4	4	no	—
47	C₂×#88322	—	2	4	no	—
48	C₂×#88322	—	4	2	no	—
49	C₂³×C₅₃	—	4	4	no	—
50	C₂³×C₅₃	—	8	8	no	—
51	C₂³×C₅₃	—	8	4	no	—
52	C₂³×C₅₃	#124	1	1	yes	characteristic
53	D₄×C₅₃	—	4	8	no	—
54	D₄×C₅₃	—	8	8	no	—
55	D₄×C₅₃	—	8	8	no	—
56	D₄×C₅₃	—	8	4	no	—
57	C₂×C₄×C₅₃	—	2	4	no	—
58	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
60	C₂×C₄×C₅₃	#121	1	2	yes	—
61	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
62	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	4	yes	—
63	C₈×C₅₃	—	2	4	no	—
64	#88322	—	4	4	no	—
65	#88322	—	8	2	no	—
66	#9780	—	53	1	no	maximal, Sylow
67	C₂²×C₅₃	—	2	2	no	—
68	C₂²×C₅₃	—	8	4	no	—
69	C₂²×C₅₃	—	8	8	no	—
70	C₂²×C₅₃	—	16	8	no	—
71	C₂²×C₅₃	#390	1	1	yes	characteristic
72	C₂²×C₅₃	#490	1	2	yes	—
73	C₄×C₅₃	—	2	2	no	—
74	C₄×C₅₃	#418	1	2	yes	—
75	C₄×C₅₃	#472	1	4	yes	—
76	C₄×C₅₃	#496	1	2	yes	—
77	C₂×#493	D₅₃	1	1	yes	characteristic
78	#1837	—	53	1	no	—
79	#1545	—	53	4	no	—
80	#1452	—	53	1	no	—
81	C₂×C₅₃	—	2	2	no	—
82	C₂×C₅₃	—	16	4	no	—
83	C₂×C₅₃	#1961	1	1	yes	characteristic
84	C₂×C₅₃	#1978	1	2	yes	—
85	C₂²×D₈	—	2	2	no	—
86	C₂×#127	C₂×D₅₃	1	2	yes	—
87	#493	—	2	2	no	—
88	#493	Dic₅₃	1	4	yes	—
89	#431	—	53	2	no	—
90	#423	—	53	2	no	—
91	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
92	#341	—	106	2	no	—
93	#331	—	106	8	no	—
94	#322	—	106	2	no	—
95	C₅₃	#9780	1	1	yes	characteristic, complemented, Sylow
96	C₂²×D₄	—	2	4	no	—
97	C₂²×D₄	—	2	2	no	—
98	C₂×D₈	—	2	8	no	—
99	C₂×D₈	—	4	8	no	—
100	C₂×M₁₆	—	106	2	no	—
101	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
102	#127	—	2	4	no	—
103	#127	—	2	2	no	—
104	#127	C₂×Dic₅₃	1	4	yes	—
105	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
106	C₄⋊C₈	—	53	4	no	—
107	C₄⋊C₈	—	106	4	no	—
108	C₂²⋊C₈	—	106	2	no	—
109	C₄×C₈	—	106	2	no	—
110	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
111	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
112	C₂⁴	—	4	2	no	—
113	C₂×D₄	—	2	4	no	—
114	C₂×D₄	—	2	8	no	—
115	C₂×D₄	—	4	4	no	—
116	C₂×D₄	—	4	2	no	—
117	C₂×D₄	—	4	8	no	—
118	C₂×D₄	—	8	8	no	—
119	C₂²×C₄	#11719947	1	2	yes	—
120	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
121	D₈	—	4	16	no	—
122	M₁₆	—	212	4	no	—
123	C₂×C₈	—	2	4	no	—
124	C₂×C₈	—	106	4	no	—
125	C₂×C₈	—	212	2	no	—
126	C₂×C₈	C₂²⋊Dic₅₃	1	4	yes	—
127	C₄²	D₄×D₅₃	1	2	yes	—
128	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
129	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
130	C₂³	—	4	4	no	—
131	C₂³	—	8	8	no	—
132	C₂³	—	8	4	no	—
133	C₂³	#13133204	1	1	yes	characteristic
134	D₄	—	4	8	no	—
135	D₄	—	8	8	no	—
136	D₄	—	8	4	no	—
137	D₄	—	8	8	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
140	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
141	C₂×C₄	#13133188	1	2	yes	—
142	C₂×C₄	#13133213	1	1	yes	characteristic
143	C₂×C₄	#13133243	1	4	yes	—
144	C₈	—	2	4	no	—
145	C₈	—	424	2	no	—
146	C₈	—	212	4	no	—
147	C₂²	—	2	2	no	—
148	C₂²	—	8	4	no	—
149	C₂²	—	8	8	no	—
150	C₂²	—	16	8	no	—
151	C₂²	#16810526	1	1	yes	characteristic
152	C₂²	#16810987	1	2	yes	—
153	C₄	—	2	2	no	—
154	C₄	#16810628	1	2	yes	—
155	C₄	#16810890	1	4	yes	—
156	C₄	#16811010	1	2	yes	—
157	C₂	—	2	2	no	—
158	C₂	—	16	4	no	—
159	C₂	#20275540	1	1	yes	characteristic
160	C₂	#20275626	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20274853

C₂

yes

maximal, characteristic, complemented

#20273383

C₂

yes

maximal, complemented

#20273007

C₂

yes

maximal, characteristic, complemented

C₂×C₅₃×#493

C₂

yes

maximal, characteristic

#16810676

C₂²

yes

—

#16810647

C₂²

yes

—

#16810388

—

#16810367

—

#16810345

—

C₂²×D₈×C₅₃

—

C₂×#127×C₅₃

C₂²

yes

—

C₅₃×#493

—

C₅₃×#493

C₄

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

—

—

D₄

yes

—

C₂²⋊#88322

—

C₄×#88322

—

C₂²×D₄×C₅₃

—

C₂²×D₄×C₅₃

—

C₂×D₈×C₅₃

—

C₂×D₈×C₅₃

—

C₂²×C₈×C₅₃

D₄

yes

—

#127×C₅₃

—

#127×C₅₃

—

#127×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

D₄

yes

—

C₂⁴×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

—

D₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

—

#11719917

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

#124

yes

characteristic

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₄×D₄

yes

—

C₂×C₄×C₅₃

#121

yes

—

C₂×C₄×C₅₃

#127

yes

characteristic

C₂×C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₈×C₅₃

—

—

—

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#390

yes

characteristic

C₂²×C₅₃

#490

yes

—

C₄×C₅₃

—

C₄×C₅₃

#418

yes

—

C₄×C₅₃

#472

yes

—

C₄×C₅₃

#496

yes

—

C₂×#493

D₅₃

yes

—

—

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#1961

yes

characteristic

C₂×C₅₃

#1978

yes

—

C₂²×D₈

—

C₂×#127

C₂×D₅₃

yes

—

#493

—

#493

Dic₅₃

yes

—

#431

—

#423

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

#341

—

106

—

#331

—

106

—

#322

—

106

—

C₅₃

#9780

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂²×D₄

—

C₂×D₈

—

C₂×D₈

—

100

C₂×M₁₆

—

106

—

101

C₂²×C₈

C₂²⋊D₅₃

yes

—

102

#127

—

103

#127

—

104

#127

C₂×Dic₅₃

yes

—

105

C₂×C₄²

C₂²×D₅₃

yes

characteristic

106

C₄⋊C₈

—

107

C₄⋊C₈

—

106

—

108

C₂²⋊C₈

—

106

—

109

C₄×C₈

—

106

—

110

C₄×C₈

C₂×Dic₅₃

yes

—

111

C₄×C₈

C₂²⋊D₅₃

yes

—

112

C₂⁴

—

113

C₂×D₄

—

114

C₂×D₄

—

115

C₂×D₄

—

116

C₂×D₄

—

117

C₂×D₄

—

118

C₂×D₄

—

119

C₂²×C₄

#11719947

yes

—

120

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

121

D₈

—

122

M₁₆

—

212

—

123

C₂×C₈

—

124

C₂×C₈

—

106

—

125

C₂×C₈

—

212

—

126

C₂×C₈

C₂²⋊Dic₅₃

yes

—

127

C₄²

D₄×D₅₃

yes

—

128

C₄²

C₂²×Dic₅₃

yes

characteristic

129

C₄²

C₂×C₂²⋊D₅₃

yes

characteristic

130

C₂³

—

131

C₂³

—

132

C₂³

—

133

C₂³

#13133204

yes

characteristic

134

D₄

—

135

D₄

—

136

D₄

—

137

D₄

—

138

C₂×C₄

—

139

C₂×C₄

C₂×C₂²⋊Dic₅₃

yes

characteristic

140

C₂×C₄

D₄×Dic₅₃

yes

—

141

C₂×C₄

#13133188

yes

—

142

C₂×C₄

#13133213

yes

characteristic

143

C₂×C₄

#13133243

yes

—

144

C₈

—

145

C₈

—

424

—

146

C₈

—

212

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

#16810526

yes

characteristic

152

C₂²

#16810987

yes

—

153

C₄

—

154

C₄

#16810628

yes

—

155

C₄

#16810890

yes

—

156

C₄

#16811010

yes

—

157

C₂

—

158

C₂

—

159

C₂

#20275540

yes

characteristic

160

C₂

#20275626

yes

—

	subgroup	G/H
center	C₂²	#16810526
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting, prime radical	C₂×C₅₃×#493	C₂
socle	C₂²×C₅₃	#390
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×#493	D₅₃
hypocenter	C₅₃	#9780
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#493

subgroup

G/H

center

C₂²

#16810526

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting, prime radical

C₂×C₅₃×#493

C₂

socle

C₂²×C₅₃

#390

cosocle

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#493

N	H ≅ G/N	type
1. #20274853	C₂	semidirect
2. #20273383	C₂	semidirect
3. #20273007	C₂	semidirect
95. C₅₃	#9780	semidirect

H ≅ G/N

type

1. #20274853

C₂

semidirect

2. #20273383

C₂

semidirect

3. #20273007

C₂

semidirect

95. C₅₃

#9780

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20274853	2
#20273383	2
#20273007	2
C₂×C₅₃×#493	2
#9780	53

subgroup

index

C₂×C₅₃×#493

Subgroups — #27283597

Conjugacy classes of subgroups

Complements

Maximal subgroups