Group #27220038 — subgroups

Subgroups — C₅₃×#12526 #27220038

Order 13568.

296 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 102 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	C₅₃×#2248	C₂	1	2	yes	maximal
2	C₅₃×#2241	C₂	1	2	yes	maximal, complemented
3	C₅₃×#1713	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1706	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1312	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×SD₃₂×C₅₃	—	2	4	no	—
7	C₅₃×#500	—	2	1	no	—
8	C₅₃×#500	—	2	1	no	—
9	C₅₃×#500	C₂²	1	1	yes	characteristic
10	C₅₃×#494	C₂²	1	2	yes	—
11	C₅₃×#493	C₂²	1	1	yes	characteristic
12	C₅₃×#492	—	2	1	no	—
13	C₅₃×#486	—	2	1	no	—
14	C₅₃×#358	—	2	4	no	—
15	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
16	C₈²×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×Q₁₆×C₅₃	—	2	2	no	—
18	C₂×Q₁₆×C₅₃	—	2	2	no	—
19	C₂×SD₁₆×C₅₃	—	4	2	no	—
20	C₂×D₈×C₅₃	—	2	2	no	—
21	C₄⋊Q₈×C₅₃	—	2	1	no	—
22	C₄⋊Q₈×C₅₃	—	2	2	no	—
23	#127×C₅₃	—	2	1	no	—
24	SD₃₂×C₅₃	—	4	8	no	—
25	C₂×C₁₆×C₅₃	D₄	1	4	yes	—
26	#107×C₅₃	—	2	2	no	—
27	#107×C₅₃	—	4	2	no	—
28	#107×C₅₃	—	4	2	no	—
29	#102×C₅₃	—	4	2	no	—
30	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
31	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
32	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
33	C₂×Q₈×C₅₃	—	4	2	no	—
34	C₂×Q₈×C₅₃	—	4	1	no	—
35	C₂×D₄×C₅₃	—	4	1	no	—
36	C₂×D₄×C₅₃	—	4	1	no	—
37	C₂×D₄×C₅₃	—	4	1	no	—
38	Q₁₆×C₅₃	—	4	4	no	—
39	Q₁₆×C₅₃	—	4	4	no	—
40	SD₁₆×C₅₃	—	8	4	no	—
41	D₈×C₅₃	—	4	4	no	—
42	C₂×C₈×C₅₃	D₈	1	2	yes	—
43	C₂×C₈×C₅₃	D₈	1	2	yes	—
44	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
45	C₄⋊C₄×C₅₃	—	4	1	no	—
46	C₄⋊C₄×C₅₃	—	4	1	no	—
47	C₄⋊C₄×C₅₃	—	4	2	no	—
48	C₄⋊C₄×C₅₃	—	4	2	no	—
49	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
50	C₁₆×C₅₃	—	2	4	no	—
51	C₂³×C₅₃	—	8	1	no	—
52	Q₈×C₅₃	—	8	2	no	—
53	Q₈×C₅₃	—	8	2	no	—
54	Q₈×C₅₃	—	8	2	no	—
55	D₄×C₅₃	—	8	2	no	—
56	D₄×C₅₃	—	8	2	no	—
57	D₄×C₅₃	—	8	2	no	—
58	C₂×C₄×C₅₃	—	8	2	no	—
59	C₂×C₄×C₅₃	—	8	1	no	—
60	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
61	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
62	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
63	C₈×C₅₃	—	2	2	no	—
64	C₈×C₅₃	SD₃₂	1	4	yes	complemented
65	C₈×C₅₃	AGL(1,ℤ/8)	1	4	yes	—
66	#12526	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₂²×C₅₃	—	16	1	no	—
68	C₂²×C₅₃	—	16	1	no	—
69	C₂²×C₅₃	—	16	1	no	—
70	C₂²×C₅₃	#493	1	1	yes	characteristic
71	C₄×C₅₃	—	16	2	no	—
72	C₄×C₅₃	—	16	1	no	—
73	C₄×C₅₃	#496	1	2	yes	—
74	C₄×C₅₃	C₂×SD₃₂	1	2	yes	—
75	C₄×C₅₃	#510	1	2	yes	—
76	#2248	C₂×C₅₃	1	2	yes	—
77	#2241	C₂×C₅₃	1	2	yes	complemented
78	#1713	C₂×C₅₃	1	1	yes	characteristic, complemented
79	#1706	C₂×C₅₃	1	1	yes	characteristic
80	#1312	C₂×C₅₃	1	1	yes	characteristic, complemented
81	C₂×C₅₃	—	32	1	no	—
82	C₂×C₅₃	#1721	1	1	yes	characteristic
83	C₂×C₅₃	#2248	1	1	yes	characteristic
84	C₂×C₅₃	#2251	1	1	yes	characteristic
85	C₂×SD₃₂	—	2	4	no	—
86	#500	—	2	1	no	—
87	#500	—	2	1	no	—
88	#500	C₂²×C₅₃	1	1	yes	characteristic
89	#494	C₂²×C₅₃	1	2	yes	—
90	#493	C₂²×C₅₃	1	1	yes	characteristic
91	#492	—	2	1	no	—
92	#486	—	2	1	no	—
93	#358	—	2	4	no	—
94	C₄×C₁₆	C₂²×C₅₃	1	2	yes	—
95	C₈²	C₂²×C₅₃	1	1	yes	characteristic
96	C₅₃	#12526	1	1	yes	characteristic, direct product factor, Sylow
97	C₂×Q₁₆	—	2	2	no	—
98	C₂×Q₁₆	—	2	2	no	—
99	C₂×SD₁₆	—	4	2	no	—
100	C₂×D₈	—	2	2	no	—
101	C₄⋊Q₈	—	2	1	no	—
102	C₄⋊Q₈	—	2	2	no	—
103	#127	—	2	1	no	—
104	SD₃₂	—	4	8	no	—
105	C₂×C₁₆	D₄×C₅₃	1	4	yes	—
106	#107	—	2	2	no	—
107	#107	—	4	2	no	—
108	#107	—	4	2	no	—
109	#102	—	4	2	no	—
110	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
111	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
112	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
113	C₂×Q₈	—	4	2	no	—
114	C₂×Q₈	—	4	1	no	—
115	C₂×D₄	—	4	1	no	—
116	C₂×D₄	—	4	1	no	—
117	C₂×D₄	—	4	1	no	—
118	Q₁₆	—	4	4	no	—
119	Q₁₆	—	4	4	no	—
120	SD₁₆	—	8	4	no	—
121	D₈	—	4	4	no	—
122	C₂×C₈	D₈×C₅₃	1	2	yes	—
123	C₂×C₈	D₈×C₅₃	1	2	yes	—
124	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
125	C₄⋊C₄	—	4	1	no	—
126	C₄⋊C₄	—	4	1	no	—
127	C₄⋊C₄	—	4	2	no	—
128	C₄⋊C₄	—	4	2	no	—
129	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
130	C₁₆	—	2	4	no	—
131	C₂³	—	8	1	no	—
132	Q₈	—	8	2	no	—
133	Q₈	—	8	2	no	—
134	Q₈	—	8	2	no	—
135	D₄	—	8	2	no	—
136	D₄	—	8	2	no	—
137	D₄	—	8	2	no	—
138	C₂×C₄	—	8	2	no	—
139	C₂×C₄	—	8	1	no	—
140	C₂×C₄	#127×C₅₃	1	1	yes	characteristic
141	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
142	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
143	C₈	—	2	2	no	—
144	C₈	SD₃₂×C₅₃	1	4	yes	complemented
145	C₈	AGL(1,ℤ/8)×C₅₃	1	4	yes	—
146	C₂²	—	16	1	no	—
147	C₂²	—	16	1	no	—
148	C₂²	—	16	1	no	—
149	C₂²	C₅₃×#493	1	1	yes	characteristic
150	C₄	—	16	2	no	—
151	C₄	—	16	1	no	—
152	C₄	C₅₃×#496	1	2	yes	—
153	C₄	C₂×SD₃₂×C₅₃	1	2	yes	—
154	C₄	C₅₃×#510	1	2	yes	—
155	C₂	—	32	1	no	—
156	C₂	C₅₃×#1721	1	1	yes	characteristic
157	C₂	C₅₃×#2248	1	1	yes	characteristic
158	C₂	C₅₃×#2251	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#2248

C₂

yes

maximal

C₅₃×#2241

C₂

yes

maximal, complemented

C₅₃×#1713

C₂

yes

maximal, characteristic, complemented

C₅₃×#1706

C₂

yes

maximal, characteristic

C₅₃×#1312

C₂

yes

maximal, characteristic, complemented

C₂×SD₃₂×C₅₃

—

C₅₃×#500

—

C₅₃×#500

—

C₅₃×#500

C₂²

yes

characteristic

C₅₃×#494

C₂²

yes

—

C₅₃×#493

C₂²

yes

characteristic

C₅₃×#492

—

C₅₃×#486

—

C₅₃×#358

—

C₄×C₁₆×C₅₃

C₂²

yes

—

C₈²×C₅₃

C₂²

yes

characteristic

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₂×SD₁₆×C₅₃

—

C₂×D₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

#127×C₅₃

—

SD₃₂×C₅₃

—

C₂×C₁₆×C₅₃

D₄

yes

—

#107×C₅₃

—

#107×C₅₃

—

#107×C₅₃

—

#102×C₅₃

—

C₄×C₈×C₅₃

D₄

yes

characteristic

C₄×C₈×C₅₃

D₄

yes

characteristic

C₄×C₈×C₅₃

C₂³

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

Q₁₆×C₅₃

—

Q₁₆×C₅₃

—

SD₁₆×C₅₃

—

D₈×C₅₃

—

C₂×C₈×C₅₃

D₈

yes

—

C₂×C₈×C₅₃

D₈

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₁₆×C₅₃

—

C₂³×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#127

yes

characteristic

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

SD₃₂

yes

complemented

C₈×C₅₃

AGL(1,ℤ/8)

yes

—

#12526

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#493

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#496

yes

—

C₄×C₅₃

C₂×SD₃₂

yes

—

C₄×C₅₃

#510

yes

—

#2248

C₂×C₅₃

yes

—

#2241

C₂×C₅₃

yes

complemented

#1713

C₂×C₅₃

yes

characteristic, complemented

#1706

C₂×C₅₃

yes

characteristic

#1312

C₂×C₅₃

yes

characteristic, complemented

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×SD₃₂

—

#500

—

#500

—

#500

C₂²×C₅₃

yes

characteristic

#494

C₂²×C₅₃

yes

—

#493

C₂²×C₅₃

yes

—

—

—

C₄×C₁₆

C₂²×C₅₃

yes

—

C₈²

C₂²×C₅₃

yes

characteristic

C₅₃

#12526

yes

characteristic, direct product factor, Sylow

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×SD₁₆

—

100

C₂×D₈

—

101

C₄⋊Q₈

—

102

C₄⋊Q₈

—

103

#127

—

104

SD₃₂

—

105

C₂×C₁₆

D₄×C₅₃

yes

—

106

#107

—

107

#107

—

108

#107

—

109

#102

—

110

C₄×C₈

D₄×C₅₃

yes

characteristic

111

C₄×C₈

D₄×C₅₃

yes

characteristic

112

C₄×C₈

C₂³×C₅₃

yes

characteristic

113

C₂×Q₈

—

114

C₂×Q₈

—

115

C₂×D₄

—

116

C₂×D₄

—

117

C₂×D₄

—

118

Q₁₆

—

119

Q₁₆

—

120

SD₁₆

—

121

D₈

—

122

C₂×C₈

D₈×C₅₃

yes

—

123

C₂×C₈

D₈×C₅₃

yes

—

124

C₂×C₈

C₂×D₄×C₅₃

yes

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

128

C₄⋊C₄

—

129

C₄²

C₂×D₄×C₅₃

yes

characteristic

130

C₁₆

—

131

C₂³

—

132

Q₈

—

133

Q₈

—

134

Q₈

—

135

D₄

—

136

D₄

—

137

D₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

#127×C₅₃

yes

characteristic

141

C₂×C₄

C₂×D₈×C₅₃

yes

characteristic

142

C₂×C₄

C₂×D₈×C₅₃

yes

characteristic

143

C₈

—

144

C₈

SD₃₂×C₅₃

yes

complemented

145

C₈

AGL(1,ℤ/8)×C₅₃

yes

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

C₅₃×#493

yes

characteristic

150

C₄

—

151

C₄

—

152

C₄

C₅₃×#496

yes

—

153

C₄

C₂×SD₃₂×C₅₃

yes

—

154

C₄

C₅₃×#510

yes

—

155

C₂

—

156

C₂

C₅₃×#1721

yes

characteristic

157

C₂

C₅₃×#2248

yes

characteristic

158

C₂

C₅₃×#2251

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	#493
commutator, Frattini, cosocle	C₄×C₈	C₂³×C₅₃
prime radical	C₅₃×#493	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₄²×C₅₃ ◁ C₈²×C₅₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₅₃

#493

commutator, Frattini, cosocle

C₄×C₈

C₂³×C₅₃

prime radical

C₅₃×#493

C₂²

LCS

G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₅₃ ◁ C₄²×C₅₃ ◁ C₈²×C₅₃ ◁ G

N	H ≅ G/N	type
2. C₅₃×#2241	C₂	semidirect
3. C₅₃×#1713	C₂	semidirect
5. C₅₃×#1312	C₂	semidirect
64. C₈×C₅₃	SD₃₂	semidirect
66. #12526	C₅₃	direct (N × H)
77. #2241	C₂×C₅₃	semidirect
78. #1713	C₂×C₅₃	semidirect
80. #1312	C₂×C₅₃	semidirect
96. C₅₃	#12526	direct (N × H)
144. C₈	SD₃₂×C₅₃	semidirect

H ≅ G/N

type

2. C₅₃×#2241

C₂

semidirect

3. C₅₃×#1713

C₂

semidirect

5. C₅₃×#1312

C₂

semidirect

64. C₈×C₅₃

SD₃₂

semidirect

66. #12526

C₅₃

direct (N × H)

77. #2241

C₂×C₅₃

semidirect

78. #1713

C₂×C₅₃

semidirect

80. #1312

C₂×C₅₃

semidirect

96. C₅₃

#12526

direct (N × H)

144. C₈

SD₃₂×C₅₃

semidirect

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

subgroup	index
C₅₃×#2248	2
C₅₃×#2241	2
C₅₃×#1713	2
C₅₃×#1706	2
C₅₃×#1312	2
#12526	53

subgroup

index

C₅₃×#2248

C₅₃×#2241

C₅₃×#1713

C₅₃×#1706

C₅₃×#1312

#12526

Subgroups — C₅₃×#12526 #27220038

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#12526 #27220038

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12526 #27220038