Group #27392492 — subgroups

Subgroups — #27392492

Order 13568.

1266 conjugacy classes of subgroups; only the 307 normal subgroups are shown.

normal subgroups only

This group has 1266 conjugacy classes of subgroups — too many to draw. Only its 307 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	C₂×C₈×D₂₁₂	C₂	1	6	yes	maximal, complemented
2	#20277996	C₂	1	1	yes	maximal, characteristic
3	#20275513	C₂	1	6	yes	maximal, complemented
4	#20274869	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1726	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₈×D₅₃	C₂²	1	4	yes	—
7	C₂×C₄×D₂₁₂	C₂²	1	6	yes	—
8	C₈×D₂₁₂	—	2	24	no	—
9	C₂×C₄⋊#88322	C₂²	1	6	yes	—
10	C₂×#13133129	C₂²	1	12	yes	—
11	#16810560	C₄	1	4	yes	—
12	#16810533	C₄	1	4	yes	—
13	C₂×#97×C₅₃	C₂²	1	3	yes	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	3	yes	—
15	C₅₃×#376	C₂²	1	1	yes	characteristic
16	C₂²×D₂₁₂	C₂×C₄	1	6	yes	—
17	C₂²×C₄×D₅₃	C₂³	1	4	yes	—
18	C₂²×#88322	C₂³	1	4	yes	—
19	C₂×C₈×D₅₃	—	2	24	no	—
20	C₄×D₂₁₂	—	2	12	no	—
21	C₄×D₂₁₂	—	2	12	no	—
22	C₂×C₄⋊Dic₅₃	C₂×C₄	1	6	yes	—
23	C₂×#11719921	C₂×C₄	1	12	yes	—
24	C₄⋊#88322	—	2	12	no	—
25	#13133129	—	2	24	no	—
26	C₂²×C₈×C₅₃	C₂³	1	3	yes	—
27	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
28	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
29	#97×C₅₃	—	2	6	no	—
30	C₄×C₈×C₅₃	D₄	1	12	yes	complemented
31	#95×C₅₃	C₂×C₄	1	4	yes	—
32	C₂³×D₅₃	C₂²×C₄	1	4	yes	—
33	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
34	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
35	C₂²×Dic₅₃	C₂²×C₄	1	4	yes	—
36	C₂×D₂₁₂	—	2	24	no	—
37	C₂×D₂₁₂	—	2	12	no	—
38	C₂×C₄×D₅₃	—	2	12	no	—
39	C₂×C₄×D₅₃	—	2	12	no	—
40	C₂×C₈×C₅₃	—	2	6	no	—
41	C₂×C₈×C₅₃	C₂×D₄	1	6	yes	—
42	C₂×C₈×C₅₃	C₂×D₄	1	6	yes	—
43	C₄²×C₅₃	C₂×D₄	1	6	yes	—
44	C₄²×C₅₃	#41	1	6	yes	—
45	#11719921	—	2	24	no	—
46	C₄⋊Dic₅₃	—	2	12	no	—
47	C₂×#88322	—	2	12	no	—
48	C₈×D₅₃	—	4	16	no	—
49	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
50	C₂²×D₅₃	—	2	12	no	—
51	C₂²×D₅₃	—	2	12	no	—
52	C₂²×D₅₃	#131	1	8	yes	—
53	C₂×C₄×C₅₃	—	2	12	no	—
54	C₂×C₄×C₅₃	C₄×D₄	1	12	yes	—
55	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
56	C₂×C₄×C₅₃	C₂×#41	1	3	yes	—
57	C₂×Dic₅₃	—	2	12	no	—
58	D₂₁₂	—	2	48	no	—
59	C₄×D₅₃	—	4	4	no	—
60	C₄×D₅₃	—	4	12	no	—
61	C₈×C₅₃	—	2	6	no	—
62	C₈×C₅₃	#127	1	4	yes	—
63	#88322	—	4	4	no	—
64	#21683	—	53	1	no	maximal, Sylow
65	C₂²×C₅₃	C₂×C₄×D₄	1	3	yes	—
66	C₂²×C₅₃	C₂×#131	1	4	yes	—
67	C₂×D₅₃	—	2	24	no	—
68	C₂×D₅₃	—	4	4	no	—
69	C₂×D₅₃	—	4	12	no	—
70	C₄×C₅₃	—	2	12	no	—
71	C₄×C₅₃	C₂×#127	1	1	yes	characteristic
72	C₄×C₅₃	#532	1	3	yes	—
73	Dic₅₃	—	4	4	no	—
74	C₂×C₈×D₄	—	53	6	no	—
75	#2397	—	53	1	no	—
76	#1955	—	53	6	no	—
77	#1843	—	53	1	no	—
78	#1726	D₅₃	1	1	yes	characteristic, complemented
79	C₂×C₅₃	C₄×#127	1	1	yes	characteristic
80	C₂×C₅₃	#2929	1	6	yes	—
81	D₅₃	—	4	8	no	—
82	C₂³×C₈	—	53	4	no	—
83	C₂×C₄×D₄	—	53	6	no	—
84	C₈×D₄	—	106	24	no	—
85	C₂×C₄⋊C₈	—	53	6	no	—
86	C₂×C₂²⋊C₈	—	53	12	no	—
87	C₂×#97	C₂×D₅₃	1	3	yes	—
88	C₂×C₄×C₈	C₂×D₅₃	1	3	yes	—
89	#399	—	53	4	no	—
90	#392	—	53	4	no	—
91	#376	C₂×D₅₃	1	1	yes	characteristic
92	C₅₃	#21683	1	1	yes	characteristic, complemented, Sylow
93	C₂²×D₄	—	53	6	no	—
94	C₂³×C₄	—	53	4	no	—
95	C₂²×C₈	—	53	4	no	—
96	C₂²×C₈	—	106	24	no	—
97	C₂²×C₈	C₂²×D₅₃	1	3	yes	—
98	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
99	C₄×D₄	—	106	12	no	—
100	C₄×D₄	—	106	12	no	—
101	C₂×C₄⋊C₄	—	53	6	no	—
102	C₂×C₂²⋊C₄	—	53	12	no	—
103	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
104	C₄⋊C₈	—	106	12	no	—
105	C₂²⋊C₈	—	106	24	no	—
106	#97	—	2	6	no	—
107	C₄×C₈	D₂₁₂	1	12	yes	complemented
108	#95	C₄×D₅₃	1	4	yes	—
109	C₂⁴	—	53	4	no	—
110	C₂×D₄	—	106	24	no	—
111	C₂×D₄	—	106	12	no	—
112	C₂²×C₄	—	53	4	no	—
113	C₂²×C₄	—	106	12	no	—
114	C₂²×C₄	—	106	12	no	—
115	C₂²×C₄	C₂×C₄×D₅₃	1	6	yes	—
116	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
117	C₂×C₈	—	2	6	no	—
118	C₂×C₈	—	106	12	no	—
119	C₂×C₈	—	212	16	no	—
120	C₂×C₈	C₂×D₂₁₂	1	6	yes	—
121	C₂×C₈	C₂×D₂₁₂	1	6	yes	—
122	C₄⋊C₄	—	106	12	no	—
123	C₂²⋊C₄	—	106	24	no	—
124	C₄²	C₂×D₂₁₂	1	6	yes	—
125	C₄²	#11719945	1	6	yes	—
126	C₂³	—	53	8	no	—
127	C₂³	—	106	12	no	—
128	C₂³	—	106	12	no	—
129	C₂³	C₂²×C₄×D₅₃	1	1	yes	characteristic
130	D₄	—	106	48	no	—
131	C₂×C₄	—	2	12	no	—
132	C₂×C₄	—	106	12	no	—
133	C₂×C₄	—	212	4	no	—
134	C₂×C₄	—	212	12	no	—
135	C₂×C₄	C₄×D₂₁₂	1	12	yes	—
136	C₂×C₄	C₂²×D₂₁₂	1	3	yes	—
137	C₂×C₄	C₂×#11719945	1	3	yes	—
138	C₈	—	2	6	no	—
139	C₈	—	212	4	no	—
140	C₈	#13133214	1	4	yes	—
141	C₂²	—	106	24	no	—
142	C₂²	—	212	12	no	—
143	C₂²	—	212	4	no	—
144	C₂²	C₂×C₄×D₂₁₂	1	3	yes	—
145	C₂²	#131×D₅₃	1	4	yes	—
146	C₄	—	2	12	no	—
147	C₄	—	212	4	no	—
148	C₄	C₂×#13133214	1	1	yes	characteristic
149	C₄	#16811173	1	3	yes	—
150	C₂	—	212	8	no	—
151	C₂	C₄×#13133214	1	1	yes	characteristic
152	C₂	#20283411	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₈×D₂₁₂

C₂

yes

maximal, complemented

#20277996

C₂

yes

maximal, characteristic

#20275513

C₂

yes

maximal, complemented

#20274869

C₂

yes

maximal, characteristic, complemented

C₅₃×#1726

C₂

yes

maximal, characteristic, complemented

C₂²×C₈×D₅₃

C₂²

yes

—

C₂×C₄×D₂₁₂

C₂²

yes

—

C₈×D₂₁₂

—

C₂×C₄⋊#88322

C₂²

yes

—

C₂×#13133129

C₂²

yes

—

#16810560

C₄

yes

—

#16810533

C₄

yes

—

C₂×#97×C₅₃

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

—

C₅₃×#376

C₂²

yes

characteristic

C₂²×D₂₁₂

C₂×C₄

yes

—

C₂²×C₄×D₅₃

C₂³

yes

—

C₂²×#88322

C₂³

yes

—

C₂×C₈×D₅₃

—

C₄×D₂₁₂

—

C₄×D₂₁₂

—

C₂×C₄⋊Dic₅₃

C₂×C₄

yes

—

C₂×#11719921

C₂×C₄

yes

—

C₄⋊#88322

—

#13133129

—

C₂²×C₈×C₅₃

C₂³

yes

—

C₂²×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₄²×C₅₃

C₂³

yes

—

#97×C₅₃

—

C₄×C₈×C₅₃

D₄

yes

complemented

#95×C₅₃

C₂×C₄

yes

—

C₂³×D₅₃

C₂²×C₄

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

—

C₂²×C₄×C₅₃

C₂⁴

yes

characteristic

C₂²×Dic₅₃

C₂²×C₄

yes

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

#41

yes

—

#11719921

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₈×D₅₃

—

C₂³×C₅₃

C₂³×C₄

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

#131

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄×D₄

yes

—

C₂×C₄×C₅₃

C₂²×D₄

yes

—

C₂×C₄×C₅₃

C₂×#41

yes

—

C₂×Dic₅₃

—

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

C₈×C₅₃

—

C₈×C₅₃

#127

yes

—

#88322

—

#21683

—

maximal, Sylow

C₂²×C₅₃

C₂×C₄×D₄

yes

—

C₂²×C₅₃

C₂×#131

yes

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂×#127

yes

characteristic

C₄×C₅₃

#532

yes

—

Dic₅₃

—

C₂×C₈×D₄

—

—

—

—

D₅₃

yes

characteristic, complemented

C₂×C₅₃

C₄×#127

yes

characteristic

C₂×C₅₃

#2929

yes

—

D₅₃

—

C₂³×C₈

—

C₂×C₄×D₄

—

C₈×D₄

—

106

—

C₂×C₄⋊C₈

—

C₂×C₂²⋊C₈

—

C₂×#97

C₂×D₅₃

yes

—

C₂×C₄×C₈

C₂×D₅₃

yes

—

#399

—

#392

—

#376

C₂×D₅₃

yes

characteristic

C₅₃

#21683

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂³×C₄

—

C₂²×C₈

—

C₂²×C₈

—

106

—

C₂²×C₈

C₂²×D₅₃

yes

—

C₂²×C₈

C₂²×D₅₃

yes

characteristic

C₄×D₄

—

106

—

100

C₄×D₄

—

106

—

101

C₂×C₄⋊C₄

—

102

C₂×C₂²⋊C₄

—

103

C₂×C₄²

C₂²×D₅₃

yes

—

104

C₄⋊C₈

—

106

—

105

C₂²⋊C₈

—

106

—

106

#97

—

107

C₄×C₈

D₂₁₂

yes

complemented

108

#95

C₄×D₅₃

yes

—

109

C₂⁴

—

110

C₂×D₄

—

106

—

111

C₂×D₄

—

106

—

112

C₂²×C₄

—

113

C₂²×C₄

—

106

—

114

C₂²×C₄

—

106

—

115

C₂²×C₄

C₂×C₄×D₅₃

yes

—

116

C₂²×C₄

C₂³×D₅₃

yes

characteristic

117

C₂×C₈

—

118

C₂×C₈

—

106

—

119

C₂×C₈

—

212

—

120

C₂×C₈

C₂×D₂₁₂

yes

—

121

C₂×C₈

C₂×D₂₁₂

yes

—

122

C₄⋊C₄

—

106

—

123

C₂²⋊C₄

—

106

—

124

C₄²

C₂×D₂₁₂

yes

—

125

C₄²

#11719945

yes

—

126

C₂³

—

127

C₂³

—

106

—

128

C₂³

—

106

—

129

C₂³

C₂²×C₄×D₅₃

yes

characteristic

130

D₄

—

106

—

131

C₂×C₄

—

132

C₂×C₄

—

106

—

133

C₂×C₄

—

212

—

134

C₂×C₄

—

212

—

135

C₂×C₄

C₄×D₂₁₂

yes

—

136

C₂×C₄

C₂²×D₂₁₂

yes

—

137

C₂×C₄

C₂×#11719945

yes

—

138

C₈

—

139

C₈

—

212

—

140

C₈

#13133214

yes

—

141

C₂²

—

106

—

142

C₂²

—

212

—

143

C₂²

—

212

—

144

C₂²

C₂×C₄×D₂₁₂

yes

—

145

C₂²

#131×D₅₃

yes

—

146

C₄

—

147

C₄

—

212

—

148

C₄

C₂×#13133214

yes

characteristic

149

C₄

#16811173

yes

—

150

C₂

—

212

—

151

C₂

C₄×#13133214

yes

characteristic

152

C₂

#20283411

yes

—

	subgroup	G/H
center	C₂²×C₈	C₂²×D₅₃
commutator, socle	C₂³×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₅₃×#1726	C₂
prime radical	#20277996	C₂
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1726	D₅₃
hypocenter	C₅₃	#21683
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ #1726

subgroup

G/H

center

C₂²×C₈

C₂²×D₅₃

commutator, socle

C₂³×C₅₃

C₂³×C₄

Frattini

C₂²×C₄

C₂³×D₅₃

Fitting

C₅₃×#1726

C₂

prime radical

#20277996

C₂

cosocle

C₂²×C₄×C₅₃

C₂⁴

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₅₃

C₁ ◁ C₂²×C₈ ◁ #1726

N	H ≅ G/N	type
1. C₂×C₈×D₂₁₂	C₂	semidirect
3. #20275513	C₂	semidirect
4. #20274869	C₂	semidirect
5. C₅₃×#1726	C₂	semidirect
30. C₄×C₈×C₅₃	D₄	semidirect
78. #1726	D₅₃	semidirect
92. C₅₃	#21683	semidirect
107. C₄×C₈	D₂₁₂	semidirect

H ≅ G/N

type

1. C₂×C₈×D₂₁₂

C₂

semidirect

3. #20275513

C₂

semidirect

4. #20274869

C₂

semidirect

5. C₅₃×#1726

C₂

semidirect

30. C₄×C₈×C₅₃

D₄

semidirect

78. #1726

D₅₃

semidirect

92. C₅₃

#21683

semidirect

107. C₄×C₈

D₂₁₂

semidirect

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

subgroup	index
C₂×C₈×D₂₁₂	2
#20277996	2
#20275513	2
#20274869	2
C₅₃×#1726	2
#21683	53

subgroup

index

C₂×C₈×D₂₁₂

C₅₃×#1726

Subgroups — #27392492

Conjugacy classes of subgroups

Complements

Maximal subgroups