Group #27300441 — subgroups

Subgroups — #27300441

Order 13568.

322 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 151 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×C₄⋊#11719908	C₂	1	1	yes	maximal, characteristic
2	#20274401	C₂	1	1	yes	maximal, characteristic
3	#20272792	C₂	1	4	yes	maximal
4	C₂×C₅₃×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719908	C₂²	1	2	yes	—
6	C₂×#13133128	C₄	1	1	yes	characteristic
7	C₂×#13133127	C₄	1	2	yes	—
8	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
9	#16810469	C₄	1	1	yes	characteristic
10	C₄⋊#11719908	—	2	2	no	—
11	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×C₄⋊C₁₆	—	2	2	no	—
14	C₂²×#88322	C₂×C₄	1	2	yes	—
15	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
16	C₂×#11719908	—	2	2	no	—
17	C₂×#11719908	—	2	4	no	—
18	#13133128	—	2	2	no	—
19	#13133127	—	2	4	no	—
20	C₈×Dic₅₃	—	2	2	no	—
21	C₈×Dic₅₃	—	2	2	no	—
22	#13133124	C₈	1	2	yes	—
23	#13133124	C₈	1	2	yes	—
24	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
26	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆×C₅₃	—	2	2	no	—
28	C₂×C₁₆×C₅₃	—	2	4	no	—
29	C₄×C₈×C₅₃	D₄	1	2	yes	—
30	C₄×C₈×C₅₃	Q₈	1	2	yes	—
31	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
34	C₂²×Dic₅₃	C₂×C₈	1	4	yes	—
35	C₂×C₈×C₅₃	—	2	2	no	—
36	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
37	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
39	C₂×C₈×C₅₃	#41	1	2	yes	—
40	C₂×C₈×C₅₃	#41	1	2	yes	—
41	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
42	C₄²×C₅₃	M₁₆	1	2	yes	—
43	C₄×Dic₅₃	—	2	2	no	—
44	C₄×Dic₅₃	—	2	2	no	—
45	C₂×#88322	—	2	2	no	—
46	C₂×#88322	—	2	2	no	—
47	C₂×#88322	M₁₆	1	4	yes	—
48	C₁₆×C₅₃	—	4	2	no	—
49	#11719908	—	4	2	no	—
50	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
51	C₂×C₄×C₅₃	—	2	2	no	—
52	C₂×C₄×C₅₃	—	2	2	no	—
53	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
54	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
55	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₅₃	#117	1	2	yes	—
57	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
58	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
59	C₂×Dic₅₃	—	2	4	no	—
60	C₂×Dic₅₃	—	2	4	no	—
61	C₂×Dic₅₃	—	2	4	no	—
62	C₈×C₅₃	—	2	2	no	—
63	C₈×C₅₃	#124	1	1	yes	characteristic
64	C₈×C₅₃	#124	1	1	yes	characteristic
65	C₈×C₅₃	#125	1	2	yes	—
66	#88322	—	2	4	no	—
67	#12880	—	53	1	no	maximal, Sylow
68	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
69	C₂²×C₅₃	#431	1	2	yes	—
70	C₂²×C₅₃	#504	1	2	yes	—
71	C₂²×C₅₃	#504	1	2	yes	—
72	C₄×C₅₃	—	2	2	no	—
73	C₄×C₅₃	—	2	2	no	—
74	C₄×C₅₃	#382	1	1	yes	characteristic
75	C₄×C₅₃	#423	1	1	yes	characteristic
76	C₄×C₅₃	#432	1	2	yes	—
77	Dic₅₃	—	4	4	no	—
78	C₂×C₄⋊C₁₆	—	53	1	no	—
79	C₂×C₄⋊C₁₆	D₅₃	1	1	yes	characteristic
80	#1726	—	53	1	no	—
81	#1374	—	53	4	no	—
82	C₂×C₅₃	#1831	1	1	yes	characteristic
83	C₂×C₅₃	#2151	1	1	yes	characteristic
84	C₂×C₅₃	#2151	1	1	yes	characteristic
85	C₂×C₅₃	#2164	1	4	yes	—
86	C₂²×C₁₆	—	53	2	no	—
87	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
88	C₂×#97	—	53	2	no	—
89	C₂×#97	—	53	1	no	—
90	C₂×C₄×C₈	—	53	2	no	—
91	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
92	#376	—	53	1	no	—
93	C₄⋊C₁₆	—	2	2	no	—
94	C₄⋊C₁₆	—	106	2	no	—
95	C₅₃	#12880	1	1	yes	characteristic, complemented, Sylow
96	C₂²×C₈	—	53	2	no	—
97	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
98	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
99	C₂×C₄²	—	53	2	no	—
100	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
101	C₂×C₁₆	—	2	4	no	—
102	C₂×C₁₆	—	2	2	no	—
103	C₂×C₁₆	—	106	4	no	—
104	C₂×C₁₆	—	106	2	no	—
105	#97	—	106	4	no	—
106	#97	—	106	2	no	—
107	C₄×C₈	—	106	2	no	—
108	C₄×C₈	—	106	2	no	—
109	C₄×C₈	Dic₁₀₆	1	2	yes	—
110	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
111	#95	—	53	2	no	—
112	#95	—	53	2	no	—
113	C₂²×C₄	—	53	4	no	—
114	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
115	C₂²×C₄	C₈×D₅₃	1	1	yes	characteristic
116	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
117	C₂×C₈	—	2	2	no	—
118	C₂×C₈	—	53	4	no	—
119	C₂×C₈	—	106	2	no	—
120	C₂×C₈	—	106	2	no	—
121	C₂×C₈	#11719919	1	2	yes	—
122	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
123	C₂×C₈	#11719947	1	2	yes	—
124	C₂×C₈	#11719949	1	2	yes	—
125	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
126	C₄²	—	106	2	no	—
127	C₄²	—	106	2	no	—
128	C₄²	#11719912	1	2	yes	—
129	C₄²	#11719919	1	2	yes	—
130	C₁₆	—	4	2	no	—
131	C₁₆	—	212	2	no	—
132	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	106	4	no	—
136	C₂×C₄	—	106	4	no	—
137	C₂×C₄	—	106	4	no	—
138	C₂×C₄	#13133146	1	2	yes	—
139	C₂×C₄	#13133146	1	2	yes	—
140	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
141	C₂×C₄	#13133175	1	2	yes	—
142	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
143	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
144	C₈	—	2	2	no	—
145	C₈	—	106	4	no	—
146	C₈	#13133204	1	1	yes	characteristic
147	C₈	#13133206	1	1	yes	characteristic
148	C₈	#13133208	1	2	yes	—
149	C₂²	C₂×#13133146	1	1	yes	characteristic
150	C₂²	#16810675	1	2	yes	—
151	C₂²	#16811039	1	2	yes	—
152	C₂²	#16811041	1	2	yes	—
153	C₄	—	2	2	no	—
154	C₄	—	2	2	no	—
155	C₄	—	212	4	no	—
156	C₄	#16810488	1	1	yes	characteristic
157	C₄	#16810646	1	1	yes	characteristic
158	C₄	#16810679	1	2	yes	—
159	C₂	#20274831	1	1	yes	characteristic
160	C₂	#20276473	1	1	yes	characteristic
161	C₂	#20276475	1	1	yes	characteristic
162	C₂	#20276521	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#11719908

C₂

yes

maximal, characteristic

#20274401

C₂

yes

maximal, characteristic

#20272792

C₂

yes

maximal

C₂×C₅₃×C₄⋊C₁₆

C₂

yes

maximal, characteristic

C₂²×#11719908

C₂²

yes

—

C₂×#13133128

C₄

yes

characteristic

C₂×#13133127

C₄

yes

—

C₂×C₈×Dic₅₃

C₂²

yes

—

#16810469

C₄

yes

characteristic

C₄⋊#11719908

—

C₂²×C₁₆×C₅₃

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×C₄⋊C₁₆

—

C₂²×#88322

C₂×C₄

yes

—

C₂×C₄×Dic₅₃

C₂×C₄

yes

—

—

—

—

—

C₈×Dic₅₃

—

C₈×Dic₅₃

—

#13133124

C₈

yes

—

#13133124

C₈

yes

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

—

C₄×C₈×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

Q₈

yes

—

C₂²×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×Dic₅₃

C₂×C₈

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₃

#41

yes

—

C₂×C₈×C₅₃

#41

yes

—

C₄²×C₅₃

C₄⋊C₄

yes

—

C₄²×C₅₃

M₁₆

yes

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

M₁₆

yes

—

C₁₆×C₅₃

—

#11719908

—

C₂³×C₅₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄⋊C₈

yes

—

C₂×C₄×C₅₃

C₄⋊C₈

yes

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

#117

yes

—

C₂×C₄×C₅₃

C₂×M₁₆

yes

—

C₂×C₄×C₅₃

C₂×M₁₆

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

C₈×C₅₃

yes

C₈×C₅₃

yes

C₈×C₅₃

yes

—

#88322

—

#12880

—

maximal, Sylow

C₂²×C₅₃

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₅₃

#431

yes

—

C₂²×C₅₃

#504

yes

—

C₂²×C₅₃

#504

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

Dic₅₃

—

C₂×C₄⋊C₁₆

—

C₂×C₄⋊C₁₆

D₅₃

yes

characteristic

#1726

—

#1374

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₅₃

yes

—

C₂×#97

—

C₂×#97

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

#376

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

106

—

C₅₃

#12880

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₄×D₅₃

yes

characteristic

C₂²×C₈

C₂²×D₅₃

yes

characteristic

C₂×C₄²

—

100

C₂×C₄²

C₄×D₅₃

yes

characteristic

101

C₂×C₁₆

—

102

C₂×C₁₆

—

103

C₂×C₁₆

—

106

—

104

C₂×C₁₆

—

106

—

105

#97

—

106

—

106

#97

—

106

—

107

C₄×C₈

—

106

—

108

C₄×C₈

—

106

—

109

C₄×C₈

Dic₁₀₆

yes

—

110

C₄×C₈

C₂²⋊D₅₃

yes

—

111

#95

—

112

#95

—

113

C₂²×C₄

—

114

C₂²×C₄

C₈×D₅₃

yes

characteristic

115

C₂²×C₄

C₈×D₅₃

yes

characteristic

116

C₂²×C₄

C₂×C₄×D₅₃

yes

characteristic

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

106

—

120

C₂×C₈

—

106

—

121

C₂×C₈

#11719919

yes

—

122

C₂×C₈

C₂×Dic₁₀₆

yes

characteristic

123

C₂×C₈

#11719947

yes

—

124

C₂×C₈

#11719949

yes

—

125

C₂×C₈

C₂×C₂²⋊D₅₃

yes

characteristic

126

C₄²

—

106

—

127

C₄²

—

106

—

128

C₄²

#11719912

yes

—

129

C₄²

#11719919

yes

—

130

C₁₆

—

131

C₁₆

—

212

—

132

C₂³

C₂×C₈×D₅₃

yes

characteristic

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

106

—

136

C₂×C₄

—

106

—

137

C₂×C₄

—

106

—

138

C₂×C₄

#13133146

yes

—

139

C₂×C₄

#13133146

yes

—

140

C₂×C₄

C₂×#11719919

yes

characteristic

141

C₂×C₄

#13133175

yes

—

142

C₂×C₄

M₁₆×D₅₃

yes

—

143

C₂×C₄

C₂×#11719912

yes

characteristic

144

C₈

—

145

C₈

—

106

—

146

C₈

#13133204

yes

characteristic

147

C₈

#13133206

yes

characteristic

148

C₈

#13133208

yes

—

149

C₂²

C₂×#13133146

yes

characteristic

150

C₂²

#16810675

yes

—

151

C₂²

#16811039

yes

—

152

C₂²

#16811041

yes

—

153

C₄

—

154

C₄

—

155

C₄

—

212

—

156

C₄

#16810488

yes

characteristic

157

C₄

#16810646

yes

characteristic

158

C₄

#16810679

yes

—

159

C₂

#20274831

yes

characteristic

160

C₂

#20276473

yes

characteristic

161

C₂

#20276475

yes

characteristic

162

C₂

#20276521

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂²×D₅₃
commutator, socle, prime radical	C₂³×C₅₃	C₂²×C₈
Fitting	C₂×C₅₃×C₄⋊C₁₆	C₂
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×C₄⋊C₁₆	D₅₃
hypocenter	C₅₃	#12880
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

subgroup

G/H

center, Frattini

C₂²×C₈

C₂²×D₅₃

commutator, socle, prime radical

C₂³×C₅₃

C₂²×C₈

Fitting

C₂×C₅₃×C₄⋊C₁₆

C₂

cosocle

C₂²×C₈×C₅₃

C₂³

hypercenter

C₂×C₄⋊C₁₆

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₅₃

C₁ ◁ C₂²×C₈ ◁ C₂×C₄⋊C₁₆

N	H ≅ G/N	type
95. C₅₃	#12880	semidirect

H ≅ G/N

type

95. C₅₃

#12880

semidirect

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

subgroup	index
C₂×C₄⋊#11719908	2
#20274401	2
#20272792	2
C₂×C₅₃×C₄⋊C₁₆	2
#12880	53

subgroup

index

C₂×C₅₃×C₄⋊C₁₆

Subgroups — #27300441

Conjugacy classes of subgroups

Complements

Maximal subgroups