Group #27300508 — subgroups

Subgroups — #27300508

Order 13568.

382 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 167 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×#16810443	C₂	1	4	yes	maximal
2	#20276471	C₂	1	1	yes	maximal, characteristic
3	C₄×C₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#2150	C₂	1	1	yes	maximal, characteristic
5	C₂×#13133157	C₂²	1	2	yes	—
6	C₂×C₄×#88322	C₄	1	1	yes	characteristic
7	C₂×C₈×Dic₅₃	C₄	1	2	yes	—
8	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
9	C₄²×Dic₅₃	C₄	1	1	yes	characteristic
10	#16810443	—	2	8	no	—
11	C₄⋊#11719908	—	2	2	no	—
12	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
13	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×C₄⋊C₁₆	—	2	2	no	—
15	C₂²×#88322	C₂×C₄	1	2	yes	—
16	C₂×C₄×Dic₅₃	—	2	2	no	—
17	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
18	#13133157	—	2	4	no	—
19	C₂×#11719908	—	2	2	no	—
20	C₄×#88322	C₈	1	2	yes	—
21	C₄×#88322	C₈	1	2	yes	—
22	C₈×Dic₅₃	—	2	4	no	—
23	C₈×Dic₅₃	D₄	1	4	yes	—
24	C₈×Dic₅₃	Q₈	1	4	yes	—
25	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
27	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
28	M₃₂×C₅₃	—	2	4	no	—
29	C₂×C₁₆×C₅₃	—	2	2	no	—
30	C₄×C₈×C₅₃	D₄	1	2	yes	—
31	C₄×C₈×C₅₃	Q₈	1	2	yes	—
32	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
33	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
34	C₂²×Dic₅₃	—	2	2	no	—
35	C₂×C₈×C₅₃	—	2	2	no	—
36	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
37	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
38	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
41	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
42	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
43	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
44	C₄×Dic₅₃	—	2	8	no	—
45	C₄×Dic₅₃	—	2	2	no	—
46	C₄×Dic₅₃	C₄⋊C₄	1	4	yes	—
47	C₂×#88322	C₄⋊C₄	1	4	yes	—
48	C₂×#88322	C₂×C₈	1	4	yes	—
49	C₂×#88322	M₁₆	1	4	yes	—
50	C₁₆×C₅₃	—	4	2	no	—
51	#11719908	—	4	2	no	—
52	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
53	C₂×C₄×C₅₃	—	2	4	no	—
54	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
55	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
56	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
57	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
59	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
60	C₂×Dic₅₃	—	2	8	no	—
61	C₂×Dic₅₃	—	2	4	no	—
62	C₈×C₅₃	—	2	2	no	—
63	C₈×C₅₃	#127	1	1	yes	characteristic
64	C₈×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
65	C₈×C₅₃	C₄⋊Q₈	1	2	yes	—
66	#88322	C₄⋊C₈	1	8	yes	—
67	#12900	—	53	1	no	maximal, Sylow
68	C₂²×C₅₃	—	2	2	no	—
69	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
70	C₂²×C₅₃	#423	1	2	yes	—
71	C₄×C₅₃	—	2	2	no	—
72	C₄×C₅₃	#364	1	4	yes	complemented
73	C₄×C₅₃	#384	1	1	yes	characteristic
74	C₄×C₅₃	C₂×C₄⋊C₈	1	2	yes	—
75	C₄×C₅₃	#423	1	1	yes	characteristic
76	Dic₅₃	—	2	8	no	—
77	C₂×#364	—	53	4	no	—
78	#2150	—	53	1	no	—
79	#2150	D₅₃	1	1	yes	characteristic
80	C₄²×C₈	—	53	1	no	—
81	C₂×C₅₃	—	2	2	no	—
82	C₂×C₅₃	#1842	1	1	yes	characteristic
83	C₂×C₅₃	C₂×#364	1	2	yes	—
84	C₂×M₃₂	—	53	2	no	—
85	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
86	C₂×C₄×C₈	—	53	2	no	—
87	C₂×C₄×C₈	—	53	1	no	—
88	C₂×C₄×C₈	—	53	2	no	—
89	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
90	C₄³	—	53	1	no	—
91	#364	—	106	8	no	—
92	C₄⋊C₁₆	—	2	2	no	—
93	C₄⋊C₁₆	—	106	2	no	—
94	C₅₃	#12900	1	1	yes	characteristic, complemented, Sylow
95	C₂²×C₈	—	53	2	no	—
96	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
97	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
98	C₂×C₄²	—	53	2	no	—
99	C₂×C₄²	—	106	2	no	—
100	C₂×C₄²	C₄×D₅₃	1	1	yes	characteristic
101	M₃₂	—	2	4	no	—
102	M₃₂	—	106	4	no	—
103	C₂×C₁₆	—	2	2	no	—
104	C₂×C₁₆	—	106	2	no	—
105	C₄×C₈	—	53	4	no	—
106	C₄×C₈	—	53	4	no	—
107	C₄×C₈	—	53	2	no	—
108	C₄×C₈	—	53	2	no	—
109	C₄×C₈	—	106	4	no	—
110	C₄×C₈	Dic₁₀₆	1	2	yes	—
111	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
112	C₂²×C₄	—	106	2	no	—
113	C₂²×C₄	#11719912	1	2	yes	—
114	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	53	4	no	—
117	C₂×C₈	—	53	4	no	—
118	C₂×C₈	—	53	4	no	—
119	C₂×C₈	#11719919	1	2	yes	—
120	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
121	C₂×C₈	D₄×D₅₃	1	2	yes	—
122	C₂×C₈	Q₈×D₅₃	1	2	yes	—
123	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
124	C₄²	—	53	4	no	—
125	C₄²	—	106	8	no	—
126	C₄²	—	106	2	no	—
127	C₄²	C₈×D₅₃	1	1	yes	characteristic
128	C₄²	C₈×D₅₃	1	1	yes	characteristic
129	C₄²	#11719919	1	2	yes	—
130	C₁₆	—	4	2	no	—
131	C₁₆	—	212	2	no	—
132	C₂³	C₂×#11719912	1	1	yes	characteristic
133	C₂×C₄	—	2	4	no	—
134	C₂×C₄	—	106	8	no	—
135	C₂×C₄	—	106	4	no	—
136	C₂×C₄	#13133146	1	2	yes	—
137	C₂×C₄	#13133146	1	2	yes	—
138	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
139	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
140	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
141	C₂×C₄	M₁₆×D₅₃	1	2	yes	—
142	C₈	—	2	2	no	—
143	C₈	—	53	8	no	—
144	C₈	#13133213	1	1	yes	characteristic
145	C₈	#13133215	1	1	yes	characteristic
146	C₈	#13133216	1	2	yes	—
147	C₂²	—	2	2	no	—
148	C₂²	C₂×#13133146	1	1	yes	characteristic
149	C₂²	#16810645	1	2	yes	—
150	C₄	—	2	2	no	—
151	C₄	—	106	8	no	—
152	C₄	#16810443	1	4	yes	complemented
153	C₄	#16810492	1	1	yes	characteristic
154	C₄	C₄⋊C₈×D₅₃	1	2	yes	—
155	C₄	#16810646	1	1	yes	characteristic
156	C₂	—	2	2	no	—
157	C₂	#20274866	1	1	yes	characteristic
158	C₂	C₂×#16810443	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16810443

C₂

yes

maximal

#20276471

C₂

yes

maximal, characteristic

C₄×C₈×Dic₅₃

C₂

yes

maximal, characteristic

C₅₃×#2150

C₂

yes

maximal, characteristic

C₂×#13133157

C₂²

yes

—

C₂×C₄×#88322

C₄

yes

characteristic

C₂×C₈×Dic₅₃

C₄

yes

—

C₂×C₈×Dic₅₃

C₂²

yes

—

C₄²×Dic₅₃

C₄

yes

characteristic

#16810443

—

C₄⋊#11719908

—

C₂×M₃₂×C₅₃

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×C₄⋊C₁₆

—

C₂²×#88322

C₂×C₄

yes

—

C₂×C₄×Dic₅₃

—

C₂×C₄×Dic₅₃

C₂×C₄

yes

—

#13133157

—

C₂×#11719908

—

C₄×#88322

C₈

yes

—

C₄×#88322

C₈

yes

—

C₈×Dic₅₃

—

C₈×Dic₅₃

D₄

yes

—

C₈×Dic₅₃

Q₈

yes

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

M₃₂×C₅₃

—

C₂×C₁₆×C₅₃

—

C₄×C₈×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

Q₈

yes

—

C₂²×C₄×C₅₃

M₁₆

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×Dic₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₃

C₂×Q₈

yes

—

C₄²×C₅₃

C₄⋊C₄

yes

—

C₄²×C₅₃

C₂×C₈

yes

characteristic

C₄²×C₅₃

C₂×C₈

yes

characteristic

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

C₄⋊C₄

yes

—

C₂×#88322

C₄⋊C₄

yes

—

C₂×#88322

C₂×C₈

yes

—

C₂×#88322

M₁₆

yes

—

C₁₆×C₅₃

—

#11719908

—

C₂³×C₅₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄⋊C₈

yes

—

C₂×C₄×C₅₃

C₄⋊C₈

yes

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂×M₁₆

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₈×C₅₃

—

C₈×C₅₃

#127

yes

characteristic

C₈×C₅₃

C₄⋊Q₈

yes

characteristic

C₈×C₅₃

C₄⋊Q₈

yes

—

#88322

C₄⋊C₈

yes

—

#12900

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₅₃

#423

yes

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

C₂×C₄⋊C₈

yes

—

C₄×C₅₃

#423

yes

characteristic

Dic₅₃

—

C₂×#364

—

#2150

—

#2150

D₅₃

yes

characteristic

C₄²×C₈

—

C₂×C₅₃

—

C₂×C₅₃

#1842

yes

characteristic

C₂×C₅₃

C₂×#364

yes

—

C₂×M₃₂

—

C₂×M₃₂

C₂×D₅₃

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

C₄³

—

#364

—

106

—

C₄⋊C₁₆

—

C₄⋊C₁₆

—

106

—

C₅₃

#12900

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₄×D₅₃

yes

characteristic

C₂²×C₈

C₂²×D₅₃

yes

characteristic

C₂×C₄²

—

C₂×C₄²

—

106

—

100

C₂×C₄²

C₄×D₅₃

yes

characteristic

101

M₃₂

—

102

M₃₂

—

106

—

103

C₂×C₁₆

—

104

C₂×C₁₆

—

106

—

105

C₄×C₈

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₄×C₈

—

109

C₄×C₈

—

106

—

110

C₄×C₈

Dic₁₀₆

yes

—

111

C₄×C₈

C₂²⋊D₅₃

yes

—

112

C₂²×C₄

—

106

—

113

C₂²×C₄

#11719912

yes

—

114

C₂²×C₄

C₂×C₄×D₅₃

yes

characteristic

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

#11719919

yes

—

120

C₂×C₈

C₂×Dic₁₀₆

yes

characteristic

121

C₂×C₈

D₄×D₅₃

yes

—

122

C₂×C₈

Q₈×D₅₃

yes

—

123

C₂×C₈

C₂×C₂²⋊D₅₃

yes

characteristic

124

C₄²

—

125

C₄²

—

106

—

126

C₄²

—

106

—

127

C₄²

C₈×D₅₃

yes

characteristic

128

C₄²

C₈×D₅₃

yes

characteristic

129

C₄²

#11719919

yes

—

130

C₁₆

—

131

C₁₆

—

212

—

132

C₂³

C₂×#11719912

yes

characteristic

133

C₂×C₄

—

134

C₂×C₄

—

106

—

135

C₂×C₄

—

106

—

136

C₂×C₄

#13133146

yes

—

137

C₂×C₄

#13133146

yes

—

138

C₂×C₄

C₄⋊C₄×D₅₃

yes

—

139

C₂×C₄

C₂×#11719919

yes

characteristic

140

C₂×C₄

C₂×C₈×D₅₃

yes

characteristic

141

C₂×C₄

M₁₆×D₅₃

yes

—

142

C₈

—

143

C₈

—

144

C₈

#13133213

yes

characteristic

145

C₈

#13133215

yes

characteristic

146

C₈

#13133216

yes

—

147

C₂²

—

148

C₂²

C₂×#13133146

yes

characteristic

149

C₂²

#16810645

yes

—

150

C₄

—

151

C₄

—

106

—

152

C₄

#16810443

yes

complemented

153

C₄

#16810492

yes

characteristic

154

C₄

C₄⋊C₈×D₅₃

yes

—

155

C₄

#16810646

yes

characteristic

156

C₂

—

157

C₂

#20274866

yes

characteristic

158

C₂

C₂×#16810443

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#2150	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₈
prime radical	C₂³×C₅₃	C₂×M₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#2150	D₅₃
hypocenter	C₅₃	#12900
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

subgroup

G/H

center

C₂×C₈

C₂×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₅₃×#2150

C₂

socle

C₂²×C₅₃

C₂×C₄⋊C₈

prime radical

C₂³×C₅₃

C₂×M₁₆

cosocle

C₂²×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #2150

N	H ≅ G/N	type
72. C₄×C₅₃	#364	semidirect
94. C₅₃	#12900	semidirect
152. C₄	#16810443	semidirect

H ≅ G/N

type

72. C₄×C₅₃

#364

semidirect

94. C₅₃

#12900

semidirect

152. C₄

#16810443

semidirect

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

subgroup	index
C₂×#16810443	2
#20276471	2
C₄×C₈×Dic₅₃	2
C₅₃×#2150	2
#12900	53

subgroup

index

C₂×#16810443

#20276471

C₄×C₈×Dic₅₃

C₅₃×#2150

#12900

Subgroups — #27300508

Conjugacy classes of subgroups

Complements

Maximal subgroups