Group #28307758 — subgroups

Subgroups — #28307758

Order 13568.

330 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 176 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	C₄×#13133292	C₂	1	1	yes	maximal, characteristic
2	C₄×D₅₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
3	#20289627	C₂	1	2	yes	maximal, complemented
4	C₈×#11719910	C₂	1	2	yes	maximal, complemented
5	#20273000	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133292	C₄	1	2	yes	—
7	C₂×D₅₃⋊C₁₆	C₄	1	2	yes	—
8	#16811512	C₄	1	2	yes	—
9	#16811509	C₄	1	2	yes	—
10	C₄×#11719910	C₂²	1	1	yes	characteristic
11	C₄×#11719910	C₂²	1	2	yes	—
12	C₄×#11719910	C₂²	1	1	yes	characteristic
13	#16810674	C₄	1	2	yes	—
14	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
15	#16810344	C₄	1	1	yes	characteristic
16	C₈×#88322	C₂²	1	2	yes	—
17	C₅₃×#322	C₄	1	1	yes	characteristic
18	#13133292	C₈	1	8	yes	complemented
19	D₅₃⋊C₁₆	C₈	1	8	yes	complemented
20	C₂×#11719910	C₂×C₄	1	2	yes	—
21	C₂×#11719910	C₂×C₄	1	2	yes	—
22	C₂×#11719910	C₂×C₄	1	4	yes	—
23	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
25	C₄²×D₅₃	C₂×C₄	1	1	yes	characteristic
26	#13133146	C₈	1	4	yes	—
27	#13133129	C₈	1	4	yes	—
28	C₄×#88322	C₂×C₄	1	2	yes	—
29	C₄×#88322	C₂³	1	1	yes	characteristic
30	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
31	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
34	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
35	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
36	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
37	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
38	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
39	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
40	C₄×Dic₅₃	C₄²	1	1	yes	characteristic
41	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
42	C₂×#88322	C₂×C₈	1	4	yes	—
43	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
44	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
45	C₈×D₅₃	C₂×C₈	1	4	yes	—
46	C₈×D₅₃	C₂×C₈	1	4	yes	—
47	#11719910	C₂×C₈	1	4	yes	—
48	#11719910	C₂×C₈	1	4	yes	—
49	#11719910	C₂×C₈	1	8	yes	—
50	C₂²×D₅₃	C₄×C₈	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
53	C₂×Dic₅₃	C₄×C₈	1	2	yes	—
54	C₂×Dic₅₃	C₄×C₈	1	1	yes	characteristic
55	C₄×D₅₃	—	2	4	no	—
56	C₄×D₅₃	C₄×C₈	1	2	yes	—
57	C₄×D₅₃	C₄×C₈	1	2	yes	—
58	C₈×C₅₃	—	2	4	no	—
59	C₈×C₅₃	C₄×C₈	1	2	yes	—
60	C₈×C₅₃	C₄×C₈	1	2	yes	—
61	#88322	—	2	4	no	—
62	#88322	C₂²×C₈	1	2	yes	—
63	#88322	C₂²×C₈	1	2	yes	—
64	#11399	—	53	1	no	maximal, Sylow
65	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
66	C₂×D₅₃	—	2	2	no	—
67	C₂×D₅₃	C₈²	1	2	yes	—
68	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
69	C₄×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
70	C₄×C₅₃	#504	1	4	yes	—
71	Dic₅₃	—	2	2	no	—
72	Dic₅₃	C₈²	1	2	yes	—
73	C₄×M₃₂	—	53	1	no	—
74	C₂×C₄×C₁₆	—	53	1	no	—
75	#1450	—	53	1	no	—
76	#1311	—	53	2	no	—
77	C₈×C₁₆	—	53	2	no	—
78	C₂×C₅₃	C₂×C₈²	1	1	yes	characteristic
79	C₂×C₅₃	#2108	1	2	yes	—
80	D₅₃	—	2	2	no	—
81	C₂×M₃₂	—	53	2	no	—
82	C₂²×C₁₆	—	53	2	no	—
83	#431	—	53	2	no	—
84	C₂×C₄×C₈	—	53	1	no	—
85	#346	—	53	2	no	—
86	C₄×C₁₆	—	53	1	no	—
87	C₄×C₁₆	—	53	2	no	—
88	C₄×C₁₆	—	53	1	no	—
89	C₄×C₁₆	—	53	2	no	—
90	#322	—	53	1	no	—
91	#322	#6157	1	1	yes	characteristic
92	C₈²	—	53	2	no	—
93	C₅₃	#11399	1	1	yes	characteristic, complemented, Sylow
94	C₂²×C₈	—	53	1	no	—
95	C₂²×C₈	—	53	1	no	—
96	C₂×C₄²	—	53	1	no	—
97	M₃₂	—	53	8	no	—
98	C₂×C₁₆	—	53	2	no	—
99	C₂×C₁₆	—	53	2	no	—
100	C₂×C₁₆	—	53	4	no	—
101	C₂×C₁₆	—	53	8	no	—
102	C₄⋊C₈	—	53	4	no	—
103	C₂²⋊C₈	—	53	4	no	—
104	C₄×C₈	—	53	1	no	—
105	C₄×C₈	—	53	1	no	—
106	C₄×C₈	—	53	1	no	—
107	C₄×C₈	—	53	2	no	—
108	C₄×C₈	C₂×#6157	1	1	yes	characteristic
109	C₄×C₈	C₂×#6157	1	2	yes	—
110	C₂²×C₄	—	53	1	no	—
111	C₂²×C₄	—	53	2	no	—
112	C₂×C₈	—	53	1	no	—
113	C₂×C₈	—	53	1	no	—
114	C₂×C₈	—	53	4	no	—
115	C₂×C₈	—	53	4	no	—
116	C₂×C₈	—	53	4	no	—
117	C₂×C₈	D₅₃⋊C₈	1	4	yes	—
118	C₂×C₈	C₄×#6157	1	1	yes	characteristic
119	C₂×C₈	C₄×#6157	1	1	yes	characteristic
120	C₄²	—	53	1	no	—
121	C₄²	—	53	2	no	—
122	C₄²	C₂²×#6157	1	1	yes	characteristic
123	C₁₆	—	53	4	no	—
124	C₁₆	—	53	4	no	—
125	C₁₆	—	53	8	no	—
126	C₂³	—	53	1	no	—
127	C₂×C₄	—	53	1	no	—
128	C₂×C₄	—	53	2	no	—
129	C₂×C₄	—	53	2	no	—
130	C₂×C₄	—	53	2	no	—
131	C₂×C₄	—	106	4	no	—
132	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
133	C₂×C₄	C₂×D₅₃⋊C₈	1	2	yes	—
134	C₈	—	2	4	no	—
135	C₈	—	53	2	no	—
136	C₈	—	53	2	no	—
137	C₈	—	106	4	no	—
138	C₈	C₈×#6157	1	2	yes	—
139	C₈	C₈×#6157	1	2	yes	—
140	C₂²	—	53	2	no	—
141	C₂²	—	106	2	no	—
142	C₂²	C₄×D₅₃⋊C₈	1	1	yes	characteristic
143	C₄	—	53	2	no	—
144	C₄	—	106	2	no	—
145	C₄	C₂×C₈×#6157	1	1	yes	characteristic
146	C₄	C₂×C₈×#6157	1	1	yes	characteristic
147	C₄	#16811682	1	4	yes	—
148	C₂	—	106	2	no	—
149	C₂	C₈×D₅₃⋊C₈	1	1	yes	characteristic
150	C₂	#20290925	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×#13133292

C₂

yes

maximal, characteristic

C₄×D₅₃⋊C₁₆

C₂

yes

maximal, characteristic

#20289627

C₂

yes

maximal, complemented

C₈×#11719910

C₂

yes

maximal, complemented

#20273000

C₂

yes

maximal, characteristic

C₂×#13133292

C₄

yes

—

C₂×D₅₃⋊C₁₆

C₄

yes

—

#16811512

C₄

yes

—

#16811509

C₄

yes

—

C₄×#11719910

C₂²

yes

characteristic

C₄×#11719910

C₂²

yes

—

C₄×#11719910

C₂²

yes

characteristic

#16810674

C₄

yes

—

C₄×C₈×D₅₃

C₂²

yes

characteristic

#16810344

C₄

yes

characteristic

C₈×#88322

C₂²

yes

—

C₅₃×#322

C₄

yes

characteristic

#13133292

C₈

yes

complemented

D₅₃⋊C₁₆

C₈

yes

complemented

C₂×#11719910

C₂×C₄

yes

—

C₂×#11719910

C₂×C₄

yes

—

C₂×#11719910

C₂×C₄

yes

—

C₂×C₈×D₅₃

C₂×C₄

yes

characteristic

C₂×C₈×D₅₃

C₂×C₄

yes

characteristic

C₄²×D₅₃

C₂×C₄

yes

characteristic

#13133146

C₈

yes

—

#13133129

C₈

yes

—

C₄×#88322

C₂×C₄

yes

—

C₄×#88322

C₂³

yes

characteristic

C₈×Dic₅₃

C₂×C₄

yes

characteristic

C₈×Dic₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₄×D₅₃

C₄²

yes

characteristic

C₂×C₄×D₅₃

C₂×C₈

yes

—

C₂×C₈×C₅₃

C₄²

yes

characteristic

C₂×C₈×C₅₃

C₄²

yes

characteristic

C₂×C₈×C₅₃

C₂×C₈

yes

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄×Dic₅₃

C₄²

yes

characteristic

C₄×Dic₅₃

C₂×C₈

yes

—

C₂×#88322

C₂×C₈

yes

—

C₂×#88322

C₂²×C₄

yes

characteristic

C₂×#88322

C₂²×C₄

yes

characteristic

C₈×D₅₃

C₂×C₈

yes

—

C₈×D₅₃

C₂×C₈

yes

—

#11719910

C₂×C₈

yes

—

#11719910

C₂×C₈

yes

—

#11719910

C₂×C₈

yes

—

C₂²×D₅₃

C₄×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₈

yes

—

C₂×Dic₅₃

C₄×C₈

yes

—

C₂×Dic₅₃

C₄×C₈

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

C₄×C₈

yes

—

C₄×D₅₃

C₄×C₈

yes

—

C₈×C₅₃

—

C₈×C₅₃

C₄×C₈

yes

—

C₈×C₅₃

C₄×C₈

yes

—

#88322

—

#88322

C₂²×C₈

yes

—

#88322

C₂²×C₈

yes

—

#11399

—

maximal, Sylow

C₂²×C₅₃

C₂×C₄×C₈

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

C₈²

yes

—

C₄×C₅₃

C₂×C₄×C₈

yes

characteristic

C₄×C₅₃

C₂×C₄×C₈

yes

characteristic

C₄×C₅₃

#504

yes

—

Dic₅₃

—

Dic₅₃

C₈²

yes

—

C₄×M₃₂

—

C₂×C₄×C₁₆

—

#1450

—

#1311

—

C₈×C₁₆

—

C₂×C₅₃

C₂×C₈²

yes

characteristic

C₂×C₅₃

#2108

yes

—

D₅₃

—

C₂×M₃₂

—

C₂²×C₁₆

—

#431

—

C₂×C₄×C₈

—

#346

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

—

—

yes

C₈²

—

C₅₃

#11399

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄²

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

100

C₂×C₁₆

—

101

C₂×C₁₆

—

102

C₄⋊C₈

—

103

C₂²⋊C₈

—

104

C₄×C₈

—

105

C₄×C₈

—

106

C₄×C₈

—

107

C₄×C₈

—

108

C₄×C₈

C₂×#6157

yes

characteristic

109

C₄×C₈

C₂×#6157

yes

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

D₅₃⋊C₈

yes

—

118

C₂×C₈

C₄×#6157

yes

characteristic

119

C₂×C₈

C₄×#6157

yes

characteristic

120

C₄²

—

121

C₄²

—

122

C₄²

C₂²×#6157

yes

characteristic

123

C₁₆

—

124

C₁₆

—

125

C₁₆

—

126

C₂³

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

106

—

132

C₂×C₄

C₂×C₄×#6157

yes

characteristic

133

C₂×C₄

C₂×D₅₃⋊C₈

yes

—

134

C₈

—

135

C₈

—

136

C₈

—

137

C₈

—

106

—

138

C₈

C₈×#6157

yes

—

139

C₈

C₈×#6157

yes

—

140

C₂²

—

141

C₂²

—

106

—

142

C₂²

C₄×D₅₃⋊C₈

yes

characteristic

143

C₄

—

144

C₄

—

106

—

145

C₄

C₂×C₈×#6157

yes

characteristic

146

C₄

C₂×C₈×#6157

yes

characteristic

147

C₄

#16811682

yes

—

148

C₂

—

106

—

149

C₂

C₈×D₅₃⋊C₈

yes

characteristic

150

C₂

#20290925

yes

—

	subgroup	G/H
center, Frattini	C₄²	C₂²×#6157
commutator	C₂×C₅₃	C₂×C₈²
Fitting	C₅₃×#322	C₄
socle	C₂²×C₅₃	C₂×C₄×C₈
prime radical	C₂²×D₅₃	C₄×C₈
cosocle	C₄×#88322	C₂³
hypercenter	#322	#6157
hypocenter	C₅₃	#11399
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ #322

subgroup

G/H

center, Frattini

C₄²

C₂²×#6157

commutator

C₂×C₅₃

C₂×C₈²

Fitting

C₅₃×#322

C₄

socle

C₂²×C₅₃

C₂×C₄×C₈

prime radical

C₂²×D₅₃

C₄×C₈

C₂³

C₅₃

G ▷ C₂×C₅₃ ▷ C₅₃

C₁ ◁ C₄² ◁ #322

N	H ≅ G/N	type
3. #20289627	C₂	semidirect
4. C₈×#11719910	C₂	semidirect
18. #13133292	C₈	semidirect
19. D₅₃⋊C₁₆	C₈	semidirect
93. C₅₃	#11399	semidirect

H ≅ G/N

type

3. #20289627

C₂

semidirect

4. C₈×#11719910

C₂

semidirect

18. #13133292

C₈

semidirect

19. D₅₃⋊C₁₆

C₈

semidirect

93. C₅₃

#11399

semidirect

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

subgroup	index
C₄×#13133292	2
C₄×D₅₃⋊C₁₆	2
#20289627	2
C₈×#11719910	2
#20273000	2
#11399	53

subgroup

index

C₄×D₅₃⋊C₁₆

Subgroups — #28307758

Conjugacy classes of subgroups

Complements

Maximal subgroups