Group #28298567 — subgroups

Subgroups — #28298567

Order 13568.

244 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 244 conjugacy classes of subgroups — too many to draw. Only its 58 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	#20290599	C₂	1	2	yes	maximal
2	#20276328	C₂	1	1	yes	maximal, characteristic
3	D₅₃⋊#108	—	2	2	no	—
4	#16811609	—	2	2	no	—
5	#16811492	C₄	1	4	yes	—
6	C₂×#13133223	C₂²	1	1	yes	characteristic
7	C₂×#13133157	C₄	1	1	yes	characteristic
8	#16810402	—	2	2	no	—
9	C₂²×C₁₆×C₅₃	C₄	1	1	yes	characteristic
10	D₅₃⋊M₁₆	—	2	4	no	—
11	#13133288	—	4	2	no	—
12	#13133287	—	4	2	no	—
13	#13133278	—	4	2	no	—
14	#13133277	—	4	2	no	—
15	C₂×#11719945	D₄	1	1	yes	characteristic
16	#13133223	—	4	1	no	—
17	#13133223	—	4	1	no	—
18	C₂×#11719917	Q₈	1	1	yes	characteristic
19	C₂×#11719912	C₂×C₄	1	2	yes	—
20	C₂×C₈×D₅₃	D₄	1	2	yes	—
21	#13133157	—	4	1	no	—
22	C₂×#11719908	—	2	1	no	—
23	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂×C₁₆×C₅₃	—	2	1	no	—
25	C₂×C₁₆×C₅₃	—	4	1	no	—
26	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
27	C₂×C₂²⋊D₅₃	SD₁₆	1	2	yes	—
28	#11719945	—	4	1	no	—
29	#11719945	—	4	1	no	—
30	C₂×D₂₁₂	Q₁₆	1	1	yes	characteristic
31	C₂×C₄×D₅₃	C₂²⋊C₄	1	2	yes	—
32	C₂×Dic₁₀₆	D₈	1	1	yes	characteristic
33	#11719940	—	4	4	no	—
34	#11719936	—	4	4	no	—
35	D₅₃⋊C₈	—	4	4	no	—
36	C₂×C₈×C₅₃	—	2	1	no	—
37	C₂×C₈×C₅₃	—	2	1	no	—
38	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
39	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
40	#11719917	—	4	1	no	—
41	C₂×#88322	C₄⋊C₄	1	2	yes	—
42	#11719912	—	4	2	no	—
43	C₈×D₅₃	—	2	2	no	—
44	C₈×D₅₃	—	2	2	no	—
45	C₁₆×C₅₃	—	4	1	no	—
46	#11719908	—	4	1	no	—
47	C₂³×C₅₃	#106	1	1	yes	characteristic
48	C₂²×D₅₃	#103	1	2	yes	—
49	C₂×C₄×C₅₃	—	2	1	no	—
50	C₂×C₄×C₅₃	—	2	1	no	—
51	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
52	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
53	C₂²⋊D₅₃	—	4	2	no	—
54	C₂×Dic₅₃	#102	1	2	yes	—
55	D₂₁₂	—	4	1	no	—
56	C₄×D₅₃	—	2	2	no	—
57	C₄×D₅₃	—	2	2	no	—
58	Dic₁₀₆	—	4	1	no	—
59	#88324	—	8	4	no	—
60	C₈×C₅₃	—	2	1	no	—
61	C₈×C₅₃	#99	1	1	yes	characteristic
62	C₈×C₅₃	#99	1	1	yes	characteristic
63	#88322	—	2	2	no	—
64	#7229	—	53	1	no	maximal, Sylow
65	C₂²×C₅₃	—	2	1	no	—
66	C₂²×C₅₃	—	4	1	no	—
67	C₂²×C₅₃	#340	1	1	yes	characteristic
68	C₂×D₅₃	—	2	2	no	—
69	C₂×D₅₃	—	4	2	no	—
70	C₄×C₅₃	—	2	1	no	—
71	C₄×C₅₃	H(3,ℤ/4)	1	1	yes	characteristic
72	C₄×C₅₃	#342	1	1	yes	characteristic
73	Dic₅₃	—	2	2	no	—
74	#2113	—	53	1	no	—
75	#1813	—	53	2	no	—
76	C₂×C₅₃	—	2	1	no	—
77	C₂×C₅₃	—	4	1	no	—
78	C₂×C₅₃	#1289	1	1	yes	characteristic
79	D₅₃	—	4	2	no	—
80	C₂×#131	—	53	1	no	—
81	C₂×M₃₂	—	53	1	no	—
82	C₂²×C₁₆	#6157	1	1	yes	characteristic
83	C₂×#108	—	106	2	no	—
84	#413	—	106	2	no	—
85	C₂²⋊C₁₆	—	106	2	no	—
86	#338	—	53	4	no	—
87	C₅₃	#7229	1	1	yes	characteristic, complemented, Sylow
88	C₂×#41	—	53	1	no	—
89	#131	—	212	1	no	—
90	#131	—	212	1	no	—
91	C₂×M₁₆	—	53	1	no	—
92	C₂×M₁₆	—	53	2	no	—
93	C₂×M₁₆	—	106	4	no	—
94	C₂²×C₈	—	53	2	no	—
95	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
96	M₃₂	—	212	1	no	—
97	C₂×C₁₆	—	2	1	no	—
98	C₂×C₁₆	—	4	1	no	—
99	C₂×C₁₆	—	106	1	no	—
100	#108	—	212	2	no	—
101	#108	—	212	2	no	—
102	#101	—	212	2	no	—
103	#100	—	212	2	no	—
104	#41	—	212	1	no	—
105	#41	—	212	1	no	—
106	C₂×Q₈	—	53	1	no	—
107	C₂×D₄	—	53	2	no	—
108	C₂×D₄	—	53	1	no	—
109	C₂²×C₄	—	53	2	no	—
110	C₂²×C₄	C₄⋊#6157	1	1	yes	characteristic
111	M₁₆	—	212	2	no	—
112	M₁₆	—	212	1	no	—
113	M₁₆	—	212	4	no	—
114	M₁₆	—	212	4	no	—
115	C₂×C₈	—	2	1	no	—
116	C₂×C₈	—	2	1	no	—
117	C₂×C₈	—	53	2	no	—
118	C₂×C₈	—	106	2	no	—
119	C₂×C₈	—	106	2	no	—
120	C₂×C₈	—	212	4	no	—
121	C₂×C₈	C₄×#6157	1	1	yes	characteristic
122	C₂×C₈	C₂²⋊#6157	1	1	yes	characteristic
123	C₁₆	—	4	1	no	—
124	C₁₆	—	212	1	no	—
125	C₂³	—	53	2	no	—
126	C₂³	#13133285	1	1	yes	characteristic
127	Q₈	—	212	1	no	—
128	D₄	—	212	1	no	—
129	D₄	—	212	2	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	1	no	—
132	C₂×C₄	—	53	2	no	—
133	C₂×C₄	—	106	2	no	—
134	C₂×C₄	—	106	2	no	—
135	C₂×C₄	(C₂×#6157)⋊C₄	1	1	yes	characteristic
136	C₂×C₄	#13133286	1	1	yes	characteristic
137	C₈	—	2	1	no	—
138	C₈	—	106	2	no	—
139	C₈	—	424	4	no	—
140	C₈	#13133275	1	1	yes	characteristic
141	C₈	#13133276	1	1	yes	characteristic
142	C₂²	—	2	1	no	—
143	C₂²	—	4	1	no	—
144	C₂²	—	106	2	no	—
145	C₂²	—	212	2	no	—
146	C₂²	#16811497	1	1	yes	characteristic
147	C₄	—	2	1	no	—
148	C₄	—	106	2	no	—
149	C₄	#16811491	1	1	yes	characteristic
150	C₄	#16811501	1	1	yes	characteristic
151	C₂	—	2	1	no	—
152	C₂	—	4	1	no	—
153	C₂	—	212	2	no	—
154	C₂	#20289578	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20290599

C₂

yes

maximal

#20276328

C₂

yes

maximal, characteristic

D₅₃⋊#108

—

#16811609

—

#16811492

C₄

yes

—

C₂×#13133223

C₂²

yes

characteristic

C₂×#13133157

C₄

yes

characteristic

#16810402

—

C₂²×C₁₆×C₅₃

C₄

yes

characteristic

D₅₃⋊M₁₆

—

—

—

—

—

D₄

yes

—

—

Q₈

yes

characteristic

C₂×#11719912

C₂×C₄

yes

—

C₂×C₈×D₅₃

D₄

yes

—

#13133157

—

C₂×#11719908

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

—

C₂²×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₂²⋊D₅₃

SD₁₆

yes

—

#11719945

—

#11719945

—

C₂×D₂₁₂

Q₁₆

yes

characteristic

C₂×C₄×D₅₃

C₂²⋊C₄

yes

—

C₂×Dic₁₀₆

D₈

yes

characteristic

#11719940

—

#11719936

—

D₅₃⋊C₈

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄²

yes

characteristic

C₂×C₈×C₅₃

C₂²⋊C₄

yes

characteristic

#11719917

—

C₂×#88322

C₄⋊C₄

yes

—

#11719912

—

C₈×D₅₃

—

C₈×D₅₃

—

C₁₆×C₅₃

—

#11719908

—

C₂³×C₅₃

#106

yes

characteristic

C₂²×D₅₃

#103

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂²⋊D₅₃

—

C₂×Dic₅₃

#102

yes

—

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

Dic₁₀₆

—

#88324

—

C₈×C₅₃

—

C₈×C₅₃

yes

C₈×C₅₃

yes

—

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#340

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

#2113

—

#1813

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#1289

yes

characteristic

D₅₃

—

C₂×#131

—

C₂×M₃₂

—

C₂²×C₁₆

#6157

yes

characteristic

C₂×#108

—

106

—

#413

—

106

—

C₂²⋊C₁₆

—

106

—

#338

—

C₅₃

#7229

yes

characteristic, complemented, Sylow

C₂×#41

—

#131

—

212

—

#131

—

212

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

106

—

C₂²×C₈

—

C₂²×C₈

C₂×#6157

yes

characteristic

M₃₂

—

212

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

106

—

100

#108

—

212

—

101

#108

—

212

—

102

#101

—

212

—

103

#100

—

212

—

104

#41

—

212

—

105

#41

—

212

—

106

C₂×Q₈

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂²×C₄

—

110

C₂²×C₄

C₄⋊#6157

yes

characteristic

111

M₁₆

—

212

—

112

M₁₆

—

212

—

113

M₁₆

—

212

—

114

M₁₆

—

212

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

106

—

119

C₂×C₈

—

106

—

120

C₂×C₈

—

212

—

121

C₂×C₈

C₄×#6157

yes

characteristic

122

C₂×C₈

C₂²⋊#6157

yes

characteristic

123

C₁₆

—

124

C₁₆

—

212

—

125

C₂³

—

126

C₂³

#13133285

yes

characteristic

127

Q₈

—

212

—

128

D₄

—

212

—

129

D₄

—

212

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

106

—

134

C₂×C₄

—

106

—

135

C₂×C₄

(C₂×#6157)⋊C₄

yes

characteristic

136

C₂×C₄

#13133286

yes

characteristic

137

C₈

—

138

C₈

—

106

—

139

C₈

—

424

—

140

C₈

#13133275

yes

characteristic

141

C₈

#13133276

yes

characteristic

142

C₂²

—

143

C₂²

—

144

C₂²

—

106

—

145

C₂²

—

212

—

146

C₂²

#16811497

yes

characteristic

147

C₄

—

148

C₄

—

106

—

149

C₄

#16811491

yes

characteristic

150

C₄

#16811501

yes

characteristic

151

C₂

—

152

C₂

—

153

C₂

—

212

—

154

C₂

#20289578

yes

characteristic

	subgroup	G/H
center	C₄	#16811491
commutator	C₂×C₈×C₅₃	C₄²
Frattini	C₂²×C₈	C₂×#6157
Fitting	C₂²×C₁₆×C₅₃	C₄
socle	C₂×C₅₃	#1289
prime radical	C₂×#11719945	D₄
cosocle	C₂×#13133223	C₂²
hypercenter	C₂²×C₁₆	#6157
hypocenter	C₅₃	#7229
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂²×C₁₆

subgroup

G/H

center

C₄

#16811491

commutator

C₂×C₈×C₅₃

C₄²

Frattini

C₂²×C₈

C₂×#6157

Fitting

C₂²×C₁₆×C₅₃

C₄

C₂×C₅₃

D₄

C₂²

C₂²×C₁₆

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂²×C₁₆

N	H ≅ G/N	type
87. C₅₃	#7229	semidirect

H ≅ G/N

type

87. C₅₃

#7229

semidirect

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

subgroup	index
#20290599	2
#20276328	2
#7229	53

subgroup

index

#20290599

#20276328

#7229

Subgroups — #28298567

Conjugacy classes of subgroups

Complements

Maximal subgroups