Group #27298791 — subgroups

Subgroups — #27298791

Order 13568.

268 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 268 conjugacy classes of subgroups — too many to draw. Only its 83 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	#20276943	C₂	1	2	yes	maximal, complemented
2	#20276939	C₂	1	2	yes	maximal
3	#20274333	C₂	1	1	yes	maximal, characteristic, complemented
4	#20274321	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1311	C₂	1	1	yes	maximal, characteristic, complemented
6	#16811028	—	2	1	no	—
7	#16811028	C₂²	1	2	yes	—
8	#16811003	—	2	1	no	—
9	#16811003	C₂²	1	1	yes	characteristic
10	#16811000	C₂²	1	1	yes	characteristic
11	#16810970	—	2	1	no	—
12	#16810970	—	2	1	no	—
13	#16810428	—	2	4	no	—
14	#16810425	—	2	4	no	—
15	C₄×C₁₆×C₅₃	C₂²	1	2	yes	—
16	C₈²×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₁₂	—	2	2	no	—
18	C₂×Dic₂₁₂	—	4	2	no	—
19	C₂×D₄₂₄	—	2	2	no	—
20	#13133217	—	2	2	no	—
21	#13133217	—	2	1	no	—
22	#13133214	—	2	1	no	—
23	#13133151	—	2	2	no	—
24	#13133151	—	2	2	no	—
25	#13133151	—	4	2	no	—
26	#13133139	—	4	2	no	—
27	#13133139	—	4	2	no	—
28	C₂×C₁₆×C₅₃	D₄	1	4	yes	—
29	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
30	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
31	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
32	C₂×D₂₁₂	—	4	1	no	—
33	C₂×D₂₁₂	—	4	1	no	—
34	C₂×D₂₁₂	—	4	1	no	—
35	C₂×Dic₁₀₆	—	4	1	no	—
36	C₂×Dic₁₀₆	—	4	2	no	—
37	C₂×C₈×C₅₃	D₈	1	2	yes	—
38	C₂×C₈×C₅₃	D₈	1	2	yes	—
39	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
40	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
41	C₄⋊Dic₅₃	—	4	2	no	—
42	C₄⋊Dic₅₃	—	4	1	no	—
43	C₄⋊Dic₅₃	—	4	1	no	—
44	C₄⋊Dic₅₃	—	4	2	no	—
45	Dic₂₁₂	—	4	4	no	—
46	Dic₂₁₂	—	8	4	no	—
47	D₄₂₄	—	4	4	no	—
48	C₁₆×C₅₃	—	2	4	no	—
49	C₂²×D₅₃	—	8	1	no	—
50	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
51	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
52	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
53	C₂×Dic₅₃	—	8	1	no	—
54	C₂×Dic₅₃	—	8	2	no	—
55	D₂₁₂	—	8	2	no	—
56	D₂₁₂	—	8	2	no	—
57	D₂₁₂	—	8	2	no	—
58	Dic₁₀₆	—	8	2	no	—
59	Dic₁₀₆	—	8	4	no	—
60	C₈×C₅₃	—	2	2	no	—
61	C₈×C₅₃	—	2	2	no	—
62	C₈×C₅₃	#135	1	4	yes	—
63	#12512	—	53	1	no	maximal, Sylow
64	C₂²×C₅₃	#493	1	1	yes	characteristic
65	C₂×D₅₃	—	16	1	no	—
66	C₂×D₅₃	—	16	1	no	—
67	C₂×D₅₃	—	16	1	no	—
68	C₄×C₅₃	#495	1	2	yes	—
69	C₄×C₅₃	#509	1	2	yes	—
70	C₄×C₅₃	#510	1	2	yes	—
71	Dic₅₃	—	16	1	no	—
72	Dic₅₃	—	16	2	no	—
73	#2241	—	53	2	no	—
74	#2240	—	53	2	no	—
75	#1713	—	53	1	no	—
76	#1709	—	53	1	no	—
77	#1311	D₅₃	1	1	yes	characteristic, complemented
78	C₂×C₅₃	#1709	1	1	yes	characteristic
79	C₂×C₅₃	#2250	1	1	yes	characteristic
80	C₂×C₅₃	#2251	1	1	yes	characteristic
81	D₅₃	—	32	1	no	—
82	#500	—	53	2	no	—
83	#500	—	106	1	no	—
84	#494	—	53	1	no	—
85	#494	—	106	1	no	—
86	#493	—	53	1	no	—
87	#486	—	106	1	no	—
88	#486	—	106	1	no	—
89	#358	—	106	4	no	—
90	#357	—	106	4	no	—
91	C₄×C₁₆	C₂×D₅₃	1	2	yes	—
92	C₈²	C₂×D₅₃	1	1	yes	characteristic
93	C₅₃	#12512	1	1	yes	characteristic, complemented, Sylow
94	C₂×Q₁₆	—	106	2	no	—
95	C₂×Q₁₆	—	212	2	no	—
96	C₂×D₈	—	106	2	no	—
97	C₄⋊Q₈	—	106	1	no	—
98	C₄⋊Q₈	—	106	2	no	—
99	#127	—	106	1	no	—
100	C₂×C₁₆	D₂₁₂	1	4	yes	—
101	#107	—	106	2	no	—
102	#107	—	106	2	no	—
103	#107	—	212	2	no	—
104	#102	—	212	2	no	—
105	#102	—	212	2	no	—
106	C₄×C₈	D₂₁₂	1	1	yes	characteristic
107	C₄×C₈	D₂₁₂	1	1	yes	characteristic
108	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
109	C₂×Q₈	—	212	1	no	—
110	C₂×Q₈	—	212	2	no	—
111	C₂×D₄	—	212	1	no	—
112	C₂×D₄	—	212	1	no	—
113	C₂×D₄	—	212	1	no	—
114	Q₁₆	—	424	4	no	—
115	Q₁₆	—	212	4	no	—
116	D₈	—	212	4	no	—
117	C₂×C₈	D₄₂₄	1	2	yes	—
118	C₂×C₈	D₄₂₄	1	2	yes	—
119	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
120	C₄⋊C₄	—	212	2	no	—
121	C₄⋊C₄	—	212	1	no	—
122	C₄⋊C₄	—	212	1	no	—
123	C₄⋊C₄	—	212	2	no	—
124	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
125	C₁₆	—	2	4	no	—
126	C₂³	—	424	1	no	—
127	Q₈	—	424	4	no	—
128	Q₈	—	424	2	no	—
129	D₄	—	424	2	no	—
130	D₄	—	424	2	no	—
131	D₄	—	424	2	no	—
132	C₂×C₄	—	424	1	no	—
133	C₂×C₄	—	424	2	no	—
134	C₂×C₄	#13133214	1	1	yes	characteristic
135	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
136	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
137	C₈	—	2	2	no	—
138	C₈	—	2	2	no	—
139	C₈	#13133239	1	4	yes	—
140	C₂²	—	848	1	no	—
141	C₂²	—	848	1	no	—
142	C₂²	—	848	1	no	—
143	C₂²	#16811000	1	1	yes	characteristic
144	C₄	—	848	1	no	—
145	C₄	—	848	2	no	—
146	C₄	#16811007	1	2	yes	—
147	C₄	#16811061	1	2	yes	—
148	C₄	#16811066	1	2	yes	—
149	C₂	—	1696	1	no	—
150	C₂	#20274321	1	1	yes	characteristic
151	C₂	#20276981	1	1	yes	characteristic
152	C₂	#20276986	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20276943

C₂

yes

maximal, complemented

#20276939

C₂

yes

maximal

#20274333

C₂

yes

maximal, characteristic, complemented

#20274321

C₂

yes

maximal, characteristic

C₅₃×#1311

C₂

yes

maximal, characteristic, complemented

#16811028

—

#16811028

C₂²

yes

—

#16811003

—

#16811003

C₂²

yes

characteristic

#16811000

C₂²

yes

—

—

—

—

C₄×C₁₆×C₅₃

C₂²

yes

—

C₈²×C₅₃

C₂²

yes

characteristic

C₂×Dic₂₁₂

—

C₂×Dic₂₁₂

—

C₂×D₄₂₄

—

—

—

—

—

—

—

—

—

C₂×C₁₆×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

D₄

yes

characteristic

C₄×C₈×C₅₃

D₄

yes

characteristic

C₄×C₈×C₅₃

C₂³

yes

characteristic

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

D₈

yes

—

C₂×C₈×C₅₃

D₈

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

Dic₂₁₂

—

Dic₂₁₂

—

D₄₂₄

—

C₁₆×C₅₃

—

C₂²×D₅₃

—

C₂×C₄×C₅₃

#127

yes

characteristic

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

#135

yes

—

—

C₂²×C₅₃

yes

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

#495

yes

—

C₄×C₅₃

#509

yes

—

C₄×C₅₃

#510

yes

—

Dic₅₃

—

Dic₅₃

—

—

—

—

—

D₅₃

yes

characteristic, complemented

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

#500

—

#500

—

106

—

#494

—

#494

—

106

—

#493

—

#486

—

106

—

#486

—

106

—

#358

—

106

—

#357

—

106

—

C₄×C₁₆

C₂×D₅₃

yes

—

C₈²

C₂×D₅₃

yes

characteristic

C₅₃

#12512

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

106

—

C₂×Q₁₆

—

212

—

C₂×D₈

—

106

—

C₄⋊Q₈

—

106

—

C₄⋊Q₈

—

106

—

#127

—

106

—

100

C₂×C₁₆

D₂₁₂

yes

—

101

#107

—

106

—

102

#107

—

106

—

103

#107

—

212

—

104

#102

—

212

—

105

#102

—

212

—

106

C₄×C₈

D₂₁₂

yes

characteristic

107

C₄×C₈

D₂₁₂

yes

characteristic

108

C₄×C₈

C₂²×D₅₃

yes

characteristic

109

C₂×Q₈

—

212

—

110

C₂×Q₈

—

212

—

111

C₂×D₄

—

212

—

112

C₂×D₄

—

212

—

113

C₂×D₄

—

212

—

114

Q₁₆

—

424

—

115

Q₁₆

—

212

—

116

D₈

—

212

—

117

C₂×C₈

D₄₂₄

yes

—

118

C₂×C₈

D₄₂₄

yes

—

119

C₂×C₈

C₂×D₂₁₂

yes

—

120

C₄⋊C₄

—

212

—

121

C₄⋊C₄

—

212

—

122

C₄⋊C₄

—

212

—

123

C₄⋊C₄

—

212

—

124

C₄²

C₂×D₂₁₂

yes

characteristic

125

C₁₆

—

126

C₂³

—

424

—

127

Q₈

—

424

—

128

Q₈

—

424

—

129

D₄

—

424

—

130

D₄

—

424

—

131

D₄

—

424

—

132

C₂×C₄

—

424

—

133

C₂×C₄

—

424

—

134

C₂×C₄

#13133214

yes

characteristic

135

C₂×C₄

C₂×D₄₂₄

yes

characteristic

136

C₂×C₄

C₂×D₄₂₄

yes

characteristic

137

C₈

—

138

C₈

—

139

C₈

#13133239

yes

—

140

C₂²

—

848

—

141

C₂²

—

848

—

142

C₂²

—

848

—

143

C₂²

#16811000

yes

characteristic

144

C₄

—

848

—

145

C₄

—

848

—

146

C₄

#16811007

yes

—

147

C₄

#16811061

yes

—

148

C₄

#16811066

yes

—

149

C₂

—

1696

—

150

C₂

#20274321

yes

characteristic

151

C₂

#20276981

yes

characteristic

152

C₂

#20276986

yes

characteristic

	subgroup	G/H
center	C₂²	#16811000
commutator, cosocle	C₄×C₈×C₅₃	C₂³
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#1311	C₂
socle	C₂²×C₅₃	#493
prime radical	#16811000	C₂²
hypercenter	#1311	D₅₃
hypocenter	C₅₃	#12512
LCS	G ▷ C₄×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

subgroup

G/H

center

C₂²

#16811000

commutator, cosocle

C₄×C₈×C₅₃

C₂³

Frattini

C₄×C₈

C₂²×D₅₃

Fitting

C₅₃×#1311

C₂

C₂²×C₅₃

C₂²

D₅₃

C₅₃

G ▷ C₄×C₈×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1311

N	H ≅ G/N	type
1. #20276943	C₂	semidirect
3. #20274333	C₂	semidirect
5. C₅₃×#1311	C₂	semidirect
77. #1311	D₅₃	semidirect
93. C₅₃	#12512	semidirect

H ≅ G/N

type

1. #20276943

C₂

semidirect

3. #20274333

C₂

semidirect

5. C₅₃×#1311

C₂

semidirect

77. #1311

D₅₃

semidirect

93. C₅₃

#12512

semidirect

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

subgroup	index
#20276943	2
#20276939	2
#20274333	2
#20274321	2
C₅₃×#1311	2
#12512	53

subgroup

index

C₅₃×#1311

Subgroups — #27298791

Conjugacy classes of subgroups

Complements

Maximal subgroups