Group #27304539 — subgroups

Subgroups — #27304539

Order 13568.

290 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 115 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×#16810457	C₂	1	1	yes	maximal, characteristic
2	#20276499	C₂	1	1	yes	maximal, characteristic
3	#20272896	C₂	1	4	yes	maximal, complemented
4	C₂²×Q₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
5	#16810661	—	2	1	no	—
6	#16810457	—	2	2	no	—
7	C₂×#13133123	C₂²	1	2	yes	—
8	#16810451	—	2	1	no	—
9	#16810449	C₂²	1	2	yes	—
10	C₂²×Q₁₆×C₅₃	—	2	1	no	—
11	C₂×Q₃₂×C₅₃	—	2	1	no	—
12	C₂×Q₃₂×C₅₃	—	2	2	no	—
13	C₂×Q₃₂×C₅₃	C₄	1	4	yes	complemented
14	C₂×Q₃₂×C₅₃	C₂²	1	2	yes	—
15	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
16	#13133176	—	4	1	no	—
17	#13133151	—	2	2	no	—
18	#13133149	—	4	1	no	—
19	#13133123	—	2	2	no	—
20	C₂²×Q₈×C₅₃	—	4	1	no	—
21	C₂×Q₁₆×C₅₃	—	2	4	no	—
22	C₂×Q₁₆×C₅₃	—	2	2	no	—
23	C₂×Q₁₆×C₅₃	—	4	1	no	—
24	C₂×Q₁₆×C₅₃	—	4	2	no	—
25	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
26	Q₃₂×C₅₃	—	2	4	no	—
27	Q₃₂×C₅₃	—	2	2	no	—
28	Q₃₂×C₅₃	C₂×C₄	1	4	yes	—
29	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
30	C₂×C₁₆×C₅₃	D₄	1	1	yes	characteristic
31	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
32	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
33	C₂×Q₈×C₅₃	—	4	4	no	—
34	C₂×Q₈×C₅₃	—	4	2	no	—
35	C₂×Q₈×C₅₃	—	8	1	no	—
36	C₂×Q₈×C₅₃	—	8	2	no	—
37	C₂²×C₄×C₅₃	—	8	1	no	—
38	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
39	Q₁₆×C₅₃	—	2	4	no	—
40	Q₁₆×C₅₃	—	4	4	no	—
41	Q₁₆×C₅₃	—	4	2	no	—
42	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
43	C₂×C₈×C₅₃	D₈	1	1	yes	characteristic
44	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
45	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂²⋊Dic₅₃	—	8	1	no	—
47	C₄⋊Dic₅₃	—	4	2	no	—
48	C₄×Dic₅₃	—	8	1	no	—
49	C₁₆×C₅₃	C₂²⋊C₄	1	2	yes	—
50	C₁₆×C₅₃	C₂²×C₄	1	1	yes	characteristic
51	C₁₆×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂³×C₅₃	D₁₆	1	1	yes	characteristic
53	Q₈×C₅₃	—	4	4	no	—
54	Q₈×C₅₃	—	8	4	no	—
55	Q₈×C₅₃	—	8	2	no	—
56	C₂×C₄×C₅₃	—	8	4	no	—
57	C₂×C₄×C₅₃	—	8	2	no	—
58	C₂×C₄×C₅₃	#102	1	2	yes	—
59	C₂×C₄×C₅₃	D₁₆	1	1	yes	characteristic
60	C₂×C₄×C₅₃	SD₃₂	1	2	yes	—
61	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
62	C₂×Dic₅₃	—	8	2	no	—
63	C₈×C₅₃	#102	1	2	yes	—
64	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₈×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
66	#13618	—	53	1	no	maximal, Sylow
67	C₂²×C₅₃	—	2	2	no	—
68	C₂²×C₅₃	#357	1	2	yes	—
69	C₂²×C₅₃	C₂×D₁₆	1	1	yes	characteristic
70	C₄×C₅₃	—	8	4	no	—
71	C₄×C₅₃	#357	1	2	yes	—
72	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
73	C₄×C₅₃	C₂×SD₃₂	1	1	yes	characteristic
74	Dic₅₃	—	16	2	no	—
75	C₂²×Q₃₂	D₅₃	1	1	yes	characteristic
76	C₂×M₆₄	—	53	1	no	—
77	#2158	—	53	1	no	—
78	#1416	—	53	4	no	—
79	C₂×C₅₃	—	2	2	no	—
80	C₂×C₅₃	C₂×#357	1	1	yes	characteristic
81	C₂×C₅₃	#2264	1	2	yes	—
82	C₂²×Q₁₆	—	2	1	no	—
83	C₂×Q₃₂	—	2	2	no	—
84	C₂×Q₃₂	—	2	1	no	—
85	C₂×Q₃₂	Dic₅₃	1	4	yes	complemented
86	C₂×Q₃₂	C₂×D₅₃	1	2	yes	—
87	C₂²×C₁₆	C₂×D₅₃	1	1	yes	characteristic
88	#427	—	106	1	no	—
89	M₆₄	—	106	2	no	—
90	C₂×C₃₂	—	53	2	no	—
91	#367	—	106	1	no	—
92	#366	—	53	2	no	—
93	C₅₃	#13618	1	1	yes	characteristic, complemented, Sylow
94	C₂²×Q₈	—	4	1	no	—
95	C₂×Q₁₆	—	2	2	no	—
96	C₂×Q₁₆	—	2	4	no	—
97	C₂×Q₁₆	—	4	1	no	—
98	C₂×Q₁₆	—	4	2	no	—
99	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
100	#117	—	212	1	no	—
101	Q₃₂	—	2	2	no	—
102	Q₃₂	—	2	4	no	—
103	Q₃₂	C₂×Dic₅₃	1	4	yes	—
104	C₂×C₁₆	C₂×Dic₅₃	1	2	yes	—
105	C₂×C₁₆	C₂²⋊D₅₃	1	2	yes	—
106	C₂×C₁₆	C₂²⋊D₅₃	1	1	yes	characteristic
107	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
108	#107	—	106	2	no	—
109	#106	—	212	1	no	—
110	C₃₂	—	106	2	no	—
111	C₂×Q₈	—	4	4	no	—
112	C₂×Q₈	—	4	2	no	—
113	C₂×Q₈	—	8	1	no	—
114	C₂×Q₈	—	8	2	no	—
115	C₂²×C₄	—	8	1	no	—
116	C₂²×C₄	#11719922	1	1	yes	characteristic
117	Q₁₆	—	2	4	no	—
118	Q₁₆	—	4	4	no	—
119	Q₁₆	—	4	2	no	—
120	C₂×C₈	#11719922	1	1	yes	characteristic
121	C₂×C₈	#11719923	1	2	yes	—
122	C₂×C₈	C₂²⋊Dic₅₃	1	2	yes	—
123	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
124	C₄⋊C₄	—	212	2	no	—
125	C₂²⋊C₄	—	424	1	no	—
126	C₄²	—	424	1	no	—
127	C₁₆	C₂²⋊Dic₅₃	1	2	yes	—
128	C₁₆	C₂²×Dic₅₃	1	1	yes	characteristic
129	C₁₆	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
130	C₂³	#13133158	1	1	yes	characteristic
131	Q₈	—	4	4	no	—
132	Q₈	—	8	2	no	—
133	Q₈	—	8	4	no	—
134	C₂×C₄	—	8	2	no	—
135	C₂×C₄	—	8	4	no	—
136	C₂×C₄	—	424	2	no	—
137	C₂×C₄	#13133138	1	2	yes	—
138	C₂×C₄	#13133158	1	1	yes	characteristic
139	C₂×C₄	#13133161	1	2	yes	—
140	C₂×C₄	C₂×#11719922	1	1	yes	characteristic
141	C₈	#13133138	1	2	yes	—
142	C₈	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
143	C₈	C₂×#11719923	1	1	yes	characteristic
144	C₂²	—	2	2	no	—
145	C₂²	#16810424	1	2	yes	—
146	C₂²	C₂×#13133158	1	1	yes	characteristic
147	C₄	—	8	4	no	—
148	C₄	—	848	2	no	—
149	C₄	#16810424	1	2	yes	—
150	C₄	C₂×#13133138	1	1	yes	characteristic
151	C₄	C₂×#13133161	1	1	yes	characteristic
152	C₂	—	2	2	no	—
153	C₂	C₂×#16810424	1	1	yes	characteristic
154	C₂	#20277033	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16810457

C₂

yes

maximal, characteristic

#20276499

C₂

yes

maximal, characteristic

#20272896

C₂

yes

maximal, complemented

C₂²×Q₃₂×C₅₃

C₂

yes

maximal, characteristic

#16810661

—

#16810457

—

C₂×#13133123

C₂²

yes

—

#16810451

—

#16810449

C₂²

yes

—

C₂²×Q₁₆×C₅₃

—

C₂×Q₃₂×C₅₃

—

C₂×Q₃₂×C₅₃

—

C₂×Q₃₂×C₅₃

C₄

yes

complemented

C₂×Q₃₂×C₅₃

C₂²

yes

—

C₂²×C₁₆×C₅₃

C₂²

yes

—

—

—

—

C₂²×Q₈×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₂²×C₈×C₅₃

D₄

yes

characteristic

Q₃₂×C₅₃

—

Q₃₂×C₅₃

—

Q₃₂×C₅₃

C₂×C₄

yes

—

C₂×C₁₆×C₅₃

C₂×C₄

yes

—

C₂×C₁₆×C₅₃

D₄

yes

characteristic

C₂×C₁₆×C₅₃

D₄

yes

—

C₂×C₁₆×C₅₃

C₂³

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₈

yes

characteristic

Q₁₆×C₅₃

—

Q₁₆×C₅₃

—

Q₁₆×C₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

D₈

yes

characteristic

C₂×C₈×C₅₃

SD₁₆

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

—

C₁₆×C₅₃

C₂²⋊C₄

yes

—

C₁₆×C₅₃

C₂²×C₄

yes

characteristic

C₁₆×C₅₃

C₂×D₄

yes

characteristic

C₂³×C₅₃

D₁₆

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#102

yes

—

C₂×C₄×C₅₃

D₁₆

yes

characteristic

C₂×C₄×C₅₃

SD₃₂

yes

—

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×Dic₅₃

—

C₈×C₅₃

#102

yes

—

C₈×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₅₃

C₂×SD₁₆

yes

characteristic

#13618

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

#357

yes

—

C₂²×C₅₃

C₂×D₁₆

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

#357

yes

—

C₄×C₅₃

C₂×#102

yes

characteristic

C₄×C₅₃

C₂×SD₃₂

yes

characteristic

Dic₅₃

—

C₂²×Q₃₂

D₅₃

yes

characteristic

C₂×M₆₄

—

#2158

—

#1416

—

C₂×C₅₃

—

C₂×C₅₃

C₂×#357

yes

characteristic

C₂×C₅₃

#2264

yes

—

C₂²×Q₁₆

—

C₂×Q₃₂

—

C₂×Q₃₂

—

C₂×Q₃₂

Dic₅₃

yes

complemented

C₂×Q₃₂

C₂×D₅₃

yes

—

C₂²×C₁₆

C₂×D₅₃

yes

characteristic

#427

—

106

—

M₆₄

—

106

—

C₂×C₃₂

—

#367

—

106

—

#366

—

C₅₃

#13618

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

C₂²⋊D₅₃

yes

characteristic

100

#117

—

212

—

101

Q₃₂

—

102

Q₃₂

—

103

Q₃₂

C₂×Dic₅₃

yes

—

104

C₂×C₁₆

C₂×Dic₅₃

yes

—

105

C₂×C₁₆

C₂²⋊D₅₃

yes

—

106

C₂×C₁₆

C₂²⋊D₅₃

yes

characteristic

107

C₂×C₁₆

C₂²×D₅₃

yes

characteristic

108

#107

—

106

—

109

#106

—

212

—

110

C₃₂

—

106

—

111

C₂×Q₈

—

112

C₂×Q₈

—

113

C₂×Q₈

—

114

C₂×Q₈

—

115

C₂²×C₄

—

116

C₂²×C₄

#11719922

yes

characteristic

117

Q₁₆

—

118

Q₁₆

—

119

Q₁₆

—

120

C₂×C₈

#11719922

yes

characteristic

121

C₂×C₈

#11719923

yes

—

122

C₂×C₈

C₂²⋊Dic₅₃

yes

—

123

C₂×C₈

C₂×C₂²⋊D₅₃

yes

characteristic

124

C₄⋊C₄

—

212

—

125

C₂²⋊C₄

—

424

—

126

C₄²

—

424

—

127

C₁₆

C₂²⋊Dic₅₃

yes

—

128

C₁₆

C₂²×Dic₅₃

yes

characteristic

129

C₁₆

C₂×C₂²⋊D₅₃

yes

characteristic

130

C₂³

#13133158

yes

characteristic

131

Q₈

—

132

Q₈

—

133

Q₈

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

424

—

137

C₂×C₄

#13133138

yes

—

138

C₂×C₄

#13133158

yes

characteristic

139

C₂×C₄

#13133161

yes

—

140

C₂×C₄

C₂×#11719922

yes

characteristic

141

C₈

#13133138

yes

—

142

C₈

C₂×C₂²⋊Dic₅₃

yes

characteristic

143

C₈

C₂×#11719923

yes

characteristic

144

C₂²

—

145

C₂²

#16810424

yes

—

146

C₂²

C₂×#13133158

yes

characteristic

147

C₄

—

148

C₄

—

848

—

149

C₄

#16810424

yes

—

150

C₄

C₂×#13133138

yes

characteristic

151

C₄

C₂×#13133161

yes

characteristic

152

C₂

—

153

C₂

C₂×#16810424

yes

characteristic

154

C₂

#20277033

yes

—

	subgroup	G/H
center	C₂²	C₂×#13133158
commutator	C₁₆×C₅₃	C₂²×C₄
Frattini	C₂×C₁₆	C₂²×D₅₃
Fitting	C₂²×Q₃₂×C₅₃	C₂
socle	C₂²×C₅₃	C₂×D₁₆
prime radical	C₂³×C₅₃	D₁₆
cosocle	C₂×C₁₆×C₅₃	C₂³
hypercenter	C₂²×Q₃₂	D₅₃
hypocenter	C₅₃	#13618
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂²×Q₃₂

subgroup

G/H

center

C₂²

C₂×#13133158

commutator

C₁₆×C₅₃

C₂²×C₄

Frattini

C₂×C₁₆

C₂²×D₅₃

Fitting

C₂²×Q₃₂×C₅₃

C₂

socle

C₂²×C₅₃

C₂×D₁₆

prime radical

C₂³×C₅₃

D₁₆

cosocle

C₂×C₁₆×C₅₃

C₂³

hypercenter

C₂²×Q₃₂

D₅₃

hypocenter

C₅₃

#13618

LCS

G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂²×Q₃₂

N	H ≅ G/N	type
3. #20272896	C₂	semidirect
13. C₂×Q₃₂×C₅₃	C₄	semidirect
85. C₂×Q₃₂	Dic₅₃	semidirect
93. C₅₃	#13618	semidirect

H ≅ G/N

type

3. #20272896

C₂

semidirect

13. C₂×Q₃₂×C₅₃

C₄

semidirect

85. C₂×Q₃₂

Dic₅₃

semidirect

93. C₅₃

#13618

semidirect

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

subgroup	index
C₂×#16810457	2
#20276499	2
#20272896	2
C₂²×Q₃₂×C₅₃	2
#13618	53

subgroup

index

C₂²×Q₃₂×C₅₃

Subgroups — #27304539

Conjugacy classes of subgroups

Complements

Maximal subgroups