Group #27384478 — subgroups

Subgroups — #27384478

Order 13568.

1118 conjugacy classes of subgroups; only the 387 normal subgroups are shown.

normal subgroups only

This group has 1118 conjugacy classes of subgroups — too many to draw. Only its 387 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×#16810645	C₂	1	6	yes	maximal
2	#376×D₅₃	C₂	1	1	yes	maximal, characteristic
3	#20274869	C₂	1	1	yes	maximal, characteristic, complemented
4	#20274868	C₂	1	6	yes	maximal, complemented
5	C₅₃×#1843	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#11719912	C₂²	1	4	yes	—
7	C₂×C₄²×D₅₃	C₂²	1	3	yes	—
8	#16810645	—	2	24	no	—
9	C₂×C₄⋊#88322	C₂²	1	6	yes	—
10	C₂×#13133146	C₂²	1	12	yes	—
11	#16810469	C₂²	1	3	yes	—
12	#16810468	C₄	1	4	yes	—
13	#95×D₅₃	C₄	1	4	yes	—
14	C₂×C₄⋊C₈×C₅₃	C₂²	1	6	yes	—
15	C₅₃×#376	C₂²	1	1	yes	characteristic
16	C₂²×C₄×D₅₃	C₂×C₄	1	6	yes	—
17	C₂²×C₄×D₅₃	C₂³	1	1	yes	characteristic
18	C₂×#11719912	—	2	24	no	—
19	C₂²×#88322	C₂³	1	4	yes	—
20	C₂×C₄×Dic₅₃	C₂×C₄	1	6	yes	—
21	C₂×C₄×Dic₅₃	C₂³	1	3	yes	—
22	C₄²×D₅₃	D₄	1	12	yes	—
23	C₄²×D₅₃	Q₈	1	12	yes	—
24	C₄⋊#88322	—	2	12	no	—
25	#13133146	—	2	24	no	—
26	#13133124	C₂×C₄	1	12	yes	—
27	C₂²×C₈×C₅₃	C₂³	1	4	yes	—
28	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
29	C₄⋊C₈×C₅₃	—	2	12	no	—
30	#95×C₅₃	C₂×C₄	1	4	yes	—
31	C₂³×D₅₃	C₂²×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	C₂²×C₄	1	6	yes	—
33	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
34	C₂²×Dic₅₃	C₂²×C₄	1	6	yes	—
35	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×D₅₃	—	4	12	no	—
37	C₂×C₄×D₅₃	C₄⋊C₄	1	24	yes	—
38	C₂×C₄×D₅₃	C₂×D₄	1	6	yes	—
39	C₂×C₄×D₅₃	C₂×Q₈	1	6	yes	—
40	C₂×C₈×C₅₃	—	2	12	no	—
41	C₄²×C₅₃	C₂×D₄	1	6	yes	—
42	C₄²×C₅₃	C₂×Q₈	1	6	yes	—
43	C₄×Dic₅₃	C₄⋊C₄	1	24	yes	—
44	C₄×Dic₅₃	C₂×D₄	1	6	yes	—
45	C₄×Dic₅₃	C₂×Q₈	1	6	yes	—
46	C₂×#88322	—	2	12	no	—
47	#11719912	—	4	16	no	—
48	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
49	C₂²×D₅₃	—	2	4	no	—
50	C₂²×D₅₃	C₂×C₄⋊C₄	1	6	yes	—
51	C₂×C₄×C₅₃	—	2	12	no	—
52	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	12	yes	—
53	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
54	C₂×C₄×C₅₃	C₂²×Q₈	1	3	yes	—
55	C₂×Dic₅₃	—	2	12	no	—
56	C₂×Dic₅₃	C₂×C₄⋊C₄	1	12	yes	—
57	C₂×Dic₅₃	C₂×C₄⋊C₄	1	6	yes	—
58	C₄×D₅₃	—	4	24	no	—
59	C₄×D₅₃	#127	1	4	yes	—
60	C₄×D₅₃	C₄⋊Q₈	1	12	yes	—
61	C₈×C₅₃	—	4	4	no	—
62	#88322	—	4	4	no	—
63	#20792	—	53	1	no	maximal, Sylow
64	C₂²×C₅₃	C₂²×C₄⋊C₄	1	3	yes	—
65	C₂²×C₅₃	#568	1	4	yes	—
66	C₂×D₅₃	—	2	12	no	—
67	C₂×D₅₃	#384	1	4	yes	—
68	C₄×C₅₃	—	2	12	no	—
69	C₄×C₅₃	C₂×#127	1	1	yes	characteristic
70	C₄×C₅₃	C₂×C₄⋊Q₈	1	3	yes	—
71	Dic₅₃	—	2	12	no	—
72	Dic₅₃	#384	1	4	yes	—
73	C₂×#423	—	53	6	no	—
74	C₂×#376	—	53	1	no	—
75	#1843	—	53	6	no	—
76	#1843	—	53	1	no	—
77	#1843	D₅₃	1	1	yes	characteristic, complemented
78	C₂×C₅₃	C₂×#384	1	1	yes	characteristic
79	C₂×C₅₃	#2905	1	6	yes	—
80	D₅₃	—	2	4	no	—
81	C₂²×M₁₆	—	53	4	no	—
82	C₂²×C₄²	—	53	3	no	—
83	#423	—	106	24	no	—
84	C₂×C₄⋊C₈	—	53	12	no	—
85	C₂×C₄⋊C₈	—	53	6	no	—
86	C₂×C₄⋊C₈	C₂×D₅₃	1	6	yes	—
87	#376	—	53	3	no	—
88	#376	—	53	4	no	—
89	#376	C₂×D₅₃	1	1	yes	characteristic
90	C₂×#95	—	53	4	no	—
91	C₅₃	#20792	1	1	yes	characteristic, complemented, Sylow
92	C₂³×C₄	—	53	1	no	—
93	C₂³×C₄	—	53	6	no	—
94	C₂×M₁₆	—	106	24	no	—
95	C₂²×C₈	—	53	4	no	—
96	C₂²×C₈	C₂²×D₅₃	1	4	yes	—
97	C₂×C₄²	—	53	6	no	—
98	C₂×C₄²	—	53	3	no	—
99	C₂×C₄²	—	53	12	no	—
100	C₂×C₄²	—	53	12	no	—
101	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
102	C₄⋊C₈	—	2	12	no	—
103	C₄⋊C₈	—	106	24	no	—
104	C₄⋊C₈	—	106	12	no	—
105	#95	—	53	12	no	—
106	#95	C₄×D₅₃	1	4	yes	—
107	C₂⁴	—	53	1	no	—
108	C₂²×C₄	—	53	1	no	—
109	C₂²×C₄	—	53	6	no	—
110	C₂²×C₄	—	53	6	no	—
111	C₂²×C₄	—	53	6	no	—
112	C₂²×C₄	—	53	24	no	—
113	C₂²×C₄	—	212	12	no	—
114	C₂²×C₄	C₂×C₄×D₅₃	1	6	yes	—
115	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
116	M₁₆	—	212	16	no	—
117	C₂×C₈	—	2	12	no	—
118	C₂×C₈	—	106	12	no	—
119	C₄²	—	53	24	no	—
120	C₄²	—	53	6	no	—
121	C₄²	—	53	6	no	—
122	C₄²	D₄×D₅₃	1	6	yes	—
123	C₄²	Q₈×D₅₃	1	6	yes	—
124	C₂³	—	53	6	no	—
125	C₂³	—	106	4	no	—
126	C₂³	C₂²×C₄×D₅₃	1	1	yes	characteristic
127	C₂×C₄	—	2	12	no	—
128	C₂×C₄	—	53	6	no	—
129	C₂×C₄	—	53	4	no	—
130	C₂×C₄	—	53	12	no	—
131	C₂×C₄	—	53	12	no	—
132	C₂×C₄	—	106	12	no	—
133	C₂×C₄	—	212	24	no	—
134	C₂×C₄	C₄⋊C₄×D₅₃	1	12	yes	—
135	C₂×C₄	C₂×D₄×D₅₃	1	3	yes	—
136	C₂×C₄	C₂×Q₈×D₅₃	1	3	yes	—
137	C₈	—	4	4	no	—
138	C₈	—	212	4	no	—
139	C₂²	—	53	4	no	—
140	C₂²	—	106	12	no	—
141	C₂²	C₂×C₄⋊C₄×D₅₃	1	3	yes	—
142	C₂²	#16811388	1	4	yes	—
143	C₄	—	2	12	no	—
144	C₄	—	53	4	no	—
145	C₄	—	106	12	no	—
146	C₄	#127×D₅₃	1	1	yes	characteristic
147	C₄	C₄⋊Q₈×D₅₃	1	3	yes	—
148	C₂	—	106	4	no	—
149	C₂	#384×D₅₃	1	1	yes	characteristic
150	C₂	#20283276	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16810645

C₂

yes

maximal

#376×D₅₃

C₂

yes

maximal, characteristic

#20274869

C₂

yes

maximal, characteristic, complemented

#20274868

C₂

yes

maximal, complemented

C₅₃×#1843

C₂

yes

maximal, characteristic, complemented

C₂²×#11719912

C₂²

yes

—

C₂×C₄²×D₅₃

C₂²

yes

—

#16810645

—

C₂×C₄⋊#88322

C₂²

yes

—

C₂×#13133146

C₂²

yes

—

#16810469

C₂²

yes

—

#16810468

C₄

yes

—

#95×D₅₃

C₄

yes

—

C₂×C₄⋊C₈×C₅₃

C₂²

yes

—

C₅₃×#376

C₂²

yes

characteristic

C₂²×C₄×D₅₃

C₂×C₄

yes

—

C₂²×C₄×D₅₃

C₂³

yes

characteristic

C₂×#11719912

—

C₂²×#88322

C₂³

yes

—

C₂×C₄×Dic₅₃

C₂×C₄

yes

—

C₂×C₄×Dic₅₃

C₂³

yes

—

C₄²×D₅₃

D₄

yes

—

C₄²×D₅₃

Q₈

yes

—

C₄⋊#88322

—

#13133146

—

#13133124

C₂×C₄

yes

—

C₂²×C₈×C₅₃

C₂³

yes

—

C₂×C₄²×C₅₃

C₂³

yes

—

C₄⋊C₈×C₅₃

—

#95×C₅₃

C₂×C₄

yes

—

C₂³×D₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂²×C₄

yes

—

C₂²×C₄×C₅₃

C₂⁴

yes

characteristic

C₂²×Dic₅₃

C₂²×C₄

yes

—

C₂²×Dic₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

C₄⋊C₄

yes

—

C₂×C₄×D₅₃

C₂×D₄

yes

—

C₂×C₄×D₅₃

C₂×Q₈

yes

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

C₂×Q₈

yes

—

C₄×Dic₅₃

C₄⋊C₄

yes

—

C₄×Dic₅₃

C₂×D₄

yes

—

C₄×Dic₅₃

C₂×Q₈

yes

—

C₂×#88322

—

#11719912

—

C₂³×C₅₃

C₂³×C₄

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²×D₄

yes

—

C₂×C₄×C₅₃

C₂²×Q₈

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×C₄⋊C₄

yes

—

C₂×Dic₅₃

C₂×C₄⋊C₄

yes

—

C₄×D₅₃

—

C₄×D₅₃

#127

yes

—

C₄×D₅₃

C₄⋊Q₈

yes

—

C₈×C₅₃

—

#88322

—

#20792

—

maximal, Sylow

C₂²×C₅₃

C₂²×C₄⋊C₄

yes

—

C₂²×C₅₃

#568

yes

—

C₂×D₅₃

—

C₂×D₅₃

#384

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×#127

yes

characteristic

C₄×C₅₃

C₂×C₄⋊Q₈

yes

—

Dic₅₃

—

Dic₅₃

#384

yes

—

—

—

—

—

D₅₃

yes

characteristic, complemented

C₂×C₅₃

C₂×#384

yes

characteristic

C₂×C₅₃

#2905

yes

—

D₅₃

—

C₂²×M₁₆

—

C₂²×C₄²

—

#423

—

106

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

C₂×D₅₃

yes

—

#376

—

#376

—

#376

C₂×D₅₃

yes

characteristic

C₂×#95

—

C₅₃

#20792

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂³×C₄

—

C₂×M₁₆

—

106

—

C₂²×C₈

—

C₂²×C₈

C₂²×D₅₃

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

100

C₂×C₄²

—

101

C₂×C₄²

C₂²×D₅₃

yes

—

102

C₄⋊C₈

—

103

C₄⋊C₈

—

106

—

104

C₄⋊C₈

—

106

—

105

#95

—

106

#95

C₄×D₅₃

yes

—

107

C₂⁴

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

212

—

114

C₂²×C₄

C₂×C₄×D₅₃

yes

—

115

C₂²×C₄

C₂³×D₅₃

yes

characteristic

116

M₁₆

—

212

—

117

C₂×C₈

—

118

C₂×C₈

—

106

—

119

C₄²

—

120

C₄²

—

121

C₄²

—

122

C₄²

D₄×D₅₃

yes

—

123

C₄²

Q₈×D₅₃

yes

—

124

C₂³

—

125

C₂³

—

106

—

126

C₂³

C₂²×C₄×D₅₃

yes

characteristic

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

106

—

133

C₂×C₄

—

212

—

134

C₂×C₄

C₄⋊C₄×D₅₃

yes

—

135

C₂×C₄

C₂×D₄×D₅₃

yes

—

136

C₂×C₄

C₂×Q₈×D₅₃

yes

—

137

C₈

—

138

C₈

—

212

—

139

C₂²

—

140

C₂²

—

106

—

141

C₂²

C₂×C₄⋊C₄×D₅₃

yes

—

142

C₂²

#16811388

yes

—

143

C₄

—

144

C₄

—

145

C₄

—

106

—

146

C₄

#127×D₅₃

yes

characteristic

147

C₄

C₄⋊Q₈×D₅₃

yes

—

148

C₂

—

106

—

149

C₂

#384×D₅₃

yes

characteristic

150

C₂

#20283276

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator, socle	C₂³×C₅₃	C₂³×C₄
Fitting	C₅₃×#1843	C₂
prime radical	C₂³×D₅₃	C₂²×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	#1843	D₅₃
hypocenter	C₅₃	#20792
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ #1843

subgroup

G/H

center, Frattini

C₂²×C₄

C₂³×D₅₃

commutator, socle

C₂³×C₅₃

C₂³×C₄

Fitting

C₅₃×#1843

C₂

prime radical

C₂³×D₅₃

C₂²×C₄

cosocle

C₂²×C₄×C₅₃

C₂⁴

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₅₃

C₁ ◁ C₂²×C₄ ◁ #1843

N	H ≅ G/N	type
3. #20274869	C₂	semidirect
4. #20274868	C₂	semidirect
5. C₅₃×#1843	C₂	semidirect
77. #1843	D₅₃	semidirect
91. C₅₃	#20792	semidirect

H ≅ G/N

type

3. #20274869

C₂

semidirect

4. #20274868

C₂

semidirect

5. C₅₃×#1843

C₂

semidirect

77. #1843

D₅₃

semidirect

91. C₅₃

#20792

semidirect

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

subgroup	index
C₂×#16810645	2
#376×D₅₃	2
#20274869	2
#20274868	2
C₅₃×#1843	2
#20792	53

subgroup

index

#376×D₅₃

C₅₃×#1843

Subgroups — #27384478

Conjugacy classes of subgroups

Complements

Maximal subgroups