Group #27217720 — subgroups

Subgroups — C₅₃×#10208 #27217720

Order 13568.

508 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 508 conjugacy classes of subgroups — too many to draw. Only its 94 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	C₅₃×#3214	C₂	1	3	yes	maximal, complemented
2	C₅₃×#1514	C₂	1	3	yes	maximal, complemented
3	C₅₃×#1308	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#532	C₂²	1	3	yes	—
5	C₅₃×#496	—	2	3	no	—
6	C₅₃×#489	—	2	3	no	—
7	C₅₃×#461	—	2	3	no	—
8	C₅₃×#459	—	2	3	no	—
9	C₅₃×#449	—	2	6	no	—
10	C₅₃×#432	C₂²	1	3	yes	—
11	C₅₃×#383	C₂²	1	1	yes	characteristic
12	C₅₃×#327	—	2	6	no	—
13	C₂×#41×C₅₃	—	2	3	no	—
14	C₂×SD₁₆×C₅₃	—	2	6	no	—
15	C₂×D₈×C₅₃	—	2	6	no	—
16	C₄⋊Q₈×C₅₃	D₄	1	3	yes	—
17	#127×C₅₃	D₄	1	3	yes	—
18	#124×C₅₃	—	2	3	no	—
19	#121×C₅₃	—	2	3	no	—
20	#121×C₅₃	—	2	3	no	—
21	C₄×D₄×C₅₃	—	2	3	no	—
22	C₄×D₄×C₅₃	—	2	3	no	—
23	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
24	C₄⋊C₈×C₅₃	—	2	3	no	—
25	#103×C₅₃	—	2	6	no	—
26	#102×C₅₃	—	2	6	no	—
27	C₂²⋊C₈×C₅₃	D₄	1	6	yes	—
28	#97×C₅₃	—	2	3	no	—
29	#95×C₅₃	—	2	3	no	—
30	#41×C₅₃	—	8	6	no	—
31	C₂×Q₈×C₅₃	—	2	3	no	—
32	C₂×D₄×C₅₃	—	2	3	no	—
33	C₂×D₄×C₅₃	—	2	3	no	—
34	C₂×D₄×C₅₃	—	2	3	no	—
35	C₂×D₄×C₅₃	—	4	3	no	—
36	C₂²×C₄×C₅₃	—	4	3	no	—
37	C₂²×C₄×C₅₃	—	4	1	no	—
38	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
39	SD₁₆×C₅₃	—	8	6	no	—
40	D₈×C₅₃	—	8	6	no	—
41	C₂×C₈×C₅₃	—	2	6	no	—
42	C₄⋊C₄×C₅₃	—	2	3	no	—
43	C₄⋊C₄×C₅₃	—	2	3	no	—
44	C₂²⋊C₄×C₅₃	—	4	3	no	—
45	C₂²⋊C₄×C₅₃	—	4	3	no	—
46	C₄²×C₅₃	C₂×D₄	1	3	yes	—
47	C₄²×C₅₃	#41	1	1	yes	characteristic
48	C₂³×C₅₃	—	4	3	no	—
49	C₂³×C₅₃	#127	1	1	yes	characteristic
50	Q₈×C₅₃	—	4	6	no	—
51	D₄×C₅₃	—	4	6	no	—
52	D₄×C₅₃	—	4	3	no	—
53	D₄×C₅₃	—	4	3	no	—
54	D₄×C₅₃	—	8	3	no	—
55	D₄×C₅₃	—	8	6	no	—
56	C₂×C₄×C₅₃	—	4	3	no	—
57	C₂×C₄×C₅₃	—	4	3	no	—
58	C₂×C₄×C₅₃	—	8	6	no	—
59	C₂×C₄×C₅₃	—	8	3	no	—
60	C₂×C₄×C₅₃	#120	1	3	yes	—
61	C₂×C₄×C₅₃	#121	1	3	yes	—
62	C₈×C₅₃	—	4	6	no	—
63	#10208	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₂²×C₅₃	—	2	3	no	—
65	C₂²×C₅₃	—	8	3	no	—
66	C₂²×C₅₃	—	8	3	no	—
67	C₂²×C₅₃	—	8	3	no	—
68	C₂²×C₅₃	#392	1	1	yes	characteristic
69	C₄×C₅₃	—	2	3	no	—
70	C₄×C₅₃	—	2	3	no	—
71	C₄×C₅₃	—	8	3	no	—
72	C₄×C₅₃	—	16	1	no	—
73	#3214	C₂×C₅₃	1	3	yes	complemented
74	#1514	C₂×C₅₃	1	3	yes	complemented
75	#1308	C₂×C₅₃	1	1	yes	characteristic, complemented
76	C₂×C₅₃	—	4	1	no	—
77	C₂×C₅₃	—	16	3	no	—
78	C₂×C₅₃	#2008	1	3	yes	—
79	#532	C₂²×C₅₃	1	3	yes	—
80	#496	—	2	3	no	—
81	#489	—	2	3	no	—
82	#461	—	2	3	no	—
83	#459	—	2	3	no	—
84	#449	—	2	6	no	—
85	#432	C₂²×C₅₃	1	3	yes	—
86	#383	C₂²×C₅₃	1	1	yes	characteristic
87	#327	—	2	6	no	—
88	C₅₃	#10208	1	1	yes	characteristic, direct product factor, Sylow
89	C₂×#41	—	2	3	no	—
90	C₂×SD₁₆	—	2	6	no	—
91	C₂×D₈	—	2	6	no	—
92	C₄⋊Q₈	D₄×C₅₃	1	3	yes	—
93	#127	D₄×C₅₃	1	3	yes	—
94	#124	—	2	3	no	—
95	#121	—	2	3	no	—
96	#121	—	2	3	no	—
97	C₄×D₄	—	2	3	no	—
98	C₄×D₄	—	2	3	no	—
99	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
100	C₄⋊C₈	—	2	3	no	—
101	#103	—	2	6	no	—
102	#102	—	2	6	no	—
103	C₂²⋊C₈	D₄×C₅₃	1	6	yes	—
104	#97	—	2	3	no	—
105	#95	—	2	3	no	—
106	#41	—	8	6	no	—
107	C₂×Q₈	—	2	3	no	—
108	C₂×D₄	—	2	3	no	—
109	C₂×D₄	—	2	3	no	—
110	C₂×D₄	—	2	3	no	—
111	C₂×D₄	—	4	3	no	—
112	C₂²×C₄	—	4	3	no	—
113	C₂²×C₄	—	4	1	no	—
114	C₂²×C₄	C₂×D₄×C₅₃	1	3	yes	—
115	SD₁₆	—	8	6	no	—
116	D₈	—	8	6	no	—
117	C₂×C₈	—	2	6	no	—
118	C₄⋊C₄	—	2	3	no	—
119	C₄⋊C₄	—	2	3	no	—
120	C₂²⋊C₄	—	4	3	no	—
121	C₂²⋊C₄	—	4	3	no	—
122	C₄²	C₂×D₄×C₅₃	1	3	yes	—
123	C₄²	#41×C₅₃	1	1	yes	characteristic
124	C₂³	—	4	3	no	—
125	C₂³	#127×C₅₃	1	1	yes	characteristic
126	Q₈	—	4	6	no	—
127	D₄	—	4	3	no	—
128	D₄	—	4	6	no	—
129	D₄	—	4	3	no	—
130	D₄	—	8	3	no	—
131	D₄	—	8	6	no	—
132	C₂×C₄	—	4	3	no	—
133	C₂×C₄	—	4	3	no	—
134	C₂×C₄	—	8	6	no	—
135	C₂×C₄	—	8	3	no	—
136	C₂×C₄	#120×C₅₃	1	3	yes	—
137	C₂×C₄	#121×C₅₃	1	3	yes	—
138	C₈	—	4	6	no	—
139	C₂²	—	2	3	no	—
140	C₂²	—	8	3	no	—
141	C₂²	—	8	3	no	—
142	C₂²	—	8	3	no	—
143	C₂²	C₅₃×#392	1	1	yes	characteristic
144	C₄	—	2	3	no	—
145	C₄	—	2	3	no	—
146	C₄	—	8	3	no	—
147	C₄	—	16	1	no	—
148	C₂	—	4	1	no	—
149	C₂	—	16	3	no	—
150	C₂	C₅₃×#2008	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#3214

C₂

yes

maximal, complemented

C₅₃×#1514

C₂

yes

maximal, complemented

C₅₃×#1308

C₂

yes

maximal, characteristic, complemented

C₅₃×#532

C₂²

yes

—

C₅₃×#496

—

C₅₃×#489

—

C₅₃×#461

—

C₅₃×#459

—

C₅₃×#449

—

C₅₃×#432

C₂²

yes

—

C₅₃×#383

C₂²

yes

characteristic

C₅₃×#327

—

C₂×#41×C₅₃

—

C₂×SD₁₆×C₅₃

—

C₂×D₈×C₅₃

—

C₄⋊Q₈×C₅₃

D₄

yes

—

#127×C₅₃

D₄

yes

—

#124×C₅₃

—

#121×C₅₃

—

#121×C₅₃

—

C₄×D₄×C₅₃

—

C₄×D₄×C₅₃

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

#103×C₅₃

—

#102×C₅₃

—

C₂²⋊C₈×C₅₃

D₄

yes

—

#97×C₅₃

—

#95×C₅₃

—

#41×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

—

SD₁₆×C₅₃

—

D₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

#41

yes

characteristic

C₂³×C₅₃

—

C₂³×C₅₃

#127

yes

characteristic

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#120

yes

—

C₂×C₄×C₅₃

#121

yes

—

C₈×C₅₃

—

#10208

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#392

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

#3214

C₂×C₅₃

yes

complemented

#1514

C₂×C₅₃

yes

complemented

#1308

C₂×C₅₃

yes

characteristic, complemented

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#2008

yes

—

#532

C₂²×C₅₃

yes

—

—

—

—

—

—

C₂²×C₅₃

yes

—

#383

C₂²×C₅₃

yes

characteristic

#327

—

C₅₃

#10208

yes

characteristic, direct product factor, Sylow

C₂×#41

—

C₂×SD₁₆

—

C₂×D₈

—

C₄⋊Q₈

D₄×C₅₃

yes

—

#127

D₄×C₅₃

yes

—

#124

—

#121

—

#121

—

C₄×D₄

—

C₄×D₄

—

C₂×C₄²

C₂³×C₅₃

yes

characteristic

100

C₄⋊C₈

—

101

#103

—

102

#102

—

103

C₂²⋊C₈

D₄×C₅₃

yes

—

104

#97

—

105

#95

—

106

#41

—

107

C₂×Q₈

—

108

C₂×D₄

—

109

C₂×D₄

—

110

C₂×D₄

—

111

C₂×D₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

C₂×D₄×C₅₃

yes

—

115

SD₁₆

—

116

D₈

—

117

C₂×C₈

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₂²⋊C₄

—

121

C₂²⋊C₄

—

122

C₄²

C₂×D₄×C₅₃

yes

—

123

C₄²

#41×C₅₃

yes

characteristic

124

C₂³

—

125

C₂³

#127×C₅₃

yes

characteristic

126

Q₈

—

127

D₄

—

128

D₄

—

129

D₄

—

130

D₄

—

131

D₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

#120×C₅₃

yes

—

137

C₂×C₄

#121×C₅₃

yes

—

138

C₈

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

C₅₃×#392

yes

characteristic

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₂

—

149

C₂

—

150

C₂

C₅₃×#2008

yes

—

	subgroup	G/H
center, socle	C₂²×C₅₃	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₅₃

#392

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₅₃

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

N	H ≅ G/N	type
1. C₅₃×#3214	C₂	semidirect
2. C₅₃×#1514	C₂	semidirect
3. C₅₃×#1308	C₂	semidirect
63. #10208	C₅₃	direct (N × H)
73. #3214	C₂×C₅₃	semidirect
74. #1514	C₂×C₅₃	semidirect
75. #1308	C₂×C₅₃	semidirect
88. C₅₃	#10208	direct (N × H)

H ≅ G/N

type

1. C₅₃×#3214

C₂

semidirect

2. C₅₃×#1514

C₂

semidirect

3. C₅₃×#1308

C₂

semidirect

63. #10208

C₅₃

direct (N × H)

73. #3214

C₂×C₅₃

semidirect

74. #1514

C₂×C₅₃

semidirect

75. #1308

C₂×C₅₃

semidirect

88. C₅₃

#10208

direct (N × H)

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

subgroup	index
C₅₃×#3214	2
C₅₃×#1514	2
C₅₃×#1308	2
#10208	53

subgroup

index

C₅₃×#3214

C₅₃×#1514

C₅₃×#1308

#10208

Subgroups — C₅₃×#10208 #27217720

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#10208 #27217720

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10208 #27217720