Group #27371897 — subgroups

Subgroups — #27371897

Order 13568.

690 conjugacy classes of subgroups; only the 227 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 227 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	#20285301	C₂	1	4	yes	maximal
2	#20285229	C₂	1	2	yes	maximal
3	#20274392	C₂	1	8	yes	maximal, complemented
4	C₂×C₅₃×#322	C₂	1	1	yes	maximal, characteristic
5	#16811180	C₂²	1	4	yes	—
6	#16811028	C₂²	1	4	yes	—
7	#16811028	C₂²	1	4	yes	—
8	#16811025	—	2	4	no	—
9	#16811021	C₂²	1	4	yes	—
10	#16811003	C₂²	1	4	yes	—
11	#16810974	C₂²	1	8	yes	—
12	#16810864	—	2	8	no	—
13	C₄×Dic₂₁₂	—	2	8	no	—
14	#16810661	—	2	4	no	—
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
17	C₅₃×#322	C₂²	1	4	yes	—
18	C₂×Dic₂₁₂	—	2	8	no	—
19	#13133217	C₂³	1	8	yes	—
20	#13133209	—	2	4	no	—
21	#13133196	—	2	8	no	—
22	C₄×Dic₁₀₆	—	2	8	no	—
23	#13133176	—	2	4	no	—
24	#13133151	—	2	8	no	—
25	#13133151	—	2	8	no	—
26	#13133149	—	2	8	no	—
27	#13133142	—	2	16	no	—
28	C₂²×C₈×C₅₃	D₄	1	4	yes	—
29	C₂²×C₈×C₅₃	D₄	1	2	yes	—
30	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
31	C₄×C₈×C₅₃	D₄	1	2	yes	—
32	C₄×C₈×C₅₃	D₄	1	4	yes	—
33	C₄×C₈×C₅₃	C₂³	1	2	yes	—
34	C₄×C₈×C₅₃	C₂³	1	4	yes	—
35	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
36	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
37	C₂×Dic₁₀₆	—	2	8	no	—
38	C₂×C₈×C₅₃	—	4	4	no	—
39	C₂×C₈×C₅₃	Q₁₆	1	8	yes	complemented
40	C₂×C₈×C₅₃	C₂×D₄	1	8	yes	—
41	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
42	C₄²×C₅₃	C₂×D₄	1	2	yes	—
43	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
45	C₂²⋊Dic₅₃	—	4	4	no	—
46	C₄⋊Dic₅₃	—	2	8	no	—
47	C₄⋊Dic₅₃	—	2	8	no	—
48	#11719919	—	4	8	no	—
49	C₄×Dic₅₃	—	4	4	no	—
50	Dic₂₁₂	—	4	16	no	—
51	C₂³×C₅₃	#127	1	1	yes	characteristic
52	C₂×C₄×C₅₃	—	2	4	no	—
53	C₂×C₄×C₅₃	#127	1	2	yes	—
54	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
55	C₂×C₄×C₅₃	C₂×Q₁₆	1	4	yes	—
56	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
57	C₂×C₄×C₅₃	C₂²×D₄	1	2	yes	—
58	C₂×Dic₅₃	—	4	8	no	—
59	Dic₁₀₆	—	4	16	no	—
60	C₈×C₅₃	—	2	8	no	—
61	C₈×C₅₃	C₂×Q₁₆	1	8	yes	—
62	#19618	—	53	1	no	maximal, Sylow
63	C₂²×C₅₃	—	2	2	no	—
64	C₂²×C₅₃	#494	1	2	yes	—
65	C₂²×C₅₃	C₂×#127	1	1	yes	characteristic
66	C₄×C₅₃	—	2	2	no	—
67	C₄×C₅₃	#494	1	2	yes	—
68	C₄×C₅₃	C₂²×Q₁₆	1	2	yes	—
69	C₄×C₅₃	#575	1	4	yes	—
70	Dic₅₃	—	8	8	no	—
71	#3170	—	53	4	no	—
72	#3160	—	53	2	no	—
73	#1723	—	53	8	no	—
74	C₂×#322	D₅₃	1	1	yes	characteristic
75	C₂×C₅₃	—	2	2	no	—
76	C₂×C₅₃	C₂×#494	1	1	yes	characteristic
77	C₂×C₅₃	#3150	1	2	yes	—
78	#533	—	53	4	no	—
79	#500	—	53	4	no	—
80	#500	—	53	4	no	—
81	#499	—	106	4	no	—
82	#498	—	53	4	no	—
83	#494	—	53	4	no	—
84	#487	—	53	8	no	—
85	#467	—	106	8	no	—
86	C₄×Q₁₆	—	106	8	no	—
87	#427	—	106	4	no	—
88	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
89	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
90	#322	C₂×D₅₃	1	4	yes	—
91	C₅₃	#19618	1	1	yes	characteristic, complemented, Sylow
92	C₂×Q₁₆	—	106	8	no	—
93	C₂²×C₈	D₂₁₂	1	4	yes	—
94	C₂²×C₈	D₂₁₂	1	2	yes	—
95	C₄⋊Q₈	—	53	8	no	—
96	#125	—	106	4	no	—
97	C₂²⋊Q₈	—	106	8	no	—
98	C₄×Q₈	—	106	8	no	—
99	#117	—	106	4	no	—
100	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
101	#107	—	106	8	no	—
102	#107	—	106	8	no	—
103	#106	—	106	8	no	—
104	#103	—	106	16	no	—
105	C₄×C₈	D₂₁₂	1	4	yes	—
106	C₄×C₈	D₂₁₂	1	2	yes	—
107	C₄×C₈	C₂²×D₅₃	1	4	yes	—
108	C₄×C₈	C₂²×D₅₃	1	2	yes	—
109	C₂×Q₈	—	106	8	no	—
110	C₂²×C₄	C₂×D₂₁₂	1	2	yes	—
111	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
112	Q₁₆	—	212	16	no	—
113	C₂×C₈	—	4	4	no	—
114	C₂×C₈	Dic₂₁₂	1	8	yes	complemented
115	C₂×C₈	C₂×D₂₁₂	1	8	yes	—
116	C₂×C₈	C₂×D₂₁₂	1	4	yes	—
117	C₄⋊C₄	—	106	8	no	—
118	C₄⋊C₄	—	106	8	no	—
119	C₄⋊C₄	—	212	8	no	—
120	C₂²⋊C₄	—	212	4	no	—
121	C₄²	—	212	4	no	—
122	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
123	C₄²	C₂×D₂₁₂	1	2	yes	—
124	C₄²	C₂³×D₅₃	1	1	yes	characteristic
125	C₂³	#13133214	1	1	yes	characteristic
126	Q₈	—	212	16	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	—	212	8	no	—
129	C₂×C₄	#13133214	1	2	yes	—
130	C₂×C₄	#13133214	1	1	yes	characteristic
131	C₂×C₄	C₂×Dic₂₁₂	1	4	yes	—
132	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
133	C₂×C₄	C₂²×D₂₁₂	1	2	yes	—
134	C₈	—	2	8	no	—
135	C₈	C₂×Dic₂₁₂	1	8	yes	—
136	C₂²	—	2	2	no	—
137	C₂²	#16811003	1	2	yes	—
138	C₂²	C₂×#13133214	1	1	yes	characteristic
139	C₄	—	2	2	no	—
140	C₄	—	424	8	no	—
141	C₄	#16811003	1	2	yes	—
142	C₄	C₂²×Dic₂₁₂	1	2	yes	—
143	C₄	#16811423	1	4	yes	—
144	C₂	—	2	2	no	—
145	C₂	C₂×#16811003	1	1	yes	characteristic
146	C₂	#20285161	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20285301

C₂

yes

maximal

#20285229

C₂

yes

maximal

#20274392

C₂

yes

maximal, complemented

C₂×C₅₃×#322

C₂

yes

maximal, characteristic

#16811180

C₂²

yes

—

#16811028

C₂²

yes

—

#16811028

C₂²

yes

—

#16811025

—

#16811021

C₂²

yes

—

#16811003

C₂²

yes

—

#16810974

C₂²

yes

—

#16810864

—

C₄×Dic₂₁₂

—

#16810661

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄×C₈×C₅₃

C₂²

yes

—

C₅₃×#322

C₂²

yes

—

C₂×Dic₂₁₂

—

#13133217

C₂³

yes

—

#13133209

—

#13133196

—

C₄×Dic₁₀₆

—

—

—

—

—

—

C₂²×C₈×C₅₃

D₄

yes

—

C₂²×C₈×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄×C₈×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

C₂³

yes

—

C₄×C₈×C₅₃

C₂³

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

Q₁₆

yes

complemented

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

C₂⁴

yes

characteristic

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719919

—

C₄×Dic₅₃

—

Dic₂₁₂

—

C₂³×C₅₃

#127

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#127

yes

—

C₂×C₄×C₅₃

#127

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₁₆

yes

—

C₂×C₄×C₅₃

C₂²×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×D₄

yes

—

C₂×Dic₅₃

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

C₂×Q₁₆

yes

—

#19618

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

#494

yes

—

C₂²×C₅₃

C₂×#127

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

#494

yes

—

C₄×C₅₃

C₂²×Q₁₆

yes

—

C₄×C₅₃

#575

yes

—

Dic₅₃

—

—

—

—

D₅₃

yes

characteristic

C₂×C₅₃

—

C₂×C₅₃

C₂×#494

yes

characteristic

C₂×C₅₃

#3150

yes

—

—

—

—

—

106

—

—

—

—

—

106

—

C₄×Q₁₆

—

106

—

#427

—

106

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

C₂×C₄×C₈

C₂×D₅₃

yes

—

#322

C₂×D₅₃

yes

—

C₅₃

#19618

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

106

—

C₂²×C₈

D₂₁₂

yes

—

C₂²×C₈

D₂₁₂

yes

—

C₄⋊Q₈

—

#125

—

106

—

C₂²⋊Q₈

—

106

—

C₄×Q₈

—

106

—

#117

—

106

—

100

C₂×C₄²

C₂²×D₅₃

yes

characteristic

101

#107

—

106

—

102

#107

—

106

—

103

#106

—

106

—

104

#103

—

106

—

105

C₄×C₈

D₂₁₂

yes

—

106

C₄×C₈

D₂₁₂

yes

—

107

C₄×C₈

C₂²×D₅₃

yes

—

108

C₄×C₈

C₂²×D₅₃

yes

—

109

C₂×Q₈

—

106

—

110

C₂²×C₄

C₂×D₂₁₂

yes

—

111

C₂²×C₄

C₂×D₂₁₂

yes

characteristic

112

Q₁₆

—

212

—

113

C₂×C₈

—

114

C₂×C₈

Dic₂₁₂

yes

complemented

115

C₂×C₈

C₂×D₂₁₂

yes

—

116

C₂×C₈

C₂×D₂₁₂

yes

—

117

C₄⋊C₄

—

106

—

118

C₄⋊C₄

—

106

—

119

C₄⋊C₄

—

212

—

120

C₂²⋊C₄

—

212

—

121

C₄²

—

212

—

122

C₄²

C₂×D₂₁₂

yes

characteristic

123

C₄²

C₂×D₂₁₂

yes

—

124

C₄²

C₂³×D₅₃

yes

characteristic

125

C₂³

#13133214

yes

characteristic

126

Q₈

—

212

—

127

C₂×C₄

—

128

C₂×C₄

—

212

—

129

C₂×C₄

#13133214

yes

—

130

C₂×C₄

#13133214

yes

characteristic

131

C₂×C₄

C₂×Dic₂₁₂

yes

—

132

C₂×C₄

C₂²×D₂₁₂

yes

characteristic

133

C₂×C₄

C₂²×D₂₁₂

yes

—

134

C₈

—

135

C₈

C₂×Dic₂₁₂

yes

—

136

C₂²

—

137

C₂²

#16811003

yes

—

138

C₂²

C₂×#13133214

yes

characteristic

139

C₄

—

140

C₄

—

424

—

141

C₄

#16811003

yes

—

142

C₄

C₂²×Dic₂₁₂

yes

—

143

C₄

#16811423

yes

—

144

C₂

—

145

C₂

C₂×#16811003

yes

characteristic

146

C₂

#20285161

yes

—

	subgroup	G/H
center	C₂×C₄	#13133214
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₂×C₅₃×#322	C₂
socle	C₂²×C₅₃	C₂×#127
prime radical	C₂³×C₅₃	#127
hypercenter	C₂×#322	D₅₃
hypocenter	C₅₃	#19618
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

subgroup

G/H

center

C₂×C₄

#13133214

commutator, cosocle

C₄²×C₅₃

C₂⁴

Frattini

C₄²

C₂³×D₅₃

Fitting

C₂×C₅₃×#322

C₂

C₂²×C₅₃

C₂³×C₅₃

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

N	H ≅ G/N	type
3. #20274392	C₂	semidirect
39. C₂×C₈×C₅₃	Q₁₆	semidirect
91. C₅₃	#19618	semidirect
114. C₂×C₈	Dic₂₁₂	semidirect

H ≅ G/N

type

3. #20274392

C₂

semidirect

39. C₂×C₈×C₅₃

Q₁₆

semidirect

91. C₅₃

#19618

semidirect

114. C₂×C₈

Dic₂₁₂

semidirect

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

subgroup	index
#20285301	2
#20285229	2
#20274392	2
C₂×C₅₃×#322	2
#19618	53

subgroup

index

C₂×C₅₃×#322

Subgroups — #27371897

Conjugacy classes of subgroups

Complements

Maximal subgroups