Group #28311397 — subgroups

Subgroups — #28311397

Order 13568.

234 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 84 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	#20291056	C₂	1	1	yes	maximal, characteristic
2	#20291052	C₂	1	1	yes	maximal, characteristic
3	#20289769	C₂	1	2	yes	maximal
4	#20289765	C₂	1	2	yes	maximal
5	#20274469	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133291	C₂²	1	1	yes	characteristic
7	C₂×#13133291	C₂²	1	2	yes	—
8	C₂²×#11719910	C₂²	1	1	yes	characteristic
9	#16811545	—	2	2	no	—
10	#16811544	—	2	2	no	—
11	#16810581	C₂²	1	1	yes	characteristic
12	#16810579	C₂²	1	2	yes	—
13	#16810381	C₄	1	1	yes	characteristic
14	#16810379	C₄	1	2	yes	—
15	C₅₃×H(3,ℤ/4)	C₄	1	1	yes	characteristic
16	#13133291	—	2	2	no	—
17	#13133291	—	2	2	no	—
18	#13133291	—	2	2	no	—
19	C₂×#11719910	—	2	1	no	—
20	C₂×#11719910	—	2	1	no	—
21	C₂×#11719910	—	2	2	no	—
22	C₂×#11719910	—	2	2	no	—
23	C₂×#11719917	C₂×C₄	1	2	yes	—
24	C₂²×#88322	C₂³	1	1	yes	characteristic
25	#13133176	C₂×C₄	1	1	yes	characteristic
26	#13133128	—	2	2	no	—
27	#13133127	Q₈	1	2	yes	—
28	C₄×#88322	—	2	2	no	—
29	C₈×Dic₅₃	D₄	1	2	yes	—
30	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
31	#117×C₅₃	C₂×C₄	1	2	yes	—
32	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
33	M₁₆×C₅₃	—	2	2	no	—
34	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
35	C₄⋊C₄×C₅₃	—	2	2	no	—
36	C₂²⋊C₄×C₅₃	—	2	2	no	—
37	C₄²×C₅₃	—	2	2	no	—
38	C₂²⋊Dic₅₃	—	2	1	no	—
39	C₄⋊Dic₅₃	—	2	1	no	—
40	C₄×Dic₅₃	C₄⋊C₄	1	2	yes	—
41	#11719917	—	2	2	no	—
42	#11719917	—	2	2	no	—
43	C₂×#88322	—	2	2	no	—
44	C₂×#88322	C₂×D₄	1	1	yes	characteristic
45	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
46	C₂×#88322	#41	1	2	yes	—
47	C₂×#88322	#41	1	2	yes	—
48	#11719910	—	4	1	no	—
49	#11719910	—	4	1	no	—
50	#11719910	—	4	2	no	—
51	C₂³×C₅₃	#131	1	1	yes	characteristic
52	C₂×C₄×C₅₃	—	4	2	no	—
53	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₅₃	#117	1	2	yes	—
55	C₂×C₄×C₅₃	#131	1	2	yes	—
56	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
57	C₂×Dic₅₃	—	2	2	no	—
58	C₈×C₅₃	—	2	2	no	—
59	#88322	—	4	2	no	—
60	#88322	#124	1	1	yes	characteristic
61	#88322	#124	1	1	yes	characteristic
62	#88322	#125	1	2	yes	—
63	#12885	—	53	1	no	maximal, Sylow
64	C₂²×C₅₃	—	4	1	no	—
65	C₂²×C₅₃	#424	1	1	yes	characteristic
66	C₂²×C₅₃	#433	1	2	yes	—
67	C₄×C₅₃	—	8	2	no	—
68	C₄×C₅₃	#382	1	1	yes	characteristic
69	C₄×C₅₃	#424	1	1	yes	characteristic
70	C₄×C₅₃	#433	1	2	yes	—
71	Dic₅₃	—	4	2	no	—
72	#2153	—	53	1	no	—
73	#2151	—	53	1	no	—
74	#1747	—	53	1	no	—
75	#1379	—	53	2	no	—
76	#1378	—	53	2	no	—
77	C₂×C₅₃	—	2	1	no	—
78	C₂×C₅₃	—	2	2	no	—
79	C₂×C₅₃	#1832	1	1	yes	characteristic
80	C₂×M₃₂	—	53	1	no	—
81	C₂×M₃₂	—	53	2	no	—
82	C₂²×C₁₆	—	53	1	no	—
83	#405	—	53	1	no	—
84	#405	—	53	2	no	—
85	#364	—	106	2	no	—
86	C₄⋊C₁₆	—	106	2	no	—
87	#338	—	53	1	no	—
88	H(3,ℤ/4)	—	53	2	no	—
89	H(3,ℤ/4)	#6157	1	1	yes	characteristic
90	C₅₃	#12885	1	1	yes	characteristic, complemented, Sylow
91	C₂×M₁₆	—	53	2	no	—
92	C₂×M₁₆	C₂×#6157	1	1	yes	characteristic
93	C₂²×C₈	—	53	1	no	—
94	#117	—	53	1	no	—
95	#117	C₂×#6157	1	2	yes	—
96	M₃₂	—	106	2	no	—
97	M₃₂	—	106	2	no	—
98	M₃₂	—	106	2	no	—
99	C₂×C₁₆	—	106	1	no	—
100	C₂×C₁₆	—	106	1	no	—
101	C₂×C₁₆	—	106	2	no	—
102	C₂×C₁₆	—	106	2	no	—
103	#97	—	53	2	no	—
104	#97	—	106	2	no	—
105	C₄×C₈	—	53	2	no	—
106	C₄×C₈	—	106	2	no	—
107	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
108	M₁₆	—	2	2	no	—
109	M₁₆	—	106	2	no	—
110	M₁₆	—	106	2	no	—
111	C₂×C₈	—	53	1	no	—
112	C₂×C₈	—	53	1	no	—
113	C₂×C₈	—	53	2	no	—
114	C₂×C₈	—	53	2	no	—
115	C₂×C₈	—	106	2	no	—
116	C₂×C₈	C₄⋊#6157	1	2	yes	—
117	C₄⋊C₄	—	2	2	no	—
118	C₄⋊C₄	—	106	1	no	—
119	C₂²⋊C₄	—	2	2	no	—
120	C₂²⋊C₄	—	106	1	no	—
121	C₄²	—	2	2	no	—
122	C₄²	—	53	2	no	—
123	C₁₆	—	212	1	no	—
124	C₁₆	—	212	1	no	—
125	C₁₆	—	212	2	no	—
126	C₂³	#13133304	1	1	yes	characteristic
127	C₂×C₄	—	4	2	no	—
128	C₂×C₄	—	106	2	no	—
129	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
130	C₂×C₄	#13133296	1	2	yes	—
131	C₂×C₄	#13133304	1	2	yes	—
132	C₂×C₄	#13133305	1	1	yes	characteristic
133	C₈	—	2	2	no	—
134	C₈	—	53	1	no	—
135	C₈	—	53	1	no	—
136	C₈	—	53	2	no	—
137	C₈	—	212	2	no	—
138	C₂²	—	4	1	no	—
139	C₂²	#16811630	1	1	yes	characteristic
140	C₂²	#16811647	1	2	yes	—
141	C₄	—	8	2	no	—
142	C₄	—	212	2	no	—
143	C₄	#16811567	1	1	yes	characteristic
144	C₄	#16811629	1	1	yes	characteristic
145	C₄	#16811648	1	2	yes	—
146	C₂	—	2	2	no	—
147	C₂	—	2	1	no	—
148	C₂	#20290633	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20291056

C₂

yes

maximal, characteristic

#20291052

C₂

yes

maximal, characteristic

#20289769

C₂

yes

maximal

#20289765

C₂

yes

maximal

#20274469

C₂

yes

maximal, characteristic

C₂×#13133291

C₂²

yes

characteristic

C₂×#13133291

C₂²

yes

—

C₂²×#11719910

C₂²

yes

—

—

C₂²

yes

characteristic

#16810579

C₂²

yes

—

#16810381

C₄

yes

characteristic

#16810379

C₄

yes

—

C₅₃×H(3,ℤ/4)

C₄

yes

—

—

—

—

—

—

—

C₂×C₄

yes

—

C₂²×#88322

C₂³

yes

characteristic

#13133176

C₂×C₄

yes

characteristic

#13133128

—

#13133127

Q₈

yes

—

C₄×#88322

—

C₈×Dic₅₃

D₄

yes

—

C₂×M₁₆×C₅₃

C₂×C₄

yes

characteristic

#117×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

M₁₆×C₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

C₄⋊C₄

yes

—

—

—

—

C₂×D₄

yes

characteristic

C₂×#88322

C₂×Q₈

yes

characteristic

C₂×#88322

#41

yes

—

C₂×#88322

#41

yes

—

#11719910

—

#11719910

—

#11719910

—

C₂³×C₅₃

#131

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

#117

yes

—

C₂×C₄×C₅₃

#131

yes

—

C₂×C₄×C₅₃

#131

yes

characteristic

C₂×Dic₅₃

—

C₈×C₅₃

—

—

yes

yes

yes

—

#12885

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

#424

yes

characteristic

C₂²×C₅₃

#433

yes

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

Dic₅₃

—

—

—

—

—

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#1832

yes

characteristic

C₂×M₃₂

—

C₂×M₃₂

—

C₂²×C₁₆

—

#405

—

#405

—

#364

—

106

—

C₄⋊C₁₆

—

106

—

—

—

yes

C₅₃

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂×M₁₆

C₂×#6157

yes

characteristic

C₂²×C₈

—

#117

—

#117

C₂×#6157

yes

—

M₃₂

—

106

—

M₃₂

—

106

—

M₃₂

—

106

—

C₂×C₁₆

—

106

—

100

C₂×C₁₆

—

106

—

101

C₂×C₁₆

—

106

—

102

C₂×C₁₆

—

106

—

103

#97

—

104

#97

—

106

—

105

C₄×C₈

—

106

C₄×C₈

—

106

—

107

C₂²×C₄

C₂²×#6157

yes

characteristic

108

M₁₆

—

109

M₁₆

—

106

—

110

M₁₆

—

106

—

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

106

—

116

C₂×C₈

C₄⋊#6157

yes

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

106

—

119

C₂²⋊C₄

—

120

C₂²⋊C₄

—

106

—

121

C₄²

—

122

C₄²

—

123

C₁₆

—

212

—

124

C₁₆

—

212

—

125

C₁₆

—

212

—

126

C₂³

#13133304

yes

characteristic

127

C₂×C₄

—

128

C₂×C₄

—

106

—

129

C₂×C₄

C₂×C₄⋊#6157

yes

characteristic

130

C₂×C₄

#13133296

yes

—

131

C₂×C₄

#13133304

yes

—

132

C₂×C₄

#13133305

yes

characteristic

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

137

C₈

—

212

—

138

C₂²

—

139

C₂²

#16811630

yes

characteristic

140

C₂²

#16811647

yes

—

141

C₄

—

142

C₄

—

212

—

143

C₄

#16811567

yes

characteristic

144

C₄

#16811629

yes

characteristic

145

C₄

#16811648

yes

—

146

C₂

—

147

C₂

—

148

C₂

#20290633

yes

characteristic

	subgroup	G/H
center	C₄	#16811567
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₅₃×H(3,ℤ/4)	C₄
socle	C₂×C₅₃	#1832
prime radical	C₂³×C₅₃	#131
cosocle	C₂²×#88322	C₂³
hypercenter	H(3,ℤ/4)	#6157
hypocenter	C₅₃	#12885
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#16811567

commutator

C₂²×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×#6157

Fitting

C₅₃×H(3,ℤ/4)

C₄

C₂×C₅₃

C₂³×C₅₃

C₂²×#88322

C₂³

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
90. C₅₃	#12885	semidirect

H ≅ G/N

type

90. C₅₃

#12885

semidirect

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

subgroup	index
#20291056	2
#20291052	2
#20289769	2
#20289765	2
#20274469	2
#12885	53

subgroup

index

Subgroups — #28311397

Conjugacy classes of subgroups

Complements

Maximal subgroups