Group #28311432 — subgroups

Subgroups — #28311432

Order 13568.

278 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 278 conjugacy classes of subgroups — too many to draw. Only its 104 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	#20291058	C₂	1	2	yes	maximal
2	#20291056	C₂	1	1	yes	maximal, characteristic
3	#20291052	C₂	1	1	yes	maximal, characteristic
4	#20291050	C₂	1	2	yes	maximal
5	#20274469	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133291	C₂²	1	2	yes	—
7	C₂×#13133291	C₂²	1	1	yes	characteristic
8	C₂²×#11719910	C₂²	1	1	yes	characteristic
9	#16811546	—	2	4	no	—
10	#16811545	—	2	2	no	—
11	#16811544	—	2	2	no	—
12	C₄⋊#11719910	—	2	4	no	—
13	#16810581	C₂²	1	1	yes	characteristic
14	#16810579	C₂²	1	2	yes	—
15	#16810381	C₄	1	1	yes	characteristic
16	#16810379	C₄	1	2	yes	—
17	C₅₃×H(3,ℤ/4)	C₄	1	1	yes	characteristic
18	#13133291	—	2	2	no	—
19	#13133291	—	2	2	no	—
20	#13133291	—	2	2	no	—
21	C₂×#11719910	—	2	2	no	—
22	C₂×#11719910	—	2	2	no	—
23	C₂×#11719910	—	2	1	no	—
24	C₂×#11719910	—	2	1	no	—
25	C₂×#11719917	C₂×C₄	1	2	yes	—
26	C₂²×#88322	C₂³	1	1	yes	characteristic
27	#13133176	C₂×C₄	1	1	yes	characteristic
28	#13133128	Q₈	1	4	yes	—
29	#13133127	Q₈	1	2	yes	—
30	C₄×#88322	D₄	1	4	yes	—
31	C₈×Dic₅₃	D₄	1	2	yes	—
32	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
33	#117×C₅₃	C₂×C₄	1	2	yes	—
34	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
35	M₁₆×C₅₃	—	2	2	no	—
36	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
37	C₄⋊C₄×C₅₃	—	2	2	no	—
38	C₂²⋊C₄×C₅₃	—	2	2	no	—
39	C₄²×C₅₃	C₄⋊C₄	1	4	yes	—
40	C₂²⋊Dic₅₃	—	2	1	no	—
41	C₄⋊Dic₅₃	—	2	1	no	—
42	C₄×Dic₅₃	C₄⋊C₄	1	2	yes	—
43	#11719917	—	2	2	no	—
44	#11719917	—	2	2	no	—
45	C₂×#88322	C₄⋊C₄	1	4	yes	—
46	C₂×#88322	C₂×D₄	1	1	yes	characteristic
47	C₂×#88322	C₂×D₄	1	2	yes	—
48	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
49	C₂×#88322	C₂×Q₈	1	2	yes	—
50	#11719910	—	4	1	no	—
51	#11719910	—	4	2	no	—
52	#11719910	—	4	1	no	—
53	C₂³×C₅₃	#131	1	1	yes	characteristic
54	C₂×C₄×C₅₃	—	2	4	no	—
55	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
56	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₅₃	#131	1	2	yes	—
58	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
59	C₂×Dic₅₃	—	2	2	no	—
60	C₈×C₅₃	—	2	2	no	—
61	#88322	—	2	4	no	—
62	#88322	#127	1	1	yes	characteristic
63	#88322	C₄⋊Q₈	1	1	yes	characteristic
64	#88322	C₄⋊Q₈	1	2	yes	—
65	#12902	—	53	1	no	maximal, Sylow
66	C₂²×C₅₃	—	4	1	no	—
67	C₂²×C₅₃	#424	1	2	yes	—
68	C₂²×C₅₃	#424	1	1	yes	characteristic
69	C₄×C₅₃	—	4	4	no	—
70	C₄×C₅₃	#384	1	1	yes	characteristic
71	C₄×C₅₃	#424	1	1	yes	characteristic
72	C₄×C₅₃	#424	1	2	yes	—
73	Dic₅₃	—	4	2	no	—
74	#2153	—	53	1	no	—
75	#2153	—	53	2	no	—
76	#2151	—	53	2	no	—
77	#2151	—	53	1	no	—
78	#1747	—	53	1	no	—
79	C₂×C₅₃	—	2	1	no	—
80	C₂×C₅₃	—	2	2	no	—
81	C₂×C₅₃	#1843	1	1	yes	characteristic
82	C₂×M₃₂	—	53	2	no	—
83	C₂×M₃₂	—	53	1	no	—
84	C₂²×C₁₆	—	53	1	no	—
85	#405	—	53	1	no	—
86	#405	—	53	2	no	—
87	#364	—	106	2	no	—
88	#364	—	106	4	no	—
89	C₄⋊C₁₆	—	106	4	no	—
90	C₄⋊C₁₆	—	106	2	no	—
91	#338	—	53	1	no	—
92	H(3,ℤ/4)	—	53	2	no	—
93	H(3,ℤ/4)	#6157	1	1	yes	characteristic
94	C₅₃	#12902	1	1	yes	characteristic, complemented, Sylow
95	C₂×M₁₆	—	53	2	no	—
96	C₂×M₁₆	C₂×#6157	1	1	yes	characteristic
97	C₂²×C₈	—	53	1	no	—
98	#117	—	53	1	no	—
99	#117	C₂×#6157	1	2	yes	—
100	M₃₂	—	106	2	no	—
101	M₃₂	—	106	2	no	—
102	M₃₂	—	106	2	no	—
103	C₂×C₁₆	—	106	2	no	—
104	C₂×C₁₆	—	106	1	no	—
105	C₂×C₁₆	—	106	2	no	—
106	C₂×C₁₆	—	106	1	no	—
107	#97	—	53	2	no	—
108	#97	—	53	4	no	—
109	C₄×C₈	—	53	2	no	—
110	C₄×C₈	—	53	4	no	—
111	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
112	M₁₆	—	2	2	no	—
113	M₁₆	—	106	2	no	—
114	M₁₆	—	106	2	no	—
115	C₂×C₈	—	53	1	no	—
116	C₂×C₈	—	53	2	no	—
117	C₂×C₈	—	53	1	no	—
118	C₂×C₈	—	53	2	no	—
119	C₂×C₈	—	53	4	no	—
120	C₂×C₈	C₄⋊#6157	1	2	yes	—
121	C₄⋊C₄	—	2	2	no	—
122	C₄⋊C₄	—	106	1	no	—
123	C₂²⋊C₄	—	2	2	no	—
124	C₂²⋊C₄	—	106	1	no	—
125	C₄²	—	53	2	no	—
126	C₄²	C₄⋊#6157	1	4	yes	—
127	C₁₆	—	212	1	no	—
128	C₁₆	—	212	2	no	—
129	C₁₆	—	212	1	no	—
130	C₂³	#13133304	1	1	yes	characteristic
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	106	2	no	—
133	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
134	C₂×C₄	C₂×C₄⋊#6157	1	2	yes	—
135	C₂×C₄	#13133304	1	2	yes	—
136	C₂×C₄	#13133305	1	1	yes	characteristic
137	C₈	—	2	2	no	—
138	C₈	—	53	1	no	—
139	C₈	—	53	1	no	—
140	C₈	—	53	2	no	—
141	C₈	—	106	4	no	—
142	C₂²	—	4	1	no	—
143	C₂²	#16811629	1	2	yes	—
144	C₂²	#16811630	1	1	yes	characteristic
145	C₄	—	4	4	no	—
146	C₄	—	212	2	no	—
147	C₄	#16811569	1	1	yes	characteristic
148	C₄	#16811629	1	1	yes	characteristic
149	C₄	#16811630	1	2	yes	—
150	C₂	—	2	2	no	—
151	C₂	—	2	1	no	—
152	C₂	#20290649	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20291058

C₂

yes

maximal

#20291056

C₂

yes

maximal, characteristic

#20291052

C₂

yes

maximal, characteristic

#20291050

C₂

yes

maximal

#20274469

C₂

yes

maximal, characteristic

C₂×#13133291

C₂²

yes

—

C₂×#13133291

C₂²

yes

characteristic

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

Q₈

yes

—

#13133127

Q₈

yes

—

C₄×#88322

D₄

yes

—

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

C₄⋊C₄

yes

—

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

C₄⋊C₄

yes

—

#11719917

—

#11719917

—

C₂×#88322

C₄⋊C₄

yes

—

C₂×#88322

C₂×D₄

yes

characteristic

C₂×#88322

C₂×D₄

yes

—

C₂×#88322

C₂×Q₈

yes

characteristic

C₂×#88322

C₂×Q₈

yes

—

#11719910

—

#11719910

—

#11719910

—

C₂³×C₅₃

#131

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

#131

yes

—

C₂×C₄×C₅₃

#131

yes

characteristic

C₂×Dic₅₃

—

C₈×C₅₃

—

—

yes

C₄⋊Q₈

yes

characteristic

#88322

C₄⋊Q₈

yes

—

#12902

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

#424

yes

—

C₂²×C₅₃

#424

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

Dic₅₃

—

—

—

—

—

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#1843

yes

characteristic

C₂×M₃₂

—

C₂×M₃₂

—

C₂²×C₁₆

—

#405

—

#405

—

#364

—

106

—

#364

—

106

—

C₄⋊C₁₆

—

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

—

100

M₃₂

—

106

—

101

M₃₂

—

106

—

102

M₃₂

—

106

—

103

C₂×C₁₆

—

106

—

104

C₂×C₁₆

—

106

—

105

C₂×C₁₆

—

106

—

106

C₂×C₁₆

—

106

—

107

#97

—

108

#97

—

109

C₄×C₈

—

110

C₄×C₈

—

111

C₂²×C₄

C₂²×#6157

yes

characteristic

112

M₁₆

—

113

M₁₆

—

106

—

114

M₁₆

—

106

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

C₄⋊#6157

yes

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

106

—

123

C₂²⋊C₄

—

124

C₂²⋊C₄

—

106

—

125

C₄²

—

126

C₄²

C₄⋊#6157

yes

—

127

C₁₆

—

212

—

128

C₁₆

—

212

—

129

C₁₆

—

212

—

130

C₂³

#13133304

yes

characteristic

131

C₂×C₄

—

132

C₂×C₄

—

106

—

133

C₂×C₄

C₂×C₄⋊#6157

yes

characteristic

134

C₂×C₄

C₂×C₄⋊#6157

yes

—

135

C₂×C₄

#13133304

yes

—

136

C₂×C₄

#13133305

yes

characteristic

137

C₈

—

138

C₈

—

139

C₈

—

140

C₈

—

141

C₈

—

106

—

142

C₂²

—

143

C₂²

#16811629

yes

—

144

C₂²

#16811630

yes

characteristic

145

C₄

—

146

C₄

—

212

—

147

C₄

#16811569

yes

characteristic

148

C₄

#16811629

yes

characteristic

149

C₄

#16811630

yes

—

150

C₂

—

151

C₂

—

152

C₂

#20290649

yes

characteristic

	subgroup	G/H
center	C₄	#16811569
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×#6157
Fitting	C₅₃×H(3,ℤ/4)	C₄
socle	C₂×C₅₃	#1843
prime radical	C₂³×C₅₃	#131
cosocle	C₂²×#88322	C₂³
hypercenter	H(3,ℤ/4)	#6157
hypocenter	C₅₃	#12902
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#16811569

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
94. C₅₃	#12902	semidirect

H ≅ G/N

type

94. C₅₃

#12902

semidirect

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

subgroup	index
#20291058	2
#20291056	2
#20291052	2
#20291050	2
#20274469	2
#12902	53

subgroup

index

Subgroups — #28311432

Conjugacy classes of subgroups

Complements

Maximal subgroups