Group #28298600 — subgroups

Subgroups — #28298600

Order 13568.

206 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 74 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	#20290272	C₂	1	2	yes	maximal
2	C₂×#16810441	C₂	1	1	yes	maximal, characteristic
3	#16811626	—	2	2	no	—
4	#16811487	C₄	1	4	yes	—
5	#16811486	—	2	2	no	—
6	C₂²×#11719908	C₄	1	1	yes	characteristic
7	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
8	#16810441	—	2	2	no	—
9	C₂²×C₁₆×C₅₃	C₄	1	1	yes	characteristic
10	C₂×#11719940	—	4	2	no	—
11	#13133284	—	4	2	no	—
12	C₄⋊#88324	—	2	4	no	—
13	C₂²×#88322	Q₈	1	1	yes	characteristic
14	C₂×C₄×Dic₅₃	D₄	1	1	yes	characteristic
15	C₂×#11719908	—	2	1	no	—
16	C₂×#11719908	—	4	1	no	—
17	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
18	C₈×Dic₅₃	D₄	1	2	yes	—
19	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂×C₁₆×C₅₃	—	2	1	no	—
21	C₂×C₁₆×C₅₃	—	4	1	no	—
22	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
23	C₂²×Dic₅₃	D₈	1	1	yes	characteristic
24	C₂²×Dic₅₃	Q₁₆	1	1	yes	characteristic
25	#11719940	—	8	2	no	—
26	#11719940	—	8	2	no	—
27	C₂×#88324	—	4	4	no	—
28	C₂×C₈×C₅₃	—	2	1	no	—
29	C₂×C₈×C₅₃	—	2	1	no	—
30	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
31	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
32	C₄×Dic₅₃	C₂²⋊C₄	1	2	yes	—
33	C₄×Dic₅₃	SD₁₆	1	2	yes	—
34	C₂×#88322	—	4	1	no	—
35	C₂×#88322	C₄⋊C₄	1	2	yes	—
36	C₁₆×C₅₃	—	4	1	no	—
37	#11719908	—	4	1	no	—
38	C₂³×C₅₃	#107	1	1	yes	characteristic
39	C₂×C₄×C₅₃	—	2	1	no	—
40	C₂×C₄×C₅₃	—	2	1	no	—
41	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
42	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
43	C₂×Dic₅₃	—	4	1	no	—
44	C₂×Dic₅₃	#102	1	2	yes	—
45	C₂×Dic₅₃	#103	1	2	yes	—
46	C₂×Dic₅₃	D₁₆	1	1	yes	characteristic
47	C₂×Dic₅₃	SD₃₂	1	2	yes	—
48	C₂×Dic₅₃	Q₃₂	1	1	yes	characteristic
49	#88324	—	8	4	no	—
50	C₈×C₅₃	—	2	1	no	—
51	C₈×C₅₃	#100	1	1	yes	characteristic
52	C₈×C₅₃	#101	1	1	yes	characteristic
53	#88322	—	2	2	no	—
54	#7241	—	53	1	no	maximal, Sylow
55	C₂²×C₅₃	—	2	1	no	—
56	C₂²×C₅₃	—	2	1	no	—
57	C₂²×C₅₃	#340	1	1	yes	characteristic
58	C₂²×C₅₃	#366	1	1	yes	characteristic
59	C₂²×C₅₃	#367	1	1	yes	characteristic
60	C₄×C₅₃	—	2	1	no	—
61	C₄×C₅₃	#338	1	1	yes	characteristic
62	C₄×C₅₃	#343	1	1	yes	characteristic
63	Dic₅₃	—	2	2	no	—
64	Dic₅₃	#357	1	2	yes	—
65	Dic₅₃	#358	1	2	yes	—
66	C₂×C₄⋊C₁₆	—	53	1	no	—
67	#1567	—	53	2	no	—
68	C₂×C₅₃	—	2	1	no	—
69	C₂×C₅₃	—	2	1	no	—
70	C₂×C₅₃	#1292	1	1	yes	characteristic
71	C₂×C₅₃	#1380	1	1	yes	characteristic
72	C₂×C₅₃	#1385	1	1	yes	characteristic
73	C₂²×C₁₆	—	53	1	no	—
74	C₂²×C₁₆	#6157	1	1	yes	characteristic
75	#423	—	106	2	no	—
76	C₂×C₄×C₈	—	53	1	no	—
77	C₄⋊C₁₆	—	106	2	no	—
78	#335	—	53	4	no	—
79	#335	—	106	2	no	—
80	C₅₃	#7241	1	1	yes	characteristic, complemented, Sylow
81	C₂×M₁₆	—	212	2	no	—
82	C₂²×C₈	—	53	1	no	—
83	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
84	C₂×C₄²	—	53	1	no	—
85	C₂×C₁₆	—	2	1	no	—
86	C₂×C₁₆	—	4	1	no	—
87	C₂×C₁₆	—	106	1	no	—
88	C₂×C₁₆	—	212	1	no	—
89	C₄⋊C₈	—	106	4	no	—
90	C₄⋊C₈	—	212	2	no	—
91	C₄×C₈	—	53	2	no	—
92	C₄×C₈	—	53	2	no	—
93	C₂²×C₄	—	53	1	no	—
94	C₂²×C₄	—	53	1	no	—
95	C₂²×C₄	C₄⋊#6157	1	1	yes	characteristic
96	M₁₆	—	424	2	no	—
97	M₁₆	—	424	2	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	2	1	no	—
100	C₂×C₈	—	53	2	no	—
101	C₂×C₈	—	212	1	no	—
102	C₂×C₈	—	212	4	no	—
103	C₂×C₈	C₄×#6157	1	1	yes	characteristic
104	C₂×C₈	C₂²⋊#6157	1	1	yes	characteristic
105	C₄²	—	53	2	no	—
106	C₄²	—	53	2	no	—
107	C₁₆	—	4	1	no	—
108	C₁₆	—	212	1	no	—
109	C₂³	#13133286	1	1	yes	characteristic
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	—	53	1	no	—
113	C₂×C₄	—	53	1	no	—
114	C₂×C₄	—	53	2	no	—
115	C₂×C₄	—	53	2	no	—
116	C₂×C₄	—	53	2	no	—
117	C₂×C₄	—	212	1	no	—
118	C₂×C₄	(C₂×#6157)⋊C₄	1	1	yes	characteristic
119	C₂×C₄	#13133285	1	1	yes	characteristic
120	C₈	—	2	1	no	—
121	C₈	—	106	2	no	—
122	C₈	—	424	4	no	—
123	C₈	#13133277	1	1	yes	characteristic
124	C₈	#13133278	1	1	yes	characteristic
125	C₂²	—	2	1	no	—
126	C₂²	—	2	1	no	—
127	C₂²	#16811497	1	1	yes	characteristic
128	C₂²	#16811549	1	1	yes	characteristic
129	C₂²	#16811550	1	1	yes	characteristic
130	C₄	—	2	1	no	—
131	C₄	—	53	2	no	—
132	C₄	—	53	2	no	—
133	C₄	—	106	2	no	—
134	C₄	#16811492	1	1	yes	characteristic
135	C₄	#16811503	1	1	yes	characteristic
136	C₂	—	2	1	no	—
137	C₂	—	2	1	no	—
138	C₂	#20289585	1	1	yes	characteristic
139	C₂	#20289773	1	1	yes	characteristic
140	C₂	#20289785	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20290272

C₂

yes

maximal

C₂×#16810441

C₂

yes

maximal, characteristic

#16811626

—

#16811487

C₄

yes

—

#16811486

—

C₂²×#11719908

C₄

yes

characteristic

C₂×C₈×Dic₅₃

C₂²

yes

characteristic

#16810441

—

C₂²×C₁₆×C₅₃

C₄

yes

—

—

—

C₂²×#88322

Q₈

yes

characteristic

C₂×C₄×Dic₅₃

D₄

yes

characteristic

C₂×#11719908

—

C₂×#11719908

—

C₈×Dic₅₃

C₂×C₄

yes

—

C₈×Dic₅₃

D₄

yes

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

—

C₂²×C₄×C₅₃

C₄⋊C₄

yes

characteristic

C₂²×Dic₅₃

D₈

yes

characteristic

C₂²×Dic₅₃

Q₁₆

yes

—

—

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄²

yes

characteristic

C₂×C₈×C₅₃

C₂²⋊C₄

yes

characteristic

C₄×Dic₅₃

C₂²⋊C₄

yes

—

C₄×Dic₅₃

SD₁₆

yes

—

C₂×#88322

—

C₂×#88322

C₄⋊C₄

yes

—

C₁₆×C₅₃

—

#11719908

—

C₂³×C₅₃

#107

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×Dic₅₃

—

C₂×Dic₅₃

#102

yes

—

C₂×Dic₅₃

#103

yes

—

C₂×Dic₅₃

D₁₆

yes

characteristic

C₂×Dic₅₃

SD₃₂

yes

—

C₂×Dic₅₃

Q₃₂

yes

characteristic

#88324

—

C₈×C₅₃

—

C₈×C₅₃

yes

C₈×C₅₃

yes

—

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

#357

yes

—

Dic₅₃

#358

yes

—

C₂×C₄⋊C₁₆

—

#1567

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂²×C₁₆

—

C₂²×C₁₆

#6157

yes

characteristic

#423

—

106

—

C₂×C₄×C₈

—

C₄⋊C₁₆

—

106

—

#335

—

#335

—

106

—

C₅₃

#7241

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

212

—

C₂²×C₈

—

C₂²×C₈

C₂×#6157

yes

characteristic

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

106

—

C₂×C₁₆

—

212

—

C₄⋊C₈

—

106

—

C₄⋊C₈

—

212

—

C₄×C₈

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄⋊#6157

yes

characteristic

M₁₆

—

424

—

M₁₆

—

424

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

212

—

102

C₂×C₈

—

212

—

103

C₂×C₈

C₄×#6157

yes

characteristic

104

C₂×C₈

C₂²⋊#6157

yes

characteristic

105

C₄²

—

106

C₄²

—

107

C₁₆

—

108

C₁₆

—

212

—

109

C₂³

#13133286

yes

characteristic

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

212

—

118

C₂×C₄

(C₂×#6157)⋊C₄

yes

characteristic

119

C₂×C₄

#13133285

yes

characteristic

120

C₈

—

121

C₈

—

106

—

122

C₈

—

424

—

123

C₈

#13133277

yes

characteristic

124

C₈

#13133278

yes

characteristic

125

C₂²

—

126

C₂²

—

127

C₂²

#16811497

yes

characteristic

128

C₂²

#16811549

yes

characteristic

129

C₂²

#16811550

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

106

—

134

C₄

#16811492

yes

characteristic

135

C₄

#16811503

yes

characteristic

136

C₂

—

137

C₂

—

138

C₂

#20289585

yes

characteristic

139

C₂

#20289773

yes

characteristic

140

C₂

#20289785

yes

characteristic

	subgroup	G/H
center	C₂²	#16811497
commutator	C₂×C₈×C₅₃	C₄²
Frattini	C₂²×C₈	C₂×#6157
Fitting	C₂²×C₁₆×C₅₃	C₄
socle	C₂²×C₅₃	#340
prime radical	C₂³×C₅₃	#107
cosocle	C₂×C₈×Dic₅₃	C₂²
hypercenter	C₂²×C₁₆	#6157
hypocenter	C₅₃	#7241
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

subgroup

G/H

center

C₂²

#16811497

commutator

C₂×C₈×C₅₃

C₄²

Frattini

C₂²×C₈

C₂×#6157

Fitting

C₂²×C₁₆×C₅₃

C₄

C₂²×C₅₃

C₂³×C₅₃

C₂×C₈×Dic₅₃

C₂²

C₂²×C₁₆

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆

N	H ≅ G/N	type
80. C₅₃	#7241	semidirect

H ≅ G/N

type

80. C₅₃

#7241

semidirect

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

subgroup	index
#20290272	2
C₂×#16810441	2
#7241	53

subgroup

index

#20290272

C₂×#16810441

#7241

Subgroups — #28298600

Conjugacy classes of subgroups

Complements

Maximal subgroups