Group #28298253 — subgroups

Subgroups — #28298253

Order 13568.

202 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 202 conjugacy classes of subgroups — too many to draw. Only its 62 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	#20289759	C₂	1	2	yes	maximal
2	#20275681	C₂	1	1	yes	maximal, characteristic
3	C₂×#13133291	—	2	2	no	—
4	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
5	#16810489	C₄	1	1	yes	characteristic
6	#16810385	—	2	2	no	—
7	C₂×#125×C₅₃	C₄	1	1	yes	characteristic
8	#13133291	—	2	4	no	—
9	#13133291	—	4	2	no	—
10	C₂×#11719910	—	4	2	no	—
11	C₂²×#88322	D₄	1	2	yes	—
12	C₂×C₄⋊Dic₅₃	—	2	1	no	—
13	C₄×#88322	—	4	1	no	—
14	#13133124	—	4	1	no	—
15	#125×C₅₃	—	2	1	no	—
16	#125×C₅₃	—	4	1	no	—
17	#125×C₅₃	C₈	1	2	yes	—
18	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
19	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
20	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	—	2	1	no	—
22	C₂²×C₄×C₅₃	—	4	1	no	—
23	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
24	C₂²×Dic₅₃	—	4	1	no	—
25	C₄⋊C₄×C₅₃	—	2	2	no	—
26	C₄⋊C₄×C₅₃	—	4	2	no	—
27	C₄⋊C₄×C₅₃	—	4	2	no	—
28	C₄⋊C₄×C₅₃	—	4	1	no	—
29	C₄²×C₅₃	—	2	1	no	—
30	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
31	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
32	C₄⋊Dic₅₃	—	4	2	no	—
33	C₂×#88322	—	2	2	no	—
34	C₂×#88322	—	2	2	no	—
35	C₂×#88322	SD₁₆	1	2	yes	—
36	C₂×#88322	Q₁₆	1	2	yes	—
37	#11719910	—	4	4	no	—
38	C₂³×C₅₃	#100	1	1	yes	characteristic
39	C₂×C₄×C₅₃	—	2	1	no	—
40	C₂×C₄×C₅₃	—	2	1	no	—
41	C₂×C₄×C₅₃	—	4	2	no	—
42	C₂×C₄×C₅₃	—	4	2	no	—
43	C₂×C₄×C₅₃	—	8	2	no	—
44	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
45	C₂×C₄×C₅₃	#99	1	1	yes	characteristic
46	C₂×C₄×C₅₃	#103	1	2	yes	—
47	C₂×C₄×C₅₃	#104	1	2	yes	—
48	C₂×Dic₅₃	—	8	2	no	—
49	C₂×Dic₅₃	—	8	1	no	—
50	#88322	—	2	2	no	—
51	#88322	—	2	2	no	—
52	#7135	—	53	1	no	maximal, Sylow
53	C₂²×C₅₃	—	2	2	no	—
54	C₂²×C₅₃	#323	1	1	yes	characteristic
55	C₂²×C₅₃	#329	1	1	yes	characteristic
56	C₂²×C₅₃	#333	1	1	yes	characteristic
57	C₄×C₅₃	—	4	1	no	—
58	C₄×C₅₃	—	4	1	no	—
59	C₄×C₅₃	—	8	2	no	—
60	C₄×C₅₃	Q₈⋊C₈	1	2	yes	—
61	C₄×C₅₃	#328	1	2	yes	—
62	Dic₅₃	—	16	1	no	—
63	#1991	—	53	1	no	—
64	#1376	—	53	2	no	—
65	C₂×C₅₃	—	2	1	no	—
66	C₂×C₅₃	—	2	1	no	—
67	C₂×C₅₃	#1272	1	1	yes	characteristic
68	C₂×C₅₃	#1366	1	2	yes	—
69	C₂×#125	#6157	1	1	yes	characteristic
70	C₂×M₃₂	—	106	2	no	—
71	C₂×C₄×C₈	—	53	1	no	—
72	#382	—	53	1	no	—
73	#340	—	106	2	no	—
74	C₅₃	#7135	1	1	yes	characteristic, complemented, Sylow
75	C₂²×C₈	—	53	2	no	—
76	#125	—	2	1	no	—
77	#125	—	4	1	no	—
78	#125	#88324	1	2	yes	—
79	C₂×C₄⋊C₄	—	2	1	no	—
80	C₂×C₄⋊C₄	—	4	1	no	—
81	C₂×C₄⋊C₄	—	106	1	no	—
82	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
83	M₃₂	—	106	4	no	—
84	M₃₂	—	212	2	no	—
85	C₂×C₁₆	—	212	2	no	—
86	C₄×C₈	—	212	1	no	—
87	#95	—	212	1	no	—
88	C₂²×C₄	—	2	1	no	—
89	C₂²×C₄	—	4	1	no	—
90	C₂²×C₄	—	212	1	no	—
91	C₂²×C₄	C₂²⋊#6157	1	1	yes	characteristic
92	C₂×C₈	—	53	2	no	—
93	C₂×C₈	—	53	2	no	—
94	C₂×C₈	—	106	2	no	—
95	C₂×C₈	—	106	2	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	4	2	no	—
98	C₄⋊C₄	—	4	2	no	—
99	C₄⋊C₄	—	4	1	no	—
100	C₄⋊C₄	—	212	2	no	—
101	C₄²	—	2	1	no	—
102	C₄²	C₂×#88324	1	1	yes	characteristic
103	C₄²	#11719940	1	1	yes	characteristic
104	C₁₆	—	212	4	no	—
105	C₂³	#13133277	1	1	yes	characteristic
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	—	8	2	no	—
111	C₂×C₄	—	424	2	no	—
112	C₂×C₄	—	424	1	no	—
113	C₂×C₄	C₂²⋊#88324	1	1	yes	characteristic
114	C₂×C₄	#13133275	1	1	yes	characteristic
115	C₂×C₄	#13133280	1	2	yes	—
116	C₂×C₄	#13133282	1	2	yes	—
117	C₈	—	106	2	no	—
118	C₈	—	106	2	no	—
119	C₂²	—	2	2	no	—
120	C₂²	#16811466	1	1	yes	characteristic
121	C₂²	#16811479	1	1	yes	characteristic
122	C₂²	#16811483	1	1	yes	characteristic
123	C₄	—	4	1	no	—
124	C₄	—	4	1	no	—
125	C₄	—	8	2	no	—
126	C₄	—	848	1	no	—
127	C₄	Q₈⋊#88324	1	2	yes	—
128	C₄	#16811477	1	2	yes	—
129	C₂	—	2	1	no	—
130	C₂	—	2	1	no	—
131	C₂	#20289535	1	1	yes	characteristic
132	C₂	#20289736	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20289759

C₂

yes

maximal

#20275681

C₂

yes

maximal, characteristic

C₂×#13133291

—

C₂×C₄×#88322

C₂²

yes

characteristic

#16810489

C₄

yes

characteristic

#16810385

—

C₂×#125×C₅₃

C₄

yes

—

—

—

C₂²×#88322

D₄

yes

—

C₂×C₄⋊Dic₅₃

—

C₄×#88322

—

#13133124

—

#125×C₅₃

—

#125×C₅₃

—

#125×C₅₃

C₈

yes

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂²×Dic₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×C₈

yes

characteristic

C₄²×C₅₃

M₁₆

yes

characteristic

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

SD₁₆

yes

—

C₂×#88322

Q₁₆

yes

—

#11719910

—

C₂³×C₅₃

#100

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₈

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

—

C₂×C₄×C₅₃

#104

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

—

—

—

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

Q₈⋊C₈

yes

—

C₄×C₅₃

#328

yes

—

Dic₅₃

—

#1991

—

#1376

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#1272

yes

characteristic

C₂×C₅₃

#1366

yes

—

C₂×#125

#6157

yes

characteristic

C₂×M₃₂

—

106

—

C₂×C₄×C₈

—

#382

—

#340

—

106

—

C₅₃

#7135

yes

characteristic, complemented, Sylow

C₂²×C₈

—

—

—

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄²

C₂×#6157

yes

characteristic

M₃₂

—

106

—

M₃₂

—

212

—

C₂×C₁₆

—

212

—

C₄×C₈

—

212

—

#95

—

212

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

212

—

C₂²×C₄

C₂²⋊#6157

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

212

—

101

C₄²

—

102

C₄²

C₂×#88324

yes

characteristic

103

C₄²

#11719940

yes

characteristic

104

C₁₆

—

212

—

105

C₂³

#13133277

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

424

—

112

C₂×C₄

—

424

—

113

C₂×C₄

C₂²⋊#88324

yes

characteristic

114

C₂×C₄

#13133275

yes

characteristic

115

C₂×C₄

#13133280

yes

—

116

C₂×C₄

#13133282

yes

—

117

C₈

—

106

—

118

C₈

—

106

—

119

C₂²

—

120

C₂²

#16811466

yes

characteristic

121

C₂²

#16811479

yes

characteristic

122

C₂²

#16811483

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

848

—

127

C₄

Q₈⋊#88324

yes

—

128

C₄

#16811477

yes

—

129

C₂

—

130

C₂

—

131

C₂

#20289535

yes

characteristic

132

C₂

#20289736

yes

—

	subgroup	G/H
center	C₂²	#16811483
commutator	C₄²×C₅₃	C₂×C₈
Frattini	C₂×C₄²	C₂×#6157
Fitting	C₂×#125×C₅₃	C₄
socle	C₂²×C₅₃	#333
prime radical	C₂³×C₅₃	#100
cosocle	C₂×C₄×#88322	C₂²
hypercenter	C₂×#125	#6157
hypocenter	C₅₃	#7135
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂²

#16811483

commutator

C₄²×C₅₃

C₂×C₈

Frattini

C₂×C₄²

C₂×#6157

Fitting

C₂×#125×C₅₃

C₄

C₂²×C₅₃

C₂³×C₅₃

C₂×C₄×#88322

C₂²

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
74. C₅₃	#7135	semidirect

H ≅ G/N

type

74. C₅₃

#7135

semidirect

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

subgroup	index
#20289759	2
#20275681	2
#7135	53

subgroup

index

#20289759

#20275681

#7135

Subgroups — #28298253

Conjugacy classes of subgroups

Complements

Maximal subgroups