Group #27296116 — subgroups

Subgroups — #27296116

Order 13568.

226 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 69 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	#20276592	C₂	1	1	yes	maximal, characteristic
2	#20273628	C₂	1	2	yes	maximal, complemented
3	#20272681	C₂	1	2	yes	maximal, complemented
4	#20272635	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1368	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₈₄₈	C₄	1	1	yes	characteristic
7	C₄×D₄₂₄	C₂²	1	2	yes	—
8	#16810700	—	2	2	no	—
9	#16810451	C₄	1	1	yes	characteristic
10	#16810425	C₄	1	2	yes	—
11	#16810399	C₂²	1	1	yes	characteristic
12	#16810350	—	2	2	no	—
13	#16810344	C₂²	1	1	yes	characteristic
14	C₅₃×#346	C₂²	1	1	yes	characteristic
15	C₅₃×#334	C₂²	1	2	yes	—
16	C₂×D₄₂₄	C₂×C₄	1	2	yes	—
17	C₂×#11719912	—	4	2	no	—
18	C₄×D₂₁₂	—	2	2	no	—
19	D₈₄₈	—	2	2	no	—
20	C₂×#11719908	—	2	1	no	—
21	#13133151	C₂×C₄	1	2	yes	—
22	#13133139	—	2	2	no	—
23	#13133129	—	4	2	no	—
24	C₄×#88322	D₄	1	2	yes	—
25	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₄⋊C₈×C₅₃	—	2	2	no	—
28	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
29	C₂×D₂₁₂	—	2	2	no	—
30	C₂×C₄×D₅₃	—	4	2	no	—
31	C₂×C₈×C₅₃	—	4	2	no	—
32	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
34	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
35	#11719921	—	4	2	no	—
36	C₄⋊Dic₅₃	—	2	2	no	—
37	C₂×#88322	—	2	2	no	—
38	D₄₂₄	—	2	2	no	—
39	D₄₂₄	—	2	2	no	—
40	#11719912	—	8	4	no	—
41	C₁₆×C₅₃	—	2	1	no	—
42	C₁₆×C₅₃	M₁₆	1	2	yes	complemented
43	#11719908	—	4	1	no	—
44	C₂²×D₅₃	—	4	2	no	—
45	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
46	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
47	C₂×Dic₅₃	—	4	2	no	—
48	D₂₁₂	—	4	2	no	—
49	D₂₁₂	—	4	2	no	—
50	C₄×D₅₃	—	8	2	no	—
51	C₄×D₅₃	—	8	2	no	—
52	C₈×C₅₃	—	2	1	no	—
53	C₈×C₅₃	—	8	2	no	—
54	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
55	C₈×C₅₃	#131	1	1	yes	characteristic
56	#88322	—	4	2	no	—
57	#12089	—	53	1	no	maximal, Sylow
58	C₂²×C₅₃	#442	1	1	yes	characteristic
59	C₂×D₅₃	—	8	2	no	—
60	C₂×D₅₃	—	8	2	no	—
61	C₂×D₅₃	—	8	2	no	—
62	C₄×C₅₃	—	2	2	no	—
63	C₄×C₅₃	#436	1	1	yes	characteristic
64	C₄×C₅₃	#444	1	1	yes	characteristic
65	Dic₅₃	—	8	2	no	—
66	#2177	—	53	1	no	—
67	#1594	—	53	2	no	—
68	#1368	D₅₃	1	1	yes	characteristic, complemented
69	#1333	—	53	2	no	—
70	#1313	—	53	1	no	—
71	C₂×C₅₃	#1594	1	1	yes	characteristic
72	C₂×C₅₃	AGL(1,ℤ/16)	1	2	yes	—
73	D₅₃	—	16	2	no	—
74	C₂×D₁₆	—	53	1	no	—
75	C₄×D₈	—	53	2	no	—
76	#436	—	106	2	no	—
77	#367	—	53	1	no	—
78	#357	—	53	2	no	—
79	#346	—	53	1	no	—
80	#346	C₂×D₅₃	1	1	yes	characteristic
81	#334	C₂×D₅₃	1	2	yes	—
82	D₄⋊C₈	—	106	2	no	—
83	#322	—	53	1	no	—
84	C₅₃	#12089	1	1	yes	characteristic, complemented, Sylow
85	C₂×D₈	—	53	2	no	—
86	C₂×M₁₆	—	212	2	no	—
87	C₄×D₄	—	106	2	no	—
88	D₁₆	—	106	2	no	—
89	C₂×C₁₆	—	106	1	no	—
90	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
91	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
92	#107	—	53	2	no	—
93	C₄⋊C₈	—	2	2	no	—
94	#102	—	106	2	no	—
95	C₂²⋊C₈	—	212	2	no	—
96	C₄×C₈	—	53	2	no	—
97	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
98	C₂×D₄	—	106	2	no	—
99	C₂²×C₄	—	212	2	no	—
100	D₈	—	106	2	no	—
101	D₈	—	106	2	no	—
102	M₁₆	—	424	4	no	—
103	C₂×C₈	—	4	2	no	—
104	C₂×C₈	—	106	2	no	—
105	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
106	C₂×C₈	#11719949	1	1	yes	characteristic
107	C₄⋊C₄	—	106	2	no	—
108	C₂²⋊C₄	—	212	2	no	—
109	C₄²	D₄×D₅₃	1	1	yes	characteristic
110	C₁₆	—	2	1	no	—
111	C₁₆	—	212	1	no	—
112	C₁₆	#11719912	1	2	yes	complemented
113	C₂³	—	212	2	no	—
114	D₄	—	212	2	no	—
115	D₄	—	212	2	no	—
116	C₂×C₄	—	424	2	no	—
117	C₂×C₄	—	424	2	no	—
118	C₂×C₄	—	212	2	no	—
119	C₂×C₄	#13133179	1	1	yes	characteristic
120	C₂×C₄	#13133244	1	2	yes	—
121	C₈	—	2	1	no	—
122	C₈	—	8	2	no	—
123	C₈	—	212	2	no	—
124	C₈	C₂×#11719912	1	1	yes	characteristic
125	C₈	#13133225	1	1	yes	characteristic
126	C₂²	—	424	2	no	—
127	C₂²	—	424	2	no	—
128	C₂²	—	424	2	no	—
129	C₂²	#16810734	1	1	yes	characteristic
130	C₄	—	2	2	no	—
131	C₄	—	424	2	no	—
132	C₄	#16810700	1	1	yes	characteristic
133	C₄	#16810744	1	1	yes	characteristic
134	C₂	—	848	2	no	—
135	C₂	#20273628	1	1	yes	characteristic
136	C₂	#20276612	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20276592

C₂

yes

maximal, characteristic

#20273628

C₂

yes

maximal, complemented

#20272681

C₂

yes

maximal, complemented

#20272635

C₂

yes

maximal, characteristic, complemented

C₅₃×#1368

C₂

yes

maximal, characteristic, complemented

C₂×D₈₄₈

C₄

yes

characteristic

C₄×D₄₂₄

C₂²

yes

—

#16810700

—

#16810451

C₄

yes

characteristic

#16810425

C₄

yes

—

#16810399

C₂²

yes

characteristic

#16810350

—

#16810344

C₂²

yes

characteristic

C₅₃×#346

C₂²

yes

characteristic

C₅₃×#334

C₂²

yes

—

C₂×D₄₂₄

C₂×C₄

yes

—

C₂×#11719912

—

C₄×D₂₁₂

—

D₈₄₈

—

C₂×#11719908

—

#13133151

C₂×C₄

yes

—

#13133139

—

#13133129

—

C₄×#88322

D₄

yes

—

C₂×C₁₆×C₅₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₃

C₂×C₄

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄×C₈×C₅₃

C₂³

yes

characteristic

C₂×D₂₁₂

—

C₂×C₄×D₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₃

#41

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

characteristic

#11719921

—

C₄⋊Dic₅₃

—

C₂×#88322

—

D₄₂₄

—

D₄₂₄

—

#11719912

—

C₁₆×C₅₃

—

C₁₆×C₅₃

M₁₆

yes

complemented

#11719908

—

C₂²×D₅₃

—

C₂×C₄×C₅₃

C₄×D₄

yes

characteristic

C₂×C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

C₂×M₁₆

yes

C₈×C₅₃

yes

—

—

C₂²×C₅₃

yes

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

#2177

—

#1594

—

#1368

D₅₃

yes

characteristic, complemented

#1333

—

#1313

—

C₂×C₅₃

#1594

yes

characteristic

C₂×C₅₃

AGL(1,ℤ/16)

yes

—

D₅₃

—

C₂×D₁₆

—

C₄×D₈

—

#436

—

106

—

—

—

—

C₂×D₅₃

yes

characteristic

#334

C₂×D₅₃

yes

—

D₄⋊C₈

—

106

—

#322

—

C₅₃

#12089

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂×M₁₆

—

212

—

C₄×D₄

—

106

—

D₁₆

—

106

—

C₂×C₁₆

—

106

—

C₂×C₁₆

C₄×D₅₃

yes

characteristic

C₂×C₁₆

C₄×D₅₃

yes

characteristic

#107

—

C₄⋊C₈

—

#102

—

106

—

C₂²⋊C₈

—

212

—

C₄×C₈

—

C₄×C₈

C₂²×D₅₃

yes

characteristic

C₂×D₄

—

106

—

C₂²×C₄

—

212

—

100

D₈

—

106

—

101

D₈

—

106

—

102

M₁₆

—

424

—

103

C₂×C₈

—

104

C₂×C₈

—

106

—

105

C₂×C₈

C₂×C₄×D₅₃

yes

characteristic

106

C₂×C₈

#11719949

yes

characteristic

107

C₄⋊C₄

—

106

—

108

C₂²⋊C₄

—

212

—

109

C₄²

D₄×D₅₃

yes

characteristic

110

C₁₆

—

111

C₁₆

—

212

—

112

C₁₆

#11719912

yes

complemented

113

C₂³

—

212

—

114

D₄

—

212

—

115

D₄

—

212

—

116

C₂×C₄

—

424

—

117

C₂×C₄

—

424

—

118

C₂×C₄

—

212

—

119

C₂×C₄

#13133179

yes

characteristic

120

C₂×C₄

#13133244

yes

—

121

C₈

—

122

C₈

—

123

C₈

—

212

—

124

C₈

C₂×#11719912

yes

characteristic

125

C₈

#13133225

yes

characteristic

126

C₂²

—

424

—

127

C₂²

—

424

—

128

C₂²

—

424

—

129

C₂²

#16810734

yes

characteristic

130

C₄

—

131

C₄

—

424

—

132

C₄

#16810700

yes

characteristic

133

C₄

#16810744

yes

characteristic

134

C₂

—

848

—

135

C₂

#20273628

yes

characteristic

136

C₂

#20276612

yes

—

	subgroup	G/H
center	C₂²	#16810734
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#1368	C₂
socle	C₂²×C₅₃	#442
prime radical	C₂×D₈₄₈	C₄
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	#1368	D₅₃
hypocenter	C₅₃	#12089
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1368

subgroup

G/H

center

C₂²

#16810734

commutator

C₂×C₈×C₅₃

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₅₃

Fitting

C₅₃×#1368

C₂

socle

C₂²×C₅₃

#442

prime radical

C₂×D₈₄₈

C₄

cosocle

C₄×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1368

N	H ≅ G/N	type
2. #20273628	C₂	semidirect
3. #20272681	C₂	semidirect
4. #20272635	C₂	semidirect
5. C₅₃×#1368	C₂	semidirect
42. C₁₆×C₅₃	M₁₆	semidirect
68. #1368	D₅₃	semidirect
84. C₅₃	#12089	semidirect
112. C₁₆	#11719912	semidirect

H ≅ G/N

type

2. #20273628

C₂

semidirect

3. #20272681

C₂

semidirect

4. #20272635

C₂

semidirect

5. C₅₃×#1368

C₂

semidirect

42. C₁₆×C₅₃

M₁₆

semidirect

68. #1368

D₅₃

semidirect

84. C₅₃

#12089

semidirect

112. C₁₆

#11719912

semidirect

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

subgroup	index
#20276592	2
#20273628	2
#20272681	2
#20272635	2
C₅₃×#1368	2
#12089	53

subgroup

index

C₅₃×#1368

Subgroups — #27296116

Conjugacy classes of subgroups

Complements

Maximal subgroups