Group #27271259 — subgroups

Subgroups — #27271259

Order 13568.

218 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 79 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	C₂×#16810401	C₂	1	2	yes	maximal
2	C₅₃×#1836	C₂	1	1	yes	maximal, characteristic
3	C₂×#13133157	—	2	2	no	—
4	#16810401	—	2	2	no	—
5	#16810401	C₄	1	4	yes	complemented
6	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
7	C₅₃×#384	C₄	1	1	yes	characteristic
8	C₅₃×#382	C₄	1	1	yes	characteristic
9	C₅₃×#340	—	2	2	no	—
10	#13133157	—	2	4	no	—
11	#13133157	—	2	4	no	—
12	C₂×#11719908	—	2	4	no	—
13	C₂²×C₈×C₅₃	D₄	1	2	yes	—
14	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
15	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
16	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
17	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
18	#97×C₅₃	C₂×C₄	1	2	yes	—
19	#97×C₅₃	D₄	1	1	yes	characteristic
20	#97×C₅₃	Q₈	1	1	yes	characteristic
21	#95×C₅₃	—	4	1	no	—
22	C₂²×C₄×C₅₃	—	2	1	no	—
23	C₂²×C₄×C₅₃	—	4	1	no	—
24	C₂²×C₄×C₅₃	—	4	1	no	—
25	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×C₈×C₅₃	—	2	2	no	—
27	C₂×C₈×C₅₃	—	2	2	no	—
28	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
29	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
30	C₄⋊C₄×C₅₃	—	4	2	no	—
31	C₄⋊C₄×C₅₃	—	4	2	no	—
32	C₄⋊C₄×C₅₃	—	8	2	no	—
33	C₄²×C₅₃	—	2	1	no	—
34	C₄²×C₅₃	C₄²	1	1	yes	characteristic
35	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
36	#11719908	—	2	8	no	—
37	C₂³×C₅₃	#100	1	1	yes	characteristic
38	C₂×C₄×C₅₃	—	2	1	no	—
39	C₂×C₄×C₅₃	—	2	1	no	—
40	C₂×C₄×C₅₃	—	4	1	no	—
41	C₂×C₄×C₅₃	—	4	1	no	—
42	C₂×C₄×C₅₃	—	8	1	no	—
43	C₂×C₄×C₅₃	—	8	1	no	—
44	C₂×C₄×C₅₃	—	8	2	no	—
45	C₂×C₄×C₅₃	—	8	1	no	—
46	C₂×C₄×C₅₃	—	8	1	no	—
47	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
48	C₂×C₄×C₅₃	#101	1	1	yes	characteristic
49	C₂×C₄×C₅₃	#104	1	2	yes	—
50	C₂×C₄×C₅₃	#104	1	2	yes	—
51	C₈×C₅₃	—	2	2	no	—
52	C₈×C₅₃	—	2	2	no	—
53	#7266	—	53	1	no	maximal, Sylow
54	C₂²×C₅₃	—	2	1	no	—
55	C₂²×C₅₃	—	2	1	no	—
56	C₂²×C₅₃	#329	1	1	yes	characteristic
57	C₂²×C₅₃	#329	1	1	yes	characteristic
58	C₂²×C₅₃	#343	1	1	yes	characteristic
59	C₄×C₅₃	—	4	1	no	—
60	C₄×C₅₃	—	4	1	no	—
61	C₄×C₅₃	—	16	1	no	—
62	C₄×C₅₃	—	16	1	no	—
63	C₄×C₅₃	#330	1	2	yes	—
64	C₄×C₅₃	#339	1	2	yes	—
65	C₂×#347	—	53	2	no	—
66	#1836	D₅₃	1	1	yes	characteristic
67	C₂×C₅₃	—	2	1	no	—
68	C₂×C₅₃	—	2	1	no	—
69	C₂×C₅₃	#1295	1	1	yes	characteristic
70	C₂×C₅₃	#1355	1	1	yes	characteristic
71	C₂×C₅₃	#1356	1	1	yes	characteristic
72	C₂×M₃₂	—	106	2	no	—
73	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
74	#384	Dic₅₃	1	1	yes	characteristic
75	#382	Dic₅₃	1	1	yes	characteristic
76	#347	—	53	4	no	—
77	#347	—	106	2	no	—
78	#340	—	2	2	no	—
79	C₅₃	#7266	1	1	yes	characteristic, complemented, Sylow
80	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
81	C₂×C₄⋊C₄	—	2	1	no	—
82	C₂×C₄⋊C₄	—	2	1	no	—
83	C₂×C₄⋊C₄	—	4	1	no	—
84	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
85	M₃₂	—	106	4	no	—
86	M₃₂	—	106	4	no	—
87	C₂×C₁₆	—	106	4	no	—
88	#97	Dic₁₀₆	1	1	yes	characteristic
89	#97	C₄×D₅₃	1	2	yes	—
90	#97	D₂₁₂	1	1	yes	characteristic
91	#95	—	4	1	no	—
92	C₂²×C₄	—	2	1	no	—
93	C₂²×C₄	—	4	1	no	—
94	C₂²×C₄	—	4	1	no	—
95	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	#11719919	1	2	yes	—
99	C₂×C₈	#11719921	1	2	yes	—
100	C₄⋊C₄	—	4	2	no	—
101	C₄⋊C₄	—	4	2	no	—
102	C₄⋊C₄	—	8	2	no	—
103	C₄²	—	2	1	no	—
104	C₄²	C₄×Dic₅₃	1	1	yes	characteristic
105	C₄²	C₄⋊Dic₅₃	1	1	yes	characteristic
106	C₁₆	—	106	8	no	—
107	C₂³	#13133134	1	1	yes	characteristic
108	C₂×C₄	—	2	1	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	4	1	no	—
111	C₂×C₄	—	4	1	no	—
112	C₂×C₄	—	8	2	no	—
113	C₂×C₄	—	8	1	no	—
114	C₂×C₄	—	8	1	no	—
115	C₂×C₄	—	8	1	no	—
116	C₂×C₄	—	8	1	no	—
117	C₂×C₄	#13133124	1	1	yes	characteristic
118	C₂×C₄	#13133136	1	1	yes	characteristic
119	C₂×C₄	#13133144	1	2	yes	—
120	C₂×C₄	#13133144	1	2	yes	—
121	C₈	—	2	2	no	—
122	C₈	—	2	2	no	—
123	C₂²	—	2	1	no	—
124	C₂²	—	2	1	no	—
125	C₂²	#16810363	1	1	yes	characteristic
126	C₂²	#16810363	1	1	yes	characteristic
127	C₂²	#16810392	1	1	yes	characteristic
128	C₄	—	4	1	no	—
129	C₄	—	4	1	no	—
130	C₄	—	16	1	no	—
131	C₄	—	16	1	no	—
132	C₄	#16810365	1	2	yes	—
133	C₄	#16810382	1	2	yes	—
134	C₂	—	2	1	no	—
135	C₂	—	2	1	no	—
136	C₂	#20272596	1	1	yes	characteristic
137	C₂	#20272748	1	1	yes	characteristic
138	C₂	#20272750	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16810401

C₂

yes

maximal

C₅₃×#1836

C₂

yes

maximal, characteristic

C₂×#13133157

—

#16810401

—

#16810401

C₄

yes

complemented

C₂×#97×C₅₃

C₂²

yes

characteristic

C₅₃×#384

C₄

yes

characteristic

C₅₃×#382

C₄

yes

characteristic

C₅₃×#340

—

#13133157

—

#13133157

—

C₂×#11719908

—

C₂²×C₈×C₅₃

D₄

yes

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

#97×C₅₃

C₂×C₄

yes

—

#97×C₅₃

D₄

yes

characteristic

#97×C₅₃

Q₈

yes

characteristic

#95×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₄²

yes

characteristic

C₄²×C₅₃

C₄⋊C₄

yes

characteristic

#11719908

—

C₂³×C₅₃

#100

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

—

C₂×C₄×C₅₃

#104

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

#7266

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#330

yes

—

C₄×C₅₃

#339

yes

—

C₂×#347

—

#1836

D₅₃

yes

characteristic

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×M₃₂

—

106

—

C₂×#97

C₂×D₅₃

yes

characteristic

#384

Dic₅₃

yes

characteristic

#382

Dic₅₃

yes

characteristic

#347

—

#347

—

106

—

#340

—

C₅₃

#7266

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂²⋊D₅₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×Dic₅₃

yes

characteristic

M₃₂

—

106

—

M₃₂

—

106

—

C₂×C₁₆

—

106

—

#97

Dic₁₀₆

yes

characteristic

#97

C₄×D₅₃

yes

—

#97

D₂₁₂

yes

characteristic

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊Dic₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

#11719919

yes

—

C₂×C₈

#11719921

yes

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄²

—

104

C₄²

C₄×Dic₅₃

yes

characteristic

105

C₄²

C₄⋊Dic₅₃

yes

characteristic

106

C₁₆

—

106

—

107

C₂³

#13133134

yes

characteristic

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

#13133124

yes

characteristic

118

C₂×C₄

#13133136

yes

characteristic

119

C₂×C₄

#13133144

yes

—

120

C₂×C₄

#13133144

yes

—

121

C₈

—

122

C₈

—

123

C₂²

—

124

C₂²

—

125

C₂²

#16810363

yes

characteristic

126

C₂²

#16810363

yes

characteristic

127

C₂²

#16810392

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

#16810365

yes

—

133

C₄

#16810382

yes

—

134

C₂

—

135

C₂

—

136

C₂

#20272596

yes

characteristic

137

C₂

#20272748

yes

characteristic

138

C₂

#20272750

yes

characteristic

	subgroup	G/H
center	C₂²	#16810363
commutator	C₄²×C₅₃	C₄²
Frattini	C₂×#97	C₂×D₅₃
Fitting	C₅₃×#1836	C₂
socle	C₂²×C₅₃	#329
prime radical	C₂³×C₅₃	#100
cosocle	C₂×#97×C₅₃	C₂²
hypercenter	#1836	D₅₃
hypocenter	C₅₃	#7266
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1836

subgroup

G/H

center

C₂²

#16810363

commutator

C₄²×C₅₃

C₄²

Frattini

C₂×#97

C₂×D₅₃

Fitting

C₅₃×#1836

C₂

C₂²×C₅₃

C₂³×C₅₃

C₂×#97×C₅₃

C₂²

D₅₃

C₅₃

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

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1836

N	H ≅ G/N	type
5. #16810401	C₄	semidirect
79. C₅₃	#7266	semidirect

H ≅ G/N

type

5. #16810401

C₄

semidirect

79. C₅₃

#7266

semidirect

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

subgroup	index
C₂×#16810401	2
C₅₃×#1836	2
#7266	53

subgroup

index

C₂×#16810401

C₅₃×#1836

#7266

Subgroups — #27271259

Conjugacy classes of subgroups

Complements

Maximal subgroups