Group #27270872 — subgroups

Subgroups — #27270872

Order 13568.

230 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 71 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	#20272795	C₂	1	2	yes	maximal
2	C₅₃×#1960	C₂	1	1	yes	maximal, characteristic
3	C₂×#13133157	—	2	2	no	—
4	C₂×C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
5	C₂×#103×C₅₃	—	2	2	no	—
6	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
7	C₅₃×#382	C₄	1	1	yes	characteristic
8	#13133157	—	4	2	no	—
9	#13133157	—	4	2	no	—
10	C₂×#11719908	—	2	4	no	—
11	C₂²×Q₈×C₅₃	—	2	1	no	—
12	C₂²×C₈×C₅₃	D₄	1	2	yes	—
13	C₄⋊Q₈×C₅₃	—	2	1	no	—
14	C₄⋊Q₈×C₅₃	—	4	1	no	—
15	C₄⋊Q₈×C₅₃	C₈	1	2	yes	—
16	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
17	C₂×C₄⋊C₄×C₅₃	—	4	1	no	—
18	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
19	#103×C₅₃	—	4	2	no	—
20	#103×C₅₃	—	4	2	no	—
21	#97×C₅₃	—	4	1	no	—
22	#95×C₅₃	—	4	1	no	—
23	C₂×Q₈×C₅₃	—	2	2	no	—
24	C₂×Q₈×C₅₃	—	4	1	no	—
25	C₂×Q₈×C₅₃	—	4	2	no	—
26	C₂×Q₈×C₅₃	—	4	2	no	—
27	C₂²×C₄×C₅₃	—	2	1	no	—
28	C₂²×C₄×C₅₃	—	4	1	no	—
29	C₂²×C₄×C₅₃	—	4	1	no	—
30	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×C₈×C₅₃	—	2	2	no	—
32	C₂×C₈×C₅₃	—	2	2	no	—
33	C₂×C₈×C₅₃	D₈	1	2	yes	—
34	C₂×C₈×C₅₃	SD₁₆	1	2	yes	—
35	C₄⋊C₄×C₅₃	—	4	2	no	—
36	C₄⋊C₄×C₅₃	—	4	2	no	—
37	C₄⋊C₄×C₅₃	—	4	2	no	—
38	C₄²×C₅₃	—	2	1	no	—
39	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
40	C₄²×C₅₃	M₁₆	1	1	yes	characteristic
41	#11719908	—	4	4	no	—
42	C₂³×C₅₃	#100	1	1	yes	characteristic
43	Q₈×C₅₃	—	4	4	no	—
44	Q₈×C₅₃	—	8	2	no	—
45	C₂×C₄×C₅₃	—	2	1	no	—
46	C₂×C₄×C₅₃	—	2	1	no	—
47	C₂×C₄×C₅₃	—	4	2	no	—
48	C₂×C₄×C₅₃	—	4	2	no	—
49	C₂×C₄×C₅₃	—	8	1	no	—
50	C₂×C₄×C₅₃	—	8	1	no	—
51	C₂×C₄×C₅₃	—	8	2	no	—
52	C₂×C₄×C₅₃	—	8	1	no	—
53	C₂×C₄×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
54	C₂×C₄×C₅₃	#99	1	1	yes	characteristic
55	C₂×C₄×C₅₃	#102	1	2	yes	—
56	C₂×C₄×C₅₃	#104	1	2	yes	—
57	C₈×C₅₃	—	2	2	no	—
58	C₈×C₅₃	—	2	2	no	—
59	#7117	—	53	1	no	maximal, Sylow
60	C₂²×C₅₃	—	2	2	no	—
61	C₂²×C₅₃	#323	1	1	yes	characteristic
62	C₂²×C₅₃	#329	1	1	yes	characteristic
63	C₂²×C₅₃	#331	1	1	yes	characteristic
64	C₄×C₅₃	—	4	1	no	—
65	C₄×C₅₃	—	4	1	no	—
66	C₄×C₅₃	—	8	2	no	—
67	C₄×C₅₃	—	16	1	no	—
68	C₄×C₅₃	D₄⋊C₈	1	2	yes	—
69	C₄×C₅₃	#327	1	2	yes	—
70	#1960	D₅₃	1	1	yes	characteristic
71	#1375	—	53	2	no	—
72	C₂×C₅₃	—	2	1	no	—
73	C₂×C₅₃	—	2	1	no	—
74	C₂×C₅₃	#1270	1	1	yes	characteristic
75	C₂×C₅₃	#1356	1	2	yes	—
76	C₂×C₄⋊Q₈	Dic₅₃	1	1	yes	characteristic
77	C₂×M₃₂	—	106	2	no	—
78	C₂×#103	—	2	2	no	—
79	C₂×#97	C₂×D₅₃	1	1	yes	characteristic
80	#382	Dic₅₃	1	1	yes	characteristic
81	C₅₃	#7117	1	1	yes	characteristic, complemented, Sylow
82	C₂²×Q₈	—	2	1	no	—
83	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
84	C₄⋊Q₈	—	2	1	no	—
85	C₄⋊Q₈	—	4	1	no	—
86	C₄⋊Q₈	#88322	1	2	yes	—
87	C₂×C₄⋊C₄	—	2	1	no	—
88	C₂×C₄⋊C₄	—	4	1	no	—
89	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
90	M₃₂	—	212	2	no	—
91	M₃₂	—	212	2	no	—
92	C₂×C₁₆	—	106	4	no	—
93	#103	—	4	2	no	—
94	#103	—	4	2	no	—
95	#97	—	4	1	no	—
96	#95	—	4	1	no	—
97	C₂×Q₈	—	2	2	no	—
98	C₂×Q₈	—	4	1	no	—
99	C₂×Q₈	—	4	2	no	—
100	C₂×Q₈	—	4	2	no	—
101	C₂²×C₄	—	2	1	no	—
102	C₂²×C₄	—	4	1	no	—
103	C₂²×C₄	—	4	1	no	—
104	C₂²×C₄	C₂²⋊Dic₅₃	1	1	yes	characteristic
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	#11719922	1	2	yes	—
108	C₂×C₈	#11719923	1	2	yes	—
109	C₄⋊C₄	—	4	2	no	—
110	C₄⋊C₄	—	4	2	no	—
111	C₄⋊C₄	—	4	2	no	—
112	C₄²	—	2	1	no	—
113	C₄²	C₂×#88322	1	1	yes	characteristic
114	C₄²	#11719917	1	1	yes	characteristic
115	C₁₆	—	212	4	no	—
116	C₂³	#13133134	1	1	yes	characteristic
117	Q₈	—	4	4	no	—
118	Q₈	—	8	2	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	8	1	no	—
124	C₂×C₄	—	8	1	no	—
125	C₂×C₄	—	8	2	no	—
126	C₂×C₄	—	8	1	no	—
127	C₂×C₄	C₂²⋊#88322	1	1	yes	characteristic
128	C₂×C₄	#13133132	1	1	yes	characteristic
129	C₂×C₄	#13133138	1	2	yes	—
130	C₂×C₄	#13133144	1	2	yes	—
131	C₈	—	2	2	no	—
132	C₈	—	2	2	no	—
133	C₂²	—	2	2	no	—
134	C₂²	#16810347	1	1	yes	characteristic
135	C₂²	#16810363	1	1	yes	characteristic
136	C₂²	#16810367	1	1	yes	characteristic
137	C₄	—	4	1	no	—
138	C₄	—	4	1	no	—
139	C₄	—	8	2	no	—
140	C₄	—	16	1	no	—
141	C₄	D₄⋊#88322	1	2	yes	—
142	C₄	#16810357	1	2	yes	—
143	C₂	—	2	1	no	—
144	C₂	—	2	1	no	—
145	C₂	#20272536	1	1	yes	characteristic
146	C₂	#20272750	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20272795

C₂

yes

maximal

C₅₃×#1960

C₂

yes

maximal, characteristic

C₂×#13133157

—

C₂×C₄⋊Q₈×C₅₃

C₄

yes

characteristic

C₂×#103×C₅₃

—

C₂×#97×C₅₃

C₂²

yes

characteristic

C₅₃×#382

C₄

yes

—

—

—

C₂²×Q₈×C₅₃

—

C₂²×C₈×C₅₃

D₄

yes

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

C₈

yes

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

#103×C₅₃

—

#103×C₅₃

—

#97×C₅₃

—

#95×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×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

D₈

yes

—

C₂×C₈×C₅₃

SD₁₆

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×C₈

yes

characteristic

C₄²×C₅₃

M₁₆

yes

characteristic

#11719908

—

C₂³×C₅₃

#100

yes

characteristic

Q₈×C₅₃

—

Q₈×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

C₂²⋊C₈

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

—

C₂×C₄×C₅₃

#104

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

#7117

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

D₄⋊C₈

yes

—

C₄×C₅₃

#327

yes

—

#1960

D₅₃

yes

characteristic

#1375

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#1270

yes

characteristic

C₂×C₅₃

#1356

yes

—

C₂×C₄⋊Q₈

Dic₅₃

yes

characteristic

C₂×M₃₂

—

106

—

C₂×#103

—

C₂×#97

C₂×D₅₃

yes

characteristic

#382

Dic₅₃

yes

characteristic

C₅₃

#7117

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

C₂²⋊D₅₃

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

#88322

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×Dic₅₃

yes

characteristic

M₃₂

—

212

—

M₃₂

—

212

—

C₂×C₁₆

—

106

—

—

—

—

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

C₂²⋊Dic₅₃

yes

characteristic

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

#11719922

yes

—

108

C₂×C₈

#11719923

yes

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄²

—

113

C₄²

C₂×#88322

yes

characteristic

114

C₄²

#11719917

yes

characteristic

115

C₁₆

—

212

—

116

C₂³

#13133134

yes

characteristic

117

Q₈

—

118

Q₈

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

C₂²⋊#88322

yes

characteristic

128

C₂×C₄

#13133132

yes

characteristic

129

C₂×C₄

#13133138

yes

—

130

C₂×C₄

#13133144

yes

—

131

C₈

—

132

C₈

—

133

C₂²

—

134

C₂²

#16810347

yes

characteristic

135

C₂²

#16810363

yes

characteristic

136

C₂²

#16810367

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

D₄⋊#88322

yes

—

142

C₄

#16810357

yes

—

143

C₂

—

144

C₂

—

145

C₂

#20272536

yes

characteristic

146

C₂

#20272750

yes

—

	subgroup	G/H
center	C₂²	#16810363
commutator	C₄²×C₅₃	C₂×C₈
Frattini	C₂×#97	C₂×D₅₃
Fitting	C₅₃×#1960	C₂
socle	C₂²×C₅₃	#329
prime radical	C₂³×C₅₃	#100
cosocle	C₂×#97×C₅₃	C₂²
hypercenter	#1960	D₅₃
hypocenter	C₅₃	#7117
LCS	G ▷ C₄²×C₅₃ ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#97 ◁ #1960

subgroup

G/H

center

C₂²

#16810363

commutator

C₄²×C₅₃

C₂×C₈

Frattini

C₂×#97

C₂×D₅₃

Fitting

C₅₃×#1960

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 ◁ #1960

N	H ≅ G/N	type
81. C₅₃	#7117	semidirect

H ≅ G/N

type

81. C₅₃

#7117

semidirect

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

subgroup	index
#20272795	2
C₅₃×#1960	2
#7117	53

subgroup

index

#20272795

C₅₃×#1960

#7117

Subgroups — #27270872

Conjugacy classes of subgroups

Complements

Maximal subgroups