Group #27366172 — subgroups

Subgroups — #27366172

Order 13568.

876 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 876 conjugacy classes of subgroups — too many to draw. Only its 179 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	#20285091	C₂	1	3	yes	maximal
2	#20274017	C₂	1	8	yes	maximal, complemented
3	#20273310	C₂	1	3	yes	maximal, complemented
4	C₅₃×#3556	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#13133214	C₂²	1	1	yes	characteristic
6	#16810989	C₂²	1	12	yes	—
7	#16810971	—	2	6	no	—
8	#16810627	—	2	6	no	—
9	C₄×#11719917	C₂²	1	3	yes	—
10	#16810365	C₂²	1	12	yes	complemented
11	C₅₃×#564	C₂²	1	4	yes	complemented
12	C₅₃×#557	—	2	6	no	—
13	C₅₃×#541	—	2	6	no	—
14	C₅₃×#533	C₂²	1	3	yes	—
15	C₂²×D₂₁₂	—	2	3	no	—
16	C₂×#11719917	—	2	3	no	—
17	#13133214	—	2	3	no	—
18	#13133214	C₂³	1	2	yes	—
19	#13133137	—	2	24	no	—
20	#13133128	C₂³	1	6	yes	—
21	C₄×#88322	—	2	3	no	—
22	C₄⋊Q₈×C₅₃	—	2	6	no	—
23	C₄⋊Q₈×C₅₃	D₄	1	6	yes	—
24	#126×C₅₃	—	2	8	no	—
25	#125×C₅₃	—	2	6	no	—
26	#125×C₅₃	—	2	24	no	—
27	#125×C₅₃	C₂³	1	6	yes	—
28	C₂²⋊Q₈×C₅₃	—	2	6	no	—
29	C₄×Q₈×C₅₃	—	2	12	no	—
30	C₄×Q₈×C₅₃	—	2	6	no	—
31	#117×C₅₃	D₄	1	6	yes	—
32	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
33	C₂³×D₅₃	—	4	1	no	—
34	C₂×Q₈×C₅₃	—	2	6	no	—
35	C₂²×C₄×C₅₃	C₂×D₄	1	3	yes	—
36	C₂×D₂₁₂	—	2	6	no	—
37	C₂×D₂₁₂	—	4	12	no	—
38	C₂×D₂₁₂	—	4	3	no	—
39	C₄⋊C₄×C₅₃	—	2	24	no	—
40	C₄⋊C₄×C₅₃	—	2	6	no	—
41	C₄⋊C₄×C₅₃	—	2	12	no	—
42	C₄⋊C₄×C₅₃	C₂×D₄	1	12	yes	—
43	C₂²⋊C₄×C₅₃	—	2	6	no	—
44	C₄²×C₅₃	—	2	8	no	—
45	C₄²×C₅₃	—	2	6	no	—
46	C₄²×C₅₃	C₂×D₄	1	3	yes	—
47	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
48	#11719917	—	4	6	no	—
49	C₂×#88322	—	2	6	no	—
50	C₂³×C₅₃	#120	1	1	yes	characteristic
51	C₂²×D₅₃	—	4	2	no	—
52	C₂²×D₅₃	—	8	6	no	—
53	Q₈×C₅₃	—	4	12	no	—
54	C₂×C₄×C₅₃	—	2	6	no	—
55	C₂×C₄×C₅₃	—	2	12	no	—
56	C₂×C₄×C₅₃	#120	1	3	yes	—
57	C₂×C₄×C₅₃	C₂²×D₄	1	3	yes	—
58	D₂₁₂	—	4	12	no	—
59	D₂₁₂	—	8	6	no	—
60	#88322	—	4	6	no	—
61	#19083	—	53	1	no	maximal, Sylow
62	C₂²×C₅₃	—	2	3	no	—
63	C₂²×C₅₃	C₂×#120	1	1	yes	characteristic
64	C₂×D₅₃	—	8	6	no	—
65	C₂×D₅₃	—	16	4	no	—
66	C₄×C₅₃	—	2	3	no	—
67	C₄×C₅₃	—	4	12	no	—
68	C₄×C₅₃	#454	1	6	yes	—
69	#3556	D₅₃	1	1	yes	characteristic, complemented
70	#3138	—	53	3	no	—
71	#1658	—	53	8	no	—
72	#1526	—	53	3	no	—
73	C₂×C₅₃	—	4	1	no	—
74	C₂×C₅₃	C₂×#454	1	3	yes	—
75	D₅₃	—	16	2	no	—
76	#564	C₂×D₅₃	1	4	yes	complemented
77	#557	—	2	6	no	—
78	#541	—	2	6	no	—
79	#533	C₂×D₅₃	1	3	yes	—
80	C₂×#127	—	53	1	no	—
81	#490	—	53	12	no	—
82	#486	—	106	6	no	—
83	#418	—	106	6	no	—
84	C₄×M₁₆	—	53	3	no	—
85	#330	—	53	12	no	—
86	C₅₃	#19083	1	1	yes	characteristic, complemented, Sylow
87	C₂²×D₄	—	106	3	no	—
88	C₂×M₁₆	—	106	3	no	—
89	C₄⋊Q₈	—	2	6	no	—
90	C₄⋊Q₈	C₂²⋊D₅₃	1	6	yes	—
91	#127	—	53	2	no	—
92	#127	—	106	3	no	—
93	#126	—	2	8	no	—
94	#125	—	2	6	no	—
95	#125	—	2	24	no	—
96	#125	C₂²×D₅₃	1	6	yes	—
97	C₂²⋊Q₈	—	2	6	no	—
98	C₄×Q₈	—	2	12	no	—
99	C₄×Q₈	—	2	6	no	—
100	#117	C₂²⋊D₅₃	1	6	yes	—
101	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
102	#102	—	106	24	no	—
103	#97	—	53	6	no	—
104	C₄×C₈	—	106	3	no	—
105	C₂⁴	—	212	1	no	—
106	C₂×Q₈	—	2	6	no	—
107	C₂×D₄	—	106	6	no	—
108	C₂×D₄	—	212	12	no	—
109	C₂×D₄	—	212	3	no	—
110	C₂²×C₄	C₂×C₂²⋊D₅₃	1	3	yes	—
111	M₁₆	—	212	6	no	—
112	C₂×C₈	—	106	6	no	—
113	C₄⋊C₄	—	2	24	no	—
114	C₄⋊C₄	—	2	6	no	—
115	C₄⋊C₄	—	2	12	no	—
116	C₄⋊C₄	C₂×C₂²⋊D₅₃	1	12	yes	—
117	C₂²⋊C₄	—	2	6	no	—
118	C₄²	—	2	8	no	—
119	C₄²	—	2	6	no	—
120	C₄²	C₂×C₂²⋊D₅₃	1	3	yes	—
121	C₄²	C₂³×D₅₃	1	1	yes	characteristic
122	C₂³	—	424	6	no	—
123	C₂³	—	212	2	no	—
124	C₂³	#13133186	1	1	yes	characteristic
125	Q₈	—	4	12	no	—
126	D₄	—	424	6	no	—
127	D₄	—	212	12	no	—
128	C₂×C₄	—	2	6	no	—
129	C₂×C₄	—	2	12	no	—
130	C₂×C₄	#13133186	1	3	yes	—
131	C₂×C₄	C₂²×C₂²⋊D₅₃	1	3	yes	—
132	C₈	—	212	6	no	—
133	C₂²	—	2	3	no	—
134	C₂²	—	848	4	no	—
135	C₂²	—	424	6	no	—
136	C₂²	C₂×#13133186	1	1	yes	characteristic
137	C₄	—	2	3	no	—
138	C₄	—	4	12	no	—
139	C₄	#16810796	1	6	yes	—
140	C₂	—	4	1	no	—
141	C₂	—	848	2	no	—
142	C₂	C₂×#16810796	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20285091

C₂

yes

maximal

#20274017

C₂

yes

maximal, complemented

#20273310

C₂

yes

maximal, complemented

C₅₃×#3556

C₂

yes

maximal, characteristic, complemented

C₂×#13133214

C₂²

yes

characteristic

#16810989

C₂²

yes

—

#16810971

—

#16810627

—

C₄×#11719917

C₂²

yes

—

#16810365

C₂²

yes

complemented

C₅₃×#564

C₂²

yes

complemented

C₅₃×#557

—

C₅₃×#541

—

C₅₃×#533

C₂²

yes

—

C₂²×D₂₁₂

—

C₂×#11719917

—

#13133214

—

#13133214

C₂³

yes

—

#13133137

—

#13133128

C₂³

yes

—

C₄×#88322

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

D₄

yes

—

#126×C₅₃

—

#125×C₅₃

—

#125×C₅₃

—

#125×C₅₃

C₂³

yes

—

C₂²⋊Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

#117×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₂³×D₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂×D₄

yes

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

C₂⁴

yes

characteristic

#11719917

—

C₂×#88322

—

C₂³×C₅₃

#120

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#120

yes

—

C₂×C₄×C₅₃

C₂²×D₄

yes

—

D₂₁₂

—

D₂₁₂

—

#88322

—

#19083

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

C₂×#120

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#454

yes

—

#3556

D₅₃

yes

characteristic, complemented

#3138

—

#1658

—

#1526

—

C₂×C₅₃

—

C₂×C₅₃

C₂×#454

yes

—

D₅₃

—

#564

C₂×D₅₃

yes

—

—

C₂×D₅₃

yes

—

C₂×#127

—

#490

—

#486

—

106

—

#418

—

106

—

C₄×M₁₆

—

#330

—

C₅₃

#19083

yes

characteristic, complemented, Sylow

C₂²×D₄

—

106

—

C₂×M₁₆

—

106

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂²⋊D₅₃

yes

—

#127

—

#127

—

106

—

—

—

—

C₂²×D₅₃

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

100

#117

C₂²⋊D₅₃

yes

—

101

C₂×C₄²

C₂²×D₅₃

yes

characteristic

102

#102

—

106

—

103

#97

—

104

C₄×C₈

—

106

—

105

C₂⁴

—

212

—

106

C₂×Q₈

—

107

C₂×D₄

—

106

—

108

C₂×D₄

—

212

—

109

C₂×D₄

—

212

—

110

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

—

111

M₁₆

—

212

—

112

C₂×C₈

—

106

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

—

116

C₄⋊C₄

C₂×C₂²⋊D₅₃

yes

—

117

C₂²⋊C₄

—

118

C₄²

—

119

C₄²

—

120

C₄²

C₂×C₂²⋊D₅₃

yes

—

121

C₄²

C₂³×D₅₃

yes

characteristic

122

C₂³

—

424

—

123

C₂³

—

212

—

124

C₂³

#13133186

yes

characteristic

125

Q₈

—

126

D₄

—

424

—

127

D₄

—

212

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

#13133186

yes

—

131

C₂×C₄

C₂²×C₂²⋊D₅₃

yes

—

132

C₈

—

212

—

133

C₂²

—

134

C₂²

—

848

—

135

C₂²

—

424

—

136

C₂²

C₂×#13133186

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

#16810796

yes

—

140

C₂

—

141

C₂

—

848

—

142

C₂

C₂×#16810796

yes

—

	subgroup	G/H
center	C₂²	C₂×#13133186
commutator, cosocle	C₄²×C₅₃	C₂⁴
Frattini	C₄²	C₂³×D₅₃
Fitting	C₅₃×#3556	C₂
socle	C₂²×C₅₃	C₂×#120
prime radical	C₂×#13133214	C₂²
hypercenter	#3556	D₅₃
hypocenter	C₅₃	#19083
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3556

subgroup

G/H

center

C₂²

C₂×#13133186

commutator, cosocle

C₄²×C₅₃

C₂⁴

Frattini

C₄²

C₂³×D₅₃

Fitting

C₅₃×#3556

C₂

C₂²×C₅₃

C₂²

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #3556

N	H ≅ G/N	type
2. #20274017	C₂	semidirect
3. #20273310	C₂	semidirect
4. C₅₃×#3556	C₂	semidirect
10. #16810365	C₂²	semidirect
11. C₅₃×#564	C₂²	semidirect
69. #3556	D₅₃	semidirect
76. #564	C₂×D₅₃	semidirect
86. C₅₃	#19083	semidirect

H ≅ G/N

type

2. #20274017

C₂

semidirect

3. #20273310

C₂

semidirect

4. C₅₃×#3556

C₂

semidirect

10. #16810365

C₂²

semidirect

11. C₅₃×#564

C₂²

semidirect

69. #3556

D₅₃

semidirect

76. #564

C₂×D₅₃

semidirect

86. C₅₃

#19083

semidirect

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

subgroup	index
#20285091	2
#20274017	2
#20273310	2
C₅₃×#3556	2
#19083	53

subgroup

index

C₅₃×#3556

Subgroups — #27366172

Conjugacy classes of subgroups

Complements

Maximal subgroups