Group #27295852 — subgroups

Subgroups — #27295852

Order 13568.

212 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 75 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	#20276597	C₂	1	1	yes	maximal, characteristic
2	#20276551	C₂	1	2	yes	maximal, complemented
3	#20272687	C₂	1	2	yes	maximal, complemented
4	#20272631	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1373	C₂	1	1	yes	maximal, characteristic, complemented
6	#16811055	C₄	1	1	yes	characteristic
7	#16811041	—	2	2	no	—
8	#16810741	C₂²	1	2	yes	—
9	#16810453	C₄	1	1	yes	characteristic
10	#16810431	C₄	1	2	yes	—
11	#16810408	—	2	2	no	—
12	C₄×#11719908	C₂²	1	1	yes	characteristic
13	C₈×#88322	C₂²	1	1	yes	characteristic
14	C₅₃×#364	C₂²	1	2	yes	—
15	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
16	#13133239	C₂×C₄	1	2	yes	—
17	#13133223	—	2	2	no	—
18	Dic₄₂₄	C₈	1	1	yes	characteristic, complemented
19	#13133162	C₈	1	2	yes	complemented
20	D₈₄₈	C₈	1	1	yes	characteristic, complemented
21	#13133156	—	4	2	no	—
22	C₂×#11719908	D₄	1	1	yes	characteristic
23	C₂×#11719908	D₄	1	1	yes	characteristic
24	C₁₆×D₅₃	—	4	2	no	—
25	#13133153	C₂×C₄	1	2	yes	—
26	#13133145	—	2	2	no	—
27	C₄×#88322	—	2	1	no	—
28	M₃₂×C₅₃	—	2	2	no	—
29	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
32	#11719945	—	2	2	no	—
33	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
34	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
35	C₄²×C₅₃	#41	1	1	yes	characteristic
36	#11719917	—	2	2	no	—
37	C₂×#88322	—	2	2	no	—
38	Dic₂₁₂	C₂×C₈	1	2	yes	—
39	D₄₂₄	C₂×C₈	1	2	yes	—
40	#11719913	—	2	2	no	—
41	#11719912	—	4	2	no	—
42	C₈×D₅₃	—	4	2	no	—
43	C₁₆×C₅₃	—	2	1	no	—
44	C₁₆×C₅₃	—	4	2	no	—
45	C₁₆×C₅₃	C₂×C₈	1	1	yes	characteristic
46	C₁₆×C₅₃	C₂×C₈	1	1	yes	characteristic
47	#11719908	—	2	1	no	—
48	#11719908	D₈	1	2	yes	—
49	C₂×C₄×C₅₃	—	2	1	no	—
50	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
51	C₂²⋊D₅₃	—	4	2	no	—
52	D₂₁₂	—	2	2	no	—
53	C₄×D₅₃	—	4	2	no	—
54	Dic₁₀₆	—	2	2	no	—
55	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
56	C₈×C₅₃	#131	1	1	yes	characteristic
57	C₈×C₅₃	C₂×D₈	1	1	yes	characteristic
58	C₈×C₅₃	#135	1	1	yes	characteristic
59	#88322	—	4	2	no	—
60	#88322	—	4	2	no	—
61	#12047	—	53	1	no	maximal, Sylow
62	C₂²×C₅₃	#443	1	1	yes	characteristic
63	C₂×D₅₃	—	4	2	no	—
64	C₄×C₅₃	—	2	2	no	—
65	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
66	C₄×C₅₃	C₄×D₈	1	1	yes	characteristic
67	Dic₅₃	—	4	2	no	—
68	#2178	—	53	1	no	—
69	#2170	—	53	2	no	—
70	#1373	D₅₃	1	1	yes	characteristic, complemented
71	#1335	—	53	2	no	—
72	#1311	—	53	1	no	—
73	C₂×C₅₃	—	2	1	no	—
74	C₂×C₅₃	C₈×D₈	1	1	yes	characteristic
75	D₅₃	—	8	2	no	—
76	#508	—	53	1	no	—
77	#504	—	106	2	no	—
78	#443	—	53	2	no	—
79	#368	—	53	1	no	—
80	#364	C₂×D₅₃	1	2	yes	—
81	#359	—	53	2	no	—
82	#350	—	106	2	no	—
83	C₄×C₁₆	—	53	1	no	—
84	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
85	C₈²	—	53	1	no	—
86	C₅₃	#12047	1	1	yes	characteristic, complemented, Sylow
87	#135	—	53	2	no	—
88	#131	—	106	2	no	—
89	Q₃₂	—	53	1	no	—
90	SD₃₂	—	53	2	no	—
91	D₁₆	—	53	1	no	—
92	M₃₂	—	2	2	no	—
93	M₃₂	—	212	2	no	—
94	C₂×C₁₆	—	53	1	no	—
95	C₂×C₁₆	—	53	1	no	—
96	C₂×C₁₆	—	212	2	no	—
97	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
98	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
99	#108	—	53	2	no	—
100	#104	—	106	2	no	—
101	C₄×C₈	—	106	1	no	—
102	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
103	#41	—	106	2	no	—
104	Q₁₆	—	53	2	no	—
105	SD₁₆	—	106	2	no	—
106	D₈	—	53	2	no	—
107	M₁₆	—	106	2	no	—
108	M₁₆	—	212	2	no	—
109	C₂×C₈	—	106	2	no	—
110	C₂×C₈	—	212	2	no	—
111	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
112	C₂×C₈	D₄×D₅₃	1	1	yes	characteristic
113	C₄²	#11719949	1	1	yes	characteristic
114	C₁₆	—	2	1	no	—
115	C₁₆	—	4	2	no	—
116	C₁₆	—	53	2	no	—
117	C₁₆	—	106	1	no	—
118	C₁₆	C₈×D₅₃	1	1	yes	characteristic
119	C₁₆	C₈×D₅₃	1	1	yes	characteristic
120	Q₈	—	106	2	no	—
121	D₄	—	106	2	no	—
122	D₄	—	212	2	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	212	2	no	—
125	C₂×C₄	#13133179	1	1	yes	characteristic
126	C₈	—	212	2	no	—
127	C₈	—	212	2	no	—
128	C₈	C₂×C₈×D₅₃	1	1	yes	characteristic
129	C₈	#13133225	1	1	yes	characteristic
130	C₈	D₈×D₅₃	1	1	yes	characteristic
131	C₈	#13133240	1	1	yes	characteristic
132	C₂²	—	212	2	no	—
133	C₂²	#16810739	1	1	yes	characteristic
134	C₄	—	2	2	no	—
135	C₄	—	212	2	no	—
136	C₄	#16810688	1	1	yes	characteristic
137	C₄	#16810705	1	1	yes	characteristic
138	C₂	—	2	1	no	—
139	C₂	—	424	2	no	—
140	C₂	#20273509	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20276597

C₂

yes

maximal, characteristic

#20276551

C₂

yes

maximal, complemented

#20272687

C₂

yes

maximal, complemented

#20272631

C₂

yes

maximal, characteristic, complemented

C₅₃×#1373

C₂

yes

maximal, characteristic, complemented

#16811055

C₄

yes

characteristic

#16811041

—

#16810741

C₂²

yes

—

#16810453

C₄

yes

characteristic

#16810431

C₄

yes

—

#16810408

—

C₄×#11719908

C₂²

yes

characteristic

C₈×#88322

C₂²

yes

characteristic

C₅₃×#364

C₂²

yes

—

C₄×C₁₆×C₅₃

C₂²

yes

characteristic

#13133239

C₂×C₄

yes

—

#13133223

—

Dic₄₂₄

C₈

yes

characteristic, complemented

#13133162

C₈

yes

complemented

D₈₄₈

C₈

yes

characteristic, complemented

#13133156

—

C₂×#11719908

D₄

yes

characteristic

C₂×#11719908

D₄

yes

characteristic

C₁₆×D₅₃

—

#13133153

C₂×C₄

yes

—

#13133145

—

C₄×#88322

—

M₃₂×C₅₃

—

C₂×C₁₆×C₅₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

C₂³

yes

characteristic

#11719945

—

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

C₄²×C₅₃

yes

—

—

Dic₂₁₂

C₂×C₈

yes

—

D₄₂₄

C₂×C₈

yes

—

#11719913

—

#11719912

—

C₈×D₅₃

—

C₁₆×C₅₃

—

C₁₆×C₅₃

—

C₁₆×C₅₃

C₂×C₈

yes

characteristic

C₁₆×C₅₃

C₂×C₈

yes

characteristic

#11719908

—

#11719908

D₈

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄×D₄

yes

characteristic

C₂²⋊D₅₃

—

D₂₁₂

—

C₄×D₅₃

—

Dic₁₀₆

—

C₈×C₅₃

C₂²×C₈

yes

characteristic

C₈×C₅₃

#131

yes

characteristic

C₈×C₅₃

C₂×D₈

yes

C₈×C₅₃

yes

—

—

—

C₂²×C₅₃

yes

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₈×D₄

yes

characteristic

C₄×C₅₃

C₄×D₈

yes

characteristic

Dic₅₃

—

#2178

—

#2170

—

#1373

D₅₃

yes

characteristic, complemented

#1335

—

#1311

—

C₂×C₅₃

—

C₂×C₅₃

C₈×D₈

yes

characteristic

D₅₃

—

#508

—

#504

—

106

—

#443

—

#368

—

#364

C₂×D₅₃

yes

—

#359

—

#350

—

106

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₅₃

yes

characteristic

C₈²

—

C₅₃

#12047

yes

characteristic, complemented, Sylow

#135

—

#131

—

106

—

Q₃₂

—

SD₃₂

—

D₁₆

—

M₃₂

—

M₃₂

—

212

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

212

—

C₂×C₁₆

C₄×D₅₃

yes

characteristic

C₂×C₁₆

C₄×D₅₃

yes

characteristic

#108

—

100

#104

—

106

—

101

C₄×C₈

—

106

—

102

C₄×C₈

C₂²×D₅₃

yes

characteristic

103

#41

—

106

—

104

Q₁₆

—

105

SD₁₆

—

106

—

106

D₈

—

107

M₁₆

—

106

—

108

M₁₆

—

212

—

109

C₂×C₈

—

106

—

110

C₂×C₈

—

212

—

111

C₂×C₈

C₂×C₄×D₅₃

yes

characteristic

112

C₂×C₈

D₄×D₅₃

yes

characteristic

113

C₄²

#11719949

yes

characteristic

114

C₁₆

—

115

C₁₆

—

116

C₁₆

—

117

C₁₆

—

106

—

118

C₁₆

C₈×D₅₃

yes

characteristic

119

C₁₆

C₈×D₅₃

yes

characteristic

120

Q₈

—

106

—

121

D₄

—

106

—

122

D₄

—

212

—

123

C₂×C₄

—

124

C₂×C₄

—

212

—

125

C₂×C₄

#13133179

yes

characteristic

126

C₈

—

212

—

127

C₈

—

212

—

128

C₈

C₂×C₈×D₅₃

yes

characteristic

129

C₈

#13133225

yes

characteristic

130

C₈

D₈×D₅₃

yes

characteristic

131

C₈

#13133240

yes

characteristic

132

C₂²

—

212

—

133

C₂²

#16810739

yes

characteristic

134

C₄

—

135

C₄

—

212

—

136

C₄

#16810688

yes

characteristic

137

C₄

#16810705

yes

characteristic

138

C₂

—

139

C₂

—

424

—

140

C₂

#20273509

yes

characteristic

	subgroup	G/H
center	C₈	D₈×D₅₃
commutator	C₈×C₅₃	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₅₃
Fitting	C₅₃×#1373	C₂
socle	C₂×C₅₃	C₈×D₈
prime radical	#16811055	C₄
cosocle	C₄×C₈×C₅₃	C₂³
hypercenter	#1373	D₅₃
hypocenter	C₅₃	#12047
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1373

subgroup

G/H

center

C₈

D₈×D₅₃

commutator

C₈×C₅₃

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₅₃

Fitting

C₅₃×#1373

C₂

socle

C₂×C₅₃

C₈×D₈

prime radical

#16811055

C₄

cosocle

C₄×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1373

N	H ≅ G/N	type
2. #20276551	C₂	semidirect
3. #20272687	C₂	semidirect
4. #20272631	C₂	semidirect
5. C₅₃×#1373	C₂	semidirect
18. Dic₄₂₄	C₈	semidirect
19. #13133162	C₈	semidirect
20. D₈₄₈	C₈	semidirect
70. #1373	D₅₃	semidirect
86. C₅₃	#12047	semidirect

H ≅ G/N

type

2. #20276551

C₂

semidirect

3. #20272687

C₂

semidirect

4. #20272631

C₂

semidirect

5. C₅₃×#1373

C₂

semidirect

18. Dic₄₂₄

C₈

semidirect

19. #13133162

C₈

semidirect

20. D₈₄₈

C₈

semidirect

70. #1373

D₅₃

semidirect

86. C₅₃

#12047

semidirect

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

subgroup	index
#20276597	2
#20276551	2
#20272687	2
#20272631	2
C₅₃×#1373	2
#12047	53

subgroup

index

C₅₃×#1373

Subgroups — #27295852

Conjugacy classes of subgroups

Complements

Maximal subgroups