Group #27304661 — subgroups

Subgroups — #27304661

Order 13568.

184 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 105 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	#20277009	C₂	1	2	yes	maximal, complemented
2	#20272913	C₂	1	1	yes	maximal, characteristic, complemented
3	#20272857	C₂	1	2	yes	maximal, complemented
4	#20272849	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#2170	C₂	1	1	yes	maximal, characteristic
6	#16810457	—	2	2	no	—
7	#16810457	C₂²	1	2	yes	—
8	C₂×#13133123	—	2	2	no	—
9	C₂×#13133123	C₂²	1	1	yes	characteristic
10	C₂×#13133123	C₂²	1	1	yes	characteristic
11	C₅₃×#504	C₂²	1	2	yes	—
12	C₅₃×#443	C₄	1	1	yes	characteristic
13	C₅₃×#364	C₄	1	1	yes	characteristic
14	C₅₃×#350	C₄	1	2	yes	—
15	C₄×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
16	#13133123	—	2	2	no	—
17	#13133123	—	2	1	no	—
18	#13133123	D₄	1	2	yes	—
19	#135×C₅₃	C₈	1	1	yes	characteristic
20	#131×C₅₃	C₂×C₄	1	2	yes	—
21	M₃₂×C₅₃	—	2	2	no	—
22	M₃₂×C₅₃	C₂×C₄	1	2	yes	—
23	C₂×C₁₆×C₅₃	—	2	2	no	—
24	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₅₃	C₂³	1	1	yes	characteristic
26	#108×C₅₃	C₈	1	1	yes	characteristic
27	#104×C₅₃	C₈	1	2	yes	—
28	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	#41×C₅₃	C₂×C₈	1	2	yes	—
30	Q₁₆×C₅₃	C₁₆	1	1	yes	characteristic
31	SD₁₆×C₅₃	C₁₆	1	2	yes	—
32	D₈×C₅₃	C₁₆	1	1	yes	characteristic
33	M₁₆×C₅₃	—	2	2	no	—
34	M₁₆×C₅₃	C₂×C₈	1	2	yes	—
35	C₂×C₈×C₅₃	—	2	2	no	—
36	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
37	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
38	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
39	C₁₆×C₅₃	—	2	2	no	—
40	C₁₆×C₅₃	—	2	1	no	—
41	C₁₆×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₁₆×C₅₃	#41	1	1	yes	characteristic
43	Q₈×C₅₃	C₂×C₁₆	1	2	yes	—
44	D₄×C₅₃	—	2	2	no	—
45	D₄×C₅₃	C₂×C₁₆	1	2	yes	—
46	C₂×C₄×C₅₃	—	2	2	no	—
47	C₂×C₄×C₅₃	—	2	1	no	—
48	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
49	C₈×C₅₃	—	2	2	no	—
50	C₈×C₅₃	C₂×C₁₆	1	1	yes	characteristic
51	C₈×C₅₃	C₂×C₁₆	1	1	yes	characteristic
52	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
53	C₈×C₅₃	#131	1	1	yes	characteristic
54	#13648	—	53	1	no	maximal, Sylow
55	C₂²×C₅₃	—	2	2	no	—
56	C₂²×C₅₃	#504	1	1	yes	characteristic
57	C₄×C₅₃	—	2	2	no	—
58	C₄×C₅₃	—	4	1	no	—
59	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
60	C₄×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
61	#2258	—	53	2	no	—
62	#2170	D₅₃	1	1	yes	characteristic
63	#1422	—	53	1	no	—
64	#1401	—	53	2	no	—
65	#1397	—	53	1	no	—
66	C₂×C₅₃	—	2	1	no	—
67	C₂×C₅₃	—	4	2	no	—
68	C₂×C₅₃	D₄×C₁₆	1	1	yes	characteristic
69	#504	C₂×D₅₃	1	2	yes	—
70	#443	Dic₅₃	1	1	yes	characteristic
71	M₆₄	—	53	2	no	—
72	M₆₄	—	106	2	no	—
73	C₂×C₃₂	—	53	1	no	—
74	C₂×C₃₂	—	53	1	no	—
75	C₂×C₃₂	—	106	2	no	—
76	#364	Dic₅₃	1	1	yes	characteristic
77	#350	Dic₅₃	1	2	yes	—
78	C₄×C₁₆	C₂×D₅₃	1	1	yes	characteristic
79	C₅₃	#13648	1	1	yes	characteristic, complemented, Sylow
80	#135	#88322	1	1	yes	characteristic
81	#131	C₂×Dic₅₃	1	2	yes	—
82	M₃₂	—	2	2	no	—
83	M₃₂	C₂×Dic₅₃	1	2	yes	—
84	C₂×C₁₆	—	2	2	no	—
85	C₂×C₁₆	C₂×Dic₅₃	1	1	yes	characteristic
86	C₂×C₁₆	C₂²×D₅₃	1	1	yes	characteristic
87	#108	#88322	1	1	yes	characteristic
88	#104	#88322	1	2	yes	—
89	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
90	C₃₂	—	53	2	no	—
91	C₃₂	—	106	2	no	—
92	C₃₂	—	106	1	no	—
93	#41	C₂×#88322	1	2	yes	—
94	Q₁₆	#11719908	1	1	yes	characteristic
95	SD₁₆	#11719908	1	2	yes	—
96	D₈	#11719908	1	1	yes	characteristic
97	M₁₆	—	2	2	no	—
98	M₁₆	C₂×#88322	1	2	yes	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	C₂×#88322	1	1	yes	characteristic
101	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
102	C₄²	C₂×#88322	1	1	yes	characteristic
103	C₁₆	—	2	2	no	—
104	C₁₆	—	2	1	no	—
105	C₁₆	D₄×D₅₃	1	1	yes	characteristic
106	C₁₆	#11719947	1	1	yes	characteristic
107	Q₈	C₂×#11719908	1	2	yes	—
108	D₄	—	2	2	no	—
109	D₄	C₂×#11719908	1	2	yes	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
113	C₈	—	2	2	no	—
114	C₈	C₂×#11719908	1	1	yes	characteristic
115	C₈	C₂×#11719908	1	1	yes	characteristic
116	C₈	D₄×Dic₅₃	1	1	yes	characteristic
117	C₈	#13133224	1	1	yes	characteristic
118	C₂²	—	2	2	no	—
119	C₂²	#16811040	1	1	yes	characteristic
120	C₄	—	2	2	no	—
121	C₄	—	4	1	no	—
122	C₄	D₄×#88322	1	1	yes	characteristic
123	C₄	C₂²×#11719908	1	1	yes	characteristic
124	C₂	—	2	1	no	—
125	C₂	—	4	2	no	—
126	C₂	D₄×#11719908	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20277009

C₂

yes

maximal, complemented

#20272913

C₂

yes

maximal, characteristic, complemented

#20272857

C₂

yes

maximal, complemented

#20272849

C₂

yes

maximal, characteristic, complemented

C₅₃×#2170

C₂

yes

maximal, characteristic

#16810457

—

#16810457

C₂²

yes

—

C₂×#13133123

—

C₂×#13133123

C₂²

yes

characteristic

C₂×#13133123

C₂²

yes

characteristic

C₅₃×#504

C₂²

yes

—

C₅₃×#443

C₄

yes

characteristic

C₅₃×#364

C₄

yes

characteristic

C₅₃×#350

C₄

yes

—

C₄×C₁₆×C₅₃

C₂²

yes

—

—

D₄

yes

—

#135×C₅₃

C₈

yes

characteristic

#131×C₅₃

C₂×C₄

yes

—

M₃₂×C₅₃

—

M₃₂×C₅₃

C₂×C₄

yes

—

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₃

C₂³

yes

characteristic

#108×C₅₃

C₈

yes

characteristic

#104×C₅₃

C₈

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

#41×C₅₃

C₂×C₈

yes

—

Q₁₆×C₅₃

C₁₆

yes

characteristic

SD₁₆×C₅₃

C₁₆

yes

—

D₈×C₅₃

C₁₆

yes

characteristic

M₁₆×C₅₃

—

M₁₆×C₅₃

C₂×C₈

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₈

yes

characteristic

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×C₈

yes

characteristic

C₁₆×C₅₃

—

C₁₆×C₅₃

—

C₁₆×C₅₃

C₂×D₄

yes

characteristic

C₁₆×C₅₃

#41

yes

characteristic

Q₈×C₅₃

C₂×C₁₆

yes

—

D₄×C₅₃

—

D₄×C₅₃

C₂×C₁₆

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₈

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

C₂×C₁₆

yes

characteristic

C₈×C₅₃

C₂×C₁₆

yes

characteristic

C₈×C₅₃

C₄×D₄

yes

C₈×C₅₃

yes

—

C₂²×C₅₃

—

C₂²×C₅₃

#504

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₈×D₄

yes

characteristic

C₄×C₅₃

C₂²×C₁₆

yes

characteristic

#2258

—

#2170

D₅₃

yes

—

—

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

D₄×C₁₆

yes

characteristic

#504

C₂×D₅₃

yes

—

#443

Dic₅₃

yes

characteristic

M₆₄

—

M₆₄

—

106

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

106

—

#364

Dic₅₃

yes

characteristic

#350

Dic₅₃

yes

—

C₄×C₁₆

C₂×D₅₃

yes

characteristic

C₅₃

#13648

yes

characteristic, complemented, Sylow

yes

C₂×Dic₅₃

yes

—

M₃₂

—

M₃₂

C₂×Dic₅₃

yes

—

C₂×C₁₆

—

C₂×C₁₆

C₂×Dic₅₃

yes

characteristic

C₂×C₁₆

C₂²×D₅₃

yes

yes

yes

—

C₄×C₈

C₂×Dic₅₃

yes

characteristic

C₃₂

—

C₃₂

—

106

—

C₃₂

—

106

—

#41

C₂×#88322

yes

—

Q₁₆

#11719908

yes

characteristic

SD₁₆

#11719908

yes

—

D₈

#11719908

yes

characteristic

M₁₆

—

M₁₆

C₂×#88322

yes

—

C₂×C₈

—

100

C₂×C₈

C₂×#88322

yes

characteristic

101

C₂×C₈

C₂²×Dic₅₃

yes

characteristic

102

C₄²

C₂×#88322

yes

characteristic

103

C₁₆

—

104

C₁₆

—

105

C₁₆

D₄×D₅₃

yes

characteristic

106

C₁₆

#11719947

yes

characteristic

107

Q₈

C₂×#11719908

yes

—

108

D₄

—

109

D₄

C₂×#11719908

yes

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

C₂²×#88322

yes

characteristic

113

C₈

—

114

C₈

C₂×#11719908

yes

characteristic

115

C₈

C₂×#11719908

yes

characteristic

116

C₈

D₄×Dic₅₃

yes

characteristic

117

C₈

#13133224

yes

characteristic

118

C₂²

—

119

C₂²

#16811040

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

D₄×#88322

yes

characteristic

123

C₄

C₂²×#11719908

yes

characteristic

124

C₂

—

125

C₂

—

126

C₂

D₄×#11719908

yes

characteristic

	subgroup	G/H
center	C₁₆	D₄×D₅₃
commutator	C₄×C₅₃	C₂²×C₁₆
Frattini	C₂×C₁₆	C₂²×D₅₃
Fitting	C₅₃×#2170	C₂
socle	C₂×C₅₃	D₄×C₁₆
prime radical	#135×C₅₃	C₈
cosocle	C₂×C₁₆×C₅₃	C₂³
hypercenter	#2170	D₅₃
hypocenter	C₅₃	#13648
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ #2170

subgroup

G/H

center

C₁₆

D₄×D₅₃

commutator

C₄×C₅₃

C₂²×C₁₆

Frattini

C₂×C₁₆

C₂²×D₅₃

Fitting

C₅₃×#2170

C₂

socle

C₂×C₅₃

D₄×C₁₆

prime radical

#135×C₅₃

C₈

cosocle

C₂×C₁₆×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₁₆ ◁ C₂×C₁₆ ◁ #2170

N	H ≅ G/N	type
1. #20277009	C₂	semidirect
2. #20272913	C₂	semidirect
3. #20272857	C₂	semidirect
4. #20272849	C₂	semidirect
79. C₅₃	#13648	semidirect

H ≅ G/N

type

1. #20277009

C₂

semidirect

2. #20272913

C₂

semidirect

3. #20272857

C₂

semidirect

4. #20272849

C₂

semidirect

79. C₅₃

#13648

semidirect

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

subgroup	index
#20277009	2
#20272913	2
#20272857	2
#20272849	2
C₅₃×#2170	2
#13648	53

subgroup

index

C₅₃×#2170

Subgroups — #27304661

Conjugacy classes of subgroups

Complements

Maximal subgroups