Group #28312584 — subgroups

Subgroups — #28312584

Order 13568.

218 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 116 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×#16811555	C₂	1	2	yes	maximal
2	#20289873	C₂	1	4	yes	maximal, complemented
3	C₂×C₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	#16811555	—	2	4	no	—
5	C₂×#13133267	C₂²	1	4	yes	—
6	C₂²×#11719908	C₂²	1	1	yes	characteristic
7	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
8	C₁₆×Dic₅₃	C₄	1	2	yes	—
9	C₁₆×Dic₅₃	C₂²	1	2	yes	—
10	C₂²×C₁₆×C₅₃	C₄	1	1	yes	characteristic
11	#13133267	—	2	4	no	—
12	C₂²×#88322	C₈	1	1	yes	characteristic
13	C₂×C₄×Dic₅₃	C₈	1	1	yes	characteristic
14	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
15	C₂×#11719908	D₄	1	2	yes	—
16	C₂×#11719908	Q₈	1	2	yes	—
17	C₂×#11719908	C₂³	1	1	yes	characteristic
18	C₈×Dic₅₃	C₈	1	2	yes	—
19	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
20	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₆×C₅₃	—	2	2	no	—
22	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
24	C₂²×Dic₅₃	C₁₆	1	2	yes	—
25	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
26	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
27	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
28	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
29	C₄×Dic₅₃	C₁₆	1	1	yes	characteristic
30	C₄×Dic₅₃	C₁₆	1	1	yes	characteristic
31	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
32	C₂×#88322	—	2	2	no	—
33	C₂×#88322	C₂×C₈	1	2	yes	—
34	C₁₆×C₅₃	—	2	2	no	—
35	#11719908	C₄⋊C₄	1	2	yes	—
36	#11719908	C₂×D₄	1	1	yes	characteristic
37	#11719908	C₂×Q₈	1	1	yes	characteristic
38	C₂³×C₅₃	C₂×C₁₆	1	1	yes	characteristic
39	C₂×C₄×C₅₃	C₄⋊C₈	1	2	yes	—
40	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
41	C₂×C₄×C₅₃	M₃₂	1	2	yes	—
42	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
43	C₂×Dic₅₃	—	2	4	no	—
44	C₂×Dic₅₃	C₂×C₁₆	1	2	yes	—
45	C₂×Dic₅₃	C₂×C₁₆	1	2	yes	—
46	C₈×C₅₃	C₄⋊C₈	1	2	yes	—
47	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
49	#88322	—	2	2	no	—
50	#13627	—	53	1	no	maximal, Sylow
51	C₂²×C₅₃	—	2	2	no	—
52	C₂²×C₅₃	C₄⋊C₁₆	1	2	yes	—
53	C₂²×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
54	C₄×C₅₃	C₄⋊C₁₆	1	2	yes	—
55	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
56	C₄×C₅₃	C₂×M₃₂	1	1	yes	characteristic
57	Dic₅₃	—	2	2	no	—
58	Dic₅₃	M₆₄	1	4	yes	complemented
59	C₂×M₆₄	—	53	2	no	—
60	C₂×C₄×C₁₆	—	53	1	no	—
61	C₄⋊C₃₂	—	53	4	no	—
62	C₂×C₅₃	—	2	2	no	—
63	C₂×C₅₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic
64	C₂×C₅₃	C₂×M₆₄	1	2	yes	—
65	C₂²×C₁₆	—	53	1	no	—
66	C₂²×C₁₆	#6157	1	1	yes	characteristic
67	C₂×C₄×C₈	—	53	1	no	—
68	M₆₄	—	106	4	no	—
69	C₂×C₃₂	—	53	4	no	—
70	C₄×C₁₆	—	53	2	no	—
71	C₄×C₁₆	—	53	2	no	—
72	C₅₃	#13627	1	1	yes	characteristic, complemented, Sylow
73	C₂²×C₈	—	53	1	no	—
74	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
75	C₂×C₄²	—	53	1	no	—
76	C₂×C₁₆	—	2	2	no	—
77	C₂×C₁₆	—	53	2	no	—
78	C₂×C₁₆	—	53	2	no	—
79	C₂×C₁₆	—	53	1	no	—
80	C₂×C₁₆	—	53	1	no	—
81	C₂×C₁₆	C₂×#6157	1	2	yes	—
82	C₄×C₈	—	53	2	no	—
83	C₄×C₈	—	53	2	no	—
84	C₃₂	—	106	4	no	—
85	C₂²×C₄	—	53	2	no	—
86	C₂²×C₄	D₅₃⋊C₈	1	1	yes	characteristic
87	C₂×C₈	—	53	2	no	—
88	C₂×C₈	—	106	2	no	—
89	C₂×C₈	D₅₃⋊C₈	1	1	yes	characteristic
90	C₂×C₈	C₄⋊#6157	1	2	yes	—
91	C₂×C₈	#11719940	1	2	yes	—
92	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
93	C₄²	—	53	1	no	—
94	C₄²	—	53	1	no	—
95	C₄²	—	53	2	no	—
96	C₁₆	—	2	2	no	—
97	C₁₆	—	53	1	no	—
98	C₁₆	—	53	2	no	—
99	C₁₆	—	53	1	no	—
100	C₂³	D₅₃⋊C₁₆	1	1	yes	characteristic
101	C₂×C₄	—	53	2	no	—
102	C₂×C₄	—	53	2	no	—
103	C₂×C₄	—	106	4	no	—
104	C₂×C₄	#13133284	1	2	yes	—
105	C₂×C₄	D₅₃⋊C₁₆	1	1	yes	characteristic
106	C₂×C₄	#13133292	1	2	yes	—
107	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
108	C₈	—	106	2	no	—
109	C₈	#13133284	1	2	yes	—
110	C₈	C₂×C₄⋊#6157	1	1	yes	characteristic
111	C₈	C₂×#11719940	1	1	yes	characteristic
112	C₂²	—	2	2	no	—
113	C₂²	#16811544	1	2	yes	—
114	C₂²	C₂×D₅₃⋊C₁₆	1	1	yes	characteristic
115	C₄	—	53	4	no	—
116	C₄	—	106	2	no	—
117	C₄	#16811544	1	2	yes	—
118	C₄	C₂×#13133284	1	1	yes	characteristic
119	C₄	C₂×#13133292	1	1	yes	characteristic
120	C₂	—	2	2	no	—
121	C₂	C₂×#16811544	1	1	yes	characteristic
122	C₂	D₅₃⋊M₆₄	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16811555

C₂

yes

maximal

#20289873

C₂

yes

maximal, complemented

C₂×C₁₆×Dic₅₃

C₂

yes

maximal, characteristic

#16811555

—

C₂×#13133267

C₂²

yes

—

C₂²×#11719908

C₂²

yes

characteristic

C₂×C₈×Dic₅₃

C₄

yes

characteristic

C₁₆×Dic₅₃

C₄

yes

—

C₁₆×Dic₅₃

C₂²

yes

—

C₂²×C₁₆×C₅₃

C₄

yes

characteristic

#13133267

—

C₂²×#88322

C₈

yes

characteristic

C₂×C₄×Dic₅₃

C₈

yes

characteristic

C₂×#11719908

C₂×C₄

yes

characteristic

C₂×#11719908

D₄

yes

—

C₂×#11719908

Q₈

yes

—

C₂×#11719908

C₂³

yes

characteristic

C₈×Dic₅₃

C₈

yes

—

C₈×Dic₅₃

C₂×C₄

yes

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂²×Dic₅₃

C₁₆

yes

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

C₂×C₈

yes

characteristic

C₂×C₈×C₅₃

M₁₆

yes

—

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₄×Dic₅₃

C₁₆

yes

characteristic

C₄×Dic₅₃

C₁₆

yes

characteristic

C₄×Dic₅₃

C₂×C₈

yes

—

C₂×#88322

—

C₂×#88322

C₂×C₈

yes

—

C₁₆×C₅₃

—

#11719908

C₄⋊C₄

yes

—

#11719908

C₂×D₄

yes

characteristic

#11719908

C₂×Q₈

yes

characteristic

C₂³×C₅₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₅₃

C₄⋊C₈

yes

—

C₂×C₄×C₅₃

C₂×C₁₆

yes

characteristic

C₂×C₄×C₅₃

M₃₂

yes

—

C₂×C₄×C₅₃

C₂²×C₈

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×C₁₆

yes

—

C₂×Dic₅₃

C₂×C₁₆

yes

—

C₈×C₅₃

C₄⋊C₈

yes

—

C₈×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₈×C₅₃

C₂×M₁₆

yes

—

—

C₂²×C₅₃

—

C₂²×C₅₃

C₄⋊C₁₆

yes

—

C₂²×C₅₃

C₂²×C₁₆

yes

characteristic

C₄×C₅₃

C₄⋊C₁₆

yes

—

C₄×C₅₃

C₂×C₄⋊C₈

yes

characteristic

C₄×C₅₃

C₂×M₃₂

yes

characteristic

Dic₅₃

—

Dic₅₃

M₆₄

yes

complemented

C₂×M₆₄

—

C₂×C₄×C₁₆

—

C₄⋊C₃₂

—

C₂×C₅₃

—

C₂×C₅₃

C₂×C₄⋊C₁₆

yes

characteristic

C₂×C₅₃

C₂×M₆₄

yes

—

C₂²×C₁₆

—

C₂²×C₁₆

#6157

yes

characteristic

C₂×C₄×C₈

—

M₆₄

—

106

—

C₂×C₃₂

—

C₄×C₁₆

—

C₄×C₁₆

—

C₅₃

#13627

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂×#6157

yes

characteristic

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×#6157

yes

—

C₄×C₈

—

C₄×C₈

—

C₃₂

—

106

—

C₂²×C₄

—

C₂²×C₄

D₅₃⋊C₈

yes

characteristic

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

D₅₃⋊C₈

yes

characteristic

C₂×C₈

C₄⋊#6157

yes

—

C₂×C₈

#11719940

yes

—

C₂×C₈

C₂²×#6157

yes

characteristic

C₄²

—

C₄²

—

C₄²

—

C₁₆

—

C₁₆

—

C₁₆

—

C₁₆

—

100

C₂³

D₅₃⋊C₁₆

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

106

—

104

C₂×C₄

#13133284

yes

—

105

C₂×C₄

D₅₃⋊C₁₆

yes

characteristic

106

C₂×C₄

#13133292

yes

—

107

C₂×C₄

C₂×D₅₃⋊C₈

yes

characteristic

108

C₈

—

106

—

109

C₈

#13133284

yes

—

110

C₈

C₂×C₄⋊#6157

yes

characteristic

111

C₈

C₂×#11719940

yes

characteristic

112

C₂²

—

113

C₂²

#16811544

yes

—

114

C₂²

C₂×D₅₃⋊C₁₆

yes

characteristic

115

C₄

—

116

C₄

—

106

—

117

C₄

#16811544

yes

—

118

C₄

C₂×#13133284

yes

characteristic

119

C₄

C₂×#13133292

yes

characteristic

120

C₂

—

121

C₂

C₂×#16811544

yes

characteristic

122

C₂

D₅₃⋊M₆₄

yes

—

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×#6157
commutator, socle	C₂²×C₅₃	C₂²×C₁₆
Fitting	C₂²×C₁₆×C₅₃	C₄
prime radical	C₂³×C₅₃	C₂×C₁₆
cosocle	C₂×#11719908	C₂³
hypercenter	C₂²×C₁₆	#6157
hypocenter	C₅₃	#13627
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×#6157

commutator, socle

C₂²×C₅₃

C₂²×C₁₆

Fitting

C₂²×C₁₆×C₅₃

C₄

prime radical

C₂³×C₅₃

C₂×C₁₆

C₂³

C₂²×C₁₆

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆

N	H ≅ G/N	type
2. #20289873	C₂	semidirect
58. Dic₅₃	M₆₄	semidirect
72. C₅₃	#13627	semidirect

H ≅ G/N

type

2. #20289873

C₂

semidirect

58. Dic₅₃

M₆₄

semidirect

72. C₅₃

#13627

semidirect

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

subgroup	index
C₂×#16811555	2
#20289873	2
C₂×C₁₆×Dic₅₃	2
#13627	53

subgroup

index

C₂×#16811555

#20289873

C₂×C₁₆×Dic₅₃

#13627

Subgroups — #28312584

Conjugacy classes of subgroups

Complements

Maximal subgroups