Group #28312647 — subgroups

Subgroups — #28312647

Order 13568.

188 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 54 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	Q₃₂×#6157	C₂	1	2	yes	maximal
2	#20289877	C₂	1	1	yes	maximal, characteristic
3	#20289865	C₂	1	2	yes	maximal
4	#20289821	C₂	1	1	yes	maximal, characteristic
5	Q₆₄×D₅₃	C₂	1	1	yes	maximal, characteristic
6	Q₁₆×#6157	—	2	2	no	—
7	D₅₃⋊C₃₂	C₂²	1	1	yes	characteristic
8	#16811549	C₂²	1	2	yes	—
9	#16811536	—	2	2	no	—
10	C₁₆×#6157	C₂²	1	1	yes	characteristic
11	Q₃₂×D₅₃	C₂²	1	2	yes	—
12	Dic₈₄₈	C₄	1	1	yes	characteristic, complemented
13	#16810463	C₄	1	2	yes	complemented
14	C₃₂×D₅₃	C₂²	1	1	yes	characteristic
15	C₅₃×Q₆₄	C₄	1	1	yes	characteristic, complemented
16	Q₈×#6157	—	4	2	no	—
17	D₅₃⋊C₁₆	D₄	1	1	yes	characteristic
18	#13133286	—	2	2	no	—
19	#13133280	—	4	2	no	—
20	C₈×#6157	D₄	1	1	yes	characteristic
21	#13133267	—	2	1	no	—
22	Q₁₆×D₅₃	—	2	2	no	—
23	Dic₄₂₄	C₂×C₄	1	2	yes	—
24	#13133163	—	2	2	no	—
25	C₁₆×D₅₃	C₂³	1	1	yes	characteristic
26	#13133123	C₂×C₄	1	1	yes	characteristic
27	Q₃₂×C₅₃	C₂×C₄	1	2	yes	—
28	C₃₂×C₅₃	C₂×C₄	1	1	yes	characteristic
29	Q₈×D₅₃	—	4	2	no	—
30	C₄⋊#6157	—	4	2	no	—
31	C₄⋊#6157	—	8	2	no	—
32	C₄×#6157	—	8	2	no	—
33	C₄×#6157	D₈	1	1	yes	characteristic
34	D₅₃⋊C₈	D₈	1	1	yes	characteristic
35	Q₁₆×C₅₃	—	2	2	no	—
36	#11719925	—	4	2	no	—
37	Dic₂₁₂	—	2	2	no	—
38	C₈×D₅₃	C₂×D₄	1	1	yes	characteristic
39	#11719910	—	2	1	no	—
40	C₁₆×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	#11719908	#41	1	1	yes	characteristic
42	C₂×#6157	—	8	2	no	—
43	C₂×#6157	D₁₆	1	2	yes	—
44	Q₈×C₅₃	—	4	2	no	—
45	C₄×D₅₃	—	8	2	no	—
46	C₄×D₅₃	C₂×D₈	1	1	yes	characteristic
47	Dic₁₀₆	—	4	2	no	—
48	Dic₁₀₆	—	8	2	no	—
49	#88324	—	2	1	no	—
50	C₈×C₅₃	C₄×D₄	1	1	yes	characteristic
51	#88322	#135	1	1	yes	characteristic
52	#13653	—	53	1	no	maximal, Sylow
53	C₂×D₅₃	C₂×D₁₆	1	1	yes	characteristic
54	#6157	—	2	2	no	—
55	#6157	—	16	2	no	—
56	C₄×C₅₃	—	8	2	no	—
57	C₄×C₅₃	C₄×D₈	1	1	yes	characteristic
58	Dic₅₃	—	8	2	no	—
59	Dic₅₃	#508	1	1	yes	characteristic
60	C₂×Q₆₄	—	53	1	no	—
61	C₄×Q₃₂	—	53	2	no	—
62	#1424	—	53	1	no	—
63	#1416	—	53	2	no	—
64	#1397	—	53	1	no	—
65	C₂×C₅₃	C₄×D₁₆	1	1	yes	characteristic
66	D₅₃	#2264	1	2	yes	—
67	C₂×Q₃₂	—	53	2	no	—
68	C₄×Q₁₆	—	106	2	no	—
69	Q₆₄	—	53	2	no	—
70	Q₆₄	—	53	1	no	—
71	Q₆₄	#6157	1	1	yes	characteristic, complemented
72	C₂×C₃₂	—	53	1	no	—
73	C₂×C₃₂	—	53	1	no	—
74	#366	—	53	2	no	—
75	#358	—	106	2	no	—
76	C₄×C₁₆	—	53	1	no	—
77	C₅₃	#13653	1	1	yes	characteristic, complemented, Sylow
78	C₂×Q₁₆	—	106	2	no	—
79	C₄×Q₈	—	212	2	no	—
80	Q₃₂	—	53	2	no	—
81	Q₃₂	—	106	2	no	—
82	Q₃₂	C₂×#6157	1	2	yes	—
83	C₂×C₁₆	—	53	1	no	—
84	C₂×C₁₆	—	53	1	no	—
85	#107	—	106	2	no	—
86	#103	—	212	2	no	—
87	C₄×C₈	—	53	1	no	—
88	C₃₂	—	53	1	no	—
89	C₃₂	—	106	1	no	—
90	C₃₂	C₂×#6157	1	1	yes	characteristic
91	C₂×Q₈	—	212	2	no	—
92	Q₁₆	—	2	2	no	—
93	Q₁₆	—	106	2	no	—
94	Q₁₆	—	212	2	no	—
95	C₂×C₈	—	53	1	no	—
96	C₂×C₈	—	53	1	no	—
97	C₄⋊C₄	—	424	2	no	—
98	C₄⋊C₄	—	212	2	no	—
99	C₄²	—	53	1	no	—
100	C₄²	—	424	2	no	—
101	C₁₆	—	53	1	no	—
102	C₁₆	—	106	1	no	—
103	C₁₆	C₂²×#6157	1	1	yes	characteristic
104	Q₈	—	4	2	no	—
105	Q₈	—	424	2	no	—
106	Q₈	—	212	2	no	—
107	C₂×C₄	—	53	1	no	—
108	C₂×C₄	—	53	2	no	—
109	C₂×C₄	—	424	2	no	—
110	C₂×C₄	—	424	2	no	—
111	C₈	—	53	1	no	—
112	C₈	—	106	1	no	—
113	C₈	D₄×#6157	1	1	yes	characteristic
114	C₂²	—	53	1	no	—
115	C₄	—	8	2	no	—
116	C₄	—	53	1	no	—
117	C₄	—	848	2	no	—
118	C₄	—	106	2	no	—
119	C₄	—	424	2	no	—
120	C₄	D₈×#6157	1	1	yes	characteristic
121	C₂	—	53	2	no	—
122	C₂	D₁₆×#6157	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₃₂×#6157

C₂

yes

maximal

#20289877

C₂

yes

maximal, characteristic

#20289865

C₂

yes

maximal

#20289821

C₂

yes

maximal, characteristic

Q₆₄×D₅₃

C₂

yes

maximal, characteristic

Q₁₆×#6157

—

D₅₃⋊C₃₂

C₂²

yes

characteristic

#16811549

C₂²

yes

—

#16811536

—

C₁₆×#6157

C₂²

yes

characteristic

Q₃₂×D₅₃

C₂²

yes

—

Dic₈₄₈

C₄

yes

characteristic, complemented

#16810463

C₄

yes

complemented

C₃₂×D₅₃

C₂²

yes

characteristic

C₅₃×Q₆₄

C₄

yes

characteristic, complemented

Q₈×#6157

—

D₅₃⋊C₁₆

D₄

yes

—

—

D₄

yes

characteristic

#13133267

—

Q₁₆×D₅₃

—

Dic₄₂₄

C₂×C₄

yes

—

#13133163

—

C₁₆×D₅₃

C₂³

yes

characteristic

#13133123

C₂×C₄

yes

characteristic

Q₃₂×C₅₃

C₂×C₄

yes

—

C₃₂×C₅₃

C₂×C₄

yes

characteristic

Q₈×D₅₃

—

—

—

—

D₈

yes

characteristic

D₅₃⋊C₈

D₈

yes

characteristic

Q₁₆×C₅₃

—

#11719925

—

Dic₂₁₂

—

C₈×D₅₃

C₂×D₄

yes

characteristic

#11719910

—

C₁₆×C₅₃

C₂²×C₄

yes

yes

—

D₁₆

yes

—

Q₈×C₅₃

—

C₄×D₅₃

—

C₄×D₅₃

C₂×D₈

yes

characteristic

Dic₁₀₆

—

Dic₁₀₆

—

#88324

—

C₈×C₅₃

C₄×D₄

yes

yes

—

C₂×D₅₃

C₂×D₁₆

yes

characteristic

#6157

—

#6157

—

C₄×C₅₃

—

C₄×C₅₃

C₄×D₈

yes

characteristic

Dic₅₃

—

Dic₅₃

#508

yes

characteristic

C₂×Q₆₄

—

C₄×Q₃₂

—

#1424

—

#1416

—

#1397

—

C₂×C₅₃

C₄×D₁₆

yes

characteristic

D₅₃

#2264

yes

—

C₂×Q₃₂

—

C₄×Q₁₆

—

106

—

Q₆₄

—

Q₆₄

—

Q₆₄

#6157

yes

characteristic, complemented

C₂×C₃₂

—

C₂×C₃₂

—

#366

—

#358

—

106

—

C₄×C₁₆

—

C₅₃

#13653

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

106

—

C₄×Q₈

—

212

—

Q₃₂

—

Q₃₂

—

106

—

Q₃₂

C₂×#6157

yes

—

C₂×C₁₆

—

C₂×C₁₆

—

#107

—

106

—

#103

—

212

—

C₄×C₈

—

C₃₂

—

C₃₂

—

106

—

C₃₂

C₂×#6157

yes

characteristic

C₂×Q₈

—

212

—

Q₁₆

—

Q₁₆

—

106

—

Q₁₆

—

212

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

424

—

C₄⋊C₄

—

212

—

C₄²

—

100

C₄²

—

424

—

101

C₁₆

—

102

C₁₆

—

106

—

103

C₁₆

C₂²×#6157

yes

characteristic

104

Q₈

—

105

Q₈

—

424

—

106

Q₈

—

212

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

424

—

110

C₂×C₄

—

424

—

111

C₈

—

112

C₈

—

106

—

113

C₈

D₄×#6157

yes

characteristic

114

C₂²

—

115

C₄

—

116

C₄

—

117

C₄

—

848

—

118

C₄

—

106

—

119

C₄

—

424

—

120

C₄

D₈×#6157

yes

characteristic

121

C₂

—

122

C₂

D₁₆×#6157

yes

characteristic

	subgroup	G/H
center	C₂	D₁₆×#6157
commutator	C₁₆×C₅₃	C₂²×C₄
Frattini	C₁₆	C₂²×#6157
Fitting	C₅₃×Q₆₄	C₄
socle	C₂×C₅₃	C₄×D₁₆
prime radical	C₂×D₅₃	C₂×D₁₆
cosocle	C₁₆×D₅₃	C₂³
hypercenter	Q₆₄	#6157
hypocenter	C₅₃	#13653
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ Q₆₄

subgroup

G/H

center

C₂

D₁₆×#6157

commutator

C₁₆×C₅₃

C₂²×C₄

Frattini

C₁₆

C₂²×#6157

Fitting

C₅₃×Q₆₄

C₄

socle

C₂×C₅₃

C₄×D₁₆

prime radical

C₂×D₅₃

C₂×D₁₆

cosocle

C₁₆×D₅₃

C₂³

Q₆₄

C₅₃

G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂ ◁ C₄ ◁ C₈ ◁ C₁₆ ◁ Q₆₄

N	H ≅ G/N	type
12. Dic₈₄₈	C₄	semidirect
13. #16810463	C₄	semidirect
15. C₅₃×Q₆₄	C₄	semidirect
71. Q₆₄	#6157	semidirect
77. C₅₃	#13653	semidirect

H ≅ G/N

type

12. Dic₈₄₈

C₄

semidirect

13. #16810463

C₄

semidirect

15. C₅₃×Q₆₄

C₄

semidirect

71. Q₆₄

#6157

semidirect

77. C₅₃

#13653

semidirect

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

subgroup	index
Q₃₂×#6157	2
#20289877	2
#20289865	2
#20289821	2
Q₆₄×D₅₃	2
#13653	53

subgroup

index

Q₃₂×#6157

Q₆₄×D₅₃

Subgroups — #28312647

Conjugacy classes of subgroups

Complements

Maximal subgroups