Group #28298979 — subgroups

Subgroups — #28298979

Order 13568.

138 conjugacy classes of subgroups; only the 48 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 48 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	#20289820	C₂	1	1	yes	maximal, characteristic, complemented
2	#20289746	C₂	1	1	yes	maximal, characteristic, complemented
3	D₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133267	—	2	1	no	—
5	#16811486	—	2	1	no	—
6	D₈×Dic₅₃	—	2	1	no	—
7	#16810449	C₄	1	1	yes	characteristic
8	#16810424	—	2	1	no	—
9	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
10	C₂×D₁₆×C₅₃	C₄	1	1	yes	characteristic
11	C₄⋊#88324	—	4	1	no	—
12	#13133267	—	2	2	no	—
13	D₄×Dic₅₃	—	4	1	no	—
14	C₂×#11719908	D₄	1	1	yes	characteristic
15	#13133151	—	2	1	no	—
16	#13133138	—	4	1	no	—
17	C₈×Dic₅₃	D₄	1	1	yes	characteristic
18	C₂×D₈×C₅₃	—	2	1	no	—
19	D₁₆×C₅₃	—	2	1	no	—
20	D₁₆×C₅₃	C₈	1	2	yes	complemented
21	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×D₄×C₅₃	—	4	1	no	—
23	C₂²×Dic₅₃	—	8	1	no	—
24	C₂×#88324	—	8	1	no	—
25	D₈×C₅₃	—	2	2	no	—
26	D₈×C₅₃	—	4	1	no	—
27	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
28	C₂²⋊Dic₅₃	—	8	1	no	—
29	C₄⋊Dic₅₃	—	4	1	no	—
30	C₄×Dic₅₃	D₈	1	1	yes	characteristic
31	C₂×#88322	SD₁₆	1	1	yes	characteristic
32	C₁₆×C₅₃	C₂×C₈	1	1	yes	characteristic
33	C₁₆×C₅₃	M₁₆	1	1	yes	characteristic
34	#11719908	—	2	1	no	—
35	C₂³×C₅₃	—	8	1	no	—
36	D₄×C₅₃	—	4	2	no	—
37	D₄×C₅₃	—	8	1	no	—
38	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
39	C₂×Dic₅₃	—	8	1	no	—
40	C₂×Dic₅₃	—	16	1	no	—
41	C₂×Dic₅₃	—	16	1	no	—
42	C₂×Dic₅₃	D₁₆	1	1	yes	characteristic
43	C₂×Dic₅₃	SD₃₂	1	1	yes	characteristic
44	#88324	—	16	1	no	—
45	C₈×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
46	C₈×C₅₃	#104	1	1	yes	characteristic
47	#88322	—	2	1	no	—
48	#7383	—	53	1	no	maximal, Sylow
49	C₂²×C₅₃	—	8	2	no	—
50	C₂²×C₅₃	—	16	1	no	—
51	C₂²×C₅₃	—	16	1	no	—
52	C₂²×C₅₃	#357	1	1	yes	characteristic
53	C₄×C₅₃	D₄⋊C₈	1	1	yes	characteristic
54	C₄×C₅₃	#359	1	1	yes	characteristic
55	Dic₅₃	—	2	1	no	—
56	Dic₅₃	—	16	1	no	—
57	Dic₅₃	D₃₂	1	1	yes	characteristic
58	Dic₅₃	SD₆₄	1	1	yes	characteristic
59	C₄×D₁₆	—	53	1	no	—
60	C₄×C₃₂	—	53	1	no	—
61	#1371	—	53	1	no	—
62	C₂×C₅₃	—	16	2	no	—
63	C₂×C₅₃	#1331	1	1	yes	characteristic
64	C₂×C₅₃	#1415	1	1	yes	characteristic
65	C₂×C₅₃	#1417	1	1	yes	characteristic
66	C₂×D₁₆	#6157	1	1	yes	characteristic
67	C₄×D₈	—	106	1	no	—
68	C₂×C₃₂	—	106	1	no	—
69	#366	—	53	1	no	—
70	#357	—	106	1	no	—
71	C₄×C₁₆	—	53	1	no	—
72	#335	—	106	1	no	—
73	C₅₃	#7383	1	1	yes	characteristic, complemented, Sylow
74	C₂×D₈	—	2	1	no	—
75	C₄×D₄	—	212	1	no	—
76	D₁₆	—	2	1	no	—
77	D₁₆	#88324	1	2	yes	complemented
78	C₂×C₁₆	—	53	1	no	—
79	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
80	#107	—	106	1	no	—
81	C₄⋊C₈	—	212	1	no	—
82	#102	—	212	1	no	—
83	C₄×C₈	—	53	1	no	—
84	C₃₂	—	106	2	no	—
85	C₂×D₄	—	4	1	no	—
86	C₂²×C₄	—	424	1	no	—
87	D₈	—	2	2	no	—
88	D₈	—	4	1	no	—
89	C₂×C₈	—	53	1	no	—
90	C₂×C₈	—	424	1	no	—
91	C₂×C₈	C₂²⋊#6157	1	1	yes	characteristic
92	C₄⋊C₄	—	212	1	no	—
93	C₂²⋊C₄	—	424	1	no	—
94	C₄²	—	53	1	no	—
95	C₁₆	—	106	1	no	—
96	C₁₆	C₂×#88324	1	1	yes	characteristic
97	C₁₆	#11719940	1	1	yes	characteristic
98	C₂³	—	8	1	no	—
99	D₄	—	4	2	no	—
100	D₄	—	8	1	no	—
101	C₂×C₄	—	53	1	no	—
102	C₂×C₄	—	53	1	no	—
103	C₂×C₄	—	848	1	no	—
104	C₂×C₄	—	848	1	no	—
105	C₂×C₄	—	424	1	no	—
106	C₂×C₄	#13133279	1	1	yes	characteristic
107	C₈	—	848	1	no	—
108	C₈	—	106	1	no	—
109	C₈	C₂²⋊#88324	1	1	yes	characteristic
110	C₈	#13133281	1	1	yes	characteristic
111	C₂²	—	8	2	no	—
112	C₂²	—	16	1	no	—
113	C₂²	—	16	1	no	—
114	C₂²	#16811535	1	1	yes	characteristic
115	C₄	—	53	1	no	—
116	C₄	—	53	1	no	—
117	C₄	—	848	1	no	—
118	C₄	—	106	1	no	—
119	C₄	D₄⋊#88324	1	1	yes	characteristic
120	C₄	#16811537	1	1	yes	characteristic
121	C₂	—	16	2	no	—
122	C₂	#20289669	1	1	yes	characteristic
123	C₂	#20289864	1	1	yes	characteristic
124	C₂	#20289866	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20289820

C₂

yes

maximal, characteristic, complemented

#20289746

C₂

yes

maximal, characteristic, complemented

D₁₆×Dic₅₃

C₂

yes

maximal, characteristic

C₂×#13133267

—

#16811486

—

D₈×Dic₅₃

—

#16810449

C₄

yes

characteristic

#16810424

—

C₁₆×Dic₅₃

C₂²

yes

characteristic

C₂×D₁₆×C₅₃

C₄

yes

characteristic

C₄⋊#88324

—

#13133267

—

D₄×Dic₅₃

—

C₂×#11719908

D₄

yes

characteristic

#13133151

—

#13133138

—

C₈×Dic₅₃

D₄

yes

characteristic

C₂×D₈×C₅₃

—

D₁₆×C₅₃

—

D₁₆×C₅₃

C₈

yes

complemented

C₂×C₁₆×C₅₃

C₂×C₄

yes

characteristic

C₂×D₄×C₅₃

—

C₂²×Dic₅₃

—

C₂×#88324

—

D₈×C₅₃

—

D₈×C₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

characteristic

C₂²⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄×Dic₅₃

D₈

yes

characteristic

C₂×#88322

SD₁₆

yes

characteristic

C₁₆×C₅₃

C₂×C₈

yes

characteristic

C₁₆×C₅₃

M₁₆

yes

characteristic

#11719908

—

C₂³×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

#102

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

D₁₆

yes

characteristic

C₂×Dic₅₃

SD₃₂

yes

characteristic

#88324

—

C₈×C₅₃

C₂²⋊C₈

yes

C₈×C₅₃

yes

—

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#357

yes

characteristic

C₄×C₅₃

D₄⋊C₈

yes

characteristic

C₄×C₅₃

#359

yes

characteristic

Dic₅₃

—

Dic₅₃

—

Dic₅₃

D₃₂

yes

characteristic

Dic₅₃

SD₆₄

yes

characteristic

C₄×D₁₆

—

C₄×C₃₂

—

#1371

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×D₁₆

yes

C₄×D₈

—

106

—

C₂×C₃₂

—

106

—

#366

—

#357

—

106

—

C₄×C₁₆

—

#335

—

106

—

C₅₃

#7383

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₄×D₄

—

212

—

D₁₆

—

D₁₆

#88324

yes

complemented

C₂×C₁₆

—

C₂×C₁₆

C₂×#6157

yes

characteristic

#107

—

106

—

C₄⋊C₈

—

212

—

#102

—

212

—

C₄×C₈

—

C₃₂

—

106

—

C₂×D₄

—

C₂²×C₄

—

424

—

D₈

—

D₈

—

C₂×C₈

—

C₂×C₈

—

424

—

C₂×C₈

C₂²⋊#6157

yes

characteristic

C₄⋊C₄

—

212

—

C₂²⋊C₄

—

424

—

C₄²

—

C₁₆

—

106

—

C₁₆

yes

C₁₆

yes

C₂³

—

D₄

—

100

D₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

848

—

104

C₂×C₄

—

848

—

105

C₂×C₄

—

424

—

106

C₂×C₄

#13133279

yes

characteristic

107

C₈

—

848

—

108

C₈

—

106

—

109

C₈

C₂²⋊#88324

yes

characteristic

110

C₈

#13133281

yes

characteristic

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

#16811535

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

848

—

118

C₄

—

106

—

119

C₄

D₄⋊#88324

yes

characteristic

120

C₄

#16811537

yes

characteristic

121

C₂

—

122

C₂

#20289669

yes

characteristic

123

C₂

#20289864

yes

characteristic

124

C₂

#20289866

yes

characteristic

	subgroup	G/H
center	C₂²	#16811535
commutator	C₁₆×C₅₃	C₂×C₈
Frattini	C₂×C₁₆	C₂×#6157
Fitting, prime radical	C₂×D₁₆×C₅₃	C₄
socle	C₂²×C₅₃	#357
cosocle	C₁₆×Dic₅₃	C₂²
hypercenter	C₂×D₁₆	#6157
hypocenter	C₅₃	#7383
LCS	G ▷ C₁₆×C₅₃ ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×D₁₆

subgroup

G/H

center

C₂²

#16811535

commutator

C₁₆×C₅₃

C₂×C₈

Frattini

C₂×C₁₆

C₂×#6157

Fitting, prime radical

C₂×D₁₆×C₅₃

C₄

socle

C₂²×C₅₃

#357

cosocle

C₁₆×Dic₅₃

C₂²

C₂×D₁₆

C₅₃

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

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ C₂×D₁₆

N	H ≅ G/N	type
1. #20289820	C₂	semidirect
2. #20289746	C₂	semidirect
20. D₁₆×C₅₃	C₈	semidirect
73. C₅₃	#7383	semidirect
77. D₁₆	#88324	semidirect

H ≅ G/N

type

1. #20289820

C₂

semidirect

2. #20289746

C₂

semidirect

20. D₁₆×C₅₃

C₈

semidirect

73. C₅₃

#7383

semidirect

77. D₁₆

#88324

semidirect

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

subgroup	index
#20289820	2
#20289746	2
D₁₆×Dic₅₃	2
#7383	53

subgroup

index

#20289820

#20289746

D₁₆×Dic₅₃

#7383

Subgroups — #28298979

Conjugacy classes of subgroups

Complements

Maximal subgroups