Group #28310076 — subgroups

Subgroups — #28310076

Order 13568.

204 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 66 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	#20291107	C₂	1	2	yes	maximal, complemented
2	#20289742	C₂	1	1	yes	maximal, characteristic, complemented
3	#20289680	C₂	1	2	yes	maximal, complemented
4	#20289631	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276600	C₂	1	1	yes	maximal, characteristic
6	#16811682	—	2	2	no	—
7	#16811545	C₂²	1	2	yes	—
8	#16811522	—	2	2	no	—
9	#16811512	C₂²	1	1	yes	characteristic
10	#16811464	C₂²	1	1	yes	characteristic
11	#16811055	C₄	1	1	yes	characteristic
12	#16810739	C₂²	1	2	yes	—
13	#16810429	C₄	1	2	yes	—
14	#16810398	C₂²	1	1	yes	characteristic
15	C₅₃×#368	C₄	1	1	yes	characteristic
16	#13133292	—	4	2	no	—
17	#13133291	—	2	2	no	—
18	D₅₃⋊C₁₆	—	4	2	no	—
19	C₂×#11719910	D₄	1	2	yes	—
20	C₄×#88324	—	2	1	no	—
21	#13133239	C₂×C₄	1	2	yes	—
22	#13133225	—	2	2	no	—
23	Dic₄₂₄	C₈	1	1	yes	characteristic, complemented
24	#13133162	C₈	1	2	yes	complemented
25	D₈₄₈	C₈	1	1	yes	characteristic, complemented
26	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
27	#13133143	—	2	2	no	—
28	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
29	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
30	#108×C₅₃	C₂×C₄	1	2	yes	—
31	#11719945	—	2	2	no	—
32	C₂×#88324	—	4	1	no	—
33	M₁₆×C₅₃	—	2	2	no	—
34	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
35	C₄×Dic₅₃	#41	1	1	yes	characteristic
36	C₂×#88322	C₂×D₄	1	1	yes	characteristic
37	Dic₂₁₂	C₂×C₈	1	2	yes	—
38	D₄₂₄	C₂×C₈	1	2	yes	—
39	#11719913	—	2	2	no	—
40	#11719912	—	4	2	no	—
41	C₈×D₅₃	—	4	2	no	—
42	#11719910	—	2	2	no	—
43	#11719910	—	4	2	no	—
44	C₁₆×C₅₃	C₂×C₈	1	1	yes	characteristic
45	C₁₆×C₅₃	C₂×C₈	1	1	yes	characteristic
46	#11719908	—	2	1	no	—
47	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
48	C₂²⋊D₅₃	—	4	2	no	—
49	C₂×Dic₅₃	—	2	1	no	—
50	D₂₁₂	—	2	2	no	—
51	C₄×D₅₃	—	4	2	no	—
52	Dic₁₀₆	—	2	2	no	—
53	#88324	—	8	1	no	—
54	C₈×C₅₃	—	4	2	no	—
55	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
56	C₈×C₅₃	#131	1	1	yes	characteristic
57	#88322	AGL(1,ℤ/8)	1	2	yes	—
58	#12068	—	53	1	no	maximal, Sylow
59	C₂²×C₅₃	#444	1	1	yes	characteristic
60	C₂×D₅₃	—	4	2	no	—
61	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
62	C₄×C₅₃	#442	1	1	yes	characteristic
63	Dic₅₃	—	4	1	no	—
64	Dic₅₃	—	4	2	no	—
65	#2179	—	53	1	no	—
66	#2171	—	53	2	no	—
67	#1369	—	53	1	no	—
68	#1336	—	53	2	no	—
69	#1313	—	53	1	no	—
70	C₂×C₅₃	—	2	1	no	—
71	C₂×C₅₃	#1580	1	1	yes	characteristic
72	D₅₃	—	8	2	no	—
73	#508	—	53	1	no	—
74	#504	—	106	2	no	—
75	#443	—	53	2	no	—
76	#368	#6157	1	1	yes	characteristic
77	#364	—	53	2	no	—
78	#359	—	53	2	no	—
79	#350	—	106	2	no	—
80	#346	—	53	1	no	—
81	#346	—	53	1	no	—
82	#322	—	53	1	no	—
83	C₅₃	#12068	1	1	yes	characteristic, complemented, Sylow
84	#135	—	53	2	no	—
85	#131	—	106	2	no	—
86	Q₃₂	—	53	1	no	—
87	SD₃₂	—	53	2	no	—
88	D₁₆	—	53	1	no	—
89	M₃₂	—	106	2	no	—
90	M₃₂	—	212	2	no	—
91	C₂×C₁₆	—	53	2	no	—
92	C₂×C₁₆	—	53	1	no	—
93	C₂×C₁₆	—	212	2	no	—
94	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
95	#108	C₂×#6157	1	2	yes	—
96	#104	—	106	2	no	—
97	C₄×C₈	—	53	1	no	—
98	C₄×C₈	—	106	1	no	—
99	#41	—	106	2	no	—
100	Q₁₆	—	53	2	no	—
101	SD₁₆	—	106	2	no	—
102	D₈	—	53	2	no	—
103	M₁₆	—	2	2	no	—
104	M₁₆	—	212	2	no	—
105	C₂×C₈	—	53	1	no	—
106	C₂×C₈	—	212	2	no	—
107	C₂×C₈	—	212	1	no	—
108	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
109	C₄²	—	53	1	no	—
110	C₁₆	—	106	2	no	—
111	C₁₆	—	106	1	no	—
112	C₁₆	—	212	2	no	—
113	C₁₆	D₅₃⋊C₈	1	1	yes	characteristic
114	C₁₆	D₅₃⋊C₈	1	1	yes	characteristic
115	Q₈	—	106	2	no	—
116	D₄	—	106	2	no	—
117	D₄	—	212	2	no	—
118	C₂×C₄	—	106	1	no	—
119	C₂×C₄	—	212	2	no	—
120	C₂×C₄	D₄×#6157	1	1	yes	characteristic
121	C₈	—	4	2	no	—
122	C₈	—	53	2	no	—
123	C₈	—	424	1	no	—
124	C₈	C₂×D₅₃⋊C₈	1	1	yes	characteristic
125	C₈	#13133305	1	1	yes	characteristic
126	C₂²	—	212	2	no	—
127	C₂²	#16811669	1	1	yes	characteristic
128	C₄	—	212	1	no	—
129	C₄	—	212	2	no	—
130	C₄	#16811651	1	1	yes	characteristic
131	C₄	#16811664	1	1	yes	characteristic
132	C₂	—	2	1	no	—
133	C₂	—	424	2	no	—
134	C₂	#20290313	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20291107

C₂

yes

maximal, complemented

#20289742

C₂

yes

maximal, characteristic, complemented

#20289680

C₂

yes

maximal, complemented

#20289631

C₂

yes

maximal, characteristic, complemented

#20276600

C₂

yes

maximal, characteristic

#16811682

—

#16811545

C₂²

yes

—

#16811522

—

#16811512

C₂²

yes

characteristic

#16811464

C₂²

yes

characteristic

#16811055

C₄

yes

characteristic

#16810739

C₂²

yes

—

#16810429

C₄

yes

—

#16810398

C₂²

yes

characteristic

C₅₃×#368

C₄

yes

characteristic

#13133292

—

#13133291

—

D₅₃⋊C₁₆

—

C₂×#11719910

D₄

yes

—

C₄×#88324

—

#13133239

C₂×C₄

yes

—

#13133225

—

Dic₄₂₄

C₈

yes

characteristic, complemented

#13133162

C₈

yes

complemented

D₈₄₈

C₈

yes

characteristic, complemented

C₂×#11719908

C₂×C₄

yes

characteristic

#13133143

—

C₈×Dic₅₃

C₂³

yes

characteristic

C₂×C₁₆×C₅₃

C₂×C₄

yes

characteristic

#108×C₅₃

C₂×C₄

yes

—

#11719945

—

C₂×#88324

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

C₂²×C₄

yes

C₄×Dic₅₃

yes

C₂×D₄

yes

characteristic

Dic₂₁₂

C₂×C₈

yes

—

D₄₂₄

C₂×C₈

yes

—

#11719913

—

#11719912

—

C₈×D₅₃

—

#11719910

—

#11719910

—

C₁₆×C₅₃

C₂×C₈

yes

characteristic

C₁₆×C₅₃

C₂×C₈

yes

characteristic

#11719908

—

C₂×C₄×C₅₃

C₄×D₄

yes

characteristic

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

C₄×D₅₃

—

Dic₁₀₆

—

#88324

—

C₈×C₅₃

—

C₈×C₅₃

C₂²×C₈

yes

C₈×C₅₃

yes

yes

—

—

C₂²×C₅₃

yes

C₂×D₅₃

—

C₄×C₅₃

C₈×D₄

yes

characteristic

C₄×C₅₃

#442

yes

characteristic

Dic₅₃

—

Dic₅₃

—

—

—

—

—

—

C₂×C₅₃

—

C₂×C₅₃

#1580

yes

characteristic

D₅₃

—

#508

—

#504

—

106

—

—

yes

—

—

—

106

—

#346

—

#346

—

#322

—

C₅₃

#12068

yes

characteristic, complemented, Sylow

#135

—

#131

—

106

—

Q₃₂

—

SD₃₂

—

D₁₆

—

M₃₂

—

106

—

M₃₂

—

212

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

212

—

C₂×C₁₆

yes

yes

—

#104

—

106

—

C₄×C₈

—

C₄×C₈

—

106

—

#41

—

106

—

100

Q₁₆

—

101

SD₁₆

—

106

—

102

D₈

—

103

M₁₆

—

104

M₁₆

—

212

—

105

C₂×C₈

—

106

C₂×C₈

—

212

—

107

C₂×C₈

—

212

—

108

C₂×C₈

C₂²×#6157

yes

characteristic

109

C₄²

—

110

C₁₆

—

106

—

111

C₁₆

—

106

—

112

C₁₆

—

212

—

113

C₁₆

D₅₃⋊C₈

yes

characteristic

114

C₁₆

D₅₃⋊C₈

yes

characteristic

115

Q₈

—

106

—

116

D₄

—

106

—

117

D₄

—

212

—

118

C₂×C₄

—

106

—

119

C₂×C₄

—

212

—

120

C₂×C₄

D₄×#6157

yes

characteristic

121

C₈

—

122

C₈

—

123

C₈

—

424

—

124

C₈

C₂×D₅₃⋊C₈

yes

characteristic

125

C₈

#13133305

yes

characteristic

126

C₂²

—

212

—

127

C₂²

#16811669

yes

characteristic

128

C₄

—

212

—

129

C₄

—

212

—

130

C₄

#16811651

yes

characteristic

131

C₄

#16811664

yes

characteristic

132

C₂

—

133

C₂

—

424

—

134

C₂

#20290313

yes

characteristic

	subgroup	G/H
center	C₄	#16811664
commutator	C₈×C₅₃	C₂²×C₈
Frattini	C₂×C₈	C₂²×#6157
Fitting	C₅₃×#368	C₄
socle	C₂×C₅₃	#1580
prime radical	#16811055	C₄
cosocle	C₈×Dic₅₃	C₂³
hypercenter	#368	#6157
hypocenter	C₅₃	#12068
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #368

subgroup

G/H

center

C₄

#16811664

commutator

C₈×C₅₃

C₂²×C₈

Frattini

C₂×C₈

C₂²×#6157

Fitting

C₅₃×#368

C₄

C₂×C₅₃

C₄

C₈×Dic₅₃

C₂³

C₅₃

G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #368

N	H ≅ G/N	type
1. #20291107	C₂	semidirect
2. #20289742	C₂	semidirect
3. #20289680	C₂	semidirect
4. #20289631	C₂	semidirect
23. Dic₄₂₄	C₈	semidirect
24. #13133162	C₈	semidirect
25. D₈₄₈	C₈	semidirect
83. C₅₃	#12068	semidirect

H ≅ G/N

type

1. #20291107

C₂

semidirect

2. #20289742

C₂

semidirect

3. #20289680

C₂

semidirect

4. #20289631

C₂

semidirect

23. Dic₄₂₄

C₈

semidirect

24. #13133162

C₈

semidirect

25. D₈₄₈

C₈

semidirect

83. C₅₃

#12068

semidirect

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

subgroup	index
#20291107	2
#20289742	2
#20289680	2
#20289631	2
#20276600	2
#12068	53

subgroup

index

Subgroups — #28310076

Conjugacy classes of subgroups

Complements

Maximal subgroups