Group #28309929 — subgroups

Subgroups — #28309929

Order 13568.

180 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 68 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	#20291090	C₂	1	2	yes	maximal
2	#20289751	C₂	1	2	yes	maximal, complemented
3	#20289743	C₂	1	2	yes	maximal, complemented
4	#20276313	C₂	1	1	yes	maximal, characteristic
5	#16811545	C₂²	1	4	yes	—
6	#16811520	—	2	2	no	—
7	#16811515	—	2	2	no	—
8	C₂×#13133156	C₄	1	1	yes	characteristic
9	C₂×C₁₆×D₅₃	C₄	1	1	yes	characteristic
10	#16810577	C₂²	1	1	yes	characteristic
11	#16810398	C₂²	1	1	yes	characteristic
12	C₄×#11719908	C₄	1	1	yes	characteristic
13	C₁₆×Dic₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×#346	C₄	1	1	yes	characteristic
15	#13133291	—	2	4	no	—
16	C₂×#11719912	D₄	1	1	yes	characteristic
17	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
18	#13133174	Q₈	1	1	yes	characteristic
19	#13133156	—	2	2	no	—
20	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
21	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
22	C₁₆×D₅₃	—	2	2	no	—
23	#13133128	D₄	1	1	yes	characteristic
24	#13133127	Q₈	1	1	yes	characteristic
25	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
26	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×D₅₃	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
32	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
33	#11719921	—	2	1	no	—
34	#11719919	—	2	1	no	—
35	C₄×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
36	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
37	C₂×#88322	C₂×D₄	1	1	yes	characteristic
38	#11719912	—	2	2	no	—
39	C₈×D₅₃	—	2	1	no	—
40	C₈×D₅₃	—	2	1	no	—
41	#11719910	—	4	4	no	—
42	C₁₆×C₅₃	—	2	1	no	—
43	C₁₆×C₅₃	—	2	1	no	—
44	#11719908	M₁₆	1	2	yes	—
45	#11719908	M₁₆	1	2	yes	—
46	C₂²×D₅₃	#108	1	1	yes	characteristic
47	C₂×C₄×C₅₃	—	2	1	no	—
48	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂×Dic₅₃	—	2	1	no	—
50	C₂×Dic₅₃	#108	1	1	yes	characteristic
51	C₄×D₅₃	—	2	1	no	—
52	C₄×D₅₃	—	2	1	no	—
53	C₈×C₅₃	—	2	1	no	—
54	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
55	C₈×C₅₃	C₂×M₁₆	1	1	yes	characteristic
56	#88322	—	2	1	no	—
57	#88322	#135	1	2	yes	—
58	#12027	—	53	1	no	maximal, Sylow
59	C₂²×C₅₃	C₂×#108	1	1	yes	characteristic
60	C₂×D₅₃	—	2	1	no	—
61	C₂×D₅₃	—	4	1	no	—
62	C₄×C₅₃	—	4	1	no	—
63	C₄×C₅₃	#423	1	1	yes	characteristic
64	C₄×C₅₃	#427	1	1	yes	characteristic
65	Dic₅₃	—	2	1	no	—
66	Dic₅₃	—	4	1	no	—
67	#2166	—	53	2	no	—
68	#2108	—	53	1	no	—
69	#1373	—	53	2	no	—
70	#1369	—	53	2	no	—
71	C₂×C₅₃	—	2	1	no	—
72	C₂×C₅₃	#1568	1	1	yes	characteristic
73	D₅₃	—	4	1	no	—
74	C₂×M₃₂	—	53	1	no	—
75	C₂²×C₁₆	—	53	1	no	—
76	#405	—	53	1	no	—
77	#364	—	53	4	no	—
78	#349	—	106	2	no	—
79	#347	—	106	2	no	—
80	#346	—	53	1	no	—
81	#346	#6157	1	1	yes	characteristic
82	C₄×C₁₆	—	53	1	no	—
83	C₄×C₁₆	—	53	1	no	—
84	C₅₃	#12027	1	1	yes	characteristic, complemented, Sylow
85	C₂×M₁₆	—	53	1	no	—
86	C₂²×C₈	—	53	1	no	—
87	#117	—	53	1	no	—
88	M₃₂	—	106	2	no	—
89	M₃₂	—	106	4	no	—
90	C₂×C₁₆	—	53	1	no	—
91	C₂×C₁₆	—	53	1	no	—
92	C₂×C₁₆	—	106	2	no	—
93	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
94	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
95	#97	—	53	1	no	—
96	#97	—	53	1	no	—
97	C₄×C₈	—	53	1	no	—
98	C₄×C₈	C₂×#6157	1	1	yes	characteristic
99	C₂²×C₄	—	53	1	no	—
100	M₁₆	—	106	2	no	—
101	C₂×C₈	—	53	1	no	—
102	C₂×C₈	—	53	1	no	—
103	C₂×C₈	—	106	1	no	—
104	C₂×C₈	—	106	1	no	—
105	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
106	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
107	C₄⋊C₄	—	106	1	no	—
108	C₂²⋊C₄	—	106	1	no	—
109	C₄²	—	53	1	no	—
110	C₄²	C₄⋊#6157	1	1	yes	characteristic
111	C₁₆	—	2	1	no	—
112	C₁₆	—	2	1	no	—
113	C₁₆	—	53	2	no	—
114	C₁₆	—	53	2	no	—
115	C₁₆	—	212	4	no	—
116	C₂³	—	53	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	53	1	no	—
119	C₂×C₄	—	106	1	no	—
120	C₂×C₄	—	106	1	no	—
121	C₂×C₄	—	106	1	no	—
122	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
123	C₈	—	2	1	no	—
124	C₈	—	53	2	no	—
125	C₈	—	106	1	no	—
126	C₈	D₅₃⋊M₁₆	1	1	yes	characteristic
127	C₈	D₅₃⋊M₁₆	1	1	yes	characteristic
128	C₂²	—	106	1	no	—
129	C₂²	—	212	1	no	—
130	C₂²	D₅₃⋊#108	1	1	yes	characteristic
131	C₄	—	4	1	no	—
132	C₄	—	106	1	no	—
133	C₄	—	212	1	no	—
134	C₄	#16811628	1	1	yes	characteristic
135	C₄	#16811633	1	1	yes	characteristic
136	C₂	—	2	1	no	—
137	C₂	—	212	1	no	—
138	C₂	#20290278	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20291090

C₂

yes

maximal

#20289751

C₂

yes

maximal, complemented

#20289743

C₂

yes

maximal, complemented

#20276313

C₂

yes

maximal, characteristic

#16811545

C₂²

yes

—

#16811520

—

#16811515

—

C₂×#13133156

C₄

yes

characteristic

C₂×C₁₆×D₅₃

C₄

yes

characteristic

#16810577

C₂²

yes

characteristic

#16810398

C₂²

yes

characteristic

C₄×#11719908

C₄

yes

characteristic

C₁₆×Dic₅₃

C₂²

yes

characteristic

C₅₃×#346

C₄

yes

characteristic

#13133291

—

C₂×#11719912

D₄

yes

characteristic

C₂×C₈×D₅₃

C₂×C₄

yes

characteristic

#13133174

Q₈

yes

characteristic

#13133156

—

C₂×#11719908

C₂×C₄

yes

characteristic

C₂×#11719908

C₂×C₄

yes

characteristic

C₁₆×D₅₃

—

#13133128

D₄

yes

characteristic

#13133127

Q₈

yes

characteristic

C₈×Dic₅₃

C₂³

yes

characteristic

C₂×C₁₆×C₅₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₄⋊C₄

yes

characteristic

#11719921

—

#11719919

—

C₄×Dic₅₃

C₂×Q₈

yes

characteristic

C₂×#88322

C₄⋊C₄

yes

characteristic

C₂×#88322

C₂×D₄

yes

characteristic

#11719912

—

C₈×D₅₃

—

C₈×D₅₃

—

#11719910

—

C₁₆×C₅₃

—

C₁₆×C₅₃

—

#11719908

M₁₆

yes

—

#11719908

M₁₆

yes

—

C₂²×D₅₃

#108

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

#108

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

—

C₈×C₅₃

—

C₈×C₅₃

C₂×M₁₆

yes

characteristic

C₈×C₅₃

C₂×M₁₆

yes

—

yes

—

—

C₂²×C₅₃

yes

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

—

—

—

—

—

C₂×C₅₃

—

C₂×C₅₃

#1568

yes

characteristic

D₅₃

—

C₂×M₃₂

—

C₂²×C₁₆

—

#405

—

#364

—

#349

—

106

—

#347

—

106

—

—

yes

C₄×C₁₆

—

C₄×C₁₆

—

C₅₃

#12027

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

#117

—

M₃₂

—

106

—

M₃₂

—

106

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

106

—

C₂×C₁₆

yes

C₂×C₁₆

yes

—

—

C₄×C₈

—

C₄×C₈

C₂×#6157

yes

characteristic

C₂²×C₄

—

100

M₁₆

—

106

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

106

—

104

C₂×C₈

—

106

—

105

C₂×C₈

C₄⋊#6157

yes

characteristic

106

C₂×C₈

C₂²×#6157

yes

characteristic

107

C₄⋊C₄

—

106

—

108

C₂²⋊C₄

—

106

—

109

C₄²

—

110

C₄²

C₄⋊#6157

yes

characteristic

111

C₁₆

—

112

C₁₆

—

113

C₁₆

—

114

C₁₆

—

115

C₁₆

—

212

—

116

C₂³

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

106

—

120

C₂×C₄

—

106

—

121

C₂×C₄

—

106

—

122

C₂×C₄

C₂×C₄⋊#6157

yes

characteristic

123

C₈

—

124

C₈

—

125

C₈

—

106

—

126

C₈

D₅₃⋊M₁₆

yes

characteristic

127

C₈

D₅₃⋊M₁₆

yes

characteristic

128

C₂²

—

106

—

129

C₂²

—

212

—

130

C₂²

D₅₃⋊#108

yes

characteristic

131

C₄

—

132

C₄

—

106

—

133

C₄

—

212

—

134

C₄

#16811628

yes

characteristic

135

C₄

#16811633

yes

characteristic

136

C₂

—

137

C₂

—

212

—

138

C₂

#20290278

yes

characteristic

	subgroup	G/H
center	C₄	#16811633
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×#6157
Fitting	C₅₃×#346	C₄
socle	C₂×C₅₃	#1568
prime radical	C₂²×D₅₃	#108
cosocle	C₈×Dic₅₃	C₂³
hypercenter	#346	#6157
hypocenter	C₅₃	#12027
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₄² ◁ C₄×C₈ ◁ #346

subgroup

G/H

center

C₄

#16811633

commutator

C₂×C₈×C₅₃

C₂²×C₄

Frattini

C₂×C₈

C₂²×#6157

Fitting

C₅₃×#346

C₄

C₂×C₅₃

C₂²×D₅₃

C₈×Dic₅₃

C₂³

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄ ◁ C₄² ◁ C₄×C₈ ◁ #346

N	H ≅ G/N	type
2. #20289751	C₂	semidirect
3. #20289743	C₂	semidirect
84. C₅₃	#12027	semidirect

H ≅ G/N

type

2. #20289751

C₂

semidirect

3. #20289743

C₂

semidirect

84. C₅₃

#12027

semidirect

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

subgroup	index
#20291090	2
#20289751	2
#20289743	2
#20276313	2
#12027	53

subgroup

index

Subgroups — #28309929

Conjugacy classes of subgroups

Complements

Maximal subgroups