Group #28309989 — subgroups

Subgroups — #28309989

Order 13568.

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

normal subgroups only

This group has 212 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	#20291104	C₂	1	2	yes	maximal, complemented
2	#20289750	C₂	1	1	yes	maximal, characteristic, complemented
3	#20289678	C₂	1	2	yes	maximal, complemented
4	#20289626	C₂	1	1	yes	maximal, characteristic, complemented
5	#20276595	C₂	1	1	yes	maximal, characteristic
6	#16811682	—	2	2	no	—
7	#16811545	C₂²	1	2	yes	—
8	#16811522	—	2	2	no	—
9	#16811509	C₂²	1	1	yes	characteristic
10	C₈×#88324	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	C₁₆×Dic₅₃	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	1	yes	characteristic
20	C₂×#11719910	D₄	1	1	yes	characteristic
21	C₄×#88324	—	2	1	no	—
22	#13133239	C₂×C₄	1	2	yes	—
23	#13133225	—	2	2	no	—
24	Dic₄₂₄	C₈	1	1	yes	characteristic, complemented
25	#13133162	C₈	1	2	yes	complemented
26	D₈₄₈	C₈	1	1	yes	characteristic, complemented
27	C₂×#11719908	C₂×C₄	1	1	yes	characteristic
28	#13133143	—	2	2	no	—
29	C₈×Dic₅₃	C₂³	1	1	yes	characteristic
30	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
31	#108×C₅₃	C₂×C₄	1	2	yes	—
32	#11719945	—	2	2	no	—
33	C₂×#88324	—	2	2	no	—
34	M₁₆×C₅₃	—	2	2	no	—
35	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
36	C₄×Dic₅₃	#41	1	1	yes	characteristic
37	C₂×#88322	C₂×D₄	1	1	yes	characteristic
38	Dic₂₁₂	C₂×C₈	1	2	yes	—
39	D₄₂₄	C₂×C₈	1	2	yes	—
40	#11719913	—	2	2	no	—
41	#11719912	—	4	2	no	—
42	C₈×D₅₃	—	4	2	no	—
43	#11719910	—	2	1	no	—
44	#11719910	—	4	2	no	—
45	#11719910	D₈	1	2	yes	—
46	C₁₆×C₅₃	C₂×C₈	1	1	yes	characteristic
47	C₁₆×C₅₃	C₂×C₈	1	1	yes	characteristic
48	#11719908	—	2	1	no	—
49	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
50	C₂²⋊D₅₃	—	4	2	no	—
51	C₂×Dic₅₃	—	2	1	no	—
52	D₂₁₂	—	2	2	no	—
53	C₄×D₅₃	—	4	2	no	—
54	Dic₁₀₆	—	2	2	no	—
55	#88324	—	4	2	no	—
56	C₈×C₅₃	—	4	2	no	—
57	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
58	C₈×C₅₃	#131	1	1	yes	characteristic
59	#88322	C₂×D₈	1	1	yes	characteristic
60	#88322	#135	1	1	yes	characteristic
61	#12047	—	53	1	no	maximal, Sylow
62	C₂²×C₅₃	#443	1	1	yes	characteristic
63	C₂×D₅₃	—	4	2	no	—
64	C₄×C₅₃	C₈×D₄	1	1	yes	characteristic
65	C₄×C₅₃	C₄×D₈	1	1	yes	characteristic
66	Dic₅₃	—	2	2	no	—
67	Dic₅₃	—	4	2	no	—
68	#2178	—	53	1	no	—
69	#2170	—	53	2	no	—
70	#1373	—	53	1	no	—
71	#1335	—	53	2	no	—
72	#1311	—	53	1	no	—
73	C₂×C₅₃	—	2	1	no	—
74	C₂×C₅₃	C₈×D₈	1	1	yes	characteristic
75	D₅₃	—	8	2	no	—
76	#508	—	53	1	no	—
77	#504	—	106	2	no	—
78	#443	—	53	2	no	—
79	#368	#6157	1	1	yes	characteristic
80	#364	—	53	2	no	—
81	#359	—	53	2	no	—
82	#350	—	106	2	no	—
83	C₄×C₁₆	—	53	1	no	—
84	C₄×C₁₆	—	53	1	no	—
85	C₈²	—	53	1	no	—
86	C₅₃	#12047	1	1	yes	characteristic, complemented, Sylow
87	#135	—	53	2	no	—
88	#131	—	106	2	no	—
89	Q₃₂	—	53	1	no	—
90	SD₃₂	—	53	2	no	—
91	D₁₆	—	53	1	no	—
92	M₃₂	—	106	2	no	—
93	M₃₂	—	212	2	no	—
94	C₂×C₁₆	—	53	1	no	—
95	C₂×C₁₆	—	53	1	no	—
96	C₂×C₁₆	—	53	1	no	—
97	C₂×C₁₆	—	212	2	no	—
98	C₂×C₁₆	C₂×#6157	1	1	yes	characteristic
99	#108	C₂×#6157	1	2	yes	—
100	#104	—	106	2	no	—
101	C₄×C₈	—	53	1	no	—
102	C₄×C₈	—	106	1	no	—
103	#41	—	106	2	no	—
104	Q₁₆	—	53	2	no	—
105	SD₁₆	—	106	2	no	—
106	D₈	—	53	2	no	—
107	M₁₆	—	2	2	no	—
108	M₁₆	—	212	2	no	—
109	C₂×C₈	—	53	1	no	—
110	C₂×C₈	—	106	2	no	—
111	C₂×C₈	—	212	2	no	—
112	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
113	C₄²	—	53	1	no	—
114	C₁₆	—	53	2	no	—
115	C₁₆	—	106	1	no	—
116	C₁₆	—	106	1	no	—
117	C₁₆	—	212	2	no	—
118	C₁₆	D₅₃⋊C₈	1	1	yes	characteristic
119	C₁₆	D₅₃⋊C₈	1	1	yes	characteristic
120	Q₈	—	106	2	no	—
121	D₄	—	106	2	no	—
122	D₄	—	212	2	no	—
123	C₂×C₄	—	106	1	no	—
124	C₂×C₄	—	212	2	no	—
125	C₂×C₄	D₄×#6157	1	1	yes	characteristic
126	C₈	—	4	2	no	—
127	C₈	—	53	1	no	—
128	C₈	—	53	1	no	—
129	C₈	—	212	2	no	—
130	C₈	C₂×D₅₃⋊C₈	1	1	yes	characteristic
131	C₈	#13133305	1	1	yes	characteristic
132	C₂²	—	212	2	no	—
133	C₂²	#16811667	1	1	yes	characteristic
134	C₄	—	106	2	no	—
135	C₄	—	212	2	no	—
136	C₄	#16811651	1	1	yes	characteristic
137	C₄	D₈×#6157	1	1	yes	characteristic
138	C₂	—	2	1	no	—
139	C₂	—	424	2	no	—
140	C₂	#20290296	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20291104

C₂

yes

maximal, complemented

#20289750

C₂

yes

maximal, characteristic, complemented

#20289678

C₂

yes

maximal, complemented

#20289626

C₂

yes

maximal, characteristic, complemented

#20276595

C₂

yes

maximal, characteristic

#16811682

—

#16811545

C₂²

yes

—

#16811522

—

#16811509

C₂²

yes

characteristic

C₈×#88324

C₂²

yes

characteristic

#16811055

C₄

yes

characteristic

#16810739

C₂²

yes

—

#16810429

C₄

yes

—

C₁₆×Dic₅₃

C₂²

yes

characteristic

C₅₃×#368

C₄

yes

characteristic

#13133292

—

#13133291

—

D₅₃⋊C₁₆

—

C₂×#11719910

D₄

yes

characteristic

C₂×#11719910

D₄

yes

characteristic

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

—

#11719910

D₈

yes

—

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

C₂×D₈

yes

yes

—

C₂²×C₅₃

yes

C₂×D₅₃

—

C₄×C₅₃

C₈×D₄

yes

characteristic

C₄×C₅₃

C₄×D₈

yes

characteristic

Dic₅₃

—

Dic₅₃

—

—

—

—

—

—

C₂×C₅₃

—

C₂×C₅₃

C₈×D₈

yes

characteristic

D₅₃

—

#508

—

#504

—

106

—

—

yes

—

—

—

106

—

C₄×C₁₆

—

C₄×C₁₆

—

C₈²

—

C₅₃

#12047

yes

characteristic, complemented, Sylow

#135

—

#131

—

106

—

Q₃₂

—

SD₃₂

—

D₁₆

—

M₃₂

—

106

—

M₃₂

—

212

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

212

—

C₂×C₁₆

yes

yes

—

100

#104

—

106

—

101

C₄×C₈

—

102

C₄×C₈

—

106

—

103

#41

—

106

—

104

Q₁₆

—

105

SD₁₆

—

106

—

106

D₈

—

107

M₁₆

—

108

M₁₆

—

212

—

109

C₂×C₈

—

110

C₂×C₈

—

106

—

111

C₂×C₈

—

212

—

112

C₂×C₈

C₂²×#6157

yes

characteristic

113

C₄²

—

114

C₁₆

—

115

C₁₆

—

106

—

116

C₁₆

—

106

—

117

C₁₆

—

212

—

118

C₁₆

D₅₃⋊C₈

yes

characteristic

119

C₁₆

D₅₃⋊C₈

yes

characteristic

120

Q₈

—

106

—

121

D₄

—

106

—

122

D₄

—

212

—

123

C₂×C₄

—

106

—

124

C₂×C₄

—

212

—

125

C₂×C₄

D₄×#6157

yes

characteristic

126

C₈

—

127

C₈

—

128

C₈

—

129

C₈

—

212

—

130

C₈

C₂×D₅₃⋊C₈

yes

characteristic

131

C₈

#13133305

yes

characteristic

132

C₂²

—

212

—

133

C₂²

#16811667

yes

characteristic

134

C₄

—

106

—

135

C₄

—

212

—

136

C₄

#16811651

yes

characteristic

137

C₄

D₈×#6157

yes

characteristic

138

C₂

—

139

C₂

—

424

—

140

C₂

#20290296

yes

characteristic

	subgroup	G/H
center	C₄	D₈×#6157
commutator	C₈×C₅₃	C₂²×C₈
Frattini	C₂×C₈	C₂²×#6157
Fitting	C₅₃×#368	C₄
socle	C₂×C₅₃	C₈×D₈
prime radical	#16811055	C₄
cosocle	C₈×Dic₅₃	C₂³
hypercenter	#368	#6157
hypocenter	C₅₃	#12047
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #368

subgroup

G/H

center

C₄

D₈×#6157

commutator

C₈×C₅₃

C₂²×C₈

Frattini

C₂×C₈

C₂²×#6157

Fitting

C₅₃×#368

C₄

socle

C₂×C₅₃

C₈×D₈

prime radical

#16811055

C₄

cosocle

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. #20291104	C₂	semidirect
2. #20289750	C₂	semidirect
3. #20289678	C₂	semidirect
4. #20289626	C₂	semidirect
24. Dic₄₂₄	C₈	semidirect
25. #13133162	C₈	semidirect
26. D₈₄₈	C₈	semidirect
86. C₅₃	#12047	semidirect

H ≅ G/N

type

1. #20291104

C₂

semidirect

2. #20289750

C₂

semidirect

3. #20289678

C₂

semidirect

4. #20289626

C₂

semidirect

24. Dic₄₂₄

C₈

semidirect

25. #13133162

C₈

semidirect

26. D₈₄₈

C₈

semidirect

86. C₅₃

#12047

semidirect

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

subgroup	index
#20291104	2
#20289750	2
#20289678	2
#20289626	2
#20276595	2
#12047	53

subgroup

index

Subgroups — #28309989

Conjugacy classes of subgroups

Complements

Maximal subgroups