Group #28311403 — subgroups

Subgroups — #28311403

Order 13568.

310 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 108 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	#20290652	C₂	1	2	yes	maximal
2	#20289773	C₂	1	4	yes	maximal
3	C₄×C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133286	C₂²	1	4	yes	—
5	C₂×#13133285	—	2	2	no	—
6	#16811567	—	2	2	no	—
7	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
8	C₄×C₈×D₅₃	C₂²	1	1	yes	characteristic
9	C₄×#11719908	C₄	1	1	yes	characteristic, complemented
10	C₁₆×Dic₅₃	C₄	1	2	yes	complemented
11	C₄×C₁₆×C₅₃	C₄	1	1	yes	characteristic, complemented
12	C₂×C₄⋊#6157	—	2	4	no	—
13	#13133286	—	2	8	no	—
14	#13133285	—	4	4	no	—
15	(C₂×#6157)⋊C₄	—	4	4	no	—
16	C₂×C₈×D₅₃	D₄	1	1	yes	characteristic
17	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
18	C₄²×D₅₃	D₄	1	1	yes	characteristic
19	C₂×#11719908	C₂×C₄	1	2	yes	—
20	C₁₆×D₅₃	—	2	4	no	—
21	C₄×#88322	Q₈	1	1	yes	characteristic
22	C₈×Dic₅₃	C₂×C₄	1	1	yes	characteristic
23	C₈×Dic₅₃	Q₈	1	1	yes	characteristic
24	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
26	C₂²×#6157	—	4	4	no	—
27	C₂×C₄×D₅₃	D₈	1	2	yes	—
28	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
29	C₄⋊#6157	—	4	4	no	—
30	C₄⋊#6157	—	4	4	no	—
31	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
33	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
34	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
35	C₄×Dic₅₃	Q₁₆	1	2	yes	—
36	C₂×#88322	C₄⋊C₄	1	1	yes	characteristic
37	C₂×#88322	C₂×Q₈	1	1	yes	characteristic
38	C₈×D₅₃	SD₁₆	1	4	yes	—
39	C₈×D₅₃	#41	1	4	yes	—
40	C₁₆×C₅₃	—	2	2	no	—
41	#11719908	—	2	2	no	—
42	C₂²×D₅₃	C₂×D₈	1	1	yes	characteristic
43	C₂×#6157	—	8	4	no	—
44	C₂×#6157	—	8	8	no	—
45	C₂×C₄×C₅₃	#107	1	2	yes	—
46	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂×Dic₅₃	#107	1	2	yes	—
48	C₂×Dic₅₃	C₂×Q₁₆	1	1	yes	characteristic
49	C₄×D₅₃	—	2	4	no	—
50	C₄×D₅₃	#124	1	2	yes	—
51	C₄×D₅₃	C₂×SD₁₆	1	2	yes	—
52	C₈×C₅₃	#106	1	2	yes	—
53	C₈×C₅₃	#117	1	2	yes	—
54	#88322	#106	1	2	yes	—
55	#88322	#125	1	2	yes	—
56	#12889	—	53	1	no	maximal, Sylow
57	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
58	C₂×D₅₃	#486	1	2	yes	—
59	C₂×D₅₃	#508	1	4	yes	—
60	#6157	—	16	4	no	—
61	C₄×C₅₃	—	2	2	no	—
62	C₄×C₅₃	#382	1	1	yes	characteristic
63	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
64	Dic₅₃	—	2	2	no	—
65	Dic₅₃	#487	1	2	yes	—
66	C₂×C₄×C₁₆	—	53	1	no	—
67	#1846	—	53	2	no	—
68	#1380	—	53	4	no	—
69	C₂×C₅₃	#1834	1	1	yes	characteristic
70	C₂×C₅₃	#2158	1	2	yes	—
71	D₅₃	#2242	1	4	yes	—
72	C₂²×C₁₆	—	53	2	no	—
73	C₂×#107	—	53	4	no	—
74	C₂×#106	—	106	2	no	—
75	C₂×C₄×C₈	—	53	1	no	—
76	#382	—	106	2	no	—
77	C₄×C₁₆	—	53	2	no	—
78	C₄×C₁₆	—	53	1	no	—
79	C₄×C₁₆	#6157	1	1	yes	characteristic, complemented
80	C₅₃	#12889	1	1	yes	characteristic, complemented, Sylow
81	C₂²×C₈	—	53	1	no	—
82	C₂²×C₈	—	53	1	no	—
83	C₂×C₄⋊C₄	—	106	4	no	—
84	C₂×C₄²	—	53	1	no	—
85	C₂×C₁₆	—	53	2	no	—
86	C₂×C₁₆	—	106	4	no	—
87	C₂×C₁₆	C₂×#6157	1	2	yes	—
88	#107	—	106	8	no	—
89	#106	—	212	4	no	—
90	C₄×C₈	—	53	1	no	—
91	C₄×C₈	—	53	1	no	—
92	C₄×C₈	—	53	1	no	—
93	C₄×C₈	C₂×#6157	1	1	yes	characteristic
94	#95	—	212	4	no	—
95	C₂²×C₄	—	53	1	no	—
96	C₂²×C₄	—	53	2	no	—
97	C₂²×C₄	—	212	4	no	—
98	C₂×C₈	—	53	1	no	—
99	C₂×C₈	—	53	1	no	—
100	C₂×C₈	—	53	4	no	—
101	C₂×C₈	—	53	4	no	—
102	C₂×C₈	C₄⋊#6157	1	1	yes	characteristic
103	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
104	C₄⋊C₄	—	212	4	no	—
105	C₄⋊C₄	—	212	4	no	—
106	C₄²	—	53	2	no	—
107	C₄²	—	53	1	no	—
108	C₄²	C₄⋊#6157	1	1	yes	characteristic
109	C₁₆	—	2	2	no	—
110	C₁₆	—	106	2	no	—
111	C₂³	—	53	1	no	—
112	C₂×C₄	—	53	1	no	—
113	C₂×C₄	—	53	2	no	—
114	C₂×C₄	—	53	2	no	—
115	C₂×C₄	—	53	2	no	—
116	C₂×C₄	—	106	4	no	—
117	C₂×C₄	—	424	4	no	—
118	C₂×C₄	—	424	8	no	—
119	C₂×C₄	#13133286	1	2	yes	—
120	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
121	C₈	—	53	2	no	—
122	C₈	—	53	2	no	—
123	C₈	#13133285	1	2	yes	—
124	C₈	#13133296	1	2	yes	—
125	C₂²	—	53	2	no	—
126	C₂²	—	53	4	no	—
127	C₂²	C₂×#13133286	1	1	yes	characteristic
128	C₄	—	2	2	no	—
129	C₄	—	53	2	no	—
130	C₄	—	848	4	no	—
131	C₄	—	106	2	no	—
132	C₄	#16811567	1	1	yes	characteristic
133	C₄	C₂×#13133285	1	1	yes	characteristic
134	C₂	—	53	4	no	—
135	C₂	#20290636	1	1	yes	characteristic
136	C₂	#20291066	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20290652

C₂

yes

maximal

#20289773

C₂

yes

maximal

C₄×C₁₆×D₅₃

C₂

yes

maximal, characteristic

C₂×#13133286

C₂²

yes

—

C₂×#13133285

—

#16811567

—

C₂×C₁₆×D₅₃

C₂²

yes

—

C₄×C₈×D₅₃

C₂²

yes

characteristic

C₄×#11719908

C₄

yes

characteristic, complemented

C₁₆×Dic₅₃

C₄

yes

complemented

C₄×C₁₆×C₅₃

C₄

yes

characteristic, complemented

C₂×C₄⋊#6157

—

#13133286

—

#13133285

—

(C₂×#6157)⋊C₄

—

C₂×C₈×D₅₃

D₄

yes

characteristic

C₂×C₈×D₅₃

C₂³

yes

characteristic

C₄²×D₅₃

D₄

yes

characteristic

C₂×#11719908

C₂×C₄

yes

—

C₁₆×D₅₃

—

C₄×#88322

Q₈

yes

characteristic

C₈×Dic₅₃

C₂×C₄

yes

characteristic

C₈×Dic₅₃

Q₈

yes

characteristic

C₂×C₁₆×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×#6157

—

C₂×C₄×D₅₃

D₈

yes

—

C₂×C₄×D₅₃

C₂×D₄

yes

characteristic

C₄⋊#6157

—

C₄⋊#6157

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₄⋊C₄

yes

characteristic

C₄×Dic₅₃

C₄⋊C₄

yes

characteristic

C₄×Dic₅₃

Q₁₆

yes

—

C₂×#88322

C₄⋊C₄

yes

characteristic

C₂×#88322

C₂×Q₈

yes

characteristic

C₈×D₅₃

SD₁₆

yes

—

C₈×D₅₃

#41

yes

—

C₁₆×C₅₃

—

#11719908

—

C₂²×D₅₃

C₂×D₈

yes

characteristic

C₂×#6157

—

C₂×#6157

—

C₂×C₄×C₅₃

#107

yes

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×Dic₅₃

#107

yes

—

C₂×Dic₅₃

C₂×Q₁₆

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

#124

yes

—

C₄×D₅₃

C₂×SD₁₆

yes

—

C₈×C₅₃

#106

yes

—

C₈×C₅₃

#117

yes

—

#88322

#106

yes

—

#88322

#125

yes

—

—

C₂²×C₅₃

yes

C₂×D₅₃

yes

—

C₂×D₅₃

#508

yes

—

#6157

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

Dic₅₃

#487

yes

—

C₂×C₄×C₁₆

—

#1846

—

#1380

—

C₂×C₅₃

#1834

yes

characteristic

C₂×C₅₃

#2158

yes

—

D₅₃

#2242

yes

—

C₂²×C₁₆

—

C₂×#107

—

C₂×#106

—

106

—

C₂×C₄×C₈

—

#382

—

106

—

C₄×C₁₆

—

C₄×C₁₆

—

C₄×C₁₆

#6157

yes

characteristic, complemented

C₅₃

#12889

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

106

—

C₂×C₁₆

C₂×#6157

yes

—

#107

—

106

—

#106

—

212

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×#6157

yes

characteristic

#95

—

212

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

212

—

C₂×C₈

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

C₄⋊#6157

yes

characteristic

103

C₂×C₈

C₂²×#6157

yes

characteristic

104

C₄⋊C₄

—

212

—

105

C₄⋊C₄

—

212

—

106

C₄²

—

107

C₄²

—

108

C₄²

C₄⋊#6157

yes

characteristic

109

C₁₆

—

110

C₁₆

—

106

—

111

C₂³

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

106

—

117

C₂×C₄

—

424

—

118

C₂×C₄

—

424

—

119

C₂×C₄

#13133286

yes

—

120

C₂×C₄

C₂×C₄⋊#6157

yes

characteristic

121

C₈

—

122

C₈

—

123

C₈

#13133285

yes

—

124

C₈

#13133296

yes

—

125

C₂²

—

126

C₂²

—

127

C₂²

C₂×#13133286

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

848

—

131

C₄

—

106

—

132

C₄

#16811567

yes

characteristic

133

C₄

C₂×#13133285

yes

characteristic

134

C₂

—

135

C₂

#20290636

yes

characteristic

136

C₂

#20291066

yes

—

	subgroup	G/H
center	C₂²	C₂×#13133286
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂×C₈	C₂²×#6157
Fitting	C₄×C₁₆×C₅₃	C₄
socle	C₂²×C₅₃	C₂×#107
prime radical	C₂²×D₅₃	C₂×D₈
cosocle	C₂×C₈×D₅₃	C₂³
hypercenter	C₄×C₁₆	#6157
hypocenter	C₅₃	#12889
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

subgroup

G/H

center

C₂²

C₂×#13133286

commutator

C₂×C₈×C₅₃

C₂²×C₄

Frattini

C₂×C₈

C₂²×#6157

Fitting

C₄×C₁₆×C₅₃

C₄

socle

C₂²×C₅₃

C₂×#107

prime radical

C₂²×D₅₃

C₂×D₈

cosocle

C₂×C₈×D₅₃

C₂³

C₄×C₁₆

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆

N	H ≅ G/N	type
9. C₄×#11719908	C₄	semidirect
10. C₁₆×Dic₅₃	C₄	semidirect
11. C₄×C₁₆×C₅₃	C₄	semidirect
79. C₄×C₁₆	#6157	semidirect
80. C₅₃	#12889	semidirect

H ≅ G/N

type

9. C₄×#11719908

C₄

semidirect

10. C₁₆×Dic₅₃

C₄

semidirect

11. C₄×C₁₆×C₅₃

C₄

semidirect

79. C₄×C₁₆

#6157

semidirect

80. C₅₃

#12889

semidirect

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

subgroup	index
#20290652	2
#20289773	2
C₄×C₁₆×D₅₃	2
#12889	53

subgroup

index

#20290652

#20289773

C₄×C₁₆×D₅₃

#12889

Subgroups — #28311403

Conjugacy classes of subgroups

Complements

Maximal subgroups