Group #27300544 — subgroups

Subgroups — #27300544

Order 13568.

306 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 179 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	C₂×#16810453	C₂	1	4	yes	maximal
2	#20274887	C₂	1	2	yes	maximal
3	C₂×C₅₃×#346	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133153	—	2	2	no	—
5	C₂×#13133153	C₂²	1	4	yes	—
6	#16810647	—	2	2	no	—
7	#16810453	—	2	8	no	—
8	C₂²×C₁₆×C₅₃	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₅₃×#346	C₄	1	4	yes	—
11	C₂×#11719917	—	2	4	no	—
12	#13133153	—	2	8	no	—
13	#13133153	—	4	4	no	—
14	C₄⋊#88322	—	4	4	no	—
15	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
16	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
17	C₂×C₄²×C₅₃	D₄	1	1	yes	characteristic
18	C₂×C₁₆×C₅₃	C₂×C₄	1	4	yes	—
19	C₂×C₁₆×C₅₃	D₄	1	4	yes	—
20	C₂×C₁₆×C₅₃	Q₈	1	4	yes	—
21	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₅₃	Q₈	1	2	yes	—
23	C₂²×C₄×C₅₃	Q₁₆	1	2	yes	—
24	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
25	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
26	C₂×C₈×C₅₃	D₈	1	2	yes	—
27	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
28	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
30	C₂×C₈×C₅₃	C₂×Q₈	1	2	yes	—
31	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
32	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
33	C₄²×C₅₃	D₈	1	2	yes	—
34	#11719917	—	4	4	no	—
35	#11719917	—	4	4	no	—
36	C₂×#88322	—	4	4	no	—
37	C₁₆×C₅₃	C₄⋊C₄	1	8	yes	—
38	C₂³×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
39	C₂×C₄×C₅₃	—	4	2	no	—
40	C₂×C₄×C₅₃	#107	1	4	yes	—
41	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
43	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
44	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
45	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
46	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
47	C₈×C₅₃	#107	1	4	yes	—
48	C₈×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
49	C₈×C₅₃	C₄⋊Q₈	1	2	yes	—
50	#88322	—	8	4	no	—
51	#12910	—	53	1	no	maximal, Sylow
52	C₂²×C₅₃	—	2	2	no	—
53	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
54	C₂²×C₅₃	#494	1	1	yes	characteristic
55	C₂²×C₅₃	#500	1	1	yes	characteristic
56	C₄×C₅₃	—	2	4	no	—
57	C₄×C₅₃	#384	1	1	yes	characteristic
58	C₄×C₅₃	C₂×#107	1	1	yes	characteristic
59	C₄×C₅₃	#493	1	1	yes	characteristic
60	C₄×C₅₃	#500	1	1	yes	characteristic
61	C₂×#368	—	53	4	no	—
62	C₂×#346	D₅₃	1	1	yes	characteristic
63	#1848	—	53	2	no	—
64	C₂×C₅₃	—	2	1	no	—
65	C₂×C₅₃	—	2	1	no	—
66	C₂×C₅₃	#1845	1	1	yes	characteristic
67	C₂×C₅₃	#2161	1	2	yes	—
68	C₂²×C₁₆	C₂×D₅₃	1	2	yes	—
69	C₂×#108	—	53	4	no	—
70	C₂×#108	—	106	2	no	—
71	#423	—	106	2	no	—
72	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
73	#368	—	106	8	no	—
74	#346	Dic₅₃	1	4	yes	—
75	C₅₃	#12910	1	1	yes	characteristic, complemented, Sylow
76	C₂×M₁₆	—	106	4	no	—
77	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
78	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
79	C₂×C₄²	D₂₁₂	1	1	yes	characteristic
80	C₂×C₁₆	Dic₁₀₆	1	4	yes	—
81	C₂×C₁₆	D₂₁₂	1	4	yes	—
82	C₂×C₁₆	C₂×Dic₅₃	1	4	yes	—
83	#108	—	106	8	no	—
84	#108	—	212	4	no	—
85	C₄⋊C₈	—	212	4	no	—
86	C₄×C₈	Dic₁₀₆	1	2	yes	—
87	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
88	C₂²×C₄	Dic₂₁₂	1	2	yes	—
89	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
90	M₁₆	—	212	4	no	—
91	M₁₆	—	212	4	no	—
92	C₂×C₈	—	212	4	no	—
93	C₂×C₈	D₄₂₄	1	2	yes	—
94	C₂×C₈	Dic₂₁₂	1	2	yes	—
95	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
96	C₂×C₈	C₂×Dic₁₀₆	1	2	yes	—
97	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
98	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
99	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
100	C₄²	D₄₂₄	1	2	yes	—
101	C₄²	C₄⋊Dic₅₃	1	2	yes	—
102	C₁₆	C₄⋊Dic₅₃	1	8	yes	—
103	C₂³	C₂×Dic₂₁₂	1	1	yes	characteristic
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	#13133151	1	4	yes	—
106	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
107	C₂×C₄	#13133214	1	1	yes	characteristic
108	C₂×C₄	#13133217	1	1	yes	characteristic
109	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
110	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
111	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
112	C₈	—	424	4	no	—
113	C₈	#13133151	1	4	yes	—
114	C₈	C₂×C₄⋊Dic₅₃	1	2	yes	—
115	C₈	#13133217	1	2	yes	—
116	C₂²	—	2	2	no	—
117	C₂²	C₂×#13133151	1	1	yes	characteristic
118	C₂²	#16811003	1	1	yes	characteristic
119	C₂²	#16811028	1	1	yes	characteristic
120	C₄	—	2	4	no	—
121	C₄	#16810493	1	1	yes	characteristic
122	C₄	C₂×#13133151	1	1	yes	characteristic
123	C₄	#16811000	1	1	yes	characteristic
124	C₄	#16811028	1	1	yes	characteristic
125	C₂	—	2	1	no	—
126	C₂	—	2	1	no	—
127	C₂	#20274875	1	1	yes	characteristic
128	C₂	#20276510	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16810453

C₂

yes

maximal

#20274887

C₂

yes

maximal

C₂×C₅₃×#346

C₂

yes

maximal, characteristic

C₂×#13133153

—

C₂×#13133153

C₂²

yes

—

#16810647

—

#16810453

—

C₂²×C₁₆×C₅₃

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#346

C₄

yes

—

—

—

—

—

C₂²×C₈×C₅₃

D₄

yes

characteristic

C₂²×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₄²×C₅₃

D₄

yes

characteristic

C₂×C₁₆×C₅₃

C₂×C₄

yes

—

C₂×C₁₆×C₅₃

D₄

yes

—

C₂×C₁₆×C₅₃

Q₈

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

Q₈

yes

—

C₂²×C₄×C₅₃

Q₁₆

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

D₈

yes

—

C₂×C₈×C₅₃

Q₁₆

yes

—

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₂×C₈×C₅₃

C₂×Q₈

yes

—

C₂×C₈×C₅₃

C₂×Q₈

yes

characteristic

C₄²×C₅₃

C₄⋊C₄

yes

—

C₄²×C₅₃

D₈

yes

—

#11719917

—

#11719917

—

C₂×#88322

—

C₁₆×C₅₃

C₄⋊C₄

yes

—

C₂³×C₅₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#107

yes

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

#127

yes

characteristic

C₂×C₄×C₅₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₁₆

yes

characteristic

C₈×C₅₃

#107

yes

—

C₈×C₅₃

C₂×C₄⋊C₄

yes

—

C₈×C₅₃

C₄⋊Q₈

yes

—

#88322

—

#12910

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

D₅₃

yes

characteristic

#1848

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#1845

yes

characteristic

C₂×C₅₃

#2161

yes

—

C₂²×C₁₆

C₂×D₅₃

yes

—

C₂×#108

—

C₂×#108

—

106

—

#423

—

106

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

#368

—

106

—

#346

Dic₅₃

yes

—

C₅₃

#12910

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

106

—

C₂²×C₈

D₂₁₂

yes

characteristic

C₂²×C₈

C₂²×D₅₃

yes

characteristic

C₂×C₄²

D₂₁₂

yes

characteristic

C₂×C₁₆

Dic₁₀₆

yes

—

C₂×C₁₆

D₂₁₂

yes

—

C₂×C₁₆

C₂×Dic₅₃

yes

—

#108

—

106

—

#108

—

212

—

C₄⋊C₈

—

212

—

C₄×C₈

Dic₁₀₆

yes

—

C₄×C₈

C₂×Dic₅₃

yes

—

C₂²×C₄

Dic₂₁₂

yes

—

C₂²×C₄

C₂×D₂₁₂

yes

characteristic

M₁₆

—

212

—

M₁₆

—

212

—

C₂×C₈

—

212

—

C₂×C₈

D₄₂₄

yes

—

C₂×C₈

Dic₂₁₂

yes

—

C₂×C₈

C₄⋊Dic₅₃

yes

—

C₂×C₈

C₂×Dic₁₀₆

yes

—

C₂×C₈

C₂×Dic₁₀₆

yes

characteristic

C₂×C₈

C₂×D₂₁₂

yes

—

C₂×C₈

C₂²×Dic₅₃

yes

characteristic

100

C₄²

D₄₂₄

yes

—

101

C₄²

C₄⋊Dic₅₃

yes

—

102

C₁₆

C₄⋊Dic₅₃

yes

—

103

C₂³

C₂×Dic₂₁₂

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

#13133151

yes

—

106

C₂×C₄

C₂×C₄⋊Dic₅₃

yes

characteristic

107

C₂×C₄

#13133214

yes

characteristic

108

C₂×C₄

#13133217

yes

characteristic

109

C₂×C₄

C₂×D₄₂₄

yes

characteristic

110

C₂×C₄

C₂×D₄₂₄

yes

characteristic

111

C₂×C₄

C₂×Dic₂₁₂

yes

characteristic

112

C₈

—

424

—

113

C₈

#13133151

yes

—

114

C₈

C₂×C₄⋊Dic₅₃

yes

—

115

C₈

#13133217

yes

—

116

C₂²

—

117

C₂²

C₂×#13133151

yes

characteristic

118

C₂²

#16811003

yes

characteristic

119

C₂²

#16811028

yes

characteristic

120

C₄

—

121

C₄

#16810493

yes

characteristic

122

C₄

C₂×#13133151

yes

characteristic

123

C₄

#16811000

yes

characteristic

124

C₄

#16811028

yes

characteristic

125

C₂

—

126

C₂

—

127

C₂

#20274875

yes

characteristic

128

C₂

#20276510

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×D₄₂₄
commutator	C₂×C₈×C₅₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#346	C₂
socle	C₂²×C₅₃	C₂×#107
prime radical	C₂³×C₅₃	C₂×Q₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#346	D₅₃
hypocenter	C₅₃	#12910
LCS	G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

subgroup

G/H

center

C₂×C₄

C₂×D₄₂₄

commutator

C₂×C₈×C₅₃

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₂×C₅₃×#346

C₂

socle

C₂²×C₅₃

C₂×#107

prime radical

C₂³×C₅₃

C₂×Q₁₆

cosocle

C₂²×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂×C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

N	H ≅ G/N	type
75. C₅₃	#12910	semidirect

H ≅ G/N

type

75. C₅₃

#12910

semidirect

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

subgroup	index
C₂×#16810453	2
#20274887	2
C₂×C₅₃×#346	2
#12910	53

subgroup

index

C₂×#16810453

#20274887

C₂×C₅₃×#346

#12910

Subgroups — #27300544

Conjugacy classes of subgroups

Complements

Maximal subgroups