Group #27277896 — subgroups

Subgroups — #27277896

Order 13568.

354 conjugacy classes of subgroups; only the 203 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 203 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	#20274887	C₂	1	2	yes	maximal
2	#20273474	C₂	1	4	yes	maximal
3	C₂×C₈²×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133153	—	2	4	no	—
5	#16810647	C₂²	1	4	yes	—
6	#16810376	—	2	8	no	—
7	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₈²×C₅₃	C₄	1	1	yes	characteristic
10	C₈²×C₅₃	C₄	1	2	yes	—
11	C₈²×C₅₃	C₄	1	1	yes	characteristic
12	C₂×#11719917	—	2	4	no	—
13	#13133153	—	4	8	no	—
14	C₄⋊#88322	—	2	8	no	—
15	C₂²×C₈×C₅₃	D₄	1	2	yes	—
16	C₂²×C₈×C₅₃	Q₈	1	4	yes	—
17	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
18	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₅₃	D₄	1	4	yes	—
23	C₄×C₈×C₅₃	Q₈	1	2	yes	—
24	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
25	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
26	C₂×C₈×C₅₃	—	2	8	no	—
27	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
28	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
29	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
30	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
31	C₂×C₈×C₅₃	D₈	1	4	yes	—
32	C₂×C₈×C₅₃	Q₁₆	1	4	yes	—
33	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
34	C₄²×C₅₃	C₂×D₄	1	2	yes	—
35	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
36	#11719917	—	4	8	no	—
37	C₂×#88322	—	4	4	no	—
38	C₂³×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
39	C₂×C₄×C₅₃	—	2	4	no	—
40	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
42	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
43	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
44	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
45	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
46	C₈×C₅₃	—	2	4	no	—
47	C₈×C₅₃	#107	1	4	yes	—
48	C₈×C₅₃	#107	1	4	yes	—
49	C₈×C₅₃	#108	1	8	yes	complemented
50	#88322	—	8	4	no	—
51	#8823	—	53	1	no	maximal, Sylow
52	C₂²×C₅₃	—	2	2	no	—
53	C₂²×C₅₃	#384	1	1	yes	characteristic
54	C₂²×C₅₃	#500	1	2	yes	—
55	C₄×C₅₃	—	2	2	no	—
56	C₄×C₅₃	C₂×#107	1	2	yes	—
57	C₄×C₅₃	C₂×#108	1	4	yes	—
58	C₄×C₅₃	#493	1	1	yes	characteristic
59	C₄×C₅₃	#494	1	1	yes	characteristic
60	#1848	—	53	2	no	—
61	#1567	—	53	4	no	—
62	C₂×C₈²	D₅₃	1	1	yes	characteristic
63	C₂×C₅₃	—	2	2	no	—
64	C₂×C₅₃	#1845	1	1	yes	characteristic
65	C₂×C₅₃	#1848	1	2	yes	—
66	C₂×#108	—	106	4	no	—
67	#423	—	53	4	no	—
68	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
69	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
70	#335	—	106	8	no	—
71	C₈²	Dic₅₃	1	1	yes	characteristic
72	C₈²	Dic₅₃	1	2	yes	—
73	C₈²	Dic₅₃	1	1	yes	characteristic
74	C₅₃	#8823	1	1	yes	characteristic, complemented, Sylow
75	C₂×M₁₆	—	106	4	no	—
76	C₂²×C₈	Dic₁₀₆	1	4	yes	—
77	C₂²×C₈	D₂₁₂	1	2	yes	—
78	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
79	#108	—	212	8	no	—
80	C₄⋊C₈	—	106	8	no	—
81	C₄×C₈	Dic₁₀₆	1	2	yes	—
82	C₄×C₈	D₂₁₂	1	4	yes	—
83	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
84	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
85	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
86	C₄×C₈	C₂×Dic₅₃	1	2	yes	—
87	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
88	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
89	M₁₆	—	212	8	no	—
90	C₂×C₈	—	2	8	no	—
91	C₂×C₈	—	212	4	no	—
92	C₂×C₈	D₄₂₄	1	4	yes	—
93	C₂×C₈	Dic₂₁₂	1	4	yes	—
94	C₂×C₈	C₄⋊Dic₅₃	1	4	yes	—
95	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
96	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
97	C₂×C₈	C₄⋊Dic₅₃	1	4	yes	—
98	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
99	C₄²	C₂×D₂₁₂	1	2	yes	—
100	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
101	C₂³	#13133217	1	1	yes	characteristic
102	C₂×C₄	—	2	4	no	—
103	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
104	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
105	C₂×C₄	#13133214	1	1	yes	characteristic
106	C₂×C₄	#13133217	1	2	yes	—
107	C₂×C₄	C₂×D₄₂₄	1	2	yes	—
108	C₂×C₄	C₂×Dic₂₁₂	1	2	yes	—
109	C₈	—	2	4	no	—
110	C₈	—	424	4	no	—
111	C₈	#13133151	1	4	yes	—
112	C₈	#13133151	1	4	yes	—
113	C₈	#13133153	1	8	yes	complemented
114	C₂²	—	2	2	no	—
115	C₂²	#16810493	1	1	yes	characteristic
116	C₂²	#16811028	1	2	yes	—
117	C₄	—	2	2	no	—
118	C₄	C₂×#13133151	1	2	yes	—
119	C₄	C₂×#13133153	1	4	yes	—
120	C₄	#16811000	1	1	yes	characteristic
121	C₄	#16811003	1	1	yes	characteristic
122	C₂	—	2	2	no	—
123	C₂	#20274875	1	1	yes	characteristic
124	C₂	#20274887	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20274887

C₂

yes

maximal

#20273474

C₂

yes

maximal

C₂×C₈²×C₅₃

C₂

yes

maximal, characteristic

C₂×#13133153

—

#16810647

C₂²

yes

—

#16810376

—

C₂×C₄×C₈×C₅₃

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₈²×C₅₃

C₄

yes

characteristic

C₈²×C₅₃

C₄

yes

—

C₈²×C₅₃

C₄

yes

—

—

—

C₂²×C₈×C₅₃

D₄

yes

—

C₂²×C₈×C₅₃

Q₈

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

Q₈

yes

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×Q₈

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

D₈

yes

—

C₂×C₈×C₅₃

Q₁₆

yes

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

C₂×Q₈

yes

characteristic

#11719917

—

C₂×#88322

—

C₂³×C₅₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₅₃

#127

yes

characteristic

C₂×C₄×C₅₃

C₄⋊Q₈

yes

—

C₂×C₄×C₅₃

C₂×D₈

yes

—

C₂×C₄×C₅₃

C₂×Q₁₆

yes

—

C₈×C₅₃

—

C₈×C₅₃

#107

yes

—

C₈×C₅₃

#107

yes

—

C₈×C₅₃

yes

—

—

C₂²×C₅₃

—

C₂²×C₅₃

#384

yes

characteristic

C₂²×C₅₃

#500

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂×#107

yes

—

C₄×C₅₃

C₂×#108

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

—

—

C₂×C₈²

D₅₃

yes

characteristic

C₂×C₅₃

—

C₂×C₅₃

#1845

yes

characteristic

C₂×C₅₃

#1848

yes

—

C₂×#108

—

106

—

#423

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

C₂×C₄×C₈

C₂×D₅₃

yes

—

#335

—

106

—

C₈²

Dic₅₃

yes

characteristic

C₈²

Dic₅₃

yes

—

C₈²

Dic₅₃

yes

characteristic

C₅₃

#8823

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

106

—

C₂²×C₈

Dic₁₀₆

yes

—

C₂²×C₈

D₂₁₂

yes

—

C₂×C₄²

C₂²×D₅₃

yes

characteristic

#108

—

212

—

C₄⋊C₈

—

106

—

C₄×C₈

Dic₁₀₆

yes

—

C₄×C₈

D₂₁₂

yes

—

C₄×C₈

C₂×Dic₅₃

yes

—

C₄×C₈

C₂×Dic₅₃

yes

characteristic

C₄×C₈

C₂×Dic₅₃

yes

characteristic

C₄×C₈

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂×Dic₁₀₆

yes

—

C₂²×C₄

C₂×D₂₁₂

yes

characteristic

M₁₆

—

212

—

C₂×C₈

—

C₂×C₈

—

212

—

C₂×C₈

D₄₂₄

yes

—

C₂×C₈

Dic₂₁₂

yes

—

C₂×C₈

C₄⋊Dic₅₃

yes

—

C₂×C₈

C₄⋊Dic₅₃

yes

—

C₂×C₈

C₄⋊Dic₅₃

yes

—

C₂×C₈

C₄⋊Dic₅₃

yes

—

C₄²

C₂×Dic₁₀₆

yes

characteristic

C₄²

C₂×D₂₁₂

yes

—

100

C₄²

C₂²×Dic₅₃

yes

characteristic

101

C₂³

#13133217

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

C₂×C₄⋊Dic₅₃

yes

characteristic

104

C₂×C₄

C₂×C₄⋊Dic₅₃

yes

—

105

C₂×C₄

#13133214

yes

characteristic

106

C₂×C₄

#13133217

yes

—

107

C₂×C₄

C₂×D₄₂₄

yes

—

108

C₂×C₄

C₂×Dic₂₁₂

yes

—

109

C₈

—

110

C₈

—

424

—

111

C₈

#13133151

yes

—

112

C₈

#13133151

yes

—

113

C₈

#13133153

yes

complemented

114

C₂²

—

115

C₂²

#16810493

yes

characteristic

116

C₂²

#16811028

yes

—

117

C₄

—

118

C₄

C₂×#13133151

yes

—

119

C₄

C₂×#13133153

yes

—

120

C₄

#16811000

yes

characteristic

121

C₄

#16811003

yes

characteristic

122

C₂

—

123

C₂

#20274875

yes

characteristic

124

C₂

#20274887

yes

—

	subgroup	G/H
center	C₂×C₄	#13133214
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₈²×C₅₃	C₂
socle	C₂²×C₅₃	#384
prime radical	C₂³×C₅₃	C₄⋊Q₈
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×C₈²	D₅₃
hypocenter	C₅₃	#8823
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈²

subgroup

G/H

center

C₂×C₄

#13133214

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₂×C₈²×C₅₃

C₂

socle

C₂²×C₅₃

#384

prime radical

C₂³×C₅₃

C₄⋊Q₈

cosocle

C₂×C₄²×C₅₃

C₂³

hypercenter

C₂×C₈²

D₅₃

hypocenter

C₅₃

#8823

LCS

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈²

N	H ≅ G/N	type
49. C₈×C₅₃	#108	semidirect
74. C₅₃	#8823	semidirect
113. C₈	#13133153	semidirect

H ≅ G/N

type

49. C₈×C₅₃

#108

semidirect

74. C₅₃

#8823

semidirect

113. C₈

#13133153

semidirect

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

subgroup	index
#20274887	2
#20273474	2
C₂×C₈²×C₅₃	2
#8823	53

subgroup

index

#20274887

#20273474

C₂×C₈²×C₅₃

#8823

Subgroups — #27277896

Conjugacy classes of subgroups

Complements

Maximal subgroups