Group #27277848 — subgroups

Subgroups — #27277848

Order 13568.

402 conjugacy classes of subgroups; only the 211 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 211 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	#20274867	C₂	1	2	yes	maximal
2	C₂×#16810376	C₂	1	4	yes	maximal
3	C₅₃×#1744	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊#88322	—	2	4	no	—
5	C₂×C₄⋊#88322	C₂²	1	4	yes	—
6	#16810376	—	2	8	no	—
7	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
8	C₄³×C₅₃	C₂²	1	1	yes	characteristic
9	C₅₃×#336	C₄	1	4	yes	—
10	C₂²×#88322	—	2	4	no	—
11	C₄⋊#88322	—	2	8	no	—
12	C₄⋊#88322	—	4	8	no	—
13	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
14	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
15	C₂×C₄²×C₅₃	D₄	1	2	yes	—
16	C₂×C₄²×C₅₃	Q₈	1	2	yes	—
17	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
18	C₄×C₈×C₅₃	D₄	1	4	yes	—
19	C₄×C₈×C₅₃	Q₈	1	4	yes	—
20	C₂²×C₄×C₅₃	C₄⋊C₄	1	4	yes	—
21	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
23	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
24	C₂×C₈×C₅₃	—	2	8	no	—
25	C₂×C₈×C₅₃	C₄⋊C₄	1	8	yes	—
26	C₄²×C₅₃	—	2	4	no	—
27	C₄²×C₅₃	—	4	2	no	—
28	C₄²×C₅₃	D₈	1	4	yes	—
29	C₄²×C₅₃	Q₁₆	1	4	yes	—
30	C₄²×C₅₃	C₂×D₄	1	2	yes	—
31	C₄²×C₅₃	C₂×Q₈	1	2	yes	—
32	C₂×#88322	—	4	4	no	—
33	C₂×#88322	—	4	8	no	—
34	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₅₃	—	2	4	no	—
36	C₂×C₄×C₅₃	—	2	4	no	—
37	C₂×C₄×C₅₃	—	2	2	no	—
38	C₂×C₄×C₅₃	#107	1	8	yes	—
39	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
40	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
41	C₂×C₄×C₅₃	C₄⋊Q₈	1	2	yes	—
42	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
43	C₂×C₄×C₅₃	#131	1	4	yes	—
44	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
45	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
46	C₈×C₅₃	—	2	8	no	—
47	#88322	—	8	4	no	—
48	#8810	—	53	1	no	maximal, Sylow
49	C₂²×C₅₃	#384	1	1	yes	characteristic
50	C₂²×C₅₃	#424	1	2	yes	—
51	C₂²×C₅₃	C₂×#107	1	2	yes	—
52	C₂²×C₅₃	#430	1	2	yes	—
53	C₄×C₅₃	—	2	8	no	—
54	C₄×C₅₃	—	2	2	no	—
55	C₄×C₅₃	#424	1	4	yes	—
56	C₄×C₅₃	#493	1	1	yes	characteristic
57	C₄×C₅₃	#494	1	1	yes	characteristic
58	C₄×C₅₃	#500	1	2	yes	—
59	#1842	—	53	2	no	—
60	#1744	D₅₃	1	1	yes	characteristic
61	C₂×#335	—	53	4	no	—
62	C₂×C₅₃	#1571	1	4	yes	—
63	C₂×C₅₃	#1843	1	1	yes	characteristic
64	C₂×C₅₃	#1845	1	1	yes	characteristic
65	C₂×C₅₃	#1849	1	1	yes	characteristic
66	C₂×C₄⋊C₈	—	53	4	no	—
67	C₂×C₄⋊C₈	—	106	4	no	—
68	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
69	C₄³	C₂×D₅₃	1	1	yes	characteristic
70	#336	Dic₅₃	1	4	yes	—
71	#335	—	106	8	no	—
72	C₅₃	#8810	1	1	yes	characteristic, complemented, Sylow
73	C₂²×C₈	—	106	4	no	—
74	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
75	C₂×C₄²	Dic₁₀₆	1	2	yes	—
76	C₂×C₄²	D₂₁₂	1	2	yes	—
77	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
78	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
79	C₄⋊C₈	—	106	8	no	—
80	C₄⋊C₈	—	212	8	no	—
81	C₄×C₈	Dic₁₀₆	1	4	yes	—
82	C₄×C₈	D₂₁₂	1	4	yes	—
83	C₂²×C₄	C₄⋊Dic₅₃	1	4	yes	—
84	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
85	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
86	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
87	C₂×C₈	—	2	8	no	—
88	C₂×C₈	—	212	4	no	—
89	C₂×C₈	—	212	8	no	—
90	C₂×C₈	C₄⋊Dic₅₃	1	8	yes	—
91	C₄²	—	2	4	no	—
92	C₄²	—	4	2	no	—
93	C₄²	D₄₂₄	1	4	yes	—
94	C₄²	Dic₂₁₂	1	4	yes	—
95	C₄²	C₂×Dic₁₀₆	1	2	yes	—
96	C₄²	C₂×D₂₁₂	1	2	yes	—
97	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
98	C₂×C₄	—	2	4	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	#13133151	1	8	yes	—
102	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
103	C₂×C₄	#13133214	1	1	yes	characteristic
104	C₂×C₄	#13133217	1	2	yes	—
105	C₂×C₄	#13133217	1	1	yes	characteristic
106	C₂×C₄	#13133224	1	4	yes	—
107	C₂×C₄	C₂×D₄₂₄	1	2	yes	—
108	C₂×C₄	C₂×Dic₂₁₂	1	2	yes	—
109	C₈	—	2	8	no	—
110	C₈	—	424	4	no	—
111	C₂²	#16810493	1	1	yes	characteristic
112	C₂²	#16810650	1	2	yes	—
113	C₂²	C₂×#13133151	1	2	yes	—
114	C₂²	#16810672	1	2	yes	—
115	C₄	—	2	8	no	—
116	C₄	—	2	2	no	—
117	C₄	#16810650	1	4	yes	—
118	C₄	#16811000	1	1	yes	characteristic
119	C₄	#16811003	1	1	yes	characteristic
120	C₄	#16811028	1	2	yes	—
121	C₂	#20273490	1	4	yes	—
122	C₂	#20274869	1	1	yes	characteristic
123	C₂	#20274875	1	1	yes	characteristic
124	C₂	#20274890	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20274867

C₂

yes

maximal

C₂×#16810376

C₂

yes

maximal

C₅₃×#1744

C₂

yes

maximal, characteristic

C₂×C₄⋊#88322

—

C₂×C₄⋊#88322

C₂²

yes

—

#16810376

—

C₂×C₄×C₈×C₅₃

C₂²

yes

—

C₄³×C₅₃

C₂²

yes

characteristic

C₅₃×#336

C₄

yes

—

C₂²×#88322

—

C₄⋊#88322

—

C₄⋊#88322

—

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

Q₈

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄×C₈×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

Q₈

yes

—

C₂²×C₄×C₅₃

C₄⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

D₈

yes

—

C₄²×C₅₃

Q₁₆

yes

—

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

C₂×Q₈

yes

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#107

yes

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₅₃

#127

yes

characteristic

C₂×C₄×C₅₃

C₄⋊Q₈

yes

—

C₂×C₄×C₅₃

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

#131

yes

—

C₂×C₄×C₅₃

C₂×D₈

yes

—

C₂×C₄×C₅₃

C₂×Q₁₆

yes

—

C₈×C₅₃

—

—

—

C₂²×C₅₃

yes

C₂²×C₅₃

yes

—

C₂²×C₅₃

C₂×#107

yes

—

C₂²×C₅₃

#430

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#424

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

#1842

—

#1744

D₅₃

yes

characteristic

C₂×#335

—

C₂×C₅₃

#1571

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

106

—

C₂×C₄×C₈

C₂×D₅₃

yes

—

C₄³

C₂×D₅₃

yes

characteristic

#336

Dic₅₃

yes

—

#335

—

106

—

C₅₃

#8810

yes

characteristic, complemented, Sylow

C₂²×C₈

—

106

—

C₂²×C₈

C₂×Dic₅₃

yes

—

C₂×C₄²

Dic₁₀₆

yes

—

C₂×C₄²

D₂₁₂

yes

—

C₂×C₄²

C₂×Dic₅₃

yes

—

C₂×C₄²

C₂²×D₅₃

yes

characteristic

C₄⋊C₈

—

106

—

C₄⋊C₈

—

212

—

C₄×C₈

Dic₁₀₆

yes

—

C₄×C₈

D₂₁₂

yes

—

C₂²×C₄

C₄⋊Dic₅₃

yes

—

C₂²×C₄

C₂×Dic₁₀₆

yes

characteristic

C₂²×C₄

C₂×D₂₁₂

yes

characteristic

C₂²×C₄

C₂²×Dic₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

212

—

C₂×C₈

—

212

—

C₂×C₈

C₄⋊Dic₅₃

yes

—

C₄²

—

C₄²

—

C₄²

D₄₂₄

yes

—

C₄²

Dic₂₁₂

yes

—

C₄²

C₂×Dic₁₀₆

yes

—

C₄²

C₂×D₂₁₂

yes

—

C₂³

C₂×C₄⋊Dic₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

#13133151

yes

—

102

C₂×C₄

C₂×C₄⋊Dic₅₃

yes

—

103

C₂×C₄

#13133214

yes

characteristic

104

C₂×C₄

#13133217

yes

—

105

C₂×C₄

#13133217

yes

characteristic

106

C₂×C₄

#13133224

yes

—

107

C₂×C₄

C₂×D₄₂₄

yes

—

108

C₂×C₄

C₂×Dic₂₁₂

yes

—

109

C₈

—

110

C₈

—

424

—

111

C₂²

#16810493

yes

characteristic

112

C₂²

#16810650

yes

—

113

C₂²

C₂×#13133151

yes

—

114

C₂²

#16810672

yes

—

115

C₄

—

116

C₄

—

117

C₄

#16810650

yes

—

118

C₄

#16811000

yes

characteristic

119

C₄

#16811003

yes

characteristic

120

C₄

#16811028

yes

—

121

C₂

#20273490

yes

—

122

C₂

#20274869

yes

characteristic

123

C₂

#20274875

yes

characteristic

124

C₂

#20274890

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁₂
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1744	C₂
socle, prime radical	C₂³×C₅₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1744	D₅₃
hypocenter	C₅₃	#8810
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

subgroup

G/H

center

C₂²×C₄

C₂×D₂₁₂

commutator

C₂²×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1744

C₂

socle, prime radical

C₂³×C₅₃

C₂×C₄⋊C₄

cosocle

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂²×C₄ ◁ C₄³ ◁ #1744

N	H ≅ G/N	type
72. C₅₃	#8810	semidirect

H ≅ G/N

type

72. C₅₃

#8810

semidirect

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

subgroup	index
#20274867	2
C₂×#16810376	2
C₅₃×#1744	2
#8810	53

subgroup

index

#20274867

C₂×#16810376

C₅₃×#1744

#8810

Subgroups — #27277848

Conjugacy classes of subgroups

Complements

Maximal subgroups