Group #27277780 — subgroups

Subgroups — #27277780

Order 13568.

262 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 107 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	#20274853	C₂	1	1	yes	maximal, characteristic
2	#20272998	C₂	1	1	yes	maximal, characteristic
3	#20272573	C₂	1	4	yes	maximal
4	C₅₃×#1835	C₂	1	1	yes	maximal, characteristic
5	#16810676	C₂²	1	2	yes	—
6	#16810647	C₂²	1	2	yes	—
7	#16810388	—	2	2	no	—
8	#16810344	C₄	1	2	yes	complemented
9	C₈×#88322	C₄	1	2	yes	complemented
10	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₅₃×#382	C₂²	1	2	yes	—
12	C₅₃×#340	—	2	2	no	—
13	C₂×#11719917	—	2	2	no	—
14	C₄⋊#88322	—	2	2	no	—
15	C₄⋊#88322	—	2	4	no	—
16	C₂²⋊#88322	—	2	2	no	—
17	C₄×#88322	C₂×C₄	1	4	yes	—
18	C₂²×C₈×C₅₃	D₄	1	2	yes	—
19	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
20	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
21	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
23	C₄×C₈×C₅₃	Q₈	1	2	yes	—
24	#95×C₅₃	—	2	4	no	—
25	C₂²×C₄×C₅₃	—	4	2	no	—
26	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
27	C₂²×C₄×C₅₃	#41	1	2	yes	—
28	C₂×C₈×C₅₃	—	2	4	no	—
29	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
30	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
31	C₄⋊C₄×C₅₃	—	4	4	no	—
32	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
33	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
34	C₄²×C₅₃	#41	1	2	yes	—
35	#11719917	—	4	4	no	—
36	C₂×#88322	—	2	4	no	—
37	C₂×#88322	—	4	2	no	—
38	C₂³×C₅₃	#124	1	1	yes	characteristic
39	C₂×C₄×C₅₃	—	4	2	no	—
40	C₂×C₄×C₅₃	—	8	4	no	—
41	C₂×C₄×C₅₃	—	8	2	no	—
42	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₅₃	#117	1	2	yes	—
44	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
45	C₂×C₄×C₅₃	#125	1	2	yes	—
46	C₂×C₄×C₅₃	#135	1	4	yes	—
47	C₈×C₅₃	—	2	2	no	—
48	C₈×C₅₃	—	2	2	no	—
49	#88322	—	4	4	no	—
50	#88322	—	8	2	no	—
51	#8792	—	53	1	no	maximal, Sylow
52	C₂²×C₅₃	—	2	2	no	—
53	C₂²×C₅₃	#382	1	1	yes	characteristic
54	C₂²×C₅₃	#488	1	2	yes	—
55	C₄×C₅₃	—	2	2	no	—
56	C₄×C₅₃	—	2	2	no	—
57	C₄×C₅₃	—	16	2	no	—
58	C₄×C₅₃	#427	1	2	yes	—
59	C₄×C₅₃	#488	1	2	yes	—
60	#1837	—	53	1	no	—
61	#1835	D₅₃	1	1	yes	characteristic
62	#1450	—	53	1	no	—
63	#1286	—	53	4	no	—
64	C₂×C₅₃	—	2	2	no	—
65	C₂×C₅₃	#1833	1	2	yes	—
66	C₂×C₅₃	#1835	1	1	yes	characteristic
67	#431	—	53	2	no	—
68	#423	—	53	2	no	—
69	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
70	#382	C₂×D₅₃	1	2	yes	—
71	#341	—	106	2	no	—
72	#340	—	2	2	no	—
73	#322	—	53	2	no	—
74	C₈²	—	53	2	no	—
75	C₅₃	#8792	1	1	yes	characteristic, complemented, Sylow
76	C₂×M₁₆	—	106	2	no	—
77	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
78	C₂×C₄⋊C₄	—	2	2	no	—
79	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
80	C₄⋊C₈	—	106	2	no	—
81	C₄⋊C₈	—	106	4	no	—
82	C₂²⋊C₈	—	106	2	no	—
83	C₄×C₈	—	53	4	no	—
84	C₄×C₈	Dic₁₀₆	1	2	yes	—
85	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
86	C₄×C₈	C₄×D₅₃	1	1	yes	characteristic
87	#95	—	2	4	no	—
88	C₂²×C₄	—	4	2	no	—
89	C₂²×C₄	#11719947	1	2	yes	—
90	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
91	M₁₆	—	212	4	no	—
92	C₂×C₈	—	2	4	no	—
93	C₂×C₈	—	106	4	no	—
94	C₂×C₈	—	212	2	no	—
95	C₂×C₈	#11719919	1	2	yes	—
96	C₂×C₈	#11719919	1	2	yes	—
97	C₄⋊C₄	—	4	4	no	—
98	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
99	C₄²	C₂×C₄×D₅₃	1	1	yes	characteristic
100	C₄²	#11719949	1	2	yes	—
101	C₂³	#13133204	1	1	yes	characteristic
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	8	4	no	—
104	C₂×C₄	—	8	2	no	—
105	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
106	C₂×C₄	#13133175	1	2	yes	—
107	C₂×C₄	#13133206	1	1	yes	characteristic
108	C₂×C₄	#13133208	1	2	yes	—
109	C₂×C₄	#13133237	1	4	yes	—
110	C₈	—	2	2	no	—
111	C₈	—	2	2	no	—
112	C₈	—	424	2	no	—
113	C₈	—	212	4	no	—
114	C₂²	—	2	2	no	—
115	C₂²	#16810488	1	1	yes	characteristic
116	C₂²	#16810977	1	2	yes	—
117	C₄	—	2	2	no	—
118	C₄	—	2	2	no	—
119	C₄	—	16	2	no	—
120	C₄	#16810659	1	2	yes	—
121	C₄	#16810980	1	2	yes	—
122	C₂	—	2	2	no	—
123	C₂	#20274838	1	2	yes	—
124	C₂	#20274845	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20274853

C₂

yes

maximal, characteristic

#20272998

C₂

yes

maximal, characteristic

#20272573

C₂

yes

maximal

C₅₃×#1835

C₂

yes

maximal, characteristic

#16810676

C₂²

yes

—

#16810647

C₂²

yes

—

#16810388

—

#16810344

C₄

yes

complemented

C₈×#88322

C₄

yes

complemented

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#382

C₂²

yes

—

C₅₃×#340

—

—

—

—

—

C₂×C₄

yes

—

C₂²×C₈×C₅₃

D₄

yes

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

Q₈

yes

—

#95×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×Q₈

yes

characteristic

C₄²×C₅₃

#41

yes

—

#11719917

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

#124

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

#117

yes

—

C₂×C₄×C₅₃

#124

yes

characteristic

C₂×C₄×C₅₃

#125

yes

—

C₂×C₄×C₅₃

#135

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

—

—

—

C₂²×C₅₃

—

C₂²×C₅₃

#382

yes

characteristic

C₂²×C₅₃

#488

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#427

yes

—

C₄×C₅₃

#488

yes

—

#1837

—

#1835

D₅₃

yes

characteristic

#1450

—

#1286

—

C₂×C₅₃

—

C₂×C₅₃

#1833

yes

—

C₂×C₅₃

yes

—

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

#382

C₂×D₅₃

yes

—

#341

—

106

—

#340

—

#322

—

C₈²

—

C₅₃

#8792

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

106

—

C₂²×C₈

C₂²⋊D₅₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₅₃

yes

characteristic

C₄⋊C₈

—

106

—

C₄⋊C₈

—

106

—

C₂²⋊C₈

—

106

—

C₄×C₈

—

C₄×C₈

Dic₁₀₆

yes

—

C₄×C₈

C₄×D₅₃

yes

characteristic

C₄×C₈

C₄×D₅₃

yes

characteristic

#95

—

C₂²×C₄

—

C₂²×C₄

#11719947

yes

—

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

M₁₆

—

212

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

212

—

C₂×C₈

#11719919

yes

—

C₂×C₈

#11719919

yes

—

C₄⋊C₄

—

C₄²

C₂×Dic₁₀₆

yes

characteristic

C₄²

C₂×C₄×D₅₃

yes

characteristic

100

C₄²

#11719949

yes

—

101

C₂³

#13133204

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

C₂×#11719919

yes

characteristic

106

C₂×C₄

#13133175

yes

—

107

C₂×C₄

#13133206

yes

characteristic

108

C₂×C₄

#13133208

yes

—

109

C₂×C₄

#13133237

yes

—

110

C₈

—

111

C₈

—

112

C₈

—

424

—

113

C₈

—

212

—

114

C₂²

—

115

C₂²

#16810488

yes

characteristic

116

C₂²

#16810977

yes

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

#16810659

yes

—

121

C₄

#16810980

yes

—

122

C₂

—

123

C₂

#20274838

yes

—

124

C₂

#20274845

yes

characteristic

	subgroup	G/H
center	C₂²	#16810488
commutator	C₄²×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1835	C₂
socle	C₂²×C₅₃	#382
prime radical	C₂³×C₅₃	#124
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	#1835	D₅₃
hypocenter	C₅₃	#8792
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1835

subgroup

G/H

center

C₂²

#16810488

commutator

C₄²×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1835

C₂

C₂²×C₅₃

C₂³×C₅₃

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1835

N	H ≅ G/N	type
8. #16810344	C₄	semidirect
9. C₈×#88322	C₄	semidirect
75. C₅₃	#8792	semidirect

H ≅ G/N

type

8. #16810344

C₄

semidirect

9. C₈×#88322

C₄

semidirect

75. C₅₃

#8792

semidirect

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

subgroup	index
#20274853	2
#20272998	2
#20272573	2
C₅₃×#1835	2
#8792	53

subgroup

index

C₅₃×#1835

Subgroups — #27277780

Conjugacy classes of subgroups

Complements

Maximal subgroups