Group #27300456 — subgroups

Subgroups — #27300456

Order 13568.

302 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 302 conjugacy classes of subgroups — too many to draw. Only its 155 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	#20276471	C₂	1	2	yes	maximal
2	#20272797	C₂	1	4	yes	maximal
3	C₅₃×#1726	C₂	1	1	yes	maximal, characteristic
4	C₂×#13133157	C₂²	1	4	yes	—
5	C₄⋊#11719908	—	2	4	no	—
6	C₂×#97×C₅₃	C₄	1	2	yes	—
7	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
8	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
9	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
10	C₅₃×#376	C₄	1	1	yes	characteristic
11	#13133157	—	2	8	no	—
12	C₂×#11719908	—	2	4	no	—
13	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
15	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
16	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
17	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
18	#97×C₅₃	—	2	4	no	—
19	#97×C₅₃	C₈	1	4	yes	—
20	C₄×C₈×C₅₃	—	2	4	no	—
21	C₄×C₈×C₅₃	D₄	1	2	yes	—
22	C₄×C₈×C₅₃	Q₈	1	2	yes	—
23	#95×C₅₃	—	2	2	no	—
24	C₂²×C₄×C₅₃	—	2	2	no	—
25	C₂²×C₄×C₅₃	M₁₆	1	2	yes	—
26	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₈×C₅₃	—	2	2	no	—
28	C₂×C₈×C₅₃	—	2	2	no	—
29	C₂×C₈×C₅₃	—	2	2	no	—
30	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
31	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
32	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
33	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
34	C₂×C₈×C₅₃	#41	1	4	yes	—
35	C₄²×C₅₃	—	2	2	no	—
36	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
37	C₄²×C₅₃	C₂×C₈	1	2	yes	—
38	C₄²×C₅₃	M₁₆	1	4	yes	—
39	#11719908	—	4	4	no	—
40	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
41	C₂×C₄×C₅₃	—	2	4	no	—
42	C₂×C₄×C₅₃	—	4	2	no	—
43	C₂×C₄×C₅₃	—	4	4	no	—
44	C₂×C₄×C₅₃	C₄⋊C₈	1	4	yes	—
45	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₅₃	#117	1	2	yes	—
47	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
48	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
49	C₈×C₅₃	—	2	2	no	—
50	C₈×C₅₃	—	2	4	no	—
51	C₈×C₅₃	#124	1	2	yes	—
52	C₈×C₅₃	#125	1	2	yes	—
53	#12884	—	53	1	no	maximal, Sylow
54	C₂²×C₅₃	—	2	2	no	—
55	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
56	C₂²×C₅₃	#432	1	2	yes	—
57	C₄×C₅₃	—	2	4	no	—
58	C₄×C₅₃	—	4	4	no	—
59	C₄×C₅₃	#382	1	1	yes	characteristic
60	C₄×C₅₃	#423	1	1	yes	characteristic
61	C₄×C₅₃	#431	1	2	yes	—
62	#2150	—	53	2	no	—
63	#1726	D₅₃	1	1	yes	characteristic
64	#1376	—	53	4	no	—
65	C₂×C₅₃	—	2	2	no	—
66	C₂×C₅₃	#1831	1	1	yes	characteristic
67	C₂×C₅₃	#2153	1	2	yes	—
68	C₂×M₃₂	—	53	4	no	—
69	C₂×#97	Dic₅₃	1	2	yes	—
70	C₂×#97	Dic₅₃	1	1	yes	characteristic
71	C₂×C₄×C₈	C₂×D₅₃	1	2	yes	—
72	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
73	#376	Dic₅₃	1	1	yes	characteristic
74	C₄⋊C₁₆	—	106	4	no	—
75	C₅₃	#12884	1	1	yes	characteristic, complemented, Sylow
76	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
77	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
78	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
79	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
80	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
81	M₃₂	—	106	8	no	—
82	C₂×C₁₆	—	106	4	no	—
83	#97	—	2	4	no	—
84	#97	#88322	1	4	yes	—
85	C₄×C₈	—	2	4	no	—
86	C₄×C₈	Dic₁₀₆	1	2	yes	—
87	C₄×C₈	D₂₁₂	1	2	yes	—
88	#95	—	2	2	no	—
89	C₂²×C₄	—	2	2	no	—
90	C₂²×C₄	#11719917	1	2	yes	—
91	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	C₂×#88322	1	4	yes	—
96	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
97	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
98	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
99	C₂×C₈	#11719945	1	4	yes	—
100	C₄²	—	2	2	no	—
101	C₄²	C₂×#88322	1	2	yes	—
102	C₄²	#11719917	1	4	yes	—
103	C₄²	C₄⋊Dic₅₃	1	2	yes	—
104	C₁₆	—	212	4	no	—
105	C₂³	C₂×#11719917	1	1	yes	characteristic
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	4	4	no	—
109	C₂×C₄	C₄⋊#88322	1	4	yes	—
110	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
111	C₂×C₄	#13133176	1	2	yes	—
112	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
113	C₂×C₄	C₂×#11719917	1	2	yes	—
114	C₈	—	2	2	no	—
115	C₈	—	2	4	no	—
116	C₈	#13133205	1	2	yes	—
117	C₈	#13133209	1	2	yes	—
118	C₂²	—	2	2	no	—
119	C₂²	C₂×C₄⋊#88322	1	1	yes	characteristic
120	C₂²	#16810680	1	2	yes	—
121	C₄	—	2	4	no	—
122	C₄	—	4	4	no	—
123	C₄	#16810489	1	1	yes	characteristic
124	C₄	#16810647	1	1	yes	characteristic
125	C₄	#16810676	1	2	yes	—
126	C₂	—	2	2	no	—
127	C₂	#20274832	1	1	yes	characteristic
128	C₂	#20276481	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20276471

C₂

yes

maximal

#20272797

C₂

yes

maximal

C₅₃×#1726

C₂

yes

maximal, characteristic

C₂×#13133157

C₂²

yes

—

C₄⋊#11719908

—

C₂×#97×C₅₃

C₄

yes

—

C₂×#97×C₅₃

C₄

yes

characteristic

C₂×C₄×C₈×C₅₃

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#376

C₄

yes

characteristic

#13133157

—

C₂×#11719908

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂²×C₈×C₅₃

C₂³

yes

characteristic

C₂×C₄²×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

#97×C₅₃

—

#97×C₅₃

C₈

yes

—

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

Q₈

yes

—

#95×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

M₁₆

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

—

C₂×C₈×C₅₃

C₂×C₈

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₃

#41

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

C₄⋊C₄

yes

—

C₄²×C₅₃

C₂×C₈

yes

—

C₄²×C₅₃

M₁₆

yes

—

#11719908

—

C₂³×C₅₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄⋊C₈

yes

—

C₂×C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₅₃

#117

yes

—

C₂×C₄×C₅₃

C₂²×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂×M₁₆

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

#124

yes

—

C₈×C₅₃

#125

yes

—

#12884

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₅₃

#432

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

#2150

—

#1726

D₅₃

yes

characteristic

#1376

—

C₂×C₅₃

—

C₂×C₅₃

#1831

yes

characteristic

C₂×C₅₃

#2153

yes

—

C₂×M₃₂

—

C₂×#97

Dic₅₃

yes

—

C₂×#97

Dic₅₃

yes

characteristic

C₂×C₄×C₈

C₂×D₅₃

yes

—

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

#376

Dic₅₃

yes

characteristic

C₄⋊C₁₆

—

106

—

C₅₃

#12884

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂×Dic₅₃

yes

—

C₂²×C₈

C₂×Dic₅₃

yes

characteristic

C₂²×C₈

C₂²×D₅₃

yes

characteristic

C₂×C₄²

C₂×Dic₅₃

yes

—

C₂×C₄²

C₂×Dic₅₃

yes

characteristic

M₃₂

—

106

—

C₂×C₁₆

—

106

—

#97

—

#97

#88322

yes

—

C₄×C₈

—

C₄×C₈

Dic₁₀₆

yes

—

C₄×C₈

D₂₁₂

yes

—

#95

—

C₂²×C₄

—

C₂²×C₄

#11719917

yes

—

C₂²×C₄

C₂²×Dic₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×#88322

yes

—

C₂×C₈

C₄⋊Dic₅₃

yes

—

C₂×C₈

C₂×Dic₁₀₆

yes

characteristic

C₂×C₈

C₂×D₂₁₂

yes

characteristic

C₂×C₈

#11719945

yes

—

100

C₄²

—

101

C₄²

C₂×#88322

yes

—

102

C₄²

#11719917

yes

—

103

C₄²

C₄⋊Dic₅₃

yes

—

104

C₁₆

—

212

—

105

C₂³

C₂×#11719917

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₄⋊#88322

yes

—

110

C₂×C₄

C₂×C₄⋊Dic₅₃

yes

characteristic

111

C₂×C₄

#13133176

yes

—

112

C₂×C₄

C₂²×#88322

yes

characteristic

113

C₂×C₄

C₂×#11719917

yes

—

114

C₈

—

115

C₈

—

116

C₈

#13133205

yes

—

117

C₈

#13133209

yes

—

118

C₂²

—

119

C₂²

C₂×C₄⋊#88322

yes

characteristic

120

C₂²

#16810680

yes

—

121

C₄

—

122

C₄

—

123

C₄

#16810489

yes

characteristic

124

C₄

#16810647

yes

characteristic

125

C₄

#16810676

yes

—

126

C₂

—

127

C₂

#20274832

yes

characteristic

128

C₂

#20276481

yes

—

	subgroup	G/H
center	C₂×C₈	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#1726	C₂
socle	C₂²×C₅₃	C₂×C₄⋊C₈
prime radical	C₂³×C₅₃	C₂×M₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#1726	D₅₃
hypocenter	C₅₃	#12884
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #1726

subgroup

G/H

center

C₂×C₈

C₂×D₂₁₂

commutator

C₂×C₄×C₅₃

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₅₃×#1726

C₂

socle

C₂²×C₅₃

C₂×C₄⋊C₈

prime radical

C₂³×C₅₃

C₂×M₁₆

cosocle

C₂²×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ #1726

N	H ≅ G/N	type
75. C₅₃	#12884	semidirect

H ≅ G/N

type

75. C₅₃

#12884

semidirect

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

subgroup	index
#20276471	2
#20272797	2
C₅₃×#1726	2
#12884	53

subgroup

index

#20276471

#20272797

C₅₃×#1726

#12884

Subgroups — #27300456

Conjugacy classes of subgroups

Complements

Maximal subgroups