Group #27220392 — subgroups

Subgroups — C₅₃×#12880 #27220392

Order 13568.

322 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 182 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×C₅₃×C₄⋊C₁₆	C₂	1	2	yes	maximal
2	C₅₃×#1726	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1374	C₂	1	4	yes	maximal
4	C₂²×C₁₆×C₅₃	C₂²	1	4	yes	—
5	C₂×#97×C₅₃	C₄	1	2	yes	—
6	C₂×#97×C₅₃	C₄	1	1	yes	characteristic
7	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
8	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₅₃×#376	C₄	1	1	yes	characteristic
10	C₅₃×C₄⋊C₁₆	—	2	4	no	—
11	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
12	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
13	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
14	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
15	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂×C₁₆×C₅₃	—	2	4	no	—
17	C₂×C₁₆×C₅₃	—	2	8	no	—
18	#97×C₅₃	—	2	4	no	—
19	#97×C₅₃	—	2	2	no	—
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	C₈	1	4	yes	—
24	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
25	C₂²×C₄×C₅₃	C₂×C₈	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	M₁₆	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	—	2	2	no	—
37	C₄²×C₅₃	C₄⋊C₄	1	2	yes	—
38	C₄²×C₅₃	M₁₆	1	2	yes	—
39	C₁₆×C₅₃	—	4	4	no	—
40	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
41	C₂×C₄×C₅₃	—	2	4	no	—
42	C₂×C₄×C₅₃	—	2	4	no	—
43	C₂×C₄×C₅₃	—	2	8	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	C₂×M₁₆	1	2	yes	—
48	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
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	#12880	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
54	C₂²×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
55	C₂²×C₅₃	#431	1	2	yes	—
56	C₂²×C₅₃	#504	1	4	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	#432	1	2	yes	—
62	C₂×C₄⋊C₁₆	C₂×C₅₃	1	2	yes	—
63	#1726	C₂×C₅₃	1	1	yes	characteristic
64	#1374	C₂×C₅₃	1	4	yes	—
65	C₂×C₅₃	#1831	1	1	yes	characteristic
66	C₂×C₅₃	#2151	1	2	yes	—
67	C₂×C₅₃	#2164	1	4	yes	—
68	C₂²×C₁₆	C₂²×C₅₃	1	4	yes	—
69	C₂×#97	C₄×C₅₃	1	2	yes	—
70	C₂×#97	C₄×C₅₃	1	1	yes	characteristic
71	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
72	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
73	#376	C₄×C₅₃	1	1	yes	characteristic
74	C₄⋊C₁₆	—	2	4	no	—
75	C₅₃	#12880	1	1	yes	characteristic, direct product factor, Sylow
76	C₂²×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
77	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
78	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
79	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
80	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
81	C₂×C₁₆	—	2	4	no	—
82	C₂×C₁₆	—	2	8	no	—
83	#97	—	2	4	no	—
84	#97	—	2	2	no	—
85	C₄×C₈	—	2	4	no	—
86	C₄×C₈	D₄×C₅₃	1	2	yes	—
87	C₄×C₈	Q₈×C₅₃	1	2	yes	—
88	#95	C₈×C₅₃	1	4	yes	—
89	C₂²×C₄	C₂×C₈×C₅₃	1	4	yes	—
90	C₂²×C₄	C₂×C₈×C₅₃	1	2	yes	—
91	C₂²×C₄	C₂²×C₄×C₅₃	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	C₄⋊C₄×C₅₃	1	2	yes	—
96	C₂×C₈	M₁₆×C₅₃	1	4	yes	—
97	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
98	C₂×C₈	C₂×Q₈×C₅₃	1	1	yes	characteristic
99	C₂×C₈	#41×C₅₃	1	4	yes	—
100	C₄²	—	2	2	no	—
101	C₄²	—	2	2	no	—
102	C₄²	C₄⋊C₄×C₅₃	1	2	yes	—
103	C₄²	M₁₆×C₅₃	1	2	yes	—
104	C₁₆	—	4	4	no	—
105	C₂³	C₂²×C₈×C₅₃	1	1	yes	characteristic
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	2	8	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	C₄⋊C₈×C₅₃	1	4	yes	—
110	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
111	C₂×C₄	#117×C₅₃	1	2	yes	—
112	C₂×C₄	C₂×M₁₆×C₅₃	1	2	yes	—
113	C₂×C₄	C₂×M₁₆×C₅₃	1	1	yes	characteristic
114	C₈	—	2	2	no	—
115	C₈	—	2	4	no	—
116	C₈	#124×C₅₃	1	2	yes	—
117	C₈	#125×C₅₃	1	2	yes	—
118	C₂²	C₂×C₄⋊C₈×C₅₃	1	1	yes	characteristic
119	C₂²	C₅₃×#431	1	2	yes	—
120	C₂²	C₅₃×#504	1	4	yes	—
121	C₄	—	2	4	no	—
122	C₄	—	4	4	no	—
123	C₄	C₅₃×#382	1	1	yes	characteristic
124	C₄	C₅₃×#423	1	1	yes	characteristic
125	C₄	C₅₃×#432	1	2	yes	—
126	C₂	C₅₃×#1831	1	1	yes	characteristic
127	C₂	C₅₃×#2151	1	2	yes	—
128	C₂	C₅₃×#2164	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₅₃×C₄⋊C₁₆

C₂

yes

maximal

C₅₃×#1726

C₂

yes

maximal, characteristic

C₅₃×#1374

C₂

yes

maximal

C₂²×C₁₆×C₅₃

C₂²

yes

—

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

C₅₃×C₄⋊C₁₆

—

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

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

—

#97×C₅₃

—

#97×C₅₃

—

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

D₄

yes

—

C₄×C₈×C₅₃

Q₈

yes

—

#95×C₅₃

C₈

yes

—

C₂²×C₄×C₅₃

C₂×C₈

yes

—

C₂²×C₄×C₅₃

C₂×C₈

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

M₁₆

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₅₃

#41

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₄⋊C₄

yes

—

C₄²×C₅₃

M₁₆

yes

—

C₁₆×C₅₃

—

C₂³×C₅₃

C₂²×C₈

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

C₂×M₁₆

yes

—

C₂×C₄×C₅₃

C₂×M₁₆

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

#124

yes

—

C₈×C₅₃

#125

yes

—

#12880

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

C₂×C₄⋊C₈

yes

characteristic

C₂²×C₅₃

#431

yes

—

C₂²×C₅₃

#504

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

yes

—

C₂×C₄⋊C₁₆

C₂×C₅₃

yes

—

#1726

C₂×C₅₃

yes

characteristic

#1374

C₂×C₅₃

yes

—

C₂×C₅₃

#1831

yes

characteristic

C₂×C₅₃

#2151

yes

—

C₂×C₅₃

#2164

yes

—

C₂²×C₁₆

C₂²×C₅₃

yes

—

C₂×#97

C₄×C₅₃

yes

—

C₂×#97

C₄×C₅₃

yes

characteristic

C₂×C₄×C₈

C₂²×C₅₃

yes

—

C₂×C₄×C₈

C₂²×C₅₃

yes

characteristic

#376

C₄×C₅₃

yes

characteristic

C₄⋊C₁₆

—

C₅₃

#12880

yes

characteristic, direct product factor, Sylow

C₂²×C₈

C₂×C₄×C₅₃

yes

characteristic

C₂²×C₈

C₂×C₄×C₅₃

yes

—

C₂²×C₈

C₂³×C₅₃

yes

characteristic

C₂×C₄²

C₂×C₄×C₅₃

yes

—

C₂×C₄²

C₂×C₄×C₅₃

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

D₄×C₅₃

yes

—

C₄×C₈

Q₈×C₅₃

yes

—

#95

C₈×C₅₃

yes

—

C₂²×C₄

C₂×C₈×C₅₃

yes

—

C₂²×C₄

C₂×C₈×C₅₃

yes

—

C₂²×C₄

C₂²×C₄×C₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊C₄×C₅₃

yes

—

C₂×C₈

M₁₆×C₅₃

yes

—

C₂×C₈

C₂×D₄×C₅₃

yes

characteristic

C₂×C₈

C₂×Q₈×C₅₃

yes

characteristic

C₂×C₈

#41×C₅₃

yes

—

100

C₄²

—

101

C₄²

—

102

C₄²

C₄⋊C₄×C₅₃

yes

—

103

C₄²

M₁₆×C₅₃

yes

—

104

C₁₆

—

105

C₂³

C₂²×C₈×C₅₃

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₄⋊C₈×C₅₃

yes

—

110

C₂×C₄

C₂×C₄⋊C₄×C₅₃

yes

characteristic

111

C₂×C₄

#117×C₅₃

yes

—

112

C₂×C₄

C₂×M₁₆×C₅₃

yes

—

113

C₂×C₄

C₂×M₁₆×C₅₃

yes

characteristic

114

C₈

—

115

C₈

—

116

C₈

#124×C₅₃

yes

—

117

C₈

#125×C₅₃

yes

—

118

C₂²

C₂×C₄⋊C₈×C₅₃

yes

characteristic

119

C₂²

C₅₃×#431

yes

—

120

C₂²

C₅₃×#504

yes

—

121

C₄

—

122

C₄

—

123

C₄

C₅₃×#382

yes

characteristic

124

C₄

C₅₃×#423

yes

characteristic

125

C₄

C₅₃×#432

yes

—

126

C₂

C₅₃×#1831

yes

characteristic

127

C₂

C₅₃×#2151

yes

—

128

C₂

C₅₃×#2164

yes

—

	subgroup	G/H
center	C₂²×C₈×C₅₃	C₂³
commutator	C₂³	C₂²×C₈×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
socle, prime radical	C₂³×C₅₃	C₂²×C₈
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₈×C₅₃ ◁ G

subgroup

G/H

center

C₂²×C₈×C₅₃

C₂³

commutator

C₂³

C₂²×C₈×C₅₃

Frattini, cosocle

C₂²×C₈

C₂³×C₅₃

socle, prime radical

C₂³×C₅₃

C₂²×C₈

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂²×C₈×C₅₃ ◁ G

N	H ≅ G/N	type
53. #12880	C₅₃	direct (N × H)
75. C₅₃	#12880	direct (N × H)

H ≅ G/N

type

53. #12880

C₅₃

direct (N × H)

75. C₅₃

#12880

direct (N × H)

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

subgroup	index
C₂×C₅₃×C₄⋊C₁₆	2
C₅₃×#1726	2
C₅₃×#1374	2
#12880	53

subgroup

index

C₂×C₅₃×C₄⋊C₁₆

C₅₃×#1726

C₅₃×#1374

#12880

Subgroups — C₅₃×#12880 #27220392

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#12880 #27220392

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12880 #27220392