Group #27216308 — subgroups

Subgroups — C₅₃×#8796 #27216308

Order 13568.

228 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 118 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	C₅₃×#1568	C₂	1	2	yes	maximal
2	C₅₃×#1450	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1287	C₂	1	4	yes	maximal
4	C₅₃×#432	C₂²	1	4	yes	—
5	C₅₃×#431	C₂²	1	2	yes	—
6	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
7	C₅₃×#335	—	2	2	no	—
8	C₅₃×#334	—	2	2	no	—
9	C₅₃×#322	C₄	1	2	yes	—
10	C₈²×C₅₃	C₄	1	2	yes	—
11	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
12	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
13	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
14	C₄⋊C₈×C₅₃	—	2	4	no	—
15	C₄⋊C₈×C₅₃	—	2	2	no	—
16	C₂²⋊C₈×C₅₃	—	2	2	no	—
17	C₂²⋊C₈×C₅₃	—	2	4	no	—
18	#97×C₅₃	—	2	4	no	—
19	C₄×C₈×C₅₃	C₂×C₄	1	4	yes	—
20	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
23	C₄×C₈×C₅₃	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
25	C₂²×C₄×C₅₃	#41	1	2	yes	—
26	C₂×C₈×C₅₃	—	2	2	no	—
27	C₂×C₈×C₅₃	—	2	2	no	—
28	C₂×C₈×C₅₃	—	2	4	no	—
29	C₂×C₈×C₅₃	—	4	4	no	—
30	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
32	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
33	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
34	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
35	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
36	C₄²×C₅₃	#41	1	2	yes	—
37	C₂³×C₅₃	#125	1	1	yes	characteristic
38	C₂×C₄×C₅₃	—	2	1	no	—
39	C₂×C₄×C₅₃	—	2	1	no	—
40	C₂×C₄×C₅₃	—	4	2	no	—
41	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₅₃	#117	1	2	yes	—
43	C₂×C₄×C₅₃	#124	1	2	yes	—
44	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
45	C₈×C₅₃	—	2	1	no	—
46	C₈×C₅₃	—	2	1	no	—
47	C₈×C₅₃	—	4	4	no	—
48	C₈×C₅₃	—	8	4	no	—
49	C₈×C₅₃	#108	1	2	yes	—
50	C₈×C₅₃	#108	1	2	yes	—
51	#8796	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂²×C₅₃	—	2	2	no	—
53	C₂²×C₅₃	—	2	1	no	—
54	C₂²×C₅₃	#382	1	1	yes	characteristic
55	C₄×C₅₃	—	2	2	no	—
56	C₄×C₅₃	—	2	2	no	—
57	C₄×C₅₃	—	2	1	no	—
58	C₄×C₅₃	C₂×#108	1	1	yes	characteristic
59	C₄×C₅₃	C₂×#108	1	1	yes	characteristic
60	#1568	C₂×C₅₃	1	2	yes	—
61	#1450	C₂×C₅₃	1	1	yes	characteristic
62	#1287	C₂×C₅₃	1	4	yes	—
63	C₂×C₅₃	—	4	1	no	—
64	C₂×C₅₃	#1833	1	2	yes	—
65	C₂×C₅₃	#1837	1	1	yes	characteristic
66	#432	C₂²×C₅₃	1	4	yes	—
67	#431	C₂²×C₅₃	1	2	yes	—
68	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
69	#335	—	2	2	no	—
70	#334	—	2	2	no	—
71	#322	C₄×C₅₃	1	2	yes	—
72	C₈²	C₄×C₅₃	1	2	yes	—
73	C₅₃	#8796	1	1	yes	characteristic, direct product factor, Sylow
74	C₂²×C₈	Q₈×C₅₃	1	1	yes	characteristic
75	C₂²×C₈	Q₈×C₅₃	1	1	yes	characteristic
76	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
77	C₄⋊C₈	—	2	4	no	—
78	C₄⋊C₈	—	2	2	no	—
79	C₂²⋊C₈	—	2	2	no	—
80	C₂²⋊C₈	—	2	4	no	—
81	#97	—	2	4	no	—
82	C₄×C₈	C₂×C₄×C₅₃	1	4	yes	—
83	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
84	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
85	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
86	C₄×C₈	D₄×C₅₃	1	1	yes	characteristic
87	C₂²×C₄	C₂×Q₈×C₅₃	1	1	yes	characteristic
88	C₂²×C₄	#41×C₅₃	1	2	yes	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	2	4	no	—
92	C₂×C₈	—	4	4	no	—
93	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
94	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
95	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
96	C₂×C₈	C₄⋊C₄×C₅₃	1	1	yes	characteristic
97	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
98	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
99	C₄²	#41×C₅₃	1	2	yes	—
100	C₂³	#125×C₅₃	1	1	yes	characteristic
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	C₂×C₄⋊C₄×C₅₃	1	1	yes	characteristic
105	C₂×C₄	#117×C₅₃	1	2	yes	—
106	C₂×C₄	#124×C₅₃	1	2	yes	—
107	C₂×C₄	#125×C₅₃	1	1	yes	characteristic
108	C₈	—	2	1	no	—
109	C₈	—	2	1	no	—
110	C₈	—	4	4	no	—
111	C₈	—	8	4	no	—
112	C₈	#108×C₅₃	1	2	yes	—
113	C₈	#108×C₅₃	1	2	yes	—
114	C₂²	—	2	2	no	—
115	C₂²	—	2	1	no	—
116	C₂²	C₅₃×#382	1	1	yes	characteristic
117	C₄	—	2	2	no	—
118	C₄	—	2	2	no	—
119	C₄	—	2	1	no	—
120	C₄	C₂×#108×C₅₃	1	1	yes	characteristic
121	C₄	C₂×#108×C₅₃	1	1	yes	characteristic
122	C₂	—	4	1	no	—
123	C₂	C₅₃×#1833	1	2	yes	—
124	C₂	C₅₃×#1837	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#1568

C₂

yes

maximal

C₅₃×#1450

C₂

yes

maximal, characteristic

C₅₃×#1287

C₂

yes

maximal

C₅₃×#432

C₂²

yes

—

C₅₃×#431

C₂²

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#335

—

C₅₃×#334

—

C₅₃×#322

C₄

yes

—

C₈²×C₅₃

C₄

yes

—

C₂²×C₈×C₅₃

Q₈

yes

characteristic

C₂²×C₈×C₅₃

Q₈

yes

characteristic

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

#97×C₅₃

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

D₄

yes

characteristic

C₄×C₈×C₅₃

D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₂×C₈×C₅₃

C₄⋊C₄

yes

characteristic

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₄²×C₅₃

C₂×D₄

yes

characteristic

C₄²×C₅₃

#41

yes

—

C₂³×C₅₃

#125

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

—

C₂×C₄×C₅₃

#125

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

#108

yes

—

C₈×C₅₃

#108

yes

—

#8796

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#382

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₂×C₅₃

yes

—

#1450

C₂×C₅₃

yes

characteristic

#1287

C₂×C₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

#1833

yes

—

C₂×C₅₃

#1837

yes

characteristic

#432

C₂²×C₅₃

yes

—

#431

C₂²×C₅₃

yes

—

C₂×C₄×C₈

C₂²×C₅₃

yes

—

—

C₄×C₅₃

yes

—

C₈²

C₄×C₅₃

yes

—

C₅₃

#8796

yes

characteristic, direct product factor, Sylow

C₂²×C₈

Q₈×C₅₃

yes

characteristic

C₂²×C₈

Q₈×C₅₃

yes

characteristic

C₂×C₄²

C₂³×C₅₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

C₂×C₄×C₅₃

yes

—

C₄×C₈

C₂×C₄×C₅₃

yes

characteristic

C₄×C₈

C₂×C₄×C₅₃

yes

characteristic

C₄×C₈

D₄×C₅₃

yes

characteristic

C₄×C₈

D₄×C₅₃

yes

characteristic

C₂²×C₄

C₂×Q₈×C₅₃

yes

characteristic

C₂²×C₄

#41×C₅₃

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄⋊C₄×C₅₃

yes

characteristic

C₂×C₈

C₄⋊C₄×C₅₃

yes

characteristic

C₂×C₈

C₄⋊C₄×C₅₃

yes

characteristic

C₂×C₈

C₄⋊C₄×C₅₃

yes

characteristic

C₄²

C₂²×C₄×C₅₃

yes

characteristic

C₄²

C₂×D₄×C₅₃

yes

characteristic

C₄²

#41×C₅₃

yes

—

100

C₂³

#125×C₅₃

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₂×C₄⋊C₄×C₅₃

yes

characteristic

105

C₂×C₄

#117×C₅₃

yes

—

106

C₂×C₄

#124×C₅₃

yes

—

107

C₂×C₄

#125×C₅₃

yes

characteristic

108

C₈

—

109

C₈

—

110

C₈

—

111

C₈

—

112

C₈

#108×C₅₃

yes

—

113

C₈

#108×C₅₃

yes

—

114

C₂²

—

115

C₂²

—

116

C₂²

C₅₃×#382

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

C₂×#108×C₅₃

yes

characteristic

121

C₄

C₂×#108×C₅₃

yes

characteristic

122

C₂

—

123

C₂

C₅₃×#1833

yes

—

124

C₂

C₅₃×#1837

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	#382
commutator	C₄²	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₂³×C₅₃	#125
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₅₃

#382

commutator

C₄²

C₂²×C₄×C₅₃

Frattini, cosocle

C₂×C₄²

C₂³×C₅₃

C₂³×C₅₃

G ▷ C₄² ▷ C₂² ▷ C₁

C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

N	H ≅ G/N	type
51. #8796	C₅₃	direct (N × H)
73. C₅₃	#8796	direct (N × H)

H ≅ G/N

type

51. #8796

C₅₃

direct (N × H)

73. C₅₃

#8796

direct (N × H)

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

subgroup	index
C₅₃×#1568	2
C₅₃×#1450	2
C₅₃×#1287	2
#8796	53

subgroup

index

C₅₃×#1568

C₅₃×#1450

C₅₃×#1287

#8796

Subgroups — C₅₃×#8796 #27216308

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#8796 #27216308

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8796 #27216308