Group #27216314 — subgroups

Subgroups — C₅₃×#8802 #27216314

Order 13568.

258 conjugacy classes of subgroups; only the 110 normal subgroups are shown.

normal subgroups only

This group has 258 conjugacy classes of subgroups — too many to draw. Only its 110 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×#1745	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1281	C₂	1	2	yes	maximal
3	C₅₃×#1276	C₂	1	2	yes	maximal
4	C₅₃×#1275	C₂	1	2	yes	maximal
5	C₅₃×#431	C₂²	1	4	yes	—
6	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
7	C₂×#97×C₅₃	C₂²	1	1	yes	characteristic
8	C₄³×C₅₃	C₂²	1	1	yes	characteristic
9	C₅₃×#336	C₄	1	4	yes	—
10	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
11	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
12	C₂×C₄²×C₅₃	—	2	2	no	—
13	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
14	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
15	C₄⋊C₈×C₅₃	—	2	4	no	—
16	C₂²⋊C₈×C₅₃	—	2	4	no	—
17	#97×C₅₃	—	2	2	no	—
18	#97×C₅₃	—	2	2	no	—
19	C₄×C₈×C₅₃	—	2	4	no	—
20	C₂²×C₄×C₅₃	—	2	2	no	—
21	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₅₃	#41	1	2	yes	—
23	C₂×C₈×C₅₃	—	2	2	no	—
24	C₂×C₈×C₅₃	—	2	2	no	—
25	C₂×C₈×C₅₃	—	2	4	no	—
26	C₂×C₈×C₅₃	—	2	4	no	—
27	C₂×C₈×C₅₃	—	4	4	no	—
28	C₄²×C₅₃	—	2	4	no	—
29	C₄²×C₅₃	—	4	4	no	—
30	C₄²×C₅₃	#41	1	2	yes	—
31	C₄²×C₅₃	#41	1	2	yes	—
32	C₂³×C₅₃	#117	1	1	yes	characteristic
33	C₂×C₄×C₅₃	—	2	4	no	—
34	C₂×C₄×C₅₃	—	4	2	no	—
35	C₂×C₄×C₅₃	—	4	2	no	—
36	C₂×C₄×C₅₃	—	4	2	no	—
37	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
38	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
39	C₂×C₄×C₅₃	#126	1	4	yes	—
40	C₂×C₄×C₅₃	#131	1	4	yes	—
41	C₈×C₅₃	—	4	2	no	—
42	C₈×C₅₃	—	4	2	no	—
43	C₈×C₅₃	—	8	4	no	—
44	#8802	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
45	C₂²×C₅₃	—	2	2	no	—
46	C₂²×C₅₃	#383	1	1	yes	characteristic
47	C₂²×C₅₃	#433	1	2	yes	—
48	C₄×C₅₃	—	2	4	no	—
49	C₄×C₅₃	—	4	4	no	—
50	C₄×C₅₃	#433	1	2	yes	—
51	C₄×C₅₃	#433	1	2	yes	—
52	#1745	C₂×C₅₃	1	1	yes	characteristic
53	#1281	C₂×C₅₃	1	2	yes	—
54	#1276	C₂×C₅₃	1	2	yes	—
55	#1275	C₂×C₅₃	1	2	yes	—
56	C₂×C₅₃	—	2	2	no	—
57	C₂×C₅₃	#1840	1	1	yes	characteristic
58	C₂×C₅₃	#1841	1	2	yes	—
59	#431	C₂²×C₅₃	1	4	yes	—
60	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
61	C₂×#97	C₂²×C₅₃	1	1	yes	characteristic
62	C₄³	C₂²×C₅₃	1	1	yes	characteristic
63	#336	C₄×C₅₃	1	4	yes	—
64	C₅₃	#8802	1	1	yes	characteristic, direct product factor, Sylow
65	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
66	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
67	C₂×C₄²	—	2	2	no	—
68	C₂×C₄²	C₂×C₄×C₅₃	1	2	yes	—
69	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
70	C₄⋊C₈	—	2	4	no	—
71	C₂²⋊C₈	—	2	4	no	—
72	#97	—	2	2	no	—
73	#97	—	2	2	no	—
74	C₄×C₈	—	2	4	no	—
75	C₂²×C₄	—	2	2	no	—
76	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
77	C₂²×C₄	#41×C₅₃	1	2	yes	—
78	C₂×C₈	—	2	2	no	—
79	C₂×C₈	—	2	2	no	—
80	C₂×C₈	—	2	4	no	—
81	C₂×C₈	—	2	4	no	—
82	C₂×C₈	—	4	4	no	—
83	C₄²	—	2	4	no	—
84	C₄²	—	4	4	no	—
85	C₄²	#41×C₅₃	1	2	yes	—
86	C₄²	#41×C₅₃	1	2	yes	—
87	C₂³	#117×C₅₃	1	1	yes	characteristic
88	C₂×C₄	—	2	4	no	—
89	C₂×C₄	—	4	2	no	—
90	C₂×C₄	—	4	2	no	—
91	C₂×C₄	—	4	2	no	—
92	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
93	C₂×C₄	#117×C₅₃	1	1	yes	characteristic
94	C₂×C₄	#126×C₅₃	1	4	yes	—
95	C₂×C₄	#131×C₅₃	1	4	yes	—
96	C₈	—	4	2	no	—
97	C₈	—	4	2	no	—
98	C₈	—	8	4	no	—
99	C₂²	—	2	2	no	—
100	C₂²	C₅₃×#383	1	1	yes	characteristic
101	C₂²	C₅₃×#433	1	2	yes	—
102	C₄	—	2	4	no	—
103	C₄	—	4	4	no	—
104	C₄	C₅₃×#433	1	2	yes	—
105	C₄	C₅₃×#433	1	2	yes	—
106	C₂	—	2	2	no	—
107	C₂	C₅₃×#1840	1	1	yes	characteristic
108	C₂	C₅₃×#1841	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#1745

C₂

yes

maximal, characteristic

C₅₃×#1281

C₂

yes

maximal

C₅₃×#1276

C₂

yes

maximal

C₅₃×#1275

C₂

yes

maximal

C₅₃×#431

C₂²

yes

—

C₂×#97×C₅₃

C₂²

yes

characteristic

C₂×#97×C₅₃

C₂²

yes

characteristic

C₄³×C₅₃

C₂²

yes

characteristic

C₅₃×#336

C₄

yes

—

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

#97×C₅₃

—

#97×C₅₃

—

C₄×C₈×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

#41

yes

—

C₄²×C₅₃

#41

yes

—

C₂³×C₅₃

#117

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

—

C₂×C₄×C₅₃

#131

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

#8802

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

#383

yes

characteristic

C₂²×C₅₃

#433

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#433

yes

—

C₄×C₅₃

#433

yes

—

#1745

C₂×C₅₃

yes

characteristic

#1281

C₂×C₅₃

yes

—

#1276

C₂×C₅₃

yes

—

#1275

C₂×C₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

#1840

yes

characteristic

C₂×C₅₃

#1841

yes

—

#431

C₂²×C₅₃

yes

—

C₂×#97

C₂²×C₅₃

yes

characteristic

C₂×#97

C₂²×C₅₃

yes

characteristic

C₄³

C₂²×C₅₃

yes

characteristic

#336

C₄×C₅₃

yes

—

C₅₃

#8802

yes

characteristic, direct product factor, Sylow

C₂²×C₈

C₂×C₄×C₅₃

yes

—

C₂²×C₈

C₂×C₄×C₅₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×C₄×C₅₃

yes

—

C₂×C₄²

C₂³×C₅₃

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

C₄×C₈

—

C₂²×C₄

—

C₂²×C₄

C₂²×C₄×C₅₃

yes

characteristic

C₂²×C₄

#41×C₅₃

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

—

C₄²

—

C₄²

#41×C₅₃

yes

—

C₄²

#41×C₅₃

yes

—

C₂³

#117×C₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#117×C₅₃

yes

characteristic

C₂×C₄

#117×C₅₃

yes

characteristic

C₂×C₄

#126×C₅₃

yes

—

C₂×C₄

#131×C₅₃

yes

—

C₈

—

C₈

—

C₈

—

C₂²

—

100

C₂²

C₅₃×#383

yes

characteristic

101

C₂²

C₅₃×#433

yes

—

102

C₄

—

103

C₄

—

104

C₄

C₅₃×#433

yes

—

105

C₄

C₅₃×#433

yes

—

106

C₂

—

107

C₂

C₅₃×#1840

yes

characteristic

108

C₂

C₅₃×#1841

yes

—

	subgroup	G/H
center	C₂×C₄×C₅₃	#117
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
socle	C₂²×C₅₃	#383
prime radical	C₂³×C₅₃	#117
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×#97×C₅₃ ◁ G

subgroup

G/H

center

C₂×C₄×C₅₃

#117

commutator

C₂²×C₄

C₂²×C₄×C₅₃

Frattini, cosocle

C₂×C₄²

C₂³×C₅₃

C₂²×C₅₃

C₂³×C₅₃

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

C₁ ◁ C₂×C₄×C₅₃ ◁ C₂×#97×C₅₃ ◁ G

N	H ≅ G/N	type
44. #8802	C₅₃	direct (N × H)
64. C₅₃	#8802	direct (N × H)

H ≅ G/N

type

44. #8802

C₅₃

direct (N × H)

64. C₅₃

#8802

direct (N × H)

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

subgroup	index
C₅₃×#1745	2
C₅₃×#1281	2
C₅₃×#1276	2
C₅₃×#1275	2
#8802	53

subgroup

index

C₅₃×#1745

C₅₃×#1281

C₅₃×#1276

C₅₃×#1275

#8802

Subgroups — C₅₃×#8802 #27216314

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#8802 #27216314

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#8802 #27216314