Group #27218914 — subgroups

Subgroups — C₅₃×#11402 #27218914

Order 13568.

322 conjugacy classes of subgroups; only the 246 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 246 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	C₄×M₃₂×C₅₃	C₂	1	2	yes	maximal
2	C₂×C₈²×C₅₃	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1312	C₂	1	4	yes	maximal, complemented
4	C₂×M₃₂×C₅₃	C₄	1	4	yes	—
5	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	—
6	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
7	C₅₃×#346	C₄	1	4	yes	—
8	C₄×C₁₆×C₅₃	C₂²	1	4	yes	—
9	C₈²×C₅₃	C₄	1	1	yes	characteristic
10	C₈²×C₅₃	C₄	1	1	yes	characteristic
11	C₈²×C₅₃	C₂²	1	2	yes	—
12	C₂²×C₈×C₅₃	C₈	1	4	yes	—
13	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
14	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
15	M₃₂×C₅₃	—	2	8	no	—
16	C₂×C₁₆×C₅₃	C₂×C₄	1	8	yes	—
17	C₄×C₈×C₅₃	C₈	1	4	yes	—
18	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
21	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
23	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
24	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
25	C₂×C₈×C₅₃	—	2	8	no	—
26	C₂×C₈×C₅₃	C₄²	1	2	yes	—
27	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
28	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	—
29	C₂×C₈×C₅₃	M₁₆	1	8	yes	—
30	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
31	C₄²×C₅₃	C₄²	1	1	yes	characteristic
32	C₄²×C₅₃	C₂×C₈	1	2	yes	—
33	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
34	C₁₆×C₅₃	—	2	8	no	—
35	C₂³×C₅₃	C₄×C₈	1	1	yes	characteristic
36	C₂×C₄×C₅₃	—	2	4	no	—
37	C₂×C₄×C₅₃	C₄×C₈	1	2	yes	—
38	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
39	C₂×C₄×C₅₃	#97	1	4	yes	—
40	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
41	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
42	C₈×C₅₃	—	2	4	no	—
43	C₈×C₅₃	#97	1	4	yes	—
44	C₈×C₅₃	M₃₂	1	8	yes	complemented
45	C₈×C₅₃	C₂×M₁₆	1	4	yes	—
46	#11402	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
47	C₂²×C₅₃	—	2	2	no	—
48	C₂²×C₅₃	#322	1	2	yes	—
49	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
50	C₄×C₅₃	—	2	2	no	—
51	C₄×C₅₃	#322	1	2	yes	—
52	C₄×C₅₃	C₂×#97	1	2	yes	—
53	C₄×C₅₃	C₂×M₃₂	1	4	yes	—
54	C₄×M₃₂	C₂×C₅₃	1	2	yes	—
55	C₂×C₈²	C₂×C₅₃	1	1	yes	characteristic
56	#1312	C₂×C₅₃	1	4	yes	complemented
57	C₂×C₅₃	—	2	2	no	—
58	C₂×C₅₃	C₂×#322	1	1	yes	characteristic
59	C₂×C₅₃	C₄×M₃₂	1	2	yes	—
60	C₂×M₃₂	C₄×C₅₃	1	4	yes	—
61	C₂×C₄×C₈	C₄×C₅₃	1	2	yes	—
62	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
63	#346	C₄×C₅₃	1	4	yes	—
64	C₄×C₁₆	C₂²×C₅₃	1	4	yes	—
65	C₈²	C₄×C₅₃	1	1	yes	characteristic
66	C₈²	C₄×C₅₃	1	1	yes	characteristic
67	C₈²	C₂²×C₅₃	1	2	yes	—
68	C₅₃	#11402	1	1	yes	characteristic, direct product factor, Sylow
69	C₂²×C₈	C₈×C₅₃	1	4	yes	—
70	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
71	C₂×C₄²	C₂×C₄×C₅₃	1	1	yes	characteristic
72	M₃₂	—	2	8	no	—
73	C₂×C₁₆	C₂×C₄×C₅₃	1	8	yes	—
74	C₄×C₈	C₈×C₅₃	1	4	yes	—
75	C₄×C₈	C₂×C₄×C₅₃	1	1	yes	characteristic
76	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
77	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
78	C₄×C₈	C₂×C₄×C₅₃	1	2	yes	—
79	C₄×C₈	C₂³×C₅₃	1	1	yes	characteristic
80	C₂²×C₄	C₄²×C₅₃	1	1	yes	characteristic
81	C₂²×C₄	C₂×C₈×C₅₃	1	2	yes	—
82	C₂×C₈	—	2	8	no	—
83	C₂×C₈	C₄²×C₅₃	1	2	yes	—
84	C₂×C₈	C₂×C₈×C₅₃	1	4	yes	—
85	C₂×C₈	C₂×C₈×C₅₃	1	4	yes	—
86	C₂×C₈	M₁₆×C₅₃	1	8	yes	—
87	C₂×C₈	C₂²×C₄×C₅₃	1	2	yes	—
88	C₄²	C₄²×C₅₃	1	1	yes	characteristic
89	C₄²	C₂×C₈×C₅₃	1	2	yes	—
90	C₄²	C₂²×C₄×C₅₃	1	1	yes	characteristic
91	C₁₆	—	2	8	no	—
92	C₂³	C₄×C₈×C₅₃	1	1	yes	characteristic
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	C₄×C₈×C₅₃	1	2	yes	—
95	C₂×C₄	C₄×C₈×C₅₃	1	1	yes	characteristic
96	C₂×C₄	#97×C₅₃	1	4	yes	—
97	C₂×C₄	C₂×C₄²×C₅₃	1	1	yes	characteristic
98	C₂×C₄	C₂²×C₈×C₅₃	1	2	yes	—
99	C₈	—	2	4	no	—
100	C₈	#97×C₅₃	1	4	yes	—
101	C₈	M₃₂×C₅₃	1	8	yes	complemented
102	C₈	C₂×M₁₆×C₅₃	1	4	yes	—
103	C₂²	—	2	2	no	—
104	C₂²	C₅₃×#322	1	2	yes	—
105	C₂²	C₂×C₄×C₈×C₅₃	1	1	yes	characteristic
106	C₄	—	2	2	no	—
107	C₄	C₅₃×#322	1	2	yes	—
108	C₄	C₂×#97×C₅₃	1	2	yes	—
109	C₄	C₂×M₃₂×C₅₃	1	4	yes	—
110	C₂	—	2	2	no	—
111	C₂	C₂×C₅₃×#322	1	1	yes	characteristic
112	C₂	C₄×M₃₂×C₅₃	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×M₃₂×C₅₃

C₂

yes

maximal

C₂×C₈²×C₅₃

C₂

yes

maximal, characteristic

C₅₃×#1312

C₂

yes

maximal, complemented

C₂×M₃₂×C₅₃

C₄

yes

—

C₂×C₄×C₈×C₅₃

C₄

yes

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#346

C₄

yes

—

C₄×C₁₆×C₅₃

C₂²

yes

—

C₈²×C₅₃

C₄

yes

characteristic

C₈²×C₅₃

C₄

yes

characteristic

C₈²×C₅₃

C₂²

yes

—

C₂²×C₈×C₅₃

C₈

yes

—

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

characteristic

M₃₂×C₅₃

—

C₂×C₁₆×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₈

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

characteristic

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂×C₄

yes

—

C₄×C₈×C₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

C₄²

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₈

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₄²

yes

—

C₂×C₈×C₅₃

C₂×C₈

yes

—

C₂×C₈×C₅₃

C₂×C₈

yes

—

C₂×C₈×C₅₃

M₁₆

yes

—

C₂×C₈×C₅₃

C₂²×C₄

yes

—

C₄²×C₅₃

C₄²

yes

characteristic

C₄²×C₅₃

C₂×C₈

yes

—

C₄²×C₅₃

C₂²×C₄

yes

characteristic

C₁₆×C₅₃

—

C₂³×C₅₃

C₄×C₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄×C₈

yes

—

C₂×C₄×C₅₃

C₄×C₈

yes

characteristic

C₂×C₄×C₅₃

#97

yes

—

C₂×C₄×C₅₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₈

yes

—

C₈×C₅₃

—

C₈×C₅₃

#97

yes

—

C₈×C₅₃

M₃₂

yes

complemented

C₈×C₅₃

C₂×M₁₆

yes

—

#11402

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

#322

yes

—

C₂²×C₅₃

C₂×C₄×C₈

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

#322

yes

—

C₄×C₅₃

C₂×#97

yes

—

C₄×C₅₃

C₂×M₃₂

yes

—

C₄×M₃₂

C₂×C₅₃

yes

—

C₂×C₈²

C₂×C₅₃

yes

characteristic

#1312

C₂×C₅₃

yes

complemented

C₂×C₅₃

—

C₂×C₅₃

C₂×#322

yes

characteristic

C₂×C₅₃

C₄×M₃₂

yes

—

C₂×M₃₂

C₄×C₅₃

yes

—

C₂×C₄×C₈

C₄×C₅₃

yes

—

C₂×C₄×C₈

C₂²×C₅₃

yes

characteristic

#346

C₄×C₅₃

yes

—

C₄×C₁₆

C₂²×C₅₃

yes

—

C₈²

C₄×C₅₃

yes

characteristic

C₈²

C₄×C₅₃

yes

characteristic

C₈²

C₂²×C₅₃

yes

—

C₅₃

#11402

yes

characteristic, direct product factor, Sylow

C₂²×C₈

C₈×C₅₃

yes

—

C₂²×C₈

C₂×C₄×C₅₃

yes

—

C₂×C₄²

C₂×C₄×C₅₃

yes

characteristic

M₃₂

—

C₂×C₁₆

C₂×C₄×C₅₃

yes

—

C₄×C₈

C₈×C₅₃

yes

—

C₄×C₈

C₂×C₄×C₅₃

yes

characteristic

C₄×C₈

C₂×C₄×C₅₃

yes

—

C₄×C₈

C₂×C₄×C₅₃

yes

—

C₄×C₈

C₂×C₄×C₅₃

yes

—

C₄×C₈

C₂³×C₅₃

yes

characteristic

C₂²×C₄

C₄²×C₅₃

yes

characteristic

C₂²×C₄

C₂×C₈×C₅₃

yes

—

C₂×C₈

—

C₂×C₈

C₄²×C₅₃

yes

—

C₂×C₈

C₂×C₈×C₅₃

yes

—

C₂×C₈

C₂×C₈×C₅₃

yes

—

C₂×C₈

M₁₆×C₅₃

yes

—

C₂×C₈

C₂²×C₄×C₅₃

yes

—

C₄²

C₄²×C₅₃

yes

characteristic

C₄²

C₂×C₈×C₅₃

yes

—

C₄²

C₂²×C₄×C₅₃

yes

characteristic

C₁₆

—

C₂³

C₄×C₈×C₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

C₄×C₈×C₅₃

yes

—

C₂×C₄

C₄×C₈×C₅₃

yes

characteristic

C₂×C₄

#97×C₅₃

yes

—

C₂×C₄

C₂×C₄²×C₅₃

yes

characteristic

C₂×C₄

C₂²×C₈×C₅₃

yes

—

C₈

—

100

C₈

#97×C₅₃

yes

—

101

C₈

M₃₂×C₅₃

yes

complemented

102

C₈

C₂×M₁₆×C₅₃

yes

—

103

C₂²

—

104

C₂²

C₅₃×#322

yes

—

105

C₂²

C₂×C₄×C₈×C₅₃

yes

characteristic

106

C₄

—

107

C₄

C₅₃×#322

yes

—

108

C₄

C₂×#97×C₅₃

yes

—

109

C₄

C₂×M₃₂×C₅₃

yes

—

110

C₂

—

111

C₂

C₂×C₅₃×#322

yes

characteristic

112

C₂

C₄×M₃₂×C₅₃

yes

—

	subgroup	G/H
center	C₄×C₈×C₅₃	C₂³
commutator	C₂²	C₂×C₄×C₈×C₅₃
Frattini, cosocle	C₄×C₈	C₂³×C₅₃
socle	C₂²×C₅₃	C₂×C₄×C₈
prime radical	C₂³×C₅₃	C₄×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄×C₈×C₅₃ ◁ G

subgroup

G/H

center

C₄×C₈×C₅₃

C₂³

commutator

C₂²

C₂×C₄×C₈×C₅₃

Frattini, cosocle

C₄×C₈

C₂³×C₅₃

socle

C₂²×C₅₃

C₂×C₄×C₈

prime radical

C₂³×C₅₃

C₄×C₈

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₄×C₈×C₅₃ ◁ G

N	H ≅ G/N	type
3. C₅₃×#1312	C₂	semidirect
44. C₈×C₅₃	M₃₂	semidirect
46. #11402	C₅₃	direct (N × H)
56. #1312	C₂×C₅₃	semidirect
68. C₅₃	#11402	direct (N × H)
101. C₈	M₃₂×C₅₃	semidirect

H ≅ G/N

type

3. C₅₃×#1312

C₂

semidirect

44. C₈×C₅₃

M₃₂

semidirect

46. #11402

C₅₃

direct (N × H)

56. #1312

C₂×C₅₃

semidirect

68. C₅₃

#11402

direct (N × H)

101. C₈

M₃₂×C₅₃

semidirect

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

subgroup	index
C₄×M₃₂×C₅₃	2
C₂×C₈²×C₅₃	2
C₅₃×#1312	2
#11402	53

subgroup

index

C₄×M₃₂×C₅₃

C₂×C₈²×C₅₃

C₅₃×#1312

#11402

Subgroups — C₅₃×#11402 #27218914

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#11402 #27218914

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#11402 #27218914