Group #27384342 — subgroups

Subgroups — #27384342

Order 13568.

1062 conjugacy classes of subgroups; only the 411 normal subgroups are shown.

normal subgroups only

This group has 1062 conjugacy classes of subgroups — too many to draw. Only its 411 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	#20274858	C₂	1	8	yes	maximal, complemented
2	#20274409	C₂	1	6	yes	maximal
3	C₂×C₄³×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂²×#11719917	C₂²	1	4	yes	—
5	#16810378	C₂²	1	24	yes	—
6	C₂²×C₄²×C₅₃	C₄	1	4	yes	—
7	C₂²×C₄²×C₅₃	C₂²	1	3	yes	—
8	C₄³×C₅₃	C₄	1	4	yes	—
9	C₄³×C₅₃	C₂²	1	4	yes	—
10	C₂×#11719917	—	2	24	no	—
11	C₂²×#88322	C₂³	1	8	yes	—
12	C₂³×C₄×C₅₃	C₂×C₄	1	6	yes	—
13	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
14	C₂×C₄²×C₅₃	—	2	16	no	—
15	C₂×C₄²×C₅₃	—	2	12	no	—
16	C₂×C₄²×C₅₃	C₂×C₄	1	12	yes	—
17	C₂×C₄²×C₅₃	C₂×C₄	1	6	yes	—
18	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
19	C₂×C₄²×C₅₃	C₂³	1	6	yes	—
20	C₂⁴×C₅₃	C₂²×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₅₃	—	2	12	no	—
22	C₂²×C₄×C₅₃	—	2	24	no	—
23	C₂²×C₄×C₅₃	C₂²×C₄	1	12	yes	—
24	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₅₃	#41	1	12	yes	—
26	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
27	C₄²×C₅₃	—	2	12	no	—
28	C₄²×C₅₃	—	2	24	no	—
29	C₄²×C₅₃	—	2	12	no	—
30	C₄²×C₅₃	M₁₆	1	16	yes	complemented
31	#11719917	—	4	16	no	—
32	C₂×#88322	—	2	24	no	—
33	C₂³×C₅₃	—	2	4	no	—
34	C₂³×C₅₃	#117	1	6	yes	—
35	C₂³×C₅₃	C₂³×C₄	1	1	yes	characteristic
36	C₂×C₄×C₅₃	—	2	48	no	—
37	C₂×C₄×C₅₃	—	2	12	no	—
38	C₂×C₄×C₅₃	—	2	12	no	—
39	C₂×C₄×C₅₃	#117	1	6	yes	—
40	C₂×C₄×C₅₃	#126	1	16	yes	—
41	C₂×C₄×C₅₃	C₂×M₁₆	1	24	yes	—
42	C₂×C₄×C₅₃	C₂×#41	1	6	yes	—
43	#88322	—	4	8	no	—
44	#20772	—	53	1	no	maximal, Sylow
45	C₂²×C₅₃	—	2	12	no	—
46	C₂²×C₅₃	#383	1	4	yes	—
47	C₂²×C₅₃	C₂×#117	1	3	yes	—
48	C₂²×C₅₃	C₂²×M₁₆	1	4	yes	—
49	C₄×C₅₃	—	2	24	no	—
50	C₄×C₅₃	#383	1	4	yes	—
51	C₄×C₅₃	C₂×#126	1	4	yes	—
52	C₂×C₄³	D₅₃	1	1	yes	characteristic
53	#1839	—	53	8	no	—
54	#1728	—	53	6	no	—
55	C₂×C₅₃	—	2	4	no	—
56	C₂×C₅₃	C₂×#383	1	1	yes	characteristic
57	C₂×C₅₃	#2920	1	6	yes	—
58	C₂²×M₁₆	—	53	4	no	—
59	C₂²×C₄²	Dic₅₃	1	4	yes	—
60	C₂²×C₄²	C₂×D₅₃	1	3	yes	—
61	C₄³	Dic₅₃	1	4	yes	—
62	C₄³	C₂×D₅₃	1	4	yes	—
63	#336	—	53	24	no	—
64	C₅₃	#20772	1	1	yes	characteristic, complemented, Sylow
65	C₂³×C₄	C₂×Dic₅₃	1	6	yes	—
66	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
67	C₂×M₁₆	—	106	24	no	—
68	C₂²×C₈	—	53	8	no	—
69	C₂×C₄²	—	2	12	no	—
70	C₂×C₄²	—	2	16	no	—
71	C₂×C₄²	C₂×Dic₅₃	1	6	yes	—
72	C₂×C₄²	C₂×Dic₅₃	1	12	yes	—
73	C₂×C₄²	C₂×Dic₅₃	1	4	yes	—
74	C₂×C₄²	C₂²×D₅₃	1	6	yes	—
75	C₂⁴	C₂²×Dic₅₃	1	1	yes	characteristic
76	C₂²×C₄	—	2	12	no	—
77	C₂²×C₄	—	2	24	no	—
78	C₂²×C₄	#11719945	1	12	yes	—
79	C₂²×C₄	C₂²×Dic₅₃	1	12	yes	—
80	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
81	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
82	M₁₆	—	212	16	no	—
83	C₂×C₈	—	106	24	no	—
84	C₄²	—	2	12	no	—
85	C₄²	—	2	24	no	—
86	C₄²	—	2	12	no	—
87	C₄²	#11719917	1	16	yes	complemented
88	C₂³	—	2	4	no	—
89	C₂³	#13133176	1	6	yes	—
90	C₂³	C₂³×Dic₅₃	1	1	yes	characteristic
91	C₂×C₄	—	2	48	no	—
92	C₂×C₄	—	2	12	no	—
93	C₂×C₄	—	2	12	no	—
94	C₂×C₄	#13133176	1	6	yes	—
95	C₂×C₄	#13133211	1	16	yes	—
96	C₂×C₄	C₂×#11719917	1	24	yes	—
97	C₂×C₄	C₂×#11719945	1	6	yes	—
98	C₈	—	212	8	no	—
99	C₂²	—	2	12	no	—
100	C₂²	#16810491	1	4	yes	—
101	C₂²	C₂×#13133176	1	3	yes	—
102	C₂²	C₂²×#11719917	1	4	yes	—
103	C₄	—	2	24	no	—
104	C₄	#16810491	1	4	yes	—
105	C₄	C₂×#13133211	1	4	yes	—
106	C₂	—	2	4	no	—
107	C₂	C₂×#16810491	1	1	yes	characteristic
108	C₂	#20283348	1	6	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20274858

C₂

yes

maximal, complemented

#20274409

C₂

yes

maximal

C₂×C₄³×C₅₃

C₂

yes

maximal, characteristic

C₂²×#11719917

C₂²

yes

—

#16810378

C₂²

yes

—

C₂²×C₄²×C₅₃

C₄

yes

—

C₂²×C₄²×C₅₃

C₂²

yes

—

C₄³×C₅₃

C₄

yes

—

C₄³×C₅₃

C₂²

yes

—

C₂×#11719917

—

C₂²×#88322

C₂³

yes

—

C₂³×C₄×C₅₃

C₂×C₄

yes

—

C₂³×C₄×C₅₃

C₂³

yes

characteristic

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

—

C₂⁴×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

—

C₂²×C₄×C₅₃

C₂⁴

yes

characteristic

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

M₁₆

yes

complemented

#11719917

—

C₂×#88322

—

C₂³×C₅₃

—

C₂³×C₅₃

#117

yes

—

C₂³×C₅₃

C₂³×C₄

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#117

yes

—

C₂×C₄×C₅₃

#126

yes

—

C₂×C₄×C₅₃

C₂×M₁₆

yes

—

C₂×C₄×C₅₃

C₂×#41

yes

—

#88322

—

#20772

—

maximal, Sylow

C₂²×C₅₃

—

C₂²×C₅₃

#383

yes

—

C₂²×C₅₃

C₂×#117

yes

—

C₂²×C₅₃

C₂²×M₁₆

yes

—

C₄×C₅₃

—

C₄×C₅₃

#383

yes

—

C₄×C₅₃

C₂×#126

yes

—

C₂×C₄³

D₅₃

yes

characteristic

#1839

—

#1728

—

C₂×C₅₃

—

C₂×C₅₃

C₂×#383

yes

characteristic

C₂×C₅₃

#2920

yes

—

C₂²×M₁₆

—

C₂²×C₄²

Dic₅₃

yes

—

C₂²×C₄²

C₂×D₅₃

yes

—

C₄³

Dic₅₃

yes

—

C₄³

C₂×D₅₃

yes

—

#336

—

C₅₃

#20772

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×Dic₅₃

yes

—

C₂³×C₄

C₂²×D₅₃

yes

characteristic

C₂×M₁₆

—

106

—

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×Dic₅₃

yes

—

C₂×C₄²

C₂×Dic₅₃

yes

—

C₂×C₄²

C₂×Dic₅₃

yes

—

C₂×C₄²

C₂²×D₅₃

yes

—

C₂⁴

C₂²×Dic₅₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#11719945

yes

—

C₂²×C₄

C₂²×Dic₅₃

yes

—

C₂²×C₄

C₂²×Dic₅₃

yes

characteristic

C₂²×C₄

C₂³×D₅₃

yes

characteristic

M₁₆

—

212

—

C₂×C₈

—

106

—

C₄²

—

C₄²

—

C₄²

—

C₄²

#11719917

yes

complemented

C₂³

—

C₂³

#13133176

yes

—

C₂³

C₂³×Dic₅₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#13133176

yes

—

C₂×C₄

#13133211

yes

—

C₂×C₄

C₂×#11719917

yes

—

C₂×C₄

C₂×#11719945

yes

—

C₈

—

212

—

C₂²

—

100

C₂²

#16810491

yes

—

101

C₂²

C₂×#13133176

yes

—

102

C₂²

C₂²×#11719917

yes

—

103

C₄

—

104

C₄

#16810491

yes

—

105

C₄

C₂×#13133211

yes

—

106

C₂

—

107

C₂

C₂×#16810491

yes

characteristic

108

C₂

#20283348

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₅₃
commutator, socle	C₂³×C₅₃	C₂³×C₄
Fitting	C₂×C₄³×C₅₃	C₂
prime radical	C₂⁴×C₅₃	C₂²×C₄
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×C₄³	D₅₃
hypocenter	C₅₃	#20772
LCS	G ▷ C₂³×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³

subgroup

G/H

center, Frattini

C₂²×C₄

C₂³×D₅₃

commutator, socle

C₂³×C₅₃

C₂³×C₄

Fitting

C₂×C₄³×C₅₃

C₂

prime radical

C₂⁴×C₅₃

C₂²×C₄

cosocle

C₂²×C₄×C₅₃

C₂⁴

hypercenter

C₂×C₄³

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₅₃

C₁ ◁ C₂²×C₄ ◁ C₂×C₄³

N	H ≅ G/N	type
1. #20274858	C₂	semidirect
30. C₄²×C₅₃	M₁₆	semidirect
64. C₅₃	#20772	semidirect
87. C₄²	#11719917	semidirect

H ≅ G/N

type

1. #20274858

C₂

semidirect

30. C₄²×C₅₃

M₁₆

semidirect

64. C₅₃

#20772

semidirect

87. C₄²

#11719917

semidirect

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

subgroup	index
#20274858	2
#20274409	2
C₂×C₄³×C₅₃	2
#20772	53

subgroup

index

#20274858

#20274409

C₂×C₄³×C₅₃

#20772

Subgroups — #27384342

Conjugacy classes of subgroups

Complements

Maximal subgroups