Group #27305127 — subgroups

Subgroups — D₅₃×M₁₂₈ #27305127

Order 13568.

134 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 134 conjugacy classes of subgroups — too many to draw. Only its 81 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×C₃₂×D₅₃	C₂	1	1	yes	maximal, characteristic
2	#20272926	C₂	1	1	yes	maximal, characteristic, complemented
3	#20272925	C₂	1	2	yes	maximal, complemented
4	C₆₄×D₅₃	C₂	1	2	yes	maximal, complemented
5	C₅₃×M₁₂₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₆×D₅₃	C₄	1	1	yes	characteristic
7	C₂×#13133123	C₄	1	1	yes	characteristic
8	C₃₂×D₅₃	C₄	1	1	yes	characteristic
9	C₃₂×D₅₃	C₄	1	1	yes	characteristic
10	C₃₂×D₅₃	C₂²	1	1	yes	characteristic
11	C₃₂×D₅₃	C₂²	1	1	yes	characteristic
12	#16810342	C₂²	1	2	yes	complemented
13	C₂×C₃₂×C₅₃	C₂²	1	1	yes	characteristic
14	C₅₃×C₆₄	C₂²	1	2	yes	complemented
15	C₂×C₈×D₅₃	C₈	1	1	yes	characteristic
16	C₂×#11719908	C₈	1	1	yes	characteristic
17	C₁₆×D₅₃	C₈	1	1	yes	characteristic
18	C₁₆×D₅₃	C₈	1	1	yes	characteristic
19	C₁₆×D₅₃	C₂×C₄	1	1	yes	characteristic
20	C₁₆×D₅₃	C₂×C₄	1	1	yes	characteristic
21	#13133123	C₂×C₄	1	1	yes	characteristic
22	#13133123	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₃₂×C₅₃	C₂×C₄	1	1	yes	characteristic
25	C₃₂×C₅₃	C₂³	1	1	yes	characteristic
26	C₂×C₄×D₅₃	C₁₆	1	1	yes	characteristic
27	C₂×C₈×C₅₃	C₂×C₈	1	1	yes	characteristic
28	C₂×#88322	C₁₆	1	1	yes	characteristic
29	C₈×D₅₃	C₁₆	1	1	yes	characteristic
30	C₈×D₅₃	C₁₆	1	1	yes	characteristic
31	C₈×D₅₃	C₂×C₈	1	1	yes	characteristic
32	C₈×D₅₃	C₂×C₈	1	1	yes	characteristic
33	C₁₆×C₅₃	C₂×C₈	1	1	yes	characteristic
34	C₁₆×C₅₃	C₂²×C₄	1	1	yes	characteristic
35	#11719908	C₂×C₈	1	1	yes	characteristic
36	#11719908	C₂×C₈	1	1	yes	characteristic
37	C₂²×D₅₃	C₃₂	1	1	yes	characteristic
38	C₂×C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
39	C₂×Dic₅₃	C₃₂	1	1	yes	characteristic
40	C₄×D₅₃	C₃₂	1	1	yes	characteristic
41	C₄×D₅₃	C₃₂	1	1	yes	characteristic
42	C₄×D₅₃	C₂×C₁₆	1	1	yes	characteristic
43	C₄×D₅₃	C₂×C₁₆	1	1	yes	characteristic
44	C₈×C₅₃	C₂×C₁₆	1	1	yes	characteristic
45	C₈×C₅₃	C₂²×C₈	1	1	yes	characteristic
46	#88322	C₂×C₁₆	1	1	yes	characteristic
47	#88322	C₂×C₁₆	1	1	yes	characteristic
48	C₂×M₁₂₈	—	53	1	no	maximal, Sylow
49	C₂²×C₅₃	C₂×C₃₂	1	1	yes	characteristic
50	C₂×D₅₃	—	2	2	no	—
51	C₂×D₅₃	C₂×C₃₂	1	1	yes	characteristic
52	C₂×D₅₃	C₂×C₃₂	1	1	yes	characteristic
53	C₄×C₅₃	C₂×C₃₂	1	1	yes	characteristic
54	C₄×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
55	Dic₅₃	C₂×C₃₂	1	1	yes	characteristic
56	Dic₅₃	C₂×C₃₂	1	1	yes	characteristic
57	C₂²×C₃₂	—	53	1	no	—
58	M₁₂₈	—	53	1	no	—
59	M₁₂₈	—	53	2	no	—
60	M₁₂₈	D₅₃	1	1	yes	characteristic, direct product factor
61	C₂×C₆₄	—	53	2	no	—
62	C₂×C₅₃	—	2	1	no	—
63	C₂×C₅₃	C₂²×C₃₂	1	1	yes	characteristic
64	D₅₃	—	2	1	no	—
65	D₅₃	M₁₂₈	1	2	yes	direct product factor
66	C₂²×C₁₆	—	53	1	no	—
67	C₂×C₃₂	—	53	1	no	—
68	C₂×C₃₂	—	53	1	no	—
69	C₂×C₃₂	—	53	1	no	—
70	C₂×C₃₂	—	53	1	no	—
71	C₂×C₃₂	—	53	1	no	—
72	C₂×C₃₂	C₂×D₅₃	1	1	yes	characteristic
73	C₆₄	—	53	2	no	—
74	C₆₄	C₂×D₅₃	1	2	yes	complemented
75	C₅₃	C₂×M₁₂₈	1	1	yes	characteristic, complemented, Sylow
76	C₂²×C₈	—	53	1	no	—
77	C₂×C₁₆	—	53	1	no	—
78	C₂×C₁₆	—	53	1	no	—
79	C₂×C₁₆	—	53	1	no	—
80	C₂×C₁₆	—	53	1	no	—
81	C₂×C₁₆	—	53	1	no	—
82	C₂×C₁₆	C₄×D₅₃	1	1	yes	characteristic
83	C₃₂	—	53	1	no	—
84	C₃₂	—	53	1	no	—
85	C₃₂	C₄×D₅₃	1	1	yes	characteristic
86	C₃₂	C₂²×D₅₃	1	1	yes	characteristic
87	C₂²×C₄	—	53	1	no	—
88	C₂×C₈	—	53	1	no	—
89	C₂×C₈	—	53	1	no	—
90	C₂×C₈	—	53	1	no	—
91	C₂×C₈	—	53	1	no	—
92	C₂×C₈	—	53	1	no	—
93	C₂×C₈	C₈×D₅₃	1	1	yes	characteristic
94	C₁₆	—	53	1	no	—
95	C₁₆	—	53	1	no	—
96	C₁₆	C₈×D₅₃	1	1	yes	characteristic
97	C₁₆	C₂×C₄×D₅₃	1	1	yes	characteristic
98	C₂³	—	53	1	no	—
99	C₂×C₄	—	53	1	no	—
100	C₂×C₄	—	53	1	no	—
101	C₂×C₄	—	53	1	no	—
102	C₂×C₄	—	53	1	no	—
103	C₂×C₄	—	53	1	no	—
104	C₂×C₄	C₁₆×D₅₃	1	1	yes	characteristic
105	C₈	—	53	1	no	—
106	C₈	—	53	1	no	—
107	C₈	C₁₆×D₅₃	1	1	yes	characteristic
108	C₈	C₂×C₈×D₅₃	1	1	yes	characteristic
109	C₂²	—	53	1	no	—
110	C₂²	—	53	1	no	—
111	C₂²	—	106	2	no	—
112	C₂²	C₃₂×D₅₃	1	1	yes	characteristic
113	C₄	—	53	1	no	—
114	C₄	—	53	1	no	—
115	C₄	C₃₂×D₅₃	1	1	yes	characteristic
116	C₄	C₂×C₁₆×D₅₃	1	1	yes	characteristic
117	C₂	—	2	1	no	—
118	C₂	—	53	2	no	—
119	C₂	—	106	1	no	—
120	C₂	C₂×C₃₂×D₅₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₃₂×D₅₃

C₂

yes

maximal, characteristic

#20272926

C₂

yes

maximal, characteristic, complemented

#20272925

C₂

yes

maximal, complemented

C₆₄×D₅₃

C₂

yes

maximal, complemented

C₅₃×M₁₂₈

C₂

yes

maximal, characteristic, complemented

C₂×C₁₆×D₅₃

C₄

yes

characteristic

C₂×#13133123

C₄

yes

characteristic

C₃₂×D₅₃

C₄

yes

characteristic

C₃₂×D₅₃

C₄

yes

characteristic

C₃₂×D₅₃

C₂²

yes

characteristic

C₃₂×D₅₃

C₂²

yes

characteristic

#16810342

C₂²

yes

complemented

C₂×C₃₂×C₅₃

C₂²

yes

characteristic

C₅₃×C₆₄

C₂²

yes

complemented

C₂×C₈×D₅₃

C₈

yes

characteristic

C₂×#11719908

C₈

yes

characteristic

C₁₆×D₅₃

C₈

yes

characteristic

C₁₆×D₅₃

C₈

yes

characteristic

C₁₆×D₅₃

C₂×C₄

yes

characteristic

C₁₆×D₅₃

C₂×C₄

yes

characteristic

#13133123

C₂×C₄

yes

characteristic

#13133123

C₂×C₄

yes

characteristic

C₂×C₁₆×C₅₃

C₂×C₄

yes

characteristic

C₃₂×C₅₃

C₂×C₄

yes

characteristic

C₃₂×C₅₃

C₂³

yes

characteristic

C₂×C₄×D₅₃

C₁₆

yes

characteristic

C₂×C₈×C₅₃

C₂×C₈

yes

characteristic

C₂×#88322

C₁₆

yes

characteristic

C₈×D₅₃

C₁₆

yes

characteristic

C₈×D₅₃

C₁₆

yes

characteristic

C₈×D₅₃

C₂×C₈

yes

characteristic

C₈×D₅₃

C₂×C₈

yes

characteristic

C₁₆×C₅₃

C₂×C₈

yes

characteristic

C₁₆×C₅₃

C₂²×C₄

yes

characteristic

#11719908

C₂×C₈

yes

characteristic

#11719908

C₂×C₈

yes

characteristic

C₂²×D₅₃

C₃₂

yes

characteristic

C₂×C₄×C₅₃

C₂×C₁₆

yes

characteristic

C₂×Dic₅₃

C₃₂

yes

characteristic

C₄×D₅₃

C₃₂

yes

characteristic

C₄×D₅₃

C₃₂

yes

characteristic

C₄×D₅₃

C₂×C₁₆

yes

characteristic

C₄×D₅₃

C₂×C₁₆

yes

characteristic

C₈×C₅₃

C₂×C₁₆

yes

characteristic

C₈×C₅₃

C₂²×C₈

yes

characteristic

#88322

C₂×C₁₆

yes

characteristic

#88322

C₂×C₁₆

yes

characteristic

C₂×M₁₂₈

—

maximal, Sylow

C₂²×C₅₃

C₂×C₃₂

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

C₂×C₃₂

yes

characteristic

C₂×D₅₃

C₂×C₃₂

yes

characteristic

C₄×C₅₃

C₂×C₃₂

yes

characteristic

C₄×C₅₃

C₂²×C₁₆

yes

characteristic

Dic₅₃

C₂×C₃₂

yes

characteristic

Dic₅₃

C₂×C₃₂

yes

characteristic

C₂²×C₃₂

—

M₁₂₈

—

M₁₂₈

—

M₁₂₈

D₅₃

yes

characteristic, direct product factor

C₂×C₆₄

—

C₂×C₅₃

—

C₂×C₅₃

C₂²×C₃₂

yes

characteristic

D₅₃

—

D₅₃

M₁₂₈

yes

direct product factor

C₂²×C₁₆

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₅₃

yes

characteristic

C₆₄

—

C₆₄

C₂×D₅₃

yes

complemented

C₅₃

C₂×M₁₂₈

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₄×D₅₃

yes

characteristic

C₃₂

—

C₃₂

—

C₃₂

C₄×D₅₃

yes

characteristic

C₃₂

C₂²×D₅₃

yes

characteristic

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₈×D₅₃

yes

characteristic

C₁₆

—

C₁₆

—

C₁₆

C₈×D₅₃

yes

characteristic

C₁₆

C₂×C₄×D₅₃

yes

characteristic

C₂³

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₁₆×D₅₃

yes

characteristic

105

C₈

—

106

C₈

—

107

C₈

C₁₆×D₅₃

yes

characteristic

108

C₈

C₂×C₈×D₅₃

yes

characteristic

109

C₂²

—

110

C₂²

—

111

C₂²

—

106

—

112

C₂²

C₃₂×D₅₃

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

C₃₂×D₅₃

yes

characteristic

116

C₄

C₂×C₁₆×D₅₃

yes

characteristic

117

C₂

—

118

C₂

—

119

C₂

—

106

—

120

C₂

C₂×C₃₂×D₅₃

yes

characteristic

	subgroup	G/H
center, Frattini	C₃₂	C₂²×D₅₃
commutator, socle	C₂×C₅₃	C₂²×C₃₂
Fitting	C₅₃×M₁₂₈	C₂
prime radical	C₂²×D₅₃	C₃₂
cosocle	C₃₂×C₅₃	C₂³
hypercenter	M₁₂₈	D₅₃
hypocenter	C₅₃	C₂×M₁₂₈
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₃₂ ◁ M₁₂₈

subgroup

G/H

center, Frattini

C₃₂

C₂²×D₅₃

commutator, socle

C₂×C₅₃

C₂²×C₃₂

Fitting

C₅₃×M₁₂₈

C₂

prime radical

C₂²×D₅₃

C₃₂

cosocle

C₃₂×C₅₃

C₂³

hypercenter

M₁₂₈

D₅₃

hypocenter

C₅₃

C₂×M₁₂₈

LCS

G ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₃₂ ◁ M₁₂₈

N	H ≅ G/N	type
2. #20272926	C₂	semidirect
3. #20272925	C₂	semidirect
4. C₆₄×D₅₃	C₂	semidirect
5. C₅₃×M₁₂₈	C₂	semidirect
12. #16810342	C₂²	semidirect
14. C₅₃×C₆₄	C₂²	semidirect
60. M₁₂₈	D₅₃	direct (N × H)
65. D₅₃	M₁₂₈	direct (N × H)
74. C₆₄	C₂×D₅₃	semidirect
75. C₅₃	C₂×M₁₂₈	semidirect

H ≅ G/N

type

2. #20272926

C₂

semidirect

3. #20272925

C₂

semidirect

4. C₆₄×D₅₃

C₂

semidirect

5. C₅₃×M₁₂₈

C₂

semidirect

12. #16810342

C₂²

semidirect

14. C₅₃×C₆₄

C₂²

semidirect

60. M₁₂₈

D₅₃

direct (N × H)

65. D₅₃

M₁₂₈

direct (N × H)

74. C₆₄

C₂×D₅₃

semidirect

75. C₅₃

C₂×M₁₂₈

semidirect

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

subgroup	index
C₂×C₃₂×D₅₃	2
#20272926	2
#20272925	2
C₆₄×D₅₃	2
C₅₃×M₁₂₈	2
C₂×M₁₂₈	53

subgroup

index

C₂×C₃₂×D₅₃

#20272926

#20272925

C₆₄×D₅₃

C₅₃×M₁₂₈

C₂×M₁₂₈

Subgroups — D₅₃×M₁₂₈ #27305127

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D53×M128 #27305127

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — D₅₃×M₁₂₈ #27305127