Group #20275518 — subgroups

Subgroups — #20275518

Order 6784.

332 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 332 conjugacy classes of subgroups — too many to draw. Only its 145 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×#13133141	C₂	1	4	yes	maximal
2	C₂×C₄×#88322	C₂	1	1	yes	maximal, characteristic
3	#16810493	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#88322	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₅₃	—	2	2	no	—
7	C₂×C₄⋊Dic₅₃	C₂²	1	2	yes	—
8	#13133141	—	2	8	no	—
9	C₄×#88322	—	2	2	no	—
10	C₂²×Q₈×C₅₃	C₂²	1	2	yes	—
11	C₄⋊Q₈×C₅₃	—	2	2	no	—
12	C₄⋊Q₈×C₅₃	C₄	1	4	yes	complemented
13	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
14	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×Q₈×C₅₃	—	2	2	no	—
16	C₂×Q₈×C₅₃	—	2	4	no	—
17	C₂×Q₈×C₅₃	—	2	4	no	—
18	C₂×Q₈×C₅₃	C₂×C₄	1	4	yes	—
19	C₂²×C₄×C₅₃	—	2	2	no	—
20	C₂²×C₄×C₅₃	D₄	1	2	yes	—
21	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
22	C₂²×Dic₅₃	—	2	2	no	—
23	C₄⋊C₄×C₅₃	—	2	8	no	—
24	C₄²×C₅₃	C₂×C₄	1	2	yes	—
25	C₄²×C₅₃	D₄	1	2	yes	—
26	C₄⋊Dic₅₃	—	2	4	no	—
27	C₄⋊Dic₅₃	—	4	4	no	—
28	C₂×#88322	—	2	2	no	—
29	C₂×#88322	—	2	2	no	—
30	C₂×#88322	D₄	1	4	yes	—
31	C₂³×C₅₃	C₂×D₄	1	1	yes	characteristic
32	Q₈×C₅₃	—	2	8	no	—
33	Q₈×C₅₃	—	4	4	no	—
34	C₂×C₄×C₅₃	—	2	4	no	—
35	C₂×C₄×C₅₃	—	4	2	no	—
36	C₂×C₄×C₅₃	—	4	2	no	—
37	C₂×C₄×C₅₃	C₂²⋊C₄	1	4	yes	—
38	C₂×C₄×C₅₃	SD₁₆	1	4	yes	—
39	C₂×C₄×C₅₃	Q₁₆	1	4	yes	—
40	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
42	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₅₃	#41	1	2	yes	—
44	C₂×Dic₅₃	—	4	2	no	—
45	C₂×Dic₅₃	—	4	4	no	—
46	#88322	—	2	4	no	—
47	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
48	C₂²×C₅₃	#124	1	1	yes	characteristic
49	C₂²×C₅₃	#127	1	1	yes	characteristic
50	C₂²×C₅₃	C₂×SD₁₆	1	2	yes	—
51	C₂²×C₅₃	C₂×Q₁₆	1	2	yes	—
52	C₄×C₅₃	—	4	4	no	—
53	C₄×C₅₃	#103	1	8	yes	complemented
54	C₄×C₅₃	C₄×D₄	1	2	yes	—
55	C₄×C₅₃	#121	1	2	yes	—
56	Dic₅₃	—	8	2	no	—
57	#1956	—	53	1	no	maximal, Sylow
58	C₂×C₅₃	#390	1	1	yes	characteristic
59	C₂×C₅₃	C₂×#103	1	2	yes	—
60	C₂×C₅₃	#487	1	2	yes	—
61	C₂×C₅₃	#492	1	1	yes	characteristic
62	C₂×C₅₃	#494	1	1	yes	characteristic
63	C₂×C₄⋊Q₈	D₅₃	1	1	yes	characteristic
64	C₂×#103	—	53	4	no	—
65	C₂×C₄×C₈	—	53	1	no	—
66	#384	—	53	1	no	—
67	C₅₃	#1956	1	1	yes	characteristic, complemented, Sylow
68	C₂²×Q₈	C₂×D₅₃	1	2	yes	—
69	C₂²×C₈	—	53	2	no	—
70	C₄⋊Q₈	—	2	2	no	—
71	C₄⋊Q₈	Dic₅₃	1	4	yes	complemented
72	C₂×C₄⋊C₄	—	2	2	no	—
73	C₂×C₄⋊C₄	—	53	2	no	—
74	C₂×C₄⋊C₄	—	106	2	no	—
75	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
76	#103	—	106	8	no	—
77	C₄×C₈	—	106	2	no	—
78	C₂×Q₈	—	2	2	no	—
79	C₂×Q₈	—	2	4	no	—
80	C₂×Q₈	—	2	4	no	—
81	C₂×Q₈	C₂×Dic₅₃	1	4	yes	—
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	—	106	2	no	—
84	C₂²×C₄	C₂²⋊D₅₃	1	2	yes	—
85	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
86	C₂×C₈	—	53	4	no	—
87	C₂×C₈	—	106	2	no	—
88	C₂×C₈	—	106	2	no	—
89	C₄⋊C₄	—	2	8	no	—
90	C₄⋊C₄	—	106	4	no	—
91	C₄⋊C₄	—	212	4	no	—
92	C₄²	C₂×Dic₅₃	1	2	yes	—
93	C₄²	C₂²⋊D₅₃	1	2	yes	—
94	C₂³	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
95	Q₈	—	2	8	no	—
96	Q₈	—	4	4	no	—
97	C₂×C₄	—	2	4	no	—
98	C₂×C₄	—	4	2	no	—
99	C₂×C₄	—	4	2	no	—
100	C₂×C₄	—	212	2	no	—
101	C₂×C₄	—	212	4	no	—
102	C₂×C₄	#11719924	1	4	yes	—
103	C₂×C₄	#11719925	1	4	yes	—
104	C₂×C₄	C₂²⋊Dic₅₃	1	4	yes	—
105	C₂×C₄	D₄×D₅₃	1	2	yes	—
106	C₂×C₄	#11719947	1	2	yes	—
107	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
108	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
109	C₈	—	106	4	no	—
110	C₂²	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
111	C₂²	#13133204	1	1	yes	characteristic
112	C₂²	#13133213	1	1	yes	characteristic
113	C₂²	C₂×#11719924	1	2	yes	—
114	C₂²	C₂×#11719925	1	2	yes	—
115	C₄	—	4	4	no	—
116	C₄	—	424	2	no	—
117	C₄	#13133141	1	8	yes	complemented
118	C₄	D₄×Dic₅₃	1	2	yes	—
119	C₄	#13133188	1	2	yes	—
120	C₂	#16810526	1	1	yes	characteristic
121	C₂	C₂×#13133141	1	2	yes	—
122	C₂	#16810973	1	2	yes	—
123	C₂	#16810997	1	1	yes	characteristic
124	C₂	#16811002	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13133141

C₂

yes

maximal

C₂×C₄×#88322

C₂

yes

maximal, characteristic

#16810493

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₅₃

C₂

yes

maximal, characteristic

C₂²×#88322

C₂²

yes

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

C₂²

yes

—

#13133141

—

C₄×#88322

—

C₂²×Q₈×C₅₃

C₂²

yes

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

C₄

yes

complemented

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

D₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂²×Dic₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

D₄

yes

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

D₄

yes

—

C₂³×C₅₃

C₂×D₄

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

SD₁₆

yes

—

C₂×C₄×C₅₃

Q₁₆

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

#88322

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

C₂²×C₅₃

yes

C₂²×C₅₃

yes

C₂²×C₅₃

C₂×SD₁₆

yes

—

C₂²×C₅₃

C₂×Q₁₆

yes

—

C₄×C₅₃

—

C₄×C₅₃

#103

yes

complemented

C₄×C₅₃

C₄×D₄

yes

—

C₄×C₅₃

#121

yes

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

—

C₂×C₅₃

#487

yes

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₄⋊Q₈

D₅₃

yes

characteristic

C₂×#103

—

C₂×C₄×C₈

—

#384

—

C₅₃

#1956

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₅₃

yes

—

C₂²×C₈

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₅₃

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

106

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

#103

—

106

—

C₄×C₈

—

106

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₅₃

yes

—

C₂²×C₄

—

C₂²×C₄

—

106

—

C₂²×C₄

C₂²⋊D₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₄⋊C₄

—

C₄⋊C₄

—

106

—

C₄⋊C₄

—

212

—

C₄²

C₂×Dic₅₃

yes

—

C₄²

C₂²⋊D₅₃

yes

—

C₂³

C₂×C₂²⋊D₅₃

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

212

—

101

C₂×C₄

—

212

—

102

C₂×C₄

#11719924

yes

—

103

C₂×C₄

#11719925

yes

—

104

C₂×C₄

C₂²⋊Dic₅₃

yes

—

105

C₂×C₄

D₄×D₅₃

yes

—

106

C₂×C₄

#11719947

yes

—

107

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

108

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

109

C₈

—

106

—

110

C₂²

C₂×C₂²⋊Dic₅₃

yes

characteristic

111

C₂²

#13133204

yes

characteristic

112

C₂²

#13133213

yes

characteristic

113

C₂²

C₂×#11719924

yes

—

114

C₂²

C₂×#11719925

yes

—

115

C₄

—

116

C₄

—

424

—

117

C₄

#13133141

yes

complemented

118

C₄

D₄×Dic₅₃

yes

—

119

C₄

#13133188

yes

—

120

C₂

#16810526

yes

characteristic

121

C₂

C₂×#13133141

yes

—

122

C₂

#16810973

yes

—

123

C₂

#16810997

yes

characteristic

124

C₂

#16811002

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×C₄⋊Q₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₅₃
hypocenter	C₅₃	#1956
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₅₃

commutator

C₂×C₄×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂×C₄⋊Q₈×C₅₃

C₂

socle

C₂³×C₅₃

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₅₃

hypocenter

C₅₃

#1956

LCS

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
12. C₄⋊Q₈×C₅₃	C₄	semidirect
53. C₄×C₅₃	#103	semidirect
67. C₅₃	#1956	semidirect
71. C₄⋊Q₈	Dic₅₃	semidirect
117. C₄	#13133141	semidirect

H ≅ G/N

type

12. C₄⋊Q₈×C₅₃

C₄

semidirect

53. C₄×C₅₃

#103

semidirect

67. C₅₃

#1956

semidirect

71. C₄⋊Q₈

Dic₅₃

semidirect

117. C₄

#13133141

semidirect

Subgroups — #20275518

Conjugacy classes of subgroups

Complements