Group #20275560 — subgroups

Subgroups — #20275560

Order 6784.

292 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 105 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	#16810647	C₂	1	1	yes	maximal, characteristic
2	C₂×#13133136	C₂	1	4	yes	maximal
3	C₄×#11719917	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#11719917	C₂²	1	2	yes	—
6	C₂×#11719917	C₂²	1	2	yes	—
7	C₄⋊#88322	—	2	2	no	—
8	#13133136	—	2	8	no	—
9	#13133128	—	2	1	no	—
10	C₄×#88322	—	2	1	no	—
11	C₂²×Q₈×C₅₃	C₂²	1	2	yes	—
12	C₄⋊Q₈×C₅₃	—	2	1	no	—
13	C₄⋊Q₈×C₅₃	—	2	1	no	—
14	C₄⋊Q₈×C₅₃	—	2	2	no	—
15	C₂×C₄⋊C₄×C₅₃	C₄	1	4	yes	—
16	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₅₃	—	2	2	no	—
18	C₂×Q₈×C₅₃	—	2	2	no	—
19	C₂×Q₈×C₅₃	—	2	4	no	—
20	C₂×Q₈×C₅₃	—	2	4	no	—
21	C₂²×C₄×C₅₃	C₂×C₄	1	4	yes	—
22	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
24	C₄⋊C₄×C₅₃	—	2	4	no	—
25	C₄⋊C₄×C₅₃	—	2	4	no	—
26	C₄²×C₅₃	D₄	1	1	yes	characteristic
27	C₄²×C₅₃	D₄	1	1	yes	characteristic
28	C₄²×C₅₃	D₄	1	2	yes	—
29	#11719917	—	2	2	no	—
30	#11719917	—	2	4	no	—
31	#11719917	D₄	1	4	yes	—
32	C₂×#88322	—	2	2	no	—
33	C₂×#88322	—	2	2	no	—
34	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
35	Q₈×C₅₃	—	4	4	no	—
36	Q₈×C₅₃	—	4	2	no	—
37	Q₈×C₅₃	—	4	2	no	—
38	C₂×C₄×C₅₃	—	2	4	no	—
39	C₂×C₄×C₅₃	—	2	4	no	—
40	C₂×C₄×C₅₃	—	2	8	no	—
41	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
42	C₂×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₅₃	#41	1	2	yes	—
47	#88322	—	2	4	no	—
48	#88322	—	4	2	no	—
49	C₂²×C₅₃	—	2	2	no	—
50	C₂²×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₅₃	C₄×D₄	1	2	yes	—
52	C₄×C₅₃	—	2	2	no	—
53	C₄×C₅₃	—	4	4	no	—
54	C₄×C₅₃	#101	1	4	yes	complemented
55	C₄×C₅₃	#121	1	2	yes	—
56	C₄×C₅₃	#124	1	1	yes	characteristic
57	C₄×C₅₃	#127	1	1	yes	characteristic
58	#1966	—	53	1	no	maximal, Sylow
59	C₂×C₅₃	—	2	2	no	—
60	C₂×C₅₃	#390	1	1	yes	characteristic
61	C₂×C₅₃	C₂×#101	1	2	yes	—
62	C₂×C₄⋊Q₈	D₅₃	1	1	yes	characteristic
63	#423	—	53	1	no	—
64	C₂×#101	—	53	4	no	—
65	C₄×M₁₆	—	53	1	no	—
66	C₅₃	#1966	1	1	yes	characteristic, complemented, Sylow
67	C₂²×Q₈	C₂×D₅₃	1	2	yes	—
68	C₂×M₁₆	—	53	2	no	—
69	C₂×M₁₆	—	53	2	no	—
70	C₄⋊Q₈	—	2	1	no	—
71	C₄⋊Q₈	—	2	1	no	—
72	C₄⋊Q₈	—	2	2	no	—
73	C₂×C₄⋊C₄	Dic₅₃	1	4	yes	—
74	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
75	C₄⋊C₈	—	106	2	no	—
76	#101	—	106	8	no	—
77	#97	—	106	1	no	—
78	C₄×C₈	—	106	1	no	—
79	C₂×Q₈	—	2	4	no	—
80	C₂×Q₈	—	2	4	no	—
81	C₂×Q₈	—	2	2	no	—
82	C₂×Q₈	—	2	2	no	—
83	C₂²×C₄	C₂×Dic₅₃	1	4	yes	—
84	C₂²×C₄	C₂×Dic₅₃	1	2	yes	—
85	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
86	M₁₆	—	53	4	no	—
87	M₁₆	—	106	2	no	—
88	M₁₆	—	106	4	no	—
89	C₂×C₈	—	106	2	no	—
90	C₂×C₈	—	106	2	no	—
91	C₄⋊C₄	—	2	4	no	—
92	C₄⋊C₄	—	2	4	no	—
93	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
94	C₄²	C₂²⋊D₅₃	1	1	yes	characteristic
95	C₄²	C₂²⋊D₅₃	1	2	yes	—
96	C₂³	C₂²×Dic₅₃	1	1	yes	characteristic
97	Q₈	—	4	4	no	—
98	Q₈	—	4	2	no	—
99	Q₈	—	4	2	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	2	8	no	—
103	C₂×C₄	C₂²⋊Dic₅₃	1	2	yes	—
104	C₂×C₄	C₂²⋊Dic₅₃	1	2	yes	—
105	C₂×C₄	D₄×D₅₃	1	2	yes	—
106	C₂×C₄	#11719947	1	2	yes	—
107	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
108	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
109	C₈	—	106	4	no	—
110	C₈	—	212	2	no	—
111	C₂²	—	2	2	no	—
112	C₂²	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
113	C₂²	D₄×Dic₅₃	1	2	yes	—
114	C₄	—	2	2	no	—
115	C₄	—	4	4	no	—
116	C₄	#13133136	1	4	yes	complemented
117	C₄	#13133188	1	2	yes	—
118	C₄	#13133204	1	1	yes	characteristic
119	C₄	#13133213	1	1	yes	characteristic
120	C₂	—	2	2	no	—
121	C₂	#16810526	1	1	yes	characteristic
122	C₂	C₂×#13133136	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16810647

C₂

yes

maximal, characteristic

C₂×#13133136

C₂

yes

maximal

C₄×#11719917

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₅₃

C₂

yes

maximal, characteristic

C₂×#11719917

C₂²

yes

—

C₂×#11719917

C₂²

yes

—

—

—

—

—

C₂²×Q₈×C₅₃

C₂²

yes

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₄

yes

—

C₂×C₄²×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

D₄

yes

characteristic

C₄²×C₅₃

D₄

yes

—

#11719917

—

#11719917

—

#11719917

D₄

yes

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₅₃

C₄×D₄

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#101

yes

complemented

C₄×C₅₃

#121

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

—

C₂×C₅₃

—

C₂×C₅₃

#390

yes

characteristic

C₂×C₅₃

C₂×#101

yes

—

C₂×C₄⋊Q₈

D₅₃

yes

characteristic

#423

—

C₂×#101

—

C₄×M₁₆

—

C₅₃

#1966

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₅₃

yes

—

C₂×M₁₆

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

Dic₅₃

yes

—

C₂×C₄²

C₂×D₅₃

yes

characteristic

C₄⋊C₈

—

106

—

#101

—

106

—

#97

—

106

—

C₄×C₈

—

106

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂×Dic₅₃

yes

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

M₁₆

—

106

—

M₁₆

—

106

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²⋊D₅₃

yes

characteristic

C₄²

C₂²⋊D₅₃

yes

characteristic

C₄²

C₂²⋊D₅₃

yes

—

C₂³

C₂²×Dic₅₃

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

C₂²⋊Dic₅₃

yes

—

104

C₂×C₄

C₂²⋊Dic₅₃

yes

—

105

C₂×C₄

D₄×D₅₃

yes

—

106

C₂×C₄

#11719947

yes

—

107

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

108

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

109

C₈

—

106

—

110

C₈

—

212

—

111

C₂²

—

112

C₂²

C₂×C₂²⋊Dic₅₃

yes

characteristic

113

C₂²

D₄×Dic₅₃

yes

—

114

C₄

—

115

C₄

—

116

C₄

#13133136

yes

complemented

117

C₄

#13133188

yes

—

118

C₄

#13133204

yes

characteristic

119

C₄

#13133213

yes

characteristic

120

C₂

—

121

C₂

#16810526

yes

characteristic

122

C₂

C₂×#13133136

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₅₃
commutator	C₂³×C₅₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₂×C₄⋊Q₈×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₂²⋊C₄
hypercenter	C₂×C₄⋊Q₈	D₅₃
hypocenter	C₅₃	#1966
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₅₃

commutator

C₂³×C₅₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₂×C₄⋊Q₈×C₅₃

C₂

socle

C₂²×C₅₃

C₂×C₂²⋊C₄

hypercenter

C₂×C₄⋊Q₈

D₅₃

hypocenter

C₅₃

#1966

LCS

G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
54. C₄×C₅₃	#101	semidirect
66. C₅₃	#1966	semidirect
116. C₄	#13133136	semidirect

H ≅ G/N

type

54. C₄×C₅₃

#101

semidirect

66. C₅₃

#1966

semidirect

116. C₄

#13133136

semidirect

Subgroups — #20275560

Conjugacy classes of subgroups

Complements