Group #20276269 — subgroups

Subgroups — #20276269

Order 6784.

210 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 67 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	#16810672	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810650	C₂	1	2	yes	maximal, complemented
3	#16810641	C₂	1	2	yes	maximal
4	#16810591	C₂	1	1	yes	maximal, characteristic
5	C₅₃×H(3,ℤ/4)	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11719945	C₂²	1	1	yes	characteristic
7	C₂×#11719917	C₂²	1	2	yes	—
8	C₂²×#88322	C₂²	1	1	yes	characteristic
9	#13133153	—	2	2	no	—
10	C₄⋊#88322	—	2	4	no	—
11	#13133145	—	2	4	no	—
12	#13133129	—	2	2	no	—
13	C₂×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	#117×C₅₃	C₂²	1	2	yes	—
15	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
16	C₂×C₂²⋊D₅₃	—	2	1	no	—
17	#11719945	—	2	2	no	—
18	#11719945	—	2	2	no	—
19	C₂×D₂₁₂	Q₈	1	1	yes	characteristic
20	C₂×C₄×D₅₃	—	2	1	no	—
21	C₂×Dic₁₀₆	Q₈	1	1	yes	characteristic
22	M₁₆×C₅₃	—	2	2	no	—
23	C₂×C₈×C₅₃	D₄	1	2	yes	—
24	C₄⋊C₄×C₅₃	—	2	2	no	—
25	C₂²⋊C₄×C₅₃	—	2	2	no	—
26	C₄²×C₅₃	D₄	1	4	yes	—
27	#11719917	—	2	2	no	—
28	#11719917	—	2	2	no	—
29	C₂×#88322	—	2	1	no	—
30	C₂×#88322	—	2	2	no	—
31	C₂×#88322	—	2	2	no	—
32	C₂³×C₅₃	#41	1	1	yes	characteristic
33	C₂²×D₅₃	—	2	1	no	—
34	C₂×C₄×C₅₃	—	2	4	no	—
35	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
36	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×C₅₃	#41	1	2	yes	—
39	C₂²⋊D₅₃	—	4	2	no	—
40	C₂×Dic₅₃	—	2	1	no	—
41	D₂₁₂	—	2	2	no	—
42	C₄×D₅₃	—	4	1	no	—
43	C₄×D₅₃	—	4	1	no	—
44	Dic₁₀₆	—	2	2	no	—
45	C₈×C₅₃	—	2	2	no	—
46	#88322	—	4	1	no	—
47	#88322	—	4	2	no	—
48	C₂²×C₅₃	—	4	1	no	—
49	C₂²×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₅₃	#123	1	2	yes	—
51	C₂×D₅₃	—	4	2	no	—
52	C₂×D₅₃	—	4	1	no	—
53	C₄×C₅₃	—	4	4	no	—
54	C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
55	C₄×C₅₃	#123	1	1	yes	characteristic
56	C₄×C₅₃	#127	1	1	yes	characteristic
57	Dic₅₃	—	4	1	no	—
58	#2098	—	53	1	no	maximal, Sylow
59	C₂×C₅₃	—	2	2	no	—
60	C₂×C₅₃	—	2	1	no	—
61	C₂×C₅₃	#399	1	1	yes	characteristic
62	D₅₃	—	8	1	no	—
63	#430	—	53	1	no	—
64	#424	—	53	2	no	—
65	#421	—	53	2	no	—
66	#408	—	53	1	no	—
67	H(3,ℤ/4)	D₅₃	1	1	yes	characteristic, complemented
68	C₅₃	#2098	1	1	yes	characteristic, complemented, Sylow
69	C₂×#41	—	53	1	no	—
70	C₂×M₁₆	—	53	2	no	—
71	C₂×M₁₆	C₂×D₅₃	1	1	yes	characteristic
72	C₂²×C₈	—	53	1	no	—
73	#117	C₂×D₅₃	1	2	yes	—
74	#108	—	106	2	no	—
75	C₄⋊C₈	—	106	4	no	—
76	#104	—	106	4	no	—
77	C₂²⋊C₈	—	106	2	no	—
78	#41	—	106	2	no	—
79	#41	—	106	2	no	—
80	C₂×Q₈	—	53	1	no	—
81	C₂×D₄	—	53	1	no	—
82	C₂×D₄	—	106	1	no	—
83	C₂²×C₄	—	106	1	no	—
84	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
85	M₁₆	—	2	2	no	—
86	M₁₆	—	106	2	no	—
87	M₁₆	—	106	2	no	—
88	C₂×C₈	—	106	1	no	—
89	C₂×C₈	—	106	2	no	—
90	C₂×C₈	—	106	2	no	—
91	C₂×C₈	D₂₁₂	1	2	yes	—
92	C₄⋊C₄	—	2	2	no	—
93	C₂²⋊C₄	—	2	2	no	—
94	C₄²	D₂₁₂	1	4	yes	—
95	C₂³	—	106	1	no	—
96	C₂³	#11719947	1	1	yes	characteristic
97	Q₈	—	106	2	no	—
98	D₄	—	106	2	no	—
99	D₄	—	212	2	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	106	1	no	—
102	C₂×C₄	—	212	1	no	—
103	C₂×C₄	—	212	1	no	—
104	C₂×C₄	C₂×D₂₁₂	1	2	yes	—
105	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
106	C₂×C₄	#11719947	1	2	yes	—
107	C₂×C₄	Q₈×D₅₃	1	1	yes	characteristic
108	C₈	—	2	2	no	—
109	C₈	—	212	1	no	—
110	C₈	—	212	2	no	—
111	C₂²	—	4	1	no	—
112	C₂²	—	212	2	no	—
113	C₂²	—	212	1	no	—
114	C₂²	#13133197	1	1	yes	characteristic
115	C₂²	#13133201	1	2	yes	—
116	C₄	—	4	4	no	—
117	C₄	—	212	1	no	—
118	C₄	#13133197	1	2	yes	—
119	C₄	#13133201	1	1	yes	characteristic
120	C₄	#13133214	1	1	yes	characteristic
121	C₂	—	2	2	no	—
122	C₂	—	2	1	no	—
123	C₂	—	424	1	no	—
124	C₂	#16810560	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810672

C₂

yes

maximal, characteristic, complemented

#16810650

C₂

yes

maximal, complemented

#16810641

C₂

yes

maximal

#16810591

C₂

yes

maximal, characteristic

C₅₃×H(3,ℤ/4)

C₂

yes

maximal, characteristic, complemented

C₂×#11719945

C₂²

yes

characteristic

C₂×#11719917

C₂²

yes

—

C₂²×#88322

C₂²

yes

—

—

—

—

C₂×M₁₆×C₅₃

C₂²

yes

characteristic

#117×C₅₃

C₂²

yes

—

C₂²×C₄×C₅₃

C₂³

yes

characteristic

C₂×C₂²⋊D₅₃

—

#11719945

—

#11719945

—

C₂×D₂₁₂

Q₈

yes

characteristic

C₂×C₄×D₅₃

—

C₂×Dic₁₀₆

Q₈

yes

characteristic

M₁₆×C₅₃

—

C₂×C₈×C₅₃

D₄

yes

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

D₄

yes

—

—

—

—

—

—

C₂³×C₅₃

#41

yes

characteristic

C₂²×D₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₅₃

#41

yes

—

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

C₄×D₅₃

—

C₄×D₅₃

—

Dic₁₀₆

—

C₈×C₅₃

—

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂²⋊Q₈

yes

characteristic

C₂²×C₅₃

#123

yes

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₂²⋊Q₈

yes

—

C₄×C₅₃

yes

C₄×C₅₃

yes

Dic₅₃

—

#2098

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

#399

yes

characteristic

D₅₃

—

—

—

—

—

D₅₃

yes

characteristic, complemented

C₅₃

#2098

yes

characteristic, complemented, Sylow

C₂×#41

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₅₃

yes

characteristic

C₂²×C₈

—

#117

C₂×D₅₃

yes

—

#108

—

106

—

C₄⋊C₈

—

106

—

#104

—

106

—

C₂²⋊C₈

—

106

—

#41

—

106

—

#41

—

106

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

106

—

C₂²×C₄

—

106

—

C₂²×C₄

C₂²×D₅₃

yes

characteristic

M₁₆

—

M₁₆

—

106

—

M₁₆

—

106

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

—

106

—

C₂×C₈

D₂₁₂

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

D₂₁₂

yes

—

C₂³

—

106

—

C₂³

#11719947

yes

characteristic

Q₈

—

106

—

D₄

—

106

—

D₄

—

212

—

100

C₂×C₄

—

101

C₂×C₄

—

106

—

102

C₂×C₄

—

212

—

103

C₂×C₄

—

212

—

104

C₂×C₄

C₂×D₂₁₂

yes

—

105

C₂×C₄

C₂×D₂₁₂

yes

characteristic

106

C₂×C₄

#11719947

yes

—

107

C₂×C₄

Q₈×D₅₃

yes

characteristic

108

C₈

—

109

C₈

—

212

—

110

C₈

—

212

—

111

C₂²

—

112

C₂²

—

212

—

113

C₂²

—

212

—

114

C₂²

#13133197

yes

characteristic

115

C₂²

#13133201

yes

—

116

C₄

—

117

C₄

—

212

—

118

C₄

#13133197

yes

—

119

C₄

#13133201

yes

characteristic

120

C₄

#13133214

yes

characteristic

121

C₂

—

122

C₂

—

123

C₂

—

424

—

124

C₂

#16810560

yes

characteristic

	subgroup	G/H
center	C₄	#13133214
commutator	C₂²×C₄×C₅₃	C₂³
Frattini	C₂²×C₄	C₂²×D₅₃
Fitting	C₅₃×H(3,ℤ/4)	C₂
socle	C₂×C₅₃	#399
hypercenter	H(3,ℤ/4)	D₅₃
hypocenter	C₅₃	#2098
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#13133214

commutator

C₂²×C₄×C₅₃

C₂³

Frattini

C₂²×C₄

C₂²×D₅₃

Fitting

C₅₃×H(3,ℤ/4)

C₂

C₂×C₅₃

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
1. #16810672	C₂	semidirect
2. #16810650	C₂	semidirect
5. C₅₃×H(3,ℤ/4)	C₂	semidirect
67. H(3,ℤ/4)	D₅₃	semidirect
68. C₅₃	#2098	semidirect

H ≅ G/N

type

1. #16810672

C₂

semidirect

2. #16810650

C₂

semidirect

5. C₅₃×H(3,ℤ/4)

C₂

semidirect

67. H(3,ℤ/4)

D₅₃

semidirect

68. C₅₃

#2098

semidirect

Subgroups — #20276269

Conjugacy classes of subgroups

Complements