Group #20274368 — subgroups

Subgroups — #20274368

Order 6784.

186 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 186 conjugacy classes of subgroups — too many to draw. Only its 61 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	#16811028	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810996	C₂	1	1	yes	maximal, characteristic
3	#16810974	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810974	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810970	C₂	1	1	yes	maximal, characteristic
6	#16810970	C₂	1	1	yes	maximal, characteristic
7	C₅₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#11719913	—	2	2	no	—
9	#13133217	C₂²	1	1	yes	characteristic
10	#13133217	C₂²	1	1	yes	characteristic
11	#13133217	C₂²	1	1	yes	characteristic
12	#13133214	C₂²	1	1	yes	characteristic
13	#13133151	—	2	2	no	—
14	#13133142	—	2	2	no	—
15	#13133142	—	2	2	no	—
16	#13133139	—	2	2	no	—
17	#13133139	—	2	2	no	—
18	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
19	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×D₂₁₂	—	2	1	no	—
22	C₂×D₂₁₂	—	2	1	no	—
23	C₂×D₂₁₂	—	2	1	no	—
24	C₂×Dic₁₀₆	—	2	1	no	—
25	C₂×Dic₁₀₆	—	2	1	no	—
26	C₂×Dic₁₀₆	—	2	1	no	—
27	C₂×C₈×C₅₃	D₄	1	2	yes	—
28	C₂×C₈×C₅₃	D₄	1	2	yes	—
29	C₂×C₈×C₅₃	D₄	1	2	yes	—
30	C₄²×C₅₃	C₂³	1	1	yes	characteristic
31	C₄⋊Dic₅₃	—	2	1	no	—
32	C₄⋊Dic₅₃	—	2	1	no	—
33	C₄⋊Dic₅₃	—	2	1	no	—
34	C₄⋊Dic₅₃	—	2	1	no	—
35	C₄⋊Dic₅₃	—	2	1	no	—
36	C₄⋊Dic₅₃	—	2	1	no	—
37	#11719913	—	4	4	no	—
38	C₂²×D₅₃	—	4	1	no	—
39	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂×Dic₅₃	—	4	1	no	—
43	C₂×Dic₅₃	—	4	1	no	—
44	C₂×Dic₅₃	—	4	1	no	—
45	D₂₁₂	—	4	2	no	—
46	D₂₁₂	—	4	2	no	—
47	D₂₁₂	—	4	2	no	—
48	Dic₁₀₆	—	4	2	no	—
49	Dic₁₀₆	—	4	2	no	—
50	Dic₁₀₆	—	4	2	no	—
51	C₈×C₅₃	—	2	2	no	—
52	C₈×C₅₃	—	2	2	no	—
53	C₈×C₅₃	—	2	2	no	—
54	C₂²×C₅₃	#127	1	1	yes	characteristic
55	C₂×D₅₃	—	8	1	no	—
56	C₂×D₅₃	—	8	1	no	—
57	C₂×D₅₃	—	8	1	no	—
58	C₄×C₅₃	#135	1	2	yes	—
59	C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
60	C₄×C₅₃	#137	1	2	yes	—
61	Dic₅₃	—	8	1	no	—
62	Dic₅₃	—	8	1	no	—
63	Dic₅₃	—	8	1	no	—
64	#1719	—	53	1	no	maximal, Sylow
65	C₂×C₅₃	#495	1	1	yes	characteristic
66	C₂×C₅₃	#496	1	1	yes	characteristic
67	C₂×C₅₃	#497	1	1	yes	characteristic
68	D₅₃	—	16	1	no	—
69	#500	—	53	1	no	—
70	#492	—	53	1	no	—
71	#487	—	53	1	no	—
72	#487	—	53	1	no	—
73	#486	—	53	1	no	—
74	#486	—	53	1	no	—
75	#322	D₅₃	1	1	yes	characteristic, complemented
76	C₅₃	#1719	1	1	yes	characteristic, complemented, Sylow
77	C₂×SD₁₆	—	106	2	no	—
78	C₄⋊Q₈	—	53	1	no	—
79	C₄⋊Q₈	—	53	1	no	—
80	C₄⋊Q₈	—	53	1	no	—
81	#127	—	53	1	no	—
82	#107	—	106	2	no	—
83	#103	—	106	2	no	—
84	#103	—	106	2	no	—
85	#102	—	106	2	no	—
86	#102	—	106	2	no	—
87	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
88	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
89	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
90	C₂×Q₈	—	106	1	no	—
91	C₂×Q₈	—	106	1	no	—
92	C₂×Q₈	—	106	1	no	—
93	C₂×D₄	—	106	1	no	—
94	C₂×D₄	—	106	1	no	—
95	C₂×D₄	—	106	1	no	—
96	SD₁₆	—	212	4	no	—
97	C₂×C₈	D₂₁₂	1	2	yes	—
98	C₂×C₈	D₂₁₂	1	2	yes	—
99	C₂×C₈	D₂₁₂	1	2	yes	—
100	C₄⋊C₄	—	106	1	no	—
101	C₄⋊C₄	—	106	1	no	—
102	C₄⋊C₄	—	106	1	no	—
103	C₄⋊C₄	—	106	1	no	—
104	C₄⋊C₄	—	106	1	no	—
105	C₄⋊C₄	—	106	1	no	—
106	C₄²	C₂²×D₅₃	1	1	yes	characteristic
107	C₂³	—	212	1	no	—
108	Q₈	—	212	2	no	—
109	Q₈	—	212	2	no	—
110	Q₈	—	212	2	no	—
111	D₄	—	212	2	no	—
112	D₄	—	212	2	no	—
113	D₄	—	212	2	no	—
114	C₂×C₄	—	212	1	no	—
115	C₂×C₄	—	212	1	no	—
116	C₂×C₄	—	212	1	no	—
117	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
118	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
119	C₂×C₄	C₂×D₂₁₂	1	1	yes	characteristic
120	C₈	—	2	2	no	—
121	C₈	—	2	2	no	—
122	C₈	—	2	2	no	—
123	C₂²	—	424	1	no	—
124	C₂²	—	424	1	no	—
125	C₂²	—	424	1	no	—
126	C₂²	#13133214	1	1	yes	characteristic
127	C₄	—	424	1	no	—
128	C₄	—	424	1	no	—
129	C₄	—	424	1	no	—
130	C₄	#13133239	1	2	yes	—
131	C₄	#13133245	1	2	yes	—
132	C₄	#13133250	1	2	yes	—
133	C₂	—	848	1	no	—
134	C₂	#16811007	1	1	yes	characteristic
135	C₂	#16811012	1	1	yes	characteristic
136	C₂	#16811017	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811028

C₂

yes

maximal, characteristic, complemented

#16810996

C₂

yes

maximal, characteristic

#16810974

C₂

yes

maximal, characteristic, complemented

#16810974

C₂

yes

maximal, characteristic, complemented

#16810970

C₂

yes

maximal, characteristic

#16810970

C₂

yes

maximal, characteristic

C₅₃×#322

C₂

yes

maximal, characteristic, complemented

C₂×#11719913

—

#13133217

C₂²

yes

characteristic

#13133217

C₂²

yes

characteristic

#13133217

C₂²

yes

characteristic

#13133214

C₂²

yes

—

—

—

—

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×D₂₁₂

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

D₄

yes

—

C₂×C₈×C₅₃

D₄

yes

—

C₂×C₈×C₅₃

D₄

yes

—

C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

#11719913

—

C₂²×D₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

D₂₁₂

—

D₂₁₂

—

D₂₁₂

—

Dic₁₀₆

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

C₈×C₅₃

—

C₈×C₅₃

—

C₂²×C₅₃

#127

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

#135

yes

—

C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₄×C₅₃

#137

yes

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

—

—

—

—

—

—

D₅₃

yes

characteristic, complemented

C₅₃

#1719

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

106

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

#127

—

#107

—

106

—

#103

—

106

—

#103

—

106

—

#102

—

106

—

#102

—

106

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₄×C₈

C₂×D₅₃

yes

characteristic

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂×Q₈

—

106

—

C₂×Q₈

—

106

—

C₂×Q₈

—

106

—

C₂×D₄

—

106

—

C₂×D₄

—

106

—

C₂×D₄

—

106

—

SD₁₆

—

212

—

C₂×C₈

D₂₁₂

yes

—

C₂×C₈

D₂₁₂

yes

—

C₂×C₈

D₂₁₂

yes

—

100

C₄⋊C₄

—

106

—

101

C₄⋊C₄

—

106

—

102

C₄⋊C₄

—

106

—

103

C₄⋊C₄

—

106

—

104

C₄⋊C₄

—

106

—

105

C₄⋊C₄

—

106

—

106

C₄²

C₂²×D₅₃

yes

characteristic

107

C₂³

—

212

—

108

Q₈

—

212

—

109

Q₈

—

212

—

110

Q₈

—

212

—

111

D₄

—

212

—

112

D₄

—

212

—

113

D₄

—

212

—

114

C₂×C₄

—

212

—

115

C₂×C₄

—

212

—

116

C₂×C₄

—

212

—

117

C₂×C₄

C₂×D₂₁₂

yes

characteristic

118

C₂×C₄

C₂×D₂₁₂

yes

characteristic

119

C₂×C₄

C₂×D₂₁₂

yes

characteristic

120

C₈

—

121

C₈

—

122

C₈

—

123

C₂²

—

424

—

124

C₂²

—

424

—

125

C₂²

—

424

—

126

C₂²

#13133214

yes

characteristic

127

C₄

—

424

—

128

C₄

—

424

—

129

C₄

—

424

—

130

C₄

#13133239

yes

—

131

C₄

#13133245

yes

—

132

C₄

#13133250

yes

—

133

C₂

—

848

—

134

C₂

#16811007

yes

characteristic

135

C₂

#16811012

yes

characteristic

136

C₂

#16811017

yes

characteristic

	subgroup	G/H
center	C₂²	#13133214
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#322	C₂
socle	C₂²×C₅₃	#127
hypercenter	#322	D₅₃
hypocenter	C₅₃	#1719
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #322

subgroup

G/H

center

C₂²

#13133214

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#322

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #322

N	H ≅ G/N	type
1. #16811028	C₂	semidirect
3. #16810974	C₂	semidirect
4. #16810974	C₂	semidirect
7. C₅₃×#322	C₂	semidirect
75. #322	D₅₃	semidirect
76. C₅₃	#1719	semidirect

H ≅ G/N

type

1. #16811028

C₂

semidirect

3. #16810974

C₂

semidirect

4. #16810974

C₂

semidirect

7. C₅₃×#322

C₂

semidirect

75. #322

D₅₃

semidirect

76. C₅₃

#1719

semidirect

Subgroups — #20274368

Conjugacy classes of subgroups

Complements