Group #20274326 — subgroups

Subgroups — #20274326

Order 6784.

178 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 178 conjugacy classes of subgroups — too many to draw. Only its 57 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	#16810975	C₂	1	1	yes	maximal, characteristic, complemented
2	#16810974	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810973	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810969	C₂	1	1	yes	maximal, characteristic
5	#16810969	C₂	1	1	yes	maximal, characteristic
6	C₈×#88322	C₂	1	1	yes	maximal, characteristic, complemented
7	C₅₃×#486	C₂	1	1	yes	maximal, characteristic
8	#13133217	C₂²	1	1	yes	characteristic
9	#13133217	C₂²	1	1	yes	characteristic
10	#13133142	—	2	2	no	—
11	#13133141	—	2	2	no	—
12	#13133140	—	2	2	no	—
13	#13133138	—	2	2	no	—
14	#13133138	—	2	2	no	—
15	C₄×#88322	C₂²	1	1	yes	characteristic
16	C₄×#88322	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
18	#127×C₅₃	C₂²	1	1	yes	characteristic
19	#102×C₅₃	—	2	2	no	—
20	C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂×Q₈×C₅₃	—	2	1	no	—
22	C₂×D₄×C₅₃	—	2	1	no	—
23	C₂×D₄×C₅₃	—	2	1	no	—
24	C₂×D₄×C₅₃	—	2	1	no	—
25	C₂×Dic₁₀₆	—	2	1	no	—
26	C₂×Dic₁₀₆	—	2	1	no	—
27	C₂×C₈×C₅₃	D₄	1	2	yes	—
28	C₄⋊C₄×C₅₃	—	2	1	no	—
29	C₄⋊C₄×C₅₃	—	2	1	no	—
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	C₂×#88322	D₄	1	2	yes	—
36	C₂×#88322	D₄	1	2	yes	—
37	C₂³×C₅₃	—	4	1	no	—
38	Q₈×C₅₃	—	4	2	no	—
39	D₄×C₅₃	—	4	2	no	—
40	D₄×C₅₃	—	4	2	no	—
41	D₄×C₅₃	—	4	2	no	—
42	C₂×C₄×C₅₃	—	4	1	no	—
43	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
46	C₂×Dic₅₃	—	4	1	no	—
47	C₂×Dic₅₃	—	4	1	no	—
48	Dic₁₀₆	—	4	2	no	—
49	Dic₁₀₆	—	4	2	no	—
50	C₈×C₅₃	—	2	2	no	—
51	#88322	—	2	2	no	—
52	#88322	—	2	2	no	—
53	C₂²×C₅₃	—	8	1	no	—
54	C₂²×C₅₃	—	8	1	no	—
55	C₂²×C₅₃	—	8	1	no	—
56	C₂²×C₅₃	#127	1	1	yes	characteristic
57	C₄×C₅₃	—	8	1	no	—
58	C₄×C₅₃	#135	1	2	yes	—
59	C₄×C₅₃	#135	1	2	yes	—
60	C₄×C₅₃	#135	1	2	yes	—
61	Dic₅₃	—	8	1	no	—
62	Dic₅₃	—	8	1	no	—
63	#1711	—	53	1	no	maximal, Sylow
64	C₂×C₅₃	—	16	1	no	—
65	C₂×C₅₃	#495	1	1	yes	characteristic
66	C₂×C₅₃	#495	1	1	yes	characteristic
67	C₂×C₅₃	#495	1	1	yes	characteristic
68	#487	—	53	1	no	—
69	#487	—	53	1	no	—
70	#487	—	53	1	no	—
71	#486	—	53	1	no	—
72	#486	—	53	1	no	—
73	#486	D₅₃	1	1	yes	characteristic
74	C₈²	—	53	1	no	—
75	C₅₃	#1711	1	1	yes	characteristic, complemented, Sylow
76	C₄⋊Q₈	—	53	1	no	—
77	C₄⋊Q₈	—	53	1	no	—
78	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
79	#127	C₂×D₅₃	1	1	yes	characteristic
80	#103	—	106	2	no	—
81	#103	—	106	2	no	—
82	#103	—	106	2	no	—
83	#102	—	2	2	no	—
84	#102	—	106	2	no	—
85	#102	—	106	2	no	—
86	C₄×C₈	—	53	1	no	—
87	C₄×C₈	—	53	1	no	—
88	C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
89	C₂×Q₈	—	2	1	no	—
90	C₂×Q₈	—	106	1	no	—
91	C₂×Q₈	—	106	1	no	—
92	C₂×D₄	—	2	1	no	—
93	C₂×D₄	—	2	1	no	—
94	C₂×D₄	—	2	1	no	—
95	C₂×C₈	—	53	2	no	—
96	C₂×C₈	—	53	2	no	—
97	C₂×C₈	C₂²⋊D₅₃	1	2	yes	—
98	C₄⋊C₄	—	2	1	no	—
99	C₄⋊C₄	—	2	1	no	—
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	C₄²	C₂²×D₅₃	1	1	yes	characteristic
105	C₂³	—	4	1	no	—
106	Q₈	—	4	2	no	—
107	Q₈	—	212	2	no	—
108	Q₈	—	212	2	no	—
109	D₄	—	4	2	no	—
110	D₄	—	4	2	no	—
111	D₄	—	4	2	no	—
112	C₂×C₄	—	4	1	no	—
113	C₂×C₄	—	212	1	no	—
114	C₂×C₄	—	212	1	no	—
115	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
116	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
117	C₂×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
118	C₈	—	2	2	no	—
119	C₈	—	106	2	no	—
120	C₈	—	106	2	no	—
121	C₂²	—	8	1	no	—
122	C₂²	—	8	1	no	—
123	C₂²	—	8	1	no	—
124	C₂²	#13133213	1	1	yes	characteristic
125	C₄	—	8	1	no	—
126	C₄	—	424	1	no	—
127	C₄	—	424	1	no	—
128	C₄	#13133236	1	2	yes	—
129	C₄	#13133237	1	2	yes	—
130	C₄	#13133238	1	2	yes	—
131	C₂	—	16	1	no	—
132	C₂	#16811004	1	1	yes	characteristic
133	C₂	#16811005	1	1	yes	characteristic
134	C₂	#16811006	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16810975

C₂

yes

maximal, characteristic, complemented

#16810974

C₂

yes

maximal, characteristic, complemented

#16810973

C₂

yes

maximal, characteristic, complemented

#16810969

C₂

yes

maximal, characteristic

#16810969

C₂

yes

maximal, characteristic

C₈×#88322

C₂

yes

maximal, characteristic, complemented

C₅₃×#486

C₂

yes

maximal, characteristic

#13133217

C₂²

yes

characteristic

#13133217

C₂²

yes

—

—

—

—

—

C₂²

yes

characteristic

C₄×#88322

C₂²

yes

characteristic

C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

#127×C₅₃

C₂²

yes

characteristic

#102×C₅₃

—

C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×Q₈×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

D₄

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

D₄

yes

—

C₂×#88322

D₄

yes

—

C₂³×C₅₃

—

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

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

—

Dic₁₀₆

—

Dic₁₀₆

—

C₈×C₅₃

—

#88322

—

#88322

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#127

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

#135

yes

—

C₄×C₅₃

#135

yes

—

C₄×C₅₃

#135

yes

—

Dic₅₃

—

Dic₅₃

—

#1711

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

—

—

—

—

—

D₅₃

yes

characteristic

C₈²

—

C₅₃

#1711

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₅₃

yes

characteristic

#127

C₂×D₅₃

yes

characteristic

#103

—

106

—

#103

—

106

—

#103

—

106

—

#102

—

#102

—

106

—

#102

—

106

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₅₃

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

106

—

C₂×Q₈

—

106

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₅₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

106

—

101

C₄⋊C₄

—

106

—

102

C₄⋊C₄

—

106

—

103

C₄⋊C₄

—

106

—

104

C₄²

C₂²×D₅₃

yes

characteristic

105

C₂³

—

106

Q₈

—

107

Q₈

—

212

—

108

Q₈

—

212

—

109

D₄

—

110

D₄

—

111

D₄

—

112

C₂×C₄

—

113

C₂×C₄

—

212

—

114

C₂×C₄

—

212

—

115

C₂×C₄

D₄×D₅₃

yes

characteristic

116

C₂×C₄

D₄×D₅₃

yes

characteristic

117

C₂×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

118

C₈

—

119

C₈

—

106

—

120

C₈

—

106

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

#13133213

yes

characteristic

125

C₄

—

126

C₄

—

424

—

127

C₄

—

424

—

128

C₄

#13133236

yes

—

129

C₄

#13133237

yes

—

130

C₄

#13133238

yes

—

131

C₂

—

132

C₂

#16811004

yes

characteristic

133

C₂

#16811005

yes

characteristic

134

C₂

#16811006

yes

characteristic

	subgroup	G/H
center	C₂²	#13133213
commutator	C₄²×C₅₃	C₂³
Frattini	C₄²	C₂²×D₅₃
Fitting	C₅₃×#486	C₂
socle	C₂²×C₅₃	#127
hypercenter	#486	D₅₃
hypocenter	C₅₃	#1711
LCS	G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #486

subgroup

G/H

center

C₂²

#13133213

commutator

C₄²×C₅₃

C₂³

Frattini

C₄²

C₂²×D₅₃

Fitting

C₅₃×#486

C₂

C₂²×C₅₃

D₅₃

C₅₃

G ▷ C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #486

N	H ≅ G/N	type
1. #16810975	C₂	semidirect
2. #16810974	C₂	semidirect
3. #16810973	C₂	semidirect
6. C₈×#88322	C₂	semidirect
75. C₅₃	#1711	semidirect

H ≅ G/N

type

1. #16810975

C₂

semidirect

2. #16810974

C₂

semidirect

3. #16810973

C₂

semidirect

6. C₈×#88322

C₂

semidirect

75. C₅₃

#1711

semidirect

Subgroups — #20274326

Conjugacy classes of subgroups

Complements