Group #16811344 — subgroups

Subgroups — #16811344

Order 3392.

212 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 97 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	#13133216	C₂	1	1	yes	maximal, characteristic, complemented
2	#13133212	C₂	1	2	yes	maximal, complemented
3	#13133210	C₂	1	2	yes	maximal, complemented
4	#13133203	C₂	1	2	yes	maximal
5	#13133193	C₂	1	1	yes	maximal, characteristic, complemented
6	Q₈×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
7	#13133182	C₂	1	2	yes	maximal, complemented
8	C₄×C₂²⋊D₅₃	C₂	1	1	yes	maximal, characteristic
9	#13133176	C₂	1	1	yes	maximal, characteristic, complemented
10	#13133175	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂²⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×C₄×C₅₃	C₂²	1	1	yes	characteristic
14	C₂×C₂²⋊D₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×D₅₃	C₂²	1	1	yes	characteristic
16	C₂×Dic₁₀₆	C₂²	1	2	yes	—
17	C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
19	C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
20	C₂²⋊Dic₅₃	C₂²	1	2	yes	—
21	C₂²⋊Dic₅₃	C₂²	1	1	yes	characteristic
22	#11719921	C₂²	1	2	yes	—
23	#11719921	C₂²	1	1	yes	characteristic
24	#11719921	C₂²	1	2	yes	—
25	C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
26	C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
27	C₄⋊Dic₅₃	C₂²	1	2	yes	—
28	#11719919	C₂²	1	1	yes	characteristic
29	#11719919	C₂²	1	2	yes	—
30	#11719919	C₂²	1	2	yes	—
31	C₄×Dic₅₃	C₂²	1	2	yes	—
32	C₄×Dic₅₃	C₂²	1	2	yes	—
33	C₄×Dic₅₃	C₂²	1	1	yes	characteristic
34	C₄×Dic₅₃	C₂²	1	2	yes	—
35	C₂³×C₅₃	C₂³	1	1	yes	characteristic
36	C₂²×D₅₃	C₂³	1	1	yes	characteristic
37	Q₈×C₅₃	—	2	2	no	—
38	C₂×C₄×C₅₃	—	2	2	no	—
39	C₂×C₄×C₅₃	C₂³	1	2	yes	—
40	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
41	C₂×C₄×C₅₃	C₂³	1	2	yes	—
42	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
43	C₂²⋊D₅₃	—	2	2	no	—
44	C₂×Dic₅₃	C₂³	1	2	yes	—
45	C₂×Dic₅₃	C₂³	1	1	yes	characteristic
46	C₂×Dic₅₃	C₂³	1	2	yes	—
47	C₂×Dic₅₃	C₂³	1	1	yes	characteristic
48	C₂×Dic₅₃	C₂³	1	1	yes	characteristic
49	C₄×D₅₃	—	2	2	no	—
50	Dic₁₀₆	—	2	4	no	—
51	C₂²×C₅₃	—	2	1	no	—
52	C₂²×C₅₃	—	2	1	no	—
53	C₂²×C₅₃	—	2	1	no	—
54	C₂²×C₅₃	C₂⁴	1	1	yes	characteristic
55	C₂×D₅₃	—	2	1	no	—
56	C₂×D₅₃	—	2	1	no	—
57	C₂×D₅₃	—	2	1	no	—
58	C₄×C₅₃	—	2	2	no	—
59	C₄×C₅₃	—	2	1	no	—
60	C₄×C₅₃	—	2	2	no	—
61	C₄×C₅₃	#41	1	2	yes	—
62	Dic₅₃	—	2	2	no	—
63	Dic₅₃	—	2	1	no	—
64	Dic₅₃	—	2	2	no	—
65	Dic₅₃	—	2	1	no	—
66	Dic₅₃	#41	1	2	yes	—
67	C₂×C₅₃	—	4	1	no	—
68	C₂×C₅₃	C₂×#41	1	1	yes	characteristic
69	C₂×C₅₃	C₂×#41	1	1	yes	characteristic
70	C₂×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
71	D₅₃	—	4	1	no	—
72	#556	—	53	1	no	maximal, Sylow
73	C₅₃	#556	1	1	yes	characteristic, complemented, Sylow
74	C₄⋊Q₈	—	53	1	no	—
75	#126	—	53	2	no	—
76	#126	—	53	2	no	—
77	#124	—	53	2	no	—
78	C₂²⋊Q₈	—	53	1	no	—
79	C₂²⋊Q₈	D₅₃	1	1	yes	characteristic, complemented
80	C₄×Q₈	—	53	1	no	—
81	C₄×Q₈	—	53	2	no	—
82	C₄×D₄	—	53	1	no	—
83	#117	—	53	1	no	—
84	#117	—	53	1	no	—
85	C₂×Q₈	—	53	2	no	—
86	C₂×Q₈	C₂×D₅₃	1	1	yes	characteristic
87	C₂×D₄	—	53	1	no	—
88	C₂²×C₄	—	53	1	no	—
89	C₂²×C₄	C₂×D₅₃	1	1	yes	characteristic
90	C₄⋊C₄	—	53	1	no	—
91	C₄⋊C₄	—	53	1	no	—
92	C₄⋊C₄	—	53	1	no	—
93	C₄⋊C₄	—	53	2	no	—
94	C₄⋊C₄	—	53	2	no	—
95	C₄⋊C₄	—	53	2	no	—
96	C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
97	C₄⋊C₄	C₂×D₅₃	1	2	yes	—
98	C₂²⋊C₄	—	53	2	no	—
99	C₂²⋊C₄	—	53	1	no	—
100	C₂²⋊C₄	—	53	2	no	—
101	C₂²⋊C₄	—	53	1	no	—
102	C₂²⋊C₄	—	53	2	no	—
103	C₂²⋊C₄	C₂×D₅₃	1	2	yes	—
104	C₄²	—	53	2	no	—
105	C₄²	—	53	2	no	—
106	C₄²	—	53	1	no	—
107	C₄²	—	53	2	no	—
108	C₂³	—	53	1	no	—
109	C₂³	C₂²×D₅₃	1	1	yes	characteristic
110	Q₈	—	2	2	no	—
111	Q₈	—	106	4	no	—
112	D₄	—	106	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	—	53	1	no	—
115	C₂×C₄	—	53	2	no	—
116	C₂×C₄	—	53	1	no	—
117	C₂×C₄	—	53	1	no	—
118	C₂×C₄	—	53	2	no	—
119	C₂×C₄	—	106	2	no	—
120	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
121	C₂×C₄	C₂²×D₅₃	1	2	yes	—
122	C₂×C₄	C₂²×D₅₃	1	2	yes	—
123	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
124	C₂²	—	2	1	no	—
125	C₂²	—	2	1	no	—
126	C₂²	—	2	1	no	—
127	C₂²	—	106	1	no	—
128	C₂²	—	106	1	no	—
129	C₂²	—	106	1	no	—
130	C₂²	C₂³×D₅₃	1	1	yes	characteristic
131	C₄	—	2	2	no	—
132	C₄	—	2	1	no	—
133	C₄	—	2	2	no	—
134	C₄	—	53	2	no	—
135	C₄	—	106	2	no	—
136	C₄	—	106	1	no	—
137	C₄	—	106	2	no	—
138	C₄	—	106	1	no	—
139	C₄	#11719947	1	2	yes	—
140	C₂	—	4	1	no	—
141	C₂	—	212	1	no	—
142	C₂	#41×D₅₃	1	1	yes	characteristic
143	C₂	C₂×#11719947	1	1	yes	characteristic
144	C₂	#13133265	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13133216

C₂

yes

maximal, characteristic, complemented

#13133212

C₂

yes

maximal, complemented

#13133210

C₂

yes

maximal, complemented

#13133203

C₂

yes

maximal

#13133193

C₂

yes

maximal, characteristic, complemented

Q₈×Dic₅₃

C₂

yes

maximal, characteristic, complemented

#13133182

C₂

yes

maximal, complemented

C₄×C₂²⋊D₅₃

C₂

yes

maximal, characteristic

#13133176

C₂

yes

maximal, characteristic, complemented

#13133175

C₂

yes

maximal, characteristic, complemented

C₂²⋊Q₈×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×C₅₃

C₂²

yes

characteristic

C₂²×C₄×C₅₃

C₂²

yes

characteristic

C₂×C₂²⋊D₅₃

C₂²

yes

characteristic

C₂×C₄×D₅₃

C₂²

yes

characteristic

C₂×Dic₁₀₆

C₂²

yes

—

C₄⋊C₄×C₅₃

C₂²

yes

characteristic

C₄⋊C₄×C₅₃

C₂²

yes

—

C₂²⋊C₄×C₅₃

C₂²

yes

—

C₂²⋊Dic₅₃

C₂²

yes

—

C₂²⋊Dic₅₃

C₂²

yes

characteristic

#11719921

C₂²

yes

—

#11719921

C₂²

yes

characteristic

#11719921

C₂²

yes

—

C₄⋊Dic₅₃

C₂²

yes

characteristic

C₄⋊Dic₅₃

C₂²

yes

characteristic

C₄⋊Dic₅₃

C₂²

yes

—

#11719919

C₂²

yes

characteristic

#11719919

C₂²

yes

—

#11719919

C₂²

yes

—

C₄×Dic₅₃

C₂²

yes

—

C₄×Dic₅₃

C₂²

yes

—

C₄×Dic₅₃

C₂²

yes

characteristic

C₄×Dic₅₃

C₂²

yes

—

C₂³×C₅₃

C₂³

yes

characteristic

C₂²×D₅₃

C₂³

yes

characteristic

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂³

yes

—

C₂×C₄×C₅₃

C₂³

yes

characteristic

C₂×C₄×C₅₃

C₂³

yes

—

C₂×C₄×C₅₃

C₂³

yes

characteristic

C₂²⋊D₅₃

—

C₂×Dic₅₃

C₂³

yes

—

C₂×Dic₅₃

C₂³

yes

characteristic

C₂×Dic₅₃

C₂³

yes

—

C₂×Dic₅₃

C₂³

yes

characteristic

C₂×Dic₅₃

C₂³

yes

characteristic

C₄×D₅₃

—

Dic₁₀₆

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂⁴

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#41

yes

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

#41

yes

—

C₂×C₅₃

—

C₂×C₅₃

C₂×#41

yes

characteristic

C₂×C₅₃

C₂×#41

yes

characteristic

C₂×C₅₃

2₋¹⁺⁴

yes

characteristic

D₅₃

—

#556

—

maximal, Sylow

C₅₃

#556

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

#126

—

#126

—

#124

—

C₂²⋊Q₈

—

C₂²⋊Q₈

D₅₃

yes

characteristic, complemented

C₄×Q₈

—

C₄×Q₈

—

C₄×D₄

—

#117

—

#117

—

C₂×Q₈

—

C₂×Q₈

C₂×D₅₃

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₅₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₅₃

yes

characteristic

C₄⋊C₄

C₂×D₅₃

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

100

C₂²⋊C₄

—

101

C₂²⋊C₄

—

102

C₂²⋊C₄

—

103

C₂²⋊C₄

C₂×D₅₃

yes

—

104

C₄²

—

105

C₄²

—

106

C₄²

—

107

C₄²

—

108

C₂³

—

109

C₂³

C₂²×D₅₃

yes

characteristic

110

Q₈

—

111

Q₈

—

106

—

112

D₄

—

106

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

106

—

120

C₂×C₄

C₂²×D₅₃

yes

characteristic

121

C₂×C₄

C₂²×D₅₃

yes

—

122

C₂×C₄

C₂²×D₅₃

yes

—

123

C₂×C₄

C₂²×D₅₃

yes

characteristic

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

106

—

128

C₂²

—

106

—

129

C₂²

—

106

—

130

C₂²

C₂³×D₅₃

yes

characteristic

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

106

—

136

C₄

—

106

—

137

C₄

—

106

—

138

C₄

—

106

—

139

C₄

#11719947

yes

—

140

C₂

—

141

C₂

—

212

—

142

C₂

#41×D₅₃

yes

characteristic

143

C₂

C₂×#11719947

yes

characteristic

144

C₂

#13133265

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅₃
commutator, socle, cosocle	C₂²×C₅₃	C₂⁴
Fitting	C₂²⋊Q₈×C₅₃	C₂
prime radical	C₂×C₂²⋊D₅₃	C₂²
hypercenter	C₂²⋊Q₈	D₅₃
hypocenter	C₅₃	#556
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂²⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₅₃

commutator, socle, cosocle

C₂²×C₅₃

C₂⁴

Fitting

C₂²⋊Q₈×C₅₃

C₂

prime radical

C₂×C₂²⋊D₅₃

C₂²

hypercenter

C₂²⋊Q₈

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂² ◁ C₂²⋊Q₈

N	H ≅ G/N	type
1. #13133216	C₂	semidirect
2. #13133212	C₂	semidirect
3. #13133210	C₂	semidirect
5. #13133193	C₂	semidirect
6. Q₈×Dic₅₃	C₂	semidirect
7. #13133182	C₂	semidirect
9. #13133176	C₂	semidirect
10. #13133175	C₂	semidirect
11. C₂²⋊Q₈×C₅₃	C₂	semidirect
73. C₅₃	#556	semidirect
79. C₂²⋊Q₈	D₅₃	semidirect

H ≅ G/N

type

1. #13133216

C₂

semidirect

2. #13133212

C₂

semidirect

3. #13133210

C₂

semidirect

5. #13133193

C₂

semidirect

6. Q₈×Dic₅₃

C₂

semidirect

7. #13133182

C₂

semidirect

9. #13133176

C₂

semidirect

10. #13133175

C₂

semidirect

11. C₂²⋊Q₈×C₅₃

C₂

semidirect

73. C₅₃

#556

semidirect

79. C₂²⋊Q₈

D₅₃

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#13133216	2
#13133212	2
#13133210	2
#13133203	2
#13133193	2
Q₈×Dic₅₃	2
#13133182	2
C₄×C₂²⋊D₅₃	2
#13133176	2
#13133175	2
C₂²⋊Q₈×C₅₃	2
#556	53

subgroup

index

Q₈×Dic₅₃

C₄×C₂²⋊D₅₃

C₂²⋊Q₈×C₅₃

Subgroups — #16811344

Conjugacy classes of subgroups

Complements

Maximal subgroups