Group #16811240 — subgroups

Subgroups — #16811240

Order 3392.

216 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 95 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	#13133210	C₂	1	2	yes	maximal, complemented
3	#13133204	C₂	1	1	yes	maximal, characteristic, complemented
4	#13133204	C₂	1	2	yes	maximal, complemented
5	#13133198	C₂	1	2	yes	maximal, complemented
6	#13133196	C₂	1	1	yes	maximal, characteristic, complemented
7	#13133194	C₂	1	2	yes	maximal, complemented
8	#13133182	C₂	1	1	yes	maximal, characteristic, complemented
9	D₄×Dic₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#13133176	C₂	1	1	yes	maximal, characteristic, complemented
11	#121×C₅₃	C₂	1	1	yes	maximal, characteristic
12	C₂×D₄×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×D₄×C₅₃	C₂²	1	2	yes	—
14	C₂²×C₄×C₅₃	C₂²	1	1	yes	characteristic
15	C₂²×Dic₅₃	C₂²	1	2	yes	—
16	C₂×Dic₁₀₆	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₀₆	C₂²	1	2	yes	—
18	C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
19	C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
20	C₂²⋊Dic₅₃	C₂²	1	1	yes	characteristic
21	C₂²⋊Dic₅₃	C₂²	1	1	yes	characteristic
22	C₂²⋊Dic₅₃	C₂²	1	2	yes	—
23	C₂²⋊Dic₅₃	C₂²	1	2	yes	—
24	C₂²⋊Dic₅₃	C₂²	1	2	yes	—
25	C₂²⋊Dic₅₃	C₂²	1	2	yes	—
26	C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
27	C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
28	C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
29	#11719919	C₂²	1	2	yes	—
30	#11719919	C₂²	1	2	yes	—
31	#11719919	C₂²	1	2	yes	—
32	C₄×Dic₅₃	C₂²	1	1	yes	characteristic
33	C₄×Dic₅₃	C₂²	1	1	yes	characteristic
34	C₄×Dic₅₃	C₂²	1	2	yes	—
35	C₂³×C₅₃	C₂³	1	2	yes	—
36	C₂³×C₅₃	C₂³	1	1	yes	characteristic
37	D₄×C₅₃	—	2	2	no	—
38	D₄×C₅₃	—	4	2	no	—
39	C₂×C₄×C₅₃	—	2	2	no	—
40	C₂×C₄×C₅₃	C₂³	1	2	yes	—
41	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
42	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
43	C₂×Dic₅₃	—	4	2	no	—
44	C₂×Dic₅₃	C₂³	1	1	yes	characteristic
45	C₂×Dic₅₃	C₂³	1	2	yes	—
46	C₂×Dic₅₃	C₂³	1	2	yes	—
47	C₂×Dic₅₃	C₂³	1	1	yes	characteristic
48	C₂×Dic₅₃	C₂³	1	1	yes	characteristic
49	C₂×Dic₅₃	C₂³	1	1	yes	characteristic
50	Dic₁₀₆	—	2	2	no	—
51	Dic₁₀₆	—	4	2	no	—
52	C₂²×C₅₃	—	2	1	no	—
53	C₂²×C₅₃	—	2	2	no	—
54	C₂²×C₅₃	—	2	2	no	—
55	C₂²×C₅₃	—	2	1	no	—
56	C₂²×C₅₃	—	2	2	no	—
57	C₂²×C₅₃	—	2	1	no	—
58	C₂²×C₅₃	C₂⁴	1	1	yes	characteristic
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	—	2	1	no	—
67	Dic₅₃	—	2	1	no	—
68	C₂×C₅₃	—	4	1	no	—
69	C₂×C₅₃	—	4	2	no	—
70	C₂×C₅₃	C₂×#41	1	1	yes	characteristic
71	C₂×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
72	C₂×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
73	#542	—	53	1	no	maximal, Sylow
74	C₅₃	#542	1	1	yes	characteristic, complemented, Sylow
75	C₄⋊Q₈	—	53	1	no	—
76	#126	—	53	2	no	—
77	#124	—	53	1	no	—
78	#124	—	53	2	no	—
79	#123	—	53	2	no	—
80	C₂²⋊Q₈	—	53	1	no	—
81	C₂²⋊Q₈	—	53	2	no	—
82	#121	D₅₃	1	1	yes	characteristic
83	C₄×Q₈	—	53	1	no	—
84	C₄×D₄	—	53	1	no	—
85	#117	—	53	1	no	—
86	C₂×Q₈	—	53	1	no	—
87	C₂×Q₈	—	53	2	no	—
88	C₂×D₄	C₂×D₅₃	1	1	yes	characteristic
89	C₂×D₄	C₂×D₅₃	1	2	yes	—
90	C₂²×C₄	—	53	2	no	—
91	C₂²×C₄	C₂×D₅₃	1	1	yes	characteristic
92	C₄⋊C₄	—	53	1	no	—
93	C₄⋊C₄	—	53	1	no	—
94	C₄⋊C₄	—	53	1	no	—
95	C₄⋊C₄	—	53	2	no	—
96	C₄⋊C₄	—	53	2	no	—
97	C₄⋊C₄	—	53	2	no	—
98	C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
99	C₂²⋊C₄	—	53	1	no	—
100	C₂²⋊C₄	—	53	1	no	—
101	C₂²⋊C₄	—	53	2	no	—
102	C₂²⋊C₄	—	53	2	no	—
103	C₂²⋊C₄	—	53	2	no	—
104	C₂²⋊C₄	—	53	2	no	—
105	C₂²⋊C₄	C₂×D₅₃	1	2	yes	—
106	C₄²	—	53	1	no	—
107	C₄²	—	53	1	no	—
108	C₄²	—	53	2	no	—
109	C₂³	C₂²×D₅₃	1	1	yes	characteristic
110	C₂³	C₂²×D₅₃	1	2	yes	—
111	Q₈	—	106	2	no	—
112	Q₈	—	212	2	no	—
113	D₄	—	2	2	no	—
114	D₄	—	4	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	53	1	no	—
117	C₂×C₄	—	53	1	no	—
118	C₂×C₄	—	53	2	no	—
119	C₂×C₄	—	53	1	no	—
120	C₂×C₄	—	53	1	no	—
121	C₂×C₄	—	53	2	no	—
122	C₂×C₄	—	212	2	no	—
123	C₂×C₄	C₂²×D₅₃	1	2	yes	—
124	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
125	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
126	C₂²	—	2	1	no	—
127	C₂²	—	2	2	no	—
128	C₂²	—	2	2	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	—	2	2	no	—
132	C₂²	C₂³×D₅₃	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	2	2	no	—
135	C₄	—	106	2	no	—
136	C₄	—	106	1	no	—
137	C₄	—	106	2	no	—
138	C₄	—	106	1	no	—
139	C₄	—	106	1	no	—
140	C₄	—	106	1	no	—
141	C₄	#11719947	1	2	yes	—
142	C₂	—	4	1	no	—
143	C₂	—	4	2	no	—
144	C₂	C₂×#11719947	1	1	yes	characteristic
145	C₂	#13133262	1	1	yes	characteristic
146	C₂	#13133264	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13133216

C₂

yes

maximal, characteristic, complemented

#13133210

C₂

yes

maximal, complemented

#13133204

C₂

yes

maximal, characteristic, complemented

#13133204

C₂

yes

maximal, complemented

#13133198

C₂

yes

maximal, complemented

#13133196

C₂

yes

maximal, characteristic, complemented

#13133194

C₂

yes

maximal, complemented

#13133182

C₂

yes

maximal, characteristic, complemented

D₄×Dic₅₃

C₂

yes

maximal, characteristic, complemented

#13133176

C₂

yes

maximal, characteristic, complemented

#121×C₅₃

C₂

yes

maximal, characteristic

C₂×D₄×C₅₃

C₂²

yes

characteristic

C₂×D₄×C₅₃

C₂²

yes

—

C₂²×C₄×C₅₃

C₂²

yes

characteristic

C₂²×Dic₅₃

C₂²

yes

—

C₂×Dic₁₀₆

C₂²

yes

characteristic

C₂×Dic₁₀₆

C₂²

yes

—

C₄⋊C₄×C₅₃

C₂²

yes

characteristic

C₂²⋊C₄×C₅₃

C₂²

yes

—

C₂²⋊Dic₅₃

C₂²

yes

characteristic

C₂²⋊Dic₅₃

C₂²

yes

characteristic

C₂²⋊Dic₅₃

C₂²

yes

—

C₂²⋊Dic₅₃

C₂²

yes

—

C₂²⋊Dic₅₃

C₂²

yes

—

C₂²⋊Dic₅₃

C₂²

yes

—

C₄⋊Dic₅₃

C₂²

yes

characteristic

C₄⋊Dic₅₃

C₂²

yes

characteristic

C₄⋊Dic₅₃

C₂²

yes

characteristic

#11719919

C₂²

yes

—

#11719919

C₂²

yes

—

#11719919

C₂²

yes

—

C₄×Dic₅₃

C₂²

yes

characteristic

C₄×Dic₅₃

C₂²

yes

characteristic

C₄×Dic₅₃

C₂²

yes

—

C₂³×C₅₃

C₂³

yes

—

C₂³×C₅₃

C₂³

yes

characteristic

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂³

yes

—

C₂×C₄×C₅₃

C₂³

yes

characteristic

C₂×C₄×C₅₃

C₂³

yes

characteristic

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂³

yes

characteristic

C₂×Dic₅₃

C₂³

yes

—

C₂×Dic₅₃

C₂³

yes

—

C₂×Dic₅₃

C₂³

yes

characteristic

C₂×Dic₅₃

C₂³

yes

characteristic

C₂×Dic₅₃

C₂³

yes

characteristic

Dic₁₀₆

—

Dic₁₀₆

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂⁴

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#41

yes

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

C₂×#41

yes

characteristic

C₂×C₅₃

H(5,𝔽₂)

yes

characteristic

C₂×C₅₃

2₋¹⁺⁴

yes

—

C₅₃

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

—

—

—

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

#121

D₅₃

yes

characteristic

C₄×Q₈

—

C₄×D₄

—

#117

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

C₂×D₅₃

yes

characteristic

C₂×D₄

C₂×D₅₃

yes

—

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

C₂²⋊C₄

—

100

C₂²⋊C₄

—

101

C₂²⋊C₄

—

102

C₂²⋊C₄

—

103

C₂²⋊C₄

—

104

C₂²⋊C₄

—

105

C₂²⋊C₄

C₂×D₅₃

yes

—

106

C₄²

—

107

C₄²

—

108

C₄²

—

109

C₂³

C₂²×D₅₃

yes

characteristic

110

C₂³

C₂²×D₅₃

yes

—

111

Q₈

—

106

—

112

Q₈

—

212

—

113

D₄

—

114

D₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

212

—

123

C₂×C₄

C₂²×D₅₃

yes

—

124

C₂×C₄

C₂²×D₅₃

yes

characteristic

125

C₂×C₄

C₂²×D₅₃

yes

characteristic

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂³×D₅₃

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

106

—

136

C₄

—

106

—

137

C₄

—

106

—

138

C₄

—

106

—

139

C₄

—

106

—

140

C₄

—

106

—

141

C₄

#11719947

yes

—

142

C₂

—

143

C₂

—

144

C₂

C₂×#11719947

yes

characteristic

145

C₂

#13133262

yes

characteristic

146

C₂

#13133264

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅₃
commutator, socle, cosocle	C₂²×C₅₃	C₂⁴
Fitting, prime radical	#121×C₅₃	C₂
hypercenter	#121	D₅₃
hypocenter	C₅₃	#542
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #121

subgroup

G/H

center, Frattini

C₂²

C₂³×D₅₃

commutator, socle, cosocle

C₂²×C₅₃

C₂⁴

Fitting, prime radical

#121×C₅₃

C₂

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂² ◁ #121

N	H ≅ G/N	type
1. #13133216	C₂	semidirect
2. #13133210	C₂	semidirect
3. #13133204	C₂	semidirect
4. #13133204	C₂	semidirect
5. #13133198	C₂	semidirect
6. #13133196	C₂	semidirect
7. #13133194	C₂	semidirect
8. #13133182	C₂	semidirect
9. D₄×Dic₅₃	C₂	semidirect
10. #13133176	C₂	semidirect
74. C₅₃	#542	semidirect

H ≅ G/N

type

1. #13133216

C₂

semidirect

2. #13133210

C₂

semidirect

3. #13133204

C₂

semidirect

4. #13133204

C₂

semidirect

5. #13133198

C₂

semidirect

6. #13133196

C₂

semidirect

7. #13133194

C₂

semidirect

8. #13133182

C₂

semidirect

9. D₄×Dic₅₃

C₂

semidirect

10. #13133176

C₂

semidirect

74. C₅₃

#542

semidirect

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

subgroup	index
#13133216	2
#13133210	2
#13133204	2
#13133198	2
#13133196	2
#13133194	2
#13133182	2
D₄×Dic₅₃	2
#13133176	2
#121×C₅₃	2
#542	53

subgroup

index

D₄×Dic₅₃

#121×C₅₃

Subgroups — #16811240

Conjugacy classes of subgroups

Complements

Maximal subgroups