Group #16811345 — subgroups

Subgroups — #16811345

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	#13133215	C₂	1	1	yes	maximal, characteristic, complemented
2	#13133212	C₂	1	2	yes	maximal, complemented
3	#13133211	C₂	1	2	yes	maximal, complemented
4	#13133206	C₂	1	1	yes	maximal, characteristic
5	#13133205	C₂	1	1	yes	maximal, characteristic
6	#13133195	C₂	1	2	yes	maximal, complemented
7	C₄×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic, complemented
8	#13133182	C₂	1	1	yes	maximal, characteristic, complemented
9	#13133179	C₂	1	1	yes	maximal, characteristic
10	#13133174	C₂	1	2	yes	maximal, complemented
11	C₄×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×D₂₁₂	C₂²	1	1	yes	characteristic
14	C₂×C₄×D₅₃	C₂²	1	2	yes	—
15	C₂×Dic₁₀₆	C₂²	1	1	yes	characteristic
16	C₂×Dic₁₀₆	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
18	C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
19	C₄²×C₅₃	C₂²	1	2	yes	—
20	C₄²×C₅₃	C₂²	1	1	yes	characteristic
21	#11719921	C₂²	1	2	yes	—
22	#11719921	C₂²	1	2	yes	—
23	#11719921	C₂²	1	2	yes	—
24	#11719921	C₂²	1	2	yes	—
25	#11719921	C₂²	1	2	yes	—
26	C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
27	#11719919	C₂²	1	1	yes	characteristic
28	#11719919	C₂²	1	2	yes	—
29	#11719919	C₂²	1	2	yes	—
30	#11719919	C₂²	1	1	yes	characteristic
31	#11719919	C₂²	1	2	yes	—
32	C₄×Dic₅₃	C₂²	1	2	yes	—
33	C₄×Dic₅₃	C₂²	1	1	yes	characteristic
34	C₄×Dic₅₃	C₂²	1	1	yes	characteristic
35	C₂²×D₅₃	C₂³	1	2	yes	—
36	Q₈×C₅₃	—	2	2	no	—
37	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₅₃	C₂³	1	2	yes	—
39	C₂×C₄×C₅₃	C₂³	1	2	yes	—
40	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
41	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
42	C₂×Dic₅₃	C₂³	1	2	yes	—
43	C₂×Dic₅₃	C₂³	1	1	yes	characteristic
44	C₂×Dic₅₃	C₂³	1	1	yes	characteristic
45	C₂×Dic₅₃	C₂³	1	1	yes	characteristic
46	C₂×Dic₅₃	C₂³	1	1	yes	characteristic
47	D₂₁₂	—	2	2	no	—
48	C₄×D₅₃	—	2	4	no	—
49	Dic₁₀₆	—	2	2	no	—
50	Dic₁₀₆	—	2	2	no	—
51	C₂²×C₅₃	C₂⁴	1	1	yes	characteristic
52	C₂×D₅₃	—	2	2	no	—
53	C₂×D₅₃	—	2	2	no	—
54	C₂×D₅₃	—	2	2	no	—
55	C₄×C₅₃	—	2	1	no	—
56	C₄×C₅₃	—	2	2	no	—
57	C₄×C₅₃	—	2	2	no	—
58	C₄×C₅₃	—	2	1	no	—
59	C₄×C₅₃	#41	1	2	yes	—
60	Dic₅₃	—	2	1	no	—
61	Dic₅₃	—	2	2	no	—
62	Dic₅₃	—	2	1	no	—
63	Dic₅₃	—	2	1	no	—
64	Dic₅₃	#41	1	2	yes	—
65	C₂×C₅₃	C₂×#41	1	1	yes	characteristic
66	C₂×C₅₃	C₂×#41	1	1	yes	characteristic
67	C₂×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
68	D₅₃	—	4	2	no	—
69	#556	—	53	1	no	maximal, Sylow
70	C₅₃	#556	1	1	yes	characteristic, complemented, Sylow
71	C₄⋊Q₈	—	53	1	no	—
72	#126	—	53	2	no	—
73	#126	—	53	2	no	—
74	#124	—	53	1	no	—
75	#124	—	53	1	no	—
76	C₂²⋊Q₈	—	53	2	no	—
77	C₄×Q₈	—	53	1	no	—
78	C₄×Q₈	—	53	1	no	—
79	C₄×Q₈	D₅₃	1	1	yes	characteristic, complemented
80	C₄×D₄	—	53	1	no	—
81	#117	—	53	2	no	—
82	C₂×Q₈	—	53	1	no	—
83	C₂×Q₈	—	53	1	no	—
84	C₂×Q₈	C₂×D₅₃	1	1	yes	characteristic
85	C₂×D₄	—	53	1	no	—
86	C₂²×C₄	—	53	2	no	—
87	C₄⋊C₄	—	53	1	no	—
88	C₄⋊C₄	—	53	2	no	—
89	C₄⋊C₄	—	53	2	no	—
90	C₄⋊C₄	—	53	1	no	—
91	C₄⋊C₄	—	53	1	no	—
92	C₄⋊C₄	—	53	2	no	—
93	C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
94	C₄⋊C₄	C₂×D₅₃	1	2	yes	—
95	C₂²⋊C₄	—	53	2	no	—
96	C₂²⋊C₄	—	53	2	no	—
97	C₂²⋊C₄	—	53	2	no	—
98	C₂²⋊C₄	—	53	2	no	—
99	C₂²⋊C₄	—	53	2	no	—
100	C₄²	—	53	2	no	—
101	C₄²	—	53	1	no	—
102	C₄²	—	53	1	no	—
103	C₄²	C₂×D₅₃	1	2	yes	—
104	C₄²	C₂×D₅₃	1	1	yes	characteristic
105	C₂³	—	53	2	no	—
106	Q₈	—	2	2	no	—
107	Q₈	—	106	2	no	—
108	Q₈	—	106	2	no	—
109	D₄	—	106	2	no	—
110	C₂×C₄	—	53	1	no	—
111	C₂×C₄	—	53	1	no	—
112	C₂×C₄	—	53	1	no	—
113	C₂×C₄	—	53	1	no	—
114	C₂×C₄	—	53	2	no	—
115	C₂×C₄	—	106	4	no	—
116	C₂×C₄	C₂²×D₅₃	1	2	yes	—
117	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
118	C₂×C₄	C₂²×D₅₃	1	2	yes	—
119	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
120	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
121	C₂²	—	106	2	no	—
122	C₂²	—	106	2	no	—
123	C₂²	—	106	2	no	—
124	C₂²	C₂³×D₅₃	1	1	yes	characteristic
125	C₄	—	2	2	no	—
126	C₄	—	2	1	no	—
127	C₄	—	2	2	no	—
128	C₄	—	2	1	no	—
129	C₄	—	53	2	no	—
130	C₄	—	106	1	no	—
131	C₄	—	106	2	no	—
132	C₄	—	106	1	no	—
133	C₄	—	106	1	no	—
134	C₄	#11719945	1	2	yes	—
135	C₂	—	212	2	no	—
136	C₂	#41×D₅₃	1	1	yes	characteristic
137	C₂	C₂×#11719945	1	1	yes	characteristic
138	C₂	#13133265	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13133215

C₂

yes

maximal, characteristic, complemented

#13133212

C₂

yes

maximal, complemented

#13133211

C₂

yes

maximal, complemented

#13133206

C₂

yes

maximal, characteristic

#13133205

C₂

yes

maximal, characteristic

#13133195

C₂

yes

maximal, complemented

C₄×Dic₁₀₆

C₂

yes

maximal, characteristic, complemented

#13133182

C₂

yes

maximal, characteristic, complemented

#13133179

C₂

yes

maximal, characteristic

#13133174

C₂

yes

maximal, complemented

C₄×Q₈×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×C₅₃

C₂²

yes

characteristic

C₂×D₂₁₂

C₂²

yes

characteristic

C₂×C₄×D₅₃

C₂²

yes

—

C₂×Dic₁₀₆

C₂²

yes

characteristic

C₂×Dic₁₀₆

C₂²

yes

characteristic

C₄⋊C₄×C₅₃

C₂²

yes

characteristic

C₄⋊C₄×C₅₃

C₂²

yes

—

C₄²×C₅₃

C₂²

yes

—

C₄²×C₅₃

C₂²

yes

characteristic

#11719921

C₂²

yes

—

#11719921

C₂²

yes

—

#11719921

C₂²

yes

—

#11719921

C₂²

yes

—

#11719921

C₂²

yes

—

C₄⋊Dic₅₃

C₂²

yes

characteristic

#11719919

C₂²

yes

characteristic

#11719919

C₂²

yes

—

#11719919

C₂²

yes

—

#11719919

C₂²

yes

characteristic

#11719919

C₂²

yes

—

C₄×Dic₅₃

C₂²

yes

—

C₄×Dic₅₃

C₂²

yes

characteristic

C₄×Dic₅₃

C₂²

yes

characteristic

C₂²×D₅₃

C₂³

yes

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

C₂³

yes

characteristic

C₂×C₄×C₅₃

C₂³

yes

—

C₂×C₄×C₅₃

C₂³

yes

—

C₂×C₄×C₅₃

C₂³

yes

characteristic

C₂×C₄×C₅₃

C₂³

yes

characteristic

C₂×Dic₅₃

C₂³

yes

—

C₂×Dic₅₃

C₂³

yes

characteristic

C₂×Dic₅₃

C₂³

yes

characteristic

C₂×Dic₅₃

C₂³

yes

characteristic

C₂×Dic₅₃

C₂³

yes

characteristic

D₂₁₂

—

C₄×D₅₃

—

Dic₁₀₆

—

Dic₁₀₆

—

C₂²×C₅₃

C₂⁴

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

#41

yes

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

#41

yes

—

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

—

—

—

—

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

D₅₃

yes

characteristic, complemented

C₄×D₄

—

#117

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×D₅₃

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

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

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

100

C₄²

—

101

C₄²

—

102

C₄²

—

103

C₄²

C₂×D₅₃

yes

—

104

C₄²

C₂×D₅₃

yes

characteristic

105

C₂³

—

106

Q₈

—

107

Q₈

—

106

—

108

Q₈

—

106

—

109

D₄

—

106

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

106

—

116

C₂×C₄

C₂²×D₅₃

yes

—

117

C₂×C₄

C₂²×D₅₃

yes

characteristic

118

C₂×C₄

C₂²×D₅₃

yes

—

119

C₂×C₄

C₂²×D₅₃

yes

characteristic

120

C₂×C₄

C₂²×D₅₃

yes

characteristic

121

C₂²

—

106

—

122

C₂²

—

106

—

123

C₂²

—

106

—

124

C₂²

C₂³×D₅₃

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

129

C₄

—

130

C₄

—

106

—

131

C₄

—

106

—

132

C₄

—

106

—

133

C₄

—

106

—

134

C₄

#11719945

yes

—

135

C₂

—

212

—

136

C₂

#41×D₅₃

yes

characteristic

137

C₂

C₂×#11719945

yes

characteristic

138

C₂

#13133265

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅₃
commutator, socle, cosocle	C₂²×C₅₃	C₂⁴
Fitting	C₄×Q₈×C₅₃	C₂
prime radical	C₂×D₂₁₂	C₂²
hypercenter	C₄×Q₈	D₅₃
hypocenter	C₅₃	#556
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×D₅₃

commutator, socle, cosocle

C₂²×C₅₃

C₂⁴

Fitting

C₄×Q₈×C₅₃

C₂

prime radical

C₂×D₂₁₂

C₂²

hypercenter

C₄×Q₈

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂² ◁ C₄×Q₈

N	H ≅ G/N	type
1. #13133215	C₂	semidirect
2. #13133212	C₂	semidirect
3. #13133211	C₂	semidirect
6. #13133195	C₂	semidirect
7. C₄×Dic₁₀₆	C₂	semidirect
8. #13133182	C₂	semidirect
10. #13133174	C₂	semidirect
11. C₄×Q₈×C₅₃	C₂	semidirect
70. C₅₃	#556	semidirect
79. C₄×Q₈	D₅₃	semidirect

H ≅ G/N

type

1. #13133215

C₂

semidirect

2. #13133212

C₂

semidirect

3. #13133211

C₂

semidirect

6. #13133195

C₂

semidirect

7. C₄×Dic₁₀₆

C₂

semidirect

8. #13133182

C₂

semidirect

10. #13133174

C₂

semidirect

11. C₄×Q₈×C₅₃

C₂

semidirect

70. C₅₃

#556

semidirect

79. C₄×Q₈

D₅₃

semidirect

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

subgroup	index
#13133215	2
#13133212	2
#13133211	2
#13133206	2
#13133205	2
#13133195	2
C₄×Dic₁₀₆	2
#13133182	2
#13133179	2
#13133174	2
C₄×Q₈×C₅₃	2
#556	53

subgroup

index

C₄×Dic₁₀₆

C₄×Q₈×C₅₃

Subgroups — #16811345

Conjugacy classes of subgroups

Complements

Maximal subgroups