Group #16811231 — subgroups

Subgroups — #16811231

Order 3392.

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

normal subgroups only

This group has 192 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	#13133215	C₂	1	1	yes	maximal, characteristic, complemented
2	#13133212	C₂	1	2	yes	maximal
3	#13133211	C₂	1	2	yes	maximal
4	#13133209	C₂	1	2	yes	maximal, complemented
5	#13133208	C₂	1	1	yes	maximal, characteristic, complemented
6	#13133207	C₂	1	2	yes	maximal, complemented
7	#13133197	C₂	1	1	yes	maximal, characteristic
8	#13133193	C₂	1	1	yes	maximal, characteristic
9	C₄×Dic₁₀₆	C₂	1	1	yes	maximal, characteristic, complemented
10	#13133174	C₂	1	1	yes	maximal, characteristic
11	C₄×Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₂×C₄×D₅₃	C₂²	1	1	yes	characteristic
14	C₂×Dic₁₀₆	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
16	C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
17	C₄²×C₅₃	C₂²	1	1	yes	characteristic
18	C₄²×C₅₃	C₂²	1	2	yes	—
19	#11719921	C₂²	1	1	yes	characteristic
20	#11719921	C₂²	1	1	yes	characteristic
21	#11719921	C₂²	1	2	yes	—
22	#11719921	C₂²	1	2	yes	—
23	C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
24	C₄⋊Dic₅₃	C₂²	1	2	yes	—
25	C₄⋊Dic₅₃	C₂²	1	2	yes	—
26	#11719919	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	2	yes	—
31	#11719919	C₂²	1	2	yes	—
32	#11719919	C₂²	1	2	yes	—
33	C₄×Dic₅₃	C₂²	1	1	yes	characteristic
34	C₄×Dic₅₃	C₂²	1	2	yes	—
35	C₂²×D₅₃	C₂³	1	1	yes	characteristic
36	Q₈×C₅₃	—	2	2	no	—
37	C₂×C₄×C₅₃	C₂³	1	2	yes	—
38	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
39	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
40	C₂×C₄×C₅₃	C₂³	1	2	yes	—
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	2	yes	—
47	C₄×D₅₃	—	2	2	no	—
48	Dic₁₀₆	—	2	2	no	—
49	C₂²×C₅₃	C₂⁴	1	1	yes	characteristic
50	C₂×D₅₃	—	2	1	no	—
51	C₂×D₅₃	—	2	1	no	—
52	C₂×D₅₃	—	2	1	no	—
53	C₄×C₅₃	—	2	2	no	—
54	C₄×C₅₃	—	2	1	no	—
55	C₄×C₅₃	—	2	2	no	—
56	C₄×C₅₃	—	2	1	no	—
57	C₄×C₅₃	#41	1	2	yes	—
58	Dic₅₃	—	2	2	no	—
59	Dic₅₃	—	2	1	no	—
60	Dic₅₃	—	2	1	no	—
61	Dic₅₃	—	2	1	no	—
62	Dic₅₃	—	2	2	no	—
63	C₂×C₅₃	C₂×#41	1	1	yes	characteristic
64	C₂×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
65	C₂×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
66	D₅₃	—	4	1	no	—
67	#541	—	53	1	no	maximal, Sylow
68	C₅₃	#541	1	1	yes	characteristic, complemented, Sylow
69	C₄⋊Q₈	—	53	1	no	—
70	#126	—	53	2	no	—
71	#126	—	53	2	no	—
72	#125	—	53	2	no	—
73	#125	—	53	1	no	—
74	#125	—	53	2	no	—
75	C₂²⋊Q₈	—	53	1	no	—
76	C₂²⋊Q₈	—	53	1	no	—
77	C₄×Q₈	—	53	1	no	—
78	C₄×Q₈	D₅₃	1	1	yes	characteristic, complemented
79	#117	—	53	1	no	—
80	C₂×Q₈	—	53	1	no	—
81	C₂×Q₈	C₂×D₅₃	1	1	yes	characteristic
82	C₂²×C₄	—	53	1	no	—
83	C₄⋊C₄	—	53	1	no	—
84	C₄⋊C₄	—	53	1	no	—
85	C₄⋊C₄	—	53	1	no	—
86	C₄⋊C₄	—	53	2	no	—
87	C₄⋊C₄	—	53	2	no	—
88	C₄⋊C₄	—	53	2	no	—
89	C₄⋊C₄	—	53	2	no	—
90	C₄⋊C₄	—	53	2	no	—
91	C₄⋊C₄	—	53	2	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	1	no	—
96	C₂²⋊C₄	—	53	1	no	—
97	C₂²⋊C₄	—	53	2	no	—
98	C₂²⋊C₄	—	53	2	no	—
99	C₄²	—	53	1	no	—
100	C₄²	—	53	2	no	—
101	C₄²	C₂×D₅₃	1	1	yes	characteristic
102	C₄²	C₂×D₅₃	1	2	yes	—
103	C₂³	—	53	1	no	—
104	Q₈	—	2	2	no	—
105	Q₈	—	106	2	no	—
106	C₂×C₄	—	53	1	no	—
107	C₂×C₄	—	53	1	no	—
108	C₂×C₄	—	53	2	no	—
109	C₂×C₄	—	53	2	no	—
110	C₂×C₄	—	53	1	no	—
111	C₂×C₄	—	106	2	no	—
112	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
113	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
114	C₂×C₄	C₂²×D₅₃	1	2	yes	—
115	C₂×C₄	C₂²×D₅₃	1	2	yes	—
116	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
117	C₂²	—	106	1	no	—
118	C₂²	—	106	1	no	—
119	C₂²	—	106	1	no	—
120	C₂²	C₂³×D₅₃	1	1	yes	characteristic
121	C₄	—	2	2	no	—
122	C₄	—	2	1	no	—
123	C₄	—	2	2	no	—
124	C₄	—	2	1	no	—
125	C₄	—	106	1	no	—
126	C₄	—	106	2	no	—
127	C₄	—	106	1	no	—
128	C₄	—	106	1	no	—
129	C₄	—	106	2	no	—
130	C₄	#11719945	1	2	yes	—
131	C₂	—	212	1	no	—
132	C₂	C₂×#11719945	1	1	yes	characteristic
133	C₂	#13133264	1	1	yes	characteristic
134	C₂	#13133265	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13133215

C₂

yes

maximal, characteristic, complemented

#13133212

C₂

yes

maximal

#13133211

C₂

yes

maximal

#13133209

C₂

yes

maximal, complemented

#13133208

C₂

yes

maximal, characteristic, complemented

#13133207

C₂

yes

maximal, complemented

#13133197

C₂

yes

maximal, characteristic

#13133193

C₂

yes

maximal, characteristic

C₄×Dic₁₀₆

C₂

yes

maximal, characteristic, complemented

#13133174

C₂

yes

maximal, characteristic

C₄×Q₈×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×C₅₃

C₂²

yes

characteristic

C₂×C₄×D₅₃

C₂²

yes

characteristic

C₂×Dic₁₀₆

C₂²

yes

characteristic

C₄⋊C₄×C₅₃

C₂²

yes

characteristic

C₄⋊C₄×C₅₃

C₂²

yes

—

C₄²×C₅₃

C₂²

yes

characteristic

C₄²×C₅₃

C₂²

yes

—

#11719921

C₂²

yes

characteristic

#11719921

C₂²

yes

characteristic

#11719921

C₂²

yes

—

#11719921

C₂²

yes

—

C₄⋊Dic₅₃

C₂²

yes

characteristic

C₄⋊Dic₅₃

C₂²

yes

—

C₄⋊Dic₅₃

C₂²

yes

—

#11719919

C₂²

yes

characteristic

#11719919

C₂²

yes

characteristic

#11719919

C₂²

yes

—

#11719919

C₂²

yes

—

#11719919

C₂²

yes

—

#11719919

C₂²

yes

—

#11719919

C₂²

yes

—

C₄×Dic₅₃

C₂²

yes

characteristic

C₄×Dic₅₃

C₂²

yes

—

C₂²×D₅₃

C₂³

yes

characteristic

Q₈×C₅₃

—

C₂×C₄×C₅₃

C₂³

yes

—

C₂×C₄×C₅₃

C₂³

yes

characteristic

C₂×C₄×C₅₃

C₂³

yes

characteristic

C₂×C₄×C₅₃

C₂³

yes

—

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

—

C₄×D₅₃

—

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

—

C₂×C₅₃

C₂×#41

yes

characteristic

C₂×C₅₃

2₋¹⁺⁴

yes

characteristic

C₂×C₅₃

2₋¹⁺⁴

yes

characteristic

D₅₃

—

#541

—

maximal, Sylow

C₅₃

#541

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

—

—

—

—

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

D₅₃

yes

characteristic, complemented

#117

—

C₂×Q₈

—

C₂×Q₈

C₂×D₅₃

yes

characteristic

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊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

—

C₄²

—

100

C₄²

—

101

C₄²

C₂×D₅₃

yes

characteristic

102

C₄²

C₂×D₅₃

yes

—

103

C₂³

—

104

Q₈

—

105

Q₈

—

106

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

106

—

112

C₂×C₄

C₂²×D₅₃

yes

characteristic

113

C₂×C₄

C₂²×D₅₃

yes

characteristic

114

C₂×C₄

C₂²×D₅₃

yes

—

115

C₂×C₄

C₂²×D₅₃

yes

—

116

C₂×C₄

C₂²×D₅₃

yes

characteristic

117

C₂²

—

106

—

118

C₂²

—

106

—

119

C₂²

—

106

—

120

C₂²

C₂³×D₅₃

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

106

—

126

C₄

—

106

—

127

C₄

—

106

—

128

C₄

—

106

—

129

C₄

—

106

—

130

C₄

#11719945

yes

—

131

C₂

—

212

—

132

C₂

C₂×#11719945

yes

characteristic

133

C₂

#13133264

yes

characteristic

134

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	C₂²×D₅₃	C₂³
hypercenter	C₄×Q₈	D₅₃
hypocenter	C₅₃	#541
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

C₂²×D₅₃

C₂³

hypercenter

C₄×Q₈

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂² ◁ C₄×Q₈

N	H ≅ G/N	type
1. #13133215	C₂	semidirect
4. #13133209	C₂	semidirect
5. #13133208	C₂	semidirect
6. #13133207	C₂	semidirect
9. C₄×Dic₁₀₆	C₂	semidirect
11. C₄×Q₈×C₅₃	C₂	semidirect
68. C₅₃	#541	semidirect
78. C₄×Q₈	D₅₃	semidirect

H ≅ G/N

type

1. #13133215

C₂

semidirect

4. #13133209

C₂

semidirect

5. #13133208

C₂

semidirect

6. #13133207

C₂

semidirect

9. C₄×Dic₁₀₆

C₂

semidirect

11. C₄×Q₈×C₅₃

C₂

semidirect

68. C₅₃

#541

semidirect

78. 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
#13133209	2
#13133208	2
#13133207	2
#13133197	2
#13133193	2
C₄×Dic₁₀₆	2
#13133174	2
C₄×Q₈×C₅₃	2
#541	53

subgroup

index

C₄×Dic₁₀₆

C₄×Q₈×C₅₃

Subgroups — #16811231

Conjugacy classes of subgroups

Complements

Maximal subgroups