Group #16811377 — subgroups

Subgroups — #16811377

Order 3392.

196 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 91 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	2	yes	maximal, complemented
2	#13133212	C₂	1	2	yes	maximal, complemented
3	#13133212	C₂	1	2	yes	maximal, complemented
4	#13133209	C₂	1	1	yes	maximal, characteristic, complemented
5	#13133205	C₂	1	1	yes	maximal, characteristic
6	#13133202	C₂	1	2	yes	maximal
7	#13133197	C₂	1	2	yes	maximal, complemented
8	#13133195	C₂	1	2	yes	maximal, complemented
9	#125×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×D₂₁₂	C₂²	1	1	yes	characteristic
11	C₂×C₄×D₅₃	C₂²	1	2	yes	—
12	C₂×Dic₁₀₆	C₂²	1	2	yes	—
13	C₂×Dic₁₀₆	C₂²	1	1	yes	characteristic
14	C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
15	C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
16	C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
17	C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
18	C₄²×C₅₃	C₂²	1	1	yes	characteristic
19	#11719921	C₂²	1	2	yes	—
20	#11719921	C₂²	1	2	yes	—
21	#11719921	C₂²	1	2	yes	—
22	#11719921	C₂²	1	2	yes	—
23	#11719921	C₂²	1	2	yes	—
24	C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
25	C₄⋊Dic₅₃	C₂²	1	2	yes	—
26	C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
27	#11719919	C₂²	1	2	yes	—
28	#11719919	C₂²	1	2	yes	—
29	#11719919	C₂²	1	2	yes	—
30	C₄×Dic₅₃	C₂²	1	2	yes	—
31	C₂²×D₅₃	C₂³	1	2	yes	—
32	C₂×C₄×C₅₃	C₂³	1	2	yes	—
33	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
34	C₂×C₄×C₅₃	C₂³	1	2	yes	—
35	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
36	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
37	C₂×Dic₅₃	C₂³	1	2	yes	—
38	C₂×Dic₅₃	C₂³	1	2	yes	—
39	C₂×Dic₅₃	C₂³	1	2	yes	—
40	D₂₁₂	—	4	1	no	—
41	C₄×D₅₃	—	4	2	no	—
42	Dic₁₀₆	—	4	1	no	—
43	Dic₁₀₆	—	4	2	no	—
44	C₂²×C₅₃	C₂⁴	1	1	yes	characteristic
45	C₂×D₅₃	—	2	2	no	—
46	C₂×D₅₃	—	2	2	no	—
47	C₂×D₅₃	—	2	2	no	—
48	C₄×C₅₃	—	2	2	no	—
49	C₄×C₅₃	—	2	1	no	—
50	C₄×C₅₃	—	2	2	no	—
51	C₄×C₅₃	—	2	1	no	—
52	C₄×C₅₃	—	2	1	no	—
53	Dic₅₃	—	2	2	no	—
54	Dic₅₃	—	2	2	no	—
55	Dic₅₃	—	2	2	no	—
56	C₂×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
57	C₂×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
58	C₂×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
59	D₅₃	—	4	2	no	—
60	#563	—	53	1	no	maximal, Sylow
61	C₅₃	#563	1	1	yes	characteristic, complemented, Sylow
62	C₄⋊Q₈	—	53	2	no	—
63	#126	—	53	2	no	—
64	#126	—	53	2	no	—
65	#125	—	53	1	no	—
66	#125	D₅₃	1	1	yes	characteristic, complemented
67	#124	—	53	1	no	—
68	#123	—	53	2	no	—
69	C₂²⋊Q₈	—	53	2	no	—
70	C₂²⋊Q₈	—	53	2	no	—
71	C₂×Q₈	—	53	2	no	—
72	C₂×Q₈	—	53	1	no	—
73	C₂×D₄	—	53	1	no	—
74	C₂²×C₄	—	53	2	no	—
75	C₄⋊C₄	—	53	1	no	—
76	C₄⋊C₄	—	53	2	no	—
77	C₄⋊C₄	—	53	2	no	—
78	C₄⋊C₄	—	53	2	no	—
79	C₄⋊C₄	—	53	2	no	—
80	C₄⋊C₄	—	53	1	no	—
81	C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
82	C₄⋊C₄	C₂×D₅₃	1	2	yes	—
83	C₄⋊C₄	C₂×D₅₃	1	2	yes	—
84	C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
85	C₂²⋊C₄	—	53	2	no	—
86	C₂²⋊C₄	—	53	2	no	—
87	C₂²⋊C₄	—	53	2	no	—
88	C₂²⋊C₄	—	53	2	no	—
89	C₂²⋊C₄	—	53	2	no	—
90	C₄²	—	53	2	no	—
91	C₄²	C₂×D₅₃	1	1	yes	characteristic
92	C₂³	—	53	2	no	—
93	Q₈	—	212	2	no	—
94	Q₈	—	212	1	no	—
95	D₄	—	212	1	no	—
96	C₂×C₄	—	53	2	no	—
97	C₂×C₄	—	53	2	no	—
98	C₂×C₄	—	53	2	no	—
99	C₂×C₄	—	212	2	no	—
100	C₂×C₄	C₂²×D₅₃	1	2	yes	—
101	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
102	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
103	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
104	C₂×C₄	C₂²×D₅₃	1	2	yes	—
105	C₂²	—	106	2	no	—
106	C₂²	—	106	2	no	—
107	C₂²	—	106	2	no	—
108	C₂²	C₂³×D₅₃	1	1	yes	characteristic
109	C₄	—	2	2	no	—
110	C₄	—	2	1	no	—
111	C₄	—	2	2	no	—
112	C₄	—	2	1	no	—
113	C₄	—	2	1	no	—
114	C₄	—	106	2	no	—
115	C₄	—	106	2	no	—
116	C₄	—	106	2	no	—
117	C₂	—	212	2	no	—
118	C₂	#13133263	1	1	yes	characteristic
119	C₂	#13133264	1	1	yes	characteristic
120	C₂	#13133265	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13133216

C₂

yes

maximal, complemented

#13133212

C₂

yes

maximal, complemented

#13133212

C₂

yes

maximal, complemented

#13133209

C₂

yes

maximal, characteristic, complemented

#13133205

C₂

yes

maximal, characteristic

#13133202

C₂

yes

maximal

#13133197

C₂

yes

maximal, complemented

#13133195

C₂

yes

maximal, complemented

#125×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂×D₂₁₂

C₂²

yes

characteristic

C₂×C₄×D₅₃

C₂²

yes

—

C₂×Dic₁₀₆

C₂²

yes

—

C₂×Dic₁₀₆

C₂²

yes

characteristic

C₄⋊C₄×C₅₃

C₂²

yes

characteristic

C₄⋊C₄×C₅₃

C₂²

yes

—

C₄⋊C₄×C₅₃

C₂²

yes

—

C₄⋊C₄×C₅₃

C₂²

yes

characteristic

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

C₄⋊Dic₅₃

C₂²

yes

—

C₄⋊Dic₅₃

C₂²

yes

characteristic

#11719919

C₂²

yes

—

#11719919

C₂²

yes

—

#11719919

C₂²

yes

—

C₄×Dic₅₃

C₂²

yes

—

C₂²×D₅₃

C₂³

yes

—

C₂×C₄×C₅₃

C₂³

yes

—

C₂×C₄×C₅₃

C₂³

yes

characteristic

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

—

C₂×Dic₅₃

C₂³

yes

—

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

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

C₂×C₅₃

H(5,𝔽₂)

yes

characteristic

C₂×C₅₃

2₋¹⁺⁴

yes

characteristic

C₂×C₅₃

2₋¹⁺⁴

yes

characteristic

D₅₃

—

#563

—

maximal, Sylow

C₅₃

#563

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

—

—

—

D₅₃

yes

characteristic, complemented

#124

—

#123

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂×Q₈

—

C₂×Q₈

—

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

—

C₄⋊C₄

C₂×D₅₃

yes

—

C₄⋊C₄

C₂×D₅₃

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂×D₅₃

yes

characteristic

C₂³

—

Q₈

—

212

—

Q₈

—

212

—

D₄

—

212

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

212

—

100

C₂×C₄

C₂²×D₅₃

yes

—

101

C₂×C₄

C₂²×D₅₃

yes

characteristic

102

C₂×C₄

C₂²×D₅₃

yes

characteristic

103

C₂×C₄

C₂²×D₅₃

yes

characteristic

104

C₂×C₄

C₂²×D₅₃

yes

—

105

C₂²

—

106

—

106

C₂²

—

106

—

107

C₂²

—

106

—

108

C₂²

C₂³×D₅₃

yes

characteristic

109

C₄

—

110

C₄

—

111

C₄

—

112

C₄

—

113

C₄

—

114

C₄

—

106

—

115

C₄

—

106

—

116

C₄

—

106

—

117

C₂

—

212

—

118

C₂

#13133263

yes

characteristic

119

C₂

#13133264

yes

characteristic

120

C₂

#13133265

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅₃
commutator, socle, cosocle	C₂²×C₅₃	C₂⁴
Fitting	#125×C₅₃	C₂
prime radical	C₂×D₂₁₂	C₂²
hypercenter	#125	D₅₃
hypocenter	C₅₃	#563
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #125

subgroup

G/H

center, Frattini

C₂²

C₂³×D₅₃

commutator, socle, cosocle

C₂²×C₅₃

C₂⁴

Fitting

#125×C₅₃

C₂

prime radical

C₂×D₂₁₂

C₂²

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂² ◁ #125

N	H ≅ G/N	type
1. #13133216	C₂	semidirect
2. #13133212	C₂	semidirect
3. #13133212	C₂	semidirect
4. #13133209	C₂	semidirect
7. #13133197	C₂	semidirect
8. #13133195	C₂	semidirect
9. #125×C₅₃	C₂	semidirect
61. C₅₃	#563	semidirect
66. #125	D₅₃	semidirect

H ≅ G/N

type

1. #13133216

C₂

semidirect

2. #13133212

C₂

semidirect

3. #13133212

C₂

semidirect

4. #13133209

C₂

semidirect

7. #13133197

C₂

semidirect

8. #13133195

C₂

semidirect

9. #125×C₅₃

C₂

semidirect

61. C₅₃

#563

semidirect

66. #125

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
#13133209	2
#13133205	2
#13133202	2
#13133197	2
#13133195	2
#125×C₅₃	2
#563	53

subgroup

index

#125×C₅₃

Subgroups — #16811377

Conjugacy classes of subgroups

Complements

Maximal subgroups