Group #16811252 — subgroups

Subgroups — #16811252

Order 3392.

206 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 103 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	#13133215	C₂	1	2	yes	maximal, complemented
3	#13133207	C₂	1	2	yes	maximal, complemented
4	#13133207	C₂	1	2	yes	maximal, complemented
5	#13133196	C₂	1	1	yes	maximal, characteristic
6	#13133194	C₂	1	2	yes	maximal
7	#13133173	C₂	1	2	yes	maximal
8	C₂×#11719919	C₂	1	1	yes	maximal, characteristic
9	C₂²⋊Q₈×C₅₃	C₂	1	1	yes	maximal, characteristic
10	C₂×Q₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₅₃	C₂²	1	1	yes	characteristic
12	C₂²×Dic₅₃	C₂²	1	1	yes	characteristic
13	C₂²×Dic₅₃	C₂²	1	1	yes	characteristic
14	C₂×Dic₁₀₆	C₂²	1	2	yes	—
15	C₂×Dic₁₀₆	C₂²	1	1	yes	characteristic
16	C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
18	C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
19	C₂²⋊Dic₅₃	C₂²	1	2	yes	—
20	C₄⋊Dic₅₃	C₂²	1	2	yes	—
21	C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
22	#11719919	C₂²	1	2	yes	—
23	#11719919	C₂²	1	2	yes	—
24	#11719919	C₂²	1	2	yes	—
25	#11719919	C₂²	1	2	yes	—
26	#11719919	C₂²	1	2	yes	—
27	#11719919	C₂²	1	2	yes	—
28	#11719919	C₂²	1	2	yes	—
29	C₄×Dic₅₃	C₂²	1	2	yes	—
30	C₄×Dic₅₃	C₂²	1	2	yes	—
31	C₂³×C₅₃	C₂³	1	1	yes	characteristic
32	Q₈×C₅₃	—	4	1	no	—
33	C₂×C₄×C₅₃	—	4	1	no	—
34	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
35	C₂×C₄×C₅₃	C₂³	1	2	yes	—
36	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
37	C₂×C₄×C₅₃	C₂³	1	2	yes	—
38	C₂×Dic₅₃	—	4	1	no	—
39	C₂×Dic₅₃	Q₈	1	4	yes	complemented
40	C₂×Dic₅₃	C₂³	1	2	yes	—
41	C₂×Dic₅₃	C₂³	1	2	yes	—
42	C₂×Dic₅₃	C₂³	1	2	yes	—
43	C₂×Dic₅₃	C₂³	1	2	yes	—
44	Dic₁₀₆	—	4	1	no	—
45	Dic₁₀₆	—	4	2	no	—
46	C₂²×C₅₃	—	2	1	no	—
47	C₂²×C₅₃	—	2	1	no	—
48	C₂²×C₅₃	C₂×Q₈	1	2	yes	—
49	C₂²×C₅₃	C₂⁴	1	1	yes	characteristic
50	C₄×C₅₃	—	2	1	no	—
51	C₄×C₅₃	—	2	2	no	—
52	C₄×C₅₃	—	2	1	no	—
53	C₄×C₅₃	—	2	2	no	—
54	Dic₅₃	—	2	2	no	—
55	Dic₅₃	—	2	2	no	—
56	Dic₅₃	—	2	2	no	—
57	Dic₅₃	C₂×Q₈	1	4	yes	—
58	C₂×C₅₃	—	2	2	no	—
59	C₂×C₅₃	C₂²×Q₈	1	1	yes	characteristic
60	C₂×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
61	C₂×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
62	#544	—	53	1	no	maximal, Sylow
63	C₅₃	#544	1	1	yes	characteristic, complemented, Sylow
64	C₄⋊Q₈	—	53	2	no	—
65	C₄⋊Q₈	—	53	2	no	—
66	#125	—	53	2	no	—
67	#125	—	53	2	no	—
68	C₂²⋊Q₈	—	53	1	no	—
69	C₂²⋊Q₈	—	53	2	no	—
70	C₂²⋊Q₈	D₅₃	1	1	yes	characteristic
71	#117	—	53	2	no	—
72	C₂×C₄⋊C₄	—	53	1	no	—
73	C₂×Q₈	—	53	1	no	—
74	C₂×Q₈	—	53	2	no	—
75	C₂×Q₈	C₂×D₅₃	1	1	yes	characteristic
76	C₂²×C₄	—	53	1	no	—
77	C₂²×C₄	—	53	1	no	—
78	C₂²×C₄	C₂×D₅₃	1	1	yes	characteristic
79	C₄⋊C₄	—	53	2	no	—
80	C₄⋊C₄	—	53	2	no	—
81	C₄⋊C₄	—	53	2	no	—
82	C₄⋊C₄	—	53	2	no	—
83	C₄⋊C₄	—	53	2	no	—
84	C₄⋊C₄	—	53	1	no	—
85	C₄⋊C₄	—	53	2	no	—
86	C₄⋊C₄	—	53	2	no	—
87	C₄⋊C₄	—	53	2	no	—
88	C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
89	C₄⋊C₄	C₂×D₅₃	1	2	yes	—
90	C₂²⋊C₄	—	53	2	no	—
91	C₂²⋊C₄	C₂×D₅₃	1	2	yes	—
92	C₄²	—	53	2	no	—
93	C₄²	—	53	2	no	—
94	C₂³	C₂²×D₅₃	1	1	yes	characteristic
95	Q₈	—	4	1	no	—
96	Q₈	—	212	1	no	—
97	Q₈	—	212	2	no	—
98	C₂×C₄	—	4	1	no	—
99	C₂×C₄	—	53	4	no	—
100	C₂×C₄	—	53	2	no	—
101	C₂×C₄	—	53	2	no	—
102	C₂×C₄	—	53	2	no	—
103	C₂×C₄	—	53	2	no	—
104	C₂×C₄	—	212	1	no	—
105	C₂×C₄	C₂²×D₅₃	1	2	yes	—
106	C₂×C₄	C₂²×D₅₃	1	2	yes	—
107	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
108	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
109	C₂²	—	2	1	no	—
110	C₂²	—	2	1	no	—
111	C₂²	Q₈×D₅₃	1	2	yes	—
112	C₂²	C₂³×D₅₃	1	1	yes	characteristic
113	C₄	—	2	1	no	—
114	C₄	—	2	1	no	—
115	C₄	—	2	2	no	—
116	C₄	—	2	2	no	—
117	C₄	—	53	4	no	—
118	C₄	—	106	2	no	—
119	C₄	—	106	2	no	—
120	C₄	—	106	2	no	—
121	C₂	—	2	2	no	—
122	C₂	C₂×Q₈×D₅₃	1	1	yes	characteristic
123	C₂	#13133262	1	1	yes	characteristic
124	C₂	#13133264	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13133216

C₂

yes

maximal, complemented

#13133215

C₂

yes

maximal, complemented

#13133207

C₂

yes

maximal, complemented

#13133207

C₂

yes

maximal, complemented

#13133196

C₂

yes

maximal, characteristic

#13133194

C₂

yes

maximal

#13133173

C₂

yes

maximal

C₂×#11719919

C₂

yes

maximal, characteristic

C₂²⋊Q₈×C₅₃

C₂

yes

maximal, characteristic

C₂×Q₈×C₅₃

C₂²

yes

characteristic

C₂²×C₄×C₅₃

C₂²

yes

characteristic

C₂²×Dic₅₃

C₂²

yes

characteristic

C₂²×Dic₅₃

C₂²

yes

characteristic

C₂×Dic₁₀₆

C₂²

yes

—

C₂×Dic₁₀₆

C₂²

yes

characteristic

C₄⋊C₄×C₅₃

C₂²

yes

characteristic

C₄⋊C₄×C₅₃

C₂²

yes

—

C₂²⋊C₄×C₅₃

C₂²

yes

—

C₂²⋊Dic₅₃

C₂²

yes

—

C₄⋊Dic₅₃

C₂²

yes

—

C₄⋊Dic₅₃

C₂²

yes

characteristic

#11719919

C₂²

yes

—

#11719919

C₂²

yes

—

#11719919

C₂²

yes

—

#11719919

C₂²

yes

—

#11719919

C₂²

yes

—

#11719919

C₂²

yes

—

#11719919

C₂²

yes

—

C₄×Dic₅₃

C₂²

yes

—

C₄×Dic₅₃

C₂²

yes

—

C₂³×C₅₃

C₂³

yes

characteristic

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂³

yes

characteristic

C₂×C₄×C₅₃

C₂³

yes

—

C₂×C₄×C₅₃

C₂³

yes

characteristic

C₂×C₄×C₅₃

C₂³

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

Q₈

yes

complemented

C₂×Dic₅₃

C₂³

yes

—

C₂×Dic₅₃

C₂³

yes

—

C₂×Dic₅₃

C₂³

yes

—

C₂×Dic₅₃

C₂³

yes

—

Dic₁₀₆

—

Dic₁₀₆

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂×Q₈

yes

—

C₂²×C₅₃

C₂⁴

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

C₂×Q₈

yes

—

C₂×C₅₃

—

C₂×C₅₃

C₂²×Q₈

yes

characteristic

C₂×C₅₃

H(5,𝔽₂)

yes

characteristic

C₂×C₅₃

2₋¹⁺⁴

yes

—

C₅₃

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

C₄⋊Q₈

—

#125

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₂²⋊Q₈

D₅₃

yes

characteristic

#117

—

C₂×C₄⋊C₄

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×D₅₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₅₃

yes

characteristic

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

C₂×D₅₃

yes

—

C₄²

—

C₄²

—

C₂³

C₂²×D₅₃

yes

characteristic

Q₈

—

Q₈

—

212

—

Q₈

—

212

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

212

—

105

C₂×C₄

C₂²×D₅₃

yes

—

106

C₂×C₄

C₂²×D₅₃

yes

—

107

C₂×C₄

C₂²×D₅₃

yes

characteristic

108

C₂×C₄

C₂²×D₅₃

yes

characteristic

109

C₂²

—

110

C₂²

—

111

C₂²

Q₈×D₅₃

yes

—

112

C₂²

C₂³×D₅₃

yes

characteristic

113

C₄

—

114

C₄

—

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

106

—

119

C₄

—

106

—

120

C₄

—

106

—

121

C₂

—

122

C₂

C₂×Q₈×D₅₃

yes

characteristic

123

C₂

#13133262

yes

characteristic

124

C₂

#13133264

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅₃
commutator, socle, cosocle	C₂²×C₅₃	C₂⁴
Fitting	C₂²⋊Q₈×C₅₃	C₂
prime radical	C₂³×C₅₃	C₂³
hypercenter	C₂²⋊Q₈	D₅₃
hypocenter	C₅₃	#544
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

C₂³×C₅₃

C₂³

hypercenter

C₂²⋊Q₈

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂² ◁ C₂²⋊Q₈

N	H ≅ G/N	type
1. #13133216	C₂	semidirect
2. #13133215	C₂	semidirect
3. #13133207	C₂	semidirect
4. #13133207	C₂	semidirect
39. C₂×Dic₅₃	Q₈	semidirect
63. C₅₃	#544	semidirect

H ≅ G/N

type

1. #13133216

C₂

semidirect

2. #13133215

C₂

semidirect

3. #13133207

C₂

semidirect

4. #13133207

C₂

semidirect

39. C₂×Dic₅₃

Q₈

semidirect

63. C₅₃

#544

semidirect

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

subgroup	index
#13133216	2
#13133215	2
#13133207	2
#13133196	2
#13133194	2
#13133173	2
C₂×#11719919	2
C₂²⋊Q₈×C₅₃	2
#544	53

subgroup

index

C₂²⋊Q₈×C₅₃

Subgroups — #16811252

Conjugacy classes of subgroups

Complements

Maximal subgroups