Group #16811357 — subgroups

Subgroups — #16811357

Order 3392.

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

normal subgroups only

This group has 236 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	#13133213	C₂	1	1	yes	maximal, characteristic
2	#13133203	C₂	1	2	yes	maximal, complemented
3	#13133203	C₂	1	2	yes	maximal, complemented
4	#13133202	C₂	1	2	yes	maximal, complemented
5	#13133198	C₂	1	1	yes	maximal, characteristic, complemented
6	#13133182	C₂	1	2	yes	maximal, complemented
7	C₄×C₂²⋊D₅₃	C₂	1	2	yes	maximal, complemented
8	#13133177	C₂	1	2	yes	maximal, complemented
9	#123×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×D₄×C₅₃	C₂²	1	1	yes	characteristic
11	C₂²×C₄×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×C₂²⋊D₅₃	C₂²	1	2	yes	—
13	C₂×C₂²⋊D₅₃	C₂²	1	2	yes	—
14	C₂²×Dic₅₃	C₂²	1	1	yes	characteristic
15	C₂×D₂₁₂	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₅₃	C₂²	1	2	yes	—
17	C₂×Dic₁₀₆	C₂²	1	2	yes	—
18	C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
19	C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
20	C₂²⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
21	C₂²⋊Dic₅₃	C₂²	1	2	yes	—
22	C₂²⋊Dic₅₃	C₂²	1	1	yes	characteristic
23	#11719921	C₂²	1	2	yes	—
24	#11719921	C₂²	1	2	yes	—
25	#11719921	C₂²	1	2	yes	—
26	#11719919	C₂²	1	2	yes	—
27	#11719919	C₂²	1	2	yes	—
28	C₄×Dic₅₃	C₂²	1	2	yes	—
29	C₄×Dic₅₃	C₂²	1	2	yes	—
30	C₄×Dic₅₃	C₂²	1	1	yes	characteristic
31	C₂³×C₅₃	C₂³	1	1	yes	characteristic
32	C₂³×C₅₃	C₂³	1	1	yes	characteristic
33	C₂²×D₅₃	C₂³	1	2	yes	—
34	D₄×C₅₃	—	4	1	no	—
35	C₂×C₄×C₅₃	—	4	1	no	—
36	C₂×C₄×C₅₃	C₂³	1	2	yes	—
37	C₂×C₄×C₅₃	C₂³	1	2	yes	—
38	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
39	C₂²⋊D₅₃	—	2	4	no	—
40	C₂²⋊D₅₃	—	2	4	no	—
41	C₂×Dic₅₃	—	4	1	no	—
42	C₂×Dic₅₃	C₂³	1	2	yes	—
43	C₂×Dic₅₃	C₂³	1	2	yes	—
44	C₂×Dic₅₃	C₂³	1	2	yes	—
45	D₂₁₂	—	4	1	no	—
46	C₄×D₅₃	—	4	2	no	—
47	Dic₁₀₆	—	2	4	no	—
48	C₂²×C₅₃	—	2	2	no	—
49	C₂²×C₅₃	—	2	2	no	—
50	C₂²×C₅₃	—	2	1	no	—
51	C₂²×C₅₃	—	2	1	no	—
52	C₂²×C₅₃	C₂⁴	1	1	yes	characteristic
53	C₂×D₅₃	—	2	2	no	—
54	C₂×D₅₃	—	2	2	no	—
55	C₂×D₅₃	—	2	2	no	—
56	C₄×C₅₃	—	2	2	no	—
57	C₄×C₅₃	—	2	2	no	—
58	C₄×C₅₃	—	2	1	no	—
59	Dic₅₃	—	2	2	no	—
60	Dic₅₃	—	2	2	no	—
61	Dic₅₃	#41	1	4	yes	—
62	C₂×C₅₃	—	4	1	no	—
63	C₂×C₅₃	—	4	1	no	—
64	C₂×C₅₃	C₂×#41	1	2	yes	—
65	C₂×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
66	D₅₃	—	4	2	no	—
67	#559	—	53	1	no	maximal, Sylow
68	C₅₃	#559	1	1	yes	characteristic, complemented, Sylow
69	#127	—	53	1	no	—
70	#124	—	53	2	no	—
71	#124	—	53	2	no	—
72	#123	—	53	1	no	—
73	#123	—	53	2	no	—
74	#123	D₅₃	1	1	yes	characteristic, complemented
75	C₄×Q₈	—	53	2	no	—
76	C₄×D₄	—	53	2	no	—
77	C₄×D₄	—	53	2	no	—
78	C₂×Q₈	—	53	2	no	—
79	C₂×D₄	—	53	2	no	—
80	C₂×D₄	—	53	2	no	—
81	C₂×D₄	—	53	1	no	—
82	C₂×D₄	C₂×D₅₃	1	1	yes	characteristic
83	C₂²×C₄	—	53	1	no	—
84	C₂²×C₄	—	53	2	no	—
85	C₂²×C₄	C₂×D₅₃	1	1	yes	characteristic
86	C₄⋊C₄	—	53	2	no	—
87	C₄⋊C₄	—	53	2	no	—
88	C₄⋊C₄	C₂×D₅₃	1	2	yes	—
89	C₂²⋊C₄	—	53	2	no	—
90	C₂²⋊C₄	—	53	1	no	—
91	C₂²⋊C₄	—	53	2	no	—
92	C₂²⋊C₄	—	53	2	no	—
93	C₂²⋊C₄	—	53	2	no	—
94	C₂²⋊C₄	C₂×D₅₃	1	2	yes	—
95	C₂²⋊C₄	C₂×D₅₃	1	1	yes	characteristic
96	C₄²	—	53	2	no	—
97	C₄²	—	53	2	no	—
98	C₄²	—	53	1	no	—
99	C₂³	—	53	2	no	—
100	C₂³	C₂²×D₅₃	1	1	yes	characteristic
101	C₂³	C₂²×D₅₃	1	1	yes	characteristic
102	Q₈	—	106	4	no	—
103	D₄	—	4	1	no	—
104	D₄	—	106	4	no	—
105	D₄	—	106	4	no	—
106	D₄	—	212	1	no	—
107	C₂×C₄	—	4	1	no	—
108	C₂×C₄	—	53	2	no	—
109	C₂×C₄	—	53	2	no	—
110	C₂×C₄	—	53	2	no	—
111	C₂×C₄	—	212	2	no	—
112	C₂×C₄	—	212	1	no	—
113	C₂×C₄	C₂²×D₅₃	1	2	yes	—
114	C₂×C₄	C₂²×D₅₃	1	2	yes	—
115	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
116	C₂²	—	2	2	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	2	1	no	—
119	C₂²	—	2	1	no	—
120	C₂²	—	106	2	no	—
121	C₂²	—	106	2	no	—
122	C₂²	—	106	2	no	—
123	C₂²	C₂³×D₅₃	1	1	yes	characteristic
124	C₄	—	2	2	no	—
125	C₄	—	2	2	no	—
126	C₄	—	2	1	no	—
127	C₄	—	53	4	no	—
128	C₄	—	106	2	no	—
129	C₄	—	106	2	no	—
130	C₂	—	4	1	no	—
131	C₂	—	4	1	no	—
132	C₂	—	212	2	no	—
133	C₂	#41×D₅₃	1	2	yes	—
134	C₂	#13133262	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13133213

C₂

yes

maximal, characteristic

#13133203

C₂

yes

maximal, complemented

#13133203

C₂

yes

maximal, complemented

#13133202

C₂

yes

maximal, complemented

#13133198

C₂

yes

maximal, characteristic, complemented

#13133182

C₂

yes

maximal, complemented

C₄×C₂²⋊D₅₃

C₂

yes

maximal, complemented

#13133177

C₂

yes

maximal, complemented

#123×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂×D₄×C₅₃

C₂²

yes

characteristic

C₂²×C₄×C₅₃

C₂²

yes

characteristic

C₂×C₂²⋊D₅₃

C₂²

yes

—

C₂×C₂²⋊D₅₃

C₂²

yes

—

C₂²×Dic₅₃

C₂²

yes

characteristic

C₂×D₂₁₂

C₂²

yes

characteristic

C₂×C₄×D₅₃

C₂²

yes

—

C₂×Dic₁₀₆

C₂²

yes

—

C₄⋊C₄×C₅₃

C₂²

yes

—

C₂²⋊C₄×C₅₃

C₂²

yes

—

C₂²⋊C₄×C₅₃

C₂²

yes

characteristic

C₂²⋊Dic₅₃

C₂²

yes

—

C₂²⋊Dic₅₃

C₂²

yes

characteristic

#11719921

C₂²

yes

—

#11719921

C₂²

yes

—

#11719921

C₂²

yes

—

#11719919

C₂²

yes

—

#11719919

C₂²

yes

—

C₄×Dic₅₃

C₂²

yes

—

C₄×Dic₅₃

C₂²

yes

—

C₄×Dic₅₃

C₂²

yes

characteristic

C₂³×C₅₃

C₂³

yes

characteristic

C₂³×C₅₃

C₂³

yes

characteristic

C₂²×D₅₃

C₂³

yes

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂³

yes

—

C₂×C₄×C₅₃

C₂³

yes

—

C₂×C₄×C₅₃

C₂³

yes

characteristic

C₂²⋊D₅₃

—

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂³

yes

—

C₂×Dic₅₃

C₂³

yes

—

C₂×Dic₅₃

C₂³

yes

—

D₂₁₂

—

C₄×D₅₃

—

Dic₁₀₆

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂⁴

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

#41

yes

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

C₂×#41

yes

—

C₂×C₅₃

H(5,𝔽₂)

yes

characteristic

D₅₃

—

#559

—

maximal, Sylow

C₅₃

#559

yes

characteristic, complemented, Sylow

—

—

—

—

—

D₅₃

yes

characteristic, complemented

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₅₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₅₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×D₅₃

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₅₃

yes

—

C₂²⋊C₄

C₂×D₅₃

yes

characteristic

C₄²

—

C₄²

—

C₄²

—

C₂³

—

100

C₂³

C₂²×D₅₃

yes

characteristic

101

C₂³

C₂²×D₅₃

yes

characteristic

102

Q₈

—

106

—

103

D₄

—

104

D₄

—

106

—

105

D₄

—

106

—

106

D₄

—

212

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

212

—

112

C₂×C₄

—

212

—

113

C₂×C₄

C₂²×D₅₃

yes

—

114

C₂×C₄

C₂²×D₅₃

yes

—

115

C₂×C₄

C₂²×D₅₃

yes

characteristic

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

106

—

121

C₂²

—

106

—

122

C₂²

—

106

—

123

C₂²

C₂³×D₅₃

yes

characteristic

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

106

—

129

C₄

—

106

—

130

C₂

—

131

C₂

—

132

C₂

—

212

—

133

C₂

#41×D₅₃

yes

—

134

C₂

#13133262

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅₃
commutator, socle, cosocle	C₂²×C₅₃	C₂⁴
Fitting	#123×C₅₃	C₂
prime radical	#13133213	C₂
hypercenter	#123	D₅₃
hypocenter	C₅₃	#559
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #123

subgroup

G/H

center, Frattini

C₂²

C₂³×D₅₃

commutator, socle, cosocle

C₂²×C₅₃

C₂⁴

Fitting

#123×C₅₃

C₂

C₂

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂² ◁ #123

N	H ≅ G/N	type
2. #13133203	C₂	semidirect
3. #13133203	C₂	semidirect
4. #13133202	C₂	semidirect
5. #13133198	C₂	semidirect
6. #13133182	C₂	semidirect
7. C₄×C₂²⋊D₅₃	C₂	semidirect
8. #13133177	C₂	semidirect
9. #123×C₅₃	C₂	semidirect
68. C₅₃	#559	semidirect
74. #123	D₅₃	semidirect

H ≅ G/N

type

2. #13133203

C₂

semidirect

3. #13133203

C₂

semidirect

4. #13133202

C₂

semidirect

5. #13133198

C₂

semidirect

6. #13133182

C₂

semidirect

7. C₄×C₂²⋊D₅₃

C₂

semidirect

8. #13133177

C₂

semidirect

9. #123×C₅₃

C₂

semidirect

68. C₅₃

#559

semidirect

74. #123

D₅₃

semidirect

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

subgroup	index
#13133213	2
#13133203	2
#13133202	2
#13133198	2
#13133182	2
C₄×C₂²⋊D₅₃	2
#13133177	2
#123×C₅₃	2
#559	53

subgroup

index

C₄×C₂²⋊D₅₃

#123×C₅₃

Subgroups — #16811357

Conjugacy classes of subgroups

Complements

Maximal subgroups