Group #16811218 — subgroups

Subgroups — #16811218

Order 3392.

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

normal subgroups only

This group has 218 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	#13133210	C₂	1	2	yes	maximal, complemented
2	#13133207	C₂	1	2	yes	maximal, complemented
3	#13133198	C₂	1	2	yes	maximal, complemented
4	#13133198	C₂	1	2	yes	maximal, complemented
5	#13133196	C₂	1	1	yes	maximal, characteristic, complemented
6	#13133194	C₂	1	2	yes	maximal, complemented
7	D₄×Dic₅₃	C₂	1	2	yes	maximal, complemented
8	C₂×#11719919	C₂	1	1	yes	maximal, characteristic, complemented
9	#121×C₅₃	C₂	1	1	yes	maximal, characteristic
10	C₂×D₄×C₅₃	C₂²	1	2	yes	—
11	C₂×D₄×C₅₃	C₂²	1	1	yes	characteristic
12	C₂²×C₄×C₅₃	C₂²	1	1	yes	characteristic
13	C₂²×Dic₅₃	C₂²	1	1	yes	characteristic
14	C₂²×Dic₅₃	C₂²	1	1	yes	characteristic
15	C₂²×Dic₅₃	C₂²	1	2	yes	—
16	C₂×Dic₁₀₆	C₂²	1	1	yes	characteristic
17	C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
18	C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
19	C₂²⋊Dic₅₃	C₂²	1	2	yes	—
20	C₂²⋊Dic₅₃	C₂²	1	2	yes	—
21	C₂²⋊Dic₅₃	C₂²	1	2	yes	—
22	C₂²⋊Dic₅₃	C₂²	1	2	yes	—
23	C₄⋊Dic₅₃	C₂²	1	2	yes	—
24	C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
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	#11719919	C₂²	1	2	yes	—
30	C₄×Dic₅₃	C₂²	1	2	yes	—
31	C₂³×C₅₃	C₂³	1	2	yes	—
32	C₂³×C₅₃	C₂³	1	1	yes	characteristic
33	D₄×C₅₃	—	2	4	no	—
34	D₄×C₅₃	—	4	1	no	—
35	C₂×C₄×C₅₃	—	4	1	no	—
36	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
37	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₅₃	C₂³	1	2	yes	—
39	C₂×Dic₅₃	—	2	2	no	—
40	C₂×Dic₅₃	—	4	2	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	C₂×Dic₅₃	C₂³	1	2	yes	—
46	Dic₁₀₆	—	4	1	no	—
47	C₂²×C₅₃	—	2	1	no	—
48	C₂²×C₅₃	—	2	1	no	—
49	C₂²×C₅₃	—	2	2	no	—
50	C₂²×C₅₃	—	2	2	no	—
51	C₂²×C₅₃	—	2	2	no	—
52	C₂²×C₅₃	#41	1	2	yes	—
53	C₂²×C₅₃	C₂⁴	1	1	yes	characteristic
54	C₄×C₅₃	—	2	2	no	—
55	C₄×C₅₃	—	2	1	no	—
56	C₄×C₅₃	—	2	1	no	—
57	Dic₅₃	—	2	2	no	—
58	Dic₅₃	—	2	2	no	—
59	Dic₅₃	—	2	2	no	—
60	Dic₅₃	—	2	2	no	—
61	C₂×C₅₃	—	2	2	no	—
62	C₂×C₅₃	—	4	2	no	—
63	C₂×C₅₃	C₂×#41	1	1	yes	characteristic
64	C₂×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
65	C₂×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
66	#539	—	53	1	no	maximal, Sylow
67	C₅₃	#539	1	1	yes	characteristic, complemented, Sylow
68	#126	—	53	2	no	—
69	#125	—	53	2	no	—
70	#123	—	53	2	no	—
71	#123	—	53	2	no	—
72	C₂²⋊Q₈	—	53	1	no	—
73	C₂²⋊Q₈	—	53	2	no	—
74	#121	D₅₃	1	1	yes	characteristic
75	C₄×D₄	—	53	2	no	—
76	C₂×C₄⋊C₄	—	53	1	no	—
77	C₂×Q₈	—	53	1	no	—
78	C₂×D₄	C₂×D₅₃	1	2	yes	—
79	C₂×D₄	C₂×D₅₃	1	1	yes	characteristic
80	C₂²×C₄	—	53	1	no	—
81	C₂²×C₄	—	53	1	no	—
82	C₂²×C₄	—	53	2	no	—
83	C₂²×C₄	C₂×D₅₃	1	1	yes	characteristic
84	C₄⋊C₄	—	53	2	no	—
85	C₄⋊C₄	—	53	2	no	—
86	C₄⋊C₄	—	53	2	no	—
87	C₄⋊C₄	—	53	2	no	—
88	C₄⋊C₄	—	53	2	no	—
89	C₄⋊C₄	—	53	1	no	—
90	C₄⋊C₄	—	53	2	no	—
91	C₄⋊C₄	C₂×D₅₃	1	1	yes	characteristic
92	C₂²⋊C₄	—	53	2	no	—
93	C₂²⋊C₄	—	53	2	no	—
94	C₂²⋊C₄	—	53	2	no	—
95	C₂²⋊C₄	—	53	2	no	—
96	C₂²⋊C₄	C₂×D₅₃	1	2	yes	—
97	C₄²	—	53	2	no	—
98	C₂³	C₂²×D₅₃	1	2	yes	—
99	C₂³	C₂²×D₅₃	1	1	yes	characteristic
100	Q₈	—	212	1	no	—
101	D₄	—	2	4	no	—
102	D₄	—	4	1	no	—
103	C₂×C₄	—	4	1	no	—
104	C₂×C₄	—	53	2	no	—
105	C₂×C₄	—	53	2	no	—
106	C₂×C₄	—	53	2	no	—
107	C₂×C₄	—	53	2	no	—
108	C₂×C₄	—	106	2	no	—
109	C₂×C₄	—	212	1	no	—
110	C₂×C₄	—	212	2	no	—
111	C₂×C₄	C₂²×D₅₃	1	2	yes	—
112	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
113	C₂×C₄	C₂²×D₅₃	1	1	yes	characteristic
114	C₂²	—	2	1	no	—
115	C₂²	—	2	1	no	—
116	C₂²	—	2	2	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	2	2	no	—
119	C₂²	#11719947	1	2	yes	—
120	C₂²	C₂³×D₅₃	1	1	yes	characteristic
121	C₄	—	2	2	no	—
122	C₄	—	2	1	no	—
123	C₄	—	2	1	no	—
124	C₄	—	106	2	no	—
125	C₄	—	106	2	no	—
126	C₄	—	106	2	no	—
127	C₄	—	106	2	no	—
128	C₂	—	2	2	no	—
129	C₂	—	4	2	no	—
130	C₂	C₂×#11719947	1	1	yes	characteristic
131	C₂	#13133262	1	1	yes	characteristic
132	C₂	#13133264	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13133210

C₂

yes

maximal, complemented

#13133207

C₂

yes

maximal, complemented

#13133198

C₂

yes

maximal, complemented

#13133198

C₂

yes

maximal, complemented

#13133196

C₂

yes

maximal, characteristic, complemented

#13133194

C₂

yes

maximal, complemented

D₄×Dic₅₃

C₂

yes

maximal, complemented

C₂×#11719919

C₂

yes

maximal, characteristic, complemented

#121×C₅₃

C₂

yes

maximal, characteristic

C₂×D₄×C₅₃

C₂²

yes

—

C₂×D₄×C₅₃

C₂²

yes

characteristic

C₂²×C₄×C₅₃

C₂²

yes

characteristic

C₂²×Dic₅₃

C₂²

yes

characteristic

C₂²×Dic₅₃

C₂²

yes

characteristic

C₂²×Dic₅₃

C₂²

yes

—

C₂×Dic₁₀₆

C₂²

yes

characteristic

C₄⋊C₄×C₅₃

C₂²

yes

characteristic

C₂²⋊C₄×C₅₃

C₂²

yes

—

C₂²⋊Dic₅₃

C₂²

yes

—

C₂²⋊Dic₅₃

C₂²

yes

—

C₂²⋊Dic₅₃

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

—

C₄×Dic₅₃

C₂²

yes

—

C₂³×C₅₃

C₂³

yes

—

C₂³×C₅₃

C₂³

yes

characteristic

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂³

yes

characteristic

C₂×C₄×C₅₃

C₂³

yes

characteristic

C₂×C₄×C₅₃

C₂³

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂³

yes

—

C₂×Dic₅₃

C₂³

yes

—

C₂×Dic₅₃

C₂³

yes

—

C₂×Dic₅₃

C₂³

yes

—

Dic₁₀₆

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

#41

yes

—

C₂²×C₅₃

C₂⁴

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

C₂×C₅₃

—

C₂×C₅₃

—

C₂×C₅₃

C₂×#41

yes

characteristic

C₂×C₅₃

H(5,𝔽₂)

yes

characteristic

C₂×C₅₃

2₋¹⁺⁴

yes

—

C₅₃

yes

characteristic, complemented, Sylow

—

—

—

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

#121

D₅₃

yes

characteristic

C₄×D₄

—

C₂×C₄⋊C₄

—

C₂×Q₈

—

C₂×D₄

C₂×D₅₃

yes

—

C₂×D₄

C₂×D₅₃

yes

characteristic

C₂²×C₄

—

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

C₂×D₅₃

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₅₃

yes

—

C₄²

—

C₂³

C₂²×D₅₃

yes

—

C₂³

C₂²×D₅₃

yes

characteristic

100

Q₈

—

212

—

101

D₄

—

102

D₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

106

—

109

C₂×C₄

—

212

—

110

C₂×C₄

—

212

—

111

C₂×C₄

C₂²×D₅₃

yes

—

112

C₂×C₄

C₂²×D₅₃

yes

characteristic

113

C₂×C₄

C₂²×D₅₃

yes

characteristic

114

C₂²

—

115

C₂²

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

#11719947

yes

—

120

C₂²

C₂³×D₅₃

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

106

—

125

C₄

—

106

—

126

C₄

—

106

—

127

C₄

—

106

—

128

C₂

—

129

C₂

—

130

C₂

C₂×#11719947

yes

characteristic

131

C₂

#13133262

yes

characteristic

132

C₂

#13133264

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅₃
commutator, socle, cosocle	C₂²×C₅₃	C₂⁴
Fitting, prime radical	#121×C₅₃	C₂
hypercenter	#121	D₅₃
hypocenter	C₅₃	#539
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #121

subgroup

G/H

center, Frattini

C₂²

C₂³×D₅₃

commutator, socle, cosocle

C₂²×C₅₃

C₂⁴

Fitting, prime radical

#121×C₅₃

C₂

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂² ◁ #121

N	H ≅ G/N	type
1. #13133210	C₂	semidirect
2. #13133207	C₂	semidirect
3. #13133198	C₂	semidirect
4. #13133198	C₂	semidirect
5. #13133196	C₂	semidirect
6. #13133194	C₂	semidirect
7. D₄×Dic₅₃	C₂	semidirect
8. C₂×#11719919	C₂	semidirect
67. C₅₃	#539	semidirect

H ≅ G/N

type

1. #13133210

C₂

semidirect

2. #13133207

C₂

semidirect

3. #13133198

C₂

semidirect

4. #13133198

C₂

semidirect

5. #13133196

C₂

semidirect

6. #13133194

C₂

semidirect

7. D₄×Dic₅₃

C₂

semidirect

8. C₂×#11719919

C₂

semidirect

67. C₅₃

#539

semidirect

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

subgroup	index
#13133210	2
#13133207	2
#13133198	2
#13133196	2
#13133194	2
D₄×Dic₅₃	2
C₂×#11719919	2
#121×C₅₃	2
#539	53

subgroup

index

D₄×Dic₅₃

#121×C₅₃

Subgroups — #16811218

Conjugacy classes of subgroups

Complements

Maximal subgroups