Group #16811367 — subgroups

Subgroups — #16811367

Order 3392.

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

normal subgroups only

This group has 220 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	#13133209	C₂	1	1	yes	maximal, characteristic, complemented
2	#13133205	C₂	1	1	yes	maximal, characteristic, complemented
3	#13133201	C₂	1	2	yes	maximal, complemented
4	#13133199	C₂	1	2	yes	maximal, complemented
5	#13133194	C₂	1	2	yes	maximal, complemented
6	#13133193	C₂	1	2	yes	maximal, complemented
7	#13133190	C₂	1	2	yes	maximal, complemented
8	#13133189	C₂	1	2	yes	maximal, complemented
9	#124×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×Q₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₂×D₄×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	2	yes	—
15	C₂×D₂₁₂	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₅₃	C₂²	1	2	yes	—
17	C₂×Dic₁₀₆	C₂²	1	1	yes	characteristic
18	C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
19	C₂²⋊C₄×C₅₃	C₂²	1	2	yes	—
20	C₄²×C₅₃	C₂²	1	1	yes	characteristic
21	C₂²⋊Dic₅₃	C₂²	1	2	yes	—
22	#11719921	C₂²	1	2	yes	—
23	#11719921	C₂²	1	2	yes	—
24	#11719921	C₂²	1	2	yes	—
25	C₄⋊Dic₅₃	C₂²	1	2	yes	—
26	C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
27	C₄⋊Dic₅₃	C₂²	1	1	yes	characteristic
28	#11719919	C₂²	1	2	yes	—
29	#11719919	C₂²	1	2	yes	—
30	#11719919	C₂²	1	2	yes	—
31	C₂³×C₅₃	C₂³	1	2	yes	—
32	C₂²×D₅₃	C₂³	1	2	yes	—
33	Q₈×C₅₃	—	4	1	no	—
34	D₄×C₅₃	—	4	1	no	—
35	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
36	C₂×C₄×C₅₃	C₂³	1	2	yes	—
37	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
38	C₂×C₄×C₅₃	C₂³	1	1	yes	characteristic
39	C₂²⋊D₅₃	—	4	2	no	—
40	C₂²⋊D₅₃	—	4	2	no	—
41	C₂×Dic₅₃	—	4	2	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	—	4	1	no	—
48	C₂²×C₅₃	—	2	2	no	—
49	C₂²×C₅₃	—	2	2	no	—
50	C₂²×C₅₃	—	2	2	no	—
51	C₂²×C₅₃	C₂⁴	1	1	yes	characteristic
52	C₂×D₅₃	—	2	2	no	—
53	C₂×D₅₃	—	2	2	no	—
54	C₂×D₅₃	—	2	2	no	—
55	C₄×C₅₃	—	2	1	no	—
56	C₄×C₅₃	—	2	2	no	—
57	C₄×C₅₃	—	2	1	no	—
58	C₄×C₅₃	—	2	1	no	—
59	Dic₅₃	—	2	2	no	—
60	Dic₅₃	—	2	2	no	—
61	Dic₅₃	—	2	2	no	—
62	C₂×C₅₃	—	4	2	no	—
63	C₂×C₅₃	H(5,𝔽₂)	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	D₅₃	—	4	2	no	—
67	#562	—	53	1	no	maximal, Sylow
68	C₅₃	#562	1	1	yes	characteristic, complemented, Sylow
69	#125	—	53	1	no	—
70	#124	—	53	1	no	—
71	#124	D₅₃	1	1	yes	characteristic, complemented
72	#123	—	53	2	no	—
73	#123	—	53	2	no	—
74	C₂²⋊Q₈	—	53	2	no	—
75	C₂²⋊Q₈	—	53	2	no	—
76	#121	—	53	2	no	—
77	#121	—	53	2	no	—
78	C₂×Q₈	—	53	1	no	—
79	C₂×Q₈	C₂×D₅₃	1	1	yes	characteristic
80	C₂×D₄	—	53	2	no	—
81	C₂×D₄	—	53	2	no	—
82	C₂×D₄	—	53	1	no	—
83	C₂×D₄	C₂×D₅₃	1	1	yes	characteristic
84	C₂²×C₄	—	53	2	no	—
85	C₂²×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	2	no	—
90	C₄⋊C₄	—	53	1	no	—
91	C₄⋊C₄	—	53	1	no	—
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	C₂²⋊C₄	C₂×D₅₃	1	2	yes	—
98	C₄²	C₂×D₅₃	1	1	yes	characteristic
99	C₂³	—	53	2	no	—
100	C₂³	C₂²×D₅₃	1	2	yes	—
101	Q₈	—	4	1	no	—
102	Q₈	—	212	1	no	—
103	D₄	—	4	1	no	—
104	D₄	—	212	1	no	—
105	D₄	—	212	2	no	—
106	D₄	—	212	2	no	—
107	C₂×C₄	—	53	2	no	—
108	C₂×C₄	—	53	2	no	—
109	C₂×C₄	—	53	2	no	—
110	C₂×C₄	—	212	2	no	—
111	C₂×C₄	—	212	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	1	yes	characteristic
115	C₂×C₄	C₂²×D₅₃	1	2	yes	—
116	C₂²	—	2	2	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	2	2	no	—
119	C₂²	—	106	2	no	—
120	C₂²	—	106	2	no	—
121	C₂²	—	106	2	no	—
122	C₂²	C₂³×D₅₃	1	1	yes	characteristic
123	C₄	—	2	2	no	—
124	C₄	—	2	1	no	—
125	C₄	—	2	1	no	—
126	C₄	—	2	1	no	—
127	C₄	—	106	2	no	—
128	C₄	—	106	2	no	—
129	C₄	—	106	2	no	—
130	C₂	—	4	2	no	—
131	C₂	—	212	2	no	—
132	C₂	#13133262	1	1	yes	characteristic
133	C₂	#13133263	1	1	yes	characteristic
134	C₂	#13133264	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13133209

C₂

yes

maximal, characteristic, complemented

#13133205

C₂

yes

maximal, characteristic, complemented

#13133201

C₂

yes

maximal, complemented

#13133199

C₂

yes

maximal, complemented

#13133194

C₂

yes

maximal, complemented

#13133193

C₂

yes

maximal, complemented

#13133190

C₂

yes

maximal, complemented

#13133189

C₂

yes

maximal, complemented

#124×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂×Q₈×C₅₃

C₂²

yes

characteristic

C₂×D₄×C₅₃

C₂²

yes

characteristic

C₂×C₂²⋊D₅₃

C₂²

yes

—

C₂×C₂²⋊D₅₃

C₂²

yes

—

C₂²×Dic₅₃

C₂²

yes

—

C₂×D₂₁₂

C₂²

yes

characteristic

C₂×C₄×D₅₃

C₂²

yes

—

C₂×Dic₁₀₆

C₂²

yes

characteristic

C₂²⋊C₄×C₅₃

C₂²

yes

—

C₂²⋊C₄×C₅₃

C₂²

yes

—

C₄²×C₅₃

C₂²

yes

characteristic

C₂²⋊Dic₅₃

C₂²

yes

—

#11719921

C₂²

yes

—

#11719921

C₂²

yes

—

#11719921

C₂²

yes

—

C₄⋊Dic₅₃

C₂²

yes

—

C₄⋊Dic₅₃

C₂²

yes

characteristic

C₄⋊Dic₅₃

C₂²

yes

characteristic

#11719919

C₂²

yes

—

#11719919

C₂²

yes

—

#11719919

C₂²

yes

—

C₂³×C₅₃

C₂³

yes

—

C₂²×D₅₃

C₂³

yes

—

Q₈×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

C₂³

yes

characteristic

C₂×C₄×C₅₃

C₂³

yes

—

C₂×C₄×C₅₃

C₂³

yes

characteristic

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

C₂⁴

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

C₂×C₅₃

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

2₋¹⁺⁴

yes

characteristic

D₅₃

—

#562

—

maximal, Sylow

C₅₃

#562

yes

characteristic, complemented, Sylow

#125

—

#124

—

#124

D₅₃

yes

characteristic, complemented

#123

—

#123

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

#121

—

#121

—

C₂×Q₈

—

C₂×Q₈

C₂×D₅₃

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×D₅₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

C₂×D₅₃

yes

—

C₂²⋊C₄

C₂×D₅₃

yes

—

C₄²

C₂×D₅₃

yes

characteristic

C₂³

—

100

C₂³

C₂²×D₅₃

yes

—

101

Q₈

—

102

Q₈

—

212

—

103

D₄

—

104

D₄

—

212

—

105

D₄

—

212

—

106

D₄

—

212

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

212

—

111

C₂×C₄

—

212

—

112

C₂×C₄

C₂²×D₅₃

yes

characteristic

113

C₂×C₄

C₂²×D₅₃

yes

characteristic

114

C₂×C₄

C₂²×D₅₃

yes

characteristic

115

C₂×C₄

C₂²×D₅₃

yes

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

106

—

120

C₂²

—

106

—

121

C₂²

—

106

—

122

C₂²

C₂³×D₅₃

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

106

—

128

C₄

—

106

—

129

C₄

—

106

—

130

C₂

—

131

C₂

—

212

—

132

C₂

#13133262

yes

characteristic

133

C₂

#13133263

yes

characteristic

134

C₂

#13133264

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×D₅₃
commutator, socle, cosocle	C₂²×C₅₃	C₂⁴
Fitting	#124×C₅₃	C₂
hypercenter	#124	D₅₃
hypocenter	C₅₃	#562
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #124

subgroup

G/H

center, Frattini

C₂²

C₂³×D₅₃

commutator, socle, cosocle

C₂²×C₅₃

C₂⁴

Fitting

#124×C₅₃

C₂

D₅₃

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂² ◁ #124

N	H ≅ G/N	type
1. #13133209	C₂	semidirect
2. #13133205	C₂	semidirect
3. #13133201	C₂	semidirect
4. #13133199	C₂	semidirect
5. #13133194	C₂	semidirect
6. #13133193	C₂	semidirect
7. #13133190	C₂	semidirect
8. #13133189	C₂	semidirect
9. #124×C₅₃	C₂	semidirect
68. C₅₃	#562	semidirect
71. #124	D₅₃	semidirect

H ≅ G/N

type

1. #13133209

C₂

semidirect

2. #13133205

C₂

semidirect

3. #13133201

C₂

semidirect

4. #13133199

C₂

semidirect

5. #13133194

C₂

semidirect

6. #13133193

C₂

semidirect

7. #13133190

C₂

semidirect

8. #13133189

C₂

semidirect

9. #124×C₅₃

C₂

semidirect

68. C₅₃

#562

semidirect

71. #124

D₅₃

semidirect

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

subgroup	index
#13133209	2
#13133205	2
#13133201	2
#13133199	2
#13133194	2
#13133193	2
#13133190	2
#13133189	2
#124×C₅₃	2
#562	53

subgroup

index

#124×C₅₃

Subgroups — #16811367

Conjugacy classes of subgroups

Complements

Maximal subgroups