Group #20291565 — subgroups

Subgroups — #20291565

Order 6784.

500 conjugacy classes of subgroups; only the 204 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 204 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	#16811651	C₂	1	8	yes	maximal, complemented
2	#16811639	C₂	1	2	yes	maximal, complemented
3	D₅₃⋊(C₂²⋊C₈)	C₂	1	4	yes	maximal, complemented
4	C₂×#13133179	C₂	1	1	yes	maximal, characteristic
5	C₂×D₅₃⋊C₈	C₂²	1	8	yes	complemented
6	C₄⋊#88324	C₂²	1	4	yes	complemented
7	#13133274	C₂²	1	8	yes	complemented
8	C₂²⋊#88324	C₂²	1	4	yes	—
9	C₄×#88324	C₂²	1	4	yes	complemented
10	C₂²×D₂₁₂	C₄	1	1	yes	characteristic
11	C₂²×C₄×D₅₃	C₂²	1	2	yes	—
12	#13133179	C₄	1	4	yes	—
13	#13133179	C₂²	1	4	yes	—
14	C₂×#11719921	C₄	1	2	yes	—
15	C₂×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₅₃	C₄	1	1	yes	characteristic
17	C₂³×D₅₃	C₂×C₄	1	2	yes	—
18	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
19	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
20	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
21	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
22	C₂×D₂₁₂	C₈	1	8	yes	complemented
23	C₂×D₂₁₂	C₂×C₄	1	4	yes	—
24	C₂×C₄×D₅₃	—	4	4	no	—
25	C₂×C₄×D₅₃	C₂×C₄	1	4	yes	—
26	C₂×C₄×D₅₃	C₂³	1	4	yes	—
27	C₂×#88324	C₂³	1	8	yes	—
28	D₅₃⋊C₈	—	4	8	no	—
29	C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
30	#11719921	C₂×C₄	1	8	yes	—
31	C₄×Dic₅₃	C₂×C₄	1	2	yes	—
32	C₄×Dic₅₃	C₂³	1	2	yes	—
33	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
34	C₂²×D₅₃	—	4	2	no	—
35	C₂²×D₅₃	—	4	2	no	—
36	C₂²×D₅₃	C₂×C₈	1	8	yes	complemented
37	C₂²×D₅₃	C₂²×C₄	1	4	yes	—
38	C₂×C₄×C₅₃	—	4	2	no	—
39	C₂×C₄×C₅₃	C₂×C₈	1	4	yes	complemented
40	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
41	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
42	C₂×Dic₅₃	—	2	2	no	—
43	C₂×Dic₅₃	—	4	1	no	—
44	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
46	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
47	D₂₁₂	C₂×C₈	1	16	yes	complemented
48	C₄×D₅₃	—	2	8	no	—
49	C₄×D₅₃	—	4	4	no	—
50	#88324	—	2	8	no	—
51	C₂²×C₅₃	—	2	1	no	—
52	C₂²×C₅₃	—	2	1	no	—
53	C₂²×C₅₃	C₂²×C₈	1	2	yes	—
54	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
55	C₂×D₅₃	—	2	4	no	—
56	C₂×D₅₃	—	2	4	no	—
57	C₂×D₅₃	—	4	8	no	—
58	C₂×D₅₃	C₂²×C₈	1	8	yes	—
59	C₄×C₅₃	—	2	4	no	—
60	C₄×C₅₃	C₂²×C₈	1	4	yes	—
61	Dic₅₃	—	2	1	no	—
62	Dic₅₃	—	2	2	no	—
63	Dic₅₃	H(5,𝔽₂)	1	2	yes	—
64	#2972	—	53	1	no	maximal, Sylow
65	C₂×C₅₃	—	2	2	no	—
66	C₂×C₅₃	#518	1	1	yes	characteristic
67	C₂×C₅₃	C₂³×C₈	1	1	yes	characteristic
68	C₂×C₅₃	#568	1	1	yes	characteristic
69	D₅₃	—	2	8	no	—
70	C₂×C₄×D₄	—	53	1	no	—
71	C₈×D₄	—	53	8	no	—
72	#431	—	53	2	no	—
73	C₂×C₂²⋊C₈	—	53	4	no	—
74	C₅₃	#2972	1	1	yes	characteristic, complemented, Sylow
75	C₂²×D₄	—	53	1	no	—
76	C₂³×C₄	—	53	2	no	—
77	C₂²×C₈	—	53	8	no	—
78	C₄×D₄	—	53	4	no	—
79	C₄×D₄	—	53	4	no	—
80	C₂×C₄⋊C₄	#6157	1	1	yes	characteristic
81	C₂×C₂²⋊C₄	—	53	2	no	—
82	C₂×C₄²	—	53	1	no	—
83	C₄⋊C₈	—	53	4	no	—
84	C₂²⋊C₈	—	53	4	no	—
85	C₂²⋊C₈	—	53	8	no	—
86	C₄×C₈	—	53	4	no	—
87	C₂⁴	—	53	2	no	—
88	C₂×D₄	—	53	4	no	—
89	C₂×D₄	—	53	8	no	—
90	C₂²×C₄	—	53	1	no	—
91	C₂²×C₄	—	53	1	no	—
92	C₂²×C₄	—	53	4	no	—
93	C₂²×C₄	—	53	4	no	—
94	C₂²×C₄	—	212	4	no	—
95	C₂²×C₄	C₂×#6157	1	2	yes	—
96	C₂²×C₄	C₂×#6157	1	1	yes	characteristic
97	C₂×C₈	—	53	8	no	—
98	C₂×C₈	—	212	8	no	—
99	C₄⋊C₄	C₂×#6157	1	4	yes	—
100	C₂²⋊C₄	—	53	8	no	—
101	C₄²	—	53	2	no	—
102	C₄²	—	53	2	no	—
103	C₂³	—	53	4	no	—
104	C₂³	—	53	8	no	—
105	C₂³	—	212	2	no	—
106	C₂³	—	212	2	no	—
107	C₂³	C₂²×#6157	1	1	yes	characteristic
108	D₄	—	53	16	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	—	53	1	no	—
111	C₂×C₄	—	53	2	no	—
112	C₂×C₄	—	53	1	no	—
113	C₂×C₄	—	106	8	no	—
114	C₂×C₄	—	106	2	no	—
115	C₂×C₄	—	212	4	no	—
116	C₂×C₄	—	212	1	no	—
117	C₂×C₄	D₅₃⋊C₈	1	4	yes	complemented
118	C₂×C₄	C₂²×#6157	1	4	yes	—
119	C₂×C₄	C₂²×#6157	1	2	yes	—
120	C₈	—	106	8	no	—
121	C₂²	—	2	1	no	—
122	C₂²	—	2	1	no	—
123	C₂²	—	53	8	no	—
124	C₂²	—	106	4	no	—
125	C₂²	—	106	4	no	—
126	C₂²	—	212	8	no	—
127	C₂²	C₂×D₅₃⋊C₈	1	2	yes	—
128	C₂²	C₂³×#6157	1	1	yes	characteristic
129	C₄	—	2	4	no	—
130	C₄	—	53	2	no	—
131	C₄	—	106	1	no	—
132	C₄	—	106	2	no	—
133	C₄	C₂×D₅₃⋊C₈	1	4	yes	—
134	C₂	—	2	2	no	—
135	C₂	—	106	8	no	—
136	C₂	#16811690	1	1	yes	characteristic
137	C₂	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
138	C₂	#16811703	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811651

C₂

yes

maximal, complemented

#16811639

C₂

yes

maximal, complemented

D₅₃⋊(C₂²⋊C₈)

C₂

yes

maximal, complemented

C₂×#13133179

C₂

yes

maximal, characteristic

C₂×D₅₃⋊C₈

C₂²

yes

complemented

C₄⋊#88324

C₂²

yes

complemented

#13133274

C₂²

yes

complemented

C₂²⋊#88324

C₂²

yes

—

C₄×#88324

C₂²

yes

complemented

C₂²×D₂₁₂

C₄

yes

characteristic

C₂²×C₄×D₅₃

C₂²

yes

—

#13133179

C₄

yes

—

#13133179

C₂²

yes

—

C₂×#11719921

C₄

yes

—

C₂×C₄×Dic₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₅₃

C₄

yes

characteristic

C₂³×D₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂²×Dic₅₃

C₂×C₄

yes

characteristic

C₂²×Dic₅₃

C₂³

yes

characteristic

C₂×D₂₁₂

C₈

yes

complemented

C₂×D₂₁₂

C₂×C₄

yes

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

C₂×C₄

yes

—

C₂×C₄×D₅₃

C₂³

yes

—

C₂×#88324

C₂³

yes

—

D₅₃⋊C₈

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

#11719921

C₂×C₄

yes

—

C₄×Dic₅₃

C₂×C₄

yes

—

C₄×Dic₅₃

C₂³

yes

—

C₂³×C₅₃

C₂²×C₄

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

C₂×C₈

yes

complemented

C₂²×D₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×C₈

yes

complemented

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

C₂²×C₄

yes

—

C₂×Dic₅₃

C₂⁴

yes

characteristic

D₂₁₂

C₂×C₈

yes

complemented

C₄×D₅₃

—

C₄×D₅₃

—

#88324

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂²×C₈

yes

—

C₂²×C₅₃

C₂³×C₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

C₂²×C₈

yes

—

C₄×C₅₃

—

C₄×C₅₃

C₂²×C₈

yes

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

H(5,𝔽₂)

yes

—

#2972

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

#518

yes

characteristic

C₂×C₅₃

C₂³×C₈

yes

characteristic

C₂×C₅₃

#568

yes

characteristic

D₅₃

—

C₂×C₄×D₄

—

C₈×D₄

—

#431

—

C₂×C₂²⋊C₈

—

C₅₃

#2972

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂³×C₄

—

C₂²×C₈

—

C₄×D₄

—

C₄×D₄

—

C₂×C₄⋊C₄

#6157

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

212

—

C₂²×C₄

C₂×#6157

yes

—

C₂²×C₄

C₂×#6157

yes

characteristic

C₂×C₈

—

C₂×C₈

—

212

—

C₄⋊C₄

C₂×#6157

yes

—

100

C₂²⋊C₄

—

101

C₄²

—

102

C₄²

—

103

C₂³

—

104

C₂³

—

105

C₂³

—

212

—

106

C₂³

—

212

—

107

C₂³

C₂²×#6157

yes

characteristic

108

D₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

106

—

114

C₂×C₄

—

106

—

115

C₂×C₄

—

212

—

116

C₂×C₄

—

212

—

117

C₂×C₄

D₅₃⋊C₈

yes

complemented

118

C₂×C₄

C₂²×#6157

yes

—

119

C₂×C₄

C₂²×#6157

yes

—

120

C₈

—

106

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

106

—

125

C₂²

—

106

—

126

C₂²

—

212

—

127

C₂²

C₂×D₅₃⋊C₈

yes

—

128

C₂²

C₂³×#6157

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

106

—

132

C₄

—

106

—

133

C₄

C₂×D₅₃⋊C₈

yes

—

134

C₂

—

135

C₂

—

106

—

136

C₂

#16811690

yes

characteristic

137

C₂

C₂²×D₅₃⋊C₈

yes

characteristic

138

C₂

#16811703

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#6157
commutator	C₂×C₅₃	C₂³×C₈
Fitting	C₂×C₄⋊C₄×C₅₃	C₄
socle	C₂²×C₅₃	C₂³×C₄
hypercenter	C₂×C₄⋊C₄	#6157
hypocenter	C₅₃	#2972
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂²

C₂³×#6157

commutator

C₂×C₅₃

C₂³×C₈

Fitting

C₂×C₄⋊C₄×C₅₃

C₄

socle

C₂²×C₅₃

C₂³×C₄

C₂×C₄⋊C₄

C₅₃

G ▷ C₂×C₅₃ ▷ C₅₃

C₁ ◁ C₂² ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
1. #16811651	C₂	semidirect
2. #16811639	C₂	semidirect
3. D₅₃⋊(C₂²⋊C₈)	C₂	semidirect
5. C₂×D₅₃⋊C₈	C₂²	semidirect
6. C₄⋊#88324	C₂²	semidirect
7. #13133274	C₂²	semidirect
9. C₄×#88324	C₂²	semidirect
22. C₂×D₂₁₂	C₈	semidirect
36. C₂²×D₅₃	C₂×C₈	semidirect
39. C₂×C₄×C₅₃	C₂×C₈	semidirect
47. D₂₁₂	C₂×C₈	semidirect
74. C₅₃	#2972	semidirect
117. C₂×C₄	D₅₃⋊C₈	semidirect

H ≅ G/N

type

1. #16811651

C₂

semidirect

2. #16811639

C₂

semidirect

3. D₅₃⋊(C₂²⋊C₈)

C₂

semidirect

5. C₂×D₅₃⋊C₈

C₂²

semidirect

6. C₄⋊#88324

C₂²

semidirect

7. #13133274

C₂²

semidirect

9. C₄×#88324

C₂²

semidirect

22. C₂×D₂₁₂

C₈

semidirect

36. C₂²×D₅₃

C₂×C₈

semidirect

39. C₂×C₄×C₅₃

C₂×C₈

semidirect

47. D₂₁₂

C₂×C₈

semidirect

74. C₅₃

#2972

semidirect

117. C₂×C₄

D₅₃⋊C₈

semidirect

Subgroups — #20291565

Conjugacy classes of subgroups

Complements