Group #20291507 — subgroups

Subgroups — #20291507

Order 6784.

348 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 162 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	#16811675	C₂	1	4	yes	maximal, complemented
2	#16811673	C₂	1	4	yes	maximal, complemented
3	#16811630	C₂	1	2	yes	maximal
4	#16811627	C₂	1	1	yes	maximal, characteristic
5	#16811588	C₂	1	2	yes	maximal
6	#16811582	C₂	1	1	yes	maximal, characteristic
7	#16811179	C₂	1	1	yes	maximal, characteristic
8	C₂×#11719936	C₂²	1	2	yes	—
9	C₂×D₅₃⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×D₅₃⋊C₈	C₂²	1	1	yes	characteristic
11	#13133284	C₂²	1	4	yes	—
12	C₄⋊#88324	C₂²	1	4	yes	—
13	C₄⋊#88324	C₂²	1	4	yes	—
14	#13133272	C₂²	1	2	yes	—
15	#13133272	C₂²	1	2	yes	—
16	#13133272	C₂²	1	4	yes	—
17	C₄×#88324	C₂²	1	4	yes	—
18	#13133216	C₄	1	2	yes	—
19	#13133209	C₄	1	1	yes	characteristic
20	#13133195	C₄	1	4	yes	—
21	#13133182	C₂²	1	4	yes	—
22	#13133175	C₂²	1	2	yes	—
23	C₄²×D₅₃	C₂²	1	1	yes	characteristic
24	#125×C₅₃	C₄	1	1	yes	characteristic
25	C₂×C₄×D₅₃	C₂×C₄	1	2	yes	—
26	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
27	C₂×Dic₁₀₆	C₂×C₄	1	4	yes	—
28	C₂×#88324	C₂³	1	4	yes	—
29	C₂×#88324	C₂³	1	1	yes	characteristic
30	C₂×#88324	C₂³	1	1	yes	characteristic
31	C₂×#88324	C₂³	1	2	yes	—
32	#11719936	—	2	4	no	—
33	D₅₃⋊C₈	—	2	2	no	—
34	D₅₃⋊C₈	Q₈	1	4	yes	complemented
35	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
36	C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
37	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
38	#11719921	C₂×C₄	1	4	yes	—
39	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
40	#11719919	C₂×C₄	1	4	yes	—
41	#11719919	C₂×C₄	1	4	yes	—
42	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
43	C₄×Dic₅₃	C₂³	1	2	yes	—
44	C₄×Dic₅₃	C₂³	1	4	yes	—
45	C₂²×D₅₃	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
47	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
48	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×Dic₅₃	C₂²×C₄	1	2	yes	—
50	C₂×Dic₅₃	C₂²×C₄	1	4	yes	—
51	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
52	C₄×D₅₃	—	4	2	no	—
53	C₄×D₅₃	C₂×Q₈	1	2	yes	—
54	C₄×D₅₃	#41	1	2	yes	—
55	Dic₁₀₆	—	4	4	no	—
56	#88324	—	2	4	no	—
57	#88324	—	2	1	no	—
58	#88324	—	2	1	no	—
59	#88324	C₂×Q₈	1	4	yes	—
60	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
61	C₂×D₅₃	—	2	2	no	—
62	C₂×D₅₃	C₄×Q₈	1	2	yes	—
63	C₄×C₅₃	—	2	4	no	—
64	C₄×C₅₃	—	2	2	no	—
65	C₄×C₅₃	C₄×Q₈	1	2	yes	—
66	Dic₅₃	—	2	2	no	—
67	Dic₅₃	—	2	4	no	—
68	Dic₅₃	C₂²×Q₈	1	1	yes	characteristic
69	Dic₅₃	C₂×#41	1	1	yes	characteristic
70	#2961	—	53	1	no	maximal, Sylow
71	C₂×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
72	C₂×C₅₃	#568	1	2	yes	—
73	D₅₃	—	2	2	no	—
74	#533	—	53	1	no	—
75	#446	—	53	4	no	—
76	#446	—	53	4	no	—
77	#424	—	53	2	no	—
78	#423	—	53	1	no	—
79	#405	—	53	2	no	—
80	C₂×#97	—	53	1	no	—
81	C₅₃	#2961	1	1	yes	characteristic, complemented, Sylow
82	C₂×M₁₆	—	53	2	no	—
83	C₂²×C₈	—	53	1	no	—
84	C₂²×C₈	—	53	1	no	—
85	C₄⋊Q₈	—	53	2	no	—
86	#125	—	53	1	no	—
87	#125	#6157	1	1	yes	characteristic
88	C₂²⋊Q₈	—	53	4	no	—
89	C₄×Q₈	—	53	4	no	—
90	#117	—	53	2	no	—
91	C₂×C₄²	—	53	1	no	—
92	C₄⋊C₈	—	53	4	no	—
93	C₄⋊C₈	—	53	4	no	—
94	C₄⋊C₈	—	53	4	no	—
95	#97	—	53	2	no	—
96	#97	—	53	2	no	—
97	#97	—	53	4	no	—
98	C₄×C₈	—	53	4	no	—
99	C₂×Q₈	—	53	4	no	—
100	C₂²×C₄	—	53	1	no	—
101	C₂²×C₄	—	53	2	no	—
102	M₁₆	—	106	4	no	—
103	C₂×C₈	—	53	2	no	—
104	C₂×C₈	—	53	4	no	—
105	C₂×C₈	—	53	4	no	—
106	C₂×C₈	—	53	1	no	—
107	C₂×C₈	—	53	1	no	—
108	C₂×C₈	—	106	2	no	—
109	C₄⋊C₄	—	53	2	no	—
110	C₄⋊C₄	—	53	4	no	—
111	C₄⋊C₄	—	53	4	no	—
112	C₄⋊C₄	C₂×#6157	1	2	yes	—
113	C₄⋊C₄	C₂×#6157	1	4	yes	—
114	C₂²⋊C₄	—	53	4	no	—
115	C₄²	—	53	4	no	—
116	C₄²	—	53	2	no	—
117	C₄²	—	53	1	no	—
118	C₄²	C₂×#6157	1	1	yes	characteristic
119	C₂³	—	53	1	no	—
120	Q₈	—	212	4	no	—
121	C₂×C₄	—	53	1	no	—
122	C₂×C₄	—	53	2	no	—
123	C₂×C₄	—	53	2	no	—
124	C₂×C₄	—	53	2	no	—
125	C₂×C₄	—	53	4	no	—
126	C₂×C₄	—	212	2	no	—
127	C₂×C₄	C₂²×#6157	1	2	yes	—
128	C₂×C₄	C₂²×#6157	1	4	yes	—
129	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
130	C₈	—	53	4	no	—
131	C₈	—	106	1	no	—
132	C₈	—	106	1	no	—
133	C₈	—	106	4	no	—
134	C₂²	—	53	2	no	—
135	C₂²	—	106	2	no	—
136	C₂²	C₂³×#6157	1	1	yes	characteristic
137	C₄	—	2	2	no	—
138	C₄	—	2	4	no	—
139	C₄	—	53	1	no	—
140	C₄	—	53	1	no	—
141	C₄	—	106	2	no	—
142	C₄	—	106	4	no	—
143	C₄	Q₈×#6157	1	2	yes	—
144	C₂	—	106	2	no	—
145	C₂	C₂×Q₈×#6157	1	1	yes	characteristic
146	C₂	#16811702	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16811675

C₂

yes

maximal, complemented

#16811673

C₂

yes

maximal, complemented

#16811630

C₂

yes

maximal

#16811627

C₂

yes

maximal, characteristic

#16811588

C₂

yes

maximal

#16811582

C₂

yes

maximal, characteristic

#16811179

C₂

yes

maximal, characteristic

C₂×#11719936

C₂²

yes

—

C₂×D₅₃⋊C₈

C₂²

yes

characteristic

C₂×D₅₃⋊C₈

C₂²

yes

characteristic

#13133284

C₂²

yes

—

C₄⋊#88324

C₂²

yes

—

C₄⋊#88324

C₂²

yes

—

#13133272

C₂²

yes

—

#13133272

C₂²

yes

—

#13133272

C₂²

yes

—

C₄×#88324

C₂²

yes

—

#13133216

C₄

yes

—

#13133209

C₄

yes

characteristic

#13133195

C₄

yes

—

#13133182

C₂²

yes

—

#13133175

C₂²

yes

—

C₄²×D₅₃

C₂²

yes

characteristic

#125×C₅₃

C₄

yes

characteristic

C₂×C₄×D₅₃

C₂×C₄

yes

—

C₂×C₄×D₅₃

C₂³

yes

characteristic

C₂×Dic₁₀₆

C₂×C₄

yes

—

C₂×#88324

C₂³

yes

—

C₂×#88324

C₂³

yes

characteristic

C₂×#88324

C₂³

yes

characteristic

C₂×#88324

C₂³

yes

—

#11719936

—

D₅₃⋊C₈

—

D₅₃⋊C₈

Q₈

yes

complemented

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₄²×C₅₃

C₂×C₄

yes

characteristic

#11719921

C₂×C₄

yes

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

#11719919

C₂×C₄

yes

—

#11719919

C₂×C₄

yes

—

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

C₂³

yes

—

C₄×Dic₅₃

C₂³

yes

—

C₂²×D₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

C₂²×C₄

yes

—

C₂×Dic₅₃

C₂²×C₄

yes

—

C₂×Dic₅₃

C₂⁴

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

C₂×Q₈

yes

—

C₄×D₅₃

#41

yes

—

Dic₁₀₆

—

—

—

—

C₂×Q₈

yes

—

C₂²×C₅₃

C₂³×C₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

C₄×Q₈

yes

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

C₄×Q₈

yes

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

C₂²×Q₈

yes

characteristic

Dic₅₃

C₂×#41

yes

characteristic

#2961

—

maximal, Sylow

C₂×C₅₃

C₂×C₄×Q₈

yes

characteristic

C₂×C₅₃

#568

yes

—

D₅₃

—

—

—

—

—

—

—

C₂×#97

—

C₅₃

#2961

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

—

C₄⋊Q₈

—

—

yes

C₂²⋊Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

#97

—

#97

—

C₄×C₈

—

C₂×Q₈

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

M₁₆

—

106

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

106

—

109

C₄⋊C₄

—

110

C₄⋊C₄

—

111

C₄⋊C₄

—

112

C₄⋊C₄

C₂×#6157

yes

—

113

C₄⋊C₄

C₂×#6157

yes

—

114

C₂²⋊C₄

—

115

C₄²

—

116

C₄²

—

117

C₄²

—

118

C₄²

C₂×#6157

yes

characteristic

119

C₂³

—

120

Q₈

—

212

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

212

—

127

C₂×C₄

C₂²×#6157

yes

—

128

C₂×C₄

C₂²×#6157

yes

—

129

C₂×C₄

C₂²×#6157

yes

characteristic

130

C₈

—

131

C₈

—

106

—

132

C₈

—

106

—

133

C₈

—

106

—

134

C₂²

—

135

C₂²

—

106

—

136

C₂²

C₂³×#6157

yes

characteristic

137

C₄

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

—

106

—

142

C₄

—

106

—

143

C₄

Q₈×#6157

yes

—

144

C₂

—

106

—

145

C₂

C₂×Q₈×#6157

yes

characteristic

146

C₂

#16811702

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×#6157
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	#125×C₅₃	C₄
hypercenter	#125	#6157
hypocenter	C₅₃	#2961
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #125

subgroup

G/H

center, Frattini

C₂²

C₂³×#6157

commutator, socle

C₂²×C₅₃

C₂³×C₄

Fitting

#125×C₅₃

C₄

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂² ◁ #125

N	H ≅ G/N	type
1. #16811675	C₂	semidirect
2. #16811673	C₂	semidirect
34. D₅₃⋊C₈	Q₈	semidirect
81. C₅₃	#2961	semidirect

H ≅ G/N

type

1. #16811675

C₂

semidirect

2. #16811673

C₂

semidirect

34. D₅₃⋊C₈

Q₈

semidirect

81. C₅₃

#2961

semidirect

Subgroups — #20291507

Conjugacy classes of subgroups

Complements