Group #20291629 — subgroups

Subgroups — #20291629

Order 6784.

314 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 314 conjugacy classes of subgroups — too many to draw. Only its 144 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	#16811674	C₂	1	4	yes	maximal, complemented
3	#16811648	C₂	1	4	yes	maximal
4	#16811646	C₂	1	1	yes	maximal, characteristic
5	#16811646	C₂	1	1	yes	maximal, characteristic
6	#16811178	C₂	1	1	yes	maximal, characteristic
7	#13133284	C₂²	1	2	yes	—
8	#13133284	C₂²	1	2	yes	—
9	#13133284	C₂²	1	4	yes	—
10	C₄⋊#88324	C₂²	1	4	yes	—
11	#13133274	C₂²	1	2	yes	—
12	#13133274	C₂²	1	2	yes	—
13	#13133272	C₂²	1	2	yes	—
14	#13133272	C₂²	1	2	yes	—
15	#13133272	C₂²	1	4	yes	—
16	C₄×#88324	C₂²	1	4	yes	—
17	#13133217	C₄	1	1	yes	characteristic
18	#13133208	C₄	1	2	yes	—
19	#13133197	C₄	1	2	yes	—
20	#13133193	C₄	1	2	yes	—
21	Q₈×Dic₅₃	C₂²	1	2	yes	—
22	#13133182	C₂²	1	2	yes	—
23	#13133175	C₂²	1	2	yes	—
24	C₄²×D₅₃	C₂²	1	1	yes	characteristic
25	C₄⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
26	C₂×Q₈×C₅₃	C₂×C₄	1	2	yes	—
27	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
28	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
30	C₂×Dic₁₀₆	C₂×C₄	1	2	yes	—
31	C₂×#88324	C₂³	1	4	yes	—
32	C₂×#88324	C₂³	1	4	yes	—
33	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
34	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
35	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
36	#11719921	C₂×C₄	1	4	yes	—
37	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
38	C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
39	C₄⋊Dic₅₃	C₂×C₄	1	4	yes	—
40	#11719919	C₂×C₄	1	4	yes	—
41	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
42	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
43	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
44	C₄×Dic₅₃	C₂³	1	4	yes	—
45	C₂²×D₅₃	C₂²×C₄	1	1	yes	characteristic
46	Q₈×C₅₃	—	4	2	no	—
47	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
49	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
51	C₂×Dic₅₃	C₂²×C₄	1	4	yes	—
52	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
54	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
55	C₄×D₅₃	—	2	2	no	—
56	C₄×D₅₃	—	4	1	no	—
57	C₄×D₅₃	—	4	1	no	—
58	Dic₁₀₆	—	4	2	no	—
59	#88324	—	2	4	no	—
60	#88324	—	2	4	no	—
61	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
62	C₂×D₅₃	—	2	1	no	—
63	C₂×D₅₃	—	2	1	no	—
64	C₂×D₅₃	—	2	1	no	—
65	C₄×C₅₃	—	2	1	no	—
66	C₄×C₅₃	—	2	1	no	—
67	C₄×C₅₃	—	2	1	no	—
68	C₄×C₅₃	—	2	4	no	—
69	Dic₅₃	—	2	4	no	—
70	Dic₅₃	—	2	1	no	—
71	Dic₅₃	—	2	1	no	—
72	Dic₅₃	2₋¹⁺⁴	1	2	yes	—
73	#2987	—	53	1	no	maximal, Sylow
74	C₂×C₅₃	#519	1	1	yes	characteristic
75	C₂×C₅₃	#568	1	1	yes	characteristic
76	C₂×C₅₃	#568	1	1	yes	characteristic
77	D₅₃	—	4	1	no	—
78	#533	—	53	1	no	—
79	#446	—	53	4	no	—
80	#446	—	53	4	no	—
81	#433	—	53	4	no	—
82	#432	—	53	1	no	—
83	#432	—	53	1	no	—
84	C₅₃	#2987	1	1	yes	characteristic, complemented, Sylow
85	C₄⋊Q₈	—	53	1	no	—
86	C₄⋊Q₈	#6157	1	1	yes	characteristic
87	#125	—	53	2	no	—
88	C₂²⋊Q₈	—	53	2	no	—
89	C₂²⋊Q₈	—	53	2	no	—
90	C₄×Q₈	—	53	2	no	—
91	C₄×Q₈	—	53	2	no	—
92	#117	—	53	2	no	—
93	C₂×C₄²	—	53	1	no	—
94	C₄⋊C₈	—	53	2	no	—
95	C₄⋊C₈	—	53	2	no	—
96	C₄⋊C₈	—	53	4	no	—
97	C₄⋊C₈	—	53	4	no	—
98	C₂²⋊C₈	—	53	2	no	—
99	C₂²⋊C₈	—	53	2	no	—
100	#97	—	53	2	no	—
101	#97	—	53	2	no	—
102	#97	—	53	4	no	—
103	C₄×C₈	—	53	4	no	—
104	C₂×Q₈	—	53	2	no	—
105	C₂×Q₈	C₂×#6157	1	2	yes	—
106	C₂²×C₄	—	53	1	no	—
107	C₂²×C₄	—	53	1	no	—
108	C₂²×C₄	—	53	1	no	—
109	C₂×C₈	—	53	4	no	—
110	C₂×C₈	—	53	4	no	—
111	C₄⋊C₄	—	53	2	no	—
112	C₄⋊C₄	—	53	4	no	—
113	C₄⋊C₄	—	53	4	no	—
114	C₄⋊C₄	—	53	2	no	—
115	C₄⋊C₄	C₂×#6157	1	2	yes	—
116	C₄⋊C₄	C₂×#6157	1	2	yes	—
117	C₂²⋊C₄	—	53	4	no	—
118	C₄²	—	53	4	no	—
119	C₄²	—	53	1	no	—
120	C₄²	—	53	1	no	—
121	C₄²	—	53	1	no	—
122	C₄²	C₂×#6157	1	1	yes	characteristic
123	C₂³	—	53	1	no	—
124	Q₈	—	4	2	no	—
125	Q₈	—	212	2	no	—
126	C₂×C₄	—	53	1	no	—
127	C₂×C₄	—	53	1	no	—
128	C₂×C₄	—	53	1	no	—
129	C₂×C₄	—	53	4	no	—
130	C₂×C₄	—	106	2	no	—
131	C₂×C₄	—	212	1	no	—
132	C₂×C₄	—	212	1	no	—
133	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
134	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
135	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
136	C₂×C₄	C₂²×#6157	1	4	yes	—
137	C₈	—	106	4	no	—
138	C₈	—	106	4	no	—
139	C₂²	—	106	1	no	—
140	C₂²	—	106	1	no	—
141	C₂²	—	106	1	no	—
142	C₂²	C₂³×#6157	1	1	yes	characteristic
143	C₄	—	2	1	no	—
144	C₄	—	2	1	no	—
145	C₄	—	2	4	no	—
146	C₄	—	2	1	no	—
147	C₄	—	53	2	no	—
148	C₄	—	106	1	no	—
149	C₄	—	106	1	no	—
150	C₄	—	106	4	no	—
151	C₂	—	212	1	no	—
152	C₂	#16811691	1	1	yes	characteristic
153	C₂	#16811701	1	1	yes	characteristic
154	C₂	#16811703	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811675

C₂

yes

maximal, complemented

#16811674

C₂

yes

maximal, complemented

#16811648

C₂

yes

maximal

#16811646

C₂

yes

maximal, characteristic

#16811646

C₂

yes

maximal, characteristic

#16811178

C₂

yes

maximal, characteristic

#13133284

C₂²

yes

—

#13133284

C₂²

yes

—

#13133284

C₂²

yes

—

C₄⋊#88324

C₂²

yes

—

#13133274

C₂²

yes

—

#13133274

C₂²

yes

—

#13133272

C₂²

yes

—

#13133272

C₂²

yes

—

#13133272

C₂²

yes

—

C₄×#88324

C₂²

yes

—

#13133217

C₄

yes

characteristic

#13133208

C₄

yes

—

#13133197

C₄

yes

—

#13133193

C₄

yes

—

Q₈×Dic₅₃

C₂²

yes

—

#13133182

C₂²

yes

—

#13133175

C₂²

yes

—

C₄²×D₅₃

C₂²

yes

characteristic

C₄⋊Q₈×C₅₃

C₄

yes

characteristic

C₂×Q₈×C₅₃

C₂×C₄

yes

—

C₂×C₄×D₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

C₂×C₄

yes

characteristic

C₂×C₄×D₅₃

C₂³

yes

characteristic

C₂×Dic₁₀₆

C₂×C₄

yes

—

C₂×#88324

C₂³

yes

—

C₂×#88324

C₂³

yes

—

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

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

C₄⋊Dic₅₃

C₂×C₄

yes

—

#11719919

C₂×C₄

yes

—

C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₄×Dic₅₃

C₂³

yes

characteristic

C₄×Dic₅₃

C₂³

yes

characteristic

C₄×Dic₅₃

C₂³

yes

—

C₂²×D₅₃

C₂²×C₄

yes

characteristic

Q₈×C₅₃

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₅₃

C₂²×C₄

yes

—

C₂×Dic₅₃

C₂²×C₄

yes

—

C₂×Dic₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

C₂²×C₄

yes

characteristic

C₂×Dic₅₃

C₂⁴

yes

characteristic

C₄×D₅₃

—

C₄×D₅₃

—

C₄×D₅₃

—

Dic₁₀₆

—

#88324

—

#88324

—

C₂²×C₅₃

C₂³×C₄

yes

characteristic

C₂×D₅₃

—

C₂×D₅₃

—

C₂×D₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

—

Dic₅₃

2₋¹⁺⁴

yes

—

—

C₂×C₅₃

yes

C₂×C₅₃

yes

C₂×C₅₃

yes

D₅₃

—

—

—

—

—

—

—

C₅₃

#2987

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

C₄⋊Q₈

#6157

yes

characteristic

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

—

C₄×Q₈

—

C₄×Q₈

—

#117

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

100

#97

—

101

#97

—

102

#97

—

103

C₄×C₈

—

104

C₂×Q₈

—

105

C₂×Q₈

C₂×#6157

yes

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂×C₈

—

110

C₂×C₈

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄⋊C₄

C₂×#6157

yes

—

116

C₄⋊C₄

C₂×#6157

yes

—

117

C₂²⋊C₄

—

118

C₄²

—

119

C₄²

—

120

C₄²

—

121

C₄²

—

122

C₄²

C₂×#6157

yes

characteristic

123

C₂³

—

124

Q₈

—

125

Q₈

—

212

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

106

—

131

C₂×C₄

—

212

—

132

C₂×C₄

—

212

—

133

C₂×C₄

C₂²×#6157

yes

characteristic

134

C₂×C₄

C₂²×#6157

yes

characteristic

135

C₂×C₄

C₂²×#6157

yes

characteristic

136

C₂×C₄

C₂²×#6157

yes

—

137

C₈

—

106

—

138

C₈

—

106

—

139

C₂²

—

106

—

140

C₂²

—

106

—

141

C₂²

—

106

—

142

C₂²

C₂³×#6157

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

—

148

C₄

—

106

—

149

C₄

—

106

—

150

C₄

—

106

—

151

C₂

—

212

—

152

C₂

#16811691

yes

characteristic

153

C₂

#16811701

yes

characteristic

154

C₂

#16811703

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#6157
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₄⋊Q₈×C₅₃	C₄
hypercenter	C₄⋊Q₈	#6157
hypocenter	C₅₃	#2987
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×#6157

commutator, socle

C₂²×C₅₃

C₂³×C₄

Fitting

C₄⋊Q₈×C₅₃

C₄

C₄⋊Q₈

C₅₃

G ▷ C₂²×C₅₃ ▷ C₅₃

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
1. #16811675	C₂	semidirect
2. #16811674	C₂	semidirect
84. C₅₃	#2987	semidirect

H ≅ G/N

type

1. #16811675

C₂

semidirect

2. #16811674

C₂

semidirect

84. C₅₃

#2987

semidirect

Subgroups — #20291629

Conjugacy classes of subgroups

Complements