Group #20246306 — subgroups

Subgroups — #20246306

Order 6016.

552 conjugacy classes of subgroups; only the 275 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 275 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	C₂×Q₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic
2	C₂×Q₈×Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	#15689021	C₂	1	3	yes	maximal
4	#15689019	C₂	1	3	yes	maximal
5	#15688988	C₂	1	3	yes	maximal
6	C₄×Dic₄₇⋊C₄	C₂	1	3	yes	maximal
7	C₂×C₄×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
8	Q₈×Dic₄₇	—	2	4	no	—
9	Q₈×Dic₄₇	C₄	1	8	yes	complemented
10	C₂×C₄⋊Dic₄₇	C₂²	1	3	yes	—
11	C₂×C₄⋊Dic₄₇	C₂²	1	3	yes	—
12	C₂×Dic₄₇⋊C₄	C₂²	1	3	yes	—
13	C₂×Dic₄₇⋊C₄	C₂²	1	3	yes	—
14	C₂×C₄×Dic₄₇	C₂²	1	3	yes	—
15	C₂×C₄×Dic₄₇	C₂²	1	3	yes	—
16	C₄₇⋊#95	C₂²	1	1	yes	characteristic
17	C₄₇⋊#95	C₂²	1	3	yes	—
18	C₄₇⋊#95	C₂²	1	6	yes	—
19	C₂²×Q₈×C₄₇	C₂²	1	1	yes	characteristic
20	C₄×Q₈×C₄₇	—	2	4	no	—
21	C₂×C₄⋊C₄×C₄₇	C₂²	1	3	yes	—
22	C₂×C₄²×C₄₇	C₂²	1	3	yes	—
23	C₂×Q₈×C₄₇	C₂×C₄	1	4	yes	—
24	C₂×Q₈×C₄₇	D₄	1	4	yes	—
25	C₂×Q₈×C₄₇	Q₈	1	4	yes	—
26	C₂²×C₄×C₄₇	C₂³	1	3	yes	—
27	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₄₇	C₂³	1	3	yes	—
29	C₂²×Dic₄₇	C₂³	1	3	yes	—
30	C₂²×Dic₄₇	C₂³	1	1	yes	characteristic
31	C₂²×Dic₄₇	C₂³	1	3	yes	—
32	C₂²×Dic₄₇	C₂³	1	1	yes	characteristic
33	C₄⋊C₄×C₄₇	—	2	6	no	—
34	C₄²×C₄₇	—	2	6	no	—
35	C₄⋊Dic₄₇	—	2	6	no	—
36	C₄⋊Dic₄₇	C₂×C₄	1	12	yes	—
37	Dic₄₇⋊C₄	—	2	6	no	—
38	Dic₄₇⋊C₄	—	2	6	no	—
39	C₄×Dic₄₇	—	2	6	no	—
40	C₄×Dic₄₇	C₂×C₄	1	12	yes	—
41	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
42	Q₈×C₄₇	C₄⋊C₄	1	16	yes	complemented
43	C₂×C₄×C₄₇	—	2	1	no	—
44	C₂×C₄×C₄₇	—	2	1	no	—
45	C₂×C₄×C₄₇	—	2	3	no	—
46	C₂×C₄×C₄₇	—	2	3	no	—
47	C₂×C₄×C₄₇	—	2	3	no	—
48	C₂×C₄×C₄₇	C₂²×C₄	1	6	yes	—
49	C₂×C₄×C₄₇	C₂×D₄	1	6	yes	—
50	C₂×C₄×C₄₇	C₂×Q₈	1	6	yes	—
51	C₂×C₄×C₄₇	#41	1	2	yes	—
52	C₂×Dic₄₇	—	2	1	no	—
53	C₂×Dic₄₇	—	2	3	no	—
54	C₂×Dic₄₇	—	2	3	no	—
55	C₂×Dic₄₇	—	2	1	no	—
56	C₂×Dic₄₇	—	2	1	no	—
57	C₂×Dic₄₇	—	2	1	no	—
58	C₂×Dic₄₇	—	2	3	no	—
59	C₂×Dic₄₇	—	2	3	no	—
60	C₂×Dic₄₇	—	2	3	no	—
61	C₂×Dic₄₇	C₂²×C₄	1	2	yes	—
62	C₂×Dic₄₇	C₂²×C₄	1	6	yes	—
63	C₂×Dic₄₇	#41	1	2	yes	—
64	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
65	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
66	C₂²×C₄₇	C₂²×Q₈	1	1	yes	characteristic
67	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
68	C₂²×C₄₇	C₂×#41	1	1	yes	characteristic
69	C₂²×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
70	C₂²×C₄₇	2₋¹⁺⁴	1	1	yes	characteristic
71	C₄×C₄₇	—	2	2	no	—
72	C₄×C₄₇	—	4	3	no	—
73	C₄×C₄₇	C₂×C₄⋊C₄	1	12	yes	—
74	Dic₄₇	—	2	6	no	—
75	Dic₄₇	—	2	2	no	—
76	Dic₄₇	—	2	2	no	—
77	Dic₄₇	—	4	3	no	—
78	#2351	—	47	1	no	maximal, Sylow
79	C₂×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
80	C₂×C₄₇	C₄×#41	1	1	yes	characteristic
81	C₂×C₄₇	#519	1	1	yes	characteristic
82	C₂×C₄₇	#548	1	1	yes	characteristic
83	C₂×C₄₇	#548	1	1	yes	characteristic
84	C₂×C₄₇	#557	1	1	yes	characteristic
85	C₂×C₄₇	#557	1	1	yes	characteristic
86	C₂×C₄×Q₈	—	47	1	no	—
87	C₂×C₄×Q₈	—	47	1	no	—
88	C₂×C₄×Q₈	D₄₇	1	1	yes	characteristic
89	#389	—	47	3	no	—
90	#389	—	47	3	no	—
91	#382	—	47	3	no	—
92	C₄×C₄⋊C₄	—	47	3	no	—
93	C₄₇	#2351	1	1	yes	characteristic, complemented, Sylow
94	C₂²×Q₈	C₂×D₄₇	1	1	yes	characteristic
95	C₄×Q₈	—	2	4	no	—
96	C₄×Q₈	—	47	8	no	—
97	C₄×Q₈	—	94	4	no	—
98	C₂×C₄⋊C₄	—	47	3	no	—
99	C₂×C₄⋊C₄	—	47	3	no	—
100	C₂×C₄⋊C₄	—	47	3	no	—
101	C₂×C₄⋊C₄	—	47	3	no	—
102	C₂×C₄⋊C₄	C₂×D₄₇	1	3	yes	—
103	C₂×C₄²	—	47	3	no	—
104	C₂×C₄²	—	47	3	no	—
105	C₂×C₄²	C₂×D₄₇	1	3	yes	—
106	#95	—	47	1	no	—
107	#95	—	47	3	no	—
108	#95	—	47	6	no	—
109	C₂×Q₈	Dic₉₄	1	4	yes	—
110	C₂×Q₈	C₄×D₄₇	1	4	yes	—
111	C₂×Q₈	C₂²⋊D₄₇	1	4	yes	—
112	C₂²×C₄	—	47	1	no	—
113	C₂²×C₄	—	47	3	no	—
114	C₂²×C₄	—	47	1	no	—
115	C₂²×C₄	—	47	3	no	—
116	C₂²×C₄	C₂²×D₄₇	1	3	yes	—
117	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
118	C₂²×C₄	C₂²×D₄₇	1	3	yes	—
119	C₄⋊C₄	—	2	6	no	—
120	C₄⋊C₄	—	47	12	no	—
121	C₄⋊C₄	—	94	6	no	—
122	C₄⋊C₄	—	94	6	no	—
123	C₄⋊C₄	—	94	6	no	—
124	C₄²	—	2	6	no	—
125	C₄²	—	47	12	no	—
126	C₄²	—	94	6	no	—
127	C₂³	C₂³×D₄₇	1	1	yes	characteristic
128	Q₈	Dic₄₇⋊C₄	1	16	yes	complemented
129	C₂×C₄	—	2	1	no	—
130	C₂×C₄	—	2	1	no	—
131	C₂×C₄	—	2	3	no	—
132	C₂×C₄	—	2	3	no	—
133	C₂×C₄	—	2	3	no	—
134	C₂×C₄	—	47	2	no	—
135	C₂×C₄	—	47	6	no	—
136	C₂×C₄	—	47	2	no	—
137	C₂×C₄	—	94	1	no	—
138	C₂×C₄	—	94	3	no	—
139	C₂×C₄	—	94	3	no	—
140	C₂×C₄	—	94	1	no	—
141	C₂×C₄	—	94	1	no	—
142	C₂×C₄	—	94	1	no	—
143	C₂×C₄	—	94	3	no	—
144	C₂×C₄	—	94	3	no	—
145	C₂×C₄	—	94	3	no	—
146	C₂×C₄	C₂×Dic₉₄	1	6	yes	—
147	C₂×C₄	C₂×C₄×D₄₇	1	6	yes	—
148	C₂×C₄	#10625354	1	2	yes	—
149	C₂×C₄	C₂×C₂²⋊D₄₇	1	6	yes	—
150	C₂²	C₂²×C₄×D₄₇	1	1	yes	characteristic
151	C₂²	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
152	C₂²	C₂²×Dic₉₄	1	1	yes	characteristic
153	C₂²	#41×D₄₇	1	1	yes	characteristic
154	C₂²	C₂×#10625354	1	1	yes	characteristic
155	C₂²	#13094071	1	1	yes	characteristic
156	C₂²	#13094072	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	—	4	3	no	—
159	C₄	—	94	2	no	—
160	C₄	—	94	6	no	—
161	C₄	—	94	2	no	—
162	C₄	—	188	3	no	—
163	C₄	C₂×Dic₄₇⋊C₄	1	12	yes	—
164	C₂	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
165	C₂	C₄×#10625354	1	1	yes	characteristic
166	C₂	#15689599	1	1	yes	characteristic
167	C₂	#15689780	1	1	yes	characteristic
168	C₂	#15689784	1	1	yes	characteristic
169	C₂	#15689849	1	1	yes	characteristic
170	C₂	#15689852	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×Dic₄₇

C₂

yes

maximal, characteristic

C₂×Q₈×Dic₄₇

C₂

yes

maximal, characteristic

#15689021

C₂

yes

maximal

#15689019

C₂

yes

maximal

#15688988

C₂

yes

maximal

C₄×Dic₄₇⋊C₄

C₂

yes

maximal

C₂×C₄×Q₈×C₄₇

C₂

yes

maximal, characteristic

Q₈×Dic₄₇

—

Q₈×Dic₄₇

C₄

yes

complemented

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×Dic₄₇⋊C₄

C₂²

yes

—

C₂×Dic₄₇⋊C₄

C₂²

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

characteristic

C₄₇⋊#95

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₂²×Q₈×C₄₇

C₂²

yes

characteristic

C₄×Q₈×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

—

C₂×Q₈×C₄₇

C₂×C₄

yes

—

C₂×Q₈×C₄₇

D₄

yes

—

C₂×Q₈×C₄₇

Q₈

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

characteristic

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

C₂×C₄

yes

—

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₄×Dic₄₇

C₂×C₄

yes

—

C₂³×C₄₇

C₂⁴

yes

characteristic

Q₈×C₄₇

C₄⋊C₄

yes

complemented

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×C₄×C₄₇

#41

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂²×C₄

yes

—

C₂×Dic₄₇

C₂²×C₄

yes

—

C₂×Dic₄₇

#41

yes

—

C₂²×C₄₇

C₂³×C₄

yes

characteristic

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₂²×C₄₇

C₂²×Q₈

yes

characteristic

C₂²×C₄₇

C₂×#41

yes

characteristic

C₂²×C₄₇

C₂×#41

yes

characteristic

C₂²×C₄₇

2₋¹⁺⁴

yes

characteristic

C₂²×C₄₇

2₋¹⁺⁴

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×C₄⋊C₄

yes

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

#2351

—

maximal, Sylow

C₂×C₄₇

C₂²×C₄⋊C₄

yes

characteristic

C₂×C₄₇

C₄×#41

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄×Q₈

—

C₂×C₄×Q₈

—

C₂×C₄×Q₈

D₄₇

yes

—

—

—

C₄×C₄⋊C₄

—

C₄₇

#2351

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₄₇

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

100

C₂×C₄⋊C₄

—

101

C₂×C₄⋊C₄

—

102

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

103

C₂×C₄²

—

104

C₂×C₄²

—

105

C₂×C₄²

C₂×D₄₇

yes

—

106

#95

—

107

#95

—

108

#95

—

109

C₂×Q₈

Dic₉₄

yes

—

110

C₂×Q₈

C₄×D₄₇

yes

—

111

C₂×Q₈

C₂²⋊D₄₇

yes

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

—

115

C₂²×C₄

—

116

C₂²×C₄

C₂²×D₄₇

yes

—

117

C₂²×C₄

C₂²×D₄₇

yes

characteristic

118

C₂²×C₄

C₂²×D₄₇

yes

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄²

—

125

C₄²

—

126

C₄²

—

127

C₂³

C₂³×D₄₇

yes

characteristic

128

Q₈

Dic₄₇⋊C₄

yes

complemented

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

C₂×Dic₉₄

yes

—

147

C₂×C₄

C₂×C₄×D₄₇

yes

—

148

C₂×C₄

#10625354

yes

—

149

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

150

C₂²

C₂²×C₄×D₄₇

yes

characteristic

151

C₂²

C₂²×C₂²⋊D₄₇

yes

characteristic

152

C₂²

C₂²×Dic₉₄

yes

characteristic

153

C₂²

#41×D₄₇

yes

characteristic

154

C₂²

C₂×#10625354

yes

characteristic

155

C₂²

#13094071

yes

characteristic

156

C₂²

#13094072

yes

characteristic

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

—

162

C₄

—

188

—

163

C₄

C₂×Dic₄₇⋊C₄

yes

—

164

C₂

C₂²×Dic₄₇⋊C₄

yes

characteristic

165

C₂

C₄×#10625354

yes

characteristic

166

C₂

#15689599

yes

characteristic

167

C₂

#15689780

yes

characteristic

168

C₂

#15689784

yes

characteristic

169

C₂

#15689849

yes

characteristic

170

C₂

#15689852

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator	C₂²×C₄₇	C₂³×C₄
Fitting	C₂×C₄×Q₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂⁴
hypercenter	C₂×C₄×Q₈	D₄₇
hypocenter	C₄₇	#2351
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₇

commutator

C₂²×C₄₇

C₂³×C₄

Fitting

C₂×C₄×Q₈×C₄₇

C₂

socle

C₂³×C₄₇

C₂⁴

hypercenter

C₂×C₄×Q₈

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₄₇

C₁ ◁ C₂³ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
9. Q₈×Dic₄₇	C₄	semidirect
42. Q₈×C₄₇	C₄⋊C₄	semidirect
93. C₄₇	#2351	semidirect
128. Q₈	Dic₄₇⋊C₄	semidirect

H ≅ G/N

type

9. Q₈×Dic₄₇

C₄

semidirect

42. Q₈×C₄₇

C₄⋊C₄

semidirect

93. C₄₇

#2351

semidirect

128. Q₈

Dic₄₇⋊C₄

semidirect

Subgroups — #20246306

Conjugacy classes of subgroups

Complements