Group #20243403 — subgroups

Subgroups — #20243403

Order 6016.

292 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 125 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×C₄⋊(C₄₇⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₂×#13093948	C₂	1	2	yes	maximal
3	#15688989	C₂	1	1	yes	maximal, characteristic
4	#15688885	C₂	1	2	yes	maximal
5	C₂×C₄×Q₈×C₄₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₇⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
8	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
9	#13093948	—	2	4	no	—
10	C₄₇⋊#95	—	2	1	no	—
11	C₄₇⋊#95	—	2	1	no	—
12	C₂²×Q₈×C₄₇	C₂²	1	1	yes	characteristic
13	C₄×Q₈×C₄₇	—	2	2	no	—
14	C₄×Q₈×C₄₇	C₄	1	4	yes	complemented
15	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
16	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×C₄²×C₄₇	—	2	1	no	—
18	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
19	C₂×Q₈×C₄₇	—	2	1	no	—
20	C₂×Q₈×C₄₇	—	2	1	no	—
21	C₂×Q₈×C₄₇	—	2	1	no	—
22	C₂×Q₈×C₄₇	C₂×C₄	1	2	yes	—
23	C₂×Q₈×C₄₇	D₄	1	2	yes	—
24	C₂×Q₈×C₄₇	Q₈	1	2	yes	—
25	C₂²×C₄×C₄₇	—	2	1	no	—
26	C₂²×C₄×C₄₇	—	2	1	no	—
27	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
30	C₂²×Dic₄₇	—	2	2	no	—
31	C₄⋊C₄×C₄₇	—	2	4	no	—
32	C₄⋊C₄×C₄₇	—	2	1	no	—
33	C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
34	C₄²×C₄₇	—	2	4	no	—
35	C₄²×C₄₇	C₂×C₄	1	2	yes	—
36	C₄²×C₄₇	D₄	1	2	yes	—
37	C₄⋊Dic₄₇	—	2	2	no	—
38	C₄⋊Dic₄₇	—	2	2	no	—
39	C₂×C₄₇⋊C₈	—	2	2	no	—
40	C₂×C₄₇⋊C₈	—	2	2	no	—
41	C₂×C₄₇⋊C₈	—	2	2	no	—
42	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
43	Q₈×C₄₇	—	2	2	no	—
44	Q₈×C₄₇	—	2	2	no	—
45	Q₈×C₄₇	—	2	2	no	—
46	Q₈×C₄₇	C₄⋊C₄	1	4	yes	—
47	C₂×C₄×C₄₇	—	2	2	no	—
48	C₂×C₄×C₄₇	—	2	2	no	—
49	C₂×C₄×C₄₇	—	2	2	no	—
50	C₂×C₄×C₄₇	—	2	2	no	—
51	C₂×C₄×C₄₇	—	2	2	no	—
52	C₂×C₄×C₄₇	—	2	2	no	—
53	C₂×C₄×C₄₇	—	4	2	no	—
54	C₂×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
55	C₂×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
56	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
59	C₂×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
60	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
61	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
62	C₂×Dic₄₇	—	4	2	no	—
63	C₂×Dic₄₇	—	4	2	no	—
64	C₂×Dic₄₇	—	4	2	no	—
65	C₄₇⋊C₈	—	4	2	no	—
66	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
67	C₂²×C₄₇	#121	1	1	yes	characteristic
68	C₂²×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
69	C₂²×C₄₇	#135	1	2	yes	—
70	C₂²×C₄₇	#137	1	2	yes	—
71	C₄×C₄₇	—	2	4	no	—
72	C₄×C₄₇	—	2	2	no	—
73	C₄×C₄₇	—	2	2	no	—
74	C₄×C₄₇	—	4	2	no	—
75	C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
76	C₄×C₄₇	#117	1	1	yes	characteristic
77	C₄×C₄₇	#121	1	1	yes	characteristic
78	C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
79	Dic₄₇	—	8	2	no	—
80	#1805	—	47	1	no	maximal, Sylow
81	C₂×C₄₇	#380	1	1	yes	characteristic
82	C₂×C₄₇	#416	1	1	yes	characteristic
83	C₂×C₄₇	#419	1	1	yes	characteristic
84	C₂×C₄₇	#464	1	2	yes	—
85	C₂×C₄₇	#479	1	2	yes	—
86	C₂×C₄×Q₈	D₄₇	1	1	yes	characteristic
87	C₂×C₄⋊C₈	—	47	1	no	—
88	C₂×#103	—	47	2	no	—
89	#382	—	47	1	no	—
90	#340	—	47	2	no	—
91	C₄₇	#1805	1	1	yes	characteristic, complemented, Sylow
92	C₂²×Q₈	C₂×D₄₇	1	1	yes	characteristic
93	C₂²×C₈	—	47	2	no	—
94	C₄×Q₈	—	2	2	no	—
95	C₄×Q₈	Dic₄₇	1	4	yes	complemented
96	C₂×C₄⋊C₄	—	2	1	no	—
97	C₂×C₄⋊C₄	—	47	2	no	—
98	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
99	C₂×C₄²	—	2	1	no	—
100	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
101	C₄⋊C₈	—	94	2	no	—
102	#103	—	94	4	no	—
103	#95	—	94	1	no	—
104	#95	—	94	1	no	—
105	C₂×Q₈	—	2	1	no	—
106	C₂×Q₈	—	2	1	no	—
107	C₂×Q₈	—	2	1	no	—
108	C₂×Q₈	Dic₉₄	1	2	yes	—
109	C₂×Q₈	D₁₈₈	1	2	yes	—
110	C₂×Q₈	C₂×Dic₄₇	1	2	yes	—
111	C₂²×C₄	—	2	1	no	—
112	C₂²×C₄	—	2	1	no	—
113	C₂²×C₄	—	94	2	no	—
114	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
115	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
116	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
117	C₂×C₈	—	94	2	no	—
118	C₂×C₈	—	94	2	no	—
119	C₂×C₈	—	94	2	no	—
120	C₄⋊C₄	—	2	4	no	—
121	C₄⋊C₄	—	2	1	no	—
122	C₄⋊C₄	—	94	2	no	—
123	C₄⋊C₄	—	94	2	no	—
124	C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
125	C₄²	—	2	4	no	—
126	C₄²	C₂×Dic₄₇	1	2	yes	—
127	C₄²	C₂²⋊D₄₇	1	2	yes	—
128	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
129	Q₈	—	2	2	no	—
130	Q₈	—	2	2	no	—
131	Q₈	—	2	2	no	—
132	Q₈	C₄⋊Dic₄₇	1	4	yes	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	2	no	—
139	C₂×C₄	—	4	2	no	—
140	C₂×C₄	—	188	2	no	—
141	C₂×C₄	—	188	2	no	—
142	C₂×C₄	—	188	2	no	—
143	C₂×C₄	C₂²⋊Dic₄₇	1	2	yes	—
144	C₂×C₄	C₂²⋊Dic₄₇	1	2	yes	—
145	C₂×C₄	C₂×Dic₉₄	1	1	yes	characteristic
146	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
147	C₂×C₄	#10625350	1	1	yes	characteristic
148	C₂×C₄	#10625350	1	1	yes	characteristic
149	C₂×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
150	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
151	C₈	—	188	2	no	—
152	C₂²	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
153	C₂²	#13093998	1	1	yes	characteristic
154	C₂²	#13094003	1	1	yes	characteristic
155	C₂²	#13094044	1	2	yes	—
156	C₂²	#13094055	1	2	yes	—
157	C₄	—	2	4	no	—
158	C₄	—	2	2	no	—
159	C₄	—	2	2	no	—
160	C₄	—	4	2	no	—
161	C₄	—	376	2	no	—
162	C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
163	C₄	#13093983	1	1	yes	characteristic
164	C₄	#13093998	1	1	yes	characteristic
165	C₄	#13094003	1	1	yes	characteristic
166	C₂	#15688983	1	1	yes	characteristic
167	C₂	#15689119	1	1	yes	characteristic
168	C₂	#15689132	1	1	yes	characteristic
169	C₂	#15689345	1	2	yes	—
170	C₂	#15689430	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₄₇⋊C₈)

C₂

yes

maximal, characteristic

C₂×#13093948

C₂

yes

maximal

#15688989

C₂

yes

maximal, characteristic

#15688885

C₂

yes

maximal

C₂×C₄×Q₈×C₄₇

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₄⋊(C₄₇⋊C₈)

—

#13093948

—

C₄₇⋊#95

—

C₄₇⋊#95

—

C₂²×Q₈×C₄₇

C₂²

yes

characteristic

C₄×Q₈×C₄₇

—

C₄×Q₈×C₄₇

C₄

yes

complemented

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

characteristic

C₂×C₄²×C₄₇

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

C₂×C₄

yes

—

C₂×Q₈×C₄₇

D₄

yes

—

C₂×Q₈×C₄₇

Q₈

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

C₂×C₄

yes

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂×C₄

yes

—

C₄²×C₄₇

D₄

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

C₄⋊C₄

yes

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×Q₈

yes

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₇

#121

yes

characteristic

C₂²×C₄₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₄₇

#135

yes

—

C₂²×C₄₇

#137

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×C₄⋊C₄

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄×C₄₇

C₂²⋊Q₈

yes

characteristic

Dic₄₇

—

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×C₄₇

#479

yes

—

C₂×C₄×Q₈

D₄₇

yes

characteristic

C₂×C₄⋊C₈

—

C₂×#103

—

#382

—

#340

—

C₄₇

#1805

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₄₇

yes

characteristic

C₂²×C₈

—

C₄×Q₈

—

C₄×Q₈

Dic₄₇

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₄₇

yes

characteristic

C₂×C₄²

—

100

C₂×C₄²

C₂×D₄₇

yes

characteristic

101

C₄⋊C₈

—

102

#103

—

103

#95

—

104

#95

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂×Q₈

Dic₉₄

yes

—

109

C₂×Q₈

D₁₈₈

yes

—

110

C₂×Q₈

C₂×Dic₄₇

yes

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

115

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

116

C₂²×C₄

C₂²×D₄₇

yes

characteristic

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₄⋊C₄

C₂×Dic₄₇

yes

—

125

C₄²

—

126

C₄²

C₂×Dic₄₇

yes

—

127

C₄²

C₂²⋊D₄₇

yes

—

128

C₂³

C₂×C₂²⋊D₄₇

yes

characteristic

129

Q₈

—

130

Q₈

—

131

Q₈

—

132

Q₈

C₄⋊Dic₄₇

yes

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

188

—

141

C₂×C₄

—

188

—

142

C₂×C₄

—

188

—

143

C₂×C₄

C₂²⋊Dic₄₇

yes

—

144

C₂×C₄

C₂²⋊Dic₄₇

yes

—

145

C₂×C₄

C₂×Dic₉₄

yes

characteristic

146

C₂×C₄

C₂×D₁₈₈

yes

characteristic

147

C₂×C₄

#10625350

yes

characteristic

148

C₂×C₄

#10625350

yes

characteristic

149

C₂×C₄

C₂²×Dic₄₇

yes

characteristic

150

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

151

C₈

—

188

—

152

C₂²

C₂×C₂²⋊Dic₄₇

yes

characteristic

153

C₂²

#13093998

yes

characteristic

154

C₂²

#13094003

yes

characteristic

155

C₂²

#13094044

yes

—

156

C₂²

#13094055

yes

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

—

376

—

162

C₄

C₂×C₄⋊Dic₄₇

yes

characteristic

163

C₄

#13093983

yes

characteristic

164

C₄

#13093998

yes

characteristic

165

C₄

#13094003

yes

characteristic

166

C₂

#15688983

yes

characteristic

167

C₂

#15689119

yes

characteristic

168

C₂

#15689132

yes

characteristic

169

C₂

#15689345

yes

—

170

C₂

#15689430

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₂×C₄×Q₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂×D₄
hypercenter	C₂×C₄×Q₈	D₄₇
hypocenter	C₄₇	#1805
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₄₇

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₂×C₄×Q₈×C₄₇

C₂

socle

C₂³×C₄₇

C₂×D₄

hypercenter

C₂×C₄×Q₈

D₄₇

hypocenter

C₄₇

#1805

LCS

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
14. C₄×Q₈×C₄₇	C₄	semidirect
91. C₄₇	#1805	semidirect
95. C₄×Q₈	Dic₄₇	semidirect

H ≅ G/N

type

14. C₄×Q₈×C₄₇

C₄

semidirect

91. C₄₇

#1805

semidirect

95. C₄×Q₈

Dic₄₇

semidirect

Subgroups — #20243403

Conjugacy classes of subgroups

Complements