Group #20243080 — subgroups

Subgroups — #20243080

Order 6016.

480 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 165 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	C₂²×#10625324	C₂	1	2	yes	maximal
2	C₂×C₄²×D₄₇	C₂	1	1	yes	maximal, characteristic
3	#15688878	C₂	1	1	yes	maximal, characteristic, complemented
4	#15688877	C₂	1	2	yes	maximal, complemented
5	C₄₇×#336	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₄₇	C₄	1	2	yes	—
7	C₂²×C₄×D₄₇	C₂²	1	1	yes	characteristic
8	C₂×#10625324	—	2	4	no	—
9	C₂×#10625324	—	2	4	no	—
10	C₂×#10625324	C₄	1	8	yes	complemented
11	C₂²×C₄₇⋊C₈	C₂²	1	2	yes	—
12	C₂×C₄×Dic₄₇	C₄	1	2	yes	—
13	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
14	C₄²×D₄₇	—	2	4	no	—
15	C₂²×C₈×C₄₇	C₂²	1	2	yes	—
16	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
17	C₂³×D₄₇	C₂×C₄	1	1	yes	characteristic
18	C₂²×C₄×C₄₇	C₂×C₄	1	2	yes	—
19	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
20	C₂²×Dic₄₇	C₂×C₄	1	2	yes	—
21	C₂²×Dic₄₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×D₄₇	—	2	4	no	—
23	C₂×C₄×D₄₇	—	2	4	no	—
24	C₂×C₄×D₄₇	—	2	4	no	—
25	C₂×C₄×D₄₇	C₂×C₄	1	4	yes	—
26	C₂×C₄×D₄₇	D₄	1	2	yes	—
27	C₂×C₄×D₄₇	D₄	1	4	yes	—
28	C₂×C₄×D₄₇	Q₈	1	2	yes	—
29	C₂×C₈×C₄₇	—	2	2	no	—
30	C₂×C₈×C₄₇	—	2	2	no	—
31	C₂×C₈×C₄₇	C₂×C₄	1	4	yes	complemented
32	C₄²×C₄₇	—	2	2	no	—
33	C₄×Dic₄₇	—	2	2	no	—
34	C₄×Dic₄₇	—	2	4	no	—
35	C₂×C₄₇⋊C₈	—	2	2	no	—
36	C₂×C₄₇⋊C₈	—	2	2	no	—
37	C₂×C₄₇⋊C₈	C₂×C₄	1	4	yes	complemented
38	#10625324	—	2	16	no	—
39	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
40	C₂²×D₄₇	—	2	2	no	—
41	C₂²×D₄₇	—	2	2	no	—
42	C₂²×D₄₇	C₄²	1	2	yes	—
43	C₂²×D₄₇	C₂²⋊C₄	1	2	yes	—
44	C₂²×D₄₇	C₄⋊C₄	1	2	yes	—
45	C₂×C₄×C₄₇	—	2	2	no	—
46	C₂×C₄×C₄₇	—	2	2	no	—
47	C₂×C₄×C₄₇	M₁₆	1	4	yes	complemented
48	C₂×C₄×C₄₇	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
52	C₂×Dic₄₇	—	2	2	no	—
53	C₂×Dic₄₇	—	2	2	no	—
54	C₂×Dic₄₇	C₄²	1	2	yes	—
55	C₂×Dic₄₇	C₂²⋊C₄	1	2	yes	—
56	C₂×Dic₄₇	C₄⋊C₄	1	2	yes	—
57	C₂×Dic₄₇	M₁₆	1	4	yes	complemented
58	C₄×D₄₇	—	2	16	no	—
59	C₄×D₄₇	C₂²⋊C₄	1	8	yes	—
60	C₄×D₄₇	C₄⋊C₄	1	8	yes	—
61	C₈×C₄₇	—	2	4	no	—
62	C₄₇⋊C₈	—	2	4	no	—
63	C₂²×C₄₇	C₂×C₄²	1	1	yes	characteristic
64	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
66	C₂²×C₄₇	C₂×M₁₆	1	2	yes	—
67	C₂²×C₄₇	C₂×M₁₆	1	2	yes	—
68	C₂×D₄₇	—	2	2	no	—
69	C₂×D₄₇	—	2	2	no	—
70	C₂×D₄₇	—	2	4	no	—
71	C₂×D₄₇	—	2	4	no	—
72	C₂×D₄₇	#95	1	4	yes	—
73	C₄×C₄₇	—	2	4	no	—
74	C₄×C₄₇	C₂×C₂²⋊C₄	1	2	yes	—
75	C₄×C₄₇	C₂×C₄⋊C₄	1	2	yes	—
76	Dic₄₇	—	2	4	no	—
77	Dic₄₇	#95	1	4	yes	—
78	#1728	—	47	1	no	maximal, Sylow
79	C₂×C₄₇	C₂×#95	1	1	yes	characteristic
80	C₂×C₄₇	C₄×M₁₆	1	1	yes	characteristic
81	C₂×C₄₇	C₄×M₁₆	1	1	yes	characteristic
82	C₂×C₄₇	#407	1	2	yes	—
83	C₂×C₄₇	#423	1	2	yes	—
84	D₄₇	—	2	4	no	—
85	C₂²×M₁₆	—	47	2	no	—
86	C₂²×C₄²	—	47	1	no	—
87	#336	—	47	2	no	—
88	#336	—	47	1	no	—
89	#336	D₄₇	1	1	yes	characteristic, complemented
90	C₄₇	#1728	1	1	yes	characteristic, complemented, Sylow
91	C₂³×C₄	—	47	1	no	—
92	C₂³×C₄	—	47	2	no	—
93	C₂×M₁₆	—	47	8	no	—
94	C₂×M₁₆	—	94	4	no	—
95	C₂×M₁₆	—	94	4	no	—
96	C₂²×C₈	—	47	2	no	—
97	C₂²×C₈	C₂×D₄₇	1	2	yes	—
98	C₂×C₄²	—	47	2	no	—
99	C₂×C₄²	—	47	1	no	—
100	C₂×C₄²	—	94	4	no	—
101	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
102	C₂⁴	—	47	1	no	—
103	C₂²×C₄	—	47	1	no	—
104	C₂²×C₄	—	47	4	no	—
105	C₂²×C₄	—	47	4	no	—
106	C₂²×C₄	—	47	2	no	—
107	C₂²×C₄	—	47	2	no	—
108	C₂²×C₄	—	47	2	no	—
109	C₂²×C₄	—	94	4	no	—
110	C₂²×C₄	—	94	4	no	—
111	C₂²×C₄	—	94	4	no	—
112	C₂²×C₄	C₄×D₄₇	1	2	yes	—
113	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
114	M₁₆	—	94	16	no	—
115	C₂×C₈	—	2	2	no	—
116	C₂×C₈	—	2	2	no	—
117	C₂×C₈	—	47	4	no	—
118	C₂×C₈	—	94	2	no	—
119	C₂×C₈	—	94	2	no	—
120	C₂×C₈	C₄×D₄₇	1	4	yes	complemented
121	C₄²	—	2	2	no	—
122	C₄²	—	94	4	no	—
123	C₄²	—	94	2	no	—
124	C₂³	—	47	2	no	—
125	C₂³	—	47	2	no	—
126	C₂³	—	47	2	no	—
127	C₂³	—	94	2	no	—
128	C₂³	—	94	2	no	—
129	C₂³	C₂×C₄×D₄₇	1	1	yes	characteristic
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	2	no	—
132	C₂×C₄	—	47	2	no	—
133	C₂×C₄	—	47	2	no	—
134	C₂×C₄	—	47	2	no	—
135	C₂×C₄	—	47	8	no	—
136	C₂×C₄	—	47	8	no	—
137	C₂×C₄	—	47	4	no	—
138	C₂×C₄	—	94	16	no	—
139	C₂×C₄	—	94	2	no	—
140	C₂×C₄	—	94	2	no	—
141	C₂×C₄	#10625324	1	4	yes	complemented
142	C₂×C₄	C₂×C₄×D₄₇	1	2	yes	—
143	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
144	C₂×C₄	D₄×D₄₇	1	2	yes	—
145	C₂×C₄	Q₈×D₄₇	1	1	yes	characteristic
146	C₈	—	2	4	no	—
147	C₈	—	94	4	no	—
148	C₂²	—	47	4	no	—
149	C₂²	—	94	2	no	—
150	C₂²	—	94	2	no	—
151	C₂²	—	94	4	no	—
152	C₂²	—	94	4	no	—
153	C₂²	C₄²×D₄₇	1	1	yes	characteristic
154	C₂²	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
155	C₂²	C₄⋊C₄×D₄₇	1	1	yes	characteristic
156	C₂²	M₁₆×D₄₇	1	2	yes	—
157	C₂²	C₂×#10625324	1	2	yes	—
158	C₄	—	2	4	no	—
159	C₄	—	47	4	no	—
160	C₄	—	94	4	no	—
161	C₄	C₂²⋊C₄×D₄₇	1	2	yes	—
162	C₄	C₄⋊C₄×D₄₇	1	2	yes	—
163	C₂	—	94	4	no	—
164	C₂	#95×D₄₇	1	1	yes	characteristic
165	C₂	C₄×#10625324	1	1	yes	characteristic
166	C₂	#15689075	1	1	yes	characteristic
167	C₂	#15689085	1	2	yes	—
168	C₂	#15689145	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#10625324

C₂

yes

maximal

C₂×C₄²×D₄₇

C₂

yes

maximal, characteristic

#15688878

C₂

yes

maximal, characteristic, complemented

#15688877

C₂

yes

maximal, complemented

C₄₇×#336

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₄₇

C₄

yes

—

C₂²×C₄×D₄₇

C₂²

yes

—

—

C₄

yes

complemented

C₂²×C₄₇⋊C₈

C₂²

yes

—

C₂×C₄×Dic₄₇

C₄

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

characteristic

C₄²×D₄₇

—

C₂²×C₈×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂³×D₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

C₂×C₄

yes

—

C₂²×Dic₄₇

C₂×C₄

yes

characteristic

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

C₂×C₄

yes

—

C₂×C₄×D₄₇

D₄

yes

—

C₂×C₄×D₄₇

D₄

yes

—

C₂×C₄×D₄₇

Q₈

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂×C₄

yes

complemented

C₄²×C₄₇

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₂×C₄

yes

complemented

#10625324

—

C₂³×C₄₇

C₂²×C₄

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

C₄²

yes

—

C₂²×D₄₇

C₂²⋊C₄

yes

—

C₂²×D₄₇

C₄⋊C₄

yes

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

M₁₆

yes

complemented

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₄²

yes

—

C₂×Dic₄₇

C₂²⋊C₄

yes

—

C₂×Dic₄₇

C₄⋊C₄

yes

—

C₂×Dic₄₇

M₁₆

yes

complemented

C₄×D₄₇

—

C₄×D₄₇

C₂²⋊C₄

yes

—

C₄×D₄₇

C₄⋊C₄

yes

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

C₂×C₄²

yes

characteristic

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₄₇

C₂×M₁₆

yes

—

C₂²×C₄₇

C₂×M₁₆

yes

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

#95

yes

—

C₄×C₄₇

—

C₄×C₄₇

C₂×C₂²⋊C₄

yes

—

C₄×C₄₇

C₂×C₄⋊C₄

yes

—

Dic₄₇

—

Dic₄₇

#95

yes

—

#1728

—

maximal, Sylow

C₂×C₄₇

C₂×#95

yes

characteristic

C₂×C₄₇

C₄×M₁₆

yes

characteristic

C₂×C₄₇

C₄×M₁₆

yes

characteristic

C₂×C₄₇

#407

yes

—

C₂×C₄₇

#423

yes

—

D₄₇

—

C₂²×M₁₆

—

C₂²×C₄²

—

#336

—

#336

—

#336

D₄₇

yes

characteristic, complemented

C₄₇

#1728

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂³×C₄

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂²×C₈

C₂×D₄₇

yes

—

C₂×C₄²

—

C₂×C₄²

—

100

C₂×C₄²

—

101

C₂×C₄²

C₂×D₄₇

yes

characteristic

102

C₂⁴

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

C₄×D₄₇

yes

—

113

C₂²×C₄

C₂²×D₄₇

yes

characteristic

114

M₁₆

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

C₄×D₄₇

yes

complemented

121

C₄²

—

122

C₄²

—

123

C₄²

—

124

C₂³

—

125

C₂³

—

126

C₂³

—

127

C₂³

—

128

C₂³

—

129

C₂³

C₂×C₄×D₄₇

yes

characteristic

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

#10625324

yes

complemented

142

C₂×C₄

C₂×C₄×D₄₇

yes

—

143

C₂×C₄

D₄×D₄₇

yes

characteristic

144

C₂×C₄

D₄×D₄₇

yes

—

145

C₂×C₄

Q₈×D₄₇

yes

characteristic

146

C₈

—

147

C₈

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

—

153

C₂²

C₄²×D₄₇

yes

characteristic

154

C₂²

C₂²⋊C₄×D₄₇

yes

characteristic

155

C₂²

C₄⋊C₄×D₄₇

yes

characteristic

156

C₂²

M₁₆×D₄₇

yes

—

157

C₂²

C₂×#10625324

yes

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

C₂²⋊C₄×D₄₇

yes

—

162

C₄

C₄⋊C₄×D₄₇

yes

—

163

C₂

—

164

C₂

#95×D₄₇

yes

characteristic

165

C₂

C₄×#10625324

yes

characteristic

166

C₂

#15689075

yes

characteristic

167

C₂

#15689085

yes

—

168

C₂

#15689145

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₄₇
commutator	C₂²×C₄₇	C₂×C₄²
Fitting	C₄₇×#336	C₂
socle	C₂³×C₄₇	C₂²×C₄
hypercenter	#336	D₄₇
hypocenter	C₄₇	#1728
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₄₇

commutator

C₂²×C₄₇

C₂×C₄²

Fitting

C₄₇×#336

C₂

socle

C₂³×C₄₇

C₂²×C₄

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₄₇

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
3. #15688878	C₂	semidirect
4. #15688877	C₂	semidirect
5. C₄₇×#336	C₂	semidirect
10. C₂×#10625324	C₄	semidirect
31. C₂×C₈×C₄₇	C₂×C₄	semidirect
37. C₂×C₄₇⋊C₈	C₂×C₄	semidirect
47. C₂×C₄×C₄₇	M₁₆	semidirect
57. C₂×Dic₄₇	M₁₆	semidirect
89. #336	D₄₇	semidirect
90. C₄₇	#1728	semidirect
120. C₂×C₈	C₄×D₄₇	semidirect
141. C₂×C₄	#10625324	semidirect

H ≅ G/N

type

3. #15688878

C₂

semidirect

4. #15688877

C₂

semidirect

5. C₄₇×#336

C₂

semidirect

10. C₂×#10625324

C₄

semidirect

31. C₂×C₈×C₄₇

C₂×C₄

semidirect

37. C₂×C₄₇⋊C₈

C₂×C₄

semidirect

47. C₂×C₄×C₄₇

M₁₆

semidirect

57. C₂×Dic₄₇

M₁₆

semidirect

89. #336

D₄₇

semidirect

90. C₄₇

#1728

semidirect

120. C₂×C₈

C₄×D₄₇

semidirect

141. C₂×C₄

#10625324

semidirect

Subgroups — #20243080

Conjugacy classes of subgroups

Complements