Group #20241730 — subgroups

Subgroups — #20241730

Order 6016.

236 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 69 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×#13093989	C₂	1	1	yes	maximal, characteristic
2	#15689177	C₂	1	2	yes	maximal
3	#15688849	C₂	1	1	yes	maximal, characteristic
4	#15688848	C₂	1	2	yes	maximal
5	C₄₇×#324	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₉₄	C₄	1	1	yes	characteristic
7	#13093989	—	2	2	no	—
8	#13093989	—	2	2	no	—
9	C₂×Dic₄₇⋊C₄	C₄	1	1	yes	characteristic
10	C₂×Dic₄₇⋊C₄	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₄₇	C₄	1	1	yes	characteristic
12	C₂×C₄×Dic₄₇	C₄	1	1	yes	characteristic
13	C₂×C₄×Dic₄₇	C₂²	1	1	yes	characteristic
14	Dic₄₇⋊C₈	—	2	2	no	—
15	C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
16	#13093934	—	2	2	no	—
17	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
18	C₂²⋊C₈×C₄₇	C₂²	1	2	yes	—
19	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
22	C₂²×Dic₄₇	C₂×C₄	1	1	yes	characteristic
23	C₂²×Dic₄₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×Dic₄₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×Dic₄₇	C₂×C₄	1	1	yes	characteristic
26	C₂×Dic₉₄	—	2	2	no	—
27	C₂×Dic₉₄	—	2	2	no	—
28	C₂×Dic₉₄	—	2	2	no	—
29	C₂×C₈×C₄₇	—	2	2	no	—
30	C₄⋊C₄×C₄₇	—	2	2	no	—
31	Dic₄₇⋊C₄	—	2	2	no	—
32	Dic₄₇⋊C₄	—	2	2	no	—
33	C₄×Dic₄₇	—	2	1	no	—
34	C₄×Dic₄₇	—	2	1	no	—
35	C₄×Dic₄₇	—	2	1	no	—
36	C₄×Dic₄₇	—	2	1	no	—
37	C₄×Dic₄₇	D₄	1	2	yes	—
38	C₄×Dic₄₇	D₄	1	2	yes	—
39	C₂×C₄₇⋊C₈	—	2	2	no	—
40	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₄₇	—	2	1	no	—
42	C₂×C₄×C₄₇	—	2	1	no	—
43	C₂×C₄×C₄₇	—	2	1	no	—
44	C₂×C₄×C₄₇	—	4	1	no	—
45	C₂×C₄×C₄₇	—	4	1	no	—
46	C₂×C₄×C₄₇	M₁₆	1	2	yes	—
47	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
48	C₂×Dic₄₇	—	2	1	no	—
49	C₂×Dic₄₇	—	2	1	no	—
50	C₂×Dic₄₇	—	2	1	no	—
51	C₂×Dic₄₇	—	2	1	no	—
52	C₂×Dic₄₇	—	2	2	no	—
53	C₂×Dic₄₇	—	2	1	no	—
54	C₂×Dic₄₇	—	4	1	no	—
55	C₂×Dic₄₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×Dic₄₇	C₂²⋊C₄	1	2	yes	—
57	C₂×Dic₄₇	C₂²⋊C₄	1	1	yes	characteristic
58	C₂×Dic₄₇	M₁₆	1	2	yes	—
59	Dic₉₄	—	2	8	no	—
60	C₈×C₄₇	—	4	2	no	—
61	C₄₇⋊C₈	—	4	2	no	—
62	C₂²×C₄₇	—	2	1	no	—
63	C₂²×C₄₇	—	2	1	no	—
64	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄₇	C₂×M₁₆	1	1	yes	characteristic
66	C₂²×C₄₇	C₂×M₁₆	1	1	yes	characteristic
67	C₄×C₄₇	—	2	2	no	—
68	C₄×C₄₇	—	2	2	no	—
69	C₄×C₄₇	—	4	1	no	—
70	Dic₄₇	—	2	2	no	—
71	Dic₄₇	—	2	2	no	—
72	Dic₄₇	—	2	1	no	—
73	Dic₄₇	—	4	1	no	—
74	Dic₄₇	#101	1	2	yes	—
75	#1467	—	47	1	no	maximal, Sylow
76	C₂×C₄₇	—	2	1	no	—
77	C₂×C₄₇	—	2	1	no	—
78	C₂×C₄₇	#407	1	1	yes	characteristic
79	C₂×C₄₇	#410	1	1	yes	characteristic
80	C₂×C₄₇	C₂×#101	1	1	yes	characteristic
81	C₂×C₄×Q₈	—	47	1	no	—
82	#432	—	47	2	no	—
83	#324	—	47	1	no	—
84	#324	—	47	2	no	—
85	#324	D₄₇	1	1	yes	characteristic
86	C₄₇	#1467	1	1	yes	characteristic, complemented, Sylow
87	C₂²×Q₈	—	47	1	no	—
88	C₄×Q₈	—	94	2	no	—
89	C₄×Q₈	—	94	2	no	—
90	C₂×C₄⋊C₄	—	47	1	no	—
91	C₂×C₄⋊C₄	—	47	1	no	—
92	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
93	C₂×C₄²	—	47	1	no	—
94	C₂×C₄²	—	47	1	no	—
95	C₂×C₄²	—	47	1	no	—
96	C₄⋊C₈	—	94	2	no	—
97	C₂²⋊C₈	—	47	2	no	—
98	C₂²⋊C₈	C₂×D₄₇	1	2	yes	—
99	#97	—	94	2	no	—
100	C₂×Q₈	—	94	2	no	—
101	C₂×Q₈	—	94	2	no	—
102	C₂×Q₈	—	94	2	no	—
103	C₂²×C₄	—	47	1	no	—
104	C₂²×C₄	—	47	1	no	—
105	C₂²×C₄	—	47	1	no	—
106	C₂²×C₄	—	47	1	no	—
107	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
108	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
109	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	94	2	no	—
112	C₄⋊C₄	—	2	2	no	—
113	C₄⋊C₄	—	94	2	no	—
114	C₄⋊C₄	—	94	2	no	—
115	C₄²	—	47	2	no	—
116	C₄²	—	47	2	no	—
117	C₄²	—	94	1	no	—
118	C₄²	—	94	1	no	—
119	C₄²	—	94	1	no	—
120	C₄²	—	94	1	no	—
121	C₂³	C₂×C₄×D₄₇	1	1	yes	characteristic
122	Q₈	—	94	8	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	4	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	47	1	no	—
129	C₂×C₄	—	47	1	no	—
130	C₂×C₄	—	47	2	no	—
131	C₂×C₄	—	47	2	no	—
132	C₂×C₄	—	94	2	no	—
133	C₂×C₄	—	94	1	no	—
134	C₂×C₄	—	94	1	no	—
135	C₂×C₄	—	94	1	no	—
136	C₂×C₄	—	94	1	no	—
137	C₂×C₄	—	94	1	no	—
138	C₂×C₄	—	188	1	no	—
139	C₂×C₄	#10625324	1	2	yes	—
140	C₂×C₄	D₄×D₄₇	1	2	yes	—
141	C₈	—	4	2	no	—
142	C₈	—	188	2	no	—
143	C₂²	—	2	1	no	—
144	C₂²	—	2	1	no	—
145	C₂²	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
146	C₂²	M₁₆×D₄₇	1	1	yes	characteristic
147	C₂²	C₂×#10625324	1	1	yes	characteristic
148	C₄	—	2	2	no	—
149	C₄	—	2	2	no	—
150	C₄	—	4	1	no	—
151	C₄	—	47	2	no	—
152	C₄	—	94	2	no	—
153	C₄	—	94	2	no	—
154	C₄	—	94	1	no	—
155	C₄	—	188	1	no	—
156	C₂	—	2	1	no	—
157	C₂	—	2	1	no	—
158	C₂	#15689085	1	1	yes	characteristic
159	C₂	#15689095	1	1	yes	characteristic
160	C₂	#101×D₄₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13093989

C₂

yes

maximal, characteristic

#15689177

C₂

yes

maximal

#15688849

C₂

yes

maximal, characteristic

#15688848

C₂

yes

maximal

C₄₇×#324

C₂

yes

maximal, characteristic

C₂²×Dic₉₄

C₄

yes

characteristic

#13093989

—

#13093989

—

C₂×Dic₄₇⋊C₄

C₄

yes

characteristic

C₂×Dic₄₇⋊C₄

C₂²

yes

characteristic

C₂×C₄×Dic₄₇

C₄

yes

characteristic

C₂×C₄×Dic₄₇

C₄

yes

characteristic

C₂×C₄×Dic₄₇

C₂²

yes

characteristic

Dic₄₇⋊C₈

—

C₂²⋊(C₄₇⋊C₈)

C₂²

yes

—

#13093934

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

characteristic

C₂²⋊C₈×C₄₇

C₂²

yes

—

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

C₂×C₄

yes

characteristic

C₂²×Dic₄₇

C₂×C₄

yes

characteristic

C₂²×Dic₄₇

C₂×C₄

yes

characteristic

C₂²×Dic₄₇

C₂×C₄

yes

characteristic

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×C₈×C₄₇

—

C₄⋊C₄×C₄₇

—

Dic₄₇⋊C₄

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₄×Dic₄₇

—

C₄×Dic₄₇

D₄

yes

—

C₄×Dic₄₇

D₄

yes

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

M₁₆

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₄₇

C₂²⋊C₄

yes

—

C₂×Dic₄₇

C₂²⋊C₄

yes

characteristic

C₂×Dic₄₇

M₁₆

yes

—

Dic₉₄

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₇

C₂×M₁₆

yes

characteristic

C₂²×C₄₇

C₂×M₁₆

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

#101

yes

—

#1467

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄×Q₈

—

—

—

—

D₄₇

yes

characteristic

C₄₇

#1467

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₄₇

yes

characteristic

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

C₂×D₄₇

yes

—

#97

—

100

C₂×Q₈

—

101

C₂×Q₈

—

102

C₂×Q₈

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

C₄×D₄₇

yes

characteristic

108

C₂²×C₄

C₄×D₄₇

yes

characteristic

109

C₂²×C₄

C₂²×D₄₇

yes

characteristic

110

C₂×C₈

—

111

C₂×C₈

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₄²

—

116

C₄²

—

117

C₄²

—

118

C₄²

—

119

C₄²

—

120

C₄²

—

121

C₂³

C₂×C₄×D₄₇

yes

characteristic

122

Q₈

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

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

—

188

—

139

C₂×C₄

#10625324

yes

—

140

C₂×C₄

D₄×D₄₇

yes

—

141

C₈

—

142

C₈

—

188

—

143

C₂²

—

144

C₂²

—

145

C₂²

C₂²⋊C₄×D₄₇

yes

characteristic

146

C₂²

M₁₆×D₄₇

yes

characteristic

147

C₂²

C₂×#10625324

yes

characteristic

148

C₄

—

149

C₄

—

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

—

155

C₄

—

188

—

156

C₂

—

157

C₂

—

158

C₂

#15689085

yes

characteristic

159

C₂

#15689095

yes

characteristic

160

C₂

#101×D₄₇

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₄₇
commutator	C₂³×C₄₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₄₇×#324	C₂
socle	C₂²×C₄₇	C₂×C₂²⋊C₄
hypercenter	#324	D₄₇
hypocenter	C₄₇	#1467
LCS	G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #324

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₄₇

commutator

C₂³×C₄₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₄₇×#324

C₂

socle

C₂²×C₄₇

C₂×C₂²⋊C₄

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #324

N	H ≅ G/N	type
86. C₄₇	#1467	semidirect

H ≅ G/N

type

86. C₄₇

#1467

semidirect

Subgroups — #20241730

Conjugacy classes of subgroups

Complements