Group #20245007 — subgroups

Subgroups — #20245007

Order 6016.

220 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 75 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×#13094032	C₂	1	1	yes	maximal, characteristic
2	M₃₂×D₄₇	C₂	1	2	yes	maximal, complemented
3	#15688905	C₂	1	1	yes	maximal, characteristic, complemented
4	#15688904	C₂	1	2	yes	maximal, complemented
5	C₄₇×#349	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10625350	C₄	1	1	yes	characteristic
7	#13094032	—	2	2	no	—
8	#13094032	—	2	2	no	—
9	C₂×#10625324	C₄	1	1	yes	characteristic
10	C₂×#10625324	C₂²	1	1	yes	characteristic
11	C₂²×C₄₇⋊C₈	C₄	1	1	yes	characteristic
12	C₂×C₈×D₄₇	C₄	1	1	yes	characteristic
13	C₂×C₈×D₄₇	C₂²	1	1	yes	characteristic
14	#13093964	C₂²	1	2	yes	—
15	#13093963	—	2	2	no	—
16	C₁₆×D₄₇	—	2	2	no	—
17	C₂×M₁₆×C₄₇	C₂²	1	1	yes	characteristic
18	M₃₂×C₄₇	C₂²	1	2	yes	—
19	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
20	C₂×C₂²⋊D₄₇	C₈	1	1	yes	characteristic
21	C₂×C₂²⋊D₄₇	C₈	1	1	yes	characteristic
22	#10625350	—	2	2	no	—
23	#10625350	—	2	2	no	—
24	C₂×D₁₈₈	C₈	1	1	yes	characteristic
25	C₂×C₄×D₄₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₄×D₄₇	C₂×C₄	1	1	yes	characteristic
27	C₂×Dic₉₄	C₈	1	1	yes	characteristic
28	M₁₆×C₄₇	—	2	2	no	—
29	C₂×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
31	C₂×C₄₇⋊C₈	—	2	1	no	—
32	C₂×C₄₇⋊C₈	—	2	1	no	—
33	C₂×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
34	C₂×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
35	#10625324	—	2	2	no	—
36	#10625324	—	2	2	no	—
37	C₈×D₄₇	—	2	1	no	—
38	C₈×D₄₇	—	2	1	no	—
39	C₈×D₄₇	D₄	1	2	yes	—
40	C₈×D₄₇	D₄	1	2	yes	—
41	C₁₆×C₄₇	—	2	2	no	—
42	C₄₇⋊C₁₆	—	2	2	no	—
43	C₂³×C₄₇	C₂×C₈	1	1	yes	characteristic
44	C₂²×D₄₇	C₂×C₈	1	1	yes	characteristic
45	C₂²×D₄₇	C₂×C₈	1	1	yes	characteristic
46	C₂×C₄×C₄₇	—	2	1	no	—
47	C₂×C₄×C₄₇	—	2	1	no	—
48	C₂×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
49	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
50	C₂²⋊D₄₇	—	2	2	no	—
51	C₂²⋊D₄₇	—	2	2	no	—
52	C₂×Dic₄₇	C₂×C₈	1	1	yes	characteristic
53	C₂×Dic₄₇	C₂×C₈	1	1	yes	characteristic
54	D₁₈₈	—	2	2	no	—
55	C₄×D₄₇	—	2	1	no	—
56	C₄×D₄₇	—	2	1	no	—
57	C₄×D₄₇	C₂²⋊C₄	1	2	yes	—
58	C₄×D₄₇	M₁₆	1	2	yes	—
59	Dic₉₄	—	2	2	no	—
60	C₈×C₄₇	—	2	1	no	—
61	C₈×C₄₇	C₂×D₄	1	2	yes	—
62	C₄₇⋊C₈	—	2	1	no	—
63	C₄₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
64	C₄₇⋊C₈	C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄₇	—	2	1	no	—
66	C₂²×C₄₇	—	4	1	no	—
67	C₂²×C₄₇	C₂²×C₈	1	1	yes	characteristic
68	C₂×D₄₇	—	2	1	no	—
69	C₂×D₄₇	—	4	1	no	—
70	C₂×D₄₇	—	4	1	no	—
71	C₂×D₄₇	C₂²⋊C₈	1	2	yes	—
72	C₄×C₄₇	—	2	1	no	—
73	C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₄×C₄₇	C₂×M₁₆	1	1	yes	characteristic
75	Dic₄₇	—	2	1	no	—
76	Dic₄₇	C₂²⋊C₈	1	2	yes	—
77	#2115	—	47	1	no	maximal, Sylow
78	C₂×C₄₇	—	2	1	no	—
79	C₂×C₄₇	—	4	1	no	—
80	C₂×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
81	D₄₇	—	2	2	no	—
82	D₄₇	—	4	1	no	—
83	C₂×#131	—	47	1	no	—
84	C₂×M₃₂	—	47	2	no	—
85	#349	—	47	1	no	—
86	#349	—	47	2	no	—
87	#349	D₄₇	1	1	yes	characteristic, complemented
88	C₄₇	#2115	1	1	yes	characteristic, complemented, Sylow
89	C₂×#41	—	47	1	no	—
90	#131	—	94	2	no	—
91	#131	—	94	2	no	—
92	C₂×M₁₆	—	47	1	no	—
93	C₂×M₁₆	—	47	1	no	—
94	C₂×M₁₆	C₂×D₄₇	1	1	yes	characteristic
95	C₂²×C₈	—	47	1	no	—
96	C₂²×C₈	—	47	1	no	—
97	C₂²×C₈	—	47	1	no	—
98	M₃₂	—	47	2	no	—
99	M₃₂	—	94	2	no	—
100	M₃₂	C₂×D₄₇	1	2	yes	—
101	C₂×C₁₆	—	94	2	no	—
102	#41	—	94	2	no	—
103	#41	—	94	2	no	—
104	C₂×Q₈	—	47	1	no	—
105	C₂×D₄	—	47	1	no	—
106	C₂×D₄	—	47	1	no	—
107	C₂×D₄	—	47	1	no	—
108	C₂²×C₄	—	47	1	no	—
109	C₂²×C₄	—	47	1	no	—
110	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
111	M₁₆	—	2	2	no	—
112	M₁₆	—	94	2	no	—
113	M₁₆	—	94	2	no	—
114	C₂×C₈	—	47	1	no	—
115	C₂×C₈	—	47	2	no	—
116	C₂×C₈	—	47	2	no	—
117	C₂×C₈	—	47	1	no	—
118	C₂×C₈	—	94	1	no	—
119	C₂×C₈	—	94	1	no	—
120	C₂×C₈	—	94	1	no	—
121	C₂×C₈	—	94	1	no	—
122	C₂×C₈	C₄×D₄₇	1	1	yes	characteristic
123	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
124	C₁₆	—	2	2	no	—
125	C₁₆	—	94	2	no	—
126	C₂³	—	47	1	no	—
127	C₂³	—	47	1	no	—
128	C₂³	C₈×D₄₇	1	1	yes	characteristic
129	Q₈	—	94	2	no	—
130	D₄	—	94	2	no	—
131	D₄	—	94	2	no	—
132	D₄	—	94	2	no	—
133	C₂×C₄	—	2	1	no	—
134	C₂×C₄	—	2	1	no	—
135	C₂×C₄	—	47	1	no	—
136	C₂×C₄	—	47	2	no	—
137	C₂×C₄	—	47	2	no	—
138	C₂×C₄	—	47	1	no	—
139	C₂×C₄	—	94	1	no	—
140	C₂×C₄	—	94	1	no	—
141	C₂×C₄	C₈×D₄₇	1	1	yes	characteristic
142	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
143	C₈	—	2	1	no	—
144	C₈	—	47	1	no	—
145	C₈	—	47	1	no	—
146	C₈	—	94	1	no	—
147	C₈	D₄×D₄₇	1	2	yes	—
148	C₂²	—	2	1	no	—
149	C₂²	—	4	1	no	—
150	C₂²	—	47	2	no	—
151	C₂²	—	94	1	no	—
152	C₂²	—	188	1	no	—
153	C₂²	—	188	1	no	—
154	C₂²	C₂×C₈×D₄₇	1	1	yes	characteristic
155	C₄	—	2	1	no	—
156	C₄	—	47	2	no	—
157	C₄	—	94	1	no	—
158	C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
159	C₄	M₁₆×D₄₇	1	1	yes	characteristic
160	C₂	—	2	1	no	—
161	C₂	—	4	1	no	—
162	C₂	—	94	2	no	—
163	C₂	—	188	1	no	—
164	C₂	C₂²⋊C₈×D₄₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094032

C₂

yes

maximal, characteristic

M₃₂×D₄₇

C₂

yes

maximal, complemented

#15688905

C₂

yes

maximal, characteristic, complemented

#15688904

C₂

yes

maximal, complemented

C₄₇×#349

C₂

yes

maximal, characteristic, complemented

C₂×#10625350

C₄

yes

—

—

C₄

yes

characteristic

C₂×#10625324

C₂²

yes

characteristic

C₂²×C₄₇⋊C₈

C₄

yes

characteristic

C₂×C₈×D₄₇

C₄

yes

characteristic

C₂×C₈×D₄₇

C₂²

yes

characteristic

#13093964

C₂²

yes

—

#13093963

—

C₁₆×D₄₇

—

C₂×M₁₆×C₄₇

C₂²

yes

characteristic

M₃₂×C₄₇

C₂²

yes

—

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂×C₂²⋊D₄₇

C₈

yes

characteristic

C₂×C₂²⋊D₄₇

C₈

yes

characteristic

#10625350

—

#10625350

—

C₂×D₁₈₈

C₈

yes

characteristic

C₂×C₄×D₄₇

C₂×C₄

yes

characteristic

C₂×C₄×D₄₇

C₂×C₄

yes

characteristic

C₂×Dic₉₄

C₈

yes

characteristic

M₁₆×C₄₇

—

C₂×C₈×C₄₇

C₂×C₄

yes

characteristic

C₂×C₈×C₄₇

C₂³

yes

characteristic

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₄₇⋊C₈

C₂×C₄

yes

characteristic

#10625324

—

#10625324

—

C₈×D₄₇

—

C₈×D₄₇

—

C₈×D₄₇

D₄

yes

—

C₈×D₄₇

D₄

yes

—

C₁₆×C₄₇

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

C₂×C₈

yes

characteristic

C₂²×D₄₇

C₂×C₈

yes

characteristic

C₂²×D₄₇

C₂×C₈

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₈

yes

characteristic

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂²⋊D₄₇

—

C₂²⋊D₄₇

—

C₂×Dic₄₇

C₂×C₈

yes

characteristic

C₂×Dic₄₇

C₂×C₈

yes

characteristic

D₁₈₈

—

C₄×D₄₇

—

C₄×D₄₇

—

C₄×D₄₇

C₂²⋊C₄

yes

—

C₄×D₄₇

M₁₆

yes

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

C₂×D₄

yes

—

C₄₇⋊C₈

—

C₄₇⋊C₈

C₂²⋊C₄

yes

characteristic

C₄₇⋊C₈

C₂²⋊C₄

yes

characteristic

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂²×C₈

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

C₂²⋊C₈

yes

—

C₄×C₄₇

—

C₄×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₄₇

C₂×M₁₆

yes

characteristic

Dic₄₇

—

Dic₄₇

C₂²⋊C₈

yes

—

#2115

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

C₂×C₂²⋊C₈

yes

characteristic

D₄₇

—

D₄₇

—

C₂×#131

—

C₂×M₃₂

—

#349

—

#349

—

#349

D₄₇

yes

characteristic, complemented

C₄₇

#2115

yes

characteristic, complemented, Sylow

C₂×#41

—

#131

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₄₇

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

M₃₂

—

M₃₂

—

100

M₃₂

C₂×D₄₇

yes

—

101

C₂×C₁₆

—

102

#41

—

103

#41

—

104

C₂×Q₈

—

105

C₂×D₄

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂²×C₄

—

109

C₂²×C₄

—

110

C₂²×C₄

C₄×D₄₇

yes

characteristic

111

M₁₆

—

112

M₁₆

—

113

M₁₆

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

—

121

C₂×C₈

—

122

C₂×C₈

C₄×D₄₇

yes

characteristic

123

C₂×C₈

C₂²×D₄₇

yes

characteristic

124

C₁₆

—

125

C₁₆

—

126

C₂³

—

127

C₂³

—

128

C₂³

C₈×D₄₇

yes

characteristic

129

Q₈

—

130

D₄

—

131

D₄

—

132

D₄

—

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

C₈×D₄₇

yes

characteristic

142

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

143

C₈

—

144

C₈

—

145

C₈

—

146

C₈

—

147

C₈

D₄×D₄₇

yes

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

—

188

—

153

C₂²

—

188

—

154

C₂²

C₂×C₈×D₄₇

yes

characteristic

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

C₂²⋊C₄×D₄₇

yes

characteristic

159

C₄

M₁₆×D₄₇

yes

characteristic

160

C₂

—

161

C₂

—

162

C₂

—

163

C₂

—

188

—

164

C₂

C₂²⋊C₈×D₄₇

yes

characteristic

	subgroup	G/H
center	C₄	C₂²⋊C₄×D₄₇
commutator	C₂²×C₄₇	C₂²×C₈
Frattini	C₂×C₈	C₂²×D₄₇
Fitting	C₄₇×#349	C₂
socle	C₂×C₄₇	C₂×C₂²⋊C₈
hypercenter	#349	D₄₇
hypocenter	C₄₇	#2115
LCS	G ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #349

subgroup

G/H

center

C₄

C₂²⋊C₄×D₄₇

commutator

C₂²×C₄₇

C₂²×C₈

Frattini

C₂×C₈

C₂²×D₄₇

Fitting

C₄₇×#349

C₂

socle

C₂×C₄₇

C₂×C₂²⋊C₈

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ #349

N	H ≅ G/N	type
2. M₃₂×D₄₇	C₂	semidirect
3. #15688905	C₂	semidirect
4. #15688904	C₂	semidirect
5. C₄₇×#349	C₂	semidirect
87. #349	D₄₇	semidirect
88. C₄₇	#2115	semidirect

H ≅ G/N

type

2. M₃₂×D₄₇

C₂

semidirect

3. #15688905

C₂

semidirect

4. #15688904

C₂

semidirect

5. C₄₇×#349

C₂

semidirect

87. #349

D₄₇

semidirect

88. C₄₇

#2115

semidirect

Subgroups — #20245007

Conjugacy classes of subgroups

Complements