Group #20245210 — subgroups

Subgroups — #20245210

Order 6016.

164 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 71 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×#13093963	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₄₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×C₂²⋊D₄₇	C₂	1	1	yes	maximal, characteristic
4	Dic₄₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
5	(C₂×D₄₇)⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
6	#15688898	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₈×D₄₇	C₂²	1	1	yes	characteristic
9	C₄×C₂²⋊D₄₇	C₄	1	1	yes	characteristic
10	#13093963	—	2	2	no	—
11	C₂×C₄₇⋊C₁₆	—	2	2	no	—
12	C₂×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
14	Dic₄₇⋊C₈	C₄	1	1	yes	characteristic
15	C₂²⋊(C₄₇⋊C₈)	C₄	1	1	yes	characteristic
16	(C₂×D₄₇)⋊C₈	C₄	1	1	yes	characteristic
17	C₈×Dic₄₇	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₄₇	C₂²	1	1	yes	characteristic
19	C₂×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
20	C₂×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
21	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
22	C₂×C₂²⋊D₄₇	C₈	1	1	yes	characteristic
23	C₂×C₄×D₄₇	C₂×C₄	1	1	yes	characteristic
24	C₂×C₈×C₄₇	—	2	1	no	—
25	C₂×C₈×C₄₇	—	2	1	no	—
26	C₂×C₈×C₄₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
28	C₂²⋊Dic₄₇	C₈	1	1	yes	characteristic
29	(C₂×D₄₇)⋊C₄	C₈	1	1	yes	characteristic
30	Dic₄₇⋊C₄	C₈	1	1	yes	characteristic
31	C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
32	C₂×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
33	C₂×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
34	C₈×D₄₇	—	2	1	no	—
35	C₈×D₄₇	—	2	1	no	—
36	C₁₆×C₄₇	—	2	1	no	—
37	C₁₆×C₄₇	—	2	1	no	—
38	C₄₇⋊C₁₆	—	2	1	no	—
39	C₄₇⋊C₁₆	D₄	1	2	yes	complemented
40	C₂³×C₄₇	C₂×C₈	1	1	yes	characteristic
41	C₂²×D₄₇	C₂×C₈	1	1	yes	characteristic
42	C₂×C₄×C₄₇	—	2	1	no	—
43	C₂×C₄×C₄₇	—	2	1	no	—
44	C₂×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
45	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
46	C₂²⋊D₄₇	—	2	2	no	—
47	C₂×Dic₄₇	C₂×C₈	1	1	yes	characteristic
48	C₂×Dic₄₇	C₂×C₈	1	1	yes	characteristic
49	C₂×Dic₄₇	C₂×C₈	1	1	yes	characteristic
50	C₄×D₄₇	—	2	1	no	—
51	C₄×D₄₇	—	2	1	no	—
52	C₈×C₄₇	—	2	1	no	—
53	C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
54	C₈×C₄₇	#41	1	1	yes	characteristic
55	C₄₇⋊C₈	—	2	1	no	—
56	C₄₇⋊C₈	—	2	1	no	—
57	C₂²×C₄₇	—	2	1	no	—
58	C₂²×C₄₇	—	2	1	no	—
59	C₂²×C₄₇	M₃₂	1	2	yes	complemented
60	C₂²×C₄₇	C₂²×C₈	1	1	yes	characteristic
61	C₂×D₄₇	—	2	1	no	—
62	C₂×D₄₇	—	2	1	no	—
63	C₂×D₄₇	—	2	1	no	—
64	C₄×C₄₇	—	2	1	no	—
65	C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
66	C₄×C₄₇	#131	1	1	yes	characteristic
67	Dic₄₇	—	2	1	no	—
68	Dic₄₇	—	2	1	no	—
69	Dic₄₇	—	2	1	no	—
70	#2168	—	47	1	no	maximal, Sylow
71	C₂×C₄₇	—	2	2	no	—
72	C₂×C₄₇	C₈×D₄	1	1	yes	characteristic
73	C₂×C₄₇	C₂×M₃₂	1	1	yes	characteristic
74	C₂×C₄₇	#504	1	1	yes	characteristic
75	D₄₇	—	4	1	no	—
76	C₂×M₃₂	—	47	1	no	—
77	C₂²×C₁₆	—	47	1	no	—
78	C₈×D₄	—	47	1	no	—
79	C₄⋊C₁₆	—	47	1	no	—
80	C₂²⋊C₁₆	—	47	1	no	—
81	C₂²⋊C₁₆	D₄₇	1	1	yes	characteristic, complemented
82	#346	—	47	1	no	—
83	C₄₇	#2168	1	1	yes	characteristic, complemented, Sylow
84	C₂²×C₈	—	47	1	no	—
85	C₂²×C₈	C₂×D₄₇	1	1	yes	characteristic
86	C₄×D₄	—	47	1	no	—
87	M₃₂	—	94	2	no	—
88	C₂×C₁₆	—	47	1	no	—
89	C₂×C₁₆	—	47	1	no	—
90	C₂×C₁₆	—	94	2	no	—
91	C₂×C₁₆	C₂×D₄₇	1	1	yes	characteristic
92	C₂×C₁₆	C₂×D₄₇	1	1	yes	characteristic
93	C₄⋊C₈	—	47	1	no	—
94	C₂²⋊C₈	—	47	1	no	—
95	C₂²⋊C₈	—	47	1	no	—
96	C₄×C₈	—	47	1	no	—
97	C₂×D₄	—	47	1	no	—
98	C₂²×C₄	—	47	1	no	—
99	C₂²×C₄	C₄×D₄₇	1	1	yes	characteristic
100	C₂×C₈	—	2	1	no	—
101	C₂×C₈	—	2	1	no	—
102	C₂×C₈	—	47	1	no	—
103	C₂×C₈	—	47	1	no	—
104	C₂×C₈	—	94	1	no	—
105	C₂×C₈	—	94	1	no	—
106	C₂×C₈	C₄×D₄₇	1	1	yes	characteristic
107	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
108	C₄⋊C₄	—	47	1	no	—
109	C₂²⋊C₄	—	47	1	no	—
110	C₂²⋊C₄	—	47	1	no	—
111	C₄²	—	47	1	no	—
112	C₁₆	—	2	1	no	—
113	C₁₆	—	2	1	no	—
114	C₁₆	—	47	2	no	—
115	C₁₆	—	94	1	no	—
116	C₂³	—	47	1	no	—
117	C₂³	C₈×D₄₇	1	1	yes	characteristic
118	D₄	—	94	2	no	—
119	C₂×C₄	—	2	1	no	—
120	C₂×C₄	—	2	1	no	—
121	C₂×C₄	—	47	1	no	—
122	C₂×C₄	—	47	1	no	—
123	C₂×C₄	—	47	1	no	—
124	C₂×C₄	—	94	1	no	—
125	C₂×C₄	—	94	1	no	—
126	C₂×C₄	C₈×D₄₇	1	1	yes	characteristic
127	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
128	C₈	—	2	1	no	—
129	C₈	—	94	1	no	—
130	C₈	—	94	1	no	—
131	C₈	D₄×D₄₇	1	1	yes	characteristic
132	C₈	#10625352	1	1	yes	characteristic
133	C₂²	—	2	1	no	—
134	C₂²	—	2	1	no	—
135	C₂²	—	94	1	no	—
136	C₂²	—	94	1	no	—
137	C₂²	—	94	1	no	—
138	C₂²	#13093963	1	2	yes	complemented
139	C₂²	C₂×C₈×D₄₇	1	1	yes	characteristic
140	C₄	—	2	1	no	—
141	C₄	—	94	1	no	—
142	C₄	—	94	1	no	—
143	C₄	—	94	1	no	—
144	C₄	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
145	C₄	#13094032	1	1	yes	characteristic
146	C₂	—	2	2	no	—
147	C₂	—	188	1	no	—
148	C₂	(C₂²⋊D₄₇)⋊C₈	1	1	yes	characteristic
149	C₂	C₂×#13093963	1	1	yes	characteristic
150	C₂	#15689541	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13093963

C₂

yes

maximal, characteristic, complemented

C₂²×C₄₇⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₈×C₂²⋊D₄₇

C₂

yes

maximal, characteristic

Dic₄₇⋊C₁₆

C₂

yes

maximal, characteristic, complemented

(C₂×D₄₇)⋊C₁₆

C₂

yes

maximal, characteristic, complemented

#15688898

C₂

yes

maximal, characteristic, complemented

C₄₇×C₂²⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₄₇

C₂²

yes

characteristic

C₄×C₂²⋊D₄₇

C₄

yes

characteristic

#13093963

—

C₂×C₄₇⋊C₁₆

—

C₂×C₄₇⋊C₁₆

C₂²

yes

characteristic

C₂×C₄₇⋊C₁₆

C₂²

yes

characteristic

Dic₄₇⋊C₈

C₄

yes

characteristic

C₂²⋊(C₄₇⋊C₈)

C₄

yes

characteristic

(C₂×D₄₇)⋊C₈

C₄

yes

characteristic

C₈×Dic₄₇

C₂²

yes

characteristic

C₂²×C₈×C₄₇

C₂²

yes

characteristic

C₂×C₁₆×C₄₇

C₂²

yes

characteristic

C₂×C₁₆×C₄₇

C₂²

yes

characteristic

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂×C₂²⋊D₄₇

C₈

yes

characteristic

C₂×C₄×D₄₇

C₂×C₄

yes

characteristic

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂×C₄

yes

characteristic

C₂×C₈×C₄₇

C₂³

yes

characteristic

C₂²⋊Dic₄₇

C₈

yes

characteristic

(C₂×D₄₇)⋊C₄

C₈

yes

characteristic

Dic₄₇⋊C₄

C₈

yes

characteristic

C₄×Dic₄₇

C₂×C₄

yes

characteristic

C₂×C₄₇⋊C₈

C₂×C₄

yes

characteristic

C₂×C₄₇⋊C₈

C₂×C₄

yes

characteristic

C₈×D₄₇

—

C₈×D₄₇

—

C₁₆×C₄₇

—

C₁₆×C₄₇

—

C₄₇⋊C₁₆

—

C₄₇⋊C₁₆

D₄

yes

complemented

C₂³×C₄₇

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

—

C₂×Dic₄₇

C₂×C₈

yes

characteristic

C₂×Dic₄₇

C₂×C₈

yes

characteristic

C₂×Dic₄₇

C₂×C₈

yes

characteristic

C₄×D₄₇

—

C₄×D₄₇

—

C₈×C₄₇

—

C₈×C₄₇

C₂×D₄

yes

characteristic

C₈×C₄₇

#41

yes

characteristic

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

M₃₂

yes

complemented

C₂²×C₄₇

C₂²×C₈

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₄×D₄

yes

characteristic

C₄×C₄₇

#131

yes

characteristic

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

#2168

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

C₈×D₄

yes

characteristic

C₂×C₄₇

C₂×M₃₂

yes

characteristic

C₂×C₄₇

#504

yes

characteristic

D₄₇

—

C₂×M₃₂

—

C₂²×C₁₆

—

C₈×D₄

—

C₄⋊C₁₆

—

C₂²⋊C₁₆

—

C₂²⋊C₁₆

D₄₇

yes

characteristic, complemented

#346

—

C₄₇

#2168

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

C₂×D₄₇

yes

characteristic

C₄×D₄

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₄₇

yes

characteristic

C₂×C₁₆

C₂×D₄₇

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₄₇

yes

characteristic

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

106

C₂×C₈

C₄×D₄₇

yes

characteristic

107

C₂×C₈

C₂²×D₄₇

yes

characteristic

108

C₄⋊C₄

—

109

C₂²⋊C₄

—

110

C₂²⋊C₄

—

111

C₄²

—

112

C₁₆

—

113

C₁₆

—

114

C₁₆

—

115

C₁₆

—

116

C₂³

—

117

C₂³

C₈×D₄₇

yes

characteristic

118

D₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

C₈×D₄₇

yes

characteristic

127

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

128

C₈

—

129

C₈

—

130

C₈

—

131

C₈

D₄×D₄₇

yes

characteristic

132

C₈

#10625352

yes

characteristic

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

138

C₂²

#13093963

yes

complemented

139

C₂²

C₂×C₈×D₄₇

yes

characteristic

140

C₄

—

141

C₄

—

142

C₄

—

143

C₄

—

144

C₄

(C₂²⋊D₄₇)⋊C₄

yes

characteristic

145

C₄

#13094032

yes

characteristic

146

C₂

—

147

C₂

—

188

—

148

C₂

(C₂²⋊D₄₇)⋊C₈

yes

characteristic

149

C₂

C₂×#13093963

yes

characteristic

150

C₂

#15689541

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₈	C₂²×D₄₇
commutator, socle	C₂²×C₄₇	C₂²×C₈
Fitting	C₄₇×C₂²⋊C₁₆	C₂
hypercenter	C₂²⋊C₁₆	D₄₇
hypocenter	C₄₇	#2168
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₈ ◁ C₂²⋊C₁₆

subgroup

G/H

center, Frattini

C₂×C₈

C₂²×D₄₇

commutator, socle

C₂²×C₄₇

C₂²×C₈

Fitting

C₄₇×C₂²⋊C₁₆

C₂

hypercenter

C₂²⋊C₁₆

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₄₇

C₁ ◁ C₂×C₈ ◁ C₂²⋊C₁₆

N	H ≅ G/N	type
1. C₂×#13093963	C₂	semidirect
2. C₂²×C₄₇⋊C₁₆	C₂	semidirect
4. Dic₄₇⋊C₁₆	C₂	semidirect
5. (C₂×D₄₇)⋊C₁₆	C₂	semidirect
6. #15688898	C₂	semidirect
7. C₄₇×C₂²⋊C₁₆	C₂	semidirect
39. C₄₇⋊C₁₆	D₄	semidirect
59. C₂²×C₄₇	M₃₂	semidirect
81. C₂²⋊C₁₆	D₄₇	semidirect
83. C₄₇	#2168	semidirect
138. C₂²	#13093963	semidirect

H ≅ G/N

type

1. C₂×#13093963

C₂

semidirect

2. C₂²×C₄₇⋊C₁₆

C₂

semidirect

4. Dic₄₇⋊C₁₆

C₂

semidirect

5. (C₂×D₄₇)⋊C₁₆

C₂

semidirect

6. #15688898

C₂

semidirect

7. C₄₇×C₂²⋊C₁₆

C₂

semidirect

39. C₄₇⋊C₁₆

D₄

semidirect

59. C₂²×C₄₇

M₃₂

semidirect

81. C₂²⋊C₁₆

D₄₇

semidirect

83. C₄₇

#2168

semidirect

138. C₂²

#13093963

semidirect

Subgroups — #20245210

Conjugacy classes of subgroups

Complements