Group #20251954 — subgroups

Subgroups — #20251954

Order 6016.

484 conjugacy classes of subgroups; only the 255 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 255 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×#13094031	C₂	1	1	yes	maximal, characteristic
2	#41×Dic₄₇	C₂	1	1	yes	maximal, characteristic
3	#15689151	C₂	1	3	yes	maximal, complemented
4	#15689149	C₂	1	3	yes	maximal, complemented
5	#15689146	C₂	1	3	yes	maximal, complemented
6	C₂×Dic₄₇⋊C₈	C₂	1	3	yes	maximal, complemented
7	C₂×#131×C₄₇	C₂	1	1	yes	maximal, characteristic
8	#13094031	—	2	4	no	—
9	C₂×#10625329	C₂²	1	3	yes	—
10	C₂²×C₄₇⋊C₈	C₂²	1	3	yes	—
11	Q₈×Dic₄₇	C₄	1	1	yes	characteristic
12	Q₈×Dic₄₇	C₄	1	1	yes	characteristic
13	D₄×Dic₄₇	C₄	1	3	yes	—
14	D₄×Dic₄₇	C₄	1	3	yes	—
15	#13093983	C₂²	1	3	yes	—
16	C₂×C₄×Dic₄₇	C₂²	1	3	yes	—
17	Dic₄₇⋊C₈	C₂²	1	1	yes	characteristic
18	Dic₄₇⋊C₈	C₂²	1	3	yes	—
19	Dic₄₇⋊C₈	C₂²	1	3	yes	—
20	Dic₄₇⋊C₈	C₂²	1	3	yes	—
21	Dic₄₇⋊C₈	C₂²	1	6	yes	—
22	C₂×#41×C₄₇	C₂²	1	1	yes	characteristic
23	#131×C₄₇	—	2	4	no	—
24	C₂×M₁₆×C₄₇	C₂²	1	3	yes	—
25	C₂²×C₈×C₄₇	C₂²	1	3	yes	—
26	#41×C₄₇	D₄	1	4	yes	—
27	#41×C₄₇	Q₈	1	4	yes	—
28	C₂×Q₈×C₄₇	C₂×C₄	1	1	yes	characteristic
29	C₂×D₄×C₄₇	C₂×C₄	1	3	yes	—
30	C₂²×C₄×C₄₇	C₂³	1	3	yes	—
31	C₂²×Dic₄₇	C₂×C₄	1	3	yes	—
32	C₂²×Dic₄₇	C₂×C₄	1	3	yes	—
33	M₁₆×C₄₇	—	2	6	no	—
34	C₂×C₈×C₄₇	—	2	6	no	—
35	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
36	C₂×C₈×C₄₇	C₂³	1	3	yes	—
37	C₂²⋊Dic₄₇	C₂×C₄	1	6	yes	—
38	C₄⋊Dic₄₇	C₂×C₄	1	3	yes	—
39	C₄⋊Dic₄₇	C₂×C₄	1	3	yes	—
40	C₄×Dic₄₇	C₂×C₄	1	3	yes	—
41	C₄×Dic₄₇	C₂×C₄	1	3	yes	—
42	C₄×Dic₄₇	C₂³	1	1	yes	characteristic
43	C₄×Dic₄₇	C₂³	1	3	yes	—
44	#10625329	—	2	6	no	—
45	C₂×C₄₇⋊C₈	—	2	6	no	—
46	C₂×C₄₇⋊C₈	C₂³	1	1	yes	characteristic
47	C₂×C₄₇⋊C₈	C₂³	1	3	yes	—
48	C₂³×C₄₇	C₂²×C₄	1	3	yes	—
49	Q₈×C₄₇	C₄⋊C₄	1	4	yes	—
50	D₄×C₄₇	C₄⋊C₄	1	12	yes	—
51	C₂×C₄×C₄₇	C₂²×C₄	1	3	yes	—
52	C₂×C₄×C₄₇	C₂×D₄	1	6	yes	—
53	C₂×C₄×C₄₇	C₂×Q₈	1	6	yes	—
54	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
55	C₂×Dic₄₇	—	2	6	no	—
56	C₂×Dic₄₇	—	4	3	no	—
57	C₂×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
58	C₂×Dic₄₇	C₂²×C₄	1	3	yes	—
59	C₂×Dic₄₇	C₂²×C₄	1	3	yes	—
60	C₂×Dic₄₇	C₂²×C₄	1	1	yes	characteristic
61	C₈×C₄₇	—	2	1	no	—
62	C₈×C₄₇	—	2	3	no	—
63	C₄₇⋊C₈	—	2	1	no	—
64	C₄₇⋊C₈	—	2	3	no	—
65	C₂²×C₄₇	—	2	3	no	—
66	C₂²×C₄₇	—	2	3	no	—
67	C₂²×C₄₇	C₂×C₄⋊C₄	1	6	yes	—
68	C₂²×C₄₇	C₂³×C₄	1	1	yes	characteristic
69	C₄×C₄₇	C₂×C₄⋊C₄	1	6	yes	—
70	C₄×C₄₇	C₂²×D₄	1	1	yes	characteristic
71	C₄×C₄₇	C₂²×Q₈	1	1	yes	characteristic
72	Dic₄₇	—	2	3	no	—
73	Dic₄₇	—	2	1	no	—
74	Dic₄₇	—	2	3	no	—
75	Dic₄₇	#131	1	2	yes	complemented
76	#2906	—	47	1	no	maximal, Sylow
77	C₂×C₄₇	—	2	6	no	—
78	C₂×C₄₇	C₂²×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄₇	C₂×#131	1	1	yes	characteristic
80	C₂×C₄₇	#568	1	1	yes	characteristic
81	C₂×#131	—	47	1	no	—
82	C₂×#131	D₄₇	1	1	yes	characteristic
83	C₄×#41	—	47	1	no	—
84	#424	—	47	3	no	—
85	#424	—	47	3	no	—
86	#423	—	47	3	no	—
87	C₂×C₄⋊C₈	—	47	3	no	—
88	C₄₇	#2906	1	1	yes	characteristic, complemented, Sylow
89	C₂×#41	C₂×D₄₇	1	1	yes	characteristic
90	#131	—	2	4	no	—
91	#131	—	94	4	no	—
92	C₂×M₁₆	—	47	3	no	—
93	C₂×M₁₆	C₂×D₄₇	1	3	yes	—
94	C₂²×C₈	—	47	3	no	—
95	C₂²×C₈	C₂×D₄₇	1	3	yes	—
96	C₄×Q₈	—	47	1	no	—
97	C₄×Q₈	—	47	1	no	—
98	C₄×D₄	—	47	3	no	—
99	C₄×D₄	—	47	3	no	—
100	#117	—	47	3	no	—
101	C₂×C₄²	—	47	3	no	—
102	C₄⋊C₈	—	47	1	no	—
103	C₄⋊C₈	—	47	3	no	—
104	C₄⋊C₈	—	47	3	no	—
105	C₄⋊C₈	—	47	3	no	—
106	C₄⋊C₈	—	47	6	no	—
107	#41	Dic₉₄	1	4	yes	—
108	#41	C₂²⋊D₄₇	1	4	yes	—
109	C₂×Q₈	C₄×D₄₇	1	1	yes	characteristic
110	C₂×D₄	C₄×D₄₇	1	3	yes	—
111	C₂²×C₄	—	47	3	no	—
112	C₂²×C₄	—	47	3	no	—
113	C₂²×C₄	C₂²×D₄₇	1	3	yes	—
114	M₁₆	—	2	6	no	—
115	M₁₆	—	94	6	no	—
116	C₂×C₈	—	2	6	no	—
117	C₂×C₈	—	47	1	no	—
118	C₂×C₈	—	47	3	no	—
119	C₂×C₈	—	94	6	no	—
120	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
121	C₂×C₈	C₂²×D₄₇	1	3	yes	—
122	C₄⋊C₄	—	47	3	no	—
123	C₄⋊C₄	—	47	3	no	—
124	C₂²⋊C₄	—	47	6	no	—
125	C₄²	—	47	3	no	—
126	C₄²	—	47	3	no	—
127	C₄²	—	47	1	no	—
128	C₄²	—	47	3	no	—
129	C₂³	C₂×C₄×D₄₇	1	3	yes	—
130	Q₈	Dic₄₇⋊C₄	1	4	yes	—
131	D₄	Dic₄₇⋊C₄	1	12	yes	—
132	C₂×C₄	—	47	1	no	—
133	C₂×C₄	—	47	1	no	—
134	C₂×C₄	—	47	3	no	—
135	C₂×C₄	—	47	3	no	—
136	C₂×C₄	—	94	6	no	—
137	C₂×C₄	—	188	3	no	—
138	C₂×C₄	C₂×Dic₉₄	1	6	yes	—
139	C₂×C₄	C₂×C₄×D₄₇	1	3	yes	—
140	C₂×C₄	C₂×C₂²⋊D₄₇	1	6	yes	—
141	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
142	C₈	—	2	1	no	—
143	C₈	—	2	3	no	—
144	C₈	—	94	1	no	—
145	C₈	—	94	3	no	—
146	C₂²	—	2	3	no	—
147	C₂²	—	2	3	no	—
148	C₂²	C₂×Dic₄₇⋊C₄	1	6	yes	—
149	C₂²	C₂²×C₄×D₄₇	1	1	yes	characteristic
150	C₄	—	47	2	no	—
151	C₄	—	94	3	no	—
152	C₄	—	94	1	no	—
153	C₄	—	94	3	no	—
154	C₄	C₂×Dic₄₇⋊C₄	1	6	yes	—
155	C₄	C₂²×C₂²⋊D₄₇	1	1	yes	characteristic
156	C₄	C₂²×Dic₉₄	1	1	yes	characteristic
157	C₂	—	2	6	no	—
158	C₂	C₂²×Dic₄₇⋊C₄	1	1	yes	characteristic
159	C₂	#131×D₄₇	1	1	yes	characteristic
160	C₂	#15689888	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094031

C₂

yes

maximal, characteristic

#41×Dic₄₇

C₂

yes

maximal, characteristic

#15689151

C₂

yes

maximal, complemented

#15689149

C₂

yes

maximal, complemented

#15689146

C₂

yes

maximal, complemented

C₂×Dic₄₇⋊C₈

C₂

yes

maximal, complemented

C₂×#131×C₄₇

C₂

yes

maximal, characteristic

#13094031

—

C₂×#10625329

C₂²

yes

—

C₂²×C₄₇⋊C₈

C₂²

yes

—

Q₈×Dic₄₇

C₄

yes

characteristic

Q₈×Dic₄₇

C₄

yes

characteristic

D₄×Dic₄₇

C₄

yes

—

D₄×Dic₄₇

C₄

yes

—

#13093983

C₂²

yes

—

C₂×C₄×Dic₄₇

C₂²

yes

—

Dic₄₇⋊C₈

C₂²

yes

characteristic

Dic₄₇⋊C₈

C₂²

yes

—

Dic₄₇⋊C₈

C₂²

yes

—

Dic₄₇⋊C₈

C₂²

yes

—

Dic₄₇⋊C₈

C₂²

yes

—

C₂×#41×C₄₇

C₂²

yes

characteristic

#131×C₄₇

—

C₂×M₁₆×C₄₇

C₂²

yes

—

C₂²×C₈×C₄₇

C₂²

yes

—

#41×C₄₇

D₄

yes

—

#41×C₄₇

Q₈

yes

—

C₂×Q₈×C₄₇

C₂×C₄

yes

characteristic

C₂×D₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂×C₄

yes

—

C₂²×Dic₄₇

C₂×C₄

yes

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂³

yes

characteristic

C₂×C₈×C₄₇

C₂³

yes

—

C₂²⋊Dic₄₇

C₂×C₄

yes

—

C₄⋊Dic₄₇

C₂×C₄

yes

—

C₄⋊Dic₄₇

C₂×C₄

yes

—

C₄×Dic₄₇

C₂×C₄

yes

—

C₄×Dic₄₇

C₂×C₄

yes

—

C₄×Dic₄₇

C₂³

yes

characteristic

C₄×Dic₄₇

C₂³

yes

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₂³

yes

characteristic

C₂×C₄₇⋊C₈

C₂³

yes

—

C₂³×C₄₇

C₂²×C₄

yes

—

Q₈×C₄₇

C₄⋊C₄

yes

—

D₄×C₄₇

C₄⋊C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂²×C₄

yes

characteristic

C₂×Dic₄₇

C₂²×C₄

yes

—

C₂×Dic₄₇

C₂²×C₄

yes

—

C₂×Dic₄₇

C₂²×C₄

yes

characteristic

C₈×C₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₄⋊C₄

yes

—

C₂²×C₄₇

C₂³×C₄

yes

characteristic

C₄×C₄₇

C₂×C₄⋊C₄

yes

—

C₄×C₄₇

C₂²×D₄

yes

characteristic

C₄×C₄₇

C₂²×Q₈

yes

characteristic

Dic₄₇

—

Dic₄₇

—

Dic₄₇

—

Dic₄₇

yes

—

C₂×C₄₇

—

C₂×C₄₇

C₂²×C₄⋊C₄

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

D₄₇

yes

characteristic

C₄×#41

—

#424

—

#424

—

#423

—

C₂×C₄⋊C₈

—

C₄₇

#2906

yes

characteristic, complemented, Sylow

C₂×#41

C₂×D₄₇

yes

characteristic

#131

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₄₇

yes

—

C₂²×C₈

—

C₂²×C₈

C₂×D₄₇

yes

—

C₄×Q₈

—

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

—

100

#117

—

101

C₂×C₄²

—

102

C₄⋊C₈

—

103

C₄⋊C₈

—

104

C₄⋊C₈

—

105

C₄⋊C₈

—

106

C₄⋊C₈

—

107

#41

Dic₉₄

yes

—

108

#41

C₂²⋊D₄₇

yes

—

109

C₂×Q₈

C₄×D₄₇

yes

characteristic

110

C₂×D₄

C₄×D₄₇

yes

—

111

C₂²×C₄

—

112

C₂²×C₄

—

113

C₂²×C₄

C₂²×D₄₇

yes

—

114

M₁₆

—

115

M₁₆

—

116

C₂×C₈

—

117

C₂×C₈

—

118

C₂×C₈

—

119

C₂×C₈

—

120

C₂×C₈

C₂²×D₄₇

yes

characteristic

121

C₂×C₈

C₂²×D₄₇

yes

—

122

C₄⋊C₄

—

123

C₄⋊C₄

—

124

C₂²⋊C₄

—

125

C₄²

—

126

C₄²

—

127

C₄²

—

128

C₄²

—

129

C₂³

C₂×C₄×D₄₇

yes

—

130

Q₈

Dic₄₇⋊C₄

yes

—

131

D₄

Dic₄₇⋊C₄

yes

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

188

—

138

C₂×C₄

C₂×Dic₉₄

yes

—

139

C₂×C₄

C₂×C₄×D₄₇

yes

—

140

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

141

C₂×C₄

C₂³×D₄₇

yes

characteristic

142

C₈

—

143

C₈

—

144

C₈

—

145

C₈

—

146

C₂²

—

147

C₂²

—

148

C₂²

C₂×Dic₄₇⋊C₄

yes

—

149

C₂²

C₂²×C₄×D₄₇

yes

characteristic

150

C₄

—

151

C₄

—

152

C₄

—

153

C₄

—

154

C₄

C₂×Dic₄₇⋊C₄

yes

—

155

C₄

C₂²×C₂²⋊D₄₇

yes

characteristic

156

C₄

C₂²×Dic₉₄

yes

characteristic

157

C₂

—

158

C₂

C₂²×Dic₄₇⋊C₄

yes

characteristic

159

C₂

#131×D₄₇

yes

characteristic

160

C₂

#15689888

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₄₇
commutator, socle	C₂²×C₄₇	C₂³×C₄
Fitting	C₂×#131×C₄₇	C₂
hypercenter	C₂×#131	D₄₇
hypocenter	C₄₇	#2906
LCS	G ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#131

subgroup

G/H

center, Frattini

C₂×C₄

C₂³×D₄₇

commutator, socle

C₂²×C₄₇

C₂³×C₄

Fitting

C₂×#131×C₄₇

C₂

D₄₇

C₄₇

G ▷ C₂²×C₄₇ ▷ C₄₇

C₁ ◁ C₂×C₄ ◁ C₂×#131

N	H ≅ G/N	type
3. #15689151	C₂	semidirect
4. #15689149	C₂	semidirect
5. #15689146	C₂	semidirect
6. C₂×Dic₄₇⋊C₈	C₂	semidirect
75. Dic₄₇	#131	semidirect
88. C₄₇	#2906	semidirect

H ≅ G/N

type

3. #15689151

C₂

semidirect

4. #15689149

C₂

semidirect

5. #15689146

C₂

semidirect

6. C₂×Dic₄₇⋊C₈

C₂

semidirect

75. Dic₄₇

#131

semidirect

88. C₄₇

#2906

semidirect

Subgroups — #20251954

Conjugacy classes of subgroups

Complements