Group #20241896 — subgroups

Subgroups — #20241896

Order 6016.

272 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 81 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	#15689180	C₂	1	1	yes	maximal, characteristic, complemented
2	#15688969	C₂	1	1	yes	maximal, characteristic, complemented
3	#15688857	C₂	1	4	yes	maximal, complemented
4	C₄₇×#532	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×Dic₄₇	—	2	1	no	—
6	C₄⋊(C₄₇⋊C₈)	—	2	1	no	—
7	C₂²⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
8	#13093935	—	2	1	no	—
9	C₄₇⋊#95	—	2	1	no	—
10	C₄₇⋊#95	C₂²	1	2	yes	—
11	C₂×#41×C₄₇	—	2	1	no	—
12	C₄⋊Q₈×C₄₇	C₄	1	1	yes	characteristic, complemented
13	#127×C₄₇	C₄	1	1	yes	characteristic, complemented
14	#124×C₄₇	—	2	1	no	—
15	#121×C₄₇	—	2	1	no	—
16	#121×C₄₇	C₂²	1	2	yes	—
17	C₄×D₄×C₄₇	—	2	1	no	—
18	C₄×D₄×C₄₇	C₄	1	2	yes	complemented
19	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
20	#41×C₄₇	—	4	4	no	—
21	C₂×Q₈×C₄₇	—	2	1	no	—
22	C₂×D₄×C₄₇	—	2	2	no	—
23	C₂×D₄×C₄₇	—	2	1	no	—
24	C₂×D₄×C₄₇	—	2	1	no	—
25	C₂×D₄×C₄₇	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₄₇	—	2	2	no	—
27	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
28	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
29	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
30	C₂²×Dic₄₇	—	2	2	no	—
31	C₄⋊C₄×C₄₇	—	2	1	no	—
32	C₄⋊C₄×C₄₇	C₂×C₄	1	2	yes	—
33	C₂²⋊C₄×C₄₇	—	2	2	no	—
34	C₂²⋊C₄×C₄₇	—	2	2	no	—
35	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
36	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
37	C₄²×C₄₇	D₄	1	1	yes	characteristic
38	C₄²×C₄₇	D₄	1	1	yes	characteristic
39	C₄×Dic₄₇	—	2	4	no	—
40	C₂×C₄₇⋊C₈	—	2	2	no	—
41	C₂³×C₄₇	—	2	2	no	—
42	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
43	Q₈×C₄₇	—	4	2	no	—
44	D₄×C₄₇	—	4	2	no	—
45	D₄×C₄₇	—	4	2	no	—
46	D₄×C₄₇	—	4	4	no	—
47	D₄×C₄₇	—	4	2	no	—
48	C₂×C₄×C₄₇	—	2	2	no	—
49	C₂×C₄×C₄₇	—	2	2	no	—
50	C₂×C₄×C₄₇	—	2	2	no	—
51	C₂×C₄×C₄₇	—	4	1	no	—
52	C₂×C₄×C₄₇	—	4	4	no	—
53	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
56	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
57	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
59	C₂×Dic₄₇	—	2	4	no	—
60	C₂×Dic₄₇	—	4	2	no	—
61	C₂×Dic₄₇	—	4	2	no	—
62	C₄₇⋊C₈	—	4	2	no	—
63	C₂²×C₄₇	—	2	1	no	—
64	C₂²×C₄₇	—	2	1	no	—
65	C₂²×C₄₇	—	4	2	no	—
66	C₂²×C₄₇	—	4	2	no	—
67	C₂²×C₄₇	—	4	2	no	—
68	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
69	C₂²×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
70	C₄×C₄₇	—	2	1	no	—
71	C₄×C₄₇	—	2	1	no	—
72	C₄×C₄₇	—	2	1	no	—
73	C₄×C₄₇	—	2	1	no	—
74	C₄×C₄₇	—	2	1	no	—
75	C₄×C₄₇	—	4	2	no	—
76	C₄×C₄₇	#104	1	2	yes	—
77	Dic₄₇	—	4	4	no	—
78	#1505	—	47	1	no	maximal, Sylow
79	C₂×C₄₇	—	2	2	no	—
80	C₂×C₄₇	—	8	2	no	—
81	C₂×C₄₇	#410	1	1	yes	characteristic
82	C₂×C₄₇	#418	1	1	yes	characteristic
83	C₂×C₄₇	C₂×#104	1	1	yes	characteristic
84	#532	D₄₇	1	1	yes	characteristic
85	#432	—	47	1	no	—
86	#376	—	47	1	no	—
87	#327	—	47	4	no	—
88	C₄₇	#1505	1	1	yes	characteristic, complemented, Sylow
89	C₂×#41	—	2	1	no	—
90	C₄⋊Q₈	Dic₄₇	1	1	yes	characteristic, complemented
91	#127	Dic₄₇	1	1	yes	characteristic, complemented
92	#124	—	2	1	no	—
93	#121	—	2	1	no	—
94	#121	C₂×D₄₇	1	2	yes	—
95	C₄×D₄	—	2	1	no	—
96	C₄×D₄	Dic₄₇	1	2	yes	complemented
97	C₂×C₄²	—	94	1	no	—
98	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
99	C₄⋊C₈	—	94	1	no	—
100	C₂²⋊C₈	—	47	2	no	—
101	#97	—	94	1	no	—
102	#95	—	47	2	no	—
103	#95	—	94	1	no	—
104	#41	—	4	4	no	—
105	C₂×Q₈	—	2	1	no	—
106	C₂×D₄	—	2	2	no	—
107	C₂×D₄	—	2	1	no	—
108	C₂×D₄	—	2	1	no	—
109	C₂×D₄	C₂×Dic₄₇	1	2	yes	—
110	C₂²×C₄	—	2	2	no	—
111	C₂²×C₄	—	94	2	no	—
112	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
113	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
114	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
115	C₂×C₈	—	94	2	no	—
116	C₄⋊C₄	—	2	1	no	—
117	C₄⋊C₄	C₂×Dic₄₇	1	2	yes	—
118	C₂²⋊C₄	—	2	2	no	—
119	C₂²⋊C₄	—	2	2	no	—
120	C₄²	—	94	4	no	—
121	C₄²	C₂×Dic₄₇	1	1	yes	characteristic
122	C₄²	C₂×Dic₄₇	1	1	yes	characteristic
123	C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
124	C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
125	C₂³	—	2	2	no	—
126	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
127	Q₈	—	4	2	no	—
128	D₄	—	4	2	no	—
129	D₄	—	4	2	no	—
130	D₄	—	4	4	no	—
131	D₄	—	4	2	no	—
132	C₂×C₄	—	2	2	no	—
133	C₂×C₄	—	2	2	no	—
134	C₂×C₄	—	2	2	no	—
135	C₂×C₄	—	4	1	no	—
136	C₂×C₄	—	4	4	no	—
137	C₂×C₄	—	94	4	no	—
138	C₂×C₄	—	188	2	no	—
139	C₂×C₄	—	188	2	no	—
140	C₂×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
141	C₂×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
142	C₂×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
143	C₂×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
144	C₂×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
145	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
146	C₈	—	188	2	no	—
147	C₂²	—	2	1	no	—
148	C₂²	—	2	1	no	—
149	C₂²	—	4	2	no	—
150	C₂²	—	4	2	no	—
151	C₂²	—	4	2	no	—
152	C₂²	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
153	C₂²	#13094050	1	2	yes	—
154	C₄	—	2	1	no	—
155	C₄	—	2	1	no	—
156	C₄	—	2	1	no	—
157	C₄	—	2	1	no	—
158	C₄	—	2	1	no	—
159	C₄	—	4	2	no	—
160	C₄	—	188	4	no	—
161	C₄	#13093951	1	2	yes	—
162	C₂	—	2	2	no	—
163	C₂	—	8	2	no	—
164	C₂	#15689097	1	1	yes	characteristic
165	C₂	#15689128	1	1	yes	characteristic
166	C₂	C₂×#13093951	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689180

C₂

yes

maximal, characteristic, complemented

#15688969

C₂

yes

maximal, characteristic, complemented

#15688857

C₂

yes

maximal, complemented

C₄₇×#532

C₂

yes

maximal, characteristic

C₂×C₄×Dic₄₇

—

C₄⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

C₂²

yes

—

#13093935

—

C₄₇⋊#95

—

C₄₇⋊#95

C₂²

yes

—

C₂×#41×C₄₇

—

C₄⋊Q₈×C₄₇

C₄

yes

characteristic, complemented

#127×C₄₇

C₄

yes

characteristic, complemented

#124×C₄₇

—

#121×C₄₇

—

#121×C₄₇

C₂²

yes

—

C₄×D₄×C₄₇

—

C₄×D₄×C₄₇

C₄

yes

complemented

C₂×C₄²×C₄₇

C₂²

yes

characteristic

#41×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

C₂×C₄

yes

—

C₂²⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

D₄

yes

characteristic

C₄×Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

Q₈×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₇

AGL(1,ℤ/8)

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#104

yes

—

Dic₄₇

—

#1505

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

yes

—

—

—

C₄₇

#1505

yes

characteristic, complemented, Sylow

C₂×#41

—

C₄⋊Q₈

Dic₄₇

yes

characteristic, complemented

#127

Dic₄₇

yes

characteristic, complemented

#124

—

#121

—

#121

C₂×D₄₇

yes

—

C₄×D₄

—

C₄×D₄

Dic₄₇

yes

complemented

C₂×C₄²

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₄⋊C₈

—

100

C₂²⋊C₈

—

101

#97

—

102

#95

—

103

#95

—

104

#41

—

105

C₂×Q₈

—

106

C₂×D₄

—

107

C₂×D₄

—

108

C₂×D₄

—

109

C₂×D₄

C₂×Dic₄₇

yes

—

110

C₂²×C₄

—

111

C₂²×C₄

—

112

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

113

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

114

C₂²×C₄

C₂²×D₄₇

yes

characteristic

115

C₂×C₈

—

116

C₄⋊C₄

—

117

C₄⋊C₄

C₂×Dic₄₇

yes

—

118

C₂²⋊C₄

—

119

C₂²⋊C₄

—

120

C₄²

—

121

C₄²

C₂×Dic₄₇

yes

characteristic

122

C₄²

C₂×Dic₄₇

yes

characteristic

123

C₄²

C₂²⋊D₄₇

yes

characteristic

124

C₄²

C₂²⋊D₄₇

yes

characteristic

125

C₂³

—

126

C₂³

C₂×C₂²⋊D₄₇

yes

characteristic

127

Q₈

—

128

D₄

—

129

D₄

—

130

D₄

—

131

D₄

—

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

—

188

—

140

C₂×C₄

C₂²⋊Dic₄₇

yes

characteristic

141

C₂×C₄

C₂²⋊Dic₄₇

yes

characteristic

142

C₂×C₄

C₂²⋊Dic₄₇

yes

characteristic

143

C₂×C₄

C₂²⋊Dic₄₇

yes

characteristic

144

C₂×C₄

C₂²×Dic₄₇

yes

characteristic

145

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

146

C₈

—

188

—

147

C₂²

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

C₂×C₂²⋊Dic₄₇

yes

characteristic

153

C₂²

#13094050

yes

—

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

188

—

161

C₄

#13093951

yes

—

162

C₂

—

163

C₂

—

164

C₂

#15689097

yes

characteristic

165

C₂

#15689128

yes

characteristic

166

C₂

C₂×#13093951

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₄₇×#532	C₂
socle	C₂²×C₄₇	C₂×C₂²⋊C₄
hypercenter	#532	D₄₇
hypocenter	C₄₇	#1505
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₄₇

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₄₇×#532

C₂

socle

C₂²×C₄₇

C₂×C₂²⋊C₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #532

N	H ≅ G/N	type
1. #15689180	C₂	semidirect
2. #15688969	C₂	semidirect
3. #15688857	C₂	semidirect
12. C₄⋊Q₈×C₄₇	C₄	semidirect
13. #127×C₄₇	C₄	semidirect
18. C₄×D₄×C₄₇	C₄	semidirect
88. C₄₇	#1505	semidirect
90. C₄⋊Q₈	Dic₄₇	semidirect
91. #127	Dic₄₇	semidirect
96. C₄×D₄	Dic₄₇	semidirect

H ≅ G/N

type

1. #15689180

C₂

semidirect

2. #15688969

C₂

semidirect

3. #15688857

C₂

semidirect

12. C₄⋊Q₈×C₄₇

C₄

semidirect

13. #127×C₄₇

C₄

semidirect

18. C₄×D₄×C₄₇

C₄

semidirect

88. C₄₇

#1505

semidirect

90. C₄⋊Q₈

Dic₄₇

semidirect

91. #127

Dic₄₇

semidirect

96. C₄×D₄

Dic₄₇

semidirect

Subgroups — #20241896

Conjugacy classes of subgroups

Complements