Group #20243415 — subgroups

Subgroups — #20243415

Order 6016.

336 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 125 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	C₂³×C₄₇⋊C₈	C₂	1	1	yes	maximal, characteristic
2	C₂×#13093957	C₂	1	2	yes	maximal, complemented
3	#15688983	C₂	1	1	yes	maximal, characteristic
4	#15688885	C₂	1	2	yes	maximal, complemented
5	C₄₇×#380	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₄₇⋊C₈	—	2	4	no	—
7	C₂²×C₄₇⋊C₈	C₄	1	4	yes	complemented
8	C₂²×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
9	C₂²×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
11	C₂×C₂²⋊Dic₄₇	—	2	1	no	—
12	#13093957	—	2	4	no	—
13	C₄₇⋊#95	—	2	1	no	—
14	C₂³×C₄×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
16	C₂×C₂²⋊C₄×C₄₇	—	2	1	no	—
17	#95×C₄₇	—	2	1	no	—
18	C₂⁴×C₄₇	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₄₇	—	2	2	no	—
20	C₂²×C₄×C₄₇	—	2	2	no	—
21	C₂²×C₄×C₄₇	—	2	2	no	—
22	C₂²×C₄×C₄₇	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₄₇	Q₈	1	2	yes	—
25	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
26	C₂²×Dic₄₇	—	2	2	no	—
27	C₄⋊C₄×C₄₇	—	2	2	no	—
28	C₄⋊C₄×C₄₇	—	2	2	no	—
29	C₂²⋊C₄×C₄₇	—	4	2	no	—
30	C₂²⋊Dic₄₇	—	4	2	no	—
31	C₄⋊Dic₄₇	—	2	2	no	—
32	C₄⋊Dic₄₇	—	2	2	no	—
33	C₂×C₄₇⋊C₈	—	2	2	no	—
34	C₂×C₄₇⋊C₈	—	2	8	no	—
35	C₂×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
36	C₂×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
37	C₂×C₄₇⋊C₈	D₄	1	4	yes	—
38	C₂³×C₄₇	—	2	2	no	—
39	C₂³×C₄₇	—	2	2	no	—
40	C₂³×C₄₇	C₄⋊C₄	1	2	yes	—
41	C₂³×C₄₇	D₈	1	2	yes	—
42	C₂³×C₄₇	Q₁₆	1	2	yes	—
43	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₄₇	—	2	2	no	—
45	C₂×C₄×C₄₇	—	2	2	no	—
46	C₂×C₄×C₄₇	—	2	2	no	—
47	C₂×C₄×C₄₇	—	2	4	no	—
48	C₂×C₄×C₄₇	—	4	2	no	—
49	C₂×C₄×C₄₇	—	4	2	no	—
50	C₂×C₄×C₄₇	—	4	2	no	—
51	C₂×C₄×C₄₇	C₄⋊C₄	1	2	yes	—
52	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
54	C₂×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₄×C₄₇	#41	1	2	yes	—
56	C₂×Dic₄₇	—	4	2	no	—
57	C₂×Dic₄₇	—	4	2	no	—
58	C₂×Dic₄₇	—	4	2	no	—
59	C₄₇⋊C₈	—	2	2	no	—
60	C₄₇⋊C₈	C₂²⋊C₄	1	4	yes	complemented
61	C₂²×C₄₇	—	2	2	no	—
62	C₂²×C₄₇	—	2	2	no	—
63	C₂²×C₄₇	—	2	4	no	—
64	C₂²×C₄₇	—	2	4	no	—
65	C₂²×C₄₇	#107	1	4	yes	complemented
66	C₂²×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
67	C₂²×C₄₇	#121	1	2	yes	—
68	C₂²×C₄₇	C₂×D₈	1	1	yes	characteristic
69	C₂²×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
70	C₂²×C₄₇	#135	1	2	yes	—
71	C₄×C₄₇	—	2	2	no	—
72	C₄×C₄₇	—	8	2	no	—
73	C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
74	C₄×C₄₇	#117	1	1	yes	characteristic
75	C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
76	Dic₄₇	—	8	2	no	—
77	#1808	—	47	1	no	maximal, Sylow
78	C₂×C₄₇	—	2	4	no	—
79	C₂×C₄₇	#380	1	1	yes	characteristic
80	C₂×C₄₇	C₂×#107	1	1	yes	characteristic
81	C₂×C₄₇	#427	1	1	yes	characteristic
82	C₂×C₄₇	#466	1	2	yes	—
83	C₂×C₄₇	#467	1	2	yes	—
84	C₂³×C₈	—	47	1	no	—
85	C₂×#107	—	47	2	no	—
86	#380	—	47	1	no	—
87	#380	D₄₇	1	1	yes	characteristic
88	#340	—	47	2	no	—
89	C₄₇	#1808	1	1	yes	characteristic, complemented, Sylow
90	C₂³×C₄	C₂×D₄₇	1	1	yes	characteristic
91	C₂²×C₈	—	47	1	no	—
92	C₂²×C₈	—	47	4	no	—
93	C₂²×C₈	—	47	1	no	—
94	C₂²×C₈	—	94	4	no	—
95	C₂×C₄⋊C₄	—	47	2	no	—
96	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
97	C₂×C₂²⋊C₄	—	2	1	no	—
98	C₂×C₂²⋊C₄	—	94	1	no	—
99	#107	—	94	4	no	—
100	#95	—	2	1	no	—
101	#95	—	94	1	no	—
102	C₂⁴	C₂²⋊D₄₇	1	1	yes	characteristic
103	C₂²×C₄	—	2	2	no	—
104	C₂²×C₄	—	2	2	no	—
105	C₂²×C₄	—	2	2	no	—
106	C₂²×C₄	—	94	2	no	—
107	C₂²×C₄	Dic₉₄	1	2	yes	—
108	C₂²×C₄	C₄×D₄₇	1	2	yes	—
109	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
110	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
111	C₂×C₈	—	47	2	no	—
112	C₂×C₈	—	47	4	no	—
113	C₂×C₈	—	47	2	no	—
114	C₂×C₈	—	94	8	no	—
115	C₂×C₈	—	94	2	no	—
116	C₄⋊C₄	—	2	2	no	—
117	C₄⋊C₄	—	2	2	no	—
118	C₄⋊C₄	—	94	2	no	—
119	C₄⋊C₄	—	94	2	no	—
120	C₂²⋊C₄	—	4	2	no	—
121	C₂²⋊C₄	—	188	2	no	—
122	C₂³	—	2	2	no	—
123	C₂³	—	2	2	no	—
124	C₂³	Dic₄₇⋊C₄	1	2	yes	—
125	C₂³	#10625334	1	2	yes	—
126	C₂³	#10625337	1	2	yes	—
127	C₂³	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	2	2	no	—
130	C₂×C₄	—	2	2	no	—
131	C₂×C₄	—	2	4	no	—
132	C₂×C₄	—	4	2	no	—
133	C₂×C₄	—	4	2	no	—
134	C₂×C₄	—	4	2	no	—
135	C₂×C₄	—	188	2	no	—
136	C₂×C₄	—	188	2	no	—
137	C₂×C₄	—	188	2	no	—
138	C₂×C₄	Dic₄₇⋊C₄	1	2	yes	—
139	C₂×C₄	C₂×Dic₉₄	1	1	yes	characteristic
140	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
141	C₂×C₄	D₄×D₄₇	1	2	yes	—
142	C₂×C₄	#10625352	1	2	yes	—
143	C₈	—	47	4	no	—
144	C₈	—	94	2	no	—
145	C₂²	—	2	2	no	—
146	C₂²	—	2	4	no	—
147	C₂²	—	2	4	no	—
148	C₂²	—	2	2	no	—
149	C₂²	#13093957	1	4	yes	complemented
150	C₂²	C₂×Dic₄₇⋊C₄	1	1	yes	characteristic
151	C₂²	#13093996	1	2	yes	—
152	C₂²	C₂×#10625334	1	1	yes	characteristic
153	C₂²	C₂×#10625337	1	1	yes	characteristic
154	C₂²	#13094044	1	2	yes	—
155	C₄	—	2	2	no	—
156	C₄	—	8	2	no	—
157	C₄	—	376	2	no	—
158	C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
159	C₄	#13093980	1	1	yes	characteristic
160	C₄	#13094001	1	2	yes	—
161	C₂	—	2	4	no	—
162	C₂	#15688981	1	1	yes	characteristic
163	C₂	C₂×#13093957	1	1	yes	characteristic
164	C₂	#15689159	1	1	yes	characteristic
165	C₂	#15689357	1	2	yes	—
166	C₂	#15689362	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×C₄₇⋊C₈

C₂

yes

maximal, characteristic

C₂×#13093957

C₂

yes

maximal, complemented

#15688983

C₂

yes

maximal, characteristic

#15688885

C₂

yes

maximal, complemented

C₄₇×#380

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₈

—

C₂²×C₄₇⋊C₈

C₄

yes

complemented

C₂²×C₄₇⋊C₈

C₂²

yes

characteristic

C₂²×C₄₇⋊C₈

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×C₂²⋊Dic₄₇

—

#13093957

—

C₄₇⋊#95

—

C₂³×C₄×C₄₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₂²⋊C₄×C₄₇

—

#95×C₄₇

—

C₂⁴×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

Q₈

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

C₂×C₄

yes

—

C₂×C₄₇⋊C₈

C₂×C₄

yes

—

C₂×C₄₇⋊C₈

D₄

yes

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₄⋊C₄

yes

—

C₂³×C₄₇

D₈

yes

—

C₂³×C₄₇

Q₁₆

yes

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₄⋊C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₄₇

#41

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

C₂²⋊C₄

yes

complemented

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#107

yes

complemented

C₂²×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₄₇

#121

yes

—

C₂²×C₄₇

C₂×D₈

yes

characteristic

C₂²×C₄₇

C₂×Q₁₆

yes

characteristic

C₂²×C₄₇

#135

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₄₇

#117

yes

characteristic

C₄×C₄₇

C₂²⋊Q₈

yes

—

Dic₄₇

—

#1808

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×C₄₇

#467

yes

—

C₂³×C₈

—

C₂×#107

—

#380

—

#380

D₄₇

yes

characteristic

#340

—

C₄₇

#1808

yes

characteristic, complemented, Sylow

C₂³×C₄

C₂×D₄₇

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

C₂×C₂²⋊C₄

—

C₂×C₂²⋊C₄

—

#107

—

100

#95

—

101

#95

—

102

C₂⁴

C₂²⋊D₄₇

yes

characteristic

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

Dic₉₄

yes

—

108

C₂²×C₄

C₄×D₄₇

yes

—

109

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

110

C₂²×C₄

C₂²×D₄₇

yes

characteristic

111

C₂×C₈

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

—

115

C₂×C₈

—

116

C₄⋊C₄

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₂²⋊C₄

—

121

C₂²⋊C₄

—

188

—

122

C₂³

—

123

C₂³

—

124

C₂³

Dic₄₇⋊C₄

yes

—

125

C₂³

#10625334

yes

—

126

C₂³

#10625337

yes

—

127

C₂³

C₂×C₂²⋊D₄₇

yes

characteristic

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

188

—

136

C₂×C₄

—

188

—

137

C₂×C₄

—

188

—

138

C₂×C₄

Dic₄₇⋊C₄

yes

—

139

C₂×C₄

C₂×Dic₉₄

yes

characteristic

140

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

141

C₂×C₄

D₄×D₄₇

yes

—

142

C₂×C₄

#10625352

yes

—

143

C₈

—

144

C₈

—

145

C₂²

—

146

C₂²

—

147

C₂²

—

148

C₂²

—

149

C₂²

#13093957

yes

complemented

150

C₂²

C₂×Dic₄₇⋊C₄

yes

characteristic

151

C₂²

#13093996

yes

—

152

C₂²

C₂×#10625334

yes

characteristic

153

C₂²

C₂×#10625337

yes

characteristic

154

C₂²

#13094044

yes

—

155

C₄

—

156

C₄

—

157

C₄

—

376

—

158

C₄

C₂²⋊C₄×D₄₇

yes

characteristic

159

C₄

#13093980

yes

characteristic

160

C₄

#13094001

yes

—

161

C₂

—

162

C₂

#15688981

yes

characteristic

163

C₂

C₂×#13093957

yes

characteristic

164

C₂

#15689159

yes

characteristic

165

C₂

#15689357

yes

—

166

C₂

#15689362

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₄₇×#380	C₂
socle	C₂³×C₄₇	C₂×D₄
hypercenter	#380	D₄₇
hypocenter	C₄₇	#1808
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #380

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₄₇

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₄₇×#380

C₂

socle

C₂³×C₄₇

C₂×D₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ #380

N	H ≅ G/N	type
2. C₂×#13093957	C₂	semidirect
4. #15688885	C₂	semidirect
7. C₂²×C₄₇⋊C₈	C₄	semidirect
60. C₄₇⋊C₈	C₂²⋊C₄	semidirect
65. C₂²×C₄₇	#107	semidirect
89. C₄₇	#1808	semidirect
149. C₂²	#13093957	semidirect

H ≅ G/N

type

2. C₂×#13093957

C₂

semidirect

4. #15688885

C₂

semidirect

7. C₂²×C₄₇⋊C₈

C₄

semidirect

60. C₄₇⋊C₈

C₂²⋊C₄

semidirect

65. C₂²×C₄₇

#107

semidirect

89. C₄₇

#1808

semidirect

149. C₂²

#13093957

semidirect

Subgroups — #20243415

Conjugacy classes of subgroups

Complements