Group #20243412 — subgroups

Subgroups — #20243412

Order 6016.

336 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 109 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	C₂²×C₈×D₄₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#13093955	C₂	1	2	yes	maximal, complemented
3	#15688980	C₂	1	2	yes	maximal
4	#15688886	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#340	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₄₇	C₂²	1	1	yes	characteristic
7	C₂²×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
8	C₂×C₈×D₄₇	—	2	4	no	—
9	C₂×C₈×D₄₇	C₄	1	4	yes	complemented
10	C₂×C₄⋊Dic₄₇	C₂²	1	2	yes	—
11	C₂×(C₂×D₄₇)⋊C₄	—	2	2	no	—
12	#13093955	—	2	4	no	—
13	C₄₇⋊#95	—	2	2	no	—
14	C₂²×C₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
16	C₂³×D₄₇	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₄₇	—	2	2	no	—
18	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
19	C₂²×Dic₄₇	—	2	2	no	—
20	C₂²×Dic₄₇	D₄	1	1	yes	characteristic
21	C₂×C₄×D₄₇	—	2	2	no	—
22	C₂×C₄×D₄₇	—	2	2	no	—
23	C₂×C₄×D₄₇	C₂×C₄	1	2	yes	—
24	C₂×C₄×D₄₇	Q₈	1	2	yes	—
25	C₂×C₈×C₄₇	—	2	2	no	—
26	C₂×C₈×C₄₇	C₂×C₄	1	2	yes	—
27	C₄⋊C₄×C₄₇	—	2	2	no	—
28	C₄⋊C₄×C₄₇	—	2	2	no	—
29	(C₂×D₄₇)⋊C₄	—	4	4	no	—
30	C₄⋊Dic₄₇	—	2	2	no	—
31	C₄⋊Dic₄₇	—	2	2	no	—
32	C₂×C₄₇⋊C₈	C₂×C₄	1	2	yes	—
33	C₂×C₄₇⋊C₈	D₄	1	4	yes	—
34	C₈×D₄₇	—	2	8	no	—
35	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
36	C₂²×D₄₇	—	2	2	no	—
37	C₂²×D₄₇	—	2	1	no	—
38	C₂²×D₄₇	—	2	1	no	—
39	C₂²×D₄₇	C₄⋊C₄	1	2	yes	—
40	C₂²×D₄₇	SD₁₆	1	4	yes	—
41	C₂×C₄×C₄₇	—	4	2	no	—
42	C₂×C₄×C₄₇	—	4	2	no	—
43	C₂×C₄×C₄₇	—	4	2	no	—
44	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₄₇	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₄₇	#41	1	2	yes	—
48	C₂×Dic₄₇	—	2	2	no	—
49	C₂×Dic₄₇	—	4	2	no	—
50	C₂×Dic₄₇	—	4	2	no	—
51	C₂×Dic₄₇	—	4	2	no	—
52	C₂×Dic₄₇	C₄⋊C₄	1	2	yes	—
53	C₄×D₄₇	—	2	2	no	—
54	C₄×D₄₇	—	2	2	no	—
55	C₄×D₄₇	—	2	4	no	—
56	C₈×C₄₇	—	2	2	no	—
57	C₄₇⋊C₈	C₂²⋊C₄	1	4	yes	complemented
58	C₂²×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
59	C₂²×C₄₇	#121	1	2	yes	—
60	C₂²×C₄₇	C₂×SD₁₆	1	2	yes	—
61	C₂²×C₄₇	#135	1	1	yes	characteristic
62	C₂²×C₄₇	#135	1	1	yes	characteristic
63	C₂×D₄₇	—	2	2	no	—
64	C₂×D₄₇	—	2	2	no	—
65	C₂×D₄₇	—	2	4	no	—
66	C₂×D₄₇	—	2	4	no	—
67	C₂×D₄₇	#106	1	4	yes	complemented
68	C₄×C₄₇	—	8	2	no	—
69	C₄×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
70	C₄×C₄₇	#117	1	1	yes	characteristic
71	C₄×C₄₇	C₂²⋊Q₈	1	2	yes	—
72	Dic₄₇	—	2	2	no	—
73	Dic₄₇	—	8	2	no	—
74	#1807	—	47	1	no	maximal, Sylow
75	C₂×C₄₇	#380	1	1	yes	characteristic
76	C₂×C₄₇	C₂×#106	1	1	yes	characteristic
77	C₂×C₄₇	#427	1	1	yes	characteristic
78	C₂×C₄₇	#465	1	2	yes	—
79	C₂×C₄₇	#465	1	2	yes	—
80	D₄₇	—	2	4	no	—
81	C₂³×C₈	—	47	1	no	—
82	C₂×#106	—	47	2	no	—
83	#380	—	47	2	no	—
84	#340	—	47	1	no	—
85	#340	D₄₇	1	1	yes	characteristic, complemented
86	C₄₇	#1807	1	1	yes	characteristic, complemented, Sylow
87	C₂³×C₄	—	47	1	no	—
88	C₂²×C₈	—	47	1	no	—
89	C₂²×C₈	—	47	4	no	—
90	C₂²×C₈	—	94	4	no	—
91	C₂²×C₈	C₂×D₄₇	1	1	yes	characteristic
92	C₂×C₄⋊C₄	—	47	2	no	—
93	C₂×C₄⋊C₄	C₂×D₄₇	1	2	yes	—
94	C₂×C₂²⋊C₄	—	94	2	no	—
95	#106	—	94	4	no	—
96	#95	—	94	2	no	—
97	C₂⁴	—	47	1	no	—
98	C₂²×C₄	—	2	2	no	—
99	C₂²×C₄	—	47	1	no	—
100	C₂²×C₄	—	47	2	no	—
101	C₂²×C₄	—	47	2	no	—
102	C₂²×C₄	—	94	2	no	—
103	C₂²×C₄	—	94	2	no	—
104	C₂²×C₄	—	94	2	no	—
105	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
106	C₂×C₈	—	2	2	no	—
107	C₂×C₈	—	47	2	no	—
108	C₂×C₈	—	47	4	no	—
109	C₂×C₈	—	94	8	no	—
110	C₂×C₈	C₄×D₄₇	1	2	yes	—
111	C₄⋊C₄	—	2	2	no	—
112	C₄⋊C₄	—	2	2	no	—
113	C₄⋊C₄	—	94	2	no	—
114	C₄⋊C₄	—	94	2	no	—
115	C₂²⋊C₄	—	188	4	no	—
116	C₂³	—	47	2	no	—
117	C₂³	—	47	4	no	—
118	C₂³	—	94	2	no	—
119	C₂³	—	94	1	no	—
120	C₂³	—	94	1	no	—
121	C₂³	D₄×D₄₇	1	1	yes	characteristic
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	—	4	2	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	—	47	2	no	—
126	C₂×C₄	—	94	2	no	—
127	C₂×C₄	—	94	2	no	—
128	C₂×C₄	—	94	2	no	—
129	C₂×C₄	—	94	4	no	—
130	C₂×C₄	—	188	2	no	—
131	C₂×C₄	—	188	2	no	—
132	C₂×C₄	—	188	2	no	—
133	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
134	C₂×C₄	#10625350	1	2	yes	—
135	C₂×C₄	D₄×D₄₇	1	2	yes	—
136	C₂×C₄	Q₈×D₄₇	1	1	yes	characteristic
137	C₈	—	2	2	no	—
138	C₈	—	47	4	no	—
139	C₂²	—	47	4	no	—
140	C₂²	—	94	2	no	—
141	C₂²	—	94	4	no	—
142	C₂²	—	94	4	no	—
143	C₂²	—	94	2	no	—
144	C₂²	C₄⋊C₄×D₄₇	1	1	yes	characteristic
145	C₂²	#13093997	1	2	yes	—
146	C₂²	SD₁₆×D₄₇	1	2	yes	—
147	C₂²	#13094043	1	1	yes	characteristic
148	C₂²	#13094047	1	1	yes	characteristic
149	C₄	—	8	2	no	—
150	C₄	—	94	2	no	—
151	C₄	—	376	2	no	—
152	C₄	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
153	C₄	#13093981	1	1	yes	characteristic
154	C₄	#13094002	1	2	yes	—
155	C₂	—	94	4	no	—
156	C₂	#15688982	1	1	yes	characteristic
157	C₂	#106×D₄₇	1	1	yes	characteristic
158	C₂	#15689160	1	1	yes	characteristic
159	C₂	#15689351	1	2	yes	—
160	C₂	#15689355	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₈×D₄₇

C₂

yes

maximal, characteristic

C₂×#13093955

C₂

yes

maximal, complemented

#15688980

C₂

yes

maximal

#15688886

C₂

yes

maximal, characteristic, complemented

C₄₇×#340

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₄₇

C₂²

yes

characteristic

C₂²×C₄₇⋊C₈

C₂²

yes

characteristic

C₂×C₈×D₄₇

—

C₂×C₈×D₄₇

C₄

yes

complemented

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×(C₂×D₄₇)⋊C₄

—

#13093955

—

C₄₇⋊#95

—

C₂²×C₈×C₄₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂³×D₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₂²×Dic₄₇

D₄

yes

characteristic

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

C₂×C₄

yes

—

C₂×C₄×D₄₇

Q₈

yes

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂×C₄

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

C₂×C₄

yes

—

C₂×C₄₇⋊C₈

D₄

yes

—

C₈×D₄₇

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

C₄⋊C₄

yes

—

C₂²×D₄₇

SD₁₆

yes

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

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

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₄⋊C₄

yes

—

C₄×D₄₇

—

C₄×D₄₇

—

C₄×D₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

C₂²⋊C₄

yes

complemented

C₂²×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₄₇

#121

yes

—

C₂²×C₄₇

C₂×SD₁₆

yes

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

#106

yes

complemented

C₄×C₄₇

—

C₄×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₄₇

#117

yes

characteristic

C₄×C₄₇

C₂²⋊Q₈

yes

—

Dic₄₇

—

Dic₄₇

—

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

C₂×C₄₇

#465

yes

—

D₄₇

—

C₂³×C₈

—

—

—

—

D₄₇

yes

characteristic, complemented

C₄₇

#1807

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×D₄₇

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

C₂×C₂²⋊C₄

—

#106

—

#95

—

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

C₂²×D₄₇

yes

characteristic

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

C₄×D₄₇

yes

—

111

C₄⋊C₄

—

112

C₄⋊C₄

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₂²⋊C₄

—

188

—

116

C₂³

—

117

C₂³

—

118

C₂³

—

119

C₂³

—

120

C₂³

—

121

C₂³

D₄×D₄₇

yes

characteristic

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

188

—

131

C₂×C₄

—

188

—

132

C₂×C₄

—

188

—

133

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

134

C₂×C₄

#10625350

yes

—

135

C₂×C₄

D₄×D₄₇

yes

—

136

C₂×C₄

Q₈×D₄₇

yes

characteristic

137

C₈

—

138

C₈

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

—

144

C₂²

C₄⋊C₄×D₄₇

yes

characteristic

145

C₂²

#13093997

yes

—

146

C₂²

SD₁₆×D₄₇

yes

—

147

C₂²

#13094043

yes

characteristic

148

C₂²

#13094047

yes

characteristic

149

C₄

—

150

C₄

—

151

C₄

—

376

—

152

C₄

C₂²⋊C₄×D₄₇

yes

characteristic

153

C₄

#13093981

yes

characteristic

154

C₄

#13094002

yes

—

155

C₂

—

156

C₂

#15688982

yes

characteristic

157

C₂

#106×D₄₇

yes

characteristic

158

C₂

#15689160

yes

characteristic

159

C₂

#15689351

yes

—

160

C₂

#15689355

yes

—

	subgroup	G/H
center	C₂³	D₄×D₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₄₇×#340	C₂
socle	C₂³×C₄₇	C₂×D₄
hypercenter	#340	D₄₇
hypocenter	C₄₇	#1807
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

subgroup

G/H

center

C₂³

D₄×D₄₇

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₄₇×#340

C₂

socle

C₂³×C₄₇

C₂×D₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ #340

N	H ≅ G/N	type
2. C₂×#13093955	C₂	semidirect
4. #15688886	C₂	semidirect
5. C₄₇×#340	C₂	semidirect
9. C₂×C₈×D₄₇	C₄	semidirect
57. C₄₇⋊C₈	C₂²⋊C₄	semidirect
67. C₂×D₄₇	#106	semidirect
85. #340	D₄₇	semidirect
86. C₄₇	#1807	semidirect

H ≅ G/N

type

2. C₂×#13093955

C₂

semidirect

4. #15688886

C₂

semidirect

5. C₄₇×#340

C₂

semidirect

9. C₂×C₈×D₄₇

C₄

semidirect

57. C₄₇⋊C₈

C₂²⋊C₄

semidirect

67. C₂×D₄₇

#106

semidirect

85. #340

D₄₇

semidirect

86. C₄₇

#1807

semidirect

Subgroups — #20243412

Conjugacy classes of subgroups

Complements