Group #20243470 — subgroups

Subgroups — #20243470

Order 6016.

272 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 77 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×#13093982	C₂	1	1	yes	maximal, characteristic
2	#15689086	C₂	1	2	yes	maximal
3	#15688892	C₂	1	1	yes	maximal, characteristic, complemented
4	#15688891	C₂	1	2	yes	maximal, complemented
5	C₄₇×#343	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₄₇	C₂²	1	1	yes	characteristic
7	C₂×#10625329	C₂²	1	2	yes	—
8	#13093982	—	2	4	no	—
9	C₂×C₄⋊Dic₄₇	C₂²	1	1	yes	characteristic
10	C₂×(C₂×D₄₇)⋊C₄	C₄	1	2	yes	—
11	C₂×C₄×Dic₄₇	C₄	1	2	yes	—
12	(C₂×D₄₇)⋊C₈	—	2	4	no	—
13	C₂×M₁₆×C₄₇	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄×C₄₇	C₂²	1	1	yes	characteristic
15	C₂³×D₄₇	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₄₇	C₂×C₄	1	2	yes	—
17	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
18	C₂²×Dic₄₇	C₂×C₄	1	1	yes	characteristic
19	C₂²×Dic₄₇	C₂×C₄	1	2	yes	—
20	C₂×C₄×D₄₇	—	2	2	no	—
21	C₂×C₄×D₄₇	—	2	1	no	—
22	C₂×C₄×D₄₇	—	2	1	no	—
23	C₂×C₄×D₄₇	D₄	1	4	yes	—
24	M₁₆×C₄₇	—	2	2	no	—
25	M₁₆×C₄₇	—	2	2	no	—
26	C₂×C₈×C₄₇	—	2	2	no	—
27	C₄⋊C₄×C₄₇	—	2	2	no	—
28	(C₂×D₄₇)⋊C₄	—	2	4	no	—
29	C₄⋊Dic₄₇	—	2	2	no	—
30	C₄×Dic₄₇	—	2	4	no	—
31	#10625329	—	2	2	no	—
32	#10625329	—	2	2	no	—
33	C₂×C₄₇⋊C₈	—	2	2	no	—
34	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
35	C₂²×D₄₇	—	2	1	no	—
36	C₂²×D₄₇	—	2	1	no	—
37	C₂²×D₄₇	—	4	1	no	—
38	C₂²×D₄₇	—	4	1	no	—
39	C₂²×D₄₇	C₂²⋊C₄	1	2	yes	—
40	C₂×C₄×C₄₇	—	2	2	no	—
41	C₂×C₄×C₄₇	—	4	2	no	—
42	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
43	C₂×C₄×C₄₇	#41	1	2	yes	—
44	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
45	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
46	C₂×Dic₄₇	—	2	1	no	—
47	C₂×Dic₄₇	—	2	1	no	—
48	C₂×Dic₄₇	—	2	2	no	—
49	C₂×Dic₄₇	—	4	2	no	—
50	C₂×Dic₄₇	C₂²⋊C₄	1	2	yes	—
51	C₄×D₄₇	—	2	4	no	—
52	C₄×D₄₇	—	2	4	no	—
53	C₈×C₄₇	—	4	2	no	—
54	C₄₇⋊C₈	—	4	2	no	—
55	C₂²×C₄₇	—	2	1	no	—
56	C₂²×C₄₇	—	2	1	no	—
57	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄₇	#117	1	1	yes	characteristic
59	C₂²×C₄₇	#117	1	1	yes	characteristic
60	C₂×D₄₇	—	2	2	no	—
61	C₂×D₄₇	—	2	2	no	—
62	C₂×D₄₇	—	2	2	no	—
63	C₂×D₄₇	—	4	2	no	—
64	C₂×D₄₇	—	4	2	no	—
65	C₄×C₄₇	—	4	2	no	—
66	C₄×C₄₇	#123	1	2	yes	—
67	C₄×C₄₇	#123	1	2	yes	—
68	Dic₄₇	—	2	2	no	—
69	Dic₄₇	—	4	2	no	—
70	#1822	—	47	1	no	maximal, Sylow
71	C₂×C₄₇	—	2	1	no	—
72	C₂×C₄₇	—	2	1	no	—
73	C₂×C₄₇	#381	1	1	yes	characteristic
74	C₂×C₄₇	#413	1	1	yes	characteristic
75	C₂×C₄₇	#413	1	1	yes	characteristic
76	D₄₇	—	4	2	no	—
77	C₂×#117	—	47	1	no	—
78	#407	—	47	2	no	—
79	#343	—	47	2	no	—
80	#343	—	47	1	no	—
81	#343	D₄₇	1	1	yes	characteristic, complemented
82	C₄₇	#1822	1	1	yes	characteristic, complemented, Sylow
83	C₂³×C₄	—	47	1	no	—
84	C₂×M₁₆	—	47	2	no	—
85	C₂×M₁₆	C₂×D₄₇	1	2	yes	—
86	#117	—	94	4	no	—
87	C₂×C₄⋊C₄	—	47	1	no	—
88	C₂×C₄⋊C₄	C₂×D₄₇	1	1	yes	characteristic
89	C₂×C₂²⋊C₄	—	47	2	no	—
90	C₂×C₄²	—	47	2	no	—
91	C₂²⋊C₈	—	94	4	no	—
92	C₂⁴	—	47	1	no	—
93	C₂²×C₄	—	47	1	no	—
94	C₂²×C₄	—	47	4	no	—
95	C₂²×C₄	—	47	2	no	—
96	C₂²×C₄	—	94	2	no	—
97	C₂²×C₄	—	94	1	no	—
98	C₂²×C₄	—	94	1	no	—
99	C₂²×C₄	C₄×D₄₇	1	2	yes	—
100	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
101	M₁₆	—	2	2	no	—
102	M₁₆	—	2	2	no	—
103	M₁₆	—	94	2	no	—
104	M₁₆	—	94	2	no	—
105	C₂×C₈	—	2	2	no	—
106	C₂×C₈	—	94	2	no	—
107	C₄⋊C₄	—	2	2	no	—
108	C₄⋊C₄	—	94	2	no	—
109	C₂²⋊C₄	—	94	4	no	—
110	C₄²	—	94	4	no	—
111	C₂³	—	47	2	no	—
112	C₂³	—	94	1	no	—
113	C₂³	—	94	1	no	—
114	C₂³	—	188	1	no	—
115	C₂³	—	188	1	no	—
116	C₂³	C₂×C₄×D₄₇	1	1	yes	characteristic
117	C₂×C₄	—	2	2	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	—	47	2	no	—
120	C₂×C₄	—	94	1	no	—
121	C₂×C₄	—	94	1	no	—
122	C₂×C₄	—	94	4	no	—
123	C₂×C₄	—	94	4	no	—
124	C₂×C₄	—	94	2	no	—
125	C₂×C₄	—	188	2	no	—
126	C₂×C₄	#10625350	1	2	yes	—
127	C₂×C₄	D₄×D₄₇	1	2	yes	—
128	C₂×C₄	#10625352	1	1	yes	characteristic
129	C₂×C₄	#10625354	1	1	yes	characteristic
130	C₈	—	4	2	no	—
131	C₈	—	188	2	no	—
132	C₂²	—	2	1	no	—
133	C₂²	—	2	1	no	—
134	C₂²	—	94	2	no	—
135	C₂²	—	94	2	no	—
136	C₂²	—	94	2	no	—
137	C₂²	—	188	2	no	—
138	C₂²	—	188	2	no	—
139	C₂²	C₂²⋊C₄×D₄₇	1	1	yes	characteristic
140	C₂²	#13093981	1	1	yes	characteristic
141	C₂²	#13093982	1	1	yes	characteristic
142	C₄	—	4	2	no	—
143	C₄	—	94	2	no	—
144	C₄	—	188	2	no	—
145	C₄	#13094006	1	2	yes	—
146	C₄	#13094009	1	2	yes	—
147	C₂	—	2	1	no	—
148	C₂	—	2	1	no	—
149	C₂	—	188	2	no	—
150	C₂	#15688985	1	1	yes	characteristic
151	C₂	#15689104	1	1	yes	characteristic
152	C₂	#15689106	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13093982

C₂

yes

maximal, characteristic

#15689086

C₂

yes

maximal

#15688892

C₂

yes

maximal, characteristic, complemented

#15688891

C₂

yes

maximal, complemented

C₄₇×#343

C₂

yes

maximal, characteristic, complemented

C₂²×C₄×D₄₇

C₂²

yes

characteristic

C₂×#10625329

C₂²

yes

—

#13093982

—

C₂×C₄⋊Dic₄₇

C₂²

yes

characteristic

C₂×(C₂×D₄₇)⋊C₄

C₄

yes

—

C₂×C₄×Dic₄₇

C₄

yes

—

(C₂×D₄₇)⋊C₈

—

C₂×M₁₆×C₄₇

C₂²

yes

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

characteristic

C₂³×D₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

C₂×C₄

yes

characteristic

C₂²×Dic₄₇

C₂×C₄

yes

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

—

C₂×C₄×D₄₇

D₄

yes

—

M₁₆×C₄₇

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₄⋊C₄×C₄₇

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

—

C₄×Dic₄₇

—

#10625329

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂²×C₄

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

C₂²⋊C₄

yes

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

#41

yes

—

C₂×C₄×C₄₇

yes

C₂×C₄×C₄₇

yes

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

C₂²⋊C₄

yes

—

C₄×D₄₇

—

C₄×D₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#123

yes

—

C₄×C₄₇

#123

yes

—

Dic₄₇

—

Dic₄₇

—

#1822

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

—

—

—

—

—

D₄₇

yes

characteristic, complemented

C₄₇

#1822

yes

characteristic, complemented, Sylow

C₂³×C₄

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₄₇

yes

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₄₇

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂²⋊C₈

—

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₄₇

yes

—

100

C₂²×C₄

C₂²×D₄₇

yes

characteristic

101

M₁₆

—

102

M₁₆

—

103

M₁₆

—

104

M₁₆

—

105

C₂×C₈

—

106

C₂×C₈

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

109

C₂²⋊C₄

—

110

C₄²

—

111

C₂³

—

112

C₂³

—

113

C₂³

—

114

C₂³

—

188

—

115

C₂³

—

188

—

116

C₂³

C₂×C₄×D₄₇

yes

characteristic

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

188

—

126

C₂×C₄

#10625350

yes

—

127

C₂×C₄

D₄×D₄₇

yes

—

128

C₂×C₄

#10625352

yes

characteristic

129

C₂×C₄

#10625354

yes

characteristic

130

C₈

—

131

C₈

—

188

—

132

C₂²

—

133

C₂²

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

—

188

—

138

C₂²

—

188

—

139

C₂²

C₂²⋊C₄×D₄₇

yes

characteristic

140

C₂²

#13093981

yes

characteristic

141

C₂²

#13093982

yes

characteristic

142

C₄

—

143

C₄

—

144

C₄

—

188

—

145

C₄

#13094006

yes

—

146

C₄

#13094009

yes

—

147

C₂

—

148

C₂

—

149

C₂

—

188

—

150

C₂

#15688985

yes

characteristic

151

C₂

#15689104

yes

characteristic

152

C₂

#15689106

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²⋊C₄×D₄₇
commutator	C₂³×C₄₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₄₇×#343	C₂
socle	C₂²×C₄₇	C₂×C₂²⋊C₄
hypercenter	#343	D₄₇
hypocenter	C₄₇	#1822
LCS	G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

subgroup

G/H

center

C₂²

C₂²⋊C₄×D₄₇

commutator

C₂³×C₄₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₄₇×#343

C₂

socle

C₂²×C₄₇

C₂×C₂²⋊C₄

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #343

N	H ≅ G/N	type
3. #15688892	C₂	semidirect
4. #15688891	C₂	semidirect
5. C₄₇×#343	C₂	semidirect
81. #343	D₄₇	semidirect
82. C₄₇	#1822	semidirect

H ≅ G/N

type

3. #15688892

C₂

semidirect

4. #15688891

C₂

semidirect

5. C₄₇×#343

C₂

semidirect

81. #343

D₄₇

semidirect

82. C₄₇

#1822

semidirect

Subgroups — #20243470

Conjugacy classes of subgroups

Complements