Group #20239568 — subgroups

Subgroups — C₄₇×#1958 #20239568

Order 6016.

336 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 128 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×#124×C₄₇	C₂	1	1	yes	maximal, characteristic
2	C₂×#103×C₄₇	C₂	1	2	yes	maximal, complemented
3	C₂×#102×C₄₇	C₂	1	2	yes	maximal
4	C₂×C₄×C₈×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#382	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×Q₈×C₄₇	C₂²	1	1	yes	characteristic
7	C₂²×D₄×C₄₇	C₂²	1	1	yes	characteristic
8	C₂²×C₈×C₄₇	C₂²	1	2	yes	—
9	#124×C₄₇	—	2	2	no	—
10	#124×C₄₇	C₄	1	4	yes	complemented
11	C₂×C₄⋊C₄×C₄₇	C₂²	1	2	yes	—
12	C₂×C₂²⋊C₄×C₄₇	—	2	2	no	—
13	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
14	#103×C₄₇	—	2	4	no	—
15	#102×C₄₇	—	2	4	no	—
16	C₄×C₈×C₄₇	—	2	2	no	—
17	#95×C₄₇	—	2	2	no	—
18	C₂⁴×C₄₇	—	2	1	no	—
19	C₂×Q₈×C₄₇	—	2	1	no	—
20	C₂×Q₈×C₄₇	—	2	2	no	—
21	C₂×Q₈×C₄₇	—	2	2	no	—
22	C₂×Q₈×C₄₇	C₂×C₄	1	2	yes	—
23	C₂×D₄×C₄₇	—	2	2	no	—
24	C₂×D₄×C₄₇	—	2	1	no	—
25	C₂×D₄×C₄₇	—	2	2	no	—
26	C₂×D₄×C₄₇	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₄₇	—	2	1	no	—
28	C₂²×C₄×C₄₇	—	2	2	no	—
29	C₂²×C₄×C₄₇	D₄	1	2	yes	—
30	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
31	C₂×C₈×C₄₇	—	2	2	no	—
32	C₂×C₈×C₄₇	—	2	2	no	—
33	C₂×C₈×C₄₇	D₄	1	4	yes	—
34	C₄⋊C₄×C₄₇	—	2	2	no	—
35	C₄⋊C₄×C₄₇	—	2	2	no	—
36	C₂²⋊C₄×C₄₇	—	2	8	no	—
37	C₄²×C₄₇	C₂×C₄	1	2	yes	—
38	C₄²×C₄₇	D₄	1	2	yes	—
39	C₂³×C₄₇	—	2	2	no	—
40	C₂³×C₄₇	—	4	1	no	—
41	C₂³×C₄₇	—	4	1	no	—
42	C₂³×C₄₇	—	4	4	no	—
43	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
44	Q₈×C₄₇	—	2	4	no	—
45	Q₈×C₄₇	—	4	2	no	—
46	D₄×C₄₇	—	2	4	no	—
47	D₄×C₄₇	—	4	2	no	—
48	C₂×C₄×C₄₇	—	2	4	no	—
49	C₂×C₄×C₄₇	—	2	2	no	—
50	C₂×C₄×C₄₇	—	4	1	no	—
51	C₂×C₄×C₄₇	—	4	1	no	—
52	C₂×C₄×C₄₇	—	4	2	no	—
53	C₂×C₄×C₄₇	—	4	4	no	—
54	C₂×C₄×C₄₇	C₂²⋊C₄	1	4	yes	—
55	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
57	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
58	C₂×C₄×C₄₇	#41	1	2	yes	—
59	C₈×C₄₇	—	2	4	no	—
60	C₂²×C₄₇	—	4	2	no	—
61	C₂²×C₄₇	—	4	4	no	—
62	C₂²×C₄₇	—	8	2	no	—
63	C₂²×C₄₇	—	8	2	no	—
64	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
65	C₂²×C₄₇	#124	1	1	yes	characteristic
66	C₂²×C₄₇	#127	1	1	yes	characteristic
67	C₂²×C₄₇	#135	1	4	yes	—
68	C₄×C₄₇	—	2	4	no	—
69	C₄×C₄₇	—	4	2	no	—
70	C₄×C₄₇	—	8	2	no	—
71	C₄×C₄₇	C₄×D₄	1	2	yes	—
72	C₄×C₄₇	#121	1	2	yes	—
73	#1958	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂×C₄₇	—	8	2	no	—
75	C₂×C₄₇	#390	1	1	yes	characteristic
76	C₂×C₄₇	#416	1	2	yes	—
77	C₂×C₄₇	#488	1	2	yes	—
78	C₂×C₄₇	#495	1	2	yes	—
79	C₂×#124	C₂×C₄₇	1	1	yes	characteristic
80	C₂×#103	C₂×C₄₇	1	2	yes	complemented
81	C₂×#102	C₂×C₄₇	1	2	yes	—
82	C₂×C₄×C₈	C₂×C₄₇	1	1	yes	characteristic, complemented
83	#382	C₂×C₄₇	1	1	yes	characteristic, complemented
84	C₄₇	#1958	1	1	yes	characteristic, direct product factor, Sylow
85	C₂²×Q₈	C₂²×C₄₇	1	1	yes	characteristic
86	C₂²×D₄	C₂²×C₄₇	1	1	yes	characteristic
87	C₂²×C₈	C₂²×C₄₇	1	2	yes	—
88	#124	—	2	2	no	—
89	#124	C₄×C₄₇	1	4	yes	complemented
90	C₂×C₄⋊C₄	C₂²×C₄₇	1	2	yes	—
91	C₂×C₂²⋊C₄	—	2	2	no	—
92	C₂×C₄²	C₂²×C₄₇	1	1	yes	characteristic
93	#103	—	2	4	no	—
94	#102	—	2	4	no	—
95	C₄×C₈	—	2	2	no	—
96	#95	—	2	2	no	—
97	C₂⁴	—	2	1	no	—
98	C₂×Q₈	—	2	1	no	—
99	C₂×Q₈	—	2	2	no	—
100	C₂×Q₈	—	2	2	no	—
101	C₂×Q₈	C₂×C₄×C₄₇	1	2	yes	—
102	C₂×D₄	—	2	2	no	—
103	C₂×D₄	—	2	1	no	—
104	C₂×D₄	—	2	2	no	—
105	C₂×D₄	C₂×C₄×C₄₇	1	2	yes	—
106	C₂²×C₄	—	2	1	no	—
107	C₂²×C₄	—	2	2	no	—
108	C₂²×C₄	D₄×C₄₇	1	2	yes	—
109	C₂²×C₄	C₂³×C₄₇	1	1	yes	characteristic
110	C₂×C₈	—	2	2	no	—
111	C₂×C₈	—	2	2	no	—
112	C₂×C₈	D₄×C₄₇	1	4	yes	—
113	C₄⋊C₄	—	2	2	no	—
114	C₄⋊C₄	—	2	2	no	—
115	C₂²⋊C₄	—	2	8	no	—
116	C₄²	C₂×C₄×C₄₇	1	2	yes	—
117	C₄²	D₄×C₄₇	1	2	yes	—
118	C₂³	—	2	2	no	—
119	C₂³	—	4	1	no	—
120	C₂³	—	4	1	no	—
121	C₂³	—	4	4	no	—
122	C₂³	C₂×D₄×C₄₇	1	1	yes	characteristic
123	Q₈	—	2	4	no	—
124	Q₈	—	4	2	no	—
125	D₄	—	2	4	no	—
126	D₄	—	4	2	no	—
127	C₂×C₄	—	2	4	no	—
128	C₂×C₄	—	2	2	no	—
129	C₂×C₄	—	4	1	no	—
130	C₂×C₄	—	4	1	no	—
131	C₂×C₄	—	4	2	no	—
132	C₂×C₄	—	4	4	no	—
133	C₂×C₄	C₂²⋊C₄×C₄₇	1	4	yes	—
134	C₂×C₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
135	C₂×C₄	C₂×D₄×C₄₇	1	2	yes	—
136	C₂×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
137	C₂×C₄	#41×C₄₇	1	2	yes	—
138	C₈	—	2	4	no	—
139	C₂²	—	4	2	no	—
140	C₂²	—	4	4	no	—
141	C₂²	—	8	2	no	—
142	C₂²	—	8	2	no	—
143	C₂²	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
144	C₂²	#124×C₄₇	1	1	yes	characteristic
145	C₂²	#127×C₄₇	1	1	yes	characteristic
146	C₂²	#135×C₄₇	1	4	yes	—
147	C₄	—	2	4	no	—
148	C₄	—	4	2	no	—
149	C₄	—	8	2	no	—
150	C₄	C₄×D₄×C₄₇	1	2	yes	—
151	C₄	#121×C₄₇	1	2	yes	—
152	C₂	—	8	2	no	—
153	C₂	C₄₇×#390	1	1	yes	characteristic
154	C₂	C₄₇×#416	1	2	yes	—
155	C₂	C₄₇×#488	1	2	yes	—
156	C₂	C₄₇×#495	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#124×C₄₇

C₂

yes

maximal, characteristic

C₂×#103×C₄₇

C₂

yes

maximal, complemented

C₂×#102×C₄₇

C₂

yes

maximal

C₂×C₄×C₈×C₄₇

C₂

yes

maximal, characteristic, complemented

C₄₇×#382

C₂

yes

maximal, characteristic, complemented

C₂²×Q₈×C₄₇

C₂²

yes

characteristic

C₂²×D₄×C₄₇

C₂²

yes

characteristic

C₂²×C₈×C₄₇

C₂²

yes

—

#124×C₄₇

—

#124×C₄₇

C₄

yes

complemented

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₂²⋊C₄×C₄₇

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

#103×C₄₇

—

#102×C₄₇

—

C₄×C₈×C₄₇

—

#95×C₄₇

—

C₂⁴×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

C₂×C₄

yes

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

D₄

yes

—

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

D₄

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

C₂×C₄

yes

—

C₄²×C₄₇

D₄

yes

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

Q₈×C₄₇

—

Q₈×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

#41

yes

—

C₈×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₄×D₄

yes

—

C₄×C₄₇

#121

yes

—

#1958

C₄₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₄₇

—

C₂×C₄₇

#390

yes

characteristic

C₂×C₄₇

#416

yes

—

C₂×C₄₇

#488

yes

—

C₂×C₄₇

#495

yes

—

C₂×#124

C₂×C₄₇

yes

characteristic

C₂×#103

C₂×C₄₇

yes

complemented

C₂×#102

C₂×C₄₇

yes

—

C₂×C₄×C₈

C₂×C₄₇

yes

characteristic, complemented

#382

C₂×C₄₇

yes

characteristic, complemented

C₄₇

#1958

yes

characteristic, direct product factor, Sylow

C₂²×Q₈

C₂²×C₄₇

yes

characteristic

C₂²×D₄

C₂²×C₄₇

yes

characteristic

C₂²×C₈

C₂²×C₄₇

yes

—

#124

—

#124

C₄×C₄₇

yes

complemented

C₂×C₄⋊C₄

C₂²×C₄₇

yes

—

C₂×C₂²⋊C₄

—

C₂×C₄²

C₂²×C₄₇

yes

characteristic

#103

—

#102

—

C₄×C₈

—

#95

—

C₂⁴

—

C₂×Q₈

—

C₂×Q₈

—

100

C₂×Q₈

—

101

C₂×Q₈

C₂×C₄×C₄₇

yes

—

102

C₂×D₄

—

103

C₂×D₄

—

104

C₂×D₄

—

105

C₂×D₄

C₂×C₄×C₄₇

yes

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

D₄×C₄₇

yes

—

109

C₂²×C₄

C₂³×C₄₇

yes

characteristic

110

C₂×C₈

—

111

C₂×C₈

—

112

C₂×C₈

D₄×C₄₇

yes

—

113

C₄⋊C₄

—

114

C₄⋊C₄

—

115

C₂²⋊C₄

—

116

C₄²

C₂×C₄×C₄₇

yes

—

117

C₄²

D₄×C₄₇

yes

—

118

C₂³

—

119

C₂³

—

120

C₂³

—

121

C₂³

—

122

C₂³

C₂×D₄×C₄₇

yes

characteristic

123

Q₈

—

124

Q₈

—

125

D₄

—

126

D₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

—

133

C₂×C₄

C₂²⋊C₄×C₄₇

yes

—

134

C₂×C₄

C₂²×C₄×C₄₇

yes

characteristic

135

C₂×C₄

C₂×D₄×C₄₇

yes

—

136

C₂×C₄

C₂×D₄×C₄₇

yes

characteristic

137

C₂×C₄

#41×C₄₇

yes

—

138

C₈

—

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

—

143

C₂²

C₂×C₂²⋊C₄×C₄₇

yes

characteristic

144

C₂²

#124×C₄₇

yes

characteristic

145

C₂²

#127×C₄₇

yes

characteristic

146

C₂²

#135×C₄₇

yes

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

C₄×D₄×C₄₇

yes

—

151

C₄

#121×C₄₇

yes

—

152

C₂

—

153

C₂

C₄₇×#390

yes

characteristic

154

C₂

C₄₇×#416

yes

—

155

C₂

C₄₇×#488

yes

—

156

C₂

C₄₇×#495

yes

—

	subgroup	G/H
center, socle	C₂³×C₄₇	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₄₇
Frattini, cosocle	C₂²×C₄	C₂³×C₄₇
prime radical	C₂²×D₄×C₄₇	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄₇ ◁ C₂×C₄²×C₄₇ ◁ G

subgroup

G/H

center, socle

C₂³×C₄₇

C₂×D₄

commutator

C₂×C₄

C₂²×C₄×C₄₇

Frattini, cosocle

C₂²×C₄

C₂³×C₄₇

prime radical

C₂²×D₄×C₄₇

C₂²

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₄₇ ◁ C₂×C₄²×C₄₇ ◁ G

N	H ≅ G/N	type
2. C₂×#103×C₄₇	C₂	semidirect
4. C₂×C₄×C₈×C₄₇	C₂	semidirect
5. C₄₇×#382	C₂	semidirect
10. #124×C₄₇	C₄	semidirect
73. #1958	C₄₇	direct (N × H)
80. C₂×#103	C₂×C₄₇	semidirect
82. C₂×C₄×C₈	C₂×C₄₇	semidirect
83. #382	C₂×C₄₇	semidirect
84. C₄₇	#1958	direct (N × H)
89. #124	C₄×C₄₇	semidirect

H ≅ G/N

type

2. C₂×#103×C₄₇

C₂

semidirect

4. C₂×C₄×C₈×C₄₇

C₂

semidirect

5. C₄₇×#382

C₂

semidirect

10. #124×C₄₇

C₄

semidirect

73. #1958

C₄₇

direct (N × H)

80. C₂×#103

C₂×C₄₇

semidirect

82. C₂×C₄×C₈

C₂×C₄₇

semidirect

83. #382

C₂×C₄₇

semidirect

84. C₄₇

#1958

direct (N × H)

89. #124

C₄×C₄₇

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×#124×C₄₇	2
C₂×#103×C₄₇	2
C₂×#102×C₄₇	2
C₂×C₄×C₈×C₄₇	2
C₄₇×#382	2
#1958	47

subgroup

index

C₂×#124×C₄₇

C₂×#103×C₄₇

C₂×#102×C₄₇

C₂×C₄×C₈×C₄₇

C₄₇×#382

#1958

Subgroups — C₄₇×#1958 #20239568

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C47×#1958 #20239568

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#1958 #20239568