Group #20239277 — subgroups

Subgroups — C₄₇×#1667 #20239277

Order 6016.

204 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 72 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	C₄₇×#493	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄₇×#459	C₂	1	2	yes	maximal, complemented
3	C₈×D₄×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₇×#335	C₂	1	1	yes	maximal, characteristic, complemented
5	C₄₇×#331	C₂	1	2	yes	maximal, complemented
6	C₂×D₈×C₄₇	—	2	2	no	—
7	C₂×D₈×C₄₇	—	2	1	no	—
8	C₂×D₈×C₄₇	—	2	1	no	—
9	C₂²×C₈×C₄₇	—	2	1	no	—
10	#127×C₄₇	C₂²	1	2	yes	—
11	C₄×D₄×C₄₇	C₂²	1	1	yes	characteristic
12	C₄⋊C₈×C₄₇	C₂²	1	2	yes	—
13	C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
14	#102×C₄₇	—	2	2	no	—
15	C₂²⋊C₈×C₄₇	—	2	1	no	—
16	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×D₄×C₄₇	—	2	2	no	—
18	C₂×D₄×C₄₇	—	2	2	no	—
19	C₂×D₄×C₄₇	—	2	2	no	—
20	C₂×D₄×C₄₇	D₄	1	1	yes	characteristic
21	C₂²×C₄×C₄₇	—	2	1	no	—
22	D₈×C₄₇	—	4	2	no	—
23	D₈×C₄₇	—	4	4	no	—
24	D₈×C₄₇	—	4	2	no	—
25	C₂×C₈×C₄₇	—	2	1	no	—
26	C₂×C₈×C₄₇	—	2	2	no	—
27	C₂×C₈×C₄₇	—	4	2	no	—
28	C₂×C₈×C₄₇	D₄	1	1	yes	characteristic
29	C₂×C₈×C₄₇	D₄	1	1	yes	characteristic
30	C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
31	C₂²⋊C₄×C₄₇	—	2	1	no	—
32	C₄²×C₄₇	C₂³	1	1	yes	characteristic
33	C₂³×C₄₇	—	2	1	no	—
34	C₂³×C₄₇	—	4	2	no	—
35	D₄×C₄₇	—	2	1	no	—
36	D₄×C₄₇	—	4	2	no	—
37	D₄×C₄₇	—	4	2	no	—
38	D₄×C₄₇	—	4	2	no	—
39	D₄×C₄₇	—	4	2	no	—
40	D₄×C₄₇	—	8	2	no	—
41	D₄×C₄₇	D₈	1	2	yes	complemented
42	C₂×C₄×C₄₇	—	2	1	no	—
43	C₂×C₄×C₄₇	—	4	1	no	—
44	C₂×C₄×C₄₇	—	4	1	no	—
45	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
48	C₈×C₄₇	—	2	1	no	—
49	C₈×C₄₇	—	4	1	no	—
50	C₈×C₄₇	—	4	2	no	—
51	C₈×C₄₇	D₈	1	2	yes	complemented
52	C₂²×C₄₇	—	2	2	no	—
53	C₂²×C₄₇	—	4	1	no	—
54	C₂²×C₄₇	—	4	1	no	—
55	C₂²×C₄₇	—	8	2	no	—
56	C₂²×C₄₇	—	8	2	no	—
57	C₂²×C₄₇	—	8	2	no	—
58	C₂²×C₄₇	#121	1	1	yes	characteristic
59	C₄×C₄₇	—	2	1	no	—
60	C₄×C₄₇	—	4	1	no	—
61	C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
62	C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
63	C₄×C₄₇	#135	1	1	yes	characteristic
64	C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
65	#1667	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂×C₄₇	—	4	2	no	—
67	C₂×C₄₇	—	16	2	no	—
68	C₂×C₄₇	#459	1	1	yes	characteristic
69	C₂×C₄₇	#466	1	1	yes	characteristic
70	C₂×C₄₇	#472	1	1	yes	characteristic
71	#493	C₂×C₄₇	1	1	yes	characteristic, complemented
72	#459	C₂×C₄₇	1	2	yes	complemented
73	C₈×D₄	C₂×C₄₇	1	1	yes	characteristic, complemented
74	#335	C₂×C₄₇	1	1	yes	characteristic, complemented
75	#331	C₂×C₄₇	1	2	yes	complemented
76	C₄₇	#1667	1	1	yes	characteristic, direct product factor, Sylow
77	C₂×D₈	—	2	2	no	—
78	C₂×D₈	—	2	1	no	—
79	C₂×D₈	—	2	1	no	—
80	C₂²×C₈	—	2	1	no	—
81	#127	C₂²×C₄₇	1	2	yes	—
82	C₄×D₄	C₂²×C₄₇	1	1	yes	characteristic
83	C₄⋊C₈	C₂²×C₄₇	1	1	yes	characteristic
84	C₄⋊C₈	C₂²×C₄₇	1	2	yes	—
85	#102	—	2	2	no	—
86	C₂²⋊C₈	—	2	1	no	—
87	C₄×C₈	C₂²×C₄₇	1	1	yes	characteristic
88	C₂×D₄	—	2	2	no	—
89	C₂×D₄	—	2	2	no	—
90	C₂×D₄	—	2	2	no	—
91	C₂×D₄	D₄×C₄₇	1	1	yes	characteristic
92	C₂²×C₄	—	2	1	no	—
93	D₈	—	4	2	no	—
94	D₈	—	4	4	no	—
95	D₈	—	4	2	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	2	2	no	—
98	C₂×C₈	—	4	2	no	—
99	C₂×C₈	D₄×C₄₇	1	1	yes	characteristic
100	C₂×C₈	D₄×C₄₇	1	1	yes	characteristic
101	C₄⋊C₄	D₄×C₄₇	1	1	yes	characteristic
102	C₂²⋊C₄	—	2	1	no	—
103	C₄²	C₂³×C₄₇	1	1	yes	characteristic
104	C₂³	—	2	1	no	—
105	C₂³	—	4	2	no	—
106	D₄	—	2	1	no	—
107	D₄	—	4	2	no	—
108	D₄	—	4	2	no	—
109	D₄	—	4	2	no	—
110	D₄	—	4	2	no	—
111	D₄	—	8	2	no	—
112	D₄	D₈×C₄₇	1	2	yes	complemented
113	C₂×C₄	—	2	1	no	—
114	C₂×C₄	—	4	1	no	—
115	C₂×C₄	—	4	1	no	—
116	C₂×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
117	C₂×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
118	C₂×C₄	#41×C₄₇	1	1	yes	characteristic
119	C₈	—	2	1	no	—
120	C₈	—	4	1	no	—
121	C₈	—	4	2	no	—
122	C₈	D₈×C₄₇	1	2	yes	complemented
123	C₂²	—	2	2	no	—
124	C₂²	—	4	1	no	—
125	C₂²	—	4	1	no	—
126	C₂²	—	8	2	no	—
127	C₂²	—	8	2	no	—
128	C₂²	—	8	2	no	—
129	C₂²	#121×C₄₇	1	1	yes	characteristic
130	C₄	—	2	1	no	—
131	C₄	—	4	1	no	—
132	C₄	C₂×D₈×C₄₇	1	1	yes	characteristic
133	C₄	C₂×D₈×C₄₇	1	1	yes	characteristic
134	C₄	#135×C₄₇	1	1	yes	characteristic
135	C₄	AGL(1,ℤ/8)×C₄₇	1	1	yes	characteristic
136	C₂	—	4	2	no	—
137	C₂	—	16	2	no	—
138	C₂	C₄₇×#459	1	1	yes	characteristic
139	C₂	C₄₇×#466	1	1	yes	characteristic
140	C₂	C₄₇×#472	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄₇×#493

C₂

yes

maximal, characteristic, complemented

C₄₇×#459

C₂

yes

maximal, complemented

C₈×D₄×C₄₇

C₂

yes

maximal, characteristic, complemented

C₄₇×#335

C₂

yes

maximal, characteristic, complemented

C₄₇×#331

C₂

yes

maximal, complemented

C₂×D₈×C₄₇

—

C₂×D₈×C₄₇

—

C₂×D₈×C₄₇

—

C₂²×C₈×C₄₇

—

#127×C₄₇

C₂²

yes

—

C₄×D₄×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

C₂²

yes

—

C₄⋊C₈×C₄₇

C₂²

yes

characteristic

#102×C₄₇

—

C₂²⋊C₈×C₄₇

—

C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

—

D₈×C₄₇

—

D₈×C₄₇

—

D₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

D₄

yes

characteristic

C₂×C₈×C₄₇

D₄

yes

characteristic

C₄⋊C₄×C₄₇

D₄

yes

characteristic

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

C₂³

yes

characteristic

C₂³×C₄₇

—

C₂³×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

D₈

yes

complemented

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

#41

yes

characteristic

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

D₈

yes

complemented

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#121

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×D₈

yes

characteristic

C₄×C₄₇

C₂×D₈

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

characteristic, complemented

#459

C₂×C₄₇

yes

complemented

C₈×D₄

C₂×C₄₇

yes

characteristic, complemented

#335

C₂×C₄₇

yes

characteristic, complemented

#331

C₂×C₄₇

yes

complemented

C₄₇

#1667

yes

characteristic, direct product factor, Sylow

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂²×C₈

—

#127

C₂²×C₄₇

yes

—

C₄×D₄

C₂²×C₄₇

yes

characteristic

C₄⋊C₈

C₂²×C₄₇

yes

characteristic

C₄⋊C₈

C₂²×C₄₇

yes

—

#102

—

C₂²⋊C₈

—

C₄×C₈

C₂²×C₄₇

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

D₄×C₄₇

yes

characteristic

C₂²×C₄

—

D₈

—

D₈

—

D₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₄×C₄₇

yes

characteristic

100

C₂×C₈

D₄×C₄₇

yes

characteristic

101

C₄⋊C₄

D₄×C₄₇

yes

characteristic

102

C₂²⋊C₄

—

103

C₄²

C₂³×C₄₇

yes

characteristic

104

C₂³

—

105

C₂³

—

106

D₄

—

107

D₄

—

108

D₄

—

109

D₄

—

110

D₄

—

111

D₄

—

112

D₄

D₈×C₄₇

yes

complemented

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

C₂×D₄×C₄₇

yes

characteristic

117

C₂×C₄

C₂×D₄×C₄₇

yes

characteristic

118

C₂×C₄

#41×C₄₇

yes

characteristic

119

C₈

—

120

C₈

—

121

C₈

—

122

C₈

D₈×C₄₇

yes

complemented

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

#121×C₄₇

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

C₂×D₈×C₄₇

yes

characteristic

133

C₄

C₂×D₈×C₄₇

yes

characteristic

134

C₄

#135×C₄₇

yes

characteristic

135

C₄

AGL(1,ℤ/8)×C₄₇

yes

characteristic

136

C₂

—

137

C₂

—

138

C₂

C₄₇×#459

yes

characteristic

139

C₂

C₄₇×#466

yes

characteristic

140

C₂

C₄₇×#472

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₄₇	#121
commutator, Frattini, cosocle	C₄²	C₂³×C₄₇
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄₇ ◁ C₄²×C₄₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₄₇

#121

commutator, Frattini, cosocle

C₄²

C₂³×C₄₇

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₄₇ ◁ C₄²×C₄₇ ◁ G

N	H ≅ G/N	type
1. C₄₇×#493	C₂	semidirect
2. C₄₇×#459	C₂	semidirect
3. C₈×D₄×C₄₇	C₂	semidirect
4. C₄₇×#335	C₂	semidirect
5. C₄₇×#331	C₂	semidirect
41. D₄×C₄₇	D₈	semidirect
51. C₈×C₄₇	D₈	semidirect
65. #1667	C₄₇	direct (N × H)
71. #493	C₂×C₄₇	semidirect
72. #459	C₂×C₄₇	semidirect
73. C₈×D₄	C₂×C₄₇	semidirect
74. #335	C₂×C₄₇	semidirect
75. #331	C₂×C₄₇	semidirect
76. C₄₇	#1667	direct (N × H)
112. D₄	D₈×C₄₇	semidirect
122. C₈	D₈×C₄₇	semidirect

H ≅ G/N

type

1. C₄₇×#493

C₂

semidirect

2. C₄₇×#459

C₂

semidirect

3. C₈×D₄×C₄₇

C₂

semidirect

4. C₄₇×#335

C₂

semidirect

5. C₄₇×#331

C₂

semidirect

41. D₄×C₄₇

D₈

semidirect

51. C₈×C₄₇

D₈

semidirect

65. #1667

C₄₇

direct (N × H)

71. #493

C₂×C₄₇

semidirect

72. #459

C₂×C₄₇

semidirect

73. C₈×D₄

C₂×C₄₇

semidirect

74. #335

C₂×C₄₇

semidirect

75. #331

C₂×C₄₇

semidirect

76. C₄₇

#1667

direct (N × H)

112. D₄

D₈×C₄₇

semidirect

122. C₈

D₈×C₄₇

semidirect

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

subgroup	index
C₄₇×#493	2
C₄₇×#459	2
C₈×D₄×C₄₇	2
C₄₇×#335	2
C₄₇×#331	2
#1667	47

subgroup

index

C₄₇×#493

C₄₇×#459

C₈×D₄×C₄₇

C₄₇×#335

C₄₇×#331

#1667

Subgroups — C₄₇×#1667 #20239277

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C47×#1667 #20239277

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#1667 #20239277