Group #20239573 — subgroups

Subgroups — C₄₇×#1963 #20239573

Order 6016.

320 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 112 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×#532	C₂	1	1	yes	maximal, characteristic
2	C₄₇×#423	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#104×C₄₇	C₂	1	4	yes	maximal, complemented
4	C₄₇×#376	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#41×C₄₇	C₂²	1	2	yes	—
6	C₂×M₁₆×C₄₇	C₂²	1	2	yes	—
7	C₄⋊Q₈×C₄₇	C₄	1	1	yes	characteristic, complemented
8	#127×C₄₇	C₄	1	1	yes	characteristic, complemented
9	#124×C₄₇	C₄	1	2	yes	complemented
10	#121×C₄₇	—	2	2	no	—
11	C₄×D₄×C₄₇	—	2	2	no	—
12	C₂×C₄²×C₄₇	C₂²	1	2	yes	—
13	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₄₇	—	2	2	no	—
15	#104×C₄₇	—	2	8	no	—
16	#95×C₄₇	—	2	2	no	—
17	#41×C₄₇	—	2	4	no	—
18	#41×C₄₇	—	2	4	no	—
19	C₂×Q₈×C₄₇	C₂×C₄	1	2	yes	—
20	C₂×D₄×C₄₇	—	2	2	no	—
21	C₂×D₄×C₄₇	—	2	2	no	—
22	C₂×D₄×C₄₇	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₄₇	—	2	2	no	—
24	C₂²×C₄×C₄₇	—	2	2	no	—
25	C₂²×C₄×C₄₇	D₄	1	2	yes	—
26	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
27	M₁₆×C₄₇	—	2	4	no	—
28	C₂×C₈×C₄₇	—	2	2	no	—
29	C₄⋊C₄×C₄₇	—	2	2	no	—
30	C₂²⋊C₄×C₄₇	—	2	4	no	—
31	C₄²×C₄₇	—	2	2	no	—
32	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
33	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
34	C₄²×C₄₇	D₄	1	2	yes	—
35	C₄²×C₄₇	D₄	1	4	yes	complemented
36	C₂³×C₄₇	—	2	2	no	—
37	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
38	Q₈×C₄₇	—	2	4	no	—
39	D₄×C₄₇	—	2	4	no	—
40	D₄×C₄₇	—	4	4	no	—
41	D₄×C₄₇	—	4	4	no	—
42	C₂×C₄×C₄₇	—	2	2	no	—
43	C₂×C₄×C₄₇	—	2	4	no	—
44	C₂×C₄×C₄₇	—	4	2	no	—
45	C₂×C₄×C₄₇	—	4	4	no	—
46	C₂×C₄×C₄₇	—	4	2	no	—
47	C₂×C₄×C₄₇	—	4	2	no	—
48	C₂×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
49	C₂×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
50	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
51	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₄₇	#41	1	2	yes	—
54	C₈×C₄₇	—	4	2	no	—
55	C₂²×C₄₇	—	2	2	no	—
56	C₂²×C₄₇	—	4	2	no	—
57	C₂²×C₄₇	—	4	4	no	—
58	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
59	C₂²×C₄₇	#121	1	2	yes	—
60	C₄×C₄₇	—	2	2	no	—
61	C₄×C₄₇	—	2	2	no	—
62	C₄×C₄₇	—	4	4	no	—
63	C₄×C₄₇	—	4	2	no	—
64	C₄×C₄₇	C₄×D₄	1	2	yes	—
65	C₄×C₄₇	#124	1	1	yes	characteristic
66	C₄×C₄₇	#127	1	1	yes	characteristic
67	#1963	C₄₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂×C₄₇	—	2	2	no	—
69	C₂×C₄₇	—	8	2	no	—
70	C₂×C₄₇	#390	1	1	yes	characteristic
71	C₂×C₄₇	#421	1	2	yes	—
72	#532	C₂×C₄₇	1	1	yes	characteristic
73	#423	C₂×C₄₇	1	1	yes	characteristic, complemented
74	C₂×#104	C₂×C₄₇	1	4	yes	complemented
75	#376	C₂×C₄₇	1	1	yes	characteristic, complemented
76	C₄₇	#1963	1	1	yes	characteristic, direct product factor, Sylow
77	C₂×#41	C₂²×C₄₇	1	2	yes	—
78	C₂×M₁₆	C₂²×C₄₇	1	2	yes	—
79	C₄⋊Q₈	C₄×C₄₇	1	1	yes	characteristic, complemented
80	#127	C₄×C₄₇	1	1	yes	characteristic, complemented
81	#124	C₄×C₄₇	1	2	yes	complemented
82	#121	—	2	2	no	—
83	C₄×D₄	—	2	2	no	—
84	C₂×C₄²	C₂²×C₄₇	1	2	yes	—
85	C₂×C₄²	C₂²×C₄₇	1	1	yes	characteristic
86	C₄⋊C₈	—	2	2	no	—
87	#104	—	2	8	no	—
88	#95	—	2	2	no	—
89	#41	—	2	4	no	—
90	#41	—	2	4	no	—
91	C₂×Q₈	C₂×C₄×C₄₇	1	2	yes	—
92	C₂×D₄	—	2	2	no	—
93	C₂×D₄	—	2	2	no	—
94	C₂×D₄	C₂×C₄×C₄₇	1	2	yes	—
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	—	2	2	no	—
97	C₂²×C₄	D₄×C₄₇	1	2	yes	—
98	C₂²×C₄	C₂³×C₄₇	1	1	yes	characteristic
99	M₁₆	—	2	4	no	—
100	C₂×C₈	—	2	2	no	—
101	C₄⋊C₄	—	2	2	no	—
102	C₂²⋊C₄	—	2	4	no	—
103	C₄²	—	2	2	no	—
104	C₄²	C₂×C₄×C₄₇	1	1	yes	characteristic
105	C₄²	C₂×C₄×C₄₇	1	1	yes	characteristic
106	C₄²	D₄×C₄₇	1	2	yes	—
107	C₄²	D₄×C₄₇	1	4	yes	complemented
108	C₂³	—	2	2	no	—
109	C₂³	C₂×D₄×C₄₇	1	1	yes	characteristic
110	Q₈	—	2	4	no	—
111	D₄	—	2	4	no	—
112	D₄	—	4	4	no	—
113	D₄	—	4	4	no	—
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	4	2	no	—
117	C₂×C₄	—	4	4	no	—
118	C₂×C₄	—	4	2	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	C₂²⋊C₄×C₄₇	1	2	yes	—
121	C₂×C₄	C₂²⋊C₄×C₄₇	1	2	yes	—
122	C₂×C₄	C₂²×C₄×C₄₇	1	1	yes	characteristic
123	C₂×C₄	C₂×D₄×C₄₇	1	2	yes	—
124	C₂×C₄	C₂×D₄×C₄₇	1	1	yes	characteristic
125	C₂×C₄	#41×C₄₇	1	2	yes	—
126	C₈	—	4	2	no	—
127	C₂²	—	2	2	no	—
128	C₂²	—	4	2	no	—
129	C₂²	—	4	4	no	—
130	C₂²	C₂×C₂²⋊C₄×C₄₇	1	1	yes	characteristic
131	C₂²	#121×C₄₇	1	2	yes	—
132	C₄	—	2	2	no	—
133	C₄	—	2	2	no	—
134	C₄	—	4	4	no	—
135	C₄	—	4	2	no	—
136	C₄	C₄×D₄×C₄₇	1	2	yes	—
137	C₄	#124×C₄₇	1	1	yes	characteristic
138	C₄	#127×C₄₇	1	1	yes	characteristic
139	C₂	—	2	2	no	—
140	C₂	—	8	2	no	—
141	C₂	C₄₇×#390	1	1	yes	characteristic
142	C₂	C₄₇×#421	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₄₇×#532

C₂

yes

maximal, characteristic

C₄₇×#423

C₂

yes

maximal, characteristic, complemented

C₂×#104×C₄₇

C₂

yes

maximal, complemented

C₄₇×#376

C₂

yes

maximal, characteristic, complemented

C₂×#41×C₄₇

C₂²

yes

—

C₂×M₁₆×C₄₇

C₂²

yes

—

C₄⋊Q₈×C₄₇

C₄

yes

characteristic, complemented

#127×C₄₇

C₄

yes

characteristic, complemented

#124×C₄₇

C₄

yes

complemented

#121×C₄₇

—

C₄×D₄×C₄₇

—

C₂×C₄²×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

—

#104×C₄₇

—

#95×C₄₇

—

#41×C₄₇

—

#41×C₄₇

—

C₂×Q₈×C₄₇

C₂×C₄

yes

—

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

M₁₆×C₄₇

—

C₂×C₈×C₄₇

—

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

D₄

yes

—

C₄²×C₄₇

D₄

yes

complemented

C₂³×C₄₇

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

Q₈×C₄₇

—

D₄×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

—

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

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₇

#121

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₄×D₄

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

C₄₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#390

yes

characteristic

C₂×C₄₇

#421

yes

—

#532

C₂×C₄₇

yes

characteristic

#423

C₂×C₄₇

yes

characteristic, complemented

C₂×#104

C₂×C₄₇

yes

complemented

#376

C₂×C₄₇

yes

characteristic, complemented

C₄₇

#1963

yes

characteristic, direct product factor, Sylow

C₂×#41

C₂²×C₄₇

yes

—

C₂×M₁₆

C₂²×C₄₇

yes

—

C₄⋊Q₈

C₄×C₄₇

yes

characteristic, complemented

#127

C₄×C₄₇

yes

characteristic, complemented

#124

C₄×C₄₇

yes

complemented

#121

—

C₄×D₄

—

C₂×C₄²

C₂²×C₄₇

yes

—

C₂×C₄²

C₂²×C₄₇

yes

characteristic

C₄⋊C₈

—

—

—

—

—

C₂×Q₈

C₂×C₄×C₄₇

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂×C₄×C₄₇

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₄×C₄₇

yes

—

C₂²×C₄

C₂³×C₄₇

yes

characteristic

M₁₆

—

100

C₂×C₈

—

101

C₄⋊C₄

—

102

C₂²⋊C₄

—

103

C₄²

—

104

C₄²

C₂×C₄×C₄₇

yes

characteristic

105

C₄²

C₂×C₄×C₄₇

yes

characteristic

106

C₄²

D₄×C₄₇

yes

—

107

C₄²

D₄×C₄₇

yes

complemented

108

C₂³

—

109

C₂³

C₂×D₄×C₄₇

yes

characteristic

110

Q₈

—

111

D₄

—

112

D₄

—

113

D₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

C₂²⋊C₄×C₄₇

yes

—

121

C₂×C₄

C₂²⋊C₄×C₄₇

yes

—

122

C₂×C₄

C₂²×C₄×C₄₇

yes

characteristic

123

C₂×C₄

C₂×D₄×C₄₇

yes

—

124

C₂×C₄

C₂×D₄×C₄₇

yes

characteristic

125

C₂×C₄

#41×C₄₇

yes

—

126

C₈

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

130

C₂²

C₂×C₂²⋊C₄×C₄₇

yes

characteristic

131

C₂²

#121×C₄₇

yes

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

C₄×D₄×C₄₇

yes

—

137

C₄

#124×C₄₇

yes

characteristic

138

C₄

#127×C₄₇

yes

characteristic

139

C₂

—

140

C₂

—

141

C₂

C₄₇×#390

yes

characteristic

142

C₂

C₄₇×#421

yes

—

	subgroup	G/H
center	C₂×C₄×C₄₇	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₄₇
Frattini, cosocle	C₂²×C₄	C₂³×C₄₇
socle	C₂²×C₄₇	C₂×C₂²⋊C₄
prime radical	C₄₇×#532	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₄₇ ◁ C₂×C₄²×C₄₇ ◁ G

subgroup

G/H

center

C₂×C₄×C₄₇

C₂×D₄

commutator

C₂×C₄

C₂²×C₄×C₄₇

Frattini, cosocle

C₂²×C₄

C₂³×C₄₇

socle

C₂²×C₄₇

C₂×C₂²⋊C₄

prime radical

C₄₇×#532

C₂

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₄₇ ◁ C₂×C₄²×C₄₇ ◁ G

N	H ≅ G/N	type
2. C₄₇×#423	C₂	semidirect
3. C₂×#104×C₄₇	C₂	semidirect
4. C₄₇×#376	C₂	semidirect
7. C₄⋊Q₈×C₄₇	C₄	semidirect
8. #127×C₄₇	C₄	semidirect
9. #124×C₄₇	C₄	semidirect
35. C₄²×C₄₇	D₄	semidirect
67. #1963	C₄₇	direct (N × H)
73. #423	C₂×C₄₇	semidirect
74. C₂×#104	C₂×C₄₇	semidirect
75. #376	C₂×C₄₇	semidirect
76. C₄₇	#1963	direct (N × H)
79. C₄⋊Q₈	C₄×C₄₇	semidirect
80. #127	C₄×C₄₇	semidirect
81. #124	C₄×C₄₇	semidirect
107. C₄²	D₄×C₄₇	semidirect

H ≅ G/N

type

2. C₄₇×#423

C₂

semidirect

3. C₂×#104×C₄₇

C₂

semidirect

4. C₄₇×#376

C₂

semidirect

7. C₄⋊Q₈×C₄₇

C₄

semidirect

8. #127×C₄₇

C₄

semidirect

9. #124×C₄₇

C₄

semidirect

35. C₄²×C₄₇

D₄

semidirect

67. #1963

C₄₇

direct (N × H)

73. #423

C₂×C₄₇

semidirect

74. C₂×#104

C₂×C₄₇

semidirect

75. #376

C₂×C₄₇

semidirect

76. C₄₇

#1963

direct (N × H)

79. C₄⋊Q₈

C₄×C₄₇

semidirect

80. #127

C₄×C₄₇

semidirect

81. #124

C₄×C₄₇

semidirect

107. C₄²

D₄×C₄₇

semidirect

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

subgroup	index
C₄₇×#532	2
C₄₇×#423	2
C₂×#104×C₄₇	2
C₄₇×#376	2
#1963	47

subgroup

index

C₄₇×#532

C₄₇×#423

C₂×#104×C₄₇

C₄₇×#376

#1963

Subgroups — C₄₇×#1963 #20239573

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C47×#1963 #20239573

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄₇×#1963 #20239573