Group #20242743 — subgroups

Subgroups — #20242743

Order 6016.

204 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 61 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	#15689500	C₂	1	1	yes	maximal, characteristic, complemented
2	#15689318	C₂	1	2	yes	maximal, complemented
3	#15688876	C₂	1	1	yes	maximal, characteristic, complemented
4	#15688866	C₂	1	2	yes	maximal, complemented
5	C₈×D₄×C₄₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₃₇₆	—	2	1	no	—
7	C₂×D₃₇₆	—	2	1	no	—
8	C₂×#10625334	—	2	2	no	—
9	#13094021	C₂²	1	2	yes	—
10	C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
11	#13093944	—	2	2	no	—
12	C₂²×C₈×C₄₇	—	2	1	no	—
13	C₄×D₄×C₄₇	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₂²⋊C₈×C₄₇	—	2	1	no	—
16	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×D₄×C₄₇	D₄	1	1	yes	characteristic
18	C₂²×C₄×C₄₇	—	2	1	no	—
19	C₂×D₁₈₈	—	2	2	no	—
20	C₂×D₁₈₈	—	2	2	no	—
21	C₂×D₁₈₈	—	2	2	no	—
22	C₂×C₈×C₄₇	—	2	1	no	—
23	C₂×C₈×C₄₇	—	4	2	no	—
24	C₂×C₈×C₄₇	D₄	1	1	yes	characteristic
25	C₂×C₈×C₄₇	D₄	1	1	yes	characteristic
26	C₄⋊C₄×C₄₇	D₄	1	1	yes	characteristic
27	C₂²⋊C₄×C₄₇	—	2	1	no	—
28	C₄²×C₄₇	C₂³	1	1	yes	characteristic
29	#10625334	—	4	4	no	—
30	C₂×C₄₇⋊C₈	—	2	2	no	—
31	D₃₇₆	—	4	2	no	—
32	D₃₇₆	—	4	2	no	—
33	C₂³×C₄₇	—	2	1	no	—
34	C₂²×D₄₇	—	4	2	no	—
35	D₄×C₄₇	—	2	1	no	—
36	D₄×C₄₇	D₈	1	2	yes	complemented
37	C₂×C₄×C₄₇	—	2	1	no	—
38	C₂×C₄×C₄₇	—	4	1	no	—
39	C₂×C₄×C₄₇	—	4	1	no	—
40	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
43	D₁₈₈	—	4	2	no	—
44	D₁₈₈	—	4	2	no	—
45	D₁₈₈	—	4	2	no	—
46	D₁₈₈	—	4	2	no	—
47	D₁₈₈	—	8	2	no	—
48	C₈×C₄₇	—	2	1	no	—
49	C₈×C₄₇	—	4	1	no	—
50	C₈×C₄₇	D₈	1	2	yes	complemented
51	C₄₇⋊C₈	—	4	2	no	—
52	C₂²×C₄₇	—	2	2	no	—
53	C₂²×C₄₇	—	4	1	no	—
54	C₂²×C₄₇	—	4	1	no	—
55	C₂²×C₄₇	#121	1	1	yes	characteristic
56	C₂×D₄₇	—	8	2	no	—
57	C₂×D₄₇	—	8	2	no	—
58	C₂×D₄₇	—	8	2	no	—
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	—	47	1	no	maximal, Sylow
66	C₂×C₄₇	—	4	2	no	—
67	C₂×C₄₇	#459	1	1	yes	characteristic
68	C₂×C₄₇	#466	1	1	yes	characteristic
69	C₂×C₄₇	#472	1	1	yes	characteristic
70	D₄₇	—	16	2	no	—
71	#493	—	47	1	no	—
72	#459	—	47	2	no	—
73	C₈×D₄	D₄₇	1	1	yes	characteristic, complemented
74	#335	—	47	1	no	—
75	#331	—	47	2	no	—
76	C₄₇	#1667	1	1	yes	characteristic, complemented, Sylow
77	C₂×D₈	—	94	2	no	—
78	C₂×D₈	—	94	1	no	—
79	C₂×D₈	—	94	1	no	—
80	C₂²×C₈	—	2	1	no	—
81	#127	—	47	2	no	—
82	C₄×D₄	C₂×D₄₇	1	1	yes	characteristic
83	C₄⋊C₈	—	47	2	no	—
84	C₄⋊C₈	C₂×D₄₇	1	1	yes	characteristic
85	#102	—	94	2	no	—
86	C₂²⋊C₈	—	2	1	no	—
87	C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
88	C₂×D₄	—	94	2	no	—
89	C₂×D₄	—	94	2	no	—
90	C₂×D₄	—	94	2	no	—
91	C₂×D₄	D₁₈₈	1	1	yes	characteristic
92	C₂²×C₄	—	2	1	no	—
93	D₈	—	188	2	no	—
94	D₈	—	188	4	no	—
95	D₈	—	188	2	no	—
96	C₂×C₈	—	2	1	no	—
97	C₂×C₈	—	4	2	no	—
98	C₂×C₈	—	94	2	no	—
99	C₂×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
100	C₂×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
101	C₄⋊C₄	D₁₈₈	1	1	yes	characteristic
102	C₂²⋊C₄	—	2	1	no	—
103	C₄²	C₂²×D₄₇	1	1	yes	characteristic
104	C₂³	—	2	1	no	—
105	C₂³	—	188	2	no	—
106	D₄	—	2	1	no	—
107	D₄	—	376	2	no	—
108	D₄	—	188	2	no	—
109	D₄	—	188	2	no	—
110	D₄	—	188	2	no	—
111	D₄	—	188	2	no	—
112	D₄	D₃₇₆	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×D₁₈₈	1	1	yes	characteristic
117	C₂×C₄	#10625350	1	1	yes	characteristic
118	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
119	C₈	—	2	1	no	—
120	C₈	—	4	1	no	—
121	C₈	—	188	2	no	—
122	C₈	#10625334	1	2	yes	complemented
123	C₂²	—	2	2	no	—
124	C₂²	—	4	1	no	—
125	C₂²	—	4	1	no	—
126	C₂²	—	376	2	no	—
127	C₂²	—	376	2	no	—
128	C₂²	—	376	2	no	—
129	C₂²	#13093998	1	1	yes	characteristic
130	C₄	—	2	1	no	—
131	C₄	—	4	1	no	—
132	C₄	C₂×#10625334	1	1	yes	characteristic
133	C₄	C₂×D₃₇₆	1	1	yes	characteristic
134	C₄	#13094046	1	1	yes	characteristic
135	C₄	#13094049	1	1	yes	characteristic
136	C₂	—	4	2	no	—
137	C₂	—	752	2	no	—
138	C₂	#15689318	1	1	yes	characteristic
139	C₂	#15689359	1	1	yes	characteristic
140	C₂	#15689393	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689500

C₂

yes

maximal, characteristic, complemented

#15689318

C₂

yes

maximal, complemented

#15688876

C₂

yes

maximal, characteristic, complemented

#15688866

C₂

yes

maximal, complemented

C₈×D₄×C₄₇

C₂

yes

maximal, characteristic, complemented

C₂×D₃₇₆

—

C₂×D₃₇₆

—

C₂×#10625334

—

#13094021

C₂²

yes

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

#13093944

—

C₂²×C₈×C₄₇

—

C₄×D₄×C₄₇

C₂²

yes

characteristic

C₄⋊C₈×C₄₇

C₂²

yes

characteristic

C₂²⋊C₈×C₄₇

—

C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×D₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

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

#10625334

—

C₂×C₄₇⋊C₈

—

D₃₇₆

—

D₃₇₆

—

C₂³×C₄₇

—

C₂²×D₄₇

—

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

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

D₁₈₈

—

C₈×C₄₇

—

C₈×C₄₇

—

C₈×C₄₇

D₈

yes

complemented

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#121

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×D₈

yes

characteristic

C₄×C₄₇

C₂×D₈

yes

C₄×C₄₇

yes

C₄×C₄₇

yes

—

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

—

#493

—

#459

—

C₈×D₄

D₄₇

yes

characteristic, complemented

#335

—

#331

—

C₄₇

#1667

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂²×C₈

—

#127

—

C₄×D₄

C₂×D₄₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂×D₄₇

yes

characteristic

#102

—

C₂²⋊C₈

—

C₄×C₈

C₂×D₄₇

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

D₁₈₈

yes

characteristic

C₂²×C₄

—

D₈

—

188

—

D₈

—

188

—

D₈

—

188

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₄₇

yes

characteristic

100

C₂×C₈

C₂²⋊D₄₇

yes

characteristic

101

C₄⋊C₄

D₁₈₈

yes

characteristic

102

C₂²⋊C₄

—

103

C₄²

C₂²×D₄₇

yes

characteristic

104

C₂³

—

105

C₂³

—

188

—

106

D₄

—

107

D₄

—

376

—

108

D₄

—

188

—

109

D₄

—

188

—

110

D₄

—

188

—

111

D₄

—

188

—

112

D₄

D₃₇₆

yes

complemented

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

C₂×D₁₈₈

yes

characteristic

117

C₂×C₄

#10625350

yes

characteristic

118

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

119

C₈

—

120

C₈

—

121

C₈

—

188

—

122

C₈

#10625334

yes

complemented

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

376

—

127

C₂²

—

376

—

128

C₂²

—

376

—

129

C₂²

#13093998

yes

characteristic

130

C₄

—

131

C₄

—

132

C₄

C₂×#10625334

yes

characteristic

133

C₄

C₂×D₃₇₆

yes

characteristic

134

C₄

#13094046

yes

characteristic

135

C₄

#13094049

yes

characteristic

136

C₂

—

137

C₂

—

752

—

138

C₂

#15689318

yes

characteristic

139

C₂

#15689359

yes

characteristic

140

C₂

#15689393

yes

characteristic

	subgroup	G/H
center	C₂²	#13093998
commutator	C₄²×C₄₇	C₂³
Frattini	C₄²	C₂²×D₄₇
Fitting	C₈×D₄×C₄₇	C₂
socle	C₂²×C₄₇	#121
hypercenter	C₈×D₄	D₄₇
hypocenter	C₄₇	#1667
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₄

subgroup

G/H

center

C₂²

#13093998

commutator

C₄²×C₄₇

C₂³

Frattini

C₄²

C₂²×D₄₇

Fitting

C₈×D₄×C₄₇

C₂

socle

C₂²×C₄₇

#121

hypercenter

C₈×D₄

D₄₇

hypocenter

C₄₇

#1667

LCS

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₄

N	H ≅ G/N	type
1. #15689500	C₂	semidirect
2. #15689318	C₂	semidirect
3. #15688876	C₂	semidirect
4. #15688866	C₂	semidirect
5. C₈×D₄×C₄₇	C₂	semidirect
36. D₄×C₄₇	D₈	semidirect
50. C₈×C₄₇	D₈	semidirect
73. C₈×D₄	D₄₇	semidirect
76. C₄₇	#1667	semidirect
112. D₄	D₃₇₆	semidirect
122. C₈	#10625334	semidirect

H ≅ G/N

type

1. #15689500

C₂

semidirect

2. #15689318

C₂

semidirect

3. #15688876

C₂

semidirect

4. #15688866

C₂

semidirect

5. C₈×D₄×C₄₇

C₂

semidirect

36. D₄×C₄₇

D₈

semidirect

50. C₈×C₄₇

D₈

semidirect

73. C₈×D₄

D₄₇

semidirect

76. C₄₇

#1667

semidirect

112. D₄

D₃₇₆

semidirect

122. C₈

#10625334

semidirect

Subgroups — #20242743

Conjugacy classes of subgroups

Complements