Group #20241709 — subgroups

Subgroups — #20241709

Order 6016.

312 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 312 conjugacy classes of subgroups — too many to draw. Only its 113 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	#15689176	C₂	1	2	yes	maximal, complemented
2	#15688847	C₂	1	4	yes	maximal, complemented
3	C₂×C₄×D₄×C₄₇	C₂	1	1	yes	maximal, characteristic
4	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
5	C₂²⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
6	C₄×C₄₇⋊C₈	—	2	2	no	—
7	C₂²×D₄×C₄₇	C₄	1	1	yes	characteristic
8	C₂³×C₄×C₄₇	C₄	1	2	yes	—
9	C₄×D₄×C₄₇	—	2	2	no	—
10	C₄×D₄×C₄₇	—	2	2	no	—
11	C₂×C₄⋊C₄×C₄₇	C₄	1	1	yes	characteristic
12	C₂×C₂²⋊C₄×C₄₇	C₂²	1	2	yes	—
13	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
14	C₂⁴×C₄₇	C₂×C₄	1	2	yes	—
15	C₂×D₄×C₄₇	—	2	2	no	—
16	C₂×D₄×C₄₇	—	2	2	no	—
17	C₂×D₄×C₄₇	C₈	1	4	yes	complemented
18	C₂²×C₄×C₄₇	—	2	2	no	—
19	C₂²×C₄×C₄₇	—	2	2	no	—
20	C₂²×C₄×C₄₇	—	4	4	no	—
21	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₄₇	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
25	C₄⋊C₄×C₄₇	—	2	2	no	—
26	C₂²⋊C₄×C₄₇	—	2	4	no	—
27	C₄²×C₄₇	D₄	1	2	yes	—
28	C₄²×C₄₇	D₄	1	2	yes	—
29	C₂×C₄₇⋊C₈	—	2	4	no	—
30	C₂³×C₄₇	—	2	2	no	—
31	C₂³×C₄₇	—	2	2	no	—
32	C₂³×C₄₇	—	4	2	no	—
33	C₂³×C₄₇	—	4	2	no	—
34	C₂³×C₄₇	C₂×C₈	1	4	yes	—
35	C₂³×C₄₇	C₂²×C₄	1	1	yes	characteristic
36	D₄×C₄₇	—	2	8	no	—
37	C₂×C₄×C₄₇	—	2	2	no	—
38	C₂×C₄×C₄₇	—	2	2	no	—
39	C₂×C₄×C₄₇	—	4	1	no	—
40	C₂×C₄×C₄₇	—	4	4	no	—
41	C₂×C₄×C₄₇	—	4	4	no	—
42	C₂×C₄×C₄₇	—	4	2	no	—
43	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
44	C₂×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
45	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₄₇	C₂×C₈	1	2	yes	—
47	C₂×C₄×C₄₇	M₁₆	1	2	yes	—
48	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
49	C₄₇⋊C₈	—	4	4	no	—
50	C₂²×C₄₇	—	2	1	no	—
51	C₂²×C₄₇	—	2	1	no	—
52	C₂²×C₄₇	—	2	4	no	—
53	C₂²×C₄₇	—	4	4	no	—
54	C₂²×C₄₇	—	4	4	no	—
55	C₂²×C₄₇	—	4	2	no	—
56	C₂²×C₄₇	—	4	2	no	—
57	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
58	C₂²×C₄₇	C₂²×C₈	1	1	yes	characteristic
59	C₂²×C₄₇	C₂×M₁₆	1	1	yes	characteristic
60	C₄×C₄₇	—	2	2	no	—
61	C₄×C₄₇	—	2	1	no	—
62	C₄×C₄₇	—	4	2	no	—
63	C₄×C₄₇	C₂²⋊C₈	1	4	yes	—
64	C₄×C₄₇	#100	1	2	yes	—
65	#1460	—	47	1	no	maximal, Sylow
66	C₂×C₄₇	—	2	1	no	—
67	C₂×C₄₇	—	2	1	no	—
68	C₂×C₄₇	—	4	4	no	—
69	C₂×C₄₇	C₂×C₂²⋊C₈	1	1	yes	characteristic
70	C₂×C₄₇	#410	1	1	yes	characteristic
71	C₂×C₄₇	C₂×#100	1	1	yes	characteristic
72	C₂×C₄×D₄	D₄₇	1	1	yes	characteristic
73	#431	—	47	2	no	—
74	#323	—	47	4	no	—
75	C₄₇	#1460	1	1	yes	characteristic, complemented, Sylow
76	C₂²×D₄	Dic₄₇	1	1	yes	characteristic
77	C₂³×C₄	Dic₄₇	1	2	yes	—
78	C₄×D₄	—	2	2	no	—
79	C₄×D₄	—	2	2	no	—
80	C₂×C₄⋊C₄	Dic₄₇	1	1	yes	characteristic
81	C₂×C₂²⋊C₄	C₂×D₄₇	1	2	yes	—
82	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
83	C₄⋊C₈	—	94	2	no	—
84	C₂²⋊C₈	—	47	4	no	—
85	C₄×C₈	—	94	2	no	—
86	C₂⁴	C₂×Dic₄₇	1	2	yes	—
87	C₂×D₄	—	2	2	no	—
88	C₂×D₄	—	2	2	no	—
89	C₂×D₄	C₄₇⋊C₈	1	4	yes	complemented
90	C₂²×C₄	—	2	2	no	—
91	C₂²×C₄	—	2	2	no	—
92	C₂²×C₄	—	4	4	no	—
93	C₂²×C₄	C₂×Dic₄₇	1	1	yes	characteristic
94	C₂²×C₄	C₂×Dic₄₇	1	2	yes	—
95	C₂²×C₄	C₂×Dic₄₇	1	1	yes	characteristic
96	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
97	C₂×C₈	—	94	4	no	—
98	C₄⋊C₄	—	2	2	no	—
99	C₂²⋊C₄	—	2	4	no	—
100	C₄²	C₂²⋊D₄₇	1	2	yes	—
101	C₄²	C₂²⋊D₄₇	1	2	yes	—
102	C₂³	—	2	2	no	—
103	C₂³	—	2	2	no	—
104	C₂³	—	4	2	no	—
105	C₂³	—	4	2	no	—
106	C₂³	C₂×C₄₇⋊C₈	1	4	yes	—
107	C₂³	C₂²×Dic₄₇	1	1	yes	characteristic
108	D₄	—	2	8	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	4	1	no	—
112	C₂×C₄	—	4	4	no	—
113	C₂×C₄	—	4	4	no	—
114	C₂×C₄	—	4	2	no	—
115	C₂×C₄	C₂×C₄₇⋊C₈	1	2	yes	—
116	C₂×C₄	#10625329	1	2	yes	—
117	C₂×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
118	C₂×C₄	C₂²⋊Dic₄₇	1	2	yes	—
119	C₂×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
120	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
121	C₈	—	188	4	no	—
122	C₂²	—	2	1	no	—
123	C₂²	—	2	1	no	—
124	C₂²	—	2	4	no	—
125	C₂²	—	4	4	no	—
126	C₂²	—	4	4	no	—
127	C₂²	—	4	2	no	—
128	C₂²	—	4	2	no	—
129	C₂²	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
130	C₂²	C₂²×C₄₇⋊C₈	1	1	yes	characteristic
131	C₂²	C₂×#10625329	1	1	yes	characteristic
132	C₄	—	2	2	no	—
133	C₄	—	2	1	no	—
134	C₄	—	4	2	no	—
135	C₄	C₂²⋊(C₄₇⋊C₈)	1	4	yes	—
136	C₄	#13093941	1	2	yes	—
137	C₂	—	2	1	no	—
138	C₂	—	2	1	no	—
139	C₂	—	4	4	no	—
140	C₂	C₂×C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
141	C₂	#15689097	1	1	yes	characteristic
142	C₂	C₂×#13093941	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689176

C₂

yes

maximal, complemented

#15688847

C₂

yes

maximal, complemented

C₂×C₄×D₄×C₄₇

C₂

yes

maximal, characteristic

C₄⋊(C₄₇⋊C₈)

—

C₂²⋊(C₄₇⋊C₈)

C₂²

yes

—

C₄×C₄₇⋊C₈

—

C₂²×D₄×C₄₇

C₄

yes

characteristic

C₂³×C₄×C₄₇

C₄

yes

—

C₄×D₄×C₄₇

—

C₄×D₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₄

yes

characteristic

C₂×C₂²⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂⁴×C₄₇

C₂×C₄

yes

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

C₈

yes

complemented

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂×C₄

yes

—

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₄⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

D₄

yes

—

C₄²×C₄₇

D₄

yes

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

C₂×C₈

yes

—

C₂³×C₄₇

C₂²×C₄

yes

characteristic

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

characteristic

C₂×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×C₈

yes

—

C₂×C₄×C₄₇

M₁₆

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₄₇

C₂²×C₈

yes

characteristic

C₂²×C₄₇

C₂×M₁₆

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂²⋊C₈

yes

—

C₄×C₄₇

#100

yes

—

#1460

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

C₂×C₂²⋊C₈

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄×D₄

D₄₇

yes

characteristic

#431

—

#323

—

C₄₇

#1460

yes

characteristic, complemented, Sylow

C₂²×D₄

Dic₄₇

yes

characteristic

C₂³×C₄

Dic₄₇

yes

—

C₄×D₄

—

C₄×D₄

—

C₂×C₄⋊C₄

Dic₄₇

yes

characteristic

C₂×C₂²⋊C₄

C₂×D₄₇

yes

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₂⁴

C₂×Dic₄₇

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₄₇⋊C₈

yes

complemented

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₄₇

yes

characteristic

C₂²×C₄

C₂×Dic₄₇

yes

—

C₂²×C₄

C₂×Dic₄₇

yes

characteristic

C₂²×C₄

C₂²×D₄₇

yes

characteristic

C₂×C₈

—

C₄⋊C₄

—

C₂²⋊C₄

—

100

C₄²

C₂²⋊D₄₇

yes

—

101

C₄²

C₂²⋊D₄₇

yes

—

102

C₂³

—

103

C₂³

—

104

C₂³

—

105

C₂³

—

106

C₂³

C₂×C₄₇⋊C₈

yes

—

107

C₂³

C₂²×Dic₄₇

yes

characteristic

108

D₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

C₂×C₄₇⋊C₈

yes

—

116

C₂×C₄

#10625329

yes

—

117

C₂×C₄

C₂²⋊Dic₄₇

yes

characteristic

118

C₂×C₄

C₂²⋊Dic₄₇

yes

—

119

C₂×C₄

C₂²⋊Dic₄₇

yes

characteristic

120

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

121

C₈

—

188

—

122

C₂²

—

123

C₂²

—

124

C₂²

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

—

129

C₂²

C₂×C₂²⋊Dic₄₇

yes

characteristic

130

C₂²

C₂²×C₄₇⋊C₈

yes

characteristic

131

C₂²

C₂×#10625329

yes

characteristic

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

C₂²⋊(C₄₇⋊C₈)

yes

—

136

C₄

#13093941

yes

—

137

C₂

—

138

C₂

—

139

C₂

—

140

C₂

C₂×C₂²⋊(C₄₇⋊C₈)

yes

characteristic

141

C₂

#15689097

yes

characteristic

142

C₂

C₂×#13093941

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊Dic₄₇
commutator	C₂²×C₄₇	C₂²×C₈
Frattini	C₂²×C₄	C₂²×D₄₇
Fitting	C₂×C₄×D₄×C₄₇	C₂
socle	C₂²×C₄₇	C₂×C₂²⋊C₄
hypercenter	C₂×C₄×D₄	D₄₇
hypocenter	C₄₇	#1460
LCS	G ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×D₄

subgroup

G/H

center

C₂×C₄

C₂²⋊Dic₄₇

commutator

C₂²×C₄₇

C₂²×C₈

Frattini

C₂²×C₄

C₂²×D₄₇

Fitting

C₂×C₄×D₄×C₄₇

C₂

socle

C₂²×C₄₇

C₂×C₂²⋊C₄

hypercenter

C₂×C₄×D₄

D₄₇

hypocenter

C₄₇

#1460

LCS

G ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×D₄

N	H ≅ G/N	type
1. #15689176	C₂	semidirect
2. #15688847	C₂	semidirect
17. C₂×D₄×C₄₇	C₈	semidirect
75. C₄₇	#1460	semidirect
89. C₂×D₄	C₄₇⋊C₈	semidirect

H ≅ G/N

type

1. #15689176

C₂

semidirect

2. #15688847

C₂

semidirect

17. C₂×D₄×C₄₇

C₈

semidirect

75. C₄₇

#1460

semidirect

89. C₂×D₄

C₄₇⋊C₈

semidirect

Subgroups — #20241709

Conjugacy classes of subgroups

Complements