Group #20241211 — subgroups

Subgroups — #20241211

Order 6016.

158 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 65 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	#15688878	C₂	1	1	yes	maximal, characteristic
2	#15688878	C₂	1	1	yes	maximal, characteristic
3	C₄₇×#389	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₄₇⋊C₈	—	2	1	no	—
5	C₂²×C₄₇⋊C₈	—	2	1	no	—
6	C₂×C₄⋊C₄×C₄₇	—	2	1	no	—
7	C₂×C₄⋊C₄×C₄₇	C₄	1	1	yes	characteristic
8	C₂×C₄⋊C₄×C₄₇	C₄	1	1	yes	characteristic
9	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
10	#95×C₄₇	—	2	1	no	—
11	C₂²×C₄×C₄₇	—	2	1	no	—
12	C₂²×C₄×C₄₇	—	2	1	no	—
13	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
14	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
15	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
16	C₄⋊C₄×C₄₇	—	2	1	no	—
17	C₄⋊C₄×C₄₇	—	4	1	no	—
18	C₄⋊C₄×C₄₇	—	4	1	no	—
19	C₄⋊C₄×C₄₇	—	4	1	no	—
20	C₄⋊C₄×C₄₇	C₈	1	2	yes	complemented
21	C₄²×C₄₇	—	4	1	no	—
22	C₂×C₄₇⋊C₈	—	2	2	no	—
23	C₂×C₄₇⋊C₈	—	2	2	no	—
24	C₂×C₄₇⋊C₈	—	4	1	no	—
25	C₂×C₄₇⋊C₈	—	4	1	no	—
26	C₂×C₄₇⋊C₈	—	4	1	no	—
27	C₂×C₄₇⋊C₈	—	4	1	no	—
28	C₂³×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₄₇	—	2	1	no	—
30	C₂×C₄×C₄₇	—	2	1	no	—
31	C₂×C₄×C₄₇	—	2	1	no	—
32	C₂×C₄×C₄₇	—	2	1	no	—
33	C₂×C₄×C₄₇	—	2	1	no	—
34	C₂×C₄×C₄₇	—	2	1	no	—
35	C₂×C₄×C₄₇	—	2	2	no	—
36	C₂×C₄×C₄₇	—	4	1	no	—
37	C₂×C₄×C₄₇	—	4	1	no	—
38	C₂×C₄×C₄₇	—	4	1	no	—
39	C₂×C₄×C₄₇	—	4	1	no	—
40	C₂×C₄×C₄₇	—	4	1	no	—
41	C₂×C₄×C₄₇	C₂×C₈	1	1	yes	characteristic
42	C₂×C₄×C₄₇	M₁₆	1	1	yes	characteristic
43	C₂×C₄×C₄₇	D₈	1	1	yes	characteristic
44	C₂×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
45	C₂×C₄×C₄₇	SD₁₆	1	1	yes	characteristic
46	C₂×C₄×C₄₇	Q₁₆	1	1	yes	characteristic
47	C₄₇⋊C₈	—	4	2	no	—
48	C₄₇⋊C₈	—	4	2	no	—
49	C₂²×C₄₇	C₂²⋊C₈	1	1	yes	characteristic
50	C₂²×C₄₇	#99	1	1	yes	characteristic
51	C₂²×C₄₇	#101	1	1	yes	characteristic
52	C₂²×C₄₇	#102	1	1	yes	characteristic
53	C₂²×C₄₇	#103	1	1	yes	characteristic
54	C₂²×C₄₇	#104	1	1	yes	characteristic
55	C₂²×C₄₇	#104	1	1	yes	characteristic
56	C₄×C₄₇	—	2	1	no	—
57	C₄×C₄₇	—	2	1	no	—
58	C₄×C₄₇	—	2	1	no	—
59	C₄×C₄₇	—	2	1	no	—
60	C₄×C₄₇	—	2	1	no	—
61	C₄×C₄₇	—	2	1	no	—
62	C₄×C₄₇	—	4	2	no	—
63	C₄×C₄₇	—	8	1	no	—
64	#1271	—	47	1	no	maximal, Sylow
65	C₂×C₄₇	#324	1	1	yes	characteristic
66	C₂×C₄₇	D₄⋊C₈	1	1	yes	characteristic
67	C₂×C₄₇	Q₈⋊C₈	1	1	yes	characteristic
68	C₂×C₄₇	#327	1	1	yes	characteristic
69	C₂×C₄₇	#328	1	1	yes	characteristic
70	C₂×C₄₇	#330	1	1	yes	characteristic
71	C₂×C₄₇	#332	1	1	yes	characteristic
72	#389	D₄₇	1	1	yes	characteristic
73	#336	—	47	1	no	—
74	#336	—	47	1	no	—
75	C₄₇	#1271	1	1	yes	characteristic, complemented, Sylow
76	C₂²×C₈	—	94	1	no	—
77	C₂²×C₈	—	94	1	no	—
78	C₂×C₄⋊C₄	—	2	1	no	—
79	C₂×C₄⋊C₄	Dic₄₇	1	1	yes	characteristic
80	C₂×C₄⋊C₄	Dic₄₇	1	1	yes	characteristic
81	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
82	#95	—	2	1	no	—
83	C₂²×C₄	—	2	1	no	—
84	C₂²×C₄	—	2	1	no	—
85	C₂²×C₄	C₂×Dic₄₇	1	1	yes	characteristic
86	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
87	C₂²×C₄	C₂²⋊D₄₇	1	1	yes	characteristic
88	C₂×C₈	—	94	2	no	—
89	C₂×C₈	—	94	2	no	—
90	C₂×C₈	—	188	1	no	—
91	C₂×C₈	—	188	1	no	—
92	C₂×C₈	—	188	1	no	—
93	C₂×C₈	—	188	1	no	—
94	C₄⋊C₄	—	2	1	no	—
95	C₄⋊C₄	—	4	1	no	—
96	C₄⋊C₄	—	4	1	no	—
97	C₄⋊C₄	—	4	1	no	—
98	C₄⋊C₄	C₄₇⋊C₈	1	2	yes	complemented
99	C₄²	—	4	1	no	—
100	C₂³	C₂²⋊Dic₄₇	1	1	yes	characteristic
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	—	2	1	no	—
105	C₂×C₄	—	2	1	no	—
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	4	1	no	—
109	C₂×C₄	—	4	1	no	—
110	C₂×C₄	—	4	1	no	—
111	C₂×C₄	—	4	1	no	—
112	C₂×C₄	—	4	1	no	—
113	C₂×C₄	C₂×C₄₇⋊C₈	1	1	yes	characteristic
114	C₂×C₄	#10625329	1	1	yes	characteristic
115	C₂×C₄	#10625334	1	1	yes	characteristic
116	C₂×C₄	#10625335	1	1	yes	characteristic
117	C₂×C₄	#10625336	1	1	yes	characteristic
118	C₂×C₄	#10625337	1	1	yes	characteristic
119	C₈	—	188	2	no	—
120	C₈	—	188	2	no	—
121	C₂²	C₂²⋊(C₄₇⋊C₈)	1	1	yes	characteristic
122	C₂²	#13093939	1	1	yes	characteristic
123	C₂²	#13093943	1	1	yes	characteristic
124	C₂²	#13093945	1	1	yes	characteristic
125	C₂²	#13093948	1	1	yes	characteristic
126	C₂²	#13093951	1	1	yes	characteristic
127	C₂²	#13093951	1	1	yes	characteristic
128	C₄	—	2	1	no	—
129	C₄	—	2	1	no	—
130	C₄	—	2	1	no	—
131	C₄	—	2	1	no	—
132	C₄	—	2	1	no	—
133	C₄	—	2	1	no	—
134	C₄	—	4	2	no	—
135	C₄	—	8	1	no	—
136	C₂	#15688849	1	1	yes	characteristic
137	C₂	D₄⋊(C₄₇⋊C₈)	1	1	yes	characteristic
138	C₂	Q₈⋊(C₄₇⋊C₈)	1	1	yes	characteristic
139	C₂	#15688857	1	1	yes	characteristic
140	C₂	#15688860	1	1	yes	characteristic
141	C₂	#15688865	1	1	yes	characteristic
142	C₂	#15688869	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15688878

C₂

yes

maximal, characteristic

#15688878

C₂

yes

maximal, characteristic

C₄₇×#389

C₂

yes

maximal, characteristic

C₂²×C₄₇⋊C₈

—

C₂²×C₄₇⋊C₈

—

C₂×C₄⋊C₄×C₄₇

—

C₂×C₄⋊C₄×C₄₇

C₄

yes

characteristic

C₂×C₄⋊C₄×C₄₇

C₄

yes

characteristic

C₂×C₄²×C₄₇

C₂²

yes

characteristic

#95×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

—

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

C₈

yes

complemented

C₄²×C₄₇

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₈

yes

characteristic

C₂×C₄×C₄₇

M₁₆

yes

characteristic

C₂×C₄×C₄₇

D₈

yes

characteristic

C₂×C₄×C₄₇

SD₁₆

yes

characteristic

C₂×C₄×C₄₇

SD₁₆

yes

characteristic

C₂×C₄×C₄₇

Q₁₆

yes

characteristic

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

C₂²⋊C₈

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₂²×C₄₇

yes

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

—

C₂×C₄₇

yes

C₂×C₄₇

D₄⋊C₈

yes

characteristic

C₂×C₄₇

Q₈⋊C₈

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

yes

characteristic

#336

—

#336

—

C₄₇

#1271

yes

characteristic, complemented, Sylow

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

Dic₄₇

yes

characteristic

C₂×C₄⋊C₄

Dic₄₇

yes

characteristic

C₂×C₄²

C₂×D₄₇

yes

characteristic

#95

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×Dic₄₇

yes

characteristic

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

C₂²×C₄

C₂²⋊D₄₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

188

—

C₂×C₈

—

188

—

C₂×C₈

—

188

—

C₂×C₈

—

188

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₄₇⋊C₈

yes

complemented

C₄²

—

100

C₂³

C₂²⋊Dic₄₇

yes

characteristic

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂×C₄₇⋊C₈

yes

characteristic

114

C₂×C₄

#10625329

yes

characteristic

115

C₂×C₄

#10625334

yes

characteristic

116

C₂×C₄

#10625335

yes

characteristic

117

C₂×C₄

#10625336

yes

characteristic

118

C₂×C₄

#10625337

yes

characteristic

119

C₈

—

188

—

120

C₈

—

188

—

121

C₂²

C₂²⋊(C₄₇⋊C₈)

yes

characteristic

122

C₂²

#13093939

yes

characteristic

123

C₂²

#13093943

yes

characteristic

124

C₂²

#13093945

yes

characteristic

125

C₂²

#13093948

yes

characteristic

126

C₂²

#13093951

yes

characteristic

127

C₂²

#13093951

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

134

C₄

—

135

C₄

—

136

C₂

#15688849

yes

characteristic

137

C₂

D₄⋊(C₄₇⋊C₈)

yes

characteristic

138

C₂

Q₈⋊(C₄₇⋊C₈)

yes

characteristic

139

C₂

#15688857

yes

characteristic

140

C₂

#15688860

yes

characteristic

141

C₂

#15688865

yes

characteristic

142

C₂

#15688869

yes

characteristic

	subgroup	G/H
center	C₂³	C₂²⋊Dic₄₇
commutator	C₂×C₄×C₄₇	C₂×C₈
Frattini	C₂×C₄²	C₂×D₄₇
Fitting	C₄₇×#389	C₂
socle	C₂³×C₄₇	C₂²⋊C₄
hypercenter	#389	D₄₇
hypocenter	C₄₇	#1271
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

subgroup

G/H

center

C₂³

C₂²⋊Dic₄₇

commutator

C₂×C₄×C₄₇

C₂×C₈

Frattini

C₂×C₄²

C₂×D₄₇

Fitting

C₄₇×#389

C₂

socle

C₂³×C₄₇

C₂²⋊C₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #389

N	H ≅ G/N	type
20. C₄⋊C₄×C₄₇	C₈	semidirect
75. C₄₇	#1271	semidirect
98. C₄⋊C₄	C₄₇⋊C₈	semidirect

H ≅ G/N

type

20. C₄⋊C₄×C₄₇

C₈

semidirect

75. C₄₇

#1271

semidirect

98. C₄⋊C₄

C₄₇⋊C₈

semidirect

Subgroups — #20241211

Conjugacy classes of subgroups

Complements