Group #20245228 — subgroups

Subgroups — #20245228

Order 6016.

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

normal subgroups only

This group has 144 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	#15689539	C₂	1	1	yes	maximal, characteristic, complemented
2	#15689238	C₂	1	1	yes	maximal, characteristic
3	#15688943	C₂	1	1	yes	maximal, characteristic, complemented
4	#15688908	C₂	1	1	yes	maximal, characteristic, complemented
5	#15688907	C₂	1	1	yes	maximal, characteristic, complemented
6	#15688898	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#504	C₂	1	1	yes	maximal, characteristic, complemented
8	#13094044	C₄	1	1	yes	characteristic
9	#13094030	C₂²	1	1	yes	characteristic
10	#13093964	C₂²	1	1	yes	characteristic
11	#13093963	—	2	1	no	—
12	C₂×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
13	C₁₆×D₄₇	—	2	1	no	—
14	#13093959	C₄	1	1	yes	characteristic
15	#13093951	C₄	1	1	yes	characteristic
16	#13093950	C₄	1	1	yes	characteristic
17	C₈×Dic₄₇	C₂²	1	1	yes	characteristic
18	#131×C₄₇	C₂²	1	1	yes	characteristic
19	M₃₂×C₄₇	—	2	1	no	—
20	M₃₂×C₄₇	C₂²	1	1	yes	characteristic
21	C₂×C₁₆×C₄₇	—	2	1	no	—
22	C₂×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
23	#41×C₄₇	C₂×C₄	1	1	yes	characteristic
24	#10625350	C₂×C₄	1	1	yes	characteristic
25	M₁₆×C₄₇	—	2	1	no	—
26	M₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₈×C₄₇	—	2	1	no	—
28	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
29	#10625337	C₈	1	1	yes	characteristic
30	#10625336	C₈	1	1	yes	characteristic
31	#10625335	C₈	1	1	yes	characteristic
32	#10625334	C₈	1	1	yes	characteristic
33	C₄×Dic₄₇	C₂×C₄	1	1	yes	characteristic
34	#10625329	C₂×C₄	1	1	yes	characteristic
35	C₂×C₄₇⋊C₈	C₂×C₄	1	1	yes	characteristic
36	#10625324	—	2	1	no	—
37	C₈×D₄₇	—	2	1	no	—
38	C₁₆×C₄₇	—	2	1	no	—
39	C₁₆×C₄₇	D₄	1	1	yes	characteristic
40	C₁₆×C₄₇	D₄	1	1	yes	characteristic
41	C₄₇⋊C₁₆	—	2	1	no	—
42	C₄₇⋊C₁₆	—	2	1	no	—
43	Q₈×C₄₇	C₂×C₈	1	1	yes	characteristic
44	D₄×C₄₇	—	2	1	no	—
45	D₄×C₄₇	C₂×C₈	1	1	yes	characteristic
46	C₂×C₄×C₄₇	—	2	1	no	—
47	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
48	C₂²⋊D₄₇	—	2	1	no	—
49	C₂×Dic₄₇	—	2	1	no	—
50	D₁₈₈	C₂×C₈	1	1	yes	characteristic
51	C₄×D₄₇	—	2	1	no	—
52	Dic₉₄	C₂×C₈	1	1	yes	characteristic
53	C₈×C₄₇	—	2	1	no	—
54	C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
55	C₈×C₄₇	#41	1	1	yes	characteristic
56	C₄₇⋊C₈	—	2	1	no	—
57	C₄₇⋊C₈	C₂×C₈	1	1	yes	characteristic
58	C₄₇⋊C₈	C₂×C₈	1	1	yes	characteristic
59	C₂²×C₄₇	—	2	1	no	—
60	C₂²×C₄₇	#131	1	1	yes	characteristic
61	C₂×D₄₇	—	2	1	no	—
62	C₄×C₄₇	—	2	1	no	—
63	C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
64	C₄×C₄₇	C₂²×C₈	1	1	yes	characteristic
65	Dic₄₇	—	2	1	no	—
66	Dic₄₇	—	4	1	no	—
67	#2171	—	47	1	no	maximal, Sylow
68	C₂×C₄₇	—	2	1	no	—
69	C₂×C₄₇	—	4	1	no	—
70	C₂×C₄₇	C₈×D₄	1	1	yes	characteristic
71	D₄₇	—	4	1	no	—
72	#504	—	47	1	no	—
73	#504	D₄₇	1	1	yes	characteristic, complemented
74	#443	—	47	1	no	—
75	#364	—	47	1	no	—
76	#350	—	47	1	no	—
77	#350	—	47	1	no	—
78	#346	—	47	1	no	—
79	C₄₇	#2171	1	1	yes	characteristic, complemented, Sylow
80	#135	—	47	1	no	—
81	#131	—	47	1	no	—
82	#131	C₂×D₄₇	1	1	yes	characteristic
83	M₃₂	—	2	1	no	—
84	M₃₂	—	47	1	no	—
85	M₃₂	—	94	1	no	—
86	M₃₂	C₂×D₄₇	1	1	yes	characteristic
87	C₂×C₁₆	—	2	1	no	—
88	C₂×C₁₆	—	47	1	no	—
89	C₂×C₁₆	—	94	1	no	—
90	C₂×C₁₆	C₂×D₄₇	1	1	yes	characteristic
91	#108	—	47	1	no	—
92	#104	—	47	1	no	—
93	#104	—	47	1	no	—
94	C₄×C₈	—	47	1	no	—
95	#41	—	47	1	no	—
96	#41	C₄×D₄₇	1	1	yes	characteristic
97	Q₁₆	—	47	1	no	—
98	SD₁₆	—	47	1	no	—
99	SD₁₆	—	47	1	no	—
100	D₈	—	47	1	no	—
101	M₁₆	—	2	1	no	—
102	M₁₆	—	47	1	no	—
103	M₁₆	—	94	1	no	—
104	M₁₆	C₄×D₄₇	1	1	yes	characteristic
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	—	47	1	no	—
107	C₂×C₈	—	94	1	no	—
108	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
109	C₄²	—	47	1	no	—
110	C₁₆	—	2	1	no	—
111	C₁₆	—	94	1	no	—
112	C₁₆	—	94	1	no	—
113	C₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
114	C₁₆	C₂²⋊D₄₇	1	1	yes	characteristic
115	Q₈	—	47	1	no	—
116	Q₈	C₈×D₄₇	1	1	yes	characteristic
117	D₄	—	2	1	no	—
118	D₄	—	47	1	no	—
119	D₄	—	94	1	no	—
120	D₄	C₈×D₄₇	1	1	yes	characteristic
121	C₂×C₄	—	2	1	no	—
122	C₂×C₄	—	94	1	no	—
123	C₂×C₄	—	94	1	no	—
124	C₂×C₄	C₂×C₄×D₄₇	1	1	yes	characteristic
125	C₈	—	2	1	no	—
126	C₈	—	47	1	no	—
127	C₈	—	47	1	no	—
128	C₈	—	94	1	no	—
129	C₈	#10625350	1	1	yes	characteristic
130	C₈	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
131	C₂²	—	2	1	no	—
132	C₂²	—	94	1	no	—
133	C₂²	#13094030	1	1	yes	characteristic
134	C₄	—	2	1	no	—
135	C₄	—	94	1	no	—
136	C₄	—	188	1	no	—
137	C₄	C₄×C₂²⋊D₄₇	1	1	yes	characteristic
138	C₄	C₂×C₈×D₄₇	1	1	yes	characteristic
139	C₂	—	2	1	no	—
140	C₂	—	4	1	no	—
141	C₂	—	188	1	no	—
142	C₂	C₈×C₂²⋊D₄₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689539

C₂

yes

maximal, characteristic, complemented

#15689238

C₂

yes

maximal, characteristic

#15688943

C₂

yes

maximal, characteristic, complemented

#15688908

C₂

yes

maximal, characteristic, complemented

#15688907

C₂

yes

maximal, characteristic, complemented

#15688898

C₂

yes

maximal, characteristic, complemented

C₄₇×#504

C₂

yes

maximal, characteristic, complemented

#13094044

C₄

yes

characteristic

#13094030

C₂²

yes

characteristic

#13093964

C₂²

yes

characteristic

#13093963

—

C₂×C₄₇⋊C₁₆

C₂²

yes

characteristic

C₁₆×D₄₇

—

#13093959

C₄

yes

characteristic

#13093951

C₄

yes

characteristic

#13093950

C₄

yes

characteristic

C₈×Dic₄₇

C₂²

yes

characteristic

#131×C₄₇

C₂²

yes

characteristic

M₃₂×C₄₇

—

M₃₂×C₄₇

C₂²

yes

characteristic

C₂×C₁₆×C₄₇

—

C₂×C₁₆×C₄₇

C₂²

yes

characteristic

#41×C₄₇

C₂×C₄

yes

characteristic

#10625350

C₂×C₄

yes

characteristic

M₁₆×C₄₇

—

M₁₆×C₄₇

C₂×C₄

yes

characteristic

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₂³

yes

characteristic

#10625337

C₈

yes

characteristic

#10625336

C₈

yes

characteristic

#10625335

C₈

yes

characteristic

#10625334

C₈

yes

characteristic

C₄×Dic₄₇

C₂×C₄

yes

characteristic

#10625329

C₂×C₄

yes

characteristic

C₂×C₄₇⋊C₈

C₂×C₄

yes

characteristic

#10625324

—

C₈×D₄₇

—

C₁₆×C₄₇

—

C₁₆×C₄₇

D₄

yes

characteristic

C₁₆×C₄₇

D₄

yes

characteristic

C₄₇⋊C₁₆

—

C₄₇⋊C₁₆

—

Q₈×C₄₇

C₂×C₈

yes

characteristic

D₄×C₄₇

—

D₄×C₄₇

C₂×C₈

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂²⋊D₄₇

—

C₂×Dic₄₇

—

D₁₈₈

C₂×C₈

yes

characteristic

C₄×D₄₇

—

Dic₉₄

C₂×C₈

yes

characteristic

C₈×C₄₇

—

C₈×C₄₇

C₂×D₄

yes

characteristic

C₈×C₄₇

#41

yes

characteristic

C₄₇⋊C₈

—

C₄₇⋊C₈

C₂×C₈

yes

characteristic

C₄₇⋊C₈

C₂×C₈

yes

characteristic

C₂²×C₄₇

—

C₂²×C₄₇

#131

yes

characteristic

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₄×D₄

yes

characteristic

C₄×C₄₇

C₂²×C₈

yes

characteristic

Dic₄₇

—

Dic₄₇

—

#2171

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

C₈×D₄

yes

characteristic

D₄₇

—

#504

—

#504

D₄₇

yes

characteristic, complemented

—

—

—

—

—

C₄₇

#2171

yes

characteristic, complemented, Sylow

#135

—

#131

—

#131

C₂×D₄₇

yes

characteristic

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

C₂×D₄₇

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₄₇

yes

—

—

—

C₄×C₈

—

#41

—

#41

C₄×D₄₇

yes

characteristic

Q₁₆

—

SD₁₆

—

SD₁₆

—

100

D₈

—

101

M₁₆

—

102

M₁₆

—

103

M₁₆

—

104

M₁₆

C₄×D₄₇

yes

characteristic

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

C₂²×D₄₇

yes

characteristic

109

C₄²

—

110

C₁₆

—

111

C₁₆

—

112

C₁₆

—

113

C₁₆

C₂²⋊D₄₇

yes

characteristic

114

C₁₆

C₂²⋊D₄₇

yes

characteristic

115

Q₈

—

116

Q₈

C₈×D₄₇

yes

characteristic

117

D₄

—

118

D₄

—

119

D₄

—

120

D₄

C₈×D₄₇

yes

characteristic

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

C₂×C₄×D₄₇

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

129

C₈

#10625350

yes

characteristic

130

C₈

C₂×C₂²⋊D₄₇

yes

characteristic

131

C₂²

—

132

C₂²

—

133

C₂²

#13094030

yes

characteristic

134

C₄

—

135

C₄

—

136

C₄

—

188

—

137

C₄

C₄×C₂²⋊D₄₇

yes

characteristic

138

C₄

C₂×C₈×D₄₇

yes

characteristic

139

C₂

—

140

C₂

—

141

C₂

—

188

—

142

C₂

C₈×C₂²⋊D₄₇

yes

characteristic

	subgroup	G/H
center	C₈	C₂×C₂²⋊D₄₇
commutator	C₄×C₄₇	C₂²×C₈
Frattini	C₂×C₈	C₂²×D₄₇
Fitting	C₄₇×#504	C₂
socle	C₂×C₄₇	C₈×D₄
hypercenter	#504	D₄₇
hypocenter	C₄₇	#2171
LCS	G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₈ ◁ C₂×C₁₆ ◁ #504

subgroup

G/H

center

C₈

C₂×C₂²⋊D₄₇

commutator

C₄×C₄₇

C₂²×C₈

Frattini

C₂×C₈

C₂²×D₄₇

Fitting

C₄₇×#504

C₂

socle

C₂×C₄₇

C₈×D₄

D₄₇

C₄₇

G ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₈ ◁ C₂×C₁₆ ◁ #504

N	H ≅ G/N	type
1. #15689539	C₂	semidirect
3. #15688943	C₂	semidirect
4. #15688908	C₂	semidirect
5. #15688907	C₂	semidirect
6. #15688898	C₂	semidirect
7. C₄₇×#504	C₂	semidirect
73. #504	D₄₇	semidirect
79. C₄₇	#2171	semidirect

H ≅ G/N

type

1. #15689539

C₂

semidirect

3. #15688943

C₂

semidirect

4. #15688908

C₂

semidirect

5. #15688907

C₂

semidirect

6. #15688898

C₂

semidirect

7. C₄₇×#504

C₂

semidirect

73. #504

D₄₇

semidirect

79. C₄₇

#2171

semidirect

Subgroups — #20245228

Conjugacy classes of subgroups

Complements