Group #20245266 — subgroups

Subgroups — #20245266

Order 6016.

146 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 146 conjugacy classes of subgroups — too many to draw. Only its 53 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	#15689553	C₂	1	1	yes	maximal, characteristic
2	#15689240	C₂	1	1	yes	maximal, characteristic, complemented
3	#15689239	C₂	1	1	yes	maximal, characteristic, complemented
4	#15688952	C₂	1	1	yes	maximal, characteristic, complemented
5	#15688931	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₆×Dic₄₇	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#359	C₂	1	1	yes	maximal, characteristic, complemented
8	#13094046	C₂²	1	1	yes	characteristic
9	#13094032	—	2	1	no	—
10	#13094031	—	2	1	no	—
11	#13093970	C₄	1	1	yes	characteristic, complemented
12	#13093968	C₄	1	1	yes	characteristic, complemented
13	#13093967	C₄	1	1	yes	characteristic, complemented
14	#13093965	C₄	1	1	yes	characteristic, complemented
15	C₂×C₄₇⋊C₁₆	C₂²	1	1	yes	characteristic
16	#13093960	C₂²	1	1	yes	characteristic
17	#13093951	—	2	1	no	—
18	#13093950	—	2	1	no	—
19	C₈×Dic₄₇	C₂²	1	1	yes	characteristic
20	#135×C₄₇	C₂²	1	1	yes	characteristic
21	C₂×C₁₆×C₄₇	C₂²	1	1	yes	characteristic
22	#108×C₄₇	C₂²	1	1	yes	characteristic
23	#41×C₄₇	—	2	1	no	—
24	#10625350	—	2	1	no	—
25	Q₁₆×C₄₇	C₂×C₄	1	1	yes	characteristic
26	SD₁₆×C₄₇	—	2	1	no	—
27	D₈×C₄₇	C₂×C₄	1	1	yes	characteristic
28	M₁₆×C₄₇	—	2	1	no	—
29	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
30	C₄×Dic₄₇	D₄	1	1	yes	characteristic
31	#10625329	—	2	1	no	—
32	#10625329	—	4	1	no	—
33	C₂×C₄₇⋊C₈	—	4	1	no	—
34	C₂×C₄₇⋊C₈	D₄	1	1	yes	characteristic
35	Dic₁₈₈	C₂×C₄	1	1	yes	characteristic
36	D₃₇₆	C₂×C₄	1	1	yes	characteristic
37	#10625325	—	2	1	no	—
38	#10625324	—	4	1	no	—
39	C₈×D₄₇	—	4	1	no	—
40	C₁₆×C₄₇	—	2	1	no	—
41	C₄₇⋊C₁₆	C₂×C₄	1	1	yes	characteristic
42	C₄₇⋊C₁₆	C₂×C₄	1	1	yes	characteristic
43	Q₈×C₄₇	—	2	1	no	—
44	D₄×C₄₇	—	2	1	no	—
45	D₄×C₄₇	—	4	1	no	—
46	C₂×C₄×C₄₇	—	4	1	no	—
47	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
48	C₂²⋊D₄₇	—	4	1	no	—
49	C₂×Dic₄₇	—	2	1	no	—
50	D₁₈₈	—	2	1	no	—
51	C₄×D₄₇	—	4	1	no	—
52	Dic₉₄	—	2	1	no	—
53	C₈×C₄₇	—	4	1	no	—
54	C₈×C₄₇	C₂²×C₄	1	1	yes	characteristic
55	C₈×C₄₇	#41	1	1	yes	characteristic
56	C₄₇⋊C₈	—	4	1	no	—
57	C₄₇⋊C₈	D₈	1	1	yes	characteristic
58	C₄₇⋊C₈	D₈	1	1	yes	characteristic
59	C₂²×C₄₇	—	4	1	no	—
60	C₂²×C₄₇	#135	1	1	yes	characteristic
61	C₂×D₄₇	—	4	1	no	—
62	C₄×C₄₇	—	4	1	no	—
63	C₄×C₄₇	C₄×D₄	1	1	yes	characteristic
64	C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
65	Dic₄₇	—	2	2	no	—
66	Dic₄₇	—	4	1	no	—
67	#2178	—	47	1	no	maximal, Sylow
68	C₂×C₄₇	—	2	1	no	—
69	C₂×C₄₇	—	8	1	no	—
70	C₂×C₄₇	C₄×D₈	1	1	yes	characteristic
71	D₄₇	—	8	1	no	—
72	#508	—	47	1	no	—
73	#443	—	47	1	no	—
74	#443	—	47	1	no	—
75	#368	—	47	1	no	—
76	#359	—	47	1	no	—
77	#359	D₄₇	1	1	yes	characteristic, complemented
78	C₄×C₁₆	—	47	1	no	—
79	C₄₇	#2178	1	1	yes	characteristic, complemented, Sylow
80	#135	—	47	1	no	—
81	#135	C₂×D₄₇	1	1	yes	characteristic
82	#131	—	94	1	no	—
83	#131	—	94	1	no	—
84	Q₃₂	—	47	1	no	—
85	SD₃₂	—	47	1	no	—
86	SD₃₂	—	47	1	no	—
87	D₁₆	—	47	1	no	—
88	C₂×C₁₆	—	47	1	no	—
89	C₂×C₁₆	C₂×D₄₇	1	1	yes	characteristic
90	#108	—	47	1	no	—
91	#108	C₂×D₄₇	1	1	yes	characteristic
92	#104	—	94	1	no	—
93	#104	—	94	1	no	—
94	C₄×C₈	—	47	1	no	—
95	#41	—	2	1	no	—
96	#41	—	94	1	no	—
97	Q₁₆	—	47	1	no	—
98	Q₁₆	C₄×D₄₇	1	1	yes	characteristic
99	SD₁₆	—	2	1	no	—
100	SD₁₆	—	94	1	no	—
101	D₈	—	47	1	no	—
102	D₈	C₄×D₄₇	1	1	yes	characteristic
103	M₁₆	—	2	1	no	—
104	M₁₆	—	94	1	no	—
105	M₁₆	—	188	1	no	—
106	M₁₆	—	188	1	no	—
107	C₂×C₈	—	47	1	no	—
108	C₂×C₈	—	188	1	no	—
109	C₂×C₈	—	188	1	no	—
110	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
111	C₄²	—	47	1	no	—
112	C₁₆	—	2	1	no	—
113	C₁₆	—	47	1	no	—
114	C₁₆	—	47	1	no	—
115	Q₈	—	2	1	no	—
116	Q₈	—	94	1	no	—
117	D₄	—	2	1	no	—
118	D₄	—	4	1	no	—
119	D₄	—	94	1	no	—
120	D₄	—	188	1	no	—
121	C₂×C₄	—	4	1	no	—
122	C₂×C₄	—	94	1	no	—
123	C₂×C₄	—	188	1	no	—
124	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
125	C₈	—	4	1	no	—
126	C₈	—	47	1	no	—
127	C₈	—	47	1	no	—
128	C₈	—	188	1	no	—
129	C₈	C₂×C₄×D₄₇	1	1	yes	characteristic
130	C₈	#10625352	1	1	yes	characteristic
131	C₂²	—	4	1	no	—
132	C₂²	—	188	1	no	—
133	C₂²	#13094045	1	1	yes	characteristic
134	C₄	—	4	1	no	—
135	C₄	—	94	2	no	—
136	C₄	—	188	1	no	—
137	C₄	(C₂²⋊D₄₇)⋊C₄	1	1	yes	characteristic
138	C₄	D₈×D₄₇	1	1	yes	characteristic
139	C₂	—	2	1	no	—
140	C₂	—	8	1	no	—
141	C₂	—	376	1	no	—
142	C₂	#15689203	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689553

C₂

yes

maximal, characteristic

#15689240

C₂

yes

maximal, characteristic, complemented

#15689239

C₂

yes

maximal, characteristic, complemented

#15688952

C₂

yes

maximal, characteristic, complemented

#15688931

C₂

yes

maximal, characteristic, complemented

C₁₆×Dic₄₇

C₂

yes

maximal, characteristic, complemented

C₄₇×#359

C₂

yes

maximal, characteristic, complemented

#13094046

C₂²

yes

—

—

C₄

yes

characteristic, complemented

#13093968

C₄

yes

characteristic, complemented

#13093967

C₄

yes

characteristic, complemented

#13093965

C₄

yes

characteristic, complemented

C₂×C₄₇⋊C₁₆

C₂²

yes

characteristic

#13093960

C₂²

yes

characteristic

#13093951

—

#13093950

—

C₈×Dic₄₇

C₂²

yes

characteristic

#135×C₄₇

C₂²

yes

characteristic

C₂×C₁₆×C₄₇

C₂²

yes

characteristic

#108×C₄₇

C₂²

yes

characteristic

#41×C₄₇

—

#10625350

—

Q₁₆×C₄₇

C₂×C₄

yes

characteristic

SD₁₆×C₄₇

—

D₈×C₄₇

C₂×C₄

yes

characteristic

M₁₆×C₄₇

—

C₂×C₈×C₄₇

C₂³

yes

characteristic

C₄×Dic₄₇

D₄

yes

characteristic

#10625329

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

D₄

yes

characteristic

Dic₁₈₈

C₂×C₄

yes

characteristic

D₃₇₆

C₂×C₄

yes

characteristic

#10625325

—

#10625324

—

C₈×D₄₇

—

C₁₆×C₄₇

—

C₄₇⋊C₁₆

C₂×C₄

yes

characteristic

C₄₇⋊C₁₆

C₂×C₄

yes

characteristic

Q₈×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²⋊D₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

C₄×D₄₇

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

C₂²×C₄

yes

characteristic

C₈×C₄₇

#41

yes

characteristic

C₄₇⋊C₈

—

C₄₇⋊C₈

D₈

yes

characteristic

C₄₇⋊C₈

D₈

yes

characteristic

C₂²×C₄₇

—

C₂²×C₄₇

#135

yes

characteristic

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₄×D₄

yes

characteristic

C₄×C₄₇

C₂×D₈

yes

characteristic

Dic₄₇

—

Dic₄₇

—

#2178

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

C₄×D₈

yes

characteristic

D₄₇

—

—

—

—

—

—

D₄₇

yes

characteristic, complemented

C₄×C₁₆

—

C₄₇

#2178

yes

characteristic, complemented, Sylow

#135

—

#135

C₂×D₄₇

yes

characteristic

#131

—

#131

—

Q₃₂

—

SD₃₂

—

SD₃₂

—

D₁₆

—

C₂×C₁₆

—

C₂×C₁₆

C₂×D₄₇

yes

characteristic

#108

—

#108

C₂×D₄₇

yes

characteristic

#104

—

#104

—

C₄×C₈

—

#41

—

#41

—

Q₁₆

—

Q₁₆

C₄×D₄₇

yes

characteristic

SD₁₆

—

100

SD₁₆

—

101

D₈

—

102

D₈

C₄×D₄₇

yes

characteristic

103

M₁₆

—

104

M₁₆

—

105

M₁₆

—

188

—

106

M₁₆

—

188

—

107

C₂×C₈

—

108

C₂×C₈

—

188

—

109

C₂×C₈

—

188

—

110

C₂×C₈

C₂²×D₄₇

yes

characteristic

111

C₄²

—

112

C₁₆

—

113

C₁₆

—

114

C₁₆

—

115

Q₈

—

116

Q₈

—

117

D₄

—

118

D₄

—

119

D₄

—

120

D₄

—

188

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

188

—

124

C₂×C₄

D₄×D₄₇

yes

characteristic

125

C₈

—

126

C₈

—

127

C₈

—

128

C₈

—

188

—

129

C₈

C₂×C₄×D₄₇

yes

characteristic

130

C₈

#10625352

yes

characteristic

131

C₂²

—

132

C₂²

—

188

—

133

C₂²

#13094045

yes

characteristic

134

C₄

—

135

C₄

—

136

C₄

—

188

—

137

C₄

(C₂²⋊D₄₇)⋊C₄

yes

characteristic

138

C₄

D₈×D₄₇

yes

characteristic

139

C₂

—

140

C₂

—

141

C₂

—

376

—

142

C₂

#15689203

yes

characteristic

	subgroup	G/H
center	C₄	D₈×D₄₇
commutator	C₈×C₄₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₄₇
Fitting	C₄₇×#359	C₂
socle	C₂×C₄₇	C₄×D₈
hypercenter	#359	D₄₇
hypocenter	C₄₇	#2178
LCS	G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #359

subgroup

G/H

center

C₄

D₈×D₄₇

commutator

C₈×C₄₇

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₄₇

Fitting

C₄₇×#359

C₂

socle

C₂×C₄₇

C₄×D₈

D₄₇

C₄₇

G ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #359

N	H ≅ G/N	type
2. #15689240	C₂	semidirect
3. #15689239	C₂	semidirect
4. #15688952	C₂	semidirect
5. #15688931	C₂	semidirect
6. C₁₆×Dic₄₇	C₂	semidirect
7. C₄₇×#359	C₂	semidirect
11. #13093970	C₄	semidirect
12. #13093968	C₄	semidirect
13. #13093967	C₄	semidirect
14. #13093965	C₄	semidirect
77. #359	D₄₇	semidirect
79. C₄₇	#2178	semidirect

H ≅ G/N

type

2. #15689240

C₂

semidirect

3. #15689239

C₂

semidirect

4. #15688952

C₂

semidirect

5. #15688931

C₂

semidirect

6. C₁₆×Dic₄₇

C₂

semidirect

7. C₄₇×#359

C₂

semidirect

11. #13093970

C₄

semidirect

12. #13093968

C₄

semidirect

13. #13093967

C₄

semidirect

14. #13093965

C₄

semidirect

77. #359

D₄₇

semidirect

79. C₄₇

#2178

semidirect

Subgroups — #20245266

Conjugacy classes of subgroups

Complements