Group #20245601 — subgroups

Subgroups — #20245601

Order 6016.

134 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 134 conjugacy classes of subgroups — too many to draw. Only its 41 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	#15689373	C₂	1	1	yes	maximal, characteristic
2	#15689170	C₂	1	1	yes	maximal, characteristic, complemented
3	#15688947	C₂	1	1	yes	maximal, characteristic, complemented
4	#15688946	C₂	1	1	yes	maximal, characteristic, complemented
5	#15688936	C₂	1	1	yes	maximal, characteristic, complemented
6	#15688904	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#361	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#10625324	C₂²	1	1	yes	characteristic
9	M₁₆×D₄₇	—	2	1	no	—
10	#13093995	—	2	1	no	—
11	#13093964	C₂²	1	1	yes	characteristic
12	#13093960	C₂²	1	1	yes	characteristic
13	#13093959	—	2	1	no	—
14	#13093956	C₂²	1	1	yes	characteristic
15	#13093945	—	2	1	no	—
16	C₂×D₈×C₄₇	C₂²	1	1	yes	characteristic
17	M₃₂×C₄₇	C₂²	1	1	yes	characteristic
18	#108×C₄₇	C₂²	1	1	yes	characteristic
19	C₂×D₄×C₄₇	—	2	1	no	—
20	C₂×C₂²⋊D₄₇	—	4	1	no	—
21	C₂×C₄×D₄₇	D₄	1	1	yes	characteristic
22	D₈×C₄₇	—	2	1	no	—
23	D₈×C₄₇	—	2	1	no	—
24	M₁₆×C₄₇	—	2	1	no	—
25	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
26	C₂²⋊Dic₄₇	—	4	1	no	—
27	C₄⋊Dic₄₇	—	2	1	no	—
28	#10625329	—	2	1	no	—
29	C₂×C₄₇⋊C₈	D₄	1	1	yes	characteristic
30	#10625324	—	2	1	no	—
31	#10625324	—	2	1	no	—
32	#10625324	—	4	1	no	—
33	C₈×D₄₇	—	4	1	no	—
34	C₁₆×C₄₇	—	2	1	no	—
35	C₄₇⋊C₁₆	—	2	1	no	—
36	C₂³×C₄₇	—	4	1	no	—
37	C₂²×D₄₇	SD₁₆	1	1	yes	characteristic
38	D₄×C₄₇	—	4	1	no	—
39	D₄×C₄₇	—	4	1	no	—
40	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
41	C₂²⋊D₄₇	—	8	2	no	—
42	C₂×Dic₄₇	—	4	1	no	—
43	C₂×Dic₄₇	SD₁₆	1	1	yes	characteristic
44	C₄×D₄₇	—	2	1	no	—
45	C₄×D₄₇	—	2	1	no	—
46	C₈×C₄₇	—	4	1	no	—
47	C₈×C₄₇	#41	1	1	yes	characteristic
48	C₈×C₄₇	#41	1	1	yes	characteristic
49	C₄₇⋊C₈	—	2	1	no	—
50	C₄₇⋊C₈	—	4	1	no	—
51	C₂²×C₄₇	—	8	1	no	—
52	C₂²×C₄₇	—	8	1	no	—
53	C₂²×C₄₇	—	8	1	no	—
54	C₂²×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
55	C₂×D₄₇	—	2	1	no	—
56	C₂×D₄₇	—	4	1	no	—
57	C₄×C₄₇	#123	1	1	yes	characteristic
58	C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
59	Dic₄₇	—	2	1	no	—
60	Dic₄₇	—	8	1	no	—
61	#2238	—	47	1	no	maximal, Sylow
62	C₂×C₄₇	—	2	1	no	—
63	C₂×C₄₇	—	16	1	no	—
64	C₂×C₄₇	#481	1	1	yes	characteristic
65	D₄₇	—	4	1	no	—
66	#469	—	47	1	no	—
67	#430	—	47	1	no	—
68	#365	—	47	1	no	—
69	#365	—	47	1	no	—
70	#361	—	47	1	no	—
71	#361	D₄₇	1	1	yes	characteristic, complemented
72	#349	—	47	1	no	—
73	C₄₇	#2238	1	1	yes	characteristic, complemented, Sylow
74	C₂×D₈	C₂×D₄₇	1	1	yes	characteristic
75	C₂×M₁₆	—	47	1	no	—
76	C₂×M₁₆	—	94	1	no	—
77	#121	—	94	1	no	—
78	M₃₂	—	47	1	no	—
79	M₃₂	C₂×D₄₇	1	1	yes	characteristic
80	#108	—	47	1	no	—
81	#108	—	94	1	no	—
82	#108	C₂×D₄₇	1	1	yes	characteristic
83	#106	—	47	1	no	—
84	#102	—	94	1	no	—
85	C₂×D₄	—	2	1	no	—
86	C₂×D₄	—	188	1	no	—
87	C₂²×C₄	—	47	1	no	—
88	D₈	—	2	1	no	—
89	D₈	—	2	1	no	—
90	M₁₆	—	2	1	no	—
91	M₁₆	—	94	1	no	—
92	M₁₆	—	94	1	no	—
93	M₁₆	—	94	1	no	—
94	M₁₆	—	188	1	no	—
95	C₂×C₈	—	47	1	no	—
96	C₂×C₈	—	188	1	no	—
97	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
98	C₄⋊C₄	—	94	1	no	—
99	C₂²⋊C₄	—	188	1	no	—
100	C₁₆	—	2	1	no	—
101	C₁₆	—	94	1	no	—
102	C₂³	—	4	1	no	—
103	C₂³	—	47	1	no	—
104	D₄	—	4	1	no	—
105	D₄	—	4	1	no	—
106	D₄	—	376	2	no	—
107	C₂×C₄	—	47	1	no	—
108	C₂×C₄	—	94	1	no	—
109	C₂×C₄	—	94	1	no	—
110	C₂×C₄	—	188	1	no	—
111	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
112	C₈	—	4	1	no	—
113	C₈	—	94	1	no	—
114	C₈	—	188	1	no	—
115	C₈	#10625350	1	1	yes	characteristic
116	C₈	#10625352	1	1	yes	characteristic
117	C₂²	—	8	1	no	—
118	C₂²	—	8	1	no	—
119	C₂²	—	8	1	no	—
120	C₂²	—	94	1	no	—
121	C₂²	—	188	1	no	—
122	C₂²	SD₁₆×D₄₇	1	1	yes	characteristic
123	C₄	—	94	1	no	—
124	C₄	—	376	1	no	—
125	C₄	#13094006	1	1	yes	characteristic
126	C₄	#13094048	1	1	yes	characteristic
127	C₂	—	2	1	no	—
128	C₂	—	16	1	no	—
129	C₂	—	188	1	no	—
130	C₂	#15689440	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689373

C₂

yes

maximal, characteristic

#15689170

C₂

yes

maximal, characteristic, complemented

#15688947

C₂

yes

maximal, characteristic, complemented

#15688946

C₂

yes

maximal, characteristic, complemented

#15688936

C₂

yes

maximal, characteristic, complemented

#15688904

C₂

yes

maximal, characteristic, complemented

C₄₇×#361

C₂

yes

maximal, characteristic, complemented

C₂×#10625324

C₂²

yes

characteristic

M₁₆×D₄₇

—

#13093995

—

#13093964

C₂²

yes

characteristic

#13093960

C₂²

yes

characteristic

#13093959

—

#13093956

C₂²

yes

characteristic

#13093945

—

C₂×D₈×C₄₇

C₂²

yes

characteristic

M₃₂×C₄₇

C₂²

yes

characteristic

#108×C₄₇

C₂²

yes

characteristic

C₂×D₄×C₄₇

—

C₂×C₂²⋊D₄₇

—

C₂×C₄×D₄₇

D₄

yes

characteristic

D₈×C₄₇

—

D₈×C₄₇

—

M₁₆×C₄₇

—

C₂×C₈×C₄₇

C₂³

yes

characteristic

C₂²⋊Dic₄₇

—

C₄⋊Dic₄₇

—

#10625329

—

C₂×C₄₇⋊C₈

D₄

yes

—

—

—

C₈×D₄₇

—

C₁₆×C₄₇

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

—

C₂²×D₄₇

SD₁₆

yes

characteristic

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²⋊D₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

SD₁₆

yes

characteristic

C₄×D₄₇

—

C₄×D₄₇

—

C₈×C₄₇

—

C₈×C₄₇

yes

C₈×C₄₇

yes

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×SD₁₆

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

Dic₄₇

—

#2238

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

#481

yes

characteristic

D₄₇

—

—

—

—

—

—

D₄₇

yes

characteristic, complemented

#349

—

C₄₇

#2238

yes

characteristic, complemented, Sylow

C₂×D₈

C₂×D₄₇

yes

characteristic

C₂×M₁₆

—

C₂×M₁₆

—

#121

—

M₃₂

—

M₃₂

C₂×D₄₇

yes

—

—

C₂×D₄₇

yes

characteristic

#106

—

#102

—

C₂×D₄

—

C₂×D₄

—

188

—

C₂²×C₄

—

D₈

—

D₈

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

188

—

C₂×C₈

—

C₂×C₈

—

188

—

C₂×C₈

C₂²×D₄₇

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

188

—

100

C₁₆

—

101

C₁₆

—

102

C₂³

—

103

C₂³

—

104

D₄

—

105

D₄

—

106

D₄

—

376

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

188

—

111

C₂×C₄

D₄×D₄₇

yes

characteristic

112

C₈

—

113

C₈

—

114

C₈

—

188

—

115

C₈

#10625350

yes

characteristic

116

C₈

#10625352

yes

characteristic

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

—

188

—

122

C₂²

SD₁₆×D₄₇

yes

characteristic

123

C₄

—

124

C₄

—

376

—

125

C₄

#13094006

yes

characteristic

126

C₄

#13094048

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

188

—

130

C₂

#15689440

yes

characteristic

	subgroup	G/H
center	C₂	#15689440
commutator	C₂×C₈×C₄₇	C₂³
Frattini	C₂×C₈	C₂²×D₄₇
Fitting	C₄₇×#361	C₂
socle	C₂×C₄₇	#481
hypercenter	#361	D₄₇
hypocenter	C₄₇	#2238
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #361

subgroup

G/H

center

C₂

#15689440

commutator

C₂×C₈×C₄₇

C₂³

Frattini

C₂×C₈

C₂²×D₄₇

Fitting

C₄₇×#361

C₂

C₂×C₄₇

D₄₇

C₄₇

G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₈ ◁ #361

N	H ≅ G/N	type
2. #15689170	C₂	semidirect
3. #15688947	C₂	semidirect
4. #15688946	C₂	semidirect
5. #15688936	C₂	semidirect
6. #15688904	C₂	semidirect
7. C₄₇×#361	C₂	semidirect
71. #361	D₄₇	semidirect
73. C₄₇	#2238	semidirect

H ≅ G/N

type

2. #15689170

C₂

semidirect

3. #15688947

C₂

semidirect

4. #15688946

C₂

semidirect

5. #15688936

C₂

semidirect

6. #15688904

C₂

semidirect

7. C₄₇×#361

C₂

semidirect

71. #361

D₄₇

semidirect

73. C₄₇

#2238

semidirect

Subgroups — #20245601

Conjugacy classes of subgroups

Complements