Group #20242864 — subgroups

Subgroups — #20242864

Order 6016.

198 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 55 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	#15689315	C₂	1	2	yes	maximal, complemented
2	#15689202	C₂	1	1	yes	maximal, characteristic, complemented
3	#15688874	C₂	1	1	yes	maximal, characteristic, complemented
4	#15688867	C₂	1	2	yes	maximal, complemented
5	C₄₇×#493	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#10625334	—	2	2	no	—
7	C₂×#10625324	—	2	1	no	—
8	C₄×D₁₈₈	C₂²	1	1	yes	characteristic
9	C₄⋊(C₄₇⋊C₈)	C₂²	1	2	yes	—
10	C₄⋊(C₄₇⋊C₈)	C₂²	1	1	yes	characteristic
11	#13093945	—	2	2	no	—
12	(C₂×D₄₇)⋊C₈	—	2	1	no	—
13	C₂×D₈×C₄₇	—	2	1	no	—
14	C₂×D₈×C₄₇	—	2	1	no	—
15	#127×C₄₇	C₂²	1	2	yes	—
16	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
17	C₂×D₄×C₄₇	—	2	2	no	—
18	C₂×D₄×C₄₇	—	2	2	no	—
19	C₂×D₄×C₄₇	—	2	2	no	—
20	C₂×D₁₈₈	D₄	1	1	yes	characteristic
21	C₂×C₄×D₄₇	—	2	1	no	—
22	D₈×C₄₇	—	4	2	no	—
23	D₈×C₄₇	—	4	2	no	—
24	C₂×C₈×C₄₇	D₄	1	1	yes	characteristic
25	C₂×C₈×C₄₇	D₄	1	1	yes	characteristic
26	C₄²×C₄₇	C₂³	1	1	yes	characteristic
27	#10625334	—	4	4	no	—
28	(C₂×D₄₇)⋊C₄	—	2	1	no	—
29	C₄⋊Dic₄₇	D₄	1	1	yes	characteristic
30	C₂×C₄₇⋊C₈	—	2	1	no	—
31	C₂×C₄₇⋊C₈	—	2	2	no	—
32	#10625324	—	4	2	no	—
33	C₂³×C₄₇	—	4	2	no	—
34	C₂²×D₄₇	—	2	1	no	—
35	D₄×C₄₇	—	4	2	no	—
36	D₄×C₄₇	—	4	2	no	—
37	D₄×C₄₇	—	4	2	no	—
38	D₄×C₄₇	—	4	2	no	—
39	D₄×C₄₇	—	8	2	no	—
40	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₄₇	#41	1	1	yes	characteristic
43	C₂×Dic₄₇	—	2	1	no	—
44	D₁₈₈	—	2	2	no	—
45	C₄×D₄₇	—	4	1	no	—
46	C₄×D₄₇	—	4	1	no	—
47	C₈×C₄₇	—	2	1	no	—
48	C₈×C₄₇	D₈	1	2	yes	complemented
49	C₄₇⋊C₈	—	4	1	no	—
50	C₄₇⋊C₈	—	4	2	no	—
51	C₂²×C₄₇	—	8	2	no	—
52	C₂²×C₄₇	—	8	2	no	—
53	C₂²×C₄₇	—	8	2	no	—
54	C₂²×C₄₇	#121	1	1	yes	characteristic
55	C₂×D₄₇	—	4	1	no	—
56	C₂×D₄₇	—	4	1	no	—
57	C₂×D₄₇	—	4	1	no	—
58	C₄×C₄₇	—	2	1	no	—
59	C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
60	C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
61	C₄×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
62	Dic₄₇	—	4	1	no	—
63	#1687	—	47	1	no	maximal, Sylow
64	C₂×C₄₇	—	16	2	no	—
65	C₂×C₄₇	#459	1	1	yes	characteristic
66	C₂×C₄₇	#469	1	1	yes	characteristic
67	C₂×C₄₇	#472	1	1	yes	characteristic
68	D₄₇	—	8	1	no	—
69	#493	D₄₇	1	1	yes	characteristic, complemented
70	#459	—	47	2	no	—
71	#436	—	47	1	no	—
72	#334	—	47	1	no	—
73	#331	—	47	2	no	—
74	C₄₇	#1687	1	1	yes	characteristic, complemented, Sylow
75	C₂×D₈	—	2	1	no	—
76	C₂×D₈	—	2	1	no	—
77	C₂×D₈	—	94	2	no	—
78	C₂×M₁₆	—	94	1	no	—
79	#127	C₂×D₄₇	1	2	yes	—
80	C₄×D₄	—	47	1	no	—
81	C₄⋊C₈	—	47	1	no	—
82	C₄⋊C₈	—	47	2	no	—
83	#102	—	94	2	no	—
84	C₂²⋊C₈	—	94	1	no	—
85	C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
86	C₂×D₄	—	2	2	no	—
87	C₂×D₄	—	2	2	no	—
88	C₂×D₄	—	2	2	no	—
89	C₂×D₄	—	47	1	no	—
90	C₂²×C₄	—	94	1	no	—
91	D₈	—	4	2	no	—
92	D₈	—	4	2	no	—
93	D₈	—	188	4	no	—
94	M₁₆	—	188	2	no	—
95	C₂×C₈	—	94	2	no	—
96	C₂×C₈	—	94	1	no	—
97	C₂×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
98	C₂×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
99	C₄⋊C₄	—	47	1	no	—
100	C₂²⋊C₄	—	94	1	no	—
101	C₄²	C₂²×D₄₇	1	1	yes	characteristic
102	C₂³	—	4	2	no	—
103	C₂³	—	94	1	no	—
104	D₄	—	4	2	no	—
105	D₄	—	4	2	no	—
106	D₄	—	4	2	no	—
107	D₄	—	4	2	no	—
108	D₄	—	8	2	no	—
109	D₄	—	94	2	no	—
110	C₂×C₄	—	94	1	no	—
111	C₂×C₄	—	188	1	no	—
112	C₂×C₄	—	188	1	no	—
113	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
114	C₂×C₄	#10625352	1	1	yes	characteristic
115	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
116	C₈	—	2	1	no	—
117	C₈	—	188	1	no	—
118	C₈	—	188	2	no	—
119	C₈	#10625334	1	2	yes	complemented
120	C₂²	—	8	2	no	—
121	C₂²	—	8	2	no	—
122	C₂²	—	8	2	no	—
123	C₂²	—	188	1	no	—
124	C₂²	—	188	1	no	—
125	C₂²	—	188	1	no	—
126	C₂²	#13093995	1	1	yes	characteristic
127	C₄	—	2	1	no	—
128	C₄	—	188	1	no	—
129	C₄	C₂×#10625334	1	1	yes	characteristic
130	C₄	#13094048	1	2	yes	—
131	C₄	#13094050	1	1	yes	characteristic
132	C₂	—	16	2	no	—
133	C₂	—	376	1	no	—
134	C₂	#15689315	1	1	yes	characteristic
135	C₂	#15689373	1	1	yes	characteristic
136	C₂	#15689390	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689315

C₂

yes

maximal, complemented

#15689202

C₂

yes

maximal, characteristic, complemented

#15688874

C₂

yes

maximal, characteristic, complemented

#15688867

C₂

yes

maximal, complemented

C₄₇×#493

C₂

yes

maximal, characteristic, complemented

C₂×#10625334

—

C₂×#10625324

—

C₄×D₁₈₈

C₂²

yes

characteristic

C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

characteristic

#13093945

—

(C₂×D₄₇)⋊C₈

—

C₂×D₈×C₄₇

—

C₂×D₈×C₄₇

—

#127×C₄₇

C₂²

yes

—

C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₁₈₈

D₄

yes

characteristic

C₂×C₄×D₄₇

—

D₈×C₄₇

—

D₈×C₄₇

—

C₂×C₈×C₄₇

D₄

yes

characteristic

C₂×C₈×C₄₇

D₄

yes

characteristic

C₄²×C₄₇

C₂³

yes

characteristic

#10625334

—

(C₂×D₄₇)⋊C₄

—

C₄⋊Dic₄₇

D₄

yes

characteristic

C₂×C₄₇⋊C₈

—

C₂×C₄₇⋊C₈

—

#10625324

—

C₂³×C₄₇

—

C₂²×D₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

#41

yes

characteristic

C₂×Dic₄₇

—

D₁₈₈

—

C₄×D₄₇

—

C₄×D₄₇

—

C₈×C₄₇

—

C₈×C₄₇

D₈

yes

complemented

C₄₇⋊C₈

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#121

yes

characteristic

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂×D₈

yes

characteristic

C₄×C₄₇

AGL(1,ℤ/8)

yes

—

C₄×C₄₇

AGL(1,ℤ/8)

yes

characteristic

Dic₄₇

—

#1687

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

D₄₇

—

#493

D₄₇

yes

characteristic, complemented

—

—

—

—

C₄₇

#1687

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

C₂×M₁₆

—

#127

C₂×D₄₇

yes

—

C₄×D₄

—

C₄⋊C₈

—

C₄⋊C₈

—

#102

—

C₂²⋊C₈

—

C₄×C₈

C₂×D₄₇

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

D₈

—

D₈

—

D₈

—

188

—

M₁₆

—

188

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₄₇

yes

characteristic

C₂×C₈

C₂²⋊D₄₇

yes

characteristic

C₄⋊C₄

—

100

C₂²⋊C₄

—

101

C₄²

C₂²×D₄₇

yes

characteristic

102

C₂³

—

103

C₂³

—

104

D₄

—

105

D₄

—

106

D₄

—

107

D₄

—

108

D₄

—

109

D₄

—

110

C₂×C₄

—

111

C₂×C₄

—

188

—

112

C₂×C₄

—

188

—

113

C₂×C₄

D₄×D₄₇

yes

characteristic

114

C₂×C₄

#10625352

yes

characteristic

115

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

116

C₈

—

117

C₈

—

188

—

118

C₈

—

188

—

119

C₈

#10625334

yes

complemented

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

188

—

124

C₂²

—

188

—

125

C₂²

—

188

—

126

C₂²

#13093995

yes

characteristic

127

C₄

—

128

C₄

—

188

—

129

C₄

C₂×#10625334

yes

characteristic

130

C₄

#13094048

yes

—

131

C₄

#13094050

yes

characteristic

132

C₂

—

133

C₂

—

376

—

134

C₂

#15689315

yes

characteristic

135

C₂

#15689373

yes

characteristic

136

C₂

#15689390

yes

characteristic

	subgroup	G/H
center	C₂²	#13093995
commutator	C₄²×C₄₇	C₂³
Frattini	C₄²	C₂²×D₄₇
Fitting	C₄₇×#493	C₂
socle	C₂²×C₄₇	#121
hypercenter	#493	D₄₇
hypocenter	C₄₇	#1687
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #493

subgroup

G/H

center

C₂²

#13093995

commutator

C₄²×C₄₇

C₂³

Frattini

C₄²

C₂²×D₄₇

Fitting

C₄₇×#493

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #493

N	H ≅ G/N	type
1. #15689315	C₂	semidirect
2. #15689202	C₂	semidirect
3. #15688874	C₂	semidirect
4. #15688867	C₂	semidirect
5. C₄₇×#493	C₂	semidirect
48. C₈×C₄₇	D₈	semidirect
69. #493	D₄₇	semidirect
74. C₄₇	#1687	semidirect
119. C₈	#10625334	semidirect

H ≅ G/N

type

1. #15689315

C₂

semidirect

2. #15689202

C₂

semidirect

3. #15688874

C₂

semidirect

4. #15688867

C₂

semidirect

5. C₄₇×#493

C₂

semidirect

48. C₈×C₄₇

D₈

semidirect

69. #493

D₄₇

semidirect

74. C₄₇

#1687

semidirect

119. C₈

#10625334

semidirect

Subgroups — #20242864

Conjugacy classes of subgroups

Complements