Group #20243008 — subgroups

Subgroups — #20243008

Order 6016.

198 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 61 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	#15689526	C₂	1	1	yes	maximal, characteristic, complemented
2	#15689519	C₂	1	1	yes	maximal, characteristic, complemented
3	#15689495	C₂	1	1	yes	maximal, characteristic
4	#15689474	C₂	1	1	yes	maximal, characteristic, complemented
5	#15689469	C₂	1	1	yes	maximal, characteristic
6	#15688845	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄₇×#492	C₂	1	1	yes	maximal, characteristic
8	C₂×#10625335	—	2	2	no	—
9	#13094024	C₂²	1	1	yes	characteristic
10	#13094024	C₂²	1	1	yes	characteristic
11	#13093957	—	2	2	no	—
12	#13093955	—	2	2	no	—
13	#13093949	—	2	2	no	—
14	#13093945	—	2	2	no	—
15	C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×C₄₇⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×SD₁₆×C₄₇	—	2	2	no	—
18	C₄⋊Q₈×C₄₇	C₂²	1	1	yes	characteristic
19	#127×C₄₇	C₂²	1	1	yes	characteristic
20	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
21	C₂×Q₈×C₄₇	—	2	1	no	—
22	C₂×D₄×C₄₇	—	2	1	no	—
23	C₂×D₄×C₄₇	—	2	1	no	—
24	C₂×D₄×C₄₇	—	2	1	no	—
25	C₂×Dic₉₄	—	2	1	no	—
26	C₂×Dic₉₄	—	2	1	no	—
27	SD₁₆×C₄₇	—	4	4	no	—
28	C₂×C₈×C₄₇	D₄	1	2	yes	—
29	C₄⋊C₄×C₄₇	—	2	1	no	—
30	C₄⋊C₄×C₄₇	—	2	1	no	—
31	C₄²×C₄₇	C₂³	1	1	yes	characteristic
32	#10625335	—	4	4	no	—
33	C₄⋊Dic₄₇	—	2	1	no	—
34	C₄⋊Dic₄₇	—	2	1	no	—
35	C₄⋊Dic₄₇	—	2	1	no	—
36	C₄⋊Dic₄₇	—	2	1	no	—
37	C₂×C₄₇⋊C₈	D₄	1	2	yes	—
38	C₂×C₄₇⋊C₈	D₄	1	2	yes	—
39	C₂³×C₄₇	—	4	1	no	—
40	Q₈×C₄₇	—	4	2	no	—
41	D₄×C₄₇	—	4	2	no	—
42	D₄×C₄₇	—	4	2	no	—
43	D₄×C₄₇	—	4	2	no	—
44	C₂×C₄×C₄₇	—	4	1	no	—
45	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
48	C₂×Dic₄₇	—	4	1	no	—
49	C₂×Dic₄₇	—	4	1	no	—
50	Dic₉₄	—	4	2	no	—
51	Dic₉₄	—	4	2	no	—
52	C₈×C₄₇	—	2	2	no	—
53	C₄₇⋊C₈	—	2	2	no	—
54	C₄₇⋊C₈	SD₁₆	1	4	yes	complemented
55	C₂²×C₄₇	—	8	1	no	—
56	C₂²×C₄₇	—	8	1	no	—
57	C₂²×C₄₇	—	8	1	no	—
58	C₂²×C₄₇	#127	1	1	yes	characteristic
59	C₄×C₄₇	—	8	1	no	—
60	C₄×C₄₇	C₂×SD₁₆	1	2	yes	—
61	C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
62	C₄×C₄₇	#137	1	2	yes	—
63	Dic₄₇	—	8	1	no	—
64	Dic₄₇	—	8	1	no	—
65	#1714	—	47	1	no	maximal, Sylow
66	C₂×C₄₇	—	16	1	no	—
67	C₂×C₄₇	#492	1	1	yes	characteristic
68	C₂×C₄₇	#496	1	1	yes	characteristic
69	C₂×C₄₇	#497	1	1	yes	characteristic
70	#500	—	47	1	no	—
71	#498	—	47	1	no	—
72	#492	—	47	1	no	—
73	#492	D₄₇	1	1	yes	characteristic
74	#487	—	47	1	no	—
75	#486	—	47	1	no	—
76	#322	—	47	1	no	—
77	C₄₇	#1714	1	1	yes	characteristic, complemented, Sylow
78	C₂×SD₁₆	—	2	2	no	—
79	C₂×SD₁₆	—	94	2	no	—
80	C₄⋊Q₈	—	47	1	no	—
81	C₄⋊Q₈	—	47	1	no	—
82	C₄⋊Q₈	C₂×D₄₇	1	1	yes	characteristic
83	#127	C₂×D₄₇	1	1	yes	characteristic
84	#107	—	94	2	no	—
85	#106	—	94	2	no	—
86	#103	—	94	2	no	—
87	#102	—	94	2	no	—
88	C₄×C₈	—	47	1	no	—
89	C₄×C₈	—	47	1	no	—
90	C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
91	C₂×Q₈	—	2	1	no	—
92	C₂×Q₈	—	94	1	no	—
93	C₂×Q₈	—	94	1	no	—
94	C₂×D₄	—	2	1	no	—
95	C₂×D₄	—	2	1	no	—
96	C₂×D₄	—	2	1	no	—
97	SD₁₆	—	4	4	no	—
98	SD₁₆	—	188	4	no	—
99	C₂×C₈	—	47	2	no	—
100	C₂×C₈	—	47	2	no	—
101	C₂×C₈	C₂²⋊D₄₇	1	2	yes	—
102	C₄⋊C₄	—	2	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	94	1	no	—
105	C₄⋊C₄	—	94	1	no	—
106	C₄⋊C₄	—	94	1	no	—
107	C₄⋊C₄	—	94	1	no	—
108	C₄²	C₂²×D₄₇	1	1	yes	characteristic
109	C₂³	—	4	1	no	—
110	Q₈	—	4	2	no	—
111	Q₈	—	188	2	no	—
112	Q₈	—	188	2	no	—
113	D₄	—	4	2	no	—
114	D₄	—	4	2	no	—
115	D₄	—	4	2	no	—
116	C₂×C₄	—	4	1	no	—
117	C₂×C₄	—	188	1	no	—
118	C₂×C₄	—	188	1	no	—
119	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
120	C₂×C₄	D₄×D₄₇	1	1	yes	characteristic
121	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
122	C₈	—	2	2	no	—
123	C₈	—	47	4	no	—
124	C₈	—	94	2	no	—
125	C₂²	—	8	1	no	—
126	C₂²	—	8	1	no	—
127	C₂²	—	8	1	no	—
128	C₂²	#13094020	1	1	yes	characteristic
129	C₄	—	8	1	no	—
130	C₄	—	376	1	no	—
131	C₄	—	376	1	no	—
132	C₄	SD₁₆×D₄₇	1	2	yes	—
133	C₄	#13094050	1	2	yes	—
134	C₄	#13094053	1	2	yes	—
135	C₂	—	16	1	no	—
136	C₂	#15689494	1	1	yes	characteristic
137	C₂	#15689510	1	1	yes	characteristic
138	C₂	#15689514	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689526

C₂

yes

maximal, characteristic, complemented

#15689519

C₂

yes

maximal, characteristic, complemented

#15689495

C₂

yes

maximal, characteristic

#15689474

C₂

yes

maximal, characteristic, complemented

#15689469

C₂

yes

maximal, characteristic

#15688845

C₂

yes

maximal, characteristic, complemented

C₄₇×#492

C₂

yes

maximal, characteristic

C₂×#10625335

—

#13094024

C₂²

yes

characteristic

#13094024

C₂²

yes

—

—

—

—

C₄×C₄₇⋊C₈

C₂²

yes

characteristic

C₄×C₄₇⋊C₈

C₂²

yes

characteristic

C₂×SD₁₆×C₄₇

—

C₄⋊Q₈×C₄₇

C₂²

yes

characteristic

#127×C₄₇

C₂²

yes

characteristic

C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×Q₈×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

SD₁₆×C₄₇

—

C₂×C₈×C₄₇

D₄

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

C₂³

yes

characteristic

#10625335

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂×C₄₇⋊C₈

D₄

yes

—

C₂×C₄₇⋊C₈

D₄

yes

—

C₂³×C₄₇

—

Q₈×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

Dic₉₄

—

C₈×C₄₇

—

C₄₇⋊C₈

—

C₄₇⋊C₈

SD₁₆

yes

complemented

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#127

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

C₂×SD₁₆

yes

—

C₄×C₄₇

AGL(1,ℤ/8)

yes

—

C₄×C₄₇

#137

yes

—

Dic₄₇

—

Dic₄₇

—

#1714

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

yes

C₂×C₄₇

yes

C₂×C₄₇

yes

—

—

—

D₄₇

yes

—

—

—

C₄₇

#1714

yes

characteristic, complemented, Sylow

C₂×SD₁₆

—

C₂×SD₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₄₇

yes

characteristic

#127

C₂×D₄₇

yes

—

—

—

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₄₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

SD₁₆

—

188

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂×C₈

C₂²⋊D₄₇

yes

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄²

C₂²×D₄₇

yes

characteristic

109

C₂³

—

110

Q₈

—

111

Q₈

—

188

—

112

Q₈

—

188

—

113

D₄

—

114

D₄

—

115

D₄

—

116

C₂×C₄

—

117

C₂×C₄

—

188

—

118

C₂×C₄

—

188

—

119

C₂×C₄

D₄×D₄₇

yes

characteristic

120

C₂×C₄

D₄×D₄₇

yes

characteristic

121

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

122

C₈

—

123

C₈

—

124

C₈

—

125

C₂²

—

126

C₂²

—

127

C₂²

—

128

C₂²

#13094020

yes

characteristic

129

C₄

—

130

C₄

—

376

—

131

C₄

—

376

—

132

C₄

SD₁₆×D₄₇

yes

—

133

C₄

#13094050

yes

—

134

C₄

#13094053

yes

—

135

C₂

—

136

C₂

#15689494

yes

characteristic

137

C₂

#15689510

yes

characteristic

138

C₂

#15689514

yes

characteristic

	subgroup	G/H
center	C₂²	#13094020
commutator	C₄²×C₄₇	C₂³
Frattini	C₄²	C₂²×D₄₇
Fitting	C₄₇×#492	C₂
socle	C₂²×C₄₇	#127
hypercenter	#492	D₄₇
hypocenter	C₄₇	#1714
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#13094020

commutator

C₄²×C₄₇

C₂³

Frattini

C₄²

C₂²×D₄₇

Fitting

C₄₇×#492

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. #15689526	C₂	semidirect
2. #15689519	C₂	semidirect
4. #15689474	C₂	semidirect
6. #15688845	C₂	semidirect
54. C₄₇⋊C₈	SD₁₆	semidirect
77. C₄₇	#1714	semidirect

H ≅ G/N

type

1. #15689526

C₂

semidirect

2. #15689519

C₂

semidirect

4. #15689474

C₂

semidirect

6. #15688845

C₂

semidirect

54. C₄₇⋊C₈

SD₁₆

semidirect

77. C₄₇

#1714

semidirect

Subgroups — #20243008

Conjugacy classes of subgroups

Complements