Group #26109976 — subgroups

Subgroups — #26109976

Order 12032.

270 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 270 conjugacy classes of subgroups — too many to draw. Only its 115 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	#20243560	C₂	1	1	yes	maximal, characteristic
2	#20243519	C₂	1	1	yes	maximal, characteristic
3	#20241247	C₂	1	4	yes	maximal
4	C₄₇×#1452	C₂	1	1	yes	maximal, characteristic
5	C₂×#13093960	—	2	2	no	—
6	#15689147	C₂²	1	2	yes	—
7	#15688989	C₂²	1	2	yes	—
8	#15688886	—	2	2	no	—
9	C₄₇×#431	C₂²	1	2	yes	—
10	C₂×C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
11	C₄₇×#322	C₄	1	4	yes	complemented
12	C₂×#10625329	—	2	2	no	—
13	C₂×C₄⋊Dic₄₇	—	2	2	no	—
14	#13093960	—	4	4	no	—
15	C₄⋊(C₄₇⋊C₈)	—	2	4	no	—
16	C₄₇⋊#95	—	2	4	no	—
17	C₂²×C₈×C₄₇	D₄	1	2	yes	—
18	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
19	C₄⋊C₈×C₄₇	—	2	2	no	—
20	C₂²⋊C₈×C₄₇	—	2	2	no	—
21	C₄×C₈×C₄₇	C₂×C₄	1	4	yes	—
22	C₄×C₈×C₄₇	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₄₇	Q₈	1	2	yes	—
24	C₂²×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
25	C₂²×C₄×C₄₇	#41	1	2	yes	—
26	C₂²×Dic₄₇	—	4	2	no	—
27	C₂×C₈×C₄₇	—	2	2	no	—
28	C₂×C₈×C₄₇	—	2	2	no	—
29	C₂×C₈×C₄₇	—	2	4	no	—
30	C₂×C₈×C₄₇	C₄⋊C₄	1	4	yes	—
31	C₄²×C₄₇	C₂²×C₄	1	1	yes	characteristic
32	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
33	C₄²×C₄₇	#41	1	2	yes	—
34	C₄⋊Dic₄₇	—	4	2	no	—
35	C₄⋊Dic₄₇	—	4	2	no	—
36	#10625329	—	4	4	no	—
37	C₂×C₄₇⋊C₈	—	4	2	no	—
38	C₂³×C₄₇	#124	1	1	yes	characteristic
39	C₂×C₄×C₄₇	—	4	2	no	—
40	C₂×C₄×C₄₇	C₂×C₄⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₄₇	#117	1	2	yes	—
42	C₂×C₄×C₄₇	#124	1	1	yes	characteristic
43	C₂×C₄×C₄₇	#125	1	2	yes	—
44	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
45	C₂×C₄×C₄₇	#137	1	2	yes	—
46	C₂×Dic₄₇	—	8	4	no	—
47	C₂×Dic₄₇	—	8	2	no	—
48	C₈×C₄₇	—	2	4	no	—
49	C₈×C₄₇	—	4	4	no	—
50	C₄₇⋊C₈	—	8	2	no	—
51	#8794	—	47	1	no	maximal, Sylow
52	C₂²×C₄₇	—	2	2	no	—
53	C₂²×C₄₇	#382	1	1	yes	characteristic
54	C₂²×C₄₇	#489	1	2	yes	—
55	C₄×C₄₇	—	2	2	no	—
56	C₄×C₄₇	—	2	2	no	—
57	C₄×C₄₇	#428	1	2	yes	—
58	C₄×C₄₇	#490	1	1	yes	characteristic
59	C₄×C₄₇	#491	1	1	yes	characteristic
60	Dic₄₇	—	16	2	no	—
61	#1848	—	47	1	no	—
62	#1835	—	47	1	no	—
63	#1452	D₄₇	1	1	yes	characteristic
64	#1286	—	47	4	no	—
65	C₂×C₄₇	—	2	2	no	—
66	C₂×C₄₇	#1833	1	2	yes	—
67	C₂×C₄₇	#1836	1	1	yes	characteristic
68	#431	C₂×D₄₇	1	2	yes	—
69	C₂×#108	—	94	2	no	—
70	#423	—	47	2	no	—
71	C₂×C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
72	#382	—	47	2	no	—
73	#340	—	94	2	no	—
74	#322	Dic₄₇	1	4	yes	complemented
75	C₄₇	#8794	1	1	yes	characteristic, complemented, Sylow
76	C₂×M₁₆	—	94	2	no	—
77	C₂²×C₈	D₁₈₈	1	2	yes	—
78	C₂×C₄⋊C₄	—	94	2	no	—
79	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
80	#108	—	188	4	no	—
81	C₄⋊C₈	—	2	2	no	—
82	C₄⋊C₈	—	94	4	no	—
83	C₂²⋊C₈	—	2	2	no	—
84	C₄×C₈	Dic₉₄	1	2	yes	—
85	C₄×C₈	C₂×Dic₄₇	1	4	yes	—
86	C₄×C₈	C₂×Dic₄₇	1	2	yes	—
87	#95	—	94	4	no	—
88	C₂²×C₄	—	188	2	no	—
89	C₂²×C₄	C₂×D₁₈₈	1	1	yes	characteristic
90	C₂²×C₄	#10625350	1	2	yes	—
91	M₁₆	—	188	4	no	—
92	C₂×C₈	—	2	2	no	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	4	no	—
95	C₂×C₈	—	188	2	no	—
96	C₂×C₈	C₄⋊Dic₄₇	1	4	yes	—
97	C₄⋊C₄	—	188	2	no	—
98	C₄⋊C₄	—	188	2	no	—
99	C₄²	C₂×Dic₉₄	1	1	yes	characteristic
100	C₄²	#10625350	1	2	yes	—
101	C₄²	C₂²×Dic₄₇	1	1	yes	characteristic
102	C₂³	#13094012	1	1	yes	characteristic
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	—	376	4	no	—
105	C₂×C₄	—	376	2	no	—
106	C₂×C₄	C₂×C₄⋊Dic₄₇	1	1	yes	characteristic
107	C₂×C₄	#13093983	1	2	yes	—
108	C₂×C₄	#13094012	1	1	yes	characteristic
109	C₂×C₄	#13094016	1	2	yes	—
110	C₂×C₄	#13094052	1	2	yes	—
111	C₂×C₄	#13094057	1	2	yes	—
112	C₈	—	2	4	no	—
113	C₈	—	4	4	no	—
114	C₈	—	376	2	no	—
115	C₂²	—	2	2	no	—
116	C₂²	#15688989	1	1	yes	characteristic
117	C₂²	#15689483	1	2	yes	—
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	752	2	no	—
121	C₄	#15689165	1	2	yes	—
122	C₄	#15689488	1	1	yes	characteristic
123	C₄	#15689492	1	1	yes	characteristic
124	C₂	—	2	2	no	—
125	C₂	#20243512	1	2	yes	—
126	C₂	#20243522	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20243560

C₂

yes

maximal, characteristic

#20243519

C₂

yes

maximal, characteristic

#20241247

C₂

yes

maximal

C₄₇×#1452

C₂

yes

maximal, characteristic

C₂×#13093960

—

#15689147

C₂²

yes

—

#15688989

C₂²

yes

—

#15688886

—

C₄₇×#431

C₂²

yes

—

C₂×C₄×C₈×C₄₇

C₂²

yes

characteristic

C₄₇×#322

C₄

yes

complemented

C₂×#10625329

—

C₂×C₄⋊Dic₄₇

—

#13093960

—

C₄⋊(C₄₇⋊C₈)

—

C₄₇⋊#95

—

C₂²×C₈×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄⋊C₈×C₄₇

—

C₂²⋊C₈×C₄₇

—

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

C₂×C₄

yes

—

C₄×C₈×C₄₇

Q₈

yes

—

C₂²×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₄₇

#41

yes

—

C₂²×Dic₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

C₄⋊C₄

yes

—

C₄²×C₄₇

C₂²×C₄

yes

characteristic

C₄²×C₄₇

C₂×Q₈

yes

characteristic

C₄²×C₄₇

#41

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

#10625329

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

#124

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₄₇

#117

yes

—

C₂×C₄×C₄₇

#124

yes

characteristic

C₂×C₄×C₄₇

#125

yes

—

C₂×C₄×C₄₇

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₄₇

#137

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₈×C₄₇

—

C₈×C₄₇

—

C₄₇⋊C₈

—

#8794

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

#382

yes

characteristic

C₂²×C₄₇

#489

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#428

yes

—

C₄×C₄₇

yes

C₄×C₄₇

yes

Dic₄₇

—

#1848

—

#1835

—

#1452

D₄₇

yes

characteristic

#1286

—

C₂×C₄₇

—

C₂×C₄₇

#1833

yes

—

C₂×C₄₇

#1836

yes

characteristic

#431

C₂×D₄₇

yes

—

C₂×#108

—

#423

—

C₂×C₄×C₈

C₂×D₄₇

yes

—

—

Dic₄₇

yes

complemented

C₄₇

#8794

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₂²×C₈

D₁₈₈

yes

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₄₇

yes

characteristic

#108

—

188

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

Dic₉₄

yes

—

C₄×C₈

C₂×Dic₄₇

yes

—

C₄×C₈

C₂×Dic₄₇

yes

—

#95

—

C₂²×C₄

—

188

—

C₂²×C₄

C₂×D₁₈₈

yes

characteristic

C₂²×C₄

#10625350

yes

—

M₁₆

—

188

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

188

—

C₂×C₈

C₄⋊Dic₄₇

yes

—

C₄⋊C₄

—

188

—

C₄⋊C₄

—

188

—

C₄²

C₂×Dic₉₄

yes

characteristic

100

C₄²

#10625350

yes

—

101

C₄²

C₂²×Dic₄₇

yes

characteristic

102

C₂³

#13094012

yes

characteristic

103

C₂×C₄

—

104

C₂×C₄

—

376

—

105

C₂×C₄

—

376

—

106

C₂×C₄

C₂×C₄⋊Dic₄₇

yes

characteristic

107

C₂×C₄

#13093983

yes

—

108

C₂×C₄

#13094012

yes

characteristic

109

C₂×C₄

#13094016

yes

—

110

C₂×C₄

#13094052

yes

—

111

C₂×C₄

#13094057

yes

—

112

C₈

—

113

C₈

—

114

C₈

—

376

—

115

C₂²

—

116

C₂²

#15688989

yes

characteristic

117

C₂²

#15689483

yes

—

118

C₄

—

119

C₄

—

120

C₄

—

752

—

121

C₄

#15689165

yes

—

122

C₄

#15689488

yes

characteristic

123

C₄

#15689492

yes

characteristic

124

C₂

—

125

C₂

#20243512

yes

—

126

C₂

#20243522

yes

characteristic

	subgroup	G/H
center	C₂²	#15688989
commutator	C₄²×C₄₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1452	C₂
socle	C₂²×C₄₇	#382
prime radical	C₂³×C₄₇	#124
cosocle	C₂×C₄²×C₄₇	C₂³
hypercenter	#1452	D₄₇
hypocenter	C₄₇	#8794
LCS	G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452

subgroup

G/H

center

C₂²

#15688989

commutator

C₄²×C₄₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1452

C₂

C₂²×C₄₇

C₂³×C₄₇

C₂×C₄²×C₄₇

C₂³

D₄₇

C₄₇

G ▷ C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452

N	H ≅ G/N	type
11. C₄₇×#322	C₄	semidirect
74. #322	Dic₄₇	semidirect
75. C₄₇	#8794	semidirect

H ≅ G/N

type

11. C₄₇×#322

C₄

semidirect

74. #322

Dic₄₇

semidirect

75. C₄₇

#8794

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20243560	2
#20243519	2
#20241247	2
C₄₇×#1452	2
#8794	47

subgroup

index

C₄₇×#1452

Subgroups — #26109976

Conjugacy classes of subgroups

Complements

Maximal subgroups