Group #26121353 — subgroups

Subgroups — #26121353

Order 12032.

478 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 141 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	#20244194	C₂	1	3	yes	maximal
2	#20243516	C₂	1	3	yes	maximal, complemented
3	C₄₇×#1273	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#13094021	C₂²	1	1	yes	characteristic
5	C₂×#13093946	—	2	6	no	—
6	#15688993	C₂²	1	3	yes	—
7	#15688886	—	2	6	no	—
8	C₂×C₄×C₈×C₄₇	C₂²	1	3	yes	—
9	C₂²×D₁₈₈	—	2	3	no	—
10	#13094021	—	2	3	no	—
11	#13094021	Q₈	1	2	yes	—
12	C₂×C₄⋊Dic₄₇	—	2	3	no	—
13	C₂×C₄⋊Dic₄₇	—	2	3	no	—
14	C₂×C₄⋊Dic₄₇	—	2	3	no	—
15	#13093946	—	4	12	no	—
16	C₂²×C₈×C₄₇	D₄	1	6	yes	—
17	C₂×C₄²×C₄₇	C₂³	1	1	yes	characteristic
18	C₄×C₈×C₄₇	—	2	6	no	—
19	C₂³×D₄₇	—	4	1	no	—
20	C₂²×C₄×C₄₇	C₂×D₄	1	3	yes	—
21	C₂²×Dic₄₇	—	4	3	no	—
22	C₂×D₁₈₈	—	2	6	no	—
23	C₂×D₁₈₈	—	4	6	no	—
24	C₂×D₁₈₈	—	4	6	no	—
25	C₂×D₁₈₈	—	4	3	no	—
26	C₂×C₈×C₄₇	—	2	6	no	—
27	C₂×C₈×C₄₇	—	2	6	no	—
28	C₂×C₈×C₄₇	—	2	6	no	—
29	C₄²×C₄₇	C₂×Q₈	1	1	yes	characteristic
30	C₄²×C₄₇	#41	1	3	yes	—
31	C₄⋊Dic₄₇	—	4	6	no	—
32	C₄⋊Dic₄₇	—	4	6	no	—
33	C₄⋊Dic₄₇	—	4	6	no	—
34	C₂³×C₄₇	#127	1	1	yes	characteristic
35	C₂²×D₄₇	—	4	2	no	—
36	C₂²×D₄₇	—	8	3	no	—
37	C₂²×D₄₇	—	8	3	no	—
38	C₂×C₄×C₄₇	C₂²⋊Q₈	1	3	yes	—
39	C₂×C₄×C₄₇	#123	1	3	yes	—
40	C₂×C₄×C₄₇	#135	1	6	yes	—
41	C₂×C₄×C₄₇	AGL(1,ℤ/8)	1	6	yes	—
42	C₂×Dic₄₇	—	8	3	no	—
43	C₂×Dic₄₇	—	8	3	no	—
44	C₂×Dic₄₇	—	8	3	no	—
45	D₁₈₈	—	4	12	no	—
46	D₁₈₈	—	8	6	no	—
47	C₈×C₄₇	—	4	6	no	—
48	#10678	—	47	1	no	maximal, Sylow
49	C₂²×C₄₇	#399	1	1	yes	characteristic
50	C₂²×C₄₇	#495	1	3	yes	—
51	C₂²×C₄₇	#496	1	3	yes	—
52	C₂×D₄₇	—	8	6	no	—
53	C₂×D₄₇	—	16	1	no	—
54	C₂×D₄₇	—	16	3	no	—
55	C₄×C₄₇	#478	1	6	yes	—
56	C₄×C₄₇	#482	1	6	yes	—
57	Dic₄₇	—	16	3	no	—
58	#1957	—	47	3	no	—
59	#1834	—	47	3	no	—
60	#1273	D₄₇	1	1	yes	characteristic, complemented
61	C₂×C₄₇	#1711	1	1	yes	characteristic
62	C₂×C₄₇	#1717	1	3	yes	—
63	C₂×C₄₇	#2092	1	3	yes	—
64	D₄₇	—	16	2	no	—
65	C₂×#127	—	47	1	no	—
66	C₂×#102	—	94	6	no	—
67	C₂×C₄×C₈	C₂×D₄₇	1	3	yes	—
68	#384	—	47	3	no	—
69	#340	—	94	6	no	—
70	C₄₇	#10678	1	1	yes	characteristic, complemented, Sylow
71	C₂²×D₄	—	94	3	no	—
72	C₂²×C₈	D₁₈₈	1	6	yes	—
73	#127	—	47	2	no	—
74	#127	—	94	3	no	—
75	C₂×C₄⋊C₄	—	94	3	no	—
76	C₂×C₄⋊C₄	—	94	3	no	—
77	C₂×C₄⋊C₄	—	94	3	no	—
78	C₂×C₄²	C₂²×D₄₇	1	1	yes	characteristic
79	#102	—	188	12	no	—
80	C₄×C₈	—	2	6	no	—
81	C₂⁴	—	188	1	no	—
82	C₂×D₄	—	94	6	no	—
83	C₂×D₄	—	188	3	no	—
84	C₂×D₄	—	188	6	no	—
85	C₂×D₄	—	188	6	no	—
86	C₂²×C₄	—	188	3	no	—
87	C₂²×C₄	C₂×D₁₈₈	1	3	yes	—
88	C₂×C₈	—	2	6	no	—
89	C₂×C₈	—	2	6	no	—
90	C₂×C₈	—	2	6	no	—
91	C₄⋊C₄	—	188	6	no	—
92	C₄⋊C₄	—	188	6	no	—
93	C₄⋊C₄	—	188	6	no	—
94	C₄²	#10625352	1	3	yes	—
95	C₄²	Q₈×D₄₇	1	1	yes	characteristic
96	C₂³	—	376	3	no	—
97	C₂³	—	376	3	no	—
98	C₂³	—	188	2	no	—
99	C₂³	#13094021	1	1	yes	characteristic
100	D₄	—	376	6	no	—
101	D₄	—	188	12	no	—
102	C₂×C₄	—	376	3	no	—
103	C₂×C₄	—	376	3	no	—
104	C₂×C₄	—	376	3	no	—
105	C₂×C₄	#13094004	1	3	yes	—
106	C₂×C₄	#13094008	1	3	yes	—
107	C₂×C₄	#13094046	1	6	yes	—
108	C₂×C₄	#13094052	1	6	yes	—
109	C₈	—	4	6	no	—
110	C₂²	—	376	6	no	—
111	C₂²	—	752	1	no	—
112	C₂²	—	752	3	no	—
113	C₂²	#15689060	1	1	yes	characteristic
114	C₂²	#15689507	1	3	yes	—
115	C₂²	#15689512	1	3	yes	—
116	C₄	—	752	3	no	—
117	C₄	#15689424	1	6	yes	—
118	C₄	#15689448	1	6	yes	—
119	C₂	—	752	2	no	—
120	C₂	#20243000	1	1	yes	characteristic
121	C₂	#20243027	1	3	yes	—
122	C₂	#20244908	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20244194

C₂

yes

maximal

#20243516

C₂

yes

maximal, complemented

C₄₇×#1273

C₂

yes

maximal, characteristic, complemented

C₂×#13094021

C₂²

yes

characteristic

C₂×#13093946

—

#15688993

C₂²

yes

—

#15688886

—

C₂×C₄×C₈×C₄₇

C₂²

yes

—

C₂²×D₁₈₈

—

#13094021

—

#13094021

Q₈

yes

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

—

#13093946

—

C₂²×C₈×C₄₇

D₄

yes

—

C₂×C₄²×C₄₇

C₂³

yes

characteristic

C₄×C₈×C₄₇

—

C₂³×D₄₇

—

C₂²×C₄×C₄₇

C₂×D₄

yes

—

C₂²×Dic₄₇

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×D₁₈₈

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₂×C₈×C₄₇

—

C₄²×C₄₇

C₂×Q₈

yes

characteristic

C₄²×C₄₇

#41

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₂³×C₄₇

#127

yes

characteristic

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂²×D₄₇

—

C₂×C₄×C₄₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₄₇

#123

yes

—

C₂×C₄×C₄₇

#135

yes

—

C₂×C₄×C₄₇

AGL(1,ℤ/8)

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

D₁₈₈

—

D₁₈₈

—

C₈×C₄₇

—

—

C₂²×C₄₇

yes

C₂²×C₄₇

yes

—

C₂²×C₄₇

#496

yes

—

C₂×D₄₇

—

C₂×D₄₇

—

C₂×D₄₇

—

C₄×C₄₇

#478

yes

—

C₄×C₄₇

#482

yes

—

Dic₄₇

—

#1957

—

#1834

—

#1273

D₄₇

yes

characteristic, complemented

C₂×C₄₇

#1711

yes

characteristic

C₂×C₄₇

#1717

yes

—

C₂×C₄₇

#2092

yes

—

D₄₇

—

C₂×#127

—

C₂×#102

—

C₂×C₄×C₈

C₂×D₄₇

yes

—

#384

—

#340

—

C₄₇

#10678

yes

characteristic, complemented, Sylow

C₂²×D₄

—

C₂²×C₈

D₁₈₈

yes

—

#127

—

#127

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₄₇

yes

characteristic

#102

—

188

—

C₄×C₈

—

C₂⁴

—

188

—

C₂×D₄

—

C₂×D₄

—

188

—

C₂×D₄

—

188

—

C₂×D₄

—

188

—

C₂²×C₄

—

188

—

C₂²×C₄

C₂×D₁₈₈

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

188

—

C₄⋊C₄

—

188

—

C₄⋊C₄

—

188

—

C₄²

#10625352

yes

—

C₄²

Q₈×D₄₇

yes

characteristic

C₂³

—

376

—

C₂³

—

376

—

C₂³

—

188

—

C₂³

#13094021

yes

characteristic

100

D₄

—

376

—

101

D₄

—

188

—

102

C₂×C₄

—

376

—

103

C₂×C₄

—

376

—

104

C₂×C₄

—

376

—

105

C₂×C₄

#13094004

yes

—

106

C₂×C₄

#13094008

yes

—

107

C₂×C₄

#13094046

yes

—

108

C₂×C₄

#13094052

yes

—

109

C₈

—

110

C₂²

—

376

—

111

C₂²

—

752

—

112

C₂²

—

752

—

113

C₂²

#15689060

yes

characteristic

114

C₂²

#15689507

yes

—

115

C₂²

#15689512

yes

—

116

C₄

—

752

—

117

C₄

#15689424

yes

—

118

C₄

#15689448

yes

—

119

C₂

—

752

—

120

C₂

#20243000

yes

characteristic

121

C₂

#20243027

yes

—

122

C₂

#20244908

yes

—

	subgroup	G/H
center	C₂³	#13094021
commutator, cosocle	C₂×C₄²×C₄₇	C₂³
Frattini	C₂×C₄²	C₂²×D₄₇
Fitting	C₄₇×#1273	C₂
socle	C₂³×C₄₇	#127
prime radical	C₂×#13094021	C₂²
hypercenter	#1273	D₄₇
hypocenter	C₄₇	#10678
LCS	G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1273

subgroup

G/H

center

C₂³

#13094021

commutator, cosocle

C₂×C₄²×C₄₇

C₂³

Frattini

C₂×C₄²

C₂²×D₄₇

Fitting

C₄₇×#1273

C₂

C₂³×C₄₇

C₂²

D₄₇

C₄₇

G ▷ C₂×C₄²×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1273

N	H ≅ G/N	type
2. #20243516	C₂	semidirect
3. C₄₇×#1273	C₂	semidirect
60. #1273	D₄₇	semidirect
70. C₄₇	#10678	semidirect

H ≅ G/N

type

2. #20243516

C₂

semidirect

3. C₄₇×#1273

C₂

semidirect

60. #1273

D₄₇

semidirect

70. C₄₇

#10678

semidirect

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

subgroup	index
#20244194	2
#20243516	2
C₄₇×#1273	2
#10678	47

subgroup

index

#20244194

#20243516

C₄₇×#1273

#10678

Subgroups — #26121353

Conjugacy classes of subgroups

Complements

Maximal subgroups