Group #26130981 — subgroups

Subgroups — #26130981

Order 12032.

290 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 290 conjugacy classes of subgroups — too many to draw. Only its 79 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	#20245611	C₂	1	4	yes	maximal, complemented
2	#20242998	C₂	1	1	yes	maximal, characteristic, complemented
3	#20241305	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₇×#1712	C₂	1	1	yes	maximal, characteristic
5	#15689528	C₂²	1	2	yes	—
6	#15689474	—	2	2	no	—
7	#15688924	—	2	8	no	—
8	C₄×C₄₇⋊C₁₆	C₂²	1	2	yes	—
9	C₄₇×#493	—	2	2	no	—
10	C₄₇×#493	C₂²	1	2	yes	—
11	C₈²×C₄₇	C₂²	1	1	yes	characteristic
12	#13094024	—	2	2	no	—
13	C₂×C₄₇⋊C₁₆	D₄	1	4	yes	—
14	#13093958	—	2	4	no	—
15	#13093949	—	4	4	no	—
16	C₂×D₈×C₄₇	—	2	4	no	—
17	C₂×D₈×C₄₇	—	4	4	no	—
18	#127×C₄₇	—	2	2	no	—
19	C₄×C₈×C₄₇	D₄	1	2	yes	—
20	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
21	C₂×D₄×C₄₇	—	4	2	no	—
22	C₂×D₄×C₄₇	—	4	4	no	—
23	C₂×Dic₉₄	—	4	2	no	—
24	D₈×C₄₇	—	4	8	no	—
25	D₈×C₄₇	—	8	8	no	—
26	C₂×C₈×C₄₇	D₈	1	4	yes	—
27	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
28	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
29	C₄⋊Dic₄₇	—	4	2	no	—
30	C₄⋊Dic₄₇	—	4	2	no	—
31	C₄₇⋊C₁₆	—	2	4	no	—
32	C₂³×C₄₇	—	8	2	no	—
33	D₄×C₄₇	—	8	4	no	—
34	D₄×C₄₇	—	8	8	no	—
35	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
36	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
37	C₂×Dic₄₇	—	8	2	no	—
38	Dic₉₄	—	8	4	no	—
39	C₈×C₄₇	—	2	4	no	—
40	C₈×C₄₇	#135	1	4	yes	—
41	#12513	—	47	1	no	maximal, Sylow
42	C₂²×C₄₇	—	16	2	no	—
43	C₂²×C₄₇	—	16	4	no	—
44	C₂²×C₄₇	#493	1	1	yes	characteristic
45	C₄×C₄₇	#495	1	2	yes	—
46	C₄×C₄₇	#509	1	4	yes	—
47	Dic₄₇	—	16	2	no	—
48	#2240	—	47	4	no	—
49	#1712	D₄₇	1	1	yes	characteristic
50	#1710	—	47	1	no	—
51	#1311	—	47	1	no	—
52	C₂×C₄₇	—	32	2	no	—
53	C₂×C₄₇	#1709	1	1	yes	characteristic
54	C₂×C₄₇	#2250	1	2	yes	—
55	#500	—	47	2	no	—
56	#493	—	2	2	no	—
57	#493	C₂×D₄₇	1	2	yes	—
58	#487	—	94	2	no	—
59	#357	—	94	8	no	—
60	C₄×C₁₆	—	47	2	no	—
61	C₈²	C₂×D₄₇	1	1	yes	characteristic
62	C₄₇	#12513	1	1	yes	characteristic, complemented, Sylow
63	C₂×D₈	—	2	4	no	—
64	C₂×D₈	—	4	4	no	—
65	C₄⋊Q₈	—	94	2	no	—
66	#127	—	2	2	no	—
67	C₂×C₁₆	—	47	4	no	—
68	#107	—	94	4	no	—
69	#103	—	188	4	no	—
70	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
71	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
72	C₂×Q₈	—	188	2	no	—
73	C₂×D₄	—	4	2	no	—
74	C₂×D₄	—	4	4	no	—
75	D₈	—	4	8	no	—
76	D₈	—	8	8	no	—
77	C₂×C₈	#10625334	1	4	yes	—
78	C₂×C₈	D₄×D₄₇	1	2	yes	—
79	C₄⋊C₄	—	188	2	no	—
80	C₄⋊C₄	—	188	2	no	—
81	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
82	C₁₆	—	94	4	no	—
83	C₂³	—	8	2	no	—
84	Q₈	—	376	4	no	—
85	D₄	—	8	4	no	—
86	D₄	—	8	8	no	—
87	C₂×C₄	—	376	2	no	—
88	C₂×C₄	#13094020	1	1	yes	characteristic
89	C₂×C₄	C₂×#10625334	1	2	yes	—
90	C₈	—	2	4	no	—
91	C₈	#13094043	1	4	yes	—
92	C₂²	—	16	2	no	—
93	C₂²	—	16	4	no	—
94	C₂²	#15689499	1	1	yes	characteristic
95	C₄	—	752	2	no	—
96	C₄	#15689506	1	2	yes	—
97	C₄	#15689559	1	4	yes	—
98	C₂	—	32	2	no	—
99	C₂	#20242993	1	1	yes	characteristic
100	C₂	#20245652	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20245611

C₂

yes

maximal, complemented

#20242998

C₂

yes

maximal, characteristic, complemented

#20241305

C₂

yes

maximal, characteristic, complemented

C₄₇×#1712

C₂

yes

maximal, characteristic

#15689528

C₂²

yes

—

#15689474

—

#15688924

—

C₄×C₄₇⋊C₁₆

C₂²

yes

—

C₄₇×#493

—

C₄₇×#493

C₂²

yes

—

C₈²×C₄₇

C₂²

yes

characteristic

#13094024

—

C₂×C₄₇⋊C₁₆

D₄

yes

—

#13093958

—

#13093949

—

C₂×D₈×C₄₇

—

C₂×D₈×C₄₇

—

#127×C₄₇

—

C₄×C₈×C₄₇

D₄

yes

—

C₄×C₈×C₄₇

C₂³

yes

characteristic

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

—

C₂×Dic₉₄

—

D₈×C₄₇

—

D₈×C₄₇

—

C₂×C₈×C₄₇

D₈

yes

—

C₂×C₈×C₄₇

C₂×D₄

yes

—

C₄²×C₄₇

C₂×D₄

yes

characteristic

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄₇⋊C₁₆

—

C₂³×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

—

C₂×Dic₄₇

—

Dic₉₄

—

C₈×C₄₇

—

C₈×C₄₇

#135

yes

—

#12513

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#493

yes

characteristic

C₄×C₄₇

#495

yes

—

C₄×C₄₇

#509

yes

—

Dic₄₇

—

#2240

—

#1712

D₄₇

yes

characteristic

#1710

—

#1311

—

C₂×C₄₇

—

C₂×C₄₇

#1709

yes

characteristic

C₂×C₄₇

#2250

yes

—

#500

—

#493

—

#493

C₂×D₄₇

yes

—

#487

—

#357

—

C₄×C₁₆

—

C₈²

C₂×D₄₇

yes

characteristic

C₄₇

#12513

yes

characteristic, complemented, Sylow

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

C₂×C₁₆

—

#107

—

#103

—

188

—

C₄×C₈

C₂²⋊D₄₇

yes

—

C₄×C₈

C₂²×D₄₇

yes

characteristic

C₂×Q₈

—

188

—

C₂×D₄

—

C₂×D₄

—

D₈

—

D₈

—

C₂×C₈

#10625334

yes

—

C₂×C₈

D₄×D₄₇

yes

—

C₄⋊C₄

—

188

—

C₄⋊C₄

—

188

—

C₄²

C₂×C₂²⋊D₄₇

yes

characteristic

C₁₆

—

C₂³

—

Q₈

—

376

—

D₄

—

D₄

—

C₂×C₄

—

376

—

C₂×C₄

#13094020

yes

characteristic

C₂×C₄

C₂×#10625334

yes

—

C₈

—

C₈

#13094043

yes

—

C₂²

—

C₂²

—

C₂²

#15689499

yes

characteristic

C₄

—

752

—

C₄

#15689506

yes

—

C₄

#15689559

yes

—

C₂

—

C₂

#20242993

yes

characteristic

100

C₂

#20245652

yes

—

	subgroup	G/H
center	C₂²	#15689499
commutator, cosocle	C₄×C₈×C₄₇	C₂³
Frattini	C₄×C₈	C₂²×D₄₇
Fitting, prime radical	C₄₇×#1712	C₂
socle	C₂²×C₄₇	#493
hypercenter	#1712	D₄₇
hypocenter	C₄₇	#12513
LCS	G ▷ C₄×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1712

subgroup

G/H

center

C₂²

#15689499

commutator, cosocle

C₄×C₈×C₄₇

C₂³

Frattini

C₄×C₈

C₂²×D₄₇

Fitting, prime radical

C₄₇×#1712

C₂

C₂²×C₄₇

D₄₇

C₄₇

G ▷ C₄×C₈×C₄₇ ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1712

N	H ≅ G/N	type
1. #20245611	C₂	semidirect
2. #20242998	C₂	semidirect
3. #20241305	C₂	semidirect
62. C₄₇	#12513	semidirect

H ≅ G/N

type

1. #20245611

C₂

semidirect

2. #20242998

C₂

semidirect

3. #20241305

C₂

semidirect

62. C₄₇

#12513

semidirect

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

subgroup	index
#20245611	2
#20242998	2
#20241305	2
C₄₇×#1712	2
#12513	47

subgroup

index

C₄₇×#1712

Subgroups — #26130981

Conjugacy classes of subgroups

Complements

Maximal subgroups