Group #26130995 — subgroups

Subgroups — #26130995

Order 12032.

354 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 103 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	#20245643	C₂	1	4	yes	maximal, complemented
2	#20243006	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×C₄₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₄₇×#1712	C₂	1	1	yes	maximal, characteristic
5	C₂×#13093968	—	2	8	no	—
6	#15689528	—	2	2	no	—
7	#15689503	C₂²	1	2	yes	—
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	C₂×Dic₁₈₈	—	2	4	no	—
13	#13094024	—	2	2	no	—
14	#13093968	—	4	16	no	—
15	C₂×C₄₇⋊C₁₆	D₄	1	4	yes	—
16	#13093958	—	4	4	no	—
17	C₂×D₈×C₄₇	—	2	4	no	—
18	C₂×D₈×C₄₇	—	4	4	no	—
19	#127×C₄₇	—	2	2	no	—
20	C₄×C₈×C₄₇	D₄	1	2	yes	—
21	C₄×C₈×C₄₇	C₂³	1	1	yes	characteristic
22	C₂×D₄×C₄₇	—	4	2	no	—
23	C₂×D₄×C₄₇	—	4	4	no	—
24	C₂×Dic₉₄	—	4	2	no	—
25	D₈×C₄₇	—	4	8	no	—
26	D₈×C₄₇	—	8	8	no	—
27	C₂×C₈×C₄₇	D₈	1	4	yes	—
28	C₂×C₈×C₄₇	C₂×D₄	1	2	yes	—
29	C₄²×C₄₇	C₂×D₄	1	1	yes	characteristic
30	C₄⋊Dic₄₇	—	4	4	no	—
31	Dic₁₈₈	—	4	8	no	—
32	C₄₇⋊C₁₆	D₈	1	8	yes	complemented
33	C₂³×C₄₇	—	8	2	no	—
34	D₄×C₄₇	—	8	4	no	—
35	D₄×C₄₇	—	8	8	no	—
36	C₂×C₄×C₄₇	#127	1	1	yes	characteristic
37	C₂×C₄×C₄₇	C₂×D₈	1	2	yes	—
38	C₂×Dic₄₇	—	8	2	no	—
39	Dic₉₄	—	8	4	no	—
40	C₈×C₄₇	SD₃₂	1	8	yes	complemented
41	C₈×C₄₇	C₂×D₈	1	4	yes	—
42	#12517	—	47	1	no	maximal, Sylow
43	C₂²×C₄₇	—	16	2	no	—
44	C₂²×C₄₇	—	16	4	no	—
45	C₂²×C₄₇	#493	1	1	yes	characteristic
46	C₄×C₄₇	#493	1	2	yes	—
47	C₄×C₄₇	C₂×SD₃₂	1	4	yes	—
48	Dic₄₇	—	16	2	no	—
49	#2248	—	47	4	no	—
50	#1713	—	47	1	no	—
51	#1712	D₄₇	1	1	yes	characteristic
52	C₈×C₁₆	—	47	1	no	—
53	C₂×C₄₇	—	32	2	no	—
54	C₂×C₄₇	#1712	1	1	yes	characteristic
55	C₂×C₄₇	#2248	1	2	yes	—
56	C₂×SD₃₂	—	94	8	no	—
57	#500	—	94	2	no	—
58	#494	—	47	2	no	—
59	#493	—	2	2	no	—
60	#493	C₂×D₄₇	1	2	yes	—
61	C₄×C₁₆	—	47	2	no	—
62	C₈²	C₂×D₄₇	1	1	yes	characteristic
63	C₄₇	#12517	1	1	yes	characteristic, complemented, Sylow
64	C₂×Q₁₆	—	94	4	no	—
65	C₂×D₈	—	2	4	no	—
66	C₂×D₈	—	4	4	no	—
67	C₄⋊Q₈	—	94	2	no	—
68	#127	—	2	2	no	—
69	SD₃₂	—	188	16	no	—
70	C₂×C₁₆	—	47	4	no	—
71	#107	—	188	4	no	—
72	C₄×C₈	C₂²⋊D₄₇	1	2	yes	—
73	C₄×C₈	C₂²×D₄₇	1	1	yes	characteristic
74	C₂×Q₈	—	188	2	no	—
75	C₂×D₄	—	4	2	no	—
76	C₂×D₄	—	4	4	no	—
77	Q₁₆	—	188	8	no	—
78	D₈	—	4	8	no	—
79	D₈	—	8	8	no	—
80	C₂×C₈	#10625334	1	4	yes	—
81	C₂×C₈	D₄×D₄₇	1	2	yes	—
82	C₄⋊C₄	—	188	4	no	—
83	C₄²	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
84	C₁₆	—	47	8	no	—
85	C₂³	—	8	2	no	—
86	Q₈	—	376	4	no	—
87	D₄	—	8	4	no	—
88	D₄	—	8	8	no	—
89	C₂×C₄	—	376	2	no	—
90	C₂×C₄	#13094020	1	1	yes	characteristic
91	C₂×C₄	C₂×#10625334	1	2	yes	—
92	C₈	#13093968	1	8	yes	complemented
93	C₈	D₈×D₄₇	1	4	yes	—
94	C₂²	—	16	2	no	—
95	C₂²	—	16	4	no	—
96	C₂²	#15689499	1	1	yes	characteristic
97	C₄	—	752	2	no	—
98	C₄	#15689498	1	2	yes	—
99	C₄	C₂×#13093968	1	4	yes	—
100	C₂	—	32	2	no	—
101	C₂	#20243002	1	1	yes	characteristic
102	C₂	#20245643	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#20245643

C₂

yes

maximal, complemented

#20243006

C₂

yes

maximal, characteristic, complemented

C₈×C₄₇⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₄₇×#1712

C₂

yes

maximal, characteristic

C₂×#13093968

—

#15689528

—

#15689503

C₂²

yes

—

C₄×C₄₇⋊C₁₆

C₂²

yes

—

C₄₇×#493

—

C₄₇×#493

C₂²

yes

—

C₈²×C₄₇

C₂²

yes

characteristic

C₂×Dic₁₈₈

—

#13094024

—

#13093968

—

C₂×C₄₇⋊C₁₆

D₄

yes

—

#13093958

—

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

—

Dic₁₈₈

—

C₄₇⋊C₁₆

D₈

yes

complemented

C₂³×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

#127

yes

characteristic

C₂×C₄×C₄₇

C₂×D₈

yes

—

C₂×Dic₄₇

—

Dic₉₄

—

C₈×C₄₇

SD₃₂

yes

complemented

C₈×C₄₇

C₂×D₈

yes

—

#12517

—

maximal, Sylow

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#493

yes

characteristic

C₄×C₄₇

#493

yes

—

C₄×C₄₇

C₂×SD₃₂

yes

—

Dic₄₇

—

#2248

—

#1713

—

#1712

D₄₇

yes

characteristic

C₈×C₁₆

—

C₂×C₄₇

—

C₂×C₄₇

#1712

yes

characteristic

C₂×C₄₇

#2248

yes

—

C₂×SD₃₂

—

—

—

—

C₂×D₄₇

yes

—

C₄×C₁₆

—

C₈²

C₂×D₄₇

yes

characteristic

C₄₇

#12517

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₂×D₈

—

C₂×D₈

—

C₄⋊Q₈

—

#127

—

SD₃₂

—

188

—

C₂×C₁₆

—

#107

—

188

—

C₄×C₈

C₂²⋊D₄₇

yes

—

C₄×C₈

C₂²×D₄₇

yes

characteristic

C₂×Q₈

—

188

—

C₂×D₄

—

C₂×D₄

—

Q₁₆

—

188

—

D₈

—

D₈

—

C₂×C₈

#10625334

yes

—

C₂×C₈

D₄×D₄₇

yes

—

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

#13093968

yes

complemented

C₈

D₈×D₄₇

yes

—

C₂²

—

C₂²

—

C₂²

#15689499

yes

characteristic

C₄

—

752

—

C₄

#15689498

yes

—

C₄

C₂×#13093968

yes

—

100

C₂

—

101

C₂

#20243002

yes

characteristic

102

C₂

#20245643

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	#12517
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. #20245643	C₂	semidirect
2. #20243006	C₂	semidirect
3. C₈×C₄₇⋊C₁₆	C₂	semidirect
32. C₄₇⋊C₁₆	D₈	semidirect
40. C₈×C₄₇	SD₃₂	semidirect
63. C₄₇	#12517	semidirect
92. C₈	#13093968	semidirect

H ≅ G/N

type

1. #20245643

C₂

semidirect

2. #20243006

C₂

semidirect

3. C₈×C₄₇⋊C₁₆

C₂

semidirect

32. C₄₇⋊C₁₆

D₈

semidirect

40. C₈×C₄₇

SD₃₂

semidirect

63. C₄₇

#12517

semidirect

92. C₈

#13093968

semidirect

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

subgroup	index
#20245643	2
#20243006	2
C₈×C₄₇⋊C₁₆	2
C₄₇×#1712	2
#12517	47

subgroup

index

#20245643

#20243006

C₈×C₄₇⋊C₁₆

C₄₇×#1712

#12517

Subgroups — #26130995

Conjugacy classes of subgroups

Complements

Maximal subgroups