Group #26137297 — subgroups

Subgroups — #26137297

Order 12032.

158 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 158 conjugacy classes of subgroups — too many to draw. Only its 67 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	#20245662	C₂	1	2	yes	maximal
2	#20241588	C₂	1	1	yes	maximal, characteristic
3	#20241587	C₂	1	2	yes	maximal
4	C₃₂×Dic₄₇	C₂	1	1	yes	maximal, characteristic
5	C₄₇×#1423	C₂	1	1	yes	maximal, characteristic
6	#15689527	—	2	2	no	—
7	C₂×C₄₇⋊C₃₂	C₂²	1	1	yes	characteristic
8	#15688949	C₂²	1	2	yes	—
9	#15688948	—	2	2	no	—
10	C₁₆×Dic₄₇	C₂²	1	1	yes	characteristic
11	C₂×C₃₂×C₄₇	C₂²	1	1	yes	characteristic
12	C₄₇×#366	C₂²	1	2	yes	—
13	#13094023	—	4	2	no	—
14	C₂×C₄₇⋊C₁₆	D₄	1	1	yes	characteristic
15	#13093958	—	2	2	no	—
16	#13093957	—	4	2	no	—
17	C₈×Dic₄₇	D₄	1	1	yes	characteristic
18	C₄₇⋊C₃₂	Q₈	1	2	yes	complemented
19	C₂×C₁₆×C₄₇	C₂³	1	1	yes	characteristic
20	#107×C₄₇	—	2	2	no	—
21	C₃₂×C₄₇	Q₈	1	2	yes	complemented
22	C₂×Dic₉₄	—	8	2	no	—
23	C₂×C₈×C₄₇	C₂×D₄	1	1	yes	characteristic
24	C₄⋊C₄×C₄₇	—	4	2	no	—
25	C₄⋊Dic₄₇	—	4	2	no	—
26	Dic₄₇⋊C₄	—	8	2	no	—
27	C₄×Dic₄₇	D₈	1	1	yes	characteristic
28	C₂×C₄₇⋊C₈	D₈	1	1	yes	characteristic
29	C₁₆×C₄₇	C₂×Q₈	1	1	yes	characteristic
30	C₁₆×C₄₇	C₂×Q₈	1	1	yes	characteristic
31	C₄₇⋊C₁₆	Q₁₆	1	2	yes	—
32	C₂×C₄×C₄₇	—	8	2	no	—
33	C₂×C₄×C₄₇	C₂×D₈	1	1	yes	characteristic
34	C₂×Dic₄₇	—	8	2	no	—
35	C₂×Dic₄₇	D₁₆	1	1	yes	characteristic
36	C₂×Dic₄₇	D₁₆	1	1	yes	characteristic
37	Dic₉₄	—	16	4	no	—
38	C₈×C₄₇	C₄⋊Q₈	1	1	yes	characteristic
39	C₈×C₄₇	C₂×Q₁₆	1	1	yes	characteristic
40	C₄₇⋊C₈	Q₃₂	1	2	yes	—
41	#13731	—	47	1	no	maximal, Sylow
42	C₂²×C₄₇	C₂×D₁₆	1	1	yes	characteristic
43	C₄×C₄₇	—	16	2	no	—
44	C₄×C₄₇	#500	1	1	yes	characteristic
45	C₄×C₄₇	C₂×Q₃₂	1	1	yes	characteristic
46	Dic₄₇	—	16	2	no	—
47	Dic₄₇	D₃₂	1	2	yes	—
48	Dic₄₇	Q₆₄	1	2	yes	—
49	#2252	—	47	2	no	—
50	#1423	—	47	1	no	—
51	#1423	—	47	2	no	—
52	#1423	D₄₇	1	1	yes	characteristic
53	C₄×C₃₂	—	47	1	no	—
54	C₂×C₄₇	#2252	1	1	yes	characteristic
55	C₂×C₄₇	C₂×D₃₂	1	1	yes	characteristic
56	C₂×C₄₇	C₂×Q₆₄	1	1	yes	characteristic
57	#500	—	94	2	no	—
58	C₂×C₃₂	—	47	1	no	—
59	C₂×C₃₂	C₂×D₄₇	1	1	yes	characteristic
60	#366	—	47	2	no	—
61	#366	—	94	2	no	—
62	#366	C₂×D₄₇	1	2	yes	—
63	C₄×C₁₆	—	47	1	no	—
64	C₄₇	#13731	1	1	yes	characteristic, complemented, Sylow
65	C₄⋊Q₈	—	188	2	no	—
66	C₂×C₁₆	—	47	1	no	—
67	C₂×C₁₆	C₂²×D₄₇	1	1	yes	characteristic
68	#107	—	2	2	no	—
69	#107	—	94	2	no	—
70	#107	—	188	2	no	—
71	C₄×C₈	—	47	1	no	—
72	C₃₂	—	47	2	no	—
73	C₃₂	Dic₉₄	1	2	yes	complemented
74	C₂×Q₈	—	376	2	no	—
75	C₂×C₈	—	47	1	no	—
76	C₂×C₈	D₄×D₄₇	1	1	yes	characteristic
77	C₄⋊C₄	—	4	2	no	—
78	C₄⋊C₄	—	376	2	no	—
79	C₄⋊C₄	—	188	2	no	—
80	C₄²	—	47	1	no	—
81	C₁₆	—	47	2	no	—
82	C₁₆	C₂×Dic₉₄	1	1	yes	characteristic
83	C₁₆	Q₈×D₄₇	1	1	yes	characteristic
84	Q₈	—	752	4	no	—
85	C₂×C₄	—	8	2	no	—
86	C₂×C₄	—	47	1	no	—
87	C₂×C₄	—	47	1	no	—
88	C₂×C₄	—	376	2	no	—
89	C₂×C₄	D₈×D₄₇	1	1	yes	characteristic
90	C₈	—	47	2	no	—
91	C₈	#13094023	1	1	yes	characteristic
92	C₈	Q₁₆×D₄₇	1	1	yes	characteristic
93	C₂²	D₁₆×D₄₇	1	1	yes	characteristic
94	C₄	—	16	2	no	—
95	C₄	—	47	2	no	—
96	C₄	—	47	2	no	—
97	C₄	—	752	2	no	—
98	C₄	#15689527	1	1	yes	characteristic
99	C₄	Q₃₂×D₄₇	1	1	yes	characteristic
100	C₂	#20245662	1	1	yes	characteristic
101	C₂	D₃₂×D₄₇	1	1	yes	characteristic
102	C₂	Q₆₄×D₄₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20245662

C₂

yes

maximal

#20241588

C₂

yes

maximal, characteristic

#20241587

C₂

yes

maximal

C₃₂×Dic₄₇

C₂

yes

maximal, characteristic

C₄₇×#1423

C₂

yes

maximal, characteristic

#15689527

—

C₂×C₄₇⋊C₃₂

C₂²

yes

characteristic

#15688949

C₂²

yes

—

#15688948

—

C₁₆×Dic₄₇

C₂²

yes

characteristic

C₂×C₃₂×C₄₇

C₂²

yes

characteristic

C₄₇×#366

C₂²

yes

—

#13094023

—

C₂×C₄₇⋊C₁₆

D₄

yes

characteristic

#13093958

—

#13093957

—

C₈×Dic₄₇

D₄

yes

characteristic

C₄₇⋊C₃₂

Q₈

yes

complemented

C₂×C₁₆×C₄₇

C₂³

yes

characteristic

#107×C₄₇

—

C₃₂×C₄₇

Q₈

yes

complemented

C₂×Dic₉₄

—

C₂×C₈×C₄₇

C₂×D₄

yes

characteristic

C₄⋊C₄×C₄₇

—

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

C₄×Dic₄₇

D₈

yes

characteristic

C₂×C₄₇⋊C₈

D₈

yes

characteristic

C₁₆×C₄₇

C₂×Q₈

yes

characteristic

C₁₆×C₄₇

C₂×Q₈

yes

characteristic

C₄₇⋊C₁₆

Q₁₆

yes

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₈

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

D₁₆

yes

characteristic

C₂×Dic₄₇

D₁₆

yes

characteristic

Dic₉₄

—

C₈×C₄₇

C₄⋊Q₈

yes

characteristic

C₈×C₄₇

C₂×Q₁₆

yes

characteristic

C₄₇⋊C₈

Q₃₂

yes

—

#13731

—

maximal, Sylow

C₂²×C₄₇

C₂×D₁₆

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

#500

yes

characteristic

C₄×C₄₇

C₂×Q₃₂

yes

characteristic

Dic₄₇

—

Dic₄₇

D₃₂

yes

—

Dic₄₇

Q₆₄

yes

—

—

—

—

D₄₇

yes

characteristic

C₄×C₃₂

—

C₂×C₄₇

#2252

yes

characteristic

C₂×C₄₇

C₂×D₃₂

yes

characteristic

C₂×C₄₇

C₂×Q₆₄

yes

characteristic

#500

—

C₂×C₃₂

—

C₂×C₃₂

C₂×D₄₇

yes

—

—

C₂×D₄₇

yes

—

C₄×C₁₆

—

C₄₇

#13731

yes

characteristic, complemented, Sylow

C₄⋊Q₈

—

188

—

C₂×C₁₆

—

C₂×C₁₆

C₂²×D₄₇

yes

—

—

—

188

—

C₄×C₈

—

C₃₂

—

C₃₂

Dic₉₄

yes

complemented

C₂×Q₈

—

376

—

C₂×C₈

—

C₂×C₈

D₄×D₄₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

376

—

C₄⋊C₄

—

188

—

C₄²

—

C₁₆

—

C₁₆

C₂×Dic₉₄

yes

characteristic

C₁₆

Q₈×D₄₇

yes

characteristic

Q₈

—

752

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

376

—

C₂×C₄

D₈×D₄₇

yes

characteristic

C₈

—

C₈

#13094023

yes

characteristic

C₈

Q₁₆×D₄₇

yes

characteristic

C₂²

D₁₆×D₄₇

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

752

—

C₄

#15689527

yes

characteristic

C₄

Q₃₂×D₄₇

yes

characteristic

100

C₂

#20245662

yes

characteristic

101

C₂

D₃₂×D₄₇

yes

characteristic

102

C₂

Q₆₄×D₄₇

yes

characteristic

	subgroup	G/H
center	C₂²	D₁₆×D₄₇
commutator, cosocle	C₂×C₁₆×C₄₇	C₂³
Frattini	C₂×C₁₆	C₂²×D₄₇
Fitting	C₄₇×#1423	C₂
socle, prime radical	C₂²×C₄₇	C₂×D₁₆
hypercenter	#1423	D₄₇
hypocenter	C₄₇	#13731
LCS	G ▷ C₂×C₁₆×C₄₇ ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1423

subgroup

G/H

center

C₂²

D₁₆×D₄₇

commutator, cosocle

C₂×C₁₆×C₄₇

C₂³

Frattini

C₂×C₁₆

C₂²×D₄₇

Fitting

C₄₇×#1423

C₂

socle, prime radical

C₂²×C₄₇

C₂×D₁₆

D₄₇

C₄₇

G ▷ C₂×C₁₆×C₄₇ ▷ C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆ ◁ #1423

N	H ≅ G/N	type
18. C₄₇⋊C₃₂	Q₈	semidirect
21. C₃₂×C₄₇	Q₈	semidirect
64. C₄₇	#13731	semidirect
73. C₃₂	Dic₉₄	semidirect

H ≅ G/N

type

18. C₄₇⋊C₃₂

Q₈

semidirect

21. C₃₂×C₄₇

Q₈

semidirect

64. C₄₇

#13731

semidirect

73. C₃₂

Dic₉₄

semidirect

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

subgroup	index
#20245662	2
#20241588	2
#20241587	2
C₃₂×Dic₄₇	2
C₄₇×#1423	2
#13731	47

subgroup

index

#20245662

#20241588

#20241587

C₃₂×Dic₄₇

C₄₇×#1423

#13731

Subgroups — #26137297

Conjugacy classes of subgroups

Complements

Maximal subgroups