Group #20242126 — subgroups

Subgroups — #20242126

Order 6016.

210 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 85 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	#15689147	C₂	1	2	yes	maximal
2	#15688869	C₂	1	4	yes	maximal, complemented
3	C₄₇×#533	C₂	1	1	yes	maximal, characteristic
4	C₂×#10625329	—	2	2	no	—
5	C₄⋊(C₄₇⋊C₈)	—	2	2	no	—
6	C₄⋊(C₄₇⋊C₈)	C₂²	1	4	yes	—
7	C₄⋊Q₈×C₄₇	C₂²	1	2	yes	—
8	#125×C₄₇	C₄	1	2	yes	—
9	C₂²⋊Q₈×C₄₇	—	2	2	no	—
10	C₄×Q₈×C₄₇	—	2	2	no	—
11	#117×C₄₇	C₄	1	2	yes	—
12	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
13	C₂×Q₈×C₄₇	—	2	2	no	—
14	C₂²×C₄×C₄₇	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₄₇	D₄	1	2	yes	—
16	C₄⋊C₄×C₄₇	—	2	4	no	—
17	C₄⋊C₄×C₄₇	—	2	2	no	—
18	C₄⋊C₄×C₄₇	C₂×C₄	1	4	yes	—
19	C₂²⋊C₄×C₄₇	—	2	2	no	—
20	C₄²×C₄₇	—	2	2	no	—
21	C₄²×C₄₇	C₂×C₄	1	1	yes	characteristic
22	C₄²×C₄₇	D₄	1	2	yes	—
23	C₄²×C₄₇	C₂³	1	1	yes	characteristic
24	#10625329	—	4	4	no	—
25	C₂×C₄₇⋊C₈	—	2	4	no	—
26	C₂³×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
27	Q₈×C₄₇	—	4	2	no	—
28	Q₈×C₄₇	—	4	2	no	—
29	C₂×C₄×C₄₇	—	2	2	no	—
30	C₂×C₄×C₄₇	—	2	2	no	—
31	C₂×C₄×C₄₇	—	2	2	no	—
32	C₂×C₄×C₄₇	—	2	4	no	—
33	C₂×C₄×C₄₇	C₂²⋊C₄	1	2	yes	—
34	C₂×C₄×C₄₇	C₂²⋊C₄	1	1	yes	characteristic
35	C₂×C₄×C₄₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
37	C₄₇⋊C₈	—	4	4	no	—
38	C₂²×C₄₇	—	2	1	no	—
39	C₂²×C₄₇	—	2	2	no	—
40	C₂²×C₄₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
41	C₄×C₄₇	—	2	1	no	—
42	C₄×C₄₇	—	2	2	no	—
43	C₄×C₄₇	—	4	4	no	—
44	C₄×C₄₇	#101	1	2	yes	—
45	C₄×C₄₇	AGL(1,ℤ/8)	1	2	yes	—
46	C₄×C₄₇	#137	1	2	yes	—
47	#1561	—	47	1	no	maximal, Sylow
48	C₂×C₄₇	—	4	1	no	—
49	C₂×C₄₇	C₂×#101	1	1	yes	characteristic
50	C₂×C₄₇	#417	1	2	yes	—
51	#533	D₄₇	1	1	yes	characteristic
52	#423	—	47	2	no	—
53	#332	—	47	4	no	—
54	C₄₇	#1561	1	1	yes	characteristic, complemented, Sylow
55	C₂×M₁₆	—	94	2	no	—
56	C₄⋊Q₈	C₂×D₄₇	1	2	yes	—
57	#125	Dic₄₇	1	2	yes	—
58	C₂²⋊Q₈	—	2	2	no	—
59	C₄×Q₈	—	2	2	no	—
60	#117	Dic₄₇	1	2	yes	—
61	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
62	C₄⋊C₈	—	47	4	no	—
63	C₄⋊C₈	—	94	2	no	—
64	C₂×Q₈	—	2	2	no	—
65	C₂²×C₄	C₂×Dic₄₇	1	1	yes	characteristic
66	C₂²×C₄	C₂²⋊D₄₇	1	2	yes	—
67	M₁₆	—	188	4	no	—
68	C₂×C₈	—	94	4	no	—
69	C₄⋊C₄	—	2	4	no	—
70	C₄⋊C₄	—	2	2	no	—
71	C₄⋊C₄	C₂×Dic₄₇	1	4	yes	—
72	C₂²⋊C₄	—	2	2	no	—
73	C₄²	—	2	2	no	—
74	C₄²	C₂×Dic₄₇	1	1	yes	characteristic
75	C₄²	C₂²⋊D₄₇	1	2	yes	—
76	C₄²	C₂²×D₄₇	1	1	yes	characteristic
77	C₂³	C₂²⋊Dic₄₇	1	1	yes	characteristic
78	Q₈	—	4	2	no	—
79	Q₈	—	4	2	no	—
80	C₂×C₄	—	2	2	no	—
81	C₂×C₄	—	2	2	no	—
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	2	4	no	—
84	C₂×C₄	C₂²⋊Dic₄₇	1	2	yes	—
85	C₂×C₄	C₂²⋊Dic₄₇	1	1	yes	characteristic
86	C₂×C₄	C₂²×Dic₄₇	1	1	yes	characteristic
87	C₂×C₄	C₂×C₂²⋊D₄₇	1	2	yes	—
88	C₈	—	188	4	no	—
89	C₂²	—	2	1	no	—
90	C₂²	—	2	2	no	—
91	C₂²	C₂×C₂²⋊Dic₄₇	1	1	yes	characteristic
92	C₄	—	2	1	no	—
93	C₄	—	2	2	no	—
94	C₄	—	4	4	no	—
95	C₄	#13093943	1	2	yes	—
96	C₄	#13094049	1	2	yes	—
97	C₄	#13094054	1	2	yes	—
98	C₂	—	4	1	no	—
99	C₂	C₂×#13093943	1	1	yes	characteristic
100	C₂	#15689124	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#15689147

C₂

yes

maximal

#15688869

C₂

yes

maximal, complemented

C₄₇×#533

C₂

yes

maximal, characteristic

C₂×#10625329

—

C₄⋊(C₄₇⋊C₈)

—

C₄⋊(C₄₇⋊C₈)

C₂²

yes

—

C₄⋊Q₈×C₄₇

C₂²

yes

—

#125×C₄₇

C₄

yes

—

C₂²⋊Q₈×C₄₇

—

C₄×Q₈×C₄₇

—

#117×C₄₇

C₄

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂×Q₈×C₄₇

—

C₂²×C₄×C₄₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

C₂×C₄

yes

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

—

C₄²×C₄₇

C₂×C₄

yes

characteristic

C₄²×C₄₇

D₄

yes

—

C₄²×C₄₇

C₂³

yes

characteristic

#10625329

—

C₂×C₄₇⋊C₈

—

C₂³×C₄₇

C₂²⋊C₄

yes

characteristic

Q₈×C₄₇

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

—

C₂×C₄×C₄₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₄₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

#101

yes

—

C₄×C₄₇

AGL(1,ℤ/8)

yes

—

C₄×C₄₇

#137

yes

—

#1561

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

C₂×#101

yes

characteristic

C₂×C₄₇

#417

yes

—

#533

D₄₇

yes

characteristic

#423

—

#332

—

C₄₇

#1561

yes

characteristic, complemented, Sylow

C₂×M₁₆

—

C₄⋊Q₈

C₂×D₄₇

yes

—

#125

Dic₄₇

yes

—

C₂²⋊Q₈

—

C₄×Q₈

—

#117

Dic₄₇

yes

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₂×Q₈

—

C₂²×C₄

C₂×Dic₄₇

yes

characteristic

C₂²×C₄

C₂²⋊D₄₇

yes

—

M₁₆

—

188

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×Dic₄₇

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂×Dic₄₇

yes

characteristic

C₄²

C₂²⋊D₄₇

yes

—

C₄²

C₂²×D₄₇

yes

characteristic

C₂³

C₂²⋊Dic₄₇

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Dic₄₇

yes

—

C₂×C₄

C₂²⋊Dic₄₇

yes

characteristic

C₂×C₄

C₂²×Dic₄₇

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

C₈

—

188

—

C₂²

—

C₂²

—

C₂²

C₂×C₂²⋊Dic₄₇

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

#13093943

yes

—

C₄

#13094049

yes

—

C₄

#13094054

yes

—

C₂

—

C₂

C₂×#13093943

yes

characteristic

100

C₂

#15689124

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₄₇
commutator	C₂×C₄×C₄₇	C₂²×C₄
Frattini	C₄²	C₂²×D₄₇
Fitting	C₄₇×#533	C₂
socle	C₂²×C₄₇	C₂×C₂²⋊C₄
hypercenter	#533	D₄₇
hypocenter	C₄₇	#1561
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₄₇

commutator

C₂×C₄×C₄₇

C₂²×C₄

Frattini

C₄²

C₂²×D₄₇

Fitting

C₄₇×#533

C₂

socle

C₂²×C₄₇

C₂×C₂²⋊C₄

D₄₇

C₄₇

G ▷ C₂×C₄×C₄₇ ▷ C₂²×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #533

N	H ≅ G/N	type
2. #15688869	C₂	semidirect
54. C₄₇	#1561	semidirect

H ≅ G/N

type

2. #15688869

C₂

semidirect

54. C₄₇

#1561

semidirect

Subgroups — #20242126

Conjugacy classes of subgroups

Complements