Group #20245405 — subgroups

Subgroups — #20245405

Order 6016.

248 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 45 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	#15689295	C₂	1	3	yes	maximal, complemented
2	#15688920	C₂	1	3	yes	maximal, complemented
3	C₄₇×H(3,𝔽₄)	C₂	1	1	yes	maximal, characteristic, complemented
4	#13094072	C₂²	1	1	yes	characteristic, complemented
5	#13094050	—	2	3	no	—
6	#13093952	—	2	3	no	—
7	#13093943	C₂²	1	3	yes	complemented
8	#124×C₄₇	—	2	3	no	—
9	#124×C₄₇	C₂²	1	3	yes	—
10	#120×C₄₇	—	2	3	no	—
11	C₂⁴×C₄₇	—	2	1	no	—
12	C₂×Q₈×C₄₇	—	2	1	no	—
13	C₂×Q₈×C₄₇	C₂³	1	1	yes	characteristic
14	C₂×D₄×C₄₇	—	2	3	no	—
15	C₂×D₄×C₄₇	D₄	1	3	yes	—
16	#10625354	—	4	1	no	—
17	Q₈×D₄₇	—	4	1	no	—
18	#10625350	—	2	3	no	—
19	C₂²⋊C₄×C₄₇	—	2	3	no	—
20	C₂²⋊C₄×C₄₇	—	2	6	no	—
21	C₄²×C₄₇	D₄	1	3	yes	—
22	#10625335	—	4	3	no	—
23	#10625334	—	4	3	no	—
24	#10625329	—	2	3	no	—
25	C₂³×C₄₇	—	2	3	no	—
26	C₂³×C₄₇	—	8	2	no	—
27	C₂³×C₄₇	—	8	1	no	—
28	Q₈×C₄₇	—	4	1	no	—
29	Q₈×C₄₇	—	8	1	no	—
30	D₄×C₄₇	—	4	3	no	—
31	D₄×C₄₇	—	8	3	no	—
32	C₂×C₄×C₄₇	—	2	3	no	—
33	C₂×C₄×C₄₇	C₂×D₄	1	3	yes	—
34	C₂²⋊D₄₇	—	4	1	no	—
35	D₁₈₈	—	2	3	no	—
36	C₄×D₄₇	—	4	3	no	—
37	Dic₉₄	—	2	3	no	—
38	C₄₇⋊C₈	—	4	3	no	—
39	C₂²×C₄₇	—	4	6	no	—
40	C₂²×C₄₇	—	4	3	no	—
41	C₂²×C₄₇	—	8	4	no	—
42	C₂²×C₄₇	#120	1	1	yes	characteristic
43	C₂×D₄₇	—	4	1	no	—
44	C₄×C₄₇	—	2	3	no	—
45	C₄×C₄₇	—	4	3	no	—
46	Dic₄₇	—	4	1	no	—
47	#2202	—	47	1	no	maximal, Sylow
48	C₂×C₄₇	—	2	1	no	—
49	C₂×C₄₇	—	8	3	no	—
50	C₂×C₄₇	B(4,𝔽₂)	1	1	yes	characteristic
51	D₄₇	—	8	1	no	—
52	H(3,𝔽₄)	D₄₇	1	1	yes	characteristic, complemented
53	#454	—	47	3	no	—
54	#355	—	47	3	no	—
55	C₄₇	#2202	1	1	yes	characteristic, complemented, Sylow
56	2₋¹⁺⁴	—	47	1	no	—
57	AGL(1,ℤ/8)	—	94	3	no	—
58	#124	—	2	3	no	—
59	#124	C₂×D₄₇	1	3	yes	—
60	#120	—	2	3	no	—
61	#104	—	94	3	no	—
62	#101	—	47	3	no	—
63	C₂⁴	—	2	1	no	—
64	#41	—	94	3	no	—
65	#41	—	188	1	no	—
66	C₂×Q₈	—	2	1	no	—
67	C₂×Q₈	—	188	1	no	—
68	C₂×Q₈	C₂²×D₄₇	1	1	yes	characteristic
69	C₂×D₄	—	2	3	no	—
70	C₂×D₄	C₂²⋊D₄₇	1	3	yes	—
71	SD₁₆	—	188	3	no	—
72	D₈	—	188	3	no	—
73	M₁₆	—	94	3	no	—
74	C₂²⋊C₄	—	2	3	no	—
75	C₂²⋊C₄	—	2	6	no	—
76	C₄²	C₂²⋊D₄₇	1	3	yes	—
77	C₂³	—	2	3	no	—
78	C₂³	—	8	2	no	—
79	C₂³	—	8	1	no	—
80	Q₈	—	4	1	no	—
81	Q₈	—	8	1	no	—
82	Q₈	—	94	3	no	—
83	D₄	—	4	3	no	—
84	D₄	—	8	3	no	—
85	D₄	—	94	3	no	—
86	D₄	—	188	1	no	—
87	C₂×C₄	—	2	3	no	—
88	C₂×C₄	—	188	3	no	—
89	C₂×C₄	C₂×C₂²⋊D₄₇	1	3	yes	—
90	C₈	—	188	3	no	—
91	C₂²	—	4	6	no	—
92	C₂²	—	4	3	no	—
93	C₂²	—	8	4	no	—
94	C₂²	—	188	1	no	—
95	C₂²	#13093993	1	1	yes	characteristic
96	C₄	—	2	3	no	—
97	C₄	—	4	3	no	—
98	C₄	—	188	1	no	—
99	C₂	—	2	1	no	—
100	C₂	—	8	3	no	—
101	C₂	—	376	1	no	—
102	C₂	#15689310	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689295

C₂

yes

maximal, complemented

#15688920

C₂

yes

maximal, complemented

C₄₇×H(3,𝔽₄)

C₂

yes

maximal, characteristic, complemented

#13094072

C₂²

yes

characteristic, complemented

#13094050

—

#13093952

—

#13093943

C₂²

yes

complemented

#124×C₄₇

—

#124×C₄₇

C₂²

yes

—

#120×C₄₇

—

C₂⁴×C₄₇

—

C₂×Q₈×C₄₇

—

C₂×Q₈×C₄₇

C₂³

yes

characteristic

C₂×D₄×C₄₇

—

C₂×D₄×C₄₇

D₄

yes

—

#10625354

—

Q₈×D₄₇

—

#10625350

—

C₂²⋊C₄×C₄₇

—

C₂²⋊C₄×C₄₇

—

C₄²×C₄₇

D₄

yes

—

#10625335

—

#10625334

—

#10625329

—

C₂³×C₄₇

—

C₂³×C₄₇

—

C₂³×C₄₇

—

Q₈×C₄₇

—

Q₈×C₄₇

—

D₄×C₄₇

—

D₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂²⋊D₄₇

—

D₁₈₈

—

C₄×D₄₇

—

Dic₉₄

—

C₄₇⋊C₈

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

—

C₂²×C₄₇

#120

yes

characteristic

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

Dic₄₇

—

#2202

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

—

C₂×C₄₇

B(4,𝔽₂)

yes

characteristic

D₄₇

—

H(3,𝔽₄)

D₄₇

yes

characteristic, complemented

#454

—

#355

—

C₄₇

#2202

yes

characteristic, complemented, Sylow

2₋¹⁺⁴

—

AGL(1,ℤ/8)

—

#124

—

#124

C₂×D₄₇

yes

—

#120

—

#104

—

#101

—

C₂⁴

—

#41

—

#41

—

188

—

C₂×Q₈

—

C₂×Q₈

—

188

—

C₂×Q₈

C₂²×D₄₇

yes

characteristic

C₂×D₄

—

C₂×D₄

C₂²⋊D₄₇

yes

—

SD₁₆

—

188

—

D₈

—

188

—

M₁₆

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂²⋊D₄₇

yes

—

C₂³

—

C₂³

—

C₂³

—

Q₈

—

Q₈

—

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

188

—

C₂×C₄

—

C₂×C₄

—

188

—

C₂×C₄

C₂×C₂²⋊D₄₇

yes

—

C₈

—

188

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

188

—

C₂²

#13093993

yes

characteristic

C₄

—

C₄

—

C₄

—

188

—

C₂

—

100

C₂

—

101

C₂

—

376

—

102

C₂

#15689310

yes

characteristic

	subgroup	G/H
center	C₂	#15689310
commutator	C₂×Q₈×C₄₇	C₂³
Frattini	C₂×Q₈	C₂²×D₄₇
Fitting	C₄₇×H(3,𝔽₄)	C₂
socle	C₂×C₄₇	B(4,𝔽₂)
hypercenter	H(3,𝔽₄)	D₄₇
hypocenter	C₄₇	#2202
LCS	G ▷ C₂×Q₈×C₄₇ ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ H(3,𝔽₄)

subgroup

G/H

center

C₂

#15689310

commutator

C₂×Q₈×C₄₇

C₂³

Frattini

C₂×Q₈

C₂²×D₄₇

Fitting

C₄₇×H(3,𝔽₄)

C₂

C₂×C₄₇

D₄₇

C₄₇

G ▷ C₂×Q₈×C₄₇ ▷ C₂²×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂ ◁ C₂² ◁ C₂×Q₈ ◁ H(3,𝔽₄)

N	H ≅ G/N	type
1. #15689295	C₂	semidirect
2. #15688920	C₂	semidirect
3. C₄₇×H(3,𝔽₄)	C₂	semidirect
4. #13094072	C₂²	semidirect
7. #13093943	C₂²	semidirect
52. H(3,𝔽₄)	D₄₇	semidirect
55. C₄₇	#2202	semidirect

H ≅ G/N

type

1. #15689295

C₂

semidirect

2. #15688920

C₂

semidirect

3. C₄₇×H(3,𝔽₄)

C₂

semidirect

4. #13094072

C₂²

semidirect

7. #13093943

C₂²

semidirect

52. H(3,𝔽₄)

D₄₇

semidirect

55. C₄₇

#2202

semidirect

Subgroups — #20245405

Conjugacy classes of subgroups

Complements