Group #20245556 — subgroups

Subgroups — #20245556

Order 6016.

126 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 126 conjugacy classes of subgroups — too many to draw. Only its 47 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	#15689241	C₂	1	1	yes	maximal, characteristic
2	#15688945	C₂	1	2	yes	maximal, complemented
3	#15688944	C₂	1	1	yes	maximal, characteristic, complemented
4	#15688932	C₂	1	2	yes	maximal
5	C₄₇×#498	C₂	1	1	yes	maximal, characteristic, complemented
6	#13094046	C₂²	1	1	yes	characteristic
7	#13094030	—	2	1	no	—
8	#13093964	C₂²	1	2	yes	—
9	#13093960	C₂²	1	1	yes	characteristic
10	#13093952	—	2	1	no	—
11	C₄⋊Q₈×C₄₇	—	2	1	no	—
12	#106×C₄₇	—	2	1	no	—
13	#106×C₄₇	C₂²	1	2	yes	—
14	C₄×C₈×C₄₇	C₂²	1	1	yes	characteristic
15	C₂×Q₈×C₄₇	—	4	1	no	—
16	#10625350	—	2	1	no	—
17	C₂×C₈×C₄₇	D₄	1	1	yes	characteristic
18	C₂×C₈×C₄₇	C₂³	1	1	yes	characteristic
19	C₄⋊C₄×C₄₇	—	2	2	no	—
20	C₄⋊C₄×C₄₇	—	4	1	no	—
21	C₄²×C₄₇	D₄	1	1	yes	characteristic
22	#10625329	—	2	1	no	—
23	Dic₁₈₈	Q₈	1	1	yes	characteristic, complemented
24	D₃₇₆	Q₈	1	1	yes	characteristic, complemented
25	#10625325	—	2	1	no	—
26	#10625324	—	4	1	no	—
27	C₈×D₄₇	—	4	1	no	—
28	C₄₇⋊C₁₆	—	2	2	no	—
29	Q₈×C₄₇	—	8	2	no	—
30	C₂×C₄×C₄₇	—	2	1	no	—
31	C₂×C₄×C₄₇	—	4	2	no	—
32	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
33	C₂²⋊D₄₇	—	4	1	no	—
34	D₁₈₈	—	2	1	no	—
35	C₄×D₄₇	—	4	1	no	—
36	Dic₉₄	—	2	1	no	—
37	C₈×C₄₇	SD₁₆	1	1	yes	characteristic
38	C₈×C₄₇	SD₁₆	1	1	yes	characteristic
39	C₈×C₄₇	C₂×Q₈	1	1	yes	characteristic
40	C₈×C₄₇	#41	1	1	yes	characteristic
41	C₄₇⋊C₈	—	4	1	no	—
42	C₂²×C₄₇	AGL(1,ℤ/8)	1	1	yes	characteristic
43	C₂×D₄₇	—	4	1	no	—
44	C₄×C₄₇	—	2	2	no	—
45	C₄×C₄₇	—	8	2	no	—
46	C₄×C₄₇	C₂²⋊Q₈	1	1	yes	characteristic
47	C₄×C₄₇	C₂×SD₁₆	1	1	yes	characteristic
48	Dic₄₇	—	4	1	no	—
49	#2230	—	47	1	no	maximal, Sylow
50	C₂×C₄₇	—	2	1	no	—
51	C₂×C₄₇	#476	1	1	yes	characteristic
52	D₄₇	—	8	1	no	—
53	#498	D₄₇	1	1	yes	characteristic, complemented
54	#443	—	47	1	no	—
55	#365	—	47	2	no	—
56	#364	—	47	1	no	—
57	#360	—	47	2	no	—
58	C₄₇	#2230	1	1	yes	characteristic, complemented, Sylow
59	#135	—	47	1	no	—
60	#131	—	94	1	no	—
61	C₄⋊Q₈	—	2	1	no	—
62	M₃₂	—	47	2	no	—
63	#108	—	47	1	no	—
64	#106	—	2	1	no	—
65	#106	C₂×D₄₇	1	2	yes	—
66	#104	—	94	1	no	—
67	C₄×C₈	C₂×D₄₇	1	1	yes	characteristic
68	#41	—	94	1	no	—
69	C₂×Q₈	—	4	1	no	—
70	Q₁₆	—	47	1	no	—
71	SD₁₆	—	94	1	no	—
72	D₈	—	47	1	no	—
73	M₁₆	—	94	1	no	—
74	M₁₆	—	188	1	no	—
75	C₂×C₈	—	188	1	no	—
76	C₂×C₈	C₂²⋊D₄₇	1	1	yes	characteristic
77	C₂×C₈	C₂²×D₄₇	1	1	yes	characteristic
78	C₄⋊C₄	—	2	2	no	—
79	C₄⋊C₄	—	4	1	no	—
80	C₄²	C₂²⋊D₄₇	1	1	yes	characteristic
81	C₁₆	—	94	2	no	—
82	Q₈	—	8	2	no	—
83	Q₈	—	94	1	no	—
84	D₄	—	94	1	no	—
85	D₄	—	188	1	no	—
86	C₂×C₄	—	2	1	no	—
87	C₂×C₄	—	4	2	no	—
88	C₂×C₄	—	188	1	no	—
89	C₂×C₄	C₂×C₂²⋊D₄₇	1	1	yes	characteristic
90	C₈	—	188	1	no	—
91	C₈	#10625336	1	1	yes	characteristic
92	C₈	#10625336	1	1	yes	characteristic
93	C₈	Q₈×D₄₇	1	1	yes	characteristic
94	C₈	#10625354	1	1	yes	characteristic
95	C₂²	—	188	1	no	—
96	C₂²	#13094050	1	1	yes	characteristic
97	C₄	—	2	2	no	—
98	C₄	—	8	2	no	—
99	C₄	—	188	1	no	—
100	C₄	#13094000	1	1	yes	characteristic
101	C₄	C₂×#10625336	1	1	yes	characteristic
102	C₂	—	2	1	no	—
103	C₂	—	376	1	no	—
104	C₂	#15689410	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#15689241

C₂

yes

maximal, characteristic

#15688945

C₂

yes

maximal, complemented

#15688944

C₂

yes

maximal, characteristic, complemented

#15688932

C₂

yes

maximal

C₄₇×#498

C₂

yes

maximal, characteristic, complemented

#13094046

C₂²

yes

characteristic

#13094030

—

#13093964

C₂²

yes

—

#13093960

C₂²

yes

characteristic

#13093952

—

C₄⋊Q₈×C₄₇

—

#106×C₄₇

—

#106×C₄₇

C₂²

yes

—

C₄×C₈×C₄₇

C₂²

yes

characteristic

C₂×Q₈×C₄₇

—

#10625350

—

C₂×C₈×C₄₇

D₄

yes

characteristic

C₂×C₈×C₄₇

C₂³

yes

characteristic

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

D₄

yes

characteristic

#10625329

—

Dic₁₈₈

Q₈

yes

characteristic, complemented

D₃₇₆

Q₈

yes

characteristic, complemented

#10625325

—

#10625324

—

C₈×D₄₇

—

C₄₇⋊C₁₆

—

Q₈×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂²⋊D₄₇

—

D₁₈₈

—

C₄×D₄₇

—

Dic₉₄

—

C₈×C₄₇

SD₁₆

yes

characteristic

C₈×C₄₇

SD₁₆

yes

characteristic

C₈×C₄₇

C₂×Q₈

yes

characteristic

C₈×C₄₇

#41

yes

characteristic

C₄₇⋊C₈

—

C₂²×C₄₇

AGL(1,ℤ/8)

yes

characteristic

C₂×D₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

C₂²⋊Q₈

yes

characteristic

C₄×C₄₇

C₂×SD₁₆

yes

characteristic

Dic₄₇

—

#2230

—

maximal, Sylow

C₂×C₄₇

—

C₂×C₄₇

#476

yes

characteristic

D₄₇

—

#498

D₄₇

yes

characteristic, complemented

—

—

—

—

C₄₇

#2230

yes

characteristic, complemented, Sylow

#135

—

#131

—

C₄⋊Q₈

—

M₃₂

—

#108

—

#106

—

#106

C₂×D₄₇

yes

—

#104

—

C₄×C₈

C₂×D₄₇

yes

characteristic

#41

—

C₂×Q₈

—

Q₁₆

—

SD₁₆

—

D₈

—

M₁₆

—

M₁₆

—

188

—

C₂×C₈

—

188

—

C₂×C₈

C₂²⋊D₄₇

yes

characteristic

C₂×C₈

C₂²×D₄₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²⋊D₄₇

yes

characteristic

C₁₆

—

Q₈

—

Q₈

—

D₄

—

D₄

—

188

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

188

—

C₂×C₄

C₂×C₂²⋊D₄₇

yes

characteristic

C₈

—

188

—

C₈

yes

C₈

yes

C₈

Q₈×D₄₇

yes

characteristic

C₈

#10625354

yes

characteristic

C₂²

—

188

—

C₂²

#13094050

yes

characteristic

C₄

—

C₄

—

C₄

—

188

—

100

C₄

#13094000

yes

characteristic

101

C₄

C₂×#10625336

yes

characteristic

102

C₂

—

103

C₂

—

376

—

104

C₂

#15689410

yes

characteristic

	subgroup	G/H
center	C₂	#15689410
commutator	C₂×C₈×C₄₇	C₂³
Frattini	C₂×C₈	C₂²×D₄₇
Fitting	C₄₇×#498	C₂
socle	C₂×C₄₇	#476
hypercenter	#498	D₄₇
hypocenter	C₄₇	#2230
LCS	G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #498

subgroup

G/H

center

C₂

#15689410

commutator

C₂×C₈×C₄₇

C₂³

Frattini

C₂×C₈

C₂²×D₄₇

Fitting

C₄₇×#498

C₂

C₂×C₄₇

D₄₇

C₄₇

G ▷ C₂×C₈×C₄₇ ▷ C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₄×C₈ ◁ #498

N	H ≅ G/N	type
2. #15688945	C₂	semidirect
3. #15688944	C₂	semidirect
5. C₄₇×#498	C₂	semidirect
23. Dic₁₈₈	Q₈	semidirect
24. D₃₇₆	Q₈	semidirect
53. #498	D₄₇	semidirect
58. C₄₇	#2230	semidirect

H ≅ G/N

type

2. #15688945

C₂

semidirect

3. #15688944

C₂

semidirect

5. C₄₇×#498

C₂

semidirect

23. Dic₁₈₈

Q₈

semidirect

24. D₃₇₆

Q₈

semidirect

53. #498

D₄₇

semidirect

58. C₄₇

#2230

semidirect

Subgroups — #20245556

Conjugacy classes of subgroups

Complements