Group #20251657 — subgroups

Subgroups — #20251657

Order 6016.

348 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 133 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	#15689062	C₂	1	3	yes	maximal
2	#15689056	C₂	1	6	yes	maximal
3	#15689043	C₂	1	3	yes	maximal
4	#15688991	C₂	1	2	yes	maximal
5	C₄₇×#384	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₉₄	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₄₇	C₂²	1	3	yes	—
8	C₂×Dic₄₇⋊C₄	C₂²	1	6	yes	—
9	C₄₇⋊#95	C₂²	1	6	yes	—
10	C₄₇⋊#95	C₂²	1	6	yes	—
11	C₄₇⋊#95	C₂²	1	3	yes	—
12	C₄₇⋊#95	C₂²	1	3	yes	—
13	C₂×C₄⋊C₄×C₄₇	C₂²	1	3	yes	—
14	C₂×C₄⋊C₄×C₄₇	C₂²	1	3	yes	—
15	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
16	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
17	C₂²×C₄×C₄₇	C₂³	1	3	yes	—
18	C₂²×C₄×C₄₇	C₂³	1	3	yes	—
19	C₂²×Dic₄₇	C₂³	1	6	yes	—
20	C₂²×Dic₄₇	C₂³	1	2	yes	—
21	C₂×Dic₉₄	—	4	3	no	—
22	C₄⋊C₄×C₄₇	—	4	3	no	—
23	C₄⋊C₄×C₄₇	—	4	3	no	—
24	C₄²×C₄₇	Q₈	1	4	yes	complemented
25	C₄⋊Dic₄₇	—	4	3	no	—
26	Dic₄₇⋊C₄	—	4	6	no	—
27	C₂³×C₄₇	C₂⁴	1	1	yes	characteristic
28	C₂×C₄×C₄₇	—	2	3	no	—
29	C₂×C₄×C₄₇	—	2	3	no	—
30	C₂×C₄×C₄₇	—	2	6	no	—
31	C₂×C₄×C₄₇	—	2	3	no	—
32	C₂×C₄×C₄₇	—	2	3	no	—
33	C₂×C₄×C₄₇	C₂×Q₈	1	6	yes	—
34	C₂×Dic₄₇	—	2	6	no	—
35	C₂×Dic₄₇	—	2	6	no	—
36	C₂×Dic₄₇	—	2	12	no	—
37	Dic₉₄	—	4	4	no	—
38	C₂²×C₄₇	C₂²×Q₈	1	1	yes	characteristic
39	C₂²×C₄₇	H(5,𝔽₂)	1	3	yes	—
40	C₂²×C₄₇	2₋¹⁺⁴	1	3	yes	—
41	C₄×C₄₇	—	2	6	no	—
42	C₄×C₄₇	—	4	1	no	—
43	C₄×C₄₇	—	4	3	no	—
44	Dic₄₇	—	4	2	no	—
45	Dic₄₇	—	4	6	no	—
46	#2843	—	47	1	no	maximal, Sylow
47	C₂×C₄₇	#544	1	3	yes	—
48	C₂×C₄₇	#560	1	1	yes	characteristic
49	C₂×C₄₇	#563	1	3	yes	—
50	#400	—	47	3	no	—
51	#398	—	47	6	no	—
52	#395	—	47	3	no	—
53	#384	D₄₇	1	1	yes	characteristic
54	#383	—	47	2	no	—
55	C₄₇	#2843	1	1	yes	characteristic, complemented, Sylow
56	C₂²×Q₈	—	47	1	no	—
57	C₂×C₄⋊C₄	—	47	6	no	—
58	C₂×C₄⋊C₄	—	47	3	no	—
59	C₂×C₄⋊C₄	C₂×D₄₇	1	3	yes	—
60	C₂×C₄⋊C₄	C₂×D₄₇	1	3	yes	—
61	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
62	#95	—	47	6	no	—
63	#95	—	47	6	no	—
64	#95	—	47	3	no	—
65	#95	—	47	3	no	—
66	C₂×Q₈	—	188	3	no	—
67	C₂²×C₄	—	47	6	no	—
68	C₂²×C₄	—	47	2	no	—
69	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
70	C₂²×C₄	C₂²×D₄₇	1	3	yes	—
71	C₂²×C₄	C₂²×D₄₇	1	3	yes	—
72	C₄⋊C₄	—	4	3	no	—
73	C₄⋊C₄	—	4	3	no	—
74	C₄⋊C₄	—	188	6	no	—
75	C₄⋊C₄	—	188	3	no	—
76	C₄²	Dic₉₄	1	4	yes	complemented
77	C₂³	C₂³×D₄₇	1	1	yes	characteristic
78	Q₈	—	188	4	no	—
79	C₂×C₄	—	2	3	no	—
80	C₂×C₄	—	2	3	no	—
81	C₂×C₄	—	2	6	no	—
82	C₂×C₄	—	2	3	no	—
83	C₂×C₄	—	2	3	no	—
84	C₂×C₄	—	94	6	no	—
85	C₂×C₄	—	94	6	no	—
86	C₂×C₄	—	94	12	no	—
87	C₂×C₄	C₂×Dic₉₄	1	6	yes	—
88	C₂²	C₂²×Dic₉₄	1	1	yes	characteristic
89	C₂²	#13094069	1	3	yes	—
90	C₂²	#13094072	1	3	yes	—
91	C₄	—	2	6	no	—
92	C₄	—	4	1	no	—
93	C₄	—	4	3	no	—
94	C₄	—	188	2	no	—
95	C₄	—	188	6	no	—
96	C₂	#15689754	1	3	yes	—
97	C₂	#15689860	1	1	yes	characteristic
98	C₂	#15689872	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#15689062

C₂

yes

maximal

#15689056

C₂

yes

maximal

#15689043

C₂

yes

maximal

#15688991

C₂

yes

maximal

C₄₇×#384

C₂

yes

maximal, characteristic

C₂²×Dic₉₄

C₂²

yes

characteristic

C₂×C₄⋊Dic₄₇

C₂²

yes

—

C₂×Dic₄₇⋊C₄

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₄₇⋊#95

C₂²

yes

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄⋊C₄×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×C₄×C₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂²×Dic₄₇

C₂³

yes

—

C₂×Dic₉₄

—

C₄⋊C₄×C₄₇

—

C₄⋊C₄×C₄₇

—

C₄²×C₄₇

Q₈

yes

complemented

C₄⋊Dic₄₇

—

Dic₄₇⋊C₄

—

C₂³×C₄₇

C₂⁴

yes

characteristic

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

—

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

C₂²×C₄₇

C₂²×Q₈

yes

characteristic

C₂²×C₄₇

H(5,𝔽₂)

yes

—

C₂²×C₄₇

2₋¹⁺⁴

yes

—

C₄×C₄₇

—

C₄×C₄₇

—

C₄×C₄₇

—

Dic₄₇

—

Dic₄₇

—

#2843

—

maximal, Sylow

C₂×C₄₇

#544

yes

—

C₂×C₄₇

#560

yes

characteristic

C₂×C₄₇

#563

yes

—

—

—

—

D₄₇

yes

characteristic

#383

—

C₄₇

#2843

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

C₂×C₄⋊C₄

C₂×D₄₇

yes

—

C₂×C₄²

C₂×D₄₇

yes

—

—

—

—

C₂×Q₈

—

188

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₄₇

yes

characteristic

C₂²×C₄

C₂²×D₄₇

yes

—

C₂²×C₄

C₂²×D₄₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

188

—

C₄⋊C₄

—

188

—

C₄²

Dic₉₄

yes

complemented

C₂³

C₂³×D₄₇

yes

characteristic

Q₈

—

188

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×Dic₉₄

yes

—

C₂²

C₂²×Dic₉₄

yes

characteristic

C₂²

#13094069

yes

—

C₂²

#13094072

yes

—

C₄

—

C₄

—

C₄

—

C₄

—

188

—

C₄

—

188

—

C₂

#15689754

yes

—

C₂

#15689860

yes

characteristic

C₂

#15689872

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₄₇
commutator, socle	C₂³×C₄₇	C₂⁴
Fitting	C₄₇×#384	C₂
hypercenter	#384	D₄₇
hypocenter	C₄₇	#2843
LCS	G ▷ C₂³×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ #384

subgroup

G/H

center, Frattini

C₂³

C₂³×D₄₇

commutator, socle

C₂³×C₄₇

C₂⁴

Fitting

C₄₇×#384

C₂

D₄₇

C₄₇

G ▷ C₂³×C₄₇ ▷ C₄₇

C₁ ◁ C₂³ ◁ #384

N	H ≅ G/N	type
24. C₄²×C₄₇	Q₈	semidirect
55. C₄₇	#2843	semidirect
76. C₄²	Dic₉₄	semidirect

H ≅ G/N

type

24. C₄²×C₄₇

Q₈

semidirect

55. C₄₇

#2843

semidirect

76. C₄²

Dic₉₄

semidirect

Subgroups — #20251657

Conjugacy classes of subgroups

Complements