Group #21454088 — subgroups

Subgroups — #21454088

Order 7808.

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	#16839701	C₂	1	3	yes	maximal
2	#16839695	C₂	1	6	yes	maximal
3	#16839682	C₂	1	3	yes	maximal
4	#16839630	C₂	1	2	yes	maximal
5	C₆₁×#384	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₂₂	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊Dic₆₁	C₂²	1	3	yes	—
8	C₂×#11758196	C₂²	1	6	yes	—
9	#14519684	C₂²	1	6	yes	—
10	#14519684	C₂²	1	3	yes	—
11	#14519684	C₂²	1	3	yes	—
12	#14519684	C₂²	1	6	yes	—
13	C₂×C₄⋊C₄×C₆₁	C₂²	1	3	yes	—
14	C₂×C₄⋊C₄×C₆₁	C₂²	1	3	yes	—
15	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
16	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
17	C₂²×C₄×C₆₁	C₂³	1	3	yes	—
18	C₂²×C₄×C₆₁	C₂³	1	3	yes	—
19	C₂²×Dic₆₁	C₂³	1	2	yes	—
20	C₂²×Dic₆₁	C₂³	1	6	yes	—
21	C₂×Dic₁₂₂	—	4	3	no	—
22	C₄⋊C₄×C₆₁	—	4	3	no	—
23	C₄⋊C₄×C₆₁	—	4	3	no	—
24	C₄²×C₆₁	Q₈	1	4	yes	complemented
25	C₄⋊Dic₆₁	—	4	3	no	—
26	#11758196	—	4	6	no	—
27	C₂³×C₆₁	C₂⁴	1	1	yes	characteristic
28	C₂×C₄×C₆₁	—	2	3	no	—
29	C₂×C₄×C₆₁	—	2	3	no	—
30	C₂×C₄×C₆₁	—	2	6	no	—
31	C₂×C₄×C₆₁	—	2	3	no	—
32	C₂×C₄×C₆₁	—	2	3	no	—
33	C₂×C₄×C₆₁	C₂×Q₈	1	6	yes	—
34	C₂×Dic₆₁	—	2	6	no	—
35	C₂×Dic₆₁	—	2	6	no	—
36	C₂×Dic₆₁	—	2	12	no	—
37	Dic₁₂₂	—	4	4	no	—
38	C₂²×C₆₁	C₂²×Q₈	1	1	yes	characteristic
39	C₂²×C₆₁	H(5,𝔽₂)	1	3	yes	—
40	C₂²×C₆₁	2₋¹⁺⁴	1	3	yes	—
41	C₄×C₆₁	—	2	6	no	—
42	C₄×C₆₁	—	4	1	no	—
43	C₄×C₆₁	—	4	3	no	—
44	Dic₆₁	—	4	2	no	—
45	Dic₆₁	—	4	6	no	—
46	#2843	—	61	1	no	maximal, Sylow
47	C₂×C₆₁	#544	1	3	yes	—
48	C₂×C₆₁	#560	1	1	yes	characteristic
49	C₂×C₆₁	#563	1	3	yes	—
50	#400	—	61	3	no	—
51	#398	—	61	6	no	—
52	#395	—	61	3	no	—
53	#384	D₆₁	1	1	yes	characteristic
54	#383	—	61	2	no	—
55	C₆₁	#2843	1	1	yes	characteristic, complemented, Sylow
56	C₂²×Q₈	—	61	1	no	—
57	C₂×C₄⋊C₄	—	61	6	no	—
58	C₂×C₄⋊C₄	—	61	3	no	—
59	C₂×C₄⋊C₄	C₂×D₆₁	1	3	yes	—
60	C₂×C₄⋊C₄	C₂×D₆₁	1	3	yes	—
61	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
62	#95	—	61	6	no	—
63	#95	—	61	6	no	—
64	#95	—	61	3	no	—
65	#95	—	61	3	no	—
66	C₂×Q₈	—	244	3	no	—
67	C₂²×C₄	—	61	6	no	—
68	C₂²×C₄	—	61	2	no	—
69	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
70	C₂²×C₄	C₂²×D₆₁	1	3	yes	—
71	C₂²×C₄	C₂²×D₆₁	1	3	yes	—
72	C₄⋊C₄	—	4	3	no	—
73	C₄⋊C₄	—	4	3	no	—
74	C₄⋊C₄	—	244	6	no	—
75	C₄⋊C₄	—	244	3	no	—
76	C₄²	Dic₁₂₂	1	4	yes	complemented
77	C₂³	C₂³×D₆₁	1	1	yes	characteristic
78	Q₈	—	244	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	—	122	6	no	—
85	C₂×C₄	—	122	6	no	—
86	C₂×C₄	—	122	12	no	—
87	C₂×C₄	C₂×Dic₁₂₂	1	6	yes	—
88	C₂²	C₂²×Dic₁₂₂	1	1	yes	characteristic
89	C₂²	#14519822	1	3	yes	—
90	C₂²	#14519825	1	3	yes	—
91	C₄	—	2	6	no	—
92	C₄	—	4	3	no	—
93	C₄	—	4	1	no	—
94	C₄	—	244	2	no	—
95	C₄	—	244	6	no	—
96	C₂	#16840393	1	3	yes	—
97	C₂	#16840499	1	1	yes	characteristic
98	C₂	#16840511	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16839701

C₂

yes

maximal

#16839695

C₂

yes

maximal

#16839682

C₂

yes

maximal

#16839630

C₂

yes

maximal

C₆₁×#384

C₂

yes

maximal, characteristic

C₂²×Dic₁₂₂

C₂²

yes

characteristic

C₂×C₄⋊Dic₆₁

C₂²

yes

—

C₂×#11758196

C₂²

yes

—

#14519684

C₂²

yes

—

#14519684

C₂²

yes

—

#14519684

C₂²

yes

—

#14519684

C₂²

yes

—

C₂×C₄⋊C₄×C₆₁

C₂²

yes

—

C₂×C₄⋊C₄×C₆₁

C₂²

yes

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×C₄×C₆₁

C₂³

yes

—

C₂²×C₄×C₆₁

C₂³

yes

—

C₂²×Dic₆₁

C₂³

yes

—

C₂²×Dic₆₁

C₂³

yes

—

C₂×Dic₁₂₂

—

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

Q₈

yes

complemented

C₄⋊Dic₆₁

—

#11758196

—

C₂³×C₆₁

C₂⁴

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×Q₈

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

C₂²×C₆₁

C₂²×Q₈

yes

characteristic

C₂²×C₆₁

H(5,𝔽₂)

yes

—

C₂²×C₆₁

2₋¹⁺⁴

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

Dic₆₁

—

Dic₆₁

—

#2843

—

maximal, Sylow

C₂×C₆₁

#544

yes

—

C₂×C₆₁

#560

yes

characteristic

C₂×C₆₁

#563

yes

—

—

—

—

D₆₁

yes

characteristic

#383

—

C₆₁

#2843

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₆₁

yes

—

C₂×C₄⋊C₄

C₂×D₆₁

yes

—

C₂×C₄²

C₂×D₆₁

yes

—

—

—

—

C₂×Q₈

—

244

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₆₁

yes

characteristic

C₂²×C₄

C₂²×D₆₁

yes

—

C₂²×C₄

C₂²×D₆₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

244

—

C₄⋊C₄

—

244

—

C₄²

Dic₁₂₂

yes

complemented

C₂³

C₂³×D₆₁

yes

characteristic

Q₈

—

244

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

122

—

C₂×C₄

—

122

—

C₂×C₄

—

122

—

C₂×C₄

C₂×Dic₁₂₂

yes

—

C₂²

C₂²×Dic₁₂₂

yes

characteristic

C₂²

#14519822

yes

—

C₂²

#14519825

yes

—

C₄

—

C₄

—

C₄

—

C₄

—

244

—

C₄

—

244

—

C₂

#16840393

yes

—

C₂

#16840499

yes

characteristic

C₂

#16840511

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₆₁
commutator, socle	C₂³×C₆₁	C₂⁴
Fitting	C₆₁×#384	C₂
hypercenter	#384	D₆₁
hypocenter	C₆₁	#2843
LCS	G ▷ C₂³×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ #384

subgroup

G/H

center, Frattini

C₂³

C₂³×D₆₁

commutator, socle

C₂³×C₆₁

C₂⁴

Fitting

C₆₁×#384

C₂

D₆₁

C₆₁

G ▷ C₂³×C₆₁ ▷ C₆₁

C₁ ◁ C₂³ ◁ #384

N	H ≅ G/N	type
24. C₄²×C₆₁	Q₈	semidirect
55. C₆₁	#2843	semidirect
76. C₄²	Dic₁₂₂	semidirect

H ≅ G/N

type

24. C₄²×C₆₁

Q₈

semidirect

55. C₆₁

#2843

semidirect

76. C₄²

Dic₁₂₂

semidirect

Subgroups — #21454088

Conjugacy classes of subgroups

Complements