Group #21451769 — subgroups

Subgroups — #21451769

Order 7808.

416 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 149 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	C₂×#14519769	C₂	1	1	yes	maximal, characteristic
2	#16839700	C₂	1	8	yes	maximal
3	#16839669	C₂	1	4	yes	maximal
4	#16839608	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Q₈×C₆₁	C₂	1	1	yes	maximal, characteristic
6	#14519769	—	2	4	no	—
7	C₂×C₄⋊Dic₆₁	C₂²	1	2	yes	—
8	C₂×#11758196	C₂²	1	4	yes	—
9	C₂×C₄×Dic₆₁	C₂²	1	2	yes	—
10	#14519684	C₂²	1	4	yes	—
11	#14519684	C₂²	1	16	yes	—
12	C₂²×Q₈×C₆₁	C₂²	1	2	yes	—
13	C₄⋊Q₈×C₆₁	—	2	4	no	—
14	C₂×C₄⋊C₄×C₆₁	C₂²	1	4	yes	—
15	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×Q₈×C₆₁	—	2	4	no	—
17	C₂×Q₈×C₆₁	—	2	8	no	—
18	C₂²×C₄×C₆₁	C₂³	1	4	yes	—
19	C₂²×C₄×C₆₁	C₂³	1	2	yes	—
20	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
21	C₂²×Dic₆₁	C₂³	1	4	yes	—
22	C₂²×Dic₆₁	C₂³	1	4	yes	—
23	C₄⋊C₄×C₆₁	—	2	8	no	—
24	C₄²×C₆₁	D₄	1	4	yes	—
25	C₄⋊Dic₆₁	—	2	4	no	—
26	#11758196	—	2	8	no	—
27	C₄×Dic₆₁	—	2	4	no	—
28	C₂³×C₆₁	C₂⁴	1	1	yes	characteristic
29	Q₈×C₆₁	—	4	8	no	—
30	C₂×C₄×C₆₁	—	2	4	no	—
31	C₂×C₄×C₆₁	—	2	4	no	—
32	C₂×C₄×C₆₁	—	2	8	no	—
33	C₂×C₄×C₆₁	C₂×D₄	1	4	yes	—
34	C₂×C₄×C₆₁	C₂×D₄	1	2	yes	—
35	C₂×C₄×C₆₁	#41	1	4	yes	—
36	C₂×Dic₆₁	—	2	4	no	—
37	C₂×Dic₆₁	—	2	4	no	—
38	C₂×Dic₆₁	—	2	8	no	—
39	C₂×Dic₆₁	—	2	8	no	—
40	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
41	C₂²×C₆₁	C₂×#41	1	2	yes	—
42	C₂²×C₆₁	2₋¹⁺⁴	1	4	yes	—
43	C₄×C₆₁	—	2	4	no	—
44	C₄×C₆₁	—	4	4	no	—
45	C₄×C₆₁	#124	1	4	yes	—
46	Dic₆₁	—	4	4	no	—
47	Dic₆₁	—	4	4	no	—
48	#2642	—	61	1	no	maximal, Sylow
49	C₂×C₆₁	C₂×#124	1	1	yes	characteristic
50	C₂×C₆₁	#536	1	2	yes	—
51	C₂×C₆₁	#541	1	4	yes	—
52	C₂×C₄⋊Q₈	D₆₁	1	1	yes	characteristic
53	C₂×#125	—	61	1	no	—
54	#400	—	61	8	no	—
55	#391	—	61	4	no	—
56	#376	—	61	1	no	—
57	C₆₁	#2642	1	1	yes	characteristic, complemented, Sylow
58	C₂²×Q₈	C₂×D₆₁	1	2	yes	—
59	C₄⋊Q₈	—	2	4	no	—
60	#125	—	122	4	no	—
61	C₂×C₄⋊C₄	—	61	2	no	—
62	C₂×C₄⋊C₄	—	61	4	no	—
63	C₂×C₄⋊C₄	C₂×D₆₁	1	4	yes	—
64	C₂×C₄²	—	61	2	no	—
65	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
66	#95	—	61	4	no	—
67	#95	—	61	16	no	—
68	C₂×Q₈	—	2	8	no	—
69	C₂×Q₈	—	2	4	no	—
70	C₂²×C₄	—	61	4	no	—
71	C₂²×C₄	—	61	4	no	—
72	C₂²×C₄	C₂²×D₆₁	1	4	yes	—
73	C₂²×C₄	C₂²×D₆₁	1	2	yes	—
74	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
75	C₄⋊C₄	—	2	8	no	—
76	C₄⋊C₄	—	122	8	no	—
77	C₄⋊C₄	—	122	4	no	—
78	C₄²	—	122	4	no	—
79	C₄²	C₂²⋊D₆₁	1	4	yes	—
80	C₂³	C₂³×D₆₁	1	1	yes	characteristic
81	Q₈	—	4	8	no	—
82	C₂×C₄	—	2	4	no	—
83	C₂×C₄	—	2	4	no	—
84	C₂×C₄	—	2	8	no	—
85	C₂×C₄	—	122	4	no	—
86	C₂×C₄	—	122	4	no	—
87	C₂×C₄	—	122	8	no	—
88	C₂×C₄	—	122	8	no	—
89	C₂×C₄	#11758224	1	4	yes	—
90	C₂×C₄	C₂×C₂²⋊D₆₁	1	4	yes	—
91	C₂×C₄	C₂×C₂²⋊D₆₁	1	2	yes	—
92	C₂²	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
93	C₂²	C₂×#11758224	1	2	yes	—
94	C₂²	#14519825	1	4	yes	—
95	C₄	—	2	4	no	—
96	C₄	—	4	4	no	—
97	C₄	—	244	4	no	—
98	C₄	—	244	4	no	—
99	C₄	#14519764	1	4	yes	—
100	C₂	C₂×#14519764	1	1	yes	characteristic
101	C₂	#16840336	1	2	yes	—
102	C₂	#16840373	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14519769

C₂

yes

maximal, characteristic

#16839700

C₂

yes

maximal

#16839669

C₂

yes

maximal

#16839608

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₆₁

C₂

yes

maximal, characteristic

#14519769

—

C₂×C₄⋊Dic₆₁

C₂²

yes

—

C₂×#11758196

C₂²

yes

—

C₂×C₄×Dic₆₁

C₂²

yes

—

#14519684

C₂²

yes

—

#14519684

C₂²

yes

—

C₂²×Q₈×C₆₁

C₂²

yes

—

C₄⋊Q₈×C₆₁

—

C₂×C₄⋊C₄×C₆₁

C₂²

yes

—

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×Q₈×C₆₁

—

C₂²×C₄×C₆₁

C₂³

yes

—

C₂²×C₄×C₆₁

C₂³

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×Dic₆₁

C₂³

yes

—

C₂²×Dic₆₁

C₂³

yes

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

D₄

yes

—

C₄⋊Dic₆₁

—

#11758196

—

C₄×Dic₆₁

—

C₂³×C₆₁

C₂⁴

yes

characteristic

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

#41

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₂²×C₆₁

C₂×#41

yes

—

C₂²×C₆₁

2₋¹⁺⁴

yes

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

#124

yes

—

Dic₆₁

—

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

yes

—

C₂×C₆₁

#541

yes

—

C₂×C₄⋊Q₈

D₆₁

yes

—

—

—

—

C₆₁

#2642

yes

characteristic, complemented, Sylow

C₂²×Q₈

C₂×D₆₁

yes

—

C₄⋊Q₈

—

#125

—

122

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₆₁

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

#95

—

#95

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₆₁

yes

—

C₂²×C₄

C₂²×D₆₁

yes

—

C₂²×C₄

C₂²×D₆₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

122

—

C₄²

—

122

—

C₄²

C₂²⋊D₆₁

yes

—

C₂³

C₂³×D₆₁

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

122

—

C₂×C₄

—

122

—

C₂×C₄

—

122

—

C₂×C₄

—

122

—

C₂×C₄

#11758224

yes

—

C₂×C₄

C₂×C₂²⋊D₆₁

yes

—

C₂×C₄

C₂×C₂²⋊D₆₁

yes

—

C₂²

C₂²×C₂²⋊D₆₁

yes

characteristic

C₂²

C₂×#11758224

yes

—

C₂²

#14519825

yes

—

C₄

—

C₄

—

C₄

—

244

—

C₄

—

244

—

C₄

#14519764

yes

—

100

C₂

C₂×#14519764

yes

characteristic

101

C₂

#16840336

yes

—

102

C₂

#16840373

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₆₁
commutator, socle	C₂³×C₆₁	C₂⁴
Fitting	C₂×C₄⋊Q₈×C₆₁	C₂
hypercenter	C₂×C₄⋊Q₈	D₆₁
hypocenter	C₆₁	#2642
LCS	G ▷ C₂³×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₆₁

commutator, socle

C₂³×C₆₁

C₂⁴

Fitting

C₂×C₄⋊Q₈×C₆₁

C₂

hypercenter

C₂×C₄⋊Q₈

D₆₁

C₆₁

G ▷ C₂³×C₆₁ ▷ C₆₁

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
57. C₆₁	#2642	semidirect

H ≅ G/N

type

57. C₆₁

#2642

semidirect

Subgroups — #21451769

Conjugacy classes of subgroups

Complements