Group #21447971 — subgroups

Subgroups — #21447971

Order 7808.

130 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 130 conjugacy classes of subgroups — too many to draw. Only its 57 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	#16840167	C₂	1	1	yes	maximal, characteristic
2	#16839587	C₂	1	2	yes	maximal
3	C₄⋊#11758185	C₂	1	1	yes	maximal, characteristic
4	#16839566	C₂	1	2	yes	maximal
5	C₄×Q₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic
6	#14519777	—	2	1	no	—
7	C₂×#11758185	C₂²	1	2	yes	—
8	#14519711	—	2	1	no	—
9	#14519711	C₂²	1	2	yes	—
10	C₂×Q₁₆×C₆₁	C₂²	1	1	yes	characteristic
11	C₄×Q₈×C₆₁	—	2	1	no	—
12	#107×C₆₁	C₂²	1	1	yes	characteristic
13	#103×C₆₁	—	2	1	no	—
14	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×Q₈×C₆₁	—	2	1	no	—
16	C₂×Dic₁₂₂	—	4	1	no	—
17	Q₁₆×C₆₁	—	2	1	no	—
18	Q₁₆×C₆₁	Q₈	1	2	yes	complemented
19	C₂×C₈×C₆₁	D₄	1	1	yes	characteristic
20	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
21	C₄⋊C₄×C₆₁	—	2	1	no	—
22	C₄⋊C₄×C₆₁	—	4	1	no	—
23	C₄²×C₆₁	—	4	1	no	—
24	C₄²×C₆₁	D₄	1	1	yes	characteristic
25	C₄⋊Dic₆₁	—	2	2	no	—
26	C₄⋊Dic₆₁	—	4	1	no	—
27	#11758185	—	2	2	no	—
28	Q₈×C₆₁	—	2	2	no	—
29	Q₈×C₆₁	—	4	1	no	—
30	C₂×C₄×C₆₁	—	4	1	no	—
31	C₂×C₄×C₆₁	—	4	1	no	—
32	C₂×C₄×C₆₁	D₈	1	1	yes	characteristic
33	C₂×C₄×C₆₁	D₈	1	1	yes	characteristic
34	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
35	C₂×Dic₆₁	—	4	2	no	—
36	Dic₁₂₂	—	8	2	no	—
37	C₈×C₆₁	—	2	1	no	—
38	C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
39	C₈×C₆₁	#41	1	1	yes	characteristic
40	C₂²×C₆₁	C₂×D₈	1	1	yes	characteristic
41	C₄×C₆₁	—	2	1	no	—
42	C₄×C₆₁	—	4	2	no	—
43	C₄×C₆₁	—	8	1	no	—
44	C₄×C₆₁	Q₃₂	1	2	yes	—
45	C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
46	C₄×C₆₁	#137	1	1	yes	characteristic
47	Dic₆₁	—	8	2	no	—
48	#2227	—	61	1	no	maximal, Sylow
49	C₂×C₆₁	#474	1	1	yes	characteristic
50	C₂×C₆₁	C₂×Q₃₂	1	1	yes	characteristic
51	C₂×C₆₁	#509	1	1	yes	characteristic
52	#500	—	61	1	no	—
53	C₄×Q₁₆	D₆₁	1	1	yes	characteristic
54	#366	—	61	2	no	—
55	C₄⋊C₁₆	—	61	1	no	—
56	#358	—	61	2	no	—
57	C₆₁	#2227	1	1	yes	characteristic, complemented, Sylow
58	C₂×Q₁₆	C₂×D₆₁	1	1	yes	characteristic
59	C₄⋊Q₈	—	122	1	no	—
60	C₄×Q₈	—	2	1	no	—
61	C₂×C₁₆	—	61	2	no	—
62	#107	—	61	2	no	—
63	#107	—	122	1	no	—
64	#107	C₂×D₆₁	1	1	yes	characteristic
65	#103	—	2	1	no	—
66	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
67	C₂×Q₈	—	2	1	no	—
68	C₂×Q₈	—	244	1	no	—
69	Q₁₆	—	2	1	no	—
70	Q₁₆	Dic₁₂₂	1	2	yes	complemented
71	C₂×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
72	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
73	C₄⋊C₄	—	2	1	no	—
74	C₄⋊C₄	—	4	1	no	—
75	C₄⋊C₄	—	122	2	no	—
76	C₄⋊C₄	—	244	1	no	—
77	C₄²	—	4	1	no	—
78	C₄²	C₂²⋊D₆₁	1	1	yes	characteristic
79	C₁₆	—	122	2	no	—
80	Q₈	—	2	2	no	—
81	Q₈	—	4	1	no	—
82	Q₈	—	488	2	no	—
83	C₂×C₄	—	4	1	no	—
84	C₂×C₄	—	4	1	no	—
85	C₂×C₄	—	244	2	no	—
86	C₂×C₄	#11758199	1	1	yes	characteristic
87	C₂×C₄	#11758199	1	1	yes	characteristic
88	C₂×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
89	C₈	—	2	1	no	—
90	C₈	C₂×Dic₁₂₂	1	1	yes	characteristic
91	C₈	#11758222	1	1	yes	characteristic
92	C₂²	C₂×#11758199	1	1	yes	characteristic
93	C₄	—	2	1	no	—
94	C₄	—	4	2	no	—
95	C₄	—	8	1	no	—
96	C₄	—	488	2	no	—
97	C₄	#14519723	1	2	yes	—
98	C₄	#14519756	1	1	yes	characteristic
99	C₄	#14519807	1	1	yes	characteristic
100	C₂	#16840042	1	1	yes	characteristic
101	C₂	C₂×#14519723	1	1	yes	characteristic
102	C₂	#16840197	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840167

C₂

yes

maximal, characteristic

#16839587

C₂

yes

maximal

C₄⋊#11758185

C₂

yes

maximal, characteristic

#16839566

C₂

yes

maximal

C₄×Q₁₆×C₆₁

C₂

yes

maximal, characteristic

#14519777

—

C₂×#11758185

C₂²

yes

—

#14519711

—

#14519711

C₂²

yes

—

C₂×Q₁₆×C₆₁

C₂²

yes

characteristic

C₄×Q₈×C₆₁

—

#107×C₆₁

C₂²

yes

characteristic

#103×C₆₁

—

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂×Q₈×C₆₁

—

C₂×Dic₁₂₂

—

Q₁₆×C₆₁

—

Q₁₆×C₆₁

Q₈

yes

complemented

C₂×C₈×C₆₁

D₄

yes

characteristic

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₄⋊C₄×C₆₁

—

C₄⋊C₄×C₆₁

—

C₄²×C₆₁

—

C₄²×C₆₁

D₄

yes

characteristic

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

#11758185

—

Q₈×C₆₁

—

Q₈×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

D₈

yes

characteristic

C₂×C₄×C₆₁

D₈

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×Dic₆₁

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

C₂×Q₈

yes

characteristic

C₈×C₆₁

#41

yes

characteristic

C₂²×C₆₁

C₂×D₈

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

—

C₄×C₆₁

Q₃₂

yes

—

C₄×C₆₁

C₂²⋊Q₈

yes

characteristic

C₄×C₆₁

#137

yes

characteristic

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

C₂×Q₃₂

yes

C₂×C₆₁

yes

—

C₄×Q₁₆

D₆₁

yes

characteristic

#366

—

C₄⋊C₁₆

—

#358

—

C₆₁

#2227

yes

characteristic, complemented, Sylow

C₂×Q₁₆

C₂×D₆₁

yes

characteristic

C₄⋊Q₈

—

122

—

C₄×Q₈

—

C₂×C₁₆

—

#107

—

#107

—

122

—

#107

C₂×D₆₁

yes

characteristic

#103

—

C₄×C₈

C₂×D₆₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

244

—

Q₁₆

—

Q₁₆

Dic₁₂₂

yes

complemented

C₂×C₈

C₂²⋊D₆₁

yes

characteristic

C₂×C₈

C₂²×D₆₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

244

—

C₄²

—

C₄²

C₂²⋊D₆₁

yes

characteristic

C₁₆

—

122

—

Q₈

—

Q₈

—

Q₈

—

488

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

244

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂×C₂²⋊D₆₁

yes

characteristic

C₈

—

C₈

C₂×Dic₁₂₂

yes

C₈

yes

C₂²

yes

C₄

—

C₄

—

C₄

—

C₄

—

488

—

C₄

#14519723

yes

—

C₄

yes

C₄

yes

100

C₂

#16840042

yes

characteristic

101

C₂

C₂×#14519723

yes

characteristic

102

C₂

#16840197

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#11758199
commutator	C₂×C₈×C₆₁	C₂³
Frattini	C₂×C₈	C₂²×D₆₁
Fitting	C₄×Q₁₆×C₆₁	C₂
socle	C₂²×C₆₁	C₂×D₈
hypercenter	C₄×Q₁₆	D₆₁
hypocenter	C₆₁	#2227
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₁₆

subgroup

G/H

center

C₂²

C₂×#11758199

commutator

C₂×C₈×C₆₁

C₂³

Frattini

C₂×C₈

C₂²×D₆₁

Fitting

C₄×Q₁₆×C₆₁

C₂

socle

C₂²×C₆₁

C₂×D₈

hypercenter

C₄×Q₁₆

D₆₁

hypocenter

C₆₁

#2227

LCS

G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×Q₁₆

N	H ≅ G/N	type
18. Q₁₆×C₆₁	Q₈	semidirect
57. C₆₁	#2227	semidirect
70. Q₁₆	Dic₁₂₂	semidirect

H ≅ G/N

type

18. Q₁₆×C₆₁

Q₈

semidirect

57. C₆₁

#2227

semidirect

70. Q₁₆

Dic₁₂₂

semidirect

Subgroups — #21447971

Conjugacy classes of subgroups

Complements