Group #21447962 — subgroups

Subgroups — #21447962

Order 7808.

130 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 130 conjugacy classes of subgroups — too many to draw. Only its 55 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	C₄×Dic₂₄₄	C₂	1	1	yes	maximal, characteristic
3	#16839590	C₂	1	2	yes	maximal
4	#16839567	C₂	1	2	yes	maximal
5	C₆₁×C₄⋊C₁₆	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₄₄	C₂²	1	1	yes	characteristic
7	#14519777	—	2	1	no	—
8	C₄×Dic₁₂₂	—	2	1	no	—
9	#14519711	—	2	1	no	—
10	#14519711	C₂²	1	2	yes	—
11	#14519711	C₂²	1	1	yes	characteristic
12	#14519702	—	2	1	no	—
13	C₂×C₁₆×C₆₁	C₂²	1	2	yes	—
14	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×Dic₁₂₂	—	2	1	no	—
16	C₂×Dic₁₂₂	—	4	1	no	—
17	C₂×C₈×C₆₁	D₄	1	1	yes	characteristic
18	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
19	C₄²×C₆₁	D₄	1	1	yes	characteristic
20	C₄⋊Dic₆₁	—	2	1	no	—
21	C₄⋊Dic₆₁	—	2	2	no	—
22	C₄⋊Dic₆₁	—	4	1	no	—
23	#11758196	—	4	1	no	—
24	C₄×Dic₆₁	—	4	1	no	—
25	Dic₂₄₄	—	2	1	no	—
26	Dic₂₄₄	Q₈	1	2	yes	complemented
27	C₁₆×C₆₁	—	2	2	no	—
28	C₂×C₄×C₆₁	D₈	1	1	yes	characteristic
29	C₂×C₄×C₆₁	D₈	1	1	yes	characteristic
30	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
31	C₂×Dic₆₁	—	4	1	no	—
32	C₂×Dic₆₁	—	4	1	no	—
33	C₂×Dic₆₁	—	4	2	no	—
34	Dic₁₂₂	—	2	2	no	—
35	Dic₁₂₂	—	4	1	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	SD₃₂	1	2	yes	—
43	C₄×C₆₁	C₂²⋊Q₈	1	1	yes	characteristic
44	C₄×C₆₁	#137	1	1	yes	characteristic
45	Dic₆₁	—	4	2	no	—
46	Dic₆₁	—	8	2	no	—
47	Dic₆₁	—	8	1	no	—
48	#2225	—	61	1	no	maximal, Sylow
49	C₂×C₆₁	#474	1	1	yes	characteristic
50	C₂×C₆₁	C₂×SD₃₂	1	1	yes	characteristic
51	C₂×C₆₁	#510	1	1	yes	characteristic
52	#500	—	61	1	no	—
53	C₄×Q₁₆	—	61	1	no	—
54	#367	—	61	2	no	—
55	C₄⋊C₁₆	D₆₁	1	1	yes	characteristic
56	#358	—	61	2	no	—
57	C₆₁	#2225	1	1	yes	characteristic, complemented, Sylow
58	C₂×Q₁₆	—	61	1	no	—
59	C₄⋊Q₈	—	122	1	no	—
60	C₄×Q₈	—	122	1	no	—
61	C₂×C₁₆	C₂×D₆₁	1	2	yes	—
62	#107	—	61	2	no	—
63	#107	—	61	1	no	—
64	#107	—	122	1	no	—
65	#103	—	122	1	no	—
66	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
67	C₂×Q₈	—	122	1	no	—
68	C₂×Q₈	—	244	1	no	—
69	Q₁₆	—	61	2	no	—
70	Q₁₆	—	122	1	no	—
71	C₂×C₈	D₂₄₄	1	1	yes	characteristic
72	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
73	C₄⋊C₄	—	122	1	no	—
74	C₄⋊C₄	—	122	2	no	—
75	C₄⋊C₄	—	244	1	no	—
76	C₄⋊C₄	—	244	1	no	—
77	C₄²	—	244	1	no	—
78	C₄²	D₂₄₄	1	1	yes	characteristic
79	C₁₆	—	2	2	no	—
80	Q₈	—	122	2	no	—
81	Q₈	—	488	2	no	—
82	Q₈	—	244	1	no	—
83	C₂×C₄	—	244	1	no	—
84	C₂×C₄	—	244	2	no	—
85	C₂×C₄	—	244	1	no	—
86	C₂×C₄	D₄₈₈	1	1	yes	characteristic
87	C₂×C₄	D₄₈₈	1	1	yes	characteristic
88	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
89	C₈	—	2	1	no	—
90	C₈	#11758224	1	1	yes	characteristic
91	C₈	Q₈×D₆₁	1	1	yes	characteristic
92	C₂²	C₂×D₄₈₈	1	1	yes	characteristic
93	C₄	—	2	1	no	—
94	C₄	—	488	1	no	—
95	C₄	—	488	2	no	—
96	C₄	—	244	2	no	—
97	C₄	#14519722	1	2	yes	—
98	C₄	#14519757	1	1	yes	characteristic
99	C₄	#14519810	1	1	yes	characteristic
100	C₂	#16840043	1	1	yes	characteristic
101	C₂	C₂×#14519722	1	1	yes	characteristic
102	C₂	#16840205	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16840167

C₂

yes

maximal, characteristic

C₄×Dic₂₄₄

C₂

yes

maximal, characteristic

#16839590

C₂

yes

maximal

#16839567

C₂

yes

maximal

C₆₁×C₄⋊C₁₆

C₂

yes

maximal, characteristic

C₂×Dic₂₄₄

C₂²

yes

characteristic

#14519777

—

C₄×Dic₁₂₂

—

#14519711

—

#14519711

C₂²

yes

—

#14519711

C₂²

yes

characteristic

#14519702

—

C₂×C₁₆×C₆₁

C₂²

yes

—

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂×Dic₁₂₂

—

C₂×Dic₁₂₂

—

C₂×C₈×C₆₁

D₄

yes

characteristic

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₄²×C₆₁

D₄

yes

characteristic

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

—

#11758196

—

C₄×Dic₆₁

—

Dic₂₄₄

—

Dic₂₄₄

Q₈

yes

complemented

C₁₆×C₆₁

—

C₂×C₄×C₆₁

D₈

yes

characteristic

C₂×C₄×C₆₁

D₈

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

Dic₁₂₂

—

Dic₁₂₂

—

C₈×C₆₁

—

C₈×C₆₁

C₂×Q₈

yes

characteristic

C₈×C₆₁

#41

yes

characteristic

C₂²×C₆₁

C₂×D₈

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

SD₃₂

yes

—

C₄×C₆₁

C₂²⋊Q₈

yes

characteristic

C₄×C₆₁

#137

yes

characteristic

Dic₆₁

—

Dic₆₁

—

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

C₂×SD₃₂

yes

C₂×C₆₁

yes

—

C₄×Q₁₆

—

#367

—

C₄⋊C₁₆

D₆₁

yes

characteristic

#358

—

C₆₁

#2225

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄⋊Q₈

—

122

—

C₄×Q₈

—

122

—

C₂×C₁₆

C₂×D₆₁

yes

—

#107

—

#107

—

#107

—

122

—

#103

—

122

—

C₄×C₈

C₂×D₆₁

yes

characteristic

C₂×Q₈

—

122

—

C₂×Q₈

—

244

—

Q₁₆

—

Q₁₆

—

122

—

C₂×C₈

D₂₄₄

yes

characteristic

C₂×C₈

C₂²×D₆₁

yes

characteristic

C₄⋊C₄

—

122

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

244

—

C₄⋊C₄

—

244

—

C₄²

—

244

—

C₄²

D₂₄₄

yes

characteristic

C₁₆

—

Q₈

—

122

—

Q₈

—

488

—

Q₈

—

244

—

C₂×C₄

—

244

—

C₂×C₄

—

244

—

C₂×C₄

—

244

—

C₂×C₄

D₄₈₈

yes

characteristic

C₂×C₄

D₄₈₈

yes

characteristic

C₂×C₄

C₂×D₂₄₄

yes

characteristic

C₈

—

C₈

#11758224

yes

characteristic

C₈

Q₈×D₆₁

yes

characteristic

C₂²

C₂×D₄₈₈

yes

characteristic

C₄

—

C₄

—

488

—

C₄

—

488

—

C₄

—

244

—

C₄

#14519722

yes

—

C₄

yes

C₄

yes

100

C₂

#16840043

yes

characteristic

101

C₂

C₂×#14519722

yes

characteristic

102

C₂

#16840205

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄₈₈
commutator	C₂×C₈×C₆₁	C₂³
Frattini	C₂×C₈	C₂²×D₆₁
Fitting	C₆₁×C₄⋊C₁₆	C₂
socle	C₂²×C₆₁	C₂×D₈
hypercenter	C₄⋊C₁₆	D₆₁
hypocenter	C₆₁	#2225
LCS	G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

subgroup

G/H

center

C₂²

C₂×D₄₈₈

commutator

C₂×C₈×C₆₁

C₂³

Frattini

C₂×C₈

C₂²×D₆₁

Fitting

C₆₁×C₄⋊C₁₆

C₂

socle

C₂²×C₆₁

C₂×D₈

hypercenter

C₄⋊C₁₆

D₆₁

hypocenter

C₆₁

#2225

LCS

G ▷ C₂×C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
26. Dic₂₄₄	Q₈	semidirect
57. C₆₁	#2225	semidirect

H ≅ G/N

type

26. Dic₂₄₄

Q₈

semidirect

57. C₆₁

#2225

semidirect

Subgroups — #21447962

Conjugacy classes of subgroups

Complements