Group #21456323 — subgroups

Subgroups — C₂×#16840164 #21456323

Order 7808.

360 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 181 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	2	yes	maximal
2	#16840164	C₂	1	8	yes	maximal, direct product factor
3	C₂×#14519711	C₂	1	2	yes	maximal
4	C₂×#14519709	C₂	1	2	yes	maximal
5	C₂×C₄×C₈×C₆₁	C₂	1	1	yes	maximal, characteristic
6	#14519769	—	2	4	no	—
7	#14519769	C₂²	1	8	yes	—
8	C₂×C₄⋊Dic₆₁	C₂²	1	4	yes	—
9	C₂×#11758196	—	2	4	no	—
10	#14519711	C₂²	1	8	yes	—
11	#14519709	C₂²	1	8	yes	—
12	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
13	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
14	C₂×C₄²×C₆₁	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₆₁	C₂²	1	4	yes	—
16	C₂²×C₄×C₆₁	D₄	1	2	yes	—
17	C₂²×C₄×C₆₁	C₂³	1	1	yes	characteristic
18	C₂²×Dic₆₁	—	2	4	no	—
19	C₂×C₈×C₆₁	Q₈	1	4	yes	—
20	C₂×C₈×C₆₁	Q₈	1	4	yes	—
21	C₂×C₈×C₆₁	C₂³	1	2	yes	—
22	C₂×C₈×C₆₁	C₂³	1	2	yes	—
23	C₄²×C₆₁	D₄	1	2	yes	—
24	C₄²×C₆₁	C₂³	1	2	yes	—
25	C₄⋊Dic₆₁	—	2	4	no	—
26	C₄⋊Dic₆₁	C₂³	1	8	yes	—
27	#11758196	—	2	16	no	—
28	C₂³×C₆₁	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₆₁	—	2	4	no	—
30	C₂×C₄×C₆₁	C₂×D₄	1	4	yes	—
31	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₆₁	C₂×Q₈	1	2	yes	—
33	C₂×C₄×C₆₁	C₂×Q₈	1	2	yes	—
34	C₂×C₄×C₆₁	C₂⁴	1	1	yes	characteristic
35	C₂×Dic₆₁	—	2	8	no	—
36	C₂×Dic₆₁	—	4	8	no	—
37	C₈×C₆₁	C₂×Q₈	1	4	yes	—
38	C₈×C₆₁	C₂×Q₈	1	4	yes	—
39	C₂²×C₆₁	C₄⋊Q₈	1	2	yes	—
40	C₂²×C₆₁	#135	1	4	yes	—
41	C₂²×C₆₁	C₂²×D₄	1	1	yes	characteristic
42	C₄×C₆₁	—	2	4	no	—
43	C₄×C₆₁	C₄⋊Q₈	1	2	yes	—
44	C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
45	C₄×C₆₁	C₂²×Q₈	1	1	yes	characteristic
46	Dic₆₁	—	4	8	no	—
47	C₂×#499	—	61	1	no	maximal, Sylow
48	C₂×C₆₁	#499	1	4	yes	complemented
49	C₂×C₆₁	C₂×C₄⋊Q₈	1	1	yes	characteristic
50	C₂×C₆₁	C₂×#135	1	2	yes	—
51	C₂×#125	—	61	2	no	—
52	#499	—	61	8	no	—
53	C₂×#107	—	61	2	no	—
54	C₂×#106	—	61	2	no	—
55	C₂×C₄×C₈	D₆₁	1	1	yes	characteristic
56	C₆₁	C₂×#499	1	1	yes	characteristic, complemented, Sylow
57	C₂²×C₈	C₂×D₆₁	1	1	yes	characteristic
58	C₂²×C₈	C₂×D₆₁	1	1	yes	characteristic
59	#125	—	61	8	no	—
60	#125	—	122	4	no	—
61	C₂×C₄⋊C₄	—	61	4	no	—
62	C₂×C₄⋊C₄	—	122	4	no	—
63	C₂×C₄²	C₂×D₆₁	1	1	yes	characteristic
64	#107	—	61	8	no	—
65	#106	—	61	8	no	—
66	C₄×C₈	C₂×D₆₁	1	4	yes	—
67	C₂²×C₄	—	122	4	no	—
68	C₂²×C₄	D₂₄₄	1	2	yes	—
69	C₂²×C₄	C₂²×D₆₁	1	1	yes	characteristic
70	C₂×C₈	Dic₁₂₂	1	4	yes	—
71	C₂×C₈	Dic₁₂₂	1	4	yes	—
72	C₂×C₈	C₂²×D₆₁	1	2	yes	—
73	C₂×C₈	C₂²×D₆₁	1	2	yes	—
74	C₄⋊C₄	—	61	8	no	—
75	C₄⋊C₄	—	122	4	no	—
76	C₄⋊C₄	—	122	16	no	—
77	C₄²	D₂₄₄	1	2	yes	—
78	C₄²	C₂²×D₆₁	1	2	yes	—
79	C₂³	C₂×D₂₄₄	1	1	yes	characteristic
80	C₂×C₄	—	2	4	no	—
81	C₂×C₄	—	122	8	no	—
82	C₂×C₄	—	244	8	no	—
83	C₂×C₄	C₂×Dic₁₂₂	1	2	yes	—
84	C₂×C₄	C₂×Dic₁₂₂	1	2	yes	—
85	C₂×C₄	C₂×D₂₄₄	1	4	yes	—
86	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
87	C₂×C₄	C₂³×D₆₁	1	1	yes	characteristic
88	C₈	C₂×Dic₁₂₂	1	4	yes	—
89	C₈	C₂×Dic₁₂₂	1	4	yes	—
90	C₂²	#14519777	1	2	yes	—
91	C₂²	#14519799	1	4	yes	—
92	C₂²	C₂²×D₂₄₄	1	1	yes	characteristic
93	C₄	—	2	4	no	—
94	C₄	—	244	8	no	—
95	C₄	#14519777	1	2	yes	—
96	C₄	C₂²×Dic₁₂₂	1	1	yes	characteristic
97	C₄	C₂²×Dic₁₂₂	1	1	yes	characteristic
98	C₂	#16840164	1	4	yes	direct product factor
99	C₂	C₂×#14519777	1	1	yes	characteristic
100	C₂	C₂×#14519799	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14519769

C₂

yes

maximal

#16840164

C₂

yes

maximal, direct product factor

C₂×#14519711

C₂

yes

maximal

C₂×#14519709

C₂

yes

maximal

C₂×C₄×C₈×C₆₁

C₂

yes

maximal, characteristic

#14519769

—

#14519769

C₂²

yes

—

C₂×C₄⋊Dic₆₁

C₂²

yes

—

C₂×#11758196

—

#14519711

C₂²

yes

—

#14519709

C₂²

yes

—

C₂²×C₈×C₆₁

C₂²

yes

characteristic

C₂²×C₈×C₆₁

C₂²

yes

characteristic

C₂×C₄²×C₆₁

C₂²

yes

characteristic

C₄×C₈×C₆₁

C₂²

yes

—

C₂²×C₄×C₆₁

D₄

yes

—

C₂²×C₄×C₆₁

C₂³

yes

characteristic

C₂²×Dic₆₁

—

C₂×C₈×C₆₁

Q₈

yes

—

C₂×C₈×C₆₁

Q₈

yes

—

C₂×C₈×C₆₁

C₂³

yes

—

C₂×C₈×C₆₁

C₂³

yes

—

C₄²×C₆₁

D₄

yes

—

C₄²×C₆₁

C₂³

yes

—

C₄⋊Dic₆₁

—

C₄⋊Dic₆₁

C₂³

yes

—

#11758196

—

C₂³×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₂×D₄

yes

—

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

C₂×Q₈

yes

—

C₂×C₄×C₆₁

C₂×Q₈

yes

—

C₂×C₄×C₆₁

C₂⁴

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

C₈×C₆₁

C₂×Q₈

yes

—

C₈×C₆₁

C₂×Q₈

yes

—

C₂²×C₆₁

C₄⋊Q₈

yes

—

C₂²×C₆₁

#135

yes

—

C₂²×C₆₁

C₂²×D₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

C₄⋊Q₈

yes

—

C₄×C₆₁

C₂²×Q₈

yes

characteristic

C₄×C₆₁

C₂²×Q₈

yes

characteristic

Dic₆₁

—

—

C₂×C₆₁

yes

C₂×C₆₁

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₆₁

C₂×#135

yes

—

—

—

—

—

C₂×C₄×C₈

D₆₁

yes

characteristic

C₆₁

C₂×#499

yes

characteristic, complemented, Sylow

C₂²×C₈

C₂×D₆₁

yes

characteristic

C₂²×C₈

C₂×D₆₁

yes

characteristic

#125

—

#125

—

122

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

122

—

C₂×C₄²

C₂×D₆₁

yes

characteristic

#107

—

#106

—

C₄×C₈

C₂×D₆₁

yes

—

C₂²×C₄

—

122

—

C₂²×C₄

D₂₄₄

yes

—

C₂²×C₄

C₂²×D₆₁

yes

characteristic

C₂×C₈

Dic₁₂₂

yes

—

C₂×C₈

Dic₁₂₂

yes

—

C₂×C₈

C₂²×D₆₁

yes

—

C₂×C₈

C₂²×D₆₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

122

—

C₄⋊C₄

—

122

—

C₄²

D₂₄₄

yes

—

C₄²

C₂²×D₆₁

yes

—

C₂³

C₂×D₂₄₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

122

—

C₂×C₄

—

244

—

C₂×C₄

C₂×Dic₁₂₂

yes

—

C₂×C₄

C₂×Dic₁₂₂

yes

—

C₂×C₄

C₂×D₂₄₄

yes

—

C₂×C₄

C₂×D₂₄₄

yes

characteristic

C₂×C₄

C₂³×D₆₁

yes

characteristic

C₈

C₂×Dic₁₂₂

yes

—

C₈

C₂×Dic₁₂₂

yes

—

C₂²

#14519777

yes

—

C₂²

#14519799

yes

—

C₂²

C₂²×D₂₄₄

yes

characteristic

C₄

—

C₄

—

244

—

C₄

#14519777

yes

—

C₄

C₂²×Dic₁₂₂

yes

characteristic

C₄

C₂²×Dic₁₂₂

yes

characteristic

C₂

#16840164

yes

direct product factor

C₂

C₂×#14519777

yes

characteristic

100

C₂

C₂×#14519799

yes

—

	subgroup	G/H
center	C₂³	C₂×D₂₄₄
commutator	C₂×C₄×C₆₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₁
Fitting	C₂×C₄×C₈×C₆₁	C₂
socle	C₂³×C₆₁	C₂×D₄
hypercenter	C₂×C₄×C₈	D₆₁
hypocenter	C₆₁	C₂×#499
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₂₄₄

commutator

C₂×C₄×C₆₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₁

Fitting

C₂×C₄×C₈×C₆₁

C₂

socle

C₂³×C₆₁

C₂×D₄

hypercenter

C₂×C₄×C₈

D₆₁

hypocenter

C₆₁

C₂×#499

LCS

G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #16840164	C₂	direct (N × H)
48. C₂×C₆₁	#499	semidirect
56. C₆₁	C₂×#499	semidirect
98. C₂	#16840164	direct (N × H)

H ≅ G/N

type

2. #16840164

C₂

direct (N × H)

48. C₂×C₆₁

#499

semidirect

56. C₆₁

C₂×#499

semidirect

98. C₂

#16840164

direct (N × H)

Subgroups — C₂×#16840164 #21456323

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16840164 #21456323

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16840164 #21456323