Group #21444627 — subgroups

Subgroups — C₈×Dic₂₄₄ #21444627

Order 7808.

164 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 164 conjugacy classes of subgroups — too many to draw. Only its 77 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	C₈×Dic₁₂₂	C₂	1	2	yes	maximal
2	C₄×Dic₂₄₄	C₂	1	1	yes	maximal, characteristic
3	#16839515	C₂	1	1	yes	maximal, characteristic
4	#16839494	C₂	1	2	yes	maximal
5	C₈²×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₄₄	C₄	1	1	yes	characteristic
7	C₄×Dic₁₂₂	C₂²	1	2	yes	—
8	#14519711	C₄	1	1	yes	characteristic
9	C₄⋊#92418	C₂²	1	2	yes	—
10	#14519706	—	2	2	no	—
11	#14519702	C₄	1	2	yes	—
12	C₈×Dic₆₁	—	2	2	no	—
13	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
14	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×Dic₁₂₂	C₂×C₄	1	2	yes	—
17	C₂×C₈×C₆₁	—	2	1	no	—
18	C₂×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₆₁	D₄	1	1	yes	characteristic
21	C₂×C₈×C₆₁	D₄	1	1	yes	characteristic
22	C₄²×C₆₁	C₂³	1	1	yes	characteristic
23	C₄⋊Dic₆₁	C₂×C₄	1	2	yes	—
24	#11758196	—	2	2	no	—
25	C₄×Dic₆₁	—	2	2	no	—
26	C₂×#92418	—	2	2	no	—
27	Dic₂₄₄	C₈	1	4	yes	direct product factor
28	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
30	C₂×C₄×C₆₁	#41	1	1	yes	characteristic
31	C₂×Dic₆₁	—	2	2	no	—
32	C₂×Dic₆₁	—	2	2	no	—
33	Dic₁₂₂	—	2	2	no	—
34	Dic₁₂₂	C₂×C₈	1	4	yes	—
35	C₈×C₆₁	—	2	2	no	—
36	C₈×C₆₁	—	2	1	no	—
37	C₈×C₆₁	—	2	1	no	—
38	C₈×C₆₁	C₂×C₈	1	2	yes	—
39	C₈×C₆₁	Q₁₆	1	2	yes	complemented
40	#92418	—	4	2	no	—
41	C₂²×C₆₁	C₄×D₄	1	1	yes	characteristic
42	C₄×C₆₁	—	2	1	no	—
43	C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
44	C₄×C₆₁	#131	1	1	yes	characteristic
45	C₄×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
46	C₄×C₆₁	#135	1	1	yes	characteristic
47	Dic₆₁	—	2	4	no	—
48	Dic₆₁	—	4	2	no	—
49	C₈×Q₁₆	—	61	1	no	maximal, Sylow
50	C₂×C₆₁	C₈×D₄	1	1	yes	characteristic
51	C₂×C₆₁	C₄×Q₁₆	1	1	yes	characteristic
52	C₂×C₆₁	#443	1	1	yes	characteristic
53	C₈×Q₈	—	61	2	no	—
54	C₄×Q₁₆	—	61	1	no	—
55	#335	—	61	1	no	—
56	Q₈⋊C₈	—	61	2	no	—
57	C₈²	D₆₁	1	1	yes	characteristic
58	C₆₁	C₈×Q₁₆	1	1	yes	characteristic, complemented, Sylow
59	C₂×Q₁₆	—	61	1	no	—
60	C₄×Q₈	—	61	2	no	—
61	#107	—	61	1	no	—
62	C₄⋊C₈	—	61	2	no	—
63	C₄⋊C₈	—	122	2	no	—
64	#103	—	61	2	no	—
65	C₄×C₈	—	122	2	no	—
66	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
67	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
68	C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
69	C₂×Q₈	—	61	2	no	—
70	Q₁₆	—	61	4	no	—
71	C₂×C₈	—	2	1	no	—
72	C₂×C₈	—	122	2	no	—
73	C₂×C₈	C₄×D₆₁	1	1	yes	characteristic
74	C₂×C₈	C₄×D₆₁	1	1	yes	characteristic
75	C₂×C₈	D₂₄₄	1	1	yes	characteristic
76	C₂×C₈	D₂₄₄	1	1	yes	characteristic
77	C₄⋊C₄	—	61	2	no	—
78	C₄⋊C₄	—	122	2	no	—
79	C₄²	—	122	2	no	—
80	C₄²	C₂²×D₆₁	1	1	yes	characteristic
81	Q₈	—	61	4	no	—
82	Q₈	—	122	2	no	—
83	C₂×C₄	—	122	2	no	—
84	C₂×C₄	—	122	2	no	—
85	C₂×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
86	C₂×C₄	C₂×D₂₄₄	1	1	yes	characteristic
87	C₂×C₄	#11758222	1	1	yes	characteristic
88	C₈	—	2	2	no	—
89	C₈	—	2	1	no	—
90	C₈	—	2	1	no	—
91	C₈	—	244	2	no	—
92	C₈	C₈×D₆₁	1	2	yes	—
93	C₈	Dic₂₄₄	1	2	yes	direct product factor
94	C₂²	C₄×D₂₄₄	1	1	yes	characteristic
95	C₄	—	2	1	no	—
96	C₄	—	122	4	no	—
97	C₄	—	244	2	no	—
98	C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
99	C₄	#14519783	1	1	yes	characteristic
100	C₄	C₂×Dic₂₄₄	1	1	yes	characteristic
101	C₄	#14519799	1	1	yes	characteristic
102	C₂	C₈×D₂₄₄	1	1	yes	characteristic
103	C₂	C₄×Dic₂₄₄	1	1	yes	characteristic
104	C₂	#16839880	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₈×Dic₁₂₂

C₂

yes

maximal

C₄×Dic₂₄₄

C₂

yes

maximal, characteristic

#16839515

C₂

yes

maximal, characteristic

#16839494

C₂

yes

maximal

C₈²×C₆₁

C₂

yes

maximal, characteristic

C₂×Dic₂₄₄

C₄

yes

characteristic

C₄×Dic₁₂₂

C₂²

yes

—

#14519711

C₄

yes

characteristic

C₄⋊#92418

C₂²

yes

—

#14519706

—

#14519702

C₄

yes

—

C₈×Dic₆₁

—

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₄×C₈×C₆₁

C₂²

yes

characteristic

C₂×Dic₁₂₂

C₂×C₄

yes

—

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂×C₈×C₆₁

D₄

yes

characteristic

C₂×C₈×C₆₁

D₄

yes

characteristic

C₄²×C₆₁

C₂³

yes

characteristic

C₄⋊Dic₆₁

C₂×C₄

yes

—

#11758196

—

C₄×Dic₆₁

—

C₂×#92418

—

Dic₂₄₄

C₈

yes

direct product factor

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₆₁

C₂×D₄

yes

characteristic

C₂×C₄×C₆₁

#41

yes

characteristic

C₂×Dic₆₁

—

C₂×Dic₆₁

—

Dic₁₂₂

—

Dic₁₂₂

C₂×C₈

yes

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

—

C₈×C₆₁

C₂×C₈

yes

—

C₈×C₆₁

Q₁₆

yes

complemented

#92418

—

C₂²×C₆₁

C₄×D₄

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

C₂²×C₈

yes

characteristic

C₄×C₆₁

#131

yes

characteristic

C₄×C₆₁

C₂×Q₁₆

yes

characteristic

C₄×C₆₁

#135

yes

characteristic

Dic₆₁

—

Dic₆₁

—

C₈×Q₁₆

—

maximal, Sylow

C₂×C₆₁

C₈×D₄

yes

characteristic

C₂×C₆₁

C₄×Q₁₆

yes

characteristic

C₂×C₆₁

#443

yes

characteristic

C₈×Q₈

—

C₄×Q₁₆

—

#335

—

Q₈⋊C₈

—

C₈²

D₆₁

yes

characteristic

C₆₁

C₈×Q₁₆

yes

characteristic, complemented, Sylow

C₂×Q₁₆

—

C₄×Q₈

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

122

—

#103

—

C₄×C₈

—

122

—

C₄×C₈

C₂×D₆₁

yes

characteristic

C₄×C₈

C₂×D₆₁

yes

characteristic

C₄×C₈

C₂×D₆₁

yes

characteristic

C₂×Q₈

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

122

—

C₂×C₈

C₄×D₆₁

yes

characteristic

C₂×C₈

C₄×D₆₁

yes

characteristic

C₂×C₈

D₂₄₄

yes

characteristic

C₂×C₈

D₂₄₄

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

122

—

C₄²

—

122

—

C₄²

C₂²×D₆₁

yes

characteristic

Q₈

—

Q₈

—

122

—

C₂×C₄

—

122

—

C₂×C₄

—

122

—

C₂×C₄

C₂×C₄×D₆₁

yes

characteristic

C₂×C₄

C₂×D₂₄₄

yes

characteristic

C₂×C₄

#11758222

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

244

—

C₈

C₈×D₆₁

yes

—

C₈

Dic₂₄₄

yes

direct product factor

C₂²

C₄×D₂₄₄

yes

characteristic

C₄

—

C₄

—

122

—

C₄

—

244

—

C₄

C₂×C₈×D₆₁

yes

characteristic

C₄

#14519783

yes

characteristic

100

C₄

C₂×Dic₂₄₄

yes

characteristic

101

C₄

#14519799

yes

characteristic

102

C₂

C₈×D₂₄₄

yes

characteristic

103

C₂

C₄×Dic₂₄₄

yes

characteristic

104

C₂

#16839880

yes

characteristic

	subgroup	G/H
center	C₂×C₈	D₂₄₄
commutator	C₄×C₆₁	C₂²×C₈
Frattini	C₄²	C₂²×D₆₁
Fitting	C₈²×C₆₁	C₂
socle	C₂²×C₆₁	C₄×D₄
hypercenter	C₈²	D₆₁
hypocenter	C₆₁	C₈×Q₁₆
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈²

subgroup

G/H

center

C₂×C₈

D₂₄₄

commutator

C₄×C₆₁

C₂²×C₈

Frattini

C₄²

C₂²×D₆₁

Fitting

C₈²×C₆₁

C₂

socle

C₂²×C₆₁

C₄×D₄

hypercenter

C₈²

D₆₁

hypocenter

C₆₁

C₈×Q₁₆

LCS

G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈²

N	H ≅ G/N	type
27. Dic₂₄₄	C₈	direct (N × H)
39. C₈×C₆₁	Q₁₆	semidirect
58. C₆₁	C₈×Q₁₆	semidirect
93. C₈	Dic₂₄₄	direct (N × H)

H ≅ G/N

type

27. Dic₂₄₄

C₈

direct (N × H)

39. C₈×C₆₁

Q₁₆

semidirect

58. C₆₁

C₈×Q₁₆

semidirect

93. C₈

Dic₂₄₄

direct (N × H)

Subgroups — C₈×Dic₂₄₄ #21444627

Conjugacy classes of subgroups

Complements

Subgroups — C8×Dic244 #21444627

Conjugacy classes of subgroups

Complements

Subgroups — C₈×Dic₂₄₄ #21444627