Group #22601225 — subgroups

Subgroups — Q₈×Dic₂₈₄ #22601225

Order 9088.

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

normal subgroups only

This group has 336 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	Q₈×Dic₁₄₂	C₂	1	2	yes	maximal
2	#17986404	C₂	1	3	yes	maximal
3	#17986295	C₂	1	6	yes	maximal
4	C₄×Dic₂₈₄	C₂	1	3	yes	maximal
5	C₈×Q₈×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₈₄	C₂²	1	1	yes	characteristic
7	#14556128	C₂²	1	6	yes	—
8	#14556127	—	2	6	no	—
9	Q₈×Dic₇₁	—	2	2	no	—
10	C₄×Dic₁₄₂	C₂²	1	6	yes	—
11	#14556062	C₂²	1	3	yes	—
12	#14556062	C₂²	1	6	yes	—
13	#14556053	C₂²	1	6	yes	—
14	C₄×Q₈×C₇₁	C₂²	1	1	yes	characteristic
15	C₄⋊C₈×C₇₁	C₂²	1	3	yes	—
16	C₄×C₈×C₇₁	C₂²	1	3	yes	—
17	C₂×Dic₁₄₂	—	2	6	no	—
18	C₂×Dic₁₄₂	C₂³	1	2	yes	—
19	Dic₂₈₄	Q₈	1	4	yes	direct product factor
20	C₄⋊Dic₇₁	—	2	6	no	—
21	C₄⋊Dic₇₁	C₂³	1	6	yes	—
22	Dic₇₁⋊C₄	—	2	6	no	—
23	C₄×Dic₇₁	—	2	6	no	—
24	C₂×Q₈×C₇₁	D₄	1	1	yes	characteristic
25	C₂×C₈×C₇₁	C₂³	1	3	yes	—
26	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₇₁	D₄	1	3	yes	—
28	C₄²×C₇₁	C₂³	1	3	yes	—
29	Q₈×C₇₁	Q₁₆	1	4	yes	complemented
30	C₂×C₄×C₇₁	C₂×D₄	1	3	yes	—
31	C₂×C₄×C₇₁	C₂×D₄	1	3	yes	—
32	C₂×C₄×C₇₁	C₂⁴	1	1	yes	characteristic
33	C₂×Dic₇₁	—	2	2	no	—
34	C₂×Dic₇₁	—	2	6	no	—
35	Dic₁₄₂	—	2	2	no	—
36	Dic₁₄₂	—	8	6	no	—
37	Dic₁₄₂	C₂×Q₈	1	4	yes	—
38	C₈×C₇₁	—	2	3	no	—
39	C₈×C₇₁	C₂×Q₈	1	2	yes	—
40	C₂²×C₇₁	C₂²×D₄	1	1	yes	characteristic
41	C₄×C₇₁	—	2	3	no	—
42	C₄×C₇₁	C₂×Q₁₆	1	6	yes	—
43	C₄×C₇₁	C₂²×Q₈	1	1	yes	characteristic
44	C₄×C₇₁	2₋¹⁺⁴	1	1	yes	characteristic
45	Dic₇₁	—	2	4	no	—
46	Dic₇₁	—	4	6	no	—
47	C₂×C₇₁	D₄×Q₈	1	1	yes	characteristic
48	C₂×C₇₁	C₂²×Q₁₆	1	1	yes	characteristic
49	C₂×C₇₁	#576	1	1	yes	characteristic
50	Q₈×Q₁₆	—	71	1	no	maximal, Sylow
51	C₇₁	Q₈×Q₁₆	1	1	yes	characteristic, complemented, Sylow
52	Q₈²	—	71	2	no	—
53	#500	—	71	3	no	—
54	#477	—	71	6	no	—
55	C₈×Q₈	D₇₁	1	1	yes	characteristic
56	C₄×Q₁₆	—	71	3	no	—
57	C₂×Q₁₆	—	71	1	no	—
58	C₄⋊Q₈	—	71	6	no	—
59	C₄⋊Q₈	—	142	6	no	—
60	C₄×Q₈	—	71	6	no	—
61	C₄×Q₈	—	142	2	no	—
62	C₄×Q₈	C₂×D₇₁	1	1	yes	characteristic
63	#107	—	71	3	no	—
64	#107	—	71	6	no	—
65	C₄⋊C₈	C₂×D₇₁	1	3	yes	—
66	#103	—	71	6	no	—
67	C₄×C₈	C₂×D₇₁	1	3	yes	—
68	C₂×Q₈	—	71	2	no	—
69	C₂×Q₈	—	142	6	no	—
70	C₂×Q₈	D₂₈₄	1	1	yes	characteristic
71	Q₁₆	—	71	4	no	—
72	C₂×C₈	C₂²×D₇₁	1	3	yes	—
73	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
74	C₄⋊C₄	—	71	6	no	—
75	C₄⋊C₄	—	142	6	no	—
76	C₄⋊C₄	—	142	6	no	—
77	C₄⋊C₄	D₂₈₄	1	3	yes	—
78	C₄²	—	142	6	no	—
79	C₄²	C₂²×D₇₁	1	3	yes	—
80	Q₈	—	568	6	no	—
81	Q₈	—	71	4	no	—
82	Q₈	—	142	2	no	—
83	Q₈	Dic₂₈₄	1	4	yes	direct product factor
84	C₂×C₄	—	142	6	no	—
85	C₂×C₄	—	142	2	no	—
86	C₂×C₄	C₂×D₂₈₄	1	3	yes	—
87	C₂×C₄	C₂×D₂₈₄	1	3	yes	—
88	C₂×C₄	C₂³×D₇₁	1	1	yes	characteristic
89	C₈	—	2	3	no	—
90	C₈	Q₈×D₇₁	1	2	yes	—
91	C₂²	C₂²×D₂₈₄	1	1	yes	characteristic
92	C₄	—	2	3	no	—
93	C₄	—	284	6	no	—
94	C₄	—	142	4	no	—
95	C₄	C₂×Dic₂₈₄	1	6	yes	—
96	C₄	C₂×Q₈×D₇₁	1	1	yes	characteristic
97	C₄	#14556175	1	1	yes	characteristic
98	C₂	Q₈×D₂₈₄	1	1	yes	characteristic
99	C₂	C₂²×Dic₂₈₄	1	1	yes	characteristic
100	C₂	#17986804	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×Dic₁₄₂

C₂

yes

maximal

#17986404

C₂

yes

maximal

#17986295

C₂

yes

maximal

C₄×Dic₂₈₄

C₂

yes

maximal

C₈×Q₈×C₇₁

C₂

yes

maximal, characteristic

C₂×Dic₂₈₄

C₂²

yes

characteristic

#14556128

C₂²

yes

—

#14556127

—

Q₈×Dic₇₁

—

C₄×Dic₁₄₂

C₂²

yes

—

#14556062

C₂²

yes

—

#14556062

C₂²

yes

—

#14556053

C₂²

yes

—

C₄×Q₈×C₇₁

C₂²

yes

characteristic

C₄⋊C₈×C₇₁

C₂²

yes

—

C₄×C₈×C₇₁

C₂²

yes

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

C₂³

yes

—

Dic₂₈₄

Q₈

yes

direct product factor

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

C₂³

yes

—

Dic₇₁⋊C₄

—

C₄×Dic₇₁

—

C₂×Q₈×C₇₁

D₄

yes

characteristic

C₂×C₈×C₇₁

C₂³

yes

—

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₄⋊C₄×C₇₁

D₄

yes

—

C₄²×C₇₁

C₂³

yes

—

Q₈×C₇₁

Q₁₆

yes

complemented

C₂×C₄×C₇₁

C₂×D₄

yes

—

C₂×C₄×C₇₁

C₂×D₄

yes

—

C₂×C₄×C₇₁

C₂⁴

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

Dic₁₄₂

C₂×Q₈

yes

—

C₈×C₇₁

—

C₈×C₇₁

C₂×Q₈

yes

—

C₂²×C₇₁

C₂²×D₄

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

C₂×Q₁₆

yes

—

C₄×C₇₁

C₂²×Q₈

yes

characteristic

C₄×C₇₁

2₋¹⁺⁴

yes

characteristic

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

D₄×Q₈

yes

characteristic

C₂×C₇₁

C₂²×Q₁₆

yes

characteristic

C₂×C₇₁

#576

yes

characteristic

Q₈×Q₁₆

—

maximal, Sylow

C₇₁

Q₈×Q₁₆

yes

characteristic, complemented, Sylow

Q₈²

—

#500

—

#477

—

C₈×Q₈

D₇₁

yes

characteristic

C₄×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

142

—

C₄×Q₈

—

C₄×Q₈

—

142

—

C₄×Q₈

C₂×D₇₁

yes

characteristic

#107

—

#107

—

C₄⋊C₈

C₂×D₇₁

yes

—

#103

—

C₄×C₈

C₂×D₇₁

yes

—

C₂×Q₈

—

C₂×Q₈

—

142

—

C₂×Q₈

D₂₈₄

yes

characteristic

Q₁₆

—

C₂×C₈

C₂²×D₇₁

yes

—

C₂×C₈

C₂²×D₇₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

142

—

C₄⋊C₄

—

142

—

C₄⋊C₄

D₂₈₄

yes

—

C₄²

—

142

—

C₄²

C₂²×D₇₁

yes

—

Q₈

—

568

—

Q₈

—

Q₈

—

142

—

Q₈

Dic₂₈₄

yes

direct product factor

C₂×C₄

—

142

—

C₂×C₄

—

142

—

C₂×C₄

C₂×D₂₈₄

yes

—

C₂×C₄

C₂×D₂₈₄

yes

—

C₂×C₄

C₂³×D₇₁

yes

characteristic

C₈

—

C₈

Q₈×D₇₁

yes

—

C₂²

C₂²×D₂₈₄

yes

characteristic

C₄

—

C₄

—

284

—

C₄

—

142

—

C₄

C₂×Dic₂₈₄

yes

—

C₄

C₂×Q₈×D₇₁

yes

characteristic

C₄

#14556175

yes

characteristic

C₂

Q₈×D₂₈₄

yes

characteristic

C₂

C₂²×Dic₂₈₄

yes

characteristic

100

C₂

#17986804

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×D₂₈₄
commutator	C₂×C₄×C₇₁	C₂⁴
Frattini	C₂×C₄	C₂³×D₇₁
Fitting	C₈×Q₈×C₇₁	C₂
socle	C₂²×C₇₁	C₂²×D₄
hypercenter	C₈×Q₈	D₇₁
hypocenter	C₇₁	Q₈×Q₁₆
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

subgroup

G/H

center

C₂²

C₂²×D₂₈₄

commutator

C₂×C₄×C₇₁

C₂⁴

Frattini

C₂×C₄

C₂³×D₇₁

Fitting

C₈×Q₈×C₇₁

C₂

socle

C₂²×C₇₁

C₂²×D₄

hypercenter

C₈×Q₈

D₇₁

hypocenter

C₇₁

Q₈×Q₁₆

LCS

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ C₈×Q₈

N	H ≅ G/N	type
19. Dic₂₈₄	Q₈	direct (N × H)
29. Q₈×C₇₁	Q₁₆	semidirect
51. C₇₁	Q₈×Q₁₆	semidirect
83. Q₈	Dic₂₈₄	direct (N × H)

H ≅ G/N

type

19. Dic₂₈₄

Q₈

direct (N × H)

29. Q₈×C₇₁

Q₁₆

semidirect

51. C₇₁

Q₈×Q₁₆

semidirect

83. Q₈

Dic₂₈₄

direct (N × H)

Subgroups — Q₈×Dic₂₈₄ #22601225

Conjugacy classes of subgroups

Complements

Subgroups — Q8×Dic284 #22601225

Conjugacy classes of subgroups

Complements

Subgroups — Q₈×Dic₂₈₄ #22601225