Group #22586384 — subgroups

Subgroups — #22586384

Order 9088.

102 conjugacy classes of subgroups; only the 39 normal subgroups are shown.

normal subgroups only

This group has 102 conjugacy classes of subgroups — too many to draw. Only its 39 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	#17986383	C₂	1	1	yes	maximal, characteristic
2	#17985750	C₂	1	1	yes	maximal, characteristic, complemented
3	C₄⋊C₁₆×C₇₁	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×Dic₂₈₄	C₄	1	1	yes	characteristic
5	C₂×#11765854	—	2	1	no	—
6	C₂×D₅₆₈	C₄	1	1	yes	characteristic
7	#14556116	—	2	1	no	—
8	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
9	C₂×C₁₆×C₇₁	—	2	1	no	—
10	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
11	C₂×Dic₁₄₂	—	2	1	no	—
12	C₂×D₂₈₄	—	2	1	no	—
13	Dic₂₈₄	—	4	1	no	—
14	#11765854	—	4	1	no	—
15	#11765854	—	4	1	no	—
16	D₅₆₈	—	4	1	no	—
17	C₂×C₇₁⋊C₈	—	4	1	no	—
18	(C₂×D₇₁)⋊C₄	—	4	1	no	—
19	(C₂×D₇₁)⋊C₄	—	4	1	no	—
20	C₂×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
22	C₄²×C₇₁	D₄	1	1	yes	characteristic
23	C₁₆×C₇₁	—	4	1	no	—
24	C₂²×D₇₁	—	4	1	no	—
25	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
26	C₂×C₄×C₇₁	D₈	1	1	yes	characteristic
27	C₂×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
28	C₂×Dic₇₁	—	4	1	no	—
29	D₂₈₄	—	4	1	no	—
30	D₂₈₄	—	4	1	no	—
31	Dic₁₄₂	—	4	1	no	—
32	Dic₁₄₂	—	4	1	no	—
33	C₈×C₇₁	—	2	1	no	—
34	C₈×C₇₁	D₈	1	1	yes	characteristic
35	C₈×C₇₁	SD₁₆	1	1	yes	characteristic
36	C₇₁⋊C₈	—	8	1	no	—
37	C₂²×C₇₁	#102	1	1	yes	characteristic
38	C₂×D₇₁	—	8	1	no	—
39	C₂×D₇₁	—	8	1	no	—
40	C₂×D₇₁	—	8	1	no	—
41	C₄×C₇₁	—	2	1	no	—
42	C₄×C₇₁	—	2	1	no	—
43	C₄×C₇₁	#100	1	1	yes	characteristic
44	C₄×C₇₁	#102	1	1	yes	characteristic
45	Dic₇₁	—	8	1	no	—
46	C₂×C₇₁	#331	1	1	yes	characteristic
47	C₂×C₇₁	#359	1	1	yes	characteristic
48	C₂×C₇₁	#360	1	1	yes	characteristic
49	D₇₁	—	16	1	no	—
50	#1360	—	71	1	no	maximal, Sylow
51	C₇₁	#1360	1	1	yes	characteristic, complemented, Sylow
52	#495	—	71	1	no	—
53	C₄⋊C₁₆	D₇₁	1	1	yes	characteristic, complemented
54	#334	—	71	1	no	—
55	C₂×Q₁₆	—	71	1	no	—
56	C₂×SD₁₆	—	142	1	no	—
57	C₂×D₈	—	71	1	no	—
58	#124	—	142	1	no	—
59	C₂×C₁₆	—	2	1	no	—
60	C₄⋊C₈	—	142	1	no	—
61	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
62	C₂×Q₈	—	142	1	no	—
63	C₂×D₄	—	142	1	no	—
64	Q₁₆	—	284	1	no	—
65	SD₁₆	—	284	1	no	—
66	SD₁₆	—	284	1	no	—
67	D₈	—	284	1	no	—
68	C₂×C₈	—	284	1	no	—
69	C₂×C₈	C₄×D₇₁	1	1	yes	characteristic
70	C₂×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
71	C₂²⋊C₄	—	284	1	no	—
72	C₂²⋊C₄	—	284	1	no	—
73	C₄²	D₂₈₄	1	1	yes	characteristic
74	C₁₆	—	4	1	no	—
75	C₂³	—	284	1	no	—
76	Q₈	—	284	1	no	—
77	Q₈	—	284	1	no	—
78	D₄	—	284	1	no	—
79	D₄	—	284	1	no	—
80	C₂×C₄	—	284	1	no	—
81	C₂×C₄	(C₂×D₇₁)⋊C₄	1	1	yes	characteristic
82	C₂×C₄	D₅₆₈	1	1	yes	characteristic
83	C₂×C₄	#11765854	1	1	yes	characteristic
84	C₈	—	2	1	no	—
85	C₈	—	568	1	no	—
86	C₈	#11765850	1	1	yes	characteristic
87	C₈	#11765852	1	1	yes	characteristic
88	C₂²	—	568	1	no	—
89	C₂²	—	568	1	no	—
90	C₂²	—	568	1	no	—
91	C₂²	#14556050	1	1	yes	characteristic
92	C₄	—	2	1	no	—
93	C₄	—	2	1	no	—
94	C₄	—	568	1	no	—
95	C₄	#14556044	1	1	yes	characteristic
96	C₄	#14556048	1	1	yes	characteristic
97	C₂	—	1136	1	no	—
98	C₂	#17985742	1	1	yes	characteristic
99	C₂	#17985807	1	1	yes	characteristic
100	C₂	#17985810	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17986383

C₂

yes

maximal, characteristic

#17985750

C₂

yes

maximal, characteristic, complemented

C₄⋊C₁₆×C₇₁

C₂

yes

maximal, characteristic, complemented

C₂×Dic₂₈₄

C₄

yes

characteristic

C₂×#11765854

—

C₂×D₅₆₈

C₄

yes

characteristic

#14556116

—

C₄⋊(C₇₁⋊C₈)

—

C₂×C₁₆×C₇₁

—

C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

—

C₂×D₂₈₄

—

Dic₂₈₄

—

#11765854

—

#11765854

—

D₅₆₈

—

C₂×C₇₁⋊C₈

—

(C₂×D₇₁)⋊C₄

—

(C₂×D₇₁)⋊C₄

—

C₂×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂×C₈×C₇₁

D₄

yes

characteristic

C₄²×C₇₁

D₄

yes

characteristic

C₁₆×C₇₁

—

C₂²×D₇₁

—

C₂×C₄×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

D₈

yes

characteristic

C₂×C₄×C₇₁

SD₁₆

yes

characteristic

C₂×Dic₇₁

—

D₂₈₄

—

D₂₈₄

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

D₈

yes

characteristic

C₈×C₇₁

SD₁₆

yes

characteristic

C₇₁⋊C₈

—

C₂²×C₇₁

#102

yes

characteristic

C₂×D₇₁

—

C₂×D₇₁

—

C₂×D₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

D₇₁

—

#1360

—

maximal, Sylow

C₇₁

#1360

yes

characteristic, complemented, Sylow

#495

—

C₄⋊C₁₆

D₇₁

yes

characteristic, complemented

#334

—

C₂×Q₁₆

—

C₂×SD₁₆

—

142

—

C₂×D₈

—

#124

—

142

—

C₂×C₁₆

—

C₄⋊C₈

—

142

—

C₄×C₈

C₂×D₇₁

yes

characteristic

C₂×Q₈

—

142

—

C₂×D₄

—

142

—

Q₁₆

—

284

—

SD₁₆

—

284

—

SD₁₆

—

284

—

D₈

—

284

—

C₂×C₈

—

284

—

C₂×C₈

C₄×D₇₁

yes

characteristic

C₂×C₈

C₂²⋊D₇₁

yes

characteristic

C₂²⋊C₄

—

284

—

C₂²⋊C₄

—

284

—

C₄²

D₂₈₄

yes

characteristic

C₁₆

—

C₂³

—

284

—

Q₈

—

284

—

Q₈

—

284

—

D₄

—

284

—

D₄

—

284

—

C₂×C₄

—

284

—

C₂×C₄

(C₂×D₇₁)⋊C₄

yes

characteristic

C₂×C₄

D₅₆₈

yes

characteristic

C₂×C₄

#11765854

yes

characteristic

C₈

—

C₈

—

568

—

C₈

yes

C₈

yes

C₂²

—

568

—

C₂²

—

568

—

C₂²

—

568

—

C₂²

#14556050

yes

characteristic

C₄

—

C₄

—

C₄

—

568

—

C₄

yes

C₄

yes

C₂

—

1136

—

C₂

yes

C₂

yes

100

C₂

#17985810

yes

characteristic

	subgroup	G/H
center	C₂²	#14556050
commutator	C₂×C₈×C₇₁	C₂×C₄
Frattini	C₄×C₈	C₂×D₇₁
Fitting	C₄⋊C₁₆×C₇₁	C₂
socle	C₂²×C₇₁	#102
hypercenter	C₄⋊C₁₆	D₇₁
hypocenter	C₇₁	#1360
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

subgroup

G/H

center

C₂²

#14556050

commutator

C₂×C₈×C₇₁

C₂×C₄

Frattini

C₄×C₈

C₂×D₇₁

Fitting

C₄⋊C₁₆×C₇₁

C₂

socle

C₂²×C₇₁

#102

hypercenter

C₄⋊C₁₆

D₇₁

hypocenter

C₇₁

#1360

LCS

G ▷ C₂×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄⋊C₁₆

N	H ≅ G/N	type
2. #17985750	C₂	semidirect
3. C₄⋊C₁₆×C₇₁	C₂	semidirect
51. C₇₁	#1360	semidirect
53. C₄⋊C₁₆	D₇₁	semidirect

H ≅ G/N

type

2. #17985750

C₂

semidirect

3. C₄⋊C₁₆×C₇₁

C₂

semidirect

51. C₇₁

#1360

semidirect

53. C₄⋊C₁₆

D₇₁

semidirect

Subgroups — #22586384

Conjugacy classes of subgroups

Complements