Group #22590479 — subgroups

Subgroups — #22590479

Order 9088.

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	C₄×Dic₂₈₄	C₂	1	1	yes	maximal, characteristic
2	#17985826	C₂	1	2	yes	maximal
3	C₄⋊(C₇₁⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
4	#17985802	C₂	1	2	yes	maximal
5	#500×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₂₈₄	C₂²	1	1	yes	characteristic
7	C₄×Dic₁₄₂	—	2	1	no	—
8	C₂×C₇₁⋊C₁₆	C₂²	1	2	yes	—
9	#14556062	C₂²	1	1	yes	characteristic
10	#14556053	—	2	1	no	—
11	C₄⋊Q₈×C₇₁	—	2	1	no	—
12	#107×C₇₁	—	2	1	no	—
13	#107×C₇₁	C₂²	1	2	yes	—
14	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
15	C₂×Dic₁₄₂	—	2	1	no	—
16	Dic₂₈₄	—	2	1	no	—
17	Dic₂₈₄	Q₈	1	2	yes	complemented
18	C₄⋊Dic₇₁	—	2	1	no	—
19	Dic₇₁⋊C₄	—	4	1	no	—
20	C₄×Dic₇₁	—	4	1	no	—
21	C₇₁⋊C₁₆	—	2	2	no	—
22	C₂×Q₈×C₇₁	—	4	1	no	—
23	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
24	C₂×C₈×C₇₁	C₂³	1	1	yes	characteristic
25	C₄⋊C₄×C₇₁	—	2	2	no	—
26	C₄⋊C₄×C₇₁	—	4	1	no	—
27	C₄²×C₇₁	D₄	1	1	yes	characteristic
28	Q₈×C₇₁	—	8	2	no	—
29	C₂×C₄×C₇₁	—	4	2	no	—
30	C₂×C₄×C₇₁	D₈	1	1	yes	characteristic
31	C₂×C₄×C₇₁	D₈	1	1	yes	characteristic
32	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
33	C₂×Dic₇₁	—	4	1	no	—
34	C₂×Dic₇₁	—	4	1	no	—
35	Dic₁₄₂	—	2	2	no	—
36	Dic₁₄₂	—	4	1	no	—
37	C₈×C₇₁	—	2	1	no	—
38	C₈×C₇₁	C₂×Q₈	1	1	yes	characteristic
39	C₈×C₇₁	#41	1	1	yes	characteristic
40	C₂²×C₇₁	C₂×D₈	1	1	yes	characteristic
41	C₄×C₇₁	—	2	1	no	—
42	C₄×C₇₁	—	8	2	no	—
43	C₄×C₇₁	SD₃₂	1	2	yes	—
44	C₄×C₇₁	C₂²⋊Q₈	1	1	yes	characteristic
45	C₄×C₇₁	#137	1	1	yes	characteristic
46	Dic₇₁	—	4	2	no	—
47	Dic₇₁	—	8	1	no	—
48	C₂×C₇₁	#474	1	1	yes	characteristic
49	C₂×C₇₁	C₂×SD₃₂	1	1	yes	characteristic
50	C₂×C₇₁	#510	1	1	yes	characteristic
51	#2225	—	71	1	no	maximal, Sylow
52	C₇₁	#2225	1	1	yes	characteristic, complemented, Sylow
53	#500	D₇₁	1	1	yes	characteristic
54	C₄×Q₁₆	—	71	1	no	—
55	#367	—	71	2	no	—
56	C₄⋊C₁₆	—	71	1	no	—
57	#358	—	71	2	no	—
58	C₂×Q₁₆	—	71	1	no	—
59	C₄⋊Q₈	—	2	1	no	—
60	C₄×Q₈	—	142	1	no	—
61	C₂×C₁₆	—	71	2	no	—
62	#107	—	2	1	no	—
63	#107	—	71	1	no	—
64	#107	C₂×D₇₁	1	2	yes	—
65	#103	—	142	1	no	—
66	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
67	C₂×Q₈	—	4	1	no	—
68	C₂×Q₈	—	142	1	no	—
69	Q₁₆	—	71	2	no	—
70	Q₁₆	—	142	1	no	—
71	C₂×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
72	C₂×C₈	C₂²×D₇₁	1	1	yes	characteristic
73	C₄⋊C₄	—	2	2	no	—
74	C₄⋊C₄	—	4	1	no	—
75	C₄⋊C₄	—	284	1	no	—
76	C₄⋊C₄	—	142	1	no	—
77	C₄²	—	284	1	no	—
78	C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
79	C₁₆	—	142	2	no	—
80	Q₈	—	8	2	no	—
81	Q₈	—	284	1	no	—
82	Q₈	—	142	2	no	—
83	C₂×C₄	—	4	2	no	—
84	C₂×C₄	—	284	1	no	—
85	C₂×C₄	—	284	1	no	—
86	C₂×C₄	#11765850	1	1	yes	characteristic
87	C₂×C₄	#11765850	1	1	yes	characteristic
88	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
89	C₈	—	2	1	no	—
90	C₈	Q₈×D₇₁	1	1	yes	characteristic
91	C₈	#11765866	1	1	yes	characteristic
92	C₂²	C₂×#11765850	1	1	yes	characteristic
93	C₄	—	2	1	no	—
94	C₄	—	8	2	no	—
95	C₄	—	284	2	no	—
96	C₄	—	568	1	no	—
97	C₄	#14556072	1	2	yes	—
98	C₄	#14556104	1	1	yes	characteristic
99	C₄	#14556159	1	1	yes	characteristic
100	C₂	#17986276	1	1	yes	characteristic
101	C₂	C₂×#14556072	1	1	yes	characteristic
102	C₂	#17986440	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄×Dic₂₈₄

C₂

yes

maximal, characteristic

#17985826

C₂

yes

maximal

C₄⋊(C₇₁⋊C₁₆)

C₂

yes

maximal, characteristic

#17985802

C₂

yes

maximal

#500×C₇₁

C₂

yes

maximal, characteristic

C₂×Dic₂₈₄

C₂²

yes

characteristic

C₄×Dic₁₄₂

—

C₂×C₇₁⋊C₁₆

C₂²

yes

—

#14556062

C₂²

yes

characteristic

#14556053

—

C₄⋊Q₈×C₇₁

—

#107×C₇₁

—

#107×C₇₁

C₂²

yes

—

C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×Dic₁₄₂

—

Dic₂₈₄

—

Dic₂₈₄

Q₈

yes

complemented

C₄⋊Dic₇₁

—

Dic₇₁⋊C₄

—

C₄×Dic₇₁

—

C₇₁⋊C₁₆

—

C₂×Q₈×C₇₁

—

C₂×C₈×C₇₁

D₄

yes

characteristic

C₂×C₈×C₇₁

C₂³

yes

characteristic

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄²×C₇₁

D₄

yes

characteristic

Q₈×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

D₈

yes

characteristic

C₂×C₄×C₇₁

D₈

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

Dic₁₄₂

—

Dic₁₄₂

—

C₈×C₇₁

—

C₈×C₇₁

C₂×Q₈

yes

characteristic

C₈×C₇₁

#41

yes

characteristic

C₂²×C₇₁

C₂×D₈

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

SD₃₂

yes

—

C₄×C₇₁

C₂²⋊Q₈

yes

characteristic

C₄×C₇₁

#137

yes

characteristic

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

#474

yes

characteristic

C₂×C₇₁

C₂×SD₃₂

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

#500

D₇₁

yes

characteristic

C₄×Q₁₆

—

#367

—

C₄⋊C₁₆

—

#358

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄×Q₈

—

142

—

C₂×C₁₆

—

#107

—

#107

—

#107

C₂×D₇₁

yes

—

#103

—

142

—

C₄×C₈

C₂×D₇₁

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

142

—

Q₁₆

—

Q₁₆

—

142

—

C₂×C₈

C₂²⋊D₇₁

yes

characteristic

C₂×C₈

C₂²×D₇₁

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

284

—

C₄⋊C₄

—

142

—

C₄²

—

284

—

C₄²

C₂²⋊D₇₁

yes

characteristic

C₁₆

—

142

—

Q₈

—

Q₈

—

284

—

Q₈

—

142

—

C₂×C₄

—

C₂×C₄

—

284

—

C₂×C₄

—

284

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

C₈

—

C₈

Q₈×D₇₁

yes

C₈

yes

C₂²

yes

C₄

—

C₄

—

C₄

—

284

—

C₄

—

568

—

C₄

#14556072

yes

—

C₄

yes

C₄

yes

100

C₂

#17986276

yes

characteristic

101

C₂

C₂×#14556072

yes

characteristic

102

C₂

#17986440

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#11765850
commutator	C₂×C₈×C₇₁	C₂³
Frattini	C₂×C₈	C₂²×D₇₁
Fitting	#500×C₇₁	C₂
socle	C₂²×C₇₁	C₂×D₈
hypercenter	#500	D₇₁
hypocenter	C₇₁	#2225
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #500

subgroup

G/H

center

C₂²

C₂×#11765850

commutator

C₂×C₈×C₇₁

C₂³

Frattini

C₂×C₈

C₂²×D₇₁

Fitting

#500×C₇₁

C₂

socle

C₂²×C₇₁

C₂×D₈

D₇₁

C₇₁

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #500

N	H ≅ G/N	type
17. Dic₂₈₄	Q₈	semidirect
52. C₇₁	#2225	semidirect

H ≅ G/N

type

17. Dic₂₈₄

Q₈

semidirect

52. C₇₁

#2225

semidirect

Subgroups — #22590479

Conjugacy classes of subgroups

Complements