Group #22588504 — subgroups

Subgroups — #22588504

Order 9088.

248 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 141 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×#14556062	C₂	1	2	yes	maximal
2	C₂×#14556060	C₂	1	2	yes	maximal
3	#17985865	C₂	1	2	yes	maximal
4	C₂×C₄×C₈×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Dic₇₁	C₂²	1	4	yes	—
6	#14556062	—	2	4	no	—
7	#14556060	—	2	4	no	—
8	C₇₁⋊#95	—	2	4	no	—
9	C₂²×C₈×C₇₁	C₂²	1	1	yes	characteristic
10	C₂²×C₈×C₇₁	C₂²	1	1	yes	characteristic
11	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
12	C₄×C₈×C₇₁	C₄	1	4	yes	complemented
13	C₂²×Dic₇₁	—	2	4	no	—
14	C₄⋊Dic₇₁	—	2	4	no	—
15	C₄⋊Dic₇₁	—	2	4	no	—
16	C₂²×C₄×C₇₁	D₄	1	2	yes	—
17	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
18	C₂×C₈×C₇₁	C₂×C₄	1	2	yes	—
19	C₂×C₈×C₇₁	C₂×C₄	1	2	yes	—
20	C₂×C₈×C₇₁	D₄	1	2	yes	—
21	C₂×C₈×C₇₁	D₄	1	2	yes	—
22	C₂×C₈×C₇₁	Q₈	1	2	yes	—
23	C₂×C₈×C₇₁	Q₈	1	2	yes	—
24	C₄²×C₇₁	C₂×C₄	1	2	yes	—
25	C₄²×C₇₁	Q₈	1	2	yes	—
26	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
27	C₂×C₄×C₇₁	—	2	4	no	—
28	C₂×C₄×C₇₁	C₄⋊C₄	1	4	yes	—
29	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
33	C₂×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
34	C₂×C₄×C₇₁	C₂×Q₈	1	1	yes	characteristic
35	C₂×Dic₇₁	—	4	4	no	—
36	C₂×Dic₇₁	—	4	8	no	—
37	C₈×C₇₁	C₄⋊C₄	1	4	yes	—
38	C₈×C₇₁	C₄⋊C₄	1	4	yes	—
39	C₂²×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
40	C₂²×C₇₁	#127	1	1	yes	characteristic
41	C₂²×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
42	C₂²×C₇₁	#135	1	4	yes	—
43	C₄×C₇₁	—	2	4	no	—
44	C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₄×C₇₁	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
47	C₄×C₇₁	C₄⋊Q₈	1	1	yes	characteristic
48	Dic₇₁	—	8	4	no	—
49	C₂×C₇₁	#384	1	1	yes	characteristic
50	C₂×C₇₁	#427	1	2	yes	—
51	C₂×C₇₁	#495	1	2	yes	—
52	C₂×C₇₁	#499	1	2	yes	—
53	#1846	—	71	1	no	maximal, Sylow
54	C₇₁	#1846	1	1	yes	characteristic, complemented, Sylow
55	C₂×#107	—	71	2	no	—
56	C₂×#106	—	71	2	no	—
57	C₂×C₄×C₈	D₇₁	1	1	yes	characteristic
58	#382	—	71	2	no	—
59	C₂²×C₈	C₂×D₇₁	1	1	yes	characteristic
60	C₂²×C₈	C₂×D₇₁	1	1	yes	characteristic
61	C₂×C₄⋊C₄	—	71	4	no	—
62	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
63	#107	—	142	4	no	—
64	#106	—	142	4	no	—
65	C₄×C₈	Dic₇₁	1	4	yes	complemented
66	#95	—	142	4	no	—
67	C₂²×C₄	—	142	4	no	—
68	C₂²×C₄	D₂₈₄	1	2	yes	—
69	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
70	C₂×C₈	Dic₁₄₂	1	2	yes	—
71	C₂×C₈	Dic₁₄₂	1	2	yes	—
72	C₂×C₈	D₂₈₄	1	2	yes	—
73	C₂×C₈	D₂₈₄	1	2	yes	—
74	C₂×C₈	C₂×Dic₇₁	1	2	yes	—
75	C₂×C₈	C₂×Dic₇₁	1	2	yes	—
76	C₄⋊C₄	—	142	4	no	—
77	C₄⋊C₄	—	142	4	no	—
78	C₄²	Dic₁₄₂	1	2	yes	—
79	C₄²	C₂×Dic₇₁	1	2	yes	—
80	C₂³	C₂×D₂₈₄	1	1	yes	characteristic
81	C₂×C₄	—	2	4	no	—
82	C₂×C₄	—	284	4	no	—
83	C₂×C₄	—	284	8	no	—
84	C₂×C₄	C₄⋊Dic₇₁	1	4	yes	—
85	C₂×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
86	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
87	C₂×C₄	C₂×D₂₈₄	1	1	yes	characteristic
88	C₂×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
89	C₂×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
90	C₂×C₄	C₂×Dic₁₄₂	1	1	yes	characteristic
91	C₈	C₄⋊Dic₇₁	1	4	yes	—
92	C₈	C₄⋊Dic₇₁	1	4	yes	—
93	C₂²	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
94	C₂²	#14556125	1	1	yes	characteristic
95	C₂²	#14556128	1	1	yes	characteristic
96	C₂²	#14556150	1	4	yes	—
97	C₄	—	2	4	no	—
98	C₄	—	568	4	no	—
99	C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
100	C₄	C₂×C₄⋊Dic₇₁	1	1	yes	characteristic
101	C₄	#14556128	1	1	yes	characteristic
102	C₄	#14556128	1	1	yes	characteristic
103	C₂	#17985869	1	1	yes	characteristic
104	C₂	#17986037	1	2	yes	—
105	C₂	#17986383	1	2	yes	—
106	C₂	#17986401	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556062

C₂

yes

maximal

C₂×#14556060

C₂

yes

maximal

#17985865

C₂

yes

maximal

C₂×C₄×C₈×C₇₁

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₇₁

C₂²

yes

—

#14556062

—

#14556060

—

C₇₁⋊#95

—

C₂²×C₈×C₇₁

C₂²

yes

characteristic

C₂²×C₈×C₇₁

C₂²

yes

characteristic

C₂×C₄²×C₇₁

C₂²

yes

characteristic

C₄×C₈×C₇₁

C₄

yes

complemented

C₂²×Dic₇₁

—

C₄⋊Dic₇₁

—

C₄⋊Dic₇₁

—

C₂²×C₄×C₇₁

D₄

yes

—

C₂²×C₄×C₇₁

C₂³

yes

characteristic

C₂×C₈×C₇₁

C₂×C₄

yes

—

C₂×C₈×C₇₁

C₂×C₄

yes

—

C₂×C₈×C₇₁

D₄

yes

—

C₂×C₈×C₇₁

D₄

yes

—

C₂×C₈×C₇₁

Q₈

yes

—

C₂×C₈×C₇₁

Q₈

yes

—

C₄²×C₇₁

C₂×C₄

yes

—

C₄²×C₇₁

Q₈

yes

—

C₂³×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₄⋊C₄

yes

—

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₈

yes

characteristic

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₈×C₇₁

C₄⋊C₄

yes

—

C₈×C₇₁

C₄⋊C₄

yes

—

C₂²×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₇₁

#127

yes

characteristic

C₂²×C₇₁

C₄⋊Q₈

yes

characteristic

C₂²×C₇₁

#135

yes

—

C₄×C₇₁

—

C₄×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₇₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₇₁

C₄⋊Q₈

yes

characteristic

C₄×C₇₁

C₄⋊Q₈

yes

characteristic

Dic₇₁

—

C₂×C₇₁

#384

yes

characteristic

C₂×C₇₁

#427

yes

—

C₂×C₇₁

#495

yes

—

C₂×C₇₁

#499

yes

—

#1846

—

maximal, Sylow

C₇₁

#1846

yes

characteristic, complemented, Sylow

C₂×#107

—

C₂×#106

—

C₂×C₄×C₈

D₇₁

yes

characteristic

#382

—

C₂²×C₈

C₂×D₇₁

yes

characteristic

C₂²×C₈

C₂×D₇₁

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₇₁

yes

characteristic

#107

—

142

—

#106

—

142

—

C₄×C₈

Dic₇₁

yes

complemented

#95

—

142

—

C₂²×C₄

—

142

—

C₂²×C₄

D₂₈₄

yes

—

C₂²×C₄

C₂²×D₇₁

yes

characteristic

C₂×C₈

Dic₁₄₂

yes

—

C₂×C₈

Dic₁₄₂

yes

—

C₂×C₈

D₂₈₄

yes

—

C₂×C₈

D₂₈₄

yes

—

C₂×C₈

C₂×Dic₇₁

yes

—

C₂×C₈

C₂×Dic₇₁

yes

—

C₄⋊C₄

—

142

—

C₄⋊C₄

—

142

—

C₄²

Dic₁₄₂

yes

—

C₄²

C₂×Dic₇₁

yes

—

C₂³

C₂×D₂₈₄

yes

characteristic

C₂×C₄

—

C₂×C₄

—

284

—

C₂×C₄

—

284

—

C₂×C₄

C₄⋊Dic₇₁

yes

—

C₂×C₄

C₂²×Dic₇₁

yes

characteristic

C₂×C₄

C₂×D₂₈₄

yes

characteristic

C₂×C₄

C₂×D₂₈₄

yes

characteristic

C₂×C₄

C₂×Dic₁₄₂

yes

characteristic

C₂×C₄

C₂×Dic₁₄₂

yes

characteristic

C₂×C₄

C₂×Dic₁₄₂

yes

characteristic

C₈

C₄⋊Dic₇₁

yes

—

C₈

C₄⋊Dic₇₁

yes

—

C₂²

C₂×C₄⋊Dic₇₁

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₄

—

C₄

—

568

—

C₄

C₂×C₄⋊Dic₇₁

yes

characteristic

100

C₄

C₂×C₄⋊Dic₇₁

yes

characteristic

101

C₄

#14556128

yes

characteristic

102

C₄

#14556128

yes

characteristic

103

C₂

#17985869

yes

characteristic

104

C₂

#17986037

yes

—

105

C₂

#17986383

yes

—

106

C₂

#17986401

yes

—

	subgroup	G/H
center	C₂³	C₂×D₂₈₄
commutator	C₂×C₄×C₇₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₁
Fitting	C₂×C₄×C₈×C₇₁	C₂
socle	C₂³×C₇₁	C₂×D₄
hypercenter	C₂×C₄×C₈	D₇₁
hypocenter	C₇₁	#1846
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₂₈₄

commutator

C₂×C₄×C₇₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₁

Fitting

C₂×C₄×C₈×C₇₁

C₂

socle

C₂³×C₇₁

C₂×D₄

hypercenter

C₂×C₄×C₈

D₇₁

hypocenter

C₇₁

#1846

LCS

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
12. C₄×C₈×C₇₁	C₄	semidirect
54. C₇₁	#1846	semidirect
65. C₄×C₈	Dic₇₁	semidirect

H ≅ G/N

type

12. C₄×C₈×C₇₁

C₄

semidirect

54. C₇₁

#1846

semidirect

65. C₄×C₈

Dic₇₁

semidirect

Subgroups — #22588504

Conjugacy classes of subgroups

Complements