Group #22588490 — subgroups

Subgroups — #22588490

Order 9088.

162 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 65 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	M₁₆×Dic₇₁	C₂	1	1	yes	maximal, characteristic
2	#17985771	C₂	1	2	yes	maximal
3	#17985760	C₂	1	2	yes	maximal
4	#17985756	C₂	1	1	yes	maximal, characteristic
5	#338×C₇₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#11765849	C₂²	1	1	yes	characteristic
7	#14556087	C₂²	1	2	yes	—
8	C₂×C₄×Dic₇₁	C₂²	1	1	yes	characteristic
9	#14556038	C₄	1	2	yes	complemented
10	C₈×Dic₇₁	C₄	1	2	yes	complemented
11	C₂×M₁₆×C₇₁	C₂²	1	2	yes	—
12	C₂×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
13	C₂²×Dic₇₁	—	2	1	no	—
14	#11765849	—	2	2	no	—
15	C₂×C₇₁⋊C₈	C₂×C₄	1	2	yes	—
16	C₄⋊Dic₇₁	—	2	2	no	—
17	C₂²⋊Dic₇₁	—	2	2	no	—
18	C₄×Dic₇₁	—	2	1	no	—
19	C₄×Dic₇₁	—	2	2	no	—
20	C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
21	C₄×Dic₇₁	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
23	M₁₆×C₇₁	—	2	2	no	—
24	M₁₆×C₇₁	—	2	2	no	—
25	M₁₆×C₇₁	—	2	2	no	—
26	C₂×C₈×C₇₁	—	2	2	no	—
27	C₂×C₈×C₇₁	C₂×C₄	1	2	yes	—
28	C₂³×C₇₁	#41	1	1	yes	characteristic
29	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₄×C₇₁	#41	1	2	yes	—
31	C₂×C₄×C₇₁	#41	1	1	yes	characteristic
32	C₂×C₄×C₇₁	#41	1	2	yes	—
33	C₂×Dic₇₁	—	2	1	no	—
34	C₂×Dic₇₁	—	2	1	no	—
35	C₂×Dic₇₁	—	4	2	no	—
36	C₂×Dic₇₁	—	4	2	no	—
37	C₈×C₇₁	—	2	2	no	—
38	C₈×C₇₁	—	4	2	no	—
39	C₇₁⋊C₈	—	2	2	no	—
40	C₂²×C₇₁	—	4	1	no	—
41	C₂²×C₇₁	#117	1	1	yes	characteristic
42	C₂²×C₇₁	#126	1	2	yes	—
43	C₄×C₇₁	#117	1	1	yes	characteristic
44	C₄×C₇₁	#117	1	1	yes	characteristic
45	C₄×C₇₁	#126	1	2	yes	—
46	Dic₇₁	—	2	2	no	—
47	Dic₇₁	—	4	1	no	—
48	Dic₇₁	—	8	2	no	—
49	C₂×C₇₁	—	2	1	no	—
50	C₂×C₇₁	—	2	2	no	—
51	C₂×C₇₁	#383	1	1	yes	characteristic
52	#1841	—	71	1	no	maximal, Sylow
53	C₇₁	#1841	1	1	yes	characteristic, complemented, Sylow
54	C₄×M₁₆	—	71	1	no	—
55	#344	—	71	2	no	—
56	#339	—	71	2	no	—
57	#338	D₇₁	1	1	yes	characteristic
58	H(3,ℤ/4)	—	71	1	no	—
59	C₂×M₁₆	—	71	1	no	—
60	C₂×M₁₆	C₂×D₇₁	1	2	yes	—
61	C₂×M₁₆	C₂×D₇₁	1	1	yes	characteristic
62	#117	—	71	2	no	—
63	C₂×C₄²	—	71	1	no	—
64	#97	—	71	2	no	—
65	C₄×C₈	—	71	2	no	—
66	C₂²×C₄	—	142	1	no	—
67	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
68	M₁₆	—	2	2	no	—
69	M₁₆	—	2	2	no	—
70	M₁₆	—	2	2	no	—
71	M₁₆	—	142	2	no	—
72	C₂×C₈	—	2	2	no	—
73	C₂×C₈	—	71	2	no	—
74	C₂×C₈	C₄×D₇₁	1	2	yes	—
75	C₄⋊C₄	—	142	2	no	—
76	C₂²⋊C₄	—	142	2	no	—
77	C₄²	—	71	1	no	—
78	C₄²	—	71	1	no	—
79	C₄²	—	142	2	no	—
80	C₄²	—	142	1	no	—
81	C₂³	#11765866	1	1	yes	characteristic
82	C₂×C₄	—	284	2	no	—
83	C₂×C₄	—	284	2	no	—
84	C₂×C₄	—	142	1	no	—
85	C₂×C₄	—	142	1	no	—
86	C₂×C₄	C₂×C₄×D₇₁	1	1	yes	characteristic
87	C₂×C₄	#11765864	1	2	yes	—
88	C₂×C₄	#11765864	1	1	yes	characteristic
89	C₂×C₄	#11765865	1	2	yes	—
90	C₈	—	2	2	no	—
91	C₈	—	4	2	no	—
92	C₈	—	142	2	no	—
93	C₂²	—	4	1	no	—
94	C₂²	#14556086	1	1	yes	characteristic
95	C₂²	#14556123	1	2	yes	—
96	C₄	—	284	1	no	—
97	C₄	—	568	2	no	—
98	C₄	—	142	2	no	—
99	C₄	#14556084	1	1	yes	characteristic
100	C₄	#14556085	1	1	yes	characteristic
101	C₄	#14556121	1	2	yes	—
102	C₂	—	2	1	no	—
103	C₂	—	2	2	no	—
104	C₂	#17985866	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

M₁₆×Dic₇₁

C₂

yes

maximal, characteristic

#17985771

C₂

yes

maximal

#17985760

C₂

yes

maximal

#17985756

C₂

yes

maximal, characteristic

#338×C₇₁

C₂

yes

maximal, characteristic

C₂×#11765849

C₂²

yes

characteristic

#14556087

C₂²

yes

—

C₂×C₄×Dic₇₁

C₂²

yes

characteristic

#14556038

C₄

yes

complemented

C₈×Dic₇₁

C₄

yes

complemented

C₂×M₁₆×C₇₁

C₂²

yes

—

C₂×M₁₆×C₇₁

C₂²

yes

characteristic

C₂²×Dic₇₁

—

#11765849

—

C₂×C₇₁⋊C₈

C₂×C₄

yes

—

C₄⋊Dic₇₁

—

C₂²⋊Dic₇₁

—

C₄×Dic₇₁

—

C₄×Dic₇₁

—

C₄×Dic₇₁

C₂×C₄

yes

characteristic

C₄×Dic₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

C₂³

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

—

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂×C₄

yes

—

C₂³×C₇₁

#41

yes

characteristic

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

#41

yes

—

C₂×C₄×C₇₁

#41

yes

characteristic

C₂×C₄×C₇₁

#41

yes

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

#117

yes

characteristic

C₂²×C₇₁

#126

yes

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

yes

—

Dic₇₁

—

Dic₇₁

—

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

C₄×M₁₆

—

#344

—

#339

—

#338

D₇₁

yes

characteristic

H(3,ℤ/4)

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₇₁

yes

—

C₂×M₁₆

C₂×D₇₁

yes

characteristic

#117

—

C₂×C₄²

—

#97

—

C₄×C₈

—

C₂²×C₄

—

142

—

C₂²×C₄

C₂²×D₇₁

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

142

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₇₁

yes

—

C₄⋊C₄

—

142

—

C₂²⋊C₄

—

142

—

C₄²

—

C₄²

—

C₄²

—

142

—

C₄²

—

142

—

C₂³

#11765866

yes

characteristic

C₂×C₄

—

284

—

C₂×C₄

—

284

—

C₂×C₄

—

142

—

C₂×C₄

—

142

—

C₂×C₄

C₂×C₄×D₇₁

yes

characteristic

C₂×C₄

#11765864

yes

—

C₂×C₄

#11765864

yes

characteristic

C₂×C₄

#11765865

yes

—

C₈

—

C₈

—

C₈

—

142

—

C₂²

—

C₂²

#14556086

yes

characteristic

C₂²

#14556123

yes

—

C₄

—

284

—

C₄

—

568

—

C₄

—

142

—

C₄

#14556084

yes

characteristic

100

C₄

#14556085

yes

characteristic

101

C₄

#14556121

yes

—

102

C₂

—

103

C₂

—

104

C₂

#17985866

yes

characteristic

	subgroup	G/H
center	C₄	#14556085
commutator	C₂×C₄×C₇₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₁
Fitting	#338×C₇₁	C₂
socle	C₂×C₇₁	#383
hypercenter	#338	D₇₁
hypocenter	C₇₁	#1841
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

subgroup

G/H

center

C₄

#14556085

commutator

C₂×C₄×C₇₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₁

Fitting

#338×C₇₁

C₂

C₂×C₇₁

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₄ ◁ C₂×M₁₆ ◁ #338

N	H ≅ G/N	type
9. #14556038	C₄	semidirect
10. C₈×Dic₇₁	C₄	semidirect
53. C₇₁	#1841	semidirect

H ≅ G/N

type

9. #14556038

C₄

semidirect

10. C₈×Dic₇₁

C₄

semidirect

53. C₇₁

#1841

semidirect

Subgroups — #22588490

Conjugacy classes of subgroups

Complements