Group #22589306 — subgroups

Subgroups — #22589306

Order 9088.

252 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 113 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×C₄×C₇₁⋊C₈	C₂	1	1	yes	maximal, characteristic
2	#17985865	C₂	1	1	yes	maximal, characteristic
3	#17985761	C₂	1	4	yes	maximal
4	C₂×#125×C₇₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₇₁⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₇₁	C₂²	1	2	yes	—
7	C₄×C₇₁⋊C₈	—	2	2	no	—
8	C₇₁⋊#95	—	2	2	no	—
9	#125×C₇₁	—	2	2	no	—
10	#125×C₇₁	C₄	1	4	yes	complemented
11	C₂×C₄⋊C₄×C₇₁	—	2	2	no	—
12	C₂×C₄⋊C₄×C₇₁	C₂²	1	2	yes	—
13	C₂×C₄²×C₇₁	C₂²	1	1	yes	characteristic
14	C₂²×Dic₇₁	—	2	2	no	—
15	C₂×C₇₁⋊C₈	—	2	2	no	—
16	C₂×C₇₁⋊C₈	—	2	2	no	—
17	C₂×C₇₁⋊C₈	Q₈	1	4	yes	—
18	C₄⋊Dic₇₁	—	2	4	no	—
19	C₂²×C₄×C₇₁	—	2	2	no	—
20	C₂²×C₄×C₇₁	D₄	1	2	yes	—
21	C₂²×C₄×C₇₁	C₂³	1	1	yes	characteristic
22	C₄⋊C₄×C₇₁	—	2	8	no	—
23	C₄⋊C₄×C₇₁	—	2	2	no	—
24	C₄⋊C₄×C₇₁	C₂×C₄	1	4	yes	—
25	C₄²×C₇₁	C₂×C₄	1	2	yes	—
26	C₄²×C₇₁	D₄	1	2	yes	—
27	C₂³×C₇₁	C₂×D₄	1	1	yes	characteristic
28	C₂×C₄×C₇₁	—	2	4	no	—
29	C₂×C₄×C₇₁	—	2	4	no	—
30	C₂×C₄×C₇₁	—	4	4	no	—
31	C₂×C₄×C₇₁	C₂²⋊C₄	1	4	yes	—
32	C₂×C₄×C₇₁	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
34	C₂×C₄×C₇₁	C₂×Q₈	1	2	yes	—
35	C₂×C₄×C₇₁	#41	1	2	yes	—
36	C₂×Dic₇₁	—	4	2	no	—
37	C₂×Dic₇₁	—	4	4	no	—
38	C₇₁⋊C₈	—	2	4	no	—
39	C₂²×C₇₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
40	C₂²×C₇₁	#124	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	—	4	4	no	—
45	C₄×C₇₁	C₄×Q₈	1	2	yes	—
46	C₄×C₇₁	C₂²⋊Q₈	1	2	yes	—
47	Dic₇₁	—	8	2	no	—
48	C₂×C₇₁	#391	1	1	yes	characteristic
49	C₂×C₇₁	#416	1	2	yes	—
50	C₂×C₇₁	#488	1	2	yes	—
51	C₂×C₇₁	#499	1	2	yes	—
52	#1991	—	71	1	no	maximal, Sylow
53	C₇₁	#1991	1	1	yes	characteristic, complemented, Sylow
54	C₂×#125	D₇₁	1	1	yes	characteristic
55	C₂×C₄×C₈	—	71	1	no	—
56	#382	—	71	1	no	—
57	#340	—	71	4	no	—
58	C₂²×C₈	—	71	2	no	—
59	#125	—	2	2	no	—
60	#125	Dic₇₁	1	4	yes	complemented
61	C₂×C₄⋊C₄	—	2	2	no	—
62	C₂×C₄⋊C₄	—	71	2	no	—
63	C₂×C₄⋊C₄	C₂×D₇₁	1	2	yes	—
64	C₂×C₄²	C₂×D₇₁	1	1	yes	characteristic
65	C₄×C₈	—	142	2	no	—
66	#95	—	142	2	no	—
67	C₂²×C₄	—	2	2	no	—
68	C₂²×C₄	—	142	2	no	—
69	C₂²×C₄	C₂²⋊D₇₁	1	2	yes	—
70	C₂²×C₄	C₂²×D₇₁	1	1	yes	characteristic
71	C₂×C₈	—	71	4	no	—
72	C₂×C₈	—	142	2	no	—
73	C₂×C₈	—	142	2	no	—
74	C₄⋊C₄	—	2	8	no	—
75	C₄⋊C₄	—	2	2	no	—
76	C₄⋊C₄	—	142	4	no	—
77	C₄⋊C₄	C₂×Dic₇₁	1	4	yes	—
78	C₄²	C₂×Dic₇₁	1	2	yes	—
79	C₄²	C₂²⋊D₇₁	1	2	yes	—
80	C₂³	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
81	C₂×C₄	—	2	4	no	—
82	C₂×C₄	—	2	4	no	—
83	C₂×C₄	—	4	4	no	—
84	C₂×C₄	—	284	2	no	—
85	C₂×C₄	—	284	4	no	—
86	C₂×C₄	C₂²⋊Dic₇₁	1	4	yes	—
87	C₂×C₄	C₂²×Dic₇₁	1	1	yes	characteristic
88	C₂×C₄	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
89	C₂×C₄	Q₈×D₇₁	1	2	yes	—
90	C₂×C₄	#11765866	1	2	yes	—
91	C₈	—	142	4	no	—
92	C₂²	C₂×C₂²⋊Dic₇₁	1	1	yes	characteristic
93	C₂²	#14556117	1	1	yes	characteristic
94	C₂²	#14556126	1	1	yes	characteristic
95	C₂²	#14556148	1	4	yes	—
96	C₄	—	2	4	no	—
97	C₄	—	4	4	no	—
98	C₄	—	568	2	no	—
99	C₄	Q₈×Dic₇₁	1	2	yes	—
100	C₄	#14556104	1	2	yes	—
101	C₂	#17985906	1	1	yes	characteristic
102	C₂	#17985995	1	2	yes	—
103	C₂	#17986356	1	2	yes	—
104	C₂	#17986398	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×C₇₁⋊C₈

C₂

yes

maximal, characteristic

#17985865

C₂

yes

maximal, characteristic

#17985761

C₂

yes

maximal

C₂×#125×C₇₁

C₂

yes

maximal, characteristic

C₂²×C₇₁⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₇₁

C₂²

yes

—

C₄×C₇₁⋊C₈

—

C₇₁⋊#95

—

#125×C₇₁

—

#125×C₇₁

C₄

yes

complemented

C₂×C₄⋊C₄×C₇₁

—

C₂×C₄⋊C₄×C₇₁

C₂²

yes

—

C₂×C₄²×C₇₁

C₂²

yes

characteristic

C₂²×Dic₇₁

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

Q₈

yes

—

C₄⋊Dic₇₁

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

D₄

yes

—

C₂²×C₄×C₇₁

C₂³

yes

characteristic

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

C₂×C₄

yes

—

C₄²×C₇₁

C₂×C₄

yes

—

C₄²×C₇₁

D₄

yes

—

C₂³×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²⋊C₄

yes

—

C₂×C₄×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×D₄

yes

characteristic

C₂×C₄×C₇₁

C₂×Q₈

yes

—

C₂×C₄×C₇₁

#41

yes

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₇₁⋊C₈

—

C₂²×C₇₁

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₇₁

#124

yes

characteristic

C₂²×C₇₁

C₄⋊Q₈

yes

characteristic

C₂²×C₇₁

#135

yes

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

C₄×Q₈

yes

—

C₄×C₇₁

C₂²⋊Q₈

yes

—

Dic₇₁

—

C₂×C₇₁

#391

yes

characteristic

C₂×C₇₁

#416

yes

—

C₂×C₇₁

#488

yes

—

C₂×C₇₁

#499

yes

—

#1991

—

maximal, Sylow

C₇₁

#1991

yes

characteristic, complemented, Sylow

C₂×#125

D₇₁

yes

characteristic

C₂×C₄×C₈

—

#382

—

#340

—

C₂²×C₈

—

#125

—

#125

Dic₇₁

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₇₁

yes

—

C₂×C₄²

C₂×D₇₁

yes

characteristic

C₄×C₈

—

142

—

#95

—

142

—

C₂²×C₄

—

C₂²×C₄

—

142

—

C₂²×C₄

C₂²⋊D₇₁

yes

—

C₂²×C₄

C₂²×D₇₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

142

—

C₂×C₈

—

142

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

142

—

C₄⋊C₄

C₂×Dic₇₁

yes

—

C₄²

C₂×Dic₇₁

yes

—

C₄²

C₂²⋊D₇₁

yes

—

C₂³

C₂×C₂²⋊D₇₁

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

284

—

C₂×C₄

—

284

—

C₂×C₄

C₂²⋊Dic₇₁

yes

—

C₂×C₄

C₂²×Dic₇₁

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₇₁

yes

characteristic

C₂×C₄

Q₈×D₇₁

yes

—

C₂×C₄

#11765866

yes

—

C₈

—

142

—

C₂²

C₂×C₂²⋊Dic₇₁

yes

C₂²

yes

C₂²

yes

C₂²

yes

—

C₄

—

C₄

—

C₄

—

568

—

C₄

Q₈×Dic₇₁

yes

—

100

C₄

#14556104

yes

—

101

C₂

#17985906

yes

characteristic

102

C₂

#17985995

yes

—

103

C₂

#17986356

yes

—

104

C₂

#17986398

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₇₁
commutator	C₂×C₄×C₇₁	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₁
Fitting	C₂×#125×C₇₁	C₂
socle	C₂³×C₇₁	C₂×D₄
hypercenter	C₂×#125	D₇₁
hypocenter	C₇₁	#1991
LCS	G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₇₁

commutator

C₂×C₄×C₇₁

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₁

Fitting

C₂×#125×C₇₁

C₂

socle

C₂³×C₇₁

C₂×D₄

D₇₁

C₇₁

G ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#125

N	H ≅ G/N	type
10. #125×C₇₁	C₄	semidirect
53. C₇₁	#1991	semidirect
60. #125	Dic₇₁	semidirect

H ≅ G/N

type

10. #125×C₇₁

C₄

semidirect

53. C₇₁

#1991

semidirect

60. #125

Dic₇₁

semidirect

Subgroups — #22589306

Conjugacy classes of subgroups

Complements