Group #22586461 — subgroups

Subgroups — #22586461

Order 9088.

166 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 53 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	#17985973	C₂	1	2	yes	maximal
2	#408×C₇₁	C₂	1	1	yes	maximal, characteristic
3	#14556087	—	2	2	no	—
4	#14556043	—	2	2	no	—
5	#14556043	C₄	1	4	yes	complemented
6	C₂×#41×C₇₁	C₂²	1	1	yes	characteristic
7	C₂×M₁₆×C₇₁	C₄	1	1	yes	characteristic, complemented
8	C₂²×C₈×C₇₁	C₄	1	1	yes	characteristic, complemented
9	C₂²⋊C₈×C₇₁	—	2	2	no	—
10	C₄⋊Dic₇₁	—	4	2	no	—
11	C₂²⋊Dic₇₁	—	2	4	no	—
12	C₄×Dic₇₁	—	4	2	no	—
13	#41×C₇₁	—	4	1	no	—
14	#41×C₇₁	—	4	1	no	—
15	C₂×Q₈×C₇₁	Q₈	1	1	yes	characteristic
16	C₂×D₄×C₇₁	C₂×C₄	1	2	yes	—
17	C₂×D₄×C₇₁	D₄	1	1	yes	characteristic
18	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₄×C₇₁	D₄	1	2	yes	—
20	M₁₆×C₇₁	—	4	1	no	—
21	C₂×C₈×C₇₁	—	2	1	no	—
22	C₂×C₈×C₇₁	—	2	1	no	—
23	C₂×C₈×C₇₁	—	4	1	no	—
24	C₂³×C₇₁	C₄²	1	1	yes	characteristic
25	C₂³×C₇₁	C₂²⋊C₄	1	2	yes	—
26	Q₈×C₇₁	—	4	1	no	—
27	D₄×C₇₁	—	4	2	no	—
28	D₄×C₇₁	—	4	1	no	—
29	C₂×C₄×C₇₁	—	2	1	no	—
30	C₂×C₄×C₇₁	—	2	1	no	—
31	C₂×C₄×C₇₁	—	2	2	no	—
32	C₂×C₄×C₇₁	—	2	2	no	—
33	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₄×C₇₁	C₄⋊C₄	1	2	yes	—
35	C₂×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
36	C₂×Dic₇₁	—	4	4	no	—
37	C₈×C₇₁	—	4	1	no	—
38	C₈×C₇₁	—	4	1	no	—
39	C₂²×C₇₁	—	2	2	no	—
40	C₂²×C₇₁	—	2	1	no	—
41	C₂²×C₇₁	—	4	1	no	—
42	C₂²×C₇₁	—	4	2	no	—
43	C₂²×C₇₁	#95	1	1	yes	characteristic
44	C₄×C₇₁	—	2	2	no	—
45	C₄×C₇₁	—	2	1	no	—
46	C₄×C₇₁	#99	1	1	yes	characteristic
47	C₄×C₇₁	#99	1	1	yes	characteristic
48	Dic₇₁	—	8	4	no	—
49	C₂×C₇₁	—	2	1	no	—
50	C₂×C₇₁	—	4	2	no	—
51	C₂×C₇₁	—	4	1	no	—
52	C₂×C₇₁	#342	1	1	yes	characteristic
53	#1391	—	71	1	no	maximal, Sylow
54	C₇₁	#1391	1	1	yes	characteristic, complemented, Sylow
55	#410	—	71	2	no	—
56	#408	D₇₁	1	1	yes	characteristic
57	C₂×#41	C₂×D₇₁	1	1	yes	characteristic
58	C₂×M₁₆	Dic₇₁	1	1	yes	characteristic, complemented
59	C₂²×C₈	Dic₇₁	1	1	yes	characteristic, complemented
60	#117	—	142	2	no	—
61	#99	—	71	4	no	—
62	#99	—	142	2	no	—
63	C₂²⋊C₈	—	2	2	no	—
64	#41	—	4	1	no	—
65	#41	—	4	1	no	—
66	C₂×Q₈	Dic₁₄₂	1	1	yes	characteristic
67	C₂×D₄	C₄×D₇₁	1	2	yes	—
68	C₂×D₄	D₂₈₄	1	1	yes	characteristic
69	C₂²×C₄	C₂×Dic₇₁	1	1	yes	characteristic
70	C₂²×C₄	C₂²⋊D₇₁	1	2	yes	—
71	M₁₆	—	4	1	no	—
72	C₂×C₈	—	2	1	no	—
73	C₂×C₈	—	2	1	no	—
74	C₂×C₈	—	4	1	no	—
75	C₄⋊C₄	—	284	2	no	—
76	C₂²⋊C₄	—	142	4	no	—
77	C₄²	—	284	2	no	—
78	C₂³	C₄×Dic₇₁	1	1	yes	characteristic
79	C₂³	(C₂×D₇₁)⋊C₄	1	2	yes	—
80	Q₈	—	4	1	no	—
81	D₄	—	4	2	no	—
82	D₄	—	4	1	no	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	2	2	no	—
86	C₂×C₄	—	2	1	no	—
87	C₂×C₄	—	284	4	no	—
88	C₂×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
89	C₂×C₄	Dic₇₁⋊C₄	1	2	yes	—
90	C₂×C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
91	C₈	—	4	1	no	—
92	C₈	—	4	1	no	—
93	C₂²	—	2	2	no	—
94	C₂²	—	2	1	no	—
95	C₂²	—	4	1	no	—
96	C₂²	—	4	2	no	—
97	C₂²	C₇₁⋊#95	1	1	yes	characteristic
98	C₄	—	2	2	no	—
99	C₄	—	2	1	no	—
100	C₄	—	568	4	no	—
101	C₄	#14556043	1	1	yes	characteristic
102	C₄	#14556043	1	1	yes	characteristic
103	C₂	—	2	1	no	—
104	C₂	—	4	2	no	—
105	C₂	—	4	1	no	—
106	C₂	#17985766	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17985973

C₂

yes

maximal

#408×C₇₁

C₂

yes

maximal, characteristic

#14556087

—

#14556043

—

#14556043

C₄

yes

complemented

C₂×#41×C₇₁

C₂²

yes

characteristic

C₂×M₁₆×C₇₁

C₄

yes

characteristic, complemented

C₂²×C₈×C₇₁

C₄

yes

characteristic, complemented

C₂²⋊C₈×C₇₁

—

C₄⋊Dic₇₁

—

C₂²⋊Dic₇₁

—

C₄×Dic₇₁

—

#41×C₇₁

—

#41×C₇₁

—

C₂×Q₈×C₇₁

Q₈

yes

characteristic

C₂×D₄×C₇₁

C₂×C₄

yes

—

C₂×D₄×C₇₁

D₄

yes

characteristic

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₁

D₄

yes

—

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂³×C₇₁

C₄²

yes

characteristic

C₂³×C₇₁

C₂²⋊C₄

yes

—

Q₈×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₄⋊C₄

yes

—

C₂×C₄×C₇₁

C₄⋊C₄

yes

characteristic

C₂×Dic₇₁

—

C₈×C₇₁

—

C₈×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#95

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

Dic₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

#410

—

#408

D₇₁

yes

characteristic

C₂×#41

C₂×D₇₁

yes

characteristic

C₂×M₁₆

Dic₇₁

yes

characteristic, complemented

C₂²×C₈

Dic₇₁

yes

characteristic, complemented

#117

—

142

—

#99

—

#99

—

142

—

C₂²⋊C₈

—

#41

—

#41

—

C₂×Q₈

Dic₁₄₂

yes

characteristic

C₂×D₄

C₄×D₇₁

yes

—

C₂×D₄

D₂₈₄

yes

characteristic

C₂²×C₄

C₂×Dic₇₁

yes

characteristic

C₂²×C₄

C₂²⋊D₇₁

yes

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

284

—

C₂²⋊C₄

—

142

—

C₄²

—

284

—

C₂³

C₄×Dic₇₁

yes

characteristic

C₂³

(C₂×D₇₁)⋊C₄

yes

—

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

284

—

C₂×C₄

C₂²⋊Dic₇₁

yes

characteristic

C₂×C₄

Dic₇₁⋊C₄

yes

—

C₂×C₄

C₄⋊Dic₇₁

yes

characteristic

C₈

—

C₈

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₇₁⋊#95

yes

characteristic

C₄

—

C₄

—

100

C₄

—

568

—

101

C₄

#14556043

yes

characteristic

102

C₄

#14556043

yes

characteristic

103

C₂

—

104

C₂

—

105

C₂

—

106

C₂

#17985766

yes

characteristic

	subgroup	G/H
center	C₄	#14556043
commutator	C₂³×C₇₁	C₄²
Frattini	C₂×#41	C₂×D₇₁
Fitting	#408×C₇₁	C₂
socle	C₂×C₇₁	#342
hypercenter	#408	D₇₁
hypocenter	C₇₁	#1391
LCS	G ▷ C₂³×C₇₁ ▷ C₂²×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×#41 ◁ #408

subgroup

G/H

center

C₄

#14556043

commutator

C₂³×C₇₁

C₄²

Frattini

C₂×#41

C₂×D₇₁

Fitting

#408×C₇₁

C₂

C₂×C₇₁

D₇₁

C₇₁

G ▷ C₂³×C₇₁ ▷ C₂²×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×#41 ◁ #408

N	H ≅ G/N	type
5. #14556043	C₄	semidirect
7. C₂×M₁₆×C₇₁	C₄	semidirect
8. C₂²×C₈×C₇₁	C₄	semidirect
54. C₇₁	#1391	semidirect
58. C₂×M₁₆	Dic₇₁	semidirect
59. C₂²×C₈	Dic₇₁	semidirect

H ≅ G/N

type

5. #14556043

C₄

semidirect

7. C₂×M₁₆×C₇₁

C₄

semidirect

8. C₂²×C₈×C₇₁

C₄

semidirect

54. C₇₁

#1391

semidirect

58. C₂×M₁₆

Dic₇₁

semidirect

59. C₂²×C₈

Dic₇₁

semidirect

Subgroups — #22586461

Conjugacy classes of subgroups

Complements