Group #22586383 — subgroups

Subgroups — #22586383

Order 9088.

102 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 102 conjugacy classes of subgroups — too many to draw. Only its 41 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⋊(C₇₁⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
2	#17985750	C₂	1	1	yes	maximal, characteristic, complemented
3	#495×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₇₁⋊C₁₆	—	2	1	no	—
5	C₄⋊(C₇₁⋊C₈)	—	2	1	no	—
6	C₂×Q₁₆×C₇₁	C₄	1	1	yes	characteristic
7	C₂×SD₁₆×C₇₁	—	2	1	no	—
8	C₂×D₈×C₇₁	C₄	1	1	yes	characteristic
9	#124×C₇₁	—	2	1	no	—
10	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
11	C₂×C₇₁⋊C₈	—	4	1	no	—
12	C₇₁⋊C₁₆	—	4	1	no	—
13	C₂×Q₈×C₇₁	—	2	1	no	—
14	C₂×D₄×C₇₁	—	2	1	no	—
15	Q₁₆×C₇₁	—	4	1	no	—
16	SD₁₆×C₇₁	—	4	1	no	—
17	SD₁₆×C₇₁	—	4	1	no	—
18	D₈×C₇₁	—	4	1	no	—
19	C₂×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
20	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
21	C₂²⋊C₄×C₇₁	—	4	1	no	—
22	C₂²⋊C₄×C₇₁	—	4	1	no	—
23	C₄²×C₇₁	D₄	1	1	yes	characteristic
24	C₂³×C₇₁	—	4	1	no	—
25	Q₈×C₇₁	—	4	1	no	—
26	Q₈×C₇₁	—	4	1	no	—
27	D₄×C₇₁	—	4	1	no	—
28	D₄×C₇₁	—	4	1	no	—
29	C₂×C₄×C₇₁	—	4	1	no	—
30	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
31	C₂×C₄×C₇₁	D₈	1	1	yes	characteristic
32	C₂×C₄×C₇₁	SD₁₆	1	1	yes	characteristic
33	C₈×C₇₁	—	2	1	no	—
34	C₈×C₇₁	D₈	1	1	yes	characteristic
35	C₈×C₇₁	SD₁₆	1	1	yes	characteristic
36	C₇₁⋊C₈	—	8	1	no	—
37	C₂²×C₇₁	—	8	1	no	—
38	C₂²×C₇₁	—	8	1	no	—
39	C₂²×C₇₁	—	8	1	no	—
40	C₂²×C₇₁	#102	1	1	yes	characteristic
41	C₄×C₇₁	—	2	1	no	—
42	C₄×C₇₁	—	2	1	no	—
43	C₄×C₇₁	—	8	1	no	—
44	C₄×C₇₁	#100	1	1	yes	characteristic
45	C₄×C₇₁	#102	1	1	yes	characteristic
46	C₂×C₇₁	—	16	1	no	—
47	C₂×C₇₁	#331	1	1	yes	characteristic
48	C₂×C₇₁	#359	1	1	yes	characteristic
49	C₂×C₇₁	#360	1	1	yes	characteristic
50	#1360	—	71	1	no	maximal, Sylow
51	C₇₁	#1360	1	1	yes	characteristic, complemented, Sylow
52	#495	D₇₁	1	1	yes	characteristic
53	C₄⋊C₁₆	—	71	1	no	—
54	#334	—	71	1	no	—
55	C₂×Q₁₆	Dic₇₁	1	1	yes	characteristic
56	C₂×SD₁₆	—	2	1	no	—
57	C₂×D₈	Dic₇₁	1	1	yes	characteristic
58	#124	—	2	1	no	—
59	C₂×C₁₆	—	142	1	no	—
60	C₄⋊C₈	—	142	1	no	—
61	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
62	C₂×Q₈	—	2	1	no	—
63	C₂×D₄	—	2	1	no	—
64	Q₁₆	—	4	1	no	—
65	SD₁₆	—	4	1	no	—
66	SD₁₆	—	4	1	no	—
67	D₈	—	4	1	no	—
68	C₂×C₈	—	284	1	no	—
69	C₂×C₈	C₂×Dic₇₁	1	1	yes	characteristic
70	C₂×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
71	C₂²⋊C₄	—	4	1	no	—
72	C₂²⋊C₄	—	4	1	no	—
73	C₄²	C₂²⋊D₇₁	1	1	yes	characteristic
74	C₁₆	—	284	1	no	—
75	C₂³	—	4	1	no	—
76	Q₈	—	4	1	no	—
77	Q₈	—	4	1	no	—
78	D₄	—	4	1	no	—
79	D₄	—	4	1	no	—
80	C₂×C₄	—	4	1	no	—
81	C₂×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
82	C₂×C₄	#11765850	1	1	yes	characteristic
83	C₂×C₄	#11765853	1	1	yes	characteristic
84	C₈	—	2	1	no	—
85	C₈	—	568	1	no	—
86	C₈	#11765850	1	1	yes	characteristic
87	C₈	#11765853	1	1	yes	characteristic
88	C₂²	—	8	1	no	—
89	C₂²	—	8	1	no	—
90	C₂²	—	8	1	no	—
91	C₂²	#14556049	1	1	yes	characteristic
92	C₄	—	2	1	no	—
93	C₄	—	2	1	no	—
94	C₄	—	8	1	no	—
95	C₄	#14556045	1	1	yes	characteristic
96	C₄	#14556049	1	1	yes	characteristic
97	C₂	—	16	1	no	—
98	C₂	#17985743	1	1	yes	characteristic
99	C₂	#17985806	1	1	yes	characteristic
100	C₂	#17985809	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄⋊(C₇₁⋊C₁₆)

C₂

yes

maximal, characteristic, complemented

#17985750

C₂

yes

maximal, characteristic, complemented

#495×C₇₁

C₂

yes

maximal, characteristic

C₂×C₇₁⋊C₁₆

—

C₄⋊(C₇₁⋊C₈)

—

C₂×Q₁₆×C₇₁

C₄

yes

characteristic

C₂×SD₁₆×C₇₁

—

C₂×D₈×C₇₁

C₄

yes

characteristic

#124×C₇₁

—

C₄×C₈×C₇₁

C₂²

yes

characteristic

C₂×C₇₁⋊C₈

—

C₇₁⋊C₁₆

—

C₂×Q₈×C₇₁

—

C₂×D₄×C₇₁

—

Q₁₆×C₇₁

—

SD₁₆×C₇₁

—

SD₁₆×C₇₁

—

D₈×C₇₁

—

C₂×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂×C₈×C₇₁

D₄

yes

characteristic

C₂²⋊C₄×C₇₁

—

C₂²⋊C₄×C₇₁

—

C₄²×C₇₁

D₄

yes

characteristic

C₂³×C₇₁

—

Q₈×C₇₁

—

Q₈×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

D₈

yes

characteristic

C₂×C₄×C₇₁

SD₁₆

yes

characteristic

C₈×C₇₁

—

C₈×C₇₁

D₈

yes

characteristic

C₈×C₇₁

SD₁₆

yes

characteristic

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#102

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₂×C₇₁

—

C₂×C₇₁

yes

C₂×C₇₁

yes

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

#495

D₇₁

yes

characteristic

C₄⋊C₁₆

—

#334

—

C₂×Q₁₆

Dic₇₁

yes

characteristic

C₂×SD₁₆

—

C₂×D₈

Dic₇₁

yes

characteristic

#124

—

C₂×C₁₆

—

142

—

C₄⋊C₈

—

142

—

C₄×C₈

C₂×D₇₁

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

Q₁₆

—

SD₁₆

—

SD₁₆

—

D₈

—

C₂×C₈

—

284

—

C₂×C₈

C₂×Dic₇₁

yes

characteristic

C₂×C₈

C₂²⋊D₇₁

yes

characteristic

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

C₂²⋊D₇₁

yes

characteristic

C₁₆

—

284

—

C₂³

—

Q₈

—

Q₈

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Dic₇₁

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

—

568

—

C₈

yes

C₈

yes

C₂²

—

C₂²

—

C₂²

—

C₂²

#14556049

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

yes

C₄

yes

C₂

—

C₂

yes

C₂

yes

100

C₂

#17985809

yes

characteristic

	subgroup	G/H
center	C₂²	#14556049
commutator	C₂×C₈×C₇₁	C₂×C₄
Frattini	C₄×C₈	C₂×D₇₁
Fitting	#495×C₇₁	C₂
socle	C₂²×C₇₁	#102
hypercenter	#495	D₇₁
hypocenter	C₇₁	#1360
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #495

subgroup

G/H

center

C₂²

#14556049

commutator

C₂×C₈×C₇₁

C₂×C₄

Frattini

C₄×C₈

C₂×D₇₁

Fitting

#495×C₇₁

C₂

C₂²×C₇₁

D₇₁

C₇₁

G ▷ C₂×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #495

N	H ≅ G/N	type
1. C₄⋊(C₇₁⋊C₁₆)	C₂	semidirect
2. #17985750	C₂	semidirect
51. C₇₁	#1360	semidirect

H ≅ G/N

type

1. C₄⋊(C₇₁⋊C₁₆)

C₂

semidirect

2. #17985750

C₂

semidirect

51. C₇₁

#1360

semidirect

Subgroups — #22586383

Conjugacy classes of subgroups

Complements