Group #22586331 — subgroups

Subgroups — #22586331

Order 9088.

110 conjugacy classes of subgroups; only the 33 normal subgroups are shown.

normal subgroups only

This group has 110 conjugacy classes of subgroups — too many to draw. Only its 33 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	#17985781	C₂	1	1	yes	maximal, characteristic, complemented
2	C₇₁⋊#338	C₂	1	1	yes	maximal, characteristic, complemented
3	#469×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#11765849	—	2	1	no	—
5	#14556068	—	2	1	no	—
6	C₂×D₈×C₇₁	C₄	1	1	yes	characteristic
7	C₂×M₁₆×C₇₁	C₂²	1	1	yes	characteristic
8	#121×C₇₁	—	2	1	no	—
9	#106×C₇₁	C₄	1	1	yes	characteristic
10	#102×C₇₁	—	2	1	no	—
11	#11765849	—	4	1	no	—
12	#11765849	—	4	1	no	—
13	C₂×C₇₁⋊C₈	—	2	2	no	—
14	C₇₁⋊C₁₆	—	4	1	no	—
15	C₂×D₄×C₇₁	—	2	1	no	—
16	C₂×D₄×C₇₁	—	4	1	no	—
17	C₂²×C₄×C₇₁	D₄	1	1	yes	characteristic
18	D₈×C₇₁	—	4	1	no	—
19	M₁₆×C₇₁	—	4	1	no	—
20	C₂×C₈×C₇₁	C₂×C₄	1	1	yes	characteristic
21	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
22	C₄⋊C₄×C₇₁	—	2	1	no	—
23	C₂²⋊C₄×C₇₁	—	4	1	no	—
24	C₂³×C₇₁	—	4	1	no	—
25	C₂³×C₇₁	SD₁₆	1	1	yes	characteristic
26	D₄×C₇₁	—	4	1	no	—
27	D₄×C₇₁	—	4	1	no	—
28	D₄×C₇₁	—	8	2	no	—
29	C₂×C₄×C₇₁	—	2	1	no	—
30	C₂×C₄×C₇₁	—	2	1	no	—
31	C₂×C₄×C₇₁	—	4	1	no	—
32	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₄×C₇₁	D₈	1	1	yes	characteristic
34	C₈×C₇₁	—	2	1	no	—
35	C₈×C₇₁	—	2	1	no	—
36	C₇₁⋊C₈	—	4	2	no	—
37	C₂²×C₇₁	—	2	1	no	—
38	C₂²×C₇₁	—	4	1	no	—
39	C₂²×C₇₁	—	8	1	no	—
40	C₂²×C₇₁	—	8	1	no	—
41	C₂²×C₇₁	—	8	1	no	—
42	C₂²×C₇₁	#102	1	1	yes	characteristic
43	C₄×C₇₁	—	2	1	no	—
44	C₄×C₇₁	—	8	1	no	—
45	C₄×C₇₁	#99	1	1	yes	characteristic
46	C₄×C₇₁	#104	1	1	yes	characteristic
47	C₂×C₇₁	—	2	1	no	—
48	C₂×C₇₁	—	4	1	no	—
49	C₂×C₇₁	—	16	1	no	—
50	C₂×C₇₁	#327	1	1	yes	characteristic
51	#1341	—	71	1	no	maximal, Sylow
52	C₇₁	#1341	1	1	yes	characteristic, complemented, Sylow
53	#469	D₇₁	1	1	yes	characteristic
54	#349	—	71	1	no	—
55	#338	—	71	1	no	—
56	C₂×D₈	Dic₇₁	1	1	yes	characteristic
57	C₂×M₁₆	—	142	1	no	—
58	C₂×M₁₆	C₂×D₇₁	1	1	yes	characteristic
59	#121	—	2	1	no	—
60	M₃₂	—	142	1	no	—
61	#106	Dic₇₁	1	1	yes	characteristic
62	#102	—	2	1	no	—
63	C₂×D₄	—	2	1	no	—
64	C₂×D₄	—	4	1	no	—
65	C₂²×C₄	C₂²⋊D₇₁	1	1	yes	characteristic
66	D₈	—	4	1	no	—
67	M₁₆	—	4	1	no	—
68	M₁₆	—	284	1	no	—
69	M₁₆	—	284	1	no	—
70	C₂×C₈	—	142	2	no	—
71	C₂×C₈	C₂×Dic₇₁	1	1	yes	characteristic
72	C₂×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
73	C₄⋊C₄	—	2	1	no	—
74	C₂²⋊C₄	—	4	1	no	—
75	C₁₆	—	284	1	no	—
76	C₂³	—	4	1	no	—
77	C₂³	#11765853	1	1	yes	characteristic
78	D₄	—	4	1	no	—
79	D₄	—	4	1	no	—
80	D₄	—	8	2	no	—
81	C₂×C₄	—	2	1	no	—
82	C₂×C₄	—	2	1	no	—
83	C₂×C₄	—	4	1	no	—
84	C₂×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
85	C₂×C₄	#11765850	1	1	yes	characteristic
86	C₈	—	2	1	no	—
87	C₈	—	2	1	no	—
88	C₈	—	284	2	no	—
89	C₂²	—	2	1	no	—
90	C₂²	—	4	1	no	—
91	C₂²	—	8	1	no	—
92	C₂²	—	8	1	no	—
93	C₂²	—	8	1	no	—
94	C₂²	#14556049	1	1	yes	characteristic
95	C₄	—	2	1	no	—
96	C₄	—	8	1	no	—
97	C₄	#14556043	1	1	yes	characteristic
98	C₄	#14556055	1	1	yes	characteristic
99	C₂	—	2	1	no	—
100	C₂	—	4	1	no	—
101	C₂	—	16	1	no	—
102	C₂	#17985733	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17985781

C₂

yes

maximal, characteristic, complemented

C₇₁⋊#338

C₂

yes

maximal, characteristic, complemented

#469×C₇₁

C₂

yes

maximal, characteristic

C₂×#11765849

—

#14556068

—

C₂×D₈×C₇₁

C₄

yes

characteristic

C₂×M₁₆×C₇₁

C₂²

yes

characteristic

#121×C₇₁

—

#106×C₇₁

C₄

yes

characteristic

#102×C₇₁

—

#11765849

—

#11765849

—

C₂×C₇₁⋊C₈

—

C₇₁⋊C₁₆

—

C₂×D₄×C₇₁

—

C₂×D₄×C₇₁

—

C₂²×C₄×C₇₁

D₄

yes

characteristic

D₈×C₇₁

—

M₁₆×C₇₁

—

C₂×C₈×C₇₁

C₂×C₄

yes

characteristic

C₂×C₈×C₇₁

D₄

yes

characteristic

C₄⋊C₄×C₇₁

—

C₂²⋊C₄×C₇₁

—

C₂³×C₇₁

—

C₂³×C₇₁

SD₁₆

yes

characteristic

D₄×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

D₈

yes

characteristic

C₈×C₇₁

—

C₈×C₇₁

—

C₇₁⋊C₈

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

#102

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

—

C₂×C₇₁

yes

—

C₇₁

yes

characteristic, complemented, Sylow

#469

D₇₁

yes

characteristic

#349

—

#338

—

C₂×D₈

Dic₇₁

yes

characteristic

C₂×M₁₆

—

142

—

C₂×M₁₆

C₂×D₇₁

yes

characteristic

#121

—

M₃₂

—

142

—

#106

Dic₇₁

yes

characteristic

#102

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

C₂²⋊D₇₁

yes

characteristic

D₈

—

M₁₆

—

M₁₆

—

284

—

M₁₆

—

284

—

C₂×C₈

—

142

—

C₂×C₈

C₂×Dic₇₁

yes

characteristic

C₂×C₈

C₂²⋊D₇₁

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₁₆

—

284

—

C₂³

—

C₂³

#11765853

yes

characteristic

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Dic₇₁

yes

characteristic

C₂×C₄

#11765850

yes

characteristic

C₈

—

C₈

—

C₈

—

284

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#14556049

yes

characteristic

C₄

—

C₄

—

C₄

yes

C₄

yes

C₂

—

100

C₂

—

101

C₂

—

102

C₂

#17985733

yes

characteristic

	subgroup	G/H
center	C₂	#17985733
commutator	C₂×C₈×C₇₁	C₂×C₄
Frattini	C₂×M₁₆	C₂×D₇₁
Fitting	#469×C₇₁	C₂
socle	C₂×C₇₁	#327
hypercenter	#469	D₇₁
hypocenter	C₇₁	#1341
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₂×C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #469

subgroup

G/H

center

C₂

#17985733

commutator

C₂×C₈×C₇₁

C₂×C₄

Frattini

C₂×M₁₆

C₂×D₇₁

Fitting

#469×C₇₁

C₂

C₂×C₇₁

D₇₁

C₇₁

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

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #469

N	H ≅ G/N	type
1. #17985781	C₂	semidirect
2. C₇₁⋊#338	C₂	semidirect
52. C₇₁	#1341	semidirect

H ≅ G/N

type

1. #17985781

C₂

semidirect

2. C₇₁⋊#338

C₂

semidirect

52. C₇₁

#1341

semidirect

Subgroups — #22586331

Conjugacy classes of subgroups

Complements