Group #22586418 — subgroups

Subgroups — #22586418

Order 9088.

160 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 105 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	C₂²×C₇₁⋊C₁₆	C₂	1	2	yes	maximal
2	C₂×C₄×C₈×C₇₁	C₂	1	1	yes	maximal, characteristic
3	C₂×C₇₁⋊C₁₆	—	2	2	no	—
4	C₂×C₇₁⋊C₁₆	—	2	2	no	—
5	C₂×C₇₁⋊C₁₆	C₄	1	4	yes	complemented
6	C₂²×C₈×C₇₁	C₄	1	1	yes	characteristic
7	C₂²×C₈×C₇₁	C₂²	1	1	yes	characteristic
8	C₂×C₄²×C₇₁	C₄	1	1	yes	characteristic
9	C₄×C₈×C₇₁	—	2	2	no	—
10	C₇₁⋊C₁₆	—	2	4	no	—
11	C₂²×C₄×C₇₁	C₈	1	2	yes	—
12	C₂²×C₄×C₇₁	C₂×C₄	1	1	yes	characteristic
13	C₂×C₈×C₇₁	—	2	1	no	—
14	C₂×C₈×C₇₁	—	2	1	no	—
15	C₂×C₈×C₇₁	C₈	1	2	yes	—
16	C₂×C₈×C₇₁	C₂×C₄	1	2	yes	—
17	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
18	C₂×C₈×C₇₁	D₄	1	2	yes	—
19	C₂×C₈×C₇₁	Q₈	1	1	yes	characteristic
20	C₄²×C₇₁	—	2	1	no	—
21	C₄²×C₇₁	—	2	1	no	—
22	C₂³×C₇₁	C₂×C₈	1	1	yes	characteristic
23	C₂×C₄×C₇₁	—	2	2	no	—
24	C₂×C₄×C₇₁	—	2	2	no	—
25	C₂×C₄×C₇₁	C₁₆	1	4	yes	complemented
26	C₂×C₄×C₇₁	C₄²	1	1	yes	characteristic
27	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₇₁	C₄⋊C₄	1	1	yes	characteristic
29	C₂×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
30	C₂×C₄×C₇₁	M₁₆	1	1	yes	characteristic
31	C₂×C₄×C₇₁	M₁₆	1	1	yes	characteristic
32	C₈×C₇₁	—	2	2	no	—
33	C₈×C₇₁	C₂²⋊C₄	1	2	yes	—
34	C₈×C₇₁	C₄⋊C₄	1	2	yes	—
35	C₂²×C₇₁	C₄×C₈	1	1	yes	characteristic
36	C₂²×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
37	C₂²×C₇₁	C₄⋊C₈	1	1	yes	characteristic
38	C₂²×C₇₁	C₂×C₁₆	1	2	yes	—
39	C₂²×C₇₁	M₃₂	1	2	yes	—
40	C₄×C₇₁	—	2	4	no	—
41	C₄×C₇₁	#95	1	1	yes	characteristic
42	C₄×C₇₁	#97	1	1	yes	characteristic
43	C₄×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
44	C₄×C₇₁	C₄⋊C₈	1	1	yes	characteristic
45	C₂×C₇₁	#336	1	1	yes	characteristic
46	C₂×C₇₁	C₄×C₁₆	1	1	yes	characteristic
47	C₂×C₇₁	#346	1	1	yes	characteristic
48	C₂×C₇₁	C₂²⋊C₁₆	1	2	yes	—
49	C₂×C₇₁	C₄⋊C₁₆	1	2	yes	—
50	#1374	—	71	1	no	maximal, Sylow
51	C₇₁	#1374	1	1	yes	characteristic, complemented, Sylow
52	C₂²×C₁₆	—	71	2	no	—
53	C₂×C₄×C₈	D₇₁	1	1	yes	characteristic
54	C₂²×C₈	Dic₇₁	1	1	yes	characteristic
55	C₂²×C₈	C₂×D₇₁	1	1	yes	characteristic
56	C₂×C₄²	Dic₇₁	1	1	yes	characteristic
57	C₂×C₁₆	—	71	4	no	—
58	C₂×C₁₆	—	142	2	no	—
59	C₂×C₁₆	—	142	2	no	—
60	C₄×C₈	—	2	2	no	—
61	C₂²×C₄	C₇₁⋊C₈	1	2	yes	—
62	C₂²×C₄	C₂×Dic₇₁	1	1	yes	characteristic
63	C₂×C₈	—	2	1	no	—
64	C₂×C₈	—	2	1	no	—
65	C₂×C₈	C₇₁⋊C₈	1	2	yes	—
66	C₂×C₈	Dic₁₄₂	1	1	yes	characteristic
67	C₂×C₈	C₄×D₇₁	1	2	yes	—
68	C₂×C₈	D₂₈₄	1	1	yes	characteristic
69	C₂×C₈	C₂²⋊D₇₁	1	2	yes	—
70	C₄²	—	2	1	no	—
71	C₄²	—	2	1	no	—
72	C₁₆	—	142	4	no	—
73	C₂³	C₂×C₇₁⋊C₈	1	1	yes	characteristic
74	C₂×C₄	—	2	2	no	—
75	C₂×C₄	—	2	2	no	—
76	C₂×C₄	C₇₁⋊C₁₆	1	4	yes	complemented
77	C₂×C₄	C₄×Dic₇₁	1	1	yes	characteristic
78	C₂×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
79	C₂×C₄	C₄⋊Dic₇₁	1	1	yes	characteristic
80	C₂×C₄	C₂×C₇₁⋊C₈	1	1	yes	characteristic
81	C₂×C₄	#11765849	1	1	yes	characteristic
82	C₂×C₄	#11765849	1	1	yes	characteristic
83	C₈	—	2	2	no	—
84	C₈	(C₂×D₇₁)⋊C₄	1	2	yes	—
85	C₈	Dic₇₁⋊C₄	1	2	yes	—
86	C₂²	C₄×C₇₁⋊C₈	1	1	yes	characteristic
87	C₂²	C₂²⋊(C₇₁⋊C₈)	1	1	yes	characteristic
88	C₂²	C₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
89	C₂²	C₂×C₇₁⋊C₁₆	1	2	yes	—
90	C₂²	#14556068	1	2	yes	—
91	C₄	—	2	4	no	—
92	C₄	C₇₁⋊#95	1	1	yes	characteristic
93	C₄	#14556039	1	1	yes	characteristic
94	C₄	C₂²⋊(C₇₁⋊C₈)	1	1	yes	characteristic
95	C₄	C₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
96	C₂	#17985754	1	1	yes	characteristic
97	C₂	C₄×C₇₁⋊C₁₆	1	1	yes	characteristic
98	C₂	#17985775	1	1	yes	characteristic
99	C₂	C₂²⋊(C₇₁⋊C₁₆)	1	2	yes	—
100	C₂	C₄⋊(C₇₁⋊C₁₆)	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₇₁⋊C₁₆

C₂

yes

maximal

C₂×C₄×C₈×C₇₁

C₂

yes

maximal, characteristic

C₂×C₇₁⋊C₁₆

—

C₂×C₇₁⋊C₁₆

—

C₂×C₇₁⋊C₁₆

C₄

yes

complemented

C₂²×C₈×C₇₁

C₄

yes

characteristic

C₂²×C₈×C₇₁

C₂²

yes

characteristic

C₂×C₄²×C₇₁

C₄

yes

characteristic

C₄×C₈×C₇₁

—

C₇₁⋊C₁₆

—

C₂²×C₄×C₇₁

C₈

yes

—

C₂²×C₄×C₇₁

C₂×C₄

yes

characteristic

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₈

yes

—

C₂×C₈×C₇₁

C₂×C₄

yes

—

C₂×C₈×C₇₁

D₄

yes

characteristic

C₂×C₈×C₇₁

D₄

yes

—

C₂×C₈×C₇₁

Q₈

yes

characteristic

C₄²×C₇₁

—

C₄²×C₇₁

—

C₂³×C₇₁

C₂×C₈

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₁₆

yes

complemented

C₂×C₄×C₇₁

C₄²

yes

characteristic

C₂×C₄×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₄⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×C₈

yes

characteristic

C₂×C₄×C₇₁

M₁₆

yes

characteristic

C₂×C₄×C₇₁

M₁₆

yes

characteristic

C₈×C₇₁

—

C₈×C₇₁

C₂²⋊C₄

yes

—

C₈×C₇₁

C₄⋊C₄

yes

—

C₂²×C₇₁

C₄×C₈

yes

characteristic

C₂²×C₇₁

C₂²⋊C₈

yes

characteristic

C₂²×C₇₁

C₄⋊C₈

yes

characteristic

C₂²×C₇₁

C₂×C₁₆

yes

—

C₂²×C₇₁

M₃₂

yes

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

C₂²⋊C₈

yes

characteristic

C₄×C₇₁

C₄⋊C₈

yes

characteristic

C₂×C₇₁

#336

yes

characteristic

C₂×C₇₁

C₄×C₁₆

yes

characteristic

C₂×C₇₁

#346

yes

characteristic

C₂×C₇₁

C₂²⋊C₁₆

yes

—

C₂×C₇₁

C₄⋊C₁₆

yes

—

#1374

—

maximal, Sylow

C₇₁

#1374

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂×C₄×C₈

D₇₁

yes

characteristic

C₂²×C₈

Dic₇₁

yes

characteristic

C₂²×C₈

C₂×D₇₁

yes

characteristic

C₂×C₄²

Dic₇₁

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

142

—

C₂×C₁₆

—

142

—

C₄×C₈

—

C₂²×C₄

C₇₁⋊C₈

yes

—

C₂²×C₄

C₂×Dic₇₁

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₇₁⋊C₈

yes

—

C₂×C₈

Dic₁₄₂

yes

characteristic

C₂×C₈

C₄×D₇₁

yes

—

C₂×C₈

D₂₈₄

yes

characteristic

C₂×C₈

C₂²⋊D₇₁

yes

—

C₄²

—

C₄²

—

C₁₆

—

142

—

C₂³

C₂×C₇₁⋊C₈

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₇₁⋊C₁₆

yes

complemented

C₂×C₄

C₄×Dic₇₁

yes

characteristic

C₂×C₄

C₂²⋊Dic₇₁

yes

characteristic

C₂×C₄

C₄⋊Dic₇₁

yes

characteristic

C₂×C₄

C₂×C₇₁⋊C₈

yes

C₂×C₄

yes

C₂×C₄

yes

C₈

—

C₈

(C₂×D₇₁)⋊C₄

yes

—

C₈

Dic₇₁⋊C₄

yes

—

C₂²

C₄×C₇₁⋊C₈

yes

characteristic

C₂²

C₂²⋊(C₇₁⋊C₈)

yes

characteristic

C₂²

C₄⋊(C₇₁⋊C₈)

yes

characteristic

C₂²

C₂×C₇₁⋊C₁₆

yes

—

C₂²

#14556068

yes

—

C₄

—

C₄

C₇₁⋊#95

yes

characteristic

C₄

#14556039

yes

characteristic

C₄

C₂²⋊(C₇₁⋊C₈)

yes

characteristic

C₄

C₄⋊(C₇₁⋊C₈)

yes

characteristic

C₂

#17985754

yes

characteristic

C₂

C₄×C₇₁⋊C₁₆

yes

characteristic

C₂

#17985775

yes

characteristic

C₂

C₂²⋊(C₇₁⋊C₁₆)

yes

—

100

C₂

C₄⋊(C₇₁⋊C₁₆)

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₈	C₂×D₇₁
commutator	C₂×C₇₁	C₄×C₁₆
Fitting	C₂×C₄×C₈×C₇₁	C₂
socle	C₂³×C₇₁	C₂×C₈
hypercenter	C₂×C₄×C₈	D₇₁
hypocenter	C₇₁	#1374
LCS	G ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

subgroup

G/H

center, Frattini

C₂²×C₈

C₂×D₇₁

commutator

C₂×C₇₁

C₄×C₁₆

Fitting

C₂×C₄×C₈×C₇₁

C₂

socle

C₂³×C₇₁

C₂×C₈

hypercenter

C₂×C₄×C₈

D₇₁

C₇₁

G ▷ C₂×C₇₁ ▷ C₇₁

C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
5. C₂×C₇₁⋊C₁₆	C₄	semidirect
25. C₂×C₄×C₇₁	C₁₆	semidirect
51. C₇₁	#1374	semidirect
76. C₂×C₄	C₇₁⋊C₁₆	semidirect

H ≅ G/N

type

5. C₂×C₇₁⋊C₁₆

C₄

semidirect

25. C₂×C₄×C₇₁

C₁₆

semidirect

51. C₇₁

#1374

semidirect

76. C₂×C₄

C₇₁⋊C₁₆

semidirect

Subgroups — #22586418

Conjugacy classes of subgroups

Complements