Group #22586295 — subgroups

Subgroups — D₄⋊(C₇₁⋊C₁₆) #22586295

Order 9088.

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

normal subgroups only

This group has 110 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	C₄⋊(C₇₁⋊C₁₆)	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×C₇₁⋊C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈×D₄×C₇₁	C₂	1	1	yes	maximal, characteristic
4	C₂×C₇₁⋊C₁₆	—	2	1	no	—
5	C₂×C₇₁⋊C₁₆	—	2	1	no	—
6	C₂²×C₈×C₇₁	—	2	1	no	—
7	C₄×D₄×C₇₁	C₄	1	1	yes	characteristic
8	C₄⋊C₈×C₇₁	C₄	1	1	yes	characteristic
9	C₂²⋊C₈×C₇₁	—	2	1	no	—
10	C₄×C₈×C₇₁	C₂²	1	1	yes	characteristic
11	C₇₁⋊C₁₆	—	2	2	no	—
12	C₇₁⋊C₁₆	—	4	1	no	—
13	C₂×D₄×C₇₁	C₈	1	1	yes	characteristic
14	C₂²×C₄×C₇₁	—	2	1	no	—
15	C₂×C₈×C₇₁	—	2	1	no	—
16	C₂×C₈×C₇₁	—	4	1	no	—
17	C₂×C₈×C₇₁	—	4	1	no	—
18	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
19	C₂×C₈×C₇₁	D₄	1	1	yes	characteristic
20	C₄⋊C₄×C₇₁	C₈	1	1	yes	characteristic
21	C₂²⋊C₄×C₇₁	—	2	1	no	—
22	C₄²×C₇₁	C₂×C₄	1	1	yes	characteristic
23	C₂³×C₇₁	—	2	1	no	—
24	D₄×C₇₁	—	2	1	no	—
25	D₄×C₇₁	C₁₆	1	2	yes	complemented
26	C₂×C₄×C₇₁	—	2	1	no	—
27	C₂×C₄×C₇₁	—	4	1	no	—
28	C₂×C₄×C₇₁	—	4	1	no	—
29	C₂×C₄×C₇₁	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₇₁	C₂×C₈	1	1	yes	characteristic
31	C₂×C₄×C₇₁	M₁₆	1	1	yes	characteristic
32	C₈×C₇₁	—	2	1	no	—
33	C₈×C₇₁	—	4	1	no	—
34	C₈×C₇₁	D₈	1	1	yes	characteristic
35	C₈×C₇₁	SD₁₆	1	1	yes	characteristic
36	C₂²×C₇₁	—	2	2	no	—
37	C₂²×C₇₁	—	4	1	no	—
38	C₂²×C₇₁	—	4	1	no	—
39	C₂²×C₇₁	C₂²⋊C₈	1	1	yes	characteristic
40	C₄×C₇₁	—	2	1	no	—
41	C₄×C₇₁	—	4	1	no	—
42	C₄×C₇₁	#102	1	1	yes	characteristic
43	C₄×C₇₁	#104	1	1	yes	characteristic
44	C₄×C₇₁	C₂×C₁₆	1	1	yes	characteristic
45	C₄×C₇₁	M₃₂	1	1	yes	characteristic
46	C₂×C₇₁	—	4	2	no	—
47	C₂×C₇₁	D₄⋊C₈	1	1	yes	characteristic
48	C₂×C₇₁	C₂²⋊C₁₆	1	1	yes	characteristic
49	C₂×C₇₁	#350	1	1	yes	characteristic
50	D₄⋊C₁₆	—	71	1	no	maximal, Sylow
51	C₇₁	D₄⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
52	C₈×D₄	D₇₁	1	1	yes	characteristic
53	C₄⋊C₁₆	—	71	1	no	—
54	C₄×C₁₆	—	71	1	no	—
55	C₂²×C₈	—	2	1	no	—
56	C₄×D₄	Dic₇₁	1	1	yes	characteristic
57	C₂×C₁₆	—	142	1	no	—
58	C₂×C₁₆	—	142	1	no	—
59	C₄⋊C₈	Dic₇₁	1	1	yes	characteristic
60	C₂²⋊C₈	—	2	1	no	—
61	C₄×C₈	C₂×D₇₁	1	1	yes	characteristic
62	C₂×D₄	C₇₁⋊C₈	1	1	yes	characteristic
63	C₂²×C₄	—	2	1	no	—
64	C₂×C₈	—	2	1	no	—
65	C₂×C₈	—	4	1	no	—
66	C₂×C₈	—	4	1	no	—
67	C₂×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
68	C₂×C₈	C₂²⋊D₇₁	1	1	yes	characteristic
69	C₄⋊C₄	C₇₁⋊C₈	1	1	yes	characteristic
70	C₂²⋊C₄	—	2	1	no	—
71	C₄²	C₂×Dic₇₁	1	1	yes	characteristic
72	C₁₆	—	284	1	no	—
73	C₁₆	—	142	2	no	—
74	C₂³	—	2	1	no	—
75	D₄	—	2	1	no	—
76	D₄	C₇₁⋊C₁₆	1	2	yes	complemented
77	C₂×C₄	—	2	1	no	—
78	C₂×C₄	—	4	1	no	—
79	C₂×C₄	—	4	1	no	—
80	C₂×C₄	C₂²⋊Dic₇₁	1	1	yes	characteristic
81	C₂×C₄	C₂×C₇₁⋊C₈	1	1	yes	characteristic
82	C₂×C₄	#11765849	1	1	yes	characteristic
83	C₈	—	2	1	no	—
84	C₈	—	4	1	no	—
85	C₈	#11765850	1	1	yes	characteristic
86	C₈	#11765853	1	1	yes	characteristic
87	C₂²	—	2	2	no	—
88	C₂²	—	4	1	no	—
89	C₂²	—	4	1	no	—
90	C₂²	C₂²⋊(C₇₁⋊C₈)	1	1	yes	characteristic
91	C₄	—	2	1	no	—
92	C₄	—	4	1	no	—
93	C₄	#14556049	1	1	yes	characteristic
94	C₄	#14556055	1	1	yes	characteristic
95	C₄	C₂×C₇₁⋊C₁₆	1	1	yes	characteristic
96	C₄	#14556068	1	1	yes	characteristic
97	C₂	—	4	2	no	—
98	C₂	D₄⋊(C₇₁⋊C₈)	1	1	yes	characteristic
99	C₂	C₂²⋊(C₇₁⋊C₁₆)	1	1	yes	characteristic
100	C₂	#17985783	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄⋊(C₇₁⋊C₁₆)

C₂

yes

maximal, characteristic, complemented

C₄×C₇₁⋊C₁₆

C₂

yes

maximal, characteristic, complemented

C₈×D₄×C₇₁

C₂

yes

maximal, characteristic

C₂×C₇₁⋊C₁₆

—

C₂×C₇₁⋊C₁₆

—

C₂²×C₈×C₇₁

—

C₄×D₄×C₇₁

C₄

yes

characteristic

C₄⋊C₈×C₇₁

C₄

yes

characteristic

C₂²⋊C₈×C₇₁

—

C₄×C₈×C₇₁

C₂²

yes

characteristic

C₇₁⋊C₁₆

—

C₇₁⋊C₁₆

—

C₂×D₄×C₇₁

C₈

yes

characteristic

C₂²×C₄×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

D₄

yes

characteristic

C₂×C₈×C₇₁

D₄

yes

characteristic

C₄⋊C₄×C₇₁

C₈

yes

characteristic

C₂²⋊C₄×C₇₁

—

C₄²×C₇₁

C₂×C₄

yes

characteristic

C₂³×C₇₁

—

D₄×C₇₁

—

D₄×C₇₁

C₁₆

yes

complemented

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₁

C₂×C₈

yes

characteristic

C₂×C₄×C₇₁

M₁₆

yes

characteristic

C₈×C₇₁

—

C₈×C₇₁

—

C₈×C₇₁

D₈

yes

characteristic

C₈×C₇₁

SD₁₆

yes

characteristic

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

—

C₂²×C₇₁

C₂²⋊C₈

yes

characteristic

C₄×C₇₁

—

C₄×C₇₁

—

C₄×C₇₁

yes

C₄×C₇₁

yes

C₄×C₇₁

C₂×C₁₆

yes

characteristic

C₄×C₇₁

M₃₂

yes

characteristic

C₂×C₇₁

—

C₂×C₇₁

D₄⋊C₈

yes

characteristic

C₂×C₇₁

C₂²⋊C₁₆

yes

characteristic

C₂×C₇₁

#350

yes

characteristic

D₄⋊C₁₆

—

maximal, Sylow

C₇₁

D₄⋊C₁₆

yes

characteristic, complemented, Sylow

C₈×D₄

D₇₁

yes

characteristic

C₄⋊C₁₆

—

C₄×C₁₆

—

C₂²×C₈

—

C₄×D₄

Dic₇₁

yes

characteristic

C₂×C₁₆

—

142

—

C₂×C₁₆

—

142

—

C₄⋊C₈

Dic₇₁

yes

characteristic

C₂²⋊C₈

—

C₄×C₈

C₂×D₇₁

yes

characteristic

C₂×D₄

C₇₁⋊C₈

yes

characteristic

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₇₁

yes

characteristic

C₂×C₈

C₂²⋊D₇₁

yes

characteristic

C₄⋊C₄

C₇₁⋊C₈

yes

characteristic

C₂²⋊C₄

—

C₄²

C₂×Dic₇₁

yes

characteristic

C₁₆

—

284

—

C₁₆

—

142

—

C₂³

—

D₄

—

D₄

C₇₁⋊C₁₆

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊Dic₇₁

yes

characteristic

C₂×C₄

C₂×C₇₁⋊C₈

yes

characteristic

C₂×C₄

#11765849

yes

characteristic

C₈

—

C₈

—

C₈

yes

C₈

yes

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂²⋊(C₇₁⋊C₈)

yes

characteristic

C₄

—

C₄

—

C₄

yes

C₄

yes

C₄

C₂×C₇₁⋊C₁₆

yes

characteristic

C₄

#14556068

yes

characteristic

C₂

—

C₂

D₄⋊(C₇₁⋊C₈)

yes

characteristic

C₂

C₂²⋊(C₇₁⋊C₁₆)

yes

characteristic

100

C₂

#17985783

yes

characteristic

	subgroup	G/H
center	C₂×C₈	C₂²⋊D₇₁
commutator	C₄×C₇₁	C₂×C₁₆
Frattini	C₄×C₈	C₂×D₇₁
Fitting	C₈×D₄×C₇₁	C₂
socle	C₂²×C₇₁	C₂²⋊C₈
hypercenter	C₈×D₄	D₇₁
hypocenter	C₇₁	D₄⋊C₁₆
LCS	G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈×D₄

subgroup

G/H

center

C₂×C₈

C₂²⋊D₇₁

commutator

C₄×C₇₁

C₂×C₁₆

Frattini

C₄×C₈

C₂×D₇₁

Fitting

C₈×D₄×C₇₁

C₂

socle

C₂²×C₇₁

C₂²⋊C₈

hypercenter

C₈×D₄

D₇₁

hypocenter

C₇₁

D₄⋊C₁₆

LCS

G ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ C₈×D₄

N	H ≅ G/N	type
1. C₄⋊(C₇₁⋊C₁₆)	C₂	semidirect
2. C₄×C₇₁⋊C₁₆	C₂	semidirect
25. D₄×C₇₁	C₁₆	semidirect
51. C₇₁	D₄⋊C₁₆	semidirect
76. D₄	C₇₁⋊C₁₆	semidirect

H ≅ G/N

type

1. C₄⋊(C₇₁⋊C₁₆)

C₂

semidirect

2. C₄×C₇₁⋊C₁₆

C₂

semidirect

25. D₄×C₇₁

C₁₆

semidirect

51. C₇₁

D₄⋊C₁₆

semidirect

76. D₄

C₇₁⋊C₁₆

semidirect

Subgroups — D₄⋊(C₇₁⋊C₁₆) #22586295

Conjugacy classes of subgroups

Complements

Subgroups — D4⋊(C71⋊C16) #22586295

Conjugacy classes of subgroups

Complements

Subgroups — D₄⋊(C₇₁⋊C₁₆) #22586295