Group #20289665 — subgroups

Subgroups — D₄⋊#11719910 #20289665

Order 6784.

110 conjugacy classes of subgroups; only the 46 normal subgroups are shown.

normal subgroups only

This group has 110 conjugacy classes of subgroups — too many to draw. Only its 46 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⋊#11719910	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#11719910	C₂	1	1	yes	maximal, characteristic, complemented
3	D₄×#88322	C₂	1	1	yes	maximal, characteristic
4	C₂×#11719910	—	2	1	no	—
5	C₂×#11719910	—	2	1	no	—
6	C₂²×#88322	—	2	1	no	—
7	C₄⋊#88322	C₄	1	1	yes	characteristic
8	C₂²⋊#88322	—	2	1	no	—
9	C₄×#88322	C₂²	1	1	yes	characteristic
10	C₄×D₄×C₅₃	C₄	1	1	yes	characteristic
11	C₂×D₄×C₅₃	C₈	1	1	yes	characteristic
12	C₂²×C₄×C₅₃	—	2	1	no	—
13	C₄⋊C₄×C₅₃	C₈	1	1	yes	characteristic
14	C₂²⋊C₄×C₅₃	—	2	1	no	—
15	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
16	C₂×#88322	—	2	1	no	—
17	C₂×#88322	—	4	1	no	—
18	C₂×#88322	—	4	1	no	—
19	C₂×#88322	D₄	1	1	yes	characteristic
20	C₂×#88322	D₄	1	1	yes	characteristic
21	#11719910	—	2	2	no	—
22	#11719910	—	4	1	no	—
23	C₂³×C₅₃	—	2	1	no	—
24	D₄×C₅₃	—	2	1	no	—
25	D₄×C₅₃	C₁₆	1	2	yes	complemented
26	C₂×C₄×C₅₃	—	2	1	no	—
27	C₂×C₄×C₅₃	—	4	1	no	—
28	C₂×C₄×C₅₃	—	4	1	no	—
29	C₂×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
30	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
31	C₂×C₄×C₅₃	M₁₆	1	1	yes	characteristic
32	#88322	—	2	1	no	—
33	#88322	—	4	1	no	—
34	#88322	D₈	1	1	yes	characteristic
35	#88322	SD₁₆	1	1	yes	characteristic
36	C₂²×C₅₃	—	2	2	no	—
37	C₂²×C₅₃	—	4	1	no	—
38	C₂²×C₅₃	—	4	1	no	—
39	C₂²×C₅₃	C₂²⋊C₈	1	1	yes	characteristic
40	C₄×C₅₃	—	2	1	no	—
41	C₄×C₅₃	—	4	1	no	—
42	C₄×C₅₃	#102	1	1	yes	characteristic
43	C₄×C₅₃	#104	1	1	yes	characteristic
44	C₄×C₅₃	C₂×C₁₆	1	1	yes	characteristic
45	C₄×C₅₃	M₃₂	1	1	yes	characteristic
46	D₄⋊C₁₆	—	53	1	no	maximal, Sylow
47	C₂×C₅₃	—	4	2	no	—
48	C₂×C₅₃	D₄⋊C₈	1	1	yes	characteristic
49	C₂×C₅₃	C₂²⋊C₁₆	1	1	yes	characteristic
50	C₂×C₅₃	#350	1	1	yes	characteristic
51	C₈×D₄	—	53	1	no	—
52	C₄⋊C₁₆	—	53	1	no	—
53	C₄×C₁₆	—	53	1	no	—
54	C₅₃	D₄⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
55	C₂²×C₈	—	106	1	no	—
56	C₄×D₄	#6157	1	1	yes	characteristic
57	C₂×C₁₆	—	106	1	no	—
58	C₂×C₁₆	—	106	1	no	—
59	C₄⋊C₈	—	53	1	no	—
60	C₂²⋊C₈	—	106	1	no	—
61	C₄×C₈	—	53	1	no	—
62	C₂×D₄	#88324	1	1	yes	characteristic
63	C₂²×C₄	—	2	1	no	—
64	C₂×C₈	—	53	1	no	—
65	C₂×C₈	—	53	1	no	—
66	C₂×C₈	—	106	1	no	—
67	C₂×C₈	—	212	1	no	—
68	C₂×C₈	—	212	1	no	—
69	C₄⋊C₄	#88324	1	1	yes	characteristic
70	C₂²⋊C₄	—	2	1	no	—
71	C₄²	C₂×#6157	1	1	yes	characteristic
72	C₁₆	—	106	2	no	—
73	C₁₆	—	212	1	no	—
74	C₂³	—	2	1	no	—
75	D₄	—	2	1	no	—
76	D₄	#11719910	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	C₂×#88324	1	1	yes	characteristic
81	C₂×C₄	#11719940	1	1	yes	characteristic
82	C₂×C₄	C₂²⋊#6157	1	1	yes	characteristic
83	C₈	—	53	1	no	—
84	C₈	—	53	1	no	—
85	C₈	—	106	1	no	—
86	C₈	—	212	1	no	—
87	C₂²	—	2	2	no	—
88	C₂²	—	4	1	no	—
89	C₂²	—	4	1	no	—
90	C₂²	C₂²⋊#88324	1	1	yes	characteristic
91	C₄	—	2	1	no	—
92	C₄	—	4	1	no	—
93	C₄	#13133279	1	1	yes	characteristic
94	C₄	#13133281	1	1	yes	characteristic
95	C₄	C₂×#11719910	1	1	yes	characteristic
96	C₄	#13133291	1	1	yes	characteristic
97	C₂	—	4	2	no	—
98	C₂	D₄⋊#88324	1	1	yes	characteristic
99	C₂	C₂²⋊#11719910	1	1	yes	characteristic
100	C₂	#16811521	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₄⋊#11719910

C₂

yes

maximal, characteristic, complemented

C₄×#11719910

C₂

yes

maximal, characteristic, complemented

D₄×#88322

C₂

yes

maximal, characteristic

C₂×#11719910

—

C₂×#11719910

—

C₂²×#88322

—

C₄⋊#88322

C₄

yes

characteristic

C₂²⋊#88322

—

C₄×#88322

C₂²

yes

characteristic

C₄×D₄×C₅₃

C₄

yes

characteristic

C₂×D₄×C₅₃

C₈

yes

characteristic

C₂²×C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₈

yes

characteristic

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×C₄

yes

—

—

—

D₄

yes

characteristic

C₂×#88322

D₄

yes

characteristic

#11719910

—

#11719910

—

C₂³×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

C₁₆

yes

complemented

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

M₁₆

yes

—

—

D₈

yes

characteristic

#88322

SD₁₆

yes

characteristic

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

—

C₂²×C₅₃

C₂²⋊C₈

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

—

C₄×C₅₃

yes

C₄×C₅₃

yes

C₄×C₅₃

C₂×C₁₆

yes

characteristic

C₄×C₅₃

M₃₂

yes

characteristic

D₄⋊C₁₆

—

maximal, Sylow

C₂×C₅₃

—

C₂×C₅₃

D₄⋊C₈

yes

characteristic

C₂×C₅₃

C₂²⋊C₁₆

yes

characteristic

C₂×C₅₃

#350

yes

characteristic

C₈×D₄

—

C₄⋊C₁₆

—

C₄×C₁₆

—

C₅₃

D₄⋊C₁₆

yes

characteristic, complemented, Sylow

C₂²×C₈

—

106

—

C₄×D₄

#6157

yes

characteristic

C₂×C₁₆

—

106

—

C₂×C₁₆

—

106

—

C₄⋊C₈

—

C₂²⋊C₈

—

106

—

C₄×C₈

—

C₂×D₄

#88324

yes

characteristic

C₂²×C₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

106

—

C₂×C₈

—

212

—

C₂×C₈

—

212

—

C₄⋊C₄

#88324

yes

characteristic

C₂²⋊C₄

—

C₄²

C₂×#6157

yes

characteristic

C₁₆

—

106

—

C₁₆

—

212

—

C₂³

—

D₄

—

D₄

#11719910

yes

complemented

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

yes

C₂×C₄

yes

C₂×C₄

C₂²⋊#6157

yes

characteristic

C₈

—

C₈

—

C₈

—

106

—

C₈

—

212

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂²⋊#88324

yes

characteristic

C₄

—

C₄

—

C₄

yes

C₄

yes

C₄

yes

C₄

yes

C₂

—

C₂

yes

C₂

yes

100

C₂

#16811521

yes

characteristic

	subgroup	G/H
center	C₂×C₄	C₂²⋊#6157
commutator	C₄×C₅₃	C₂×C₁₆
Frattini	C₄²	C₂×#6157
Fitting	C₄×D₄×C₅₃	C₄
socle	C₂²×C₅₃	C₂²⋊C₈
hypercenter	C₄×D₄	#6157
hypocenter	C₅₃	D₄⋊C₁₆
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×D₄

subgroup

G/H

center

C₂×C₄

C₂²⋊#6157

commutator

C₄×C₅₃

C₂×C₁₆

Frattini

C₄²

C₂×#6157

Fitting

C₄×D₄×C₅₃

C₄

socle

C₂²×C₅₃

C₂²⋊C₈

hypercenter

C₄×D₄

#6157

hypocenter

C₅₃

D₄⋊C₁₆

LCS

G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ C₄×D₄

N	H ≅ G/N	type
1. C₄⋊#11719910	C₂	semidirect
2. C₄×#11719910	C₂	semidirect
25. D₄×C₅₃	C₁₆	semidirect
54. C₅₃	D₄⋊C₁₆	semidirect
76. D₄	#11719910	semidirect

H ≅ G/N

type

1. C₄⋊#11719910

C₂

semidirect

2. C₄×#11719910

C₂

semidirect

25. D₄×C₅₃

C₁₆

semidirect

54. C₅₃

D₄⋊C₁₆

semidirect

76. D₄

#11719910

semidirect

Subgroups — D₄⋊#11719910 #20289665

Conjugacy classes of subgroups

Complements

Subgroups — D4⋊#11719910 #20289665

Conjugacy classes of subgroups

Complements

Subgroups — D₄⋊#11719910 #20289665