Group #20276468 — subgroups

Subgroups — C₂×C₄⋊#11719908 #20276468

Order 6784.

184 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 141 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×#11719908	C₂	1	2	yes	maximal
2	C₄⋊#11719908	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
4	C₂×#11719908	—	2	4	no	—
5	C₂×#11719908	C₂²	1	4	yes	—
6	C₂²×C₈×C₅₃	C₄	1	1	yes	characteristic
7	C₂²×C₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×C₄²×C₅₃	C₄	1	1	yes	characteristic
9	C₄×C₈×C₅₃	C₄	1	2	yes	—
10	C₄×C₈×C₅₃	C₂²	1	2	yes	—
11	C₂²×C₄×C₅₃	C₈	1	1	yes	characteristic
12	C₂²×C₄×C₅₃	C₈	1	1	yes	characteristic
13	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₅₃	—	2	2	no	—
15	C₂×C₈×C₅₃	C₂×C₄	1	2	yes	—
16	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×C₅₃	D₄	1	2	yes	—
18	C₂×C₈×C₅₃	Q₈	1	2	yes	—
19	C₂×C₈×C₅₃	C₂³	1	1	yes	characteristic
20	C₄²×C₅₃	C₈	1	2	yes	—
21	C₄²×C₅₃	C₂×C₄	1	2	yes	—
22	#11719908	—	2	4	no	—
23	C₂³×C₅₃	C₂×C₈	1	1	yes	characteristic
24	C₂×C₄×C₅₃	—	2	2	no	—
25	C₂×C₄×C₅₃	C₁₆	1	4	yes	complemented
26	C₂×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
27	C₂×C₄×C₅₃	C₂×C₈	1	2	yes	—
28	C₂×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
29	C₂×C₄×C₅₃	C₂×C₈	1	2	yes	—
30	C₂×C₄×C₅₃	M₁₆	1	2	yes	—
31	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
32	C₈×C₅₃	—	2	2	no	—
33	C₈×C₅₃	C₄⋊C₄	1	2	yes	—
34	C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
35	C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
36	C₂²×C₅₃	C₄⋊C₈	1	2	yes	—
37	C₂²×C₅₃	C₂×C₁₆	1	2	yes	—
38	C₂²×C₅₃	M₃₂	1	2	yes	—
39	C₂²×C₅₃	C₂²×C₈	1	1	yes	characteristic
40	C₄×C₅₃	—	2	2	no	—
41	C₄×C₅₃	C₄⋊C₈	1	2	yes	—
42	C₄×C₅₃	C₂×C₁₆	1	4	yes	complemented
43	C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
45	C₂×C₄⋊C₁₆	—	53	1	no	maximal, Sylow
46	C₂×C₅₃	C₄⋊C₁₆	1	4	yes	complemented
47	C₂×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
48	C₂×C₅₃	C₂²×C₁₆	1	1	yes	characteristic
49	C₂×C₅₃	C₂×M₃₂	1	1	yes	characteristic
50	C₂²×C₁₆	—	53	2	no	—
51	C₂×C₄×C₈	D₅₃	1	1	yes	characteristic
52	C₄⋊C₁₆	—	53	4	no	—
53	C₅₃	C₂×C₄⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
54	C₂²×C₈	Dic₅₃	1	1	yes	characteristic
55	C₂²×C₈	C₂×D₅₃	1	1	yes	characteristic
56	C₂×C₄²	Dic₅₃	1	1	yes	characteristic
57	C₂×C₁₆	—	53	4	no	—
58	C₂×C₁₆	—	106	4	no	—
59	C₄×C₈	Dic₅₃	1	2	yes	—
60	C₄×C₈	C₂×D₅₃	1	2	yes	—
61	C₂²×C₄	#88322	1	1	yes	characteristic
62	C₂²×C₄	#88322	1	1	yes	characteristic
63	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
64	C₂×C₈	—	2	2	no	—
65	C₂×C₈	Dic₁₀₆	1	2	yes	—
66	C₂×C₈	D₂₁₂	1	2	yes	—
67	C₂×C₈	C₂×Dic₅₃	1	2	yes	—
68	C₂×C₈	C₂×Dic₅₃	1	1	yes	characteristic
69	C₂×C₈	C₂²×D₅₃	1	1	yes	characteristic
70	C₄²	#88322	1	2	yes	—
71	C₄²	C₂×Dic₅₃	1	2	yes	—
72	C₁₆	—	106	4	no	—
73	C₂³	C₂×#88322	1	1	yes	characteristic
74	C₂×C₄	—	2	2	no	—
75	C₂×C₄	#11719908	1	4	yes	complemented
76	C₂×C₄	C₂×#88322	1	2	yes	—
77	C₂×C₄	C₂×#88322	1	1	yes	characteristic
78	C₂×C₄	C₂×#88322	1	2	yes	—
79	C₂×C₄	#11719917	1	2	yes	—
80	C₂×C₄	C₄⋊Dic₅₃	1	2	yes	—
81	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
82	C₈	—	2	2	no	—
83	C₈	C₄⋊Dic₅₃	1	2	yes	—
84	C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
85	C₈	C₂×D₂₁₂	1	1	yes	characteristic
86	C₂²	C₄⋊#88322	1	2	yes	—
87	C₂²	C₂×#11719908	1	2	yes	—
88	C₂²	#13133157	1	2	yes	—
89	C₂²	C₂²×#88322	1	1	yes	characteristic
90	C₄	—	2	2	no	—
91	C₄	C₄⋊#88322	1	2	yes	—
92	C₄	C₂×#11719908	1	4	yes	complemented
93	C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
94	C₄	C₂×#11719917	1	1	yes	characteristic
95	C₂	C₄⋊#11719908	1	4	yes	direct product factor
96	C₂	C₂×C₄⋊#88322	1	1	yes	characteristic
97	C₂	C₂²×#11719908	1	1	yes	characteristic
98	C₂	C₂×#13133157	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11719908

C₂

yes

maximal

C₄⋊#11719908

C₂

yes

maximal, direct product factor

C₂×C₄×C₈×C₅₃

C₂

yes

maximal, characteristic

C₂×#11719908

—

C₂×#11719908

C₂²

yes

—

C₂²×C₈×C₅₃

C₄

yes

characteristic

C₂²×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄²×C₅₃

C₄

yes

characteristic

C₄×C₈×C₅₃

C₄

yes

—

C₄×C₈×C₅₃

C₂²

yes

—

C₂²×C₄×C₅₃

C₈

yes

characteristic

C₂²×C₄×C₅₃

C₈

yes

characteristic

C₂²×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×C₄

yes

—

C₂×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂×C₈×C₅₃

D₄

yes

—

C₂×C₈×C₅₃

Q₈

yes

—

C₂×C₈×C₅₃

C₂³

yes

characteristic

C₄²×C₅₃

C₈

yes

—

C₄²×C₅₃

C₂×C₄

yes

—

#11719908

—

C₂³×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₁₆

yes

complemented

C₂×C₄×C₅₃

C₄⋊C₄

yes

—

C₂×C₄×C₅₃

C₂×C₈

yes

—

C₂×C₄×C₅₃

C₂×C₈

yes

characteristic

C₂×C₄×C₅₃

C₂×C₈

yes

—

C₂×C₄×C₅₃

M₁₆

yes

—

C₂×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₈×C₅₃

—

C₈×C₅₃

C₄⋊C₄

yes

—

C₈×C₅₃

C₂×D₄

yes

characteristic

C₈×C₅₃

C₂×Q₈

yes

characteristic

C₂²×C₅₃

C₄⋊C₈

yes

—

C₂²×C₅₃

C₂×C₁₆

yes

—

C₂²×C₅₃

M₃₂

yes

—

C₂²×C₅₃

C₂²×C₈

yes

characteristic

C₄×C₅₃

—

C₄×C₅₃

C₄⋊C₈

yes

—

C₄×C₅₃

C₂×C₁₆

yes

complemented

C₄×C₅₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₅₃

C₂×M₁₆

yes

characteristic

C₂×C₄⋊C₁₆

—

maximal, Sylow

C₂×C₅₃

C₄⋊C₁₆

yes

complemented

C₂×C₅₃

C₂×C₄⋊C₈

yes

characteristic

C₂×C₅₃

C₂²×C₁₆

yes

characteristic

C₂×C₅₃

C₂×M₃₂

yes

characteristic

C₂²×C₁₆

—

C₂×C₄×C₈

D₅₃

yes

characteristic

C₄⋊C₁₆

—

C₅₃

C₂×C₄⋊C₁₆

yes

characteristic, complemented, Sylow

C₂²×C₈

Dic₅₃

yes

characteristic

C₂²×C₈

C₂×D₅₃

yes

characteristic

C₂×C₄²

Dic₅₃

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

106

—

C₄×C₈

Dic₅₃

yes

—

C₄×C₈

C₂×D₅₃

yes

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

C₂×Dic₅₃

yes

characteristic

C₂×C₈

—

C₂×C₈

Dic₁₀₆

yes

—

C₂×C₈

D₂₁₂

yes

—

C₂×C₈

C₂×Dic₅₃

yes

—

C₂×C₈

C₂×Dic₅₃

yes

characteristic

C₂×C₈

C₂²×D₅₃

yes

characteristic

C₄²

#88322

yes

—

C₄²

C₂×Dic₅₃

yes

—

C₁₆

—

106

—

C₂³

C₂×#88322

yes

characteristic

C₂×C₄

—

C₂×C₄

#11719908

yes

complemented

C₂×C₄

C₂×#88322

yes

—

C₂×C₄

C₂×#88322

yes

characteristic

C₂×C₄

C₂×#88322

yes

—

C₂×C₄

#11719917

yes

—

C₂×C₄

C₄⋊Dic₅₃

yes

—

C₂×C₄

C₂²×Dic₅₃

yes

characteristic

C₈

—

C₈

C₄⋊Dic₅₃

yes

—

C₈

C₂×Dic₁₀₆

yes

characteristic

C₈

C₂×D₂₁₂

yes

characteristic

C₂²

C₄⋊#88322

yes

—

C₂²

C₂×#11719908

yes

—

C₂²

#13133157

yes

—

C₂²

C₂²×#88322

yes

characteristic

C₄

—

C₄

C₄⋊#88322

yes

—

C₄

C₂×#11719908

yes

complemented

C₄

C₂×C₄⋊Dic₅₃

yes

C₄

yes

C₂

yes

direct product factor

C₂

C₂×C₄⋊#88322

yes

C₂

yes

C₂

yes

	subgroup	G/H
center	C₂²×C₈	C₂×D₅₃
commutator	C₂×C₅₃	C₂²×C₁₆
Frattini	C₂×C₈	C₂²×D₅₃
Fitting	C₂×C₄×C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₂×C₈
hypercenter	C₂×C₄×C₈	D₅₃
hypocenter	C₅₃	C₂×C₄⋊C₁₆
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂²×C₈

C₂×D₅₃

commutator

C₂×C₅₃

C₂²×C₁₆

Frattini

C₂×C₈

C₂²×D₅₃

Fitting

C₂×C₄×C₈×C₅₃

C₂

socle

C₂³×C₅₃

C₂×C₈

hypercenter

C₂×C₄×C₈

D₅₃

hypocenter

C₅₃

C₂×C₄⋊C₁₆

LCS

G ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. C₄⋊#11719908	C₂	direct (N × H)
25. C₂×C₄×C₅₃	C₁₆	semidirect
42. C₄×C₅₃	C₂×C₁₆	semidirect
46. C₂×C₅₃	C₄⋊C₁₆	semidirect
53. C₅₃	C₂×C₄⋊C₁₆	semidirect
75. C₂×C₄	#11719908	semidirect
92. C₄	C₂×#11719908	semidirect
95. C₂	C₄⋊#11719908	direct (N × H)

H ≅ G/N

type

2. C₄⋊#11719908

C₂

direct (N × H)

25. C₂×C₄×C₅₃

C₁₆

semidirect

42. C₄×C₅₃

C₂×C₁₆

semidirect

46. C₂×C₅₃

C₄⋊C₁₆

semidirect

53. C₅₃

C₂×C₄⋊C₁₆

semidirect

75. C₂×C₄

#11719908

semidirect

92. C₄

C₂×#11719908

semidirect

95. C₂

C₄⋊#11719908

direct (N × H)

Subgroups — C₂×C₄⋊#11719908 #20276468

Conjugacy classes of subgroups

Complements

Subgroups — C2×C4⋊#11719908 #20276468

Conjugacy classes of subgroups

Complements

Subgroups — C₂×C₄⋊#11719908 #20276468