Group #21447572 — subgroups

Subgroups — C₂×C₄⋊#11758185 #21447572

Order 7808.

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×#11758185	C₂	1	2	yes	maximal
2	C₄⋊#11758185	C₂	1	4	yes	maximal, direct product factor
3	C₂×C₄×C₈×C₆₁	C₂	1	1	yes	maximal, characteristic
4	C₂×#11758185	—	2	4	no	—
5	C₂×#11758185	C₂²	1	4	yes	—
6	C₂²×C₈×C₆₁	C₄	1	1	yes	characteristic
7	C₂²×C₈×C₆₁	C₂²	1	1	yes	characteristic
8	C₂×C₄²×C₆₁	C₄	1	1	yes	characteristic
9	C₄×C₈×C₆₁	C₄	1	2	yes	—
10	C₄×C₈×C₆₁	C₂²	1	2	yes	—
11	C₂²×C₄×C₆₁	C₈	1	1	yes	characteristic
12	C₂²×C₄×C₆₁	C₈	1	1	yes	characteristic
13	C₂²×C₄×C₆₁	C₂×C₄	1	1	yes	characteristic
14	C₂×C₈×C₆₁	—	2	2	no	—
15	C₂×C₈×C₆₁	C₂×C₄	1	2	yes	—
16	C₂×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
17	C₂×C₈×C₆₁	D₄	1	2	yes	—
18	C₂×C₈×C₆₁	Q₈	1	2	yes	—
19	C₂×C₈×C₆₁	C₂³	1	1	yes	characteristic
20	C₄²×C₆₁	C₈	1	2	yes	—
21	C₄²×C₆₁	C₂×C₄	1	2	yes	—
22	#11758185	—	2	4	no	—
23	C₂³×C₆₁	C₂×C₈	1	1	yes	characteristic
24	C₂×C₄×C₆₁	—	2	2	no	—
25	C₂×C₄×C₆₁	C₁₆	1	4	yes	complemented
26	C₂×C₄×C₆₁	C₄⋊C₄	1	2	yes	—
27	C₂×C₄×C₆₁	C₂×C₈	1	2	yes	—
28	C₂×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
29	C₂×C₄×C₆₁	C₂×C₈	1	2	yes	—
30	C₂×C₄×C₆₁	M₁₆	1	2	yes	—
31	C₂×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
32	C₈×C₆₁	—	2	2	no	—
33	C₈×C₆₁	C₄⋊C₄	1	2	yes	—
34	C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
35	C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
36	C₂²×C₆₁	C₄⋊C₈	1	2	yes	—
37	C₂²×C₆₁	C₂×C₁₆	1	2	yes	—
38	C₂²×C₆₁	M₃₂	1	2	yes	—
39	C₂²×C₆₁	C₂²×C₈	1	1	yes	characteristic
40	C₄×C₆₁	—	2	2	no	—
41	C₄×C₆₁	C₄⋊C₈	1	2	yes	—
42	C₄×C₆₁	C₂×C₁₆	1	4	yes	complemented
43	C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
44	C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
45	C₂×C₄⋊C₁₆	—	61	1	no	maximal, Sylow
46	C₂×C₆₁	C₄⋊C₁₆	1	4	yes	complemented
47	C₂×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
48	C₂×C₆₁	C₂²×C₁₆	1	1	yes	characteristic
49	C₂×C₆₁	C₂×M₃₂	1	1	yes	characteristic
50	C₂²×C₁₆	—	61	2	no	—
51	C₂×C₄×C₈	D₆₁	1	1	yes	characteristic
52	C₄⋊C₁₆	—	61	4	no	—
53	C₆₁	C₂×C₄⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
54	C₂²×C₈	Dic₆₁	1	1	yes	characteristic
55	C₂²×C₈	C₂×D₆₁	1	1	yes	characteristic
56	C₂×C₄²	Dic₆₁	1	1	yes	characteristic
57	C₂×C₁₆	—	61	4	no	—
58	C₂×C₁₆	—	122	4	no	—
59	C₄×C₈	Dic₆₁	1	2	yes	—
60	C₄×C₈	C₂×D₆₁	1	2	yes	—
61	C₂²×C₄	#92418	1	1	yes	characteristic
62	C₂²×C₄	#92418	1	1	yes	characteristic
63	C₂²×C₄	C₂×Dic₆₁	1	1	yes	characteristic
64	C₂×C₈	—	2	2	no	—
65	C₂×C₈	Dic₁₂₂	1	2	yes	—
66	C₂×C₈	D₂₄₄	1	2	yes	—
67	C₂×C₈	C₂×Dic₆₁	1	2	yes	—
68	C₂×C₈	C₂×Dic₆₁	1	1	yes	characteristic
69	C₂×C₈	C₂²×D₆₁	1	1	yes	characteristic
70	C₄²	#92418	1	2	yes	—
71	C₄²	C₂×Dic₆₁	1	2	yes	—
72	C₁₆	—	122	4	no	—
73	C₂³	C₂×#92418	1	1	yes	characteristic
74	C₂×C₄	—	2	2	no	—
75	C₂×C₄	#11758185	1	4	yes	complemented
76	C₂×C₄	C₂×#92418	1	2	yes	—
77	C₂×C₄	C₂×#92418	1	1	yes	characteristic
78	C₂×C₄	C₂×#92418	1	2	yes	—
79	C₂×C₄	#11758194	1	2	yes	—
80	C₂×C₄	C₄⋊Dic₆₁	1	2	yes	—
81	C₂×C₄	C₂²×Dic₆₁	1	1	yes	characteristic
82	C₈	—	2	2	no	—
83	C₈	C₄⋊Dic₆₁	1	2	yes	—
84	C₈	C₂×Dic₁₂₂	1	1	yes	characteristic
85	C₈	C₂×D₂₄₄	1	1	yes	characteristic
86	C₂²	C₄⋊#92418	1	2	yes	—
87	C₂²	C₂×#11758185	1	2	yes	—
88	C₂²	#14519717	1	2	yes	—
89	C₂²	C₂²×#92418	1	1	yes	characteristic
90	C₄	—	2	2	no	—
91	C₄	C₄⋊#92418	1	2	yes	—
92	C₄	C₂×#11758185	1	4	yes	complemented
93	C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
94	C₄	C₂×#11758194	1	1	yes	characteristic
95	C₂	C₄⋊#11758185	1	4	yes	direct product factor
96	C₂	C₂×C₄⋊#92418	1	1	yes	characteristic
97	C₂	C₂²×#11758185	1	1	yes	characteristic
98	C₂	C₂×#14519717	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11758185

C₂

yes

maximal

C₄⋊#11758185

C₂

yes

maximal, direct product factor

C₂×C₄×C₈×C₆₁

C₂

yes

maximal, characteristic

C₂×#11758185

—

C₂×#11758185

C₂²

yes

—

C₂²×C₈×C₆₁

C₄

yes

characteristic

C₂²×C₈×C₆₁

C₂²

yes

characteristic

C₂×C₄²×C₆₁

C₄

yes

characteristic

C₄×C₈×C₆₁

C₄

yes

—

C₄×C₈×C₆₁

C₂²

yes

—

C₂²×C₄×C₆₁

C₈

yes

characteristic

C₂²×C₄×C₆₁

C₈

yes

characteristic

C₂²×C₄×C₆₁

C₂×C₄

yes

characteristic

C₂×C₈×C₆₁

—

C₂×C₈×C₆₁

C₂×C₄

yes

—

C₂×C₈×C₆₁

C₂×C₄

yes

characteristic

C₂×C₈×C₆₁

D₄

yes

—

C₂×C₈×C₆₁

Q₈

yes

—

C₂×C₈×C₆₁

C₂³

yes

characteristic

C₄²×C₆₁

C₈

yes

—

C₄²×C₆₁

C₂×C₄

yes

—

#11758185

—

C₂³×C₆₁

C₂×C₈

yes

characteristic

C₂×C₄×C₆₁

—

C₂×C₄×C₆₁

C₁₆

yes

complemented

C₂×C₄×C₆₁

C₄⋊C₄

yes

—

C₂×C₄×C₆₁

C₂×C₈

yes

—

C₂×C₄×C₆₁

C₂×C₈

yes

characteristic

C₂×C₄×C₆₁

C₂×C₈

yes

—

C₂×C₄×C₆₁

M₁₆

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

characteristic

C₈×C₆₁

—

C₈×C₆₁

C₄⋊C₄

yes

—

C₈×C₆₁

C₂×D₄

yes

characteristic

C₈×C₆₁

C₂×Q₈

yes

characteristic

C₂²×C₆₁

C₄⋊C₈

yes

—

C₂²×C₆₁

C₂×C₁₆

yes

—

C₂²×C₆₁

M₃₂

yes

—

C₂²×C₆₁

C₂²×C₈

yes

characteristic

C₄×C₆₁

—

C₄×C₆₁

C₄⋊C₈

yes

—

C₄×C₆₁

C₂×C₁₆

yes

complemented

C₄×C₆₁

C₂×C₄⋊C₄

yes

characteristic

C₄×C₆₁

C₂×M₁₆

yes

characteristic

C₂×C₄⋊C₁₆

—

maximal, Sylow

C₂×C₆₁

C₄⋊C₁₆

yes

complemented

C₂×C₆₁

C₂×C₄⋊C₈

yes

characteristic

C₂×C₆₁

C₂²×C₁₆

yes

characteristic

C₂×C₆₁

C₂×M₃₂

yes

characteristic

C₂²×C₁₆

—

C₂×C₄×C₈

D₆₁

yes

characteristic

C₄⋊C₁₆

—

C₆₁

C₂×C₄⋊C₁₆

yes

characteristic, complemented, Sylow

C₂²×C₈

Dic₆₁

yes

characteristic

C₂²×C₈

C₂×D₆₁

yes

characteristic

C₂×C₄²

Dic₆₁

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

—

122

—

C₄×C₈

Dic₆₁

yes

—

C₄×C₈

C₂×D₆₁

yes

—

C₂²×C₄

yes

C₂²×C₄

yes

C₂²×C₄

C₂×Dic₆₁

yes

characteristic

C₂×C₈

—

C₂×C₈

Dic₁₂₂

yes

—

C₂×C₈

D₂₄₄

yes

—

C₂×C₈

C₂×Dic₆₁

yes

—

C₂×C₈

C₂×Dic₆₁

yes

characteristic

C₂×C₈

C₂²×D₆₁

yes

characteristic

C₄²

#92418

yes

—

C₄²

C₂×Dic₆₁

yes

—

C₁₆

—

122

—

C₂³

C₂×#92418

yes

characteristic

C₂×C₄

—

C₂×C₄

#11758185

yes

complemented

C₂×C₄

C₂×#92418

yes

—

C₂×C₄

C₂×#92418

yes

characteristic

C₂×C₄

C₂×#92418

yes

—

C₂×C₄

#11758194

yes

—

C₂×C₄

C₄⋊Dic₆₁

yes

—

C₂×C₄

C₂²×Dic₆₁

yes

characteristic

C₈

—

C₈

C₄⋊Dic₆₁

yes

—

C₈

C₂×Dic₁₂₂

yes

characteristic

C₈

C₂×D₂₄₄

yes

characteristic

C₂²

C₄⋊#92418

yes

—

C₂²

C₂×#11758185

yes

—

C₂²

#14519717

yes

—

C₂²

C₂²×#92418

yes

characteristic

C₄

—

C₄

C₄⋊#92418

yes

—

C₄

C₂×#11758185

yes

complemented

C₄

C₂×C₄⋊Dic₆₁

yes

C₄

yes

C₂

yes

direct product factor

C₂

C₂×C₄⋊#92418

yes

C₂

yes

C₂

yes

	subgroup	G/H
center	C₂²×C₈	C₂×D₆₁
commutator	C₂×C₆₁	C₂²×C₁₆
Frattini	C₂×C₈	C₂²×D₆₁
Fitting	C₂×C₄×C₈×C₆₁	C₂
socle	C₂³×C₆₁	C₂×C₈
hypercenter	C₂×C₄×C₈	D₆₁
hypocenter	C₆₁	C₂×C₄⋊C₁₆
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂²×C₈

C₂×D₆₁

commutator

C₂×C₆₁

C₂²×C₁₆

Frattini

C₂×C₈

C₂²×D₆₁

Fitting

C₂×C₄×C₈×C₆₁

C₂

socle

C₂³×C₆₁

C₂×C₈

hypercenter

C₂×C₄×C₈

D₆₁

hypocenter

C₆₁

C₂×C₄⋊C₁₆

LCS

G ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. C₄⋊#11758185	C₂	direct (N × H)
25. C₂×C₄×C₆₁	C₁₆	semidirect
42. C₄×C₆₁	C₂×C₁₆	semidirect
46. C₂×C₆₁	C₄⋊C₁₆	semidirect
53. C₆₁	C₂×C₄⋊C₁₆	semidirect
75. C₂×C₄	#11758185	semidirect
92. C₄	C₂×#11758185	semidirect
95. C₂	C₄⋊#11758185	direct (N × H)

H ≅ G/N

type

2. C₄⋊#11758185

C₂

direct (N × H)

25. C₂×C₄×C₆₁

C₁₆

semidirect

42. C₄×C₆₁

C₂×C₁₆

semidirect

46. C₂×C₆₁

C₄⋊C₁₆

semidirect

53. C₆₁

C₂×C₄⋊C₁₆

semidirect

75. C₂×C₄

#11758185

semidirect

92. C₄

C₂×#11758185

semidirect

95. C₂

C₄⋊#11758185

direct (N × H)

Subgroups — C₂×C₄⋊#11758185 #21447572

Conjugacy classes of subgroups

Complements

Subgroups — C2×C4⋊#11758185 #21447572

Conjugacy classes of subgroups

Complements

Subgroups — C₂×C₄⋊#11758185 #21447572