Group #21462600 — subgroups

Subgroups — C₂×Q₈×#92420 #21462600

Order 7808.

440 conjugacy classes of subgroups; only the 274 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 274 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	Q₈×#92420	C₂	1	8	yes	maximal, direct product factor
2	C₂×C₄⋊#92420	C₂	1	3	yes	maximal
3	C₂×C₄×#92420	C₂	1	3	yes	maximal
4	C₂×Q₈×Dic₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂²×#92420	C₂²	1	3	yes	—
6	C₂²×#92420	C₂²	1	1	yes	characteristic
7	C₄⋊#92420	C₂²	1	12	yes	—
8	C₄×#92420	C₂²	1	12	yes	—
9	Q₈×Dic₆₁	C₄	1	4	yes	—
10	Q₈×Dic₆₁	C₂²	1	4	yes	—
11	C₂×C₄⋊Dic₆₁	C₄	1	3	yes	—
12	C₂×C₄×Dic₆₁	C₂²	1	3	yes	—
13	C₂²×Q₈×C₆₁	C₄	1	1	yes	characteristic
14	C₂×Q₈×C₆₁	C₈	1	8	yes	complemented
15	C₂×Q₈×C₆₁	C₂×C₄	1	4	yes	—
16	C₂²×C₄×C₆₁	C₂×C₄	1	3	yes	—
17	C₂²×Dic₆₁	C₂×C₄	1	3	yes	—
18	C₂²×Dic₆₁	C₂³	1	1	yes	characteristic
19	C₂×#92420	—	2	6	no	—
20	C₂×#92420	Q₈	1	4	yes	direct product factor
21	C₂×#92420	C₂³	1	6	yes	—
22	C₂×#92420	C₂³	1	2	yes	—
23	C₄⋊Dic₆₁	C₂×C₄	1	12	yes	—
24	C₄×Dic₆₁	C₂×C₄	1	6	yes	—
25	C₄×Dic₆₁	C₂³	1	6	yes	—
26	C₂³×C₆₁	C₂²×C₄	1	1	yes	characteristic
27	Q₈×C₆₁	C₂×C₈	1	16	yes	complemented
28	C₂×C₄×C₆₁	C₂×C₈	1	12	yes	—
29	C₂×C₄×C₆₁	C₂²×C₄	1	6	yes	—
30	C₂×Dic₆₁	—	2	6	no	—
31	C₂×Dic₆₁	C₂²×C₄	1	6	yes	—
32	C₂×Dic₆₁	C₂²×C₄	1	1	yes	characteristic
33	C₂×Dic₆₁	C₂×Q₈	1	2	yes	—
34	C₂×Dic₆₁	#41	1	2	yes	—
35	C₂×Dic₆₁	C₂⁴	1	1	yes	characteristic
36	#92420	—	2	6	no	—
37	#92420	C₂×Q₈	1	4	yes	direct product factor
38	C₂²×C₆₁	C₄×Q₈	1	2	yes	—
39	C₂²×C₆₁	C₂²×C₈	1	2	yes	—
40	C₂²×C₆₁	#131	1	2	yes	—
41	C₂²×C₆₁	C₂³×C₄	1	1	yes	characteristic
42	C₄×C₆₁	C₂²×C₈	1	12	yes	—
43	Dic₆₁	—	2	6	no	—
44	Dic₆₁	C₄×Q₈	1	2	yes	—
45	Dic₆₁	C₂²×Q₈	1	1	yes	characteristic
46	Dic₆₁	C₂×#41	1	1	yes	characteristic
47	C₂×C₈×Q₈	—	61	1	no	maximal, Sylow
48	C₂×C₆₁	C₈×Q₈	1	4	yes	complemented
49	C₂×C₆₁	C₂×C₄×Q₈	1	1	yes	characteristic
50	C₂×C₆₁	C₂³×C₈	1	1	yes	characteristic
51	C₂×C₆₁	C₂×#131	1	1	yes	characteristic
52	C₂×C₄×Q₈	—	61	1	no	—
53	C₈×Q₈	—	61	8	no	—
54	C₂×C₄⋊C₈	—	61	3	no	—
55	C₂×C₄×C₈	—	61	3	no	—
56	C₆₁	C₂×C₈×Q₈	1	1	yes	characteristic, complemented, Sylow
57	C₂²×Q₈	#6923	1	1	yes	characteristic
58	C₂²×C₈	—	61	1	no	—
59	C₂²×C₈	—	61	3	no	—
60	C₄×Q₈	—	61	4	no	—
61	C₄×Q₈	—	61	4	no	—
62	C₂×C₄⋊C₄	—	61	3	no	—
63	C₂×C₄²	—	61	3	no	—
64	C₄⋊C₈	—	61	12	no	—
65	C₄×C₈	—	61	12	no	—
66	C₂×Q₈	#92420	1	8	yes	direct product factor
67	C₂×Q₈	C₂×#6923	1	4	yes	—
68	C₂²×C₄	—	61	1	no	—
69	C₂²×C₄	—	61	3	no	—
70	C₂²×C₄	C₂×#6923	1	3	yes	—
71	C₂×C₈	—	61	2	no	—
72	C₂×C₈	—	61	4	no	—
73	C₂×C₈	—	61	6	no	—
74	C₂×C₈	—	122	6	no	—
75	C₄⋊C₄	—	61	12	no	—
76	C₄²	—	61	6	no	—
77	C₄²	—	61	6	no	—
78	C₂³	C₂²×#6923	1	1	yes	characteristic
79	Q₈	C₂×#92420	1	16	yes	direct product factor
80	C₂×C₄	—	61	2	no	—
81	C₂×C₄	—	61	2	no	—
82	C₂×C₄	—	61	1	no	—
83	C₂×C₄	—	61	1	no	—
84	C₂×C₄	—	61	6	no	—
85	C₂×C₄	—	122	6	no	—
86	C₂×C₄	C₂×#92420	1	12	yes	—
87	C₂×C₄	C₂²×#6923	1	6	yes	—
88	C₈	—	61	4	no	—
89	C₈	—	122	6	no	—
90	C₂²	Q₈×#6923	1	2	yes	—
91	C₂²	C₂²×#92420	1	2	yes	—
92	C₂²	#14519865	1	2	yes	—
93	C₂²	C₂³×#6923	1	1	yes	characteristic
94	C₄	—	61	2	no	—
95	C₄	—	61	1	no	—
96	C₄	—	61	1	no	—
97	C₄	—	122	6	no	—
98	C₄	C₂²×#92420	1	12	yes	—
99	C₂	Q₈×#92420	1	4	yes	direct product factor
100	C₂	C₂×Q₈×#6923	1	1	yes	characteristic
101	C₂	C₂³×#92420	1	1	yes	characteristic
102	C₂	C₂×#14519865	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×#92420

C₂

yes

maximal, direct product factor

C₂×C₄⋊#92420

C₂

yes

maximal

C₂×C₄×#92420

C₂

yes

maximal

C₂×Q₈×Dic₆₁

C₂

yes

maximal, characteristic

C₂²×#92420

C₂²

yes

—

C₂²×#92420

C₂²

yes

characteristic

C₄⋊#92420

C₂²

yes

—

C₄×#92420

C₂²

yes

—

Q₈×Dic₆₁

C₄

yes

—

Q₈×Dic₆₁

C₂²

yes

—

C₂×C₄⋊Dic₆₁

C₄

yes

—

C₂×C₄×Dic₆₁

C₂²

yes

—

C₂²×Q₈×C₆₁

C₄

yes

characteristic

C₂×Q₈×C₆₁

C₈

yes

complemented

C₂×Q₈×C₆₁

C₂×C₄

yes

—

C₂²×C₄×C₆₁

C₂×C₄

yes

—

C₂²×Dic₆₁

C₂×C₄

yes

—

C₂²×Dic₆₁

C₂³

yes

characteristic

C₂×#92420

—

C₂×#92420

Q₈

yes

direct product factor

C₂×#92420

C₂³

yes

—

C₂×#92420

C₂³

yes

—

C₄⋊Dic₆₁

C₂×C₄

yes

—

C₄×Dic₆₁

C₂×C₄

yes

—

C₄×Dic₆₁

C₂³

yes

—

C₂³×C₆₁

C₂²×C₄

yes

characteristic

Q₈×C₆₁

C₂×C₈

yes

complemented

C₂×C₄×C₆₁

C₂×C₈

yes

—

C₂×C₄×C₆₁

C₂²×C₄

yes

—

C₂×Dic₆₁

—

C₂×Dic₆₁

C₂²×C₄

yes

—

C₂×Dic₆₁

C₂²×C₄

yes

characteristic

C₂×Dic₆₁

C₂×Q₈

yes

—

C₂×Dic₆₁

#41

yes

—

C₂×Dic₆₁

C₂⁴

yes

characteristic

#92420

—

#92420

C₂×Q₈

yes

direct product factor

C₂²×C₆₁

C₄×Q₈

yes

—

C₂²×C₆₁

C₂²×C₈

yes

—

C₂²×C₆₁

#131

yes

—

C₂²×C₆₁

C₂³×C₄

yes

characteristic

C₄×C₆₁

C₂²×C₈

yes

—

Dic₆₁

—

Dic₆₁

C₄×Q₈

yes

—

Dic₆₁

C₂²×Q₈

yes

characteristic

Dic₆₁

C₂×#41

yes

characteristic

C₂×C₈×Q₈

—

maximal, Sylow

C₂×C₆₁

C₈×Q₈

yes

complemented

C₂×C₆₁

C₂×C₄×Q₈

yes

characteristic

C₂×C₆₁

C₂³×C₈

yes

characteristic

C₂×C₆₁

C₂×#131

yes

characteristic

C₂×C₄×Q₈

—

C₈×Q₈

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

—

C₆₁

C₂×C₈×Q₈

yes

characteristic, complemented, Sylow

C₂²×Q₈

#6923

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₄⋊C₈

—

C₄×C₈

—

C₂×Q₈

#92420

yes

direct product factor

C₂×Q₈

C₂×#6923

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×#6923

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

122

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₂³

C₂²×#6923

yes

characteristic

Q₈

C₂×#92420

yes

direct product factor

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

122

—

C₂×C₄

C₂×#92420

yes

—

C₂×C₄

C₂²×#6923

yes

—

C₈

—

C₈

—

122

—

C₂²

Q₈×#6923

yes

—

C₂²

C₂²×#92420

yes

—

C₂²

#14519865

yes

—

C₂²

C₂³×#6923

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

—

122

—

C₄

C₂²×#92420

yes

—

C₂

Q₈×#92420

yes

direct product factor

100

C₂

C₂×Q₈×#6923

yes

characteristic

101

C₂

C₂³×#92420

yes

characteristic

102

C₂

C₂×#14519865

yes

characteristic

	subgroup	G/H
center	C₂³	C₂²×#6923
commutator	C₂×C₆₁	C₂³×C₈
Frattini	C₂²	C₂³×#6923
Fitting	C₂²×Q₈×C₆₁	C₄
socle	C₂³×C₆₁	C₂²×C₄
hypercenter	C₂²×Q₈	#6923
hypocenter	C₆₁	C₂×C₈×Q₈
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂²×Q₈

subgroup

G/H

center

C₂³

C₂²×#6923

commutator

C₂×C₆₁

C₂³×C₈

Frattini

C₂²

C₂³×#6923

Fitting

C₂²×Q₈×C₆₁

C₄

socle

C₂³×C₆₁

C₂²×C₄

hypercenter

C₂²×Q₈

#6923

hypocenter

C₆₁

C₂×C₈×Q₈

LCS

G ▷ C₂×C₆₁ ▷ C₆₁

UCS

C₁ ◁ C₂³ ◁ C₂²×Q₈

N	H ≅ G/N	type
1. Q₈×#92420	C₂	direct (N × H)
14. C₂×Q₈×C₆₁	C₈	semidirect
20. C₂×#92420	Q₈	direct (N × H)
27. Q₈×C₆₁	C₂×C₈	semidirect
37. #92420	C₂×Q₈	direct (N × H)
48. C₂×C₆₁	C₈×Q₈	semidirect
56. C₆₁	C₂×C₈×Q₈	semidirect
66. C₂×Q₈	#92420	direct (N × H)
79. Q₈	C₂×#92420	direct (N × H)
99. C₂	Q₈×#92420	direct (N × H)

H ≅ G/N

type

1. Q₈×#92420

C₂

direct (N × H)

14. C₂×Q₈×C₆₁

C₈

semidirect

20. C₂×#92420

Q₈

direct (N × H)

27. Q₈×C₆₁

C₂×C₈

semidirect

37. #92420

C₂×Q₈

direct (N × H)

48. C₂×C₆₁

C₈×Q₈

semidirect

56. C₆₁

C₂×C₈×Q₈

semidirect

66. C₂×Q₈

#92420

direct (N × H)

79. Q₈

C₂×#92420

direct (N × H)

99. C₂

Q₈×#92420

direct (N × H)

Subgroups — C₂×Q₈×#92420 #21462600

Conjugacy classes of subgroups

Complements

Subgroups — C2×Q8×#92420 #21462600

Conjugacy classes of subgroups

Complements

Subgroups — C₂×Q₈×#92420 #21462600