Group #44528632 — subgroups

Subgroups — #450×C₄₈₇ #44528632

Order 31168.

188 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 62 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×D₄×C₄₈₇	C₂	1	1	yes	maximal, characteristic
2	C₂×SD₁₆×C₄₈₇	C₂	1	2	yes	maximal, complemented
3	C₂²⋊Q₈×C₄₈₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#102×C₄₈₇	C₂	1	2	yes	maximal, complemented
5	C₂²⋊C₈×C₄₈₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂⁴×C₄₈₇	—	2	1	no	—
7	C₂×Q₈×C₄₈₇	C₂²	1	1	yes	characteristic, complemented
8	C₂×D₄×C₄₈₇	—	2	4	no	—
9	C₂×D₄×C₄₈₇	—	2	1	no	—
10	C₂×D₄×C₄₈₇	C₂²	1	2	yes	—
11	C₂²×C₄×C₄₈₇	C₂²	1	1	yes	characteristic
12	SD₁₆×C₄₈₇	—	2	4	no	—
13	C₂×C₈×C₄₈₇	C₂²	1	2	yes	complemented
14	C₄⋊C₄×C₄₈₇	—	2	1	no	—
15	C₄⋊C₄×C₄₈₇	C₂²	1	1	yes	characteristic, complemented
16	C₂²⋊C₄×C₄₈₇	—	2	1	no	—
17	C₂³×C₄₈₇	—	2	4	no	—
18	C₂³×C₄₈₇	—	2	2	no	—
19	C₂³×C₄₈₇	—	4	2	no	—
20	C₂³×C₄₈₇	—	4	2	no	—
21	C₂³×C₄₈₇	D₄	1	1	yes	characteristic
22	Q₈×C₄₈₇	—	2	2	no	—
23	D₄×C₄₈₇	—	2	4	no	—
24	D₄×C₄₈₇	—	2	2	no	—
25	D₄×C₄₈₇	D₄	1	4	yes	—
26	C₂×C₄×C₄₈₇	—	2	2	no	—
27	C₂×C₄×C₄₈₇	—	2	1	no	—
28	C₂×C₄×C₄₈₇	—	2	1	no	—
29	C₂×C₄×C₄₈₇	D₄	1	1	yes	characteristic
30	C₂×C₄×C₄₈₇	C₂³	1	1	yes	characteristic
31	C₈×C₄₈₇	—	2	2	no	—
32	C₂²×C₄₈₇	—	2	1	no	—
33	C₂²×C₄₈₇	—	2	1	no	—
34	C₂²×C₄₈₇	—	2	4	no	—
35	C₂²×C₄₈₇	—	4	8	no	—
36	C₂²×C₄₈₇	—	4	2	no	—
37	C₂²×C₄₈₇	—	4	2	no	—
38	C₂²×C₄₈₇	SD₁₆	1	2	yes	complemented
39	C₂²×C₄₈₇	C₂×D₄	1	1	yes	characteristic
40	C₄×C₄₈₇	—	2	1	no	—
41	C₄×C₄₈₇	—	4	1	no	—
42	C₄×C₄₈₇	—	4	1	no	—
43	C₄×C₄₈₇	C₂×D₄	1	2	yes	—
44	C₂×C₄₈₇	—	2	2	no	—
45	C₂×C₄₈₇	—	4	4	no	—
46	C₂×C₄₈₇	#120	1	1	yes	characteristic
47	C₂×C₄₈₇	C₂×SD₁₆	1	1	yes	characteristic
48	C₂×C₄₈₇	AGL(1,ℤ/8)	1	1	yes	characteristic
49	C₄₈₇	#450	1	1	yes	characteristic, direct product factor, Sylow
50	#450	C₄₈₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
51	C₂²×D₄	C₂×C₄₈₇	1	1	yes	characteristic
52	C₂×SD₁₆	C₂×C₄₈₇	1	2	yes	complemented
53	C₂²⋊Q₈	C₂×C₄₈₇	1	1	yes	characteristic, complemented
54	#102	C₂×C₄₈₇	1	2	yes	complemented
55	C₂²⋊C₈	C₂×C₄₈₇	1	1	yes	characteristic, complemented
56	C₂⁴	—	2	1	no	—
57	C₂×Q₈	C₂²×C₄₈₇	1	1	yes	characteristic, complemented
58	C₂×D₄	—	2	4	no	—
59	C₂×D₄	—	2	1	no	—
60	C₂×D₄	C₂²×C₄₈₇	1	2	yes	—
61	C₂²×C₄	C₂²×C₄₈₇	1	1	yes	characteristic
62	SD₁₆	—	2	4	no	—
63	C₂×C₈	C₂²×C₄₈₇	1	2	yes	complemented
64	C₄⋊C₄	—	2	1	no	—
65	C₄⋊C₄	C₂²×C₄₈₇	1	1	yes	characteristic, complemented
66	C₂²⋊C₄	—	2	1	no	—
67	C₂³	—	2	4	no	—
68	C₂³	—	2	2	no	—
69	C₂³	—	4	2	no	—
70	C₂³	—	4	2	no	—
71	C₂³	D₄×C₄₈₇	1	1	yes	characteristic
72	Q₈	—	2	2	no	—
73	D₄	—	2	4	no	—
74	D₄	—	2	2	no	—
75	D₄	D₄×C₄₈₇	1	4	yes	—
76	C₂×C₄	—	2	2	no	—
77	C₂×C₄	—	2	1	no	—
78	C₂×C₄	—	2	1	no	—
79	C₂×C₄	D₄×C₄₈₇	1	1	yes	characteristic
80	C₂×C₄	C₂³×C₄₈₇	1	1	yes	characteristic
81	C₈	—	2	2	no	—
82	C₂²	—	2	1	no	—
83	C₂²	—	2	1	no	—
84	C₂²	—	2	4	no	—
85	C₂²	—	4	8	no	—
86	C₂²	—	4	2	no	—
87	C₂²	—	4	2	no	—
88	C₂²	SD₁₆×C₄₈₇	1	2	yes	complemented
89	C₂²	C₂×D₄×C₄₈₇	1	1	yes	characteristic
90	C₄	—	2	1	no	—
91	C₄	—	4	1	no	—
92	C₄	—	4	1	no	—
93	C₄	C₂×D₄×C₄₈₇	1	2	yes	—
94	C₂	—	2	2	no	—
95	C₂	—	4	4	no	—
96	C₂	#120×C₄₈₇	1	1	yes	characteristic
97	C₂	C₂×SD₁₆×C₄₈₇	1	1	yes	characteristic
98	C₂	AGL(1,ℤ/8)×C₄₈₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₄×C₄₈₇

C₂

yes

maximal, characteristic

C₂×SD₁₆×C₄₈₇

C₂

yes

maximal, complemented

C₂²⋊Q₈×C₄₈₇

C₂

yes

maximal, characteristic, complemented

#102×C₄₈₇

C₂

yes

maximal, complemented

C₂²⋊C₈×C₄₈₇

C₂

yes

maximal, characteristic, complemented

C₂⁴×C₄₈₇

—

C₂×Q₈×C₄₈₇

C₂²

yes

characteristic, complemented

C₂×D₄×C₄₈₇

—

C₂×D₄×C₄₈₇

—

C₂×D₄×C₄₈₇

C₂²

yes

—

C₂²×C₄×C₄₈₇

C₂²

yes

characteristic

SD₁₆×C₄₈₇

—

C₂×C₈×C₄₈₇

C₂²

yes

complemented

C₄⋊C₄×C₄₈₇

—

C₄⋊C₄×C₄₈₇

C₂²

yes

characteristic, complemented

C₂²⋊C₄×C₄₈₇

—

C₂³×C₄₈₇

—

C₂³×C₄₈₇

—

C₂³×C₄₈₇

—

C₂³×C₄₈₇

—

C₂³×C₄₈₇

D₄

yes

characteristic

Q₈×C₄₈₇

—

D₄×C₄₈₇

—

D₄×C₄₈₇

—

D₄×C₄₈₇

D₄

yes

—

C₂×C₄×C₄₈₇

—

C₂×C₄×C₄₈₇

—

C₂×C₄×C₄₈₇

—

C₂×C₄×C₄₈₇

D₄

yes

characteristic

C₂×C₄×C₄₈₇

C₂³

yes

characteristic

C₈×C₄₈₇

—

C₂²×C₄₈₇

—

C₂²×C₄₈₇

—

C₂²×C₄₈₇

—

C₂²×C₄₈₇

—

C₂²×C₄₈₇

—

C₂²×C₄₈₇

—

C₂²×C₄₈₇

SD₁₆

yes

complemented

C₂²×C₄₈₇

C₂×D₄

yes

characteristic

C₄×C₄₈₇

—

C₄×C₄₈₇

—

C₄×C₄₈₇

—

C₄×C₄₈₇

C₂×D₄

yes

—

C₂×C₄₈₇

—

C₂×C₄₈₇

—

C₂×C₄₈₇

#120

yes

characteristic

C₂×C₄₈₇

C₂×SD₁₆

yes

C₂×C₄₈₇

yes

C₄₈₇

yes

characteristic, direct product factor, Sylow

#450

C₄₈₇

yes

maximal, characteristic, direct product factor, Sylow

C₂²×D₄

C₂×C₄₈₇

yes

characteristic

C₂×SD₁₆

C₂×C₄₈₇

yes

complemented

C₂²⋊Q₈

C₂×C₄₈₇

yes

characteristic, complemented

#102

C₂×C₄₈₇

yes

complemented

C₂²⋊C₈

C₂×C₄₈₇

yes

characteristic, complemented

C₂⁴

—

C₂×Q₈

C₂²×C₄₈₇

yes

characteristic, complemented

C₂×D₄

—

C₂×D₄

—

C₂×D₄

C₂²×C₄₈₇

yes

—

C₂²×C₄

C₂²×C₄₈₇

yes

characteristic

SD₁₆

—

C₂×C₈

C₂²×C₄₈₇

yes

complemented

C₄⋊C₄

—

C₄⋊C₄

C₂²×C₄₈₇

yes

characteristic, complemented

C₂²⋊C₄

—

C₂³

—

C₂³

—

C₂³

—

C₂³

—

C₂³

D₄×C₄₈₇

yes

characteristic

Q₈

—

D₄

—

D₄

—

D₄

D₄×C₄₈₇

yes

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₄×C₄₈₇

yes

characteristic

C₂×C₄

C₂³×C₄₈₇

yes

characteristic

C₈

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

SD₁₆×C₄₈₇

yes

complemented

C₂²

C₂×D₄×C₄₈₇

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

C₂×D₄×C₄₈₇

yes

—

C₂

—

C₂

—

C₂

#120×C₄₈₇

yes

characteristic

C₂

C₂×SD₁₆×C₄₈₇

yes

characteristic

C₂

AGL(1,ℤ/8)×C₄₈₇

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₄₈₇	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂³×C₄₈₇
prime radical	C₂²×D₄×C₄₈₇	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄₈₇ ◁ C₂²×C₄×C₄₈₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₄₈₇

C₂×D₄

commutator, Frattini, cosocle

C₂×C₄

C₂³×C₄₈₇

prime radical

C₂²×D₄×C₄₈₇

C₂

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₄₈₇ ◁ C₂²×C₄×C₄₈₇ ◁ G

N	H ≅ G/N	type
2. C₂×SD₁₆×C₄₈₇	C₂	semidirect
3. C₂²⋊Q₈×C₄₈₇	C₂	semidirect
4. #102×C₄₈₇	C₂	semidirect
5. C₂²⋊C₈×C₄₈₇	C₂	semidirect
7. C₂×Q₈×C₄₈₇	C₂²	semidirect
13. C₂×C₈×C₄₈₇	C₂²	semidirect
15. C₄⋊C₄×C₄₈₇	C₂²	semidirect
38. C₂²×C₄₈₇	SD₁₆	semidirect
49. C₄₈₇	#450	direct (N × H)
50. #450	C₄₈₇	direct (N × H)
52. C₂×SD₁₆	C₂×C₄₈₇	semidirect
53. C₂²⋊Q₈	C₂×C₄₈₇	semidirect
54. #102	C₂×C₄₈₇	semidirect
55. C₂²⋊C₈	C₂×C₄₈₇	semidirect
57. C₂×Q₈	C₂²×C₄₈₇	semidirect
63. C₂×C₈	C₂²×C₄₈₇	semidirect
65. C₄⋊C₄	C₂²×C₄₈₇	semidirect
88. C₂²	SD₁₆×C₄₈₇	semidirect

H ≅ G/N

type

2. C₂×SD₁₆×C₄₈₇

C₂

semidirect

3. C₂²⋊Q₈×C₄₈₇

C₂

semidirect

4. #102×C₄₈₇

C₂

semidirect

5. C₂²⋊C₈×C₄₈₇

C₂

semidirect

7. C₂×Q₈×C₄₈₇

C₂²

semidirect

13. C₂×C₈×C₄₈₇

C₂²

semidirect

15. C₄⋊C₄×C₄₈₇

C₂²

semidirect

38. C₂²×C₄₈₇

SD₁₆

semidirect

49. C₄₈₇

#450

direct (N × H)

50. #450

C₄₈₇

direct (N × H)

52. C₂×SD₁₆

C₂×C₄₈₇

semidirect

53. C₂²⋊Q₈

C₂×C₄₈₇

semidirect

54. #102

C₂×C₄₈₇

semidirect

55. C₂²⋊C₈

C₂×C₄₈₇

semidirect

57. C₂×Q₈

C₂²×C₄₈₇

semidirect

63. C₂×C₈

C₂²×C₄₈₇

semidirect

65. C₄⋊C₄

C₂²×C₄₈₇

semidirect

88. C₂²

SD₁₆×C₄₈₇

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×D₄×C₄₈₇	2
C₂×SD₁₆×C₄₈₇	2
C₂²⋊Q₈×C₄₈₇	2
#102×C₄₈₇	2
C₂²⋊C₈×C₄₈₇	2
#450	487

subgroup

index

C₂²×D₄×C₄₈₇

C₂×SD₁₆×C₄₈₇

C₂²⋊Q₈×C₄₈₇

#102×C₄₈₇

C₂²⋊C₈×C₄₈₇

#450

487

Subgroups — #450×C₄₈₇ #44528632

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #450×C487 #44528632

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #450×C₄₈₇ #44528632