Group #20253896 — subgroups

Subgroups — C₂×#15689528 #20253896

Order 6016.

424 conjugacy classes of subgroups; only the 213 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 213 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	C₂×#13094024	C₂	1	2	yes	maximal
2	#15689528	C₂	1	8	yes	maximal, direct product factor
3	C₂×#13093958	C₂	1	4	yes	maximal
4	C₂×C₄×C₈×C₄₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₉₄	—	2	2	no	—
6	#13094024	—	2	4	no	—
7	#13094024	C₂²	1	8	yes	—
8	C₂×C₄⋊Dic₄₇	—	2	2	no	—
9	C₂×C₄⋊Dic₄₇	C₂²	1	4	yes	—
10	#13093958	C₂²	1	16	yes	—
11	C₂²×C₈×C₄₇	C₂²	1	2	yes	—
12	C₂×C₄²×C₄₇	C₂²	1	1	yes	characteristic
13	C₄×C₈×C₄₇	C₂²	1	4	yes	—
14	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
15	C₂²×C₄×C₄₇	D₄	1	1	yes	characteristic
16	C₂²×C₄×C₄₇	C₂³	1	1	yes	characteristic
17	C₂²×Dic₄₇	—	2	4	no	—
18	C₂×Dic₉₄	—	2	8	no	—
19	C₂×Dic₉₄	—	4	8	no	—
20	C₂×C₈×C₄₇	Q₈	1	8	yes	complemented
21	C₂×C₈×C₄₇	C₂³	1	4	yes	—
22	C₄²×C₄₇	D₄	1	2	yes	—
23	C₄²×C₄₇	C₂³	1	2	yes	—
24	C₄⋊Dic₄₇	—	2	8	no	—
25	C₄⋊Dic₄₇	—	2	4	no	—
26	C₄⋊Dic₄₇	C₂³	1	8	yes	—
27	C₂³×C₄₇	C₂×D₄	1	1	yes	characteristic
28	C₂×C₄×C₄₇	D₈	1	4	yes	—
29	C₂×C₄×C₄₇	Q₁₆	1	4	yes	—
30	C₂×C₄×C₄₇	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
32	C₂×C₄×C₄₇	C₂×D₄	1	2	yes	—
33	C₂×C₄×C₄₇	C₂×Q₈	1	4	yes	—
34	C₂×C₄×C₄₇	C₂⁴	1	1	yes	characteristic
35	C₂×Dic₄₇	—	2	8	no	—
36	C₂×Dic₄₇	—	4	8	no	—
37	Dic₉₄	—	4	16	no	—
38	C₈×C₄₇	C₂×Q₈	1	8	yes	complemented
39	C₂²×C₄₇	C₄⋊Q₈	1	2	yes	—
40	C₂²×C₄₇	C₂×D₈	1	2	yes	—
41	C₂²×C₄₇	C₂×Q₁₆	1	2	yes	—
42	C₂²×C₄₇	C₂²×D₄	1	1	yes	characteristic
43	C₄×C₄₇	C₄⋊Q₈	1	2	yes	—
44	C₄×C₄₇	C₂×D₈	1	4	yes	—
45	C₄×C₄₇	C₂×Q₁₆	1	4	yes	—
46	C₄×C₄₇	C₂²×Q₈	1	2	yes	—
47	Dic₄₇	—	4	8	no	—
48	C₂×#500	—	47	1	no	maximal, Sylow
49	C₂×C₄₇	#500	1	4	yes	complemented
50	C₂×C₄₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
51	C₂×C₄₇	C₂²×D₈	1	1	yes	characteristic
52	C₂×C₄₇	C₂²×Q₁₆	1	1	yes	characteristic
53	C₂×C₄⋊Q₈	—	47	2	no	—
54	#500	—	47	8	no	—
55	C₂×#107	—	47	4	no	—
56	C₂×C₄×C₈	D₄₇	1	1	yes	characteristic
57	C₄₇	C₂×#500	1	1	yes	characteristic, complemented, Sylow
58	C₂²×Q₈	—	94	2	no	—
59	C₂²×C₈	C₂×D₄₇	1	2	yes	—
60	C₄⋊Q₈	—	47	8	no	—
61	C₄⋊Q₈	—	94	4	no	—
62	C₂×C₄⋊C₄	—	47	4	no	—
63	C₂×C₄⋊C₄	—	94	2	no	—
64	C₂×C₄²	C₂×D₄₇	1	1	yes	characteristic
65	#107	—	47	16	no	—
66	C₄×C₈	C₂×D₄₇	1	4	yes	—
67	C₂×Q₈	—	94	8	no	—
68	C₂×Q₈	—	188	8	no	—
69	C₂²×C₄	—	94	4	no	—
70	C₂²×C₄	D₁₈₈	1	1	yes	characteristic
71	C₂²×C₄	D₁₈₈	1	1	yes	characteristic
72	C₂²×C₄	C₂²×D₄₇	1	1	yes	characteristic
73	C₂×C₈	Dic₉₄	1	8	yes	complemented
74	C₂×C₈	C₂²×D₄₇	1	4	yes	—
75	C₄⋊C₄	—	47	8	no	—
76	C₄⋊C₄	—	94	4	no	—
77	C₄⋊C₄	—	94	8	no	—
78	C₄²	D₁₈₈	1	2	yes	—
79	C₄²	C₂²×D₄₇	1	2	yes	—
80	C₂³	C₂×D₁₈₈	1	1	yes	characteristic
81	Q₈	—	188	16	no	—
82	C₂×C₄	—	94	8	no	—
83	C₂×C₄	—	188	8	no	—
84	C₂×C₄	D₃₇₆	1	4	yes	—
85	C₂×C₄	Dic₁₈₈	1	4	yes	—
86	C₂×C₄	C₂×Dic₉₄	1	4	yes	—
87	C₂×C₄	C₂×D₁₈₈	1	1	yes	characteristic
88	C₂×C₄	C₂×D₁₈₈	1	2	yes	—
89	C₂×C₄	C₂×D₁₈₈	1	2	yes	—
90	C₂×C₄	C₂³×D₄₇	1	1	yes	characteristic
91	C₈	C₂×Dic₉₄	1	8	yes	complemented
92	C₂²	#13094024	1	2	yes	—
93	C₂²	C₂×D₃₇₆	1	2	yes	—
94	C₂²	C₂×Dic₁₈₈	1	2	yes	—
95	C₂²	C₂²×D₁₈₈	1	1	yes	characteristic
96	C₄	—	188	8	no	—
97	C₄	#13094024	1	2	yes	—
98	C₄	C₂×D₃₇₆	1	4	yes	—
99	C₄	C₂×Dic₁₈₈	1	4	yes	—
100	C₄	C₂²×Dic₉₄	1	2	yes	—
101	C₂	#15689528	1	4	yes	direct product factor
102	C₂	C₂×#13094024	1	1	yes	characteristic
103	C₂	C₂²×D₃₇₆	1	1	yes	characteristic
104	C₂	C₂²×Dic₁₈₈	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#13094024

C₂

yes

maximal

#15689528

C₂

yes

maximal, direct product factor

C₂×#13093958

C₂

yes

maximal

C₂×C₄×C₈×C₄₇

C₂

yes

maximal, characteristic

C₂²×Dic₉₄

—

#13094024

—

#13094024

C₂²

yes

—

C₂×C₄⋊Dic₄₇

—

C₂×C₄⋊Dic₄₇

C₂²

yes

—

#13093958

C₂²

yes

—

C₂²×C₈×C₄₇

C₂²

yes

—

C₂×C₄²×C₄₇

C₂²

yes

characteristic

C₄×C₈×C₄₇

C₂²

yes

—

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

D₄

yes

characteristic

C₂²×C₄×C₄₇

C₂³

yes

characteristic

C₂²×Dic₄₇

—

C₂×Dic₉₄

—

C₂×Dic₉₄

—

C₂×C₈×C₄₇

Q₈

yes

complemented

C₂×C₈×C₄₇

C₂³

yes

—

C₄²×C₄₇

D₄

yes

—

C₄²×C₄₇

C₂³

yes

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

—

C₄⋊Dic₄₇

C₂³

yes

—

C₂³×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

D₈

yes

—

C₂×C₄×C₄₇

Q₁₆

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

characteristic

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×D₄

yes

—

C₂×C₄×C₄₇

C₂×Q₈

yes

—

C₂×C₄×C₄₇

C₂⁴

yes

characteristic

C₂×Dic₄₇

—

C₂×Dic₄₇

—

Dic₉₄

—

C₈×C₄₇

C₂×Q₈

yes

complemented

C₂²×C₄₇

C₄⋊Q₈

yes

—

C₂²×C₄₇

C₂×D₈

yes

—

C₂²×C₄₇

C₂×Q₁₆

yes

—

C₂²×C₄₇

C₂²×D₄

yes

characteristic

C₄×C₄₇

C₄⋊Q₈

yes

—

C₄×C₄₇

C₂×D₈

yes

—

C₄×C₄₇

C₂×Q₁₆

yes

—

C₄×C₄₇

C₂²×Q₈

yes

—

Dic₄₇

—

—

C₂×C₄₇

yes

C₂×C₄₇

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₄₇

C₂²×D₈

yes

characteristic

C₂×C₄₇

C₂²×Q₁₆

yes

characteristic

C₂×C₄⋊Q₈

—

#500

—

C₂×#107

—

C₂×C₄×C₈

D₄₇

yes

characteristic

C₄₇

C₂×#500

yes

characteristic, complemented, Sylow

C₂²×Q₈

—

C₂²×C₈

C₂×D₄₇

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₄₇

yes

characteristic

#107

—

C₄×C₈

C₂×D₄₇

yes

—

C₂×Q₈

—

C₂×Q₈

—

188

—

C₂²×C₄

—

C₂²×C₄

D₁₈₈

yes

characteristic

C₂²×C₄

D₁₈₈

yes

characteristic

C₂²×C₄

C₂²×D₄₇

yes

characteristic

C₂×C₈

Dic₉₄

yes

complemented

C₂×C₈

C₂²×D₄₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

D₁₈₈

yes

—

C₄²

C₂²×D₄₇

yes

—

C₂³

C₂×D₁₈₈

yes

characteristic

Q₈

—

188

—

C₂×C₄

—

C₂×C₄

—

188

—

C₂×C₄

D₃₇₆

yes

—

C₂×C₄

Dic₁₈₈

yes

—

C₂×C₄

C₂×Dic₉₄

yes

—

C₂×C₄

C₂×D₁₈₈

yes

characteristic

C₂×C₄

C₂×D₁₈₈

yes

—

C₂×C₄

C₂×D₁₈₈

yes

—

C₂×C₄

C₂³×D₄₇

yes

characteristic

C₈

C₂×Dic₉₄

yes

complemented

C₂²

#13094024

yes

—

C₂²

C₂×D₃₇₆

yes

—

C₂²

C₂×Dic₁₈₈

yes

—

C₂²

C₂²×D₁₈₈

yes

characteristic

C₄

—

188

—

C₄

#13094024

yes

—

C₄

C₂×D₃₇₆

yes

—

C₄

C₂×Dic₁₈₈

yes

—

100

C₄

C₂²×Dic₉₄

yes

—

101

C₂

#15689528

yes

direct product factor

102

C₂

C₂×#13094024

yes

characteristic

103

C₂

C₂²×D₃₇₆

yes

characteristic

104

C₂

C₂²×Dic₁₈₈

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₁₈₈
commutator	C₂×C₄×C₄₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₄₇
Fitting	C₂×C₄×C₈×C₄₇	C₂
socle	C₂³×C₄₇	C₂×D₄
hypercenter	C₂×C₄×C₈	D₄₇
hypocenter	C₄₇	C₂×#500
LCS	G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₁₈₈

commutator

C₂×C₄×C₄₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₄₇

Fitting

C₂×C₄×C₈×C₄₇

C₂

socle

C₂³×C₄₇

C₂×D₄

hypercenter

C₂×C₄×C₈

D₄₇

hypocenter

C₄₇

C₂×#500

LCS

G ▷ C₂×C₄×C₄₇ ▷ C₂×C₄₇ ▷ C₄₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #15689528	C₂	direct (N × H)
20. C₂×C₈×C₄₇	Q₈	semidirect
38. C₈×C₄₇	C₂×Q₈	semidirect
49. C₂×C₄₇	#500	semidirect
57. C₄₇	C₂×#500	semidirect
73. C₂×C₈	Dic₉₄	semidirect
91. C₈	C₂×Dic₉₄	semidirect
101. C₂	#15689528	direct (N × H)

H ≅ G/N

type

2. #15689528

C₂

direct (N × H)

20. C₂×C₈×C₄₇

Q₈

semidirect

38. C₈×C₄₇

C₂×Q₈

semidirect

49. C₂×C₄₇

#500

semidirect

57. C₄₇

C₂×#500

semidirect

73. C₂×C₈

Dic₉₄

semidirect

91. C₈

C₂×Dic₉₄

semidirect

101. C₂

#15689528

direct (N × H)

Subgroups — C₂×#15689528 #20253896

Conjugacy classes of subgroups

Complements

Subgroups — C2×#15689528 #20253896

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#15689528 #20253896