Group #15683905 — subgroups

Subgroups — C₂×#13092822 #15683905

Order 2944.

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×#10625040	C₂	1	2	yes	maximal
2	#13092822	C₂	1	8	yes	maximal, direct product factor
3	C₂×#10625000	C₂	1	4	yes	maximal
4	C₂×C₄×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₄₆	—	2	2	no	—
6	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
7	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₂₃	—	2	2	no	—
9	C₂×C₄⋊Dic₂₃	C₂²	1	4	yes	—
10	#10625040	—	2	4	no	—
11	#10625040	C₂²	1	8	yes	—
12	C₄×C₈×C₂₃	C₂²	1	4	yes	—
13	#10625000	C₂²	1	16	yes	—
14	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
15	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
16	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
17	C₂²×Dic₂₃	—	2	4	no	—
18	C₂×Dic₄₆	—	2	8	no	—
19	C₂×Dic₄₆	—	4	8	no	—
20	C₂×C₈×C₂₃	Q₈	1	8	yes	complemented
21	C₂×C₈×C₂₃	C₂³	1	4	yes	—
22	C₄²×C₂₃	D₄	1	2	yes	—
23	C₄²×C₂₃	C₂³	1	2	yes	—
24	C₄⋊Dic₂₃	—	2	4	no	—
25	C₄⋊Dic₂₃	—	2	8	no	—
26	C₄⋊Dic₂₃	C₂³	1	8	yes	—
27	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
28	C₂×C₄×C₂₃	D₈	1	4	yes	—
29	C₂×C₄×C₂₃	Q₁₆	1	4	yes	—
30	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
32	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
33	C₂×C₄×C₂₃	C₂×Q₈	1	4	yes	—
34	C₂×C₄×C₂₃	C₂⁴	1	1	yes	characteristic
35	C₂×Dic₂₃	—	2	8	no	—
36	C₂×Dic₂₃	—	4	8	no	—
37	Dic₄₆	—	4	16	no	—
38	C₈×C₂₃	C₂×Q₈	1	8	yes	complemented
39	C₂×#500	—	23	1	no	maximal, Sylow
40	C₂²×C₂₃	C₄⋊Q₈	1	2	yes	—
41	C₂²×C₂₃	C₂×D₈	1	2	yes	—
42	C₂²×C₂₃	C₂×Q₁₆	1	2	yes	—
43	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
44	C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
45	C₄×C₂₃	C₂×D₈	1	4	yes	—
46	C₄×C₂₃	C₂×Q₁₆	1	4	yes	—
47	C₄×C₂₃	C₂²×Q₈	1	2	yes	—
48	Dic₂₃	—	4	8	no	—
49	C₂×C₄⋊Q₈	—	23	2	no	—
50	#500	—	23	8	no	—
51	C₂×#107	—	23	4	no	—
52	C₂×C₄×C₈	D₂₃	1	1	yes	characteristic
53	C₂×C₂₃	#500	1	4	yes	complemented
54	C₂×C₂₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
55	C₂×C₂₃	C₂²×D₈	1	1	yes	characteristic
56	C₂×C₂₃	C₂²×Q₁₆	1	1	yes	characteristic
57	C₂²×Q₈	—	46	2	no	—
58	C₂²×C₈	C₂×D₂₃	1	2	yes	—
59	C₄⋊Q₈	—	23	8	no	—
60	C₄⋊Q₈	—	46	4	no	—
61	C₂×C₄⋊C₄	—	23	4	no	—
62	C₂×C₄⋊C₄	—	46	2	no	—
63	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
64	#107	—	23	16	no	—
65	C₄×C₈	C₂×D₂₃	1	4	yes	—
66	C₂₃	C₂×#500	1	1	yes	characteristic, complemented, Sylow
67	C₂×Q₈	—	46	8	no	—
68	C₂×Q₈	—	92	8	no	—
69	C₂²×C₄	—	46	4	no	—
70	C₂²×C₄	D₉₂	1	1	yes	characteristic
71	C₂²×C₄	D₉₂	1	1	yes	characteristic
72	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
73	C₂×C₈	Dic₄₆	1	8	yes	complemented
74	C₂×C₈	C₂²×D₂₃	1	4	yes	—
75	C₄⋊C₄	—	23	8	no	—
76	C₄⋊C₄	—	46	4	no	—
77	C₄⋊C₄	—	46	8	no	—
78	C₄²	D₉₂	1	2	yes	—
79	C₄²	C₂²×D₂₃	1	2	yes	—
80	C₂³	C₂×D₉₂	1	1	yes	characteristic
81	Q₈	—	92	16	no	—
82	C₂×C₄	—	46	8	no	—
83	C₂×C₄	—	92	8	no	—
84	C₂×C₄	D₁₈₄	1	4	yes	—
85	C₂×C₄	Dic₉₂	1	4	yes	—
86	C₂×C₄	C₂×Dic₄₆	1	4	yes	—
87	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
88	C₂×C₄	C₂×D₉₂	1	2	yes	—
89	C₂×C₄	C₂×D₉₂	1	2	yes	—
90	C₂×C₄	C₂³×D₂₃	1	1	yes	characteristic
91	C₈	C₂×Dic₄₆	1	8	yes	complemented
92	C₂²	#10625040	1	2	yes	—
93	C₂²	C₂×D₁₈₄	1	2	yes	—
94	C₂²	C₂×Dic₉₂	1	2	yes	—
95	C₂²	C₂²×D₉₂	1	1	yes	characteristic
96	C₄	—	92	8	no	—
97	C₄	#10625040	1	2	yes	—
98	C₄	C₂×D₁₈₄	1	4	yes	—
99	C₄	C₂×Dic₉₂	1	4	yes	—
100	C₄	C₂²×Dic₄₆	1	2	yes	—
101	C₂	#13092822	1	4	yes	direct product factor
102	C₂	C₂×#10625040	1	1	yes	characteristic
103	C₂	C₂²×D₁₈₄	1	1	yes	characteristic
104	C₂	C₂²×Dic₉₂	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10625040

C₂

yes

maximal

#13092822

C₂

yes

maximal, direct product factor

C₂×#10625000

C₂

yes

maximal

C₂×C₄×C₈×C₂₃

C₂

yes

maximal, characteristic

C₂²×Dic₄₆

—

C₂²×C₈×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₃

—

C₂×C₄⋊Dic₂₃

C₂²

yes

—

#10625040

—

#10625040

C₂²

yes

—

C₄×C₈×C₂₃

C₂²

yes

—

#10625000

C₂²

yes

—

C₂²×C₄×C₂₃

D₄

yes

characteristic

C₂²×C₄×C₂₃

D₄

yes

characteristic

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

C₂×Dic₄₆

—

C₂×Dic₄₆

—

C₂×C₈×C₂₃

Q₈

yes

complemented

C₂×C₈×C₂₃

C₂³

yes

—

C₄²×C₂₃

D₄

yes

—

C₄²×C₂₃

C₂³

yes

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

C₂³

yes

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

D₈

yes

—

C₂×C₄×C₂₃

Q₁₆

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×Q₈

yes

—

C₂×C₄×C₂₃

C₂⁴

yes

characteristic

C₂×Dic₂₃

—

C₂×Dic₂₃

—

Dic₄₆

—

C₈×C₂₃

C₂×Q₈

yes

complemented

C₂×#500

—

maximal, Sylow

C₂²×C₂₃

C₄⋊Q₈

yes

—

C₂²×C₂₃

C₂×D₈

yes

—

C₂²×C₂₃

C₂×Q₁₆

yes

—

C₂²×C₂₃

C₂²×D₄

yes

characteristic

C₄×C₂₃

C₄⋊Q₈

yes

—

C₄×C₂₃

C₂×D₈

yes

—

C₄×C₂₃

C₂×Q₁₆

yes

—

C₄×C₂₃

C₂²×Q₈

yes

—

Dic₂₃

—

C₂×C₄⋊Q₈

—

#500

—

C₂×#107

—

C₂×C₄×C₈

D₂₃

yes

characteristic

C₂×C₂₃

#500

yes

complemented

C₂×C₂₃

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₂₃

C₂²×D₈

yes

characteristic

C₂×C₂₃

C₂²×Q₁₆

yes

characteristic

C₂²×Q₈

—

C₂²×C₈

C₂×D₂₃

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

#107

—

C₄×C₈

C₂×D₂₃

yes

—

C₂₃

C₂×#500

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

D₉₂

yes

characteristic

C₂²×C₄

D₉₂

yes

characteristic

C₂²×C₄

C₂²×D₂₃

yes

characteristic

C₂×C₈

Dic₄₆

yes

complemented

C₂×C₈

C₂²×D₂₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

D₉₂

yes

—

C₄²

C₂²×D₂₃

yes

—

C₂³

C₂×D₉₂

yes

characteristic

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₁₈₄

yes

—

C₂×C₄

Dic₉₂

yes

—

C₂×C₄

C₂×Dic₄₆

yes

—

C₂×C₄

C₂×D₉₂

yes

characteristic

C₂×C₄

C₂×D₉₂

yes

—

C₂×C₄

C₂×D₉₂

yes

—

C₂×C₄

C₂³×D₂₃

yes

characteristic

C₈

C₂×Dic₄₆

yes

complemented

C₂²

#10625040

yes

—

C₂²

C₂×D₁₈₄

yes

—

C₂²

C₂×Dic₉₂

yes

—

C₂²

C₂²×D₉₂

yes

characteristic

C₄

—

C₄

#10625040

yes

—

C₄

C₂×D₁₈₄

yes

—

C₄

C₂×Dic₉₂

yes

—

100

C₄

C₂²×Dic₄₆

yes

—

101

C₂

#13092822

yes

direct product factor

102

C₂

C₂×#10625040

yes

characteristic

103

C₂

C₂²×D₁₈₄

yes

characteristic

104

C₂

C₂²×Dic₉₂

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₉₂
commutator, cosocle	C₂×C₄×C₂₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₂₃
Fitting	C₂×C₄×C₈×C₂₃	C₂
socle, prime radical	C₂³×C₂₃	C₂×D₄
hypercenter	C₂×C₄×C₈	D₂₃
hypocenter	C₂₃	C₂×#500
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₉₂

commutator, cosocle

C₂×C₄×C₂₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₂₃

Fitting

C₂×C₄×C₈×C₂₃

C₂

socle, prime radical

C₂³×C₂₃

C₂×D₄

hypercenter

C₂×C₄×C₈

D₂₃

hypocenter

C₂₃

C₂×#500

LCS

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
2. #13092822	C₂	direct (N × H)
20. C₂×C₈×C₂₃	Q₈	semidirect
38. C₈×C₂₃	C₂×Q₈	semidirect
53. C₂×C₂₃	#500	semidirect
66. C₂₃	C₂×#500	semidirect
73. C₂×C₈	Dic₄₆	semidirect
91. C₈	C₂×Dic₄₆	semidirect
101. C₂	#13092822	direct (N × H)

H ≅ G/N

type

2. #13092822

C₂

direct (N × H)

20. C₂×C₈×C₂₃

Q₈

semidirect

38. C₈×C₂₃

C₂×Q₈

semidirect

53. C₂×C₂₃

#500

semidirect

66. C₂₃

C₂×#500

semidirect

73. C₂×C₈

Dic₄₆

semidirect

91. C₈

C₂×Dic₄₆

semidirect

101. C₂

#13092822

direct (N × H)

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

subgroup	index
C₂×#10625040	2
#13092822	2
C₂×#10625000	2
C₂×C₄×C₈×C₂₃	2
C₂×#500	23

subgroup

index

C₂×C₄×C₈×C₂₃

Subgroups — C₂×#13092822 #15683905

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#13092822 #15683905

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#13092822 #15683905