Group #15673561 — subgroups

Subgroups — #15673561

Order 2944.

248 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 248 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	C₂×#10625000	C₂	1	2	yes	maximal
2	C₂×#10624999	C₂	1	2	yes	maximal
3	#13092283	C₂	1	2	yes	maximal
4	C₂×C₄×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
6	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₂₃	C₂²	1	4	yes	—
9	C₄×C₈×C₂₃	C₄	1	4	yes	complemented
10	C₂₃⋊#95	—	2	4	no	—
11	#10625000	—	2	4	no	—
12	#10624999	—	2	4	no	—
13	C₂²×C₄×C₂₃	D₄	1	2	yes	—
14	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
15	C₂²×Dic₂₃	—	2	4	no	—
16	C₂×C₈×C₂₃	C₂×C₄	1	2	yes	—
17	C₂×C₈×C₂₃	C₂×C₄	1	2	yes	—
18	C₂×C₈×C₂₃	D₄	1	2	yes	—
19	C₂×C₈×C₂₃	D₄	1	2	yes	—
20	C₂×C₈×C₂₃	Q₈	1	2	yes	—
21	C₂×C₈×C₂₃	Q₈	1	2	yes	—
22	C₄²×C₂₃	C₂×C₄	1	2	yes	—
23	C₄²×C₂₃	Q₈	1	2	yes	—
24	C₄⋊Dic₂₃	—	2	4	no	—
25	C₄⋊Dic₂₃	—	2	4	no	—
26	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
27	C₂×C₄×C₂₃	—	2	4	no	—
28	C₂×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
29	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
31	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
33	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
34	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
35	C₂×Dic₂₃	—	4	4	no	—
36	C₂×Dic₂₃	—	4	8	no	—
37	C₈×C₂₃	C₄⋊C₄	1	4	yes	—
38	C₈×C₂₃	C₄⋊C₄	1	4	yes	—
39	#1846	—	23	1	no	maximal, Sylow
40	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
41	C₂²×C₂₃	#127	1	1	yes	characteristic
42	C₂²×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
43	C₂²×C₂₃	#135	1	4	yes	—
44	C₄×C₂₃	—	2	4	no	—
45	C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
48	C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
49	Dic₂₃	—	8	4	no	—
50	C₂×#107	—	23	2	no	—
51	C₂×#106	—	23	2	no	—
52	C₂×C₄×C₈	D₂₃	1	1	yes	characteristic
53	#382	—	23	2	no	—
54	C₂×C₂₃	#384	1	1	yes	characteristic
55	C₂×C₂₃	#427	1	2	yes	—
56	C₂×C₂₃	#495	1	2	yes	—
57	C₂×C₂₃	#499	1	2	yes	—
58	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
59	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
60	C₂×C₄⋊C₄	—	23	4	no	—
61	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
62	#107	—	46	4	no	—
63	#106	—	46	4	no	—
64	C₄×C₈	Dic₂₃	1	4	yes	complemented
65	#95	—	46	4	no	—
66	C₂₃	#1846	1	1	yes	characteristic, complemented, Sylow
67	C₂²×C₄	—	46	4	no	—
68	C₂²×C₄	D₉₂	1	2	yes	—
69	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
70	C₂×C₈	Dic₄₆	1	2	yes	—
71	C₂×C₈	Dic₄₆	1	2	yes	—
72	C₂×C₈	D₉₂	1	2	yes	—
73	C₂×C₈	D₉₂	1	2	yes	—
74	C₂×C₈	C₂×Dic₂₃	1	2	yes	—
75	C₂×C₈	C₂×Dic₂₃	1	2	yes	—
76	C₄⋊C₄	—	46	4	no	—
77	C₄⋊C₄	—	46	4	no	—
78	C₄²	Dic₄₆	1	2	yes	—
79	C₄²	C₂×Dic₂₃	1	2	yes	—
80	C₂³	C₂×D₉₂	1	1	yes	characteristic
81	C₂×C₄	—	2	4	no	—
82	C₂×C₄	—	92	4	no	—
83	C₂×C₄	—	92	8	no	—
84	C₂×C₄	C₄⋊Dic₂₃	1	4	yes	—
85	C₂×C₄	C₂×Dic₄₆	1	1	yes	characteristic
86	C₂×C₄	C₂×Dic₄₆	1	1	yes	characteristic
87	C₂×C₄	C₂×Dic₄₆	1	1	yes	characteristic
88	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
89	C₂×C₄	C₂×D₉₂	1	1	yes	characteristic
90	C₂×C₄	C₂²×Dic₂₃	1	1	yes	characteristic
91	C₈	C₄⋊Dic₂₃	1	4	yes	—
92	C₈	C₄⋊Dic₂₃	1	4	yes	—
93	C₂²	#10625040	1	1	yes	characteristic
94	C₂²	#10625045	1	1	yes	characteristic
95	C₂²	#10625075	1	4	yes	—
96	C₂²	C₂×C₄⋊Dic₂₃	1	1	yes	characteristic
97	C₄	—	2	4	no	—
98	C₄	—	184	4	no	—
99	C₄	#10625040	1	1	yes	characteristic
100	C₄	#10625040	1	1	yes	characteristic
101	C₄	C₂×C₄⋊Dic₂₃	1	1	yes	characteristic
102	C₄	C₂×C₄⋊Dic₂₃	1	1	yes	characteristic
103	C₂	#13092287	1	1	yes	characteristic
104	C₂	#13092455	1	2	yes	—
105	C₂	#13092801	1	2	yes	—
106	C₂	#13092819	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10625000

C₂

yes

maximal

C₂×#10624999

C₂

yes

maximal

#13092283

C₂

yes

maximal

C₂×C₄×C₈×C₂₃

C₂

yes

maximal, characteristic

C₂²×C₈×C₂₃

C₂²

yes

characteristic

C₂²×C₈×C₂₃

C₂²

yes

characteristic

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₃

C₂²

yes

—

C₄×C₈×C₂₃

C₄

yes

complemented

C₂₃⋊#95

—

#10625000

—

#10624999

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

C₂×C₈×C₂₃

C₂×C₄

yes

—

C₂×C₈×C₂₃

C₂×C₄

yes

—

C₂×C₈×C₂₃

D₄

yes

—

C₂×C₈×C₂₃

D₄

yes

—

C₂×C₈×C₂₃

Q₈

yes

—

C₂×C₈×C₂₃

Q₈

yes

—

C₄²×C₂₃

C₂×C₄

yes

—

C₄²×C₂₃

Q₈

yes

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₄⋊C₄

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

characteristic

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₈×C₂₃

C₄⋊C₄

yes

—

C₈×C₂₃

C₄⋊C₄

yes

—

#1846

—

maximal, Sylow

C₂²×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₃

#127

yes

characteristic

C₂²×C₂₃

C₄⋊Q₈

yes

characteristic

C₂²×C₂₃

#135

yes

—

C₄×C₂₃

—

C₄×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₂₃

C₄⋊Q₈

yes

characteristic

C₄×C₂₃

C₄⋊Q₈

yes

characteristic

Dic₂₃

—

C₂×#107

—

C₂×#106

—

C₂×C₄×C₈

D₂₃

yes

characteristic

#382

—

C₂×C₂₃

#384

yes

characteristic

C₂×C₂₃

#427

yes

—

C₂×C₂₃

#495

yes

—

C₂×C₂₃

#499

yes

—

C₂²×C₈

C₂×D₂₃

yes

characteristic

C₂²×C₈

C₂×D₂₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

#107

—

#106

—

C₄×C₈

Dic₂₃

yes

complemented

#95

—

C₂₃

#1846

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

D₉₂

yes

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

C₂×C₈

Dic₄₆

yes

—

C₂×C₈

Dic₄₆

yes

—

C₂×C₈

D₉₂

yes

—

C₂×C₈

D₉₂

yes

—

C₂×C₈

C₂×Dic₂₃

yes

—

C₂×C₈

C₂×Dic₂₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

Dic₄₆

yes

—

C₄²

C₂×Dic₂₃

yes

—

C₂³

C₂×D₉₂

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄⋊Dic₂₃

yes

—

C₂×C₄

C₂×Dic₄₆

yes

characteristic

C₂×C₄

C₂×Dic₄₆

yes

characteristic

C₂×C₄

C₂×Dic₄₆

yes

characteristic

C₂×C₄

C₂×D₉₂

yes

characteristic

C₂×C₄

C₂×D₉₂

yes

characteristic

C₂×C₄

C₂²×Dic₂₃

yes

characteristic

C₈

C₄⋊Dic₂₃

yes

—

C₈

C₄⋊Dic₂₃

yes

—

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

C₂×C₄⋊Dic₂₃

yes

characteristic

C₄

—

C₄

—

184

—

C₄

#10625040

yes

characteristic

100

C₄

#10625040

yes

characteristic

101

C₄

C₂×C₄⋊Dic₂₃

yes

characteristic

102

C₄

C₂×C₄⋊Dic₂₃

yes

characteristic

103

C₂

#13092287

yes

characteristic

104

C₂

#13092455

yes

—

105

C₂

#13092801

yes

—

106

C₂

#13092819

yes

—

	subgroup	G/H
center	C₂³	C₂×D₉₂
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂×C₄×C₈×C₂₃	C₂
socle, prime radical	C₂³×C₂₃	C₂×D₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₂×C₄×C₈	D₂₃
hypocenter	C₂₃	#1846
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₉₂

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂×C₄×C₈×C₂₃

C₂

socle, prime radical

C₂³×C₂₃

C₂×D₄

cosocle

C₂²×C₄×C₂₃

C₂³

hypercenter

C₂×C₄×C₈

D₂₃

hypocenter

C₂₃

#1846

LCS

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
9. C₄×C₈×C₂₃	C₄	semidirect
64. C₄×C₈	Dic₂₃	semidirect
66. C₂₃	#1846	semidirect

H ≅ G/N

type

9. C₄×C₈×C₂₃

C₄

semidirect

64. C₄×C₈

Dic₂₃

semidirect

66. C₂₃

#1846

semidirect

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

subgroup	index
C₂×#10625000	2
C₂×#10624999	2
#13092283	2
C₂×C₄×C₈×C₂₃	2
#1846	23

subgroup

index

C₂×C₄×C₈×C₂₃

Subgroups — #15673561

Conjugacy classes of subgroups

Complements

Maximal subgroups