Group #15673556 — subgroups

Subgroups — #15673556

Order 2944.

280 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 149 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×#10624989	C₂	1	4	yes	maximal
2	C₂×C₄×C₂₃⋊C₈	C₂	1	1	yes	maximal, characteristic
3	#13092287	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#384	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
6	C₂×C₄⋊C₄×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	2	yes	—
10	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
11	#10624989	—	2	8	no	—
12	C₄×C₂₃⋊C₈	C₄	1	4	yes	complemented
13	C₂²×C₄×C₂₃	—	2	2	no	—
14	C₂²×C₄×C₂₃	D₄	1	2	yes	—
15	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
16	C₂²×Dic₂₃	—	2	2	no	—
17	C₄⋊C₄×C₂₃	—	2	4	no	—
18	C₄⋊C₄×C₂₃	—	4	4	no	—
19	C₄²×C₂₃	C₂×C₄	1	2	yes	—
20	C₄²×C₂₃	Q₈	1	2	yes	—
21	C₄⋊Dic₂₃	—	2	4	no	—
22	C₄⋊Dic₂₃	—	4	4	no	—
23	C₂×C₂₃⋊C₈	C₂×C₄	1	4	yes	—
24	C₂×C₂₃⋊C₈	D₄	1	4	yes	—
25	C₂×C₂₃⋊C₈	Q₈	1	4	yes	—
26	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
27	C₂×C₄×C₂₃	—	4	2	no	—
28	C₂×C₄×C₂₃	—	4	4	no	—
29	C₂×C₄×C₂₃	C₄⋊C₄	1	4	yes	—
30	C₂×C₄×C₂₃	D₈	1	4	yes	—
31	C₂×C₄×C₂₃	Q₁₆	1	4	yes	—
32	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
34	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
35	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
36	C₂×Dic₂₃	—	4	2	no	—
37	C₂×Dic₂₃	—	4	4	no	—
38	C₂₃⋊C₈	C₄⋊C₄	1	8	yes	complemented
39	#1845	—	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	C₂×D₈	1	2	yes	—
44	C₂²×C₂₃	C₂×Q₁₆	1	2	yes	—
45	C₄×C₂₃	—	8	2	no	—
46	C₄×C₂₃	#107	1	8	yes	complemented
47	C₄×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
48	C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
49	Dic₂₃	—	8	2	no	—
50	C₂×#107	—	23	4	no	—
51	C₂×C₄×C₈	—	23	1	no	—
52	#384	—	23	1	no	—
53	#384	D₂₃	1	1	yes	characteristic
54	C₂×C₂₃	#384	1	1	yes	characteristic
55	C₂×C₂₃	C₂×#107	1	2	yes	—
56	C₂×C₂₃	#493	1	1	yes	characteristic
57	C₂×C₂₃	#494	1	1	yes	characteristic
58	C₂×C₂₃	#500	1	2	yes	—
59	C₂²×C₈	—	23	2	no	—
60	C₂×C₄⋊C₄	—	2	2	no	—
61	C₂×C₄⋊C₄	—	23	2	no	—
62	C₂×C₄⋊C₄	—	46	2	no	—
63	C₂×C₄⋊C₄	C₂×D₂₃	1	2	yes	—
64	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
65	#107	—	46	8	no	—
66	C₄×C₈	—	23	4	no	—
67	C₂₃	#1845	1	1	yes	characteristic, complemented, Sylow
68	C₂²×C₄	—	2	2	no	—
69	C₂²×C₄	—	46	2	no	—
70	C₂²×C₄	C₂²⋊D₂₃	1	2	yes	—
71	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
72	C₂×C₈	—	23	4	no	—
73	C₂×C₈	—	23	4	no	—
74	C₂×C₈	—	23	4	no	—
75	C₄⋊C₄	—	2	4	no	—
76	C₄⋊C₄	—	4	4	no	—
77	C₄⋊C₄	—	46	4	no	—
78	C₄⋊C₄	—	92	4	no	—
79	C₄²	Dic₄₆	1	2	yes	—
80	C₄²	C₄×D₂₃	1	2	yes	—
81	C₂³	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
82	C₂×C₄	—	4	4	no	—
83	C₂×C₄	—	4	2	no	—
84	C₂×C₄	—	92	4	no	—
85	C₂×C₄	—	92	2	no	—
86	C₂×C₄	Dic₂₃⋊C₄	1	4	yes	—
87	C₂×C₄	#67272	1	4	yes	—
88	C₂×C₄	#67275	1	4	yes	—
89	C₂×C₄	C₂×Dic₄₆	1	1	yes	characteristic
90	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
91	C₂×C₄	D₄×D₂₃	1	2	yes	—
92	C₂×C₄	Q₈×D₂₃	1	2	yes	—
93	C₈	—	23	8	no	—
94	C₂²	C₂×Dic₂₃⋊C₄	1	1	yes	characteristic
95	C₂²	C₂×#67272	1	2	yes	—
96	C₂²	#10625111	1	1	yes	characteristic
97	C₂²	C₂×#67275	1	2	yes	—
98	C₂²	#10625115	1	1	yes	characteristic
99	C₄	—	8	2	no	—
100	C₄	—	184	2	no	—
101	C₄	#10624989	1	8	yes	complemented
102	C₄	#10625059	1	2	yes	—
103	C₄	C₄⋊C₄×D₂₃	1	2	yes	—
104	C₂	#13092286	1	1	yes	characteristic
105	C₂	C₂×#10624989	1	2	yes	—
106	C₂	#13092793	1	1	yes	characteristic
107	C₂	#13092796	1	1	yes	characteristic
108	C₂	#13092820	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#10624989

C₂

yes

maximal

C₂×C₄×C₂₃⋊C₈

C₂

yes

maximal, characteristic

#13092287

C₂

yes

maximal, characteristic

C₂₃×#384

C₂

yes

maximal, characteristic

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×C₄⋊Dic₂₃

—

C₂×C₄⋊Dic₂₃

C₂²

yes

—

C₂²×C₂₃⋊C₈

C₂²

yes

—

#10624989

—

C₄×C₂₃⋊C₈

C₄

yes

complemented

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄²×C₂₃

C₂×C₄

yes

—

C₄²×C₂₃

Q₈

yes

—

C₄⋊Dic₂₃

—

C₄⋊Dic₂₃

—

C₂×C₂₃⋊C₈

C₂×C₄

yes

—

C₂×C₂₃⋊C₈

D₄

yes

—

C₂×C₂₃⋊C₈

Q₈

yes

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₄⋊C₄

yes

—

C₂×C₄×C₂₃

D₈

yes

—

C₂×C₄×C₂₃

Q₁₆

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂₃⋊C₈

C₄⋊C₄

yes

complemented

#1845

—

maximal, Sylow

C₂²×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₃

#127

yes

characteristic

C₂²×C₂₃

C₄⋊Q₈

yes

characteristic

C₂²×C₂₃

C₂×D₈

yes

—

C₂²×C₂₃

C₂×Q₁₆

yes

—

C₄×C₂₃

—

C₄×C₂₃

#107

yes

complemented

C₄×C₂₃

C₂×C₄⋊C₄

yes

—

C₄×C₂₃

C₄⋊Q₈

yes

—

Dic₂₃

—

C₂×#107

—

C₂×C₄×C₈

—

#384

—

#384

D₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₂₃

yes

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

#107

—

C₄×C₈

—

C₂₃

#1845

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊D₂₃

yes

—

C₂²×C₄

C₂²×D₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

Dic₄₆

yes

—

C₄²

C₄×D₂₃

yes

—

C₂³

C₂×C₂²⋊D₂₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

Dic₂₃⋊C₄

yes

—

C₂×C₄

#67272

yes

—

C₂×C₄

#67275

yes

—

C₂×C₄

C₂×Dic₄₆

yes

characteristic

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

C₂×C₄

D₄×D₂₃

yes

—

C₂×C₄

Q₈×D₂₃

yes

—

C₈

—

C₂²

C₂×Dic₂₃⋊C₄

yes

characteristic

C₂²

C₂×#67272

yes

—

C₂²

#10625111

yes

characteristic

C₂²

C₂×#67275

yes

—

C₂²

#10625115

yes

characteristic

C₄

—

100

C₄

—

184

—

101

C₄

#10624989

yes

complemented

102

C₄

#10625059

yes

—

103

C₄

C₄⋊C₄×D₂₃

yes

—

104

C₂

#13092286

yes

characteristic

105

C₂

C₂×#10624989

yes

—

106

C₂

#13092793

yes

characteristic

107

C₂

#13092796

yes

characteristic

108

C₂

#13092820

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₂₃
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₂₃
Fitting	C₂₃×#384	C₂
socle, prime radical	C₂³×C₂₃	C₂×D₄
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	#384	D₂₃
hypocenter	C₂₃	#1845
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₂₃

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₂₃

Fitting

C₂₃×#384

C₂

socle, prime radical

C₂³×C₂₃

C₂×D₄

cosocle

C₂²×C₄×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #384

N	H ≅ G/N	type
12. C₄×C₂₃⋊C₈	C₄	semidirect
38. C₂₃⋊C₈	C₄⋊C₄	semidirect
46. C₄×C₂₃	#107	semidirect
67. C₂₃	#1845	semidirect
101. C₄	#10624989	semidirect

H ≅ G/N

type

12. C₄×C₂₃⋊C₈

C₄

semidirect

38. C₂₃⋊C₈

C₄⋊C₄

semidirect

46. C₄×C₂₃

#107

semidirect

67. C₂₃

#1845

semidirect

101. C₄

#10624989

semidirect

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

subgroup	index
C₂×#10624989	2
C₂×C₄×C₂₃⋊C₈	2
#13092287	2
C₂₃×#384	2
#1845	23

subgroup

index

C₂×#10624989

C₂×C₄×C₂₃⋊C₈

#13092287

C₂₃×#384

#1845

Subgroups — #15673556

Conjugacy classes of subgroups

Complements

Maximal subgroups