Group #15673550 — subgroups

Subgroups — #15673550

Order 2944.

256 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 117 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×C₄⋊(C₂₃⋊C₈)	C₂	1	1	yes	maximal, characteristic
2	C₂×Dic₂₃⋊C₈	C₂	1	4	yes	maximal
3	#13092263	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
6	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
7	C₂×C₄×Dic₂₃	C₂²	1	2	yes	—
8	C₂²×C₂₃⋊C₈	C₂²	1	2	yes	—
9	C₂₃×C₄⋊C₈	—	2	2	no	—
10	C₂₃⋊#95	C₄	1	2	yes	—
11	C₂₃⋊#95	C₄	1	2	yes	—
12	Dic₂₃⋊C₈	—	2	8	no	—
13	C₄⋊(C₂₃⋊C₈)	—	2	2	no	—
14	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
15	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
17	C₂²×Dic₂₃	C₂×C₄	1	4	yes	—
18	C₂×C₈×C₂₃	—	2	2	no	—
19	C₂×C₈×C₂₃	—	2	4	no	—
20	C₄²×C₂₃	D₄	1	2	yes	—
21	C₄²×C₂₃	Q₈	1	2	yes	—
22	C₄×Dic₂₃	D₄	1	4	yes	—
23	C₄×Dic₂₃	Q₈	1	4	yes	—
24	C₂×C₂₃⋊C₈	—	2	2	no	—
25	C₂×C₂₃⋊C₈	—	2	4	no	—
26	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
27	C₂×C₄×C₂₃	—	2	2	no	—
28	C₂×C₄×C₂₃	—	2	2	no	—
29	C₂×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
30	C₂×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
31	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
32	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
33	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
34	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
35	C₂×Dic₂₃	—	2	4	no	—
36	C₂×Dic₂₃	—	2	4	no	—
37	C₂×Dic₂₃	C₄⋊C₄	1	8	yes	—
38	C₈×C₂₃	—	4	2	no	—
39	C₂₃⋊C₈	—	4	2	no	—
40	#1843	—	23	1	no	maximal, Sylow
41	C₂²×C₂₃	C₂×C₄⋊C₄	1	2	yes	—
42	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
43	C₂²×C₂₃	#131	1	2	yes	—
44	C₂²×C₂₃	#131	1	2	yes	—
45	C₄×C₂₃	—	2	2	no	—
46	C₄×C₂₃	—	2	2	no	—
47	C₄×C₂₃	#127	1	1	yes	characteristic
48	C₄×C₂₃	C₄⋊Q₈	1	2	yes	—
49	C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
50	Dic₂₃	—	2	8	no	—
51	C₂×C₄⋊C₈	—	23	4	no	—
52	C₂×C₄⋊C₈	—	23	1	no	—
53	C₂×C₄⋊C₈	D₂₃	1	1	yes	characteristic
54	#376	—	23	1	no	—
55	C₂×C₂₃	#384	1	1	yes	characteristic
56	C₂×C₂₃	#424	1	4	yes	—
57	C₂×C₂₃	#424	1	1	yes	characteristic
58	C₂×C₂₃	#424	1	1	yes	characteristic
59	C₂²×C₈	—	23	2	no	—
60	C₂²×C₈	C₂×D₂₃	1	2	yes	—
61	C₂×C₄²	—	23	2	no	—
62	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
63	C₄⋊C₈	—	2	2	no	—
64	C₄⋊C₈	—	46	8	no	—
65	C₄⋊C₈	—	46	2	no	—
66	#95	—	23	2	no	—
67	#95	—	23	2	no	—
68	C₂₃	#1843	1	1	yes	characteristic, complemented, Sylow
69	C₂²×C₄	—	23	4	no	—
70	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
71	C₂²×C₄	C₄×D₂₃	1	1	yes	characteristic
72	C₂²×C₄	C₂²×D₂₃	1	1	yes	characteristic
73	C₂×C₈	—	2	4	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	46	4	no	—
76	C₂×C₈	—	46	2	no	—
77	C₄²	—	23	4	no	—
78	C₄²	—	23	4	no	—
79	C₄²	Dic₄₆	1	2	yes	—
80	C₄²	C₂²⋊D₂₃	1	2	yes	—
81	C₂³	C₂×C₄×D₂₃	1	1	yes	characteristic
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	23	8	no	—
85	C₂×C₄	—	46	4	no	—
86	C₂×C₄	—	46	4	no	—
87	C₂×C₄	Dic₂₃⋊C₄	1	2	yes	—
88	C₂×C₄	Dic₂₃⋊C₄	1	2	yes	—
89	C₂×C₄	C₂×Dic₄₆	1	1	yes	characteristic
90	C₂×C₄	D₄×D₂₃	1	2	yes	—
91	C₂×C₄	Q₈×D₂₃	1	2	yes	—
92	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
93	C₈	—	4	2	no	—
94	C₈	—	92	2	no	—
95	C₂²	C₄⋊C₄×D₂₃	1	2	yes	—
96	C₂²	#10625072	1	2	yes	—
97	C₂²	#10625078	1	2	yes	—
98	C₂²	C₂×Dic₂₃⋊C₄	1	1	yes	characteristic
99	C₄	—	2	2	no	—
100	C₄	—	2	2	no	—
101	C₄	—	46	8	no	—
102	C₄	#10625059	1	2	yes	—
103	C₄	#10625111	1	1	yes	characteristic
104	C₄	#10625115	1	1	yes	characteristic
105	C₂	#13092286	1	1	yes	characteristic
106	C₂	#13092442	1	4	yes	—
107	C₂	#13092443	1	1	yes	characteristic
108	C₂	#13092445	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₂₃⋊C₈)

C₂

yes

maximal, characteristic

C₂×Dic₂₃⋊C₈

C₂

yes

maximal

#13092263

C₂

yes

maximal, characteristic

C₂×C₂₃×C₄⋊C₈

C₂

yes

maximal, characteristic

C₂²×C₈×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂×C₄×Dic₂₃

C₂²

yes

—

C₂²×C₂₃⋊C₈

C₂²

yes

—

C₂₃×C₄⋊C₈

—

C₂₃⋊#95

C₄

yes

—

C₂₃⋊#95

C₄

yes

—

Dic₂₃⋊C₈

—

C₄⋊(C₂₃⋊C₈)

—

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×Dic₂₃

C₂×C₄

yes

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₄²×C₂₃

D₄

yes

—

C₄²×C₂₃

Q₈

yes

—

C₄×Dic₂₃

D₄

yes

—

C₄×Dic₂₃

Q₈

yes

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₄⋊C₄

yes

—

C₂×C₄×C₂₃

C₄⋊C₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

—

C₂×Dic₂₃

—

C₂×Dic₂₃

—

C₂×Dic₂₃

C₄⋊C₄

yes

—

C₈×C₂₃

—

C₂₃⋊C₈

—

#1843

—

maximal, Sylow

C₂²×C₂₃

C₂×C₄⋊C₄

yes

—

C₂²×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₃

#131

yes

—

C₂²×C₂₃

#131

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#127

yes

characteristic

C₄×C₂₃

C₄⋊Q₈

yes

—

C₄×C₂₃

C₄⋊Q₈

yes

characteristic

Dic₂₃

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

D₂₃

yes

characteristic

#376

—

C₂×C₂₃

#384

yes

characteristic

C₂×C₂₃

#424

yes

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂²×C₈

—

C₂²×C₈

C₂×D₂₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

#95

—

#95

—

C₂₃

#1843

yes

characteristic, complemented, Sylow

C₂²×C₄

—

C₂²×C₄

C₄×D₂₃

yes

characteristic

C₂²×C₄

C₄×D₂₃

yes

characteristic

C₂²×C₄

C₂²×D₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄²

—

C₄²

—

C₄²

Dic₄₆

yes

—

C₄²

C₂²⋊D₂₃

yes

—

C₂³

C₂×C₄×D₂₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

Dic₂₃⋊C₄

yes

—

C₂×C₄

Dic₂₃⋊C₄

yes

—

C₂×C₄

C₂×Dic₄₆

yes

characteristic

C₂×C₄

D₄×D₂₃

yes

—

C₂×C₄

Q₈×D₂₃

yes

—

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

C₈

—

C₈

—

C₂²

C₄⋊C₄×D₂₃

yes

—

C₂²

#10625072

yes

—

C₂²

#10625078

yes

—

C₂²

C₂×Dic₂₃⋊C₄

yes

characteristic

C₄

—

100

C₄

—

101

C₄

—

102

C₄

#10625059

yes

—

103

C₄

#10625111

yes

characteristic

104

C₄

#10625115

yes

characteristic

105

C₂

#13092286

yes

characteristic

106

C₂

#13092442

yes

—

107

C₂

#13092443

yes

characteristic

108

C₂

#13092445

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₂₃
commutator, socle, prime radical	C₂³×C₂₃	C₂²×C₄
Fitting	C₂×C₂₃×C₄⋊C₈	C₂
cosocle	C₂²×C₄×C₂₃	C₂³
hypercenter	C₂×C₄⋊C₈	D₂₃
hypocenter	C₂₃	#1843
LCS	G ▷ C₂³×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₂₃

commutator, socle, prime radical

C₂³×C₂₃

C₂²×C₄

Fitting

C₂×C₂₃×C₄⋊C₈

C₂

cosocle

C₂²×C₄×C₂₃

C₂³

hypercenter

C₂×C₄⋊C₈

D₂₃

C₂₃

G ▷ C₂³×C₂₃ ▷ C₂₃

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
68. C₂₃	#1843	semidirect

H ≅ G/N

type

68. C₂₃

#1843

semidirect

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

subgroup	index
C₂×C₄⋊(C₂₃⋊C₈)	2
C₂×Dic₂₃⋊C₈	2
#13092263	2
C₂×C₂₃×C₄⋊C₈	2
#1843	23

subgroup

index

C₂×C₄⋊(C₂₃⋊C₈)

C₂×Dic₂₃⋊C₈

#13092263

C₂×C₂₃×C₄⋊C₈

#1843

Subgroups — #15673550

Conjugacy classes of subgroups

Complements

Maximal subgroups