Group #15672535 — subgroups

Subgroups — #15672535

Order 2944.

206 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 206 conjugacy classes of subgroups — too many to draw. Only its 57 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	#13093147	C₂	1	1	yes	maximal, characteristic
2	#13092709	C₂	1	2	yes	maximal
3	#13092166	C₂	1	1	yes	maximal, characteristic
4	#13092149	C₂	1	2	yes	maximal
5	C₂₃×#494	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₁₆×C₂₃	—	2	2	no	—
7	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
8	C₂₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	#10625115	—	2	2	no	—
10	#10625059	—	2	2	no	—
11	#10625058	—	2	2	no	—
12	C₄×Dic₄₆	C₂²	1	2	yes	—
13	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
14	#10624992	—	2	2	no	—
15	#10624989	—	2	2	no	—
16	C₄⋊(C₂₃⋊C₈)	C₂²	1	2	yes	—
17	C₂×Q₈×C₂₃	—	2	1	no	—
18	C₂×Q₈×C₂₃	D₄	1	2	yes	—
19	C₂×Dic₄₆	—	2	2	no	—
20	C₂×Dic₄₆	D₄	1	2	yes	—
21	Q₁₆×C₂₃	—	4	4	no	—
22	C₂×C₈×C₂₃	—	2	1	no	—
23	C₄⋊C₄×C₂₃	—	2	2	no	—
24	C₄⋊C₄×C₂₃	—	2	2	no	—
25	C₄²×C₂₃	C₂³	1	1	yes	characteristic
26	C₄⋊Dic₂₃	D₄	1	2	yes	—
27	Dic₂₃⋊C₄	—	2	4	no	—
28	Dic₂₃⋊C₄	—	2	2	no	—
29	C₄×Dic₂₃	—	2	2	no	—
30	C₄×Dic₂₃	—	2	2	no	—
31	C₂×C₂₃⋊C₈	—	2	2	no	—
32	Q₈×C₂₃	—	4	2	no	—
33	Q₈×C₂₃	—	4	2	no	—
34	C₂×C₄×C₂₃	—	2	2	no	—
35	C₂×C₄×C₂₃	—	4	1	no	—
36	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
37	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
38	C₂×Dic₂₃	—	2	2	no	—
39	C₂×Dic₂₃	—	2	2	no	—
40	Dic₄₆	—	2	2	no	—
41	Dic₄₆	—	8	2	no	—
42	Dic₄₆	Q₁₆	1	4	yes	complemented
43	C₈×C₂₃	—	2	2	no	—
44	C₂₃⋊C₈	—	4	2	no	—
45	#1633	—	23	1	no	maximal, Sylow
46	C₂²×C₂₃	#120	1	1	yes	characteristic
47	C₄×C₂₃	—	2	1	no	—
48	C₄×C₂₃	—	4	2	no	—
49	C₄×C₂₃	—	8	1	no	—
50	C₄×C₂₃	C₂×Q₁₆	1	2	yes	—
51	C₄×C₂₃	#137	1	2	yes	—
52	Dic₂₃	—	2	4	no	—
53	Dic₂₃	—	4	2	no	—
54	Q₈²	—	23	1	no	—
55	#494	D₂₃	1	1	yes	characteristic
56	#477	—	23	2	no	—
57	#333	—	23	1	no	—
58	Q₈⋊C₈	—	23	2	no	—
59	C₂×C₂₃	#451	1	2	yes	—
60	C₂×C₂₃	#453	1	1	yes	characteristic
61	C₂×Q₁₆	—	2	2	no	—
62	C₄⋊Q₈	—	46	2	no	—
63	C₄⋊Q₈	—	46	2	no	—
64	C₄⋊Q₈	C₂×D₂₃	1	1	yes	characteristic
65	C₄⋊Q₈	C₂×D₂₃	1	1	yes	characteristic
66	C₄×Q₈	—	23	2	no	—
67	C₄×Q₈	—	46	2	no	—
68	#107	—	46	2	no	—
69	C₄⋊C₈	—	23	2	no	—
70	#103	—	46	2	no	—
71	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
72	C₂₃	#1633	1	1	yes	characteristic, complemented, Sylow
73	C₂×Q₈	—	2	1	no	—
74	C₂×Q₈	—	23	2	no	—
75	C₂×Q₈	—	46	2	no	—
76	C₂×Q₈	C₂²⋊D₂₃	1	2	yes	—
77	Q₁₆	—	4	4	no	—
78	C₂×C₈	—	2	1	no	—
79	C₂×C₈	—	46	2	no	—
80	C₄⋊C₄	—	2	2	no	—
81	C₄⋊C₄	—	2	2	no	—
82	C₄⋊C₄	—	23	2	no	—
83	C₄⋊C₄	—	46	4	no	—
84	C₄⋊C₄	—	46	2	no	—
85	C₄²	—	46	2	no	—
86	C₄²	—	46	2	no	—
87	C₄²	C₂²×D₂₃	1	1	yes	characteristic
88	Q₈	—	4	2	no	—
89	Q₈	—	4	2	no	—
90	Q₈	—	23	4	no	—
91	Q₈	—	46	2	no	—
92	Q₈	—	184	2	no	—
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	4	1	no	—
95	C₂×C₄	—	46	2	no	—
96	C₂×C₄	—	46	2	no	—
97	C₂×C₄	D₄×D₂₃	1	2	yes	—
98	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
99	C₈	—	2	2	no	—
100	C₈	—	92	2	no	—
101	C₂²	#10625108	1	1	yes	characteristic
102	C₄	—	2	1	no	—
103	C₄	—	4	2	no	—
104	C₄	—	8	1	no	—
105	C₄	—	46	4	no	—
106	C₄	—	92	2	no	—
107	C₄	Q₁₆×D₂₃	1	2	yes	—
108	C₄	#10625089	1	2	yes	—
109	C₂	#13092571	1	2	yes	—
110	C₂	#13092584	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13093147

C₂

yes

maximal, characteristic

#13092709

C₂

yes

maximal

#13092166

C₂

yes

maximal, characteristic

#13092149

C₂

yes

maximal

C₂₃×#494

C₂

yes

maximal, characteristic

C₂×Q₁₆×C₂₃

—

C₂₃×C₄⋊Q₈

C₂²

yes

characteristic

C₂₃×C₄⋊Q₈

C₂²

yes

—

—

—

C₄×Dic₄₆

C₂²

yes

—

C₄×C₈×C₂₃

C₂²

yes

characteristic

#10624992

—

#10624989

—

C₄⋊(C₂₃⋊C₈)

C₂²

yes

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

D₄

yes

—

C₂×Dic₄₆

—

C₂×Dic₄₆

D₄

yes

—

Q₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄²×C₂₃

C₂³

yes

characteristic

C₄⋊Dic₂₃

D₄

yes

—

Dic₂₃⋊C₄

—

Dic₂₃⋊C₄

—

C₄×Dic₂₃

—

C₄×Dic₂₃

—

C₂×C₂₃⋊C₈

—

Q₈×C₂₃

—

Q₈×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×Dic₂₃

—

C₂×Dic₂₃

—

Dic₄₆

—

Dic₄₆

—

Dic₄₆

Q₁₆

yes

complemented

C₈×C₂₃

—

C₂₃⋊C₈

—

—

C₂²×C₂₃

yes

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₂×Q₁₆

yes

—

C₄×C₂₃

#137

yes

—

Dic₂₃

—

Dic₂₃

—

Q₈²

—

#494

D₂₃

yes

characteristic

#477

—

#333

—

Q₈⋊C₈

—

C₂×C₂₃

#451

yes

—

C₂×C₂₃

#453

yes

characteristic

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₂₃

yes

characteristic

C₄⋊Q₈

C₂×D₂₃

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

#107

—

C₄⋊C₈

—

#103

—

C₄×C₈

C₂×D₂₃

yes

characteristic

C₂₃

#1633

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₂₃

yes

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²×D₂₃

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

Q₈

—

Q₈

—

184

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₄×D₂₃

yes

—

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

C₈

—

100

C₈

—

101

C₂²

#10625108

yes

characteristic

102

C₄

—

103

C₄

—

104

C₄

—

105

C₄

—

106

C₄

—

107

C₄

Q₁₆×D₂₃

yes

—

108

C₄

#10625089

yes

—

109

C₂

#13092571

yes

—

110

C₂

#13092584

yes

characteristic

	subgroup	G/H
center	C₂²	#10625108
commutator, cosocle	C₄²×C₂₃	C₂³
Frattini	C₄²	C₂²×D₂₃
Fitting	C₂₃×#494	C₂
socle, prime radical	C₂²×C₂₃	#120
hypercenter	#494	D₂₃
hypocenter	C₂₃	#1633
LCS	G ▷ C₄²×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

#10625108

commutator, cosocle

C₄²×C₂₃

C₂³

Frattini

C₄²

C₂²×D₂₃

Fitting

C₂₃×#494

C₂

C₂²×C₂₃

D₂₃

C₂₃

G ▷ C₄²×C₂₃ ▷ C₂²×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
42. Dic₄₆	Q₁₆	semidirect
72. C₂₃	#1633	semidirect

H ≅ G/N

type

42. Dic₄₆

Q₁₆

semidirect

72. C₂₃

#1633

semidirect

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

subgroup	index
#13093147	2
#13092709	2
#13092166	2
#13092149	2
C₂₃×#494	2
#1633	23

subgroup

index

C₂₃×#494

Subgroups — #15672535

Conjugacy classes of subgroups

Complements

Maximal subgroups