Group #15675616 — subgroups

Subgroups — #15675616

Order 2944.

124 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 124 conjugacy classes of subgroups — too many to draw. Only its 45 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	#13092675	C₂	1	1	yes	maximal, characteristic
2	#13092241	C₂	1	2	yes	maximal, complemented
3	#13092232	C₂	1	2	yes	maximal, complemented
4	#13092199	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#430	C₂	1	1	yes	maximal, characteristic
6	C₂×M₁₆×C₂₃	—	2	1	no	—
7	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
8	#10625095	—	2	1	no	—
9	C₂×Dic₉₂	C₂²	1	1	yes	characteristic
10	C₂₃×#108	—	2	1	no	—
11	C₂₃×#108	C₂²	1	2	yes	—
12	#10624999	C₂²	1	1	yes	characteristic
13	#10624998	—	2	1	no	—
14	#10624994	C₂²	1	2	yes	—
15	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
16	C₂×Dic₄₆	—	2	1	no	—
17	M₁₆×C₂₃	—	2	2	no	—
18	M₁₆×C₂₃	—	2	2	no	—
19	M₁₆×C₂₃	—	4	1	no	—
20	C₂×C₈×C₂₃	—	4	1	no	—
21	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
22	C₂×C₈×C₂₃	C₂³	1	1	yes	characteristic
23	C₂²⋊Dic₂₃	—	4	1	no	—
24	C₄⋊Dic₂₃	—	2	1	no	—
25	Dic₂₃⋊C₄	—	4	1	no	—
26	Dic₉₂	—	2	2	no	—
27	C₂₃⋊C₁₆	—	2	2	no	—
28	C₂³×C₂₃	SD₁₆	1	1	yes	characteristic
29	C₂×C₄×C₂₃	—	2	1	no	—
30	C₂×C₄×C₂₃	—	2	1	no	—
31	C₂×C₄×C₂₃	SD₁₆	1	1	yes	characteristic
32	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
33	C₂×Dic₂₃	—	4	1	no	—
34	C₂×Dic₂₃	—	4	1	no	—
35	Dic₄₆	—	4	1	no	—
36	Dic₄₆	—	4	1	no	—
37	C₈×C₂₃	—	2	1	no	—
38	C₈×C₂₃	—	4	2	no	—
39	C₈×C₂₃	#41	1	2	yes	—
40	#2239	—	23	1	no	maximal, Sylow
41	C₂²×C₂₃	—	2	1	no	—
42	C₂²×C₂₃	—	4	1	no	—
43	C₂²×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
44	C₄×C₂₃	—	2	1	no	—
45	C₄×C₂₃	#123	1	1	yes	characteristic
46	C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
47	Dic₂₃	—	8	1	no	—
48	Dic₂₃	—	8	1	no	—
49	#470	—	23	1	no	—
50	#430	D₂₃	1	1	yes	characteristic
51	#365	—	23	2	no	—
52	#362	—	23	2	no	—
53	#349	—	23	1	no	—
54	C₂×C₂₃	—	2	1	no	—
55	C₂×C₂₃	—	4	1	no	—
56	C₂×C₂₃	#481	1	1	yes	characteristic
57	C₂×Q₁₆	—	23	1	no	—
58	C₂×M₁₆	—	2	1	no	—
59	C₂×M₁₆	C₂×D₂₃	1	1	yes	characteristic
60	C₂²⋊Q₈	—	46	1	no	—
61	M₃₂	—	23	2	no	—
62	#108	—	2	1	no	—
63	#108	C₂×D₂₃	1	2	yes	—
64	#106	—	23	1	no	—
65	#103	—	46	1	no	—
66	C₂₃	#2239	1	1	yes	characteristic, complemented, Sylow
67	C₂×Q₈	—	46	1	no	—
68	C₂²×C₄	C₂²⋊D₂₃	1	1	yes	characteristic
69	Q₁₆	—	46	2	no	—
70	M₁₆	—	2	2	no	—
71	M₁₆	—	2	2	no	—
72	M₁₆	—	4	1	no	—
73	C₂×C₈	—	4	1	no	—
74	C₂×C₈	C₂²⋊D₂₃	1	1	yes	characteristic
75	C₂×C₈	C₂²×D₂₃	1	1	yes	characteristic
76	C₄⋊C₄	—	46	1	no	—
77	C₄⋊C₄	—	92	1	no	—
78	C₂²⋊C₄	—	92	1	no	—
79	C₁₆	—	46	2	no	—
80	C₂³	#67274	1	1	yes	characteristic
81	Q₈	—	92	1	no	—
82	Q₈	—	92	1	no	—
83	C₂×C₄	—	2	1	no	—
84	C₂×C₄	—	2	1	no	—
85	C₂×C₄	—	92	1	no	—
86	C₂×C₄	—	92	1	no	—
87	C₂×C₄	#67274	1	1	yes	characteristic
88	C₂×C₄	C₂×C₂²⋊D₂₃	1	1	yes	characteristic
89	C₈	—	2	1	no	—
90	C₈	—	4	2	no	—
91	C₈	#67288	1	2	yes	—
92	C₂²	—	2	1	no	—
93	C₂²	—	4	1	no	—
94	C₂²	C₂×#67274	1	1	yes	characteristic
95	C₄	—	2	1	no	—
96	C₄	—	184	1	no	—
97	C₄	—	184	1	no	—
98	C₄	#10625100	1	1	yes	characteristic
99	C₄	#10625103	1	1	yes	characteristic
100	C₂	—	2	1	no	—
101	C₂	—	4	1	no	—
102	C₂	#13092735	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13092675

C₂

yes

maximal, characteristic

#13092241

C₂

yes

maximal, complemented

#13092232

C₂

yes

maximal, complemented

#13092199

C₂

yes

maximal, characteristic

C₂₃×#430

C₂

yes

maximal, characteristic

C₂×M₁₆×C₂₃

—

C₂×M₁₆×C₂₃

C₂²

yes

characteristic

#10625095

—

C₂×Dic₉₂

C₂²

yes

characteristic

C₂₃×#108

—

C₂₃×#108

C₂²

yes

—

#10624999

C₂²

yes

characteristic

#10624998

—

#10624994

C₂²

yes

—

C₂²×C₄×C₂₃

D₄

yes

characteristic

C₂×Dic₄₆

—

M₁₆×C₂₃

—

M₁₆×C₂₃

—

M₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

D₄

yes

characteristic

C₂×C₈×C₂₃

C₂³

yes

characteristic

C₂²⋊Dic₂₃

—

C₄⋊Dic₂₃

—

Dic₂₃⋊C₄

—

Dic₉₂

—

C₂₃⋊C₁₆

—

C₂³×C₂₃

SD₁₆

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

SD₁₆

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×Dic₂₃

—

C₂×Dic₂₃

—

Dic₄₆

—

Dic₄₆

—

C₈×C₂₃

—

C₈×C₂₃

—

C₈×C₂₃

#41

yes

—

#2239

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×SD₁₆

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

yes

C₄×C₂₃

yes

Dic₂₃

—

Dic₂₃

—

#470

—

#430

D₂₃

yes

—

—

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#481

yes

characteristic

C₂×Q₁₆

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₂₃

yes

characteristic

C₂²⋊Q₈

—

M₃₂

—

#108

—

#108

C₂×D₂₃

yes

—

#106

—

#103

—

C₂₃

#2239

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂²×C₄

C₂²⋊D₂₃

yes

characteristic

Q₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₂₃

yes

characteristic

C₂×C₈

C₂²×D₂₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₁₆

—

C₂³

#67274

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

#67274

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₂₃

yes

characteristic

C₈

—

C₈

—

C₈

#67288

yes

—

C₂²

—

C₂²

—

C₂²

C₂×#67274

yes

characteristic

C₄

—

C₄

—

184

—

C₄

—

184

—

C₄

yes

C₄

yes

100

C₂

—

101

C₂

—

102

C₂

#13092735

yes

characteristic

	subgroup	G/H
center	C₂	#13092735
commutator, cosocle	C₂×C₈×C₂₃	C₂³
Frattini	C₂×C₈	C₂²×D₂₃
Fitting	C₂₃×#430	C₂
socle	C₂×C₂₃	#481
prime radical	C₂³×C₂₃	SD₁₆
hypercenter	#430	D₂₃
hypocenter	C₂₃	#2239
LCS	G ▷ C₂×C₈×C₂₃ ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #430

subgroup

G/H

center

C₂

#13092735

commutator, cosocle

C₂×C₈×C₂₃

C₂³

Frattini

C₂×C₈

C₂²×D₂₃

Fitting

C₂₃×#430

C₂

socle

C₂×C₂₃

#481

prime radical

C₂³×C₂₃

SD₁₆

D₂₃

C₂₃

G ▷ C₂×C₈×C₂₃ ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×M₁₆ ◁ #430

N	H ≅ G/N	type
2. #13092241	C₂	semidirect
3. #13092232	C₂	semidirect
66. C₂₃	#2239	semidirect

H ≅ G/N

type

2. #13092241

C₂

semidirect

3. #13092232

C₂

semidirect

66. C₂₃

#2239

semidirect

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

subgroup	index
#13092675	2
#13092241	2
#13092232	2
#13092199	2
C₂₃×#430	2
#2239	23

subgroup

index

C₂₃×#430

Subgroups — #15675616

Conjugacy classes of subgroups

Complements

Maximal subgroups