Group #15671416 — subgroups

Subgroups — #15671416

Order 2944.

104 conjugacy classes of subgroups; only the 35 normal subgroups are shown.

normal subgroups only

This group has 104 conjugacy classes of subgroups — too many to draw. Only its 35 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	#13092658	C₂	1	1	yes	maximal, characteristic
2	#13092182	C₂	1	1	yes	maximal, characteristic
3	C₂₃×C₂²⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
5	#10625095	—	2	1	no	—
6	C₂×#67267	—	2	1	no	—
7	C₂×Dic₉₂	C₄	1	1	yes	characteristic
8	C₂×C₁₆×C₂₃	—	2	1	no	—
9	#10625000	C₄	1	1	yes	characteristic
10	#10624998	—	2	1	no	—
11	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
12	C₂×Dic₄₆	—	2	1	no	—
13	C₂×C₈×C₂₃	—	4	1	no	—
14	C₂×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
15	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
16	C₂²⋊Dic₂₃	—	4	1	no	—
17	C₄⋊Dic₂₃	—	2	1	no	—
18	Dic₂₃⋊C₄	—	4	1	no	—
19	#67267	—	2	2	no	—
20	#67267	—	4	1	no	—
21	C₂×C₂₃⋊C₈	—	4	1	no	—
22	Dic₉₂	—	4	1	no	—
23	C₁₆×C₂₃	—	4	1	no	—
24	C₂³×C₂₃	SD₁₆	1	1	yes	characteristic
25	C₂×C₄×C₂₃	—	2	1	no	—
26	C₂×C₄×C₂₃	—	2	1	no	—
27	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
28	C₂×C₄×C₂₃	D₈	1	1	yes	characteristic
29	C₂×Dic₂₃	—	4	1	no	—
30	C₂×Dic₂₃	—	4	1	no	—
31	Dic₄₆	—	4	1	no	—
32	Dic₄₆	—	4	1	no	—
33	C₈×C₂₃	—	2	1	no	—
34	C₈×C₂₃	—	2	1	no	—
35	C₂₃⋊C₈	—	4	2	no	—
36	#1350	—	23	1	no	maximal, Sylow
37	C₂²×C₂₃	—	2	1	no	—
38	C₂²×C₂₃	—	2	1	no	—
39	C₂²×C₂₃	—	2	1	no	—
40	C₂²×C₂₃	#102	1	1	yes	characteristic
41	C₄×C₂₃	—	2	1	no	—
42	C₄×C₂₃	#99	1	1	yes	characteristic
43	C₄×C₂₃	#104	1	1	yes	characteristic
44	Dic₂₃	—	8	1	no	—
45	Dic₂₃	—	8	1	no	—
46	#467	—	23	1	no	—
47	C₂²⋊C₁₆	D₂₃	1	1	yes	characteristic
48	#341	—	23	1	no	—
49	C₂×C₂₃	—	4	1	no	—
50	C₂×C₂₃	#327	1	1	yes	characteristic
51	C₂×C₂₃	#359	1	1	yes	characteristic
52	C₂×C₂₃	#362	1	1	yes	characteristic
53	C₂×Q₁₆	—	23	1	no	—
54	C₂×M₁₆	—	46	1	no	—
55	C₂²×C₈	C₂×D₂₃	1	1	yes	characteristic
56	C₂²⋊Q₈	—	46	1	no	—
57	C₂×C₁₆	—	2	1	no	—
58	#107	—	23	1	no	—
59	#103	—	46	1	no	—
60	C₂₃	#1350	1	1	yes	characteristic, complemented, Sylow
61	C₂×Q₈	—	46	1	no	—
62	C₂²×C₄	D₉₂	1	1	yes	characteristic
63	Q₁₆	—	92	1	no	—
64	M₁₆	—	46	2	no	—
65	M₁₆	—	92	1	no	—
66	C₂×C₈	—	4	1	no	—
67	C₂×C₈	—	92	1	no	—
68	C₂×C₈	C₄×D₂₃	1	1	yes	characteristic
69	C₂×C₈	C₂²⋊D₂₃	1	1	yes	characteristic
70	C₄⋊C₄	—	46	1	no	—
71	C₄⋊C₄	—	92	1	no	—
72	C₂²⋊C₄	—	92	1	no	—
73	C₁₆	—	4	1	no	—
74	C₂³	#67263	1	1	yes	characteristic
75	Q₈	—	92	1	no	—
76	Q₈	—	92	1	no	—
77	C₂×C₄	—	2	1	no	—
78	C₂×C₄	—	2	1	no	—
79	C₂×C₄	—	92	1	no	—
80	C₂×C₄	—	92	1	no	—
81	C₂×C₄	D₁₈₄	1	1	yes	characteristic
82	C₂×C₄	(C₂×D₂₃)⋊C₄	1	1	yes	characteristic
83	C₈	—	2	1	no	—
84	C₈	—	2	1	no	—
85	C₈	—	92	2	no	—
86	C₂²	—	2	1	no	—
87	C₂²	—	2	1	no	—
88	C₂²	—	2	1	no	—
89	C₂²	#10625003	1	1	yes	characteristic
90	C₄	—	2	1	no	—
91	C₄	—	184	1	no	—
92	C₄	—	184	1	no	—
93	C₄	#10624988	1	1	yes	characteristic
94	C₄	#10625007	1	1	yes	characteristic
95	C₂	—	4	1	no	—
96	C₂	#13092152	1	1	yes	characteristic
97	C₂	#13092225	1	1	yes	characteristic
98	C₂	#13092234	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13092658

C₂

yes

maximal, characteristic

#13092182

C₂

yes

maximal, characteristic

C₂₃×C₂²⋊C₁₆

C₂

yes

maximal, characteristic

C₂²×C₈×C₂₃

C₂²

yes

characteristic

#10625095

—

C₂×#67267

—

C₂×Dic₉₂

C₄

yes

characteristic

C₂×C₁₆×C₂₃

—

#10625000

C₄

yes

characteristic

#10624998

—

C₂²×C₄×C₂₃

D₄

yes

characteristic

C₂×Dic₄₆

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₂×C₄

yes

characteristic

C₂×C₈×C₂₃

D₄

yes

characteristic

C₂²⋊Dic₂₃

—

C₄⋊Dic₂₃

—

Dic₂₃⋊C₄

—

#67267

—

#67267

—

C₂×C₂₃⋊C₈

—

Dic₉₂

—

C₁₆×C₂₃

—

C₂³×C₂₃

SD₁₆

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₃

D₈

yes

characteristic

C₂×Dic₂₃

—

C₂×Dic₂₃

—

Dic₄₆

—

Dic₄₆

—

C₈×C₂₃

—

C₈×C₂₃

—

C₂₃⋊C₈

—

#1350

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#102

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

yes

C₄×C₂₃

yes

Dic₂₃

—

Dic₂₃

—

#467

—

C₂²⋊C₁₆

D₂₃

yes

characteristic

#341

—

C₂×C₂₃

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×Q₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₂₃

yes

characteristic

C₂²⋊Q₈

—

C₂×C₁₆

—

#107

—

#103

—

C₂₃

#1350

yes

characteristic, complemented, Sylow

C₂×Q₈

—

C₂²×C₄

D₉₂

yes

characteristic

Q₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₂₃

yes

characteristic

C₂×C₈

C₂²⋊D₂₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₁₆

—

C₂³

#67263

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

D₁₈₄

yes

characteristic

C₂×C₄

(C₂×D₂₃)⋊C₄

yes

characteristic

C₈

—

C₈

—

C₈

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#10625003

yes

characteristic

C₄

—

C₄

—

184

—

C₄

—

184

—

C₄

yes

C₄

yes

C₂

—

C₂

yes

C₂

yes

C₂

yes

	subgroup	G/H
center	C₂²	#10625003
commutator	C₂×C₈×C₂₃	C₂×C₄
Frattini	C₂²×C₈	C₂×D₂₃
Fitting	C₂₃×C₂²⋊C₁₆	C₂
socle	C₂²×C₂₃	#102
prime radical	C₂³×C₂₃	SD₁₆
cosocle	C₂²×C₈×C₂₃	C₂²
hypercenter	C₂²⋊C₁₆	D₂₃
hypocenter	C₂₃	#1350
LCS	G ▷ C₂×C₈×C₂₃ ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²⋊C₁₆

subgroup

G/H

center

C₂²

#10625003

commutator

C₂×C₈×C₂₃

C₂×C₄

Frattini

C₂²×C₈

C₂×D₂₃

Fitting

C₂₃×C₂²⋊C₁₆

C₂

socle

C₂²×C₂₃

#102

prime radical

C₂³×C₂₃

SD₁₆

cosocle

C₂²×C₈×C₂₃

C₂²

hypercenter

C₂²⋊C₁₆

D₂₃

hypocenter

C₂₃

#1350

LCS

G ▷ C₂×C₈×C₂₃ ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²⋊C₁₆

N	H ≅ G/N	type
60. C₂₃	#1350	semidirect

H ≅ G/N

type

60. C₂₃

#1350

semidirect

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

subgroup	index
#13092658	2
#13092182	2
C₂₃×C₂²⋊C₁₆	2
#1350	23

subgroup

index

#13092658

#13092182

C₂₃×C₂²⋊C₁₆

#1350

Subgroups — #15671416

Conjugacy classes of subgroups

Complements

Maximal subgroups