Group #15671274 — subgroups

Subgroups — #15671274

Order 2944.

184 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 61 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	#13092381	C₂	1	2	yes	maximal
2	C₂×C₂₃×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
4	C₂²×C₈×C₂₃	C₄	1	2	yes	—
5	C₂×#67267	—	2	2	no	—
6	C₂₃×C₂²⋊C₈	—	2	2	no	—
7	C₂²⋊(C₂₃⋊C₈)	—	2	2	no	—
8	C₂²⋊(C₂₃⋊C₈)	C₄	1	4	yes	—
9	C₂⁴×C₂₃	Q₈	1	1	yes	characteristic
10	C₂²×C₄×C₂₃	—	4	2	no	—
11	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
12	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
13	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
14	C₂²×C₄×C₂₃	D₄	1	2	yes	—
15	C₂×C₈×C₂₃	—	2	2	no	—
16	C₂×C₈×C₂₃	—	4	2	no	—
17	#67267	—	4	4	no	—
18	C₂×C₂₃⋊C₈	—	2	4	no	—
19	C₂³×C₂₃	—	2	2	no	—
20	C₂³×C₂₃	—	4	1	no	—
21	C₂³×C₂₃	—	4	1	no	—
22	C₂³×C₂₃	C₄⋊C₄	1	2	yes	—
23	C₂³×C₂₃	C₄⋊C₄	1	1	yes	characteristic
24	C₂×C₄×C₂₃	—	2	4	no	—
25	C₂×C₄×C₂₃	—	2	2	no	—
26	C₂×C₄×C₂₃	—	4	2	no	—
27	C₂×C₄×C₂₃	—	4	1	no	—
28	C₂×C₄×C₂₃	C₄²	1	1	yes	characteristic
29	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
30	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
31	C₈×C₂₃	—	4	2	no	—
32	C₂₃⋊C₈	—	4	4	no	—
33	#1293	—	23	1	no	maximal, Sylow
34	C₂²×C₂₃	—	2	1	no	—
35	C₂²×C₂₃	—	2	1	no	—
36	C₂²×C₂₃	—	2	2	no	—
37	C₂²×C₂₃	—	2	2	no	—
38	C₂²×C₂₃	—	2	2	no	—
39	C₂²×C₂₃	—	4	4	no	—
40	C₂²×C₂₃	#95	1	1	yes	characteristic
41	C₂²×C₂₃	#108	1	2	yes	—
42	C₄×C₂₃	—	2	2	no	—
43	C₄×C₂₃	—	2	1	no	—
44	C₄×C₂₃	#99	1	2	yes	—
45	#407	—	23	2	no	—
46	C₂×C₂²⋊C₈	D₂₃	1	1	yes	characteristic
47	C₂×C₂₃	—	2	2	no	—
48	C₂×C₂₃	—	4	2	no	—
49	C₂×C₂₃	#338	1	1	yes	characteristic
50	C₂×C₂₃	#341	1	1	yes	characteristic
51	C₂×C₂₃	#342	1	1	yes	characteristic
52	C₂³×C₄	C₂×D₂₃	1	1	yes	characteristic
53	C₂×M₁₆	—	46	2	no	—
54	C₂²×C₈	Dic₂₃	1	2	yes	—
55	C₂²⋊C₈	—	2	2	no	—
56	C₂²⋊C₈	—	23	4	no	—
57	C₂²⋊C₈	—	46	2	no	—
58	C₂₃	#1293	1	1	yes	characteristic, complemented, Sylow
59	C₂⁴	Dic₄₆	1	1	yes	characteristic
60	C₂²×C₄	—	4	2	no	—
61	C₂²×C₄	C₄×D₂₃	1	2	yes	—
62	C₂²×C₄	D₉₂	1	1	yes	characteristic
63	C₂²×C₄	C₂×Dic₂₃	1	1	yes	characteristic
64	C₂²×C₄	C₂²⋊D₂₃	1	2	yes	—
65	M₁₆	—	92	4	no	—
66	C₂×C₈	—	2	2	no	—
67	C₂×C₈	—	4	2	no	—
68	C₂×C₈	—	46	4	no	—
69	C₂³	—	2	2	no	—
70	C₂³	—	4	1	no	—
71	C₂³	—	4	1	no	—
72	C₂³	Dic₂₃⋊C₄	1	2	yes	—
73	C₂³	C₄⋊Dic₂₃	1	1	yes	characteristic
74	C₂×C₄	—	2	2	no	—
75	C₂×C₄	—	2	4	no	—
76	C₂×C₄	—	4	2	no	—
77	C₂×C₄	—	4	1	no	—
78	C₂×C₄	C₄×Dic₂₃	1	1	yes	characteristic
79	C₂×C₄	(C₂×D₂₃)⋊C₄	1	2	yes	—
80	C₂×C₄	C₂²⋊Dic₂₃	1	1	yes	characteristic
81	C₈	—	4	2	no	—
82	C₈	—	92	4	no	—
83	C₂²	—	2	1	no	—
84	C₂²	—	2	1	no	—
85	C₂²	—	2	2	no	—
86	C₂²	—	2	2	no	—
87	C₂²	—	2	2	no	—
88	C₂²	—	4	4	no	—
89	C₂²	#10625001	1	2	yes	—
90	C₂²	C₂₃⋊#95	1	1	yes	characteristic
91	C₄	—	2	2	no	—
92	C₄	—	2	1	no	—
93	C₄	#10625016	1	2	yes	—
94	C₂	—	2	2	no	—
95	C₂	—	4	2	no	—
96	C₂	C₂₃⋊#338	1	1	yes	characteristic
97	C₂	#13092182	1	1	yes	characteristic
98	C₂	#13092184	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13092381

C₂

yes

maximal

C₂×C₂₃×C₂²⋊C₈

C₂

yes

maximal, characteristic

C₂³×C₄×C₂₃

C₂²

yes

characteristic

C₂²×C₈×C₂₃

C₄

yes

—

C₂×#67267

—

C₂₃×C₂²⋊C₈

—

C₂²⋊(C₂₃⋊C₈)

—

C₂²⋊(C₂₃⋊C₈)

C₄

yes

—

C₂⁴×C₂₃

Q₈

yes

characteristic

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

D₄

yes

characteristic

C₂²×C₄×C₂₃

D₄

yes

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

#67267

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

C₄⋊C₄

yes

—

C₂³×C₂₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₄²

yes

characteristic

C₂×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

characteristic

C₈×C₂₃

—

C₂₃⋊C₈

—

#1293

—

maximal, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#95

yes

characteristic

C₂²×C₂₃

#108

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#99

yes

—

#407

—

C₂×C₂²⋊C₈

D₂₃

yes

characteristic

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂³×C₄

C₂×D₂₃

yes

characteristic

C₂×M₁₆

—

C₂²×C₈

Dic₂₃

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂₃

#1293

yes

characteristic, complemented, Sylow

C₂⁴

Dic₄₆

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₄×D₂₃

yes

—

C₂²×C₄

D₉₂

yes

characteristic

C₂²×C₄

C₂×Dic₂₃

yes

characteristic

C₂²×C₄

C₂²⋊D₂₃

yes

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂³

—

C₂³

—

C₂³

—

C₂³

Dic₂₃⋊C₄

yes

—

C₂³

C₄⋊Dic₂₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₂₃

yes

characteristic

C₂×C₄

(C₂×D₂₃)⋊C₄

yes

—

C₂×C₄

C₂²⋊Dic₂₃

yes

characteristic

C₈

—

C₈

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

#10625001

yes

—

C₂²

C₂₃⋊#95

yes

characteristic

C₄

—

C₄

—

C₄

#10625016

yes

—

C₂

—

C₂

—

C₂

C₂₃⋊#338

yes

C₂

yes

C₂

yes

	subgroup	G/H
center	C₂²	C₂₃⋊#95
commutator	C₂×C₄×C₂₃	C₄²
Frattini	C₂³×C₄	C₂×D₂₃
Fitting	C₂×C₂₃×C₂²⋊C₈	C₂
socle	C₂²×C₂₃	#95
prime radical	C₂⁴×C₂₃	Q₈
cosocle	C₂³×C₄×C₂₃	C₂²
hypercenter	C₂×C₂²⋊C₈	D₂₃
hypocenter	C₂₃	#1293
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

subgroup

G/H

center

C₂²

C₂₃⋊#95

commutator

C₂×C₄×C₂₃

C₄²

Frattini

C₂³×C₄

C₂×D₂₃

Fitting

C₂×C₂₃×C₂²⋊C₈

C₂

socle

C₂²×C₂₃

#95

prime radical

C₂⁴×C₂₃

Q₈

cosocle

C₂³×C₄×C₂₃

C₂²

hypercenter

C₂×C₂²⋊C₈

D₂₃

hypocenter

C₂₃

#1293

LCS

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×C₂²⋊C₈

N	H ≅ G/N	type
58. C₂₃	#1293	semidirect

H ≅ G/N

type

58. C₂₃

#1293

semidirect

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

subgroup	index
#13092381	2
C₂×C₂₃×C₂²⋊C₈	2
#1293	23

subgroup

index

#13092381

C₂×C₂₃×C₂²⋊C₈

#1293

Subgroups — #15671274

Conjugacy classes of subgroups

Complements

Maximal subgroups