Group #15672305 — subgroups

Subgroups — #15672305

Order 2944.

148 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 148 conjugacy classes of subgroups — too many to draw. Only its 63 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	#13092544	C₂	1	2	yes	maximal
2	C₄×Dic₉₂	C₂	1	1	yes	maximal, characteristic
3	#13092147	C₂	1	2	yes	maximal
4	#13092138	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#334	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₉₂	C₄	1	1	yes	characteristic
7	C₄×Dic₄₆	C₂²	1	2	yes	—
8	C₂₃×C₄⋊C₈	C₂²	1	2	yes	—
9	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
10	#10625000	C₄	1	1	yes	characteristic
11	#10624998	C₄	1	2	yes	—
12	#10624997	—	2	2	no	—
13	Dic₂₃⋊C₈	—	2	2	no	—
14	C₄×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
15	C₄×C₂₃⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×Dic₄₆	C₂×C₄	1	2	yes	—
17	C₂×C₈×C₂₃	—	2	2	no	—
18	C₂×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
19	C₂×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
20	C₄²×C₂₃	C₂³	1	1	yes	characteristic
21	C₄⋊Dic₂₃	C₂×C₄	1	2	yes	—
22	Dic₂₃⋊C₄	—	2	2	no	—
23	C₄×Dic₂₃	—	2	2	no	—
24	C₂×C₂₃⋊C₈	—	2	1	no	—
25	C₂×C₂₃⋊C₈	D₄	1	2	yes	—
26	Dic₉₂	—	2	2	no	—
27	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
28	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
29	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
30	C₂×Dic₂₃	—	2	2	no	—
31	C₂×Dic₂₃	—	2	2	no	—
32	Dic₄₆	—	2	2	no	—
33	Dic₄₆	—	2	2	no	—
34	C₈×C₂₃	—	2	1	no	—
35	C₈×C₂₃	—	4	2	no	—
36	C₈×C₂₃	M₁₆	1	2	yes	—
37	C₂₃⋊C₈	—	2	2	no	—
38	C₂₃⋊C₈	—	4	1	no	—
39	#1593	—	23	1	no	maximal, Sylow
40	C₂²×C₂₃	C₄×D₄	1	1	yes	characteristic
41	C₄×C₂₃	—	2	1	no	—
42	C₄×C₂₃	C₂×M₁₆	1	1	yes	characteristic
43	C₄×C₂₃	#131	1	1	yes	characteristic
44	C₄×C₂₃	#137	1	2	yes	—
45	Dic₂₃	—	4	2	no	—
46	Dic₂₃	—	4	2	no	—
47	#446	—	23	2	no	—
48	C₄×Q₁₆	—	23	1	no	—
49	#334	D₂₃	1	1	yes	characteristic
50	Q₈⋊C₈	—	23	2	no	—
51	#322	—	23	1	no	—
52	C₂×C₂₃	#436	1	1	yes	characteristic
53	C₂×C₂₃	#441	1	1	yes	characteristic
54	C₂×C₂₃	#444	1	1	yes	characteristic
55	C₂×Q₁₆	—	23	1	no	—
56	C₄×Q₈	—	23	2	no	—
57	#107	—	23	1	no	—
58	C₄⋊C₈	—	46	2	no	—
59	C₄⋊C₈	C₂×D₂₃	1	2	yes	—
60	#103	—	23	2	no	—
61	#97	—	46	2	no	—
62	C₄×C₈	—	23	1	no	—
63	C₄×C₈	—	23	1	no	—
64	C₄×C₈	C₂×D₂₃	1	1	yes	characteristic
65	C₂₃	#1593	1	1	yes	characteristic, complemented, Sylow
66	C₂×Q₈	—	23	2	no	—
67	Q₁₆	—	46	2	no	—
68	C₂×C₈	—	2	2	no	—
69	C₂×C₈	—	23	2	no	—
70	C₂×C₈	—	46	1	no	—
71	C₂×C₈	C₄×D₂₃	1	1	yes	characteristic
72	C₂×C₈	C₄×D₂₃	1	1	yes	characteristic
73	C₄⋊C₄	—	23	2	no	—
74	C₄⋊C₄	—	46	2	no	—
75	C₄²	—	46	2	no	—
76	C₄²	C₂²×D₂₃	1	1	yes	characteristic
77	Q₈	—	46	2	no	—
78	Q₈	—	46	2	no	—
79	C₂×C₄	—	46	2	no	—
80	C₂×C₄	—	46	2	no	—
81	C₂×C₄	C₂×C₄×D₂₃	1	1	yes	characteristic
82	C₂×C₄	D₄×D₂₃	1	1	yes	characteristic
83	C₂×C₄	#67292	1	1	yes	characteristic
84	C₈	—	2	1	no	—
85	C₈	—	4	2	no	—
86	C₈	—	46	2	no	—
87	C₈	—	92	1	no	—
88	C₈	#67262	1	2	yes	—
89	C₂²	#10625064	1	1	yes	characteristic
90	C₄	—	2	1	no	—
91	C₄	—	92	2	no	—
92	C₄	—	92	2	no	—
93	C₄	C₂×#67262	1	1	yes	characteristic
94	C₄	#10625078	1	1	yes	characteristic
95	C₄	#10625086	1	2	yes	—
96	C₂	#13092494	1	1	yes	characteristic
97	C₂	#13092523	1	1	yes	characteristic
98	C₂	#13092538	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#13092544

C₂

yes

maximal

C₄×Dic₉₂

C₂

yes

maximal, characteristic

#13092147

C₂

yes

maximal

#13092138

C₂

yes

maximal, characteristic

C₂₃×#334

C₂

yes

maximal, characteristic

C₂×Dic₉₂

C₄

yes

characteristic

C₄×Dic₄₆

C₂²

yes

—

C₂₃×C₄⋊C₈

C₂²

yes

—

C₄×C₈×C₂₃

C₂²

yes

characteristic

#10625000

C₄

yes

characteristic

#10624998

C₄

yes

—

#10624997

—

Dic₂₃⋊C₈

—

C₄×C₂₃⋊C₈

C₂²

yes

characteristic

C₄×C₂₃⋊C₈

C₂²

yes

characteristic

C₂×Dic₄₆

C₂×C₄

yes

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₂×C₄

yes

characteristic

C₂×C₈×C₂₃

C₂×C₄

yes

characteristic

C₄²×C₂₃

C₂³

yes

characteristic

C₄⋊Dic₂₃

C₂×C₄

yes

—

Dic₂₃⋊C₄

—

C₄×Dic₂₃

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

D₄

yes

—

Dic₉₂

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

#41

yes

characteristic

C₂×Dic₂₃

—

C₂×Dic₂₃

—

Dic₄₆

—

Dic₄₆

—

C₈×C₂₃

—

C₈×C₂₃

—

C₈×C₂₃

M₁₆

yes

—

C₂₃⋊C₈

—

C₂₃⋊C₈

—

#1593

—

maximal, Sylow

C₂²×C₂₃

C₄×D₄

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

C₂×M₁₆

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

—

Dic₂₃

—

Dic₂₃

—

#446

—

C₄×Q₁₆

—

#334

D₂₃

yes

characteristic

Q₈⋊C₈

—

#322

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×Q₁₆

—

C₄×Q₈

—

#107

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₂₃

yes

—

#103

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₂₃

yes

characteristic

C₂₃

#1593

yes

characteristic, complemented, Sylow

C₂×Q₈

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₄×D₂₃

yes

characteristic

C₂×C₈

C₄×D₂₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

C₂²×D₂₃

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×C₄×D₂₃

yes

characteristic

C₂×C₄

D₄×D₂₃

yes

characteristic

C₂×C₄

#67292

yes

characteristic

C₈

—

C₈

—

C₈

—

C₈

—

C₈

#67262

yes

—

C₂²

#10625064

yes

characteristic

C₄

—

C₄

—

C₄

—

C₄

yes

C₄

yes

C₄

yes

—

C₂

yes

C₂

yes

C₂

yes

	subgroup	G/H
center	C₂×C₄	D₄×D₂₃
commutator	C₂×C₄×C₂₃	C₂²×C₄
Frattini	C₄²	C₂²×D₂₃
Fitting	C₂₃×#334	C₂
socle, prime radical	C₂²×C₂₃	C₄×D₄
cosocle	C₄²×C₂₃	C₂³
hypercenter	#334	D₂₃
hypocenter	C₂₃	#1593
LCS	G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

subgroup

G/H

center

C₂×C₄

D₄×D₂₃

commutator

C₂×C₄×C₂₃

C₂²×C₄

Frattini

C₄²

C₂²×D₂₃

Fitting

C₂₃×#334

C₂

socle, prime radical

C₂²×C₂₃

C₄×D₄

cosocle

C₄²×C₂₃

C₂³

D₂₃

C₂₃

G ▷ C₂×C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂×C₄ ◁ C₄² ◁ #334

N	H ≅ G/N	type
65. C₂₃	#1593	semidirect

H ≅ G/N

type

65. C₂₃

#1593

semidirect

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

subgroup	index
#13092544	2
C₄×Dic₉₂	2
#13092147	2
#13092138	2
C₂₃×#334	2
#1593	23

subgroup

index

C₄×Dic₉₂

C₂₃×#334

Subgroups — #15672305

Conjugacy classes of subgroups

Complements

Maximal subgroups