Group #15671232 — subgroups

Subgroups — #15671232

Order 2944.

180 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 180 conjugacy classes of subgroups — too many to draw. Only its 113 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	C₂×C₄⋊(C₂₃⋊C₈)	C₂	1	2	yes	maximal
2	C₂×C₄×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂²×C₈×C₂₃	C₄	1	2	yes	—
4	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
5	C₂²×C₂₃⋊C₈	—	2	2	no	—
6	C₄×C₈×C₂₃	—	2	2	no	—
7	C₄⋊(C₂₃⋊C₈)	—	2	2	no	—
8	C₄⋊(C₂₃⋊C₈)	C₄	1	4	yes	—
9	C₂²×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
10	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
11	C₂²×C₄×C₂₃	Q₈	1	1	yes	characteristic
12	C₂×C₈×C₂₃	—	2	2	no	—
13	C₂×C₈×C₂₃	—	2	2	no	—
14	C₂×C₈×C₂₃	C₈	1	4	yes	complemented
15	C₄²×C₂₃	C₂×C₄	1	2	yes	—
16	C₄²×C₂₃	D₄	1	2	yes	—
17	C₂×C₂₃⋊C₈	—	2	4	no	—
18	C₂×C₂₃⋊C₈	—	4	2	no	—
19	C₂×C₂₃⋊C₈	—	4	2	no	—
20	C₂³×C₂₃	C₄⋊C₄	1	1	yes	characteristic
21	C₂×C₄×C₂₃	—	2	2	no	—
22	C₂×C₄×C₂₃	C₄²	1	1	yes	characteristic
23	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
24	C₂×C₄×C₂₃	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
26	C₂×C₄×C₂₃	C₂×C₈	1	2	yes	—
27	C₂×C₄×C₂₃	M₁₆	1	2	yes	—
28	C₂×C₄×C₂₃	D₈	1	1	yes	characteristic
29	C₂×C₄×C₂₃	SD₁₆	1	2	yes	—
30	C₂×C₄×C₂₃	Q₁₆	1	1	yes	characteristic
31	C₈×C₂₃	—	2	4	no	—
32	C₂₃⋊C₈	—	4	4	no	—
33	#1277	—	23	1	no	maximal, Sylow
34	C₂²×C₂₃	#95	1	1	yes	characteristic
35	C₂²×C₂₃	C₄⋊C₈	1	2	yes	—
36	C₂²×C₂₃	#106	1	1	yes	characteristic
37	C₂²×C₂₃	#107	1	1	yes	characteristic
38	C₂²×C₂₃	#108	1	2	yes	—
39	C₄×C₂₃	—	2	2	no	—
40	C₄×C₂₃	C₄×C₈	1	1	yes	characteristic
41	C₄×C₂₃	#97	1	1	yes	characteristic
42	C₄×C₂₃	C₂²⋊C₈	1	2	yes	—
43	C₄×C₂₃	#102	1	2	yes	—
44	C₄×C₂₃	#103	1	2	yes	—
45	C₂×C₄⋊C₈	—	23	2	no	—
46	C₂×C₄×C₈	D₂₃	1	1	yes	characteristic
47	C₂×C₂₃	#334	1	2	yes	—
48	C₂×C₂₃	#335	1	2	yes	—
49	C₂×C₂₃	#336	1	1	yes	characteristic
50	C₂×C₂₃	#340	1	1	yes	characteristic
51	C₂×C₂₃	#341	1	1	yes	characteristic
52	C₂²×C₈	—	46	2	no	—
53	C₂²×C₈	Dic₂₃	1	2	yes	—
54	C₂×C₄²	C₂×D₂₃	1	1	yes	characteristic
55	C₄⋊C₈	—	23	4	no	—
56	C₄⋊C₈	—	46	2	no	—
57	C₄×C₈	—	2	2	no	—
58	C₂₃	#1277	1	1	yes	characteristic, complemented, Sylow
59	C₂²×C₄	Dic₄₆	1	1	yes	characteristic
60	C₂²×C₄	D₉₂	1	1	yes	characteristic
61	C₂²×C₄	C₂×Dic₂₃	1	1	yes	characteristic
62	C₂×C₈	—	2	2	no	—
63	C₂×C₈	—	2	2	no	—
64	C₂×C₈	—	46	4	no	—
65	C₂×C₈	—	92	2	no	—
66	C₂×C₈	—	92	2	no	—
67	C₂×C₈	C₂₃⋊C₈	1	4	yes	complemented
68	C₄²	C₄×D₂₃	1	2	yes	—
69	C₄²	C₂²⋊D₂₃	1	2	yes	—
70	C₂³	C₄⋊Dic₂₃	1	1	yes	characteristic
71	C₂×C₄	—	2	2	no	—
72	C₂×C₄	#67263	1	2	yes	—
73	C₂×C₄	D₁₈₄	1	1	yes	characteristic
74	C₂×C₄	Dic₉₂	1	1	yes	characteristic
75	C₂×C₄	C₂×C₂₃⋊C₈	1	2	yes	—
76	C₂×C₄	#67267	1	2	yes	—
77	C₂×C₄	C₄×Dic₂₃	1	1	yes	characteristic
78	C₂×C₄	Dic₂₃⋊C₄	1	2	yes	—
79	C₂×C₄	(C₂×D₂₃)⋊C₄	1	2	yes	—
80	C₂×C₄	C₂²⋊Dic₂₃	1	1	yes	characteristic
81	C₈	—	2	4	no	—
82	C₈	—	92	4	no	—
83	C₂²	C₄⋊(C₂₃⋊C₈)	1	2	yes	—
84	C₂²	#10624999	1	1	yes	characteristic
85	C₂²	#10625000	1	1	yes	characteristic
86	C₂²	#10625001	1	2	yes	—
87	C₂²	C₂₃⋊#95	1	1	yes	characteristic
88	C₄	—	2	2	no	—
89	C₄	C₄×C₂₃⋊C₈	1	1	yes	characteristic
90	C₄	#10624985	1	1	yes	characteristic
91	C₄	#10624998	1	2	yes	—
92	C₄	#10625003	1	2	yes	—
93	C₄	C₂²⋊(C₂₃⋊C₈)	1	2	yes	—
94	C₂	#13092168	1	2	yes	—
95	C₂	#13092170	1	2	yes	—
96	C₂	#13092172	1	1	yes	characteristic
97	C₂	#13092180	1	1	yes	characteristic
98	C₂	#13092182	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊(C₂₃⋊C₈)

C₂

yes

maximal

C₂×C₄×C₈×C₂₃

C₂

yes

maximal, characteristic

C₂²×C₈×C₂₃

C₄

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂²×C₂₃⋊C₈

—

C₄×C₈×C₂₃

—

C₄⋊(C₂₃⋊C₈)

—

C₄⋊(C₂₃⋊C₈)

C₄

yes

—

C₂²×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

D₄

yes

characteristic

C₂²×C₄×C₂₃

Q₈

yes

characteristic

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₈

yes

complemented

C₄²×C₂₃

C₂×C₄

yes

—

C₄²×C₂₃

D₄

yes

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂×C₂₃⋊C₈

—

C₂³×C₂₃

C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₄²

yes

characteristic

C₂×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₃

C₄⋊C₄

yes

—

C₂×C₄×C₂₃

C₂×C₈

yes

—

C₂×C₄×C₂₃

M₁₆

yes

—

C₂×C₄×C₂₃

D₈

yes

characteristic

C₂×C₄×C₂₃

SD₁₆

yes

—

C₂×C₄×C₂₃

Q₁₆

yes

characteristic

C₈×C₂₃

—

C₂₃⋊C₈

—

—

C₂²×C₂₃

yes

C₂²×C₂₃

C₄⋊C₈

yes

—

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

—

C₄×C₂₃

—

C₄×C₂₃

C₄×C₈

yes

characteristic

C₄×C₂₃

#97

yes

characteristic

C₄×C₂₃

C₂²⋊C₈

yes

—

C₄×C₂₃

#102

yes

—

C₄×C₂₃

#103

yes

—

C₂×C₄⋊C₈

—

C₂×C₄×C₈

D₂₃

yes

characteristic

C₂×C₂₃

#334

yes

—

C₂×C₂₃

#335

yes

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂²×C₈

—

C₂²×C₈

Dic₂₃

yes

—

C₂×C₄²

C₂×D₂₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄×C₈

—

C₂₃

#1277

yes

characteristic, complemented, Sylow

C₂²×C₄

Dic₄₆

yes

characteristic

C₂²×C₄

D₉₂

yes

characteristic

C₂²×C₄

C₂×Dic₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂₃⋊C₈

yes

complemented

C₄²

C₄×D₂₃

yes

—

C₄²

C₂²⋊D₂₃

yes

—

C₂³

C₄⋊Dic₂₃

yes

characteristic

C₂×C₄

—

C₂×C₄

#67263

yes

—

C₂×C₄

D₁₈₄

yes

characteristic

C₂×C₄

Dic₉₂

yes

characteristic

C₂×C₄

C₂×C₂₃⋊C₈

yes

—

C₂×C₄

#67267

yes

—

C₂×C₄

C₄×Dic₂₃

yes

characteristic

C₂×C₄

Dic₂₃⋊C₄

yes

—

C₂×C₄

(C₂×D₂₃)⋊C₄

yes

—

C₂×C₄

C₂²⋊Dic₂₃

yes

characteristic

C₈

—

C₈

—

C₂²

C₄⋊(C₂₃⋊C₈)

yes

—

C₂²

yes

C₂²

yes

C₂²

yes

—

C₂²

C₂₃⋊#95

yes

characteristic

C₄

—

C₄

C₄×C₂₃⋊C₈

yes

C₄

yes

C₄

yes

—

C₄

#10625003

yes

—

C₄

C₂²⋊(C₂₃⋊C₈)

yes

—

C₂

#13092168

yes

—

C₂

#13092170

yes

—

C₂

yes

C₂

yes

C₂

yes

	subgroup	G/H
center	C₂²×C₄	D₉₂
commutator	C₄×C₂₃	C₄×C₈
Frattini	C₂×C₄²	C₂×D₂₃
Fitting	C₂×C₄×C₈×C₂₃	C₂
socle, prime radical	C₂³×C₂₃	C₄⋊C₄
cosocle	C₂×C₄²×C₂₃	C₂²
hypercenter	C₂×C₄×C₈	D₂₃
hypocenter	C₂₃	#1277
LCS	G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂²×C₄

D₉₂

commutator

C₄×C₂₃

C₄×C₈

Frattini

C₂×C₄²

C₂×D₂₃

Fitting

C₂×C₄×C₈×C₂₃

C₂

socle, prime radical

C₂³×C₂₃

C₄⋊C₄

cosocle

C₂×C₄²×C₂₃

C₂²

hypercenter

C₂×C₄×C₈

D₂₃

hypocenter

C₂₃

#1277

LCS

G ▷ C₄×C₂₃ ▷ C₂×C₂₃ ▷ C₂₃

UCS

C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
14. C₂×C₈×C₂₃	C₈	semidirect
58. C₂₃	#1277	semidirect
67. C₂×C₈	C₂₃⋊C₈	semidirect

H ≅ G/N

type

14. C₂×C₈×C₂₃

C₈

semidirect

58. C₂₃

#1277

semidirect

67. C₂×C₈

C₂₃⋊C₈

semidirect

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

subgroup	index
C₂×C₄⋊(C₂₃⋊C₈)	2
C₂×C₄×C₈×C₂₃	2
#1277	23

subgroup

index

C₂×C₄⋊(C₂₃⋊C₈)

C₂×C₄×C₈×C₂₃

#1277

Subgroups — #15671232

Conjugacy classes of subgroups

Complements

Maximal subgroups