Group #15669334 — subgroups

Subgroups — C₂₃×#1715 #15669334

Order 2944.

220 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 220 conjugacy classes of subgroups — too many to draw. Only its 80 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	C₂₃×#498	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂₃×#493	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#492	C₂	1	2	yes	maximal, complemented
4	C₂₃×#486	C₂	1	2	yes	maximal, complemented
5	C₂₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×SD₁₆×C₂₃	—	2	4	no	—
7	C₂×D₈×C₂₃	—	2	2	no	—
8	C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
9	C₂₃×#127	C₂²	1	1	yes	characteristic
10	C₂₃×#127	C₂²	1	1	yes	characteristic
11	C₂₃×#106	—	2	2	no	—
12	C₂₃×#102	—	2	4	no	—
13	C₄×C₈×C₂₃	C₂²	1	2	yes	—
14	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
15	C₂×Q₈×C₂₃	—	2	2	no	—
16	C₂×D₄×C₂₃	—	2	2	no	—
17	C₂×D₄×C₂₃	—	2	1	no	—
18	C₂×D₄×C₂₃	—	2	1	no	—
19	C₂×D₄×C₂₃	—	2	2	no	—
20	SD₁₆×C₂₃	—	4	8	no	—
21	D₈×C₂₃	—	4	4	no	—
22	C₂×C₈×C₂₃	D₄	1	4	yes	—
23	C₂×C₈×C₂₃	D₄	1	2	yes	—
24	C₄⋊C₄×C₂₃	—	2	2	no	—
25	C₄⋊C₄×C₂₃	—	2	2	no	—
26	C₄²×C₂₃	C₂³	1	1	yes	characteristic
27	C₂³×C₂₃	—	4	1	no	—
28	C₂³×C₂₃	—	4	1	no	—
29	Q₈×C₂₃	—	4	4	no	—
30	D₄×C₂₃	—	4	2	no	—
31	D₄×C₂₃	—	4	2	no	—
32	D₄×C₂₃	—	4	4	no	—
33	D₄×C₂₃	—	4	4	no	—
34	C₂×C₄×C₂₃	—	4	2	no	—
35	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
36	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
37	C₈×C₂₃	—	2	4	no	—
38	C₈×C₂₃	SD₁₆	1	4	yes	complemented
39	#1715	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂²×C₂₃	—	8	1	no	—
41	C₂²×C₂₃	—	8	1	no	—
42	C₂²×C₂₃	—	8	2	no	—
43	C₂²×C₂₃	—	8	2	no	—
44	C₂²×C₂₃	#127	1	1	yes	characteristic
45	C₄×C₂₃	—	8	2	no	—
46	C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
47	C₄×C₂₃	AGL(1,ℤ/8)	1	4	yes	—
48	#498	C₂×C₂₃	1	1	yes	characteristic, complemented
49	#493	C₂×C₂₃	1	1	yes	characteristic
50	#492	C₂×C₂₃	1	2	yes	complemented
51	#486	C₂×C₂₃	1	2	yes	complemented
52	#322	C₂×C₂₃	1	1	yes	characteristic, complemented
53	C₂×C₂₃	—	16	1	no	—
54	C₂×C₂₃	—	16	1	no	—
55	C₂×C₂₃	#492	1	1	yes	characteristic
56	C₂×C₂₃	#496	1	2	yes	—
57	C₂×SD₁₆	—	2	4	no	—
58	C₂×D₈	—	2	2	no	—
59	C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
60	#127	C₂²×C₂₃	1	1	yes	characteristic
61	#127	C₂²×C₂₃	1	1	yes	characteristic
62	#106	—	2	2	no	—
63	#102	—	2	4	no	—
64	C₄×C₈	C₂²×C₂₃	1	2	yes	—
65	C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
66	C₂₃	#1715	1	1	yes	characteristic, direct product factor, Sylow
67	C₂×Q₈	—	2	2	no	—
68	C₂×D₄	—	2	2	no	—
69	C₂×D₄	—	2	1	no	—
70	C₂×D₄	—	2	1	no	—
71	C₂×D₄	—	2	2	no	—
72	SD₁₆	—	4	8	no	—
73	D₈	—	4	4	no	—
74	C₂×C₈	D₄×C₂₃	1	4	yes	—
75	C₂×C₈	D₄×C₂₃	1	2	yes	—
76	C₄⋊C₄	—	2	2	no	—
77	C₄⋊C₄	—	2	2	no	—
78	C₄²	C₂³×C₂₃	1	1	yes	characteristic
79	C₂³	—	4	1	no	—
80	C₂³	—	4	1	no	—
81	Q₈	—	4	4	no	—
82	D₄	—	4	2	no	—
83	D₄	—	4	2	no	—
84	D₄	—	4	4	no	—
85	D₄	—	4	4	no	—
86	C₂×C₄	—	4	2	no	—
87	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
88	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
89	C₈	—	2	4	no	—
90	C₈	SD₁₆×C₂₃	1	4	yes	complemented
91	C₂²	—	8	1	no	—
92	C₂²	—	8	1	no	—
93	C₂²	—	8	2	no	—
94	C₂²	—	8	2	no	—
95	C₂²	C₂₃×#127	1	1	yes	characteristic
96	C₄	—	8	2	no	—
97	C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
98	C₄	C₂₃×AGL(1,ℤ/8)	1	4	yes	—
99	C₂	—	16	1	no	—
100	C₂	—	16	1	no	—
101	C₂	C₂₃×#492	1	1	yes	characteristic
102	C₂	C₂₃×#496	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#498

C₂

yes

maximal, characteristic, complemented

C₂₃×#493

C₂

yes

maximal, characteristic

C₂₃×#492

C₂

yes

maximal, complemented

C₂₃×#486

C₂

yes

maximal, complemented

C₂₃×#322

C₂

yes

maximal, characteristic, complemented

C₂×SD₁₆×C₂₃

—

C₂×D₈×C₂₃

—

C₂₃×C₄⋊Q₈

C₂²

yes

—

C₂₃×#127

C₂²

yes

characteristic

C₂₃×#127

C₂²

yes

characteristic

C₂₃×#106

—

C₂₃×#102

—

C₄×C₈×C₂₃

C₂²

yes

—

C₄×C₈×C₂₃

C₂²

yes

characteristic

C₂×Q₈×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

SD₁₆×C₂₃

—

D₈×C₂₃

—

C₂×C₈×C₂₃

D₄

yes

—

C₂×C₈×C₂₃

D₄

yes

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄²×C₂₃

C₂³

yes

characteristic

C₂³×C₂₃

—

C₂³×C₂₃

—

Q₈×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₈×C₂₃

—

C₈×C₂₃

SD₁₆

yes

complemented

#1715

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#127

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

C₂×SD₁₆

yes

—

C₄×C₂₃

AGL(1,ℤ/8)

yes

—

#498

C₂×C₂₃

yes

characteristic, complemented

#493

C₂×C₂₃

yes

characteristic

#492

C₂×C₂₃

yes

complemented

#486

C₂×C₂₃

yes

complemented

#322

C₂×C₂₃

yes

characteristic, complemented

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#492

yes

characteristic

C₂×C₂₃

#496

yes

—

C₂×SD₁₆

—

C₂×D₈

—

C₄⋊Q₈

C₂²×C₂₃

yes

—

#127

C₂²×C₂₃

yes

characteristic

#127

C₂²×C₂₃

yes

characteristic

#106

—

#102

—

C₄×C₈

C₂²×C₂₃

yes

—

C₄×C₈

C₂²×C₂₃

yes

characteristic

C₂₃

#1715

yes

characteristic, direct product factor, Sylow

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

D₈

—

C₂×C₈

D₄×C₂₃

yes

—

C₂×C₈

D₄×C₂₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂³×C₂₃

yes

characteristic

C₂³

—

C₂³

—

Q₈

—

D₄

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

C₂×C₄

C₂×D₄×C₂₃

yes

characteristic

C₂×C₄

C₂×D₄×C₂₃

yes

—

C₈

—

C₈

SD₁₆×C₂₃

yes

complemented

C₂²

—

C₂²

—

C₂²

—

C₂²

—

C₂²

C₂₃×#127

yes

characteristic

C₄

—

C₄

C₂×SD₁₆×C₂₃

yes

—

C₄

C₂₃×AGL(1,ℤ/8)

yes

—

C₂

—

100

C₂

—

101

C₂

C₂₃×#492

yes

characteristic

102

C₂

C₂₃×#496

yes

—

	subgroup	G/H
center, socle	C₂²×C₂₃	#127
commutator, Frattini, cosocle	C₄²	C₂³×C₂₃
prime radical	C₂₃×#493	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₄²×C₂₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₂₃

#127

commutator, Frattini, cosocle

C₄²

C₂³×C₂₃

prime radical

C₂₃×#493

C₂

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₂₃ ◁ C₄²×C₂₃ ◁ G

N	H ≅ G/N	type
1. C₂₃×#498	C₂	semidirect
3. C₂₃×#492	C₂	semidirect
4. C₂₃×#486	C₂	semidirect
5. C₂₃×#322	C₂	semidirect
38. C₈×C₂₃	SD₁₆	semidirect
39. #1715	C₂₃	direct (N × H)
48. #498	C₂×C₂₃	semidirect
50. #492	C₂×C₂₃	semidirect
51. #486	C₂×C₂₃	semidirect
52. #322	C₂×C₂₃	semidirect
66. C₂₃	#1715	direct (N × H)
90. C₈	SD₁₆×C₂₃	semidirect

H ≅ G/N

type

1. C₂₃×#498

C₂

semidirect

3. C₂₃×#492

C₂

semidirect

4. C₂₃×#486

C₂

semidirect

5. C₂₃×#322

C₂

semidirect

38. C₈×C₂₃

SD₁₆

semidirect

39. #1715

C₂₃

direct (N × H)

48. #498

C₂×C₂₃

semidirect

50. #492

C₂×C₂₃

semidirect

51. #486

C₂×C₂₃

semidirect

52. #322

C₂×C₂₃

semidirect

66. C₂₃

#1715

direct (N × H)

90. C₈

SD₁₆×C₂₃

semidirect

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

subgroup	index
C₂₃×#498	2
C₂₃×#493	2
C₂₃×#492	2
C₂₃×#486	2
C₂₃×#322	2
#1715	23

subgroup

index

C₂₃×#498

C₂₃×#493

C₂₃×#492

C₂₃×#486

C₂₃×#322

#1715

Subgroups — C₂₃×#1715 #15669334

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#1715 #15669334

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#1715 #15669334