Group #19155756 — subgroups

Subgroups — C₂₃×#8130 #19155756

Order 5888.

520 conjugacy classes of subgroups; only the 230 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 230 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×#2920	C₂	1	3	yes	maximal
2	C₂₃×#1309	C₂	1	4	yes	maximal, complemented
3	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
4	C₂₃×#432	—	2	6	no	—
5	C₂₃×#431	—	2	6	no	—
6	C₂₃×#423	C₂²	1	6	yes	—
7	C₂₃×#407	C₄	1	6	yes	—
8	C₄×M₁₆×C₂₃	C₄	1	6	yes	—
9	C₂³×C₄×C₂₃	C₂×C₄	1	3	yes	—
10	C₂×M₁₆×C₂₃	C₂×C₄	1	12	yes	—
11	C₂×C₄²×C₂₃	C₂×C₄	1	3	yes	—
12	C₂×C₄²×C₂₃	D₄	1	6	yes	—
13	C₂×C₄²×C₂₃	Q₈	1	2	yes	—
14	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
15	C₂₃×C₄⋊C₈	—	2	12	no	—
16	C₂₃×C₂²⋊C₈	—	2	12	no	—
17	C₂₃×#97	—	2	6	no	—
18	C₄×C₈×C₂₃	—	2	6	no	—
19	C₂⁴×C₂₃	C₄²	1	1	yes	characteristic
20	C₂²×C₄×C₂₃	—	4	6	no	—
21	C₂²×C₄×C₂₃	C₄²	1	3	yes	—
22	C₂²×C₄×C₂₃	C₂²⋊C₄	1	6	yes	—
23	C₂²×C₄×C₂₃	C₄⋊C₄	1	6	yes	—
24	C₂²×C₄×C₂₃	C₂²×C₄	1	3	yes	—
25	M₁₆×C₂₃	—	4	12	no	—
26	C₂×C₈×C₂₃	—	2	12	no	—
27	C₄²×C₂₃	C₂²⋊C₄	1	3	yes	—
28	C₄²×C₂₃	C₂²⋊C₄	1	3	yes	—
29	C₄²×C₂₃	C₄⋊C₄	1	6	yes	—
30	C₄²×C₂₃	C₂×D₄	1	3	yes	—
31	C₄²×C₂₃	C₂×Q₈	1	1	yes	characteristic
32	#8130	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
33	C₂³×C₂₃	—	4	3	no	—
34	C₂³×C₂₃	#95	1	2	yes	—
35	C₂³×C₂₃	C₂×C₄²	1	1	yes	characteristic
36	C₂×C₄×C₂₃	—	2	6	no	—
37	C₂×C₄×C₂₃	—	2	12	no	—
38	C₂×C₄×C₂₃	—	4	3	no	—
39	C₂×C₄×C₂₃	—	4	6	no	—
40	C₂×C₄×C₂₃	#95	1	6	yes	—
41	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	3	yes	—
42	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	3	yes	—
43	C₈×C₂₃	—	4	12	no	—
44	#2920	C₂×C₂₃	1	3	yes	—
45	#1309	C₂×C₂₃	1	4	yes	complemented
46	C₂²×C₂₃	—	2	6	no	—
47	C₂²×C₂₃	—	2	3	no	—
48	C₂²×C₂₃	—	4	4	no	—
49	C₂²×C₂₃	C₂×#95	1	1	yes	characteristic
50	C₄×C₂₃	—	2	6	no	—
51	C₄×C₂₃	—	2	3	no	—
52	C₄×C₂₃	#413	1	6	yes	—
53	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
54	#432	—	2	6	no	—
55	#431	—	2	6	no	—
56	#423	C₂²×C₂₃	1	6	yes	—
57	#407	C₄×C₂₃	1	6	yes	—
58	C₄×M₁₆	C₄×C₂₃	1	6	yes	—
59	C₂×C₂₃	—	4	2	no	—
60	C₂×C₂₃	—	4	1	no	—
61	C₂×C₂₃	#1742	1	3	yes	—
62	C₂³×C₄	C₂×C₄×C₂₃	1	3	yes	—
63	C₂×M₁₆	C₂×C₄×C₂₃	1	12	yes	—
64	C₂×C₄²	C₂×C₄×C₂₃	1	3	yes	—
65	C₂×C₄²	D₄×C₂₃	1	6	yes	—
66	C₂×C₄²	Q₈×C₂₃	1	2	yes	—
67	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
68	C₄⋊C₈	—	2	12	no	—
69	C₂²⋊C₈	—	2	12	no	—
70	#97	—	2	6	no	—
71	C₄×C₈	—	2	6	no	—
72	C₂₃	#8130	1	1	yes	characteristic, direct product factor, Sylow
73	C₂⁴	C₄²×C₂₃	1	1	yes	characteristic
74	C₂²×C₄	—	4	6	no	—
75	C₂²×C₄	C₄²×C₂₃	1	3	yes	—
76	C₂²×C₄	C₂²⋊C₄×C₂₃	1	6	yes	—
77	C₂²×C₄	C₄⋊C₄×C₂₃	1	6	yes	—
78	C₂²×C₄	C₂²×C₄×C₂₃	1	3	yes	—
79	M₁₆	—	4	12	no	—
80	C₂×C₈	—	2	12	no	—
81	C₄²	C₂²⋊C₄×C₂₃	1	3	yes	—
82	C₄²	C₂²⋊C₄×C₂₃	1	3	yes	—
83	C₄²	C₄⋊C₄×C₂₃	1	6	yes	—
84	C₄²	C₂×D₄×C₂₃	1	3	yes	—
85	C₄²	C₂×Q₈×C₂₃	1	1	yes	characteristic
86	C₂³	—	4	3	no	—
87	C₂³	C₂₃×#95	1	2	yes	—
88	C₂³	C₂×C₄²×C₂₃	1	1	yes	characteristic
89	C₂×C₄	—	2	6	no	—
90	C₂×C₄	—	2	12	no	—
91	C₂×C₄	—	4	3	no	—
92	C₂×C₄	—	4	6	no	—
93	C₂×C₄	C₂₃×#95	1	6	yes	—
94	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	3	yes	—
95	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	3	yes	—
96	C₈	—	4	12	no	—
97	C₂²	—	2	3	no	—
98	C₂²	—	2	6	no	—
99	C₂²	—	4	4	no	—
100	C₂²	C₂×C₂₃×#95	1	1	yes	characteristic
101	C₄	—	2	3	no	—
102	C₄	—	2	6	no	—
103	C₄	C₂₃×#413	1	6	yes	—
104	C₂	—	4	1	no	—
105	C₂	—	4	2	no	—
106	C₂	C₂₃×#1742	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#2920

C₂

yes

maximal

C₂₃×#1309

C₂

yes

maximal, complemented

C₂²×C₄²×C₂₃

C₂²

yes

characteristic

C₂₃×#432

—

C₂₃×#431

—

C₂₃×#423

C₂²

yes

—

C₂₃×#407

C₄

yes

—

C₄×M₁₆×C₂₃

C₄

yes

—

C₂³×C₄×C₂₃

C₂×C₄

yes

—

C₂×M₁₆×C₂₃

C₂×C₄

yes

—

C₂×C₄²×C₂₃

C₂×C₄

yes

—

C₂×C₄²×C₂₃

D₄

yes

—

C₂×C₄²×C₂₃

Q₈

yes

—

C₂×C₄²×C₂₃

C₂³

yes

characteristic

C₂₃×C₄⋊C₈

—

C₂₃×C₂²⋊C₈

—

C₂₃×#97

—

C₄×C₈×C₂₃

—

C₂⁴×C₂₃

C₄²

yes

characteristic

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₄²

yes

—

C₂²×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₂₃

C₄⋊C₄

yes

—

C₂²×C₄×C₂₃

C₂²×C₄

yes

—

M₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₄²×C₂₃

C₂²⋊C₄

yes

—

C₄²×C₂₃

C₂²⋊C₄

yes

—

C₄²×C₂₃

C₄⋊C₄

yes

—

C₄²×C₂₃

C₂×D₄

yes

—

C₄²×C₂₃

C₂×Q₈

yes

characteristic

#8130

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₂₃

—

C₂³×C₂₃

#95

yes

—

C₂³×C₂₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

#95

yes

—

C₂×C₄×C₂₃

C₂×C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

C₂×C₄⋊C₄

yes

—

C₈×C₂₃

—

#2920

C₂×C₂₃

yes

—

#1309

C₂×C₂₃

yes

complemented

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×#95

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#413

yes

—

C₂²×C₄²

C₂²×C₂₃

yes

—

—

C₂²×C₂₃

yes

—

#407

C₄×C₂₃

yes

—

C₄×M₁₆

C₄×C₂₃

yes

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#1742

yes

—

C₂³×C₄

C₂×C₄×C₂₃

yes

—

C₂×M₁₆

C₂×C₄×C₂₃

yes

—

C₂×C₄²

C₂×C₄×C₂₃

yes

—

C₂×C₄²

D₄×C₂₃

yes

—

C₂×C₄²

Q₈×C₂₃

yes

—

C₂×C₄²

C₂³×C₂₃

yes

characteristic

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

C₄×C₈

—

C₂₃

#8130

yes

characteristic, direct product factor, Sylow

C₂⁴

C₄²×C₂₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

C₄²×C₂₃

yes

—

C₂²×C₄

C₂²⋊C₄×C₂₃

yes

—

C₂²×C₄

C₄⋊C₄×C₂₃

yes

—

C₂²×C₄

C₂²×C₄×C₂₃

yes

—

M₁₆

—

C₂×C₈

—

C₄²

C₂²⋊C₄×C₂₃

yes

—

C₄²

C₂²⋊C₄×C₂₃

yes

—

C₄²

C₄⋊C₄×C₂₃

yes

—

C₄²

C₂×D₄×C₂₃

yes

—

C₄²

C₂×Q₈×C₂₃

yes

characteristic

C₂³

—

C₂³

C₂₃×#95

yes

—

C₂³

C₂×C₄²×C₂₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂₃×#95

yes

—

C₂×C₄

C₂×C₂²⋊C₄×C₂₃

yes

—

C₂×C₄

C₂×C₄⋊C₄×C₂₃

yes

—

C₈

—

C₂²

—

C₂²

—

C₂²

—

100

C₂²

C₂×C₂₃×#95

yes

characteristic

101

C₄

—

102

C₄

—

103

C₄

C₂₃×#413

yes

—

104

C₂

—

105

C₂

—

106

C₂

C₂₃×#1742

yes

—

	subgroup	G/H
center, socle	C₂²×C₂₃	C₂×#95
commutator	C₂³	C₂×C₄²×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
prime radical	C₂⁴×C₂₃	C₄²
LCS	G ▷ C₂³ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₂₃

C₂×#95

commutator

C₂³

C₂×C₄²×C₂₃

Frattini, cosocle

C₂×C₄²

C₂³×C₂₃

prime radical

C₂⁴×C₂₃

C₄²

LCS

G ▷ C₂³ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

N	H ≅ G/N	type
2. C₂₃×#1309	C₂	semidirect
32. #8130	C₂₃	direct (N × H)
45. #1309	C₂×C₂₃	semidirect
72. C₂₃	#8130	direct (N × H)

H ≅ G/N

type

2. C₂₃×#1309

C₂

semidirect

32. #8130

C₂₃

direct (N × H)

45. #1309

C₂×C₂₃

semidirect

72. C₂₃

#8130

direct (N × H)

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

subgroup	index
C₂₃×#2920	2
C₂₃×#1309	2
#8130	23

subgroup

index

C₂₃×#2920

C₂₃×#1309

#8130

Subgroups — C₂₃×#8130 #19155756

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#8130 #19155756

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8130 #19155756