Group #19158212 — subgroups

Subgroups — C₂₃×#10586 #19158212

Order 5888.

350 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 114 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×#2845	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1836	C₂	1	3	yes	maximal
3	C₂₃×#1295	C₂	1	3	yes	maximal
4	C₂×C₂₃×#97	C₂²	1	3	yes	—
5	C₂₃×#398	—	2	6	no	—
6	C₂₃×#384	C₂²	1	1	yes	characteristic
7	C₂₃×#382	C₂²	1	3	yes	—
8	C₂₃×#340	—	2	6	no	—
9	C₂²×C₈×C₂₃	—	2	3	no	—
10	C₂×C₄⋊C₄×C₂₃	—	2	3	no	—
11	C₂×C₄⋊C₄×C₂₃	—	2	6	no	—
12	C₂×C₄⋊C₄×C₂₃	D₄	1	6	yes	—
13	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
14	C₂₃×#97	—	2	3	no	—
15	C₂₃×#97	Q₈	1	6	yes	—
16	C₂₃×#95	—	2	2	no	—
17	C₂₃×#95	—	2	6	no	—
18	C₂²×C₄×C₂₃	—	2	6	no	—
19	C₂²×C₄×C₂₃	—	4	1	no	—
20	C₂²×C₄×C₂₃	C₂×D₄	1	3	yes	—
21	C₂×C₈×C₂₃	—	2	6	no	—
22	C₂×C₈×C₂₃	—	4	3	no	—
23	C₂×C₈×C₂₃	—	4	3	no	—
24	C₄⋊C₄×C₂₃	—	4	6	no	—
25	C₄⋊C₄×C₂₃	—	4	6	no	—
26	C₄⋊C₄×C₂₃	—	8	6	no	—
27	C₄²×C₂₃	C₂×Q₈	1	3	yes	—
28	C₄²×C₂₃	#41	1	1	yes	characteristic
29	#10586	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
30	C₂³×C₂₃	#120	1	1	yes	characteristic
31	C₂×C₄×C₂₃	—	2	3	no	—
32	C₂×C₄×C₂₃	—	2	3	no	—
33	C₂×C₄×C₂₃	—	4	6	no	—
34	C₂×C₄×C₂₃	—	4	6	no	—
35	C₂×C₄×C₂₃	—	4	6	no	—
36	C₂×C₄×C₂₃	—	8	3	no	—
37	C₂×C₄×C₂₃	C₂²⋊Q₈	1	3	yes	—
38	C₂×C₄×C₂₃	C₄⋊Q₈	1	3	yes	—
39	C₈×C₂₃	—	4	6	no	—
40	#2845	C₂×C₂₃	1	1	yes	characteristic
41	#1836	C₂×C₂₃	1	3	yes	—
42	#1295	C₂×C₂₃	1	3	yes	—
43	C₂²×C₂₃	#395	1	1	yes	characteristic
44	C₂²×C₂₃	#454	1	3	yes	—
45	C₂²×C₂₃	#455	1	3	yes	—
46	C₄×C₂₃	—	2	3	no	—
47	C₄×C₂₃	—	2	3	no	—
48	C₄×C₂₃	—	8	6	no	—
49	C₄×C₂₃	—	16	1	no	—
50	C₂×#97	C₂²×C₂₃	1	3	yes	—
51	#398	—	2	6	no	—
52	#384	C₂²×C₂₃	1	1	yes	characteristic
53	#382	C₂²×C₂₃	1	3	yes	—
54	#340	—	2	6	no	—
55	C₂×C₂₃	#1658	1	1	yes	characteristic
56	C₂×C₂₃	#1662	1	3	yes	—
57	C₂×C₂₃	#2061	1	3	yes	—
58	C₂²×C₈	—	2	3	no	—
59	C₂×C₄⋊C₄	—	2	3	no	—
60	C₂×C₄⋊C₄	—	2	6	no	—
61	C₂×C₄⋊C₄	D₄×C₂₃	1	6	yes	—
62	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
63	#97	—	2	3	no	—
64	#97	Q₈×C₂₃	1	6	yes	—
65	#95	—	2	2	no	—
66	#95	—	2	6	no	—
67	C₂₃	#10586	1	1	yes	characteristic, direct product factor, Sylow
68	C₂²×C₄	—	2	6	no	—
69	C₂²×C₄	—	4	1	no	—
70	C₂²×C₄	C₂×D₄×C₂₃	1	3	yes	—
71	C₂×C₈	—	2	6	no	—
72	C₂×C₈	—	4	3	no	—
73	C₂×C₈	—	4	3	no	—
74	C₄⋊C₄	—	4	6	no	—
75	C₄⋊C₄	—	4	6	no	—
76	C₄⋊C₄	—	8	6	no	—
77	C₄²	C₂×Q₈×C₂₃	1	3	yes	—
78	C₄²	#41×C₂₃	1	1	yes	characteristic
79	C₂³	C₂₃×#120	1	1	yes	characteristic
80	C₂×C₄	—	2	3	no	—
81	C₂×C₄	—	2	3	no	—
82	C₂×C₄	—	4	6	no	—
83	C₂×C₄	—	4	6	no	—
84	C₂×C₄	—	4	6	no	—
85	C₂×C₄	—	8	3	no	—
86	C₂×C₄	C₂₃×C₂²⋊Q₈	1	3	yes	—
87	C₂×C₄	C₂₃×C₄⋊Q₈	1	3	yes	—
88	C₈	—	4	6	no	—
89	C₂²	C₂₃×#395	1	1	yes	characteristic
90	C₂²	C₂₃×#454	1	3	yes	—
91	C₂²	C₂₃×#455	1	3	yes	—
92	C₄	—	2	3	no	—
93	C₄	—	2	3	no	—
94	C₄	—	8	6	no	—
95	C₄	—	16	1	no	—
96	C₂	C₂₃×#1658	1	1	yes	characteristic
97	C₂	C₂₃×#1662	1	3	yes	—
98	C₂	C₂₃×#2061	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#2845

C₂

yes

maximal, characteristic

C₂₃×#1836

C₂

yes

maximal

C₂₃×#1295

C₂

yes

maximal

C₂×C₂₃×#97

C₂²

yes

—

C₂₃×#398

—

C₂₃×#384

C₂²

yes

characteristic

C₂₃×#382

C₂²

yes

—

C₂₃×#340

—

C₂²×C₈×C₂₃

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

D₄

yes

—

C₂×C₄²×C₂₃

C₂³

yes

characteristic

C₂₃×#97

—

C₂₃×#97

Q₈

yes

—

C₂₃×#95

—

C₂₃×#95

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄²×C₂₃

C₂×Q₈

yes

—

C₄²×C₂₃

#41

yes

characteristic

#10586

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₂₃

#120

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₂₃

C₄⋊Q₈

yes

—

C₈×C₂₃

—

#2845

C₂×C₂₃

yes

characteristic

#1836

C₂×C₂₃

yes

—

#1295

C₂×C₂₃

yes

—

C₂²×C₂₃

#395

yes

characteristic

C₂²×C₂₃

#454

yes

—

C₂²×C₂₃

#455

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₂×#97

C₂²×C₂₃

yes

—

#398

—

#384

C₂²×C₂₃

yes

characteristic

#382

C₂²×C₂₃

yes

—

#340

—

C₂×C₂₃

#1658

yes

characteristic

C₂×C₂₃

#1662

yes

—

C₂×C₂₃

#2061

yes

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

D₄×C₂₃

yes

—

C₂×C₄²

C₂³×C₂₃

yes

characteristic

#97

—

#97

Q₈×C₂₃

yes

—

#95

—

#95

—

C₂₃

#10586

yes

characteristic, direct product factor, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×D₄×C₂₃

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂×Q₈×C₂₃

yes

—

C₄²

#41×C₂₃

yes

characteristic

C₂³

C₂₃×#120

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂₃×C₂²⋊Q₈

yes

—

C₂×C₄

C₂₃×C₄⋊Q₈

yes

—

C₈

—

C₂²

C₂₃×#395

yes

characteristic

C₂²

C₂₃×#454

yes

—

C₂²

C₂₃×#455

yes

—

C₄

—

C₄

—

C₄

—

C₄

—

C₂

C₂₃×#1658

yes

characteristic

C₂

C₂₃×#1662

yes

—

C₂

C₂₃×#2061

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₂₃

#120

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₂₃

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

N	H ≅ G/N	type
29. #10586	C₂₃	direct (N × H)
67. C₂₃	#10586	direct (N × H)

H ≅ G/N

type

29. #10586

C₂₃

direct (N × H)

67. C₂₃

#10586

direct (N × H)

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

subgroup	index
C₂₃×#2845	2
C₂₃×#1836	2
C₂₃×#1295	2
#10586	23

subgroup

index

C₂₃×#2845

C₂₃×#1836

C₂₃×#1295

#10586

Subgroups — C₂₃×#10586 #19158212

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#10586 #19158212

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#10586 #19158212