Group #19160518 — subgroups

Subgroups — C₂₃×#12892 #19160518

Order 5888.

274 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 158 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×C₂₃×#346	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1848	C₂	1	2	yes	maximal
3	C₂₃×#1381	C₂	1	4	yes	maximal
4	C₂²×C₁₆×C₂₃	C₂²	1	2	yes	—
5	C₂×C₂₃×#108	—	2	2	no	—
6	C₂×C₂₃×#108	C₂²	1	4	yes	—
7	C₂₃×#423	—	2	2	no	—
8	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₂₃×#346	C₄	1	4	yes	—
10	C₂×M₁₆×C₂₃	—	2	4	no	—
11	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
12	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
13	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
14	C₂×C₁₆×C₂₃	—	2	4	no	—
15	C₂×C₁₆×C₂₃	C₂×C₄	1	4	yes	—
16	C₂₃×#108	—	2	8	no	—
17	C₂₃×#108	—	4	4	no	—
18	C₂₃×C₄⋊C₈	—	4	4	no	—
19	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
20	C₄×C₈×C₂₃	Q₈	1	2	yes	—
21	C₂²×C₄×C₂₃	Q₁₆	1	2	yes	—
22	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
23	M₁₆×C₂₃	—	4	4	no	—
24	M₁₆×C₂₃	—	4	4	no	—
25	C₂×C₈×C₂₃	—	4	4	no	—
26	C₂×C₈×C₂₃	C₄⋊C₄	1	2	yes	—
27	C₂×C₈×C₂₃	SD₁₆	1	4	yes	—
28	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
30	C₂×C₈×C₂₃	#41	1	4	yes	—
31	C₄²×C₂₃	C₄⋊C₄	1	2	yes	—
32	C₄²×C₂₃	D₈	1	2	yes	—
33	C₁₆×C₂₃	—	2	4	no	—
34	#12892	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
35	C₂³×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
36	C₂×C₄×C₂₃	—	4	2	no	—
37	C₂×C₄×C₂₃	#107	1	4	yes	—
38	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
39	C₂×C₄×C₂₃	#124	1	2	yes	—
40	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
41	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
42	C₈×C₂₃	—	8	4	no	—
43	C₈×C₂₃	#106	1	4	yes	—
44	C₈×C₂₃	#117	1	2	yes	—
45	C₈×C₂₃	#125	1	2	yes	—
46	C₂×#346	C₂×C₂₃	1	1	yes	characteristic
47	#1848	C₂×C₂₃	1	2	yes	—
48	#1381	C₂×C₂₃	1	4	yes	—
49	C₂²×C₂₃	—	2	2	no	—
50	C₂²×C₂₃	C₂×#107	1	1	yes	characteristic
51	C₂²×C₂₃	#487	1	2	yes	—
52	C₄×C₂₃	—	2	4	no	—
53	C₄×C₂₃	#382	1	1	yes	characteristic
54	C₄×C₂₃	C₂×#106	1	1	yes	characteristic
55	C₄×C₂₃	#486	1	2	yes	—
56	C₂²×C₁₆	C₂²×C₂₃	1	2	yes	—
57	C₂×#108	—	2	2	no	—
58	C₂×#108	C₂²×C₂₃	1	4	yes	—
59	#423	—	2	2	no	—
60	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
61	#346	C₄×C₂₃	1	4	yes	—
62	C₂×C₂₃	—	2	2	no	—
63	C₂×C₂₃	#1834	1	1	yes	characteristic
64	C₂×C₂₃	#2161	1	2	yes	—
65	C₂×M₁₆	—	2	4	no	—
66	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
67	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
68	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
69	C₂×C₁₆	—	2	4	no	—
70	C₂×C₁₆	C₂×C₄×C₂₃	1	4	yes	—
71	#108	—	2	8	no	—
72	#108	—	4	4	no	—
73	C₄⋊C₈	—	4	4	no	—
74	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
75	C₄×C₈	Q₈×C₂₃	1	2	yes	—
76	C₂₃	#12892	1	1	yes	characteristic, direct product factor, Sylow
77	C₂²×C₄	Q₁₆×C₂₃	1	2	yes	—
78	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
79	M₁₆	—	4	4	no	—
80	M₁₆	—	4	4	no	—
81	C₂×C₈	—	4	4	no	—
82	C₂×C₈	C₄⋊C₄×C₂₃	1	2	yes	—
83	C₂×C₈	SD₁₆×C₂₃	1	4	yes	—
84	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
85	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
86	C₂×C₈	#41×C₂₃	1	4	yes	—
87	C₄²	C₄⋊C₄×C₂₃	1	2	yes	—
88	C₄²	D₈×C₂₃	1	2	yes	—
89	C₁₆	—	2	4	no	—
90	C₂³	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
91	C₂×C₄	—	4	2	no	—
92	C₂×C₄	C₂₃×#107	1	4	yes	—
93	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
94	C₂×C₄	C₂₃×#124	1	2	yes	—
95	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
96	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
97	C₈	—	8	4	no	—
98	C₈	C₂₃×#106	1	4	yes	—
99	C₈	C₂₃×#117	1	2	yes	—
100	C₈	C₂₃×#125	1	2	yes	—
101	C₂²	—	2	2	no	—
102	C₂²	C₂×C₂₃×#107	1	1	yes	characteristic
103	C₂²	C₂₃×#487	1	2	yes	—
104	C₄	—	2	4	no	—
105	C₄	C₂₃×#382	1	1	yes	characteristic
106	C₄	C₂×C₂₃×#106	1	1	yes	characteristic
107	C₄	C₂₃×#486	1	2	yes	—
108	C₂	—	2	2	no	—
109	C₂	C₂₃×#1834	1	1	yes	characteristic
110	C₂	C₂₃×#2161	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂₃×#346

C₂

yes

maximal, characteristic

C₂₃×#1848

C₂

yes

maximal

C₂₃×#1381

C₂

yes

maximal

C₂²×C₁₆×C₂₃

C₂²

yes

—

C₂×C₂₃×#108

—

C₂×C₂₃×#108

C₂²

yes

—

C₂₃×#423

—

C₂×C₄×C₈×C₂₃

C₂²

yes

characteristic

C₂₃×#346

C₄

yes

—

C₂×M₁₆×C₂₃

—

C₂²×C₈×C₂₃

D₄

yes

characteristic

C₂²×C₈×C₂₃

C₂³

yes

characteristic

C₂×C₄²×C₂₃

D₄

yes

characteristic

C₂×C₁₆×C₂₃

—

C₂×C₁₆×C₂₃

C₂×C₄

yes

—

C₂₃×#108

—

C₂₃×#108

—

C₂₃×C₄⋊C₈

—

C₄×C₈×C₂₃

C₂×C₄

yes

—

C₄×C₈×C₂₃

Q₈

yes

—

C₂²×C₄×C₂₃

Q₁₆

yes

—

C₂²×C₄×C₂₃

C₂×D₄

yes

characteristic

M₁₆×C₂₃

—

M₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₄⋊C₄

yes

—

C₂×C₈×C₂₃

SD₁₆

yes

—

C₂×C₈×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₈×C₂₃

#41

yes

—

C₄²×C₂₃

C₄⋊C₄

yes

—

C₄²×C₂₃

D₈

yes

—

C₁₆×C₂₃

—

#12892

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₂₃

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

#107

yes

—

C₂×C₄×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₂₃

#124

yes

—

C₂×C₄×C₂₃

C₂×D₈

yes

characteristic

C₂×C₄×C₂₃

C₂×SD₁₆

yes

—

C₈×C₂₃

—

C₈×C₂₃

#106

yes

—

C₈×C₂₃

#117

yes

—

C₈×C₂₃

#125

yes

—

C₂×#346

C₂×C₂₃

yes

characteristic

#1848

C₂×C₂₃

yes

—

#1381

C₂×C₂₃

yes

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×#107

yes

characteristic

C₂²×C₂₃

#487

yes

—

C₄×C₂₃

—

C₄×C₂₃

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

—

C₂²×C₁₆

C₂²×C₂₃

yes

—

C₂×#108

—

C₂×#108

C₂²×C₂₃

yes

—

#423

—

C₂×C₄×C₈

C₂²×C₂₃

yes

characteristic

#346

C₄×C₂₃

yes

—

C₂×C₂₃

—

C₂×C₂₃

#1834

yes

characteristic

C₂×C₂₃

#2161

yes

—

C₂×M₁₆

—

C₂²×C₈

D₄×C₂₃

yes

characteristic

C₂²×C₈

C₂³×C₂₃

yes

characteristic

C₂×C₄²

D₄×C₂₃

yes

characteristic

C₂×C₁₆

—

C₂×C₁₆

C₂×C₄×C₂₃

yes

—

#108

—

#108

—

C₄⋊C₈

—

C₄×C₈

C₂×C₄×C₂₃

yes

—

C₄×C₈

Q₈×C₂₃

yes

—

C₂₃

#12892

yes

characteristic, direct product factor, Sylow

C₂²×C₄

Q₁₆×C₂₃

yes

—

C₂²×C₄

C₂×D₄×C₂₃

yes

characteristic

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

C₄⋊C₄×C₂₃

yes

—

C₂×C₈

SD₁₆×C₂₃

yes

—

C₂×C₈

C₂²×C₄×C₂₃

yes

characteristic

C₂×C₈

C₂×Q₈×C₂₃

yes

characteristic

C₂×C₈

#41×C₂₃

yes

—

C₄²

C₄⋊C₄×C₂₃

yes

—

C₄²

D₈×C₂₃

yes

—

C₁₆

—

C₂³

C₂×Q₁₆×C₂₃

yes

characteristic

C₂×C₄

—

C₂×C₄

C₂₃×#107

yes

—

C₂×C₄

C₂×C₄⋊C₄×C₂₃

yes

characteristic

C₂×C₄

C₂₃×#124

yes

—

C₂×C₄

C₂×D₈×C₂₃

yes

characteristic

C₂×C₄

C₂×SD₁₆×C₂₃

yes

—

C₈

—

C₈

C₂₃×#106

yes

—

C₈

C₂₃×#117

yes

—

100

C₈

C₂₃×#125

yes

—

101

C₂²

—

102

C₂²

C₂×C₂₃×#107

yes

characteristic

103

C₂²

C₂₃×#487

yes

—

104

C₄

—

105

C₄

C₂₃×#382

yes

characteristic

106

C₄

C₂×C₂₃×#106

yes

characteristic

107

C₄

C₂₃×#486

yes

—

108

C₂

—

109

C₂

C₂₃×#1834

yes

characteristic

110

C₂

C₂₃×#2161

yes

—

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂×D₈
commutator	C₂×C₈	C₂²×C₄×C₂₃
Frattini, cosocle	C₂²×C₈	C₂³×C₂₃
socle	C₂²×C₂₃	C₂×#107
prime radical	C₂³×C₂₃	C₂×Q₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

subgroup

G/H

center

C₂×C₄×C₂₃

C₂×D₈

commutator

C₂×C₈

C₂²×C₄×C₂₃

Frattini, cosocle

C₂²×C₈

C₂³×C₂₃

socle

C₂²×C₂₃

C₂×#107

prime radical

C₂³×C₂₃

C₂×Q₁₆

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ C₂×C₄×C₈×C₂₃ ◁ G

N	H ≅ G/N	type
34. #12892	C₂₃	direct (N × H)
76. C₂₃	#12892	direct (N × H)

H ≅ G/N

type

34. #12892

C₂₃

direct (N × H)

76. C₂₃

#12892

direct (N × H)

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

subgroup	index
C₂×C₂₃×#346	2
C₂₃×#1848	2
C₂₃×#1381	2
#12892	23

subgroup

index

C₂×C₂₃×#346

C₂₃×#1848

C₂₃×#1381

#12892

Subgroups — C₂₃×#12892 #19160518

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#12892 #19160518

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12892 #19160518