Group #19155759 — subgroups

Subgroups — C₂₃×#8133 #19155759

Order 5888.

446 conjugacy classes of subgroups; only the 270 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 270 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×#1744	C₂	1	2	yes	maximal
2	C₄²×C₈×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1273	C₂	1	4	yes	maximal
4	C₂×C₄×C₈×C₂₃	C₄	1	2	yes	—
5	C₂×C₄×C₈×C₂₃	C₄	1	2	yes	—
6	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
7	C₂×C₄×C₈×C₂₃	C₂²	1	4	yes	—
8	C₄³×C₂₃	C₂²	1	1	yes	characteristic
9	C₂₃×#336	C₄	1	8	yes	—
10	C₂²×C₈×C₂₃	C₂×C₄	1	8	yes	—
11	C₂²×C₈×C₂₃	C₂×C₄	1	4	yes	—
12	C₂×C₄²×C₂₃	C₂×C₄	1	4	yes	—
13	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
14	C₂×C₄²×C₂₃	C₂×C₄	1	1	yes	characteristic
15	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
16	C₄×C₈×C₂₃	—	2	4	no	—
17	C₄×C₈×C₂₃	—	2	4	no	—
18	C₄×C₈×C₂₃	—	2	8	no	—
19	C₄×C₈×C₂₃	C₈	1	8	yes	complemented
20	C₂²×C₄×C₂₃	C₄²	1	2	yes	—
21	C₂²×C₄×C₂₃	C₄²	1	2	yes	—
22	C₂²×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₂₃	C₂²×C₄	1	2	yes	—
24	C₂×C₈×C₂₃	—	2	4	no	—
25	C₂×C₈×C₂₃	—	2	4	no	—
26	C₂×C₈×C₂₃	—	2	8	no	—
27	C₂×C₈×C₂₃	—	2	16	no	—
28	C₂×C₈×C₂₃	C₂×C₈	1	8	yes	—
29	C₄²×C₂₃	—	2	8	no	—
30	C₄²×C₂₃	C₂×C₈	1	4	yes	—
31	C₄²×C₂₃	M₁₆	1	4	yes	—
32	C₄²×C₂₃	#41	1	4	yes	—
33	#8133	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
34	C₂³×C₂₃	C₂×C₄²	1	1	yes	characteristic
35	C₂×C₄×C₂₃	—	2	4	no	—
36	C₂×C₄×C₂₃	—	2	4	no	—
37	C₂×C₄×C₂₃	C₄×C₈	1	4	yes	—
38	C₂×C₄×C₂₃	#97	1	4	yes	—
39	C₂×C₄×C₂₃	#117	1	2	yes	—
40	C₂×C₄×C₂₃	#117	1	4	yes	—
41	C₂×C₄×C₂₃	C₂²×C₈	1	2	yes	—
42	C₂×C₄×C₂₃	C₂×M₁₆	1	2	yes	—
43	C₂×C₄×C₂₃	#131	1	8	yes	—
44	C₈×C₂₃	—	2	8	no	—
45	C₈×C₂₃	—	4	8	no	—
46	#1744	C₂×C₂₃	1	2	yes	—
47	C₄²×C₈	C₂×C₂₃	1	1	yes	characteristic
48	#1273	C₂×C₂₃	1	4	yes	—
49	C₂²×C₂₃	#376	1	1	yes	characteristic
50	C₂²×C₂₃	C₂×C₄×C₈	1	1	yes	characteristic
51	C₂²×C₂₃	C₂×#97	1	1	yes	characteristic
52	C₂²×C₂₃	#405	1	4	yes	—
53	C₄×C₂₃	—	2	4	no	—
54	C₄×C₂₃	—	2	4	no	—
55	C₄×C₂₃	#431	1	4	yes	—
56	C₄×C₂₃	#433	1	8	yes	—
57	C₂×C₄×C₈	C₄×C₂₃	1	2	yes	—
58	C₂×C₄×C₈	C₄×C₂₃	1	2	yes	—
59	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
60	C₂×C₄×C₈	C₂²×C₂₃	1	4	yes	—
61	C₄³	C₂²×C₂₃	1	1	yes	characteristic
62	#336	C₄×C₂₃	1	8	yes	—
63	C₂×C₂₃	#1450	1	2	yes	—
64	C₂×C₂₃	#1452	1	2	yes	—
65	C₂×C₂₃	#1744	1	1	yes	characteristic
66	C₂×C₂₃	#1746	1	2	yes	—
67	C₂²×C₈	C₂×C₄×C₂₃	1	8	yes	—
68	C₂²×C₈	C₂×C₄×C₂₃	1	4	yes	—
69	C₂×C₄²	C₂×C₄×C₂₃	1	4	yes	—
70	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
71	C₂×C₄²	C₂×C₄×C₂₃	1	1	yes	characteristic
72	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
73	C₄×C₈	—	2	4	no	—
74	C₄×C₈	—	2	4	no	—
75	C₄×C₈	—	2	8	no	—
76	C₄×C₈	C₈×C₂₃	1	8	yes	complemented
77	C₂₃	#8133	1	1	yes	characteristic, direct product factor, Sylow
78	C₂²×C₄	C₄²×C₂₃	1	2	yes	—
79	C₂²×C₄	C₄²×C₂₃	1	2	yes	—
80	C₂²×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
81	C₂²×C₄	C₂²×C₄×C₂₃	1	2	yes	—
82	C₂×C₈	—	2	4	no	—
83	C₂×C₈	—	2	4	no	—
84	C₂×C₈	—	2	8	no	—
85	C₂×C₈	—	2	16	no	—
86	C₂×C₈	C₂×C₈×C₂₃	1	8	yes	—
87	C₄²	—	2	8	no	—
88	C₄²	C₂×C₈×C₂₃	1	4	yes	—
89	C₄²	M₁₆×C₂₃	1	4	yes	—
90	C₄²	#41×C₂₃	1	4	yes	—
91	C₂³	C₂×C₄²×C₂₃	1	1	yes	characteristic
92	C₂×C₄	—	2	4	no	—
93	C₂×C₄	—	2	4	no	—
94	C₂×C₄	C₄×C₈×C₂₃	1	4	yes	—
95	C₂×C₄	C₂₃×#97	1	4	yes	—
96	C₂×C₄	C₂₃×#117	1	2	yes	—
97	C₂×C₄	C₂₃×#117	1	4	yes	—
98	C₂×C₄	C₂²×C₈×C₂₃	1	2	yes	—
99	C₂×C₄	C₂×M₁₆×C₂₃	1	2	yes	—
100	C₂×C₄	C₂₃×#131	1	8	yes	—
101	C₈	—	2	8	no	—
102	C₈	—	4	8	no	—
103	C₂²	C₂₃×#376	1	1	yes	characteristic
104	C₂²	C₂×C₄×C₈×C₂₃	1	1	yes	characteristic
105	C₂²	C₂×C₂₃×#97	1	1	yes	characteristic
106	C₂²	C₂₃×#405	1	4	yes	—
107	C₄	—	2	4	no	—
108	C₄	—	2	4	no	—
109	C₄	C₂₃×#431	1	4	yes	—
110	C₄	C₂₃×#433	1	8	yes	—
111	C₂	C₂₃×#1450	1	2	yes	—
112	C₂	C₂₃×#1452	1	2	yes	—
113	C₂	C₂₃×#1744	1	1	yes	characteristic
114	C₂	C₂₃×#1746	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#1744

C₂

yes

maximal

C₄²×C₈×C₂₃

C₂

yes

maximal, characteristic

C₂₃×#1273

C₂

yes

maximal

C₂×C₄×C₈×C₂₃

C₄

yes

—

C₂×C₄×C₈×C₂₃

C₄

yes

—

C₂×C₄×C₈×C₂₃

C₂²

yes

—

C₂×C₄×C₈×C₂₃

C₂²

yes

—

C₄³×C₂₃

C₂²

yes

characteristic

C₂₃×#336

C₄

yes

—

C₂²×C₈×C₂₃

C₂×C₄

yes

—

C₂²×C₈×C₂₃

C₂×C₄

yes

—

C₂×C₄²×C₂₃

C₂×C₄

yes

—

C₂×C₄²×C₂₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₂₃

C₂×C₄

yes

characteristic

C₂×C₄²×C₂₃

C₂³

yes

characteristic

C₄×C₈×C₂₃

—

C₄×C₈×C₂₃

—

C₄×C₈×C₂₃

—

C₄×C₈×C₂₃

C₈

yes

complemented

C₂²×C₄×C₂₃

C₄²

yes

—

C₂²×C₄×C₂₃

C₄²

yes

—

C₂²×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₂₃

C₂²×C₄

yes

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₂×C₈

yes

—

C₄²×C₂₃

—

C₄²×C₂₃

C₂×C₈

yes

—

C₄²×C₂₃

M₁₆

yes

—

C₄²×C₂₃

#41

yes

—

#8133

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₂₃

C₂×C₄²

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₄×C₈

yes

—

C₂×C₄×C₂₃

#97

yes

—

C₂×C₄×C₂₃

#117

yes

—

C₂×C₄×C₂₃

#117

yes

—

C₂×C₄×C₂₃

C₂²×C₈

yes

—

C₂×C₄×C₂₃

C₂×M₁₆

yes

—

C₂×C₄×C₂₃

#131

yes

—

C₈×C₂₃

—

C₈×C₂₃

—

#1744

C₂×C₂₃

yes

—

C₄²×C₈

C₂×C₂₃

yes

characteristic

#1273

C₂×C₂₃

yes

—

C₂²×C₂₃

#376

yes

characteristic

C₂²×C₂₃

C₂×C₄×C₈

yes

characteristic

C₂²×C₂₃

C₂×#97

yes

characteristic

C₂²×C₂₃

#405

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#431

yes

—

C₄×C₂₃

#433

yes

—

C₂×C₄×C₈

C₄×C₂₃

yes

—

C₂×C₄×C₈

C₄×C₂₃

yes

—

C₂×C₄×C₈

C₂²×C₂₃

yes

—

C₂×C₄×C₈

C₂²×C₂₃

yes

—

C₄³

C₂²×C₂₃

yes

characteristic

#336

C₄×C₂₃

yes

—

C₂×C₂₃

#1450

yes

—

C₂×C₂₃

#1452

yes

—

C₂×C₂₃

#1744

yes

characteristic

C₂×C₂₃

#1746

yes

—

C₂²×C₈

C₂×C₄×C₂₃

yes

—

C₂²×C₈

C₂×C₄×C₂₃

yes

—

C₂×C₄²

C₂×C₄×C₂₃

yes

—

C₂×C₄²

C₂×C₄×C₂₃

yes

characteristic

C₂×C₄²

C₂×C₄×C₂₃

yes

characteristic

C₂×C₄²

C₂³×C₂₃

yes

characteristic

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₈×C₂₃

yes

complemented

C₂₃

#8133

yes

characteristic, direct product factor, Sylow

C₂²×C₄

C₄²×C₂₃

yes

—

C₂²×C₄

C₄²×C₂₃

yes

—

C₂²×C₄

C₂²×C₄×C₂₃

yes

characteristic

C₂²×C₄

C₂²×C₄×C₂₃

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×C₈×C₂₃

yes

—

C₄²

—

C₄²

C₂×C₈×C₂₃

yes

—

C₄²

M₁₆×C₂₃

yes

—

C₄²

#41×C₂₃

yes

—

C₂³

C₂×C₄²×C₂₃

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×C₈×C₂₃

yes

—

C₂×C₄

C₂₃×#97

yes

—

C₂×C₄

C₂₃×#117

yes

—

C₂×C₄

C₂₃×#117

yes

—

C₂×C₄

C₂²×C₈×C₂₃

yes

—

C₂×C₄

C₂×M₁₆×C₂₃

yes

—

100

C₂×C₄

C₂₃×#131

yes

—

101

C₈

—

102

C₈

—

103

C₂²

C₂₃×#376

yes

characteristic

104

C₂²

C₂×C₄×C₈×C₂₃

yes

characteristic

105

C₂²

C₂×C₂₃×#97

yes

characteristic

106

C₂²

C₂₃×#405

yes

—

107

C₄

—

108

C₄

—

109

C₄

C₂₃×#431

yes

—

110

C₄

C₂₃×#433

yes

—

111

C₂

C₂₃×#1450

yes

—

112

C₂

C₂₃×#1452

yes

—

113

C₂

C₂₃×#1744

yes

characteristic

114

C₂

C₂₃×#1746

yes

—

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

subgroup

G/H

center

C₂×C₄²×C₂₃

C₂³

commutator

C₂²

C₂×C₄×C₈×C₂₃

Frattini, cosocle

C₂×C₄²

C₂³×C₂₃

socle, prime radical

C₂³×C₂₃

C₂×C₄²

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂×C₄²×C₂₃ ◁ G

N	H ≅ G/N	type
19. C₄×C₈×C₂₃	C₈	semidirect
33. #8133	C₂₃	direct (N × H)
76. C₄×C₈	C₈×C₂₃	semidirect
77. C₂₃	#8133	direct (N × H)

H ≅ G/N

type

19. C₄×C₈×C₂₃

C₈

semidirect

33. #8133

C₂₃

direct (N × H)

76. C₄×C₈

C₈×C₂₃

semidirect

77. C₂₃

#8133

direct (N × H)

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

subgroup	index
C₂₃×#1744	2
C₄²×C₈×C₂₃	2
C₂₃×#1273	2
#8133	23

subgroup

index

C₂₃×#1744

C₄²×C₈×C₂₃

C₂₃×#1273

#8133

Subgroups — C₂₃×#8133 #19155759

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#8133 #19155759

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8133 #19155759