Group #19159692 — subgroups

Subgroups — C₂₃×#12066 #19159692

Order 5888.

214 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 106 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×#2176	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1579	C₂	1	2	yes	maximal
3	C₂₃×#1368	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#1334	C₂	1	2	yes	maximal
5	C₂₃×#1313	C₂	1	1	yes	maximal, characteristic
6	C₂×C₂₃×Q₃₂	C₄	1	1	yes	characteristic
7	C₈×Q₈×C₂₃	—	2	2	no	—
8	C₄×Q₁₆×C₂₃	C₂²	1	2	yes	—
9	C₂₃×#367	C₄	1	1	yes	characteristic
10	C₂₃×#358	C₄	1	2	yes	—
11	C₂₃×#346	C₂²	1	1	yes	characteristic
12	C₂₃×#346	C₂²	1	1	yes	characteristic
13	C₂₃×#334	C₂²	1	2	yes	—
14	C₂₃×Q₈⋊C₈	—	2	2	no	—
15	C₂₃×#322	C₂²	1	1	yes	characteristic
16	C₂×Q₁₆×C₂₃	C₂×C₄	1	2	yes	—
17	C₄×Q₈×C₂₃	—	2	2	no	—
18	C₂₃×Q₃₂	C₈	1	4	yes	complemented
19	C₂×C₁₆×C₂₃	—	2	1	no	—
20	C₂×C₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
22	C₂₃×#107	C₂×C₄	1	2	yes	—
23	C₂₃×C₄⋊C₈	—	2	2	no	—
24	C₂₃×C₄⋊C₈	—	4	2	no	—
25	C₂₃×#103	—	2	2	no	—
26	C₄×C₈×C₂₃	—	4	2	no	—
27	C₄×C₈×C₂₃	D₄	1	2	yes	—
28	C₄×C₈×C₂₃	C₂³	1	1	yes	characteristic
29	C₂×Q₈×C₂₃	—	2	2	no	—
30	Q₁₆×C₂₃	—	2	2	no	—
31	Q₁₆×C₂₃	C₂×C₈	1	4	yes	—
32	C₂×C₈×C₂₃	—	2	2	no	—
33	C₂×C₈×C₂₃	—	4	2	no	—
34	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
35	C₂×C₈×C₂₃	#41	1	1	yes	characteristic
36	C₄⋊C₄×C₂₃	—	2	2	no	—
37	C₄⋊C₄×C₂₃	—	4	2	no	—
38	C₄²×C₂₃	—	4	2	no	—
39	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
40	C₁₆×C₂₃	—	2	1	no	—
41	C₁₆×C₂₃	—	4	1	no	—
42	C₁₆×C₂₃	C₂×C₈	1	2	yes	—
43	#12066	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	Q₈×C₂₃	—	2	4	no	—
45	Q₈×C₂₃	—	4	2	no	—
46	C₂×C₄×C₂₃	—	4	2	no	—
47	C₂×C₄×C₂₃	—	4	2	no	—
48	C₂×C₄×C₂₃	C₄×D₄	1	1	yes	characteristic
49	C₂×C₄×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
50	C₈×C₂₃	—	2	1	no	—
51	C₈×C₂₃	—	4	2	no	—
52	C₈×C₂₃	—	8	2	no	—
53	C₈×C₂₃	C₂²×C₈	1	1	yes	characteristic
54	C₈×C₂₃	#131	1	1	yes	characteristic
55	#2176	C₂×C₂₃	1	1	yes	characteristic
56	#1579	C₂×C₂₃	1	2	yes	—
57	#1368	C₂×C₂₃	1	1	yes	characteristic
58	#1334	C₂×C₂₃	1	2	yes	—
59	#1313	C₂×C₂₃	1	1	yes	characteristic
60	C₂²×C₂₃	#442	1	1	yes	characteristic
61	C₄×C₂₃	—	2	2	no	—
62	C₄×C₂₃	—	4	4	no	—
63	C₄×C₂₃	—	8	2	no	—
64	C₄×C₂₃	C₈×D₄	1	1	yes	characteristic
65	C₄×C₂₃	#444	1	1	yes	characteristic
66	C₂×Q₃₂	C₄×C₂₃	1	1	yes	characteristic
67	C₈×Q₈	—	2	2	no	—
68	C₄×Q₁₆	C₂²×C₂₃	1	2	yes	—
69	#367	C₄×C₂₃	1	1	yes	characteristic
70	#358	C₄×C₂₃	1	2	yes	—
71	#346	C₂²×C₂₃	1	1	yes	characteristic
72	#346	C₂²×C₂₃	1	1	yes	characteristic
73	#334	C₂²×C₂₃	1	2	yes	—
74	Q₈⋊C₈	—	2	2	no	—
75	#322	C₂²×C₂₃	1	1	yes	characteristic
76	C₂×C₂₃	#1580	1	1	yes	characteristic
77	C₂×C₂₃	#2180	1	2	yes	—
78	C₂×Q₁₆	C₂×C₄×C₂₃	1	2	yes	—
79	C₄×Q₈	—	2	2	no	—
80	Q₃₂	C₈×C₂₃	1	4	yes	complemented
81	C₂×C₁₆	—	2	1	no	—
82	C₂×C₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
83	C₂×C₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
84	#107	C₂×C₄×C₂₃	1	2	yes	—
85	C₄⋊C₈	—	2	2	no	—
86	C₄⋊C₈	—	4	2	no	—
87	#103	—	2	2	no	—
88	C₄×C₈	—	4	2	no	—
89	C₄×C₈	D₄×C₂₃	1	2	yes	—
90	C₄×C₈	C₂³×C₂₃	1	1	yes	characteristic
91	C₂₃	#12066	1	1	yes	characteristic, direct product factor, Sylow
92	C₂×Q₈	—	2	2	no	—
93	Q₁₆	—	2	2	no	—
94	Q₁₆	C₂×C₈×C₂₃	1	4	yes	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	4	2	no	—
97	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
98	C₂×C₈	#41×C₂₃	1	1	yes	characteristic
99	C₄⋊C₄	—	2	2	no	—
100	C₄⋊C₄	—	4	2	no	—
101	C₄²	—	4	2	no	—
102	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
103	C₁₆	—	2	1	no	—
104	C₁₆	—	4	1	no	—
105	C₁₆	C₂×C₈×C₂₃	1	2	yes	—
106	Q₈	—	2	4	no	—
107	Q₈	—	4	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	C₄×D₄×C₂₃	1	1	yes	characteristic
111	C₂×C₄	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
112	C₈	—	2	1	no	—
113	C₈	—	4	2	no	—
114	C₈	—	8	2	no	—
115	C₈	C₂²×C₈×C₂₃	1	1	yes	characteristic
116	C₈	C₂₃×#131	1	1	yes	characteristic
117	C₂²	C₂₃×#442	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	4	4	no	—
120	C₄	—	8	2	no	—
121	C₄	C₈×D₄×C₂₃	1	1	yes	characteristic
122	C₄	C₂₃×#444	1	1	yes	characteristic
123	C₂	C₂₃×#1580	1	1	yes	characteristic
124	C₂	C₂₃×#2180	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#2176

C₂

yes

maximal, characteristic

C₂₃×#1579

C₂

yes

maximal

C₂₃×#1368

C₂

yes

maximal, characteristic

C₂₃×#1334

C₂

yes

maximal

C₂₃×#1313

C₂

yes

maximal, characteristic

C₂×C₂₃×Q₃₂

C₄

yes

characteristic

C₈×Q₈×C₂₃

—

C₄×Q₁₆×C₂₃

C₂²

yes

—

C₂₃×#367

C₄

yes

characteristic

C₂₃×#358

C₄

yes

—

C₂₃×#346

C₂²

yes

characteristic

C₂₃×#346

C₂²

yes

characteristic

C₂₃×#334

C₂²

yes

—

C₂₃×Q₈⋊C₈

—

C₂₃×#322

C₂²

yes

characteristic

C₂×Q₁₆×C₂₃

C₂×C₄

yes

—

C₄×Q₈×C₂₃

—

C₂₃×Q₃₂

C₈

yes

complemented

C₂×C₁₆×C₂₃

—

C₂×C₁₆×C₂₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₂₃

C₂×C₄

yes

characteristic

C₂₃×#107

C₂×C₄

yes

—

C₂₃×C₄⋊C₈

—

C₂₃×C₄⋊C₈

—

C₂₃×#103

—

C₄×C₈×C₂₃

—

C₄×C₈×C₂₃

D₄

yes

—

C₄×C₈×C₂₃

C₂³

yes

characteristic

C₂×Q₈×C₂₃

—

Q₁₆×C₂₃

—

Q₁₆×C₂₃

C₂×C₈

yes

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₂₃

#41

yes

characteristic

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

C₂×D₄

yes

characteristic

C₁₆×C₂₃

—

C₁₆×C₂₃

—

C₁₆×C₂₃

C₂×C₈

yes

—

#12066

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

Q₈×C₂₃

—

Q₈×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₄×D₄

yes

characteristic

C₂×C₄×C₂₃

AGL(1,ℤ/8)

yes

—

C₈×C₂₃

—

C₈×C₂₃

—

C₈×C₂₃

—

C₈×C₂₃

C₂²×C₈

yes

C₈×C₂₃

yes

C₂×C₂₃

yes

characteristic

#1579

C₂×C₂₃

yes

—

#1368

C₂×C₂₃

yes

characteristic

#1334

C₂×C₂₃

yes

—

#1313

C₂×C₂₃

yes

characteristic

C₂²×C₂₃

#442

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₈×D₄

yes

characteristic

C₄×C₂₃

#444

yes

characteristic

C₂×Q₃₂

C₄×C₂₃

yes

characteristic

C₈×Q₈

—

C₄×Q₁₆

C₂²×C₂₃

yes

—

#367

C₄×C₂₃

yes

characteristic

#358

C₄×C₂₃

yes

—

#346

C₂²×C₂₃

yes

characteristic

#346

C₂²×C₂₃

yes

characteristic

#334

C₂²×C₂₃

yes

—

Q₈⋊C₈

—

#322

C₂²×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

—

C₂×Q₁₆

C₂×C₄×C₂₃

yes

—

C₄×Q₈

—

Q₃₂

C₈×C₂₃

yes

complemented

C₂×C₁₆

—

C₂×C₁₆

C₂×C₄×C₂₃

yes

characteristic

C₂×C₁₆

C₂×C₄×C₂₃

yes

characteristic

#107

C₂×C₄×C₂₃

yes

—

C₄⋊C₈

—

C₄⋊C₈

—

#103

—

C₄×C₈

—

C₄×C₈

D₄×C₂₃

yes

—

C₄×C₈

C₂³×C₂₃

yes

characteristic

C₂₃

#12066

yes

characteristic, direct product factor, Sylow

C₂×Q₈

—

Q₁₆

—

Q₁₆

C₂×C₈×C₂₃

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²×C₄×C₂₃

yes

characteristic

C₂×C₈

#41×C₂₃

yes

characteristic

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄²

—

102

C₄²

C₂×D₄×C₂₃

yes

characteristic

103

C₁₆

—

104

C₁₆

—

105

C₁₆

C₂×C₈×C₂₃

yes

—

106

Q₈

—

107

Q₈

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

C₄×D₄×C₂₃

yes

characteristic

111

C₂×C₄

C₂₃×AGL(1,ℤ/8)

yes

—

112

C₈

—

113

C₈

—

114

C₈

—

115

C₈

C₂²×C₈×C₂₃

yes

characteristic

116

C₈

C₂₃×#131

yes

characteristic

117

C₂²

C₂₃×#442

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

C₈×D₄×C₂₃

yes

characteristic

122

C₄

C₂₃×#444

yes

characteristic

123

C₂

C₂₃×#1580

yes

characteristic

124

C₂

C₂₃×#2180

yes

—

	subgroup	G/H
center, socle, prime radical	C₂²×C₂₃	#442
commutator	C₈	C₂²×C₈×C₂₃
Frattini, cosocle	C₄×C₈	C₂³×C₂₃
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₂₃ ◁ C₄²×C₂₃ ◁ C₂₃×#322 ◁ G

subgroup

G/H

center, socle, prime radical

C₂²×C₂₃

#442

commutator

C₈

C₂²×C₈×C₂₃

Frattini, cosocle

C₄×C₈

C₂³×C₂₃

LCS

G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₂₃ ◁ C₄²×C₂₃ ◁ C₂₃×#322 ◁ G

N	H ≅ G/N	type
18. C₂₃×Q₃₂	C₈	semidirect
43. #12066	C₂₃	direct (N × H)
80. Q₃₂	C₈×C₂₃	semidirect
91. C₂₃	#12066	direct (N × H)

H ≅ G/N

type

18. C₂₃×Q₃₂

C₈

semidirect

43. #12066

C₂₃

direct (N × H)

80. Q₃₂

C₈×C₂₃

semidirect

91. C₂₃

#12066

direct (N × H)

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

subgroup	index
C₂₃×#2176	2
C₂₃×#1579	2
C₂₃×#1368	2
C₂₃×#1334	2
C₂₃×#1313	2
#12066	23

subgroup

index

C₂₃×#2176

C₂₃×#1579

C₂₃×#1368

C₂₃×#1334

C₂₃×#1313

#12066

Subgroups — C₂₃×#12066 #19159692

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#12066 #19159692

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12066 #19159692