Group #19160517 — subgroups

Subgroups — C₂₃×#12891 #19160517

Order 5888.

282 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 282 conjugacy classes of subgroups — too many to draw. Only its 166 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×C₄×C₁₆×C₂₃	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	C₄×C₁₆×C₂₃	C₄	1	2	yes	—
10	C₄×C₁₆×C₂₃	C₄	1	2	yes	—
11	C₂×M₁₆×C₂₃	—	2	4	no	—
12	C₂²×C₈×C₂₃	D₄	1	1	yes	characteristic
13	C₂²×C₈×C₂₃	C₂³	1	1	yes	characteristic
14	C₂×C₄²×C₂₃	D₄	1	1	yes	characteristic
15	C₂×C₁₆×C₂₃	—	2	4	no	—
16	C₂×C₁₆×C₂₃	C₂×C₄	1	4	yes	—
17	C₂₃×#108	—	2	8	no	—
18	C₂₃×#108	—	4	4	no	—
19	C₂₃×C₄⋊C₈	—	4	4	no	—
20	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
21	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
22	C₄×C₈×C₂₃	Q₈	1	1	yes	characteristic
23	C₄×C₈×C₂₃	Q₈	1	1	yes	characteristic
24	C₂²×C₄×C₂₃	Q₁₆	1	2	yes	—
25	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
26	M₁₆×C₂₃	—	4	4	no	—
27	M₁₆×C₂₃	—	4	4	no	—
28	C₂×C₈×C₂₃	—	4	4	no	—
29	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
30	C₂×C₈×C₂₃	C₄⋊C₄	1	1	yes	characteristic
31	C₂×C₈×C₂₃	SD₁₆	1	4	yes	—
32	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
33	C₂×C₈×C₂₃	C₂×Q₈	1	1	yes	characteristic
34	C₂×C₈×C₂₃	#41	1	4	yes	—
35	C₄²×C₂₃	C₄⋊C₄	1	1	yes	characteristic
36	C₄²×C₂₃	C₄⋊C₄	1	1	yes	characteristic
37	C₄²×C₂₃	D₈	1	2	yes	—
38	C₁₆×C₂₃	—	2	4	no	—
39	#12891	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂³×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
41	C₂×C₄×C₂₃	—	2	4	no	—
42	C₂×C₄×C₂₃	#107	1	2	yes	—
43	C₂×C₄×C₂₃	#107	1	2	yes	—
44	C₂×C₄×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₂₃	#124	1	2	yes	—
46	C₂×C₄×C₂₃	C₂×D₈	1	1	yes	characteristic
47	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
48	C₈×C₂₃	—	8	4	no	—
49	C₈×C₂₃	#106	1	2	yes	—
50	C₈×C₂₃	#106	1	2	yes	—
51	C₈×C₂₃	#117	1	2	yes	—
52	C₈×C₂₃	#125	1	2	yes	—
53	C₂×C₄×C₁₆	C₂×C₂₃	1	1	yes	characteristic
54	#1848	C₂×C₂₃	1	2	yes	—
55	#1381	C₂×C₂₃	1	4	yes	—
56	C₂²×C₂₃	—	2	2	no	—
57	C₂²×C₂₃	C₂×#107	1	1	yes	characteristic
58	C₂²×C₂₃	#487	1	2	yes	—
59	C₄×C₂₃	—	2	2	no	—
60	C₄×C₂₃	#368	1	4	yes	—
61	C₄×C₂₃	#382	1	1	yes	characteristic
62	C₄×C₂₃	C₂×#106	1	1	yes	characteristic
63	C₄×C₂₃	#486	1	2	yes	—
64	C₂²×C₁₆	C₂²×C₂₃	1	2	yes	—
65	C₂×#108	—	2	2	no	—
66	C₂×#108	C₂²×C₂₃	1	4	yes	—
67	#423	—	2	2	no	—
68	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
69	C₄×C₁₆	C₄×C₂₃	1	2	yes	—
70	C₄×C₁₆	C₄×C₂₃	1	2	yes	—
71	C₂×C₂₃	—	2	2	no	—
72	C₂×C₂₃	#1834	1	1	yes	characteristic
73	C₂×C₂₃	C₂×#368	1	2	yes	—
74	C₂×M₁₆	—	2	4	no	—
75	C₂²×C₈	D₄×C₂₃	1	1	yes	characteristic
76	C₂²×C₈	C₂³×C₂₃	1	1	yes	characteristic
77	C₂×C₄²	D₄×C₂₃	1	1	yes	characteristic
78	C₂×C₁₆	—	2	4	no	—
79	C₂×C₁₆	C₂×C₄×C₂₃	1	4	yes	—
80	#108	—	2	8	no	—
81	#108	—	4	4	no	—
82	C₄⋊C₈	—	4	4	no	—
83	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
84	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
85	C₄×C₈	Q₈×C₂₃	1	1	yes	characteristic
86	C₄×C₈	Q₈×C₂₃	1	1	yes	characteristic
87	C₂₃	#12891	1	1	yes	characteristic, direct product factor, Sylow
88	C₂²×C₄	Q₁₆×C₂₃	1	2	yes	—
89	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
90	M₁₆	—	4	4	no	—
91	M₁₆	—	4	4	no	—
92	C₂×C₈	—	4	4	no	—
93	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
94	C₂×C₈	C₄⋊C₄×C₂₃	1	1	yes	characteristic
95	C₂×C₈	SD₁₆×C₂₃	1	4	yes	—
96	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
97	C₂×C₈	C₂×Q₈×C₂₃	1	1	yes	characteristic
98	C₂×C₈	#41×C₂₃	1	4	yes	—
99	C₄²	C₄⋊C₄×C₂₃	1	1	yes	characteristic
100	C₄²	C₄⋊C₄×C₂₃	1	1	yes	characteristic
101	C₄²	D₈×C₂₃	1	2	yes	—
102	C₁₆	—	2	4	no	—
103	C₂³	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	C₂₃×#107	1	2	yes	—
106	C₂×C₄	C₂₃×#107	1	2	yes	—
107	C₂×C₄	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
108	C₂×C₄	C₂₃×#124	1	2	yes	—
109	C₂×C₄	C₂×D₈×C₂₃	1	1	yes	characteristic
110	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
111	C₈	—	8	4	no	—
112	C₈	C₂₃×#106	1	2	yes	—
113	C₈	C₂₃×#106	1	2	yes	—
114	C₈	C₂₃×#117	1	2	yes	—
115	C₈	C₂₃×#125	1	2	yes	—
116	C₂²	—	2	2	no	—
117	C₂²	C₂×C₂₃×#107	1	1	yes	characteristic
118	C₂²	C₂₃×#487	1	2	yes	—
119	C₄	—	2	2	no	—
120	C₄	C₂₃×#368	1	4	yes	—
121	C₄	C₂₃×#382	1	1	yes	characteristic
122	C₄	C₂×C₂₃×#106	1	1	yes	characteristic
123	C₄	C₂₃×#486	1	2	yes	—
124	C₂	—	2	2	no	—
125	C₂	C₂₃×#1834	1	1	yes	characteristic
126	C₂	C₂×C₂₃×#368	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×C₁₆×C₂₃

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

C₄×C₁₆×C₂₃

C₄

yes

—

C₄×C₁₆×C₂₃

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

characteristic

C₄×C₈×C₂₃

C₂×C₄

yes

characteristic

C₄×C₈×C₂₃

Q₈

yes

characteristic

C₄×C₈×C₂₃

Q₈

yes

characteristic

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

characteristic

C₂×C₈×C₂₃

C₄⋊C₄

yes

characteristic

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

characteristic

C₄²×C₂₃

C₄⋊C₄

yes

characteristic

C₄²×C₂₃

D₈

yes

—

C₁₆×C₂₃

—

#12891

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

#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

#106

yes

—

C₈×C₂₃

#117

yes

—

C₈×C₂₃

#125

yes

—

C₂×C₄×C₁₆

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

#368

yes

—

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

C₄×C₁₆

C₄×C₂₃

yes

—

C₄×C₁₆

C₄×C₂₃

yes

—

C₂×C₂₃

—

C₂×C₂₃

#1834

yes

characteristic

C₂×C₂₃

C₂×#368

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

characteristic

C₄×C₈

C₂×C₄×C₂₃

yes

characteristic

C₄×C₈

Q₈×C₂₃

yes

characteristic

C₄×C₈

Q₈×C₂₃

yes

characteristic

C₂₃

#12891

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

characteristic

C₂×C₈

C₄⋊C₄×C₂₃

yes

characteristic

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

characteristic

100

C₄²

C₄⋊C₄×C₂₃

yes

characteristic

101

C₄²

D₈×C₂₃

yes

—

102

C₁₆

—

103

C₂³

C₂×Q₁₆×C₂₃

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

C₂₃×#107

yes

—

106

C₂×C₄

C₂₃×#107

yes

—

107

C₂×C₄

C₂×C₄⋊C₄×C₂₃

yes

characteristic

108

C₂×C₄

C₂₃×#124

yes

—

109

C₂×C₄

C₂×D₈×C₂₃

yes

characteristic

110

C₂×C₄

C₂×SD₁₆×C₂₃

yes

—

111

C₈

—

112

C₈

C₂₃×#106

yes

—

113

C₈

C₂₃×#106

yes

—

114

C₈

C₂₃×#117

yes

—

115

C₈

C₂₃×#125

yes

—

116

C₂²

—

117

C₂²

C₂×C₂₃×#107

yes

characteristic

118

C₂²

C₂₃×#487

yes

—

119

C₄

—

120

C₄

C₂₃×#368

yes

—

121

C₄

C₂₃×#382

yes

characteristic

122

C₄

C₂×C₂₃×#106

yes

characteristic

123

C₄

C₂₃×#486

yes

—

124

C₂

—

125

C₂

C₂₃×#1834

yes

characteristic

126

C₂

C₂×C₂₃×#368

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
39. #12891	C₂₃	direct (N × H)
87. C₂₃	#12891	direct (N × H)

H ≅ G/N

type

39. #12891

C₂₃

direct (N × H)

87. C₂₃

#12891

direct (N × H)

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

subgroup	index
C₂×C₄×C₁₆×C₂₃	2
C₂₃×#1848	2
C₂₃×#1381	2
#12891	23

subgroup

index

C₂×C₄×C₁₆×C₂₃

C₂₃×#1848

C₂₃×#1381

#12891

Subgroups — C₂₃×#12891 #19160517

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#12891 #19160517

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#12891 #19160517