Group #19161274 — subgroups

Subgroups — C₂₃×#13648 #19161274

Order 5888.

184 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 118 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×#2258	C₂	1	2	yes	maximal, complemented
2	C₂₃×#2170	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#1422	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂₃×#1401	C₂	1	2	yes	maximal, complemented
5	C₂₃×#1397	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×#504	C₂²	1	2	yes	—
7	C₂₃×#443	C₄	1	1	yes	characteristic
8	C₂₃×M₆₄	—	2	2	no	—
9	C₂₃×M₆₄	C₂²	1	2	yes	—
10	C₂×C₂₃×C₃₂	—	2	2	no	—
11	C₂×C₂₃×C₃₂	C₂²	1	1	yes	characteristic
12	C₂×C₂₃×C₃₂	C₂²	1	1	yes	characteristic
13	C₂₃×#364	C₄	1	1	yes	characteristic
14	C₂₃×#350	C₄	1	2	yes	—
15	C₄×C₁₆×C₂₃	C₂²	1	1	yes	characteristic
16	C₂₃×#135	C₈	1	1	yes	characteristic
17	C₂₃×#131	C₂×C₄	1	2	yes	—
18	C₂₃×M₃₂	—	2	2	no	—
19	C₂₃×M₃₂	C₂×C₄	1	2	yes	—
20	C₂×C₁₆×C₂₃	—	2	2	no	—
21	C₂×C₁₆×C₂₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆×C₂₃	C₂³	1	1	yes	characteristic
23	C₂₃×#108	C₈	1	1	yes	characteristic
24	C₂₃×#104	C₈	1	2	yes	—
25	C₄×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
26	C₂₃×C₃₂	—	2	2	no	—
27	C₂₃×C₃₂	—	2	1	no	—
28	C₂₃×C₃₂	D₄	1	2	yes	—
29	#41×C₂₃	C₂×C₈	1	2	yes	—
30	Q₁₆×C₂₃	C₁₆	1	1	yes	characteristic
31	SD₁₆×C₂₃	C₁₆	1	2	yes	—
32	D₈×C₂₃	C₁₆	1	1	yes	characteristic
33	M₁₆×C₂₃	—	2	2	no	—
34	M₁₆×C₂₃	C₂×C₈	1	2	yes	—
35	C₂×C₈×C₂₃	—	2	2	no	—
36	C₂×C₈×C₂₃	C₂×C₈	1	1	yes	characteristic
37	C₂×C₈×C₂₃	C₂²×C₄	1	1	yes	characteristic
38	C₄²×C₂₃	C₂×C₈	1	1	yes	characteristic
39	C₁₆×C₂₃	—	2	2	no	—
40	C₁₆×C₂₃	—	2	1	no	—
41	C₁₆×C₂₃	C₂×D₄	1	1	yes	characteristic
42	C₁₆×C₂₃	#41	1	1	yes	characteristic
43	#13648	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
44	Q₈×C₂₃	C₂×C₁₆	1	2	yes	—
45	D₄×C₂₃	—	2	2	no	—
46	D₄×C₂₃	C₂×C₁₆	1	2	yes	—
47	C₂×C₄×C₂₃	—	2	2	no	—
48	C₂×C₄×C₂₃	—	2	1	no	—
49	C₂×C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
50	C₈×C₂₃	—	2	2	no	—
51	C₈×C₂₃	C₂×C₁₆	1	1	yes	characteristic
52	C₈×C₂₃	C₂×C₁₆	1	1	yes	characteristic
53	C₈×C₂₃	C₄×D₄	1	1	yes	characteristic
54	C₈×C₂₃	#131	1	1	yes	characteristic
55	#2258	C₂×C₂₃	1	2	yes	complemented
56	#2170	C₂×C₂₃	1	1	yes	characteristic
57	#1422	C₂×C₂₃	1	1	yes	characteristic, complemented
58	#1401	C₂×C₂₃	1	2	yes	complemented
59	#1397	C₂×C₂₃	1	1	yes	characteristic, complemented
60	C₂²×C₂₃	—	2	2	no	—
61	C₂²×C₂₃	#504	1	1	yes	characteristic
62	C₄×C₂₃	—	2	2	no	—
63	C₄×C₂₃	—	4	1	no	—
64	C₄×C₂₃	C₈×D₄	1	1	yes	characteristic
65	C₄×C₂₃	C₂²×C₁₆	1	1	yes	characteristic
66	#504	C₂²×C₂₃	1	2	yes	—
67	#443	C₄×C₂₃	1	1	yes	characteristic
68	M₆₄	—	2	2	no	—
69	M₆₄	C₂²×C₂₃	1	2	yes	—
70	C₂×C₃₂	—	2	2	no	—
71	C₂×C₃₂	C₂²×C₂₃	1	1	yes	characteristic
72	C₂×C₃₂	C₂²×C₂₃	1	1	yes	characteristic
73	#364	C₄×C₂₃	1	1	yes	characteristic
74	#350	C₄×C₂₃	1	2	yes	—
75	C₄×C₁₆	C₂²×C₂₃	1	1	yes	characteristic
76	C₂×C₂₃	—	2	1	no	—
77	C₂×C₂₃	—	4	2	no	—
78	C₂×C₂₃	D₄×C₁₆	1	1	yes	characteristic
79	#135	C₈×C₂₃	1	1	yes	characteristic
80	#131	C₂×C₄×C₂₃	1	2	yes	—
81	M₃₂	—	2	2	no	—
82	M₃₂	C₂×C₄×C₂₃	1	2	yes	—
83	C₂×C₁₆	—	2	2	no	—
84	C₂×C₁₆	C₂×C₄×C₂₃	1	1	yes	characteristic
85	C₂×C₁₆	C₂³×C₂₃	1	1	yes	characteristic
86	#108	C₈×C₂₃	1	1	yes	characteristic
87	#104	C₈×C₂₃	1	2	yes	—
88	C₄×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
89	C₃₂	—	2	2	no	—
90	C₃₂	—	2	1	no	—
91	C₃₂	D₄×C₂₃	1	2	yes	—
92	C₂₃	#13648	1	1	yes	characteristic, direct product factor, Sylow
93	#41	C₂×C₈×C₂₃	1	2	yes	—
94	Q₁₆	C₁₆×C₂₃	1	1	yes	characteristic
95	SD₁₆	C₁₆×C₂₃	1	2	yes	—
96	D₈	C₁₆×C₂₃	1	1	yes	characteristic
97	M₁₆	—	2	2	no	—
98	M₁₆	C₂×C₈×C₂₃	1	2	yes	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	C₂×C₈×C₂₃	1	1	yes	characteristic
101	C₂×C₈	C₂²×C₄×C₂₃	1	1	yes	characteristic
102	C₄²	C₂×C₈×C₂₃	1	1	yes	characteristic
103	C₁₆	—	2	2	no	—
104	C₁₆	—	2	1	no	—
105	C₁₆	C₂×D₄×C₂₃	1	1	yes	characteristic
106	C₁₆	#41×C₂₃	1	1	yes	characteristic
107	Q₈	C₂×C₁₆×C₂₃	1	2	yes	—
108	D₄	—	2	2	no	—
109	D₄	C₂×C₁₆×C₂₃	1	2	yes	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	1	no	—
112	C₂×C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
113	C₈	—	2	2	no	—
114	C₈	C₂×C₁₆×C₂₃	1	1	yes	characteristic
115	C₈	C₂×C₁₆×C₂₃	1	1	yes	characteristic
116	C₈	C₄×D₄×C₂₃	1	1	yes	characteristic
117	C₈	C₂₃×#131	1	1	yes	characteristic
118	C₂²	—	2	2	no	—
119	C₂²	C₂₃×#504	1	1	yes	characteristic
120	C₄	—	2	2	no	—
121	C₄	—	4	1	no	—
122	C₄	C₈×D₄×C₂₃	1	1	yes	characteristic
123	C₄	C₂²×C₁₆×C₂₃	1	1	yes	characteristic
124	C₂	—	2	1	no	—
125	C₂	—	4	2	no	—
126	C₂	D₄×C₁₆×C₂₃	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#2258

C₂

yes

maximal, complemented

C₂₃×#2170

C₂

yes

maximal, characteristic

C₂₃×#1422

C₂

yes

maximal, characteristic, complemented

C₂₃×#1401

C₂

yes

maximal, complemented

C₂₃×#1397

C₂

yes

maximal, characteristic, complemented

C₂₃×#504

C₂²

yes

—

C₂₃×#443

C₄

yes

characteristic

C₂₃×M₆₄

—

C₂₃×M₆₄

C₂²

yes

—

C₂×C₂₃×C₃₂

—

C₂×C₂₃×C₃₂

C₂²

yes

characteristic

C₂×C₂₃×C₃₂

C₂²

yes

characteristic

C₂₃×#364

C₄

yes

characteristic

C₂₃×#350

C₄

yes

—

C₄×C₁₆×C₂₃

C₂²

yes

characteristic

C₂₃×#135

C₈

yes

characteristic

C₂₃×#131

C₂×C₄

yes

—

C₂₃×M₃₂

—

C₂₃×M₃₂

C₂×C₄

yes

—

C₂×C₁₆×C₂₃

—

C₂×C₁₆×C₂₃

C₂×C₄

yes

characteristic

C₂×C₁₆×C₂₃

C₂³

yes

characteristic

C₂₃×#108

C₈

yes

characteristic

C₂₃×#104

C₈

yes

—

C₄×C₈×C₂₃

C₂×C₄

yes

characteristic

C₂₃×C₃₂

—

C₂₃×C₃₂

—

C₂₃×C₃₂

D₄

yes

—

#41×C₂₃

C₂×C₈

yes

—

Q₁₆×C₂₃

C₁₆

yes

characteristic

SD₁₆×C₂₃

C₁₆

yes

—

D₈×C₂₃

C₁₆

yes

characteristic

M₁₆×C₂₃

—

M₁₆×C₂₃

C₂×C₈

yes

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₂×C₈

yes

characteristic

C₂×C₈×C₂₃

C₂²×C₄

yes

characteristic

C₄²×C₂₃

C₂×C₈

yes

characteristic

C₁₆×C₂₃

—

C₁₆×C₂₃

—

C₁₆×C₂₃

C₂×D₄

yes

C₁₆×C₂₃

yes

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

Q₈×C₂₃

C₂×C₁₆

yes

—

D₄×C₂₃

—

D₄×C₂₃

C₂×C₁₆

yes

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²×C₈

yes

characteristic

C₈×C₂₃

—

C₈×C₂₃

C₂×C₁₆

yes

characteristic

C₈×C₂₃

C₂×C₁₆

yes

characteristic

C₈×C₂₃

C₄×D₄

yes

C₈×C₂₃

yes

C₂×C₂₃

yes

complemented

#2170

C₂×C₂₃

yes

characteristic

#1422

C₂×C₂₃

yes

characteristic, complemented

#1401

C₂×C₂₃

yes

complemented

#1397

C₂×C₂₃

yes

characteristic, complemented

C₂²×C₂₃

—

C₂²×C₂₃

#504

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₈×D₄

yes

characteristic

C₄×C₂₃

C₂²×C₁₆

yes

characteristic

#504

C₂²×C₂₃

yes

—

#443

C₄×C₂₃

yes

characteristic

M₆₄

—

M₆₄

C₂²×C₂₃

yes

—

C₂×C₃₂

—

C₂×C₃₂

C₂²×C₂₃

yes

characteristic

C₂×C₃₂

C₂²×C₂₃

yes

characteristic

#364

C₄×C₂₃

yes

characteristic

#350

C₄×C₂₃

yes

—

C₄×C₁₆

C₂²×C₂₃

yes

characteristic

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

D₄×C₁₆

yes

characteristic

#135

C₈×C₂₃

yes

characteristic

#131

C₂×C₄×C₂₃

yes

—

M₃₂

—

M₃₂

C₂×C₄×C₂₃

yes

—

C₂×C₁₆

—

C₂×C₁₆

C₂×C₄×C₂₃

yes

characteristic

C₂×C₁₆

C₂³×C₂₃

yes

characteristic

#108

C₈×C₂₃

yes

characteristic

#104

C₈×C₂₃

yes

—

C₄×C₈

C₂×C₄×C₂₃

yes

characteristic

C₃₂

—

C₃₂

—

C₃₂

D₄×C₂₃

yes

—

C₂₃

#13648

yes

characteristic, direct product factor, Sylow

#41

C₂×C₈×C₂₃

yes

—

Q₁₆

C₁₆×C₂₃

yes

characteristic

SD₁₆

C₁₆×C₂₃

yes

—

D₈

C₁₆×C₂₃

yes

characteristic

M₁₆

—

M₁₆

C₂×C₈×C₂₃

yes

—

C₂×C₈

—

100

C₂×C₈

C₂×C₈×C₂₃

yes

characteristic

101

C₂×C₈

C₂²×C₄×C₂₃

yes

characteristic

102

C₄²

C₂×C₈×C₂₃

yes

characteristic

103

C₁₆

—

104

C₁₆

—

105

C₁₆

C₂×D₄×C₂₃

yes

characteristic

106

C₁₆

#41×C₂₃

yes

characteristic

107

Q₈

C₂×C₁₆×C₂₃

yes

—

108

D₄

—

109

D₄

C₂×C₁₆×C₂₃

yes

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

C₂²×C₈×C₂₃

yes

characteristic

113

C₈

—

114

C₈

C₂×C₁₆×C₂₃

yes

characteristic

115

C₈

C₂×C₁₆×C₂₃

yes

characteristic

116

C₈

C₄×D₄×C₂₃

yes

characteristic

117

C₈

C₂₃×#131

yes

characteristic

118

C₂²

—

119

C₂²

C₂₃×#504

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

C₈×D₄×C₂₃

yes

characteristic

123

C₄

C₂²×C₁₆×C₂₃

yes

characteristic

124

C₂

—

125

C₂

—

126

C₂

D₄×C₁₆×C₂₃

yes

characteristic

	subgroup	G/H
center	C₁₆×C₂₃	C₂×D₄
commutator	C₄	C₂²×C₁₆×C₂₃
Frattini, cosocle	C₂×C₁₆	C₂³×C₂₃
socle	C₂×C₂₃	D₄×C₁₆
prime radical	C₂₃×#135	C₈
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₁₆×C₂₃ ◁ C₂×C₂₃×C₃₂ ◁ G

subgroup

G/H

center

C₁₆×C₂₃

C₂×D₄

commutator

C₄

C₂²×C₁₆×C₂₃

Frattini, cosocle

C₂×C₁₆

C₂³×C₂₃

socle

C₂×C₂₃

D₄×C₁₆

prime radical

C₂₃×#135

C₈

LCS

G ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₁₆×C₂₃ ◁ C₂×C₂₃×C₃₂ ◁ G

N	H ≅ G/N	type
1. C₂₃×#2258	C₂	semidirect
3. C₂₃×#1422	C₂	semidirect
4. C₂₃×#1401	C₂	semidirect
5. C₂₃×#1397	C₂	semidirect
43. #13648	C₂₃	direct (N × H)
55. #2258	C₂×C₂₃	semidirect
57. #1422	C₂×C₂₃	semidirect
58. #1401	C₂×C₂₃	semidirect
59. #1397	C₂×C₂₃	semidirect
92. C₂₃	#13648	direct (N × H)

H ≅ G/N

type

1. C₂₃×#2258

C₂

semidirect

3. C₂₃×#1422

C₂

semidirect

4. C₂₃×#1401

C₂

semidirect

5. C₂₃×#1397

C₂

semidirect

43. #13648

C₂₃

direct (N × H)

55. #2258

C₂×C₂₃

semidirect

57. #1422

C₂×C₂₃

semidirect

58. #1401

C₂×C₂₃

semidirect

59. #1397

C₂×C₂₃

semidirect

92. C₂₃

#13648

direct (N × H)

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

subgroup	index
C₂₃×#2258	2
C₂₃×#2170	2
C₂₃×#1422	2
C₂₃×#1401	2
C₂₃×#1397	2
#13648	23

subgroup

index

C₂₃×#2258

C₂₃×#2170

C₂₃×#1422

C₂₃×#1401

C₂₃×#1397

#13648

Subgroups — C₂₃×#13648 #19161274

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#13648 #19161274

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#13648 #19161274