Group #15669487 — subgroups

Subgroups — C₂₃×#1868 #15669487

Order 2944.

260 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 108 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×#568	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#424	C₂	1	2	yes	maximal, complemented
3	C₂₃×#410	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#408	C₂	1	1	yes	maximal, characteristic
5	C₂₃×#344	C₂	1	2	yes	maximal, complemented
6	C₂×#41×C₂₃	C₂²	1	1	yes	characteristic
7	C₂₃×#131	—	2	4	no	—
8	C₂×M₁₆×C₂₃	—	2	2	no	—
9	C₂×M₁₆×C₂₃	C₂²	1	1	yes	characteristic
10	C₂×M₁₆×C₂₃	C₂²	1	2	yes	—
11	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
12	C₂₃×#117	C₂²	1	2	yes	—
13	C₂₃×C₄⋊C₈	—	2	4	no	—
14	C₂₃×#99	C₄	1	4	yes	—
15	C₂₃×C₂²⋊C₈	—	2	2	no	—
16	#41×C₂₃	—	2	2	no	—
17	#41×C₂₃	—	2	2	no	—
18	C₂×Q₈×C₂₃	Q₈	1	1	yes	characteristic
19	C₂×D₄×C₂₃	C₂×C₄	1	2	yes	—
20	C₂×D₄×C₂₃	Q₈	1	1	yes	characteristic
21	C₂²×C₄×C₂₃	D₄	1	2	yes	—
22	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
23	M₁₆×C₂₃	—	2	4	no	—
24	M₁₆×C₂₃	—	2	2	no	—
25	M₁₆×C₂₃	—	2	2	no	—
26	M₁₆×C₂₃	—	2	2	no	—
27	C₂×C₈×C₂₃	—	2	4	no	—
28	C₂×C₈×C₂₃	—	2	2	no	—
29	C₂×C₈×C₂₃	—	2	1	no	—
30	C₂×C₈×C₂₃	—	2	1	no	—
31	C₂×C₈×C₂₃	D₄	1	4	yes	—
32	C₄⋊C₄×C₂₃	—	2	2	no	—
33	C₂²⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
34	C₄²×C₂₃	—	2	2	no	—
35	C₂³×C₂₃	C₄⋊C₄	1	2	yes	—
36	C₂³×C₂₃	C₂²×C₄	1	1	yes	characteristic
37	Q₈×C₂₃	—	2	2	no	—
38	D₄×C₂₃	—	2	4	no	—
39	D₄×C₂₃	—	2	2	no	—
40	C₂×C₄×C₂₃	—	2	2	no	—
41	C₂×C₄×C₂₃	—	2	2	no	—
42	C₂×C₄×C₂₃	—	2	4	no	—
43	C₂×C₄×C₂₃	C₄⋊C₄	1	2	yes	—
44	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₂₃	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₂₃	#41	1	2	yes	—
48	C₈×C₂₃	—	2	4	no	—
49	C₈×C₂₃	—	4	1	no	—
50	C₈×C₂₃	—	4	1	no	—
51	#1868	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
52	C₂²×C₂₃	—	2	2	no	—
53	C₂²×C₂₃	—	4	1	no	—
54	C₂²×C₂₃	—	4	2	no	—
55	C₂²×C₂₃	C₂×C₄⋊C₄	1	1	yes	characteristic
56	C₂²×C₂₃	C₄×D₄	1	2	yes	—
57	C₄×C₂₃	—	2	2	no	—
58	C₄×C₂₃	—	4	4	no	—
59	C₄×C₂₃	#120	1	1	yes	characteristic
60	C₄×C₂₃	C₂²⋊Q₈	1	2	yes	—
61	C₄×C₂₃	#123	1	1	yes	characteristic
62	#568	C₂×C₂₃	1	1	yes	characteristic
63	#424	C₂×C₂₃	1	2	yes	complemented
64	#410	C₂×C₂₃	1	1	yes	characteristic
65	#408	C₂×C₂₃	1	1	yes	characteristic
66	#344	C₂×C₂₃	1	2	yes	complemented
67	C₂×C₂₃	—	2	2	no	—
68	C₂×C₂₃	—	2	1	no	—
69	C₂×C₂₃	—	4	2	no	—
70	C₂×C₂₃	#385	1	1	yes	characteristic
71	C₂×#41	C₂²×C₂₃	1	1	yes	characteristic
72	#131	—	2	4	no	—
73	C₂×M₁₆	—	2	2	no	—
74	C₂×M₁₆	C₂²×C₂₃	1	1	yes	characteristic
75	C₂×M₁₆	C₂²×C₂₃	1	2	yes	—
76	C₂²×C₈	C₂²×C₂₃	1	1	yes	characteristic
77	#117	C₂²×C₂₃	1	2	yes	—
78	C₄⋊C₈	—	2	4	no	—
79	#99	C₄×C₂₃	1	4	yes	—
80	C₂²⋊C₈	—	2	2	no	—
81	C₂₃	#1868	1	1	yes	characteristic, direct product factor, Sylow
82	#41	—	2	2	no	—
83	#41	—	2	2	no	—
84	C₂×Q₈	Q₈×C₂₃	1	1	yes	characteristic
85	C₂×D₄	C₂×C₄×C₂₃	1	2	yes	—
86	C₂×D₄	Q₈×C₂₃	1	1	yes	characteristic
87	C₂²×C₄	D₄×C₂₃	1	2	yes	—
88	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
89	M₁₆	—	2	4	no	—
90	M₁₆	—	2	2	no	—
91	M₁₆	—	2	2	no	—
92	M₁₆	—	2	2	no	—
93	C₂×C₈	—	2	4	no	—
94	C₂×C₈	—	2	1	no	—
95	C₂×C₈	—	2	1	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	D₄×C₂₃	1	4	yes	—
98	C₄⋊C₄	—	2	2	no	—
99	C₂²⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
100	C₄²	—	2	2	no	—
101	C₂³	C₄⋊C₄×C₂₃	1	2	yes	—
102	C₂³	C₂²×C₄×C₂₃	1	1	yes	characteristic
103	Q₈	—	2	2	no	—
104	D₄	—	2	4	no	—
105	D₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	C₄⋊C₄×C₂₃	1	2	yes	—
110	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
111	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
112	C₂×C₄	C₂×Q₈×C₂₃	1	1	yes	characteristic
113	C₂×C₄	#41×C₂₃	1	2	yes	—
114	C₈	—	2	4	no	—
115	C₈	—	4	1	no	—
116	C₈	—	4	1	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	4	1	no	—
119	C₂²	—	4	2	no	—
120	C₂²	C₂×C₄⋊C₄×C₂₃	1	1	yes	characteristic
121	C₂²	C₄×D₄×C₂₃	1	2	yes	—
122	C₄	—	2	2	no	—
123	C₄	—	4	4	no	—
124	C₄	C₂₃×#120	1	1	yes	characteristic
125	C₄	C₂₃×C₂²⋊Q₈	1	2	yes	—
126	C₄	C₂₃×#123	1	1	yes	characteristic
127	C₂	—	2	2	no	—
128	C₂	—	2	1	no	—
129	C₂	—	4	2	no	—
130	C₂	C₂₃×#385	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂₃×#568

C₂

yes

maximal, characteristic

C₂₃×#424

C₂

yes

maximal, complemented

C₂₃×#410

C₂

yes

maximal, characteristic

C₂₃×#408

C₂

yes

maximal, characteristic

C₂₃×#344

C₂

yes

maximal, complemented

C₂×#41×C₂₃

C₂²

yes

characteristic

C₂₃×#131

—

C₂×M₁₆×C₂₃

—

C₂×M₁₆×C₂₃

C₂²

yes

characteristic

C₂×M₁₆×C₂₃

C₂²

yes

—

C₂²×C₈×C₂₃

C₂²

yes

characteristic

C₂₃×#117

C₂²

yes

—

C₂₃×C₄⋊C₈

—

C₂₃×#99

C₄

yes

—

C₂₃×C₂²⋊C₈

—

#41×C₂₃

—

#41×C₂₃

—

C₂×Q₈×C₂₃

Q₈

yes

characteristic

C₂×D₄×C₂₃

C₂×C₄

yes

—

C₂×D₄×C₂₃

Q₈

yes

characteristic

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

M₁₆×C₂₃

—

M₁₆×C₂₃

—

M₁₆×C₂₃

—

M₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

D₄

yes

—

C₄⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

C₂×C₄

yes

—

C₄²×C₂₃

—

C₂³×C₂₃

C₄⋊C₄

yes

—

C₂³×C₂₃

C₂²×C₄

yes

characteristic

Q₈×C₂₃

—

D₄×C₂₃

—

D₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₄⋊C₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₈×C₂₃

—

C₈×C₂₃

—

C₈×C₂₃

—

#1868

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₂₃

C₄×D₄

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#120

yes

characteristic

C₄×C₂₃

C₂²⋊Q₈

yes

—

C₄×C₂₃

#123

yes

characteristic

#568

C₂×C₂₃

yes

characteristic

#424

C₂×C₂₃

yes

complemented

#410

C₂×C₂₃

yes

characteristic

#408

C₂×C₂₃

yes

characteristic

#344

C₂×C₂₃

yes

complemented

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

—

C₂×C₂₃

#385

yes

characteristic

C₂×#41

C₂²×C₂₃

yes

characteristic

#131

—

C₂×M₁₆

—

C₂×M₁₆

C₂²×C₂₃

yes

characteristic

C₂×M₁₆

C₂²×C₂₃

yes

—

C₂²×C₈

C₂²×C₂₃

yes

characteristic

#117

C₂²×C₂₃

yes

—

C₄⋊C₈

—

#99

C₄×C₂₃

yes

—

C₂²⋊C₈

—

C₂₃

#1868

yes

characteristic, direct product factor, Sylow

#41

—

#41

—

C₂×Q₈

Q₈×C₂₃

yes

characteristic

C₂×D₄

C₂×C₄×C₂₃

yes

—

C₂×D₄

Q₈×C₂₃

yes

characteristic

C₂²×C₄

D₄×C₂₃

yes

—

C₂²×C₄

C₂³×C₂₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

M₁₆

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₄×C₂₃

yes

—

C₄⋊C₄

—

C₂²⋊C₄

C₂×C₄×C₂₃

yes

—

100

C₄²

—

101

C₂³

C₄⋊C₄×C₂₃

yes

—

102

C₂³

C₂²×C₄×C₂₃

yes

characteristic

103

Q₈

—

104

D₄

—

105

D₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₄⋊C₄×C₂₃

yes

—

110

C₂×C₄

C₂×D₄×C₂₃

yes

—

111

C₂×C₄

C₂×D₄×C₂₃

yes

characteristic

112

C₂×C₄

C₂×Q₈×C₂₃

yes

characteristic

113

C₂×C₄

#41×C₂₃

yes

—

114

C₈

—

115

C₈

—

116

C₈

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₂×C₄⋊C₄×C₂₃

yes

characteristic

121

C₂²

C₄×D₄×C₂₃

yes

—

122

C₄

—

123

C₄

—

124

C₄

C₂₃×#120

yes

characteristic

125

C₄

C₂₃×C₂²⋊Q₈

yes

—

126

C₄

C₂₃×#123

yes

characteristic

127

C₂

—

128

C₂

—

129

C₂

—

130

C₂

C₂₃×#385

yes

characteristic

	subgroup	G/H
center	C₄×C₂₃	#120
commutator	C₂³	C₂²×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂³×C₂₃
socle	C₂×C₂₃	#385
prime radical	C₂×#41×C₂₃	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ G

subgroup

G/H

center

C₄×C₂₃

#120

commutator

C₂³

C₂²×C₄×C₂₃

Frattini, cosocle

C₂²×C₄

C₂³×C₂₃

socle

C₂×C₂₃

#385

prime radical

C₂×#41×C₂₃

C₂²

LCS

G ▷ C₂³ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₄×C₂₃ ◁ C₂²×C₄×C₂₃ ◁ G

N	H ≅ G/N	type
2. C₂₃×#424	C₂	semidirect
5. C₂₃×#344	C₂	semidirect
51. #1868	C₂₃	direct (N × H)
63. #424	C₂×C₂₃	semidirect
66. #344	C₂×C₂₃	semidirect
81. C₂₃	#1868	direct (N × H)

H ≅ G/N

type

2. C₂₃×#424

C₂

semidirect

5. C₂₃×#344

C₂

semidirect

51. #1868

C₂₃

direct (N × H)

63. #424

C₂×C₂₃

semidirect

66. #344

C₂×C₂₃

semidirect

81. C₂₃

#1868

direct (N × H)

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

subgroup	index
C₂₃×#568	2
C₂₃×#424	2
C₂₃×#410	2
C₂₃×#408	2
C₂₃×#344	2
#1868	23

subgroup

index

C₂₃×#568

C₂₃×#424

C₂₃×#410

C₂₃×#408

C₂₃×#344

#1868

Subgroups — C₂₃×#1868 #15669487

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#1868 #15669487

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#1868 #15669487