Group #15669706 — subgroups

Subgroups — C₂₃×#2087 #15669706

Order 2944.

224 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 104 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×C₂₃×#107	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#389	C₂	1	2	yes	maximal
4	C₂₃×#340	C₂	1	2	yes	maximal
5	C₂₃×#340	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₂₃	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
9	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
10	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
11	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
12	C₂₃×#107	—	2	2	no	—
13	C₂₃×C₄⋊C₈	—	2	2	no	—
14	C₂₃×#95	—	2	2	no	—
15	C₂²×C₄×C₂₃	—	2	2	no	—
16	C₂²×C₄×C₂₃	—	2	2	no	—
17	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
18	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
19	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
20	C₂×C₈×C₂₃	—	2	2	no	—
21	C₂×C₈×C₂₃	—	2	1	no	—
22	C₂×C₈×C₂₃	—	2	2	no	—
23	C₂×C₈×C₂₃	D₄	1	2	yes	—
24	C₄⋊C₄×C₂₃	—	2	2	no	—
25	C₄⋊C₄×C₂₃	—	2	2	no	—
26	C₄⋊C₄×C₂₃	—	2	2	no	—
27	C₄⋊C₄×C₂₃	—	4	4	no	—
28	C₄⋊C₄×C₂₃	Q₈	1	4	yes	—
29	C₄²×C₂₃	—	2	1	no	—
30	C₄²×C₂₃	—	2	1	no	—
31	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₂₃	—	2	1	no	—
33	C₂×C₄×C₂₃	—	2	1	no	—
34	C₂×C₄×C₂₃	—	2	1	no	—
35	C₂×C₄×C₂₃	—	2	4	no	—
36	C₂×C₄×C₂₃	—	2	2	no	—
37	C₂×C₄×C₂₃	—	4	2	no	—
38	C₂×C₄×C₂₃	—	4	2	no	—
39	C₂×C₄×C₂₃	—	4	2	no	—
40	C₂×C₄×C₂₃	—	4	2	no	—
41	C₂×C₄×C₂₃	—	4	2	no	—
42	C₂×C₄×C₂₃	Q₁₆	1	2	yes	—
43	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₂₃	C₂×Q₈	1	2	yes	—
45	C₂×C₄×C₂₃	#41	1	2	yes	—
46	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
47	C₈×C₂₃	—	2	2	no	—
48	C₈×C₂₃	—	4	1	no	—
49	#2087	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
50	C₂²×C₂₃	#121	1	1	yes	characteristic
51	C₂²×C₂₃	C₂²⋊Q₈	1	2	yes	—
52	C₂²×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
53	C₂²×C₂₃	#135	1	1	yes	characteristic
54	C₂²×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
55	C₄×C₂₃	—	2	2	no	—
56	C₄×C₂₃	—	4	1	no	—
57	C₄×C₂₃	—	4	4	no	—
58	C₄×C₂₃	—	8	2	no	—
59	C₄×C₂₃	#123	1	1	yes	characteristic
60	C₄×C₂₃	#125	1	2	yes	—
61	C₄×C₂₃	C₄⋊Q₈	1	1	yes	characteristic
62	C₂×#107	C₂×C₂₃	1	1	yes	characteristic
63	C₂×C₄⋊C₈	C₂×C₂₃	1	1	yes	characteristic
64	#389	C₂×C₂₃	1	2	yes	—
65	#340	C₂×C₂₃	1	2	yes	—
66	#340	C₂×C₂₃	1	1	yes	characteristic
67	C₂×C₂₃	#398	1	1	yes	characteristic
68	C₂×C₂₃	#467	1	1	yes	characteristic
69	C₂×C₂₃	#469	1	1	yes	characteristic
70	C₂×C₂₃	#477	1	2	yes	—
71	C₂×C₂₃	#478	1	2	yes	—
72	C₂²×C₈	C₂²×C₂₃	1	1	yes	characteristic
73	C₂²×C₈	C₂²×C₂₃	1	1	yes	characteristic
74	C₂×C₄⋊C₄	—	2	2	no	—
75	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
76	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
77	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
78	#107	—	2	2	no	—
79	C₄⋊C₈	—	2	2	no	—
80	#95	—	2	2	no	—
81	C₂₃	#2087	1	1	yes	characteristic, direct product factor, Sylow
82	C₂²×C₄	—	2	2	no	—
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
85	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
86	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	D₄×C₂₃	1	2	yes	—
91	C₄⋊C₄	—	2	2	no	—
92	C₄⋊C₄	—	2	2	no	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	—	4	4	no	—
95	C₄⋊C₄	Q₈×C₂₃	1	4	yes	—
96	C₄²	—	2	1	no	—
97	C₄²	—	2	1	no	—
98	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
99	C₂×C₄	—	2	1	no	—
100	C₂×C₄	—	2	1	no	—
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	2	4	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	4	2	no	—
105	C₂×C₄	—	4	2	no	—
106	C₂×C₄	—	4	2	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	Q₁₆×C₂₃	1	2	yes	—
110	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
111	C₂×C₄	C₂×Q₈×C₂₃	1	2	yes	—
112	C₂×C₄	#41×C₂₃	1	2	yes	—
113	C₂×C₄	#41×C₂₃	1	1	yes	characteristic
114	C₈	—	2	2	no	—
115	C₈	—	4	1	no	—
116	C₂²	C₂₃×#121	1	1	yes	characteristic
117	C₂²	C₂₃×C₂²⋊Q₈	1	2	yes	—
118	C₂²	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
119	C₂²	C₂₃×#135	1	1	yes	characteristic
120	C₂²	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
121	C₄	—	2	2	no	—
122	C₄	—	4	1	no	—
123	C₄	—	4	4	no	—
124	C₄	—	8	2	no	—
125	C₄	C₂₃×#123	1	1	yes	characteristic
126	C₄	C₂₃×#125	1	2	yes	—
127	C₄	C₂₃×C₄⋊Q₈	1	1	yes	characteristic
128	C₂	C₂₃×#398	1	1	yes	characteristic
129	C₂	C₂₃×#467	1	1	yes	characteristic
130	C₂	C₂₃×#469	1	1	yes	characteristic
131	C₂	C₂₃×#477	1	2	yes	—
132	C₂	C₂₃×#478	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂₃×#107

C₂

yes

maximal, characteristic

C₂×C₂₃×C₄⋊C₈

C₂

yes

maximal, characteristic

C₂₃×#389

C₂

yes

maximal

C₂₃×#340

C₂

yes

maximal

C₂₃×#340

C₂

yes

maximal, characteristic

C₂²×C₈×C₂₃

C₂²

yes

characteristic

C₂²×C₈×C₂₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

C₂²

yes

—

C₂×C₄⋊C₄×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂₃×#107

—

C₂₃×C₄⋊C₈

—

C₂₃×#95

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

D₄

yes

characteristic

C₂²×C₄×C₂₃

D₄

yes

characteristic

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

D₄

yes

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

Q₈

yes

—

C₄²×C₂₃

—

C₄²×C₂₃

—

C₂³×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

Q₁₆

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

C₂×Q₈

yes

—

C₂×C₄×C₂₃

#41

yes

—

C₂×C₄×C₂₃

#41

yes

characteristic

C₈×C₂₃

—

C₈×C₂₃

—

#2087

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₂₃

#121

yes

characteristic

C₂²×C₂₃

C₂²⋊Q₈

yes

—

C₂²×C₂₃

C₂×Q₁₆

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#123

yes

characteristic

C₄×C₂₃

#125

yes

—

C₄×C₂₃

C₄⋊Q₈

yes

characteristic

C₂×#107

C₂×C₂₃

yes

characteristic

C₂×C₄⋊C₈

C₂×C₂₃

yes

characteristic

#389

C₂×C₂₃

yes

—

#340

C₂×C₂₃

yes

—

#340

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

—

C₂×C₂₃

#478

yes

—

C₂²×C₈

C₂²×C₂₃

yes

characteristic

C₂²×C₈

C₂²×C₂₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²×C₂₃

yes

—

C₂×C₄⋊C₄

C₂²×C₂₃

yes

—

C₂×C₄²

C₂²×C₂₃

yes

characteristic

#107

—

C₄⋊C₈

—

#95

—

C₂₃

#2087

yes

characteristic, direct product factor, Sylow

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₄×C₂₃

yes

characteristic

C₂²×C₄

D₄×C₂₃

yes

characteristic

C₂²×C₄

C₂³×C₂₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₄×C₂₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

Q₈×C₂₃

yes

—

C₄²

—

C₄²

—

C₂³

C₂×D₄×C₂₃

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

Q₁₆×C₂₃

yes

—

110

C₂×C₄

C₂×D₄×C₂₃

yes

characteristic

111

C₂×C₄

C₂×Q₈×C₂₃

yes

—

112

C₂×C₄

#41×C₂₃

yes

—

113

C₂×C₄

#41×C₂₃

yes

characteristic

114

C₈

—

115

C₈

—

116

C₂²

C₂₃×#121

yes

characteristic

117

C₂²

C₂₃×C₂²⋊Q₈

yes

—

118

C₂²

C₂×Q₁₆×C₂₃

yes

characteristic

119

C₂²

C₂₃×#135

yes

characteristic

120

C₂²

C₂₃×AGL(1,ℤ/8)

yes

—

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

C₂₃×#123

yes

characteristic

126

C₄

C₂₃×#125

yes

—

127

C₄

C₂₃×C₄⋊Q₈

yes

characteristic

128

C₂

C₂₃×#398

yes

characteristic

129

C₂

C₂₃×#467

yes

characteristic

130

C₂

C₂₃×#469

yes

characteristic

131

C₂

C₂₃×#477

yes

—

132

C₂

C₂₃×#478

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂³×C₂₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₂₃

C₂×D₄

commutator, Frattini, cosocle

C₂²×C₄

C₂³×C₂₃

LCS

G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

N	H ≅ G/N	type
49. #2087	C₂₃	direct (N × H)
81. C₂₃	#2087	direct (N × H)

H ≅ G/N

type

49. #2087

C₂₃

direct (N × H)

81. C₂₃

#2087

direct (N × H)

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

subgroup	index
C₂×C₂₃×#107	2
C₂×C₂₃×C₄⋊C₈	2
C₂₃×#389	2
C₂₃×#340	2
#2087	23

subgroup

index

C₂×C₂₃×#107

C₂×C₂₃×C₄⋊C₈

C₂₃×#389

C₂₃×#340

#2087

Subgroups — C₂₃×#2087 #15669706

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#2087 #15669706

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2087 #15669706