Group #15669443 — subgroups

Subgroups — C₂₃×#1824 #15669443

Order 2944.

344 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 344 conjugacy classes of subgroups — too many to draw. Only its 136 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	C₂²×C₄⋊C₄×C₂₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₂₃×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#380	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#340	C₂	1	4	yes	maximal, complemented
5	C₂³×C₄×C₂₃	—	2	1	no	—
6	C₂³×C₄×C₂₃	C₂²	1	1	yes	characteristic
7	C₂²×C₈×C₂₃	C₂²	1	2	yes	—
8	C₂×C₄⋊C₄×C₂₃	—	2	1	no	—
9	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
10	C₂×C₄⋊C₄×C₂₃	C₄	1	4	yes	complemented
11	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
12	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
13	C₂×C₂²⋊C₄×C₂₃	—	2	1	no	—
14	C₂₃×C₂²⋊C₈	—	2	2	no	—
15	C₂₃×#95	—	2	1	no	—
16	C₂⁴×C₂₃	D₄	1	1	yes	characteristic
17	C₂²×C₄×C₂₃	—	2	4	no	—
18	C₂²×C₄×C₂₃	—	2	4	no	—
19	C₂²×C₄×C₂₃	—	2	2	no	—
20	C₂²×C₄×C₂₃	—	2	2	no	—
21	C₂²×C₄×C₂₃	—	4	4	no	—
22	C₂²×C₄×C₂₃	C₂×C₄	1	2	yes	—
23	C₂²×C₄×C₂₃	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₂₃	D₄	1	2	yes	—
25	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
26	C₂×C₈×C₂₃	—	2	2	no	—
27	C₂×C₈×C₂₃	—	2	4	no	—
28	C₄⋊C₄×C₂₃	—	2	4	no	—
29	C₄⋊C₄×C₂₃	—	2	2	no	—
30	C₄⋊C₄×C₂₃	—	2	4	no	—
31	C₄⋊C₄×C₂₃	C₂×C₄	1	4	yes	—
32	C₂²⋊C₄×C₂₃	—	4	2	no	—
33	C₂³×C₂₃	—	2	2	no	—
34	C₂³×C₂₃	—	2	2	no	—
35	C₂³×C₂₃	C₂²⋊C₄	1	2	yes	—
36	C₂³×C₂₃	SD₁₆	1	2	yes	—
37	C₂³×C₂₃	Q₁₆	1	2	yes	—
38	C₂³×C₂₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₂₃	—	2	2	no	—
40	C₂×C₄×C₂₃	—	2	4	no	—
41	C₂×C₄×C₂₃	—	2	4	no	—
42	C₂×C₄×C₂₃	—	2	4	no	—
43	C₂×C₄×C₂₃	—	4	8	no	—
44	C₂×C₄×C₂₃	—	4	4	no	—
45	C₂×C₄×C₂₃	—	4	2	no	—
46	C₂×C₄×C₂₃	—	4	2	no	—
47	C₂×C₄×C₂₃	—	4	2	no	—
48	C₂×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
49	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₄×C₂₃	#41	1	4	yes	—
52	C₈×C₂₃	—	4	2	no	—
53	#1824	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
54	C₂²×C₂₃	—	2	2	no	—
55	C₂²×C₂₃	—	2	2	no	—
56	C₂²×C₂₃	—	2	4	no	—
57	C₂²×C₂₃	—	2	4	no	—
58	C₂²×C₂₃	#103	1	4	yes	—
59	C₂²×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₂²×C₂₃	#123	1	2	yes	—
61	C₂²×C₂₃	C₂×SD₁₆	1	1	yes	characteristic
62	C₂²×C₂₃	C₂×Q₁₆	1	1	yes	characteristic
63	C₂²×C₂₃	AGL(1,ℤ/8)	1	2	yes	—
64	C₄×C₂₃	—	2	2	no	—
65	C₄×C₂₃	—	4	4	no	—
66	C₄×C₂₃	—	8	2	no	—
67	C₄×C₂₃	#117	1	2	yes	—
68	C₄×C₂₃	#123	1	2	yes	—
69	C₂²×C₄⋊C₄	C₂×C₂₃	1	1	yes	characteristic
70	C₂×C₂²⋊C₈	C₂×C₂₃	1	1	yes	characteristic
71	#380	C₂×C₂₃	1	1	yes	characteristic
72	#340	C₂×C₂₃	1	4	yes	complemented
73	C₂×C₂₃	—	2	4	no	—
74	C₂×C₂₃	#381	1	1	yes	characteristic
75	C₂×C₂₃	C₂×#103	1	1	yes	characteristic
76	C₂×C₂₃	#418	1	1	yes	characteristic
77	C₂×C₂₃	#481	1	2	yes	—
78	C₂×C₂₃	#484	1	2	yes	—
79	C₂³×C₄	—	2	1	no	—
80	C₂³×C₄	C₂²×C₂₃	1	1	yes	characteristic
81	C₂²×C₈	C₂²×C₂₃	1	2	yes	—
82	C₂×C₄⋊C₄	—	2	1	no	—
83	C₂×C₄⋊C₄	—	2	2	no	—
84	C₂×C₄⋊C₄	C₄×C₂₃	1	4	yes	complemented
85	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
86	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
87	C₂×C₂²⋊C₄	—	2	1	no	—
88	C₂²⋊C₈	—	2	2	no	—
89	#95	—	2	1	no	—
90	C₂₃	#1824	1	1	yes	characteristic, direct product factor, Sylow
91	C₂⁴	D₄×C₂₃	1	1	yes	characteristic
92	C₂²×C₄	—	2	4	no	—
93	C₂²×C₄	—	2	4	no	—
94	C₂²×C₄	—	2	2	no	—
95	C₂²×C₄	—	2	2	no	—
96	C₂²×C₄	—	4	4	no	—
97	C₂²×C₄	C₂×C₄×C₂₃	1	2	yes	—
98	C₂²×C₄	D₄×C₂₃	1	1	yes	characteristic
99	C₂²×C₄	D₄×C₂₃	1	2	yes	—
100	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	2	4	no	—
103	C₄⋊C₄	—	2	4	no	—
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	—	2	4	no	—
106	C₄⋊C₄	C₂×C₄×C₂₃	1	4	yes	—
107	C₂²⋊C₄	—	4	2	no	—
108	C₂³	—	2	2	no	—
109	C₂³	—	2	2	no	—
110	C₂³	C₂²⋊C₄×C₂₃	1	2	yes	—
111	C₂³	SD₁₆×C₂₃	1	2	yes	—
112	C₂³	Q₁₆×C₂₃	1	2	yes	—
113	C₂³	C₂×D₄×C₂₃	1	1	yes	characteristic
114	C₂×C₄	—	2	2	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	2	4	no	—
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	—	4	8	no	—
119	C₂×C₄	—	4	4	no	—
120	C₂×C₄	—	4	2	no	—
121	C₂×C₄	—	4	2	no	—
122	C₂×C₄	—	4	2	no	—
123	C₂×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
124	C₂×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
125	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
126	C₂×C₄	#41×C₂₃	1	4	yes	—
127	C₈	—	4	2	no	—
128	C₂²	—	2	2	no	—
129	C₂²	—	2	2	no	—
130	C₂²	—	2	4	no	—
131	C₂²	—	2	4	no	—
132	C₂²	C₂₃×#103	1	4	yes	—
133	C₂²	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
134	C₂²	C₂₃×#123	1	2	yes	—
135	C₂²	C₂×SD₁₆×C₂₃	1	1	yes	characteristic
136	C₂²	C₂×Q₁₆×C₂₃	1	1	yes	characteristic
137	C₂²	C₂₃×AGL(1,ℤ/8)	1	2	yes	—
138	C₄	—	2	2	no	—
139	C₄	—	4	4	no	—
140	C₄	—	8	2	no	—
141	C₄	C₂₃×#117	1	2	yes	—
142	C₄	C₂₃×#123	1	2	yes	—
143	C₂	—	2	4	no	—
144	C₂	C₂₃×#381	1	1	yes	characteristic
145	C₂	C₂×C₂₃×#103	1	1	yes	characteristic
146	C₂	C₂₃×#418	1	1	yes	characteristic
147	C₂	C₂₃×#481	1	2	yes	—
148	C₂	C₂₃×#484	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×C₄⋊C₄×C₂₃

C₂

yes

maximal, characteristic

C₂×C₂₃×C₂²⋊C₈

C₂

yes

maximal, characteristic

C₂₃×#380

C₂

yes

maximal, characteristic

C₂₃×#340

C₂

yes

maximal, complemented

C₂³×C₄×C₂₃

—

C₂³×C₄×C₂₃

C₂²

yes

characteristic

C₂²×C₈×C₂₃

C₂²

yes

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

C₄

yes

complemented

C₂×C₄⋊C₄×C₂₃

C₂²

yes

—

C₂×C₄⋊C₄×C₂₃

C₂²

yes

—

C₂×C₂²⋊C₄×C₂₃

—

C₂₃×C₂²⋊C₈

—

C₂₃×#95

—

C₂⁴×C₂₃

D₄

yes

characteristic

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂×C₄

yes

—

C₂²×C₄×C₂₃

D₄

yes

characteristic

C₂²×C₄×C₂₃

D₄

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

C₂×C₄

yes

—

C₂²⋊C₄×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

C₂²⋊C₄

yes

—

C₂³×C₂₃

SD₁₆

yes

—

C₂³×C₂₃

Q₁₆

yes

—

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

C₂²⋊C₄

yes

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

#41

yes

—

C₈×C₂₃

—

#1824

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#103

yes

—

C₂²×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₂₃

#123

yes

—

C₂²×C₂₃

C₂×SD₁₆

yes

characteristic

C₂²×C₂₃

C₂×Q₁₆

yes

characteristic

C₂²×C₂₃

AGL(1,ℤ/8)

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#117

yes

—

C₄×C₂₃

#123

yes

—

C₂²×C₄⋊C₄

C₂×C₂₃

yes

characteristic

C₂×C₂²⋊C₈

C₂×C₂₃

yes

characteristic

#380

C₂×C₂₃

yes

characteristic

#340

C₂×C₂₃

yes

complemented

C₂×C₂₃

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

—

C₂×C₂₃

#484

yes

—

C₂³×C₄

—

C₂³×C₄

C₂²×C₂₃

yes

characteristic

C₂²×C₈

C₂²×C₂₃

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₄×C₂₃

yes

complemented

C₂×C₄⋊C₄

C₂²×C₂₃

yes

—

C₂×C₄⋊C₄

C₂²×C₂₃

yes

—

C₂×C₂²⋊C₄

—

C₂²⋊C₈

—

#95

—

C₂₃

#1824

yes

characteristic, direct product factor, Sylow

C₂⁴

D₄×C₂₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄×C₂₃

yes

—

C₂²×C₄

D₄×C₂₃

yes

characteristic

C₂²×C₄

D₄×C₂₃

yes

—

100

C₂²×C₄

C₂³×C₂₃

yes

characteristic

101

C₂×C₈

—

102

C₂×C₈

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

C₂×C₄×C₂₃

yes

—

107

C₂²⋊C₄

—

108

C₂³

—

109

C₂³

—

110

C₂³

C₂²⋊C₄×C₂₃

yes

—

111

C₂³

SD₁₆×C₂₃

yes

—

112

C₂³

Q₁₆×C₂₃

yes

—

113

C₂³

C₂×D₄×C₂₃

yes

characteristic

114

C₂×C₄

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

C₂²⋊C₄×C₂₃

yes

—

124

C₂×C₄

C₂²×C₄×C₂₃

yes

characteristic

125

C₂×C₄

C₂×D₄×C₂₃

yes

characteristic

126

C₂×C₄

#41×C₂₃

yes

—

127

C₈

—

128

C₂²

—

129

C₂²

—

130

C₂²

—

131

C₂²

—

132

C₂²

C₂₃×#103

yes

—

133

C₂²

C₂×C₂²⋊C₄×C₂₃

yes

characteristic

134

C₂²

C₂₃×#123

yes

—

135

C₂²

C₂×SD₁₆×C₂₃

yes

characteristic

136

C₂²

C₂×Q₁₆×C₂₃

yes

characteristic

137

C₂²

C₂₃×AGL(1,ℤ/8)

yes

—

138

C₄

—

139

C₄

—

140

C₄

—

141

C₄

C₂₃×#117

yes

—

142

C₄

C₂₃×#123

yes

—

143

C₂

—

144

C₂

C₂₃×#381

yes

characteristic

145

C₂

C₂×C₂₃×#103

yes

characteristic

146

C₂

C₂₃×#418

yes

characteristic

147

C₂

C₂₃×#481

yes

—

148

C₂

C₂₃×#484

yes

—

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂×D₄
commutator	C₂×C₄	C₂²×C₄×C₂₃
Frattini, cosocle	C₂²×C₄	C₂³×C₂₃
prime radical	C₂⁴×C₂₃	D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂³×C₄×C₂₃ ◁ G

subgroup

G/H

center, socle

C₂³×C₂₃

C₂×D₄

commutator

C₂×C₄

C₂²×C₄×C₂₃

Frattini, cosocle

C₂²×C₄

C₂³×C₂₃

prime radical

C₂⁴×C₂₃

D₄

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₂₃ ◁ C₂³×C₄×C₂₃ ◁ G

N	H ≅ G/N	type
4. C₂₃×#340	C₂	semidirect
10. C₂×C₄⋊C₄×C₂₃	C₄	semidirect
53. #1824	C₂₃	direct (N × H)
72. #340	C₂×C₂₃	semidirect
84. C₂×C₄⋊C₄	C₄×C₂₃	semidirect
90. C₂₃	#1824	direct (N × H)

H ≅ G/N

type

4. C₂₃×#340

C₂

semidirect

10. C₂×C₄⋊C₄×C₂₃

C₄

semidirect

53. #1824

C₂₃

direct (N × H)

72. #340

C₂×C₂₃

semidirect

84. C₂×C₄⋊C₄

C₄×C₂₃

semidirect

90. C₂₃

#1824

direct (N × H)

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

subgroup	index
C₂²×C₄⋊C₄×C₂₃	2
C₂×C₂₃×C₂²⋊C₈	2
C₂₃×#380	2
C₂₃×#340	2
#1824	23

subgroup

index

C₂²×C₄⋊C₄×C₂₃

C₂×C₂₃×C₂²⋊C₈

C₂₃×#380

C₂₃×#340

#1824

Subgroups — C₂₃×#1824 #15669443

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#1824 #15669443

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#1824 #15669443