Group #19157465 — subgroups

Subgroups — C₂₃×#9839 #19157465

Order 5888.

542 conjugacy classes of subgroups; only the 214 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 214 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×#494	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1990	C₂	1	4	yes	maximal
3	C₂₃×#1844	C₂	1	1	yes	maximal, characteristic
4	C₂×C₂₃×#322	C₂	1	1	yes	maximal, characteristic
5	C₂²×Q₁₆×C₂₃	—	2	2	no	—
6	C₂×C₂₃×C₄⋊Q₈	C₂²	1	2	yes	—
7	C₂₃×#494	—	2	2	no	—
8	C₂₃×#494	C₄	1	4	yes	complemented
9	C₂×C₂₃×#106	—	2	2	no	—
10	C₂×C₄×C₈×C₂₃	C₂²	1	2	yes	—
11	C₂×C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
12	C₂₃×#384	C₂²	1	2	yes	—
13	C₂₃×#340	—	2	8	no	—
14	C₂₃×#322	—	2	2	no	—
15	C₂²×Q₈×C₂₃	—	2	2	no	—
16	C₂×Q₁₆×C₂₃	—	2	8	no	—
17	C₂×Q₁₆×C₂₃	—	4	4	no	—
18	C₂×Q₁₆×C₂₃	—	4	4	no	—
19	C₂²×C₈×C₂₃	D₄	1	4	yes	—
20	C₂²×C₈×C₂₃	D₄	1	2	yes	—
21	C₂₃×C₄⋊Q₈	—	2	2	no	—
22	C₂₃×C₄⋊Q₈	—	2	4	no	—
23	C₂₃×C₄⋊Q₈	C₂×C₄	1	4	yes	—
24	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
25	C₂×C₄⋊C₄×C₂₃	—	2	4	no	—
26	C₂×C₄⋊C₄×C₂₃	—	2	2	no	—
27	C₂×C₄²×C₂₃	C₂³	1	1	yes	characteristic
28	C₂₃×#106	—	4	4	no	—
29	C₄×C₈×C₂₃	—	2	2	no	—
30	C₄×C₈×C₂₃	—	2	2	no	—
31	C₄×C₈×C₂₃	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₂₃	D₄	1	2	yes	—
33	C₂×Q₈×C₂₃	—	2	4	no	—
34	C₂×Q₈×C₂₃	—	4	2	no	—
35	C₂×Q₈×C₂₃	—	4	4	no	—
36	C₂×Q₈×C₂₃	—	4	4	no	—
37	C₂²×C₄×C₂₃	—	4	2	no	—
38	C₂²×C₄×C₂₃	—	4	2	no	—
39	C₂²×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
40	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
41	Q₁₆×C₂₃	—	4	16	no	—
42	C₂×C₈×C₂₃	—	2	4	no	—
43	C₂×C₈×C₂₃	—	2	8	no	—
44	C₂×C₈×C₂₃	C₂²⋊C₄	1	4	yes	—
45	C₂×C₈×C₂₃	D₈	1	4	yes	—
46	C₂×C₈×C₂₃	SD₁₆	1	4	yes	—
47	C₄⋊C₄×C₂₃	—	2	8	no	—
48	C₄⋊C₄×C₂₃	—	4	4	no	—
49	C₄⋊C₄×C₂₃	—	4	4	no	—
50	C₄⋊C₄×C₂₃	—	4	8	no	—
51	C₄²×C₂₃	C₂²×C₄	1	1	yes	characteristic
52	C₄²×C₂₃	C₂×D₄	1	1	yes	characteristic
53	C₄²×C₂₃	#41	1	2	yes	—
54	#9839	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
55	C₂³×C₂₃	#127	1	1	yes	characteristic
56	Q₈×C₂₃	—	4	8	no	—
57	Q₈×C₂₃	—	8	4	no	—
58	C₂×C₄×C₂₃	—	4	4	no	—
59	C₂×C₄×C₂₃	—	8	2	no	—
60	C₂×C₄×C₂₃	—	8	2	no	—
61	C₂×C₄×C₂₃	—	8	2	no	—
62	C₂×C₄×C₂₃	—	8	4	no	—
63	C₂×C₄×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₂₃	C₄×D₄	1	2	yes	—
65	C₂×C₄×C₂₃	#121	1	2	yes	—
66	C₂×C₄×C₂₃	#124	1	1	yes	characteristic
67	C₂×C₄×C₂₃	C₂×D₈	1	2	yes	—
68	C₂×C₄×C₂₃	C₂×SD₁₆	1	2	yes	—
69	C₂×C₄×C₂₃	#137	1	8	yes	—
70	C₈×C₂₃	—	4	4	no	—
71	C₈×C₂₃	#102	1	8	yes	—
72	C₂×#494	C₂×C₂₃	1	1	yes	characteristic
73	#1990	C₂×C₂₃	1	4	yes	—
74	#1844	C₂×C₂₃	1	1	yes	characteristic
75	C₂×#322	C₂×C₂₃	1	1	yes	characteristic
76	C₂²×C₂₃	#390	1	1	yes	characteristic
77	C₂²×C₂₃	#492	1	1	yes	characteristic
78	C₂²×C₂₃	#493	1	1	yes	characteristic
79	C₂²×C₂₃	#497	1	4	yes	—
80	C₄×C₂₃	—	8	4	no	—
81	C₄×C₂₃	—	16	2	no	—
82	C₄×C₂₃	C₂×#102	1	2	yes	—
83	C₄×C₂₃	#441	1	4	yes	—
84	C₄×C₂₃	#470	1	4	yes	—
85	C₄×C₂₃	#486	1	2	yes	—
86	C₂²×Q₁₆	—	2	2	no	—
87	C₂×C₄⋊Q₈	C₂²×C₂₃	1	2	yes	—
88	#494	—	2	2	no	—
89	#494	C₄×C₂₃	1	4	yes	complemented
90	C₂×#106	—	2	2	no	—
91	C₂×C₄×C₈	C₂²×C₂₃	1	2	yes	—
92	C₂×C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
93	#384	C₂²×C₂₃	1	2	yes	—
94	#340	—	2	8	no	—
95	#322	—	2	2	no	—
96	C₂×C₂₃	#1716	1	2	yes	—
97	C₂×C₂₃	#1722	1	2	yes	—
98	C₂×C₂₃	#1957	1	1	yes	characteristic
99	C₂×C₂₃	#1971	1	2	yes	—
100	C₂²×Q₈	—	2	2	no	—
101	C₂×Q₁₆	—	2	8	no	—
102	C₂×Q₁₆	—	4	4	no	—
103	C₂×Q₁₆	—	4	4	no	—
104	C₂²×C₈	D₄×C₂₃	1	4	yes	—
105	C₂²×C₈	D₄×C₂₃	1	2	yes	—
106	C₄⋊Q₈	—	2	2	no	—
107	C₄⋊Q₈	—	2	4	no	—
108	C₄⋊Q₈	C₂×C₄×C₂₃	1	4	yes	—
109	C₂×C₄⋊C₄	—	2	4	no	—
110	C₂×C₄⋊C₄	—	2	4	no	—
111	C₂×C₄⋊C₄	—	2	2	no	—
112	C₂×C₄²	C₂³×C₂₃	1	1	yes	characteristic
113	#106	—	4	4	no	—
114	C₄×C₈	—	2	2	no	—
115	C₄×C₈	—	2	2	no	—
116	C₄×C₈	C₂×C₄×C₂₃	1	2	yes	—
117	C₄×C₈	D₄×C₂₃	1	2	yes	—
118	C₂₃	#9839	1	1	yes	characteristic, direct product factor, Sylow
119	C₂×Q₈	—	2	4	no	—
120	C₂×Q₈	—	4	2	no	—
121	C₂×Q₈	—	4	4	no	—
122	C₂×Q₈	—	4	4	no	—
123	C₂²×C₄	—	4	2	no	—
124	C₂²×C₄	—	4	2	no	—
125	C₂²×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
126	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
127	Q₁₆	—	4	16	no	—
128	C₂×C₈	—	2	4	no	—
129	C₂×C₈	—	2	8	no	—
130	C₂×C₈	C₂²⋊C₄×C₂₃	1	4	yes	—
131	C₂×C₈	D₈×C₂₃	1	4	yes	—
132	C₂×C₈	SD₁₆×C₂₃	1	4	yes	—
133	C₄⋊C₄	—	2	8	no	—
134	C₄⋊C₄	—	4	4	no	—
135	C₄⋊C₄	—	4	4	no	—
136	C₄⋊C₄	—	4	8	no	—
137	C₄²	C₂²×C₄×C₂₃	1	1	yes	characteristic
138	C₄²	C₂×D₄×C₂₃	1	1	yes	characteristic
139	C₄²	#41×C₂₃	1	2	yes	—
140	C₂³	C₂₃×#127	1	1	yes	characteristic
141	Q₈	—	4	8	no	—
142	Q₈	—	8	4	no	—
143	C₂×C₄	—	4	4	no	—
144	C₂×C₄	—	8	2	no	—
145	C₂×C₄	—	8	2	no	—
146	C₂×C₄	—	8	2	no	—
147	C₂×C₄	—	8	4	no	—
148	C₂×C₄	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
149	C₂×C₄	C₄×D₄×C₂₃	1	2	yes	—
150	C₂×C₄	C₂₃×#121	1	2	yes	—
151	C₂×C₄	C₂₃×#124	1	1	yes	characteristic
152	C₂×C₄	C₂×D₈×C₂₃	1	2	yes	—
153	C₂×C₄	C₂×SD₁₆×C₂₃	1	2	yes	—
154	C₂×C₄	C₂₃×#137	1	8	yes	—
155	C₈	—	4	4	no	—
156	C₈	C₂₃×#102	1	8	yes	—
157	C₂²	C₂₃×#390	1	1	yes	characteristic
158	C₂²	C₂₃×#492	1	1	yes	characteristic
159	C₂²	C₂₃×#493	1	1	yes	characteristic
160	C₂²	C₂₃×#497	1	4	yes	—
161	C₄	—	8	4	no	—
162	C₄	—	16	2	no	—
163	C₄	C₂×C₂₃×#102	1	2	yes	—
164	C₄	C₂₃×#441	1	4	yes	—
165	C₄	C₂₃×#470	1	4	yes	—
166	C₄	C₂₃×#486	1	2	yes	—
167	C₂	C₂₃×#1716	1	2	yes	—
168	C₂	C₂₃×#1722	1	2	yes	—
169	C₂	C₂₃×#1957	1	1	yes	characteristic
170	C₂	C₂₃×#1971	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂₃×#494

C₂

yes

maximal, characteristic

C₂₃×#1990

C₂

yes

maximal

C₂₃×#1844

C₂

yes

maximal, characteristic

C₂×C₂₃×#322

C₂

yes

maximal, characteristic

C₂²×Q₁₆×C₂₃

—

C₂×C₂₃×C₄⋊Q₈

C₂²

yes

—

C₂₃×#494

—

C₂₃×#494

C₄

yes

complemented

C₂×C₂₃×#106

—

C₂×C₄×C₈×C₂₃

C₂²

yes

—

C₂×C₄×C₈×C₂₃

C₂²

yes

characteristic

C₂₃×#384

C₂²

yes

—

C₂₃×#340

—

C₂₃×#322

—

C₂²×Q₈×C₂₃

—

C₂×Q₁₆×C₂₃

—

C₂×Q₁₆×C₂₃

—

C₂×Q₁₆×C₂₃

—

C₂²×C₈×C₂₃

D₄

yes

—

C₂²×C₈×C₂₃

D₄

yes

—

C₂₃×C₄⋊Q₈

—

C₂₃×C₄⋊Q₈

—

C₂₃×C₄⋊Q₈

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄⋊C₄×C₂₃

—

C₂×C₄²×C₂₃

C₂³

yes

characteristic

C₂₃×#106

—

C₄×C₈×C₂₃

—

C₄×C₈×C₂₃

—

C₄×C₈×C₂₃

C₂×C₄

yes

—

C₄×C₈×C₂₃

D₄

yes

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₂₃

C₂×D₄

yes

—

Q₁₆×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₂²⋊C₄

yes

—

C₂×C₈×C₂₃

D₈

yes

—

C₂×C₈×C₂₃

SD₁₆

yes

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₄²×C₂₃

C₂²×C₄

yes

characteristic

C₄²×C₂₃

C₂×D₄

yes

characteristic

C₄²×C₂₃

#41

yes

—

#9839

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₂₃

#127

yes

characteristic

Q₈×C₂₃

—

Q₈×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₂₃

C₄×D₄

yes

—

C₂×C₄×C₂₃

#121

yes

—

C₂×C₄×C₂₃

#124

yes

characteristic

C₂×C₄×C₂₃

C₂×D₈

yes

—

C₂×C₄×C₂₃

C₂×SD₁₆

yes

—

C₂×C₄×C₂₃

#137

yes

—

C₈×C₂₃

—

C₈×C₂₃

#102

yes

—

C₂×#494

C₂×C₂₃

yes

characteristic

#1990

C₂×C₂₃

yes

—

#1844

C₂×C₂₃

yes

characteristic

C₂×#322

C₂×C₂₃

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

C₂²×C₂₃

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₂×#102

yes

—

C₄×C₂₃

#441

yes

—

C₄×C₂₃

#470

yes

—

C₄×C₂₃

#486

yes

—

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

C₂²×C₂₃

yes

—

#494

—

#494

C₄×C₂₃

yes

complemented

C₂×#106

—

C₂×C₄×C₈

C₂²×C₂₃

yes

—

C₂×C₄×C₈

C₂²×C₂₃

yes

characteristic

#384

C₂²×C₂₃

yes

—

#340

—

#322

—

C₂×C₂₃

#1716

yes

—

C₂×C₂₃

#1722

yes

—

C₂×C₂₃

#1957

yes

characteristic

C₂×C₂₃

#1971

yes

—

100

C₂²×Q₈

—

101

C₂×Q₁₆

—

102

C₂×Q₁₆

—

103

C₂×Q₁₆

—

104

C₂²×C₈

D₄×C₂₃

yes

—

105

C₂²×C₈

D₄×C₂₃

yes

—

106

C₄⋊Q₈

—

107

C₄⋊Q₈

—

108

C₄⋊Q₈

C₂×C₄×C₂₃

yes

—

109

C₂×C₄⋊C₄

—

110

C₂×C₄⋊C₄

—

111

C₂×C₄⋊C₄

—

112

C₂×C₄²

C₂³×C₂₃

yes

characteristic

113

#106

—

114

C₄×C₈

—

115

C₄×C₈

—

116

C₄×C₈

C₂×C₄×C₂₃

yes

—

117

C₄×C₈

D₄×C₂₃

yes

—

118

C₂₃

#9839

yes

characteristic, direct product factor, Sylow

119

C₂×Q₈

—

120

C₂×Q₈

—

121

C₂×Q₈

—

122

C₂×Q₈

—

123

C₂²×C₄

—

124

C₂²×C₄

—

125

C₂²×C₄

C₂×D₄×C₂₃

yes

characteristic

126

C₂²×C₄

C₂×D₄×C₂₃

yes

—

127

Q₁₆

—

128

C₂×C₈

—

129

C₂×C₈

—

130

C₂×C₈

C₂²⋊C₄×C₂₃

yes

—

131

C₂×C₈

D₈×C₂₃

yes

—

132

C₂×C₈

SD₁₆×C₂₃

yes

—

133

C₄⋊C₄

—

134

C₄⋊C₄

—

135

C₄⋊C₄

—

136

C₄⋊C₄

—

137

C₄²

C₂²×C₄×C₂₃

yes

characteristic

138

C₄²

C₂×D₄×C₂₃

yes

characteristic

139

C₄²

#41×C₂₃

yes

—

140

C₂³

C₂₃×#127

yes

characteristic

141

Q₈

—

142

Q₈

—

143

C₂×C₄

—

144

C₂×C₄

—

145

C₂×C₄

—

146

C₂×C₄

—

147

C₂×C₄

—

148

C₂×C₄

C₂×C₂²⋊C₄×C₂₃

yes

characteristic

149

C₂×C₄

C₄×D₄×C₂₃

yes

—

150

C₂×C₄

C₂₃×#121

yes

—

151

C₂×C₄

C₂₃×#124

yes

characteristic

152

C₂×C₄

C₂×D₈×C₂₃

yes

—

153

C₂×C₄

C₂×SD₁₆×C₂₃

yes

—

154

C₂×C₄

C₂₃×#137

yes

—

155

C₈

—

156

C₈

C₂₃×#102

yes

—

157

C₂²

C₂₃×#390

yes

characteristic

158

C₂²

C₂₃×#492

yes

characteristic

159

C₂²

C₂₃×#493

yes

characteristic

160

C₂²

C₂₃×#497

yes

—

161

C₄

—

162

C₄

—

163

C₄

C₂×C₂₃×#102

yes

—

164

C₄

C₂₃×#441

yes

—

165

C₄

C₂₃×#470

yes

—

166

C₄

C₂₃×#486

yes

—

167

C₂

C₂₃×#1716

yes

—

168

C₂

C₂₃×#1722

yes

—

169

C₂

C₂₃×#1957

yes

characteristic

170

C₂

C₂₃×#1971

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₂₃	#127
commutator	C₄²	C₂²×C₄×C₂₃
Frattini, cosocle	C₂×C₄²	C₂³×C₂₃
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₂₃

#127

commutator

C₄²

C₂²×C₄×C₂₃

Frattini, cosocle

C₂×C₄²

C₂³×C₂₃

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₂₃ ◁ C₂×C₄²×C₂₃ ◁ G

N	H ≅ G/N	type
8. C₂₃×#494	C₄	semidirect
54. #9839	C₂₃	direct (N × H)
89. #494	C₄×C₂₃	semidirect
118. C₂₃	#9839	direct (N × H)

H ≅ G/N

type

8. C₂₃×#494

C₄

semidirect

54. #9839

C₂₃

direct (N × H)

89. #494

C₄×C₂₃

semidirect

118. C₂₃

#9839

direct (N × H)

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

subgroup	index
C₂×C₂₃×#494	2
C₂₃×#1990	2
C₂₃×#1844	2
C₂×C₂₃×#322	2
#9839	23

subgroup

index

C₂×C₂₃×#494

C₂₃×#1990

C₂₃×#1844

C₂×C₂₃×#322

#9839

Subgroups — C₂₃×#9839 #19157465

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#9839 #19157465

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#9839 #19157465