Group #19156331 — subgroups

Subgroups — C₂₃×#8705 #19156331

Order 5888.

486 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 142 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×#383	C₂	1	1	yes	maximal, characteristic
2	C₂₃×#1817	C₂	1	2	yes	maximal
3	C₂₃×#1280	C₂	1	4	yes	maximal
4	C₂²×C₄²×C₂₃	C₂²	1	1	yes	characteristic
5	C₂×C₂₃×C₂²⋊C₈	C₂²	1	4	yes	—
6	C₂₃×#383	—	2	4	no	—
7	C₂×C₂₃×#95	C₄	1	2	yes	—
8	C₂×C₂₃×#95	C₄	1	2	yes	—
9	C₂×C₂₃×#95	C₂²	1	2	yes	—
10	C₂₃×#336	—	2	4	no	—
11	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
12	C₂³×C₄×C₂₃	C₂×C₄	1	4	yes	—
13	C₂³×C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
14	C₂³×C₄×C₂₃	C₂³	1	1	yes	characteristic
15	C₂²×C₈×C₂₃	—	2	4	no	—
16	C₂×C₄²×C₂₃	—	2	4	no	—
17	C₂×C₄²×C₂₃	D₄	1	4	yes	—
18	C₂₃×C₂²⋊C₈	—	2	8	no	—
19	C₂₃×#95	—	2	4	no	—
20	C₂₃×#95	—	2	4	no	—
21	C₂₃×#95	—	2	4	no	—
22	C₂⁴×C₂₃	C₂²×C₄	1	1	yes	characteristic
23	C₂²×C₄×C₂₃	—	2	2	no	—
24	C₂²×C₄×C₂₃	—	2	2	no	—
25	C₂²×C₄×C₂₃	—	2	4	no	—
26	C₂²×C₄×C₂₃	—	2	4	no	—
27	C₂²×C₄×C₂₃	—	2	8	no	—
28	C₂²×C₄×C₂₃	—	4	2	no	—
29	C₂²×C₄×C₂₃	—	4	2	no	—
30	C₂²×C₄×C₂₃	—	4	4	no	—
31	C₂²×C₄×C₂₃	—	4	4	no	—
32	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
33	C₂²×C₄×C₂₃	C₂²⋊C₄	1	2	yes	—
34	C₂²×C₄×C₂₃	C₂×D₄	1	2	yes	—
35	C₂²×C₄×C₂₃	#41	1	4	yes	—
36	C₂×C₈×C₂₃	—	4	8	no	—
37	C₂×C₈×C₂₃	—	4	4	no	—
38	C₄²×C₂₃	—	2	8	no	—
39	#8705	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
40	C₂³×C₂₃	—	2	2	no	—
41	C₂³×C₂₃	—	2	2	no	—
42	C₂³×C₂₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
43	C₂³×C₂₃	#117	1	2	yes	—
44	C₂³×C₂₃	#131	1	4	yes	—
45	C₂×C₄×C₂₃	—	2	4	no	—
46	C₂×C₄×C₂₃	—	2	2	no	—
47	C₂×C₄×C₂₃	—	2	2	no	—
48	C₂×C₄×C₂₃	—	2	4	no	—
49	C₂×C₄×C₂₃	—	2	4	no	—
50	C₂×C₄×C₂₃	—	4	4	no	—
51	C₂×C₄×C₂₃	—	4	4	no	—
52	C₂×C₄×C₂₃	—	4	4	no	—
53	C₂×C₄×C₂₃	—	4	4	no	—
54	C₂×C₄×C₂₃	—	4	4	no	—
55	C₂×C₄×C₂₃	—	4	4	no	—
56	C₂×C₄×C₂₃	—	8	8	no	—
57	C₂×C₄×C₂₃	#123	1	4	yes	—
58	C₈×C₂₃	—	8	4	no	—
59	C₂×#383	C₂×C₂₃	1	1	yes	characteristic
60	#1817	C₂×C₂₃	1	2	yes	—
61	#1280	C₂×C₂₃	1	4	yes	—
62	C₂²×C₂₃	—	2	2	no	—
63	C₂²×C₂₃	—	2	2	no	—
64	C₂²×C₂₃	—	2	8	no	—
65	C₂²×C₂₃	#381	1	1	yes	characteristic
66	C₂²×C₂₃	#408	1	2	yes	—
67	C₂²×C₂₃	#410	1	2	yes	—
68	C₂²×C₂₃	#413	1	2	yes	—
69	C₂²×C₂₃	#433	1	4	yes	—
70	C₄×C₂₃	—	2	4	no	—
71	C₄×C₂₃	—	4	2	no	—
72	C₄×C₂₃	—	4	2	no	—
73	C₄×C₂₃	—	8	4	no	—
74	C₂²×C₄²	C₂²×C₂₃	1	1	yes	characteristic
75	C₂×C₂²⋊C₈	C₂²×C₂₃	1	4	yes	—
76	#383	—	2	4	no	—
77	C₂×#95	C₄×C₂₃	1	2	yes	—
78	C₂×#95	C₄×C₂₃	1	2	yes	—
79	C₂×#95	C₂²×C₂₃	1	2	yes	—
80	#336	—	2	4	no	—
81	C₂×C₂₃	—	2	4	no	—
82	C₂×C₂₃	#1473	1	4	yes	—
83	C₂×C₂₃	#1818	1	1	yes	characteristic
84	C₂×C₂₃	#1820	1	1	yes	characteristic
85	C₂×C₂₃	#1822	1	1	yes	characteristic
86	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
87	C₂³×C₄	C₂×C₄×C₂₃	1	4	yes	—
88	C₂³×C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
89	C₂³×C₄	C₂³×C₂₃	1	1	yes	characteristic
90	C₂²×C₈	—	2	4	no	—
91	C₂×C₄²	—	2	4	no	—
92	C₂×C₄²	D₄×C₂₃	1	4	yes	—
93	C₂²⋊C₈	—	2	8	no	—
94	#95	—	2	4	no	—
95	#95	—	2	4	no	—
96	#95	—	2	4	no	—
97	C₂₃	#8705	1	1	yes	characteristic, direct product factor, Sylow
98	C₂⁴	C₂²×C₄×C₂₃	1	1	yes	characteristic
99	C₂²×C₄	—	2	2	no	—
100	C₂²×C₄	—	2	2	no	—
101	C₂²×C₄	—	2	4	no	—
102	C₂²×C₄	—	2	4	no	—
103	C₂²×C₄	—	2	8	no	—
104	C₂²×C₄	—	4	2	no	—
105	C₂²×C₄	—	4	2	no	—
106	C₂²×C₄	—	4	4	no	—
107	C₂²×C₄	—	4	4	no	—
108	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
109	C₂²×C₄	C₂²⋊C₄×C₂₃	1	2	yes	—
110	C₂²×C₄	C₂×D₄×C₂₃	1	2	yes	—
111	C₂²×C₄	#41×C₂₃	1	4	yes	—
112	C₂×C₈	—	4	8	no	—
113	C₂×C₈	—	4	4	no	—
114	C₄²	—	2	8	no	—
115	C₂³	—	2	2	no	—
116	C₂³	—	2	2	no	—
117	C₂³	C₂×C₂²⋊C₄×C₂₃	1	1	yes	characteristic
118	C₂³	C₂₃×#117	1	2	yes	—
119	C₂³	C₂₃×#131	1	4	yes	—
120	C₂×C₄	—	2	4	no	—
121	C₂×C₄	—	2	2	no	—
122	C₂×C₄	—	2	2	no	—
123	C₂×C₄	—	2	4	no	—
124	C₂×C₄	—	2	4	no	—
125	C₂×C₄	—	4	4	no	—
126	C₂×C₄	—	4	4	no	—
127	C₂×C₄	—	4	4	no	—
128	C₂×C₄	—	4	4	no	—
129	C₂×C₄	—	4	4	no	—
130	C₂×C₄	—	4	4	no	—
131	C₂×C₄	—	8	8	no	—
132	C₂×C₄	C₂₃×#123	1	4	yes	—
133	C₈	—	8	4	no	—
134	C₂²	—	2	8	no	—
135	C₂²	—	2	2	no	—
136	C₂²	—	2	2	no	—
137	C₂²	C₂₃×#381	1	1	yes	characteristic
138	C₂²	C₂₃×#408	1	2	yes	—
139	C₂²	C₂₃×#410	1	2	yes	—
140	C₂²	C₂₃×#413	1	2	yes	—
141	C₂²	C₂₃×#433	1	4	yes	—
142	C₄	—	2	4	no	—
143	C₄	—	4	2	no	—
144	C₄	—	4	2	no	—
145	C₄	—	8	4	no	—
146	C₂	—	2	4	no	—
147	C₂	C₂₃×#1473	1	4	yes	—
148	C₂	C₂₃×#1818	1	1	yes	characteristic
149	C₂	C₂₃×#1820	1	1	yes	characteristic
150	C₂	C₂₃×#1822	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂₃×#383

C₂

yes

maximal, characteristic

C₂₃×#1817

C₂

yes

maximal

C₂₃×#1280

C₂

yes

maximal

C₂²×C₄²×C₂₃

C₂²

yes

characteristic

C₂×C₂₃×C₂²⋊C₈

C₂²

yes

—

C₂₃×#383

—

C₂×C₂₃×#95

C₄

yes

—

C₂×C₂₃×#95

C₄

yes

—

C₂×C₂₃×#95

C₂²

yes

—

C₂₃×#336

—

C₂³×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂³×C₄×C₂₃

C₂×C₄

yes

—

C₂³×C₄×C₂₃

C₂×C₄

yes

characteristic

C₂³×C₄×C₂₃

C₂³

yes

characteristic

C₂²×C₈×C₂₃

—

C₂×C₄²×C₂₃

—

C₂×C₄²×C₂₃

D₄

yes

—

C₂₃×C₂²⋊C₈

—

C₂₃×#95

—

C₂₃×#95

—

C₂₃×#95

—

C₂⁴×C₂₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

—

C₂²×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₂₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₂₃

C₂×D₄

yes

—

C₂²×C₄×C₂₃

#41

yes

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₄²×C₂₃

—

#8705

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

C₂×C₂²⋊C₄

yes

characteristic

C₂³×C₂₃

#117

yes

—

C₂³×C₂₃

#131

yes

—

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

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

#123

yes

—

C₈×C₂₃

—

C₂×#383

C₂×C₂₃

yes

characteristic

#1817

C₂×C₂₃

yes

—

#1280

C₂×C₂₃

yes

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

#381

yes

characteristic

C₂²×C₂₃

#408

yes

—

C₂²×C₂₃

#410

yes

—

C₂²×C₂₃

#413

yes

—

C₂²×C₂₃

#433

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₂²×C₄²

C₂²×C₂₃

yes

characteristic

C₂×C₂²⋊C₈

C₂²×C₂₃

yes

—

#383

—

C₂×#95

C₄×C₂₃

yes

—

C₂×#95

C₄×C₂₃

yes

—

C₂×#95

C₂²×C₂₃

yes

—

#336

—

C₂×C₂₃

—

C₂×C₂₃

#1473

yes

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂³×C₄

C₂×C₄×C₂₃

yes

characteristic

C₂³×C₄

C₂×C₄×C₂₃

yes

—

C₂³×C₄

C₂×C₄×C₂₃

yes

characteristic

C₂³×C₄

C₂³×C₂₃

yes

characteristic

C₂²×C₈

—

C₂×C₄²

—

C₂×C₄²

D₄×C₂₃

yes

—

C₂²⋊C₈

—

#95

—

#95

—

#95

—

C₂₃

#8705

yes

characteristic, direct product factor, Sylow

C₂⁴

C₂²×C₄×C₂₃

yes

characteristic

C₂²×C₄

—

100

C₂²×C₄

—

101

C₂²×C₄

—

102

C₂²×C₄

—

103

C₂²×C₄

—

104

C₂²×C₄

—

105

C₂²×C₄

—

106

C₂²×C₄

—

107

C₂²×C₄

—

108

C₂²×C₄

C₂²⋊C₄×C₂₃

yes

—

109

C₂²×C₄

C₂²⋊C₄×C₂₃

yes

—

110

C₂²×C₄

C₂×D₄×C₂₃

yes

—

111

C₂²×C₄

#41×C₂₃

yes

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₄²

—

115

C₂³

—

116

C₂³

—

117

C₂³

C₂×C₂²⋊C₄×C₂₃

yes

characteristic

118

C₂³

C₂₃×#117

yes

—

119

C₂³

C₂₃×#131

yes

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

—

130

C₂×C₄

—

131

C₂×C₄

—

132

C₂×C₄

C₂₃×#123

yes

—

133

C₈

—

134

C₂²

—

135

C₂²

—

136

C₂²

—

137

C₂²

C₂₃×#381

yes

characteristic

138

C₂²

C₂₃×#408

yes

—

139

C₂²

C₂₃×#410

yes

—

140

C₂²

C₂₃×#413

yes

—

141

C₂²

C₂₃×#433

yes

—

142

C₄

—

143

C₄

—

144

C₄

—

145

C₄

—

146

C₂

—

147

C₂

C₂₃×#1473

yes

—

148

C₂

C₂₃×#1818

yes

characteristic

149

C₂

C₂₃×#1820

yes

characteristic

150

C₂

C₂₃×#1822

yes

characteristic

	subgroup	G/H
center, socle	C₂³×C₂₃	C₂×C₂²⋊C₄
commutator	C₂⁴	C₂²×C₄×C₂₃
Frattini, cosocle	C₂³×C₄	C₂³×C₂₃
prime radical	C₂⁴×C₂₃	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

subgroup

G/H

center, socle

C₂³×C₂₃

C₂×C₂²⋊C₄

commutator

C₂⁴

C₂²×C₄×C₂₃

Frattini, cosocle

C₂³×C₄

C₂³×C₂₃

prime radical

C₂⁴×C₂₃

C₂²×C₄

LCS

G ▷ C₂⁴ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₂₃ ◁ C₂²×C₄²×C₂₃ ◁ G

N	H ≅ G/N	type
39. #8705	C₂₃	direct (N × H)
97. C₂₃	#8705	direct (N × H)

H ≅ G/N

type

39. #8705

C₂₃

direct (N × H)

97. C₂₃

#8705

direct (N × H)

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

subgroup	index
C₂×C₂₃×#383	2
C₂₃×#1817	2
C₂₃×#1280	2
#8705	23

subgroup

index

C₂×C₂₃×#383

C₂₃×#1817

C₂₃×#1280

#8705

Subgroups — C₂₃×#8705 #19156331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#8705 #19156331

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#8705 #19156331