Group #15670259 — subgroups

Subgroups — C₂₃×#2640 #15670259

Order 2944.

500 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 200 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×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂₃×#124	C₂	1	1	yes	maximal, characteristic
3	C₂₃×#397	C₂	1	4	yes	maximal, complemented
4	C₂₃×#393	C₂	1	4	yes	maximal, complemented
5	C₂₃×#391	C₂	1	2	yes	maximal, complemented
6	C₂₃×#390	C₂	1	2	yes	maximal
7	C₂₃×#376	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×Q₈×C₂₃	C₂²	1	1	yes	characteristic
9	C₂²×Q₈×C₂₃	C₂²	1	2	yes	—
10	C₂²×D₄×C₂₃	C₂²	1	1	yes	characteristic
11	C₂₃×C₄⋊Q₈	—	2	4	no	—
12	C₂₃×#124	—	2	4	no	—
13	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
14	C₂×C₄⋊C₄×C₂₃	C₂²	1	2	yes	—
15	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
16	C₂×C₂²⋊C₄×C₂₃	C₂²	1	2	yes	—
17	C₂×C₂²⋊C₄×C₂₃	C₂²	1	8	yes	—
18	C₂×C₄²×C₂₃	C₂²	1	2	yes	—
19	C₂×C₄²×C₂₃	C₂²	1	1	yes	characteristic
20	C₂₃×#95	C₂²	1	2	yes	—
21	C₂₃×#95	C₂²	1	2	yes	—
22	C₂₃×#95	C₂²	1	8	yes	—
23	C₂⁴×C₂₃	C₂³	1	2	yes	—
24	C₂×Q₈×C₂₃	—	2	2	no	—
25	C₂×Q₈×C₂₃	—	2	4	no	—
26	C₂×Q₈×C₂₃	—	2	4	no	—
27	C₂×Q₈×C₂₃	—	4	4	no	—
28	C₂×D₄×C₂₃	—	2	2	no	—
29	C₂×D₄×C₂₃	—	2	4	no	—
30	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
31	C₂²×C₄×C₂₃	C₂³	1	4	yes	—
32	C₂²×C₄×C₂₃	C₂³	1	2	yes	—
33	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
34	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₂₃	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₂₃	—	2	4	no	—
37	C₄⋊C₄×C₂₃	—	2	4	no	—
38	C₂²⋊C₄×C₂₃	—	2	4	no	—
39	C₂²⋊C₄×C₂₃	—	2	4	no	—
40	C₂²⋊C₄×C₂₃	—	4	8	no	—
41	C₄²×C₂₃	—	2	4	no	—
42	C₄²×C₂₃	D₄	1	4	yes	—
43	C₂³×C₂₃	—	2	2	no	—
44	C₂³×C₂₃	—	2	4	no	—
45	C₂³×C₂₃	—	2	4	no	—
46	C₂³×C₂₃	—	2	4	no	—
47	C₂³×C₂₃	C₂⁴	1	1	yes	characteristic
48	Q₈×C₂₃	—	4	8	no	—
49	Q₈×C₂₃	—	4	4	no	—
50	D₄×C₂₃	—	4	4	no	—
51	C₂×C₄×C₂₃	—	2	2	no	—
52	C₂×C₄×C₂₃	—	2	2	no	—
53	C₂×C₄×C₂₃	—	2	2	no	—
54	C₂×C₄×C₂₃	—	2	4	no	—
55	C₂×C₄×C₂₃	—	2	4	no	—
56	C₂×C₄×C₂₃	—	2	4	no	—
57	C₂×C₄×C₂₃	—	2	4	no	—
58	C₂×C₄×C₂₃	—	2	8	no	—
59	C₂×C₄×C₂₃	—	2	4	no	—
60	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
61	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
62	C₂×C₄×C₂₃	C₂×D₄	1	2	yes	—
63	C₂×C₄×C₂₃	#41	1	4	yes	—
64	#2640	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
65	C₂²×C₂₃	—	4	2	no	—
66	C₂²×C₂₃	—	4	2	no	—
67	C₂²×C₂₃	—	4	2	no	—
68	C₂²×C₂₃	—	4	8	no	—
69	C₂²×C₂₃	C₂²×D₄	1	1	yes	characteristic
70	C₂²×C₂₃	C₂×#41	1	2	yes	—
71	C₂²×C₂₃	H(5,𝔽₂)	1	2	yes	—
72	C₂²×C₂₃	2₋¹⁺⁴	1	2	yes	—
73	C₄×C₂₃	—	2	2	no	—
74	C₄×C₂₃	—	2	2	no	—
75	C₄×C₂₃	—	4	2	no	—
76	C₄×C₂₃	—	4	4	no	—
77	C₄×C₂₃	—	4	4	no	—
78	C₄×C₂₃	#124	1	4	yes	—
79	C₂×C₄⋊Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
80	C₂×#124	C₂×C₂₃	1	1	yes	characteristic
81	#397	C₂×C₂₃	1	4	yes	complemented
82	#393	C₂×C₂₃	1	4	yes	complemented
83	#391	C₂×C₂₃	1	2	yes	complemented
84	#390	C₂×C₂₃	1	2	yes	—
85	#376	C₂×C₂₃	1	1	yes	characteristic, complemented
86	C₂×C₂₃	—	8	2	no	—
87	C₂×C₂₃	C₂×#124	1	1	yes	characteristic
88	C₂×C₂₃	#535	1	1	yes	characteristic
89	C₂×C₂₃	#536	1	1	yes	characteristic
90	C₂×C₂₃	#542	1	4	yes	—
91	C₂²×Q₈	C₂²×C₂₃	1	1	yes	characteristic
92	C₂²×Q₈	C₂²×C₂₃	1	2	yes	—
93	C₂²×D₄	C₂²×C₂₃	1	1	yes	characteristic
94	C₄⋊Q₈	—	2	4	no	—
95	#124	—	2	4	no	—
96	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
97	C₂×C₄⋊C₄	C₂²×C₂₃	1	2	yes	—
98	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
99	C₂×C₂²⋊C₄	C₂²×C₂₃	1	2	yes	—
100	C₂×C₂²⋊C₄	C₂²×C₂₃	1	8	yes	—
101	C₂×C₄²	C₂²×C₂₃	1	2	yes	—
102	C₂×C₄²	C₂²×C₂₃	1	1	yes	characteristic
103	#95	C₂²×C₂₃	1	2	yes	—
104	#95	C₂²×C₂₃	1	2	yes	—
105	#95	C₂²×C₂₃	1	8	yes	—
106	C₂₃	#2640	1	1	yes	characteristic, direct product factor, Sylow
107	C₂⁴	C₂³×C₂₃	1	2	yes	—
108	C₂×Q₈	—	2	2	no	—
109	C₂×Q₈	—	2	4	no	—
110	C₂×Q₈	—	2	4	no	—
111	C₂×Q₈	—	4	4	no	—
112	C₂×D₄	—	2	2	no	—
113	C₂×D₄	—	2	4	no	—
114	C₂²×C₄	C₂³×C₂₃	1	4	yes	—
115	C₂²×C₄	C₂³×C₂₃	1	2	yes	—
116	C₂²×C₄	C₂³×C₂₃	1	4	yes	—
117	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
118	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
119	C₂²×C₄	C₂³×C₂₃	1	1	yes	characteristic
120	C₄⋊C₄	—	2	4	no	—
121	C₄⋊C₄	—	2	4	no	—
122	C₂²⋊C₄	—	2	4	no	—
123	C₂²⋊C₄	—	2	4	no	—
124	C₂²⋊C₄	—	4	8	no	—
125	C₄²	—	2	4	no	—
126	C₄²	D₄×C₂₃	1	4	yes	—
127	C₂³	—	2	2	no	—
128	C₂³	—	2	4	no	—
129	C₂³	—	2	4	no	—
130	C₂³	—	2	4	no	—
131	C₂³	C₂⁴×C₂₃	1	1	yes	characteristic
132	Q₈	—	4	8	no	—
133	Q₈	—	4	4	no	—
134	D₄	—	4	4	no	—
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	2	2	no	—
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	—	2	4	no	—
140	C₂×C₄	—	2	4	no	—
141	C₂×C₄	—	2	4	no	—
142	C₂×C₄	—	2	8	no	—
143	C₂×C₄	—	2	4	no	—
144	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
145	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
146	C₂×C₄	C₂×D₄×C₂₃	1	2	yes	—
147	C₂×C₄	#41×C₂₃	1	4	yes	—
148	C₂²	—	4	2	no	—
149	C₂²	—	4	2	no	—
150	C₂²	—	4	2	no	—
151	C₂²	—	4	8	no	—
152	C₂²	C₂²×D₄×C₂₃	1	1	yes	characteristic
153	C₂²	C₂×#41×C₂₃	1	2	yes	—
154	C₂²	C₂₃×H(5,𝔽₂)	1	2	yes	—
155	C₂²	C₂₃×2₋¹⁺⁴	1	2	yes	—
156	C₄	—	2	2	no	—
157	C₄	—	2	2	no	—
158	C₄	—	4	2	no	—
159	C₄	—	4	4	no	—
160	C₄	—	4	4	no	—
161	C₄	C₂₃×#124	1	4	yes	—
162	C₂	—	8	2	no	—
163	C₂	C₂×C₂₃×#124	1	1	yes	characteristic
164	C₂	C₂₃×#535	1	1	yes	characteristic
165	C₂	C₂₃×#536	1	1	yes	characteristic
166	C₂	C₂₃×#542	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₂₃×C₄⋊Q₈

C₂

yes

maximal, characteristic, complemented

C₂×C₂₃×#124

C₂

yes

maximal, characteristic

C₂₃×#397

C₂

yes

maximal, complemented

C₂₃×#393

C₂

yes

maximal, complemented

C₂₃×#391

C₂

yes

maximal, complemented

C₂₃×#390

C₂

yes

maximal

C₂₃×#376

C₂

yes

maximal, characteristic, complemented

C₂²×Q₈×C₂₃

C₂²

yes

characteristic

C₂²×Q₈×C₂₃

C₂²

yes

—

C₂²×D₄×C₂₃

C₂²

yes

characteristic

C₂₃×C₄⋊Q₈

—

C₂₃×#124

—

C₂×C₄⋊C₄×C₂₃

C₂²

yes

—

C₂×C₄⋊C₄×C₂₃

C₂²

yes

—

C₂×C₂²⋊C₄×C₂₃

C₂²

yes

—

C₂×C₂²⋊C₄×C₂₃

C₂²

yes

—

C₂×C₂²⋊C₄×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

—

C₂×C₄²×C₂₃

C₂²

yes

characteristic

C₂₃×#95

C₂²

yes

—

C₂₃×#95

C₂²

yes

—

C₂₃×#95

C₂²

yes

—

C₂⁴×C₂₃

C₂³

yes

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×Q₈×C₂₃

—

C₂×D₄×C₂₃

—

C₂×D₄×C₂₃

—

C₂²×C₄×C₂₃

C₂³

yes

—

C₂²×C₄×C₂₃

C₂³

yes

—

C₂²×C₄×C₂₃

C₂³

yes

—

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₂²×C₄×C₂₃

C₂³

yes

characteristic

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

D₄

yes

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

—

C₂³×C₂₃

C₂⁴

yes

characteristic

Q₈×C₂₃

—

Q₈×C₂₃

—

D₄×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×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

C₂×D₄

yes

—

C₂×C₄×C₂₃

#41

yes

—

#2640

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₂²×D₄

yes

characteristic

C₂²×C₂₃

C₂×#41

yes

—

C₂²×C₂₃

H(5,𝔽₂)

yes

—

C₂²×C₂₃

2₋¹⁺⁴

yes

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

#124

yes

—

C₂×C₄⋊Q₈

C₂×C₂₃

yes

characteristic, complemented

C₂×#124

C₂×C₂₃

yes

characteristic

#397

C₂×C₂₃

yes

complemented

#393

C₂×C₂₃

yes

complemented

#391

C₂×C₂₃

yes

complemented

#390

C₂×C₂₃

yes

—

#376

C₂×C₂₃

yes

characteristic, complemented

C₂×C₂₃

—

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

—

C₂²×Q₈

C₂²×C₂₃

yes

characteristic

C₂²×Q₈

C₂²×C₂₃

yes

—

C₂²×D₄

C₂²×C₂₃

yes

characteristic

C₄⋊Q₈

—

#124

—

C₂×C₄⋊C₄

C₂²×C₂₃

yes

—

C₂×C₄⋊C₄

C₂²×C₂₃

yes

—

C₂×C₂²⋊C₄

C₂²×C₂₃

yes

—

C₂×C₂²⋊C₄

C₂²×C₂₃

yes

—

100

C₂×C₂²⋊C₄

C₂²×C₂₃

yes

—

101

C₂×C₄²

C₂²×C₂₃

yes

—

102

C₂×C₄²

C₂²×C₂₃

yes

characteristic

103

#95

C₂²×C₂₃

yes

—

104

#95

C₂²×C₂₃

yes

—

105

#95

C₂²×C₂₃

yes

—

106

C₂₃

#2640

yes

characteristic, direct product factor, Sylow

107

C₂⁴

C₂³×C₂₃

yes

—

108

C₂×Q₈

—

109

C₂×Q₈

—

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂×D₄

—

113

C₂×D₄

—

114

C₂²×C₄

C₂³×C₂₃

yes

—

115

C₂²×C₄

C₂³×C₂₃

yes

—

116

C₂²×C₄

C₂³×C₂₃

yes

—

117

C₂²×C₄

C₂³×C₂₃

yes

characteristic

118

C₂²×C₄

C₂³×C₂₃

yes

characteristic

119

C₂²×C₄

C₂³×C₂₃

yes

characteristic

120

C₄⋊C₄

—

121

C₄⋊C₄

—

122

C₂²⋊C₄

—

123

C₂²⋊C₄

—

124

C₂²⋊C₄

—

125

C₄²

—

126

C₄²

D₄×C₂₃

yes

—

127

C₂³

—

128

C₂³

—

129

C₂³

—

130

C₂³

—

131

C₂³

C₂⁴×C₂₃

yes

characteristic

132

Q₈

—

133

Q₈

—

134

D₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

C₂×D₄×C₂₃

yes

—

145

C₂×C₄

C₂×D₄×C₂₃

yes

—

146

C₂×C₄

C₂×D₄×C₂₃

yes

—

147

C₂×C₄

#41×C₂₃

yes

—

148

C₂²

—

149

C₂²

—

150

C₂²

—

151

C₂²

—

152

C₂²

C₂²×D₄×C₂₃

yes

characteristic

153

C₂²

C₂×#41×C₂₃

yes

—

154

C₂²

C₂₃×H(5,𝔽₂)

yes

—

155

C₂²

C₂₃×2₋¹⁺⁴

yes

—

156

C₄

—

157

C₄

—

158

C₄

—

159

C₄

—

160

C₄

—

161

C₄

C₂₃×#124

yes

—

162

C₂

—

163

C₂

C₂×C₂₃×#124

yes

characteristic

164

C₂

C₂₃×#535

yes

characteristic

165

C₂

C₂₃×#536

yes

characteristic

166

C₂

C₂₃×#542

yes

—

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

subgroup

G/H

center, socle

C₂³×C₂₃

C₂⁴

commutator, Frattini, cosocle

C₂³

C₂⁴×C₂₃

prime radical

C₂²×D₄×C₂₃

C₂²

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂³×C₂₃ ◁ G

N	H ≅ G/N	type
1. C₂×C₂₃×C₄⋊Q₈	C₂	semidirect
3. C₂₃×#397	C₂	semidirect
4. C₂₃×#393	C₂	semidirect
5. C₂₃×#391	C₂	semidirect
7. C₂₃×#376	C₂	semidirect
64. #2640	C₂₃	direct (N × H)
79. C₂×C₄⋊Q₈	C₂×C₂₃	semidirect
81. #397	C₂×C₂₃	semidirect
82. #393	C₂×C₂₃	semidirect
83. #391	C₂×C₂₃	semidirect
85. #376	C₂×C₂₃	semidirect
106. C₂₃	#2640	direct (N × H)

H ≅ G/N

type

1. C₂×C₂₃×C₄⋊Q₈

C₂

semidirect

3. C₂₃×#397

C₂

semidirect

4. C₂₃×#393

C₂

semidirect

5. C₂₃×#391

C₂

semidirect

7. C₂₃×#376

C₂

semidirect

64. #2640

C₂₃

direct (N × H)

79. C₂×C₄⋊Q₈

C₂×C₂₃

semidirect

81. #397

C₂×C₂₃

semidirect

82. #393

C₂×C₂₃

semidirect

83. #391

C₂×C₂₃

semidirect

85. #376

C₂×C₂₃

semidirect

106. C₂₃

#2640

direct (N × H)

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

subgroup	index
C₂×C₂₃×C₄⋊Q₈	2
C₂×C₂₃×#124	2
C₂₃×#397	2
C₂₃×#393	2
C₂₃×#391	2
C₂₃×#390	2
C₂₃×#376	2
#2640	23

subgroup

index

C₂×C₂₃×C₄⋊Q₈

C₂×C₂₃×#124

C₂₃×#397

C₂₃×#393

C₂₃×#391

C₂₃×#390

C₂₃×#376

#2640

Subgroups — C₂₃×#2640 #15670259

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×#2640 #15670259

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×#2640 #15670259