Group #15669197 — subgroups

Subgroups — C₂₃×SD₁₆⋊C₈ #15669197

Order 2944.

176 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 176 conjugacy classes of subgroups — too many to draw. Only its 100 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	C₈×Q₈×C₂₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×SD₁₆×C₂₃	C₂	1	1	yes	maximal, characteristic
3	C₈×D₄×C₂₃	C₂	1	1	yes	maximal, characteristic
4	C₂₃×#335	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂₃×Q₈⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂₃×D₄⋊C₈	C₂	1	1	yes	maximal, characteristic
7	C₂₃×#322	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×SD₁₆×C₂₃	C₄	1	1	yes	characteristic
9	C₂²×C₈×C₂₃	—	2	1	no	—
10	C₄×Q₈×C₂₃	C₂²	1	1	yes	characteristic
11	C₄×D₄×C₂₃	C₂²	1	1	yes	characteristic
12	C₂₃×#106	C₄	1	1	yes	characteristic
13	C₂₃×C₄⋊C₈	—	2	1	no	—
14	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂₃×#103	C₄	1	1	yes	characteristic
17	C₂₃×#102	C₄	1	1	yes	characteristic
18	C₂₃×C₂²⋊C₈	—	2	1	no	—
19	C₄×C₈×C₂₃	—	2	1	no	—
20	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
21	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
22	C₄×C₈×C₂₃	C₂²	1	1	yes	characteristic
23	C₂×Q₈×C₂₃	C₂×C₄	1	1	yes	characteristic
24	C₂×D₄×C₂₃	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₂₃	—	2	1	no	—
26	SD₁₆×C₂₃	C₈	1	4	yes	complemented
27	C₂×C₈×C₂₃	—	2	1	no	—
28	C₂×C₈×C₂₃	—	2	1	no	—
29	C₂×C₈×C₂₃	—	2	1	no	—
30	C₂×C₈×C₂₃	—	4	2	no	—
31	C₂×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
32	C₂×C₈×C₂₃	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
34	C₂×C₈×C₂₃	D₄	1	1	yes	characteristic
35	C₄⋊C₄×C₂₃	—	2	1	no	—
36	C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
37	C₄⋊C₄×C₂₃	C₂×C₄	1	1	yes	characteristic
38	C₂²⋊C₄×C₂₃	—	2	1	no	—
39	C₄²×C₂₃	—	2	1	no	—
40	C₄²×C₂₃	C₂³	1	1	yes	characteristic
41	C₂³×C₂₃	—	2	1	no	—
42	Q₈×C₂₃	—	2	1	no	—
43	Q₈×C₂₃	C₂×C₈	1	2	yes	complemented
44	D₄×C₂₃	—	2	1	no	—
45	D₄×C₂₃	C₂×C₈	1	2	yes	—
46	C₂×C₄×C₂₃	—	2	1	no	—
47	C₂×C₄×C₂₃	—	2	1	no	—
48	C₂×C₄×C₂₃	—	2	1	no	—
49	C₂×C₄×C₂₃	—	4	1	no	—
50	C₂×C₄×C₂₃	—	4	1	no	—
51	C₂×C₄×C₂₃	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₂₃	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₂₃	#41	1	1	yes	characteristic
54	C₈×C₂₃	—	2	1	no	—
55	C₈×C₂₃	—	2	1	no	—
56	C₈×C₂₃	—	2	1	no	—
57	C₈×C₂₃	—	4	1	no	—
58	C₈×C₂₃	—	4	1	no	—
59	C₈×C₂₃	—	4	1	no	—
60	C₈×C₂₃	C₂×C₈	1	2	yes	complemented
61	SD₁₆⋊C₈	C₂₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂²×C₂₃	—	2	2	no	—
63	C₂²×C₂₃	—	4	1	no	—
64	C₂²×C₂₃	—	4	1	no	—
65	C₂²×C₂₃	C₄×D₄	1	1	yes	characteristic
66	C₄×C₂₃	—	2	2	no	—
67	C₄×C₂₃	—	2	1	no	—
68	C₄×C₂₃	—	4	1	no	—
69	C₄×C₂₃	—	4	1	no	—
70	C₄×C₂₃	C₂²×C₈	1	1	yes	characteristic
71	C₄×C₂₃	#131	1	1	yes	characteristic
72	C₄×C₂₃	AGL(1,ℤ/8)	1	1	yes	characteristic
73	C₄×C₂₃	#137	1	1	yes	characteristic
74	C₈×Q₈	C₂×C₂₃	1	1	yes	characteristic, complemented
75	C₄×SD₁₆	C₂×C₂₃	1	1	yes	characteristic
76	C₈×D₄	C₂×C₂₃	1	1	yes	characteristic
77	#335	C₂×C₂₃	1	1	yes	characteristic, complemented
78	Q₈⋊C₈	C₂×C₂₃	1	1	yes	characteristic, complemented
79	D₄⋊C₈	C₂×C₂₃	1	1	yes	characteristic
80	#322	C₂×C₂₃	1	1	yes	characteristic, complemented
81	C₂×C₂₃	—	4	2	no	—
82	C₂×C₂₃	C₈×D₄	1	1	yes	characteristic
83	C₂×C₂₃	#440	1	1	yes	characteristic
84	C₂×C₂₃	#444	1	1	yes	characteristic
85	C₂×SD₁₆	C₄×C₂₃	1	1	yes	characteristic
86	C₂²×C₈	—	2	1	no	—
87	C₄×Q₈	C₂²×C₂₃	1	1	yes	characteristic
88	C₄×D₄	C₂²×C₂₃	1	1	yes	characteristic
89	#106	C₄×C₂₃	1	1	yes	characteristic
90	C₄⋊C₈	—	2	1	no	—
91	C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
92	C₄⋊C₈	C₂²×C₂₃	1	1	yes	characteristic
93	#103	C₄×C₂₃	1	1	yes	characteristic
94	#102	C₄×C₂₃	1	1	yes	characteristic
95	C₂²⋊C₈	—	2	1	no	—
96	C₄×C₈	—	2	1	no	—
97	C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
98	C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
99	C₄×C₈	C₂²×C₂₃	1	1	yes	characteristic
100	C₂₃	SD₁₆⋊C₈	1	1	yes	characteristic, direct product factor, Sylow
101	C₂×Q₈	C₂×C₄×C₂₃	1	1	yes	characteristic
102	C₂×D₄	C₂×C₄×C₂₃	1	1	yes	characteristic
103	C₂²×C₄	—	2	1	no	—
104	SD₁₆	C₈×C₂₃	1	4	yes	complemented
105	C₂×C₈	—	2	1	no	—
106	C₂×C₈	—	2	1	no	—
107	C₂×C₈	—	2	1	no	—
108	C₂×C₈	—	4	2	no	—
109	C₂×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
110	C₂×C₈	C₂×C₄×C₂₃	1	1	yes	characteristic
111	C₂×C₈	D₄×C₂₃	1	1	yes	characteristic
112	C₂×C₈	D₄×C₂₃	1	1	yes	characteristic
113	C₄⋊C₄	—	2	1	no	—
114	C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
115	C₄⋊C₄	C₂×C₄×C₂₃	1	1	yes	characteristic
116	C₂²⋊C₄	—	2	1	no	—
117	C₄²	—	2	1	no	—
118	C₄²	C₂³×C₂₃	1	1	yes	characteristic
119	C₂³	—	2	1	no	—
120	Q₈	—	2	1	no	—
121	Q₈	C₂×C₈×C₂₃	1	2	yes	complemented
122	D₄	—	2	1	no	—
123	D₄	C₂×C₈×C₂₃	1	2	yes	—
124	C₂×C₄	—	2	1	no	—
125	C₂×C₄	—	2	1	no	—
126	C₂×C₄	—	2	1	no	—
127	C₂×C₄	—	4	1	no	—
128	C₂×C₄	—	4	1	no	—
129	C₂×C₄	C₂²×C₄×C₂₃	1	1	yes	characteristic
130	C₂×C₄	C₂×D₄×C₂₃	1	1	yes	characteristic
131	C₂×C₄	#41×C₂₃	1	1	yes	characteristic
132	C₈	—	2	1	no	—
133	C₈	—	2	1	no	—
134	C₈	—	2	1	no	—
135	C₈	—	4	1	no	—
136	C₈	—	4	1	no	—
137	C₈	—	4	1	no	—
138	C₈	C₂×C₈×C₂₃	1	2	yes	complemented
139	C₂²	—	2	2	no	—
140	C₂²	—	4	1	no	—
141	C₂²	—	4	1	no	—
142	C₂²	C₄×D₄×C₂₃	1	1	yes	characteristic
143	C₄	—	2	2	no	—
144	C₄	—	2	1	no	—
145	C₄	—	4	1	no	—
146	C₄	—	4	1	no	—
147	C₄	C₂²×C₈×C₂₃	1	1	yes	characteristic
148	C₄	C₂₃×#131	1	1	yes	characteristic
149	C₄	C₂₃×AGL(1,ℤ/8)	1	1	yes	characteristic
150	C₄	C₂₃×#137	1	1	yes	characteristic
151	C₂	—	4	2	no	—
152	C₂	C₈×D₄×C₂₃	1	1	yes	characteristic
153	C₂	C₂₃×#440	1	1	yes	characteristic
154	C₂	C₂₃×#444	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₈×Q₈×C₂₃

C₂

yes

maximal, characteristic, complemented

C₄×SD₁₆×C₂₃

C₂

yes

maximal, characteristic

C₈×D₄×C₂₃

C₂

yes

maximal, characteristic

C₂₃×#335

C₂

yes

maximal, characteristic, complemented

C₂₃×Q₈⋊C₈

C₂

yes

maximal, characteristic, complemented

C₂₃×D₄⋊C₈

C₂

yes

maximal, characteristic

C₂₃×#322

C₂

yes

maximal, characteristic, complemented

C₂×SD₁₆×C₂₃

C₄

yes

characteristic

C₂²×C₈×C₂₃

—

C₄×Q₈×C₂₃

C₂²

yes

characteristic

C₄×D₄×C₂₃

C₂²

yes

characteristic

C₂₃×#106

C₄

yes

characteristic

C₂₃×C₄⋊C₈

—

C₂₃×C₄⋊C₈

C₂²

yes

characteristic

C₂₃×C₄⋊C₈

C₂²

yes

characteristic

C₂₃×#103

C₄

yes

characteristic

C₂₃×#102

C₄

yes

characteristic

C₂₃×C₂²⋊C₈

—

C₄×C₈×C₂₃

—

C₄×C₈×C₂₃

C₂²

yes

characteristic

C₄×C₈×C₂₃

C₂²

yes

characteristic

C₄×C₈×C₂₃

C₂²

yes

characteristic

C₂×Q₈×C₂₃

C₂×C₄

yes

characteristic

C₂×D₄×C₂₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₂₃

—

SD₁₆×C₂₃

C₈

yes

complemented

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

—

C₂×C₈×C₂₃

C₂×C₄

yes

characteristic

C₂×C₈×C₂₃

C₂×C₄

yes

characteristic

C₂×C₈×C₂₃

D₄

yes

characteristic

C₂×C₈×C₂₃

D₄

yes

characteristic

C₄⋊C₄×C₂₃

—

C₄⋊C₄×C₂₃

C₂×C₄

yes

characteristic

C₄⋊C₄×C₂₃

C₂×C₄

yes

characteristic

C₂²⋊C₄×C₂₃

—

C₄²×C₂₃

—

C₄²×C₂₃

C₂³

yes

characteristic

C₂³×C₂₃

—

Q₈×C₂₃

—

Q₈×C₂₃

C₂×C₈

yes

complemented

D₄×C₂₃

—

D₄×C₂₃

C₂×C₈

yes

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

—

C₂×C₄×C₂₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₂₃

C₂×D₄

yes

characteristic

C₂×C₄×C₂₃

#41

yes

characteristic

C₈×C₂₃

—

C₈×C₂₃

—

C₈×C₂₃

—

C₈×C₂₃

—

C₈×C₂₃

—

C₈×C₂₃

—

C₈×C₂₃

C₂×C₈

yes

complemented

SD₁₆⋊C₈

C₂₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

—

C₂²×C₂₃

C₄×D₄

yes

characteristic

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

—

C₄×C₂₃

C₂²×C₈

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

C₄×C₂₃

yes

C₈×Q₈

C₂×C₂₃

yes

characteristic, complemented

C₄×SD₁₆

C₂×C₂₃

yes

characteristic

C₈×D₄

C₂×C₂₃

yes

characteristic

#335

C₂×C₂₃

yes

characteristic, complemented

Q₈⋊C₈

C₂×C₂₃

yes

characteristic, complemented

D₄⋊C₈

C₂×C₂₃

yes

characteristic

#322

C₂×C₂₃

yes

characteristic, complemented

C₂×C₂₃

—

C₂×C₂₃

C₈×D₄

yes

C₂×C₂₃

yes

C₂×C₂₃

yes

C₂×SD₁₆

C₄×C₂₃

yes

characteristic

C₂²×C₈

—

C₄×Q₈

C₂²×C₂₃

yes

characteristic

C₄×D₄

C₂²×C₂₃

yes

characteristic

#106

C₄×C₂₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂²×C₂₃

yes

characteristic

C₄⋊C₈

C₂²×C₂₃

yes

characteristic

#103

C₄×C₂₃

yes

characteristic

#102

C₄×C₂₃

yes

characteristic

C₂²⋊C₈

—

C₄×C₈

—

C₄×C₈

C₂²×C₂₃

yes

characteristic

C₄×C₈

C₂²×C₂₃

yes

characteristic

C₄×C₈

C₂²×C₂₃

yes

characteristic

100

C₂₃

SD₁₆⋊C₈

yes

characteristic, direct product factor, Sylow

101

C₂×Q₈

C₂×C₄×C₂₃

yes

characteristic

102

C₂×D₄

C₂×C₄×C₂₃

yes

characteristic

103

C₂²×C₄

—

104

SD₁₆

C₈×C₂₃

yes

complemented

105

C₂×C₈

—

106

C₂×C₈

—

107

C₂×C₈

—

108

C₂×C₈

—

109

C₂×C₈

C₂×C₄×C₂₃

yes

characteristic

110

C₂×C₈

C₂×C₄×C₂₃

yes

characteristic

111

C₂×C₈

D₄×C₂₃

yes

characteristic

112

C₂×C₈

D₄×C₂₃

yes

characteristic

113

C₄⋊C₄

—

114

C₄⋊C₄

C₂×C₄×C₂₃

yes

characteristic

115

C₄⋊C₄

C₂×C₄×C₂₃

yes

characteristic

116

C₂²⋊C₄

—

117

C₄²

—

118

C₄²

C₂³×C₂₃

yes

characteristic

119

C₂³

—

120

Q₈

—

121

Q₈

C₂×C₈×C₂₃

yes

complemented

122

D₄

—

123

D₄

C₂×C₈×C₂₃

yes

—

124

C₂×C₄

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

—

129

C₂×C₄

C₂²×C₄×C₂₃

yes

characteristic

130

C₂×C₄

C₂×D₄×C₂₃

yes

characteristic

131

C₂×C₄

#41×C₂₃

yes

characteristic

132

C₈

—

133

C₈

—

134

C₈

—

135

C₈

—

136

C₈

—

137

C₈

—

138

C₈

C₂×C₈×C₂₃

yes

complemented

139

C₂²

—

140

C₂²

—

141

C₂²

—

142

C₂²

C₄×D₄×C₂₃

yes

characteristic

143

C₄

—

144

C₄

—

145

C₄

—

146

C₄

—

147

C₄

C₂²×C₈×C₂₃

yes

characteristic

148

C₄

C₂₃×#131

yes

characteristic

149

C₄

C₂₃×AGL(1,ℤ/8)

yes

characteristic

150

C₄

C₂₃×#137

yes

characteristic

151

C₂

—

152

C₂

C₈×D₄×C₂₃

yes

characteristic

153

C₂

C₂₃×#440

yes

characteristic

154

C₂

C₂₃×#444

yes

characteristic

	subgroup	G/H
center	C₂×C₄×C₂₃	C₂×D₄
commutator	C₄	C₂²×C₈×C₂₃
Frattini, cosocle	C₄²	C₂³×C₂₃
socle	C₂²×C₂₃	C₄×D₄
prime radical	C₂×D₄×C₂₃	C₂×C₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₂₃ ◁ C₄×C₈×C₂₃ ◁ G

subgroup

G/H

center

C₂×C₄×C₂₃

C₂×D₄

commutator

C₄

C₂²×C₈×C₂₃

Frattini, cosocle

C₄²

C₂³×C₂₃

socle

C₂²×C₂₃

C₄×D₄

prime radical

C₂×D₄×C₂₃

C₂×C₄

LCS

G ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂×C₄×C₂₃ ◁ C₄×C₈×C₂₃ ◁ G

N	H ≅ G/N	type
1. C₈×Q₈×C₂₃	C₂	semidirect
4. C₂₃×#335	C₂	semidirect
5. C₂₃×Q₈⋊C₈	C₂	semidirect
7. C₂₃×#322	C₂	semidirect
26. SD₁₆×C₂₃	C₈	semidirect
43. Q₈×C₂₃	C₂×C₈	semidirect
60. C₈×C₂₃	C₂×C₈	semidirect
61. SD₁₆⋊C₈	C₂₃	direct (N × H)
74. C₈×Q₈	C₂×C₂₃	semidirect
77. #335	C₂×C₂₃	semidirect
78. Q₈⋊C₈	C₂×C₂₃	semidirect
80. #322	C₂×C₂₃	semidirect
100. C₂₃	SD₁₆⋊C₈	direct (N × H)
104. SD₁₆	C₈×C₂₃	semidirect
121. Q₈	C₂×C₈×C₂₃	semidirect
138. C₈	C₂×C₈×C₂₃	semidirect

H ≅ G/N

type

1. C₈×Q₈×C₂₃

C₂

semidirect

4. C₂₃×#335

C₂

semidirect

5. C₂₃×Q₈⋊C₈

C₂

semidirect

7. C₂₃×#322

C₂

semidirect

26. SD₁₆×C₂₃

C₈

semidirect

43. Q₈×C₂₃

C₂×C₈

semidirect

60. C₈×C₂₃

C₂×C₈

semidirect

61. SD₁₆⋊C₈

C₂₃

direct (N × H)

74. C₈×Q₈

C₂×C₂₃

semidirect

77. #335

C₂×C₂₃

semidirect

78. Q₈⋊C₈

C₂×C₂₃

semidirect

80. #322

C₂×C₂₃

semidirect

100. C₂₃

SD₁₆⋊C₈

direct (N × H)

104. SD₁₆

C₈×C₂₃

semidirect

121. Q₈

C₂×C₈×C₂₃

semidirect

138. C₈

C₂×C₈×C₂₃

semidirect

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

subgroup	index
C₈×Q₈×C₂₃	2
C₄×SD₁₆×C₂₃	2
C₈×D₄×C₂₃	2
C₂₃×#335	2
C₂₃×Q₈⋊C₈	2
C₂₃×D₄⋊C₈	2
C₂₃×#322	2
SD₁₆⋊C₈	23

subgroup

index

C₈×Q₈×C₂₃

C₄×SD₁₆×C₂₃

C₈×D₄×C₂₃

C₂₃×#335

C₂₃×Q₈⋊C₈

C₂₃×D₄⋊C₈

C₂₃×#322

SD₁₆⋊C₈

Subgroups — C₂₃×SD₁₆⋊C₈ #15669197

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C23×SD16⋊C8 #15669197

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂₃×SD₁₆⋊C₈ #15669197