Finite Group Databasebeta

← C23×#2901 #15670520

Subgroups — C23×#2901 #15670520

Order 2944.

738 conjugacy classes of subgroups; only the 340 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 340 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#585C211yesmaximal, characteristic
2C23×#568C211yesmaximal, characteristic, complemented
3C4×#41×C23C211yesmaximal, characteristic, complemented
4C23×#421C216yesmaximal, complemented
5C2×C23×#104C216yesmaximal, complemented
6C23×2−1+4—21no—
7C23×2−1+4C414yescomplemented
8C23×H(5,𝔽2)—23no—
9C23×H(5,𝔽2)C414yescomplemented
10C2×#41×C23—24no—
11C2×#41×C23C2216yes—
12C2×#41×C23C2211yescharacteristic
13C23×#131—24no—
14C2×M16×C23C2216yescomplemented
15C4×Q8×C23—21no—
16C4×D4×C23—23no—
17C23×#117C2213yescomplemented
18C2×C42×C23C2213yescomplemented
19C23×#104C2214yescomplemented
20C23×#104C22112yescomplemented
21#41×C23—24no—
22#41×C23—212no—
23#41×C23—26no—
24#41×C23—26no—
25#41×C23C2×C414yes—
26#41×C23C2×C4112yescomplemented
27#41×C23D414yes—
28#41×C23C2314yes—
29C2×Q8×C23—24no—
30C2×Q8×C23C2×C416yescomplemented
31C2×Q8×C23D411yescharacteristic
32C2×D4×C23—26no—
33C2×D4×C23—212no—
34C2×D4×C23C2×C416yescomplemented
35C2×D4×C23D413yes—
36C22×C4×C23—26no—
37C22×C4×C23—23no—
38C22×C4×C23C2313yes—
39M16×C23—26no—
40M16×C23C2314yes—
41C2×C8×C23—26no—
42C4⋊C4×C23—23no—
43C22⋊C4×C23—23no—
44C42×C23—23no—
45C42×C23C2311yescharacteristic
46C42×C23C2313yes—
47C23×C23—26no—
48C23×C23C2×D413yes—
49Q8×C23—26no—
50Q8×C23C22⋊C414yes—
51Q8×C23C22×C414yes—
52D4×C23—212no—
53D4×C23—24no—
54D4×C23—26no—
55D4×C23C22⋊C4112yes—
56D4×C23C22×C414yes—
57C2×C4×C23—26no—
58C2×C4×C23—21no—
59C2×C4×C23—212no—
60C2×C4×C23—24no—
61C2×C4×C23—23no—
62C2×C4×C23—46no—
63C2×C4×C23C22×C416yes—
64C2×C4×C23C2×D413yes—
65C2×C4×C23C2×D416yes—
66C2×C4×C23C2411yescharacteristic
67C8×C23—24no—
68#2901C2311yesmaximal, characteristic, direct product factor, Sylow
69C22×C23—24no—
70C22×C23—43no—
71C22×C23—86no—
72C22×C23C2×C22⋊C416yes—
73C22×C23C22×D411yescharacteristic
74C4×C23—22no—
75C4×C23—24no—
76C4×C23—43no—
77C4×C23C2×C22⋊C416yes—
78C4×C23C23×C411yescharacteristic
79C4×C23C22×D411yescharacteristic
80#585C2×C2311yescharacteristic
81#568C2×C2311yescharacteristic, complemented
82C4×#41C2×C2311yescharacteristic, complemented
83#421C2×C2316yescomplemented
84C2×#104C2×C2316yescomplemented
85C2×C23—21no—
86C2×C23—26no—
87C2×C23—44no—
88C2×C23C22×C22⋊C411yescharacteristic
892−1+4—21no—
902−1+4C4×C2314yescomplemented
91H(5,𝔽2)—23no—
92H(5,𝔽2)C4×C2314yescomplemented
93C2×#41—24no—
94C2×#41C22×C2316yes—
95C2×#41C22×C2311yescharacteristic
96#131—24no—
97C2×M16C22×C2316yescomplemented
98C4×Q8—21no—
99C4×D4—23no—
100#117C22×C2313yescomplemented
101C2×C42C22×C2313yescomplemented
102#104C22×C2314yescomplemented
103#104C22×C23112yescomplemented
104C23#290111yescharacteristic, direct product factor, Sylow
105#41—24no—
106#41—212no—
107#41—26no—
108#41—26no—
109#41C2×C4×C2314yes—
110#41C2×C4×C23112yescomplemented
111#41D4×C2314yes—
112#41C23×C2314yes—
113C2×Q8—24no—
114C2×Q8C2×C4×C2316yescomplemented
115C2×Q8D4×C2311yescharacteristic
116C2×D4—26no—
117C2×D4—212no—
118C2×D4C2×C4×C2316yescomplemented
119C2×D4D4×C2313yes—
120C22×C4—26no—
121C22×C4—23no—
122C22×C4C23×C2313yes—
123M16—26no—
124M16C23×C2314yes—
125C2×C8—26no—
126C4⋊C4—23no—
127C22⋊C4—23no—
128C42—23no—
129C42C23×C2311yescharacteristic
130C42C23×C2313yes—
131C23—26no—
132C23C2×D4×C2313yes—
133Q8—26no—
134Q8C22⋊C4×C2314yes—
135Q8C22×C4×C2314yes—
136D4—212no—
137D4—24no—
138D4—26no—
139D4C22⋊C4×C23112yes—
140D4C22×C4×C2314yes—
141C2×C4—26no—
142C2×C4—21no—
143C2×C4—212no—
144C2×C4—24no—
145C2×C4—23no—
146C2×C4—46no—
147C2×C4C22×C4×C2316yes—
148C2×C4C2×D4×C2313yes—
149C2×C4C2×D4×C2316yes—
150C2×C4C24×C2311yescharacteristic
151C8—24no—
152C22—24no—
153C22—43no—
154C22—86no—
155C22C2×C22⋊C4×C2316yes—
156C22C22×D4×C2311yescharacteristic
157C4—22no—
158C4—24no—
159C4—43no—
160C4C2×C22⋊C4×C2316yes—
161C4C23×C4×C2311yescharacteristic
162C4C22×D4×C2311yescharacteristic
163C2—21no—
164C2—26no—
165C2—44no—
166C2C22×C22⋊C4×C2311yescharacteristic
subgroupG/H
centerC4×C23C22×D4
commutatorC4C23×C4×C23
Frattini, cosocleC2×C4C24×C23
socleC2×C23C22×C22⋊C4
prime radicalC23×#585C2
LCSG ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C4×C23 ◁ C2×#41×C23 ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C23×#568C2semidirect
3. C4×#41×C23C2semidirect
4. C23×#421C2semidirect
5. C2×C23×#104C2semidirect
7. C23×2−1+4C4semidirect
9. C23×H(5,𝔽2)C4semidirect
14. C2×M16×C23C22semidirect
17. C23×#117C22semidirect
18. C2×C42×C23C22semidirect
19. C23×#104C22semidirect
20. C23×#104C22semidirect
26. #41×C23C2×C4semidirect
30. C2×Q8×C23C2×C4semidirect
34. C2×D4×C23C2×C4semidirect
68. #2901C23direct (N × H)
81. #568C2×C23semidirect
82. C4×#41C2×C23semidirect
83. #421C2×C23semidirect
84. C2×#104C2×C23semidirect
90. 2−1+4C4×C23semidirect
92. H(5,𝔽2)C4×C23semidirect
97. C2×M16C22×C23semidirect
100. #117C22×C23semidirect
101. C2×C42C22×C23semidirect
102. #104C22×C23semidirect
103. #104C22×C23semidirect
104. C23#2901direct (N × H)
110. #41C2×C4×C23semidirect
114. C2×Q8C2×C4×C23semidirect
118. C2×D4C2×C4×C23semidirect

Maximal subgroups

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

subgroupindex
C23×#5852
C23×#5682
C4×#41×C232
C23×#4212
C2×C23×#1042
#290123