Finite Group Databasebeta

← C23×#13063373 #26010487

Subgroups — C23×#13063373 #26010487

Order 11008.

59806 conjugacy classes of subgroups; only the 32139 normal subgroups are shown.

normal subgroups only

This group has 59806 conjugacy classes of subgroups — too many to draw. Only its 32139 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
1C22×#13063373C21112yesmaximal, direct product factor
2C23×#10621902C214yesmaximal, complemented
3C23×#10621898C216yesmaximal, complemented
4C23×Q8×D43C214yesmaximal, complemented
5C24×Q8×C43C211yesmaximal, characteristic, complemented
6C2×#13063373C2211792yesdirect product factor
7C22×#10621902C221224yescomplemented
8C22×#10621898C221336yescomplemented
9C23×Dic86C2216yes—
10C22×Q8×D43C221224yescomplemented
11C23×D172C2216yes—
12C23×C22⋊D43C2214yes—
13C23×C4×D43C22112yes—
14C23×Q8×C43C2214yes—
15C23×Q8×C43C22156yescomplemented
16C24×C4×C43C2213yes—
17C24×D43C2314yes—
18#13063373C2314096yesdirect product factor
19C2×#10621902C2311792yescomplemented
20C2×#10621898C2312688yescomplemented
21C22×Dic86C231168yes—
22C2×Q8×D43C2311792yescomplemented
23C22×D172C231168yes—
24C22×C22⋊D43C231112yes—
25C23×Dic43C2314yes—
26C22×C4×D43C231336yes—
27C25×C43C2311yescharacteristic
28C22×Q8×C43C231112yes—
29C22×Q8×C43C231448yescomplemented
30C23×C4×C43C2316yes—
31C23×C4×C43C23184yes—
32C24×C43—28no—
33C24×C43C24114yes—
34C24×C43C2411yescharacteristic
35C23×D43—232no—
36C23×D43C24156yes—
37C2×Q8×C43C241448yes—
38C2×Q8×C43C241512yescomplemented
39C22×C4×C43C24184yes—
40C22×C4×C43C241336yes—
41C2×C22⋊D43C241448yes—
42C22×Dic43C24156yes—
43#10621902C2412048yescomplemented
44Q8×D43C2412048yescomplemented
45#10621898C2413072yescomplemented
46C2×D172C241672yes—
47C2×C4×D43C2411344yes—
48C2×Dic86C241672yes—
49C23×C43—256no—
50C23×C432−1+418yescomplemented
51C23×C43C25128yes—
52C23×C43C2517yes—
53C22×D43—2224no—
54C22×D43C251112yes—
55Q8×C43C251256yes—
56C2×C4×C43C251168yes—
57C2×C4×C43C251192yes—
58C22⋊D43C251256yes—
59C2×Dic43C251112yes—
60D172C251384yes—
61C4×D43C251768yes—
62Dic86C251384yes—
63C23×2−1+4—431nomaximal, Sylow
64C22×C43—256no—
65C22×C43C2×2−1+4128yescomplemented
66C22×C43C2618yes—
67C22×C43C2617yes—
68C2×D43—2224no—
69C2×D43C26132yes—
70C4×C43C26148yes—
71Dic43C26132yes—
72C22×2−1+4—43112no—
73C23×#41—434no—
74C23×#41—436no—
75C24×Q8—434no—
76C24×Q8D4311yescharacteristic, complemented
77C2×C43—28no—
78C2×C43C22×2−1+4114yescomplemented
79C2×C43C2711yescharacteristic
80D43—232no—
81C2×2−1+4—431792no—
82C22×#41—43224no—
83C22×#41—43336no—
84C23×Q8—43224no—
85C23×Q8—436no—
86C23×Q8C2×D4314yes—
87C23×Q8C2×D43156yescomplemented
88C23×D4—434no—
89C23×D4—436no—
90C24×C4—4312no—
91C24×C4C2×D4313yes—
92C43C23×2−1+411yescharacteristic, complemented, Sylow
93C25—434no—
94C25C22×D4311yescharacteristic
952−1+4—434096no—
96C2×#41—431792no—
97C2×#41—432688no—
98C22×Q8—431792no—
99C22×Q8—43168no—
100C22×Q8C22×D431112yes—
101C22×Q8C22×D431448yescomplemented
102C22×D4—43112no—
103C22×D4—43168no—
104C23×C4—434no—
105C23×C4—43336no—
106C23×C4C22×D43184yes—
107C23×C4C22×D4316yes—
108C24—28no—
109C24—4356no—
110C24—8632no—
111C24C23×D43114yes—
112C24C23×D4311yescharacteristic
113#41—432048no—
114#41—433072no—
115C2×Q8—432048no—
116C2×Q8—43672no—
117C2×Q8C23×D431448yes—
118C2×Q8C23×D431512yescomplemented
119C2×D4—43448no—
120C2×D4—43672no—
121C22×C4—4356no—
122C22×C4—431344no—
123C22×C4C23×D43184yes—
124C22×C4C23×D431336yes—
125C23—256no—
126C23—43112no—
127C23—86224no—
128C23#1306337318yesdirect product factor
129C23C24×D43128yes—
130C23C24×D4317yes—
131Q8—43384no—
132Q8C24×D431256yes—
133D4—43256no—
134D4—43384no—
135C2×C4—43112no—
136C2×C4—43768no—
137C2×C4C24×D431168yes—
138C2×C4C24×D431192yes—
139C22—256no—
140C22—4332no—
141C22—86224no—
142C22C2×#13063373128yesdirect product factor
143C22C25×D4318yes—
144C22C25×D4317yes—
145C4—4332no—
146C4C25×D43148yes—
147C2—28no—
148C2—8632no—
149C2C22×#13063373114yesdirect product factor
150C2C26×D4311yescharacteristic
subgroupG/H
centerC24C23×D43
commutator, cosocleC2×C43C27
FrattiniC2C26×D43
FittingC24×Q8×C43C2
socleC24×C43C24
hypercenterC24×Q8D43
hypocenterC43C23×2−1+4
LCSG ▷ C2×C43 ▷ C43
UCSC1 ◁ C24 ◁ C24×Q8

Complements

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

NH ≅ G/Ntype
1. C22×#13063373C2direct (N × H)
2. C23×#10621902C2semidirect
3. C23×#10621898C2semidirect
4. C23×Q8×D43C2semidirect
5. C24×Q8×C43C2semidirect
6. C2×#13063373C22direct (N × H)
7. C22×#10621902C22semidirect
8. C22×#10621898C22semidirect
10. C22×Q8×D43C22semidirect
15. C23×Q8×C43C22semidirect
18. #13063373C23direct (N × H)
19. C2×#10621902C23semidirect
20. C2×#10621898C23semidirect
22. C2×Q8×D43C23semidirect
29. C22×Q8×C43C23semidirect
38. C2×Q8×C43C24semidirect
43. #10621902C24semidirect
44. Q8×D43C24semidirect
45. #10621898C24semidirect
50. C23×C432−1+4semidirect
65. C22×C43C2×2−1+4semidirect
76. C24×Q8D43semidirect
78. C2×C43C22×2−1+4semidirect
87. C23×Q8C2×D43semidirect
92. C43C23×2−1+4semidirect
101. C22×Q8C22×D43semidirect
118. C2×Q8C23×D43semidirect
128. C23#13063373direct (N × H)
142. C22C2×#13063373direct (N × H)
149. C2C22×#13063373direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#130633732
C23×#106219022
C23×#106218982
C23×Q8×D432
C24×Q8×C432
C23×2−1+443