Finite Group Databasebeta

← C23×#13133265 #28297982

Subgroups — C23×#13133265 #28297982

Order 13568.

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×#13133265C21112yesmaximal, direct product factor
2C23×#11719949C214yesmaximal, complemented
3C23×#11719945C216yesmaximal, complemented
4C23×Q8×D53C214yesmaximal, complemented
5C24×Q8×C53C211yesmaximal, characteristic, complemented
6C2×#13133265C2211792yesdirect product factor
7C22×#11719949C221224yescomplemented
8C22×#11719945C221336yescomplemented
9C23×Dic106C2216yes—
10C22×Q8×D53C221224yescomplemented
11C23×D212C2216yes—
12C23×C22⋊D53C2214yes—
13C23×C4×D53C22112yes—
14C23×Q8×C53C2214yes—
15C23×Q8×C53C22156yescomplemented
16C24×C4×C53C2213yes—
17C24×D53C2314yes—
18#13133265C2314096yesdirect product factor
19C2×#11719949C2311792yescomplemented
20C2×#11719945C2312688yescomplemented
21C22×Dic106C231168yes—
22C2×Q8×D53C2311792yescomplemented
23C22×D212C231168yes—
24C22×C22⋊D53C231112yes—
25C23×Dic53C2314yes—
26C22×C4×D53C231336yes—
27C25×C53C2311yescharacteristic
28C22×Q8×C53C231112yes—
29C22×Q8×C53C231448yescomplemented
30C23×C4×C53C2316yes—
31C23×C4×C53C23184yes—
32C24×C53—28no—
33C24×C53C24114yes—
34C24×C53C2411yescharacteristic
35C23×D53—232no—
36C23×D53C24156yes—
37C2×Q8×C53C241448yes—
38C2×Q8×C53C241512yescomplemented
39C22×C4×C53C24184yes—
40C22×C4×C53C241336yes—
41C2×C22⋊D53C241448yes—
42C22×Dic53C24156yes—
43#11719949C2412048yescomplemented
44Q8×D53C2412048yescomplemented
45#11719945C2413072yescomplemented
46C2×D212C241672yes—
47C2×C4×D53C2411344yes—
48C2×Dic106C241672yes—
49C23×C53—256no—
50C23×C532−1+418yescomplemented
51C23×C53C25128yes—
52C23×C53C2517yes—
53C22×D53—2224no—
54C22×D53C251112yes—
55Q8×C53C251256yes—
56C2×C4×C53C251168yes—
57C2×C4×C53C251192yes—
58C22⋊D53C251256yes—
59C2×Dic53C251112yes—
60D212C251384yes—
61C4×D53C251768yes—
62Dic106C251384yes—
63C23×2−1+4—531nomaximal, Sylow
64C22×C53—256no—
65C22×C53C2×2−1+4128yescomplemented
66C22×C53C2618yes—
67C22×C53C2617yes—
68C2×D53—2224no—
69C2×D53C26132yes—
70C4×C53C26148yes—
71Dic53C26132yes—
72C22×2−1+4—53112no—
73C23×#41—534no—
74C23×#41—536no—
75C24×Q8—534no—
76C24×Q8D5311yescharacteristic, complemented
77C2×C53—28no—
78C2×C53C22×2−1+4114yescomplemented
79C2×C53C2711yescharacteristic
80D53—232no—
81C2×2−1+4—531792no—
82C22×#41—53224no—
83C22×#41—53336no—
84C23×Q8—53224no—
85C23×Q8—536no—
86C23×Q8C2×D5314yes—
87C23×Q8C2×D53156yescomplemented
88C23×D4—534no—
89C23×D4—536no—
90C24×C4—5312no—
91C24×C4C2×D5313yes—
92C53C23×2−1+411yescharacteristic, complemented, Sylow
93C25—534no—
94C25C22×D5311yescharacteristic
952−1+4—534096no—
96C2×#41—531792no—
97C2×#41—532688no—
98C22×Q8—531792no—
99C22×Q8—53168no—
100C22×Q8C22×D531112yes—
101C22×Q8C22×D531448yescomplemented
102C22×D4—53112no—
103C22×D4—53168no—
104C23×C4—534no—
105C23×C4—53336no—
106C23×C4C22×D53184yes—
107C23×C4C22×D5316yes—
108C24—28no—
109C24—5356no—
110C24—10632no—
111C24C23×D53114yes—
112C24C23×D5311yescharacteristic
113#41—532048no—
114#41—533072no—
115C2×Q8—532048no—
116C2×Q8—53672no—
117C2×Q8C23×D531448yes—
118C2×Q8C23×D531512yescomplemented
119C2×D4—53448no—
120C2×D4—53672no—
121C22×C4—5356no—
122C22×C4—531344no—
123C22×C4C23×D53184yes—
124C22×C4C23×D531336yes—
125C23—256no—
126C23—53112no—
127C23—106224no—
128C23#1313326518yesdirect product factor
129C23C24×D53128yes—
130C23C24×D5317yes—
131Q8—53384no—
132Q8C24×D531256yes—
133D4—53256no—
134D4—53384no—
135C2×C4—53112no—
136C2×C4—53768no—
137C2×C4C24×D531168yes—
138C2×C4C24×D531192yes—
139C22—256no—
140C22—5332no—
141C22—106224no—
142C22C2×#13133265128yesdirect product factor
143C22C25×D5318yes—
144C22C25×D5317yes—
145C4—5332no—
146C4C25×D53148yes—
147C2—28no—
148C2—10632no—
149C2C22×#13133265114yesdirect product factor
150C2C26×D5311yescharacteristic
subgroupG/H
centerC24C23×D53
commutator, cosocleC2×C53C27
FrattiniC2C26×D53
FittingC24×Q8×C53C2
socleC24×C53C24
hypercenterC24×Q8D53
hypocenterC53C23×2−1+4
LCSG ▷ C2×C53 ▷ C53
UCSC1 ◁ C24 ◁ C24×Q8

Complements

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

NH ≅ G/Ntype
1. C22×#13133265C2direct (N × H)
2. C23×#11719949C2semidirect
3. C23×#11719945C2semidirect
4. C23×Q8×D53C2semidirect
5. C24×Q8×C53C2semidirect
6. C2×#13133265C22direct (N × H)
7. C22×#11719949C22semidirect
8. C22×#11719945C22semidirect
10. C22×Q8×D53C22semidirect
15. C23×Q8×C53C22semidirect
18. #13133265C23direct (N × H)
19. C2×#11719949C23semidirect
20. C2×#11719945C23semidirect
22. C2×Q8×D53C23semidirect
29. C22×Q8×C53C23semidirect
38. C2×Q8×C53C24semidirect
43. #11719949C24semidirect
44. Q8×D53C24semidirect
45. #11719945C24semidirect
50. C23×C532−1+4semidirect
65. C22×C53C2×2−1+4semidirect
76. C24×Q8D53semidirect
78. C2×C53C22×2−1+4semidirect
87. C23×Q8C2×D53semidirect
92. C53C23×2−1+4semidirect
101. C22×Q8C22×D53semidirect
118. C2×Q8C23×D53semidirect
128. C23#13133265direct (N × H)
142. C22C2×#13133265direct (N × H)
149. C2C22×#13133265direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131332652
C23×#117199492
C23×#117199452
C23×Q8×D532
C24×Q8×C532
C23×2−1+453