Finite Group Databasebeta

← C23×#4147 #13013190

Subgroups — C23×#4147 #13013190

Order 1280.

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
1C24×C5×Q8C211yesmaximal, characteristic, complemented
2C22×#4147C21112yesmaximal, direct product factor
3C23×Q8×D5C214yesmaximal, complemented
4C23×#713C214yesmaximal, complemented
5C23×#709C216yesmaximal, complemented
6C23×C5×Q8C2214yes—
7C23×C5×Q8C22156yescomplemented
8C24×C4×C5C2213yes—
9C23×C22⋊D5C2214yes—
10C2×#4147C2211792yesdirect product factor
11C22×#713C221224yescomplemented
12C22×Q8×D5C221224yescomplemented
13C22×#709C221336yescomplemented
14C23×D20C2216yes—
15C23×C4×D5C22112yes—
16C23×Dic10C2216yes—
17C23×2−1+4—51nomaximal, Sylow
18C25×C5C2311yescharacteristic
19C24×D5C2314yes—
20C22×C5×Q8C231112yes—
21C22×C5×Q8C231448yescomplemented
22C23×C4×C5C2316yes—
23C23×C4×C5C23184yes—
24C22×C22⋊D5C231112yes—
25C23×Dic5C2314yes—
26#4147C2314096yesdirect product factor
27C2×#713C2311792yescomplemented
28C2×Q8×D5C2311792yescomplemented
29C2×#709C2312688yescomplemented
30C22×D20C231168yes—
31C22×C4×D5C231336yes—
32C22×Dic10C231168yes—
33C22×2−1+4—5112no—
34C23×#41—54no—
35C23×#41—56no—
36C24×Q8—54no—
37C24×Q8D511yescharacteristic, complemented
38C24×C5—28no—
39C24×C5C24114yes—
40C24×C5C2411yescharacteristic
41C23×D5—232no—
42C23×D5C24156yes—
43C2×C5×Q8C241448yes—
44C2×C5×Q8C241512yescomplemented
45C22×C4×C5C24184yes—
46C22×C4×C5C241336yes—
47C2×C22⋊D5C241448yes—
48C22×Dic5C24156yes—
49#713C2412048yescomplemented
50Q8×D5C2412048yescomplemented
51#709C2413072yescomplemented
52C2×D20C241672yes—
53C2×C4×D5C2411344yes—
54C2×Dic10C241672yes—
55C2×2−1+4—51792no—
56C22×#41—5224no—
57C22×#41—5336no—
58C23×Q8—5224no—
59C23×Q8—56no—
60C23×Q8C2×D514yes—
61C23×Q8C2×D5156yescomplemented
62C23×D4—54no—
63C23×D4—56no—
64C24×C4—512no—
65C24×C4C2×D513yes—
66C23×C5—256no—
67C23×C52−1+418yescomplemented
68C23×C5C25128yes—
69C23×C5C2517yes—
70C22×D5—2224no—
71C22×D5C251112yes—
72C5×Q8C251256yes—
73C2×C4×C5C251168yes—
74C2×C4×C5C251192yes—
75C22⋊D5C251256yes—
76C2×Dic5C251112yes—
77D20C251384yes—
78C4×D5C251768yes—
79Dic10C251384yes—
80C25—54no—
81C25C22×D511yescharacteristic
822−1+4—54096no—
83C2×#41—51792no—
84C2×#41—52688no—
85C22×Q8—51792no—
86C22×Q8—5168no—
87C22×Q8C22×D51112yes—
88C22×Q8C22×D51448yescomplemented
89C22×D4—5112no—
90C22×D4—5168no—
91C23×C4—54no—
92C23×C4—5336no—
93C23×C4C22×D5184yes—
94C23×C4C22×D516yes—
95C22×C5—256no—
96C22×C5C2×2−1+4128yescomplemented
97C22×C5C2618yes—
98C22×C5C2617yes—
99C2×D5—2224no—
100C2×D5C26132yes—
101C4×C5C26148yes—
102Dic5C26132yes—
103C24—28no—
104C24—556no—
105C24—1032no—
106C24C23×D5114yes—
107C24C23×D511yescharacteristic
108#41—52048no—
109#41—53072no—
110C2×Q8—52048no—
111C2×Q8—5672no—
112C2×Q8C23×D51448yes—
113C2×Q8C23×D51512yescomplemented
114C2×D4—5448no—
115C2×D4—5672no—
116C22×C4—556no—
117C22×C4—51344no—
118C22×C4C23×D5184yes—
119C22×C4C23×D51336yes—
120C2×C5—28no—
121C2×C5C22×2−1+4114yescomplemented
122C2×C5C2711yescharacteristic
123D5—232no—
124C23—256no—
125C23—5112no—
126C23—10224no—
127C23#414718yesdirect product factor
128C23C24×D5128yes—
129C23C24×D517yes—
130Q8—5384no—
131Q8C24×D51256yes—
132D4—5256no—
133D4—5384no—
134C2×C4—5112no—
135C2×C4—5768no—
136C2×C4C24×D51168yes—
137C2×C4C24×D51192yes—
138C5C23×2−1+411yescharacteristic, complemented, Sylow
139C22—256no—
140C22—532no—
141C22—10224no—
142C22C2×#4147128yesdirect product factor
143C22C25×D518yes—
144C22C25×D517yes—
145C4—532no—
146C4C25×D5148yes—
147C2—28no—
148C2—1032no—
149C2C22×#4147114yesdirect product factor
150C2C26×D511yescharacteristic
subgroupG/H
centerC24C23×D5
commutator, cosocleC2×C5C27
FrattiniC2C26×D5
FittingC24×C5×Q8C2
socleC24×C5C24
hypercenterC24×Q8D5
hypocenterC5C23×2−1+4
LCSG ▷ C2×C5 ▷ C5
UCSC1 ◁ C24 ◁ C24×Q8

Complements

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

NH ≅ G/Ntype
1. C24×C5×Q8C2semidirect
2. C22×#4147C2direct (N × H)
3. C23×Q8×D5C2semidirect
4. C23×#713C2semidirect
5. C23×#709C2semidirect
7. C23×C5×Q8C22semidirect
10. C2×#4147C22direct (N × H)
11. C22×#713C22semidirect
12. C22×Q8×D5C22semidirect
13. C22×#709C22semidirect
21. C22×C5×Q8C23semidirect
26. #4147C23direct (N × H)
27. C2×#713C23semidirect
28. C2×Q8×D5C23semidirect
29. C2×#709C23semidirect
37. C24×Q8D5semidirect
44. C2×C5×Q8C24semidirect
49. #713C24semidirect
50. Q8×D5C24semidirect
51. #709C24semidirect
61. C23×Q8C2×D5semidirect
67. C23×C52−1+4semidirect
88. C22×Q8C22×D5semidirect
96. C22×C5C2×2−1+4semidirect
113. C2×Q8C23×D5semidirect
121. C2×C5C22×2−1+4semidirect
127. C23#4147direct (N × H)
138. C5C23×2−1+4semidirect
142. C22C2×#4147direct (N × H)
149. C2C22×#4147direct (N × H)

Maximal subgroups

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

subgroupindex
C24×C5×Q82
C22×#41472
C23×Q8×D52
C23×#7132
C23×#7092
C23×2−1+45