Finite Group Databasebeta

← C23×#88254 #16776809

Subgroups — C23×#88254 #16776809

Order 3328.

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×#88254C21112yesmaximal, direct product factor
2C23×#6131C214yesmaximal, complemented
3C23×#6127C216yesmaximal, complemented
4C23×Q8×D13C214yesmaximal, complemented
5C24×Q8×C13C211yesmaximal, characteristic, complemented
6C23×Q8×C13C2214yes—
7C23×Q8×C13C22156yescomplemented
8C24×C4×C13C2213yes—
9C23×C22⋊D13C2214yes—
10C2×#88254C2211792yesdirect product factor
11C22×#6131C221224yescomplemented
12C22×Q8×D13C221224yescomplemented
13C22×#6127C221336yescomplemented
14C23×D52C2216yes—
15C23×C4×D13C22112yes—
16C23×Dic26C2216yes—
17C25×C13C2311yescharacteristic
18C24×D13C2314yes—
19C22×Q8×C13C231112yes—
20C22×Q8×C13C231448yescomplemented
21C23×C4×C13C2316yes—
22C23×C4×C13C23184yes—
23C22×C22⋊D13C231112yes—
24C23×Dic13C2314yes—
25#88254C2314096yesdirect product factor
26C2×#6131C2311792yescomplemented
27C2×Q8×D13C2311792yescomplemented
28C2×#6127C2312688yescomplemented
29C22×D52C231168yes—
30C22×C4×D13C231336yes—
31C22×Dic26C231168yes—
32C23×2−1+4—131nomaximal, Sylow
33C24×C13—28no—
34C24×C13C24114yes—
35C24×C13C2411yescharacteristic
36C23×D13—232no—
37C23×D13C24156yes—
38C2×Q8×C13C241448yes—
39C2×Q8×C13C241512yescomplemented
40C22×C4×C13C24184yes—
41C22×C4×C13C241336yes—
42C2×C22⋊D13C241448yes—
43C22×Dic13C24156yes—
44#6131C2412048yescomplemented
45Q8×D13C2412048yescomplemented
46#6127C2413072yescomplemented
47C2×D52C241672yes—
48C2×C4×D13C2411344yes—
49C2×Dic26C241672yes—
50C22×2−1+4—13112no—
51C23×#41—134no—
52C23×#41—136no—
53C24×Q8—134no—
54C24×Q8D1311yescharacteristic, complemented
55C23×C13—256no—
56C23×C132−1+418yescomplemented
57C23×C13C25128yes—
58C23×C13C2517yes—
59C22×D13—2224no—
60C22×D13C251112yes—
61Q8×C13C251256yes—
62C2×C4×C13C251168yes—
63C2×C4×C13C251192yes—
64C22⋊D13C251256yes—
65C2×Dic13C251112yes—
66D52C251384yes—
67C4×D13C251768yes—
68Dic26C251384yes—
69C2×2−1+4—131792no—
70C22×#41—13224no—
71C22×#41—13336no—
72C23×Q8—13224no—
73C23×Q8—136no—
74C23×Q8C2×D1314yes—
75C23×Q8C2×D13156yescomplemented
76C23×D4—134no—
77C23×D4—136no—
78C24×C4—1312no—
79C24×C4C2×D1313yes—
80C22×C13—256no—
81C22×C13C2×2−1+4128yescomplemented
82C22×C13C2618yes—
83C22×C13C2617yes—
84C2×D13—2224no—
85C2×D13C26132yes—
86C4×C13C26148yes—
87Dic13C26132yes—
88C25—134no—
89C25C22×D1311yescharacteristic
902−1+4—134096no—
91C2×#41—131792no—
92C2×#41—132688no—
93C22×Q8—131792no—
94C22×Q8—13168no—
95C22×Q8C22×D131112yes—
96C22×Q8C22×D131448yescomplemented
97C22×D4—13112no—
98C22×D4—13168no—
99C23×C4—134no—
100C23×C4—13336no—
101C23×C4C22×D13184yes—
102C23×C4C22×D1316yes—
103C2×C13—28no—
104C2×C13C22×2−1+4114yescomplemented
105C2×C13C2711yescharacteristic
106D13—232no—
107C24—28no—
108C24—1356no—
109C24—2632no—
110C24C23×D13114yes—
111C24C23×D1311yescharacteristic
112#41—132048no—
113#41—133072no—
114C2×Q8—132048no—
115C2×Q8—13672no—
116C2×Q8C23×D131448yes—
117C2×Q8C23×D131512yescomplemented
118C2×D4—13448no—
119C2×D4—13672no—
120C22×C4—1356no—
121C22×C4—131344no—
122C22×C4C23×D13184yes—
123C22×C4C23×D131336yes—
124C13C23×2−1+411yescharacteristic, complemented, Sylow
125C23—256no—
126C23—13112no—
127C23—26224no—
128C23#8825418yesdirect product factor
129C23C24×D13128yes—
130C23C24×D1317yes—
131Q8—13384no—
132Q8C24×D131256yes—
133D4—13256no—
134D4—13384no—
135C2×C4—13112no—
136C2×C4—13768no—
137C2×C4C24×D131168yes—
138C2×C4C24×D131192yes—
139C22—256no—
140C22—1332no—
141C22—26224no—
142C22C2×#88254128yesdirect product factor
143C22C25×D1318yes—
144C22C25×D1317yes—
145C4—1332no—
146C4C25×D13148yes—
147C2—28no—
148C2—2632no—
149C2C22×#88254114yesdirect product factor
150C2C26×D1311yescharacteristic
subgroupG/H
centerC24C23×D13
commutator, cosocleC2×C13C27
FrattiniC2C26×D13
FittingC24×Q8×C13C2
socleC24×C13C24
hypercenterC24×Q8D13
hypocenterC13C23×2−1+4
LCSG ▷ C2×C13 ▷ C13
UCSC1 ◁ C24 ◁ C24×Q8

Complements

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

NH ≅ G/Ntype
1. C22×#88254C2direct (N × H)
2. C23×#6131C2semidirect
3. C23×#6127C2semidirect
4. C23×Q8×D13C2semidirect
5. C24×Q8×C13C2semidirect
7. C23×Q8×C13C22semidirect
10. C2×#88254C22direct (N × H)
11. C22×#6131C22semidirect
12. C22×Q8×D13C22semidirect
13. C22×#6127C22semidirect
20. C22×Q8×C13C23semidirect
25. #88254C23direct (N × H)
26. C2×#6131C23semidirect
27. C2×Q8×D13C23semidirect
28. C2×#6127C23semidirect
39. C2×Q8×C13C24semidirect
44. #6131C24semidirect
45. Q8×D13C24semidirect
46. #6127C24semidirect
54. C24×Q8D13semidirect
56. C23×C132−1+4semidirect
75. C23×Q8C2×D13semidirect
81. C22×C13C2×2−1+4semidirect
96. C22×Q8C22×D13semidirect
104. C2×C13C22×2−1+4semidirect
117. C2×Q8C23×D13semidirect
124. C13C23×2−1+4semidirect
128. C23#88254direct (N × H)
142. C22C2×#88254direct (N × H)
149. C2C22×#88254direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#882542
C23×#61312
C23×#61272
C23×Q8×D132
C24×Q8×C132
C23×2−1+413