Finite Group Databasebeta

← C23×#14579209 #35202159

Subgroups — C23×#14579209 #35202159

Order 20224.

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×#14579209C21112yesmaximal, direct product factor
2C23×#11929625C214yesmaximal, complemented
3C23×#11929624C216yesmaximal, complemented
4C23×Q8×D79C214yesmaximal, complemented
5C24×Q8×C79C211yesmaximal, characteristic, complemented
6C2×#14579209C2211792yesdirect product factor
7C22×#11929625C221224yescomplemented
8C22×#11929624C221336yescomplemented
9C23×Dic158C2216yes—
10C22×Q8×D79C221224yescomplemented
11C23×D316C2216yes—
12C23×C22⋊D79C2214yes—
13C23×C4×D79C22112yes—
14C23×Q8×C79C2214yes—
15C23×Q8×C79C22156yescomplemented
16C24×C4×C79C2213yes—
17C24×D79C2314yes—
18#14579209C2314096yesdirect product factor
19C2×#11929625C2311792yescomplemented
20C2×#11929624C2312688yescomplemented
21C22×Dic158C231168yes—
22C2×Q8×D79C2311792yescomplemented
23C22×D316C231168yes—
24C22×C22⋊D79C231112yes—
25C23×Dic79C2314yes—
26C22×C4×D79C231336yes—
27C25×C79C2311yescharacteristic
28C22×Q8×C79C231112yes—
29C22×Q8×C79C231448yescomplemented
30C23×C4×C79C2316yes—
31C23×C4×C79C23184yes—
32C23×D79—232no—
33C23×D79C24156yes—
34#11929625C2412048yescomplemented
35#11929624C2413072yescomplemented
36C2×Dic158C241672yes—
37Q8×D79C2412048yescomplemented
38C2×D316C241672yes—
39C2×C22⋊D79C241448yes—
40C22×Dic79C24156yes—
41C2×C4×D79C2411344yes—
42C24×C79—28no—
43C24×C79C24114yes—
44C24×C79C2411yescharacteristic
45C2×Q8×C79C241448yes—
46C2×Q8×C79C241512yescomplemented
47C22×C4×C79C24184yes—
48C22×C4×C79C241336yes—
49C23×C79—256no—
50C23×C792−1+418yescomplemented
51C23×C79C25128yes—
52C23×C79C2517yes—
53C22×D79—2224no—
54C22×D79C251112yes—
55Q8×C79C251256yes—
56C2×C4×C79C251168yes—
57C2×C4×C79C251192yes—
58C22⋊D79C251256yes—
59C2×Dic79C251112yes—
60D316C251384yes—
61C4×D79C251768yes—
62Dic158C251384yes—
63C22×C79—256no—
64C22×C79C2×2−1+4128yescomplemented
65C22×C79C2618yes—
66C22×C79C2617yes—
67C2×D79—2224no—
68C2×D79C26132yes—
69C4×C79C26148yes—
70Dic79C26132yes—
71C23×2−1+4—791nomaximal, Sylow
72C2×C79—28no—
73C2×C79C22×2−1+4114yescomplemented
74C2×C79C2711yescharacteristic
75D79—232no—
76C22×2−1+4—79112no—
77C23×#41—794no—
78C23×#41—796no—
79C24×Q8—794no—
80C24×Q8D7911yescharacteristic, complemented
81C79C23×2−1+411yescharacteristic, complemented, Sylow
82C2×2−1+4—791792no—
83C22×#41—79224no—
84C22×#41—79336no—
85C23×Q8—79224no—
86C23×Q8—796no—
87C23×Q8C2×D7914yes—
88C23×Q8C2×D79156yescomplemented
89C23×D4—794no—
90C23×D4—796no—
91C24×C4—7912no—
92C24×C4C2×D7913yes—
93C25—794no—
94C25C22×D7911yescharacteristic
952−1+4—794096no—
96C2×#41—791792no—
97C2×#41—792688no—
98C22×Q8—791792no—
99C22×Q8—79168no—
100C22×Q8C22×D791112yes—
101C22×Q8C22×D791448yescomplemented
102C22×D4—79112no—
103C22×D4—79168no—
104C23×C4—794no—
105C23×C4—79336no—
106C23×C4C22×D79184yes—
107C23×C4C22×D7916yes—
108C24—28no—
109C24—7956no—
110C24—15832no—
111C24C23×D79114yes—
112C24C23×D7911yescharacteristic
113#41—792048no—
114#41—793072no—
115C2×Q8—792048no—
116C2×Q8—79672no—
117C2×Q8C23×D791448yes—
118C2×Q8C23×D791512yescomplemented
119C2×D4—79448no—
120C2×D4—79672no—
121C22×C4—7956no—
122C22×C4—791344no—
123C22×C4C23×D79184yes—
124C22×C4C23×D791336yes—
125C23—256no—
126C23—79112no—
127C23—158224no—
128C23#1457920918yesdirect product factor
129C23C24×D79128yes—
130C23C24×D7917yes—
131Q8—79384no—
132Q8C24×D791256yes—
133D4—79256no—
134D4—79384no—
135C2×C4—79112no—
136C2×C4—79768no—
137C2×C4C24×D791168yes—
138C2×C4C24×D791192yes—
139C22—256no—
140C22—7932no—
141C22—158224no—
142C22C2×#14579209128yesdirect product factor
143C22C25×D7918yes—
144C22C25×D7917yes—
145C4—7932no—
146C4C25×D79148yes—
147C2—28no—
148C2—15832no—
149C2C22×#14579209114yesdirect product factor
150C2C26×D7911yescharacteristic
subgroupG/H
centerC24C23×D79
commutator, cosocleC2×C79C27
FrattiniC2C26×D79
FittingC24×Q8×C79C2
socleC24×C79C24
hypercenterC24×Q8D79
hypocenterC79C23×2−1+4
LCSG ▷ C2×C79 ▷ C79
UCSC1 ◁ C24 ◁ C24×Q8

Complements

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

NH ≅ G/Ntype
1. C22×#14579209C2direct (N × H)
2. C23×#11929625C2semidirect
3. C23×#11929624C2semidirect
4. C23×Q8×D79C2semidirect
5. C24×Q8×C79C2semidirect
6. C2×#14579209C22direct (N × H)
7. C22×#11929625C22semidirect
8. C22×#11929624C22semidirect
10. C22×Q8×D79C22semidirect
15. C23×Q8×C79C22semidirect
18. #14579209C23direct (N × H)
19. C2×#11929625C23semidirect
20. C2×#11929624C23semidirect
22. C2×Q8×D79C23semidirect
29. C22×Q8×C79C23semidirect
34. #11929625C24semidirect
35. #11929624C24semidirect
37. Q8×D79C24semidirect
46. C2×Q8×C79C24semidirect
50. C23×C792−1+4semidirect
64. C22×C79C2×2−1+4semidirect
73. C2×C79C22×2−1+4semidirect
80. C24×Q8D79semidirect
81. C79C23×2−1+4semidirect
88. C23×Q8C2×D79semidirect
101. C22×Q8C22×D79semidirect
118. C2×Q8C23×D79semidirect
128. C23#14579209direct (N × H)
142. C22C2×#14579209direct (N × H)
149. C2C22×#14579209direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145792092
C23×#119296252
C23×#119296242
C23×Q8×D792
C24×Q8×C792
C23×2−1+479