Finite Group Databasebeta

← C23×#14524091 #31769721

Subgroups — C23×#14524091 #31769721

Order 17152.

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×#14524091C21112yesmaximal, direct product factor
2C23×#11761629C214yesmaximal, complemented
3C23×#11761628C216yesmaximal, complemented
4C23×Q8×D67C214yesmaximal, complemented
5C24×Q8×C67C211yesmaximal, characteristic, complemented
6C2×#14524091C2211792yesdirect product factor
7C22×#11761629C221224yescomplemented
8C22×#11761628C221336yescomplemented
9C23×Dic134C2216yes—
10C22×Q8×D67C221224yescomplemented
11C23×D268C2216yes—
12C23×C22⋊D67C2214yes—
13C23×C4×D67C22112yes—
14C23×Q8×C67C2214yes—
15C23×Q8×C67C22156yescomplemented
16C24×C4×C67C2213yes—
17C24×D67C2314yes—
18#14524091C2314096yesdirect product factor
19C2×#11761629C2311792yescomplemented
20C2×#11761628C2312688yescomplemented
21C22×Dic134C231168yes—
22C2×Q8×D67C2311792yescomplemented
23C22×D268C231168yes—
24C22×C22⋊D67C231112yes—
25C23×Dic67C2314yes—
26C22×C4×D67C231336yes—
27C25×C67C2311yescharacteristic
28C22×Q8×C67C231112yes—
29C22×Q8×C67C231448yescomplemented
30C23×C4×C67C2316yes—
31C23×C4×C67C23184yes—
32C23×D67—232no—
33C23×D67C24156yes—
34#11761629C2412048yescomplemented
35#11761628C2413072yescomplemented
36C2×Dic134C241672yes—
37Q8×D67C2412048yescomplemented
38C2×D268C241672yes—
39C2×C22⋊D67C241448yes—
40C22×Dic67C24156yes—
41C2×C4×D67C2411344yes—
42C24×C67—28no—
43C24×C67C24114yes—
44C24×C67C2411yescharacteristic
45C2×Q8×C67C241448yes—
46C2×Q8×C67C241512yescomplemented
47C22×C4×C67C24184yes—
48C22×C4×C67C241336yes—
49C23×C67—256no—
50C23×C672−1+418yescomplemented
51C23×C67C25128yes—
52C23×C67C2517yes—
53C22×D67—2224no—
54C22×D67C251112yes—
55Q8×C67C251256yes—
56C2×C4×C67C251168yes—
57C2×C4×C67C251192yes—
58C22⋊D67C251256yes—
59C2×Dic67C251112yes—
60D268C251384yes—
61C4×D67C251768yes—
62Dic134C251384yes—
63C22×C67—256no—
64C22×C67C2×2−1+4128yescomplemented
65C22×C67C2618yes—
66C22×C67C2617yes—
67C2×D67—2224no—
68C2×D67C26132yes—
69C4×C67C26148yes—
70Dic67C26132yes—
71C23×2−1+4—671nomaximal, Sylow
72C2×C67—28no—
73C2×C67C22×2−1+4114yescomplemented
74C2×C67C2711yescharacteristic
75D67—232no—
76C22×2−1+4—67112no—
77C23×#41—674no—
78C23×#41—676no—
79C24×Q8—674no—
80C24×Q8D6711yescharacteristic, complemented
81C67C23×2−1+411yescharacteristic, complemented, Sylow
82C2×2−1+4—671792no—
83C22×#41—67224no—
84C22×#41—67336no—
85C23×Q8—67224no—
86C23×Q8—676no—
87C23×Q8C2×D6714yes—
88C23×Q8C2×D67156yescomplemented
89C23×D4—674no—
90C23×D4—676no—
91C24×C4—6712no—
92C24×C4C2×D6713yes—
93C25—674no—
94C25C22×D6711yescharacteristic
952−1+4—674096no—
96C2×#41—671792no—
97C2×#41—672688no—
98C22×Q8—671792no—
99C22×Q8—67168no—
100C22×Q8C22×D671112yes—
101C22×Q8C22×D671448yescomplemented
102C22×D4—67112no—
103C22×D4—67168no—
104C23×C4—674no—
105C23×C4—67336no—
106C23×C4C22×D67184yes—
107C23×C4C22×D6716yes—
108C24—28no—
109C24—6756no—
110C24—13432no—
111C24C23×D67114yes—
112C24C23×D6711yescharacteristic
113#41—672048no—
114#41—673072no—
115C2×Q8—672048no—
116C2×Q8—67672no—
117C2×Q8C23×D671448yes—
118C2×Q8C23×D671512yescomplemented
119C2×D4—67448no—
120C2×D4—67672no—
121C22×C4—6756no—
122C22×C4—671344no—
123C22×C4C23×D67184yes—
124C22×C4C23×D671336yes—
125C23—256no—
126C23—67112no—
127C23—134224no—
128C23#1452409118yesdirect product factor
129C23C24×D67128yes—
130C23C24×D6717yes—
131Q8—67384no—
132Q8C24×D671256yes—
133D4—67256no—
134D4—67384no—
135C2×C4—67112no—
136C2×C4—67768no—
137C2×C4C24×D671168yes—
138C2×C4C24×D671192yes—
139C22—256no—
140C22—6732no—
141C22—134224no—
142C22C2×#14524091128yesdirect product factor
143C22C25×D6718yes—
144C22C25×D6717yes—
145C4—6732no—
146C4C25×D67148yes—
147C2—28no—
148C2—13432no—
149C2C22×#14524091114yesdirect product factor
150C2C26×D6711yescharacteristic
subgroupG/H
centerC24C23×D67
commutator, cosocleC2×C67C27
FrattiniC2C26×D67
FittingC24×Q8×C67C2
socleC24×C67C24
hypercenterC24×Q8D67
hypocenterC67C23×2−1+4
LCSG ▷ C2×C67 ▷ C67
UCSC1 ◁ C24 ◁ C24×Q8

Complements

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

NH ≅ G/Ntype
1. C22×#14524091C2direct (N × H)
2. C23×#11761629C2semidirect
3. C23×#11761628C2semidirect
4. C23×Q8×D67C2semidirect
5. C24×Q8×C67C2semidirect
6. C2×#14524091C22direct (N × H)
7. C22×#11761629C22semidirect
8. C22×#11761628C22semidirect
10. C22×Q8×D67C22semidirect
15. C23×Q8×C67C22semidirect
18. #14524091C23direct (N × H)
19. C2×#11761629C23semidirect
20. C2×#11761628C23semidirect
22. C2×Q8×D67C23semidirect
29. C22×Q8×C67C23semidirect
34. #11761629C24semidirect
35. #11761628C24semidirect
37. Q8×D67C24semidirect
46. C2×Q8×C67C24semidirect
50. C23×C672−1+4semidirect
64. C22×C67C2×2−1+4semidirect
73. C2×C67C22×2−1+4semidirect
80. C24×Q8D67semidirect
81. C67C23×2−1+4semidirect
88. C23×Q8C2×D67semidirect
101. C22×Q8C22×D67semidirect
118. C2×Q8C23×D67semidirect
128. C23#14524091direct (N × H)
142. C22C2×#14524091direct (N × H)
149. C2C22×#14524091direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145240912
C23×#117616292
C23×#117616282
C23×Q8×D672
C24×Q8×C672
C23×2−1+467