Finite Group Databasebeta

← C23×#14556176 #32911342

Subgroups — C23×#14556176 #32911342

Order 18176.

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×#14556176C21112yesmaximal, direct product factor
2C23×#11765866C214yesmaximal, complemented
3C23×#11765865C216yesmaximal, complemented
4C23×Q8×D71C214yesmaximal, complemented
5C24×Q8×C71C211yesmaximal, characteristic, complemented
6C2×#14556176C2211792yesdirect product factor
7C22×#11765866C221224yescomplemented
8C22×#11765865C221336yescomplemented
9C23×Dic142C2216yes—
10C22×Q8×D71C221224yescomplemented
11C23×D284C2216yes—
12C23×C22⋊D71C2214yes—
13C23×C4×D71C22112yes—
14C23×Q8×C71C2214yes—
15C23×Q8×C71C22156yescomplemented
16C24×C4×C71C2213yes—
17C24×D71C2314yes—
18#14556176C2314096yesdirect product factor
19C2×#11765866C2311792yescomplemented
20C2×#11765865C2312688yescomplemented
21C22×Dic142C231168yes—
22C2×Q8×D71C2311792yescomplemented
23C22×D284C231168yes—
24C22×C22⋊D71C231112yes—
25C23×Dic71C2314yes—
26C22×C4×D71C231336yes—
27C25×C71C2311yescharacteristic
28C22×Q8×C71C231112yes—
29C22×Q8×C71C231448yescomplemented
30C23×C4×C71C2316yes—
31C23×C4×C71C23184yes—
32C23×D71—232no—
33C23×D71C24156yes—
34#11765866C2412048yescomplemented
35#11765865C2413072yescomplemented
36C2×Dic142C241672yes—
37Q8×D71C2412048yescomplemented
38C2×D284C241672yes—
39C2×C22⋊D71C241448yes—
40C22×Dic71C24156yes—
41C2×C4×D71C2411344yes—
42C24×C71—28no—
43C24×C71C24114yes—
44C24×C71C2411yescharacteristic
45C2×Q8×C71C241448yes—
46C2×Q8×C71C241512yescomplemented
47C22×C4×C71C24184yes—
48C22×C4×C71C241336yes—
49C23×C71—256no—
50C23×C712−1+418yescomplemented
51C23×C71C25128yes—
52C23×C71C2517yes—
53C22×D71—2224no—
54C22×D71C251112yes—
55Q8×C71C251256yes—
56C2×C4×C71C251168yes—
57C2×C4×C71C251192yes—
58C22⋊D71C251256yes—
59C2×Dic71C251112yes—
60D284C251384yes—
61C4×D71C251768yes—
62Dic142C251384yes—
63C22×C71—256no—
64C22×C71C2×2−1+4128yescomplemented
65C22×C71C2618yes—
66C22×C71C2617yes—
67C2×D71—2224no—
68C2×D71C26132yes—
69C4×C71C26148yes—
70Dic71C26132yes—
71C23×2−1+4—711nomaximal, Sylow
72C2×C71—28no—
73C2×C71C22×2−1+4114yescomplemented
74C2×C71C2711yescharacteristic
75D71—232no—
76C22×2−1+4—71112no—
77C23×#41—714no—
78C23×#41—716no—
79C24×Q8—714no—
80C24×Q8D7111yescharacteristic, complemented
81C71C23×2−1+411yescharacteristic, complemented, Sylow
82C2×2−1+4—711792no—
83C22×#41—71224no—
84C22×#41—71336no—
85C23×Q8—71224no—
86C23×Q8—716no—
87C23×Q8C2×D7114yes—
88C23×Q8C2×D71156yescomplemented
89C23×D4—714no—
90C23×D4—716no—
91C24×C4—7112no—
92C24×C4C2×D7113yes—
93C25—714no—
94C25C22×D7111yescharacteristic
952−1+4—714096no—
96C2×#41—711792no—
97C2×#41—712688no—
98C22×Q8—711792no—
99C22×Q8—71168no—
100C22×Q8C22×D711112yes—
101C22×Q8C22×D711448yescomplemented
102C22×D4—71112no—
103C22×D4—71168no—
104C23×C4—714no—
105C23×C4—71336no—
106C23×C4C22×D71184yes—
107C23×C4C22×D7116yes—
108C24—28no—
109C24—7156no—
110C24—14232no—
111C24C23×D71114yes—
112C24C23×D7111yescharacteristic
113#41—712048no—
114#41—713072no—
115C2×Q8—712048no—
116C2×Q8—71672no—
117C2×Q8C23×D711448yes—
118C2×Q8C23×D711512yescomplemented
119C2×D4—71448no—
120C2×D4—71672no—
121C22×C4—7156no—
122C22×C4—711344no—
123C22×C4C23×D71184yes—
124C22×C4C23×D711336yes—
125C23—256no—
126C23—71112no—
127C23—142224no—
128C23#1455617618yesdirect product factor
129C23C24×D71128yes—
130C23C24×D7117yes—
131Q8—71384no—
132Q8C24×D711256yes—
133D4—71256no—
134D4—71384no—
135C2×C4—71112no—
136C2×C4—71768no—
137C2×C4C24×D711168yes—
138C2×C4C24×D711192yes—
139C22—256no—
140C22—7132no—
141C22—142224no—
142C22C2×#14556176128yesdirect product factor
143C22C25×D7118yes—
144C22C25×D7117yes—
145C4—7132no—
146C4C25×D71148yes—
147C2—28no—
148C2—14232no—
149C2C22×#14556176114yesdirect product factor
150C2C26×D7111yescharacteristic
subgroupG/H
centerC24C23×D71
commutator, cosocleC2×C71C27
FrattiniC2C26×D71
FittingC24×Q8×C71C2
socleC24×C71C24
hypercenterC24×Q8D71
hypocenterC71C23×2−1+4
LCSG ▷ C2×C71 ▷ C71
UCSC1 ◁ C24 ◁ C24×Q8

Complements

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

NH ≅ G/Ntype
1. C22×#14556176C2direct (N × H)
2. C23×#11765866C2semidirect
3. C23×#11765865C2semidirect
4. C23×Q8×D71C2semidirect
5. C24×Q8×C71C2semidirect
6. C2×#14556176C22direct (N × H)
7. C22×#11765866C22semidirect
8. C22×#11765865C22semidirect
10. C22×Q8×D71C22semidirect
15. C23×Q8×C71C22semidirect
18. #14556176C23direct (N × H)
19. C2×#11765866C23semidirect
20. C2×#11765865C23semidirect
22. C2×Q8×D71C23semidirect
29. C22×Q8×C71C23semidirect
34. #11765866C24semidirect
35. #11765865C24semidirect
37. Q8×D71C24semidirect
46. C2×Q8×C71C24semidirect
50. C23×C712−1+4semidirect
64. C22×C71C2×2−1+4semidirect
73. C2×C71C22×2−1+4semidirect
80. C24×Q8D71semidirect
81. C71C23×2−1+4semidirect
88. C23×Q8C2×D71semidirect
101. C22×Q8C22×D71semidirect
118. C2×Q8C23×D71semidirect
128. C23#14556176direct (N × H)
142. C22C2×#14556176direct (N × H)
149. C2C22×#14556176direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#145561762
C23×#117658662
C23×#117658652
C23×Q8×D712
C24×Q8×C712
C23×2−1+471