Finite Group Databasebeta

← C23×#14556097 #32904083

Subgroups — C23×#14556097 #32904083

Order 18176.

42862 conjugacy classes of subgroups; only the 5451 normal subgroups are shown.

normal subgroups only

This group has 42862 conjugacy classes of subgroups — too many to draw. Only its 5451 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×C22⋊D71C213yesmaximal, complemented
2C22×#14556097C2156yesmaximal, direct product factor
3C23×C22⋊Dic71C213yesmaximal, complemented
4C27×C71C211yesmaximal, characteristic, complemented
5C25×D71C2211yescharacteristic, complemented
6C23×C22⋊D71—248no—
7C23×C22⋊D71C22184yescomplemented
8C24×Dic71C2213yescomplemented
9C2×#14556097C221448yesdirect product factor
10C22×C22⋊Dic71C22184yescomplemented
11C26×C71—248no—
12C26×C71C2213yes—
13C26×C71C22128yescomplemented
14C24×D71—224no—
15C24×D71C23114yescomplemented
16C22×C22⋊D71—2672no—
17C22×C22⋊D71C231336yescomplemented
18C23×Dic71—224no—
19C23×Dic71C23142yescomplemented
20#14556097C231512yesdirect product factor
21C2×C22⋊Dic71C231336yescomplemented
22C25×C71—2512no—
23C25×C71—248no—
24C25×C71—2672no—
25C25×C71D4148yescomplemented
26C25×C71C23142yes—
27C25×C71C2311yescharacteristic
28C25×C71C231112yescomplemented
29C23×D71—2168no—
30C23×D71—464no—
31C23×D71C24128yescomplemented
32C2×C22⋊D71—21344no—
33C2×C22⋊D71C241192yescomplemented
34C22×Dic71—2168no—
35C22×Dic71C24184yescomplemented
36C22⋊Dic71C241192yescomplemented
37C24×C71—23584no—
38C24×C71—2384no—
39C24×C71—2336no—
40C24×C71—21344no—
41C24×C71C2×D4124yes—
42C24×C71C2×D41336yescomplemented
43C24×C71C24184yes—
44C24×C71C2417yes—
45C24×C71C24164yescomplemented
46C23×C71—23584no—
47C23×C71—21344no—
48C23×C71—2336no—
49C23×C71—2384no—
50C23×C71#120164yescomplemented
51C23×C71C22×D4184yes—
52C23×C71C22×D41336yescomplemented
53C23×C71C25124yes—
54C23×C71C2517yes—
55C22×D71—2168no—
56C22×D71—4224no—
57C22×D71C2518yescomplemented
58C22⋊D71—2384no—
59C2×Dic71—2168no—
60C2×Dic71C25124yescomplemented
61C22×C71—2512no—
62C22×C71—248no—
63C22×C71—2672no—
64C22×C71C2×#1201112yescomplemented
65C22×C71C23×D4142yes—
66C22×C71C23×D4148yescomplemented
67C22×C71C2611yescharacteristic
68C2×D71—224no—
69C2×D71—4112no—
70Dic71—224no—
71C23×#120—711nomaximal, Sylow
72C2×C71—248no—
73C2×C71C22×#120128yescomplemented
74C2×C71C24×D413yes—
75D71—48no—
76C27D7111yescharacteristic, complemented
77C24×D4—713no—
78C22×#120—7156no—
79C23×C22⋊C4—713no—
80C71C23×#12011yescharacteristic, complemented, Sylow
81C26—248no—
82C26—711no—
83C26C2×D7113yes—
84C26C2×D71128yescomplemented
85C23×D4—7184no—
86C23×D4—14248no—
87C24×C4—713no—
88C2×#120—71448no—
89C22×C22⋊C4—7184no—
90C25—2512no—
91C25—248no—
92C25—2672no—
93C25—7114no—
94C25—14224no—
95C25C22⋊D71148yescomplemented
96C25C22×D71142yes—
97C25C22×D7111yescharacteristic
98C25C22×D711112yescomplemented
99C22×D4—71336no—
100C22×D4—142672no—
101C23×C4—7142no—
102C23×C4—14224no—
103#120—71512no—
104C2×C22⋊C4—71336no—
105C24—23584no—
106C24—2336no—
107C24—21344no—
108C24—2384no—
109C24—28464no—
110C24—7128no—
111C24—142168no—
112C24C2×C22⋊D71124yes—
113C24C2×C22⋊D711336yescomplemented
114C24C23×D71184yes—
115C24C23×D7117yes—
116C24C23×D71164yescomplemented
117C2×D4—71192no—
118C2×D4—1421344no—
119C22×C4—7184no—
120C22×C4—142168no—
121C22⋊C4—71192no—
122C23—23584no—
123C23—2336no—
124C23—2384no—
125C23—21344no—
126C23—284224no—
127C23—718no—
128C23—142168no—
129C23#14556097164yesdirect product factor
130C23C22×C22⋊D71184yes—
131C23C22×C22⋊D711336yescomplemented
132C23C24×D71124yes—
133C23C24×D7117yes—
134D4—142384no—
135C2×C4—7124no—
136C2×C4—142168no—
137C22—2512no—
138C22—2672no—
139C22—248no—
140C22—284112no—
141C22—14224no—
142C22C2×#145560971112yesdirect product factor
143C22C23×C22⋊D71142yes—
144C22C23×C22⋊D71148yescomplemented
145C22C25×D7111yescharacteristic
146C4—14224no—
147C2—248no—
148C2—2848no—
149C2C22×#14556097128yesdirect product factor
150C2C24×C22⋊D7113yes—
subgroupG/H
centerC25C22×D71
commutator, cosocleC22×C71C26
FrattiniC22C25×D71
FittingC27×C71C2
socleC25×C71C23
hypercenterC27D71
hypocenterC71C23×#120
LCSG ▷ C22×C71 ▷ C71
UCSC1 ◁ C25 ◁ C27

Complements

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

NH ≅ G/Ntype
1. C24×C22⋊D71C2semidirect
2. C22×#14556097C2direct (N × H)
3. C23×C22⋊Dic71C2semidirect
4. C27×C71C2semidirect
5. C25×D71C22semidirect
7. C23×C22⋊D71C22semidirect
8. C24×Dic71C22semidirect
9. C2×#14556097C22direct (N × H)
10. C22×C22⋊Dic71C22semidirect
13. C26×C71C22semidirect
15. C24×D71C23semidirect
17. C22×C22⋊D71C23semidirect
19. C23×Dic71C23semidirect
20. #14556097C23direct (N × H)
21. C2×C22⋊Dic71C23semidirect
25. C25×C71D4semidirect
28. C25×C71C23semidirect
31. C23×D71C24semidirect
33. C2×C22⋊D71C24semidirect
35. C22×Dic71C24semidirect
36. C22⋊Dic71C24semidirect
42. C24×C71C2×D4semidirect
45. C24×C71C24semidirect
50. C23×C71#120semidirect
52. C23×C71C22×D4semidirect
57. C22×D71C25semidirect
60. C2×Dic71C25semidirect
64. C22×C71C2×#120semidirect
66. C22×C71C23×D4semidirect
73. C2×C71C22×#120semidirect
76. C27D71semidirect
80. C71C23×#120semidirect
84. C26C2×D71semidirect
95. C25C22⋊D71semidirect
98. C25C22×D71semidirect
113. C24C2×C22⋊D71semidirect
116. C24C23×D71semidirect
129. C23#14556097direct (N × H)
131. C23C22×C22⋊D71semidirect
142. C22C2×#14556097direct (N × H)
144. C22C23×C22⋊D71semidirect
149. C2C22×#14556097direct (N × H)

Maximal subgroups

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

subgroupindex
C24×C22⋊D712
C22×#145560972
C23×C22⋊Dic712
C27×C712
C23×#12071