Finite Group Databasebeta

← C22×#22576129 #46849580

Subgroups — C22×#22576129 #46849580

Order 34688.

4544 conjugacy classes of subgroups; only the 805 normal subgroups are shown.

normal subgroups only

This group has 4544 conjugacy classes of subgroups — too many to draw. Only its 805 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
1C23×C22⋊D271C213yesmaximal, complemented
2C2×#22576129C2124yesmaximal, direct product factor
3C22×C22⋊Dic271C213yesmaximal, complemented
4C26×C271C211yesmaximal, characteristic, complemented
5C24×D271C2211yescharacteristic, complemented
6C22×C22⋊D271—224no—
7C22×C22⋊D271C22136yescomplemented
8C23×Dic271C2213yescomplemented
9#22576129C22164yesdirect product factor
10C2×C22⋊Dic271C22136yescomplemented
11C25×C271—224no—
12C25×C271C2213yes—
13C25×C271C22112yescomplemented
14C23×D271—212no—
15C23×D271C2316yescomplemented
16C2×C22⋊D271—2144no—
17C2×C22⋊D271C23148yescomplemented
18C22×Dic271—212no—
19C22×Dic271C23118yescomplemented
20C22⋊Dic271C23148yescomplemented
21C24×C271—2128no—
22C24×C271—224no—
23C24×C271—2144no—
24C24×C271D4124yescomplemented
25C24×C271C23118yes—
26C24×C271C2311yescharacteristic
27C24×C271C23116yescomplemented
28C22×D271—236no—
29C22×D271—416no—
30C22×D271C2414yescomplemented
31C22⋊D271—296no—
32C2×Dic271—236no—
33C2×Dic271C24112yescomplemented
34C23×C271—2384no—
35C23×C271—296no—
36C23×C271—272no—
37C23×C271—296no—
38C23×C271C2×D4112yes—
39C23×C271C2×D4172yescomplemented
40C23×C271C24112yes—
41C23×C271C2413yes—
42C22×C271—2128no—
43C22×C271—2144no—
44C22×C271—224no—
45C22×C271#120116yescomplemented
46C22×C271C22×D4118yes—
47C22×C271C22×D4124yescomplemented
48C22×C271C2511yescharacteristic
49C2×D271—212no—
50C2×D271—424no—
51Dic271—212no—
52C2×C271—224no—
53C2×C271C2×#120112yescomplemented
54C2×C271C23×D413yes—
55D271—44no—
56C271C22×#12011yescharacteristic, complemented, Sylow
57C22×#120—2711nomaximal, Sylow
58C26D27111yescharacteristic, complemented
59C23×D4—2713no—
60C2×#120—27124no—
61C22×C22⋊C4—2713no—
62C25—224no—
63C25—2711no—
64C25C2×D27113yes—
65C25C2×D271112yescomplemented
66C22×D4—27136no—
67C22×D4—54224no—
68C23×C4—2713no—
69#120—27164no—
70C2×C22⋊C4—27136no—
71C24—2128no—
72C24—224no—
73C24—2144no—
74C24—2716no—
75C24—54212no—
76C24C22⋊D271124yescomplemented
77C24C22×D271118yes—
78C24C22×D27111yescharacteristic
79C24C22×D271116yescomplemented
80C2×D4—27148no—
81C2×D4—542144no—
82C22×C4—27118no—
83C22×C4—54212no—
84C22⋊C4—27148no—
85C23—2384no—
86C23—296no—
87C23—272no—
88C23—296no—
89C23—2714no—
90C23—54236no—
91C23—108416no—
92C23C2×C22⋊D271112yes—
93C23C2×C22⋊D271172yescomplemented
94C23C23×D271112yes—
95C23C23×D27113yes—
96D4—54296no—
97C2×C4—27112no—
98C2×C4—54236no—
99C22—2128no—
100C22—2144no—
101C22—224no—
102C22—54212no—
103C22—108424no—
104C22#22576129116yesdirect product factor
105C22C22×C22⋊D271118yes—
106C22C22×C22⋊D271124yescomplemented
107C22C24×D27111yescharacteristic
108C4—54212no—
109C2—224no—
110C2—10844no—
111C2C2×#22576129112yesdirect product factor
112C2C23×C22⋊D27113yes—
subgroupG/H
centerC24C22×D271
commutator, cosocleC22×C271C25
FrattiniC22C24×D271
FittingC26×C271C2
socleC24×C271C23
hypercenterC26D271
hypocenterC271C22×#120
LCSG ▷ C22×C271 ▷ C271
UCSC1 ◁ C24 ◁ C26

Complements

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

NH ≅ G/Ntype
1. C23×C22⋊D271C2semidirect
2. C2×#22576129C2direct (N × H)
3. C22×C22⋊Dic271C2semidirect
4. C26×C271C2semidirect
5. C24×D271C22semidirect
7. C22×C22⋊D271C22semidirect
8. C23×Dic271C22semidirect
9. #22576129C22direct (N × H)
10. C2×C22⋊Dic271C22semidirect
13. C25×C271C22semidirect
15. C23×D271C23semidirect
17. C2×C22⋊D271C23semidirect
19. C22×Dic271C23semidirect
20. C22⋊Dic271C23semidirect
24. C24×C271D4semidirect
27. C24×C271C23semidirect
30. C22×D271C24semidirect
33. C2×Dic271C24semidirect
39. C23×C271C2×D4semidirect
45. C22×C271#120semidirect
47. C22×C271C22×D4semidirect
53. C2×C271C2×#120semidirect
56. C271C22×#120semidirect
58. C26D271semidirect
65. C25C2×D271semidirect
76. C24C22⋊D271semidirect
79. C24C22×D271semidirect
93. C23C2×C22⋊D271semidirect
104. C22#22576129direct (N × H)
106. C22C22×C22⋊D271semidirect
111. C2C2×#22576129direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C22⋊D2712
C2×#225761292
C22×C22⋊Dic2712
C26×C2712
C22×#120271