Finite Group Databasebeta

← C23×D4×D173 #36368354

Subgroups — C23×D4×D173 #36368354

Order 22144.

10552 conjugacy classes of subgroups; only the 3557 normal subgroups are shown.

normal subgroups only

This group has 10552 conjugacy classes of subgroups — too many to draw. Only its 3557 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
1C25×D173C212yesmaximal, complemented
2C23×D692C211yesmaximal, characteristic, complemented
3C23×C22⋊D173C212yesmaximal, complemented
4C22×D4×D173C2156yesmaximal, direct product factor
5C23×C4×D173C211yesmaximal, characteristic, complemented
6C23×D4×C173C211yesmaximal, characteristic, complemented
7C24×D173—232no—
8C24×D173C2211yescharacteristic
9C24×D173C22156yescomplemented
10C24×D173C2212yescomplemented
11C22×D692C22128yescomplemented
12C22×C22⋊D173C22156yescomplemented
13C2×D4×D173C221448yesdirect product factor
14C23×Dic173C2211yescharacteristic, complemented
15C22×C4×D173C22128yescomplemented
16C25×C173C2212yescomplemented
17C22×D4×C173C22128yescomplemented
18C23×C4×C173C2211yescharacteristic, complemented
19C23×D173—2448no—
20C23×D173—216no—
21C23×D173D4116yesdirect product factor
22C23×D173C23114yes—
23C23×D173C23128yescomplemented
24C23×D173C231224yescomplemented
25C2×D692C231112yescomplemented
26C2×C22⋊D173C231224yescomplemented
27D4×D173C231512yesdirect product factor
28C22×Dic173C23114yescomplemented
29C2×C4×D173C231112yescomplemented
30C24×C173—216no—
31C24×C173C2311yescharacteristic
32C24×C173C23128yescomplemented
33C2×D4×C173C231112yescomplemented
34C22×C4×C173C23114yescomplemented
35C22×D173—2112no—
36C22×D173—2896no—
37C22×D173C2×D41112yesdirect product factor
38C22×D173C24128yes—
39C22×D173C241128yescomplemented
40C22×D173C24156yescomplemented
41D692C24164yescomplemented
42C22⋊D173C241128yescomplemented
43C2×Dic173C24128yescomplemented
44C4×D173C24164yescomplemented
45C23×C173—2112no—
46C23×C173C2×D418yescomplemented
47C23×C173C2417yes—
48C23×C173C24156yescomplemented
49D4×C173C24164yescomplemented
50C2×C4×C173C24128yescomplemented
51C22×C173—2112no—
52C22×C173C22×D4128yescomplemented
53C22×C173C2517yes—
54C22×C173C25116yescomplemented
55C2×D173—2112no—
56C2×D173—2256no—
57C2×D173C22×D41112yesdirect product factor
58C2×D173C2518yes—
59C2×D173C25116yescomplemented
60C4×C173C2518yescomplemented
61Dic173C2518yescomplemented
62C2×C173—216no—
63C2×C173C23×D4114yescomplemented
64C2×C173C2611yescharacteristic
65D173—216no—
66D173C23×D4116yesdirect product factor
67C173C24×D411yescharacteristic, complemented, Sylow
68C24×D4—1731nomaximal, Sylow
69C26—1732no—
70C23×D4—1732no—
71C23×D4—17356no—
72C23×D4—1731no—
73C23×D4D17311yescharacteristic, direct product factor
74C24×C4—1731no—
75C25—34632no—
76C25—1732no—
77C25—17356no—
78C25—1731no—
79C25C2×D17312yescomplemented
80C22×D4—17356no—
81C22×D4—173448no—
82C22×D4—17328no—
83C22×D4C2×D173128yesdirect product factor
84C23×C4—1731no—
85C23×C4—17328no—
86C23×C4C2×D17311yescharacteristic, complemented
87C24—216no—
88C24—346448no—
89C24—34616no—
90C24—17328no—
91C24—173224no—
92C24—17316no—
93C24—17314no—
94C24C22×D17311yescharacteristic
95C24C22×D173128yescomplemented
96C2×D4—173224no—
97C2×D4—173512no—
98C2×D4—173112no—
99C2×D4C22×D1731112yesdirect product factor
100C22×C4—17314no—
101C22×C4—173112no—
102C22×C4C22×D173114yescomplemented
103C23—2112no—
104C23—346896no—
105C23—346112no—
106C23—17356no—
107C23—173128no—
108C23—173112no—
109C23—17328no—
110C23D4×D17318yesdirect product factor
111C23C23×D17317yes—
112C23C23×D173156yescomplemented
113D4—173128no—
114D4—17364no—
115D4C23×D173164yesdirect product factor
116C2×C4—17328no—
117C2×C4—17364no—
118C2×C4C23×D173128yescomplemented
119C22—2112no—
120C22—346256no—
121C22—346112no—
122C22—17316no—
123C22—173112no—
124C22—1738no—
125C22C2×D4×D173128yesdirect product factor
126C22C24×D17317yes—
127C22C24×D173116yescomplemented
128C4—1738no—
129C4C24×D17318yescomplemented
130C2—216no—
131C2—34616no—
132C2—17316no—
133C2C22×D4×D173114yesdirect product factor
134C2C25×D17311yescharacteristic
subgroupG/H
centerC24C22×D173
commutator, cosocleC2×C173C26
FrattiniC2C25×D173
FittingC23×D4×C173C2
socleC24×C173C23
hypercenterC23×D4D173
hypocenterC173C24×D4
LCSG ▷ C2×C173 ▷ C173
UCSC1 ◁ C24 ◁ C23×D4

Complements

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

NH ≅ G/Ntype
1. C25×D173C2semidirect
2. C23×D692C2semidirect
3. C23×C22⋊D173C2semidirect
4. C22×D4×D173C2direct (N × H)
5. C23×C4×D173C2semidirect
6. C23×D4×C173C2semidirect
9. C24×D173C22semidirect
10. C24×D173C22semidirect
11. C22×D692C22semidirect
12. C22×C22⋊D173C22semidirect
13. C2×D4×D173C22direct (N × H)
14. C23×Dic173C22semidirect
15. C22×C4×D173C22semidirect
16. C25×C173C22semidirect
17. C22×D4×C173C22semidirect
18. C23×C4×C173C22semidirect
21. C23×D173D4direct (N × H)
23. C23×D173C23semidirect
24. C23×D173C23semidirect
25. C2×D692C23semidirect
26. C2×C22⋊D173C23semidirect
27. D4×D173C23direct (N × H)
28. C22×Dic173C23semidirect
29. C2×C4×D173C23semidirect
32. C24×C173C23semidirect
33. C2×D4×C173C23semidirect
34. C22×C4×C173C23semidirect
37. C22×D173C2×D4direct (N × H)
39. C22×D173C24semidirect
40. C22×D173C24semidirect
41. D692C24semidirect
42. C22⋊D173C24semidirect
43. C2×Dic173C24semidirect
44. C4×D173C24semidirect
46. C23×C173C2×D4semidirect
48. C23×C173C24semidirect
49. D4×C173C24semidirect
50. C2×C4×C173C24semidirect
52. C22×C173C22×D4semidirect
54. C22×C173C25semidirect
57. C2×D173C22×D4direct (N × H)
59. C2×D173C25semidirect
60. C4×C173C25semidirect
61. Dic173C25semidirect
63. C2×C173C23×D4semidirect
66. D173C23×D4direct (N × H)
67. C173C24×D4semidirect
73. C23×D4D173direct (N × H)
79. C25C2×D173semidirect
83. C22×D4C2×D173direct (N × H)
86. C23×C4C2×D173semidirect
95. C24C22×D173semidirect
99. C2×D4C22×D173direct (N × H)
102. C22×C4C22×D173semidirect
110. C23D4×D173direct (N × H)
112. C23C23×D173semidirect
115. D4C23×D173direct (N × H)
118. C2×C4C23×D173semidirect
125. C22C2×D4×D173direct (N × H)
127. C22C24×D173semidirect
129. C4C24×D173semidirect
133. C2C22×D4×D173direct (N × H)

Maximal subgroups

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

subgroupindex
C25×D1732
C23×D6922
C23×C22⋊D1732
C22×D4×D1732
C23×C4×D1732
C23×D4×C1732
C24×D4173