Finite Group Databasebeta

← D4×D1724 #42119695

Subgroups — D4×D1724 #42119695

Order 27584.

428 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 123 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×D1724C212yesmaximal, complemented
2C2×D4×D431C212yesmaximal, complemented
3#28352186C211yesmaximal, characteristic, complemented
4#28352164C212yesmaximal, complemented
5#28352163C212yesmaximal, complemented
6#28352159C214yesmaximal, complemented
7C4×D1724C211yesmaximal, characteristic, complemented
8C4×D4×C431C211yesmaximal, characteristic, complemented
9C23×D431C2214yescomplemented
10C2×D1724—28no—
11C2×D1724C2211yescharacteristic
12C2×D1724C2214yescomplemented
13C2×D1724C2212yescomplemented
14C2×D1724C2212yescomplemented
15C2×D1724C2212yescomplemented
16C2×C22⋊D431C2214yescomplemented
17D4×D431—28no—
18C2×C4×D431C2212yescomplemented
19C4⋊Dic431C2211yescharacteristic, complemented
20(C2×D431)⋊C4C2212yescomplemented
21(C2×D431)⋊C4C2214yescomplemented
22C2×D4×C431C2211yescharacteristic
23C22×C4×C431C2212yescomplemented
24C4⋊C4×C431C2211yescharacteristic, complemented
25C22⋊C4×C431C2212yescomplemented
26C42×C431C2211yescharacteristic, complemented
27C22×D431—28no—
28C22×D431—28no—
29C22×D431—44no—
30C22×D431C2314yes—
31C22×D431C2312yes—
32D1724—28no—
33D1724—24no—
34D1724—24no—
35D1724—42no—
36D1724D414yesdirect product factor
37C22⋊D431—28no—
38C2×Dic431C2312yes—
39C4×D431—24no—
40C23×C431C2312yes—
41D4×C431D414yescomplemented
42C2×C4×C431—24no—
43C2×C4×C431C2311yescharacteristic
44C2×C4×C431C2312yes—
45C2×C4×C431C2311yescharacteristic
46C2×C4×C431C2311yescharacteristic
47C22×C431—22no—
48C22×C431—22no—
49C22×C431C2×D414yescomplemented
50C22×C431C2411yescharacteristic
51C2×D431—22no—
52C2×D431—22no—
53C2×D431—24no—
54C2×D431—24no—
55C2×D431—24no—
56C2×D431—416no—
57C2×D431C2×D414yescomplemented
58C4×C431—22no—
59C4×C431—21no—
60C4×C431C2×D412yescomplemented
61C4×C431C2×D412yescomplemented
62Dic431—22no—
63C2×C431—24no—
64C2×C431C22×D411yescharacteristic
65C2×C431C22×D411yescharacteristic
66C2×C431H(5,𝔽2)11yescharacteristic
67D431—24no—
68D431—44no—
69C431D4211yescharacteristic, complemented, Sylow
70D42—4311nomaximal, Sylow
71C22×D4—4312no—
72C22×D4—4312no—
73#127—4311no—
74#121—4312no—
75#121—4312no—
76#120—4314no—
77C4×D4—4311no—
78C4×D4D43111yescharacteristic, complemented
79C24—4314no—
80C2×D4—8628no—
81C2×D4—8628no—
82C2×D4—4314no—
83C2×D4—4314no—
84C2×D4—4312no—
85C2×D4—4312no—
86C2×D4—4311no—
87C2×D4—4312no—
88C2×D4C2×D43111yescharacteristic
89C22×C4—4312no—
90C22×C4C2×D43112yescomplemented
91C4⋊C4—4311no—
92C4⋊C4C2×D43111yescharacteristic, complemented
93C22⋊C4—4312no—
94C22⋊C4—4314no—
95C22⋊C4C2×D43112yescomplemented
96C42C2×D43111yescharacteristic, complemented
97C23—8628no—
98C23—8628no—
99C23—4314no—
100C23—4312no—
101C23—17244no—
102C23C22×D43112yes—
103D4—8628no—
104D4—8628no—
105D4—8624no—
106D4—8624no—
107D4—4314no—
108D4—17242no—
109D4D172414yesdirect product factor
110C2×C4—24no—
111C2×C4—8624no—
112C2×C4—4312no—
113C2×C4C22×D43111yescharacteristic
114C2×C4C22×D43112yes—
115C2×C4C22×D43111yescharacteristic
116C2×C4C22×D43111yescharacteristic
117C22—22no—
118C22—22no—
119C22—8622no—
120C22—8622no—
121C22—8624no—
122C22—8624no—
123C22—8624no—
124C22—4314no—
125C22—172416no—
126C22C2×D172414yescomplemented
127C22C23×D43111yescharacteristic
128C4—22no—
129C4—21no—
130C4—8622no—
131C4D4×D43112yescomplemented
132C4C2×D172412yescomplemented
133C2—24no—
134C2—8624no—
135C2—17244no—
136C2C2×D4×D43111yescharacteristic
137C2C22×D172411yescharacteristic
138C2#2835223511yescharacteristic
subgroupG/H
center, FrattiniC22C23×D431
commutator, socle, cosocleC22×C431C24
FittingC4×D4×C431C2
hypercenterC4×D4D431
hypocenterC431D42
LCSG ▷ C22×C431 ▷ C431
UCSC1 ◁ C22 ◁ C4×D4

Complements

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

NH ≅ G/Ntype
1. C22×D1724C2semidirect
2. C2×D4×D431C2semidirect
3. #28352186C2semidirect
4. #28352164C2semidirect
5. #28352163C2semidirect
6. #28352159C2semidirect
7. C4×D1724C2semidirect
8. C4×D4×C431C2semidirect
9. C23×D431C22semidirect
12. C2×D1724C22semidirect
13. C2×D1724C22semidirect
14. C2×D1724C22semidirect
15. C2×D1724C22semidirect
16. C2×C22⋊D431C22semidirect
18. C2×C4×D431C22semidirect
19. C4⋊Dic431C22semidirect
20. (C2×D431)⋊C4C22semidirect
21. (C2×D431)⋊C4C22semidirect
23. C22×C4×C431C22semidirect
24. C4⋊C4×C431C22semidirect
25. C22⋊C4×C431C22semidirect
26. C42×C431C22semidirect
36. D1724D4direct (N × H)
41. D4×C431D4semidirect
49. C22×C431C2×D4semidirect
57. C2×D431C2×D4semidirect
60. C4×C431C2×D4semidirect
61. C4×C431C2×D4semidirect
69. C431D42semidirect
78. C4×D4D431semidirect
90. C22×C4C2×D431semidirect
92. C4⋊C4C2×D431semidirect
95. C22⋊C4C2×D431semidirect
96. C42C2×D431semidirect
109. D4D1724direct (N × H)
126. C22C2×D1724semidirect
131. C4D4×D431semidirect
132. C4C2×D1724semidirect

Maximal subgroups

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

subgroupindex
C22×D17242
C2×D4×D4312
#283521862
#283521642
#283521632
#283521592
C4×D17242
C4×D4×C4312
D42431