Finite Group Databasebeta

← #120×D223 #28384064

Subgroups — #120×D223 #28384064

Order 14272.

662 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 662 conjugacy classes of subgroups — too many to draw. Only its 131 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×D223C211yesmaximal, characteristic, complemented
2C2×D4×D223C213yesmaximal, complemented
3#20297274C213yesmaximal, complemented
4#20297273C211yesmaximal, characteristic, complemented
5#20297272C213yesmaximal, complemented
6C22⋊C4×D223C213yesmaximal, complemented
7#120×C223C211yesmaximal, characteristic, complemented
8C23×D223—212no—
9C23×D223C2213yes—
10C23×D223C2213yescomplemented
11C23×D223C2211yescharacteristic, complemented
12C2×D892C2213yescomplemented
13C2×C22⋊D223C2213yescomplemented
14C2×C22⋊D223C2213yescomplemented
15D4×D223—212no—
16C2×C4×D223C2213yescomplemented
17C22⋊Dic223C2213yescomplemented
18(C2×D223)⋊C4C2216yescomplemented
19C24×C223C2211yescharacteristic, complemented
20C2×D4×C223C2213yescomplemented
21C22⋊C4×C223C2213yescomplemented
22C22×D223—232no—
23C22×D223—26no—
24C22×D223—212no—
25C22×D223—212no—
26C22×D223—26no—
27C22×D223D4112yescomplemented
28C22×D223C2313yes—
29C22×D223C2311yescharacteristic
30C22×D223C2311yescharacteristic, complemented
31D892—26no—
32C22⋊D223—26no—
33C22⋊D223—26no—
34C2×Dic223C2313yescomplemented
35C4×D223—26no—
36C23×C223—26no—
37C23×C223C2313yes—
38C23×C223C2311yescharacteristic, complemented
39D4×C223—26no—
40C2×C4×C223C2313yescomplemented
41C22×C223—28no—
42C22×C223—26no—
43C22×C223—23no—
44C22×C223C2×D416yescomplemented
45C22×C223C2411yescharacteristic
46C2×D223—224no—
47C2×D223—26no—
48C2×D223—224no—
49C2×D223—23no—
50C2×D223—44no—
51C2×D223C2×D416yes—
52C2×D223C2×D416yescomplemented
53C4×C223—23no—
54Dic223—23no—
55C2×C223—26no—
56C2×C223—41no—
57C2×C223C22×D413yes—
58D223—26no—
59D223—41no—
60D223#12014yesdirect product factor
61C223C2×#12011yescharacteristic, complemented, Sylow
62C2×#120—2231nomaximal, Sylow
63C25—2231no—
64C22×D4—2233no—
65#120—2231no—
66#120—2233no—
67#120—2233no—
68#120D22311yescharacteristic, direct product factor
69C2×C22⋊C4—2233no—
70C24—44612no—
71C24—2233no—
72C24—2231no—
73C24—2233no—
74C24C2×D22311yescharacteristic, complemented
75C2×D4—44612no—
76C2×D4—2233no—
77C2×D4—2233no—
78C2×D4—2233no—
79C2×D4C2×D22313yescomplemented
80C22×C4—2233no—
81C22⋊C4—2233no—
82C22⋊C4—2236no—
83C22⋊C4C2×D22313yescomplemented
84C23—26no—
85C23—4466no—
86C23—44632no—
87C23—44612no—
88C23—44612no—
89C23—4466no—
90C23—2231no—
91C23—22312no—
92C23—2233no—
93C23—2231no—
94C23C22×D22313yes—
95C23C22×D22311yescharacteristic, complemented
96D4—26no—
97D4—4466no—
98D4—4466no—
99D4—4466no—
100C2×C4—4466no—
101C2×C4—2233no—
102C2×C4C22×D22313yescomplemented
103C22—28no—
104C22—26no—
105C22—23no—
106C22—8924no—
107C22—44624no—
108C22—44624no—
109C22—4463no—
110C22—4466no—
111C22—2236no—
112C22—2236no—
113C22D4×D22316yescomplemented
114C22C23×D22311yescharacteristic
115C4—23no—
116C4—4463no—
117C2—26no—
118C2—41no—
119C2—8921no—
120C2—4466no—
121C2—2234no—
122C2C2×D4×D22313yes—
subgroupG/H
center, FrattiniC22C23×D223
commutator, socle, cosocleC22×C223C24
Fitting#120×C223C2
hypercenter#120D223
hypocenterC223C2×#120
LCSG ▷ C22×C223 ▷ C223
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C24×D223C2semidirect
2. C2×D4×D223C2semidirect
3. #20297274C2semidirect
4. #20297273C2semidirect
5. #20297272C2semidirect
6. C22⋊C4×D223C2semidirect
7. #120×C223C2semidirect
10. C23×D223C22semidirect
11. C23×D223C22semidirect
12. C2×D892C22semidirect
13. C2×C22⋊D223C22semidirect
14. C2×C22⋊D223C22semidirect
16. C2×C4×D223C22semidirect
17. C22⋊Dic223C22semidirect
18. (C2×D223)⋊C4C22semidirect
19. C24×C223C22semidirect
20. C2×D4×C223C22semidirect
21. C22⋊C4×C223C22semidirect
27. C22×D223D4semidirect
30. C22×D223C23semidirect
34. C2×Dic223C23semidirect
38. C23×C223C23semidirect
40. C2×C4×C223C23semidirect
44. C22×C223C2×D4semidirect
52. C2×D223C2×D4semidirect
60. D223#120direct (N × H)
61. C223C2×#120semidirect
68. #120D223direct (N × H)
74. C24C2×D223semidirect
79. C2×D4C2×D223semidirect
83. C22⋊C4C2×D223semidirect
95. C23C22×D223semidirect
102. C2×C4C22×D223semidirect
113. C22D4×D223semidirect

Maximal subgroups

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

subgroupindex
C24×D2232
C2×D4×D2232
#202972742
#202972732
#202972722
C22⋊C4×D2232
#120×C2232
C2×#120223