Finite Group Databasebeta

← #120×D349 #36374628

Subgroups — #120×D349 #36374628

Order 22336.

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×D349C211yesmaximal, characteristic, complemented
2C2×D4×D349C213yesmaximal, complemented
3#26013609C213yesmaximal, complemented
4#26013608C211yesmaximal, characteristic, complemented
5#26013607C213yesmaximal, complemented
6C22⋊C4×D349C213yesmaximal, complemented
7#120×C349C211yesmaximal, characteristic, complemented
8C23×D349—212no—
9C23×D349C2213yes—
10C23×D349C2213yescomplemented
11C23×D349C2211yescharacteristic, complemented
12C2×D1396C2213yescomplemented
13C2×C22⋊D349C2213yescomplemented
14C2×C22⋊D349C2213yescomplemented
15D4×D349—212no—
16C2×C4×D349C2213yescomplemented
17C22⋊Dic349C2213yescomplemented
18#19150210C2216yescomplemented
19C24×C349C2211yescharacteristic, complemented
20C2×D4×C349C2213yescomplemented
21C22⋊C4×C349C2213yescomplemented
22C22×D349—232no—
23C22×D349—26no—
24C22×D349—212no—
25C22×D349—212no—
26C22×D349—26no—
27C22×D349D4112yescomplemented
28C22×D349C2313yes—
29C22×D349C2311yescharacteristic
30C22×D349C2311yescharacteristic, complemented
31D1396—26no—
32C22⋊D349—26no—
33C22⋊D349—26no—
34C2×Dic349C2313yescomplemented
35C4×D349—26no—
36C23×C349—26no—
37C23×C349C2313yes—
38C23×C349C2311yescharacteristic, complemented
39D4×C349—26no—
40C2×C4×C349C2313yescomplemented
41C22×C349—28no—
42C22×C349—26no—
43C22×C349—23no—
44C22×C349C2×D416yescomplemented
45C22×C349C2411yescharacteristic
46C2×D349—224no—
47C2×D349—26no—
48C2×D349—224no—
49C2×D349—23no—
50C2×D349—44no—
51C2×D349C2×D416yes—
52C2×D349C2×D416yescomplemented
53C4×C349—23no—
54Dic349—23no—
55C2×C349—26no—
56C2×C349—41no—
57C2×C349C22×D413yes—
58D349—26no—
59D349—41no—
60D349#12014yesdirect product factor
61C349C2×#12011yescharacteristic, complemented, Sylow
62C2×#120—3491nomaximal, Sylow
63C25—3491no—
64C22×D4—3493no—
65#120—3491no—
66#120—3493no—
67#120—3493no—
68#120D34911yescharacteristic, direct product factor
69C2×C22⋊C4—3493no—
70C24—3493no—
71C24—3493no—
72C24—3491no—
73C24—69812no—
74C24C2×D34911yescharacteristic, complemented
75C2×D4—3493no—
76C2×D4—3493no—
77C2×D4—3493no—
78C2×D4—69812no—
79C2×D4C2×D34913yescomplemented
80C22×C4—3493no—
81C22⋊C4—3493no—
82C22⋊C4—3496no—
83C22⋊C4C2×D34913yescomplemented
84C23—26no—
85C23—3491no—
86C23—34912no—
87C23—3493no—
88C23—3491no—
89C23—6986no—
90C23—69812no—
91C23—69812no—
92C23—69832no—
93C23—6986no—
94C23C22×D34913yes—
95C23C22×D34911yescharacteristic, complemented
96D4—26no—
97D4—6986no—
98D4—6986no—
99D4—6986no—
100C2×C4—3493no—
101C2×C4—6986no—
102C2×C4C22×D34913yescomplemented
103C22—28no—
104C22—26no—
105C22—23no—
106C22—3496no—
107C22—3496no—
108C22—13964no—
109C22—69824no—
110C22—69824no—
111C22—6983no—
112C22—6986no—
113C22D4×D34916yescomplemented
114C22C23×D34911yescharacteristic
115C4—23no—
116C4—6983no—
117C2—26no—
118C2—41no—
119C2—3494no—
120C2—13961no—
121C2—6986no—
122C2C2×D4×D34913yes—
subgroupG/H
center, FrattiniC22C23×D349
commutator, socle, cosocleC22×C349C24
Fitting#120×C349C2
hypercenter#120D349
hypocenterC349C2×#120
LCSG ▷ C22×C349 ▷ C349
UCSC1 ◁ C22 ◁ #120

Complements

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

NH ≅ G/Ntype
1. C24×D349C2semidirect
2. C2×D4×D349C2semidirect
3. #26013609C2semidirect
4. #26013608C2semidirect
5. #26013607C2semidirect
6. C22⋊C4×D349C2semidirect
7. #120×C349C2semidirect
10. C23×D349C22semidirect
11. C23×D349C22semidirect
12. C2×D1396C22semidirect
13. C2×C22⋊D349C22semidirect
14. C2×C22⋊D349C22semidirect
16. C2×C4×D349C22semidirect
17. C22⋊Dic349C22semidirect
18. #19150210C22semidirect
19. C24×C349C22semidirect
20. C2×D4×C349C22semidirect
21. C22⋊C4×C349C22semidirect
27. C22×D349D4semidirect
30. C22×D349C23semidirect
34. C2×Dic349C23semidirect
38. C23×C349C23semidirect
40. C2×C4×C349C23semidirect
44. C22×C349C2×D4semidirect
52. C2×D349C2×D4semidirect
60. D349#120direct (N × H)
61. C349C2×#120semidirect
68. #120D349direct (N × H)
74. C24C2×D349semidirect
79. C2×D4C2×D349semidirect
83. C22⋊C4C2×D349semidirect
95. C23C22×D349semidirect
102. C2×C4C22×D349semidirect
113. C22D4×D349semidirect

Maximal subgroups

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

subgroupindex
C24×D3492
C2×D4×D3492
#260136092
#260136082
#260136072
C22⋊C4×D3492
#120×C3492
C2×#120349