Finite Group Databasebeta

← #3549×C251 #44548136

Subgroups — #3549×C251 #44548136

Order 32128.

1258 conjugacy classes of subgroups; only the 768 normal subgroups are shown.

normal subgroups only

This group has 1258 conjugacy classes of subgroups — too many to draw. Only its 768 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
1H(3,𝔽4)×C251C214yesmaximal, complemented
2#555×C251C216yesmaximal, complemented
3D42×C251C216yesmaximal, complemented
4#535×C251C2112yesmaximal, complemented
5#532×C251C213yesmaximal, complemented
6C2×#41×C251C2216yescomplemented
7C22×D4×C251C22112yescomplemented
8C4⋊Q8×C251C2213yescomplemented
9#127×C251C2213yes—
10#127×C251C2218yescomplemented
11#124×C251C2216yescomplemented
12#124×C251C22124yescomplemented
13#121×C251C22112yescomplemented
14#121×C251C22124yescomplemented
15#120×C251C2218yescomplemented
16#120×C251C22124yescomplemented
17C4×D4×C251C22112yescomplemented
18#117×C251C22112yescomplemented
19C2×C42×C251C2211yescharacteristic, complemented
20C24×C251C2318yes—
21#41×C251—412no—
22C2×Q8×C251C2316yes—
23C2×D4×C251—424no—
24C2×D4×C251C23112yes—
25C2×D4×C251C2316yes—
26C2×D4×C251C23124yes—
27C2×D4×C251C23112yes—
28C22×C4×C251C2313yes—
29C22×C4×C251C23112yes—
30C4⋊C4×C251C23112yes—
31C22⋊C4×C251C23124yes—
32C22⋊C4×C251C23124yes—
33C42×C251C2313yes—
34C42×C251C2311yescharacteristic
35C42×C251C2318yes—
36C23×C251—424no—
37C23×C251C24112yes—
38C23×C251C2411yescharacteristic
39Q8×C251—46no—
40D4×C251—224no—
41D4×C251—424no—
42D4×C251—412no—
43D4×C251—46no—
44C2×C4×C251—224no—
45C2×C4×C251—26no—
46C2×C4×C251C2413yes—
47C2×C4×C251C24112yes—
48C2×C4×C251C2413yes—
49C22×C251—23no—
50C22×C251—212no—
51C22×C251—224no—
52C22×C251—816no—
53C22×C251C2511yescharacteristic
54C4×C251—212no—
55C4×C251—23no—
56C4×C251H(5,𝔽2)16yes—
57C2×C251—41no—
58C2×C251—412no—
59C2×C251C2×H(5,𝔽2)13yes—
60C251#354911yescharacteristic, direct product factor, Sylow
61#3549C25111yesmaximal, characteristic, direct product factor, Sylow
62H(3,𝔽4)C2×C25114yescomplemented
63#555C2×C25116yescomplemented
64D42C2×C25116yescomplemented
65#535C2×C251112yescomplemented
66#532C2×C25113yescomplemented
67C2×#41C22×C25116yescomplemented
68C22×D4C22×C251112yescomplemented
69C4⋊Q8C22×C25113yescomplemented
70#127C22×C25113yes—
71#127C22×C25118yescomplemented
72#124C22×C25116yescomplemented
73#124C22×C251124yescomplemented
74#121C22×C251112yescomplemented
75#121C22×C251124yescomplemented
76#120C22×C25118yescomplemented
77#120C22×C251124yescomplemented
78C4×D4C22×C251112yescomplemented
79#117C22×C251112yescomplemented
80C2×C42C22×C25111yescharacteristic, complemented
81C24C23×C25118yes—
82#41—412no—
83C2×Q8C23×C25116yes—
84C2×D4—424no—
85C2×D4C23×C251112yes—
86C2×D4C23×C25116yes—
87C2×D4C23×C251124yes—
88C2×D4C23×C251112yes—
89C22×C4C23×C25113yes—
90C22×C4C23×C251112yes—
91C4⋊C4C23×C251112yes—
92C22⋊C4C23×C251124yes—
93C22⋊C4C23×C251124yes—
94C42C23×C25113yes—
95C42C23×C25111yescharacteristic
96C42C23×C25118yes—
97C23—424no—
98C23C24×C251112yes—
99C23C24×C25111yescharacteristic
100Q8—46no—
101D4—224no—
102D4—424no—
103D4—412no—
104D4—46no—
105C2×C4—224no—
106C2×C4—26no—
107C2×C4C24×C25113yes—
108C2×C4C24×C251112yes—
109C2×C4C24×C25113yes—
110C22—23no—
111C22—212no—
112C22—224no—
113C22—816no—
114C22C25×C25111yescharacteristic
115C4—212no—
116C4—23no—
117C4H(5,𝔽2)×C25116yes—
118C2—412no—
119C2—41no—
120C2C2×H(5,𝔽2)×C25113yes—
subgroupG/H
center, socleC22×C251C25
commutator, Frattini, cosocleC22C25×C251
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C251 ◁ G

Complements

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

NH ≅ G/Ntype
1. H(3,𝔽4)×C251C2semidirect
2. #555×C251C2semidirect
3. D42×C251C2semidirect
4. #535×C251C2semidirect
5. #532×C251C2semidirect
6. C2×#41×C251C22semidirect
7. C22×D4×C251C22semidirect
8. C4⋊Q8×C251C22semidirect
10. #127×C251C22semidirect
11. #124×C251C22semidirect
12. #124×C251C22semidirect
13. #121×C251C22semidirect
14. #121×C251C22semidirect
15. #120×C251C22semidirect
16. #120×C251C22semidirect
17. C4×D4×C251C22semidirect
18. #117×C251C22semidirect
19. C2×C42×C251C22semidirect
60. C251#3549direct (N × H)
61. #3549C251direct (N × H)
62. H(3,𝔽4)C2×C251semidirect
63. #555C2×C251semidirect
64. D42C2×C251semidirect
65. #535C2×C251semidirect
66. #532C2×C251semidirect
67. C2×#41C22×C251semidirect
68. C22×D4C22×C251semidirect
69. C4⋊Q8C22×C251semidirect
71. #127C22×C251semidirect
72. #124C22×C251semidirect
73. #124C22×C251semidirect
74. #121C22×C251semidirect
75. #121C22×C251semidirect
76. #120C22×C251semidirect
77. #120C22×C251semidirect
78. C4×D4C22×C251semidirect
79. #117C22×C251semidirect
80. C2×C42C22×C251semidirect

Maximal subgroups

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

subgroupindex
H(3,𝔽4)×C2512
#555×C2512
D42×C2512
#535×C2512
#532×C2512
#3549251