Finite Group Databasebeta

← #44559767

Subgroups — #44559767

Order 32128.

772 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 772 conjugacy classes of subgroups — too many to draw. Only its 161 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
1#30632251C213yesmaximal, complemented
2#30631874C218yesmaximal, complemented
3#30631661C213yesmaximal, complemented
4#585×C251C211yesmaximal, characteristic, complemented
5C22×D1004C2211yescharacteristic, complemented
6#22548711—23no—
7#22548704—23no—
8#22548689—26no—
9#22548683C22112yescomplemented
10#22548674C2213yescomplemented
11#22548630C22112yescomplemented
122−1+4×C251—23no—
13H(5,𝔽2)×C251—23no—
14H(5,𝔽2)×C251C2214yes—
15C2×#41×C251—26no—
16C2×#41×C251C2213yes—
17C23×D251C2312yes—
18C2×D1004—23no—
19C2×D1004—23no—
20C2×C22⋊D251—212no—
21C4⋊Dic251—23no—
22C22⋊Dic251C2316yes—
23(C2×D251)⋊C4—212no—
24C4×Dic251—23no—
25#41×C251—24no—
26#41×C251—212no—
27#41×C251—26no—
28#41×C251—212no—
29#41×C251—26no—
30C2×Q8×C251—26no—
31C2×Q8×C251D413yescomplemented
32C2×D4×C251—26no—
33C2×D4×C251—212no—
34C2×D4×C251D413yescomplemented
35C2×D4×C251C2316yes—
36C22×C4×C251—24no—
37C22×C4×C251D416yescomplemented
38C22×C4×C251C2311yescharacteristic
39C22×D251—26no—
40C22×D251—82no—
41D1004—41no—
42D1004—43no—
43C22⋊D251—812no—
44C2×Dic251—26no—
45C23×C251—24no—
46C23×C251C2×D416yes—
47C23×C251C2411yescharacteristic
48Q8×C251—24no—
49Q8×C251—26no—
50D4×C251—212no—
51D4×C251—212no—
52D4×C251—26no—
53C2×C4×C251—212no—
54C2×C4×C251—26no—
55C2×C4×C251—26no—
56C2×C4×C251C2×D416yes—
57C2×C4×C251C2×D413yes—
58C2×C4×C251C2×D413yes—
59C22×C251—26no—
60C22×C251—41no—
61C22×C251—46no—
62C22×C251—84no—
63C22×C251C22×D413yes—
64C2×D251—412no—
65C2×D251—42no—
66C4×C251—26no—
67C4×C251#12011yescharacteristic
68C4×C251#12013yes—
69Dic251—46no—
70C2×C251—23no—
71C2×C251—46no—
72C2×C251C2×#12011yescharacteristic
73D251—82no—
74C251#302711yescharacteristic, complemented, Sylow
75#3027—2511nomaximal, Sylow
76#585D25111yescharacteristic, complemented
77#535—2513no—
78B(4,𝔽2)—2518no—
79#410—2513no—
802−1+4—23no—
81H(5,𝔽2)—23no—
82H(5,𝔽2)C2×D25114yes—
83C2×#41—26no—
84C2×#41C2×D25113yes—
85C22×D4—2511no—
86#127—5023no—
87#124—5023no—
88#121—5026no—
89#120—25112no—
90#117—2513no—
91#99—25112no—
92C24—2512no—
93#41—24no—
94#41—212no—
95#41—26no—
96#41—212no—
97#41—26no—
98C2×Q8—26no—
99C2×Q8C22⋊D25113yescomplemented
100C2×D4—26no—
101C2×D4—212no—
102C2×D4—50212no—
103C2×D4—5023no—
104C2×D4—5023no—
105C2×D4C22⋊D25113yescomplemented
106C2×D4C22×D25116yes—
107C22×C4—24no—
108C22×C4C22⋊D25116yescomplemented
109C22×C4C22×D25111yescharacteristic
110C4⋊C4—5023no—
111C22⋊C4—50212no—
112C22⋊C4—2516no—
113C42—5023no—
114C23—24no—
115C23—20082no—
116C23—5026no—
117C23C2×C22⋊D25116yes—
118C23C23×D25111yescharacteristic
119Q8—24no—
120Q8—26no—
121D4—212no—
122D4—212no—
123D4—26no—
124D4—200812no—
125D4—10041no—
126D4—10043no—
127C2×C4—212no—
128C2×C4—26no—
129C2×C4—26no—
130C2×C4—5026no—
131C2×C4C2×C22⋊D25116yes—
132C2×C4C2×C22⋊D25113yes—
133C2×C4C2×C22⋊D25113yes—
134C22—26no—
135C22—41no—
136C22—46no—
137C22—84no—
138C22—100412no—
139C22—10042no—
140C22C22×C22⋊D25113yes—
141C4—26no—
142C4—10046no—
143C4#2254868411yescharacteristic
144C4#2254868413yes—
145C2—23no—
146C2—46no—
147C2—20082no—
148C2C2×#2254868411yescharacteristic
subgroupG/H
centerC2C2×#22548684
commutator, cosocleC23×C251C24
FrattiniC23C23×D251
Fitting#585×C251C2
socleC2×C251C2×#120
hypercenter#585D251
hypocenterC251#3027
LCSG ▷ C23×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C2 ◁ C22×C4 ◁ #585

Complements

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

NH ≅ G/Ntype
1. #30632251C2semidirect
2. #30631874C2semidirect
3. #30631661C2semidirect
4. #585×C251C2semidirect
5. C22×D1004C22semidirect
9. #22548683C22semidirect
10. #22548674C22semidirect
11. #22548630C22semidirect
31. C2×Q8×C251D4semidirect
34. C2×D4×C251D4semidirect
37. C22×C4×C251D4semidirect
74. C251#3027semidirect
76. #585D251semidirect
99. C2×Q8C22⋊D251semidirect
105. C2×D4C22⋊D251semidirect
108. C22×C4C22⋊D251semidirect

Maximal subgroups

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

subgroupindex
#306322512
#306318742
#306316612
#585×C2512
#3027251