Finite Group Databasebeta

← #44549657

Subgroups — #44549657

Order 32128.

230 conjugacy classes of subgroups; only the 45 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 45 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#30632435C211yesmaximal, characteristic, complemented
2#30632074C212yesmaximal, complemented
3#30631429C211yesmaximal, characteristic, complemented
4#30631426C212yesmaximal, complemented
5#490×C251C211yesmaximal, characteristic, complemented
6C2×#16860798—22no—
7C2×#16860795—22no—
8#22548703C2212yes—
9#22548700—22no—
10#22548693—22no—
11#22548690—22no—
12#22548626C2212yes—
13#127×C251C2211yescharacteristic
14#125×C251C2211yescharacteristic
15#102×C251—22no—
16#97×C251C2211yescharacteristic
17C2×Dic502D412yes—
18C2×D1004—22no—
19C2×D1004D412yes—
20C2×C4×D251—22no—
21#16860798—44no—
22#16860795—44no—
23C2×C251⋊C8—22no—
24Dic251⋊C4—22no—
25(C2×D251)⋊C4—22no—
26(C2×D251)⋊C4—22no—
27(C2×D251)⋊C4—22no—
28C2×D4×C251—21no—
29C2×D4×C251—22no—
30C2×C8×C251—21no—
31C4⋊C4×C251—22no—
32C4⋊C4×C251D412yes—
33C42×C251C2311yescharacteristic
34C22×D251—22no—
35Dic502—42no—
36D1004—42no—
37D1004—82no—
38C2×Dic251—22no—
39C4×D251—82no—
40C251⋊C8—24no—
41C23×C251—41no—
42D4×C251—42no—
43D4×C251—42no—
44D4×C251—42no—
45C2×C4×C251—22no—
46C2×C4×C251C2×D412yes—
47C2×C4×C251C2×D411yescharacteristic
48C8×C251—41no—
49C22×C251—82no—
50C22×C251—81no—
51C22×C251#12011yescharacteristic
52C2×D251—42no—
53C2×D251—42no—
54C2×D251—42no—
55C4×C251—22no—
56C4×C251—21no—
57C4×C251—42no—
58Dic251—42no—
59C2×C251—161no—
60C2×C251#45312yes—
61C2×C251#45411yescharacteristic
62D251—82no—
63C251#165611yescharacteristic, complemented, Sylow
64#1656—2511nomaximal, Sylow
65#562—2511no—
66#496—2512no—
67#490D25111yescharacteristic, complemented
68#330—2511no—
69#329—2512no—
70C2×SD16—5022no—
71C2×D8—5022no—
72#127C2×D25111yescharacteristic
73#125C2×D25111yescharacteristic
74#124—2512no—
75#123—5022no—
76C22⋊Q8—5022no—
77#121—5022no—
78#102—22no—
79#97—2512no—
80#97C2×D25111yescharacteristic
81C2×Q8—2512no—
82C2×D4—22no—
83C2×D4—21no—
84C2×D4—5022no—
85C2×D4—2512no—
86C22×C4—5022no—
87SD16—10044no—
88D8—10044no—
89C2×C8—21no—
90C2×C8—5022no—
91C4⋊C4—22no—
92C4⋊C4—5022no—
93C4⋊C4C22⋊D25112yes—
94C22⋊C4—5022no—
95C22⋊C4—5022no—
96C22⋊C4—5022no—
97C42C22×D25111yescharacteristic
98C23—41no—
99C23—5022no—
100Q8—10042no—
101D4—42no—
102D4—42no—
103D4—42no—
104D4—20082no—
105D4—10042no—
106C2×C4—22no—
107C2×C4—20082no—
108C2×C4—5022no—
109C2×C4D4×D25112yes—
110C2×C4C2×C22⋊D25111yescharacteristic
111C8—41no—
112C8—5024no—
113C22—82no—
114C22—81no—
115C22—10042no—
116C22—10042no—
117C22—10042no—
118C22#2254868311yescharacteristic
119C4—22no—
120C4—21no—
121C4—42no—
122C4—10042no—
123C2—161no—
124C2—20082no—
125C2#3063185312yes—
126C2#3063185911yescharacteristic
subgroupG/H
centerC22#22548683
commutator, cosocleC42×C251C23
FrattiniC42C22×D251
Fitting#490×C251C2
socleC22×C251#120
hypercenter#490D251
hypocenterC251#1656
LCSG ▷ C42×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #490

Complements

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

NH ≅ G/Ntype
1. #30632435C2semidirect
2. #30632074C2semidirect
3. #30631429C2semidirect
4. #30631426C2semidirect
5. #490×C251C2semidirect
63. C251#1656semidirect
67. #490D251semidirect

Maximal subgroups

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

subgroupindex
#306324352
#306320742
#306314292
#306314262
#490×C2512
#1656251