Finite Group Databasebeta

← #44549658

Subgroups — #44549658

Order 32128.

230 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 49 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#30632077C212yesmaximal, complemented
2#30632053C211yesmaximal, characteristic, complemented
3#30631429C211yesmaximal, characteristic, complemented
4#30631427C212yesmaximal, complemented
5#562×C251C211yesmaximal, characteristic, complemented
6C2×#16860797—22no—
7C2×#16860795—22no—
8#22548712C2211yescharacteristic
9#22548635—22no—
10#22548626C2211yescharacteristic
11#22548626C2212yes—
12#125×C251C2211yescharacteristic
13#124×C251C2212yes—
14#123×C251—22no—
15C22⋊Q8×C251—22no—
16#121×C251—22no—
17C2×D1004—21no—
18C2×D1004—22no—
19#16860797—44no—
20#16860795—44no—
21C2×C251⋊C8—22no—
22C2×C251⋊C8—21no—
23C2×Q8×C251D412yes—
24C2×D4×C251—22no—
25C2×D4×C251D412yes—
26C22×C4×C251—22no—
27C4⋊C4×C251—22no—
28C4⋊C4×C251—22no—
29C4⋊C4×C251D412yes—
30C22⋊C4×C251—22no—
31C22⋊C4×C251—22no—
32C22⋊C4×C251—22no—
33C42×C251C2311yescharacteristic
34C22×D251—41no—
35D1004—42no—
36D1004—42no—
37D1004—42no—
38C251⋊C8—24no—
39C251⋊C8—41no—
40C23×C251—22no—
41Q8×C251—42no—
42D4×C251—42no—
43D4×C251—82no—
44C2×C4×C251—22no—
45C2×C4×C251—22no—
46C2×C4×C251—82no—
47C2×C4×C251C2×D411yescharacteristic
48C2×C4×C251C2×D412yes—
49C22×C251—42no—
50C22×C251—42no—
51C22×C251—42no—
52C22×C251#12011yescharacteristic
53C2×D251—82no—
54C2×D251—81no—
55C4×C251—22no—
56C4×C251—21no—
57C4×C251—42no—
58C4×C251—42no—
59C2×C251—82no—
60C2×C251#45312yes—
61C2×C251#45411yescharacteristic
62D251—161no—
63C251#165611yescharacteristic, complemented, Sylow
64#1656—2511nomaximal, Sylow
65#562D25111yescharacteristic, complemented
66#496—2512no—
67#490—2511no—
68#330—2511no—
69#329—2512no—
70C2×SD16—5022no—
71C2×D8—5022no—
72#127—2511no—
73#125C2×D25111yescharacteristic
74#124C2×D25112yes—
75#123—22no—
76C22⋊Q8—22no—
77#121—22no—
78#102—5022no—
79#97—2511no—
80#97—2512no—
81C2×Q8C22⋊D25112yes—
82C2×D4—22no—
83C2×D4—5021no—
84C2×D4—5022no—
85C2×D4C22⋊D25112yes—
86C22×C4—22no—
87SD16—10044no—
88D8—10044no—
89C2×C8—5021no—
90C2×C8—5022no—
91C4⋊C4—22no—
92C4⋊C4—22no—
93C4⋊C4C22⋊D25112yes—
94C22⋊C4—22no—
95C22⋊C4—22no—
96C22⋊C4—22no—
97C42C22×D25111yescharacteristic
98C23—22no—
99C23—10041no—
100Q8—42no—
101D4—42no—
102D4—82no—
103D4—10042no—
104D4—10042no—
105D4—10042no—
106C2×C4—22no—
107C2×C4—22no—
108C2×C4—82no—
109C2×C4C2×C22⋊D25111yescharacteristic
110C2×C4C2×C22⋊D25112yes—
111C8—10041no—
112C8—5024no—
113C22—42no—
114C22—42no—
115C22—42no—
116C22—20082no—
117C22—20081no—
118C22#2254868411yescharacteristic
119C4—22no—
120C4—21no—
121C4—42no—
122C4—42no—
123C2—82no—
124C2—40161no—
125C2#3063185512yes—
126C2#3063186011yescharacteristic
subgroupG/H
centerC22#22548684
commutator, cosocleC42×C251C23
FrattiniC42C22×D251
Fitting#562×C251C2
socleC22×C251#120
hypercenter#562D251
hypocenterC251#1656
LCSG ▷ C42×C251 ▷ C22×C251 ▷ C251
UCSC1 ◁ C22 ◁ C42 ◁ #562

Complements

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

NH ≅ G/Ntype
1. #30632077C2semidirect
2. #30632053C2semidirect
3. #30631429C2semidirect
4. #30631427C2semidirect
5. #562×C251C2semidirect
63. C251#1656semidirect
65. #562D251semidirect

Maximal subgroups

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

subgroupindex
#306320772
#306320532
#306314292
#306314272
#562×C2512
#1656251