Finite Group Databasebeta

← #44549688

Subgroups — #44549688

Order 32128.

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

normal subgroups only

This group has 196 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#30632057C211yesmaximal, characteristic, complemented
2#30632046C212yesmaximal, complemented
3#30631429C211yesmaximal, characteristic, complemented
4#30631427C212yesmaximal, complemented
5#562×C251C211yesmaximal, characteristic
6#22548715C2211yescharacteristic
7#22548639—22no—
8#22548638—22no—
9#22548636—22no—
10#22548626C2211yescharacteristic
11#22548626C2212yes—
12#125×C251C2211yescharacteristic
13#124×C251C2212yes—
14#123×C251—22no—
15C22⋊Q8×C251—22no—
16#121×C251—22no—
17C2×Dic502—21no—
18C2×C251⋊C8—22no—
19C2×C251⋊C8—21no—
20C4⋊Dic251—22no—
21C2×Q8×C251D412yes—
22C2×D4×C251—22no—
23C2×D4×C251D412yes—
24C22×C4×C251—22no—
25C4⋊C4×C251—22no—
26C4⋊C4×C251—22no—
27C4⋊C4×C251D412yes—
28C22⋊C4×C251—22no—
29C22⋊C4×C251—22no—
30C22⋊C4×C251—22no—
31C42×C251C2311yescharacteristic
32Dic502—42no—
33C2×Dic251—41no—
34C251⋊C8—41no—
35C251⋊C8—42no—
36C23×C251—22no—
37Q8×C251—42no—
38D4×C251—42no—
39D4×C251—82no—
40C2×C4×C251—22no—
41C2×C4×C251—22no—
42C2×C4×C251—82no—
43C2×C4×C251C2×D411yescharacteristic
44C2×C4×C251C2×D412yes—
45C22×C251—42no—
46C22×C251—42no—
47C22×C251—42no—
48C22×C251#12011yescharacteristic
49C4×C251—21no—
50C4×C251—22no—
51C4×C251—42no—
52C4×C251—42no—
53Dic251—81no—
54C2×C251—82no—
55C2×C251#45411yescharacteristic
56C2×C251#45512yes—
57C251#166211yescharacteristic, complemented, Sylow
58#1662—2511nomaximal, Sylow
59#562D25111yescharacteristic
60#491—2511no—
61#489—2512no—
62#330—2511no—
63#329—2512no—
64C4⋊Q8—2511no—
65#125C2×D25111yescharacteristic
66#124C2×D25112yes—
67#123—22no—
68C22⋊Q8—22no—
69#121—22no—
70#103—5022no—
71#103—5022no—
72#102—5022no—
73#97—2511no—
74#97—2512no—
75C2×Q8—5021no—
76C2×Q8C22⋊D25112yes—
77C2×D4—22no—
78C2×D4C22⋊D25112yes—
79C22×C4—22no—
80C2×C8—5022no—
81C2×C8—5021no—
82C4⋊C4—22no—
83C4⋊C4—22no—
84C4⋊C4—5022no—
85C4⋊C4C22⋊D25112yes—
86C22⋊C4—22no—
87C22⋊C4—22no—
88C22⋊C4—22no—
89C42C22×D25111yescharacteristic
90C23—22no—
91Q8—42no—
92Q8—10042no—
93D4—42no—
94D4—82no—
95C2×C4—22no—
96C2×C4—22no—
97C2×C4—82no—
98C2×C4—10041no—
99C2×C4C2×C22⋊D25111yescharacteristic
100C2×C4C2×C22⋊D25112yes—
101C8—10041no—
102C8—10042no—
103C22—42no—
104C22—42no—
105C22—42no—
106C22#2254868411yescharacteristic
107C4—21no—
108C4—22no—
109C4—42no—
110C4—42no—
111C4—20081no—
112C2—82no—
113C2#3063186011yescharacteristic
114C2#3063186512yes—
subgroupG/H
centerC22#22548684
commutator, cosocleC42×C251C23
FrattiniC42C22×D251
Fitting, prime radical#562×C251C2
socleC22×C251#120
hypercenter#562D251
hypocenterC251#1662
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. #30632057C2semidirect
2. #30632046C2semidirect
3. #30631429C2semidirect
4. #30631427C2semidirect
57. C251#1662semidirect

Maximal subgroups

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

subgroupindex
#306320572
#306320462
#306314292
#306314272
#562×C2512
#1662251