Finite Group Databasebeta

← #44549666

Subgroups — #44549666

Order 32128.

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

normal subgroups only

This group has 182 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
1C251⋊#564C211yesmaximal, characteristic, complemented
2#30632051C212yesmaximal
3#30631429C211yesmaximal, characteristic, complemented
4#30631428C212yesmaximal, complemented
5#490×C251C211yesmaximal, characteristic
6#22548707C2212yes—
7#22548706—22no—
8#22548705—22no—
9#22548705—22no—
10#22548636—22no—
11#22548636—22no—
12#22548626C2212yes—
13#127×C251C2211yescharacteristic
14#125×C251C2211yescharacteristic
15#102×C251—22no—
16#97×C251C2211yescharacteristic
17C2×C251⋊C8—22no—
18C4⋊Dic251—22no—
19C4⋊Dic251D412yes—
20C4⋊Dic251D412yes—
21Dic251⋊C4—22no—
22Dic251⋊C4—22no—
23Dic251⋊C4—22no—
24Dic251⋊C4—22no—
25C4×Dic251—22no—
26C2×D4×C251—21no—
27C2×D4×C251—22no—
28C2×C8×C251—21no—
29C4⋊C4×C251—22no—
30C4⋊C4×C251D412yes—
31C42×C251C2311yescharacteristic
32C2×Dic251—22no—
33C2×Dic251—22no—
34C251⋊C8—42no—
35C23×C251—41no—
36D4×C251—42no—
37D4×C251—44no—
38C2×C4×C251—22no—
39C2×C4×C251C2×D412yes—
40C2×C4×C251C2×D411yescharacteristic
41C8×C251—41no—
42C22×C251—82no—
43C22×C251—81no—
44C22×C251#12011yescharacteristic
45C4×C251—21no—
46C4×C251—22no—
47C4×C251—42no—
48Dic251—42no—
49Dic251—42no—
50C2×C251—161no—
51C2×C251#45412yes—
52C2×C251#45411yescharacteristic
53C251#165811yescharacteristic, complemented, Sylow
54#1658—2511nomaximal, Sylow
55#564—2511no—
56#490—2512no—
57#490D25111yescharacteristic
58#330—2511no—
59#330—2512no—
60#127C2×D25111yescharacteristic
61#125—5022no—
62#125—5022no—
63#125—5022no—
64#125—2512no—
65#125C2×D25111yescharacteristic
66#102—22no—
67#102—5022no—
68#102—5022no—
69#97—2512no—
70#97C2×D25111yescharacteristic
71C2×D4—22no—
72C2×D4—21no—
73C2×C8—21no—
74C2×C8—5022no—
75C4⋊C4—22no—
76C4⋊C4—5022no—
77C4⋊C4—5022no—
78C4⋊C4—5022no—
79C4⋊C4—5022no—
80C4⋊C4—5022no—
81C4⋊C4—2512no—
82C4⋊C4—2512no—
83C4⋊C4C22⋊D25112yes—
84C42—5022no—
85C42C22×D25111yescharacteristic
86C23—41no—
87D4—44no—
88D4—42no—
89C2×C4—22no—
90C2×C4—5022no—
91C2×C4—5022no—
92C2×C4D4×D25112yes—
93C2×C4C2×C22⋊D25111yescharacteristic
94C8—41no—
95C8—10042no—
96C22—82no—
97C22—81no—
98C22#2254868311yescharacteristic
99C4—22no—
100C4—21no—
101C4—42no—
102C4—10042no—
103C4—10042no—
104C2—161no—
105C2#3063185812yes—
106C2#3063185911yescharacteristic
subgroupG/H
centerC22#22548683
commutator, cosocleC42×C251C23
FrattiniC42C22×D251
Fitting#490×C251C2
socleC22×C251#120
prime radical#127×C251C22
hypercenter#490D251
hypocenterC251#1658
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. C251⋊#564C2semidirect
3. #30631429C2semidirect
4. #30631428C2semidirect
53. C251#1658semidirect

Maximal subgroups

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

subgroupindex
C251⋊#5642
#306320512
#306314292
#306314282
#490×C2512
#1658251