Finite Group Databasebeta

← #24906758

Subgroups — #24906758

Order 10624.

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#19128449C211yesmaximal, characteristic, complemented
2#19128090C212yesmaximal, complemented
3#19128067C211yesmaximal, characteristic, complemented
4#19127440C212yesmaximal, complemented
5#330×C83C211yesmaximal, characteristic, complemented
6C2×#13050579—22no—
7C2×D664—22no—
8#14581789C2211yescharacteristic
9#14581780C2212yes—
10#14581777—22no—
11#14581770—22no—
12#14581767—22no—
13#14581712—22no—
14#14581703C2211yescharacteristic
15#125×C83C2211yescharacteristic
16#97×C83C2212yes—
17C2×Dic166D412yes—
18C2×D332—22no—
19C2×D332—21no—
20C2×D332—22no—
21C2×D332D412yes—
22C2×C4×D83—22no—
23#13050579—44no—
24D664—44no—
25C2×C83⋊C8—21no—
26Dic83⋊C4—22no—
27(C2×D83)⋊C4—22no—
28(C2×D83)⋊C4—22no—
29(C2×D83)⋊C4—22no—
30C2×C8×C83—22no—
31C4⋊C4×C83—22no—
32C4⋊C4×C83D412yes—
33C42×C83C2311yescharacteristic
34C22×D83—22no—
35C22×D83—41no—
36C2×C4×C83—22no—
37C2×C4×C83C2×D412yes—
38C2×C4×C83C2×D411yescharacteristic
39C2×Dic83—22no—
40D332—42no—
41D332—42no—
42D332—42no—
43D332—42no—
44D332—82no—
45C4×D83—82no—
46Dic166—42no—
47C8×C83—24no—
48C83⋊C8—41no—
49C22×C83#12011yescharacteristic
50C2×D83—42no—
51C2×D83—42no—
52C2×D83—42no—
53C2×D83—82no—
54C2×D83—81no—
55C4×C83—22no—
56C4×C83—21no—
57C4×C83—42no—
58Dic83—42no—
59C2×C83#45312yes—
60C2×C83#45411yescharacteristic
61D83—82no—
62D83—161no—
63#1656—831nomaximal, Sylow
64C83#165611yescharacteristic, complemented, Sylow
65#562—831no—
66#496—832no—
67#490—831no—
68#330D8311yescharacteristic, complemented
69#329—832no—
70C2×SD16—1662no—
71C2×D8—1662no—
72#127—831no—
73#125C2×D8311yescharacteristic
74#124—832no—
75#123—1662no—
76C22⋊Q8—1662no—
77#121—1662no—
78#102—1662no—
79#97—831no—
80#97C2×D8312yes—
81C2×Q8—832no—
82C2×D4—832no—
83C2×D4—1662no—
84C2×D4—1662no—
85C2×D4—1661no—
86C22×C4—1662no—
87SD16—3324no—
88D8—3324no—
89C2×C8—22no—
90C2×C8—1661no—
91C4⋊C4—22no—
92C4⋊C4—1662no—
93C4⋊C4D33212yes—
94C22⋊C4—1662no—
95C22⋊C4—1662no—
96C22⋊C4—1662no—
97C42C22×D8311yescharacteristic
98C23—3321no—
99C23—1662no—
100Q8—3322no—
101D4—3322no—
102D4—3322no—
103D4—3322no—
104D4—3322no—
105D4—6642no—
106C2×C4—22no—
107C2×C4—6642no—
108C2×C4—1662no—
109C2×C4D4×D8312yes—
110C2×C4C2×D33211yescharacteristic
111C8—24no—
112C8—3321no—
113C22—3322no—
114C22—3322no—
115C22—3322no—
116C22—6642no—
117C22—6641no—
118C22#1458176211yescharacteristic
119C4—22no—
120C4—21no—
121C4—42no—
122C4—3322no—
123C2—13281no—
124C2—6642no—
125C2#1912787012yes—
126C2#1912787611yescharacteristic
subgroupG/H
centerC22#14581762
commutator, cosocleC42×C83C23
FrattiniC42C22×D83
Fitting#330×C83C2
socleC22×C83#120
hypercenter#330D83
hypocenterC83#1656
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #330

Complements

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

NH ≅ G/Ntype
1. #19128449C2semidirect
2. #19128090C2semidirect
3. #19128067C2semidirect
4. #19127440C2semidirect
5. #330×C83C2semidirect
64. C83#1656semidirect
68. #330D83semidirect

Maximal subgroups

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

subgroupindex
#191284492
#191280902
#191280672
#191274402
#330×C832
#165683