Finite Group Databasebeta

← #27271968

Subgroups — #27271968

Order 13568.

138 conjugacy classes of subgroups; only the 25 normal subgroups are shown.

normal subgroups only

This group has 138 conjugacy classes of subgroups — too many to draw. Only its 25 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#20274314C211yesmaximal, characteristic
2#20272880C211yesmaximal, characteristic
3C53×#1410C211yesmaximal, characteristic
4#16811021C411yescharacteristic, complemented
5#16810975—21no—
6#16810969—21no—
7#16810416—21no—
8C8×#88322C411yescharacteristic, complemented
9C53×#492C2211yescharacteristic
10C53×#356—21no—
11#13133217—21no—
12#13133149—41no—
13#13133140—42no—
14#13133138—42no—
15#13133132—41no—
16C4×#88322—21no—
17C2×SD16×C53—41no—
18C4⋊Q8×C53D411yescharacteristic
19#127×C53D411yescharacteristic
20#101×C53—41no—
21C4×C8×C53C2×C411yescharacteristic
22C2×Q8×C53—21no—
23C2×D4×C53—21no—
24C2×D4×C53—41no—
25C2×Dic106—41no—
26SD16×C53—82no—
27M16×C53—81no—
28C2×C8×C53—21no—
29C4⋊C4×C53—41no—
30C42×C53C22⋊C411yescharacteristic
31C22⋊Dic53—81no—
32C4⋊Dic53—41no—
33C4⋊Dic53—41no—
34C2×#88322—22no—
35C23×C53—41no—
36Q8×C53—81no—
37D4×C53—81no—
38D4×C53—82no—
39C2×C4×C53—21no—
40C2×C4×C53—41no—
41C2×C4×C53#9911yescharacteristic
42C2×Dic53—81no—
43C2×Dic53—161no—
44Dic106—82no—
45C8×C53—22no—
46C8×C53—161no—
47#88322—42no—
48#7536—531nomaximal, Sylow
49C22×C53—81no—
50C22×C53—161no—
51C22×C53#35311yescharacteristic
52C4×C53—22no—
53C4×C53—21no—
54C4×C53—81no—
55Dic53—161no—
56Dic53—321no—
57#1708—531no—
58#1410D5311yescharacteristic
59#1409—531no—
60C2×C53—21no—
61C2×C53—161no—
62C2×C53#140911yescharacteristic
63#498—531no—
64#492C2×D5311yescharacteristic
65#487—1061no—
66#486—1061no—
67#356—21no—
68#353—1061no—
69C82—531no—
70C53#753611yescharacteristic, complemented, Sylow
71C2×SD16—41no—
72C4⋊Q8—1061no—
73C4⋊Q8C22⋊D5311yescharacteristic
74#127D21211yescharacteristic
75#106—2121no—
76#103—2122no—
77#102—2122no—
78#101—41no—
79#99—2121no—
80C4×C8—1061no—
81C4×C8C4×D5311yescharacteristic
82C2×Q8—21no—
83C2×Q8—2121no—
84C2×D4—21no—
85C2×D4—41no—
86SD16—82no—
87M16—81no—
88C2×C8—21no—
89C2×C8—1062no—
90C4⋊C4—41no—
91C4⋊C4—2121no—
92C4⋊C4—2121no—
93C22⋊C4—4241no—
94C42#1171992111yescharacteristic
95C23—41no—
96Q8—81no—
97Q8—4242no—
98D4—81no—
99D4—82no—
100C2×C4—21no—
101C2×C4—41no—
102C2×C4—8481no—
103C2×C4—4241no—
104C2×C4#1313313111yescharacteristic
105C8—22no—
106C8—161no—
107C8—2122no—
108C22—81no—
109C22—161no—
110C22#1681041511yescharacteristic
111C4—22no—
112C4—21no—
113C4—81no—
114C4—8481no—
115C4—16961no—
116C2—21no—
117C2—161no—
118C2#2027287911yescharacteristic
subgroupG/H
centerC2#20272879
commutatorC4×C8×C53C2×C4
Frattini#492C2×D53
FittingC53×#1410C2
socleC2×C53#1409
prime radical#127×C53D4
cosocleC53×#492C22
hypercenter#1410D53
hypocenterC53#7536
LCSG ▷ C4×C8×C53 ▷ C42×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #492 ◁ #1410

Complements

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

NH ≅ G/Ntype
4. #16811021C4semidirect
8. C8×#88322C4semidirect
70. C53#7536semidirect

Maximal subgroups

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

subgroupindex
#202743142
#202728802
C53×#14102
#753653