Finite Group Databasebeta

← #27271965

Subgroups — #27271965

Order 13568.

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

normal subgroups only

This group has 150 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#20274301C211yesmaximal, characteristic
2#20272880C211yesmaximal, characteristic
3C53×#1410C211yesmaximal, characteristic
4#16811021C411yescharacteristic, complemented
5#16811019—21no—
6#16810995—21no—
7#16810416—21no—
8C8×#88322C411yescharacteristic, complemented
9C53×#492C2211yescharacteristic
10C53×#356—21no—
11C2×#11719923—42no—
12#13133217—21no—
13#13133149—41no—
14#13133148—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—
32#11719923—84no—
33C4⋊Dic53—41no—
34C4⋊Dic53—41no—
35C2×#88322—22no—
36C23×C53—41no—
37Q8×C53—81no—
38D4×C53—81no—
39D4×C53—82no—
40C2×C4×C53—21no—
41C2×C4×C53—41no—
42C2×C4×C53#9911yescharacteristic
43C2×Dic53—81no—
44C2×Dic53—161no—
45Dic106—82no—
46C8×C53—22no—
47C8×C53—161no—
48#88322—24no—
49#7535—531nomaximal, Sylow
50C22×C53—81no—
51C22×C53—161no—
52C22×C53#35311yescharacteristic
53C4×C53—22no—
54C4×C53—21no—
55C4×C53—81no—
56Dic53—161no—
57Dic53—321no—
58#1705—531no—
59#1410D5311yescharacteristic
60#1409—531no—
61C2×C53—21no—
62C2×C53—161no—
63C2×C53#140911yescharacteristic
64#498—531no—
65#498—1061no—
66#492—1061no—
67#492C2×D5311yescharacteristic
68#356—21no—
69#353—1061no—
70C82—531no—
71C53#753511yescharacteristic, complemented, Sylow
72C2×SD16—41no—
73C2×SD16—2122no—
74C4⋊Q8—1061no—
75C4⋊Q8C22⋊D5311yescharacteristic
76#127D21211yescharacteristic
77#106—2122no—
78#106—2121no—
79#101—41no—
80#99—2121no—
81C4×C8—1061no—
82C4×C8C4×D5311yescharacteristic
83C2×Q8—21no—
84C2×Q8—2121no—
85C2×D4—21no—
86C2×D4—41no—
87SD16—82no—
88SD16—4244no—
89M16—81no—
90C2×C8—21no—
91C2×C8—1062no—
92C4⋊C4—41no—
93C4⋊C4—2121no—
94C4⋊C4—2121no—
95C22⋊C4—4241no—
96C42#1171992111yescharacteristic
97C23—41no—
98Q8—81no—
99Q8—4242no—
100D4—81no—
101D4—82no—
102C2×C4—21no—
103C2×C4—41no—
104C2×C4—8481no—
105C2×C4—4241no—
106C2×C4#1313313111yescharacteristic
107C8—22no—
108C8—161no—
109C8—1064no—
110C22—81no—
111C22—161no—
112C22#1681041511yescharacteristic
113C4—22no—
114C4—21no—
115C4—81no—
116C4—8481no—
117C4—16961no—
118C2—21no—
119C2—161no—
120C2#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#7535
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
71. C53#7535semidirect

Maximal subgroups

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

subgroupindex
#202743012
#202728802
C53×#14102
#753553