Finite Group Databasebeta

← #28299338

Subgroups — #28299338

Order 13568.

142 conjugacy classes of subgroups; only the 22 normal subgroups are shown.

normal subgroups only

This group has 142 conjugacy classes of subgroups — too many to draw. Only its 22 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#20289852C211yesmaximal, characteristic
2#20289724C211yesmaximal, characteristic
3#20273855C211yesmaximal, characteristic
4#16811546—21no—
5#16811534—21no—
6#16811002C2211yescharacteristic
7#16810918—21no—
8#16810372C411yescharacteristic
9Q8⋊#88322—21no—
10Q82×C53C411yescharacteristic
11#13133291—41no—
12#13133278—41no—
13C2×#11719925—41no—
14#13133217D411yescharacteristic
15#13133150—41no—
16C4⋊#88322—21no—
17#13133141—41no—
18C4×#88322D411yescharacteristic
19C4⋊Q8×C53—41no—
20C4⋊Q8×C53—41no—
21C4⋊Q8×C53C2×C411yescharacteristic
22C4×Q8×C53—21no—
23C4×Q8×C53—41no—
24C2×Q8×C53—21no—
25C2×Q8×C53—21no—
26C2×Q8×C53—41no—
27C2×Dic106—21no—
28#11719940—81no—
29C4⋊C4×C53—21no—
30C4⋊C4×C53—41no—
31C4⋊C4×C53—42no—
32C4⋊C4×C53—41no—
33C42×C53—41no—
34C42×C53—41no—
35C42×C53C22⋊C411yescharacteristic
36#11719925—81no—
37#11719925—81no—
38C4⋊Dic53—41no—
39C2×#88322—21no—
40C2×#88322—41no—
41#11719910—81no—
42Q8×C53—22no—
43Q8×C53—41no—
44Q8×C53—81no—
45Q8×C53—161no—
46C2×C4×C53—21no—
47C2×C4×C53—41no—
48C2×C4×C53—41no—
49C2×C4×C53—41no—
50C2×C4×C53#9911yescharacteristic
51C2×Dic53—41no—
52Dic106—81no—
53#88324—161no—
54#88322—21no—
55#88322—21no—
56#88322—81no—
57#7522—531nomaximal, Sylow
58C22×C53AΣL(1,𝔽16)11yescharacteristic
59C4×C53—21no—
60C4×C53—21no—
61C4×C53—21no—
62C4×C53—42no—
63C4×C53—81no—
64C4×C53—81no—
65Dic53—81no—
66#1633—531no—
67#1411—531no—
68#1358—531no—
69C2×C53—21no—
70C2×C53#140211yescharacteristic
71Q82#615711yescharacteristic
72#494—531no—
73#477—1061no—
74#364—1061no—
75#356—1061no—
76#333—531no—
77Q8⋊C8—1061no—
78C53#752211yescharacteristic, complemented, Sylow
79C2×Q16—2121no—
80C4⋊Q8—41no—
81C4⋊Q8—41no—
82C4⋊Q8—531no—
83C4⋊Q8C2×#615711yescharacteristic
84C4×Q8—21no—
85C4×Q8—41no—
86M32—2121no—
87#107—2121no—
88C4⋊C8—1061no—
89#103—2121no—
90#101—2121no—
91C4×C8—531no—
92C2×Q8—21no—
93C2×Q8—21no—
94C2×Q8—41no—
95C2×Q8—1061no—
96Q16—4241no—
97Q16—4241no—
98M16—4241no—
99C2×C8—1061no—
100C2×C8—2121no—
101C4⋊C4—21no—
102C4⋊C4—41no—
103C4⋊C4—42no—
104C4⋊C4—41no—
105C4⋊C4—2121no—
106C42—41no—
107C42—41no—
108C42C22⋊#615711yescharacteristic
109C16—4241no—
110Q8—22no—
111Q8—41no—
112Q8—81no—
113Q8—161no—
114Q8—4241no—
115C2×C4—21no—
116C2×C4—41no—
117C2×C4—41no—
118C2×C4—41no—
119C2×C4—2121no—
120C2×C4#1313327511yescharacteristic
121C8—8481no—
122C8—1061no—
123C8—1061no—
124C8—4241no—
125C22#1681152311yescharacteristic
126C4—21no—
127C4—21no—
128C4—21no—
129C4—42no—
130C4—81no—
131C4—81no—
132C4—4241no—
133C2—21no—
134C2#2028983411yescharacteristic
subgroupG/H
centerC2#20289834
commutatorC4⋊Q8×C53C2×C4
FrattiniC4⋊Q8C2×#6157
FittingQ82×C53C4
socleC2×C53#1402
prime radicalC22×C53AΣL(1,𝔽16)
cosocle#16811002C22
hypercenterQ82#6157
hypocenterC53#7522
LCSG ▷ C4⋊Q8×C53 ▷ C42×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ C4⋊Q8 ◁ Q82

Complements

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

NH ≅ G/Ntype
78. C53#7522semidirect

Maximal subgroups

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

subgroupindex
#202898522
#202897242
#202738552
#752253