Finite Group Databasebeta

← #27271933

Subgroups — #27271933

Order 13568.

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

normal subgroups only

This group has 158 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#20274007C211yesmaximal, characteristic
2#20272886C211yesmaximal, characteristic, complemented
3C53×#1355C211yesmaximal, characteristic, complemented
4#16811371C411yescharacteristic, complemented
5#16811010—21no—
6#16810416—21no—
7#16810365C411yescharacteristic, complemented
8#16810362—21no—
9C53×#490C2211yescharacteristic
10C53×#347—21no—
11C2×#11719923—41no—
12C2×#11719922—41no—
13#13133205—21no—
14#13133202—41no—
15#13133195—41no—
16#13133192—41no—
17#13133132—41no—
18#13133128—21no—
19#127×C53D411yescharacteristic
20#125×C53C2×C411yescharacteristic
21M32×C53—41no—
22#102×C53—41no—
23#97×C53D411yescharacteristic
24C2×D4×C53—21no—
25C2×D4×C53—41no—
26C2×D212—21no—
27C2×D212—41no—
28C2×C4×D53—41no—
29C2×Dic106—21no—
30C2×C8×C53—21no—
31C4⋊C4×C53—21no—
32C4⋊C4×C53—41no—
33C42×C53C22⋊C411yescharacteristic
34C22⋊Dic53—81no—
35#11719923—82no—
36#11719922—82no—
37#11719921—41no—
38#11719921—41no—
39#11719921—41no—
40#11719919—41no—
41C2×#88322—41no—
42C16×C53—42no—
43C23×C53—41no—
44C22×D53—41no—
45D4×C53—81no—
46D4×C53—81no—
47D4×C53—81no—
48C2×C4×C53—21no—
49C2×C4×C53—41no—
50C2×C4×C53#9911yescharacteristic
51C2×Dic53—41no—
52C2×Dic53—161no—
53D212—81no—
54D212—161no—
55C4×D53—161no—
56Dic106—81no—
57C8×C53—41no—
58#88322—42no—
59#7523—531nomaximal, Sylow
60C22×C53—81no—
61C22×C53—161no—
62C22×C53AΣL(1,𝔽16)11yescharacteristic
63C2×D53—81no—
64C2×D53—81no—
65C2×D53—81no—
66C4×C53—21no—
67C4×C53—41no—
68C4×C53—81no—
69Dic53—81no—
70Dic53—321no—
71#1656—531no—
72#1412—531no—
73#1355D5311yescharacteristic, complemented
74C2×C53—21no—
75C2×C53—161no—
76C2×C53#140411yescharacteristic
77D53—161no—
78#562—531no—
79#496—1061no—
80#490C2×D5311yescharacteristic
81#353—1061no—
82#347—21no—
83#330—531no—
84#329—1061no—
85C53#752311yescharacteristic, complemented, Sylow
86C2×SD16—2121no—
87C2×D8—2121no—
88#127D21211yescharacteristic
89#125C4×D5311yescharacteristic
90#124—1061no—
91#123—2121no—
92C22⋊Q8—2121no—
93#121—2121no—
94M32—41no—
95#102—41no—
96#99—2121no—
97#97—1061no—
98#97C22⋊D5311yescharacteristic
99C2×Q8—1061no—
100C2×D4—21no—
101C2×D4—41no—
102C2×D4—1061no—
103C2×D4—2121no—
104C22×C4—2121no—
105SD16—4242no—
106D8—4242no—
107C2×C8—21no—
108C2×C8—2121no—
109C4⋊C4—21no—
110C4⋊C4—41no—
111C4⋊C4—2121no—
112C22⋊C4—4241no—
113C22⋊C4—2121no—
114C22⋊C4—2121no—
115C22⋊C4—2121no—
116C42#1171992111yescharacteristic
117C16—42no—
118C23—41no—
119C23—2121no—
120Q8—4241no—
121D4—81no—
122D4—81no—
123D4—81no—
124D4—8481no—
125D4—4241no—
126C2×C4—21no—
127C2×C4—41no—
128C2×C4—8481no—
129C2×C4—8481no—
130C2×C4—2121no—
131C2×C4#1313313111yescharacteristic
132C8—41no—
133C8—2122no—
134C22—81no—
135C22—161no—
136C22—4241no—
137C22—4241no—
138C22—4241no—
139C22#1681041111yescharacteristic
140C4—21no—
141C4—41no—
142C4—81no—
143C4—16961no—
144C4—4241no—
145C2—21no—
146C2—161no—
147C2—8481no—
148C2#2027286711yescharacteristic
subgroupG/H
centerC2#20272867
commutator#125×C53C2×C4
Frattini#490C2×D53
FittingC53×#1355C2
socleC2×C53#1404
prime radical#20274007C2
cosocleC53×#490C22
hypercenter#1355D53
hypocenterC53#7523
LCSG ▷ #125×C53 ▷ C42×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #490 ◁ #1355

Complements

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

NH ≅ G/Ntype
2. #20272886C2semidirect
3. C53×#1355C2semidirect
4. #16811371C4semidirect
7. #16810365C4semidirect
73. #1355D53semidirect
85. C53#7523semidirect

Maximal subgroups

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

subgroupindex
#202740072
#202728862
C53×#13552
#752353