Finite Group Databasebeta

← #27271925

Subgroups — #27271925

Order 13568.

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

normal subgroups only

This group has 148 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#20273858C211yesmaximal, characteristic
2#20272754C211yesmaximal, characteristic, complemented
3C53×#1411C211yesmaximal, characteristic, complemented
4#16811339C411yescharacteristic
5#16810910—21no—
6#16810444—21no—
7#16810372C411yescharacteristic
8#16810352—21no—
9C53×#494C2211yescharacteristic
10C53×#356—21no—
11#13133206—41no—
12#13133192—41no—
13C4×D212—21no—
14#13133175—41no—
15#13133157—41no—
16#13133148—41no—
17C4⋊#88322—21no—
18#13133137—41no—
19C2×Q16×C53—41no—
20C4⋊Q8×C53C2×C411yescharacteristic
21C4⋊Q8×C53D411yescharacteristic
22#101×C53—41no—
23C4×C8×C53D411yescharacteristic
24C2×Q8×C53—21no—
25C2×Q8×C53—21no—
26C2×D212—21no—
27C2×D212—41no—
28C2×C4×D53—41no—
29Q16×C53—81no—
30Q16×C53—81no—
31M16×C53—81no—
32C2×C8×C53—21no—
33C4⋊C4×C53—41no—
34C4⋊C4×C53—41no—
35C42×C53C22⋊C411yescharacteristic
36#11719921—42no—
37#11719921—41no—
38C4⋊Dic53—21no—
39C4×Dic53—41no—
40C2×#88322—41no—
41#11719908—81no—
42C22×D53—41no—
43Q8×C53—81no—
44Q8×C53—81no—
45C2×C4×C53—21no—
46C2×C4×C53—41no—
47C2×C4×C53—41no—
48C2×C4×C53#9911yescharacteristic
49C2×Dic53—41no—
50D212—41no—
51D212—41no—
52D212—161no—
53C4×D53—81no—
54C4×D53—81no—
55C8×C53—21no—
56C8×C53—21no—
57C8×C53—161no—
58#88322—81no—
59#7521—531nomaximal, Sylow
60C22×C53AΣL(1,𝔽16)11yescharacteristic
61C2×D53—81no—
62C2×D53—81no—
63C2×D53—81no—
64C4×C53—21no—
65C4×C53—21no—
66C4×C53—21no—
67C4×C53—81no—
68C4×C53—81no—
69Dic53—81no—
70#1634—531no—
71#1411D5311yescharacteristic, complemented
72#1358—531no—
73C2×C53—21no—
74C2×C53#140211yescharacteristic
75D53—161no—
76#555—531no—
77#494C2×D5311yescharacteristic
78#476—1061no—
79#364—1061no—
80#356—21no—
81#333—531no—
82D4⋊C8—1061no—
83C53#752111yescharacteristic, complemented, Sylow
84C2×Q16—41no—
85C4⋊Q8C4×D5311yescharacteristic
86C4⋊Q8C22⋊D5311yescharacteristic
87#124—2121no—
88#121—2121no—
89C4×D4—1061no—
90#117—2121no—
91M32—2121no—
92#106—2121no—
93C4⋊C8—1061no—
94#102—2121no—
95#101—41no—
96C4×C8D21211yescharacteristic
97C2×Q8—21no—
98C2×Q8—21no—
99C2×D4—1061no—
100C2×D4—2121no—
101C22×C4—2121no—
102Q16—81no—
103Q16—81no—
104M16—81no—
105C2×C8—21no—
106C2×C8—2121no—
107C4⋊C4—41no—
108C4⋊C4—41no—
109C4⋊C4—1061no—
110C22⋊C4—2122no—
111C22⋊C4—2121no—
112C42—2121no—
113C42#1171992111yescharacteristic
114C16—4241no—
115C23—2121no—
116Q8—81no—
117Q8—81no—
118D4—8481no—
119D4—2121no—
120D4—2121no—
121C2×C4—21no—
122C2×C4—41no—
123C2×C4—41no—
124C2×C4—4241no—
125C2×C4—4241no—
126C2×C4—2121no—
127C2×C4#1313313111yescharacteristic
128C8—21no—
129C8—21no—
130C8—161no—
131C8—4241no—
132C22—4241no—
133C22—4241no—
134C22—4241no—
135C22#1681040911yescharacteristic
136C4—21no—
137C4—21no—
138C4—21no—
139C4—81no—
140C4—81no—
141C4—4241no—
142C2—21no—
143C2—8481no—
144C2#2027285911yescharacteristic
subgroupG/H
centerC2#20272859
commutatorC4⋊Q8×C53C2×C4
Frattini#494C2×D53
FittingC53×#1411C2
socleC2×C53#1402
prime radical#16811339C4
cosocleC53×#494C22
hypercenter#1411D53
hypocenterC53#7521
LCSG ▷ C4⋊Q8×C53 ▷ C42×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #494 ◁ #1411

Complements

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

NH ≅ G/Ntype
2. #20272754C2semidirect
3. C53×#1411C2semidirect
71. #1411D53semidirect
83. C53#7521semidirect

Maximal subgroups

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

subgroupindex
#202738582
#202727542
C53×#14112
#752153