Finite Group Databasebeta

← #27271911

Subgroups — #27271911

Order 13568.

162 conjugacy classes of subgroups; only the 27 normal subgroups are shown.

normal subgroups only

This group has 162 conjugacy classes of subgroups — too many to draw. Only its 27 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#20272878C211yesmaximal, characteristic
2#20272752C211yesmaximal, characteristic
3C53×#1620C211yesmaximal, characteristic
4#16810444—21no—
5#16810416—21no—
6C53×#559C411yescharacteristic, complemented
7C53×#493C2211yescharacteristic
8C53×#460—21no—
9C53×#331C411yescharacteristic, complemented
10C53×Q8⋊C8—21no—
11#13133157—41no—
12#13133132—41no—
13C2×SD16×C53—41no—
14C2×D8×C53—41no—
15#127×C53C2×C411yescharacteristic
16#127×C53D411yescharacteristic
17#124×C53—41no—
18#123×C53—41no—
19C4×Q8×C53—21no—
20C4×D4×C53—41no—
21C4⋊C8×C53—21no—
22#102×C53—41no—
23C4×C8×C53D411yescharacteristic
24C2×Q8×C53—21no—
25C2×D4×C53—21no—
26C2×D4×C53—21no—
27C2×D4×C53—41no—
28C2×D4×C53—41no—
29C22×C4×C53—41no—
30SD16×C53—82no—
31D8×C53—81no—
32D8×C53—81no—
33C2×C8×C53—21no—
34C2×C8×C53—41no—
35C4⋊C4×C53—21no—
36C4⋊C4×C53—41no—
37C22⋊C4×C53—42no—
38C22⋊C4×C53—41no—
39C42×C53—41no—
40C42×C53C22⋊C411yescharacteristic
41C22⋊Dic53—81no—
42#11719908—81no—
43C23×C53—41no—
44C23×C53—41no—
45Q8×C53—41no—
46Q8×C53—41no—
47D4×C53—81no—
48D4×C53—81no—
49D4×C53—81no—
50D4×C53—81no—
51D4×C53—81no—
52D4×C53—81no—
53C2×C4×C53—21no—
54C2×C4×C53—41no—
55C2×C4×C53—41no—
56C2×C4×C53—161no—
57C2×C4×C53#9911yescharacteristic
58C2×Dic53—161no—
59C8×C53—21no—
60C8×C53—21no—
61C8×C53—81no—
62#7516—531nomaximal, Sylow
63C22×C53—82no—
64C22×C53—81no—
65C22×C53—81no—
66C22×C53—161no—
67C22×C53AΣL(1,𝔽16)11yescharacteristic
68C4×C53—21no—
69C4×C53—21no—
70C4×C53—21no—
71C4×C53—81no—
72C4×C53—81no—
73Dic53—321no—
74#1620D5311yescharacteristic
75#1408—531no—
76#1357—531no—
77C2×C53—21no—
78C2×C53—161no—
79C2×C53—161no—
80C2×C53#140211yescharacteristic
81#559Dic5311yescharacteristic, complemented
82#493C2×D5311yescharacteristic
83#460—21no—
84#364—1061no—
85#353—1061no—
86#331Dic5311yescharacteristic, complemented
87Q8⋊C8—21no—
88C53#751611yescharacteristic, complemented, Sylow
89C2×SD16—41no—
90C2×D8—41no—
91#127C2×Dic5311yescharacteristic
92#127C22⋊D5311yescharacteristic
93#124—41no—
94#123—41no—
95C4×Q8—21no—
96C4×D4—41no—
97M32—2121no—
98C4⋊C8—21no—
99#102—41no—
100#99—2121no—
101C4×C8C22⋊D5311yescharacteristic
102C2×Q8—21no—
103C2×D4—21no—
104C2×D4—21no—
105C2×D4—41no—
106C2×D4—41no—
107C22×C4—41no—
108SD16—82no—
109D8—81no—
110D8—81no—
111C2×C8—21no—
112C2×C8—41no—
113C4⋊C4—21no—
114C4⋊C4—41no—
115C22⋊C4—42no—
116C22⋊C4—41no—
117C22⋊C4—4241no—
118C42—41no—
119C42C22⋊Dic5311yescharacteristic
120C16—4241no—
121C23—41no—
122C23—41no—
123Q8—41no—
124Q8—41no—
125D4—81no—
126D4—81no—
127D4—81no—
128D4—81no—
129D4—81no—
130D4—81no—
131C2×C4—21no—
132C2×C4—41no—
133C2×C4—41no—
134C2×C4—161no—
135C2×C4—8481no—
136C2×C4#1313313211yescharacteristic
137C8—21no—
138C8—21no—
139C8—81no—
140C22—82no—
141C22—81no—
142C22—81no—
143C22—161no—
144C22#1681041011yescharacteristic
145C4—21no—
146C4—21no—
147C4—21no—
148C4—81no—
149C4—81no—
150C4—16961no—
151C2—21no—
152C2—161no—
153C2—161no—
154C2#2027286011yescharacteristic
subgroupG/H
centerC2#20272860
commutator#127×C53C2×C4
Frattini#493C2×D53
FittingC53×#1620C2
socleC2×C53#1402
prime radical, cosocleC53×#493C22
hypercenter#1620D53
hypocenterC53#7516
LCSG ▷ #127×C53 ▷ C42×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #493 ◁ #1620

Complements

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

NH ≅ G/Ntype
6. C53×#559C4semidirect
9. C53×#331C4semidirect
81. #559Dic53semidirect
86. #331Dic53semidirect
88. C53#7516semidirect

Maximal subgroups

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

subgroupindex
#202728782
#202727522
C53×#16202
#751653