Finite Group Databasebeta

← #27271914

Subgroups — #27271914

Order 13568.

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

normal subgroups only

This group has 196 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#20272882C211yesmaximal, characteristic, complemented
2#20272756C211yesmaximal, characteristic, complemented
3C53×#1629C211yesmaximal, characteristic
4#16810444—21no—
5#16810422—21no—
6D42×C53C411yescharacteristic
7C53×#492C2211yescharacteristic
8C53×#476—21no—
9C53×#331C411yescharacteristic
10C53×D4⋊C8—21no—
11#13133157—41no—
12#13133136—41no—
13C22×D4×C53—41no—
14C2×SD16×C53—41no—
15C4⋊Q8×C53D411yescharacteristic
16#127×C53C2×C411yescharacteristic
17#121×C53—41no—
18#120×C53—41no—
19C4×D4×C53—21no—
20#106×C53—41no—
21C4⋊C8×C53—21no—
22#102×C53—41no—
23C4×C8×C53D411yescharacteristic
24C24×C53—41no—
25C2×Q8×C53—21no—
26C2×D4×C53—21no—
27C2×D4×C53—21no—
28C2×D4×C53—41no—
29C2×D4×C53—42no—
30C2×D4×C53—82no—
31C2×D4×C53—82no—
32C22×C4×C53—41no—
33SD16×C53—81no—
34SD16×C53—81no—
35C2×C8×C53—21no—
36C2×C8×C53—41no—
37C4⋊C4×C53—21no—
38C4⋊C4×C53—41no—
39C22⋊C4×C53—41no—
40C22⋊C4×C53—41no—
41C42×C53C22⋊C411yescharacteristic
42#11719917—81no—
43#11719908—81no—
44C23×C53—41no—
45C23×C53—41no—
46C23×C53—84no—
47C23×C53—161no—
48Q8×C53—81no—
49D4×C53—22no—
50D4×C53—41no—
51D4×C53—81no—
52D4×C53—81no—
53D4×C53—81no—
54D4×C53—84no—
55C2×C4×C53—21no—
56C2×C4×C53—41no—
57C2×C4×C53—41no—
58C2×C4×C53—81no—
59C2×C4×C53—81no—
60C2×C4×C53#9911yescharacteristic
61C8×C53—21no—
62C8×C53—21no—
63C8×C53—81no—
64#88322—161no—
65#7517—531nomaximal, Sylow
66C22×C53—42no—
67C22×C53—81no—
68C22×C53—82no—
69C22×C53—81no—
70C22×C53—81no—
71C22×C53—162no—
72C22×C53—162no—
73C22×C53AΣL(1,𝔽16)11yescharacteristic
74C4×C53—21no—
75C4×C53—21no—
76C4×C53—21no—
77C4×C53—81no—
78C4×C53—81no—
79#1629D5311yescharacteristic
80#1410—531no—
81#1359—531no—
82C2×C53—21no—
83C2×C53—82no—
84C2×C53—161no—
85C2×C53#140211yescharacteristic
86D42Dic5311yescharacteristic
87#492C2×D5311yescharacteristic
88#476—21no—
89#364—1061no—
90#356—1061no—
91#331Dic5311yescharacteristic
92D4⋊C8—21no—
93C53#751711yescharacteristic, complemented, Sylow
94C22×D4—41no—
95C2×SD16—41no—
96C4⋊Q8C22⋊D5311yescharacteristic
97#127C2×Dic5311yescharacteristic
98#121—41no—
99#120—41no—
100C4×D4—21no—
101M32—2121no—
102#106—41no—
103C4⋊C8—21no—
104#102—41no—
105#101—2121no—
106C4×C8C22⋊D5311yescharacteristic
107C24—41no—
108C2×Q8—21no—
109C2×D4—21no—
110C2×D4—21no—
111C2×D4—41no—
112C2×D4—42no—
113C2×D4—82no—
114C2×D4—82no—
115C22×C4—41no—
116SD16—81no—
117SD16—81no—
118M16—4241no—
119C2×C8—21no—
120C2×C8—41no—
121C4⋊C4—21no—
122C4⋊C4—41no—
123C22⋊C4—41no—
124C22⋊C4—41no—
125C42C22⋊Dic5311yescharacteristic
126C16—4241no—
127C23—41no—
128C23—41no—
129C23—84no—
130C23—161no—
131Q8—81no—
132D4—22no—
133D4—41no—
134D4—81no—
135D4—81no—
136D4—81no—
137D4—84no—
138C2×C4—21no—
139C2×C4—41no—
140C2×C4—41no—
141C2×C4—81no—
142C2×C4—81no—
143C2×C4#1313313211yescharacteristic
144C8—21no—
145C8—21no—
146C8—81no—
147C8—8481no—
148C22—42no—
149C22—81no—
150C22—82no—
151C22—81no—
152C22—81no—
153C22—162no—
154C22—162no—
155C22#1681041011yescharacteristic
156C4—21no—
157C4—21no—
158C4—21no—
159C4—81no—
160C4—81no—
161C2—21no—
162C2—82no—
163C2—161no—
164C2#2027286011yescharacteristic
subgroupG/H
centerC2#20272860
commutator#127×C53C2×C4
Frattini#492C2×D53
FittingC53×#1629C2
socleC2×C53#1402
prime radicalD42×C53C4
cosocleC53×#492C22
hypercenter#1629D53
hypocenterC53#7517
LCSG ▷ #127×C53 ▷ C42×C53 ▷ C2×C4×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ C42 ◁ #492 ◁ #1629

Complements

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

NH ≅ G/Ntype
1. #20272882C2semidirect
2. #20272756C2semidirect
93. C53#7517semidirect

Maximal subgroups

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

subgroupindex
#202728822
#202727562
C53×#16292
#751753