Finite Group Databasebeta

← #27281634

Subgroups — #27281634

Order 13568.

422 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 99 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#20285095C211yesmaximal, characteristic
2#20273441C211yesmaximal, characteristic
3#20273265C212yesmaximal
4#20272579C212yesmaximal
5C53×#2381C211yesmaximal, characteristic
6#16811180C2211yescharacteristic
7#16810991—22no—
8#16810973C414yescomplemented
9#16810680C2212yes—
10#16810632—22no—
11C4×#11719917C2211yescharacteristic
12#16810372—22no—
13#16810359—24no—
14C2×C4⋊Q8×C53C2211yescharacteristic
15C53×#391—22no—
16C53×#387—24no—
17C53×#383C2212yes—
18C2×#11719917—21no—
19#13133217C2×C412yes—
20#13133209—21no—
21#13133196—22no—
22C4×Dic106—22no—
23#13133176—21no—
24C4⋊#88322—22no—
25#13133141—28no—
26C22⋊#88322—22no—
27#13133128—21no—
28#13133128—22no—
29C4×#88322C2×C412yes—
30C22×Q8×C53D412yes—
31C4⋊Q8×C53—22no—
32C4⋊Q8×C53C2×C412yes—
33C4⋊Q8×C53D412yes—
34C2×C4⋊C4×C53—22no—
35C2×C4⋊C4×C53—22no—
36C2×C42×C53—22no—
37C2×C42×C53C2311yescharacteristic
38#95×C53—24no—
39#95×C53—22no—
40#95×C53—22no—
41#95×C53D414yes—
42C2×Q8×C53—22no—
43C2×Q8×C53—42no—
44C2×Q8×C53—42no—
45C2×Q8×C53C22⋊C414yes—
46C22×C4×C53—22no—
47C22×C4×C53—24no—
48C22×C4×C53C2×D411yescharacteristic
49C22×C4×C53C2×D412yes—
50C2×Dic106—22no—
51C4⋊C4×C53—24no—
52C4⋊C4×C53—24no—
53C4⋊C4×C53—82no—
54C42×C53—82no—
55C42×C53C22×C411yescharacteristic
56C42×C53C2×D411yescharacteristic
57C42×C53#4112yes—
58C22⋊Dic53—41no—
59C4⋊Dic53—22no—
60C4⋊Dic53—22no—
61#11719919—42no—
62C4×Dic53—41no—
63#11719917—42no—
64C2×#88322—22no—
65C2×#88322—42no—
66C23×C53#12011yescharacteristic
67Q8×C53—44no—
68Q8×C53—82no—
69C2×C4×C53—22no—
70C2×C4×C53—24no—
71C2×C4×C53—42no—
72C2×C4×C53—42no—
73C2×C4×C53—42no—
74C2×C4×C53—48no—
75C2×C4×C53—44no—
76C2×C4×C53C2×C22⋊C411yescharacteristic
77C2×C4×C53C4×D412yes—
78C2×C4×C53#12112yes—
79C2×C4×C53#12311yescharacteristic
80C2×Dic53—42no—
81Dic106—82no—
82#88322—42no—
83#88322—82no—
84#9426—531nomaximal, Sylow
85C22×C53—22no—
86C22×C53#38611yescharacteristic
87C22×C53#45512yes—
88C4×C53—21no—
89C4×C53—21no—
90C4×C53—22no—
91C4×C53—22no—
92C4×C53—44no—
93C4×C53—84no—
94Dic53—82no—
95#3139—531no—
96#2381D5311yescharacteristic
97#1558—531no—
98#1515—532no—
99#1288—532no—
100C2×C53—22no—
101C2×C53#188511yescharacteristic
102C2×C53#190512yes—
103#533—531no—
104C2×C4⋊Q8C2×D5311yescharacteristic
105#491—1062no—
106#487—534no—
107#432—532no—
108#419—1062no—
109C4×M16—531no—
110#391—22no—
111#387—24no—
112#383C2×D5312yes—
113#333—1062no—
114#328—1064no—
115C53#942611yescharacteristic, complemented, Sylow
116C22×Q8C22⋊D5312yes—
117C2×M16—1061no—
118C4⋊Q8—22no—
119C4⋊Q8—532no—
120C4⋊Q8C4×D5312yes—
121C4⋊Q8D21212yes—
122#125—1061no—
123C22⋊Q8—1062no—
124C4×Q8—1062no—
125#117—1061no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4—22no—
128C2×C42—22no—
129C2×C42C22×D5311yescharacteristic
130C4⋊C8—1062no—
131#103—1068no—
132C22⋊C8—1062no—
133#97—1061no—
134#97—1062no—
135C4×C8—532no—
136#95—24no—
137#95—22no—
138#95—22no—
139#95C22⋊D5314yes—
140C2×Q8—22no—
141C2×Q8—42no—
142C2×Q8—42no—
143C2×Q8—1062no—
144C2×Q8#1171992114yes—
145C22×C4—22no—
146C22×C4—24no—
147C22×C4C2×C22⋊D5311yescharacteristic
148C22×C4C2×C22⋊D5312yes—
149M16—2122no—
150C2×C8—1062no—
151C2×C8—2122no—
152C4⋊C4—24no—
153C4⋊C4—24no—
154C4⋊C4—82no—
155C4⋊C4—1062no—
156C4⋊C4—1062no—
157C4⋊C4—2122no—
158C22⋊C4—2121no—
159C42—82no—
160C42—2121no—
161C42C2×C4×D5311yescharacteristic
162C42C2×D21211yescharacteristic
163C42#1171994512yes—
164C23#1313318611yescharacteristic
165Q8—44no—
166Q8—82no—
167Q8—4242no—
168C2×C4—22no—
169C2×C4—24no—
170C2×C4—42no—
171C2×C4—42no—
172C2×C4—42no—
173C2×C4—48no—
174C2×C4—44no—
175C2×C4—2122no—
176C2×C4C2×#1171992111yescharacteristic
177C2×C4C4×C22⋊D5312yes—
178C2×C4#1313319112yes—
179C2×C4#1313320011yescharacteristic
180C8—4242no—
181C8—2122no—
182C22—22no—
183C22#1681050211yescharacteristic
184C22#1681080112yes—
185C4—22no—
186C4—22no—
187C4—21no—
188C4—21no—
189C4—44no—
190C4—84no—
191C4—4242no—
192C2—22no—
193C2#2027510311yescharacteristic
194C2#2027520912yes—
subgroupG/H
centerC22#16810502
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2381C2
socleC22×C53#386
prime radicalC23×C53#120
cosocleC2×C42×C53C23
hypercenter#2381D53
hypocenterC53#9426
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2381

Complements

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

NH ≅ G/Ntype
8. #16810973C4semidirect
115. C53#9426semidirect

Maximal subgroups

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

subgroupindex
#202850952
#202734412
#202732652
#202725792
C53×#23812
#942653