Finite Group Databasebeta

← #27279049

Subgroups — #27279049

Order 13568.

502 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 502 conjugacy classes of subgroups — too many to draw. Only its 95 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#20285160C211yesmaximal, characteristic
2#20283786C211yesmaximal, characteristic, complemented
3#20273185C212yesmaximal, complemented
4#20272615C212yesmaximal, complemented
5C53×#1543C211yesmaximal, characteristic, complemented
6C2×#13133237—22no—
7#16811173C2211yescharacteristic
8#16811018C412yes—
9#16811015—21no—
10#16811013C412yes—
11#16811010—21no—
12#16810695—24no—
13#16810680C2212yes—
14#16810590—22no—
15C2×#13133128C2211yescharacteristic
16#16810352—24no—
17C53×#532C2211yescharacteristic
18C53×#431C2212yes—
19C53×#333—21no—
20C53×#331—21no—
21C2×#11719945—21no—
22#13133237—44no—
23#13133237—44no—
24C2×#11719925—22no—
25C2×#11719923—22no—
26C2×#11719924—22no—
27C2×#11719922—22no—
28C2×#11719917—22no—
29C22×#88322—22no—
30C22×#88322—21no—
31#13133217C2×C411yescharacteristic
32#13133214C2×C411yescharacteristic
33#13133205—21no—
34#13133191—22no—
35C4×D212—22no—
36C4⋊#88322D414yes—
37C22⋊#88322—24no—
38C22⋊#88322—22no—
39#13133128—21no—
40#13133128—22no—
41#13133128C2×C412yes—
42C2×#41×C53D412yes—
43C4⋊Q8×C53D411yescharacteristic
44#127×C53D411yescharacteristic
45#124×C53C2×C412yes—
46#121×C53—22no—
47C4×D4×C53—22no—
48C2×C42×C53C2311yescharacteristic
49C4⋊C8×C53—22no—
50C22⋊C8×C53—22no—
51C4×C8×C53—22no—
52#41×C53—42no—
53#41×C53—42no—
54C2×Q8×C53C22⋊C412yes—
55C2×D4×C53—22no—
56C2×D4×C53—22no—
57C2×D4×C53C22⋊C412yes—
58C22×C4×C53—22no—
59C22×C4×C53C2×D411yescharacteristic
60C22×C4×C53#4112yes—
61C2×C22⋊D53—41no—
62#11719945—42no—
63#11719945—42no—
64C2×D212—21no—
65C2×D212—22no—
66C2×C4×D53—41no—
67C2×Dic106—21no—
68C2×C8×C53—42no—
69C4⋊C4×C53—22no—
70C22⋊C4×C53—24no—
71C42×C53C22×C411yescharacteristic
72C42×C53C2×D411yescharacteristic
73C42×C53C2×D412yes—
74#11719925—44no—
75#11719924—44no—
76#11719923—44no—
77#11719922—44no—
78#11719921—42no—
79C4⋊Dic53—22no—
80#11719917—82no—
81C2×#88322—24no—
82C2×#88322—22no—
83C2×#88322—24no—
84C2×#88322—82no—
85C23×C53—22no—
86C23×C53#12311yescharacteristic
87C22×D53—41no—
88Q8×C53—42no—
89D4×C53—42no—
90D4×C53—82no—
91D4×C53—82no—
92C2×C4×C53—22no—
93C2×C4×C53—22no—
94C2×C4×C53—42no—
95C2×C4×C53—42no—
96C2×C4×C53—42no—
97C2×C4×C53C2×C22⋊C411yescharacteristic
98C2×C4×C53C4×D412yes—
99C2×C4×C53#12011yescharacteristic
100C2×C4×C53#12112yes—
101C22⋊D53—81no—
102C22⋊D53—81no—
103C2×Dic53—41no—
104D212—44no—
105D212—42no—
106C4×D53—82no—
107Dic106—42no—
108C8×C53—82no—
109#88322—24no—
110#88322—44no—
111#9007—531nomaximal, Sylow
112C22×C53—22no—
113C22×C53—21no—
114C22×C53—44no—
115C22×C53—42no—
116C22×C53#38611yescharacteristic
117C2×D53—82no—
118C2×D53—81no—
119C4×C53—22no—
120C4×C53—22no—
121C4×C53—21no—
122C4×C53—42no—
123C4×C53#45312yes—
124Dic53—81no—
125#3150—531no—
126#2980—531no—
127#1543D5311yescharacteristic, complemented
128#1496—532no—
129#1302—532no—
130C2×C53—41no—
131C2×C53—82no—
132C2×C53#188112yes—
133C2×C53#188711yescharacteristic
134D53—161no—
135C2×#135—1062no—
136#532—531no—
137#532C2×D5311yescharacteristic
138#497—532no—
139#497—1061no—
140#496—532no—
141#496—1061no—
142#435—1064no—
143#432—532no—
144#431C2×D5312yes—
145#408—1062no—
146C2×#97—531no—
147#333—21no—
148#331—21no—
149D4⋊C8—1064no—
150C53#900711yescharacteristic, complemented, Sylow
151C2×#41—1061no—
152C2×#41D21212yes—
153#135—2124no—
154#135—2124no—
155C2×Q16—1062no—
156C2×SD16—1062no—
157C2×SD16—1062no—
158C2×D8—1062no—
159C2×M16—1062no—
160C22×C8—1061no—
161C22×C8—1062no—
162C4⋊Q8—531no—
163C4⋊Q8C22⋊D5311yescharacteristic
164#127—531no—
165#127C22⋊D5311yescharacteristic
166#124—1061no—
167#124C4×D5312yes—
168#121—22no—
169#121—1062no—
170C4×D4—22no—
171C4×D4—1062no—
172C2×C42C22×D5311yescharacteristic
173C4⋊C8—22no—
174C4⋊C8—534no—
175C22⋊C8—22no—
176C22⋊C8—1064no—
177C22⋊C8—1062no—
178#97—532no—
179#97—1062no—
180#97—1061no—
181C4×C8—22no—
182#41—42no—
183#41—42no—
184#41—2122no—
185#41—2122no—
186C2×Q8—1061no—
187C2×Q8#1171992112yes—
188C2×D4—22no—
189C2×D4—22no—
190C2×D4—1061no—
191C2×D4—1062no—
192C2×D4—2121no—
193C2×D4#1171992112yes—
194C22×C4—22no—
195C22×C4—2121no—
196C22×C4C2×D21211yescharacteristic
197C22×C4#1171994712yes—
198Q16—2124no—
199SD16—2124no—
200SD16—2124no—
201D8—2124no—
202M16—4242no—
203C2×C8—42no—
204C2×C8—1064no—
205C2×C8—1062no—
206C2×C8—1064no—
207C2×C8—4242no—
208C4⋊C4—22no—
209C4⋊C4—1062no—
210C22⋊C4—24no—
211C22⋊C4—2122no—
212C42C2×C4×D5311yescharacteristic
213C42D4×D5312yes—
214C42C2×C22⋊D5311yescharacteristic
215C23—22no—
216C23—2121no—
217C23#1313320111yescharacteristic
218Q8—42no—
219Q8—2122no—
220D4—42no—
221D4—82no—
222D4—82no—
223D4—4241no—
224D4—4241no—
225D4—2122no—
226D4—2124no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—42no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—4242no—
233C2×C4—2121no—
234C2×C4C2×#1171992111yescharacteristic
235C2×C4#1313317712yes—
236C2×C4#1313318711yescharacteristic
237C2×C4#1313318912yes—
238C8—82no—
239C8—1064no—
240C8—2124no—
241C22—22no—
242C22—21no—
243C22—44no—
244C22—42no—
245C22—4242no—
246C22—4241no—
247C22#1681050011yescharacteristic
248C4—22no—
249C4—22no—
250C4—21no—
251C4—42no—
252C4—4241no—
253C4#1681079312yes—
254C2—41no—
255C2—82no—
256C2—8481no—
257C2#2027507512yes—
258C2#2027511111yescharacteristic
subgroupG/H
centerC2×C4#13133187
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1543C2
socleC22×C53#386
prime radical#20285160C2
cosocleC2×C42×C53C23
hypercenter#1543D53
hypocenterC53#9007
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1543

Complements

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

NH ≅ G/Ntype
2. #20283786C2semidirect
3. #20273185C2semidirect
4. #20272615C2semidirect
5. C53×#1543C2semidirect
127. #1543D53semidirect
150. C53#9007semidirect

Maximal subgroups

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

subgroupindex
#202851602
#202837862
#202731852
#202726152
C53×#15432
#900753