Finite Group Databasebeta

← #27284945

Subgroups — #27284945

Order 13568.

566 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 105 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#20284746C212yesmaximal
2#20275547C211yesmaximal, characteristic
3#20273144C212yesmaximal, complemented
4#20272544C211yesmaximal, characteristic, complemented
5C53×#2399C211yesmaximal, characteristic, complemented
6#16811173C2211yescharacteristic
7#16810941—22no—
8#16810908—22no—
9#16810823—22no—
10#16810762—22no—
11C4×#11719924—24no—
12#16810676C2212yes—
13#16810647C2211yescharacteristic
14C2×#13133145—22no—
15Q8⋊#88322—24no—
16C2×C4×Q8×C53C2212yes—
17C53×#398—22no—
18C53×#395—22no—
19C4×C4⋊C4×C53—22no—
20C53×#376C2211yescharacteristic
21C2×#11719945—21no—
22C2×#11719924—24no—
23C2×#11719917—21no—
24#13133217D411yescharacteristic
25#13133214D411yescharacteristic
26#13133205—21no—
27#13133191—22no—
28C4×D212—22no—
29#13133148—24no—
30C4⋊#88322—22no—
31C4⋊#88322—22no—
32#13133145—44no—
33#13133141—24no—
34#13133137—24no—
35C22⋊#88322—22no—
36C4×#88322D414yes—
37C22×Q8×C53D412yes—
38C4×Q8×C53—22no—
39C4×Q8×C53—22no—
40C4×Q8×C53—22no—
41C4×Q8×C53—22no—
42C2×C4⋊C4×C53—24no—
43C2×C4⋊C4×C53—22no—
44C2×C4⋊C4×C53D412yes—
45C2×C42×C53—22no—
46C2×C42×C53D412yes—
47C2×C42×C53C2311yescharacteristic
48#95×C53—22no—
49#95×C53—22no—
50#95×C53—22no—
51C2×Q8×C53—22no—
52C2×Q8×C53—22no—
53C2×Q8×C53—44no—
54C22×C4×C53—22no—
55C22×C4×C53—22no—
56C22×C4×C53—22no—
57C22×C4×C53C2×D412yes—
58C22×C4×C53C2×D411yescharacteristic
59C2×C22⋊D53—41no—
60#11719945—42no—
61#11719945—42no—
62C2×D212—22no—
63C2×D212—21no—
64C2×C4×D53—41no—
65C2×Dic106—21no—
66C4⋊C4×C53—22no—
67C4⋊C4×C53—22no—
68C4⋊C4×C53—48no—
69C4⋊C4×C53—42no—
70C4⋊C4×C53—42no—
71C42×C53—22no—
72C42×C53—42no—
73C42×C53—42no—
74C42×C53SD1614yescomplemented
75C42×C53C2×D411yescharacteristic
76C42×C53C2×D412yes—
77C42×C53#4111yescharacteristic
78#11719924—48no—
79#11719921—42no—
80C4⋊Dic53—22no—
81#11719917—42no—
82C2×#88322—24no—
83C2×#88322—41no—
84C23×C53#12011yescharacteristic
85C22×D53—41no—
86Q8×C53—24no—
87Q8×C53—24no—
88Q8×C53—42no—
89Q8×C53—42no—
90C2×C4×C53—24no—
91C2×C4×C53—24no—
92C2×C4×C53—42no—
93C2×C4×C53—42no—
94C2×C4×C53—44no—
95C2×C4×C53—42no—
96C2×C4×C53—44no—
97C2×C4×C53—42no—
98C2×C4×C53#12012yes—
99C2×C4×C53#12111yescharacteristic
100C2×C4×C53#12112yes—
101C2×C4×C53#12711yescharacteristic
102C2×C4×C53C2×SD1612yes—
103C2×C4×C53#13712yes—
104C22⋊D53—82no—
105C2×Dic53—41no—
106D212—44no—
107D212—42no—
108C4×D53—81no—
109C4×D53—81no—
110Dic106—42no—
111#88322—44no—
112#88322—81no—
113#10014—531nomaximal, Sylow
114C22×C53—22no—
115C22×C53#39211yescharacteristic
116C22×C53#44812yes—
117C2×D53—82no—
118C2×D53—81no—
119C4×C53—22no—
120C4×C53—22no—
121C4×C53—44no—
122C4×C53—44no—
123C4×C53—82no—
124C4×C53#46112yes—
125C4×C53#49211yescharacteristic
126C4×C53#49711yescharacteristic
127Dic53—81no—
128#3103—532no—
129#2399D5311yescharacteristic, complemented
130#1963—531no—
131#1487—532no—
132#1274—531no—
133C2×C53—22no—
134C2×C53#201012yes—
135C2×C53#201311yescharacteristic
136D53—161no—
137#532—531no—
138C2×C4×Q8C2×D5312yes—
139#481—1062no—
140#475—1062no—
141#460—1062no—
142#448—1062no—
143C4×SD16—1064no—
144#431—532no—
145#423—531no—
146C2×#104—1062no—
147#398—22no—
148#395—22no—
149C4×C4⋊C4—22no—
150#376C2×D5311yescharacteristic
151Q8⋊C8—1064no—
152C53#1001411yescharacteristic, complemented, Sylow
153C2×#41—1061no—
154C22×Q8C22⋊D5312yes—
155C2×SD16—1064no—
156C2×M16—1061no—
157C4⋊Q8—531no—
158#127—531no—
159#124—1061no—
160#121—1062no—
161C4×Q8—22no—
162C4×Q8—22no—
163C4×Q8—22no—
164C4×Q8—22no—
165C4×D4—1062no—
166C2×C4⋊C4—24no—
167C2×C4⋊C4—22no—
168C2×C4⋊C4C22⋊D5312yes—
169C2×C42—22no—
170C2×C42C22⋊D5312yes—
171C2×C42C22×D5311yescharacteristic
172#106—1064no—
173C4⋊C8—1062no—
174C4⋊C8—1062no—
175#104—2124no—
176#103—1064no—
177#102—1064no—
178C22⋊C8—1062no—
179C4×C8—534no—
180#95—22no—
181#95—22no—
182#95—22no—
183#41—2122no—
184#41—2122no—
185C2×Q8—22no—
186C2×Q8—22no—
187C2×Q8—44no—
188C2×Q8—1061no—
189C2×D4—1062no—
190C2×D4—1061no—
191C2×D4—2121no—
192C22×C4—22no—
193C22×C4—22no—
194C22×C4—22no—
195C22×C4—2121no—
196C22×C4C2×C22⋊D5312yes—
197C22×C4C2×C22⋊D5311yescharacteristic
198SD16—2128no—
199M16—2122no—
200C2×C8—1064no—
201C2×C8—2121no—
202C4⋊C4—22no—
203C4⋊C4—22no—
204C4⋊C4—48no—
205C4⋊C4—42no—
206C4⋊C4—42no—
207C4⋊C4—1062no—
208C22⋊C4—2122no—
209C42—22no—
210C42—42no—
211C42—42no—
212C42#1171992414yescomplemented
213C42D4×D5311yescharacteristic
214C42D4×D5312yes—
215C42#1171994711yescharacteristic
216C23—2121no—
217C23#1313318611yescharacteristic
218Q8—24no—
219Q8—24no—
220Q8—42no—
221Q8—42no—
222Q8—2122no—
223D4—4242no—
224D4—2124no—
225D4—2122no—
226C2×C4—24no—
227C2×C4—24no—
228C2×C4—42no—
229C2×C4—44no—
230C2×C4—42no—
231C2×C4—42no—
232C2×C4—44no—
233C2×C4—42no—
234C2×C4—4241no—
235C2×C4—4241no—
236C2×C4—2121no—
237C2×C4#1313318512yes—
238C2×C4#1313318811yescharacteristic
239C2×C4#1313318912yes—
240C2×C4#1313321311yescharacteristic
241C2×C4C2×#1171992412yes—
242C2×C4#1313324812yes—
243C8—4241no—
244C8—2124no—
245C22—22no—
246C22—4242no—
247C22—4241no—
248C22#1681053111yescharacteristic
249C22#1681076212yes—
250C4—22no—
251C4—22no—
252C4—44no—
253C4—44no—
254C4—82no—
255C4—4241no—
256C4#1681082512yes—
257C4#1681099711yescharacteristic
258C4#1681101511yescharacteristic
259C2—22no—
260C2—8481no—
261C2#2027578012yes—
262C2#2027579811yescharacteristic
subgroupG/H
centerC2×C4#13133213
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#2399C2
socleC22×C53#392
prime radical#16811173C22
hypercenter#2399D53
hypocenterC53#10014
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2399

Complements

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

NH ≅ G/Ntype
3. #20273144C2semidirect
4. #20272544C2semidirect
5. C53×#2399C2semidirect
74. C42×C53SD16semidirect
129. #2399D53semidirect
152. C53#10014semidirect
212. C42#11719924semidirect

Maximal subgroups

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

subgroupindex
#202847462
#202755472
#202731442
#202725442
C53×#23992
#1001453