Finite Group Databasebeta

← #27282899

Subgroups — #27282899

Order 13568.

558 conjugacy classes of subgroups; only the 151 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 151 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
1C4×#13133205C211yesmaximal, characteristic
2#20274832C211yesmaximal, characteristic, complemented
3C2×#16810355C212yesmaximal, complemented
4C2×#16810352C212yesmaximal
5C53×#1744C211yesmaximal, characteristic, complemented
6C2×#13133205C411yescharacteristic
7C2×C4×Dic106C2211yescharacteristic
8C2×C4×D212C2211yescharacteristic
9C2×C4⋊#88322C2212yes—
10#16810529C411yescharacteristic
11#16810525C411yescharacteristic
12#16810514—22no—
13#16810489C411yescharacteristic
14C4×#11719921—22no—
15#16810378—22no—
16#16810355—24no—
17#16810352—24no—
18C2×C4×C8×C53C2212yes—
19C43×C53C2211yescharacteristic
20C53×#336—22no—
21C22×Dic106C2×C411yescharacteristic
22C22×D212C2×C411yescharacteristic
23C22×C4×D53—21no—
24C22×#88322—22no—
25#13133205—22no—
26#13133205—22no—
27C4×Dic106—22no—
28C4×Dic106—22no—
29C4×D212—22no—
30C4×D212—22no—
31C2×C4⋊Dic53C2×C411yescharacteristic
32C2×C4⋊Dic53C2×C411yescharacteristic
33C2×#11719919—21no—
34C2×#11719921—22no—
35C2×#11719921—22no—
36C2×#11719921—21no—
37C2×C4×Dic53—21no—
38C4⋊#88322—22no—
39C4⋊#88322—22no—
40#13133124—22no—
41#13133124—22no—
42C22×C8×C53—22no—
43C2×C42×C53C2×C411yescharacteristic
44C2×C42×C53C2×C411yescharacteristic
45C2×C42×C53D412yes—
46C2×C42×C53D412yes—
47C2×C42×C53C2311yescharacteristic
48C4×C8×C53—22no—
49C4×C8×C53D414yes—
50C23×D53—21no—
51C22×C4×C53C22⋊C412yes—
52C22×C4×C53C22⋊C412yes—
53C22×C4×C53C22×C411yescharacteristic
54C22×C4×C53C2×D411yescharacteristic
55C22×C4×C53C2×D411yescharacteristic
56C22×Dic53—21no—
57C22×Dic53—21no—
58C22×Dic53—21no—
59C2×D212—22no—
60C2×D212—22no—
61C2×D212—21no—
62C2×D212M1612yes—
63C2×C4×D53—41no—
64C2×C4×D53—41no—
65C2×C4×D53—44no—
66C2×Dic106—22no—
67C2×Dic106—22no—
68C2×Dic106—21no—
69C2×Dic106M1612yes—
70C2×C8×C53—24no—
71C2×C8×C53—44no—
72C42×C53—21no—
73C42×C53—21no—
74C42×C53—22no—
75C42×C53—24no—
76C42×C53—42no—
77C42×C53C2×D412yes—
78C42×C53#4112yes—
79#11719921—28no—
80#11719921—44no—
81#11719921—42no—
82C4⋊Dic53—22no—
83C4⋊Dic53—22no—
84#11719919—42no—
85C4×Dic53—41no—
86C4×Dic53—41no—
87C2×#88322—42no—
88C2×#88322—44no—
89C23×C53C2×C22⋊C411yescharacteristic
90C22×D53—22no—
91C22×D53—41no—
92C22×D53—41no—
93C22×D53—42no—
94C22×D53—42no—
95C2×C4×C53—22no—
96C2×C4×C53—22no—
97C2×C4×C53—22no—
98C2×C4×C53—22no—
99C2×C4×C53—22no—
100C2×C4×C53—22no—
101C2×C4×C53—22no—
102C2×C4×C53C4×D412yes—
103C2×C4×C53#12112yes—
104C2×C4×C53#12411yescharacteristic
105C2×C4×C53#12711yescharacteristic
106C2×C4×C53C2×M1611yescharacteristic
107C2×C4×C53C2×M1611yescharacteristic
108C2×C4×C53#13112yes—
109C2×C4×C53#13514yes—
110C2×Dic53—22no—
111C2×Dic53—41no—
112C2×Dic53—41no—
113C2×Dic53—41no—
114C2×Dic53—41no—
115C2×Dic53—42no—
116C2×Dic53—42no—
117D212—24no—
118D212—42no—
119C4×D53—82no—
120C4×D53—82no—
121Dic106—24no—
122Dic106—42no—
123C8×C53—44no—
124#88322—82no—
125#9645—531nomaximal, Sylow
126C22×C53#39011yescharacteristic
127C22×C53#40711yescharacteristic
128C22×C53#40811yescharacteristic
129C22×C53#41612yes—
130C22×C53#42112yes—
131C2×D53—42no—
132C2×D53—44no—
133C2×D53—81no—
134C2×D53—81no—
135C2×D53—82no—
136C4×C53—22no—
137C4×C53—42no—
138C4×C53—42no—
139C4×C53#43512yes—
140C4×C53#43512yes—
141C4×C53#48812yes—
142C4×C53#49512yes—
143Dic53—42no—
144Dic53—81no—
145Dic53—81no—
146C4×#124—531no—
147#1831—531no—
148#1744D5311yescharacteristic, complemented
149C2×Q8⋊C8—532no—
150C2×D4⋊C8—532no—
151C2×C53#148212yes—
152C2×C53#148812yes—
153C2×C53#195511yescharacteristic
154C2×C53#195811yescharacteristic
155C2×C53#196311yescharacteristic
156D53—82no—
157C2×#124—531no—
158C2×C4×Q8—531no—
159C2×C4×D4—531no—
160C2×C4⋊C8—532no—
161C2×C4×C8C2×D5312yes—
162#391—531no—
163#390—531no—
164#388—1062no—
165#382—531no—
166C4×C22⋊C4—1062no—
167C43C2×D5311yescharacteristic
168#336—22no—
169#336—1062no—
170Q8⋊C8—1064no—
171D4⋊C8—1064no—
172C53#964511yescharacteristic, complemented, Sylow
173C22×Q8—531no—
174C22×D4—531no—
175C23×C4—1061no—
176C22×C8—22no—
177C22×C8—1062no—
178#124—1062no—
179#124—1062no—
180C4×Q8—1062no—
181C4×Q8—1062no—
182C4×D4—1062no—
183C4×D4—1062no—
184C2×C4⋊C4—531no—
185C2×C4⋊C4—531no—
186C2×C4⋊C4—1061no—
187C2×C22⋊C4—1062no—
188C2×C22⋊C4—1061no—
189C2×C22⋊C4—1062no—
190C2×C42—1061no—
191C2×C42C4×D5311yescharacteristic
192C2×C42C4×D5311yescharacteristic
193C2×C42D21212yes—
194C2×C42C22⋊D5312yes—
195C2×C42C22×D5311yescharacteristic
196C4⋊C8—1062no—
197C4⋊C8—1062no—
198C4×C8—22no—
199C4×C8D21214yes—
200#95—1062no—
201#95—1062no—
202C24—1061no—
203C2×Q8—532no—
204C2×Q8—1061no—
205C2×Q8—1062no—
206C2×Q8—1062no—
207C2×D4—532no—
208C2×D4—1062no—
209C2×D4—1061no—
210C2×D4—1062no—
211C22×C4—1061no—
212C22×C4—1061no—
213C22×C4—1061no—
214C22×C4—2121no—
215C22×C4—2121no—
216C22×C4—2124no—
217C22×C4#1171992112yes—
218C22×C4#1171992112yes—
219C22×C4C2×C4×D5311yescharacteristic
220C22×C4C2×D21211yescharacteristic
221C22×C4C2×C22⋊D5311yescharacteristic
222C2×C8—24no—
223C2×C8—44no—
224C2×C8—2122no—
225C2×C8—2124no—
226C4⋊C4—1062no—
227C4⋊C4—1062no—
228C4⋊C4—2122no—
229C22⋊C4—1068no—
230C22⋊C4—2124no—
231C22⋊C4—2122no—
232C42—21no—
233C42—21no—
234C42—22no—
235C42—24no—
236C42—42no—
237C42—2121no—
238C42—2121no—
239C42C2×D21212yes—
240C42#1171994512yes—
241C23—1062no—
242C23—2121no—
243C23—2121no—
244C23—2122no—
245C23—2122no—
246C23C2×#1171992111yescharacteristic
247Q8—1064no—
248Q8—2122no—
249D4—1064no—
250D4—2122no—
251C2×C4—22no—
252C2×C4—22no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—22no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—1062no—
259C2×C4—4242no—
260C2×C4—4242no—
261C2×C4—2121no—
262C2×C4—2121no—
263C2×C4—2122no—
264C2×C4—2122no—
265C2×C4—2121no—
266C2×C4—2121no—
267C2×C4C4×D21212yes—
268C2×C4#1313319112yes—
269C2×C4#1313320511yescharacteristic
270C2×C4#1313321411yescharacteristic
271C2×C4M16×D5311yescharacteristic
272C2×C4M16×D5311yescharacteristic
273C2×C4#1313322512yes—
274C2×C4#1313323914yes—
275C8—44no—
276C8—4242no—
277C22—4241no—
278C22—4241no—
279C22—4242no—
280C22—2122no—
281C22—2124no—
282C22#1681052511yescharacteristic
283C22#1681058611yescharacteristic
284C22#1681059111yescharacteristic
285C22#1681062012yes—
286C22#1681064112yes—
287C4—22no—
288C4—42no—
289C4—42no—
290C4—4241no—
291C4—4241no—
292C4—2122no—
293C4#1681069612yes—
294C4#1681069612yes—
295C4#1681097812yes—
296C4#1681100712yes—
297C2—4242no—
298C2#2027312412yes—
299C2#2027315012yes—
300C2#2027551311yescharacteristic
301C2#2027552611yescharacteristic
302C2#2027554711yescharacteristic
subgroupG/H
centerC22×C4C2×D212
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1744C2
socleC23×C53C2×C22⋊C4
prime radicalC22×D212C2×C4
cosocleC2×C42×C53C23
hypercenter#1744D53
hypocenterC53#9645
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C43 ◁ #1744

Complements

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

NH ≅ G/Ntype
2. #20274832C2semidirect
3. C2×#16810355C2semidirect
5. C53×#1744C2semidirect
148. #1744D53semidirect
172. C53#9645semidirect

Maximal subgroups

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

subgroupindex
C4×#131332052
#202748322
C2×#168103552
C2×#168103522
C53×#17442
#964553