Finite Group Databasebeta

← #27279031

Subgroups — #27279031

Order 13568.

578 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 115 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
1C2×#16811018C211yesmaximal, characteristic
2#20275687C212yesmaximal, complemented
3C2×#16810363C211yesmaximal, characteristic
4#20272596C212yesmaximal, complemented
5C53×#2379C211yesmaximal, characteristic
6C22×#11719925—21no—
7C22×#11719923—21no—
8C2×#13133217C2211yescharacteristic
9#16811018—22no—
10#16811018C414yescomplemented
11C2×#13133128C2212yes—
12C2×#13133128C2211yescharacteristic
13#16810385—22no—
14#16810385—22no—
15#16810363—22no—
16C2×#124×C53C2211yescharacteristic
17C53×#391—21no—
18C53×#388—22no—
19C53×#386—21no—
20C53×#385—22no—
21C53×#382C2212yes—
22C22×Dic106—21no—
23C2×#11719925—24no—
24C2×#11719925—42no—
25C2×#11719925—42no—
26C2×#11719923—24no—
27C2×#11719923—42no—
28C2×#11719923—42no—
29C22×#88322—21no—
30C22×#88322—22no—
31#13133217—21no—
32#13133217—22no—
33#13133217C2×C412yescomplemented
34C2×C4⋊Dic53—22no—
35#13133128—21no—
36#13133128—22no—
37#13133128—22no—
38#13133128C2×C412yescomplemented
39C22×Q8×C53D411yescharacteristic
40C22×D4×C53D411yescharacteristic
41C23×C4×C53—21no—
42#124×C53—22no—
43#124×C53C2×C412yes—
44#124×C53D412yes—
45C2×C4⋊C4×C53—21no—
46C2×C4⋊C4×C53D412yes—
47C2×C4⋊C4×C53D412yes—
48C2×C22⋊C4×C53—22no—
49C2×C22⋊C4×C53—21no—
50C2×C22⋊C4×C53—22no—
51C2×C42×C53—21no—
52C2×C42×C53C2311yescharacteristic
53#95×C53—22no—
54#95×C53—22no—
55#95×C53—22no—
56C24×C53—21no—
57C2×Q8×C53—21no—
58C2×Q8×C53—41no—
59C2×Q8×C53—41no—
60C2×Q8×C53C22⋊C412yes—
61C2×D4×C53—21no—
62C2×D4×C53—41no—
63C2×D4×C53—41no—
64C2×D4×C53C22⋊C412yes—
65C22×C4×C53—21no—
66C22×C4×C53—22no—
67C22×C4×C53—22no—
68C22×C4×C53—81no—
69C22×C4×C53—82no—
70C22×C4×C53C2×D411yescharacteristic
71C22×C4×C53C2×D412yes—
72C22×Dic53—41no—
73C2×Dic106—22no—
74C2×Dic106—41no—
75C2×Dic106—41no—
76C2×Dic106—41no—
77C2×Dic106—41no—
78C2×Dic106—41no—
79C4⋊C4×C53—42no—
80C4⋊C4×C53—42no—
81C4⋊C4×C53—81no—
82C22⋊C4×C53—24no—
83C22⋊C4×C53—24no—
84C22⋊C4×C53—81no—
85C22⋊C4×C53—82no—
86C42×C53—81no—
87C42×C53C22×C411yescharacteristic
88C42×C53C2×D411yescharacteristic
89C42×C53#4112yes—
90#11719925—48no—
91#11719923—48no—
92C4⋊Dic53—24no—
93C4⋊Dic53—42no—
94C2×#88322—22no—
95C2×#88322—22no—
96C2×#88322—22no—
97C2×#88322—24no—
98C2×#88322—42no—
99C2×#88322—42no—
100C23×C53—22no—
101C23×C53—41no—
102C23×C53—41no—
103C23×C53—42no—
104C23×C53—42no—
105C23×C53#12011yescharacteristic
106Q8×C53—42no—
107Q8×C53—81no—
108D4×C53—42no—
109D4×C53—81no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—22no—
113C2×C4×C53—22no—
114C2×C4×C53—22no—
115C2×C4×C53—41no—
116C2×C4×C53—41no—
117C2×C4×C53—42no—
118C2×C4×C53—42no—
119C2×C4×C53—42no—
120C2×C4×C53—42no—
121C2×C4×C53—42no—
122C2×C4×C53—42no—
123C2×C4×C53—84no—
124C2×C4×C53C2×C22⋊C411yescharacteristic
125C2×C4×C53C4×D412yes—
126C2×C4×C53#12112yes—
127C2×C4×C53#12311yescharacteristic
128C2×Dic53—42no—
129C2×Dic53—81no—
130C2×Dic53—81no—
131Dic106—42no—
132Dic106—42no—
133Dic106—81no—
134Dic106—81no—
135#88322—24no—
136#88322—44no—
137#9003—531nomaximal, Sylow
138C22×C53—44no—
139C22×C53—42no—
140C22×C53—81no—
141C22×C53—81no—
142C22×C53—82no—
143C22×C53#38611yescharacteristic
144C22×C53#45311yescharacteristic
145C22×C53#45311yescharacteristic
146C22×C53#45512yes—
147C22×C53#45612yes—
148C4×C53—22no—
149C4×C53—22no—
150C4×C53—21no—
151C4×C53—21no—
152C4×C53—42no—
153C4×C53—82no—
154C4×C53—82no—
155Dic53—82no—
156C2×#497—531no—
157#2379D5311yescharacteristic
158#1993—532no—
159C2×#329—531no—
160#1295—532no—
161C2×C53—82no—
162C2×C53#165911yescharacteristic
163C2×C53#166011yescharacteristic
164C2×C53#166112yes—
165C2×C53#188212yes—
166C2×C53#188711yescharacteristic
167C22×Q16—1061no—
168C22×SD16—1061no—
169C2×C4⋊Q8—531no—
170C2×#124C2×D5311yescharacteristic
171#497—534no—
172#497—1062no—
173C2×#97—531no—
174C2×#97—532no—
175#391—21no—
176#388—22no—
177#386—21no—
178#385—22no—
179#382C2×D5312yes—
180#340—1062no—
181#340—1062no—
182#329—1062no—
183C53#900311yescharacteristic, complemented, Sylow
184C22×Q8—1061no—
185C22×Q8C22⋊D5311yescharacteristic
186C22×D4C22⋊D5311yescharacteristic
187C23×C4—21no—
188C2×Q16—1064no—
189C2×Q16—2122no—
190C2×Q16—2122no—
191C2×SD16—1064no—
192C2×SD16—2122no—
193C2×SD16—2122no—
194C22×C8—1061no—
195C22×C8—1062no—
196C4⋊Q8—532no—
197C4⋊Q8—1061no—
198C4⋊Q8—1062no—
199#124—22no—
200#124C4×D5312yes—
201#124D21212yes—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—1062no—
204C2×C4⋊C4C22⋊D5312yes—
205C2×C4⋊C4C22⋊D5312yes—
206C2×C22⋊C4—22no—
207C2×C22⋊C4—22no—
208C2×C22⋊C4—21no—
209C2×C42—21no—
210C2×C42C22×D5311yescharacteristic
211#97—532no—
212#97—1061no—
213#97—1062no—
214#97—1062no—
215#95—22no—
216#95—22no—
217#95—22no—
218C24—21no—
219C2×Q8—21no—
220C2×Q8—41no—
221C2×Q8—41no—
222C2×Q8—1062no—
223C2×Q8—2121no—
224C2×Q8—2121no—
225C2×Q8—2121no—
226C2×Q8—2121no—
227C2×Q8—2121no—
228C2×Q8#1171992112yes—
229C2×D4—21no—
230C2×D4—41no—
231C2×D4—41no—
232C2×D4#1171992112yes—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—81no—
237C22×C4—82no—
238C22×C4—2121no—
239C22×C4C2×C22⋊D5311yescharacteristic
240C22×C4C2×C22⋊D5312yes—
241Q16—2128no—
242SD16—2128no—
243C2×C8—1062no—
244C2×C8—1062no—
245C2×C8—1062no—
246C2×C8—1064no—
247C2×C8—2122no—
248C2×C8—2122no—
249C4⋊C4—42no—
250C4⋊C4—42no—
251C4⋊C4—81no—
252C4⋊C4—1064no—
253C4⋊C4—2122no—
254C22⋊C4—24no—
255C22⋊C4—24no—
256C22⋊C4—81no—
257C22⋊C4—82no—
258C42—81no—
259C42C2×C4×D5311yescharacteristic
260C42C2×D21211yescharacteristic
261C42#1171994512yes—
262C23—22no—
263C23—41no—
264C23—41no—
265C23—42no—
266C23—42no—
267C23#1313318611yescharacteristic
268Q8—42no—
269Q8—81no—
270Q8—4241no—
271Q8—4241no—
272Q8—2122no—
273Q8—2122no—
274D4—42no—
275D4—81no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—84no—
290C2×C4—4241no—
291C2×C4—4241no—
292C2×C4—2122no—
293C2×C4C2×#1171992111yescharacteristic
294C2×C4C4×C22⋊D5312yes—
295C2×C4#1313319112yes—
296C2×C4#1313320011yescharacteristic
297C8—1064no—
298C8—2124no—
299C22—44no—
300C22—42no—
301C22—81no—
302C22—81no—
303C22—82no—
304C22#1681050211yescharacteristic
305C22#1681079111yescharacteristic
306C22#1681079111yescharacteristic
307C22#1681080112yes—
308C22#1681080612yes—
309C4—22no—
310C4—22no—
311C4—21no—
312C4—21no—
313C4—42no—
314C4—82no—
315C4—82no—
316C4—4242no—
317C2—82no—
318C2#2027402111yescharacteristic
319C2#2027402611yescharacteristic
320C2#2027403212yes—
321C2#2027508412yes—
322C2#2027511311yescharacteristic
subgroupG/H
centerC23#13133186
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2379C2
socleC23×C53#120
prime radicalC22×D4×C53D4
cosocleC2×C42×C53C23
hypercenter#2379D53
hypocenterC53#9003
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #2379

Complements

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

NH ≅ G/Ntype
2. #20275687C2semidirect
4. #20272596C2semidirect
10. #16811018C4semidirect
33. #13133217C2×C4semidirect
38. #13133128C2×C4semidirect
183. C53#9003semidirect

Maximal subgroups

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

subgroupindex
C2×#168110182
#202756872
C2×#168103632
#202725962
C53×#23792
#900353