Finite Group Databasebeta

← #27281736

Subgroups — #27281736

Order 13568.

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

normal subgroups only

This group has 474 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#20279164C211yesmaximal, characteristic
2#20274792C211yesmaximal, characteristic
3#20273249C212yesmaximal
4#20272569C212yesmaximal
5C53×#2921C211yesmaximal, characteristic
6C2×#13133196C2211yescharacteristic
7#16810551—21no—
8#16810551—21no—
9#16810528C411yescharacteristic, complemented
10#16810528C411yescharacteristic, complemented
11#16810507—21no—
12#16810507—21no—
13#16810492C411yescharacteristic, complemented
14#16810488C411yescharacteristic, complemented
15#16810486C2212yes—
16C4×C22⋊Dic53C2211yescharacteristic
17#16810390—21no—
18#16810390—21no—
19#16810361—24no—
20C2×#117×C53C2211yescharacteristic
21C53×#433—24no—
22C53×#423—21no—
23C53×#407C2212yes—
24C4×M16×C53—21no—
25C22×Dic106C2×C411yescharacteristic
26#13133196—22no—
27#13133196—22no—
28C2×C4⋊Dic53C2×C411yescharacteristic
29C2×#11719919—21no—
30C2×#11719919—21no—
31C2×#11719919—21no—
32C2×C22⋊Dic53—21no—
33C2×C22⋊Dic53—21no—
34C2×C22⋊Dic53—22no—
35C2×C4×Dic53C2×C411yescharacteristic
36C2×C4×Dic53C2×C411yescharacteristic
37#13133124—21no—
38#13133124—21no—
39#13133124—21no—
40#13133124—21no—
41#13133124—21no—
42#13133124—21no—
43#13133124—21no—
44#13133124—22no—
45#13133124—22no—
46C23×C4×C53C2311yescharacteristic
47C2×M16×C53—22no—
48#117×C53—22no—
49#117×C53—22no—
50C2×C4⋊C4×C53C2×C411yescharacteristic
51C2×C4⋊C4×C53C2×C411yescharacteristic
52C2×C22⋊C4×C53D411yescharacteristic
53C2×C22⋊C4×C53D411yescharacteristic
54C2×C42×C53D411yescharacteristic
55C2×C42×C53D411yescharacteristic
56C4⋊C8×C53—24no—
57C22⋊C8×C53—22no—
58C22⋊C8×C53D414yes—
59#97×C53—22no—
60C4×C8×C53—22no—
61C24×C53C2×D411yescharacteristic
62C22×C4×C53—41no—
63C22×C4×C53—41no—
64C22×C4×C53C22⋊C411yescharacteristic
65C22×C4×C53C22⋊C411yescharacteristic
66C22×C4×C53C22⋊C411yescharacteristic
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22×C411yescharacteristic
69C22×C4×C53C2×D411yescharacteristic
70C22×C4×C53C2×D412yes—
71C22×C4×C53#4112yes—
72C22×Dic53—21no—
73C22×Dic53—21no—
74C22×Dic53—21no—
75C22×Dic53—21no—
76C22×Dic53—42no—
77C2×Dic106—22no—
78C2×Dic106—41no—
79C2×Dic106—41no—
80M16×C53—42no—
81M16×C53—42no—
82C2×C8×C53—24no—
83C4⋊C4×C53—22no—
84C4⋊C4×C53—22no—
85C22⋊C4×C53—22no—
86C22⋊C4×C53—22no—
87C42×C53—21no—
88C42×C53—21no—
89C42×C53—21no—
90C42×C53—21no—
91C22⋊Dic53—42no—
92C22⋊Dic53—42no—
93C22⋊Dic53—42no—
94C22⋊Dic53—42no—
95C4⋊Dic53—22no—
96#11719919—42no—
97#11719919—42no—
98#11719919—41no—
99#11719919—41no—
100C4×Dic53—41no—
101C4×Dic53—41no—
102C23×C53—21no—
103C23×C53—21no—
104C23×C53—41no—
105C23×C53—41no—
106C23×C53C2×C22⋊C411yescharacteristic
107C23×C53#12112yes—
108C2×C4×C53—21no—
109C2×C4×C53—21no—
110C2×C4×C53—21no—
111C2×C4×C53—21no—
112C2×C4×C53—21no—
113C2×C4×C53—21no—
114C2×C4×C53—21no—
115C2×C4×C53—21no—
116C2×C4×C53—42no—
117C2×C4×C53—42no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—41no—
121C2×C4×C53—41no—
122C2×C4×C53C4×D412yes—
123C2×C4×C53#12011yescharacteristic
124C2×C4×C53#12311yescharacteristic
125C2×Dic53—22no—
126C2×Dic53—41no—
127C2×Dic53—41no—
128C2×Dic53—41no—
129C2×Dic53—41no—
130C2×Dic53—41no—
131C2×Dic53—41no—
132C2×Dic53—41no—
133C2×Dic53—41no—
134C2×Dic53—41no—
135C2×Dic53—41no—
136C2×Dic53—41no—
137C2×Dic53—82no—
138C2×Dic53—82no—
139C2×Dic53—82no—
140Dic106—82no—
141C8×C53—44no—
142#9444—531nomaximal, Sylow
143C22×C53—21no—
144C22×C53—21no—
145C22×C53—22no—
146C22×C53—22no—
147C22×C53—22no—
148C22×C53—81no—
149C22×C53—81no—
150C22×C53#38611yescharacteristic
151C22×C53#41011yescharacteristic
152C22×C53#41011yescharacteristic
153C4×C53—21no—
154C4×C53—22no—
155C4×C53—21no—
156C4×C53—41no—
157C4×C53—41no—
158C4×C53—41no—
159C4×C53—41no—
160Dic53—42no—
161Dic53—81no—
162Dic53—81no—
163Dic53—81no—
164Dic53—162no—
165#2921D5311yescharacteristic
166#2499—531no—
167#1820—531no—
168#1511—532no—
169#1284—532no—
170C2×C53—21no—
171C2×C53—21no—
172C2×C53—42no—
173C2×C53#188411yescharacteristic
174C2×C53#189911yescharacteristic
175C2×C53#190911yescharacteristic
176C2×C22⋊Q8—531no—
177C2×#117C2×D5311yescharacteristic
178#433—24no—
179#423—21no—
180#407C2×D5312yes—
181C4×M16—21no—
182#397—1061no—
183#397—1061no—
184#391—531no—
185#391—531no—
186#387—1061no—
187#387—1061no—
188#384—531no—
189#382—531no—
190#381—532no—
191C4×C22⋊C4—531no—
192#342—1061no—
193#342—1061no—
194#328—1064no—
195C53#944411yescharacteristic, complemented, Sylow
196C22×Q8—531no—
197C23×C4C22×D5311yescharacteristic
198C2×M16—22no—
199C22⋊Q8—1062no—
200C22⋊Q8—1062no—
201#117—22no—
202#117—22no—
203C2×C4⋊C4—531no—
204C2×C4⋊C4—1061no—
205C2×C4⋊C4—1061no—
206C2×C4⋊C4—1061no—
207C2×C4⋊C4C4×D5311yescharacteristic
208C2×C4⋊C4C4×D5311yescharacteristic
209C2×C22⋊C4—1061no—
210C2×C22⋊C4—1061no—
211C2×C22⋊C4—1062no—
212C2×C22⋊C4D21211yescharacteristic
213C2×C22⋊C4D21211yescharacteristic
214C2×C42—531no—
215C2×C42—531no—
216C2×C42C22⋊D5311yescharacteristic
217C2×C42C22⋊D5311yescharacteristic
218C4⋊C8—24no—
219C22⋊C8—22no—
220C22⋊C8C22⋊D5314yes—
221#97—22no—
222C4×C8—22no—
223#95—1061no—
224#95—1061no—
225#95—1061no—
226#95—1061no—
227#95—1061no—
228#95—1061no—
229#95—1061no—
230#95—1062no—
231#95—1062no—
232C24C2×D21211yescharacteristic
233C2×Q8—1062no—
234C2×Q8—2121no—
235C2×Q8—2121no—
236C22×C4—41no—
237C22×C4—41no—
238C22×C4—1061no—
239C22×C4—1061no—
240C22×C4—1061no—
241C22×C4—1061no—
242C22×C4—2122no—
243C22×C4#1171992111yescharacteristic
244C22×C4#1171992111yescharacteristic
245C22×C4#1171992111yescharacteristic
246C22×C4#1171992111yescharacteristic
247C22×C4C2×C4×D5311yescharacteristic
248C22×C4#1171994512yes—
249C22×C4C2×C22⋊D5311yescharacteristic
250C22×C4C2×C22⋊D5312yes—
251M16—42no—
252M16—42no—
253C2×C8—24no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—1062no—
257C4⋊C4—2122no—
258C4⋊C4—2122no—
259C4⋊C4—2121no—
260C4⋊C4—2121no—
261C22⋊C4—22no—
262C22⋊C4—22no—
263C22⋊C4—2122no—
264C22⋊C4—2122no—
265C22⋊C4—2122no—
266C22⋊C4—2122no—
267C42—21no—
268C42—21no—
269C42—21no—
270C42—21no—
271C42—2121no—
272C42—2121no—
273C23—21no—
274C23—21no—
275C23—41no—
276C23—41no—
277C23C2×#1171992111yescharacteristic
278C23#1313319112yes—
279Q8—4242no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—42no—
289C2×C4—42no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—1062no—
295C2×C4—4242no—
296C2×C4—4242no—
297C2×C4—4242no—
298C2×C4—2121no—
299C2×C4—2121no—
300C2×C4—2121no—
301C2×C4—2121no—
302C2×C4—2121no—
303C2×C4—2121no—
304C2×C4—2121no—
305C2×C4—2121no—
306C2×C4—2121no—
307C2×C4—2121no—
308C2×C4—2121no—
309C2×C4C4×C22⋊D5312yes—
310C2×C4#1313318611yescharacteristic
311C2×C4#1313320011yescharacteristic
312C8—44no—
313C22—21no—
314C22—21no—
315C22—22no—
316C22—22no—
317C22—22no—
318C22—81no—
319C22—81no—
320C22#1681050211yescharacteristic
321C22#1681059611yescharacteristic
322C22#1681059611yescharacteristic
323C4—21no—
324C4—22no—
325C4—21no—
326C4—41no—
327C4—41no—
328C4—41no—
329C4—41no—
330C4—8482no—
331C4—4241no—
332C4—4241no—
333C4—4241no—
334C4—2122no—
335C2—21no—
336C2—21no—
337C2—42no—
338C2#2027509811yescharacteristic
339C2#2027517311yescharacteristic
340C2#2027523711yescharacteristic
subgroupG/H
centerC22#16810502
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#2921C2
socleC22×C53#386
prime radicalC24×C53C2×D4
cosocleC23×C4×C53C23
hypercenter#2921D53
hypocenterC53#9444
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2921

Complements

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

NH ≅ G/Ntype
9. #16810528C4semidirect
10. #16810528C4semidirect
13. #16810492C4semidirect
14. #16810488C4semidirect
195. C53#9444semidirect

Maximal subgroups

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

subgroupindex
#202791642
#202747922
#202732492
#202725692
C53×#29212
#944453