Finite Group Databasebeta

← #27295029

Subgroups — #27295029

Order 13568.

324 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 79 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#20285286C211yesmaximal, characteristic
2#20276511C211yesmaximal, characteristic, complemented
3#20274844C211yesmaximal, characteristic, complemented
4#20272727C211yesmaximal, characteristic, complemented
5#20272725C211yesmaximal, characteristic
6#20272723C211yesmaximal, characteristic
7C53×#1348C211yesmaximal, characteristic, complemented
8#16811159—21no—
9#16811024C411yescharacteristic, complemented
10#16811008C411yescharacteristic, complemented
11#16810979—21no—
12#16810859C2211yescharacteristic
13#16810854—21no—
14Q16×Dic53C411yescharacteristic, complemented
15#16810708—21no—
16#16810705C411yescharacteristic, complemented
17C2×C8×Dic53C2211yescharacteristic
18#16810488—21no—
19#16810441—21no—
20C22⋊#11719908C2211yescharacteristic
21C16×Dic53—21no—
22#16810386C2211yescharacteristic
23#16810385—21no—
24C53×#467C2211yescharacteristic
25C53×C22⋊C16C2211yescharacteristic
26C53×#340C2211yescharacteristic
27C2×#11719924—21no—
28C2×D424C2×C411yescharacteristic
29C22×#88322D411yescharacteristic
30#13133212—41no—
31#13133208—21no—
32#13133206—21no—
33#13133201—41no—
34#13133191—21no—
35Q8×Dic53—21no—
36#13133179—21no—
37#13133177—41no—
38#13133175—41no—
39C2×C4⋊Dic53—21no—
40C2×C4×Dic53D411yescharacteristic
41C2×#11719908—21no—
42#13133151C2×C411yescharacteristic
43#13133148—21no—
44#13133141—21no—
45#13133139—21no—
46#13133137—21no—
47C8×Dic53C2×C411yescharacteristic
48C8×Dic53C2×C411yescharacteristic
49C8×Dic53D411yescharacteristic
50C8×Dic53D411yescharacteristic
51#13133124—41no—
52#13133124—41no—
53C2×Q16×C53C2×C411yescharacteristic
54C22×C8×C53C2311yescharacteristic
55C22⋊Q8×C53—21no—
56C2×C4⋊C4×C53—21no—
57C2×C16×C53—21no—
58#107×C53C2×C411yescharacteristic
59#103×C53—21no—
60C2×Q8×C53—21no—
61C22×C4×C53—41no—
62C22×C4×C53C2×D411yescharacteristic
63C2×C22⋊D53—41no—
64C22×Dic53—41no—
65C22×Dic53D811yescharacteristic
66C22×Dic53D811yescharacteristic
67C2×D212—21no—
68C2×C4×D53—41no—
69Q16×C53—41no—
70C2×C8×C53—41no—
71C2×C8×C53C22×C411yescharacteristic
72C2×C8×C53C2×D411yescharacteristic
73C4⋊C4×C53—21no—
74C4⋊C4×C53—41no—
75C4⋊C4×C53—41no—
76C4⋊C4×C53—41no—
77C22⋊C4×C53—41no—
78#11719924—42no—
79#11719921—41no—
80#11719921—41no—
81#11719921—41no—
82#11719921—41no—
83C4⋊Dic53—21no—
84C4⋊Dic53—41no—
85C4⋊Dic53—41no—
86C4⋊Dic53—41no—
87C4⋊Dic53—41no—
88#11719919—41no—
89C4×Dic53—41no—
90C4×Dic53C22⋊C411yescharacteristic
91C4×Dic53C22⋊C411yescharacteristic
92C4×Dic53SD1611yescharacteristic
93C4×Dic53SD1611yescharacteristic
94C2×#88322—41no—
95C2×#88322C22⋊C411yescharacteristic
96C2×#88322C22⋊C411yescharacteristic
97D424—41no—
98C16×C53—41no—
99#11719908—41no—
100C23×C53C2×D811yescharacteristic
101C22×D53—41no—
102Q8×C53—41no—
103Q8×C53—41no—
104C2×C4×C53—21no—
105C2×C4×C53—21no—
106C2×C4×C53—41no—
107C2×C4×C53—41no—
108C2×C4×C53—81no—
109C2×C4×C53—81no—
110C2×C4×C53—81no—
111C2×C4×C53C2×C22⋊C411yescharacteristic
112C2×C4×C53C2×SD1611yescharacteristic
113C22⋊D53—81no—
114C22⋊D53—81no—
115C2×Dic53—21no—
116C2×Dic53—21no—
117C2×Dic53—41no—
118C2×Dic53—41no—
119C2×Dic53—81no—
120C2×Dic53—81no—
121C2×Dic53—81no—
122C2×Dic53#10211yescharacteristic
123C2×Dic53#10211yescharacteristic
124C2×Dic53#10211yescharacteristic
125C2×Dic53#10211yescharacteristic
126D212—41no—
127D212—41no—
128C4×D53—82no—
129C8×C53—21no—
130C8×C53—21no—
131#88322—21no—
132#88322—21no—
133#11855—531nomaximal, Sylow
134C22×C53—21no—
135C22×C53—21no—
136C22×C53C2×#10211yescharacteristic
137C22×C53#50811yescharacteristic
138C22×C53#50811yescharacteristic
139C2×D53—81no—
140C2×D53—81no—
141C2×D53—81no—
142C4×C53—21no—
143C4×C53—81no—
144C4×C53—81no—
145C4×C53—161no—
146C4×C53#41011yescharacteristic
147C4×C53#42111yescharacteristic
148Dic53—21no—
149Dic53—21no—
150Dic53—21no—
151Dic53—21no—
152Dic53—81no—
153Dic53—161no—
154#3168—531no—
155#2162—531no—
156#1835—531no—
157#1348—531no—
158#1348D5311yescharacteristic, complemented
159#1347—531no—
160#1347—531no—
161C2×C53—21no—
162C2×C53—21no—
163C2×C53#151711yescharacteristic
164C2×C53#213811yescharacteristic
165C2×C53#214511yescharacteristic
166D53—161no—
167#529—1061no—
168#499—531no—
169#495—531no—
170#488—1061no—
171#467C2×D5311yescharacteristic
172#466—531no—
173#465—1061no—
174C4×Q16—531no—
175C4×SD16—1061no—
176C4×D8—531no—
177C2×C4×C8—531no—
178#382—1061no—
179C4⋊C16—1061no—
180C22⋊C16—531no—
181C22⋊C16C2×D5311yescharacteristic
182C4×C16—1061no—
183#340—531no—
184#340—1061no—
185#340C2×D5311yescharacteristic
186C53#1185511yescharacteristic, complemented, Sylow
187C2×Q16C4×D5311yescharacteristic
188C2×SD16—1061no—
189C2×D8—531no—
190C22×C8—531no—
191C22×C8C22×D5311yescharacteristic
192#126—2121no—
193#125—1061no—
194#124—1061no—
195#123—2121no—
196C22⋊Q8—21no—
197#121—1061no—
198C4×Q8—1061no—
199C4×D4—1061no—
200C4×D4—2121no—
201#117—2121no—
202C2×C4⋊C4—21no—
203C2×C4⋊C4—1061no—
204C2×C42—531no—
205C2×C16—21no—
206C2×C16—1061no—
207#107—531no—
208#107C4×D5311yescharacteristic
209#106—1061no—
210#103—21no—
211#103—1061no—
212#102—1061no—
213#102—1061no—
214C4×C8—531no—
215C4×C8—531no—
216C4×C8—531no—
217C4×C8—531no—
218#95—2121no—
219#95—2121no—
220C2×Q8—21no—
221C2×D4—1061no—
222C2×D4—2121no—
223C22×C4—41no—
224C22×C4—531no—
225C22×C4—531no—
226C22×C4—2121no—
227C22×C4—2121no—
228C22×C4D4×D5311yescharacteristic
229Q16—41no—
230SD16—2122no—
231D8—2121no—
232C2×C8—41no—
233C2×C8—531no—
234C2×C8—531no—
235C2×C8—2121no—
236C2×C8C2×C4×D5311yescharacteristic
237C2×C8D4×D5311yescharacteristic
238C4⋊C4—21no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—1061no—
243C4⋊C4—2121no—
244C4⋊C4—2121no—
245C4⋊C4—2121no—
246C4⋊C4—2121no—
247C4⋊C4—2121no—
248C22⋊C4—41no—
249C22⋊C4—2121no—
250C22⋊C4—2121no—
251C22⋊C4—2121no—
252C22⋊C4—2121no—
253C42—531no—
254C42—531no—
255C42—531no—
256C42—531no—
257C42—2121no—
258C16—41no—
259C16—2121no—
260C23—2121no—
261C23D8×D5311yescharacteristic
262Q8—41no—
263Q8—41no—
264D4—4241no—
265D4—4241no—
266D4—2121no—
267D4—2121no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—81no—
273C2×C4—81no—
274C2×C4—81no—
275C2×C4—531no—
276C2×C4—531no—
277C2×C4—531no—
278C2×C4—531no—
279C2×C4—1061no—
280C2×C4—1061no—
281C2×C4—4242no—
282C2×C4—4241no—
283C2×C4—4241no—
284C2×C4—4241no—
285C2×C4—2121no—
286C2×C4—2121no—
287C2×C4C22⋊C4×D5311yescharacteristic
288C2×C4SD16×D5311yescharacteristic
289C8—21no—
290C8—21no—
291C8—1061no—
292C8—1061no—
293C22—21no—
294C22—21no—
295C22—4241no—
296C22—4241no—
297C22—4241no—
298C22#102×D5311yescharacteristic
299C22#1681105411yescharacteristic
300C22#1681105611yescharacteristic
301C4—21no—
302C4—81no—
303C4—81no—
304C4—161no—
305C4—8481no—
306C4—1061no—
307C4—1061no—
308C4—1061no—
309C4—1061no—
310C4—4241no—
311C4#1681059511yescharacteristic
312C4#1681063811yescharacteristic
313C2—21no—
314C2—21no—
315C2—8481no—
316C2#2027327311yescharacteristic
317C2#2027642111yescharacteristic
318C2#2027644911yescharacteristic
subgroupG/H
centerC22#102×D53
commutatorC2×C8×C53C22×C4
FrattiniC22×C8C22×D53
FittingC53×#1348C2
socleC22×C53C2×#102
prime radical#16810859C22
cosocleC22×C8×C53C23
hypercenter#1348D53
hypocenterC53#11855
LCSG ▷ C2×C8×C53 ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #1348

Complements

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

NH ≅ G/Ntype
2. #20276511C2semidirect
3. #20274844C2semidirect
4. #20272727C2semidirect
7. C53×#1348C2semidirect
9. #16811024C4semidirect
10. #16811008C4semidirect
14. Q16×Dic53C4semidirect
16. #16810705C4semidirect
158. #1348D53semidirect
186. C53#11855semidirect

Maximal subgroups

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

subgroupindex
#202852862
#202765112
#202748442
#202727272
#202727252
#202727232
C53×#13482
#1185553