Finite Group Databasebeta

← #27279320

Subgroups — #27279320

Order 13568.

470 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20285275C211yesmaximal, characteristic
2#20275634C211yesmaximal, characteristic
3#20273214C211yesmaximal, characteristic, complemented
4#20273113C211yesmaximal, characteristic
5#20272554C211yesmaximal, characteristic, complemented
6#20272553C211yesmaximal, characteristic, complemented
7C53×#2962C211yesmaximal, characteristic, complemented
8#16811173C2211yescharacteristic
9#16811019C411yescharacteristic, complemented
10#16810997C411yescharacteristic, complemented
11#16810973C411yescharacteristic, complemented
12#16810971C411yescharacteristic, complemented
13#16810854—22no—
14C4×#11719924—22no—
15#16810676C2211yescharacteristic
16C2×#13133143—21no—
17C2×#13133143—21no—
18C2×C4×#88322C2211yescharacteristic
19C4×C4⋊Dic53C2211yescharacteristic
20#16810360—22no—
21#16810350—21no—
22#16810350—21no—
23C53×#533C2211yescharacteristic
24C53×#446—22no—
25C53×#446—21no—
26C8×Q8×C53—21no—
27C53×#424—21no—
28C2×C4⋊C8×C53C2211yescharacteristic
29C53×#405—21no—
30C4×M16×C53C2211yescharacteristic
31C2×#11719945—21no—
32C2×#11719924—22no—
33C22×#88322—21no—
34#13133217C2×C411yescharacteristic
35#13133214C2×C411yescharacteristic
36#13133205—21no—
37#13133191—21no—
38#13133191—21no—
39C4×D212—21no—
40C4×D212—21no—
41C2×C4⋊Dic53—21no—
42C2×C4×Dic53—21no—
43#13133148—22no—
44C4⋊#88322—21no—
45#13133143—42no—
46#13133143—42no—
47#13133141—22no—
48#13133137—22no—
49C22⋊#88322—21no—
50C4×#88322—21no—
51C4×#88322—21no—
52C4×#88322C2×C411yescharacteristic
53C4×#88322C2×C411yescharacteristic
54#13133124—21no—
55C2×M16×C53D411yescharacteristic
56C2×M16×C53D411yescharacteristic
57C22×C8×C53—21no—
58C4⋊Q8×C53C2×C411yescharacteristic
59C4⋊Q8×C53C2×C411yescharacteristic
60#125×C53—21no—
61C22⋊Q8×C53—21no—
62C22⋊Q8×C53D412yes—
63C4×Q8×C53—21no—
64C4×Q8×C53D412yes—
65#117×C53—21no—
66C2×C42×C53C2311yescharacteristic
67C4⋊C8×C53—22no—
68C4⋊C8×C53—22no—
69C4⋊C8×C53—21no—
70C4⋊C8×C53D411yescharacteristic
71C4⋊C8×C53D411yescharacteristic
72#97×C53—22no—
73#97×C53—21no—
74C4×C8×C53—22no—
75C4×C8×C53—21no—
76C2×Q8×C53—21no—
77C2×Q8×C53C22⋊C412yes—
78C22×C4×C53C2×D411yescharacteristic
79C22×C4×C53C2×D411yescharacteristic
80C22×C4×C53#4111yescharacteristic
81C2×C22⋊D53—41no—
82C22×Dic53—41no—
83#11719945—41no—
84#11719945—41no—
85#11719945—41no—
86#11719945—41no—
87C2×D212—21no—
88C2×D212—21no—
89C2×D212—21no—
90C2×C4×D53—41no—
91C2×Dic106—21no—
92M16×C53—21no—
93M16×C53—21no—
94M16×C53—21no—
95M16×C53SD1612yescomplemented
96C2×C8×C53—21no—
97C2×C8×C53—21no—
98C2×C8×C53—21no—
99C2×C8×C53—21no—
100C2×C8×C53—41no—
101C2×C8×C53—41no—
102C4⋊C4×C53—21no—
103C4⋊C4×C53—21no—
104C4⋊C4×C53—21no—
105C4⋊C4×C53—22no—
106C4⋊C4×C53—22no—
107C4⋊C4×C53C22⋊C412yes—
108C22⋊C4×C53—22no—
109C42×C53—22no—
110C42×C53C22×C411yescharacteristic
111C42×C53C2×D411yescharacteristic
112C42×C53C2×D411yescharacteristic
113C42×C53#4111yescharacteristic
114#11719924—44no—
115#11719921—41no—
116#11719921—41no—
117C4⋊Dic53—22no—
118C4⋊Dic53—21no—
119C4⋊Dic53—21no—
120C4⋊Dic53—41no—
121C4×Dic53—41no—
122C4×Dic53—41no—
123C2×#88322—21no—
124C2×#88322—21no—
125C2×#88322—42no—
126C2×#88322—41no—
127C23×C53#12111yescharacteristic
128C22×D53—41no—
129Q8×C53—42no—
130Q8×C53—41no—
131Q8×C53—41no—
132C2×C4×C53—21no—
133C2×C4×C53—21no—
134C2×C4×C53—22no—
135C2×C4×C53—22no—
136C2×C4×C53—41no—
137C2×C4×C53C2×C22⋊C411yescharacteristic
138C2×C4×C53C4×D411yescharacteristic
139C2×C4×C53C4×D411yescharacteristic
140C2×C4×C53#12011yescharacteristic
141C2×C4×C53#12111yescharacteristic
142C2×C4×C53#12311yescharacteristic
143C2×C4×C53C2×SD1611yescharacteristic
144C2×C4×C53#13511yescharacteristic
145C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
146C2×C4×C53#13711yescharacteristic
147C22⋊D53—81no—
148C22⋊D53—81no—
149C2×Dic53—42no—
150C2×Dic53—41no—
151C2×Dic53—81no—
152C2×Dic53—81no—
153D212—42no—
154D212—41no—
155D212—41no—
156D212—41no—
157D212—41no—
158C4×D53—81no—
159C4×D53—81no—
160Dic106—41no—
161Dic106—41no—
162C8×C53—22no—
163C8×C53—41no—
164C8×C53—41no—
165C8×C53—41no—
166#88322—22no—
167#88322—41no—
168#88322—81no—
169#9054—531nomaximal, Sylow
170C22×C53—21no—
171C22×C53—21no—
172C22×C53#38611yescharacteristic
173C22×C53#46011yescharacteristic
174C22×C53#46411yescharacteristic
175C2×D53—81no—
176C2×D53—81no—
177C2×D53—81no—
178C4×C53—21no—
179C4×C53—21no—
180C4×C53—21no—
181C4×C53—42no—
182C4×C53—42no—
183C4×C53C4×SD1611yescharacteristic
184C4×C53#44011yescharacteristic
185C4×C53#44311yescharacteristic
186C4×C53#44411yescharacteristic
187C4×C53#44811yescharacteristic
188C4×C53#44911yescharacteristic
189Dic53—82no—
190Dic53—81no—
191#3167—531no—
192#2962D5311yescharacteristic, complemented
193#1980—531no—
194#1503—531no—
195#1480—531no—
196#1279—531no—
197#1278—531no—
198C2×C53—21no—
199C2×C53—21no—
200C2×C53#187611yescharacteristic
201C2×C53#188011yescharacteristic
202C2×C53#188411yescharacteristic
203D53—161no—
204#533C2×D5311yescharacteristic
205#532—531no—
206#498—531no—
207#492—531no—
208#487—531no—
209#486—531no—
210#465—1062no—
211#446—22no—
212#446—21no—
213C8×Q8—21no—
214C4×SD16—1062no—
215#431—531no—
216#424—21no—
217C2×C4⋊C8C2×D5311yescharacteristic
218C2×#104—1061no—
219C2×#104—1061no—
220#405—21no—
221C4×M16C2×D5311yescharacteristic
222C2×C4×C8—531no—
223C4×C4⋊C4—531no—
224#328—1062no—
225D4⋊C8—1061no—
226D4⋊C8—1061no—
227C53#905411yescharacteristic, complemented, Sylow
228C2×#41—1061no—
229C2×SD16—1062no—
230C2×M16C22⋊D5311yescharacteristic
231C2×M16C22⋊D5311yescharacteristic
232C22×C8—21no—
233C22×C8—1061no—
234C4⋊Q8—531no—
235C4⋊Q8C4×D5311yescharacteristic
236C4⋊Q8C4×D5311yescharacteristic
237#127—531no—
238#125—21no—
239#124—1061no—
240C22⋊Q8—21no—
241C22⋊Q8D21212yes—
242#121—1061no—
243#121—1061no—
244C4×Q8—21no—
245C4×Q8C22⋊D5312yes—
246C4×D4—1061no—
247C4×D4—1061no—
248#117—21no—
249C2×C4⋊C4—1061no—
250C2×C42—1061no—
251C2×C42C22×D5311yescharacteristic
252#106—1062no—
253C4⋊C8—22no—
254C4⋊C8—22no—
255C4⋊C8—21no—
256C4⋊C8—1061no—
257C4⋊C8C22⋊D5311yescharacteristic
258C4⋊C8C22⋊D5311yescharacteristic
259#104—2122no—
260#104—2122no—
261#103—1062no—
262#102—1062no—
263C22⋊C8—1061no—
264#97—22no—
265#97—21no—
266C4×C8—22no—
267C4×C8—21no—
268C4×C8—531no—
269C4×C8—531no—
270C4×C8—1061no—
271C4×C8—1061no—
272#95—1061no—
273#41—2121no—
274#41—2121no—
275#41—2121no—
276#41—2121no—
277C2×Q8—21no—
278C2×Q8—1061no—
279C2×Q8#1171992112yes—
280C2×D4—1061no—
281C2×D4—1061no—
282C2×D4—1061no—
283C2×D4—2121no—
284C22×C4—2121no—
285C22×C4—2121no—
286C22×C4C2×D21211yescharacteristic
287C22×C4#1171994511yescharacteristic
288C22×C4C2×C22⋊D5311yescharacteristic
289SD16—2124no—
290M16—21no—
291M16—21no—
292M16—21no—
293M16#1171992412yescomplemented
294C2×C8—21no—
295C2×C8—21no—
296C2×C8—21no—
297C2×C8—21no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—1061no—
301C2×C8—1061no—
302C2×C8—2122no—
303C2×C8—2121no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—1062no—
310C4⋊C4—1061no—
311C4⋊C4—1061no—
312C4⋊C4—2121no—
313C4⋊C4#1171992112yes—
314C22⋊C4—22no—
315C22⋊C4—2121no—
316C22⋊C4—2121no—
317C42—22no—
318C42—2121no—
319C42—2121no—
320C42C2×C4×D5311yescharacteristic
321C42#1171994511yescharacteristic
322C42C2×C22⋊D5311yescharacteristic
323C42C2×C22⋊D5311yescharacteristic
324C23—2121no—
325C23#1313319111yescharacteristic
326Q8—42no—
327Q8—41no—
328Q8—41no—
329Q8—2121no—
330Q8—2121no—
331D4—4241no—
332D4—4241no—
333D4—2122no—
334D4—2121no—
335D4—2121no—
336D4—2121no—
337D4—2121no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—41no—
343C2×C4—4241no—
344C2×C4—4241no—
345C2×C4—4241no—
346C2×C4—4241no—
347C2×C4—2122no—
348C2×C4—2121no—
349C2×C4C2×#1171992111yescharacteristic
350C2×C4C4×C22⋊D5311yescharacteristic
351C2×C4C4×C22⋊D5311yescharacteristic
352C2×C4#1313318611yescharacteristic
353C2×C4#1313319111yescharacteristic
354C2×C4#1313320011yescharacteristic
355C2×C4C2×#1171992411yescharacteristic
356C2×C4#1313323711yescharacteristic
357C2×C4#1313324211yescharacteristic
358C2×C4#1313324811yescharacteristic
359C8—22no—
360C8—41no—
361C8—41no—
362C8—41no—
363C8—1062no—
364C8—4241no—
365C8—2121no—
366C22—21no—
367C22—21no—
368C22—4241no—
369C22—4241no—
370C22—4241no—
371C22#1681050211yescharacteristic
372C22#1681082311yescharacteristic
373C22#1681084511yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—42no—
378C4—42no—
379C4—4242no—
380C4—4241no—
381C4C4×#1171992411yescharacteristic
382C4#1681072111yescharacteristic
383C4#1681073811yescharacteristic
384C4#1681074311yescharacteristic
385C4#1681076211yescharacteristic
386C4#1681076811yescharacteristic
387C2—21no—
388C2—21no—
389C2—8481no—
390C2#2027504211yescharacteristic
391C2#2027507011yescharacteristic
392C2#2027509811yescharacteristic
subgroupG/H
centerC2×C4#13133186
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#2962C2
socleC22×C53#386
prime radical#16811173C22
cosocleC2×C42×C53C23
hypercenter#2962D53
hypocenterC53#9054
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #2962

Complements

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

NH ≅ G/Ntype
3. #20273214C2semidirect
5. #20272554C2semidirect
6. #20272553C2semidirect
7. C53×#2962C2semidirect
9. #16811019C4semidirect
10. #16810997C4semidirect
11. #16810973C4semidirect
12. #16810971C4semidirect
95. M16×C53SD16semidirect
192. #2962D53semidirect
227. C53#9054semidirect
293. M16#11719924semidirect

Maximal subgroups

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

subgroupindex
#202852752
#202756342
#202732142
#202731132
#202725542
#202725532
C53×#29622
#905453