Finite Group Databasebeta

← #27279318

Subgroups — #27279318

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#20283664C211yesmaximal, characteristic, complemented
3#20275633C211yesmaximal, characteristic
4#20273115C211yesmaximal, characteristic
5#20272556C211yesmaximal, characteristic, complemented
6#20272551C211yesmaximal, characteristic, complemented
7C53×#1503C211yesmaximal, characteristic, complemented
8#16811173C2211yescharacteristic
9#16811019C411yescharacteristic, complemented
10#16810997C411yescharacteristic, complemented
11#16810973C411yescharacteristic, complemented
12#16810971C411yescharacteristic, complemented
13#16810854—22no—
14#16810754—21no—
15#16810754—22no—
16Q8×#88322—21no—
17C4×#11719924—22no—
18#16810650—21no—
19C2×C4⋊#88322C2211yescharacteristic
20C2×#13133145—21no—
21C2×#13133145—21no—
22#16810579—21no—
23C4×#11719917C2211yescharacteristic
24C2×C4×#88322C2211yescharacteristic
25#16810352—21no—
26#16810352—21no—
27C53×#533C2211yescharacteristic
28C53×#431C2211yescharacteristic
29C4×C4⋊C4×C53C2211yescharacteristic
30C53×#328—22no—
31C2×#11719945—21no—
32C2×#11719924—22no—
33C2×#11719917D411yescharacteristic
34C2×#11719917D411yescharacteristic
35C22×#88322—21no—
36C22×#88322—21no—
37#13133217C2×C411yescharacteristic
38#13133214C2×C411yescharacteristic
39#13133205—21no—
40#13133191—21no—
41#13133191—21no—
42C4×D212—21no—
43C4×D212—21no—
44#13133148—22no—
45C4⋊#88322—22no—
46C4⋊#88322—22no—
47C4⋊#88322—21no—
48C4⋊#88322D411yescharacteristic
49C4⋊#88322D411yescharacteristic
50#13133145—42no—
51#13133145—42no—
52#13133141—22no—
53#13133137—22no—
54#13133128—22no—
55#13133128—21no—
56C4×#88322—22no—
57C4×#88322—21no—
58C4×#88322—21no—
59C4×#88322C2×C411yescharacteristic
60C4×#88322C2×C411yescharacteristic
61C4⋊Q8×C53C2×C411yescharacteristic
62C4⋊Q8×C53C2×C411yescharacteristic
63#125×C53—21no—
64C22⋊Q8×C53—21no—
65C22⋊Q8×C53D412yes—
66C4×Q8×C53—21no—
67C4×Q8×C53D412yes—
68#117×C53—21no—
69C2×C4⋊C4×C53—21no—
70C2×C42×C53—21no—
71C2×C42×C53C2311yescharacteristic
72C4⋊C8×C53—21no—
73C22⋊C8×C53—21no—
74C4×C8×C53—21no—
75#95×C53—21no—
76C2×Q8×C53—21no—
77C2×Q8×C53C22⋊C412yes—
78C22×C4×C53—41no—
79C22×C4×C53C2×D411yescharacteristic
80C22×C4×C53C2×D411yescharacteristic
81C22×C4×C53#4111yescharacteristic
82C2×C22⋊D53—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—
92C2×C8×C53—41no—
93C4⋊C4×C53—21no—
94C4⋊C4×C53—21no—
95C4⋊C4×C53—21no—
96C4⋊C4×C53—22no—
97C4⋊C4×C53—22no—
98C4⋊C4×C53—22no—
99C4⋊C4×C53—41no—
100C4⋊C4×C53C22⋊C412yes—
101C22⋊C4×C53—22no—
102C42×C53—22no—
103C42×C53—41no—
104C42×C53—41no—
105C42×C53C22×C411yescharacteristic
106C42×C53C2×D411yescharacteristic
107C42×C53C2×D411yescharacteristic
108C42×C53#4111yescharacteristic
109#11719924—44no—
110#11719921—41no—
111#11719921—41no—
112C4⋊Dic53—21no—
113C4⋊Dic53—21no—
114#11719917—21no—
115#11719917—21no—
116#11719917—21no—
117#11719917SD1612yescomplemented
118C2×#88322—21no—
119C2×#88322—21no—
120C2×#88322—21no—
121C2×#88322—21no—
122C2×#88322—21no—
123C2×#88322—21no—
124C2×#88322—42no—
125C2×#88322—41no—
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×C53—42no—
138C2×C4×C53—81no—
139C2×C4×C53—81no—
140C2×C4×C53C2×C22⋊C411yescharacteristic
141C2×C4×C53C4×D411yescharacteristic
142C2×C4×C53C4×D411yescharacteristic
143C2×C4×C53#12011yescharacteristic
144C2×C4×C53#12111yescharacteristic
145C2×C4×C53#12311yescharacteristic
146C2×C4×C53C2×SD1611yescharacteristic
147C2×C4×C53#13511yescharacteristic
148C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C53#13711yescharacteristic
150C22⋊D53—81no—
151C22⋊D53—81no—
152C2×Dic53—41no—
153D212—42no—
154D212—41no—
155D212—41no—
156D212—41no—
157D212—41no—
158C4×D53—81no—
159C4×D53—81no—
160Dic106—41no—
161Dic106—41no—
162C8×C53—81no—
163#88322—22no—
164#88322—22no—
165#88322—41no—
166#88322—41no—
167#88322—41no—
168#88322—41no—
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×C53—82no—
184C4×C53C4×SD1611yescharacteristic
185C4×C53#44011yescharacteristic
186C4×C53#44311yescharacteristic
187C4×C53#44411yescharacteristic
188C4×C53#44811yescharacteristic
189C4×C53#44911yescharacteristic
190Dic53—81no—
191#3167—531no—
192#2962—531no—
193#1980—531no—
194#1503D5311yescharacteristic, complemented
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—1061no—
212#446—1062no—
213C8×Q8—1061no—
214C4×SD16—1062no—
215#431C2×D5311yescharacteristic
216#424—1061no—
217C2×C4⋊C8—531no—
218C2×#104—1061no—
219C2×#104—1061no—
220#405—1061no—
221C4×M16—531no—
222C2×C4×C8—531no—
223C4×C4⋊C4C2×D5311yescharacteristic
224#328—22no—
225D4⋊C8—1061no—
226D4⋊C8—1061no—
227C53#905411yescharacteristic, complemented, Sylow
228C2×#41—1061no—
229C2×SD16—1062no—
230C2×M16—531no—
231C2×M16—531no—
232C22×C8—1061no—
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⋊Q8C22⋊D5312yes—
242#121—1061no—
243#121—1061no—
244C4×Q8—21no—
245C4×Q8D21212yes—
246C4×D4—1061no—
247C4×D4—1061no—
248#117—21no—
249C2×C4⋊C4—21no—
250C2×C42—21no—
251C2×C42C22×D5311yescharacteristic
252#106—1062no—
253C4⋊C8—21no—
254C4⋊C8—531no—
255C4⋊C8—531no—
256C4⋊C8—1062no—
257C4⋊C8—1062no—
258C4⋊C8—1061no—
259#104—2122no—
260#104—2122no—
261#103—1062no—
262#102—1062no—
263C22⋊C8—21no—
264#97—1062no—
265#97—1061no—
266C4×C8—21no—
267C4×C8—531no—
268C4×C8—531no—
269C4×C8—1062no—
270C4×C8—1061no—
271C4×C8—1061no—
272#95—21no—
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—41no—
285C22×C4—2121no—
286C22×C4D4×D5311yescharacteristic
287C22×C4#1171994711yescharacteristic
288C22×C4C2×C22⋊D5311yescharacteristic
289SD16—2124no—
290M16—532no—
291M16—1061no—
292M16—1061no—
293M16—1061no—
294C2×C8—41no—
295C2×C8—1061no—
296C2×C8—1061no—
297C2×C8—1061no—
298C2×C8—1061no—
299C2×C8—1061no—
300C2×C8—1061no—
301C2×C8—2122no—
302C2×C8—2121no—
303C2×C8—2121no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—22no—
308C4⋊C4—22no—
309C4⋊C4—22no—
310C4⋊C4—41no—
311C4⋊C4—1061no—
312C4⋊C4—1061no—
313C4⋊C4#1171992112yes—
314C22⋊C4—22no—
315C22⋊C4—2121no—
316C22⋊C4—2121no—
317C42—22no—
318C42—41no—
319C42—41no—
320C42C2×C4×D5311yescharacteristic
321C42C2×D21211yescharacteristic
322C42D4×D5311yescharacteristic
323C42#1171994711yescharacteristic
324C23—2121no—
325C23#1313318911yescharacteristic
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—42no—
344C2×C4—81no—
345C2×C4—81no—
346C2×C4—4241no—
347C2×C4—4241no—
348C2×C4—2121no—
349C2×C4C2×#1171992111yescharacteristic
350C2×C4#1313317711yescharacteristic
351C2×C4#1313317711yescharacteristic
352C2×C4#1313318711yescharacteristic
353C2×C4#1313318911yescharacteristic
354C2×C4#1313320111yescharacteristic
355C2×C4SD16×D5311yescharacteristic
356C2×C4#1313324011yescharacteristic
357C2×C4#1313324411yescharacteristic
358C2×C4#1313324911yescharacteristic
359C8—81no—
360C8—1062no—
361C8—1062no—
362C8—2121no—
363C8—2121no—
364C8—2121no—
365C8—2121no—
366C22—21no—
367C22—21no—
368C22—4241no—
369C22—4241no—
370C22—4241no—
371C22#1681050011yescharacteristic
372C22#1681082111yescharacteristic
373C22#1681084711yescharacteristic
374C4—21no—
375C4—21no—
376C4—21no—
377C4—42no—
378C4—42no—
379C4—82no—
380C4—4241no—
381C4#1681070811yescharacteristic
382C4#1681072611yescharacteristic
383C4#1681073711yescharacteristic
384C4#1681074211yescharacteristic
385C4#1681076311yescharacteristic
386C4#1681077111yescharacteristic
387C2—21no—
388C2—21no—
389C2—8481no—
390C2#2027504011yescharacteristic
391C2#2027506811yescharacteristic
392C2#2027509611yescharacteristic
subgroupG/H
centerC2×C4#13133187
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1503C2
socleC22×C53#386
prime radical#16811173C22
cosocleC2×C42×C53C23
hypercenter#1503D53
hypocenterC53#9054
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1503

Complements

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

NH ≅ G/Ntype
2. #20283664C2semidirect
5. #20272556C2semidirect
6. #20272551C2semidirect
7. C53×#1503C2semidirect
9. #16811019C4semidirect
10. #16810997C4semidirect
11. #16810973C4semidirect
12. #16810971C4semidirect
117. #11719917SD16semidirect
194. #1503D53semidirect
227. C53#9054semidirect

Maximal subgroups

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

subgroupindex
#202852752
#202836642
#202756332
#202731152
#202725562
#202725512
C53×#15032
#905453