Finite Group Databasebeta

← #27279910

Subgroups — #27279910

Order 13568.

472 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283735C211yesmaximal, characteristic, complemented
2#20274525C211yesmaximal, characteristic, complemented
3#20273307C211yesmaximal, characteristic, complemented
4#20273223C211yesmaximal, characteristic, complemented
5#20272607C211yesmaximal, characteristic, complemented
6#20272586C211yesmaximal, characteristic, complemented
7C53×#3115C211yesmaximal, characteristic
8Q8×#88322—21no—
9#16810701—22no—
10#16810701—21no—
11#16810680—21no—
12#16810680C2211yescharacteristic
13C2×#13133144—21no—
14C4×#11719917—21no—
15C4×#11719917C2211yescharacteristic
16C4×#11719917C2211yescharacteristic
17#16810469C2211yescharacteristic
18#16810382—21no—
19#16810363—21no—
20#16810363—21no—
21#16810357—22no—
22C53×#532C2211yescharacteristic
23C4×#41×C53C2211yescharacteristic
24C53×#482—21no—
25C53×#475C411yescharacteristic, complemented
26C53×#474C411yescharacteristic, complemented
27C53×#460C411yescharacteristic, complemented
28C53×#459C411yescharacteristic, complemented
29C53×#449—21no—
30C53×#442—21no—
31C53×#440—21no—
32C53×#432C2211yescharacteristic
33C2×#11719917—22no—
34C2×#11719917—21no—
35C2×#11719917—21no—
36C2×C4×Dic53—21no—
37C4⋊#88322—22no—
38C4⋊#88322—21no—
39#13133144—24no—
40C22⋊#88322—22no—
41C22⋊#88322D412yes—
42#13133128—21no—
43#13133128—21no—
44#13133128D411yescharacteristic
45#13133128D411yescharacteristic
46C4×#88322—22no—
47C4×#88322—21no—
48C4×#88322—21no—
49#13133124—21no—
50#13133124—21no—
51C2×#41×C53—21no—
52C2×#41×C53D411yescharacteristic
53C2×SD16×C53—21no—
54C2×D8×C53—21no—
55C4⋊Q8×C53C2×C411yescharacteristic
56#127×C53C2×C411yescharacteristic
57#124×C53—21no—
58#121×C53—21no—
59#121×C53—21no—
60C4×Q8×C53C2×C411yescharacteristic
61C4×Q8×C53D411yescharacteristic
62C4×D4×C53—21no—
63C4×D4×C53—21no—
64C4×D4×C53—21no—
65C4×D4×C53—21no—
66C4×D4×C53C2×C411yescharacteristic
67C4×D4×C53D411yescharacteristic
68#117×C53—21no—
69#117×C53D411yescharacteristic
70C2×C42×C53—21no—
71C2×C42×C53C2311yescharacteristic
72#107×C53—21no—
73#106×C53—21no—
74C4⋊C8×C53C2×C411yescharacteristic
75C4⋊C8×C53C2×C411yescharacteristic
76#103×C53—21no—
77#102×C53—21no—
78#102×C53—21no—
79#102×C53—21no—
80C22⋊C8×C53—21no—
81#97×C53—21no—
82#41×C53—22no—
83#41×C53—41no—
84#41×C53—82no—
85C2×Q8×C53—21no—
86C2×Q8×C53C22⋊C411yescharacteristic
87C2×D4×C53—21no—
88C2×D4×C53—21no—
89C2×D4×C53—21no—
90C2×D4×C53—21no—
91C2×D4×C53—41no—
92C2×D4×C53C22⋊C411yescharacteristic
93C22×C4×C53—21no—
94C22×C4×C53—21no—
95C22×C4×C53—21no—
96C22×C4×C53—41no—
97C22×C4×C53C2×D411yescharacteristic
98C22×C4×C53C2×D411yescharacteristic
99C22×C4×C53#4111yescharacteristic
100C22×Dic53—41no—
101SD16×C53—42no—
102D8×C53—42no—
103C2×C8×C53—21no—
104C2×C8×C53—21no—
105C4⋊C4×C53—21no—
106C4⋊C4×C53—21no—
107C4⋊C4×C53—21no—
108C4⋊C4×C53—21no—
109C4⋊C4×C53C22⋊C411yescharacteristic
110C4⋊C4×C53C22⋊C411yescharacteristic
111C22⋊C4×C53—21no—
112C22⋊C4×C53—21no—
113C22⋊C4×C53—21no—
114C22⋊C4×C53—41no—
115C22⋊C4×C53—41no—
116C42×C53—21no—
117C42×C53—21no—
118C42×C53—21no—
119C42×C53C22×C411yescharacteristic
120C42×C53C2×D411yescharacteristic
121C42×C53C2×D411yescharacteristic
122C42×C53#4111yescharacteristic
123C4×Dic53—22no—
124C4×Dic53—41no—
125#11719917—22no—
126#11719917—42no—
127#11719917—44no—
128#11719917—41no—
129C2×#88322—21no—
130C2×#88322—22no—
131C2×#88322—21no—
132C2×#88322—41no—
133C23×C53—21no—
134C23×C53—41no—
135C23×C53#12111yescharacteristic
136Q8×C53—21no—
137Q8×C53—81no—
138Q8×C53#10412yes—
139D4×C53—21no—
140D4×C53—21no—
141D4×C53—42no—
142D4×C53—41no—
143D4×C53—41no—
144D4×C53—81no—
145D4×C53—81no—
146D4×C53—82no—
147C2×C4×C53—21no—
148C2×C4×C53—21no—
149C2×C4×C53—21no—
150C2×C4×C53—21no—
151C2×C4×C53—21no—
152C2×C4×C53—41no—
153C2×C4×C53—41no—
154C2×C4×C53—42no—
155C2×C4×C53—42no—
156C2×C4×C53—41no—
157C2×C4×C53—81no—
158C2×C4×C53—82no—
159C2×C4×C53C2×C22⋊C411yescharacteristic
160C2×C4×C53C4×D411yescharacteristic
161C2×C4×C53C4×D411yescharacteristic
162C2×C4×C53#12011yescharacteristic
163C2×C4×C53#12111yescharacteristic
164C2×C4×C53#12311yescharacteristic
165C2×Dic53—42no—
166C2×Dic53—81no—
167C2×Dic53—81no—
168C8×C53—41no—
169C8×C53—41no—
170#88322—22no—
171#88322—42no—
172#88322—41no—
173#88322—42no—
174#9162—531nomaximal, Sylow
175C22×C53—21no—
176C22×C53—21no—
177C22×C53—21no—
178C22×C53—41no—
179C22×C53—41no—
180C22×C53—41no—
181C22×C53—81no—
182C22×C53—81no—
183C22×C53—81no—
184C22×C53#38611yescharacteristic
185C4×C53—21no—
186C4×C53—22no—
187C4×C53—21no—
188C4×C53—21no—
189C4×C53—21no—
190C4×C53—21no—
191C4×C53—41no—
192C4×C53—41no—
193C4×C53—81no—
194C4×C53C2×#10411yescharacteristic
195C4×C53#42111yescharacteristic
196Dic53—82no—
197#3115D5311yescharacteristic
198#2971—531no—
199#1759—531no—
200#1525—531no—
201#1505—531no—
202#1300—531no—
203#1291—531no—
204C2×C53—41no—
205C2×C53—81no—
206C2×C53—161no—
207C2×C53#187111yescharacteristic
208C2×C53#190311yescharacteristic
209C2×C53#190611yescharacteristic
210#532C2×D5311yescharacteristic
211C4×#41C2×D5311yescharacteristic
212#482—21no—
213#475Dic5311yescharacteristic, complemented
214#474Dic5311yescharacteristic, complemented
215#460Dic5311yescharacteristic, complemented
216#459Dic5311yescharacteristic, complemented
217#449—21no—
218C8×Q8—1061no—
219#442—21no—
220#440—21no—
221#436—1062no—
222#436—1061no—
223#432—531no—
224#432—1061no—
225#432C2×D5311yescharacteristic
226C2×#104—1061no—
227C4×M16—531no—
228C4×M16—531no—
229C4×M16—1061no—
230#376—531no—
231#339—1061no—
232#329—1061no—
233#329—1061no—
234#327—1062no—
235C53#916211yescharacteristic, complemented, Sylow
236C2×#41—21no—
237C2×#41C22⋊D5311yescharacteristic
238C2×SD16—21no—
239C2×D8—21no—
240C2×M16—1062no—
241C2×M16—1061no—
242C2×M16—1061no—
243C4⋊Q8C2×Dic5311yescharacteristic
244#127C2×Dic5311yescharacteristic
245#124—21no—
246#121—21no—
247#121—21no—
248C4×Q8C2×Dic5311yescharacteristic
249C4×Q8C22⋊D5311yescharacteristic
250C4×D4—21no—
251C4×D4—21no—
252C4×D4—21no—
253C4×D4—21no—
254C4×D4C2×Dic5311yescharacteristic
255C4×D4C22⋊D5311yescharacteristic
256#117—21no—
257#117C22⋊D5311yescharacteristic
258C2×C42—21no—
259C2×C42—1061no—
260C2×C42C22×D5311yescharacteristic
261#107—21no—
262#106—21no—
263C4⋊C8—1062no—
264C4⋊C8—1061no—
265C4⋊C8C2×Dic5311yescharacteristic
266C4⋊C8C2×Dic5311yescharacteristic
267#104—1064no—
268#103—21no—
269#102—21no—
270#102—21no—
271#102—21no—
272C22⋊C8—21no—
273C22⋊C8—532no—
274C22⋊C8—1062no—
275#97—21no—
276#97—531no—
277#97—531no—
278#97—1061no—
279#97—1061no—
280C4×C8—1062no—
281C4×C8—1061no—
282C4×C8—1061no—
283#95—1061no—
284#95—1061no—
285#41—22no—
286#41—41no—
287#41—82no—
288C2×Q8—21no—
289C2×Q8C22⋊Dic5311yescharacteristic
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—21no—
294C2×D4—41no—
295C2×D4C22⋊Dic5311yescharacteristic
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—21no—
299C22×C4—41no—
300C22×C4—2121no—
301C22×C4D4×D5311yescharacteristic
302C22×C4#1171994711yescharacteristic
303C22×C4C2×C22⋊D5311yescharacteristic
304SD16—42no—
305D8—42no—
306M16—1062no—
307M16—2122no—
308M16—2124no—
309M16—2121no—
310C2×C8—21no—
311C2×C8—21no—
312C2×C8—1061no—
313C2×C8—1061no—
314C2×C8—1062no—
315C2×C8—2121no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4C22⋊Dic5311yescharacteristic
321C4⋊C4C22⋊Dic5311yescharacteristic
322C22⋊C4—21no—
323C22⋊C4—21no—
324C22⋊C4—21no—
325C22⋊C4—41no—
326C22⋊C4—41no—
327C42—21no—
328C42—21no—
329C42—21no—
330C42—1062no—
331C42—2121no—
332C42D4×D5311yescharacteristic
333C42#1171994711yescharacteristic
334C42C22×Dic5311yescharacteristic
335C42C2×C22⋊D5311yescharacteristic
336C23—21no—
337C23—41no—
338C23#1313318911yescharacteristic
339Q8—21no—
340Q8—81no—
341Q8#1313314412yes—
342D4—21no—
343D4—21no—
344D4—42no—
345D4—41no—
346D4—41no—
347D4—81no—
348D4—81no—
349D4—82no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—42no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—81no—
361C2×C4—82no—
362C2×C4—4241no—
363C2×C4—4241no—
364C2×C4—2122no—
365C2×C4C2×C22⋊Dic5311yescharacteristic
366C2×C4D4×Dic5311yescharacteristic
367C2×C4D4×Dic5311yescharacteristic
368C2×C4#1313318511yescharacteristic
369C2×C4#1313318911yescharacteristic
370C2×C4#1313319811yescharacteristic
371C8—41no—
372C8—41no—
373C8—1062no—
374C8—2121no—
375C8—2122no—
376C8—2122no—
377C22—21no—
378C22—21no—
379C22—21no—
380C22—41no—
381C22—41no—
382C22—41no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22#1681050311yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—22no—
391C4—21no—
392C4—21no—
393C4—41no—
394C4—41no—
395C4—81no—
396C4—4242no—
397C4C2×#1313314411yescharacteristic
398C4#1681064011yescharacteristic
399C2—41no—
400C2—81no—
401C2—161no—
402C2#2027501811yescharacteristic
403C2#2027519711yescharacteristic
404C2#2027521811yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#3115C2
socleC22×C53#386
cosocleC2×C42×C53C23
hypercenter#3115D53
hypocenterC53#9162
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3115

Complements

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

NH ≅ G/Ntype
1. #20283735C2semidirect
2. #20274525C2semidirect
3. #20273307C2semidirect
4. #20273223C2semidirect
5. #20272607C2semidirect
6. #20272586C2semidirect
25. C53×#475C4semidirect
26. C53×#474C4semidirect
27. C53×#460C4semidirect
28. C53×#459C4semidirect
213. #475Dic53semidirect
214. #474Dic53semidirect
215. #460Dic53semidirect
216. #459Dic53semidirect
235. C53#9162semidirect

Maximal subgroups

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

subgroupindex
#202837352
#202745252
#202733072
#202732232
#202726072
#202725862
C53×#31152
#916253