Finite Group Databasebeta

← #27280905

Subgroups — #27280905

Order 13568.

514 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283143C211yesmaximal, characteristic, complemented
2#20273326C211yesmaximal, characteristic, complemented
3#20273149C211yesmaximal, characteristic, complemented
4#20273132C211yesmaximal, characteristic, complemented
5#20272613C211yesmaximal, characteristic, complemented
6#20272586C211yesmaximal, characteristic, complemented
7C53×#3115C211yesmaximal, characteristic
8C2×#13133224—21no—
9#16810680C2211yescharacteristic
10#16810676C2211yescharacteristic
11#16810579—22no—
12#16810579—21no—
13C4×#11719917—21no—
14C4×#11719917C2211yescharacteristic
15C2×#13133128—21no—
16C2×#13133128C2211yescharacteristic
17#16810363—21no—
18#16810363—21no—
19Q8⋊#88322—21no—
20D4⋊#88322—22no—
21D4⋊#88322—21no—
22C53×#532C2211yescharacteristic
23C4×#41×C53C2211yescharacteristic
24C53×#482—21no—
25C53×#475C411yescharacteristic
26C53×#474C411yescharacteristic
27C53×#460C411yescharacteristic
28C53×#459C411yescharacteristic
29C53×#449—21no—
30C53×#442—21no—
31C53×#440—21no—
32C53×#432C2211yescharacteristic
33#13133224—24no—
34#13133224—24no—
35C2×#11719917—22no—
36C2×#11719917—21no—
37C22×#88322—22no—
38C22×#88322—21no—
39C4⋊#88322—21no—
40C4⋊#88322—21no—
41C22⋊#88322—21no—
42C22⋊#88322—21no—
43#13133128—22no—
44#13133128—21no—
45#13133128—21no—
46#13133128—21no—
47#13133128D411yescharacteristic
48#13133128D411yescharacteristic
49C4×#88322—22no—
50C4×#88322—21no—
51C4×#88322D412yes—
52C2×#41×C53—21no—
53C2×#41×C53D411yescharacteristic
54C2×SD16×C53—21no—
55C2×D8×C53—21no—
56C4⋊Q8×C53C2×C411yescharacteristic
57#127×C53C2×C411yescharacteristic
58#124×C53—21no—
59#121×C53—21no—
60#121×C53—21no—
61C4×Q8×C53C2×C411yescharacteristic
62C4×Q8×C53D411yescharacteristic
63C4×D4×C53—21no—
64C4×D4×C53—21no—
65C4×D4×C53—21no—
66C4×D4×C53—21no—
67C4×D4×C53C2×C411yescharacteristic
68C4×D4×C53D411yescharacteristic
69#117×C53—21no—
70#117×C53D411yescharacteristic
71C2×C42×C53—21no—
72C2×C42×C53C2311yescharacteristic
73#107×C53—21no—
74#106×C53—21no—
75C4⋊C8×C53C2×C411yescharacteristic
76C4⋊C8×C53C2×C411yescharacteristic
77#103×C53—21no—
78#102×C53—21no—
79#102×C53—21no—
80#102×C53—21no—
81C22⋊C8×C53—21no—
82#97×C53—21no—
83#41×C53—22no—
84#41×C53—22no—
85#41×C53—82no—
86C2×Q8×C53—21no—
87C2×Q8×C53C22⋊C411yescharacteristic
88C2×D4×C53—21no—
89C2×D4×C53—21no—
90C2×D4×C53—21no—
91C2×D4×C53—21no—
92C2×D4×C53—41no—
93C2×D4×C53C22⋊C411yescharacteristic
94C22×C4×C53—21no—
95C22×C4×C53—21no—
96C22×C4×C53—21no—
97C22×C4×C53—41no—
98C22×C4×C53C2×D411yescharacteristic
99C22×C4×C53#4111yescharacteristic
100C22×C4×C53#4111yescharacteristic
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×C53C2×D411yescharacteristic
123#11719917—28no—
124#11719917—22no—
125#11719917—22no—
126C2×#88322—28no—
127C2×#88322—22no—
128C2×#88322—22no—
129C2×#88322—21no—
130C2×#88322—22no—
131C2×#88322—21no—
132C2×#88322—41no—
133C2×#88322—41no—
134C23×C53—21no—
135C23×C53—41no—
136C23×C53#12311yescharacteristic
137Q8×C53—21no—
138Q8×C53—81no—
139Q8×C53#10412yes—
140D4×C53—24no—
141D4×C53—21no—
142D4×C53—42no—
143D4×C53—41no—
144D4×C53—41no—
145D4×C53—81no—
146D4×C53—82no—
147D4×C53#10412yes—
148C2×C4×C53—24no—
149C2×C4×C53—21no—
150C2×C4×C53—21no—
151C2×C4×C53—21no—
152C2×C4×C53—21no—
153C2×C4×C53—21no—
154C2×C4×C53—41no—
155C2×C4×C53—41no—
156C2×C4×C53—41no—
157C2×C4×C53—81no—
158C2×C4×C53—81no—
159C2×C4×C53—82no—
160C2×C4×C53C2×C22⋊C411yescharacteristic
161C2×C4×C53C4×D411yescharacteristic
162C2×C4×C53C4×D411yescharacteristic
163C2×C4×C53#12011yescharacteristic
164C2×C4×C53#12111yescharacteristic
165C2×C4×C53#12111yescharacteristic
166C8×C53—41no—
167C8×C53—41no—
168#88322—22no—
169#88322—22no—
170#88322—24no—
171#88322—81no—
172#88322—81no—
173#9311—531nomaximal, Sylow
174C22×C53—21no—
175C22×C53—21no—
176C22×C53—22no—
177C22×C53—21no—
178C22×C53—41no—
179C22×C53—41no—
180C22×C53—81no—
181C22×C53—81no—
182C22×C53—81no—
183C22×C53#38611yescharacteristic
184C4×C53—21no—
185C4×C53—21no—
186C4×C53—21no—
187C4×C53—22no—
188C4×C53—21no—
189C4×C53—21no—
190C4×C53—41no—
191C4×C53—41no—
192C4×C53—81no—
193C4×C53C2×#10411yescharacteristic
194C4×C53C2×#10411yescharacteristic
195#3115D5311yescharacteristic
196#2875—531no—
197#1530—531no—
198#1488—531no—
199#1484—531no—
200#1302—531no—
201#1291—531no—
202C2×C53—41no—
203C2×C53—42no—
204C2×C53—161no—
205C2×C53#188111yescharacteristic
206C2×C53#189611yescharacteristic
207C2×C53#190311yescharacteristic
208C2×#131—1061no—
209#532C2×D5311yescharacteristic
210C4×#41C2×D5311yescharacteristic
211#482—21no—
212#475Dic5311yescharacteristic
213#474Dic5311yescharacteristic
214#460Dic5311yescharacteristic
215#459Dic5311yescharacteristic
216#449—21no—
217#442—21no—
218#440—21no—
219#432—531no—
220#432C2×D5311yescharacteristic
221#431—531no—
222#405—1062no—
223#405—1061no—
224C4×M16—531no—
225C4×M16—1061no—
226C2×#97—531no—
227C2×#97—1061no—
228#329—1061no—
229#329—1061no—
230Q8⋊C8—1061no—
231D4⋊C8—1062no—
232D4⋊C8—1061no—
233C53#931111yescharacteristic, complemented, Sylow
234C2×#41—21no—
235C2×#41C22⋊D5311yescharacteristic
236C2×SD16—21no—
237C2×D8—21no—
238#131—1064no—
239#131—1064no—
240C2×M16—1062no—
241C2×M16—1061no—
242C22×C8—1062no—
243C22×C8—1061no—
244C4⋊Q8C2×Dic5311yescharacteristic
245#127C2×Dic5311yescharacteristic
246#124—21no—
247#121—21no—
248#121—21no—
249C4×Q8C2×Dic5311yescharacteristic
250C4×Q8C22⋊D5311yescharacteristic
251C4×D4—21no—
252C4×D4—21no—
253C4×D4—21no—
254C4×D4—21no—
255C4×D4C2×Dic5311yescharacteristic
256C4×D4C22⋊D5311yescharacteristic
257#117—21no—
258#117C22⋊D5311yescharacteristic
259C2×C42—21no—
260C2×C42C22×D5311yescharacteristic
261#107—21no—
262#106—21no—
263C4⋊C8—1061no—
264C4⋊C8—1061no—
265C4⋊C8C2×Dic5311yescharacteristic
266C4⋊C8C2×Dic5311yescharacteristic
267#103—21no—
268#102—21no—
269#102—21no—
270#102—21no—
271C22⋊C8—21no—
272C22⋊C8—1061no—
273C22⋊C8—1061no—
274#97—21no—
275#97—531no—
276#97—531no—
277#97—1062no—
278#97—1061no—
279#97—1061no—
280#97—1061no—
281C4×C8—532no—
282C4×C8—1062no—
283C4×C8—1061no—
284#41—22no—
285#41—22no—
286#41—82no—
287C2×Q8—21no—
288C2×Q8C22⋊Dic5311yescharacteristic
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4—21no—
293C2×D4—41no—
294C2×D4C22⋊Dic5311yescharacteristic
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—41no—
299C22×C4#1171994711yescharacteristic
300C22×C4#1171994711yescharacteristic
301C22×C4C2×C22⋊D5311yescharacteristic
302SD16—42no—
303D8—42no—
304M16—1068no—
305M16—1062no—
306M16—1062no—
307C2×C8—21no—
308C2×C8—21no—
309C2×C8—1068no—
310C2×C8—1062no—
311C2×C8—1062no—
312C2×C8—1062no—
313C2×C8—1061no—
314C2×C8—1061no—
315C2×C8—2121no—
316C2×C8—2121no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—21no—
321C4⋊C4C22⋊Dic5311yescharacteristic
322C4⋊C4C22⋊Dic5311yescharacteristic
323C22⋊C4—21no—
324C22⋊C4—21no—
325C22⋊C4—21no—
326C22⋊C4—41no—
327C22⋊C4—41no—
328C42—21no—
329C42—21no—
330C42—21no—
331C42D4×D5311yescharacteristic
332C42D4×D5311yescharacteristic
333C42C22×Dic5311yescharacteristic
334C42C2×C22⋊D5311yescharacteristic
335C23—21no—
336C23—41no—
337C23#1313319811yescharacteristic
338Q8—21no—
339Q8—81no—
340Q8#1313314412yes—
341D4—24no—
342D4—21no—
343D4—42no—
344D4—41no—
345D4—41no—
346D4—81no—
347D4—82no—
348D4#1313314412yes—
349C2×C4—24no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—81no—
359C2×C4—81no—
360C2×C4—82no—
361C2×C4C2×C22⋊Dic5311yescharacteristic
362C2×C4D4×Dic5311yescharacteristic
363C2×C4D4×Dic5311yescharacteristic
364C2×C4#1313318511yescharacteristic
365C2×C4#1313318911yescharacteristic
366C2×C4#1313318911yescharacteristic
367C8—41no—
368C8—41no—
369C8—1062no—
370C8—1064no—
371C8—1062no—
372C8—4241no—
373C8—4241no—
374C22—21no—
375C22—21no—
376C22—22no—
377C22—21no—
378C22—41no—
379C22—41no—
380C22—81no—
381C22—81no—
382C22—81no—
383C22#1681050311yescharacteristic
384C4—21no—
385C4—21no—
386C4—21no—
387C4—22no—
388C4—21no—
389C4—21no—
390C4—41no—
391C4—41no—
392C4—81no—
393C4C2×#1313314411yescharacteristic
394C4C2×#1313314411yescharacteristic
395C2—41no—
396C2—42no—
397C2—161no—
398C2#2027507911yescharacteristic
399C2#2027515911yescharacteristic
400C2#2027519711yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC53×#3115C2
socleC22×C53#386
cosocleC2×C42×C53C23
hypercenter#3115D53
hypocenterC53#9311
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. #20283143C2semidirect
2. #20273326C2semidirect
3. #20273149C2semidirect
4. #20273132C2semidirect
5. #20272613C2semidirect
6. #20272586C2semidirect
233. C53#9311semidirect

Maximal subgroups

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

subgroupindex
#202831432
#202733262
#202731492
#202731322
#202726132
#202725862
C53×#31152
#931153