Finite Group Databasebeta

← #27281381

Subgroups — #27281381

Order 13568.

492 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 103 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#20277530C211yesmaximal, characteristic
2#20275660C211yesmaximal, characteristic, complemented
3#20273313C211yesmaximal, characteristic
4#20273266C211yesmaximal, characteristic, complemented
5#20272581C211yesmaximal, characteristic, complemented
6#20272570C211yesmaximal, characteristic, complemented
7C53×#3215C211yesmaximal, characteristic
8#16810680C2211yescharacteristic
9C4×#11719917C2211yescharacteristic
10#16810530—21no—
11#16810516—21no—
12#16810516—21no—
13#16810503—21no—
14#16810498—21no—
15#16810491C2211yescharacteristic
16C4×C4⋊Dic53C2211yescharacteristic
17C22⋊C4×Dic53—21no—
18#16810382—21no—
19#16810382—21no—
20#16810363—22no—
21#16810360—21no—
22#16810360—21no—
23C53×#533C2211yescharacteristic
24C2×#124×C53C2211yescharacteristic
25C53×#497C412yescomplemented
26C53×#489C412yescomplemented
27C53×#464—22no—
28C53×#451—21no—
29C53×#450—21no—
30C53×#432C2211yescharacteristic
31C23×Dic53—21no—
32C2×#11719917—21no—
33C2×C4⋊Dic53—21no—
34C2×C4⋊Dic53—21no—
35C2×C22⋊Dic53—21no—
36C2×C22⋊Dic53—21no—
37C2×C22⋊Dic53—21no—
38C2×C4×Dic53—21no—
39C2×C4×Dic53D411yescharacteristic
40C2×C4×Dic53D411yescharacteristic
41C4⋊#88322—21no—
42C22⋊#88322—21no—
43#13133128—21no—
44#13133128—21no—
45C4×#88322—21no—
46#13133124—21no—
47#13133124—21no—
48#13133124—21no—
49#13133124—21no—
50#13133124—21no—
51#13133124D411yescharacteristic
52#13133124D411yescharacteristic
53C22×Q8×C53D411yescharacteristic
54C22×D4×C53D411yescharacteristic
55C2×Q16×C53—22no—
56C2×SD16×C53—22no—
57C4⋊Q8×C53C2×C411yescharacteristic, complemented
58C4⋊Q8×C53C2×C411yescharacteristic, complemented
59#125×C53—21no—
60#124×C53—21no—
61#124×C53—21no—
62#124×C53C2×C412yes—
63#124×C53D412yes—
64C22⋊Q8×C53—21no—
65C22⋊Q8×C53—21no—
66C4×Q8×C53—21no—
67C4×Q8×C53—21no—
68#117×C53—21no—
69C2×C22⋊C4×C53—21no—
70C2×C22⋊C4×C53—21no—
71C2×C42×C53C2311yescharacteristic
72C4⋊C8×C53—21no—
73#103×C53—22no—
74#102×C53—22no—
75C22⋊C8×C53—21no—
76#97×C53C2×C412yescomplemented
77C24×C53—21no—
78C2×Q8×C53—21no—
79C2×Q8×C53—21no—
80C2×Q8×C53—21no—
81C2×Q8×C53—81no—
82C2×Q8×C53C22⋊C412yes—
83C2×D4×C53—21no—
84C2×D4×C53—81no—
85C2×D4×C53C22⋊C412yes—
86C22×C4×C53—21no—
87C22×C4×C53C2×D411yescharacteristic
88C22×C4×C53C2×D411yescharacteristic
89C22×C4×C53C2×D411yescharacteristic
90C22×Dic53—21no—
91C22×Dic53—21no—
92C22×Dic53—21no—
93C22×Dic53—21no—
94C22×Dic53—42no—
95C22×Dic53—81no—
96C22×Dic53—81no—
97Q16×C53—82no—
98SD16×C53—82no—
99C2×C8×C53—22no—
100C4⋊C4×C53—21no—
101C4⋊C4×C53—21no—
102C4⋊C4×C53—21no—
103C4⋊C4×C53—21no—
104C4⋊C4×C53—41no—
105C4⋊C4×C53—41no—
106C22⋊C4×C53—22no—
107C22⋊C4×C53—22no—
108C22⋊C4×C53—22no—
109C22⋊C4×C53—22no—
110C22⋊C4×C53—41no—
111C42×C53—41no—
112C42×C53C22×C411yescharacteristic
113C42×C53C2×D411yescharacteristic
114C42×C53#4111yescharacteristic
115C42×C53#4111yescharacteristic
116C22⋊Dic53—42no—
117C22⋊Dic53—81no—
118C22⋊Dic53—81no—
119C4⋊Dic53—42no—
120C4⋊Dic53—81no—
121C4×Dic53—41no—
122C4×Dic53—41no—
123C4×Dic53—81no—
124#11719917—22no—
125#11719917—41no—
126C2×#88322—41no—
127C2×#88322—41no—
128C23×C53—22no—
129C23×C53—41no—
130C23×C53—41no—
131C23×C53—41no—
132C23×C53—41no—
133C23×C53—41no—
134C23×C53—41no—
135C23×C53#12011yescharacteristic
136Q8×C53—42no—
137Q8×C53—42no—
138Q8×C53—81no—
139Q8×C53—81no—
140D4×C53—42no—
141D4×C53—81no—
142C2×C4×C53—21no—
143C2×C4×C53—21no—
144C2×C4×C53—22no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53C2×C22⋊C411yescharacteristic
152C2×C4×C53C4×D411yescharacteristic
153C2×C4×C53C4×D411yescharacteristic
154C2×C4×C53#12111yescharacteristic
155C2×C4×C53#12111yescharacteristic
156C2×C4×C53#12311yescharacteristic
157C2×Dic53—22no—
158C2×Dic53—41no—
159C2×Dic53—41no—
160C2×Dic53—41no—
161C2×Dic53—41no—
162C2×Dic53—41no—
163C2×Dic53—41no—
164C2×Dic53—41no—
165C2×Dic53—41no—
166C2×Dic53—41no—
167C2×Dic53—41no—
168C2×Dic53—41no—
169C2×Dic53—84no—
170C8×C53—42no—
171#88322—42no—
172#88322—81no—
173#9379—531nomaximal, Sylow
174C22×C53—21no—
175C22×C53—21no—
176C22×C53—42no—
177C22×C53—42no—
178C22×C53—42no—
179C22×C53—81no—
180C22×C53—81no—
181C22×C53—81no—
182C22×C53—81no—
183C22×C53#38611yescharacteristic
184C22×C53#45511yescharacteristic
185C22×C53#45611yescharacteristic
186C4×C53—21no—
187C4×C53—21no—
188C4×C53—21no—
189C4×C53—21no—
190C4×C53—21no—
191C4×C53—21no—
192C4×C53—42no—
193C4×C53—81no—
194C4×C53—81no—
195Dic53—42no—
196Dic53—81no—
197Dic53—81no—
198Dic53—81no—
199#3215D5311yescharacteristic
200#2337—531no—
201#1986—531no—
202#1527—531no—
203#1515—531no—
204#1288—531no—
205#1285—531no—
206C2×C53—21no—
207C2×C53—21no—
208C2×C53—82no—
209C2×C53#188211yescharacteristic
210C2×C53#190611yescharacteristic
211C2×C53#191111yescharacteristic
212#533C2×D5311yescharacteristic
213C2×#124C2×D5311yescharacteristic
214#497Dic5312yescomplemented
215#489Dic5312yescomplemented
216#464—22no—
217#451—21no—
218#450—21no—
219#432—531no—
220#432C2×D5311yescharacteristic
221C4×M16—531no—
222#391—1061no—
223#388—1061no—
224#388—1061no—
225#386—1061no—
226#385—1061no—
227#383—531no—
228C4×C4⋊C4—531no—
229C4×C22⋊C4—1061no—
230#339—1061no—
231#339—1061no—
232#329—1062no—
233#328—1061no—
234#328—1061no—
235C53#937911yescharacteristic, complemented, Sylow
236C22×Q8C22⋊D5311yescharacteristic
237C22×D4C22⋊D5311yescharacteristic
238C23×C4—1061no—
239C2×Q16—22no—
240C2×SD16—22no—
241C2×M16—1061no—
242C4⋊Q8C2×Dic5311yescharacteristic, complemented
243C4⋊Q8C2×Dic5311yescharacteristic, complemented
244#125—21no—
245#124—21no—
246#124—21no—
247#124C2×Dic5312yes—
248#124C22⋊D5312yes—
249C22⋊Q8—21no—
250C22⋊Q8—21no—
251C4×Q8—21no—
252C4×Q8—21no—
253#117—21no—
254C2×C4⋊C4—1061no—
255C2×C4⋊C4—1061no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—1061no—
259C2×C22⋊C4—1061no—
260C2×C22⋊C4—1061no—
261C2×C42—531no—
262C2×C42—531no—
263C2×C42—1061no—
264C2×C42C22×D5311yescharacteristic
265C4⋊C8—21no—
266C4⋊C8—1061no—
267#103—22no—
268#102—22no—
269C22⋊C8—21no—
270C22⋊C8—1061no—
271#97—1061no—
272#97—1061no—
273#97C2×Dic5312yescomplemented
274C4×C8—1061no—
275#95—531no—
276#95—531no—
277#95—1061no—
278#95—1061no—
279#95—1061no—
280#95—1061no—
281#95—1061no—
282C24—21no—
283C2×Q8—21no—
284C2×Q8—21no—
285C2×Q8—21no—
286C2×Q8—81no—
287C2×Q8C22⋊Dic5312yes—
288C2×D4—21no—
289C2×D4—81no—
290C2×D4C22⋊Dic5312yes—
291C22×C4—21no—
292C22×C4—1061no—
293C22×C4—1061no—
294C22×C4—1061no—
295C22×C4—1061no—
296C22×C4—4241no—
297C22×C4—4241no—
298C22×C4—2122no—
299C22×C4D4×D5311yescharacteristic
300C22×C4D4×D5311yescharacteristic
301C22×C4C2×C22⋊D5311yescharacteristic
302Q16—82no—
303SD16—82no—
304M16—1062no—
305M16—2121no—
306C2×C8—22no—
307C2×C8—2121no—
308C2×C8—2121no—
309C4⋊C4—21no—
310C4⋊C4—21no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4—4241no—
316C4⋊C4—2122no—
317C22⋊C4—22no—
318C22⋊C4—22no—
319C22⋊C4—22no—
320C22⋊C4—22no—
321C22⋊C4—41no—
322C22⋊C4—4241no—
323C22⋊C4—4241no—
324C22⋊C4—2122no—
325C42—41no—
326C42—4241no—
327C42—2121no—
328C42—2121no—
329C42#1171994711yescharacteristic
330C42#1171994711yescharacteristic
331C42C22×Dic5311yescharacteristic
332C42C2×C22⋊D5311yescharacteristic
333C23—22no—
334C23—41no—
335C23—41no—
336C23—41no—
337C23—41no—
338C23—41no—
339C23—41no—
340C23#1313318511yescharacteristic
341Q8—42no—
342Q8—42no—
343Q8—81no—
344Q8—81no—
345D4—42no—
346D4—81no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—22no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—1062no—
357C2×C4—4244no—
358C2×C4—2121no—
359C2×C4—2121no—
360C2×C4—2121no—
361C2×C4—2121no—
362C2×C4—2121no—
363C2×C4—2121no—
364C2×C4—2121no—
365C2×C4—2121no—
366C2×C4—2121no—
367C2×C4—2121no—
368C2×C4—2121no—
369C2×C4C2×C22⋊Dic5311yescharacteristic
370C2×C4D4×Dic5311yescharacteristic
371C2×C4D4×Dic5311yescharacteristic
372C2×C4#1313318911yescharacteristic
373C2×C4#1313318911yescharacteristic
374C2×C4#1313319811yescharacteristic
375C8—42no—
376C8—4241no—
377C8—2122no—
378C22—21no—
379C22—21no—
380C22—42no—
381C22—42no—
382C22—42no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22#1681050311yescharacteristic
388C22#1681079911yescharacteristic
389C22#1681080411yescharacteristic
390C4—21no—
391C4—21no—
392C4—21no—
393C4—21no—
394C4—21no—
395C4—21no—
396C4—42no—
397C4—81no—
398C4—81no—
399C4—4241no—
400C4—4241no—
401C4—4241no—
402C4—2122no—
403C2—21no—
404C2—21no—
405C2—82no—
406C2#2027508611yescharacteristic
407C2#2027521811yescharacteristic
408C2#2027524811yescharacteristic
subgroupG/H
centerC22#16810503
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#3215C2
socleC22×C53#386
prime radicalC22×D4×C53D4
cosocleC2×C42×C53C23
hypercenter#3215D53
hypocenterC53#9379
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3215

Complements

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

NH ≅ G/Ntype
2. #20275660C2semidirect
4. #20273266C2semidirect
5. #20272581C2semidirect
6. #20272570C2semidirect
25. C53×#497C4semidirect
26. C53×#489C4semidirect
57. C4⋊Q8×C53C2×C4semidirect
58. C4⋊Q8×C53C2×C4semidirect
76. #97×C53C2×C4semidirect
214. #497Dic53semidirect
215. #489Dic53semidirect
235. C53#9379semidirect
242. C4⋊Q8C2×Dic53semidirect
243. C4⋊Q8C2×Dic53semidirect
273. #97C2×Dic53semidirect

Maximal subgroups

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

subgroupindex
#202775302
#202756602
#202733132
#202732662
#202725812
#202725702
C53×#32152
#937953