Finite Group Databasebeta

← #27369459

Subgroups — #27369459

Order 13568.

598 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#20285851C211yesmaximal, characteristic
2#20284837C211yesmaximal, characteristic
3#20283796C211yesmaximal, characteristic, complemented
4#20274161C212yesmaximal, complemented
5#20274133C212yesmaximal, complemented
6#20274117C212yesmaximal, complemented
7#20274095C212yesmaximal, complemented
8#20273493C211yesmaximal, characteristic, complemented
9#20273425C211yesmaximal, characteristic, complemented
10#20273375C211yesmaximal, characteristic
11C53×#3104C211yesmaximal, characteristic, complemented
12#16811173C2211yescharacteristic
13#16811026C2211yescharacteristic
14#16811019C2211yescharacteristic
15#16810971C2211yescharacteristic
16#16810971C2211yescharacteristic
17#16810947—22no—
18#16810936—21no—
19#16810920—22no—
20#16810918C2212yes—
21#16810823C2212yes—
22#16810762—21no—
23Q8×#88322—22no—
24Q8×#88322C2212yes—
25Q8×#88322C2212yes—
26#16810721—22no—
27#16810680C2211yescharacteristic
28#16810676—22no—
29#16810676C2211yescharacteristic
30#16810676C2211yescharacteristic
31#16810375C2212yes—
32#16810373C2212yes—
33#16810372C2212yes—
34#16810371C2212yes—
35#16810369C2212yes—
36#16810366C2212yes—
37C53×#533C2211yescharacteristic
38C2×C4×Q8×C53C2211yescharacteristic
39C53×#484—21no—
40C53×#477C2212yes—
41C53×#462C2212yes—
42C53×#451—21no—
43C4×Q16×C53—22no—
44C53×#431C2211yescharacteristic
45C2×#11719945—21no—
46C2×#11719924—22no—
47#13133217C2311yescharacteristic
48#13133214C2311yescharacteristic
49#13133205—21no—
50#13133191—21no—
51#13133191—21no—
52C4×D212—21no—
53C4×D212—21no—
54#13133150—22no—
55#13133150—22no—
56#13133148—22no—
57C4⋊#88322—22no—
58C4⋊#88322—22no—
59C4⋊#88322D412yes—
60C4⋊#88322C2312yes—
61C4⋊#88322C2312yes—
62#13133141—22no—
63#13133137—22no—
64#13133137—22no—
65#13133137—22no—
66C22⋊#88322—21no—
67C22⋊#88322—21no—
68C22⋊#88322D412yes—
69#13133128—21no—
70C4×#88322—21no—
71C4×#88322—22no—
72C4×#88322—22no—
73C4×#88322C2311yescharacteristic
74C4×#88322C2311yescharacteristic
75C22×Q8×C53D411yescharacteristic
76C2×Q16×C53—22no—
77C4⋊Q8×C53C2311yescharacteristic
78C4⋊Q8×C53C2311yescharacteristic
79#125×C53—21no—
80C22⋊Q8×C53—21no—
81C22⋊Q8×C53—21no—
82C4×Q8×C53—21no—
83C4×Q8×C53—21no—
84C4×Q8×C53—21no—
85C4×Q8×C53—21no—
86C4×Q8×C53D412yes—
87C4×Q8×C53C2312yes—
88#117×C53—21no—
89C2×C4⋊C4×C53—21no—
90C2×C4⋊C4×C53D411yescharacteristic
91C2×C42×C53—21no—
92C2×C42×C53C2311yescharacteristic
93#107×C53—22no—
94C4⋊C8×C53C2312yes—
95#103×C53—22no—
96#103×C53—22no—
97C22⋊C8×C53—21no—
98C4×C8×C53—21no—
99C2×Q8×C53—24no—
100C2×Q8×C53—21no—
101C2×Q8×C53—21no—
102C2×Q8×C53—21no—
103C2×Q8×C53C2×D412yes—
104C22×C4×C53—21no—
105C22×C4×C53—21no—
106C22×C4×C53C2×D411yescharacteristic
107C22×C4×C53C2×D411yescharacteristic
108C22×C4×C53#4111yescharacteristic
109C2×C22⋊D53—41no—
110#11719945—82no—
111C2×D212—21no—
112C2×D212—21no—
113C2×D212—21no—
114C2×C4×D53—41no—
115C2×Dic106—21no—
116Q16×C53—44no—
117C2×C8×C53—22no—
118C4⋊C4×C53—22no—
119C4⋊C4×C53—21no—
120C4⋊C4×C53—22no—
121C4⋊C4×C53—21no—
122C4⋊C4×C53—21no—
123C4⋊C4×C53—21no—
124C4⋊C4×C53—21no—
125C4⋊C4×C53—41no—
126C4⋊C4×C53—41no—
127C4⋊C4×C53C2×D412yes—
128C22⋊C4×C53—41no—
129C42×C53—22no—
130C42×C53—22no—
131C42×C53—41no—
132C42×C53C2×D411yescharacteristic
133C42×C53C2×D411yescharacteristic
134C42×C53#4111yescharacteristic
135C42×C53C2411yescharacteristic
136#11719924—44no—
137#11719921—41no—
138#11719921—41no—
139C4⋊Dic53—21no—
140C4⋊Dic53—21no—
141C2×#88322—22no—
142C2×#88322—22no—
143C2×#88322C2×D412yes—
144C2×#88322C2×D412yes—
145C23×C53#12111yescharacteristic
146C22×D53—41no—
147Q8×C53—24no—
148Q8×C53—22no—
149Q8×C53—42no—
150Q8×C53—81no—
151Q8×C53#13514yes—
152C2×C4×C53—24no—
153C2×C4×C53—22no—
154C2×C4×C53—22no—
155C2×C4×C53—22no—
156C2×C4×C53—22no—
157C2×C4×C53—41no—
158C2×C4×C53—41no—
159C2×C4×C53—41no—
160C2×C4×C53—81no—
161C2×C4×C53#12111yescharacteristic
162C2×C4×C53#12111yescharacteristic
163C2×C4×C53#12111yescharacteristic
164C2×C4×C53C22×D411yescharacteristic
165C2×C4×C53C22×D411yescharacteristic
166C2×C4×C53C2×#4111yescharacteristic
167C22⋊D53—82no—
168C2×Dic53—41no—
169D212—42no—
170D212—42no—
171D212—81no—
172C4×D53—82no—
173Dic106—81no—
174C8×C53—42no—
175#88322—22no—
176#88322—22no—
177#88322—42no—
178#88322—42no—
179#19406—531nomaximal, Sylow
180C22×C53—21no—
181C22×C53—21no—
182C22×C53—21no—
183C22×C53C2×#12111yescharacteristic
184C2×D53—81no—
185C2×D53—81no—
186C2×D53—81no—
187C4×C53—21no—
188C4×C53—24no—
189C4×C53—21no—
190C4×C53—21no—
191C4×C53—21no—
192C4×C53—42no—
193C4×C53—81no—
194C4×C53—81no—
195C4×C53C2×#13512yes—
196C4×C53#57812yes—
197Dic53—81no—
198#3228—531no—
199#3112—531no—
200#3104D5311yescharacteristic, complemented
201#2982—531no—
202#1682—532no—
203#1678—532no—
204#1676—532no—
205#1672—532no—
206#1572—531no—
207#1554—531no—
208#1543—531no—
209C2×C53—41no—
210C2×C53#304411yescharacteristic
211C2×C53#305011yescharacteristic
212C2×C53#306811yescharacteristic
213D53—161no—
214#533C2×D5311yescharacteristic
215#532—531no—
216C2×C4×Q8C2×D5311yescharacteristic
217#500—531no—
218#498—531no—
219#486—531no—
220#486—531no—
221#484—21no—
222#482—1062no—
223#480—1061no—
224#478—1062no—
225#477—532no—
226#477C2×D5312yes—
227#462C2×D5312yes—
228#460—532no—
229#451—21no—
230#448—1061no—
231C8×Q8—532no—
232C8×Q8—532no—
233C8×Q8—1062no—
234#440—1062no—
235C4×Q16—22no—
236#432—531no—
237#431—531no—
238#431—531no—
239#431—1062no—
240#431C2×D5311yescharacteristic
241#335—532no—
242#334—532no—
243#333—532no—
244#333—532no—
245#332—532no—
246#331—532no—
247C53#1940611yescharacteristic, complemented, Sylow
248C2×#41—1061no—
249C22×Q8C22⋊D5311yescharacteristic
250C2×Q16—22no—
251C2×SD16—1062no—
252C4⋊Q8—531no—
253C4⋊Q8C22×D5311yescharacteristic
254C4⋊Q8C22×D5311yescharacteristic
255#127—531no—
256#125—21no—
257#124—1061no—
258C22⋊Q8—21no—
259C22⋊Q8—21no—
260#121—1061no—
261#121—1061no—
262C4×Q8—21no—
263C4×Q8—21no—
264C4×Q8—21no—
265C4×Q8—21no—
266C4×Q8C22⋊D5312yes—
267C4×Q8C22×D5312yes—
268C4×D4—1061no—
269C4×D4—1061no—
270#117—21no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4C22⋊D5311yescharacteristic
273C2×C42—21no—
274C2×C42C22×D5311yescharacteristic
275#107—22no—
276#107—1062no—
277#107—1062no—
278#106—1062no—
279C4⋊C8—532no—
280C4⋊C8—532no—
281C4⋊C8—532no—
282C4⋊C8—1062no—
283C4⋊C8—1062no—
284C4⋊C8C22×D5312yes—
285#103—22no—
286#103—22no—
287#103—1062no—
288#102—1062no—
289#102—1062no—
290#102—1062no—
291C22⋊C8—21no—
292C22⋊C8—532no—
293C22⋊C8—1061no—
294C22⋊C8—1061no—
295#97—1061no—
296C4×C8—21no—
297C4×C8—531no—
298C4×C8—531no—
299C4×C8—1062no—
300C4×C8—1062no—
301C4×C8—1061no—
302#41—4242no—
303C2×Q8—24no—
304C2×Q8—21no—
305C2×Q8—21no—
306C2×Q8—21no—
307C2×Q8—1061no—
308C2×Q8C2×C22⋊D5312yes—
309C2×D4—1061no—
310C2×D4—1061no—
311C2×D4—1061no—
312C2×D4—2121no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—2121no—
316C22×C4D4×D5311yescharacteristic
317C22×C4#1171994711yescharacteristic
318C22×C4C2×C22⋊D5311yescharacteristic
319Q16—44no—
320SD16—2124no—
321C2×C8—22no—
322C2×C8—532no—
323C2×C8—532no—
324C2×C8—1062no—
325C2×C8—1062no—
326C4⋊C4—22no—
327C4⋊C4—21no—
328C4⋊C4—22no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—1061no—
336C4⋊C4—1061no—
337C4⋊C4C2×C22⋊D5312yes—
338C22⋊C4—41no—
339C22⋊C4—2121no—
340C22⋊C4—2121no—
341C42—22no—
342C42—22no—
343C42—41no—
344C42D4×D5311yescharacteristic
345C42#1171994711yescharacteristic
346C42C2×C22⋊D5311yescharacteristic
347C42C23×D5311yescharacteristic
348C23—2121no—
349C23#1313318911yescharacteristic
350Q8—24no—
351Q8—22no—
352Q8—42no—
353Q8—81no—
354Q8—4241no—
355Q8#1313323714yes—
356D4—4241no—
357D4—4242no—
358D4—2122no—
359D4—2122no—
360C2×C4—24no—
361C2×C4—22no—
362C2×C4—22no—
363C2×C4—22no—
364C2×C4—22no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—81no—
369C2×C4—4242no—
370C2×C4—2121no—
371C2×C4#1313318911yescharacteristic
372C2×C4#1313318911yescharacteristic
373C2×C4#1313318911yescharacteristic
374C2×C4C2×D4×D5311yescharacteristic
375C2×C4C22×C22⋊D5311yescharacteristic
376C2×C4C2×#1171994711yescharacteristic
377C8—42no—
378C8—1062no—
379C8—1062no—
380C8—2122no—
381C8—2122no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—4241no—
386C22—4241no—
387C22—4241no—
388C22C2×#1313318911yescharacteristic
389C4—21no—
390C4—21no—
391C4—21no—
392C4—24no—
393C4—21no—
394C4—42no—
395C4—81no—
396C4—81no—
397C4—4241no—
398C4C2×#1313323712yes—
399C4#1681143612yes—
400C2—41no—
401C2—8481no—
402C2#2028424211yescharacteristic
403C2#2028429311yescharacteristic
404C2#2028444511yescharacteristic
subgroupG/H
centerC22C2×#13133189
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#3104C2
socleC22×C53C2×#121
prime radical#16811173C22
hypercenter#3104D53
hypocenterC53#19406
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3104

Complements

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

NH ≅ G/Ntype
3. #20283796C2semidirect
4. #20274161C2semidirect
5. #20274133C2semidirect
6. #20274117C2semidirect
7. #20274095C2semidirect
8. #20273493C2semidirect
9. #20273425C2semidirect
11. C53×#3104C2semidirect
200. #3104D53semidirect
247. C53#19406semidirect

Maximal subgroups

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

subgroupindex
#202858512
#202848372
#202837962
#202741612
#202741332
#202741172
#202740952
#202734932
#202734252
#202733752
C53×#31042
#1940653