Finite Group Databasebeta

← #27369462

Subgroups — #27369462

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#20285853C211yesmaximal, characteristic
2#20284837C211yesmaximal, characteristic
3#20284754C211yesmaximal, characteristic, complemented
4#20274159C212yesmaximal, complemented
5#20274131C212yesmaximal, complemented
6#20274119C212yesmaximal, complemented
7#20274097C212yesmaximal, complemented
8#20273495C211yesmaximal, characteristic, complemented
9#20273423C211yesmaximal, characteristic, complemented
10#20273375C211yesmaximal, characteristic
11C53×#2982C211yesmaximal, characteristic, complemented
12#16811180C2211yescharacteristic
13#16811173C2211yescharacteristic
14#16811028C2211yescharacteristic
15#16811021C2211yescharacteristic
16#16810970C2211yescharacteristic
17#16810970C2211yescharacteristic
18#16810958—21no—
19#16810948—22no—
20#16810936—21no—
21#16810924—22no—
22#16810918C2212yes—
23#16810918C2212yes—
24#16810833C2212yes—
25#16810823C2212yes—
26#16810779—21no—
27#16810762—21no—
28#16810721—22no—
29C4×#11719925—22no—
30#16810680C2211yescharacteristic
31#16810676C2211yescharacteristic
32#16810376C2212yes—
33#16810374C2212yes—
34#16810371C2212yes—
35#16810371C2212yes—
36#16810368C2212yes—
37#16810366C2212yes—
38C2×C4×Q8×C53C2211yescharacteristic
39C8×Q8×C53—22no—
40C8×Q8×C53C2212yes—
41C8×Q8×C53C2212yes—
42C53×#431—22no—
43C53×#431C2211yescharacteristic
44C53×#431C2211yescharacteristic
45C2×#11719945—21no—
46C2×#11719925—22no—
47C2×#11719924—22no—
48#13133217C2311yescharacteristic
49#13133217C2311yescharacteristic
50#13133217C2311yescharacteristic
51#13133214C2311yescharacteristic
52#13133209—21no—
53#13133205—21no—
54#13133196—21no—
55#13133196—21no—
56#13133191—21no—
57#13133191—21no—
58C4×Dic106—21no—
59C4×Dic106—21no—
60C4×D212—21no—
61C4×D212—21no—
62#13133176—21no—
63#13133151—22no—
64#13133150—22no—
65#13133150—22no—
66#13133149—22no—
67C4⋊#88322C2312yes—
68C4⋊#88322C2312yes—
69#13133141—22no—
70#13133141—22no—
71#13133140—22no—
72#13133139—22no—
73#13133139—22no—
74#13133137—22no—
75C22⋊#88322—21no—
76C22⋊#88322—21no—
77#13133128—21no—
78C4×#88322—21no—
79C22×Q8×C53D411yescharacteristic
80C4×Q8×C53—21no—
81C4×Q8×C53—21no—
82C4×Q8×C53D412yes—
83C4×Q8×C53C2312yes—
84C2×C4⋊C4×C53—21no—
85C2×C4⋊C4×C53D411yescharacteristic
86C2×C42×C53—21no—
87C2×C42×C53C2311yescharacteristic
88C4⋊C8×C53—22no—
89C4⋊C8×C53—22no—
90C4⋊C8×C53D412yes—
91C4⋊C8×C53C2312yes—
92C22⋊C8×C53—21no—
93C22⋊C8×C53D412yes—
94C4×C8×C53—22no—
95C4×C8×C53—22no—
96C4×C8×C53—21no—
97C4×C8×C53C2311yescharacteristic
98C4×C8×C53C2311yescharacteristic
99C2×Q8×C53—24no—
100C2×Q8×C53—21no—
101C2×Q8×C53C2×D412yes—
102C22×C4×C53—21no—
103C22×C4×C53—21no—
104C22×C4×C53C2×D411yescharacteristic
105C22×C4×C53C2×D411yescharacteristic
106C22×C4×C53#4111yescharacteristic
107C2×C22⋊D53—41no—
108#11719945—82no—
109C2×D212—21no—
110C2×D212—21no—
111C2×D212—21no—
112C2×C4×D53—41no—
113C2×Dic106—21no—
114C2×Dic106—21no—
115C2×Dic106—21no—
116C2×C8×C53—22no—
117C2×C8×C53C2×D412yes—
118C2×C8×C53C2×D412yes—
119C4⋊C4×C53—22no—
120C4⋊C4×C53—21no—
121C4⋊C4×C53—22no—
122C4⋊C4×C53C2×D412yes—
123C42×C53—22no—
124C42×C53—22no—
125C42×C53C2×D411yescharacteristic
126C42×C53C2×D411yescharacteristic
127C42×C53#4111yescharacteristic
128C42×C53C2411yescharacteristic
129C22⋊Dic53—41no—
130#11719925—44no—
131#11719924—44no—
132#11719921—41no—
133#11719921—41no—
134C4⋊Dic53—21no—
135C4⋊Dic53—21no—
136C4⋊Dic53—21no—
137C4⋊Dic53—21no—
138C4⋊Dic53—21no—
139C4⋊Dic53—21no—
140#11719919—41no—
141#11719919—41no—
142C4×Dic53—41no—
143C2×#88322—22no—
144C2×#88322—22no—
145C23×C53#12111yescharacteristic
146C22×D53—41no—
147Q8×C53—24no—
148Q8×C53—22no—
149Q8×C53#13514yes—
150C2×C4×C53—24no—
151C2×C4×C53—22no—
152C2×C4×C53—22no—
153C2×C4×C53—22no—
154C2×C4×C53—22no—
155C2×C4×C53—41no—
156C2×C4×C53—81no—
157C2×C4×C53#12111yescharacteristic
158C2×C4×C53#12111yescharacteristic
159C2×C4×C53#12111yescharacteristic
160C2×C4×C53C22×D411yescharacteristic
161C2×C4×C53C22×D411yescharacteristic
162C2×C4×C53C2×#4111yescharacteristic
163C22⋊D53—82no—
164C2×Dic53—41no—
165C2×Dic53—41no—
166C2×Dic53—41no—
167D212—42no—
168D212—42no—
169D212—81no—
170C4×D53—82no—
171Dic106—42no—
172Dic106—81no—
173Dic106—81no—
174C8×C53—22no—
175C8×C53—22no—
176C8×C53—42no—
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×C53C2×#13512yes—
194C4×C53#57812yes—
195Dic53—81no—
196Dic53—81no—
197Dic53—81no—
198#3228—531no—
199#3112—531no—
200#3104—531no—
201#2982D5311yescharacteristic, complemented
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#533—531no—
215#532—531no—
216C2×C4×Q8C2×D5311yescharacteristic
217#500—531no—
218#498—531no—
219#486—531no—
220#486—531no—
221#484—1061no—
222#482—1062no—
223#480—1061no—
224#478—1062no—
225#477—532no—
226#477—532no—
227#462—532no—
228#460—532no—
229#451—1061no—
230#448—1061no—
231C8×Q8—22no—
232C8×Q8C2×D5312yes—
233C8×Q8C2×D5312yes—
234#440—1062no—
235C4×Q16—1062no—
236#432—531no—
237#431—22no—
238#431—531no—
239#431C2×D5311yescharacteristic
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×Q8D21211yescharacteristic
250C2×Q16—1062no—
251C2×SD16—1062no—
252C4⋊Q8—531no—
253C4⋊Q8—531no—
254C4⋊Q8—531no—
255#127—531no—
256#125—1061no—
257#124—1061no—
258C22⋊Q8—1061no—
259C22⋊Q8—1061no—
260#121—1061no—
261#121—1061no—
262C4×Q8—21no—
263C4×Q8—21no—
264C4×Q8—1061no—
265C4×Q8—1061no—
266C4×Q8D21212yes—
267C4×Q8C22×D5312yes—
268C4×D4—1061no—
269C4×D4—1061no—
270#117—1061no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4D21211yescharacteristic
273C2×C42—21no—
274C2×C42C22×D5311yescharacteristic
275#107—1062no—
276#107—1062no—
277#107—1062no—
278#106—1062no—
279C4⋊C8—22no—
280C4⋊C8—22no—
281C4⋊C8—532no—
282C4⋊C8—532no—
283C4⋊C8C22⋊D5312yes—
284C4⋊C8C22×D5312yes—
285#103—1062no—
286#103—1062no—
287#103—1062no—
288#102—1062no—
289#102—1062no—
290#102—1062no—
291C22⋊C8—21no—
292C22⋊C8—1061no—
293C22⋊C8—1061no—
294C22⋊C8C22⋊D5312yes—
295#97—1061no—
296C4×C8—22no—
297C4×C8—22no—
298C4×C8—21no—
299C4×C8—1061no—
300C4×C8C22×D5311yescharacteristic
301C4×C8C22×D5311yescharacteristic
302#41—4242no—
303C2×Q8—24no—
304C2×Q8—21no—
305C2×Q8—1061no—
306C2×Q8—1061no—
307C2×Q8—1061no—
308C2×Q8C2×D21212yes—
309C2×D4—1061no—
310C2×D4—1061no—
311C2×D4—1061no—
312C2×D4—2121no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—2121no—
316C22×C4C2×D21211yescharacteristic
317C22×C4#1171994511yescharacteristic
318C22×C4C2×C22⋊D5311yescharacteristic
319Q16—2124no—
320SD16—2124no—
321C2×C8—22no—
322C2×C8—1062no—
323C2×C8—1062no—
324C2×C8C2×C22⋊D5312yes—
325C2×C8C2×C22⋊D5312yes—
326C4⋊C4—22no—
327C4⋊C4—21no—
328C4⋊C4—22no—
329C4⋊C4—1061no—
330C4⋊C4—1061no—
331C4⋊C4—1061no—
332C4⋊C4—1061no—
333C4⋊C4—1061no—
334C4⋊C4—1061no—
335C4⋊C4—2121no—
336C4⋊C4—2121no—
337C4⋊C4C2×D21212yes—
338C22⋊C4—2121no—
339C22⋊C4—2121no—
340C22⋊C4—2121no—
341C42—22no—
342C42—22no—
343C42—2121no—
344C42C2×D21211yescharacteristic
345C42#1171994511yescharacteristic
346C42C2×C22⋊D5311yescharacteristic
347C42C23×D5311yescharacteristic
348C23—2121no—
349C23#1313319111yescharacteristic
350Q8—24no—
351Q8—22no—
352Q8—4241no—
353Q8—4241no—
354Q8—2122no—
355Q8#1313323914yes—
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—81no—
367C2×C4—4242no—
368C2×C4—2121no—
369C2×C4—2121no—
370C2×C4—2121no—
371C2×C4#1313319111yescharacteristic
372C2×C4#1313319111yescharacteristic
373C2×C4#1313319111yescharacteristic
374C2×C4C22×C22⋊D5311yescharacteristic
375C2×C4C22×D21211yescharacteristic
376C2×C4C2×#1171994511yescharacteristic
377C8—22no—
378C8—22no—
379C8—42no—
380C8—2122no—
381C8—2122no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—4241no—
386C22—4241no—
387C22—4241no—
388C22C2×#1313319111yescharacteristic
389C4—21no—
390C4—21no—
391C4—21no—
392C4—24no—
393C4—21no—
394C4—42no—
395C4—4241no—
396C4—4241no—
397C4—4241no—
398C4C2×#1313323912yes—
399C4#1681143812yes—
400C2—41no—
401C2—8481no—
402C2#2028424511yescharacteristic
403C2#2028429611yescharacteristic
404C2#2028444311yescharacteristic
subgroupG/H
centerC22C2×#13133191
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
FittingC53×#2982C2
socleC22×C53C2×#121
prime radical#16811173C22
hypercenter#2982D53
hypocenterC53#19406
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2982

Complements

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

NH ≅ G/Ntype
3. #20284754C2semidirect
4. #20274159C2semidirect
5. #20274131C2semidirect
6. #20274119C2semidirect
7. #20274097C2semidirect
8. #20273495C2semidirect
9. #20273423C2semidirect
11. C53×#2982C2semidirect
201. #2982D53semidirect
247. C53#19406semidirect

Maximal subgroups

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

subgroupindex
#202858532
#202848372
#202847542
#202741592
#202741312
#202741192
#202740972
#202734952
#202734232
#202733752
C53×#29822
#1940653