Finite Group Databasebeta

← #27469610

Subgroups — #27469610

Order 13568.

726 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283020C211yesmaximal, characteristic
2#20277120C211yesmaximal, characteristic
3#20276202C212yesmaximal, complemented
4#20276201C212yesmaximal, complemented
5#20275685C211yesmaximal, characteristic, complemented
6#20274917C212yesmaximal
7#20274915C212yesmaximal
8#20274883C211yesmaximal, characteristic, complemented
9#20274881C211yesmaximal, characteristic, complemented
10#20274478C211yesmaximal, characteristic
11C53×#1992C211yesmaximal, characteristic, complemented
12C2×#13133175C2212yes—
13C2×C42×D53C2211yescharacteristic
14C2×#13133151C2212yes—
15C2×#13133150C2212yes—
16C2×#13133149C2212yes—
17C2×#13133148C2212yes—
18C2×#13133129C2212yes—
19C2×#13133128C2211yescharacteristic
20#16810519C2211yescharacteristic
21#16810519C2211yescharacteristic
22#16810504—21no—
23#16810504—21no—
24#16810494—21no—
25#16810494—21no—
26#16810493C2211yescharacteristic
27#16810489C2211yescharacteristic
28#16810488C2212yes—
29#16810480—21no—
30#16810480—21no—
31#16810480C2211yescharacteristic
32#16810480C2211yescharacteristic
33C4⋊C4×Dic53—21no—
34C4⋊C4×Dic53—21no—
35#16810385C2212yes—
36#16810385C2212yes—
37#16810377C2212yes—
38C2×#125×C53C2211yescharacteristic
39C2×#97×C53C2211yescharacteristic
40C53×#384C2211yescharacteristic
41C53×#340C2212yes—
42C53×#340C2212yes—
43C22×C4×D53D411yescharacteristic
44C22×C4×D53D411yescharacteristic
45C22×C4×D53C2311yescharacteristic
46C22×#88322C2312yes—
47#13133175—42no—
48#13133175—42no—
49C2×C4⋊Dic53—21no—
50C2×C4⋊Dic53—21no—
51C2×C4⋊Dic53C2311yescharacteristic
52C2×C4⋊Dic53C2311yescharacteristic
53C2×C4⋊Dic53C2312yes—
54C2×#11719919—21no—
55C2×#11719919—21no—
56C2×#11719921—22no—
57C2×#11719921—21no—
58C2×#11719921—21no—
59C2×C4×Dic53—22no—
60C2×C4×Dic53D411yescharacteristic
61C2×C4×Dic53D411yescharacteristic
62C2×C4×Dic53C2311yescharacteristic
63C42×D53—22no—
64C42×D53—22no—
65#13133151—42no—
66#13133150—42no—
67#13133149—42no—
68#13133148—42no—
69#13133129Q818yes—
70#13133128—22no—
71#13133124—21no—
72#13133124—21no—
73#13133124—21no—
74#13133124—21no—
75#13133124—21no—
76#13133124—21no—
77#13133124—22no—
78#13133124—22no—
79C22×C8×C53C2312yes—
80#125×C53—22no—
81#125×C53—22no—
82C2×C4⋊C4×C53—21no—
83C2×C4⋊C4×C53—21no—
84C2×C4⋊C4×C53—21no—
85C2×C4⋊C4×C53—21no—
86C2×C4⋊C4×C53C2311yescharacteristic
87C2×C4⋊C4×C53C2311yescharacteristic
88C2×C4⋊C4×C53C2312yes—
89C2×C42×C53C2311yescharacteristic
90#97×C53—22no—
91C23×D53C2×D411yescharacteristic
92C22×C4×C53—22no—
93C22×C4×C53—21no—
94C22×C4×C53—21no—
95C22×C4×C53C2×D411yescharacteristic
96C22×C4×C53C2×D411yescharacteristic
97C22×C4×C53C2411yescharacteristic
98C22×Dic53—21no—
99C22×Dic53—21no—
100C22×Dic53—22no—
101C22×Dic53C2×D411yescharacteristic
102C22×Dic53C2×D411yescharacteristic
103C22×Dic53C2×D411yescharacteristic
104C2×C4×D53—21no—
105C2×C4×D53—21no—
106C2×C4×D53—22no—
107C2×C4×D53—22no—
108C2×C4×D53—22no—
109C2×C4×D53—41no—
110C2×C4×D53—41no—
111C2×C4×D53—41no—
112C2×C4×D53—41no—
113C2×C4×D53C2×Q814yes—
114C2×C8×C53—22no—
115C2×C8×C53—22no—
116C2×C8×C53C2×Q814yes—
117C4⋊C4×C53—21no—
118C4⋊C4×C53—21no—
119C4⋊C4×C53—21no—
120C4⋊C4×C53—21no—
121C4⋊C4×C53—22no—
122C4⋊C4×C53—22no—
123C4⋊C4×C53—42no—
124C4⋊C4×C53—42no—
125C4⋊C4×C53—42no—
126C4⋊C4×C53—42no—
127C42×C53#4112yes—
128C42×C53#4112yes—
129#11719921—42no—
130#11719921—42no—
131#11719921—44no—
132C4⋊Dic53—22no—
133C4⋊Dic53—22no—
134C4⋊Dic53—22no—
135C4⋊Dic53—22no—
136C4⋊Dic53—42no—
137C4⋊Dic53—42no—
138#11719919—42no—
139#11719919—42no—
140C4×Dic53—22no—
141C4×Dic53—22no—
142C4×Dic53—21no—
143C4×Dic53—21no—
144C4×Dic53—44no—
145C2×#88322—22no—
146C2×#88322—22no—
147C2×#88322C2×Q814yes—
148C23×C53C22×D411yescharacteristic
149C22×D53—21no—
150C22×D53—21no—
151C22×D53—21no—
152C22×D53—21no—
153C22×D53—21no—
154C22×D53—21no—
155C22×D53C4⋊Q812yes—
156C2×C4×C53—21no—
157C2×C4×C53—21no—
158C2×C4×C53—21no—
159C2×C4×C53—21no—
160C2×C4×C53—41no—
161C2×C4×C53—41no—
162C2×C4×C53—42no—
163C2×C4×C53—41no—
164C2×C4×C53—41no—
165C2×C4×C53—41no—
166C2×C4×C53—41no—
167C2×C4×C53—42no—
168C2×C4×C53—42no—
169C2×C4×C53#12312yes—
170C2×C4×C53#12312yes—
171C2×C4×C53C22×Q812yes—
172C2×C4×C53C2×#4111yescharacteristic
173C2×C4×C53C2×#4111yescharacteristic
174C2×C4×C53H(5,𝔽2)12yes—
175C2×Dic53—21no—
176C2×Dic53—21no—
177C2×Dic53—21no—
178C2×Dic53—21no—
179C2×Dic53—21no—
180C2×Dic53—21no—
181C2×Dic53—21no—
182C2×Dic53—21no—
183C2×Dic53—42no—
184C2×Dic53—41no—
185C2×Dic53—41no—
186C2×Dic53—41no—
187C2×Dic53—41no—
188C2×Dic53—42no—
189C2×Dic53—42no—
190C2×Dic53—41no—
191C2×Dic53—41no—
192C2×Dic53C4⋊Q812yes—
193C4×D53—24no—
194C4×D53—24no—
195C4×D53—44no—
196C4×D53—44no—
197C8×C53—24no—
198#88322—24no—
199#27954—531nomaximal, Sylow
200C22×C53C2×#12311yescharacteristic
201C22×C53C2×C4⋊Q811yescharacteristic
202C22×C53#53511yescharacteristic
203C22×C53#57511yescharacteristic
204C22×C53#57511yescharacteristic
205C22×C53#57511yescharacteristic
206C22×C53#57511yescharacteristic
207C2×D53—22no—
208C2×D53—22no—
209C2×D53—22no—
210C2×D53—41no—
211C2×D53—41no—
212C2×D53—41no—
213C2×D53—41no—
214C4×C53—22no—
215C4×C53—22no—
216C4×C53—81no—
217C4×C53—81no—
218C4×C53—82no—
219C4×C53#55412yes—
220C4×C53#55412yes—
221Dic53—22no—
222Dic53—41no—
223Dic53—41no—
224Dic53—81no—
225Dic53—82no—
226Dic53—81no—
227#2851—531no—
228#2285—531no—
229#2085—532no—
230#2085—532no—
231#1992—531no—
232#1992D5311yescharacteristic, complemented
233#1854—532no—
234#1854—532no—
235#1847—531no—
236#1847—531no—
237#1749—531no—
238C2×C53#250211yescharacteristic
239C2×C53#309311yescharacteristic
240C2×C53#309311yescharacteristic
241C2×C53#316211yescharacteristic
242C2×C53#316611yescharacteristic
243C2×C53#321911yescharacteristic
244C2×C53#321911yescharacteristic
245D53—42no—
246C2×#125C2×D5311yescharacteristic
247C2×#117—532no—
248C22×C42—531no—
249C2×#107—532no—
250C2×#107—532no—
251C2×#106—532no—
252C2×#106—532no—
253C2×C22⋊C8—532no—
254C2×#97—531no—
255C2×#97C2×D5311yescharacteristic
256#389—531no—
257#389—531no—
258#387—1061no—
259#387—1061no—
260#385—1061no—
261#385—1061no—
262#384—531no—
263#384C2×D5311yescharacteristic
264#382—532no—
265#382—531no—
266#380—531no—
267#380—531no—
268#380—1061no—
269#380—1061no—
270C4×C4⋊C4—1061no—
271C4×C4⋊C4—1061no—
272#340—532no—
273#340—532no—
274#340C2×D5312yes—
275#340C2×D5312yes—
276#336—532no—
277C53#2795411yescharacteristic, complemented, Sylow
278C23×C4—531no—
279C23×C4—531no—
280C23×C4—531no—
281C22×C8—532no—
282C22×C8C22×D5312yes—
283#125—22no—
284#125—22no—
285#117—2122no—
286#117—2122no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—21no—
291C2×C4⋊C4—532no—
292C2×C4⋊C4—531no—
293C2×C4⋊C4—531no—
294C2×C4⋊C4—1061no—
295C2×C4⋊C4—1061no—
296C2×C4⋊C4—1061no—
297C2×C4⋊C4—1061no—
298C2×C4⋊C4C22×D5312yes—
299C2×C4⋊C4C22×D5311yescharacteristic
300C2×C4⋊C4C22×D5311yescharacteristic
301C2×C22⋊C4—1062no—
302C2×C22⋊C4—1061no—
303C2×C22⋊C4—1061no—
304C2×C42—531no—
305C2×C42—531no—
306C2×C42—531no—
307C2×C42—1062no—
308C2×C42—1062no—
309C2×C42—1062no—
310C2×C42C22×D5311yescharacteristic
311#107—2122no—
312#107—2122no—
313#106—2122no—
314#106—2122no—
315C22⋊C8—538no—
316#97—22no—
317#97—1062no—
318#95—1062no—
319#95—1062no—
320#95—1061no—
321#95—1061no—
322#95—1061no—
323#95—1061no—
324#95—1061no—
325#95—1061no—
326C24—531no—
327C22×C4—22no—
328C22×C4—21no—
329C22×C4—21no—
330C22×C4—531no—
331C22×C4—534no—
332C22×C4—531no—
333C22×C4—531no—
334C22×C4—1061no—
335C22×C4—1061no—
336C22×C4—1062no—
337C22×C4—1062no—
338C22×C4—1062no—
339C22×C4—1061no—
340C22×C4—1061no—
341C22×C4—1062no—
342C22×C4—2121no—
343C22×C4—2121no—
344C22×C4—2121no—
345C22×C4—2121no—
346C22×C4D4×D5311yescharacteristic
347C22×C4D4×D5311yescharacteristic
348C22×C4C23×D5311yescharacteristic
349C2×C8—22no—
350C2×C8—22no—
351C2×C8—534no—
352C2×C8—1062no—
353C2×C8—1062no—
354C2×C8Q8×D5314yes—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—42no—
362C4⋊C4—42no—
363C4⋊C4—42no—
364C4⋊C4—42no—
365C4⋊C4—1062no—
366C4⋊C4—1062no—
367C4⋊C4—1062no—
368C4⋊C4—1062no—
369C4⋊C4—2122no—
370C4⋊C4—2122no—
371C4⋊C4—2122no—
372C4⋊C4—2122no—
373C22⋊C4—2124no—
374C22⋊C4—2122no—
375C22⋊C4—2122no—
376C42—1062no—
377C42—1062no—
378C42—1061no—
379C42—1061no—
380C42—2124no—
381C42#1171994512yes—
382C42#1171994712yes—
383C23—532no—
384C23—1061no—
385C23—1061no—
386C23—1061no—
387C23—1061no—
388C23—1061no—
389C23—1061no—
390C23C2×D4×D5311yescharacteristic
391C2×C4—21no—
392C2×C4—21no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—41no—
396C2×C4—41no—
397C2×C4—41no—
398C2×C4—42no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—42no—
404C2×C4—532no—
405C2×C4—1061no—
406C2×C4—1061no—
407C2×C4—1064no—
408C2×C4—1064no—
409C2×C4—1061no—
410C2×C4—1061no—
411C2×C4—1061no—
412C2×C4—1061no—
413C2×C4—1061no—
414C2×C4—1061no—
415C2×C4—2124no—
416C2×C4—2124no—
417C2×C4—2121no—
418C2×C4—2121no—
419C2×C4—2121no—
420C2×C4—2121no—
421C2×C4—2122no—
422C2×C4—2122no—
423C2×C4—2122no—
424C2×C4—2121no—
425C2×C4—2121no—
426C2×C4#1313319912yes—
427C2×C4#1313319912yes—
428C2×C4C2×Q8×D5312yes—
429C2×C4C2×#1171994711yescharacteristic
430C2×C4C2×#1171994511yescharacteristic
431C2×C4#1313326312yes—
432C8—24no—
433C8—1064no—
434C22—1062no—
435C22—1062no—
436C22—1062no—
437C22—2121no—
438C22—2121no—
439C22—2121no—
440C22—2121no—
441C22C2×#1313319911yescharacteristic
442C22C4⋊Q8×D5311yescharacteristic
443C22#1681119111yescharacteristic
444C22#1681141911yescharacteristic
445C22#1681141911yescharacteristic
446C22#1681142211yescharacteristic
447C22#1681142211yescharacteristic
448C4—22no—
449C4—22no—
450C4—82no—
451C4—81no—
452C4—81no—
453C4—1062no—
454C4—4241no—
455C4—4241no—
456C4—4242no—
457C4—2121no—
458C4—2121no—
459C4#1681133112yes—
460C4#1681133312yes—
461C2—2122no—
462C2#2027919211yescharacteristic
463C2#2028464511yescharacteristic
464C2#2028464511yescharacteristic
465C2#2028523611yescharacteristic
466C2#2028526611yescharacteristic
467C2#2028575311yescharacteristic
468C2#2028575511yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1992C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
hypercenter#1992D53
hypocenterC53#27954
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1992

Complements

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

NH ≅ G/Ntype
3. #20276202C2semidirect
4. #20276201C2semidirect
5. #20275685C2semidirect
8. #20274883C2semidirect
9. #20274881C2semidirect
11. C53×#1992C2semidirect
232. #1992D53semidirect
277. C53#27954semidirect

Maximal subgroups

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

subgroupindex
#202830202
#202771202
#202762022
#202762012
#202756852
#202749172
#202749152
#202748832
#202748812
#202744782
C53×#19922
#2795453