Finite Group Databasebeta

← #27469602

Subgroups — #27469602

Order 13568.

842 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283020C211yesmaximal, characteristic
2#384×D53C211yesmaximal, characteristic
3#20276195C212yesmaximal, complemented
4#20276190C212yesmaximal, complemented
5#20275685C211yesmaximal, characteristic, complemented
6#20274909C212yesmaximal
7#20274904C212yesmaximal
8#20274883C211yesmaximal, characteristic, complemented
9#20274881C211yesmaximal, characteristic, complemented
10#20274478C211yesmaximal, characteristic
11C53×#1992C211yesmaximal, characteristic, complemented
12C2×C4⋊C4×D53—21no—
13C2×C4⋊C4×D53—21no—
14C2×C4⋊C4×D53C2212yes—
15C2×C42×D53C2211yescharacteristic
16C2×#13133151C2212yes—
17C2×#13133150C2212yes—
18C2×#13133149C2212yes—
19C2×#13133148C2212yes—
20C2×#13133129C2212yes—
21C2×#13133128C2211yescharacteristic
22#16810519C2211yescharacteristic
23#16810519C2211yescharacteristic
24#16810504—21no—
25#16810504—21no—
26#16810494—21no—
27#16810494—21no—
28#16810493C2211yescharacteristic
29#16810492—21no—
30#16810492—21no—
31#16810492C2212yes—
32#16810489C2211yescharacteristic
33#16810480C2211yescharacteristic
34#16810480C2211yescharacteristic
35#16810385C2212yes—
36#16810385C2212yes—
37#16810377C2212yes—
38C2×#125×C53C2211yescharacteristic
39C2×#97×C53C2211yescharacteristic
40C53×#384C2211yescharacteristic
41C53×#340C2212yes—
42C53×#340C2212yes—
43C22×C4×D53—22no—
44C22×C4×D53D411yescharacteristic
45C22×C4×D53D411yescharacteristic
46C22×C4×D53C2311yescharacteristic
47C22×#88322C2312yes—
48C2×C4⋊Dic53—21no—
49C2×C4⋊Dic53—21no—
50C2×C4⋊Dic53C2311yescharacteristic
51C2×C4⋊Dic53C2311yescharacteristic
52C2×C4⋊Dic53C2312yes—
53C2×#11719919—21no—
54C2×#11719919—21no—
55C2×#11719919—22no—
56C2×#11719919—22no—
57C2×#11719919—22no—
58C4⋊C4×D53—42no—
59C4⋊C4×D53—42no—
60C4⋊C4×D53—44no—
61C4⋊C4×D53—44no—
62C2×#11719921—21no—
63C2×#11719921—21no—
64C2×C4×Dic53D411yescharacteristic
65C2×C4×Dic53D411yescharacteristic
66C2×C4×Dic53C2311yescharacteristic
67C42×D53—22no—
68C42×D53—22no—
69#13133151—42no—
70#13133150—42no—
71#13133149—42no—
72#13133148—42no—
73#13133129Q818yes—
74#13133128—22no—
75#13133124—21no—
76#13133124—21no—
77#13133124—21no—
78#13133124—21no—
79#13133124—21no—
80#13133124—21no—
81C22×C8×C53C2312yes—
82#125×C53—22no—
83#125×C53—22no—
84C2×C4⋊C4×C53—21no—
85C2×C4⋊C4×C53—21no—
86C2×C4⋊C4×C53—21no—
87C2×C4⋊C4×C53—21no—
88C2×C4⋊C4×C53C2311yescharacteristic
89C2×C4⋊C4×C53C2311yescharacteristic
90C2×C4⋊C4×C53C2312yes—
91C2×C42×C53C2311yescharacteristic
92#97×C53—22no—
93C23×D53C2×D411yescharacteristic
94C22×C4×C53—21no—
95C22×C4×C53—21no—
96C22×C4×C53—22no—
97C22×C4×C53C2×D411yescharacteristic
98C22×C4×C53C2×D411yescharacteristic
99C22×C4×C53C2411yescharacteristic
100C22×Dic53—21no—
101C22×Dic53—21no—
102C22×Dic53—22no—
103C22×Dic53C2×D411yescharacteristic
104C22×Dic53C2×D411yescharacteristic
105C22×Dic53C2×D411yescharacteristic
106C2×C4×D53—22no—
107C2×C4×D53—22no—
108C2×C4×D53—22no—
109C2×C4×D53—22no—
110C2×C4×D53—21no—
111C2×C4×D53—21no—
112C2×C4×D53—22no—
113C2×C4×D53—22no—
114C2×C4×D53—22no—
115C2×C4×D53—44no—
116C2×C4×D53—44no—
117C2×C4×D53—44no—
118C2×C4×D53C2×Q814yes—
119C2×C8×C53—22no—
120C2×C8×C53—22no—
121C2×C8×C53C2×Q814yes—
122C4⋊C4×C53—21no—
123C4⋊C4×C53—21no—
124C4⋊C4×C53—21no—
125C4⋊C4×C53—21no—
126C4⋊C4×C53—22no—
127C4⋊C4×C53—22no—
128C4⋊C4×C53—42no—
129C4⋊C4×C53—42no—
130C4⋊C4×C53—42no—
131C4⋊C4×C53—42no—
132C42×C53#4112yes—
133C42×C53#4112yes—
134#11719921—42no—
135#11719921—42no—
136C4⋊Dic53—22no—
137C4⋊Dic53—22no—
138C4⋊Dic53—22no—
139C4⋊Dic53—22no—
140C4⋊Dic53—42no—
141C4⋊Dic53—42no—
142#11719919—44no—
143#11719919—44no—
144#11719919—44no—
145#11719919—42no—
146#11719919—42no—
147C4×Dic53—22no—
148C4×Dic53—22no—
149C4×Dic53—21no—
150C4×Dic53—21no—
151C2×#88322—22no—
152C2×#88322—22no—
153C2×#88322C2×Q814yes—
154C23×C53C22×D411yescharacteristic
155C22×D53—21no—
156C22×D53—21no—
157C22×D53—21no—
158C22×D53—21no—
159C22×D53C4⋊Q812yes—
160C22×D53AGL(1,ℤ/8)12yes—
161C22×D53#13712yes—
162C2×C4×C53—21no—
163C2×C4×C53—21no—
164C2×C4×C53—21no—
165C2×C4×C53—21no—
166C2×C4×C53—41no—
167C2×C4×C53—41no—
168C2×C4×C53—42no—
169C2×C4×C53—42no—
170C2×C4×C53—42no—
171C2×C4×C53—41no—
172C2×C4×C53—41no—
173C2×C4×C53—41no—
174C2×C4×C53—41no—
175C2×C4×C53#12312yes—
176C2×C4×C53#12312yes—
177C2×C4×C53C22×Q812yes—
178C2×C4×C53C2×#4111yescharacteristic
179C2×C4×C53C2×#4111yescharacteristic
180C2×C4×C53H(5,𝔽2)12yes—
181C2×Dic53—21no—
182C2×Dic53—21no—
183C2×Dic53—21no—
184C2×Dic53—21no—
185C2×Dic53—21no—
186C2×Dic53—21no—
187C2×Dic53—42no—
188C2×Dic53—41no—
189C2×Dic53—41no—
190C2×Dic53—42no—
191C2×Dic53—42no—
192C2×Dic53—41no—
193C2×Dic53—41no—
194C2×Dic53—41no—
195C2×Dic53—41no—
196C2×Dic53C4⋊Q812yes—
197C2×Dic53AGL(1,ℤ/8)12yes—
198C2×Dic53#13712yes—
199C4×D53—28no—
200C4×D53—28no—
201C4×D53—24no—
202C4×D53—24no—
203C4×D53—88no—
204C8×C53—24no—
205#88322—24no—
206#27953—531nomaximal, Sylow
207C22×C53C2×#12311yescharacteristic
208C22×C53C2×C4⋊Q811yescharacteristic
209C22×C53#53511yescharacteristic
210C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
211C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
212C22×C53C2×#13711yescharacteristic
213C22×C53C2×#13711yescharacteristic
214C2×D53—22no—
215C2×D53—22no—
216C2×D53—22no—
217C2×D53—22no—
218C2×D53—22no—
219C2×D53—22no—
220C2×D53#50114yes—
221C4×C53—22no—
222C4×C53—22no—
223C4×C53—81no—
224C4×C53—81no—
225C4×C53—82no—
226C4×C53#55412yes—
227C4×C53#55412yes—
228Dic53—22no—
229Dic53—22no—
230Dic53—22no—
231Dic53—82no—
232Dic53—81no—
233Dic53—81no—
234#2851—531no—
235C2×#384—531no—
236#2084—532no—
237#2083—532no—
238#1992—531no—
239#1992D5311yescharacteristic, complemented
240#1853—532no—
241#1852—532no—
242#1847—531no—
243#1847—531no—
244#1749—531no—
245C2×C53#250211yescharacteristic
246C2×C53#309211yescharacteristic
247C2×C53#309211yescharacteristic
248C2×C53C2×#50111yescharacteristic
249C2×C53#316511yescharacteristic
250C2×C53#321711yescharacteristic
251C2×C53#321811yescharacteristic
252D53—24no—
253C2×#125C2×D5311yescharacteristic
254C22×C4⋊C4—532no—
255C22×C4⋊C4—1061no—
256C22×C4⋊C4—1061no—
257C22×C42—531no—
258C2×#107—532no—
259C2×#107—532no—
260C2×#106—532no—
261C2×#106—532no—
262C2×C22⋊C8—532no—
263C2×#97—531no—
264C2×#97C2×D5311yescharacteristic
265#389—531no—
266#389—531no—
267#387—1061no—
268#387—1061no—
269#385—1061no—
270#385—1061no—
271#384—532no—
272#384—531no—
273#384—1061no—
274#384—1061no—
275#384C2×D5311yescharacteristic
276#382—531no—
277#380—531no—
278#380—531no—
279#340—532no—
280#340—532no—
281#340C2×D5312yes—
282#340C2×D5312yes—
283#336—532no—
284C53#2795311yescharacteristic, complemented, Sylow
285C23×C4—531no—
286C23×C4—531no—
287C23×C4—531no—
288C23×C4—1062no—
289C22×C8—532no—
290C22×C8C22×D5312yes—
291#125—22no—
292#125—22no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4—21no—
296C2×C4⋊C4—21no—
297C2×C4⋊C4—532no—
298C2×C4⋊C4—531no—
299C2×C4⋊C4—531no—
300C2×C4⋊C4—1062no—
301C2×C4⋊C4—1062no—
302C2×C4⋊C4—1062no—
303C2×C4⋊C4—1061no—
304C2×C4⋊C4—1061no—
305C2×C4⋊C4—1061no—
306C2×C4⋊C4—1061no—
307C2×C4⋊C4—2122no—
308C2×C4⋊C4—2122no—
309C2×C4⋊C4—2124no—
310C2×C4⋊C4—2124no—
311C2×C4⋊C4C22×D5312yes—
312C2×C4⋊C4C22×D5311yescharacteristic
313C2×C4⋊C4C22×D5311yescharacteristic
314C2×C22⋊C4—1061no—
315C2×C22⋊C4—1061no—
316C2×C42—531no—
317C2×C42—531no—
318C2×C42—531no—
319C2×C42—1062no—
320C2×C42—1062no—
321C2×C42C22×D5311yescharacteristic
322#107—2122no—
323#107—2122no—
324#106—2122no—
325#106—2122no—
326C22⋊C8—538no—
327#97—22no—
328#97—1062no—
329#95—1061no—
330#95—1061no—
331#95—1061no—
332#95—1061no—
333#95—1061no—
334#95—1061no—
335C24—531no—
336C22×C4—21no—
337C22×C4—21no—
338C22×C4—22no—
339C22×C4—531no—
340C22×C4—531no—
341C22×C4—531no—
342C22×C4—534no—
343C22×C4—1062no—
344C22×C4—1062no—
345C22×C4—1062no—
346C22×C4—1062no—
347C22×C4—1062no—
348C22×C4—1062no—
349C22×C4—1061no—
350C22×C4—1061no—
351C22×C4—1062no—
352C22×C4—1061no—
353C22×C4—1061no—
354C22×C4—1062no—
355C22×C4—2124no—
356C22×C4—2124no—
357C22×C4—2124no—
358C22×C4D4×D5311yescharacteristic
359C22×C4D4×D5311yescharacteristic
360C22×C4C23×D5311yescharacteristic
361C2×C8—22no—
362C2×C8—22no—
363C2×C8—534no—
364C2×C8—1062no—
365C2×C8—1062no—
366C2×C8Q8×D5314yes—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—22no—
372C4⋊C4—22no—
373C4⋊C4—42no—
374C4⋊C4—42no—
375C4⋊C4—42no—
376C4⋊C4—42no—
377C4⋊C4—1062no—
378C4⋊C4—1062no—
379C4⋊C4—1062no—
380C4⋊C4—1062no—
381C4⋊C4—2122no—
382C4⋊C4—2122no—
383C4⋊C4—2124no—
384C4⋊C4—2122no—
385C4⋊C4—2122no—
386C4⋊C4—2124no—
387C4⋊C4—2124no—
388C22⋊C4—2122no—
389C22⋊C4—2122no—
390C42—1062no—
391C42—1062no—
392C42—1061no—
393C42—1061no—
394C42#1171994512yes—
395C42#1171994712yes—
396C23—532no—
397C23—532no—
398C23—532no—
399C23—1061no—
400C23—1061no—
401C23—1061no—
402C23—1061no—
403C23C2×D4×D5311yescharacteristic
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—42no—
417C2×C4—532no—
418C2×C4—532no—
419C2×C4—532no—
420C2×C4—1068no—
421C2×C4—1068no—
422C2×C4—1061no—
423C2×C4—1061no—
424C2×C4—1061no—
425C2×C4—1061no—
426C2×C4—1061no—
427C2×C4—1061no—
428C2×C4—1064no—
429C2×C4—1064no—
430C2×C4—4248no—
431C2×C4—2121no—
432C2×C4—2122no—
433C2×C4—2122no—
434C2×C4—2121no—
435C2×C4—2121no—
436C2×C4—2121no—
437C2×C4—2122no—
438C2×C4—2121no—
439C2×C4—2121no—
440C2×C4#1313319912yes—
441C2×C4#1313319912yes—
442C2×C4C2×Q8×D5312yes—
443C2×C4C2×#1171994711yescharacteristic
444C2×C4C2×#1171994511yescharacteristic
445C2×C4#1313326312yes—
446C8—24no—
447C8—1064no—
448C22—534no—
449C22—1062no—
450C22—1062no—
451C22—1062no—
452C22—1062no—
453C22—1062no—
454C22—1062no—
455C22C2×#1313319911yescharacteristic
456C22C4⋊Q8×D5311yescharacteristic
457C22#1681119111yescharacteristic
458C22AGL(1,ℤ/8)×D5311yescharacteristic
459C22AGL(1,ℤ/8)×D5311yescharacteristic
460C22#137×D5311yescharacteristic
461C22#137×D5311yescharacteristic
462C4—22no—
463C4—22no—
464C4—82no—
465C4—81no—
466C4—81no—
467C4—1062no—
468C4—1062no—
469C4—1062no—
470C4—4242no—
471C4—4241no—
472C4—4241no—
473C4#1681133112yes—
474C4#1681133312yes—
475C2—1064no—
476C2#2027919211yescharacteristic
477C2#2028463611yescharacteristic
478C2#2028463611yescharacteristic
479C2#501×D5311yescharacteristic
480C2#2028526011yescharacteristic
481C2#2028573411yescharacteristic
482C2#2028574311yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1992C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
hypercenter#1992D53
hypocenterC53#27953
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. #20276195C2semidirect
4. #20276190C2semidirect
5. #20275685C2semidirect
8. #20274883C2semidirect
9. #20274881C2semidirect
11. C53×#1992C2semidirect
239. #1992D53semidirect
284. C53#27953semidirect

Maximal subgroups

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

subgroupindex
#202830202
#384×D532
#202761952
#202761902
#202756852
#202749092
#202749042
#202748832
#202748812
#202744782
C53×#19922
#2795353