Finite Group Databasebeta

← #27466128

Subgroups — #27466128

Order 13568.

886 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277243C211yesmaximal, characteristic
2C2×#16810492C211yesmaximal, characteristic
3#20276275C212yesmaximal, complemented
4#20276255C212yesmaximal, complemented
5#20275475C212yesmaximal, complemented
6#20275022C211yesmaximal, characteristic
7#20275020C211yesmaximal, characteristic
8#20274907C211yesmaximal, characteristic
9#20274719C212yesmaximal, complemented
10#20274693C211yesmaximal, characteristic
11C53×#1853C211yesmaximal, characteristic
12C2×#13133196C2211yescharacteristic
13C2×#13133182C2211yescharacteristic
14C22×C4⋊Dic53C2211yescharacteristic
15C22×#11719919—21no—
16C22×#11719919—21no—
17C22×C4×Dic53C2211yescharacteristic
18C2×#13133148C2212yes—
19C2×#13133146C2212yes—
20C2×#13133142C2212yes—
21C2×#13133140C2212yes—
22C2×C22⋊#88322C2211yescharacteristic
23#16810528C2211yescharacteristic
24#16810519C2211yescharacteristic
25#16810505—21no—
26#16810495—21no—
27#16810492—21no—
28#16810492—21no—
29#16810492C2212yes—
30#16810492C2212yes—
31C4⋊C4×Dic53C2211yescharacteristic
32#16810474—21no—
33C22⋊C4×Dic53—21no—
34#16810386C2212yes—
35#16810385C2212yes—
36#16810385C2212yes—
37#16810377C2212yes—
38C22×C4⋊C4×C53C2211yescharacteristic
39C2×#106×C53C2212yes—
40C2×C22⋊C8×C53C2211yescharacteristic
41C53×#380C2211yescharacteristic
42C53×#340C2212yes—
43C22×Dic106C2311yescharacteristic
44C23×Dic53—21no—
45C23×Dic53D411yescharacteristic
46C23×Dic53D411yescharacteristic
47C22×#88322C2312yes—
48#13133196—22no—
49#13133196Q814yes—
50#13133182—24no—
51C2×C4⋊Dic53—21no—
52C2×C4⋊Dic53—42no—
53C2×C4⋊Dic53C2312yes—
54C2×C4⋊Dic53C2311yescharacteristic
55C2×#11719919—21no—
56C2×#11719919—21no—
57C2×#11719919—21no—
58C2×#11719919—22no—
59C2×#11719919—22no—
60C2×#11719919—22no—
61C2×#11719919—22no—
62C2×#11719919—44no—
63C2×#11719919—44no—
64C2×C22⋊Dic53—21no—
65C2×C4×Dic53—24no—
66C2×C4×Dic53—21no—
67C2×C4×Dic53—21no—
68C2×C4×Dic53D411yescharacteristic
69C2×C4×Dic53D411yescharacteristic
70C2×C4×Dic53C2311yescharacteristic
71C2×C4×Dic53C2311yescharacteristic
72#13133148—42no—
73#13133146—24no—
74#13133142—42no—
75#13133140—42no—
76C22⋊#88322—22no—
77#13133124—21no—
78#13133124—21no—
79#13133124—21no—
80#13133124—21no—
81C23×C4×C53—21no—
82C23×C4×C53C2311yescharacteristic
83C22×C8×C53C2312yes—
84C2×C4⋊C4×C53—21no—
85C2×C4⋊C4×C53—42no—
86C2×C4⋊C4×C53C2311yescharacteristic
87C2×C4⋊C4×C53C2312yes—
88C2×C4⋊C4×C53C2311yescharacteristic
89C2×C22⋊C4×C53—21no—
90#106×C53—42no—
91C22⋊C8×C53—22no—
92#95×C53—21no—
93C24×C53C2×D411yescharacteristic
94C22×C4×C53—22no—
95C22×C4×C53—22no—
96C22×C4×C53—21no—
97C22×C4×C53—21no—
98C22×C4×C53—22no—
99C22×C4×C53—42no—
100C22×C4×C53—42no—
101C22×C4×C53—42no—
102C22×C4×C53C2×D411yescharacteristic
103C22×C4×C53C2×Q812yes—
104C22×C4×C53#4112yes—
105C22×C4×C53C2411yescharacteristic
106C22×Dic53—22no—
107C22×Dic53—22no—
108C22×Dic53—22no—
109C22×Dic53—22no—
110C22×Dic53—22no—
111C22×Dic53—21no—
112C22×Dic53—21no—
113C22×Dic53—22no—
114C22×Dic53—42no—
115C22×Dic53—42no—
116C22×Dic53—42no—
117C22×Dic53SD1614yes—
118C22×Dic53C2×D411yescharacteristic
119C22×Dic53C2×D411yescharacteristic
120C22×Dic53C2×D411yescharacteristic
121C22×Dic53C2×D411yescharacteristic
122C2×Dic106—22no—
123C2×Dic106—22no—
124C2×Dic106—21no—
125C2×Dic106C2×Q812yes—
126C2×C8×C53—22no—
127C2×C8×C53—22no—
128C2×C8×C53—22no—
129C4⋊C4×C53—22no—
130C4⋊C4×C53—22no—
131C4⋊C4×C53—22no—
132C4⋊C4×C53—22no—
133C4⋊C4×C53—42no—
134C22⋊C4×C53—42no—
135C22⋊Dic53—24no—
136C4⋊Dic53—21no—
137C4⋊Dic53—22no—
138C4⋊Dic53—22no—
139C4⋊Dic53—42no—
140C4⋊Dic53C2×Q812yes—
141#11719919—24no—
142#11719919—44no—
143#11719919—42no—
144#11719919—44no—
145#11719919—44no—
146#11719919—42no—
147#11719919—44no—
148C4×Dic53—22no—
149C4×Dic53—22no—
150C4×Dic53—22no—
151C4×Dic53—42no—
152C4×Dic53—42no—
153C4×Dic53#4114yes—
154C2×#88322—22no—
155C2×#88322—22no—
156C2×#88322—22no—
157C23×C53—21no—
158C23×C53—21no—
159C23×C53—21no—
160C23×C53—21no—
161C23×C53C22⋊Q812yes—
162C23×C53C2×SD1612yes—
163C23×C53#13712yes—
164C23×C53C22×D411yescharacteristic
165C2×C4×C53—21no—
166C2×C4×C53—21no—
167C2×C4×C53—24no—
168C2×C4×C53—24no—
169C2×C4×C53—41no—
170C2×C4×C53—42no—
171C2×C4×C53—41no—
172C2×C4×C53—42no—
173C2×C4×C53—42no—
174C2×C4×C53—41no—
175C2×C4×C53—41no—
176C2×C4×C53—41no—
177C2×C4×C53—41no—
178C2×C4×C53—84no—
179C2×C4×C53C22⋊Q812yes—
180C2×C4×C53C22×Q811yescharacteristic
181C2×C4×C53C2×#4112yes—
182C2×C4×C53C2×#4111yescharacteristic
183C2×C4×C53H(5,𝔽2)12yes—
184C2×Dic53—28no—
185C2×Dic53—21no—
186C2×Dic53—21no—
187C2×Dic53—21no—
188C2×Dic53—21no—
189C2×Dic53—21no—
190C2×Dic53—21no—
191C2×Dic53—28no—
192C2×Dic53—22no—
193C2×Dic53—22no—
194C2×Dic53—41no—
195C2×Dic53—41no—
196C2×Dic53—41no—
197C2×Dic53—41no—
198C2×Dic53—42no—
199C2×Dic53—42no—
200C2×Dic53—42no—
201C2×Dic53—84no—
202C2×Dic53#12312yes—
203C2×Dic53#12312yes—
204C2×Dic53C2×SD1612yes—
205C2×Dic53C2×SD1612yes—
206C2×Dic53AGL(1,ℤ/8)12yes—
207C2×Dic53#13712yes—
208Dic106—24no—
209Dic106—42no—
210C8×C53—42no—
211#88322—42no—
212#27680—531nomaximal, Sylow
213C22×C53—22no—
214C22×C53—22no—
215C22×C53—22no—
216C22×C53—22no—
217C22×C53—22no—
218C22×C53—22no—
219C22×C53#47514yes—
220C22×C53C2×C22⋊Q811yescharacteristic
221C22×C53C2×#12311yescharacteristic
222C22×C53#53511yescharacteristic
223C22×C53C22×SD1611yescharacteristic
224C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
225C22×C53C2×#13711yescharacteristic
226C22×C53C2×#13711yescharacteristic
227C4×C53—22no—
228C4×C53—81no—
229C4×C53—82no—
230C4×C53—81no—
231C4×C53#55312yes—
232C4×C53#55412yes—
233Dic53—22no—
234Dic53—22no—
235Dic53—22no—
236Dic53—42no—
237Dic53—42no—
238Dic53—82no—
239Dic53#48314yes—
240C4×C22⋊Q8—531no—
241C2×#384—531no—
242#2099—532no—
243#2096—532no—
244#1948—532no—
245#1872—531no—
246#1872—531no—
247#1853—531no—
248#1853D5311yescharacteristic
249#1803—532no—
250#1797—531no—
251C2×C53—24no—
252C2×C53#248711yescharacteristic
253C2×C53C2×#47511yescharacteristic
254C2×C53#307911yescharacteristic
255C2×C53C2×#48311yescharacteristic
256C2×C53#309211yescharacteristic
257C2×C53#320811yescharacteristic
258C2×C53#321811yescharacteristic
259C2×C22⋊Q8—531no—
260C2×C4×Q8—531no—
261C22×C4⋊C4—531no—
262C22×C4⋊C4—1061no—
263C22×C4⋊C4—1061no—
264C22×C4⋊C4C2×D5311yescharacteristic
265C22×C42—531no—
266C2×#106—532no—
267C2×#106C2×D5312yes—
268C2×C4⋊C8—532no—
269C2×#103—532no—
270C2×#103—532no—
271C2×C22⋊C8—531no—
272C2×C22⋊C8C2×D5311yescharacteristic
273#391—531no—
274#389—531no—
275#387—1061no—
276#385—1061no—
277#384—532no—
278#384—532no—
279#384—1061no—
280#384—1061no—
281#380C2×D5311yescharacteristic
282C4×C4⋊C4—531no—
283C4×C4⋊C4—1061no—
284C4×C22⋊C4—1061no—
285#340—532no—
286#340—532no—
287#340—532no—
288#340C2×D5312yes—
289#336—532no—
290C53#2768011yescharacteristic, complemented, Sylow
291C22×Q8—531no—
292C23×C4—21no—
293C23×C4—531no—
294C23×C4—531no—
295C23×C4—1061no—
296C23×C4C22×D5311yescharacteristic
297C22×C8—532no—
298C22×C8C22×D5312yes—
299C22⋊Q8—534no—
300C22⋊Q8—1062no—
301C4×Q8—1064no—
302C2×C4⋊C4—21no—
303C2×C4⋊C4—42no—
304C2×C4⋊C4—531no—
305C2×C4⋊C4—532no—
306C2×C4⋊C4—1061no—
307C2×C4⋊C4—1062no—
308C2×C4⋊C4—1062no—
309C2×C4⋊C4—1062no—
310C2×C4⋊C4—1062no—
311C2×C4⋊C4—1061no—
312C2×C4⋊C4—1061no—
313C2×C4⋊C4—1061no—
314C2×C4⋊C4—2122no—
315C2×C4⋊C4—2124no—
316C2×C4⋊C4—2124no—
317C2×C4⋊C4C22×D5312yes—
318C2×C4⋊C4C22×D5311yescharacteristic
319C2×C4⋊C4C22×D5311yescharacteristic
320C2×C22⋊C4—21no—
321C2×C22⋊C4—1061no—
322C2×C42—531no—
323C2×C42—531no—
324C2×C42—531no—
325C2×C42—531no—
326C2×C42—1064no—
327C2×C42—1061no—
328C2×C42—1061no—
329#106—42no—
330#106—2122no—
331C4⋊C8—1064no—
332#103—2122no—
333#103—2122no—
334C22⋊C8—22no—
335C22⋊C8—1062no—
336#95—21no—
337#95—1061no—
338#95—1061no—
339#95—1061no—
340#95—1061no—
341C24D4×D5311yescharacteristic
342C2×Q8—532no—
343C2×Q8—1061no—
344C2×Q8—1062no—
345C2×Q8—1062no—
346C22×C4—22no—
347C22×C4—22no—
348C22×C4—21no—
349C22×C4—21no—
350C22×C4—22no—
351C22×C4—42no—
352C22×C4—42no—
353C22×C4—42no—
354C22×C4—531no—
355C22×C4—534no—
356C22×C4—531no—
357C22×C4—531no—
358C22×C4—531no—
359C22×C4—1062no—
360C22×C4—1062no—
361C22×C4—1062no—
362C22×C4—1062no—
363C22×C4—1062no—
364C22×C4—1061no—
365C22×C4—1061no—
366C22×C4—1062no—
367C22×C4—2122no—
368C22×C4—2122no—
369C22×C4—2122no—
370C22×C4#1171994512yes—
371C22×C4D4×D5311yescharacteristic
372C22×C4Q8×D5312yes—
373C22×C4C23×D5311yescharacteristic
374C2×C8—22no—
375C2×C8—22no—
376C2×C8—22no—
377C2×C8—1062no—
378C2×C8—1062no—
379C2×C8—1062no—
380C4⋊C4—22no—
381C4⋊C4—22no—
382C4⋊C4—22no—
383C4⋊C4—22no—
384C4⋊C4—42no—
385C4⋊C4—532no—
386C4⋊C4—1064no—
387C4⋊C4—1061no—
388C4⋊C4—1062no—
389C4⋊C4—1062no—
390C4⋊C4—2122no—
391C4⋊C4—2124no—
392C4⋊C4—2124no—
393C4⋊C4—2124no—
394C4⋊C4—2122no—
395C4⋊C4—2122no—
396C4⋊C4—2124no—
397C22⋊C4—42no—
398C22⋊C4—1064no—
399C42—534no—
400C42—1062no—
401C42—1062no—
402C42—1062no—
403C42—2122no—
404C42—2122no—
405C23—21no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23#1313319512yes—
410C23SD16×D5312yes—
411C23#1313324612yes—
412C23C2×D4×D5311yescharacteristic
413Q8—1064no—
414Q8—2122no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—24no—
418C2×C4—24no—
419C2×C4—41no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—42no—
427C2×C4—41no—
428C2×C4—84no—
429C2×C4—532no—
430C2×C4—532no—
431C2×C4—532no—
432C2×C4—532no—
433C2×C4—532no—
434C2×C4—532no—
435C2×C4—1068no—
436C2×C4—1068no—
437C2×C4—1061no—
438C2×C4—1061no—
439C2×C4—1061no—
440C2×C4—1061no—
441C2×C4—1061no—
442C2×C4—1061no—
443C2×C4—1062no—
444C2×C4—1062no—
445C2×C4—4244no—
446C2×C4—2122no—
447C2×C4—2122no—
448C2×C4—2122no—
449C2×C4—2121no—
450C2×C4—2121no—
451C2×C4—2121no—
452C2×C4—2121no—
453C2×C4#1313319512yes—
454C2×C4C2×Q8×D5311yescharacteristic
455C2×C4#41×D5312yes—
456C2×C4C2×#1171994511yescharacteristic
457C2×C4#1313326212yes—
458C8—42no—
459C8—2122no—
460C22—22no—
461C22—22no—
462C22—22no—
463C22—22no—
464C22—22no—
465C22—22no—
466C22#1681090714yes—
467C22C2×#1313319511yescharacteristic
468C22#123×D5311yescharacteristic
469C22#1681118811yescharacteristic
470C22C2×SD16×D5311yescharacteristic
471C22AGL(1,ℤ/8)×D5311yescharacteristic
472C22#137×D5311yescharacteristic
473C22C2×#1313324611yescharacteristic
474C4—22no—
475C4—81no—
476C4—82no—
477C4—81no—
478C4—534no—
479C4—1062no—
480C4—1062no—
481C4—1062no—
482C4—4242no—
483C4—2122no—
484C4—2122no—
485C4#1681132212yes—
486C4#1681132712yes—
487C2—24no—
488C2#2027903111yescharacteristic
489C2C2×#1681090711yescharacteristic
490C2#2028452811yescharacteristic
491C2#483×D5311yescharacteristic
492C2#2028463711yescharacteristic
493C2#2028565411yescharacteristic
494C2#2028574011yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1853C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#1853D53
hypocenterC53#27680
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1853

Complements

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

NH ≅ G/Ntype
3. #20276275C2semidirect
4. #20276255C2semidirect
5. #20275475C2semidirect
9. #20274719C2semidirect
290. C53#27680semidirect

Maximal subgroups

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

subgroupindex
#202772432
C2×#168104922
#202762752
#202762552
#202754752
#202750222
#202750202
#202749072
#202747192
#202746932
C53×#18532
#2768053