Finite Group Databasebeta

← #27394652

Subgroups — #27394652

Order 13568.

902 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810678C211yesmaximal, characteristic
2#384×D53C211yesmaximal, characteristic
3#20277108C211yesmaximal, characteristic
4#20275475C212yesmaximal, complemented
5#20275470C212yesmaximal, complemented
6#20274906C212yesmaximal
7#20274901C212yesmaximal
8#20274884C211yesmaximal, characteristic, complemented
9#20274850C211yesmaximal, characteristic, complemented
10#20274849C211yesmaximal, characteristic, complemented
11C53×#1847C211yesmaximal, characteristic, complemented
12C2×C4⋊C4×D53—21no—
13C2×C4⋊C4×D53—21no—
14C2×C4⋊C4×D53C2212yes—
15C2×C42×D53C2211yescharacteristic
16#16810678C418yescomplemented
17C2×#13133150C2212yes—
18C2×#13133148C2212yes—
19C2×#13133146C2212yes—
20C2×#13133129C2212yes—
21C2×#13133128C2211yescharacteristic
22#16810493C2211yescharacteristic
23#16810492—21no—
24#16810492—21no—
25#16810492C2212yes—
26#16810490—21no—
27#16810489C2211yescharacteristic
28#16810485—21no—
29#16810480C2211yescharacteristic
30#16810480C2211yescharacteristic
31C4⋊C4×Dic53C2211yescharacteristic
32C4⋊C4×Dic53C2211yescharacteristic
33#16810470—21no—
34#16810468—21no—
35#16810386C2212yes—
36#16810386C2212yes—
37#16810385C2212yes—
38#16810385C2212yes—
39C2×#107×C53C2212yes—
40C2×#106×C53C2212yes—
41C2×#97×C53C2211yescharacteristic
42C53×#384C2211yescharacteristic
43C53×#382C2211yescharacteristic
44C22×C4×D53—22no—
45C22×C4×D53D411yescharacteristic
46C22×C4×D53D411yescharacteristic
47C22×C4×D53C2311yescharacteristic
48C22×#88322C2312yes—
49C2×C4⋊Dic53—21no—
50C2×C4⋊Dic53—21no—
51C2×C4⋊Dic53C2312yes—
52C2×C4⋊Dic53C2311yescharacteristic
53C2×C4⋊Dic53C2311yescharacteristic
54C2×#11719919—22no—
55C2×#11719919—22no—
56C2×#11719919—22no—
57C4⋊C4×D53—42no—
58C4⋊C4×D53—42no—
59C4⋊C4×D53—44no—
60C4⋊C4×D53—44no—
61C2×#11719921—21no—
62C2×#11719921—21no—
63C2×C4×Dic53—21no—
64C2×C4×Dic53—21no—
65C2×C4×Dic53D411yescharacteristic
66C2×C4×Dic53D411yescharacteristic
67C2×C4×Dic53C2311yescharacteristic
68C42×D53C2×C414yes—
69C42×D53Q814yes—
70#13133150—42no—
71#13133148—42no—
72#13133146C2×C418yescomplemented
73#13133129C2×C418yes—
74#13133128C2×C414yescomplemented
75#13133124—21no—
76#13133124—21no—
77#13133124—21no—
78#13133124—21no—
79#13133124—21no—
80#13133124—21no—
81C22×C8×C53C2312yes—
82C2×C4⋊C4×C53—21no—
83C2×C4⋊C4×C53—21no—
84C2×C4⋊C4×C53C2312yes—
85C2×C4⋊C4×C53C2311yescharacteristic
86C2×C4⋊C4×C53C2311yescharacteristic
87C2×C42×C53C2311yescharacteristic
88#107×C53—42no—
89#106×C53—42no—
90#97×C53C2×C414yescomplemented
91#95×C53—21no—
92#95×C53—21no—
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—22no—
111C2×C4×D53—22no—
112C2×C4×D53—44no—
113C2×C4×D53—44no—
114C2×C4×D53—44no—
115C2×C4×D53C4⋊C414yes—
116C2×C4×D53C4⋊C414yes—
117C2×C4×D53C22×C412yes—
118C2×C4×D53C2×Q812yes—
119C2×C8×C53—22no—
120C2×C8×C53—22no—
121C2×C8×C53C22×C414yes—
122C4⋊C4×C53—22no—
123C4⋊C4×C53—22no—
124C4⋊C4×C53—22no—
125C4⋊C4×C53—22no—
126C4⋊C4×C53—42no—
127C4⋊C4×C53—42no—
128C42×C53C22×C412yes—
129C42×C53C2×Q812yes—
130#11719921—42no—
131#11719921—42no—
132C4⋊Dic53—22no—
133C4⋊Dic53—22no—
134C4⋊Dic53—22no—
135C4⋊Dic53—22no—
136C4⋊Dic53—42no—
137C4⋊Dic53—42no—
138#11719919—44no—
139#11719919—44no—
140#11719919—44no—
141C4×Dic53—42no—
142C4×Dic53—42no—
143C4×Dic53C4⋊C414yes—
144C4×Dic53C4⋊C414yes—
145C4×Dic53C22×C412yes—
146C4×Dic53C2×Q812yes—
147C2×#88322—22no—
148C2×#88322—22no—
149C2×#88322C22×C414yes—
150C23×C53C22×D411yescharacteristic
151C22×D53—21no—
152C22×D53—21no—
153C22×D53—21no—
154C22×D53—21no—
155C22×D53C2×C4⋊C412yes—
156C22×D53AGL(1,ℤ/8)12yes—
157C22×D53#13712yes—
158C2×C4×C53—21no—
159C2×C4×C53—21no—
160C2×C4×C53—21no—
161C2×C4×C53—21no—
162C2×C4×C53—42no—
163C2×C4×C53—41no—
164C2×C4×C53—41no—
165C2×C4×C53—42no—
166C2×C4×C53—42no—
167C2×C4×C53—41no—
168C2×C4×C53—41no—
169C2×C4×C53—41no—
170C2×C4×C53—41no—
171C2×C4×C53C2×C4⋊C412yes—
172C2×C4×C53C2×C4⋊C412yes—
173C2×C4×C53C23×C411yescharacteristic
174C2×C4×C53C22×Q811yescharacteristic
175C2×C4×C53H(5,𝔽2)12yes—
176C2×C4×C532−1+412yes—
177C2×Dic53—21no—
178C2×Dic53—21no—
179C2×Dic53—21no—
180C2×Dic53—21no—
181C2×Dic53—42no—
182C2×Dic53—41no—
183C2×Dic53—41no—
184C2×Dic53—42no—
185C2×Dic53—42no—
186C2×Dic53—41no—
187C2×Dic53—41no—
188C2×Dic53—41no—
189C2×Dic53—41no—
190C2×Dic53C2×C4⋊C412yes—
191C2×Dic53C2×C4⋊C412yes—
192C2×Dic53C2×C4⋊C412yes—
193C2×Dic53AGL(1,ℤ/8)12yes—
194C2×Dic53#13712yes—
195C4×D53—28no—
196C4×D53—24no—
197C4×D53—24no—
198C4×D53—28no—
199C4×D53—88no—
200C8×C53—24no—
201#88322—24no—
202#21987—531nomaximal, Sylow
203C22×C53C22×C4⋊C411yescharacteristic
204C22×C53#53511yescharacteristic
205C22×C53#53611yescharacteristic
206C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
207C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
208C22×C53C2×#13711yescharacteristic
209C22×C53C2×#13711yescharacteristic
210C2×D53—22no—
211C2×D53—22no—
212C2×D53—22no—
213C2×D53—22no—
214C2×D53—22no—
215C2×D53—22no—
216C2×D53#42814yes—
217C4×C53—22no—
218C4×C53—22no—
219C4×C53—82no—
220C4×C53—81no—
221C4×C53—81no—
222C4×C53#52012yes—
223C4×C53#54412yes—
224Dic53—22no—
225Dic53—22no—
226Dic53—82no—
227Dic53—81no—
228Dic53—81no—
229Dic53#42814yes—
230C2×#432—531no—
231C2×#384—531no—
232#2283—531no—
233#1948—532no—
234#1947—532no—
235#1853—532no—
236#1852—532no—
237#1847—531no—
238#1847D5311yescharacteristic, complemented
239#1836—531no—
240#1836—531no—
241C2×C53#231711yescharacteristic
242C2×C53C2×#42811yescharacteristic
243C2×C53C2×#42811yescharacteristic
244C2×C53#321711yescharacteristic
245C2×C53#321811yescharacteristic
246C2×C53#323811yescharacteristic
247C2×C53#323911yescharacteristic
248D53—24no—
249C22×C4⋊C4—532no—
250C22×C4⋊C4—1061no—
251C22×C4⋊C4—1061no—
252C22×C42—531no—
253#432—538no—
254C2×#107—532no—
255C2×#107C2×D5312yes—
256C2×#106—532no—
257C2×#106C2×D5312yes—
258C2×C4⋊C8—532no—
259C2×C22⋊C8—532no—
260C2×#97—531no—
261C2×#97C2×D5311yescharacteristic
262#384—532no—
263#384—531no—
264#384—1061no—
265#384—1061no—
266#384C2×D5311yescharacteristic
267#383—1061no—
268#382—531no—
269#382C2×D5311yescharacteristic
270#381—1061no—
271#380—531no—
272#380—531no—
273C4×C4⋊C4—531no—
274C4×C4⋊C4—531no—
275C4×C22⋊C4—1061no—
276#376—1061no—
277#340—532no—
278#340—532no—
279#340—532no—
280#340—532no—
281C53#2198711yescharacteristic, complemented, Sylow
282C23×C4—531no—
283C23×C4—531no—
284C23×C4—531no—
285C23×C4—1062no—
286C22×C8—532no—
287C22×C8C22×D5312yes—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4—532no—
291C2×C4⋊C4—531no—
292C2×C4⋊C4—531no—
293C2×C4⋊C4—1062no—
294C2×C4⋊C4—1062no—
295C2×C4⋊C4—1062no—
296C2×C4⋊C4—1061no—
297C2×C4⋊C4—1061no—
298C2×C4⋊C4—2122no—
299C2×C4⋊C4—2122no—
300C2×C4⋊C4—2124no—
301C2×C4⋊C4—2124no—
302C2×C4⋊C4C22×D5312yes—
303C2×C4⋊C4C22×D5311yescharacteristic
304C2×C4⋊C4C22×D5311yescharacteristic
305C2×C22⋊C4—1061no—
306C2×C22⋊C4—1061no—
307C2×C42—531no—
308C2×C42—531no—
309C2×C42—531no—
310C2×C42—534no—
311C2×C42—534no—
312C2×C42—1061no—
313C2×C42—1061no—
314C2×C42C22×D5311yescharacteristic
315#107—42no—
316#107—2122no—
317#106—42no—
318#106—2122no—
319C4⋊C8—538no—
320C22⋊C8—538no—
321#97—534no—
322#97C4×D5314yescomplemented
323#95—21no—
324#95—21no—
325#95—1061no—
326#95—1061no—
327#95—1061no—
328#95—1061no—
329#95—1061no—
330#95—1061no—
331C24—531no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—531no—
336C22×C4—531no—
337C22×C4—531no—
338C22×C4—534no—
339C22×C4—534no—
340C22×C4—532no—
341C22×C4—532no—
342C22×C4—1062no—
343C22×C4—1062no—
344C22×C4—1062no—
345C22×C4—1062no—
346C22×C4—1062no—
347C22×C4—1062no—
348C22×C4—1061no—
349C22×C4—1061no—
350C22×C4—1062no—
351C22×C4—2124no—
352C22×C4—2124no—
353C22×C4—2124no—
354C22×C4D4×D5311yescharacteristic
355C22×C4D4×D5311yescharacteristic
356C22×C4C23×D5311yescharacteristic
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—534no—
360C2×C8—1062no—
361C2×C8—1062no—
362C2×C8C2×C4×D5314yes—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—42no—
368C4⋊C4—42no—
369C4⋊C4—1062no—
370C4⋊C4—1062no—
371C4⋊C4—1062no—
372C4⋊C4—1062no—
373C4⋊C4—2122no—
374C4⋊C4—2122no—
375C4⋊C4—2124no—
376C4⋊C4—2124no—
377C4⋊C4—2124no—
378C22⋊C4—2122no—
379C22⋊C4—2122no—
380C42—534no—
381C42—534no—
382C42—532no—
383C42—532no—
384C42—2122no—
385C42—2122no—
386C42C2×C4×D5312yes—
387C42Q8×D5312yes—
388C23—532no—
389C23—532no—
390C23—532no—
391C23—1061no—
392C23—1061no—
393C23—1061no—
394C23—1061no—
395C23C2×D4×D5311yescharacteristic
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—41no—
401C2×C4—41no—
402C2×C4—42no—
403C2×C4—42no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—532no—
410C2×C4—532no—
411C2×C4—532no—
412C2×C4—532no—
413C2×C4—532no—
414C2×C4—1068no—
415C2×C4—1068no—
416C2×C4—1061no—
417C2×C4—1061no—
418C2×C4—1061no—
419C2×C4—1061no—
420C2×C4—1064no—
421C2×C4—1064no—
422C2×C4—4248no—
423C2×C4—2121no—
424C2×C4—2121no—
425C2×C4—2122no—
426C2×C4—2122no—
427C2×C4—2121no—
428C2×C4—2121no—
429C2×C4—2122no—
430C2×C4—2121no—
431C2×C4—2121no—
432C2×C4C4⋊C4×D5312yes—
433C2×C4C4⋊C4×D5312yes—
434C2×C4C22×C4×D5311yescharacteristic
435C2×C4C2×Q8×D5311yescharacteristic
436C2×C4#1313326212yes—
437C2×C4#1313326512yes—
438C8—24no—
439C8—1064no—
440C22—534no—
441C22—1062no—
442C22—1062no—
443C22—1062no—
444C22—1062no—
445C22—1062no—
446C22—1062no—
447C22C2×C4⋊C4×D5311yescharacteristic
448C22#1681118611yescharacteristic
449C22#1681119811yescharacteristic
450C22AGL(1,ℤ/8)×D5311yescharacteristic
451C22AGL(1,ℤ/8)×D5311yescharacteristic
452C22#137×D5311yescharacteristic
453C22#137×D5311yescharacteristic
454C4—22no—
455C4—22no—
456C4—82no—
457C4—81no—
458C4—81no—
459C4—534no—
460C4—1062no—
461C4—1062no—
462C4—4242no—
463C4—4241no—
464C4—4241no—
465C4#1681110412yes—
466C4#1681125012yes—
467C2—1064no—
468C2#2027733611yescharacteristic
469C2#428×D5311yescharacteristic
470C2#428×D5311yescharacteristic
471C2#2028572911yescharacteristic
472C2#2028573811yescharacteristic
473C2#2028593711yescharacteristic
474C2#2028594611yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1847C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
cosocleC22×C4×C53C24
hypercenter#1847D53
hypocenterC53#21987
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1847

Complements

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

NH ≅ G/Ntype
4. #20275475C2semidirect
5. #20275470C2semidirect
8. #20274884C2semidirect
9. #20274850C2semidirect
10. #20274849C2semidirect
11. C53×#1847C2semidirect
16. #16810678C4semidirect
72. #13133146C2×C4semidirect
74. #13133128C2×C4semidirect
90. #97×C53C2×C4semidirect
238. #1847D53semidirect
281. C53#21987semidirect
322. #97C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168106782
#384×D532
#202771082
#202754752
#202754702
#202749062
#202749012
#202748842
#202748502
#202748492
C53×#18472
#2198753