Finite Group Databasebeta

← #27465724

Subgroups — #27465724

Order 13568.

1010 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810937C211yesmaximal, characteristic
2C2×#16810936C211yesmaximal, characteristic
3#20277221C211yesmaximal, characteristic
4C2×#16810492C211yesmaximal, characteristic, complemented
5#20276190C212yesmaximal, complemented
6#20276122C212yesmaximal, complemented
7#20275472C212yesmaximal, complemented
8#20274799C211yesmaximal, characteristic, complemented
9#20274714C212yesmaximal, complemented
10#20274693C211yesmaximal, characteristic, complemented
11C53×#1823C211yesmaximal, characteristic, complemented
12C2×#13133191C2211yescharacteristic
13C2×#13133179C2211yescharacteristic
14C2×#13133177—21no—
15C22×C4⋊Dic53C2211yescharacteristic
16C22×#11719919—21no—
17C22×#11719919—21no—
18C22×C4×Dic53C2211yescharacteristic
19#16810937—24no—
20#16810936—24no—
21C2×#13133151C2212yes—
22C2×#13133150C2212yes—
23C2×#13133146C2212yes—
24C2×#13133139C2212yes—
25C2×#13133137C2212yes—
26C2×C22⋊#88322C2211yescharacteristic
27#16810524C2211yescharacteristic
28#16810519C2211yescharacteristic
29#16810512—21no—
30#16810500—21no—
31#16810492—21no—
32#16810492—21no—
33#16810492C2212yes—
34#16810492C2212yes—
35C4⋊C4×Dic53C2211yescharacteristic
36#16810470—21no—
37#16810385C2212yes—
38#16810385C2212yes—
39#16810377C2212yes—
40C22×C4⋊C4×C53C2211yescharacteristic
41C2×C22⋊C8×C53C2211yescharacteristic
42C53×#380C2211yescharacteristic
43C53×#340C2212yes—
44C53×#340C2212yes—
45C22×D212C2311yescharacteristic
46C22×C22⋊D53—21no—
47C23×Dic53—21no—
48C23×Dic53D411yescharacteristic
49C23×Dic53D411yescharacteristic
50C22×C4×D53—21no—
51C22×#88322C2312yes—
52#13133191—22no—
53#13133191D414yes—
54#13133179—24no—
55#13133177—44no—
56C2×C4⋊Dic53—21no—
57C2×C4⋊Dic53—22no—
58C2×C4⋊Dic53—22no—
59C2×C4⋊Dic53C2312yes—
60C2×C4⋊Dic53C2311yescharacteristic
61C2×#11719919—21no—
62C2×#11719919—22no—
63C2×#11719919—22no—
64C2×#11719919—22no—
65C2×#11719919—22no—
66C2×#11719919—44no—
67C2×#11719919—44no—
68C2×#11719921—21no—
69C2×#11719921—21no—
70C2×#11719921—21no—
71C2×#11719921—21no—
72C2×C4×Dic53—24no—
73C2×C4×Dic53—21no—
74C2×C4×Dic53D411yescharacteristic
75C2×C4×Dic53D411yescharacteristic
76C2×C4×Dic53C2311yescharacteristic
77C2×C4×Dic53C2311yescharacteristic
78#13133151—24no—
79#13133150—24no—
80#13133146—24no—
81#13133139—24no—
82#13133137—24no—
83C22⋊#88322—22no—
84#13133124—21no—
85#13133124—21no—
86C23×C4×C53—21no—
87C23×C4×C53C2311yescharacteristic
88C22×C8×C53C2312yes—
89C2×C4⋊C4×C53—21no—
90C2×C4⋊C4×C53—22no—
91C2×C4⋊C4×C53—22no—
92C2×C4⋊C4×C53C2311yescharacteristic
93C2×C4⋊C4×C53C2312yes—
94C2×C4⋊C4×C53C2311yescharacteristic
95C2×C22⋊C4×C53—21no—
96C22⋊C8×C53—22no—
97#95×C53—21no—
98C24×C53C2×D411yescharacteristic
99C23×D53—21no—
100C22×C4×C53—22no—
101C22×C4×C53—22no—
102C22×C4×C53—21no—
103C22×C4×C53—21no—
104C22×C4×C53—22no—
105C22×C4×C53—42no—
106C22×C4×C53—42no—
107C22×C4×C53—42no—
108C22×C4×C53C2×D411yescharacteristic
109C22×C4×C53C2×D412yes—
110C22×C4×C53#4112yes—
111C22×C4×C53C2411yescharacteristic
112C2×C22⋊D53—24no—
113C2×C22⋊D53—42no—
114C2×C22⋊D53—42no—
115C22×Dic53—22no—
116C22×Dic53—22no—
117C22×Dic53—22no—
118C22×Dic53—22no—
119C22×Dic53—22no—
120C22×Dic53—21no—
121C22×Dic53—22no—
122C22×Dic53—42no—
123C22×Dic53—42no—
124C22×Dic53—42no—
125C22×Dic53D814yes—
126C22×Dic53C2×D411yescharacteristic
127C22×Dic53C2×D411yescharacteristic
128C22×Dic53C2×D411yescharacteristic
129C22×Dic53C2×D411yescharacteristic
130C2×D212—22no—
131C2×D212—22no—
132C2×D212—21no—
133C2×D212C2×D412yes—
134C2×C4×D53—42no—
135C2×C4×D53—42no—
136C2×C4×D53—42no—
137C2×C8×C53—22no—
138C2×C8×C53—22no—
139C2×C8×C53—22no—
140C4⋊C4×C53—22no—
141C4⋊C4×C53—22no—
142C4⋊C4×C53—22no—
143C4⋊C4×C53—22no—
144C4⋊C4×C53—42no—
145C22⋊C4×C53—42no—
146#11719921—24no—
147#11719921—42no—
148#11719921—42no—
149#11719921—42no—
150C4⋊Dic53—21no—
151C4⋊Dic53—22no—
152C4⋊Dic53—22no—
153C4⋊Dic53—42no—
154C4⋊Dic53C2×D412yes—
155#11719919—44no—
156#11719919—44no—
157#11719919—44no—
158#11719919—44no—
159#11719919—42no—
160C4×Dic53—22no—
161C4×Dic53—22no—
162C4×Dic53—22no—
163C4×Dic53—42no—
164C4×Dic53#4114yes—
165C2×#88322—22no—
166C2×#88322—22no—
167C2×#88322—22no—
168C23×C53—21no—
169C23×C53—21no—
170C23×C53—21no—
171C23×C53—21no—
172C23×C53#12112yes—
173C23×C53C2×D812yes—
174C23×C53AGL(1,ℤ/8)12yes—
175C23×C53C22×D411yescharacteristic
176C22×D53—22no—
177C22×D53—41no—
178C22×D53—41no—
179C22×D53—41no—
180C22×D53—41no—
181C22×D53—41no—
182C22×D53—41no—
183C2×C4×C53—21no—
184C2×C4×C53—21no—
185C2×C4×C53—24no—
186C2×C4×C53—24no—
187C2×C4×C53—42no—
188C2×C4×C53—41no—
189C2×C4×C53—41no—
190C2×C4×C53—41no—
191C2×C4×C53—42no—
192C2×C4×C53—41no—
193C2×C4×C53—42no—
194C2×C4×C53—41no—
195C2×C4×C53—41no—
196C2×C4×C53—84no—
197C2×C4×C53#12112yes—
198C2×C4×C53C22×D411yescharacteristic
199C2×C4×C53C2×#4112yes—
200C2×C4×C53C2×#4111yescharacteristic
201C2×C4×C532−1+412yes—
202C22⋊D53—48no—
203C2×Dic53—28no—
204C2×Dic53—21no—
205C2×Dic53—21no—
206C2×Dic53—21no—
207C2×Dic53—21no—
208C2×Dic53—21no—
209C2×Dic53—21no—
210C2×Dic53—28no—
211C2×Dic53—22no—
212C2×Dic53—41no—
213C2×Dic53—41no—
214C2×Dic53—42no—
215C2×Dic53—42no—
216C2×Dic53—42no—
217C2×Dic53—84no—
218C2×Dic53#12312yes—
219C2×Dic53#12312yes—
220C2×Dic53C2×D812yes—
221C2×Dic53C2×D812yes—
222C2×Dic53#13712yes—
223C2×Dic53#13712yes—
224D212—24no—
225D212—42no—
226C4×D53—84no—
227C8×C53—42no—
228#88322—42no—
229#27644—531nomaximal, Sylow
230C22×C53—22no—
231C22×C53—22no—
232C22×C53—22no—
233C22×C53—22no—
234C22×C53—22no—
235C22×C53—22no—
236C22×C53#45914yes—
237C22×C53C2×#12111yescharacteristic
238C22×C53C2×#12311yescharacteristic
239C22×C53#53611yescharacteristic
240C22×C53C22×D811yescharacteristic
241C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
242C22×C53C2×#13711yescharacteristic
243C22×C53C2×#13711yescharacteristic
244C2×D53—42no—
245C2×D53—42no—
246C2×D53—42no—
247C2×D53—81no—
248C2×D53—81no—
249C2×D53—81no—
250C2×D53—81no—
251C4×C53—22no—
252C4×C53—81no—
253C4×C53—81no—
254C4×C53—82no—
255C4×C53#54712yes—
256C4×C53#55212yes—
257Dic53—22no—
258Dic53—22no—
259Dic53—22no—
260Dic53—42no—
261Dic53—82no—
262Dic53#48014yescomplemented
263C2×#480—531no—
264C2×#480—531no—
265C4×#121—531no—
266C2×#384—531no—
267#2083—532no—
268#2071—532no—
269#1947—532no—
270#1823—531no—
271#1823D5311yescharacteristic, complemented
272#1802—532no—
273#1797—531no—
274C2×C53—24no—
275C2×C53#248611yescharacteristic
276C2×C53C2×#45911yescharacteristic
277C2×C53#303911yescharacteristic
278C2×C53C2×#48011yescharacteristic
279C2×C53#309211yescharacteristic
280C2×C53#322611yescharacteristic
281C2×C53#323811yescharacteristic
282D53—82no—
283C2×#121—531no—
284C2×C4×D4—531no—
285C2×C4×D4—1061no—
286C22×C4⋊C4—531no—
287C22×C4⋊C4—1061no—
288C22×C4⋊C4—1061no—
289C22×C4⋊C4C2×D5311yescharacteristic
290C22×C42—531no—
291#480—1064no—
292#480—1064no—
293C2×#107—532no—
294C2×#107—532no—
295C2×C4⋊C8—532no—
296C2×#102—532no—
297C2×#102—532no—
298C2×C22⋊C8—531no—
299C2×C22⋊C8C2×D5311yescharacteristic
300#390—531no—
301#389—531no—
302#388—1061no—
303#386—1061no—
304#384—532no—
305#384—532no—
306#384—1061no—
307#384—1061no—
308#380C2×D5311yescharacteristic
309C4×C4⋊C4—531no—
310C4×C22⋊C4—1061no—
311#340—532no—
312#340—532no—
313#340C2×D5312yes—
314#340C2×D5312yes—
315#336—532no—
316C53#2764411yescharacteristic, complemented, Sylow
317C22×D4—531no—
318C22×D4—1061no—
319C23×C4—21no—
320C23×C4—531no—
321C23×C4—531no—
322C23×C4—1061no—
323C23×C4—1061no—
324C23×C4C22×D5311yescharacteristic
325C22×C8—532no—
326C22×C8C22×D5312yes—
327#121—534no—
328#121—1062no—
329C4×D4—1064no—
330C4×D4—2124no—
331C2×C4⋊C4—22no—
332C2×C4⋊C4—22no—
333C2×C4⋊C4—21no—
334C2×C4⋊C4—531no—
335C2×C4⋊C4—532no—
336C2×C4⋊C4—1062no—
337C2×C4⋊C4—1062no—
338C2×C4⋊C4—1062no—
339C2×C4⋊C4—1062no—
340C2×C4⋊C4—1062no—
341C2×C4⋊C4—1062no—
342C2×C4⋊C4—1061no—
343C2×C4⋊C4—1061no—
344C2×C4⋊C4—2124no—
345C2×C4⋊C4—2124no—
346C2×C4⋊C4C22×D5312yes—
347C2×C4⋊C4C22×D5311yescharacteristic
348C2×C4⋊C4C22×D5311yescharacteristic
349C2×C22⋊C4—21no—
350C2×C22⋊C4—1061no—
351C2×C22⋊C4—1061no—
352C2×C22⋊C4—1061no—
353C2×C22⋊C4—1061no—
354C2×C42—531no—
355C2×C42—531no—
356C2×C42—531no—
357C2×C42—531no—
358C2×C42—1064no—
359C2×C42—1061no—
360#107—1064no—
361#107—1064no—
362C4⋊C8—1064no—
363#102—1064no—
364#102—1064no—
365C22⋊C8—22no—
366C22⋊C8—1062no—
367#95—21no—
368#95—1061no—
369#95—1061no—
370C24—1061no—
371C24D4×D5311yescharacteristic
372C2×D4—532no—
373C2×D4—1064no—
374C2×D4—1062no—
375C2×D4—1061no—
376C2×D4—1062no—
377C2×D4—2122no—
378C2×D4—2122no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—21no—
382C22×C4—21no—
383C22×C4—22no—
384C22×C4—42no—
385C22×C4—42no—
386C22×C4—42no—
387C22×C4—531no—
388C22×C4—534no—
389C22×C4—531no—
390C22×C4—531no—
391C22×C4—531no—
392C22×C4—1062no—
393C22×C4—1062no—
394C22×C4—1062no—
395C22×C4—1062no—
396C22×C4—1062no—
397C22×C4—1061no—
398C22×C4—1062no—
399C22×C4—2122no—
400C22×C4—2122no—
401C22×C4—2122no—
402C22×C4—2122no—
403C22×C4—2122no—
404C22×C4—2122no—
405C22×C4#1171994512yes—
406C22×C4D4×D5311yescharacteristic
407C22×C4D4×D5312yes—
408C22×C4C23×D5311yescharacteristic
409C2×C8—22no—
410C2×C8—22no—
411C2×C8—22no—
412C2×C8—1062no—
413C2×C8—1062no—
414C2×C8—1062no—
415C4⋊C4—22no—
416C4⋊C4—22no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—42no—
420C4⋊C4—532no—
421C4⋊C4—1061no—
422C4⋊C4—1062no—
423C4⋊C4—1062no—
424C4⋊C4—2124no—
425C4⋊C4—2124no—
426C4⋊C4—2122no—
427C4⋊C4—2124no—
428C4⋊C4—2124no—
429C4⋊C4—2122no—
430C22⋊C4—42no—
431C22⋊C4—1064no—
432C22⋊C4—2122no—
433C22⋊C4—2122no—
434C22⋊C4—2122no—
435C42—534no—
436C42—1062no—
437C42—1062no—
438C42—1062no—
439C42—2122no—
440C23—21no—
441C23—21no—
442C23—21no—
443C23—21no—
444C23—1062no—
445C23—2121no—
446C23—2121no—
447C23—2121no—
448C23—2121no—
449C23—2121no—
450C23—2121no—
451C23#1313319012yes—
452C23D8×D5312yes—
453C23#1313324412yes—
454C23C2×D4×D5311yescharacteristic
455D4—1064no—
456D4—2128no—
457D4—2122no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—24no—
461C2×C4—24no—
462C2×C4—42no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—84no—
472C2×C4—532no—
473C2×C4—532no—
474C2×C4—532no—
475C2×C4—532no—
476C2×C4—532no—
477C2×C4—532no—
478C2×C4—1068no—
479C2×C4—1068no—
480C2×C4—1061no—
481C2×C4—1061no—
482C2×C4—1061no—
483C2×C4—1061no—
484C2×C4—1061no—
485C2×C4—1061no—
486C2×C4—1062no—
487C2×C4—4244no—
488C2×C4—4244no—
489C2×C4—2122no—
490C2×C4—2122no—
491C2×C4—2122no—
492C2×C4—2121no—
493C2×C4—2121no—
494C2×C4#1313319012yes—
495C2×C4C2×D4×D5311yescharacteristic
496C2×C4#41×D5312yes—
497C2×C4C2×#1171994511yescharacteristic
498C2×C4#1313326412yes—
499C8—42no—
500C8—2122no—
501C22—22no—
502C22—22no—
503C22—22no—
504C22—22no—
505C22—22no—
506C22—22no—
507C22—4241no—
508C22—4241no—
509C22—4241no—
510C22—4241no—
511C22—2122no—
512C22—2122no—
513C22—2122no—
514C22#1681081714yes—
515C22C2×#1313319011yescharacteristic
516C22#123×D5311yescharacteristic
517C22#1681119511yescharacteristic
518C22C2×D8×D5311yescharacteristic
519C22C2×#1313324411yescharacteristic
520C22#137×D5311yescharacteristic
521C22#137×D5311yescharacteristic
522C4—22no—
523C4—81no—
524C4—81no—
525C4—82no—
526C4—534no—
527C4—1062no—
528C4—1062no—
529C4—1062no—
530C4—4242no—
531C4—2122no—
532C4#1681127112yes—
533C4#1681130612yes—
534C2—24no—
535C2—4242no—
536C2#2027902211yescharacteristic
537C2C2×#1681081711yescharacteristic
538C2#2028420111yescharacteristic
539C2#480×D5311yescharacteristic
540C2#2028463911yescharacteristic
541C2#2028583411yescharacteristic
542C2#2028593911yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1823C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
hypercenter#1823D53
hypocenterC53#27644
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1823

Complements

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

NH ≅ G/Ntype
4. C2×#16810492C2semidirect
5. #20276190C2semidirect
6. #20276122C2semidirect
7. #20275472C2semidirect
8. #20274799C2semidirect
9. #20274714C2semidirect
10. #20274693C2semidirect
11. C53×#1823C2semidirect
262. Dic53#480semidirect
271. #1823D53semidirect
316. C53#27644semidirect

Maximal subgroups

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

subgroupindex
C2×#168109372
C2×#168109362
#202772212
C2×#168104922
#202761902
#202761222
#202754722
#202747992
#202747142
#202746932
C53×#18232
#2764453