Finite Group Databasebeta

← #27438746

Subgroups — #27438746

Order 13568.

1276 conjugacy classes of subgroups.

The subgroup lattice has 510 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811029C211yesmaximal, characteristic, complemented
2C2×#16810923C211yesmaximal, characteristic, complemented
3#20279881C211yesmaximal, characteristic, complemented
4#20275901C212yesmaximal, complemented
5#20275894C212yesmaximal, complemented
6#20275664C211yesmaximal, characteristic, complemented
7#20275514C211yesmaximal, characteristic, complemented
8#20274908C212yesmaximal, complemented
9#20274903C212yesmaximal, complemented
10#20274540C211yesmaximal, characteristic, complemented
11C2×C53×#547C211yesmaximal, characteristic
12C2×#13133209C2211yescharacteristic
13#16811029—24no—
14#16810923—24no—
15C2×#13133150C2212yes—
16C2×#13133150C2211yescharacteristic
17C2×#13133148C2212yes—
18C2×#13133148C2211yescharacteristic
19C2×C4⋊#88322C2211yescharacteristic
20C2×#13133138C2211yescharacteristic
21C2×#13133138C2211yescharacteristic
22C2×#13133138C2212yes—
23C2×C22⋊#88322C2212yes—
24C2×C22⋊#88322C2212yes—
25C2×#13133128C2211yescharacteristic
26#16810561—21no—
27#16810537—21no—
28#16810517—22no—
29#16810483C2212yes—
30#16810483C2212yes—
31C4×C4⋊Dic53C2211yescharacteristic
32#16810385C2212yes—
33#16810385C2212yes—
34#16810385C2212yes—
35C22×#41×C53—21no—
36C53×#547—24no—
37C53×#547—24no—
38C2×C4⋊Q8×C53C2211yescharacteristic
39C2×#123×C53—22no—
40C2×C22⋊Q8×C53—21no—
41C2×#121×C53C2212yes—
42C2×C4×D4×C53C2211yescharacteristic
43C2×C4×D4×C53C2211yescharacteristic
44C22×C4⋊C4×C53C2212yes—
45C22×#88322C2311yescharacteristic
46C22×#88322C2311yescharacteristic
47C22×#88322C2312yes—
48#13133209—22no—
49#13133209—22no—
50C2×C4⋊Dic53C2311yescharacteristic
51C2×C4⋊Dic53C2311yescharacteristic
52C2×C4⋊Dic53C2312yes—
53C2×#11719919—21no—
54C2×#11719919—21no—
55C2×C22⋊Dic53—22no—
56C2×C22⋊Dic53—22no—
57C2×C4×Dic53—21no—
58#13133150—24no—
59#13133150—22no—
60#13133148—24no—
61#13133148—22no—
62C4⋊#88322—22no—
63#13133138—24no—
64#13133138—22no—
65#13133138—22no—
66C22⋊#88322—24no—
67C22⋊#88322—24no—
68#13133128—22no—
69#13133124—22no—
70#13133124—22no—
71#13133124—21no—
72C2×#41×C53—28no—
73C2×#41×C53—48no—
74C22×Q8×C53—21no—
75C22×D4×C53—22no—
76C22×D4×C53D411yescharacteristic
77C22×D4×C53C2311yescharacteristic
78C23×C4×C53—22no—
79C23×C4×C53—21no—
80C23×C4×C53C2312yes—
81C4⋊Q8×C53—22no—
82C4⋊Q8×C53D414yes—
83#123×C53—216no—
84C22⋊Q8×C53—28no—
85#121×C53—24no—
86#121×C53—24no—
87C4×D4×C53—22no—
88C4×D4×C53—22no—
89C4×D4×C53—22no—
90C4×D4×C53D414yes—
91C2×C4⋊C4×C53—24no—
92C2×C4⋊C4×C53—24no—
93C2×C4⋊C4×C53—22no—
94C2×C4⋊C4×C53—21no—
95C2×C4⋊C4×C53D411yescharacteristic
96C2×C4⋊C4×C53C2311yescharacteristic
97C2×C4⋊C4×C53C2312yes—
98C2×C22⋊C4×C53—21no—
99C2×C22⋊C4×C53—22no—
100C2×C22⋊C4×C53D412yes—
101C2×C42×C53C2311yescharacteristic
102C24×C53—21no—
103C24×C53C2×D412yes—
104#41×C53—432no—
105C2×Q8×C53—24no—
106C2×Q8×C53—44no—
107C2×D4×C53—24no—
108C2×D4×C53—28no—
109C2×D4×C53—22no—
110C2×D4×C53—21no—
111C2×D4×C53—22no—
112C2×D4×C53—22no—
113C2×D4×C53—44no—
114C2×D4×C53—44no—
115C2×D4×C53C2×D412yes—
116C22×C4×C53—28no—
117C22×C4×C53—21no—
118C22×C4×C53—22no—
119C22×C4×C53—24no—
120C22×C4×C53—22no—
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53—22no—
124C22×C4×C53—22no—
125C22×C4×C53—22no—
126C22×C4×C53—44no—
127C22×C4×C53—44no—
128C22×C4×C53—44no—
129C22×C4×C53C2×D412yes—
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2411yescharacteristic
133C22×Dic53—22no—
134C22×Dic53—21no—
135C22×Dic53—21no—
136C4⋊C4×C53—28no—
137C4⋊C4×C53—24no—
138C4⋊C4×C53—22no—
139C4⋊C4×C53—21no—
140C4⋊C4×C53—22no—
141C4⋊C4×C53C2×D412yes—
142C4⋊C4×C53C2×D414yes—
143C22⋊C4×C53—28no—
144C22⋊C4×C53—24no—
145C22⋊C4×C53—24no—
146C42×C53C2×D412yes—
147C42×C53C2×D412yes—
148C22⋊Dic53—44no—
149C22⋊Dic53—82no—
150C4⋊Dic53—21no—
151C4⋊Dic53—21no—
152C4⋊Dic53—21no—
153C4⋊Dic53—21no—
154C4⋊Dic53—22no—
155C4⋊Dic53—22no—
156#11719919—42no—
157#11719919—42no—
158C4×Dic53—81no—
159C2×#88322—22no—
160C2×#88322—22no—
161C2×#88322—21no—
162C2×#88322—21no—
163C2×#88322—22no—
164C2×#88322—22no—
165C2×#88322#4114yescomplemented
166C23×C53—22no—
167C23×C53—22no—
168C23×C53—22no—
169C23×C53—22no—
170C23×C53—22no—
171C23×C53—22no—
172C23×C53—22no—
173C23×C53—42no—
174C23×C53—42no—
175C23×C53—41no—
176C23×C53—41no—
177C23×C53AGL(1,ℤ/8)14yes—
178C23×C53C22×D411yescharacteristic
179Q8×C53—48no—
180D4×C53—28no—
181D4×C53—22no—
182D4×C53—22no—
183D4×C53—416no—
184D4×C53—42no—
185C2×C4×C53—22no—
186C2×C4×C53—21no—
187C2×C4×C53—21no—
188C2×C4×C53—22no—
189C2×C4×C53—24no—
190C2×C4×C53—22no—
191C2×C4×C53—22no—
192C2×C4×C53—22no—
193C2×C4×C53—48no—
194C2×C4×C53—416no—
195C2×C4×C53—42no—
196C2×C4×C53—42no—
197C2×C4×C53—42no—
198C2×C4×C53—42no—
199C2×C4×C53—42no—
200C2×C4×C53—42no—
201C2×C4×C53—42no—
202C2×C4×C53#12012yes—
203C2×C4×C53#12012yes—
204C2×C4×C53C22×D411yescharacteristic
205C2×C4×C53C22×D411yescharacteristic
206C2×C4×C53C2×#4112yes—
207C2×C4×C532−1+412yes—
208C2×Dic53—41no—
209C2×Dic53—41no—
210C2×Dic53—42no—
211C2×Dic53—42no—
212C2×Dic53—42no—
213C2×Dic53—42no—
214C2×Dic53—41no—
215C2×Dic53—41no—
216#88322—24no—
217#88322—41no—
218#88322—41no—
219#25709—531nomaximal, Sylow
220C22×C53—24no—
221C22×C53—24no—
222C22×C53—24no—
223C22×C53—42no—
224C22×C53—42no—
225C22×C53—42no—
226C22×C53—42no—
227C22×C53—42no—
228C22×C53—42no—
229C22×C53—42no—
230C22×C53—82no—
231C22×C53—82no—
232C22×C53C2×#12011yescharacteristic
233C22×C53#54712yes—
234C22×C53C2×AGL(1,ℤ/8)12yes—
235C22×C53#57711yescharacteristic
236C22×C53#57811yescharacteristic
237C4×C53—22no—
238C4×C53—22no—
239C4×C53—42no—
240C4×C53—42no—
241C4×C53—44no—
242C4×C53#53211yescharacteristic
243C4×C53#53611yescharacteristic
244C4×C53#54812yes—
245Dic53—82no—
246Dic53—81no—
247Dic53—81no—
248C2×#547D5311yescharacteristic
249C2×#501—531no—
250C2×#478—531no—
251#2562—531no—
252#2034—532no—
253#2033—532no—
254#1987—531no—
255#1955—531no—
256#1853—532no—
257#1852—532no—
258#1762—531no—
259C2×C53—44no—
260C2×C53—82no—
261C2×C53#243211yescharacteristic
262C2×C53#300311yescharacteristic
263C2×C53#301311yescharacteristic
264C2×C53#334312yes—
265C2×C53#334512yes—
266C22×#41—21no—
267#547—24no—
268#547—24no—
269C2×C4⋊Q8C2×D5311yescharacteristic
270C2×#125—531no—
271C2×#123—22no—
272C2×C22⋊Q8—21no—
273C2×#121C2×D5312yes—
274C2×C4×D4C2×D5311yescharacteristic
275C2×C4×D4C2×D5311yescharacteristic
276C22×C4⋊C4C2×D5312yes—
277#501—1064no—
278#478—1064no—
279C2×#107—531no—
280C2×#107—532no—
281C2×#106—531no—
282C2×#106—532no—
283C2×C4⋊C8—531no—
284C2×#102—531no—
285C2×#102—531no—
286C2×#102—532no—
287C2×C22⋊C8—532no—
288C2×C22⋊C8—532no—
289C2×#97—531no—
290#400—1061no—
291#394—1061no—
292#388—1062no—
293#380—532no—
294#380—532no—
295C4×C4⋊C4—531no—
296#340—532no—
297#340—532no—
298#340—532no—
299C53#2570911yescharacteristic, complemented, Sylow
300C2×#41—28no—
301C2×#41—48no—
302C22×Q8—21no—
303C22×D4—22no—
304C22×D4C22⋊D5311yescharacteristic
305C22×D4C22×D5311yescharacteristic
306C23×C4—22no—
307C23×C4—21no—
308C23×C4C22×D5312yes—
309C22×C8—532no—
310C22×C8—531no—
311C22×C8—531no—
312C4⋊Q8—22no—
313C4⋊Q8C22⋊D5314yes—
314#125—1062no—
315#125—1062no—
316#123—216no—
317C22⋊Q8—28no—
318#121—24no—
319#121—24no—
320C4×D4—22no—
321C4×D4—22no—
322C4×D4—22no—
323C4×D4C22⋊D5314yes—
324C2×C4⋊C4—22no—
325C2×C4⋊C4—21no—
326C2×C4⋊C4—24no—
327C2×C4⋊C4—24no—
328C2×C4⋊C4—531no—
329C2×C4⋊C4—531no—
330C2×C4⋊C4—532no—
331C2×C4⋊C4—1061no—
332C2×C4⋊C4—1061no—
333C2×C4⋊C4C22⋊D5311yescharacteristic
334C2×C4⋊C4C22×D5311yescharacteristic
335C2×C4⋊C4C22×D5312yes—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—1062no—
339C2×C22⋊C4—1062no—
340C2×C22⋊C4C22⋊D5312yes—
341C2×C42—1061no—
342C2×C42C22×D5311yescharacteristic
343#107—1062no—
344#107—1064no—
345#106—1062no—
346#106—1064no—
347C4⋊C8—1062no—
348#102—1064no—
349#102—1062no—
350#102—1062no—
351C22⋊C8—1064no—
352C22⋊C8—1064no—
353#97—1062no—
354#95—1061no—
355#95—1062no—
356#95—1062no—
357C24—21no—
358C24C2×C22⋊D5312yes—
359#41—432no—
360C2×Q8—24no—
361C2×Q8—44no—
362C2×D4—24no—
363C2×D4—28no—
364C2×D4—22no—
365C2×D4—21no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—44no—
369C2×D4—44no—
370C2×D4C2×C22⋊D5312yes—
371C22×C4—28no—
372C22×C4—21no—
373C22×C4—22no—
374C22×C4—24no—
375C22×C4—22no—
376C22×C4—22no—
377C22×C4—22no—
378C22×C4—22no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—44no—
382C22×C4—44no—
383C22×C4—44no—
384C22×C4—1062no—
385C22×C4—1061no—
386C22×C4—1061no—
387C22×C4C2×C22⋊D5312yes—
388C22×C4C2×C22⋊D5311yescharacteristic
389C22×C4C2×C22⋊D5311yescharacteristic
390C22×C4C23×D5311yescharacteristic
391C2×C8—534no—
392C2×C8—1062no—
393C2×C8—1062no—
394C2×C8—1061no—
395C2×C8—1061no—
396C2×C8—1062no—
397C2×C8—1062no—
398C4⋊C4—28no—
399C4⋊C4—24no—
400C4⋊C4—22no—
401C4⋊C4—21no—
402C4⋊C4—22no—
403C4⋊C4—1061no—
404C4⋊C4—1061no—
405C4⋊C4—1061no—
406C4⋊C4—1061no—
407C4⋊C4—1062no—
408C4⋊C4—1062no—
409C4⋊C4—2122no—
410C4⋊C4—2122no—
411C4⋊C4C2×C22⋊D5312yes—
412C4⋊C4C2×C22⋊D5314yes—
413C22⋊C4—28no—
414C22⋊C4—24no—
415C22⋊C4—24no—
416C22⋊C4—4242no—
417C22⋊C4—2124no—
418C42—4241no—
419C42C2×C22⋊D5312yes—
420C42C2×C22⋊D5312yes—
421C23—22no—
422C23—22no—
423C23—22no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—22no—
428C23—42no—
429C23—42no—
430C23—41no—
431C23—41no—
432C23#1313324314yes—
433C23C22×C22⋊D5311yescharacteristic
434Q8—48no—
435D4—28no—
436D4—22no—
437D4—22no—
438D4—416no—
439D4—42no—
440C2×C4—22no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—24no—
448C2×C4—48no—
449C2×C4—416no—
450C2×C4—42no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—2122no—
458C2×C4—2121no—
459C2×C4—2121no—
460C2×C4—2122no—
461C2×C4—2122no—
462C2×C4—2122no—
463C2×C4—2121no—
464C2×C4—2121no—
465C2×C4#1313318612yes—
466C2×C4#1313318612yes—
467C2×C4C22×C22⋊D5311yescharacteristic
468C2×C4C22×C22⋊D5311yescharacteristic
469C2×C4#41×D5312yes—
470C2×C4#1313326412yes—
471C8—1064no—
472C8—2121no—
473C8—2121no—
474C22—24no—
475C22—24no—
476C22—24no—
477C22—42no—
478C22—42no—
479C22—42no—
480C22—42no—
481C22—42no—
482C22—42no—
483C22—42no—
484C22—82no—
485C22—82no—
486C22C2×#1313318611yescharacteristic
487C22#1681127212yes—
488C22C2×#1313324312yes—
489C22#1681143111yescharacteristic
490C22#1681143811yescharacteristic
491C4—22no—
492C4—22no—
493C4—42no—
494C4—44no—
495C4—42no—
496C4—4241no—
497C4—4241no—
498C4—4242no—
499C4#1681116911yescharacteristic
500C4#1681119411yescharacteristic
501C4#1681128012yes—
502C2—44no—
503C2—82no—
504C2#2027833911yescharacteristic
505C2#2028394211yescharacteristic
506C2#2028403011yescharacteristic
507C2#2028714912yes—
508C2#2028718512yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
Fitting, prime radicalC2×C53×#547C2
socleC23×C53C22×D4
hypercenterC2×#547D53
hypocenterC53#25709
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. C2×#16811029C2semidirect
2. C2×#16810923C2semidirect
3. #20279881C2semidirect
4. #20275901C2semidirect
5. #20275894C2semidirect
6. #20275664C2semidirect
7. #20275514C2semidirect
8. #20274908C2semidirect
9. #20274903C2semidirect
10. #20274540C2semidirect
165. C2×#88322#41semidirect
299. C53#25709semidirect

Maximal subgroups

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

subgroupindex
C2×#168110292
C2×#168109232
#202798812
#202759012
#202758942
#202756642
#202755142
#202749082
#202749032
#202745402
C2×C53×#5472
#2570953