Finite Group Databasebeta

← #27355541

Subgroups — #27355541

Order 13568.

654 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283732C211yesmaximal, characteristic, complemented
2#20283727C211yesmaximal, characteristic, complemented
3C4×#13133239C211yesmaximal, characteristic
4#20273604C212yesmaximal, complemented
5#20273564C212yesmaximal, complemented
6#20273543C212yesmaximal, complemented
7#20273514C212yesmaximal, complemented
8#20273147C211yesmaximal, characteristic, complemented
9#20273147C211yesmaximal, characteristic, complemented
10C8×#11719917C211yesmaximal, characteristic, complemented
11C53×#1568C211yesmaximal, characteristic, complemented
12C2×#13133239C411yescharacteristic
13C4×#11719945C2211yescharacteristic
14C4×#11719945C2211yescharacteristic
15#16810750C2212yes—
16#16810747C2212yes—
17C4×Dic212C411yescharacteristic
18C4×Dic212C2211yescharacteristic
19C4×#11719913C411yescharacteristic
20C4×#11719913C411yescharacteristic
21C4×#11719913C2211yescharacteristic
22C4×#11719913C2211yescharacteristic
23C4×D424C411yescharacteristic
24C4×D424C2211yescharacteristic
25#16810700C2212yes—
26#16810698—22no—
27#16810697—22no—
28#16810688C2212yes—
29#16810678—21no—
30#16810678—21no—
31#16810676C2211yescharacteristic
32#16810661C411yescharacteristic
33#16810620C411yescharacteristic
34#16810620C411yescharacteristic
35C4×#11719917C2211yescharacteristic
36#16810575—21no—
37#97×D53—21no—
38#16810353C2212yes—
39#16810353C2212yes—
40#16810350C2212yes—
41#16810350C2212yes—
42#16810345C2212yes—
43C8×#88322C2212yes—
44C53×#432C2211yescharacteristic
45C53×#432C2211yescharacteristic
46C2×C4×C8×C53C2211yescharacteristic
47C53×#335C2212yes—
48C53×#334C2212yes—
49C2×#11719945C2×C411yescharacteristic
50C2×#11719945C2×C411yescharacteristic
51#13133239—24no—
52C2×Dic212C2×C411yescharacteristic
53C2×#11719913C2×C411yescharacteristic
54C2×#11719913C2×C411yescharacteristic
55C2×D424C2×C411yescharacteristic
56C2×#11719917D412yes—
57C2×#11719912—22no—
58C2×C8×D53—22no—
59C4×Dic106C2×C411yescharacteristic
60C4×Dic106C2×C411yescharacteristic
61C4×Dic106C2311yescharacteristic
62C4×Dic106C2311yescharacteristic
63C4×D212C2×C411yescharacteristic
64C4×D212C2×C411yescharacteristic
65C4×D212C2311yescharacteristic
66C4×D212C2311yescharacteristic
67C4×C22⋊D53—21no—
68C4×C22⋊D53—21no—
69C4×C22⋊D53—21no—
70C4×C22⋊D53—21no—
71#13133176C2×C411yescharacteristic
72#13133176C2×C411yescharacteristic
73#13133174—21no—
74#13133174—21no—
75C42×D53—21no—
76C42×D53—21no—
77#13133151C2×C411yescharacteristic
78#13133151C2×C411yescharacteristic
79#13133149C2×C411yescharacteristic
80#13133149C2×C411yescharacteristic
81C4⋊#88322—21no—
82#13133146—22no—
83#13133146—22no—
84#13133142C2×C411yescharacteristic
85#13133142C2×C411yescharacteristic
86#13133142C2×C411yescharacteristic
87#13133142C2×C411yescharacteristic
88#13133139C2×C411yescharacteristic
89#13133139C2×C411yescharacteristic
90#13133139C2×C411yescharacteristic
91#13133139C2×C411yescharacteristic
92C22⋊#88322—21no—
93#13133129—22no—
94#13133129—22no—
95#13133128D411yescharacteristic
96#13133128D411yescharacteristic
97#13133127—22no—
98C4×#88322C2312yes—
99C4×#88322C2312yes—
100C8×Dic53—22no—
101C22×C8×C53C2×C411yescharacteristic
102C22×C8×C53C2×C411yescharacteristic
103C2×C42×C53C2311yescharacteristic
104C4⋊C8×C53C2312yes—
105C4⋊C8×C53C2312yes—
106C22⋊C8×C53—21no—
107C22⋊C8×C53—21no—
108#97×C53—21no—
109#97×C53—21no—
110C4×C8×C53C2×C411yescharacteristic
111C4×C8×C53C2×C411yescharacteristic
112C4×C8×C53C2311yescharacteristic
113C4×C8×C53C2311yescharacteristic
114C22×C4×C53C22×C411yescharacteristic
115C22×C4×C53C2×D411yescharacteristic
116C22×C4×C53#4111yescharacteristic
117C2×C22⋊D53—21no—
118C2×C22⋊D53—21no—
119#11719945—22no—
120#11719945—22no—
121#11719945—41no—
122#11719945—41no—
123C2×D212C22×C411yescharacteristic
124C2×D212C22×C411yescharacteristic
125C2×C4×D53—21no—
126C2×C4×D53—21no—
127C2×C4×D53—21no—
128C2×C4×D53—21no—
129C2×C4×D53—21no—
130C2×C4×D53—21no—
131C2×Dic106C22×C411yescharacteristic
132C2×Dic106C22×C411yescharacteristic
133C2×C8×C53—22no—
134C2×C8×C53—22no—
135C2×C8×C53—22no—
136C2×C8×C53—41no—
137C2×C8×C53C22×C411yescharacteristic
138C2×C8×C53C22×C411yescharacteristic
139C2×C8×C53C22×C411yescharacteristic
140C2×C8×C53C22×C411yescharacteristic
141C42×C53C22×C411yescharacteristic
142C42×C53C2×D411yescharacteristic
143C42×C53#4111yescharacteristic
144C42×C53C2411yescharacteristic
145C22⋊Dic53—21no—
146C22⋊Dic53—21no—
147#11719921—21no—
148#11719921—21no—
149#11719921—21no—
150#11719921—21no—
151C4⋊Dic53C22×C411yescharacteristic
152C4⋊Dic53C22×C411yescharacteristic
153C4⋊Dic53C22×C411yescharacteristic
154C4⋊Dic53C22×C411yescharacteristic
155#11719919—21no—
156#11719919—21no—
157#11719919—21no—
158#11719919—21no—
159C4×Dic53—21no—
160C4×Dic53—21no—
161C4×Dic53—21no—
162C4×Dic53—21no—
163C4×Dic53—21no—
164C4×Dic53—21no—
165#11719917—24no—
166C2×#88322—22no—
167C2×#88322C2×D412yes—
168C2×#88322C2×D412yes—
169Dic212—22no—
170D424—22no—
171#11719913—22no—
172#11719913M1614yescomplemented
173#11719912—44no—
174C8×D53—44no—
175C23×C53C4×D411yescharacteristic
176C22×D53—21no—
177C22×D53—21no—
178C2×C4×C53—21no—
179C2×C4×C53—21no—
180C2×C4×C53—22no—
181C2×C4×C53—41no—
182C2×C4×C53C4×D411yescharacteristic
183C2×C4×C53C4×D411yescharacteristic
184C2×C4×C53C4×D411yescharacteristic
185C2×C4×C53C23×C411yescharacteristic
186C2×C4×C53C22×D411yescharacteristic
187C2×C4×C53C2×#4111yescharacteristic
188C22⋊D53—42no—
189C22⋊D53—42no—
190C2×Dic53—21no—
191C2×Dic53—21no—
192C2×Dic53—21no—
193C2×Dic53—21no—
194C2×Dic53—21no—
195C2×Dic53—21no—
196D212—21no—
197D212—21no—
198D212—21no—
199D212C2×M1612yes—
200C4×D53—42no—
201C4×D53—42no—
202C4×D53—42no—
203C4×D53—42no—
204C4×D53—81no—
205C4×D53—81no—
206Dic106—21no—
207Dic106—21no—
208Dic106—21no—
209Dic106C2×M1612yes—
210C8×C53—21no—
211C8×C53—21no—
212C8×C53—21no—
213C8×C53—42no—
214C8×C53—42no—
215C8×C53C2×M1612yes—
216#88322—22no—
217#88322—42no—
218#88322#13514yescomplemented
219#18037—531nomaximal, Sylow
220C22×C53—21no—
221C22×C53—21no—
222C22×C53—21no—
223C22×C53C2×C4×D411yescharacteristic
224C2×D53—22no—
225C2×D53—41no—
226C2×D53—41no—
227C2×D53—41no—
228C2×D53—41no—
229C2×D53—41no—
230C4×C53—21no—
231C4×C53—21no—
232C4×C53—21no—
233C4×C53—21no—
234C4×C53C22×M1611yescharacteristic
235C4×C53#56811yescharacteristic
236C4×C53C2×#13512yes—
237Dic53—22no—
238Dic53—41no—
239Dic53—41no—
240Dic53—41no—
241Dic53—41no—
242Dic53—41no—
243#2971—531no—
244#2970—531no—
245C4×#135—531no—
246#1590—532no—
247#1584—532no—
248#1581—532no—
249C8×SD16—532no—
250#1568D5311yescharacteristic, complemented
251#1488—531no—
252#1488—531no—
253C8×M16—531no—
254C2×C53—41no—
255C2×C53D4×M1611yescharacteristic
256C2×C53C4×#13511yescharacteristic
257C2×C53#295711yescharacteristic
258D53—42no—
259D53—81no—
260C2×#135—531no—
261C4×#41—531no—
262C4×#41—531no—
263#446—532no—
264C8×Q8—532no—
265C4×Q16—531no—
266C4×Q16—531no—
267C4×SD16—531no—
268C4×SD16—531no—
269C4×SD16—531no—
270C4×SD16—531no—
271C4×D8—531no—
272C4×D8—531no—
273#436—532no—
274#436—1062no—
275#435—1062no—
276C8×D4—532no—
277#432—1061no—
278#432—1061no—
279#432C2×D5311yescharacteristic
280#432C2×D5311yescharacteristic
281#431—531no—
282#427—531no—
283#416—531no—
284#416—531no—
285C4×M16—531no—
286C4×M16—1061no—
287C2×#97—1061no—
288C2×C4×C8C2×D5311yescharacteristic
289#335C2×D5312yes—
290#334C2×D5312yes—
291Q8⋊C8—532no—
292Q8⋊C8—532no—
293D4⋊C8—532no—
294D4⋊C8—532no—
295#322—532no—
296C82—532no—
297C53#1803711yescharacteristic, complemented, Sylow
298C2×#41—531no—
299C2×#41—531no—
300#135—1064no—
301C2×Q16—531no—
302C2×SD16—531no—
303C2×SD16—531no—
304C2×D8—531no—
305C2×M16—532no—
306C2×M16—1062no—
307C22×C8—1062no—
308C22×C8C4×D5311yescharacteristic
309C22×C8C4×D5311yescharacteristic
310C4×Q8—531no—
311C4×Q8—531no—
312C4×Q8—531no—
313C4×Q8—531no—
314C4×D4—531no—
315C4×D4—531no—
316C4×D4—531no—
317C4×D4—531no—
318C4×D4—1061no—
319C4×D4—1061no—
320C4×D4—1061no—
321C4×D4—1061no—
322#117—531no—
323#117—531no—
324#117—1061no—
325#117—1061no—
326C2×C42—1061no—
327C2×C42—1061no—
328C2×C42C22×D5311yescharacteristic
329#107—531no—
330#107—531no—
331#106—531no—
332#106—531no—
333C4⋊C8—1062no—
334C4⋊C8—1062no—
335C4⋊C8—1061no—
336C4⋊C8C22×D5312yes—
337C4⋊C8C22×D5312yes—
338#103—531no—
339#103—531no—
340#103—531no—
341#103—531no—
342#102—531no—
343#102—531no—
344#102—531no—
345#102—531no—
346C22⋊C8—21no—
347C22⋊C8—21no—
348C22⋊C8—1062no—
349C22⋊C8—1062no—
350C22⋊C8—1061no—
351#97—21no—
352#97—21no—
353#97—531no—
354#97—531no—
355#97—1062no—
356C4×C8—532no—
357C4×C8—532no—
358C4×C8—1062no—
359C4×C8C4×D5311yescharacteristic
360C4×C8C4×D5311yescharacteristic
361C4×C8C22×D5311yescharacteristic
362C4×C8C22×D5311yescharacteristic
363#41—1062no—
364#41—1062no—
365#41—2121no—
366#41—2121no—
367C2×Q8—531no—
368C2×Q8—531no—
369C2×D4—531no—
370C2×D4—531no—
371C2×D4—1061no—
372C2×D4—1061no—
373C22×C4—1061no—
374C22×C4—1061no—
375C22×C4—1061no—
376C22×C4—1061no—
377C22×C4—1061no—
378C22×C4—1061no—
379C22×C4C2×C4×D5311yescharacteristic
380C22×C4D4×D5311yescharacteristic
381C22×C4#1171994911yescharacteristic
382Q16—1062no—
383SD16—534no—
384SD16—1062no—
385D8—1062no—
386M16—1064no—
387M16—2124no—
388C2×C8—22no—
389C2×C8—22no—
390C2×C8—22no—
391C2×C8—41no—
392C2×C8—532no—
393C2×C8—532no—
394C2×C8—1062no—
395C2×C8—2124no—
396C2×C8C2×C4×D5311yescharacteristic
397C2×C8C2×C4×D5311yescharacteristic
398C2×C8C2×C4×D5311yescharacteristic
399C2×C8C2×C4×D5311yescharacteristic
400C4⋊C4—531no—
401C4⋊C4—531no—
402C4⋊C4—531no—
403C4⋊C4—531no—
404C4⋊C4—1061no—
405C4⋊C4—1061no—
406C4⋊C4—1061no—
407C4⋊C4—1061no—
408C22⋊C4—1061no—
409C22⋊C4—1061no—
410C22⋊C4—1061no—
411C22⋊C4—1061no—
412C22⋊C4—1061no—
413C22⋊C4—1061no—
414C42—1061no—
415C42—1061no—
416C42—1061no—
417C42—1061no—
418C42—1061no—
419C42—1061no—
420C42C2×C4×D5311yescharacteristic
421C42D4×D5311yescharacteristic
422C42#1171994911yescharacteristic
423C42C23×D5311yescharacteristic
424C23—1061no—
425C23—1061no—
426C23#1313317911yescharacteristic
427Q8—532no—
428Q8—1061no—
429Q8—1061no—
430Q8—1061no—
431D4—532no—
432D4—1061no—
433D4—1061no—
434D4—1061no—
435D4—2122no—
436D4—2122no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—22no—
440C2×C4—41no—
441C2×C4—1061no—
442C2×C4—1061no—
443C2×C4—1061no—
444C2×C4—1061no—
445C2×C4—1061no—
446C2×C4—1061no—
447C2×C4—4241no—
448C2×C4—4241no—
449C2×C4—2122no—
450C2×C4—2122no—
451C2×C4—2122no—
452C2×C4—2122no—
453C2×C4#1313317911yescharacteristic
454C2×C4#1313317911yescharacteristic
455C2×C4#1313317911yescharacteristic
456C2×C4C22×C4×D5311yescharacteristic
457C2×C4C2×D4×D5311yescharacteristic
458C2×C4C2×#1171994911yescharacteristic
459C8—21no—
460C8—21no—
461C8—21no—
462C8—42no—
463C8—42no—
464C8—534no—
465C8—1062no—
466C8—2122no—
467C8M16×D5312yes—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—1062no—
472C22—2121no—
473C22—2121no—
474C22—2121no—
475C22—2121no—
476C22—2121no—
477C22C2×#1313317911yescharacteristic
478C4—21no—
479C4—21no—
480C4—21no—
481C4—21no—
482C4—1062no—
483C4—2121no—
484C4—2121no—
485C4—2121no—
486C4—2121no—
487C4—2121no—
488C4C2×M16×D5311yescharacteristic
489C4#1681138911yescharacteristic
490C4#135×D5312yes—
491C2—41no—
492C2—4241no—
493C2—2122no—
494C2#2028345011yescharacteristic
495C2#2028348611yescharacteristic
496C2#2028363111yescharacteristic
subgroupG/H
centerC2×C4C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC42C23×D53
FittingC53×#1568C2
socleC22×C53C2×C4×D4
prime radicalC2×#13133239C4
cosocleC42×C53C24
hypercenter#1568D53
hypocenterC53#18037
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #1568

Complements

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

NH ≅ G/Ntype
1. #20283732C2semidirect
2. #20283727C2semidirect
4. #20273604C2semidirect
5. #20273564C2semidirect
6. #20273543C2semidirect
7. #20273514C2semidirect
8. #20273147C2semidirect
9. #20273147C2semidirect
10. C8×#11719917C2semidirect
11. C53×#1568C2semidirect
172. #11719913M16semidirect
218. #88322#135semidirect
250. #1568D53semidirect
297. C53#18037semidirect

Maximal subgroups

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

subgroupindex
#202837322
#202837272
C4×#131332392
#202736042
#202735642
#202735432
#202735142
#202731472
C8×#117199172
C53×#15682
#1803753