Finite Group Databasebeta

← #27364609

Subgroups — #27364609

Order 13568.

946 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#20285863C211yesmaximal, characteristic, complemented
2#20285637C211yesmaximal, characteristic, complemented
3C2×#16810903C211yesmaximal, characteristic, complemented
4#20273950C214yesmaximal, complemented
5#20273943C214yesmaximal, complemented
6#20273372C211yesmaximal, characteristic, complemented
7#20273114C211yesmaximal, characteristic, complemented
8#20273106C211yesmaximal, characteristic, complemented
9C53×#3506C211yesmaximal, characteristic
10C2×#13133217C2211yescharacteristic
11#16811026C2212yescomplemented
12#16810995—21no—
13#16810975—21no—
14#16810973C2212yescomplemented
15#16810956—22no—
16#16810918—22no—
17#16810903—22no—
18#16810903C2214yes—
19#16810828C2214yescomplemented
20#16810761—22no—
21#16810676C2211yescharacteristic
22#16810676C2211yescharacteristic
23C2×#13133150—21no—
24C2×C4⋊#88322C2211yescharacteristic
25C2×#13133138—21no—
26#16810372C2212yescomplemented
27#16810369C2212yescomplemented
28D4⋊#88322C2214yes—
29D4⋊#88322C2214yescomplemented
30C53×#559—22no—
31C53×#556—21no—
32C53×#555C2212yes—
33C53×#553—22no—
34C53×#550—21no—
35C53×#547C2212yes—
36C53×#538—22no—
37C53×#535—21no—
38C53×#533C2211yescharacteristic
39C53×#532C2211yescharacteristic
40C53×#525—22no—
41C53×#520—21no—
42C2×C4×D4×C53C2211yescharacteristic
43C22×Dic106—21no—
44C2×#11719923—24no—
45C22×#88322—21no—
46#13133217—21no—
47#13133217—21no—
48#13133217—21no—
49#13133217C2312yes—
50C2×C4⋊Dic53—21no—
51C2×C4⋊Dic53—21no—
52#13133150—24no—
53#13133150—24no—
54C4⋊#88322—21no—
55C4⋊#88322—21no—
56C4⋊#88322C2312yes—
57C4⋊#88322C2312yes—
58#13133141—24no—
59#13133140—24no—
60#13133138—24no—
61C22⋊#88322—21no—
62C22⋊#88322—21no—
63C4×#88322—21no—
64C4×#88322C2312yes—
65C2×#41×C53—22no—
66C2×#41×C53—21no—
67C22×D4×C53D411yescharacteristic
68C23×C4×C53—21no—
69C4⋊Q8×C53D411yescharacteristic
70C4⋊Q8×C53C2311yescharacteristic
71C4⋊Q8×C53C2311yescharacteristic
72#127×C53—22no—
73#127×C53D411yescharacteristic
74#126×C53—24no—
75#125×C53—21no—
76#124×C53—22no—
77#124×C53—21no—
78#124×C53—24no—
79#123×C53—24no—
80#123×C53—24no—
81C22⋊Q8×C53—22no—
82C22⋊Q8×C53—21no—
83C22⋊Q8×C53D412yes—
84#121×C53—24no—
85#121×C53—22no—
86#121×C53—22no—
87#121×C53—22no—
88#121×C53—21no—
89#121×C53D412yes—
90#120×C53—24no—
91C4×Q8×C53—22no—
92C4×Q8×C53—21no—
93C4×Q8×C53D412yes—
94C4×D4×C53—24no—
95C4×D4×C53—22no—
96C4×D4×C53—24no—
97C4×D4×C53—21no—
98C4×D4×C53—21no—
99C4×D4×C53—21no—
100C4×D4×C53D412yes—
101C4×D4×C53C2312yes—
102C4×D4×C53C2312yes—
103#117×C53—22no—
104#117×C53—22no—
105#117×C53—21no—
106C2×C4⋊C4×C53—22no—
107C2×C4⋊C4×C53D411yescharacteristic
108C2×C22⋊C4×C53—21no—
109C2×C42×C53C2311yescharacteristic
110C24×C53—21no—
111#41×C53—42no—
112#41×C53—42no—
113#41×C53—84no—
114C2×Q8×C53—21no—
115C2×Q8×C53—21no—
116C2×Q8×C53C2×D412yes—
117C2×D4×C53—22no—
118C2×D4×C53—24no—
119C2×D4×C53—24no—
120C2×D4×C53—21no—
121C2×D4×C53—21no—
122C2×D4×C53—21no—
123C2×D4×C53—22no—
124C2×D4×C53—22no—
125C2×D4×C53C2×D412yes—
126C2×D4×C53C2×D412yes—
127C22×C4×C53—24no—
128C22×C4×C53—21no—
129C22×C4×C53—22no—
130C22×C4×C53—22no—
131C22×C4×C53—22no—
132C22×C4×C53—82no—
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53C2×D411yescharacteristic
136C22×Dic53—41no—
137C2×Dic106—22no—
138C2×Dic106—41no—
139C2×Dic106—82no—
140C4⋊C4×C53—24no—
141C4⋊C4×C53—21no—
142C4⋊C4×C53—22no—
143C4⋊C4×C53—21no—
144C4⋊C4×C53—22no—
145C4⋊C4×C53—21no—
146C4⋊C4×C53—21no—
147C4⋊C4×C53—21no—
148C4⋊C4×C53C2×D412yes—
149C4⋊C4×C53C2×D412yes—
150C4⋊C4×C53C2×D412yes—
151C22⋊C4×C53—24no—
152C22⋊C4×C53—24no—
153C22⋊C4×C53—24no—
154C22⋊C4×C53—22no—
155C22⋊C4×C53—22no—
156C22⋊C4×C53—22no—
157C22⋊C4×C53—22no—
158C22⋊C4×C53—22no—
159C42×C53—24no—
160C42×C53—22no—
161C42×C53C2×D411yescharacteristic
162C42×C53C2×D411yescharacteristic
163C42×C53C2×D411yescharacteristic
164C42×C53C2411yescharacteristic
165#11719923—84no—
166C4⋊Dic53—22no—
167C4⋊Dic53—22no—
168C4⋊Dic53—41no—
169C4⋊Dic53—41no—
170C2×#88322—22no—
171C2×#88322—22no—
172C2×#88322—22no—
173C2×#88322—42no—
174C23×C53—22no—
175C23×C53—22no—
176C23×C53—42no—
177C23×C53—81no—
178C23×C53—81no—
179C23×C53#12011yescharacteristic
180Q8×C53—42no—
181Q8×C53—42no—
182Q8×C53—42no—
183D4×C53—22no—
184D4×C53—22no—
185D4×C53—24no—
186D4×C53—44no—
187D4×C53—42no—
188D4×C53—42no—
189D4×C53—81no—
190D4×C53—84no—
191D4×C53—84no—
192C2×C4×C53—22no—
193C2×C4×C53—22no—
194C2×C4×C53—22no—
195C2×C4×C53—22no—
196C2×C4×C53—22no—
197C2×C4×C53—41no—
198C2×C4×C53—42no—
199C2×C4×C53—42no—
200C2×C4×C53—44no—
201C2×C4×C53—82no—
202C2×C4×C53—81no—
203C2×C4×C53—84no—
204C2×C4×C53#12011yescharacteristic
205C2×C4×C53#12011yescharacteristic
206C2×C4×C53#12011yescharacteristic
207C2×C4×C53#13512yes—
208C2×C4×C53#13712yes—
209C2×C4×C53C22×D411yescharacteristic
210C2×C4×C53C22×D411yescharacteristic
211C2×C4×C53C22×D411yescharacteristic
212C2×Dic53—42no—
213C2×Dic53—82no—
214Dic106—44no—
215Dic106—82no—
216#88322—42no—
217#88322—42no—
218#88322—42no—
219#18929—531nomaximal, Sylow
220C22×C53—21no—
221C22×C53—21no—
222C22×C53—42no—
223C22×C53—42no—
224C22×C53—42no—
225C22×C53—42no—
226C22×C53—42no—
227C22×C53—42no—
228C22×C53—84no—
229C22×C53#45212yes—
230C22×C53C2×#12011yescharacteristic
231C4×C53—21no—
232C4×C53—21no—
233C4×C53—21no—
234C4×C53—42no—
235C4×C53—42no—
236C4×C53—42no—
237C4×C53—42no—
238C4×C53#45212yes—
239C4×C53C2×#13511yescharacteristic
240C4×C53C2×#13711yescharacteristic
241C4×C53#57612yes—
242Dic53—82no—
243#3506D5311yescharacteristic
244#3230—531no—
245#3206—531no—
246C2×#474—531no—
247#1647—534no—
248#1646—534no—
249#1542—531no—
250#1480—531no—
251#1478—531no—
252C2×C53—22no—
253C2×C53—82no—
254C2×C53—82no—
255C2×C53C2×#45211yescharacteristic
256C2×C53#300811yescharacteristic
257C2×C53#302211yescharacteristic
258#559—22no—
259#556—21no—
260#555C2×D5312yes—
261#553—22no—
262#550—21no—
263#547C2×D5312yes—
264#538—22no—
265#535—21no—
266#533C2×D5311yescharacteristic
267#532C2×D5311yescharacteristic
268C2×C4⋊Q8—531no—
269#525—22no—
270#520—21no—
271C2×C4×D4C2×D5311yescharacteristic
272#500—532no—
273#492—1061no—
274#487—532no—
275#487—1061no—
276#484—1062no—
277#477—1062no—
278#474—534no—
279#474—1062no—
280#461—534no—
281#448—1062no—
282#431—531no—
283#431—531no—
284C2×#107—1061no—
285C2×C4⋊C8—531no—
286C2×#102—1061no—
287#333—532no—
288#332—532no—
289D4⋊C8—534no—
290D4⋊C8—534no—
291C53#1892911yescharacteristic, complemented, Sylow
292C2×#41—22no—
293C2×#41—21no—
294C22×Q8—1061no—
295C22×D4C22⋊D5311yescharacteristic
296C23×C4—21no—
297C2×SD16—1064no—
298C22×C8—1061no—
299C4⋊Q8—532no—
300C4⋊Q8—1061no—
301C4⋊Q8—1061no—
302C4⋊Q8—1061no—
303C4⋊Q8C22⋊D5311yescharacteristic
304C4⋊Q8C22×D5311yescharacteristic
305C4⋊Q8C22×D5311yescharacteristic
306#127—22no—
307#127C22⋊D5311yescharacteristic
308#126—24no—
309#125—21no—
310#124—22no—
311#124—24no—
312#124—21no—
313#123—24no—
314#123—24no—
315C22⋊Q8—21no—
316C22⋊Q8—22no—
317C22⋊Q8C22⋊D5312yes—
318#121—24no—
319#121—22no—
320#121—22no—
321#121—22no—
322#121—21no—
323#121C22⋊D5312yes—
324#120—24no—
325C4×Q8—22no—
326C4×Q8—21no—
327C4×Q8C22⋊D5312yes—
328C4×D4—24no—
329C4×D4—22no—
330C4×D4—24no—
331C4×D4—21no—
332C4×D4—21no—
333C4×D4—21no—
334C4×D4C22⋊D5312yes—
335C4×D4C22×D5312yes—
336C4×D4C22×D5312yes—
337#117—21no—
338#117—22no—
339#117—22no—
340C2×C4⋊C4—22no—
341C2×C4⋊C4—1061no—
342C2×C4⋊C4—1061no—
343C2×C4⋊C4C22⋊D5311yescharacteristic
344C2×C22⋊C4—21no—
345C2×C42C22×D5311yescharacteristic
346#107—1064no—
347#107—1064no—
348C4⋊C8—532no—
349C4⋊C8—532no—
350C4⋊C8—1061no—
351C4⋊C8—1061no—
352#103—1064no—
353#103—1064no—
354#102—1064no—
355C22⋊C8—1061no—
356C22⋊C8—1061no—
357C4×C8—532no—
358C4×C8—1061no—
359C24—21no—
360#41—42no—
361#41—42no—
362#41—84no—
363C2×Q8—21no—
364C2×Q8—21no—
365C2×Q8—1062no—
366C2×Q8—4242no—
367C2×Q8—2121no—
368C2×Q8C2×C22⋊D5312yes—
369C2×D4—22no—
370C2×D4—24no—
371C2×D4—24no—
372C2×D4—21no—
373C2×D4—21no—
374C2×D4—21no—
375C2×D4—22no—
376C2×D4—22no—
377C2×D4C2×C22⋊D5312yes—
378C2×D4C2×C22⋊D5312yes—
379C22×C4—24no—
380C22×C4—21no—
381C22×C4—22no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—82no—
385C22×C4—2121no—
386C22×C4C2×C22⋊D5311yescharacteristic
387C22×C4C2×C22⋊D5311yescharacteristic
388C22×C4C2×C22⋊D5311yescharacteristic
389SD16—4244no—
390C2×C8—1062no—
391C2×C8—1062no—
392C2×C8—1062no—
393C2×C8—2122no—
394C4⋊C4—24no—
395C4⋊C4—21no—
396C4⋊C4—22no—
397C4⋊C4—21no—
398C4⋊C4—22no—
399C4⋊C4—21no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—1062no—
403C4⋊C4—1062no—
404C4⋊C4—2121no—
405C4⋊C4—2121no—
406C4⋊C4C2×C22⋊D5312yes—
407C4⋊C4C2×C22⋊D5312yes—
408C4⋊C4C2×C22⋊D5312yes—
409C22⋊C4—24no—
410C22⋊C4—24no—
411C22⋊C4—24no—
412C22⋊C4—22no—
413C22⋊C4—22no—
414C22⋊C4—22no—
415C22⋊C4—22no—
416C22⋊C4—22no—
417C42—24no—
418C42—22no—
419C42C2×C22⋊D5311yescharacteristic
420C42C2×C22⋊D5311yescharacteristic
421C42C2×C22⋊D5311yescharacteristic
422C42C23×D5311yescharacteristic
423C23—22no—
424C23—22no—
425C23—42no—
426C23—81no—
427C23—81no—
428C23#1313318611yescharacteristic
429Q8—42no—
430Q8—42no—
431Q8—42no—
432Q8—4242no—
433Q8—2124no—
434D4—22no—
435D4—22no—
436D4—24no—
437D4—44no—
438D4—42no—
439D4—42no—
440D4—81no—
441D4—84no—
442D4—84no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—22no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—42no—
451C2×C4—44no—
452C2×C4—82no—
453C2×C4—81no—
454C2×C4—84no—
455C2×C4—4242no—
456C2×C4—2122no—
457C2×C4#1313318611yescharacteristic
458C2×C4#1313318611yescharacteristic
459C2×C4#1313318611yescharacteristic
460C2×C4#1313323712yes—
461C2×C4#1313324712yes—
462C2×C4C22×C22⋊D5311yescharacteristic
463C2×C4C22×C22⋊D5311yescharacteristic
464C2×C4C22×C22⋊D5311yescharacteristic
465C8—2122no—
466C8—2122no—
467C8—2122no—
468C22—21no—
469C22—21no—
470C22—42no—
471C22—42no—
472C22—42no—
473C22—42no—
474C22—42no—
475C22—42no—
476C22—84no—
477C22#1681078512yes—
478C22C2×#1313318611yescharacteristic
479C4—21no—
480C4—21no—
481C4—21no—
482C4—42no—
483C4—42no—
484C4—42no—
485C4—42no—
486C4—4242no—
487C4#1681078512yes—
488C4C2×#1313323711yescharacteristic
489C4C2×#1313324711yescharacteristic
490C4#1681142612yes—
491C2—22no—
492C2—82no—
493C2—82no—
494C2C2×#1681078511yescharacteristic
495C2#2028398311yescharacteristic
496C2#2028409411yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
Fitting, prime radicalC53×#3506C2
socleC22×C53C2×#120
hypercenter#3506D53
hypocenterC53#18929
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3506

Complements

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

NH ≅ G/Ntype
1. #20285863C2semidirect
2. #20285637C2semidirect
3. C2×#16810903C2semidirect
4. #20273950C2semidirect
5. #20273943C2semidirect
6. #20273372C2semidirect
7. #20273114C2semidirect
8. #20273106C2semidirect
11. #16811026C22semidirect
14. #16810973C22semidirect
19. #16810828C22semidirect
26. #16810372C22semidirect
27. #16810369C22semidirect
29. D4⋊#88322C22semidirect
291. C53#18929semidirect

Maximal subgroups

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

subgroupindex
#202858632
#202856372
C2×#168109032
#202739502
#202739432
#202733722
#202731142
#202731062
C53×#35062
#1892953