Finite Group Databasebeta

← #27364489

Subgroups — #27364489

Order 13568.

946 conjugacy classes of subgroups.

The subgroup lattice has 500 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×#16810913C211yesmaximal, characteristic, complemented
4#20273887C214yesmaximal, complemented
5#20273842C214yesmaximal, complemented
6#20273364C211yesmaximal, characteristic, complemented
7#20273118C211yesmaximal, characteristic, complemented
8#20273106C211yesmaximal, characteristic, complemented
9C53×#3506C211yesmaximal, characteristic
10C2×#13133217C2211yescharacteristic
11#16811026—21no—
12#16810995C2212yescomplemented
13#16810975C2212yescomplemented
14#16810973—21no—
15#16810956—22no—
16#16810918C2214yescomplemented
17#16810913—22no—
18#16810913C2214yes—
19#16810828—22no—
20#16810761—22no—
21#16810676C2211yescharacteristic
22#16810676C2211yescharacteristic
23C2×#13133148—21no—
24C2×C4⋊#88322C2211yescharacteristic
25C2×#13133138—21no—
26#16810369C2212yescomplemented
27#16810367C2212yescomplemented
28Q8⋊#88322C2214yescomplemented
29D4⋊#88322C2214yes—
30C53×#559—22no—
31C53×#556C2212yes—
32C53×#555—21no—
33C53×#553—22no—
34C53×#550C2212yes—
35C53×#547—21no—
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#13133148—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×C53D411yescharacteristic
71C4⋊Q8×C53C2311yescharacteristic
72#127×C53—22no—
73#127×C53C2311yescharacteristic
74#126×C53—24no—
75#125×C53—21no—
76#124×C53—24no—
77#124×C53—21no—
78#124×C53—22no—
79#123×C53—24no—
80#123×C53—24no—
81C22⋊Q8×C53—22no—
82C22⋊Q8×C53—21no—
83C22⋊Q8×C53D412yes—
84#121×C53—22no—
85#121×C53—22no—
86#121×C53—22no—
87#121×C53—24no—
88#121×C53—21no—
89#121×C53D412yes—
90#120×C53—24no—
91C4×Q8×C53—22no—
92C4×Q8×C53—21no—
93C4×Q8×C53C2312yes—
94C4×D4×C53—24no—
95C4×D4×C53—24no—
96C4×D4×C53—22no—
97C4×D4×C53—21no—
98C4×D4×C53—21no—
99C4×D4×C53—21no—
100C4×D4×C53D412yes—
101C4×D4×C53D412yes—
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—22no—
122C2×D4×C53—22no—
123C2×D4×C53—21no—
124C2×D4×C53—21no—
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—21no—
143C4⋊C4×C53—21no—
144C4⋊C4×C53—22no—
145C4⋊C4×C53—21no—
146C4⋊C4×C53—21no—
147C4⋊C4×C53—22no—
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—24no—
181Q8×C53—41no—
182Q8×C53—41no—
183Q8×C53—81no—
184D4×C53—22no—
185D4×C53—22no—
186D4×C53—42no—
187D4×C53—44no—
188D4×C53—41no—
189D4×C53—41no—
190D4×C53—42no—
191D4×C53—42no—
192D4×C53—84no—
193D4×C53—84no—
194C2×C4×C53—22no—
195C2×C4×C53—22no—
196C2×C4×C53—22no—
197C2×C4×C53—22no—
198C2×C4×C53—22no—
199C2×C4×C53—41no—
200C2×C4×C53—44no—
201C2×C4×C53—44no—
202C2×C4×C53—82no—
203C2×C4×C53—81no—
204C2×C4×C53—84no—
205C2×C4×C53#12011yescharacteristic
206C2×C4×C53#12011yescharacteristic
207C2×C4×C53#12011yescharacteristic
208C2×C4×C53#13512yes—
209C2×C4×C53#13712yes—
210C2×C4×C53C22×D411yescharacteristic
211C2×C4×C53C22×D411yescharacteristic
212C2×C4×C53C22×D411yescharacteristic
213C2×Dic53—42no—
214C2×Dic53—82no—
215Dic106—44no—
216Dic106—82no—
217#88322—42no—
218#88322—42no—
219#88322—42no—
220#18917—531nomaximal, Sylow
221C22×C53—21no—
222C22×C53—21no—
223C22×C53—42no—
224C22×C53—42no—
225C22×C53—42no—
226C22×C53—42no—
227C22×C53—42no—
228C22×C53—42no—
229C22×C53—84no—
230C22×C53#45212yes—
231C22×C53C2×#12011yescharacteristic
232C4×C53—21no—
233C4×C53—21no—
234C4×C53—21no—
235C4×C53—42no—
236C4×C53—42no—
237C4×C53—42no—
238C4×C53—42no—
239C4×C53#45212yes—
240C4×C53C2×#13511yescharacteristic
241C4×C53C2×#13711yescharacteristic
242C4×C53#57712yes—
243Dic53—82no—
244#3506D5311yescharacteristic
245#3230—531no—
246#3206—531no—
247C2×#476—531no—
248#1638—534no—
249#1631—534no—
250#1540—531no—
251#1481—531no—
252#1478—531no—
253C2×C53—22no—
254C2×C53—82no—
255C2×C53—82no—
256C2×C53C2×#45211yescharacteristic
257C2×C53#300911yescharacteristic
258C2×C53#302111yescharacteristic
259#559—22no—
260#556C2×D5312yes—
261#555—21no—
262#553—22no—
263#550C2×D5312yes—
264#547—21no—
265#538—22no—
266#535—21no—
267#533C2×D5311yescharacteristic
268#532C2×D5311yescharacteristic
269C2×C4⋊Q8—531no—
270#525—22no—
271#520—21no—
272C2×C4×D4C2×D5311yescharacteristic
273#500—1061no—
274#492—532no—
275#487—532no—
276#487—1061no—
277#484—1062no—
278#477—534no—
279#476—534no—
280#476—1062no—
281#461—1062no—
282#448—1062no—
283#431—531no—
284#431—531no—
285C2×#106—1061no—
286C2×C4⋊C8—531no—
287C2×#102—1061no—
288#332—532no—
289#331—532no—
290Q8⋊C8—534no—
291D4⋊C8—534no—
292C53#1891711yescharacteristic, complemented, Sylow
293C2×#41—22no—
294C2×#41—21no—
295C22×Q8—1061no—
296C22×D4C22⋊D5311yescharacteristic
297C23×C4—21no—
298C2×SD16—1064no—
299C22×C8—1061no—
300C4⋊Q8—532no—
301C4⋊Q8—1061no—
302C4⋊Q8—1061no—
303C4⋊Q8—1061no—
304C4⋊Q8C22⋊D5311yescharacteristic
305C4⋊Q8C22⋊D5311yescharacteristic
306C4⋊Q8C22×D5311yescharacteristic
307#127—22no—
308#127C22×D5311yescharacteristic
309#126—24no—
310#125—21no—
311#124—24no—
312#124—22no—
313#124—21no—
314#123—24no—
315#123—24no—
316C22⋊Q8—22no—
317C22⋊Q8—21no—
318C22⋊Q8C22⋊D5312yes—
319#121—22no—
320#121—22no—
321#121—24no—
322#121—21no—
323#121—22no—
324#121C22⋊D5312yes—
325#120—24no—
326C4×Q8—22no—
327C4×Q8—21no—
328C4×Q8C22×D5312yes—
329C4×D4—24no—
330C4×D4—24no—
331C4×D4—22no—
332C4×D4—21no—
333C4×D4—21no—
334C4×D4—21no—
335C4×D4C22⋊D5312yes—
336C4×D4C22⋊D5312yes—
337C4×D4C22×D5312yes—
338#117—22no—
339#117—21no—
340#117—22no—
341C2×C4⋊C4—22no—
342C2×C4⋊C4—1061no—
343C2×C4⋊C4—1061no—
344C2×C4⋊C4C22⋊D5311yescharacteristic
345C2×C22⋊C4—21no—
346C2×C42C22×D5311yescharacteristic
347#107—1064no—
348#106—1064no—
349C4⋊C8—532no—
350C4⋊C8—532no—
351C4⋊C8—1061no—
352C4⋊C8—1061no—
353#103—1064no—
354#103—1064no—
355#102—1064no—
356C22⋊C8—1061no—
357C22⋊C8—1061no—
358C4×C8—532no—
359C4×C8—1061no—
360C24—21no—
361#41—42no—
362#41—42no—
363#41—84no—
364C2×Q8—21no—
365C2×Q8—21no—
366C2×Q8—1062no—
367C2×Q8—4242no—
368C2×Q8—2121no—
369C2×Q8C2×C22⋊D5312yes—
370C2×D4—22no—
371C2×D4—24no—
372C2×D4—24no—
373C2×D4—21no—
374C2×D4—22no—
375C2×D4—22no—
376C2×D4—21no—
377C2×D4—21no—
378C2×D4C2×C22⋊D5312yes—
379C2×D4C2×C22⋊D5312yes—
380C22×C4—24no—
381C22×C4—21no—
382C22×C4—22no—
383C22×C4—22no—
384C22×C4—22no—
385C22×C4—82no—
386C22×C4—2121no—
387C22×C4C2×C22⋊D5311yescharacteristic
388C22×C4C2×C22⋊D5311yescharacteristic
389C22×C4C2×C22⋊D5311yescharacteristic
390SD16—4244no—
391C2×C8—1062no—
392C2×C8—1062no—
393C2×C8—1062no—
394C2×C8—2122no—
395C4⋊C4—24no—
396C4⋊C4—21no—
397C4⋊C4—21no—
398C4⋊C4—21no—
399C4⋊C4—22no—
400C4⋊C4—21no—
401C4⋊C4—21no—
402C4⋊C4—22no—
403C4⋊C4—1062no—
404C4⋊C4—1062no—
405C4⋊C4—2121no—
406C4⋊C4—2121no—
407C4⋊C4C2×C22⋊D5312yes—
408C4⋊C4C2×C22⋊D5312yes—
409C4⋊C4C2×C22⋊D5312yes—
410C22⋊C4—24no—
411C22⋊C4—24no—
412C22⋊C4—24no—
413C22⋊C4—22no—
414C22⋊C4—22no—
415C22⋊C4—22no—
416C22⋊C4—22no—
417C22⋊C4—22no—
418C42—24no—
419C42—22no—
420C42C2×C22⋊D5311yescharacteristic
421C42C2×C22⋊D5311yescharacteristic
422C42C2×C22⋊D5311yescharacteristic
423C42C23×D5311yescharacteristic
424C23—22no—
425C23—22no—
426C23—42no—
427C23—81no—
428C23—81no—
429C23#1313318611yescharacteristic
430Q8—24no—
431Q8—41no—
432Q8—41no—
433Q8—81no—
434Q8—4242no—
435Q8—2124no—
436D4—22no—
437D4—22no—
438D4—42no—
439D4—44no—
440D4—42no—
441D4—41no—
442D4—41no—
443D4—42no—
444D4—84no—
445D4—84no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—22no—
449C2×C4—22no—
450C2×C4—22no—
451C2×C4—41no—
452C2×C4—44no—
453C2×C4—44no—
454C2×C4—82no—
455C2×C4—81no—
456C2×C4—84no—
457C2×C4—4242no—
458C2×C4—2122no—
459C2×C4#1313318611yescharacteristic
460C2×C4#1313318611yescharacteristic
461C2×C4#1313318611yescharacteristic
462C2×C4#1313323712yes—
463C2×C4#1313324712yes—
464C2×C4C22×C22⋊D5311yescharacteristic
465C2×C4C22×C22⋊D5311yescharacteristic
466C2×C4C22×C22⋊D5311yescharacteristic
467C8—2122no—
468C8—2122no—
469C8—2122no—
470C22—21no—
471C22—21no—
472C22—42no—
473C22—42no—
474C22—42no—
475C22—42no—
476C22—42no—
477C22—42no—
478C22—84no—
479C22#1681078512yes—
480C22C2×#1313318611yescharacteristic
481C4—21no—
482C4—21no—
483C4—21no—
484C4—42no—
485C4—42no—
486C4—42no—
487C4—42no—
488C4—4242no—
489C4#1681078512yes—
490C4C2×#1313323711yescharacteristic
491C4C2×#1313324711yescharacteristic
492C4#1681143412yes—
493C2—22no—
494C2—82no—
495C2—82no—
496C2C2×#1681078511yescharacteristic
497C2#2028399211yescharacteristic
498C2#2028408711yescharacteristic
subgroupG/H
centerC22C2×#13133186
commutator, cosocleC42×C53C24
FrattiniC42C23×D53
Fitting, prime radicalC53×#3506C2
socleC22×C53C2×#120
hypercenter#3506D53
hypocenterC53#18917
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×#16810913C2semidirect
4. #20273887C2semidirect
5. #20273842C2semidirect
6. #20273364C2semidirect
7. #20273118C2semidirect
8. #20273106C2semidirect
12. #16810995C22semidirect
13. #16810975C22semidirect
16. #16810918C22semidirect
26. #16810369C22semidirect
27. #16810367C22semidirect
28. Q8⋊#88322C22semidirect
292. C53#18917semidirect

Maximal subgroups

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

subgroupindex
#202858632
#202856372
C2×#168109132
#202738872
#202738422
#202733642
#202731182
#202731062
C53×#35062
#1891753