Finite Group Databasebeta

← #27278188

Subgroups — #27278188

Order 13568.

702 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277418C211yesmaximal, characteristic
2#20275032C211yesmaximal, characteristic
3#20274908C211yesmaximal, characteristic, complemented
4C2×#16810356C211yesmaximal, characteristic
5#20272594C211yesmaximal, characteristic, complemented
6#20272558C211yesmaximal, characteristic, complemented
7C53×#1444C211yesmaximal, characteristic, complemented
8C2×#13133191C2211yescharacteristic
9C2×C4×C22⋊D53—21no—
10C2×#13133148—21no—
11C2×#13133137—21no—
12C2×C22⋊#88322C2211yescharacteristic
13C2×C22⋊#88322C2211yescharacteristic
14#16810515—21no—
15#16810508—21no—
16#16810502—21no—
17#16810501—21no—
18#16810497—21no—
19#16810483C2211yescharacteristic
20#16810385—21no—
21#16810385—21no—
22#16810356C414yescomplemented
23C22×C4⋊C4×C53—21no—
24C22×C4⋊C4×C53C2211yescharacteristic
25C2×#95×C53—21no—
26C2×#95×C53C2211yescharacteristic
27C2×#95×C53C2211yescharacteristic
28C22×D212D411yescharacteristic
29C22×C22⋊D53—21no—
30C22×C4×D53—21no—
31C22×#88322—21no—
32C22×#88322—21no—
33#13133191—21no—
34#13133191—21no—
35#13133191C2×C412yes—
36#13133191Q812yes—
37C4×C22⋊D53—42no—
38C4×C22⋊D53—42no—
39C2×C4⋊Dic53D411yescharacteristic
40C2×C4⋊Dic53D411yescharacteristic
41C2×C4⋊Dic53D411yescharacteristic
42C2×#11719919—21no—
43C2×C22⋊Dic53—21no—
44C2×#11719921—21no—
45C2×#11719921—21no—
46C2×#11719921—21no—
47C2×#11719921—21no—
48C2×C4×Dic53—21no—
49#13133148—42no—
50#13133137—42no—
51C22⋊#88322—21no—
52C22⋊#88322—21no—
53C22⋊#88322—21no—
54C22⋊#88322C2×C412yescomplemented
55#13133124—21no—
56#13133124—21no—
57#13133124—21no—
58C23×C4×C53—21no—
59C23×C4×C53—21no—
60C23×C4×C53D411yescharacteristic
61C23×C4×C53D411yescharacteristic
62C23×C4×C53C2311yescharacteristic
63C2×C4⋊C4×C53—21no—
64C2×C4⋊C4×C53—21no—
65C2×C4⋊C4×C53—42no—
66C2×C4⋊C4×C53—42no—
67C2×C4⋊C4×C53—42no—
68C2×C4⋊C4×C53—42no—
69#95×C53—21no—
70#95×C53—21no—
71#95×C53—21no—
72#95×C53—42no—
73#95×C53C2×C412yescomplemented
74C24×C53C2×D411yescharacteristic
75C23×D53—21no—
76C22×C4×C53—22no—
77C22×C4×C53—22no—
78C22×C4×C53—22no—
79C22×C4×C53—21no—
80C22×C4×C53—21no—
81C22×C4×C53—22no—
82C22×C4×C53—21no—
83C22×C4×C53—21no—
84C22×C4×C53—21no—
85C22×C4×C53—41no—
86C22×C4×C53—41no—
87C22×C4×C53—41no—
88C22×C4×C53—41no—
89C22×C4×C53—41no—
90C22×C4×C53—41no—
91C22×C4×C53—41no—
92C22×C4×C53—41no—
93C22×C4×C53—41no—
94C22×C4×C53—41no—
95C22×C4×C53—41no—
96C22×C4×C53—41no—
97C22×C4×C53—41no—
98C22×C4×C53—41no—
99C22×C4×C53—41no—
100C22×C4×C53—41no—
101C22×C4×C53SD1612yes—
102C22×C4×C53C22×C411yescharacteristic
103C22×C4×C53C2×D411yescharacteristic
104C22×C4×C53C2×D411yescharacteristic
105C22×C4×C53C2×Q811yescharacteristic
106C22×C4×C53#4111yescharacteristic
107C22×C4×C53#4111yescharacteristic
108C2×C22⋊D53—22no—
109C2×C22⋊D53—22no—
110C2×C22⋊D53—41no—
111C2×C22⋊D53—41no—
112C2×C22⋊D53—41no—
113C2×C22⋊D53—41no—
114C22×Dic53—21no—
115C22×Dic53—21no—
116C22×Dic53—21no—
117C2×D212—21no—
118C2×D212—41no—
119C2×D212—41no—
120C2×D212C4⋊C412yescomplemented
121C2×C4×D53—42no—
122C2×C4×D53—81no—
123C2×C4×D53—81no—
124C4⋊C4×C53—22no—
125C4⋊C4×C53—22no—
126C4⋊C4×C53—41no—
127C4⋊C4×C53—41no—
128C4⋊C4×C53—48no—
129C22⋊Dic53—41no—
130C22⋊Dic53—41no—
131#11719921—22no—
132#11719921—22no—
133#11719921—42no—
134#11719921—81no—
135#11719921—81no—
136C4⋊Dic53—21no—
137C4⋊Dic53—41no—
138C4⋊Dic53—41no—
139C4⋊Dic53C4⋊C412yescomplemented
140#11719919—42no—
141C4×Dic53—42no—
142C2×#88322—22no—
143C2×#88322—41no—
144C2×#88322—41no—
145C2×#88322—41no—
146C2×#88322—41no—
147C2×#88322—41no—
148C23×C53—21no—
149C23×C53—21no—
150C23×C53—21no—
151C23×C53—21no—
152C23×C53C4×D411yescharacteristic
153C23×C53#12011yescharacteristic
154C23×C53C22⋊Q811yescharacteristic
155C23×C53C2×SD1611yescharacteristic
156C23×C53#13511yescharacteristic
157C23×C53AGL(1,ℤ/8)11yescharacteristic
158C23×C53AGL(1,ℤ/8)11yescharacteristic
159C22×D53—22no—
160C22×D53—41no—
161C22×D53—41no—
162C22×D53—41no—
163C22×D53—41no—
164C22×D53—41no—
165C22×D53—41no—
166C2×C4×C53—21no—
167C2×C4×C53—21no—
168C2×C4×C53—21no—
169C2×C4×C53—21no—
170C2×C4×C53—24no—
171C2×C4×C53—22no—
172C2×C4×C53—41no—
173C2×C4×C53—41no—
174C2×C4×C53—41no—
175C2×C4×C53—41no—
176C2×C4×C53—42no—
177C2×C4×C53—42no—
178C2×C4×C53—42no—
179C2×C4×C53—42no—
180C2×C4×C53—42no—
181C2×C4×C53—42no—
182C2×C4×C53—41no—
183C2×C4×C53—41no—
184C2×C4×C53—42no—
185C2×C4×C53—42no—
186C2×C4×C53—42no—
187C2×C4×C53—42no—
188C2×C4×C53—41no—
189C2×C4×C53—41no—
190C2×C4×C53—42no—
191C2×C4×C53—41no—
192C2×C4×C53—81no—
193C2×C4×C53—81no—
194C2×C4×C53—81no—
195C2×C4×C53—81no—
196C2×C4×C53—81no—
197C2×C4×C53—81no—
198C2×C4×C53C2×C4⋊C411yescharacteristic
199C2×C4×C53C4×D411yescharacteristic
200C2×C4×C53C22⋊Q811yescharacteristic
201C2×C4×C53#12311yescharacteristic
202C22⋊D53—42no—
203C22⋊D53—42no—
204C22⋊D53—42no—
205C22⋊D53—42no—
206C2×Dic53—22no—
207C2×Dic53—41no—
208C2×Dic53—41no—
209C2×Dic53—41no—
210C2×Dic53—41no—
211C2×Dic53—41no—
212C2×Dic53—41no—
213C2×Dic53—41no—
214C2×Dic53—41no—
215D212—42no—
216D212—81no—
217C4×D53—84no—
218#88322—42no—
219#88322—81no—
220#8873—531nomaximal, Sylow
221C22×C53—21no—
222C22×C53—21no—
223C22×C53—21no—
224C22×C53—21no—
225C22×C53—21no—
226C22×C53—21no—
227C22×C53—21no—
228C22×C53—21no—
229C22×C53—21no—
230C22×C53—21no—
231C22×C53—21no—
232C22×C53—21no—
233C22×C53#38511yescharacteristic
234C22×C53C4×SD1611yescharacteristic
235C22×C53#44211yescharacteristic
236C22×C53#44911yescharacteristic
237C22×C53#45011yescharacteristic
238C22×C53#45311yescharacteristic
239C22×C53#45611yescharacteristic
240C22×C53B(4,𝔽2)11yescharacteristic
241C22×C53#45811yescharacteristic
242C22×C53#47611yescharacteristic
243C22×C53#47811yescharacteristic
244C2×D53—42no—
245C2×D53—42no—
246C2×D53—42no—
247C2×D53—81no—
248C2×D53—81no—
249C2×D53—81no—
250C2×D53—81no—
251C4×C53—21no—
252C4×C53—21no—
253C4×C53—21no—
254C4×C53—21no—
255C4×C53—44no—
256C4×C53—42no—
257C4×C53—81no—
258C4×C53—82no—
259Dic53—42no—
260Dic53—81no—
261Dic53—81no—
262#2325—531no—
263#1874—531no—
264#1853—531no—
265C2×#327—531no—
266#1444D5311yescharacteristic, complemented
267#1294—531no—
268#1280—531no—
269C2×C53—21no—
270C2×C53—21no—
271C2×C53—21no—
272C2×C53—21no—
273C2×C53#160011yescharacteristic
274C2×C53#160411yescharacteristic
275C2×C53#160911yescharacteristic
276C2×C53#161311yescharacteristic
277C2×C53#185911yescharacteristic
278C2×C53#186411yescharacteristic
279C2×C53#186711yescharacteristic
280D53—82no—
281C2×#121—531no—
282C2×C4×D4—1061no—
283C22×C4⋊C4—21no—
284C22×C4⋊C4C2×D5311yescharacteristic
285C2×#106—1061no—
286C2×#102—1061no—
287C2×C22⋊C8—531no—
288C2×C22⋊C8—531no—
289#388—1061no—
290#387—1061no—
291#386—1061no—
292#386—1061no—
293#385—1061no—
294#380—531no—
295C2×#95—21no—
296C2×#95C2×D5311yescharacteristic
297C2×#95C2×D5311yescharacteristic
298#340—1061no—
299#340—1061no—
300#327—534no—
301C53#887311yescharacteristic, complemented, Sylow
302C22×D4—531no—
303C22×D4—1061no—
304C23×C4—21no—
305C23×C4—21no—
306C23×C4—1061no—
307C23×C4C22⋊D5311yescharacteristic
308C23×C4C22⋊D5311yescharacteristic
309C23×C4C22×D5311yescharacteristic
310C22×C8—1061no—
311C22×C8—1061no—
312#121—532no—
313#121—532no—
314#121—1061no—
315#121—1061no—
316C4×D4—2122no—
317C4×D4—2122no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—42no—
321C2×C4⋊C4—42no—
322C2×C4⋊C4—42no—
323C2×C4⋊C4—42no—
324C2×C4⋊C4—531no—
325C2×C4⋊C4—531no—
326C2×C4⋊C4—531no—
327C2×C4⋊C4—1061no—
328C2×C22⋊C4—1061no—
329C2×C22⋊C4—1061no—
330C2×C22⋊C4—1061no—
331C2×C22⋊C4—1061no—
332C2×C22⋊C4—1061no—
333C2×C42—1061no—
334#106—2122no—
335#102—2122no—
336C22⋊C8—532no—
337C22⋊C8—1061no—
338C22⋊C8—1061no—
339C22⋊C8—1061no—
340#95—21no—
341#95—21no—
342#95—21no—
343#95—42no—
344#95—1061no—
345#95—1061no—
346#95—1061no—
347#95C4×D5312yescomplemented
348C24—1061no—
349C24C2×C22⋊D5311yescharacteristic
350C2×D4—532no—
351C2×D4—1062no—
352C2×D4—1062no—
353C2×D4—1061no—
354C2×D4—2121no—
355C2×D4—2121no—
356C2×D4—2121no—
357C2×D4—2121no—
358C2×D4—2121no—
359C2×D4—2121no—
360C22×C4—22no—
361C22×C4—22no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4—21no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—41no—
372C22×C4—41no—
373C22×C4—41no—
374C22×C4—41no—
375C22×C4—41no—
376C22×C4—41no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—41no—
380C22×C4—41no—
381C22×C4—41no—
382C22×C4—41no—
383C22×C4—41no—
384C22×C4—41no—
385C22×C4—1061no—
386C22×C4—1061no—
387C22×C4—1061no—
388C22×C4—4241no—
389C22×C4—4241no—
390C22×C4—2122no—
391C22×C4#1171992412yes—
392C22×C4C2×C4×D5311yescharacteristic
393C22×C4#1171994511yescharacteristic
394C22×C4D4×D5311yescharacteristic
395C22×C4D4×D5311yescharacteristic
396C22×C4Q8×D5311yescharacteristic
397C22×C4#1171994911yescharacteristic
398C2×C8—1062no—
399C2×C8—2121no—
400C2×C8—2121no—
401C2×C8—2121no—
402C2×C8—2121no—
403C2×C8—2121no—
404C4⋊C4—22no—
405C4⋊C4—22no—
406C4⋊C4—41no—
407C4⋊C4—41no—
408C4⋊C4—48no—
409C4⋊C4—532no—
410C4⋊C4—1061no—
411C4⋊C4—2121no—
412C4⋊C4—2121no—
413C4⋊C4—2122no—
414C22⋊C4—1062no—
415C22⋊C4—1062no—
416C22⋊C4—4241no—
417C22⋊C4—4241no—
418C22⋊C4—2122no—
419C22⋊C4—2121no—
420C22⋊C4—2121no—
421C42—2122no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—21no—
426C23—1062no—
427C23—2121no—
428C23—2121no—
429C23—2121no—
430C23—2121no—
431C23—2121no—
432C23—2121no—
433C23C4×C22⋊D5311yescharacteristic
434C23#1313318511yescharacteristic
435C23#1313319311yescharacteristic
436C23C2×#1171992411yescharacteristic
437C23#1313323711yescharacteristic
438C23#1313324211yescharacteristic
439C23#1313324311yescharacteristic
440D4—4241no—
441D4—2122no—
442D4—2122no—
443D4—2122no—
444D4—2122no—
445D4—2122no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—24no—
451C2×C4—22no—
452C2×C4—41no—
453C2×C4—41no—
454C2×C4—41no—
455C2×C4—41no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—42no—
459C2×C4—42no—
460C2×C4—42no—
461C2×C4—41no—
462C2×C4—41no—
463C2×C4—42no—
464C2×C4—42no—
465C2×C4—42no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—42no—
472C2×C4—81no—
473C2×C4—81no—
474C2×C4—81no—
475C2×C4—81no—
476C2×C4—81no—
477C2×C4—81no—
478C2×C4—1062no—
479C2×C4—4244no—
480C2×C4—2121no—
481C2×C4—2121no—
482C2×C4—2121no—
483C2×C4—2121no—
484C2×C4—2121no—
485C2×C4—2121no—
486C2×C4—2121no—
487C2×C4—2121no—
488C2×C4C4⋊C4×D5311yescharacteristic
489C2×C4#1313317911yescharacteristic
490C2×C4#1313319511yescharacteristic
491C2×C4#1313320211yescharacteristic
492C8—4241no—
493C8—2122no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—21no—
501C22—21no—
502C22—21no—
503C22—21no—
504C22—21no—
505C22—21no—
506C22—4241no—
507C22—4241no—
508C22—4241no—
509C22—4241no—
510C22—2122no—
511C22—2122no—
512C22—2122no—
513C22#1681049411yescharacteristic
514C22C4×#1171992411yescharacteristic
515C22#1681073311yescharacteristic
516C22#1681076611yescharacteristic
517C22#1681077311yescharacteristic
518C22#1681079011yescharacteristic
519C22#1681080511yescharacteristic
520C22#1681080911yescharacteristic
521C22#1681081211yescharacteristic
522C22#1681091011yescharacteristic
523C22#1681092011yescharacteristic
524C4—21no—
525C4—21no—
526C4—21no—
527C4—21no—
528C4—44no—
529C4—42no—
530C4—81no—
531C4—82no—
532C4—4241no—
533C4—4241no—
534C4—2122no—
535C2—21no—
536C2—21no—
537C2—21no—
538C2—21no—
539C2—4242no—
540C2#2027365911yescharacteristic
541C2#2027368311yescharacteristic
542C2#2027371011yescharacteristic
543C2#2027373411yescharacteristic
544C2#2027494011yescharacteristic
545C2#2027497511yescharacteristic
546C2#2027499411yescharacteristic
subgroupG/H
centerC23#13133185
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1444C2
socleC23×C53#120
prime radicalC2×#13133191C22
cosocleC23×C4×C53C23
hypercenter#1444D53
hypocenterC53#8873
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1444

Complements

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

NH ≅ G/Ntype
3. #20274908C2semidirect
5. #20272594C2semidirect
6. #20272558C2semidirect
7. C53×#1444C2semidirect
22. #16810356C4semidirect
54. C22⋊#88322C2×C4semidirect
73. #95×C53C2×C4semidirect
120. C2×D212C4⋊C4semidirect
139. C4⋊Dic53C4⋊C4semidirect
266. #1444D53semidirect
301. C53#8873semidirect
347. #95C4×D53semidirect

Maximal subgroups

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

subgroupindex
#202774182
#202750322
#202749082
C2×#168103562
#202725942
#202725582
C53×#14442
#887353