Finite Group Databasebeta

← #27278167

Subgroups — #27278167

Order 13568.

658 conjugacy classes of subgroups.

The subgroup lattice has 542 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×#1442C211yesmaximal, 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×C53C2211yescharacteristic
24C2×#95×C53—21no—
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—
66#95×C53—21no—
67#95×C53—21no—
68#95×C53—21no—
69#95×C53—42no—
70#95×C53—42no—
71#95×C53C2×C412yescomplemented
72C24×C53C2×D411yescharacteristic
73C23×D53—21no—
74C22×C4×C53—21no—
75C22×C4×C53—21no—
76C22×C4×C53—22no—
77C22×C4×C53—22no—
78C22×C4×C53—21no—
79C22×C4×C53—21no—
80C22×C4×C53—21no—
81C22×C4×C53—41no—
82C22×C4×C53—41no—
83C22×C4×C53—41no—
84C22×C4×C53—41no—
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×C53SD1612yes—
100C22×C4×C53C22×C411yescharacteristic
101C22×C4×C53C2×D411yescharacteristic
102C22×C4×C53C2×D411yescharacteristic
103C22×C4×C53C2×Q811yescharacteristic
104C22×C4×C53#4111yescharacteristic
105C22×C4×C53#4111yescharacteristic
106C2×C22⋊D53—22no—
107C2×C22⋊D53—22no—
108C2×C22⋊D53—41no—
109C2×C22⋊D53—41no—
110C2×C22⋊D53—41no—
111C2×C22⋊D53—41no—
112C22×Dic53—21no—
113C22×Dic53—21no—
114C22×Dic53—21no—
115C2×D212—21no—
116C2×D212—41no—
117C2×D212—41no—
118C2×D212C4⋊C412yes—
119C2×C4×D53—42no—
120C2×C4×D53—81no—
121C2×C4×D53—81no—
122C4⋊C4×C53—22no—
123C4⋊C4×C53—22no—
124C4⋊C4×C53—41no—
125C4⋊C4×C53—41no—
126C22⋊Dic53—41no—
127C22⋊Dic53—41no—
128#11719921—22no—
129#11719921—22no—
130#11719921—42no—
131#11719921—81no—
132#11719921—81no—
133C4⋊Dic53—21no—
134C4⋊Dic53—41no—
135C4⋊Dic53—41no—
136C4⋊Dic53C4⋊C412yes—
137#11719919—42no—
138C4×Dic53—42no—
139C2×#88322—22no—
140C2×#88322—41no—
141C2×#88322—41no—
142C2×#88322—41no—
143C2×#88322—41no—
144C2×#88322—41no—
145C23×C53—21no—
146C23×C53—21no—
147C23×C53—21no—
148C23×C53—21no—
149C23×C53C4×D411yescharacteristic
150C23×C53#12011yescharacteristic
151C23×C53C22⋊Q811yescharacteristic
152C23×C53C2×SD1611yescharacteristic
153C23×C53#13511yescharacteristic
154C23×C53AGL(1,ℤ/8)11yescharacteristic
155C23×C53AGL(1,ℤ/8)11yescharacteristic
156C22×D53—22no—
157C22×D53—41no—
158C22×D53—41no—
159C22×D53—41no—
160C22×D53—41no—
161C22×D53—41no—
162C22×D53—41no—
163C2×C4×C53—21no—
164C2×C4×C53—21no—
165C2×C4×C53—21no—
166C2×C4×C53—21no—
167C2×C4×C53—22no—
168C2×C4×C53—41no—
169C2×C4×C53—41no—
170C2×C4×C53—41no—
171C2×C4×C53—41no—
172C2×C4×C53—41no—
173C2×C4×C53—41no—
174C2×C4×C53—42no—
175C2×C4×C53—42no—
176C2×C4×C53—42no—
177C2×C4×C53—42no—
178C2×C4×C53—42no—
179C2×C4×C53—42no—
180C2×C4×C53—42no—
181C2×C4×C53—41no—
182C2×C4×C53—41no—
183C2×C4×C53—42no—
184C2×C4×C53—41no—
185C2×C4×C53—81no—
186C2×C4×C53—81no—
187C2×C4×C53—81no—
188C2×C4×C53—81no—
189C2×C4×C53—81no—
190C2×C4×C53—81no—
191C2×C4×C53—81no—
192C2×C4×C53—81no—
193C2×C4×C53—81no—
194C2×C4×C53—81no—
195C2×C4×C53C2×C4⋊C411yescharacteristic
196C2×C4×C53C4×D411yescharacteristic
197C2×C4×C53C22⋊Q811yescharacteristic
198C2×C4×C53#12311yescharacteristic
199C22⋊D53—42no—
200C22⋊D53—42no—
201C22⋊D53—42no—
202C22⋊D53—42no—
203C2×Dic53—22no—
204C2×Dic53—41no—
205C2×Dic53—41no—
206C2×Dic53—41no—
207C2×Dic53—41no—
208C2×Dic53—41no—
209C2×Dic53—41no—
210C2×Dic53—41no—
211C2×Dic53—41no—
212D212—42no—
213D212—81no—
214C4×D53—84no—
215#88322—42no—
216#88322—81no—
217#8870—531nomaximal, Sylow
218C22×C53—21no—
219C22×C53—21no—
220C22×C53—21no—
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#38511yescharacteristic
231C22×C53C4×SD1611yescharacteristic
232C22×C53#44211yescharacteristic
233C22×C53#44911yescharacteristic
234C22×C53#45011yescharacteristic
235C22×C53#45411yescharacteristic
236C22×C53#45511yescharacteristic
237C22×C53B(4,𝔽2)11yescharacteristic
238C22×C53#45811yescharacteristic
239C22×C53#47611yescharacteristic
240C22×C53#47811yescharacteristic
241C2×D53—42no—
242C2×D53—42no—
243C2×D53—42no—
244C2×D53—81no—
245C2×D53—81no—
246C2×D53—81no—
247C2×D53—81no—
248C4×C53—21no—
249C4×C53—21no—
250C4×C53—21no—
251C4×C53—21no—
252C4×C53—42no—
253C4×C53—82no—
254C4×C53—82no—
255C4×C53—81no—
256Dic53—42no—
257Dic53—81no—
258Dic53—81no—
259#2325—531no—
260#1874—531no—
261#1853—531no—
262C2×#327—531no—
263#1442D5311yescharacteristic, complemented
264#1294—531no—
265#1280—531no—
266C2×C53—21no—
267C2×C53—21no—
268C2×C53—21no—
269C2×C53—21no—
270C2×C53#159711yescharacteristic
271C2×C53#160111yescharacteristic
272C2×C53#161211yescharacteristic
273C2×C53#161611yescharacteristic
274C2×C53#186011yescharacteristic
275C2×C53#186411yescharacteristic
276C2×C53#186711yescharacteristic
277D53—82no—
278C2×#121—531no—
279C2×C4×D4—1061no—
280C22×C4⋊C4C2×D5311yescharacteristic
281C2×#106—1061no—
282C2×#102—1061no—
283C2×C22⋊C8—531no—
284C2×C22⋊C8—531no—
285#388—1061no—
286#387—1061no—
287#386—1061no—
288#386—1061no—
289#385—1061no—
290#380—531no—
291C2×#95—21no—
292C2×#95—21no—
293C2×#95C2×D5311yescharacteristic
294C2×#95C2×D5311yescharacteristic
295#340—1061no—
296#340—1061no—
297#327—534no—
298C53#887011yescharacteristic, complemented, Sylow
299C22×D4—531no—
300C22×D4—1061no—
301C23×C4—21no—
302C23×C4—21no—
303C23×C4—1061no—
304C23×C4C22⋊D5311yescharacteristic
305C23×C4C22⋊D5311yescharacteristic
306C23×C4C22×D5311yescharacteristic
307C22×C8—1061no—
308C22×C8—1061no—
309#121—532no—
310#121—532no—
311#121—1061no—
312#121—1061no—
313C4×D4—2122no—
314C4×D4—2122no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—42no—
318C2×C4⋊C4—531no—
319C2×C4⋊C4—531no—
320C2×C4⋊C4—531no—
321C2×C4⋊C4—1061no—
322C2×C22⋊C4—1061no—
323C2×C22⋊C4—1061no—
324C2×C22⋊C4—1061no—
325C2×C22⋊C4—1061no—
326C2×C22⋊C4—1061no—
327C2×C42—1061no—
328#106—2122no—
329#102—2122no—
330C22⋊C8—532no—
331C22⋊C8—1061no—
332C22⋊C8—1061no—
333C22⋊C8—1061no—
334#95—21no—
335#95—21no—
336#95—21no—
337#95—42no—
338#95—42no—
339#95—1061no—
340#95—1061no—
341#95—1061no—
342#95C4×D5312yescomplemented
343C24—1061no—
344C24C2×C22⋊D5311yescharacteristic
345C2×D4—532no—
346C2×D4—1062no—
347C2×D4—1062no—
348C2×D4—1061no—
349C2×D4—2121no—
350C2×D4—2121no—
351C2×D4—2121no—
352C2×D4—2121no—
353C2×D4—2121no—
354C2×D4—2121no—
355C22×C4—21no—
356C22×C4—21no—
357C22×C4—22no—
358C22×C4—22no—
359C22×C4—21no—
360C22×C4—21no—
361C22×C4—21no—
362C22×C4—41no—
363C22×C4—41no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
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—1061no—
381C22×C4—1061no—
382C22×C4—1061no—
383C22×C4—4241no—
384C22×C4—4241no—
385C22×C4—2122no—
386C22×C4#1171992412yes—
387C22×C4C2×C4×D5311yescharacteristic
388C22×C4#1171994511yescharacteristic
389C22×C4D4×D5311yescharacteristic
390C22×C4D4×D5311yescharacteristic
391C22×C4Q8×D5311yescharacteristic
392C22×C4#1171994911yescharacteristic
393C2×C8—1062no—
394C2×C8—2121no—
395C2×C8—2121no—
396C2×C8—2121no—
397C2×C8—2121no—
398C2×C8—2121no—
399C4⋊C4—22no—
400C4⋊C4—22no—
401C4⋊C4—41no—
402C4⋊C4—41no—
403C4⋊C4—532no—
404C4⋊C4—1061no—
405C4⋊C4—2121no—
406C4⋊C4—2121no—
407C4⋊C4—2122no—
408C22⋊C4—1062no—
409C22⋊C4—1062no—
410C22⋊C4—4241no—
411C22⋊C4—4241no—
412C22⋊C4—2122no—
413C22⋊C4—2121no—
414C22⋊C4—2121no—
415C42—2122no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—1062no—
421C23—2121no—
422C23—2121no—
423C23—2121no—
424C23—2121no—
425C23—2121no—
426C23—2121no—
427C23C4×C22⋊D5311yescharacteristic
428C23#1313318511yescharacteristic
429C23#1313319311yescharacteristic
430C23C2×#1171992411yescharacteristic
431C23#1313323711yescharacteristic
432C23#1313324211yescharacteristic
433C23#1313324311yescharacteristic
434D4—4241no—
435D4—2122no—
436D4—2122no—
437D4—2122no—
438D4—2122no—
439D4—2122no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—22no—
445C2×C4—41no—
446C2×C4—41no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—42no—
452C2×C4—42no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—41no—
458C2×C4—41no—
459C2×C4—42no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4—81no—
463C2×C4—81no—
464C2×C4—81no—
465C2×C4—81no—
466C2×C4—81no—
467C2×C4—81no—
468C2×C4—81no—
469C2×C4—81no—
470C2×C4—81no—
471C2×C4—81no—
472C2×C4—1062no—
473C2×C4—4244no—
474C2×C4—2121no—
475C2×C4—2121no—
476C2×C4—2121no—
477C2×C4—2121no—
478C2×C4—2121no—
479C2×C4—2121no—
480C2×C4—2121no—
481C2×C4—2121no—
482C2×C4C4⋊C4×D5311yescharacteristic
483C2×C4#1313317911yescharacteristic
484C2×C4#1313319511yescharacteristic
485C2×C4#1313320211yescharacteristic
486C8—4241no—
487C8—2122no—
488C22—21no—
489C22—21no—
490C22—21no—
491C22—21no—
492C22—21no—
493C22—21no—
494C22—21no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—21no—
499C22—21no—
500C22—4241no—
501C22—4241no—
502C22—4241no—
503C22—4241no—
504C22—2122no—
505C22—2122no—
506C22—2122no—
507C22#1681049411yescharacteristic
508C22C4×#1171992411yescharacteristic
509C22#1681073311yescharacteristic
510C22#1681076611yescharacteristic
511C22#1681077311yescharacteristic
512C22#1681079511yescharacteristic
513C22#1681080011yescharacteristic
514C22#1681080911yescharacteristic
515C22#1681081211yescharacteristic
516C22#1681091011yescharacteristic
517C22#1681092011yescharacteristic
518C4—21no—
519C4—21no—
520C4—21no—
521C4—21no—
522C4—42no—
523C4—82no—
524C4—81no—
525C4—82no—
526C4—4241no—
527C4—4241no—
528C4—2122no—
529C2—21no—
530C2—21no—
531C2—21no—
532C2—21no—
533C2—4242no—
534C2#2027364211yescharacteristic
535C2#2027366611yescharacteristic
536C2#2027372711yescharacteristic
537C2#2027375111yescharacteristic
538C2#2027494711yescharacteristic
539C2#2027497511yescharacteristic
540C2#2027499411yescharacteristic
subgroupG/H
centerC23#13133185
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC53×#1442C2
socleC23×C53#120
prime radicalC2×#13133191C22
cosocleC23×C4×C53C23
hypercenter#1442D53
hypocenterC53#8870
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1442

Complements

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

NH ≅ G/Ntype
3. #20274908C2semidirect
5. #20272594C2semidirect
6. #20272558C2semidirect
7. C53×#1442C2semidirect
22. #16810356C4semidirect
54. C22⋊#88322C2×C4semidirect
71. #95×C53C2×C4semidirect
263. #1442D53semidirect
298. C53#8870semidirect
342. #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×#14422
#887053