Finite Group Databasebeta

← #27284314

Subgroups — #27284314

Order 13568.

812 conjugacy classes of subgroups.

The subgroup lattice has 546 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810934C211yesmaximal, characteristic, complemented
2C2×#16810548C211yesmaximal, characteristic, complemented
3#20274799C211yesmaximal, characteristic, complemented
4C2×#16810357C211yesmaximal, characteristic, complemented
5C2×#16810347C211yesmaximal, characteristic, complemented
6#20272593C211yesmaximal, characteristic, complemented
7C53×#2617C211yesmaximal, characteristic
8C22×C4⋊Dic53C2211yescharacteristic
9C22×C22⋊Dic53—21no—
10#16810934—22no—
11#16810934—22no—
12C2×#13133150—21no—
13C2×#13133138—21no—
14C2×C22⋊#88322C2211yescharacteristic
15C2×C22⋊#88322C2211yescharacteristic
16#16810548—22no—
17#16810548—22no—
18C2×#13133124—21no—
19C2×#13133124C2211yescharacteristic
20#16810385—22no—
21#16810357—22no—
22#16810347—22no—
23C2×#121×C53C2211yescharacteristic
24C2×#120×C53—21no—
25C22×C22⋊C4×C53C2211yescharacteristic
26C53×#397—21no—
27C53×#394—22no—
28C53×#392—21no—
29C53×#386—21no—
30C53×#380C2211yescharacteristic
31C23×Dic53—21no—
32C23×Dic53—21no—
33C22×#88322—21no—
34C22×#88322—21no—
35C2×C4⋊Dic53—21no—
36C2×C4⋊Dic53—21no—
37C2×C4⋊Dic53—21no—
38C2×C4⋊Dic53—21no—
39C2×C4⋊Dic53—21no—
40C2×C4⋊Dic53D412yescomplemented
41C2×C22⋊Dic53—24no—
42C2×C22⋊Dic53—42no—
43C2×C22⋊Dic53—42no—
44#13133150—24no—
45#13133138—24no—
46C22⋊#88322—21no—
47C22⋊#88322—21no—
48C22⋊#88322—21no—
49C22⋊#88322D412yescomplemented
50#13133124—24no—
51#13133124—21no—
52#13133124D412yescomplemented
53C25×C53D411yescharacteristic
54C22×D4×C53—21no—
55C22×D4×C53—22no—
56C22×D4×C53D411yescharacteristic
57C23×C4×C53D411yescharacteristic
58C23×C4×C53C2311yescharacteristic
59#121×C53—21no—
60#121×C53—21no—
61#121×C53—21no—
62#121×C53—21no—
63#120×C53—42no—
64#120×C53—42no—
65C2×C4⋊C4×C53D411yescharacteristic
66C2×C4⋊C4×C53D412yes—
67C2×C22⋊C4×C53—22no—
68C2×C22⋊C4×C53—22no—
69C2×C22⋊C4×C53—21no—
70C2×C22⋊C4×C53—21no—
71C2×C22⋊C4×C53—21no—
72C2×C22⋊C4×C53—21no—
73C2×C22⋊C4×C53—21no—
74C2×C22⋊C4×C53—21no—
75C2×C22⋊C4×C53—21no—
76C2×C22⋊C4×C53—21no—
77#95×C53—22no—
78#95×C53—21no—
79C24×C53—21no—
80C24×C53—21no—
81C24×C53—21no—
82C24×C53—21no—
83C24×C53—21no—
84C24×C53—21no—
85C24×C53—21no—
86C24×C53—42no—
87C24×C53—42no—
88C24×C53D812yes—
89C24×C53C2×D411yescharacteristic
90C2×D4×C53—21no—
91C2×D4×C53—21no—
92C2×D4×C53—42no—
93C2×D4×C53—44no—
94C2×D4×C53—41no—
95C2×D4×C53—41no—
96C2×D4×C53—41no—
97C2×D4×C53—41no—
98C2×D4×C53—41no—
99C2×D4×C53—41no—
100C2×D4×C53—82no—
101C2×D4×C53—82no—
102C22×C4×C53—21no—
103C22×C4×C53—21no—
104C22×C4×C53—22no—
105C22×C4×C53—21no—
106C22×C4×C53—21no—
107C22×C4×C53—42no—
108C22×C4×C53—42no—
109C22×C4×C53C2×D411yescharacteristic
110C22×C4×C53C2×D411yescharacteristic
111C22×C4×C53C2×D411yescharacteristic
112C22×C4×C53C2×D411yescharacteristic
113C22×C4×C53C2×D411yescharacteristic
114C22×C4×C53#4111yescharacteristic
115C22×Dic53—22no—
116C22×Dic53—22no—
117C22×Dic53—42no—
118C22×Dic53—42no—
119C22×Dic53—41no—
120C22×Dic53—41no—
121C22×Dic53—41no—
122C22×Dic53—41no—
123C22×Dic53—41no—
124C22×Dic53—41no—
125C22×Dic53—41no—
126C22×Dic53—41no—
127C4⋊C4×C53—21no—
128C4⋊C4×C53—21no—
129C4⋊C4×C53—42no—
130C22⋊C4×C53—44no—
131C22⋊C4×C53—41no—
132C22⋊C4×C53—41no—
133C22⋊C4×C53—41no—
134C22⋊C4×C53—41no—
135C22⋊C4×C53—41no—
136C22⋊C4×C53—41no—
137C22⋊C4×C53—41no—
138C22⋊C4×C53—41no—
139C22⋊C4×C53—82no—
140C22⋊Dic53—48no—
141C4⋊Dic53—22no—
142C4⋊Dic53—22no—
143C4⋊Dic53—41no—
144C4⋊Dic53—41no—
145C2×#88322—22no—
146C2×#88322—42no—
147C2×#88322—42no—
148C2×#88322—41no—
149C23×C53—22no—
150C23×C53—22no—
151C23×C53—22no—
152C23×C53—22no—
153C23×C53—22no—
154C23×C53—44no—
155C23×C53—44no—
156C23×C53—42no—
157C23×C53—42no—
158C23×C53—41no—
159C23×C53—41no—
160C23×C53—41no—
161C23×C53—42no—
162C23×C53—42no—
163C23×C53—41no—
164C23×C53—41no—
165C23×C53—41no—
166C23×C53—41no—
167C23×C53—42no—
168C23×C53—42no—
169C23×C53—82no—
170C23×C53—82no—
171C23×C53—81no—
172C23×C53—81no—
173C23×C53#12011yescharacteristic
174C23×C53#12011yescharacteristic
175C23×C53#12111yescharacteristic
176C23×C53C2×D811yescharacteristic
177C23×C53#13511yescharacteristic
178C23×C53AGL(1,ℤ/8)11yescharacteristic
179C23×C53AGL(1,ℤ/8)11yescharacteristic
180D4×C53—42no—
181D4×C53—81no—
182D4×C53—88no—
183D4×C53—81no—
184D4×C53—81no—
185D4×C53—81no—
186D4×C53—81no—
187C2×C4×C53—22no—
188C2×C4×C53—22no—
189C2×C4×C53—41no—
190C2×C4×C53—41no—
191C2×C4×C53—41no—
192C2×C4×C53—41no—
193C2×C4×C53—42no—
194C2×C4×C53—41no—
195C2×C4×C53—41no—
196C2×C4×C53—42no—
197C2×C4×C53—41no—
198C2×C4×C53—41no—
199C2×C4×C53—42no—
200C2×C4×C53—42no—
201C2×C4×C53—42no—
202C2×C4×C53—81no—
203C2×C4×C53#12011yescharacteristic
204C2×C4×C53#12111yescharacteristic
205C2×C4×C53#12111yescharacteristic
206C2×C4×C53#12711yescharacteristic
207C2×Dic53—42no—
208C2×Dic53—42no—
209C2×Dic53—42no—
210C2×Dic53—42no—
211C2×Dic53—42no—
212C2×Dic53—42no—
213C2×Dic53—82no—
214C2×Dic53—82no—
215C2×Dic53—81no—
216C2×Dic53—81no—
217C2×Dic53—81no—
218C2×Dic53—81no—
219#88322—42no—
220#88322—81no—
221#9918—531nomaximal, Sylow
222C22×C53—22no—
223C22×C53—22no—
224C22×C53—22no—
225C22×C53—22no—
226C22×C53—21no—
227C22×C53—21no—
228C22×C53—21no—
229C22×C53—21no—
230C22×C53—44no—
231C22×C53—44no—
232C22×C53—42no—
233C22×C53—82no—
234C22×C53—82no—
235C22×C53—82no—
236C22×C53—82no—
237C22×C53—81no—
238C22×C53—81no—
239C22×C53—82no—
240C22×C53—82no—
241C22×C53—81no—
242C22×C53—81no—
243C22×C53—81no—
244C22×C53#39211yescharacteristic
245C22×C53#44711yescharacteristic
246C22×C53#44711yescharacteristic
247C22×C53#44911yescharacteristic
248C22×C53#44911yescharacteristic
249C22×C53#45512yes—
250C22×C53#45812yes—
251C22×C53#46611yescharacteristic
252C22×C53#46911yescharacteristic
253C4×C53—21no—
254C4×C53—21no—
255C4×C53—21no—
256C4×C53—21no—
257C4×C53—82no—
258C4×C53—81no—
259C4×C53—81no—
260Dic53—82no—
261Dic53—82no—
262C2×#480—531no—
263#2617D5311yescharacteristic
264C2×#397—531no—
265#1823—531no—
266C2×#327—531no—
267C2×#323—531no—
268#1294—531no—
269C2×C53—21no—
270C2×C53—21no—
271C2×C53—21no—
272C2×C53—21no—
273C2×C53—82no—
274C2×C53—161no—
275C2×C53#160712yes—
276C2×C53#161212yes—
277C2×C53#199911yescharacteristic
278C2×C53#200411yescharacteristic
279C2×C53#202211yescharacteristic
280C2×#121C2×D5311yescharacteristic
281C2×#120—21no—
282C22×C4⋊C4—531no—
283C22×C22⋊C4—1061no—
284C22×C22⋊C4C2×D5311yescharacteristic
285#480—1062no—
286#480—1062no—
287C2×#107—1061no—
288C2×#102—1061no—
289C2×C22⋊C8—531no—
290C2×C22⋊C8—531no—
291#397—21no—
292#397—1062no—
293#397—1062no—
294#394—22no—
295#392—21no—
296#386—21no—
297#380C2×D5311yescharacteristic
298C2×#95—531no—
299C2×#95—1061no—
300#340—1062no—
301#327—1062no—
302#323—1062no—
303C53#991811yescharacteristic, complemented, Sylow
304C25C22⋊D5311yescharacteristic
305C22×D4—21no—
306C22×D4—22no—
307C22×D4C22⋊D5311yescharacteristic
308C23×C4—1061no—
309C23×C4—1061no—
310C23×C4C22⋊D5311yescharacteristic
311C23×C4C22×D5311yescharacteristic
312C22×C8—1061no—
313C22×C8—1061no—
314#121—21no—
315#121—21no—
316#121—21no—
317#121—21no—
318#120—42no—
319#120—42no—
320C2×C4⋊C4—532no—
321C2×C4⋊C4—1061no—
322C2×C4⋊C4—1061no—
323C2×C4⋊C4—1061no—
324C2×C4⋊C4—1061no—
325C2×C4⋊C4—1061no—
326C2×C4⋊C4C22⋊D5311yescharacteristic
327C2×C4⋊C4C22⋊D5312yes—
328C2×C22⋊C4—22no—
329C2×C22⋊C4—21no—
330C2×C22⋊C4—21no—
331C2×C22⋊C4—21no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—21no—
334C2×C22⋊C4—21no—
335C2×C22⋊C4—21no—
336C2×C22⋊C4—21no—
337C2×C22⋊C4—22no—
338C2×C22⋊C4—1064no—
339C2×C22⋊C4—2122no—
340C2×C22⋊C4—2122no—
341#107—1064no—
342#102—1064no—
343C22⋊C8—532no—
344C22⋊C8—1061no—
345C22⋊C8—1061no—
346C22⋊C8—1061no—
347#95—22no—
348#95—21no—
349#95—532no—
350#95—1064no—
351#95—1061no—
352C24—21no—
353C24—21no—
354C24—21no—
355C24—21no—
356C24—21no—
357C24—21no—
358C24—21no—
359C24—42no—
360C24—42no—
361C24#1171992212yes—
362C24C2×C22⋊D5311yescharacteristic
363C2×D4—21no—
364C2×D4—21no—
365C2×D4—42no—
366C2×D4—44no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—82no—
374C2×D4—82no—
375C22×C4—21no—
376C22×C4—21no—
377C22×C4—21no—
378C22×C4—22no—
379C22×C4—21no—
380C22×C4—42no—
381C22×C4—42no—
382C22×C4—1062no—
383C22×C4—1062no—
384C22×C4—2122no—
385C22×C4—2122no—
386C22×C4—2121no—
387C22×C4—2121no—
388C22×C4—2121no—
389C22×C4—2121no—
390C22×C4—2121no—
391C22×C4—2121no—
392C22×C4—2121no—
393C22×C4—2121no—
394C22×C4D4×D5311yescharacteristic
395C22×C4D4×D5311yescharacteristic
396C22×C4D4×D5311yescharacteristic
397C22×C4#1171994711yescharacteristic
398C22×C4C2×C22⋊D5311yescharacteristic
399C22×C4C2×C22⋊D5311yescharacteristic
400C2×C8—1062no—
401C2×C8—2122no—
402C2×C8—2121no—
403C2×C8—2122no—
404C4⋊C4—21no—
405C4⋊C4—21no—
406C4⋊C4—42no—
407C4⋊C4—1062no—
408C4⋊C4—1062no—
409C4⋊C4—2121no—
410C4⋊C4—2121no—
411C22⋊C4—44no—
412C22⋊C4—41no—
413C22⋊C4—41no—
414C22⋊C4—41no—
415C22⋊C4—41no—
416C22⋊C4—41no—
417C22⋊C4—41no—
418C22⋊C4—41no—
419C22⋊C4—41no—
420C22⋊C4—82no—
421C22⋊C4—2128no—
422C23—22no—
423C23—22no—
424C23—22no—
425C23—22no—
426C23—22no—
427C23—44no—
428C23—42no—
429C23—44no—
430C23—42no—
431C23—41no—
432C23—41no—
433C23—41no—
434C23—42no—
435C23—42no—
436C23—41no—
437C23—41no—
438C23—41no—
439C23—41no—
440C23—42no—
441C23—42no—
442C23—82no—
443C23—81no—
444C23—82no—
445C23—81no—
446C23#1313318511yescharacteristic
447C23#1313318611yescharacteristic
448C23#1313318911yescharacteristic
449C23C2×#1171992211yescharacteristic
450C23#1313323711yescharacteristic
451C23#1313324211yescharacteristic
452C23#1313324311yescharacteristic
453D4—42no—
454D4—81no—
455D4—88no—
456D4—81no—
457D4—81no—
458D4—81no—
459D4—81no—
460C2×C4—22no—
461C2×C4—22no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—41no—
466C2×C4—42no—
467C2×C4—42no—
468C2×C4—41no—
469C2×C4—41no—
470C2×C4—41no—
471C2×C4—41no—
472C2×C4—42no—
473C2×C4—42no—
474C2×C4—42no—
475C2×C4—81no—
476C2×C4—4242no—
477C2×C4—4242no—
478C2×C4—4241no—
479C2×C4—4241no—
480C2×C4—4241no—
481C2×C4—4241no—
482C2×C4—2122no—
483C2×C4—2122no—
484C2×C4—2122no—
485C2×C4—2122no—
486C2×C4—2122no—
487C2×C4—2122no—
488C2×C4#1313318511yescharacteristic
489C2×C4#1313318811yescharacteristic
490C2×C4#1313318911yescharacteristic
491C2×C4#1313321311yescharacteristic
492C8—4241no—
493C8—2122no—
494C22—21no—
495C22—21no—
496C22—22no—
497C22—22no—
498C22—22no—
499C22—22no—
500C22—21no—
501C22—21no—
502C22—44no—
503C22—44no—
504C22—42no—
505C22—82no—
506C22—82no—
507C22—82no—
508C22—81no—
509C22—82no—
510C22—81no—
511C22—81no—
512C22—82no—
513C22—82no—
514C22—81no—
515C22—81no—
516C22#1681053111yescharacteristic
517C22#1681075611yescharacteristic
518C22#1681075711yescharacteristic
519C22#1681076611yescharacteristic
520C22#1681076811yescharacteristic
521C22#1681080112yes—
522C22#1681081312yes—
523C22#1681085711yescharacteristic
524C22#1681087411yescharacteristic
525C4—21no—
526C4—21no—
527C4—21no—
528C4—21no—
529C4—81no—
530C4—82no—
531C4—81no—
532C4—4242no—
533C4—4242no—
534C2—21no—
535C2—21no—
536C2—21no—
537C2—21no—
538C2—82no—
539C2—161no—
540C2#2027370112yes—
541C2#2027372912yes—
542C2#2027571111yescharacteristic
543C2#2027574211yescharacteristic
544C2#2027584411yescharacteristic
subgroupG/H
centerC23#13133186
commutator, cosocleC23×C4×C53C23
FrattiniC23×C4C22×D53
Fitting, prime radicalC53×#2617C2
socleC23×C53#120
hypercenter#2617D53
hypocenterC53#9918
LCSG ▷ C23×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #2617

Complements

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

NH ≅ G/Ntype
1. C2×#16810934C2semidirect
2. C2×#16810548C2semidirect
3. #20274799C2semidirect
4. C2×#16810357C2semidirect
5. C2×#16810347C2semidirect
6. #20272593C2semidirect
40. C2×C4⋊Dic53D4semidirect
49. C22⋊#88322D4semidirect
52. #13133124D4semidirect
303. C53#9918semidirect

Maximal subgroups

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

subgroupindex
C2×#168109342
C2×#168105482
#202747992
C2×#168103572
C2×#168103472
#202725932
C53×#26172
#991853