Finite Group Databasebeta

← #27275905

Subgroups — #27275905

Order 13568.

1020 conjugacy classes of subgroups.

The subgroup lattice has 522 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20275023C212yesmaximal, complemented
2C2×#16810360C212yesmaximal, complemented
3C2×#16810347C211yesmaximal, characteristic
4#20272960C211yesmaximal, characteristic
5C2×C53×#543C211yesmaximal, characteristic
6C22×C4⋊Dic53C2211yescharacteristic
7C22×C22⋊Dic53—21no—
8C2×#13133141—22no—
9C2×C22⋊#88322C2212yes—
10C2×#13133124—21no—
11C2×#13133124C2211yescharacteristic
12#16810385—22no—
13#16810360—24no—
14#16810347—22no—
15C53×#543—24no—
16C53×#543—22no—
17C53×#543C414yescomplemented
18C2×C22⋊Q8×C53—22no—
19C2×C22⋊Q8×C53—21no—
20C2×C22⋊Q8×C53—22no—
21C2×C22⋊Q8×C53C2212yes—
22C22×C22⋊C4×C53—21no—
23C22×C22⋊C4×C53C2211yescharacteristic
24C23×Dic53—21no—
25C23×Dic53—21no—
26C22×#88322—22no—
27C2×C4⋊Dic53—22no—
28C2×C4⋊Dic53—42no—
29C2×C22⋊Dic53—42no—
30C2×C22⋊Dic53—42no—
31C2×C22⋊Dic53—42no—
32#13133141—44no—
33C22⋊#88322—22no—
34C22⋊#88322—22no—
35#13133124—22no—
36#13133124—42no—
37C25×C53D411yescharacteristic
38C22×Q8×C53—21no—
39C22×Q8×C53D412yes—
40C23×C4×C53—21no—
41C23×C4×C53—21no—
42C23×C4×C53D411yescharacteristic
43C23×C4×C53C2311yescharacteristic
44C22⋊Q8×C53—28no—
45C22⋊Q8×C53—28no—
46C22⋊Q8×C53—24no—
47C22⋊Q8×C53—28no—
48C22⋊Q8×C53—28no—
49C22⋊Q8×C53—24no—
50C22⋊Q8×C53—22no—
51C22⋊Q8×C53—22no—
52C22⋊Q8×C53C2×C414yes—
53C22⋊Q8×C53D414yes—
54C2×C4⋊C4×C53—21no—
55C2×C4⋊C4×C53—22no—
56C2×C4⋊C4×C53—22no—
57C2×C4⋊C4×C53D412yes—
58C2×C22⋊C4×C53—22no—
59C2×C22⋊C4×C53—24no—
60C2×C22⋊C4×C53—24no—
61C2×C22⋊C4×C53—22no—
62C2×C22⋊C4×C53—22no—
63C2×C22⋊C4×C53—22no—
64C2×C22⋊C4×C53C2×C412yes—
65C2×C22⋊C4×C53D412yes—
66C24×C53—21no—
67C24×C53—21no—
68C24×C53—21no—
69C24×C53—21no—
70C24×C53—41no—
71C24×C53—41no—
72C24×C53—41no—
73C24×C53—41no—
74C24×C53C22⋊C412yes—
75C24×C53SD1612yes—
76C24×C53Q1612yes—
77C24×C53C2×D411yescharacteristic
78C2×Q8×C53—24no—
79C2×Q8×C53—22no—
80C2×Q8×C53—42no—
81C2×Q8×C53—42no—
82C2×Q8×C53—81no—
83C2×Q8×C53—81no—
84C2×Q8×C53C22⋊C414yescomplemented
85C22×C4×C53—22no—
86C22×C4×C53—21no—
87C22×C4×C53—22no—
88C22×C4×C53—24no—
89C22×C4×C53—24no—
90C22×C4×C53—21no—
91C22×C4×C53—41no—
92C22×C4×C53—41no—
93C22×C4×C53—41no—
94C22×C4×C53—41no—
95C22×C4×C53—81no—
96C22×C4×C53—81no—
97C22×C4×C53—81no—
98C22×C4×C53—81no—
99C22×C4×C53C22⋊C412yes—
100C22×C4×C53C22×C411yescharacteristic
101C22×C4×C53C2×D412yes—
102C22×C4×C53C2×D412yes—
103C22×C4×C53C2×D411yescharacteristic
104C22×Dic53—22no—
105C22×Dic53—22no—
106C22×Dic53—41no—
107C22×Dic53—41no—
108C22×Dic53—41no—
109C22×Dic53—41no—
110C22×Dic53—41no—
111C22×Dic53—41no—
112C22×Dic53—41no—
113C22×Dic53—41no—
114C22×Dic53—41no—
115C22×Dic53—41no—
116C22×Dic53—41no—
117C22×Dic53—41no—
118C4⋊C4×C53—28no—
119C4⋊C4×C53—24no—
120C4⋊C4×C53—24no—
121C4⋊C4×C53—24no—
122C4⋊C4×C53—22no—
123C4⋊C4×C53C22⋊C414yescomplemented
124C22⋊C4×C53—28no—
125C22⋊C4×C53—24no—
126C22⋊C4×C53—24no—
127C22⋊C4×C53—24no—
128C22⋊C4×C53—24no—
129C22⋊Dic53—48no—
130C4⋊Dic53—24no—
131C4⋊Dic53—42no—
132C2×#88322—42no—
133C2×#88322—42no—
134C2×#88322—42no—
135C23×C53—21no—
136C23×C53—21no—
137C23×C53—21no—
138C23×C53—21no—
139C23×C53—22no—
140C23×C53—22no—
141C23×C53—22no—
142C23×C53—22no—
143C23×C53—22no—
144C23×C53—22no—
145C23×C53—42no—
146C23×C53—42no—
147C23×C53—42no—
148C23×C53—42no—
149C23×C53—42no—
150C23×C53—42no—
151C23×C53—41no—
152C23×C53—41no—
153C23×C53—41no—
154C23×C53—41no—
155C23×C53—81no—
156C23×C53—81no—
157C23×C53—81no—
158C23×C53—81no—
159C23×C53—81no—
160C23×C53—81no—
161C23×C53#10314yescomplemented
162C23×C53C2×C22⋊C411yescharacteristic
163C23×C53#12011yescharacteristic
164C23×C53#12011yescharacteristic
165C23×C53C2×SD1611yescharacteristic
166C23×C53C2×Q1611yescharacteristic
167C23×C53#13711yescharacteristic
168C23×C53#13711yescharacteristic
169Q8×C53—44no—
170Q8×C53—82no—
171Q8×C53—84no—
172C2×C4×C53—22no—
173C2×C4×C53—22no—
174C2×C4×C53—24no—
175C2×C4×C53—24no—
176C2×C4×C53—22no—
177C2×C4×C53—41no—
178C2×C4×C53—41no—
179C2×C4×C53—42no—
180C2×C4×C53—41no—
181C2×C4×C53—42no—
182C2×C4×C53—41no—
183C2×C4×C53—42no—
184C2×C4×C53—42no—
185C2×C4×C53—42no—
186C2×C4×C53—42no—
187C2×C4×C53—81no—
188C2×C4×C53—84no—
189C2×C4×C53—84no—
190C2×C4×C53C2×C22⋊C412yes—
191C2×C4×C53#12012yes—
192C2×Dic53—42no—
193C2×Dic53—42no—
194C2×Dic53—42no—
195C2×Dic53—42no—
196C2×Dic53—42no—
197C2×Dic53—42no—
198C2×Dic53—81no—
199C2×Dic53—81no—
200C2×Dic53—81no—
201C2×Dic53—81no—
202C2×Dic53—81no—
203C2×Dic53—81no—
204C2×Dic53—81no—
205C2×Dic53—81no—
206#88322—82no—
207#8405—531nomaximal, Sylow
208C22×C53—21no—
209C22×C53—21no—
210C22×C53—21no—
211C22×C53—21no—
212C22×C53—21no—
213C22×C53—21no—
214C22×C53—21no—
215C22×C53—21no—
216C22×C53—21no—
217C22×C53—21no—
218C22×C53—21no—
219C22×C53—21no—
220C22×C53—24no—
221C22×C53—44no—
222C22×C53—42no—
223C22×C53—42no—
224C22×C53—42no—
225C22×C53—42no—
226C22×C53—42no—
227C22×C53—42no—
228C22×C53—82no—
229C22×C53—82no—
230C22×C53—82no—
231C22×C53#37911yescharacteristic
232C22×C53C2×#10311yescharacteristic
233C22×C53#41911yescharacteristic
234C22×C53#44811yescharacteristic
235C22×C53#44811yescharacteristic
236C22×C53#45111yescharacteristic
237C22×C53#45111yescharacteristic
238C22×C53#45311yescharacteristic
239C22×C53#45511yescharacteristic
240C22×C53B(4,𝔽2)11yescharacteristic
241C22×C53#45811yescharacteristic
242C4×C53—22no—
243C4×C53—22no—
244C4×C53—42no—
245C4×C53—44no—
246C4×C53—44no—
247Dic53—82no—
248Dic53—82no—
249C2×#543D5311yescharacteristic
250#1872—532no—
251C2×#328—532no—
252C2×#323—531no—
253#1439—531no—
254C2×C53—21no—
255C2×C53—21no—
256C2×C53—21no—
257C2×C53—21no—
258C2×C53—44no—
259C2×C53#159611yescharacteristic
260C2×C53#160211yescharacteristic
261C2×C53#160811yescharacteristic
262C2×C53#161411yescharacteristic
263C2×C53#178711yescharacteristic
264C2×C53#179011yescharacteristic
265C2×C53#179211yescharacteristic
266#543—24no—
267#543—22no—
268#543Dic5314yescomplemented
269C2×C22⋊Q8—22no—
270C2×C22⋊Q8—21no—
271C2×C22⋊Q8—22no—
272C2×C22⋊Q8C2×D5312yes—
273C22×C4⋊C4—531no—
274C22×C22⋊C4—21no—
275C22×C22⋊C4—1061no—
276C22×C22⋊C4C2×D5311yescharacteristic
277C2×#103—1062no—
278C2×C22⋊C8—532no—
279C2×#95—531no—
280C2×#95—1061no—
281#340—1062no—
282#328—1064no—
283#323—1062no—
284C53#840511yescharacteristic, complemented, Sylow
285C25C22⋊D5311yescharacteristic
286C22×Q8—21no—
287C22×Q8C22⋊D5312yes—
288C23×C4—21no—
289C23×C4—21no—
290C23×C4—1061no—
291C23×C4—1061no—
292C23×C4C22⋊D5311yescharacteristic
293C23×C4C22×D5311yescharacteristic
294C22×C8—1062no—
295C22⋊Q8—28no—
296C22⋊Q8—28no—
297C22⋊Q8—24no—
298C22⋊Q8—28no—
299C22⋊Q8—22no—
300C22⋊Q8—22no—
301C22⋊Q8—24no—
302C22⋊Q8—28no—
303C22⋊Q8C2×Dic5314yes—
304C22⋊Q8C22⋊D5314yes—
305C2×C4⋊C4—21no—
306C2×C4⋊C4—22no—
307C2×C4⋊C4—22no—
308C2×C4⋊C4—1062no—
309C2×C4⋊C4—2122no—
310C2×C4⋊C4C22⋊D5312yes—
311C2×C22⋊C4—22no—
312C2×C22⋊C4—24no—
313C2×C22⋊C4—22no—
314C2×C22⋊C4—24no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—2122no—
318C2×C22⋊C4—2122no—
319C2×C22⋊C4—2122no—
320C2×C22⋊C4C2×Dic5312yes—
321C2×C22⋊C4C22⋊D5312yes—
322#103—2124no—
323C22⋊C8—1062no—
324C22⋊C8—1062no—
325#95—1062no—
326#95—2122no—
327C24—21no—
328C24—21no—
329C24—21no—
330C24—21no—
331C24—41no—
332C24—41no—
333C24—41no—
334C24—41no—
335C24#1171992412yes—
336C24#1171992512yes—
337C24C22⋊Dic5312yes—
338C24C2×C22⋊D5311yescharacteristic
339C2×Q8—24no—
340C2×Q8—22no—
341C2×Q8—42no—
342C2×Q8—42no—
343C2×Q8—81no—
344C2×Q8—81no—
345C2×Q8C22⋊Dic5314yescomplemented
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—22no—
349C22×C4—24no—
350C22×C4—24no—
351C22×C4—21no—
352C22×C4—41no—
353C22×C4—41no—
354C22×C4—41no—
355C22×C4—41no—
356C22×C4—81no—
357C22×C4—81no—
358C22×C4—81no—
359C22×C4—81no—
360C22×C4—1062no—
361C22×C4—1062no—
362C22×C4—2121no—
363C22×C4—2121no—
364C22×C4—2121no—
365C22×C4—2121no—
366C22×C4—2121no—
367C22×C4—2121no—
368C22×C4—2121no—
369C22×C4—2121no—
370C22×C4—2121no—
371C22×C4—2121no—
372C22×C4—2121no—
373C22×C4—2121no—
374C22×C4C22⋊Dic5312yes—
375C22×C4C22×Dic5311yescharacteristic
376C22×C4C2×C22⋊D5312yes—
377C22×C4C2×C22⋊D5312yes—
378C22×C4C2×C22⋊D5311yescharacteristic
379C2×C8—2122no—
380C2×C8—2122no—
381C2×C8—2122no—
382C4⋊C4—24no—
383C4⋊C4—28no—
384C4⋊C4—24no—
385C4⋊C4—24no—
386C4⋊C4—22no—
387C4⋊C4—1064no—
388C4⋊C4—2122no—
389C4⋊C4C22⋊Dic5314yescomplemented
390C22⋊C4—28no—
391C22⋊C4—24no—
392C22⋊C4—24no—
393C22⋊C4—24no—
394C22⋊C4—24no—
395C22⋊C4—2128no—
396C23—21no—
397C23—21no—
398C23—21no—
399C23—21no—
400C23—22no—
401C23—22no—
402C23—22no—
403C23—22no—
404C23—22no—
405C23—22no—
406C23—42no—
407C23—42no—
408C23—42no—
409C23—42no—
410C23—42no—
411C23—42no—
412C23—41no—
413C23—41no—
414C23—41no—
415C23—41no—
416C23—81no—
417C23—81no—
418C23—81no—
419C23—81no—
420C23—81no—
421C23—81no—
422C23#1313314114yescomplemented
423C23C2×C22⋊Dic5311yescharacteristic
424C23#1313318611yescharacteristic
425C23#1313318611yescharacteristic
426C23C2×#1171992411yescharacteristic
427C23C2×#1171992511yescharacteristic
428C23#1313324811yescharacteristic
429C23#1313324811yescharacteristic
430Q8—44no—
431Q8—82no—
432Q8—84no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—24no—
436C2×C4—22no—
437C2×C4—24no—
438C2×C4—41no—
439C2×C4—41no—
440C2×C4—42no—
441C2×C4—41no—
442C2×C4—42no—
443C2×C4—41no—
444C2×C4—42no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—81no—
449C2×C4—84no—
450C2×C4—84no—
451C2×C4—4241no—
452C2×C4—4241no—
453C2×C4—4241no—
454C2×C4—4241no—
455C2×C4—4241no—
456C2×C4—4241no—
457C2×C4—4241no—
458C2×C4—4241no—
459C2×C4—2122no—
460C2×C4—2122no—
461C2×C4—2122no—
462C2×C4—2122no—
463C2×C4—2122no—
464C2×C4—2122no—
465C2×C4C2×C22⋊Dic5312yes—
466C2×C4#1313318612yes—
467C8—4242no—
468C22—21no—
469C22—21no—
470C22—21no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—21no—
479C22—21no—
480C22—24no—
481C22—44no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—42no—
486C22—42no—
487C22—42no—
488C22—82no—
489C22—82no—
490C22—82no—
491C22#1681047911yescharacteristic
492C22C2×#1313314111yescharacteristic
493C22#1681063211yescharacteristic
494C22#1681076211yescharacteristic
495C22#1681076211yescharacteristic
496C22#1681077911yescharacteristic
497C22#1681077911yescharacteristic
498C22#1681079111yescharacteristic
499C22#1681080111yescharacteristic
500C22#1681081011yescharacteristic
501C22#1681081311yescharacteristic
502C4—22no—
503C4—22no—
504C4—44no—
505C4—42no—
506C4—44no—
507C4—4242no—
508C4—4242no—
509C2—21no—
510C2—21no—
511C2—21no—
512C2—21no—
513C2—44no—
514C2#2027363811yescharacteristic
515C2#2027367211yescharacteristic
516C2#2027370611yescharacteristic
517C2#2027374011yescharacteristic
518C2#2027465011yescharacteristic
519C2#2027466311yescharacteristic
520C2#2027466911yescharacteristic
subgroupG/H
centerC23#13133186
commutatorC22×C4×C53C22×C4
FrattiniC23×C4C22×D53
FittingC2×C53×#543C2
socleC23×C53#120
prime radicalC25×C53D4
cosocleC23×C4×C53C23
hypercenterC2×#543D53
hypocenterC53#8405
LCSG ▷ C22×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#543

Complements

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

NH ≅ G/Ntype
1. #20275023C2semidirect
2. C2×#16810360C2semidirect
17. C53×#543C4semidirect
84. C2×Q8×C53C22⋊C4semidirect
123. C4⋊C4×C53C22⋊C4semidirect
161. C23×C53#103semidirect
268. #543Dic53semidirect
284. C53#8405semidirect
345. C2×Q8C22⋊Dic53semidirect
389. C4⋊C4C22⋊Dic53semidirect
422. C23#13133141semidirect

Maximal subgroups

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

subgroupindex
#202750232
C2×#168103602
C2×#168103472
#202729602
C2×C53×#5432
#840553