Finite Group Databasebeta

← #27271050

Subgroups — #27271050

Order 13568.

678 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810347C211yesmaximal, characteristic
2#20272937C211yesmaximal, characteristic
3C2×C53×#323C211yesmaximal, characteristic
4C24×Dic53C411yescharacteristic
5C22×C22⋊Dic53C411yescharacteristic
6C2×C22⋊#88322—21no—
7C2×#13133124—21no—
8C2×#13133124—21no—
9#16810347—21no—
10#16810347C412yescomplemented
11C22×C22⋊C4×C53C2211yescharacteristic
12C2×C22⋊C8×C53—21no—
13C53×#323—21no—
14C53×#323C412yescomplemented
15C23×Dic53—21no—
16C23×Dic53—21no—
17C23×Dic53—21no—
18C23×Dic53—42no—
19C23×Dic53—42no—
20C23×Dic53—41no—
21C23×Dic53—41no—
22C23×Dic53C812yescomplemented
23C22×#88322—41no—
24C2×C22⋊Dic53—21no—
25C2×C22⋊Dic53—41no—
26C2×C22⋊Dic53—41no—
27C2×C22⋊Dic53C812yescomplemented
28C22⋊#88322—22no—
29C22⋊#88322—41no—
30#13133124—42no—
31#13133124—42no—
32C25×C53C2×C411yescharacteristic
33C23×C4×C53D411yescharacteristic
34C23×C4×C53D411yescharacteristic
35C22×C8×C53—41no—
36C2×C22⋊C4×C53—41no—
37C2×C22⋊C4×C53—41no—
38C2×C22⋊C4×C53C2×C411yescharacteristic
39C2×C22⋊C4×C53C2×C411yescharacteristic
40C2×C22⋊C4×C53D411yescharacteristic
41C2×C22⋊C4×C53Q811yescharacteristic
42C22⋊C8×C53—22no—
43C22⋊C8×C53—41no—
44C24×C53—21no—
45C24×C53—21no—
46C24×C53—21no—
47C24×C53—21no—
48C24×C53—41no—
49C24×C53—41no—
50C24×C53—41no—
51C24×C53—41no—
52C24×C53C4211yescharacteristic
53C24×C53C22⋊C411yescharacteristic
54C24×C53C4⋊C411yescharacteristic
55C24×C53C2×C811yescharacteristic
56C24×C53C2×C811yescharacteristic
57C24×C53M1611yescharacteristic
58C24×C53M1611yescharacteristic
59C22×C4×C53—21no—
60C22×C4×C53—21no—
61C22×C4×C53—21no—
62C22×C4×C53—21no—
63C22×C4×C53—41no—
64C22×C4×C53—41no—
65C22×C4×C53—41no—
66C22×C4×C53—41no—
67C22×C4×C53C22⋊C411yescharacteristic
68C22×C4×C53C22⋊C411yescharacteristic
69C22×C4×C53C4⋊C411yescharacteristic
70C22×C4×C53C4⋊C411yescharacteristic
71C22×Dic53—22no—
72C22×Dic53—22no—
73C22×Dic53—22no—
74C22×Dic53—22no—
75C22×Dic53—44no—
76C22×Dic53—44no—
77C22×Dic53—41no—
78C22×Dic53—41no—
79C22×Dic53—42no—
80C22×Dic53—48no—
81C22×Dic53—42no—
82C22×Dic53—42no—
83C22×Dic53—41no—
84C22×Dic53—41no—
85C22×Dic53—42no—
86C22×Dic53—42no—
87C22×Dic53—42no—
88C22×Dic53—41no—
89C22×Dic53—41no—
90C22×Dic53—41no—
91C22×Dic53—41no—
92C22×Dic53—41no—
93C22×Dic53—41no—
94C2×C8×C53—42no—
95C2×C8×C53—81no—
96C2×C8×C53—81no—
97C22⋊C4×C53—41no—
98C22⋊C4×C53—41no—
99C22⋊C4×C53—41no—
100C22⋊C4×C53—41no—
101C22⋊Dic53—42no—
102C22⋊Dic53—42no—
103C2×#88322—42no—
104C2×#88322—81no—
105C2×#88322—81no—
106C23×C53—21no—
107C23×C53—21no—
108C23×C53—21no—
109C23×C53—21no—
110C23×C53—21no—
111C23×C53—21no—
112C23×C53—21no—
113C23×C53—21no—
114C23×C53—21no—
115C23×C53—21no—
116C23×C53—21no—
117C23×C53—21no—
118C23×C53—21no—
119C23×C53—21no—
120C23×C53—21no—
121C23×C53—21no—
122C23×C53—41no—
123C23×C53—41no—
124C23×C53—41no—
125C23×C53—41no—
126C23×C53—41no—
127C23×C53—41no—
128C23×C53—41no—
129C23×C53—41no—
130C23×C53—41no—
131C23×C53—41no—
132C23×C53—41no—
133C23×C53—41no—
134C23×C53—42no—
135C23×C53—42no—
136C23×C53—42no—
137C23×C53—42no—
138C23×C53—42no—
139C23×C53—42no—
140C23×C53—41no—
141C23×C53—41no—
142C23×C53—41no—
143C23×C53—41no—
144C23×C53#9511yescharacteristic
145C23×C53C4×C811yescharacteristic
146C23×C53#9711yescharacteristic
147C23×C53C22⋊C811yescharacteristic
148C23×C53C22⋊C811yescharacteristic
149C23×C53#9911yescharacteristic
150C23×C53#9911yescharacteristic
151C23×C53#10011yescharacteristic
152C23×C53#10111yescharacteristic
153C23×C53C4⋊C811yescharacteristic
154C23×C53C4⋊C811yescharacteristic
155C2×C4×C53—21no—
156C2×C4×C53—21no—
157C2×C4×C53—21no—
158C2×C4×C53—21no—
159C2×C4×C53—41no—
160C2×C4×C53—41no—
161C2×C4×C53—41no—
162C2×C4×C53—41no—
163C2×C4×C53—41no—
164C2×C4×C53—41no—
165C2×C4×C53—41no—
166C2×C4×C53—41no—
167C2×C4×C53—81no—
168C2×C4×C53—81no—
169C2×Dic53—24no—
170C2×Dic53—42no—
171C2×Dic53—42no—
172C2×Dic53—42no—
173C2×Dic53—44no—
174C2×Dic53—44no—
175C2×Dic53—44no—
176C2×Dic53—42no—
177C2×Dic53—42no—
178C2×Dic53—42no—
179C2×Dic53—44no—
180C2×Dic53—42no—
181C2×Dic53—42no—
182C2×Dic53—42no—
183C2×Dic53—81no—
184C2×Dic53—81no—
185C2×Dic53—81no—
186C2×Dic53—81no—
187C8×C53—82no—
188#88322—82no—
189#7187—531nomaximal, Sylow
190C22×C53—21no—
191C22×C53—21no—
192C22×C53—21no—
193C22×C53—21no—
194C22×C53—21no—
195C22×C53—21no—
196C22×C53—21no—
197C22×C53—21no—
198C22×C53—21no—
199C22×C53—21no—
200C22×C53—21no—
201C22×C53—21no—
202C22×C53—21no—
203C22×C53—21no—
204C22×C53—21no—
205C22×C53—21no—
206C22×C53—41no—
207C22×C53—41no—
208C22×C53—41no—
209C22×C53—41no—
210C22×C53—42no—
211C22×C53—42no—
212C22×C53—42no—
213C22×C53—42no—
214C22×C53—42no—
215C22×C53—42no—
216C22×C53—41no—
217C22×C53—41no—
218C22×C53—41no—
219C22×C53—41no—
220C22×C53—41no—
221C22×C53—41no—
222C22×C53—41no—
223C22×C53—41no—
224C22×C53—41no—
225C22×C53—41no—
226C22×C53—41no—
227C22×C53—41no—
228C22×C53#32311yescharacteristic
229C22×C53#32311yescharacteristic
230C22×C53#32411yescharacteristic
231C22×C53#32411yescharacteristic
232C22×C53#33611yescharacteristic
233C22×C53#34211yescharacteristic
234C22×C53#34311yescharacteristic
235C22×C53AΣL(1,𝔽16)11yescharacteristic
236C22×C53AΣL(1,𝔽16)11yescharacteristic
237C22×C53#35211yescharacteristic
238C22×C53#35211yescharacteristic
239C4×C53—41no—
240C4×C53—41no—
241C4×C53—41no—
242C4×C53—41no—
243Dic53—44no—
244Dic53—82no—
245C2×#323—531no—
246C2×#323D5311yescharacteristic
247C22⋊#95—531no—
248C2×C53—21no—
249C2×C53—21no—
250C2×C53—21no—
251C2×C53—21no—
252C2×C53—41no—
253C2×C53—41no—
254C2×C53—41no—
255C2×C53—41no—
256C2×C53#128011yescharacteristic
257C2×C53#131611yescharacteristic
258C2×C53#131711yescharacteristic
259C2×C53#132611yescharacteristic
260C2×C53#132611yescharacteristic
261C2×C53#139011yescharacteristic
262C2×C53#139011yescharacteristic
263C24×C4—531no—
264C22×C22⋊C4—531no—
265C22×C22⋊C4C2×D5311yescharacteristic
266C2×C22⋊C8—21no—
267C2×C22⋊C8—1061no—
268C2×#95—1061no—
269C2×#95—1061no—
270#323—21no—
271#323—532no—
272#323—1061no—
273#323Dic5312yescomplemented
274C53#718711yescharacteristic, complemented, Sylow
275C25C4×D5311yescharacteristic
276C23×C4—532no—
277C23×C4—1061no—
278C23×C4—1061no—
279C23×C4—1061no—
280C23×C4—2122no—
281C23×C4—2122no—
282C23×C4—2121no—
283C23×C4—2121no—
284C23×C4D21211yescharacteristic
285C23×C4C22⋊D5311yescharacteristic
286C22×C8—41no—
287C22×C8—2121no—
288C2×C22⋊C4—41no—
289C2×C22⋊C4—41no—
290C2×C22⋊C4—532no—
291C2×C22⋊C4—1061no—
292C2×C22⋊C4—2121no—
293C2×C22⋊C4—2121no—
294C2×C22⋊C4Dic10611yescharacteristic
295C2×C22⋊C4C4×D5311yescharacteristic
296C2×C22⋊C4C2×Dic5311yescharacteristic
297C2×C22⋊C4C22⋊D5311yescharacteristic
298C22⋊C8—22no—
299C22⋊C8—41no—
300C22⋊C8—1062no—
301C22⋊C8—2121no—
302#95—2122no—
303#95—2122no—
304C24—21no—
305C24—21no—
306C24—21no—
307C24—21no—
308C24—41no—
309C24—41no—
310C24—41no—
311C24—41no—
312C24C8×D5311yescharacteristic
313C24C8×D5311yescharacteristic
314C24#1171991211yescharacteristic
315C24#1171991211yescharacteristic
316C24C4×Dic5311yescharacteristic
317C24#1171991911yescharacteristic
318C24#1171992111yescharacteristic
319C22×C4—21no—
320C22×C4—21no—
321C22×C4—21no—
322C22×C4—21no—
323C22×C4—41no—
324C22×C4—41no—
325C22×C4—41no—
326C22×C4—41no—
327C22×C4—1062no—
328C22×C4—1062no—
329C22×C4—1062no—
330C22×C4—1062no—
331C22×C4—2124no—
332C22×C4—2124no—
333C22×C4—2121no—
334C22×C4—2121no—
335C22×C4—2121no—
336C22×C4—2121no—
337C22×C4—2121no—
338C22×C4—2121no—
339C22×C4—2121no—
340C22×C4—2121no—
341C22×C4—2122no—
342C22×C4—2128no—
343C22×C4—2122no—
344C22×C4—2122no—
345C22×C4—2121no—
346C22×C4—2121no—
347C22×C4—2122no—
348C22×C4—2122no—
349C22×C4—2122no—
350C22×C4#1171991911yescharacteristic
351C22×C4C4⋊Dic5311yescharacteristic
352C22×C4#1171992111yescharacteristic
353C22×C4C22⋊Dic5311yescharacteristic
354C2×C8—42no—
355C2×C8—81no—
356C2×C8—81no—
357C2×C8—4241no—
358C2×C8—4241no—
359C2×C8—2122no—
360C22⋊C4—41no—
361C22⋊C4—41no—
362C22⋊C4—41no—
363C22⋊C4—41no—
364C22⋊C4—2122no—
365C22⋊C4—2122no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—21no—
375C23—21no—
376C23—21no—
377C23—21no—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—41no—
383C23—41no—
384C23—41no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23—41no—
392C23—41no—
393C23—41no—
394C23—42no—
395C23—42no—
396C23—42no—
397C23—42no—
398C23—42no—
399C23—42no—
400C23—41no—
401C23—41no—
402C23—41no—
403C23—41no—
404C23#1313312411yescharacteristic
405C23C8×Dic5311yescharacteristic
406C23#1313312711yescharacteristic
407C23#1313312911yescharacteristic
408C23#1313312911yescharacteristic
409C23#1313313111yescharacteristic
410C23#1313313111yescharacteristic
411C23#1313313311yescharacteristic
412C23#1313313511yescharacteristic
413C23#1313314611yescharacteristic
414C23#1313314611yescharacteristic
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—21no—
418C2×C4—21no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—41no—
424C2×C4—41no—
425C2×C4—41no—
426C2×C4—41no—
427C2×C4—81no—
428C2×C4—81no—
429C2×C4—1064no—
430C2×C4—4241no—
431C2×C4—4241no—
432C2×C4—4241no—
433C2×C4—4241no—
434C2×C4—2122no—
435C2×C4—2122no—
436C2×C4—2122no—
437C2×C4—2122no—
438C2×C4—2122no—
439C2×C4—2122no—
440C2×C4—2124no—
441C2×C4—2124no—
442C2×C4—2124no—
443C2×C4—2122no—
444C2×C4—2122no—
445C2×C4—2122no—
446C2×C4—2124no—
447C8—82no—
448C8—4242no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—21no—
464C22—21no—
465C22—41no—
466C22—41no—
467C22—41no—
468C22—41no—
469C22—42no—
470C22—42no—
471C22—42no—
472C22—42no—
473C22—42no—
474C22—42no—
475C22—41no—
476C22—41no—
477C22—41no—
478C22—41no—
479C22—41no—
480C22—41no—
481C22—41no—
482C22—41no—
483C22—41no—
484C22—41no—
485C22—41no—
486C22—41no—
487C22#1681034611yescharacteristic
488C22#1681034611yescharacteristic
489C22#1681034811yescharacteristic
490C22#1681034811yescharacteristic
491C22#1681037711yescharacteristic
492C22#1681038911yescharacteristic
493C22#1681039111yescharacteristic
494C22#1681040911yescharacteristic
495C22#1681041111yescharacteristic
496C22#1681041211yescharacteristic
497C22#1681041411yescharacteristic
498C4—41no—
499C4—41no—
500C4—41no—
501C4—41no—
502C4—4242no—
503C4—2124no—
504C2—21no—
505C2—21no—
506C2—21no—
507C2—21no—
508C2—41no—
509C2—41no—
510C2—41no—
511C2—41no—
512C2#2027255711yescharacteristic
513C2#2027264211yescharacteristic
514C2#2027264411yescharacteristic
515C2#2027266211yescharacteristic
516C2#2027266411yescharacteristic
517C2#2027283211yescharacteristic
518C2#2027283411yescharacteristic
subgroupG/H
centerC23#13133131
commutatorC23×C53C4×C8
FrattiniC22×C22⋊C4C2×D53
FittingC2×C53×#323C2
socleC23×C53#99
prime radicalC25×C53C2×C4
cosocleC22×C22⋊C4×C53C22
hypercenterC2×#323D53
hypocenterC53#7187
LCSG ▷ C23×C53 ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C24 ◁ C22×C22⋊C4 ◁ C2×#323

Complements

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

NH ≅ G/Ntype
10. #16810347C4semidirect
14. C53×#323C4semidirect
22. C23×Dic53C8semidirect
27. C2×C22⋊Dic53C8semidirect
273. #323Dic53semidirect
274. C53#7187semidirect

Maximal subgroups

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

subgroupindex
C2×#168103472
#202729372
C2×C53×#3232
#718753