Finite Group Databasebeta

← #27277550

Subgroups — #27277550

Order 13568.

622 conjugacy classes of subgroups.

The subgroup lattice has 502 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277067C211yesmaximal, characteristic
2C2×#16810346C211yesmaximal, characteristic
3C2×#16810346C211yesmaximal, characteristic
4#20272563C211yesmaximal, characteristic, complemented
5#20272562C211yesmaximal, characteristic, complemented
6#20272561C211yesmaximal, characteristic, complemented
7C53×#1282C211yesmaximal, characteristic, complemented
8C22×#11719921C2211yescharacteristic
9C2×C22⋊C4×D53C411yescharacteristic
10C2×C22⋊C4×D53C411yescharacteristic
11C2×C22⋊#88322C2211yescharacteristic
12C2×C22⋊#88322C2211yescharacteristic
13C22⋊C4×Dic53C411yescharacteristic
14C22⋊C4×Dic53C411yescharacteristic
15C4×#11719921—21no—
16C4×#11719921—21no—
17C4×C22⋊Dic53C2211yescharacteristic
18#16810470—21no—
19#16810470—21no—
20#16810346—21no—
21#16810346—21no—
22#16810346—21no—
23#16810346—21no—
24C2×C22⋊C8×C53C2211yescharacteristic
25C2×C22⋊C8×C53C2211yescharacteristic
26C4×C22⋊C4×C53C2211yescharacteristic
27C24×D53C2×C411yescharacteristic
28C23×Dic53C2×C411yescharacteristic
29C22×C4×D53—21no—
30C22×C4×D53—21no—
31C22×#88322—21no—
32C22×#88322—21no—
33C2×C22⋊Dic53C2×C411yescharacteristic
34C2×C22⋊Dic53C2×C411yescharacteristic
35C2×#11719921—21no—
36C2×#11719921—21no—
37C2×#11719921—21no—
38C2×#11719921—21no—
39C2×#11719921—21no—
40C2×#11719921—21no—
41C2×#11719921D411yescharacteristic
42C2×#11719921D411yescharacteristic
43C2×#11719921D411yescharacteristic
44C2×#11719921D411yescharacteristic
45C22⋊C4×D53—22no—
46C22⋊C4×D53—22no—
47C22⋊C4×D53—22no—
48C22⋊C4×D53C814yescomplemented
49C2×C4×Dic53—21no—
50C2×C4×Dic53—21no—
51C2×C4×Dic53—21no—
52C22⋊#88322—21no—
53C22⋊#88322—21no—
54C22⋊#88322—21no—
55C22⋊#88322—21no—
56#13133124—21no—
57#13133124—21no—
58#13133124—21no—
59C23×C4×C53C2311yescharacteristic
60C22×C8×C53—21no—
61C22×C8×C53—21no—
62C2×C22⋊C4×C53C2×C411yescharacteristic
63C2×C22⋊C4×C53C2×C411yescharacteristic
64C2×C42×C53—21no—
65C22⋊C8×C53—21no—
66C22⋊C8×C53—21no—
67C22⋊C8×C53—21no—
68C22⋊C8×C53—21no—
69#95×C53—21no—
70C24×C53C22×C411yescharacteristic
71C23×D53—21no—
72C23×D53—21no—
73C23×D53—21no—
74C23×D53—21no—
75C23×D53—81no—
76C23×D53—81no—
77C23×D53C22⋊C411yescharacteristic
78C23×D53C22⋊C411yescharacteristic
79C23×D53C2×C812yes—
80C23×D53M1612yes—
81C22×C4×C53—21no—
82C22×C4×C53—21no—
83C22×C4×C53—81no—
84C22×C4×C53C2×D411yescharacteristic
85C22×C4×C53C2×D411yescharacteristic
86C22×C4×C53#4111yescharacteristic
87C22×C4×C53#4111yescharacteristic
88C22×C4×C53#4111yescharacteristic
89C22×C4×C53#4111yescharacteristic
90C22×Dic53—21no—
91C22×Dic53—21no—
92C22×Dic53—21no—
93C22×Dic53—21no—
94C22×Dic53—81no—
95C22×Dic53C22⋊C411yescharacteristic
96C22×Dic53C22⋊C411yescharacteristic
97C2×C4×D53—24no—
98C2×C4×D53—24no—
99C2×C4×D53—81no—
100C2×C4×D53—81no—
101C2×C4×D53—81no—
102C2×C4×D53—81no—
103C2×C8×C53—41no—
104C2×C8×C53—41no—
105C2×C8×C53—81no—
106C2×C8×C53—81no—
107C22⋊C4×C53—21no—
108C22⋊C4×C53—21no—
109C22⋊C4×C53C2×C812yes—
110C22⋊C4×C53M1612yes—
111C42×C53—81no—
112C22⋊Dic53—21no—
113C22⋊Dic53—21no—
114C22⋊Dic53C2×C812yes—
115C22⋊Dic53M1612yes—
116#11719921—24no—
117#11719921—24no—
118#11719921—21no—
119#11719921—21no—
120#11719921—21no—
121#11719921—21no—
122#11719921—21no—
123#11719921—21no—
124#11719921—21no—
125#11719921—21no—
126C4×Dic53—81no—
127C4×Dic53—81no—
128C4×Dic53—81no—
129C2×#88322—41no—
130C2×#88322—41no—
131C2×#88322—81no—
132C2×#88322—81no—
133C23×C53—41no—
134C23×C53—41no—
135C23×C53C2×C22⋊C411yescharacteristic
136C23×C53#11711yescharacteristic
137C23×C53#11711yescharacteristic
138C23×C53C22×C811yescharacteristic
139C23×C53C2×M1611yescharacteristic
140C23×C53C2×M1611yescharacteristic
141C23×C53C2×M1611yescharacteristic
142C22×D53—21no—
143C22×D53—21no—
144C22×D53—21no—
145C22×D53—21no—
146C22×D53—22no—
147C22×D53—22no—
148C22×D53—22no—
149C22×D53—22no—
150C22×D53—41no—
151C22×D53—41no—
152C22×D53—41no—
153C22×D53—41no—
154C22×D53—81no—
155C22×D53—81no—
156C22×D53—81no—
157C22×D53—81no—
158C22×D53—82no—
159C22×D53—82no—
160C22×D53—82no—
161C22×D53—82no—
162C22×D53C22⋊C812yes—
163C22×D53C22⋊C812yes—
164C2×C4×C53—22no—
165C2×C4×C53—22no—
166C2×C4×C53—41no—
167C2×C4×C53—41no—
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—41no—
175C2×C4×C53—41no—
176C2×C4×C53#12311yescharacteristic
177C2×C4×C53#12311yescharacteristic
178C2×C4×C53#12311yescharacteristic
179C2×C4×C53#12311yescharacteristic
180C2×Dic53—21no—
181C2×Dic53—21no—
182C2×Dic53—22no—
183C2×Dic53—22no—
184C2×Dic53—41no—
185C2×Dic53—41no—
186C2×Dic53—41no—
187C2×Dic53—41no—
188C2×Dic53—41no—
189C2×Dic53—41no—
190C2×Dic53—81no—
191C2×Dic53—81no—
192C4×D53—84no—
193C4×D53—84no—
194C8×C53—81no—
195C8×C53—81no—
196#88322—81no—
197#88322—81no—
198#8734—531nomaximal, Sylow
199C22×C53—21no—
200C22×C53—21no—
201C22×C53—21no—
202C22×C53—21no—
203C22×C53—21no—
204C22×C53—21no—
205C22×C53—41no—
206C22×C53—41no—
207C22×C53—41no—
208C22×C53—41no—
209C22×C53#38111yescharacteristic
210C22×C53C2×C22⋊C811yescharacteristic
211C22×C53#40711yescharacteristic
212C22×C53#43111yescharacteristic
213C22×C53#43111yescharacteristic
214C22×C53#43211yescharacteristic
215C22×C53#43211yescharacteristic
216C2×D53—22no—
217C2×D53—22no—
218C2×D53—42no—
219C2×D53—42no—
220C2×D53—41no—
221C2×D53—41no—
222C2×D53—41no—
223C2×D53—41no—
224C2×D53—42no—
225C2×D53—42no—
226C2×D53—82no—
227C2×D53—82no—
228C2×D53—82no—
229C2×D53—82no—
230C4×C53—21no—
231C4×C53—21no—
232C4×C53—21no—
233C4×C53—21no—
234C4×C53—42no—
235C4×C53—42no—
236Dic53—42no—
237Dic53—42no—
238Dic53—41no—
239Dic53—41no—
240#2273—531no—
241C2×#323—531no—
242C2×#323—531no—
243#1282—531no—
244#1282—531no—
245#1282—531no—
246#1282D5311yescharacteristic, complemented
247C2×C53—21no—
248C2×C53—21no—
249C2×C53—21no—
250C2×C53—21no—
251C2×C53—21no—
252C2×C53—21no—
253C2×C53#181711yescharacteristic
254C2×C53#182811yescharacteristic
255C2×C53#182911yescharacteristic
256D53—42no—
257D53—42no—
258C22×C22⋊C4—531no—
259C22×C22⋊C4—531no—
260C22×C22⋊C4—531no—
261C2×C22⋊C8—531no—
262C2×C22⋊C8—531no—
263C2×C22⋊C8C2×D5311yescharacteristic
264C2×C22⋊C8C2×D5311yescharacteristic
265C4×C22⋊C4—531no—
266C4×C22⋊C4—531no—
267C4×C22⋊C4—531no—
268C4×C22⋊C4—1061no—
269C4×C22⋊C4—1061no—
270C4×C22⋊C4—1061no—
271C4×C22⋊C4—1061no—
272C4×C22⋊C4C2×D5311yescharacteristic
273#323—1061no—
274#323—1061no—
275#323—1061no—
276#323—1061no—
277C53#873411yescharacteristic, complemented, Sylow
278C25—531no—
279C23×C4—531no—
280C23×C4—1061no—
281C23×C4—1061no—
282C23×C4C22×D5311yescharacteristic
283C22×C8—21no—
284C22×C8—21no—
285C22×C8—1061no—
286C22×C8—1061no—
287C2×C22⋊C4—531no—
288C2×C22⋊C4—531no—
289C2×C22⋊C4—534no—
290C2×C22⋊C4—531no—
291C2×C22⋊C4—531no—
292C2×C22⋊C4—531no—
293C2×C22⋊C4—531no—
294C2×C22⋊C4—1062no—
295C2×C22⋊C4—1062no—
296C2×C22⋊C4—1061no—
297C2×C22⋊C4—1061no—
298C2×C22⋊C4—1061no—
299C2×C22⋊C4—1061no—
300C2×C22⋊C4—1061no—
301C2×C22⋊C4—1061no—
302C2×C22⋊C4—1062no—
303C2×C22⋊C4C4×D5311yescharacteristic
304C2×C22⋊C4C4×D5311yescharacteristic
305C2×C42—21no—
306C2×C42—1061no—
307C2×C42—1061no—
308C2×C42—1061no—
309C22⋊C8—21no—
310C22⋊C8—21no—
311C22⋊C8—21no—
312C22⋊C8—21no—
313C22⋊C8—1061no—
314C22⋊C8—1061no—
315C22⋊C8—1061no—
316C22⋊C8—1061no—
317#95—21no—
318#95—1061no—
319#95—1061no—
320#95—1061no—
321C24—532no—
322C24—532no—
323C24—531no—
324C24—531no—
325C24—1061no—
326C24—1061no—
327C24—1061no—
328C24—1061no—
329C24—4241no—
330C24—4241no—
331C24C2×C4×D5311yescharacteristic
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—81no—
335C22×C4—531no—
336C22×C4—531no—
337C22×C4—1064no—
338C22×C4—1064no—
339C22×C4—1061no—
340C22×C4—1061no—
341C22×C4—1061no—
342C22×C4—1061no—
343C22×C4—4241no—
344C22×C4—4241no—
345C22×C4—4241no—
346C22×C4—4241no—
347C22×C4—4241no—
348C22×C4#1171994511yescharacteristic
349C22×C4#1171994511yescharacteristic
350C22×C4D4×D5311yescharacteristic
351C22×C4D4×D5311yescharacteristic
352C22×C4#1171994711yescharacteristic
353C22×C4#1171994911yescharacteristic
354C2×C8—41no—
355C2×C8—41no—
356C2×C8—81no—
357C2×C8—81no—
358C2×C8—4241no—
359C2×C8—4241no—
360C2×C8—2121no—
361C2×C8—2121no—
362C22⋊C4—21no—
363C22⋊C4—21no—
364C22⋊C4—532no—
365C22⋊C4—532no—
366C22⋊C4—1064no—
367C22⋊C4—1064no—
368C22⋊C4—1061no—
369C22⋊C4—1061no—
370C22⋊C4—1061no—
371C22⋊C4—1061no—
372C22⋊C4—1061no—
373C22⋊C4—1061no—
374C22⋊C4—1061no—
375C22⋊C4—1061no—
376C22⋊C4—1061no—
377C22⋊C4—1061no—
378C22⋊C4C8×D5312yes—
379C22⋊C4#1171991212yes—
380C42—81no—
381C42—4241no—
382C42—4241no—
383C42—4241no—
384C23—41no—
385C23—41no—
386C23—532no—
387C23—532no—
388C23—1062no—
389C23—1062no—
390C23—1062no—
391C23—1062no—
392C23—1061no—
393C23—1061no—
394C23—1061no—
395C23—1061no—
396C23—4241no—
397C23—4241no—
398C23—4241no—
399C23—4241no—
400C23—4242no—
401C23—4242no—
402C23—4242no—
403C23—4242no—
404C23—2121no—
405C23—2121no—
406C23—2121no—
407C23—2121no—
408C23C22⋊C4×D5311yescharacteristic
409C23#1313317411yescharacteristic
410C23#1313317511yescharacteristic
411C23C2×C8×D5311yescharacteristic
412C23M16×D5311yescharacteristic
413C23M16×D5311yescharacteristic
414C23C2×#1171991211yescharacteristic
415C2×C4—22no—
416C2×C4—22no—
417C2×C4—41no—
418C2×C4—41no—
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—1062no—
428C2×C4—1062no—
429C2×C4—1061no—
430C2×C4—1061no—
431C2×C4—4244no—
432C2×C4—4244no—
433C2×C4—4241no—
434C2×C4—4241no—
435C2×C4—2121no—
436C2×C4—2121no—
437C2×C4—2121no—
438C2×C4—2121no—
439C2×C4—2121no—
440C2×C4—2121no—
441C2×C4#1313319911yescharacteristic
442C2×C4#1313319911yescharacteristic
443C2×C4#1313320211yescharacteristic
444C2×C4#1313320211yescharacteristic
445C8—81no—
446C8—81no—
447C8—4241no—
448C8—4241no—
449C22—21no—
450C22—21no—
451C22—21no—
452C22—21no—
453C22—21no—
454C22—21no—
455C22—41no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—1062no—
460C22—1062no—
461C22—4242no—
462C22—4242no—
463C22—4242no—
464C22—4242no—
465C22—2122no—
466C22—2122no—
467C22—2122no—
468C22—2122no—
469C22—2121no—
470C22—2121no—
471C22—2121no—
472C22—2121no—
473C22#1681048511yescharacteristic
474C22C22⋊C8×D5311yescharacteristic
475C22#1681058511yescharacteristic
476C22#1681067411yescharacteristic
477C22#1681067511yescharacteristic
478C22#1681067811yescharacteristic
479C22#1681067911yescharacteristic
480C4—21no—
481C4—21no—
482C4—21no—
483C4—21no—
484C4—42no—
485C4—42no—
486C4—2122no—
487C4—2122no—
488C4—2121no—
489C4—2121no—
490C2—21no—
491C2—21no—
492C2—21no—
493C2—21no—
494C2—21no—
495C2—21no—
496C2—2122no—
497C2—2122no—
498C2#2027478211yescharacteristic
499C2#2027482211yescharacteristic
500C2#2027482511yescharacteristic
subgroupG/H
centerC22#16810485
commutatorC23×C53C22×C8
FrattiniC23×C4C22×D53
FittingC53×#1282C2
socleC22×C53#381
prime radicalC24×D53C2×C4
cosocleC23×C4×C53C23
hypercenter#1282D53
hypocenterC53#8734
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1282

Complements

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

NH ≅ G/Ntype
4. #20272563C2semidirect
5. #20272562C2semidirect
6. #20272561C2semidirect
7. C53×#1282C2semidirect
48. C22⋊C4×D53C8semidirect
246. #1282D53semidirect
277. C53#8734semidirect

Maximal subgroups

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

subgroupindex
#202770672
C2×#168103462
#202725632
#202725622
#202725612
C53×#12822
#873453