Finite Group Databasebeta

← #27278781

Subgroups — #27278781

Order 13568.

626 conjugacy classes of subgroups.

The subgroup lattice has 498 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20283737C211yesmaximal, characteristic
2C4×#13133201C211yesmaximal, characteristic
3#20274781C211yesmaximal, characteristic, complemented
4#20274781C211yesmaximal, characteristic, complemented
5#20273031C211yesmaximal, characteristic
6#20272563C211yesmaximal, characteristic, complemented
7C53×#1282C211yesmaximal, characteristic, complemented
8C2×#13133201C411yescharacteristic
9C2×C4×C22⋊D53C2211yescharacteristic
10C22×C4×Dic53C2211yescharacteristic
11C8×C22⋊D53—22no—
12#16810686—22no—
13#16810673—21no—
14C2×C22⋊#88322C2211yescharacteristic
15C2×C22⋊#88322C2211yescharacteristic
16C22⋊C8×D53—21no—
17#16810515—21no—
18#16810508—21no—
19#16810500C411yescharacteristic
20#16810498C411yescharacteristic
21#16810484C411yescharacteristic
22C4×C4⋊Dic53—21no—
23C4×#11719921—21no—
24#16810377—22no—
25#16810377—22no—
26#16810348—21no—
27#16810346—21no—
28C2×C22⋊C8×C53C2211yescharacteristic
29C2×C22⋊C8×C53C2211yescharacteristic
30C4×C22⋊C4×C53C2211yescharacteristic
31C22×C22⋊D53C2×C411yescharacteristic
32C23×Dic53C2×C411yescharacteristic
33C23×Dic53C2×C411yescharacteristic
34C22×C4×D53D411yescharacteristic
35C22×#88322—21no—
36C22×#88322D412yes—
37C2×C8×D53—22no—
38#13133201—22no—
39#13133201C814yescomplemented
40C4×C22⋊D53—21no—
41C4×C22⋊D53—21no—
42C4×C22⋊D53—21no—
43C4×C22⋊D53—21no—
44C2×C4⋊Dic53—21no—
45C2×C4⋊Dic53—21no—
46C2×#11719919D411yescharacteristic
47C2×C22⋊Dic53C2×C411yescharacteristic
48C2×#11719921—21no—
49C2×#11719921—21no—
50C2×#11719921D411yescharacteristic
51C2×C4×Dic53—21no—
52C2×C4×Dic53—21no—
53C2×C4×Dic53—21no—
54C2×C4×Dic53—21no—
55C2×C4×Dic53—21no—
56C2×C4×Dic53—21no—
57C2×C4×Dic53D411yescharacteristic
58#13133146—22no—
59C22⋊#88322—21no—
60C22⋊#88322—21no—
61C22⋊#88322—21no—
62C22⋊#88322—21no—
63#13133129—22no—
64C8×Dic53—22no—
65#13133124—21no—
66#13133124—21no—
67#13133124—21no—
68#13133124—21no—
69C23×C4×C53C2311yescharacteristic
70C22×C8×C53—21no—
71C22×C8×C53D412yes—
72C2×C22⋊C4×C53C2×C411yescharacteristic
73C2×C22⋊C4×C53C2×C411yescharacteristic
74C2×C42×C53—21no—
75C22⋊C8×C53—21no—
76C22⋊C8×C53—21no—
77C22⋊C8×C53—21no—
78C22⋊C8×C53—21no—
79#95×C53—21no—
80C24×C53C22×C411yescharacteristic
81C23×D53C22⋊C411yescharacteristic
82C22×C4×C53—21no—
83C22×C4×C53—21no—
84C22×C4×C53—42no—
85C22×C4×C53C2×D411yescharacteristic
86C22×C4×C53C2×D411yescharacteristic
87C22×C4×C53C2×D411yescharacteristic
88C22×C4×C53C2×D411yescharacteristic
89C22×C4×C53#4111yescharacteristic
90C22×C4×C53#4111yescharacteristic
91C2×C22⋊D53—21no—
92C2×C22⋊D53—21no—
93C2×C22⋊D53—21no—
94C2×C22⋊D53—21no—
95C2×C22⋊D53C2×C812yes—
96C2×C22⋊D53M1612yes—
97C22×Dic53—21no—
98C22×Dic53—21no—
99C22×Dic53—21no—
100C22×Dic53—21no—
101C22×Dic53—21no—
102C22×Dic53—21no—
103C22×Dic53—21no—
104C22×Dic53—21no—
105C22×Dic53—21no—
106C22×Dic53—42no—
107C22×Dic53C22⋊C411yescharacteristic
108C22×Dic53C22⋊C411yescharacteristic
109C22×Dic53C22⋊C411yescharacteristic
110C22×Dic53C2×C812yescomplemented
111C2×C4×D53—21no—
112C2×C4×D53—21no—
113C2×C4×D53—42no—
114C2×C8×C53—22no—
115C2×C8×C53—42no—
116C2×C8×C53—42no—
117C2×C8×C53—41no—
118C22⋊C4×C53—21no—
119C22⋊C4×C53—21no—
120C22⋊C4×C53—21no—
121C22⋊C4×C53C2×C812yescomplemented
122C42×C53—42no—
123C22⋊Dic53—21no—
124C22⋊Dic53—21no—
125#11719921—24no—
126#11719921—21no—
127#11719921—21no—
128#11719921—42no—
129C4⋊Dic53—24no—
130C4⋊Dic53—42no—
131#11719919—21no—
132#11719919—21no—
133C4×Dic53—21no—
134C4×Dic53—21no—
135C4×Dic53—41no—
136C4×Dic53—41no—
137C4×Dic53—41no—
138C4×Dic53—41no—
139C2×#88322—22no—
140C2×#88322—42no—
141C2×#88322—42no—
142C2×#88322—41no—
143C8×D53—82no—
144C23×C53—41no—
145C23×C53—41no—
146C23×C53C2×C22⋊C411yescharacteristic
147C23×C53C4×D411yescharacteristic
148C23×C53C4×D411yescharacteristic
149C23×C53C22×C811yescharacteristic
150C23×C53C2×M1611yescharacteristic
151C23×C53#13111yescharacteristic
152C23×C53#13111yescharacteristic
153C22×D53—21no—
154C22×D53—21no—
155C22×D53—41no—
156C22×D53—41no—
157C22×D53C22⋊C812yes—
158C2×C4×C53—22no—
159C2×C4×C53—22no—
160C2×C4×C53—22no—
161C2×C4×C53—22no—
162C2×C4×C53—41no—
163C2×C4×C53—41no—
164C2×C4×C53—41no—
165C2×C4×C53—41no—
166C2×C4×C53—41no—
167C2×C4×C53—41no—
168C2×C4×C53—41no—
169C2×C4×C53—41no—
170C2×C4×C53#12011yescharacteristic
171C2×C4×C53#12111yescharacteristic
172C2×C4×C53#12111yescharacteristic
173C2×C4×C53#12311yescharacteristic
174C22⋊D53—22no—
175C22⋊D53—22no—
176C22⋊D53—22no—
177C22⋊D53—22no—
178C2×Dic53—21no—
179C2×Dic53—21no—
180C2×Dic53—22no—
181C2×Dic53—22no—
182C2×Dic53—22no—
183C2×Dic53—21no—
184C2×Dic53—21no—
185C2×Dic53—22no—
186C2×Dic53—22no—
187C2×Dic53—22no—
188C2×Dic53—41no—
189C2×Dic53—41no—
190C2×Dic53—41no—
191C2×Dic53—41no—
192C2×Dic53—41no—
193C2×Dic53—41no—
194C2×Dic53—41no—
195C2×Dic53—41no—
196C2×Dic53—41no—
197C2×Dic53—41no—
198C2×Dic53C22⋊C812yes—
199C4×D53—42no—
200C4×D53—81no—
201C8×C53—42no—
202C8×C53—81no—
203#88322—42no—
204#88322—81no—
205#8966—531nomaximal, Sylow
206C22×C53—21no—
207C22×C53—21no—
208C22×C53—21no—
209C22×C53—21no—
210C22×C53—21no—
211C22×C53—21no—
212C22×C53—41no—
213C22×C53—41no—
214C22×C53—41no—
215C22×C53—41no—
216C22×C53#38611yescharacteristic
217C22×C53C2×C22⋊C811yescharacteristic
218C22×C53#40811yescharacteristic
219C22×C53C8×D411yescharacteristic
220C22×C53C8×D411yescharacteristic
221C22×C53#43511yescharacteristic
222C22×C53#43511yescharacteristic
223C2×D53—22no—
224C2×D53—41no—
225C2×D53—41no—
226C2×D53—42no—
227C2×D53—81no—
228C4×C53—21no—
229C4×C53—21no—
230C4×C53—21no—
231C4×C53—42no—
232C4×C53—81no—
233C4×C53#45812yes—
234Dic53—22no—
235Dic53—42no—
236Dic53—41no—
237Dic53—42no—
238Dic53—41no—
239#2972—531no—
240C4×#123—531no—
241#1817—531no—
242#1817—531no—
243#1459—531no—
244#1282—531no—
245#1282D5311yescharacteristic, complemented
246C2×C53—21no—
247C2×C53—21no—
248C2×C53—21no—
249C2×C53—21no—
250C2×C53—21no—
251C2×C53—21no—
252C2×C53#186911yescharacteristic
253C2×C53#189011yescharacteristic
254C2×C53#189111yescharacteristic
255D53—42no—
256C2×#123—531no—
257C2×C4×D4—531no—
258C22×C42—531no—
259C8×D4—1062no—
260C8×D4—1062no—
261#431—1061no—
262C2×C22⋊C8—531no—
263C2×C22⋊C8—531no—
264C2×C22⋊C8—1061no—
265C2×C22⋊C8C2×D5311yescharacteristic
266C2×C22⋊C8C2×D5311yescharacteristic
267#388—1061no—
268#387—1061no—
269#386—531no—
270#385—531no—
271#381—531no—
272C4×C4⋊C4—1061no—
273C4×C22⋊C4—1061no—
274C4×C22⋊C4C2×D5311yescharacteristic
275#336—1062no—
276#336—1062no—
277#324—1061no—
278#323—1061no—
279C53#896611yescharacteristic, complemented, Sylow
280C22×D4—531no—
281C23×C4—531no—
282C23×C4—531no—
283C23×C4—531no—
284C23×C4C22×D5311yescharacteristic
285C22×C8—21no—
286C22×C8—532no—
287C22×C8—1062no—
288C22×C8—1061no—
289C22×C8D21212yes—
290#123—534no—
291#123—1062no—
292C4×D4—1061no—
293C4×D4—1061no—
294C4×D4—1061no—
295C4×D4—1061no—
296C2×C4⋊C4—531no—
297C2×C4⋊C4—1061no—
298C2×C4⋊C4—1061no—
299C2×C22⋊C4—531no—
300C2×C22⋊C4—531no—
301C2×C22⋊C4—1061no—
302C2×C22⋊C4—1061no—
303C2×C22⋊C4C4×D5311yescharacteristic
304C2×C22⋊C4C4×D5311yescharacteristic
305C2×C42—21no—
306C2×C42—531no—
307C2×C42—1061no—
308C2×C42—1061no—
309C2×C42—1061no—
310C2×C42—1061no—
311C2×C42—1061no—
312C2×C42—1061no—
313C4⋊C8—1062no—
314C22⋊C8—21no—
315C22⋊C8—21no—
316C22⋊C8—21no—
317C22⋊C8—21no—
318C22⋊C8—1062no—
319C22⋊C8—1061no—
320C22⋊C8—1061no—
321C22⋊C8—1061no—
322C22⋊C8—1061no—
323C4×C8—1062no—
324#95—21no—
325#95—1061no—
326#95—1061no—
327#95—1061no—
328#95—1061no—
329C24—531no—
330C24C2×C4×D5311yescharacteristic
331C2×D4—532no—
332C2×D4—532no—
333C2×D4—1061no—
334C2×D4—1061no—
335C2×D4—1061no—
336C2×D4—1061no—
337C22×C4—21no—
338C22×C4—21no—
339C22×C4—42no—
340C22×C4—531no—
341C22×C4—532no—
342C22×C4—531no—
343C22×C4—531no—
344C22×C4—1061no—
345C22×C4—1061no—
346C22×C4—1061no—
347C22×C4—1061no—
348C22×C4—1061no—
349C22×C4—1061no—
350C22×C4—1061no—
351C22×C4—1061no—
352C22×C4—1061no—
353C22×C4—1061no—
354C22×C4—1061no—
355C22×C4—2122no—
356C22×C4—2122no—
357C22×C4C2×D21211yescharacteristic
358C22×C4#1171994511yescharacteristic
359C22×C4D4×D5311yescharacteristic
360C22×C4D4×D5311yescharacteristic
361C22×C4D4×D5311yescharacteristic
362C22×C4#1171994911yescharacteristic
363C2×C8—22no—
364C2×C8—42no—
365C2×C8—42no—
366C2×C8—41no—
367C2×C8—1062no—
368C2×C8—4242no—
369C2×C8—2122no—
370C2×C8—2121no—
371C2×C8—2122no—
372C4⋊C4—1064no—
373C4⋊C4—1061no—
374C4⋊C4—1061no—
375C4⋊C4—2122no—
376C22⋊C4—21no—
377C22⋊C4—21no—
378C22⋊C4—21no—
379C22⋊C4—1064no—
380C22⋊C4—1061no—
381C22⋊C4—1061no—
382C22⋊C4—1061no—
383C22⋊C4—1061no—
384C22⋊C4—2122no—
385C22⋊C4C8×D5312yescomplemented
386C42—42no—
387C42—1061no—
388C42—1061no—
389C42—2121no—
390C42—2121no—
391C42—2121no—
392C42—2121no—
393C23—41no—
394C23—41no—
395C23—532no—
396C23—1061no—
397C23—1061no—
398C23—2121no—
399C23—2121no—
400C23C22⋊C4×D5311yescharacteristic
401C23#1313317911yescharacteristic
402C23C4×D21211yescharacteristic
403C23C2×C8×D5311yescharacteristic
404C23M16×D5311yescharacteristic
405C23#1313322311yescharacteristic
406C23#1313322511yescharacteristic
407D4—1062no—
408D4—1062no—
409D4—1062no—
410D4—1062no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—22no—
414C2×C4—22no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—532no—
424C2×C4—1061no—
425C2×C4—1061no—
426C2×C4—1062no—
427C2×C4—1062no—
428C2×C4—1062no—
429C2×C4—1061no—
430C2×C4—1061no—
431C2×C4—1062no—
432C2×C4—1062no—
433C2×C4—1062no—
434C2×C4—4241no—
435C2×C4—2121no—
436C2×C4—2121no—
437C2×C4—2121no—
438C2×C4—2121no—
439C2×C4—2121no—
440C2×C4—2121no—
441C2×C4—2121no—
442C2×C4—2121no—
443C2×C4—2122no—
444C2×C4—2121no—
445C2×C4—2121no—
446C2×C4#1313318711yescharacteristic
447C2×C4#1313319011yescharacteristic
448C2×C4#1313319211yescharacteristic
449C2×C4#1313320211yescharacteristic
450C8—42no—
451C8—81no—
452C8—4241no—
453C8—2122no—
454C22—21no—
455C22—21no—
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—1062no—
465C22—4241no—
466C22—2122no—
467C22—2121no—
468C22—2121no—
469C22#1681050111yescharacteristic
470C22C22⋊C8×D5311yescharacteristic
471C22#1681058811yescharacteristic
472C22#1681068811yescharacteristic
473C22C8×D21211yescharacteristic
474C22#1681069311yescharacteristic
475C22#1681069611yescharacteristic
476C4—21no—
477C4—21no—
478C4—21no—
479C4—42no—
480C4—81no—
481C4—1062no—
482C4—2122no—
483C4—2121no—
484C4—2122no—
485C4—2121no—
486C4#1681081412yes—
487C2—21no—
488C2—21no—
489C2—21no—
490C2—21no—
491C2—21no—
492C2—21no—
493C2—2122no—
494C2#2027500611yescharacteristic
495C2#2027512711yescharacteristic
496C2#2027513211yescharacteristic
subgroupG/H
centerC2×C4#13133187
commutatorC23×C53C22×C8
FrattiniC23×C4C22×D53
FittingC53×#1282C2
socleC22×C53#386
prime radicalC22×C22⋊D53C2×C4
cosocleC23×C4×C53C23
hypercenter#1282D53
hypocenterC53#8966
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #1282

Complements

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

NH ≅ G/Ntype
3. #20274781C2semidirect
4. #20274781C2semidirect
6. #20272563C2semidirect
7. C53×#1282C2semidirect
39. #13133201C8semidirect
110. C22×Dic53C2×C8semidirect
121. C22⋊C4×C53C2×C8semidirect
245. #1282D53semidirect
279. C53#8966semidirect
385. C22⋊C4C8×D53semidirect

Maximal subgroups

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

subgroupindex
#202837372
C4×#131332012
#202747812
#202730312
#202725632
C53×#12822
#896653