Finite Group Databasebeta

← #27278632

Subgroups — #27278632

Order 13568.

604 conjugacy classes of subgroups.

The subgroup lattice has 506 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810679C211yesmaximal, characteristic
2C4×#13133197C211yesmaximal, characteristic
3C4×#13133129C211yesmaximal, characteristic
4C2×#16810355C211yesmaximal, characteristic, complemented
5C2×#16810353C211yesmaximal, characteristic, complemented
6#20272556C211yesmaximal, characteristic, complemented
7C53×#1279C211yesmaximal, characteristic, complemented
8C2×#13133197C411yescharacteristic
9C2×C4×Dic106C2211yescharacteristic
10C2×C42×D53C2211yescharacteristic
11#16810679—22no—
12#16810679—22no—
13C2×C4⋊#88322C2211yescharacteristic
14C2×#13133129—21no—
15C2×#13133129—21no—
16C2×#13133127—21no—
17C2×C4×#88322C2211yescharacteristic
18#16810528C411yescharacteristic
19#16810522C411yescharacteristic
20#16810508—21no—
21#16810497—21no—
22#16810480C411yescharacteristic
23C4×C4⋊Dic53—21no—
24C4×#11719921—21no—
25#16810377—21no—
26#16810355—22no—
27#16810353—22no—
28C2×C4⋊C8×C53C2211yescharacteristic
29C2×C4×C8×C53C2211yescharacteristic
30C4×C4⋊C4×C53C2211yescharacteristic
31C22×Dic106C2×C411yescharacteristic
32C22×C4×D53C2×C411yescharacteristic
33C22×C4×D53D411yescharacteristic
34C22×C4×D53D411yescharacteristic
35C22×#88322—21no—
36C22×#88322—21no—
37#13133197—22no—
38#13133197—22no—
39C4×Dic106—21no—
40C4×Dic106—21no—
41C4×Dic106—21no—
42C4×Dic106—21no—
43C2×C4⋊Dic53—21no—
44C2×C4⋊Dic53—21no—
45C2×C4⋊Dic53C2×C411yescharacteristic
46C2×#11719919—21no—
47C2×#11719921—21no—
48C2×#11719921—21no—
49C2×C4×Dic53—21no—
50C2×C4×Dic53C2×C411yescharacteristic
51C2×C4×Dic53D411yescharacteristic
52C2×C4×Dic53D411yescharacteristic
53C42×D53—21no—
54C42×D53—21no—
55C42×D53—21no—
56C42×D53—21no—
57C4⋊#88322—21no—
58C4⋊#88322D412yes—
59#13133129—24no—
60#13133129—24no—
61#13133127—24no—
62C4×#88322—21no—
63C4×#88322—21no—
64#13133124—21no—
65#13133124—21no—
66#13133124—21no—
67#13133124—21no—
68C22×C8×C53—21no—
69C22×C8×C53—21no—
70C2×C4⋊C4×C53C2×C411yescharacteristic
71C2×C4⋊C4×C53C2×C411yescharacteristic
72C2×C42×C53—21no—
73C2×C42×C53C2311yescharacteristic
74C4⋊C8×C53—21no—
75C4⋊C8×C53D412yes—
76C4×C8×C53—21no—
77C4×C8×C53—21no—
78#95×C53—21no—
79C23×D53C22⋊C411yescharacteristic
80C22×C4×C53—21no—
81C22×C4×C53—21no—
82C22×C4×C53C22×C411yescharacteristic
83C22×C4×C53C2×D411yescharacteristic
84C22×C4×C53C2×D411yescharacteristic
85C22×Dic53—21no—
86C22×Dic53—21no—
87C22×Dic53C22⋊C411yescharacteristic
88C22×Dic53C22⋊C411yescharacteristic
89C22×Dic53C22⋊C411yescharacteristic
90C2×C4×D53—22no—
91C2×C4×D53—21no—
92C2×C4×D53—21no—
93C2×C4×D53—21no—
94C2×C4×D53—21no—
95C2×C4×D53—21no—
96C2×C4×D53—21no—
97C2×C4×D53—21no—
98C2×C4×D53—21no—
99C2×C4×D53—21no—
100C2×C4×D53—21no—
101C2×C4×D53—21no—
102C2×C4×D53—21no—
103C2×C4×D53—21no—
104C2×C4×D53—21no—
105C2×C4×D53—41no—
106C2×Dic106—21no—
107C2×Dic106—21no—
108C2×Dic106—21no—
109C2×Dic106—21no—
110C2×Dic106—21no—
111C2×Dic106M1612yes—
112C2×C8×C53—22no—
113C2×C8×C53—22no—
114C2×C8×C53—41no—
115C2×C8×C53—41no—
116C2×C8×C53—41no—
117C2×C8×C53—41no—
118C4⋊C4×C53—21no—
119C4⋊C4×C53—21no—
120C4⋊C4×C53—21no—
121C4⋊C4×C53M1612yes—
122C42×C53—41no—
123C42×C53—41no—
124C42×C53C2×D411yescharacteristic
125C42×C53C2×D411yescharacteristic
126C42×C53#4111yescharacteristic
127C42×C53#4111yescharacteristic
128#11719921—24no—
129#11719921—42no—
130C4⋊Dic53—24no—
131C4⋊Dic53—21no—
132C4⋊Dic53—21no—
133C4⋊Dic53—42no—
134#11719919—41no—
135#11719919—41no—
136C4×Dic53—21no—
137C4×Dic53—21no—
138C4×Dic53—21no—
139C4×Dic53—21no—
140C4×Dic53—21no—
141C4×Dic53—21no—
142C4×Dic53—41no—
143C4×Dic53—41no—
144C2×#88322—22no—
145C2×#88322—22no—
146C2×#88322—41no—
147C2×#88322—41no—
148C2×#88322—41no—
149C2×#88322—41no—
150C23×C53C2×C22⋊C411yescharacteristic
151C22×D53—21no—
152C22×D53—21no—
153C22×D53—21no—
154C22×D53—21no—
155C22×D53—21no—
156C22×D53—21no—
157C22×D53#10412yes—
158C2×C4×C53—21no—
159C2×C4×C53—21no—
160C2×C4×C53—22no—
161C2×C4×C53—41no—
162C2×C4×C53—41no—
163C2×C4×C53—41no—
164C2×C4×C53—41no—
165C2×C4×C53—41no—
166C2×C4×C53C4×D411yescharacteristic
167C2×C4×C53C4×D411yescharacteristic
168C2×C4×C53#12011yescharacteristic
169C2×C4×C53#12111yescharacteristic
170C2×C4×C53#12111yescharacteristic
171C2×C4×C53#12311yescharacteristic
172C2×C4×C53C2×M1611yescharacteristic
173C2×C4×C53C2×M1611yescharacteristic
174C2×C4×C53#13111yescharacteristic
175C2×C4×C53#13111yescharacteristic
176C2×C4×C53#13511yescharacteristic
177C2×C4×C53#13511yescharacteristic
178C2×C4×C53#13711yescharacteristic
179C2×C4×C53#13711yescharacteristic
180C2×Dic53—21no—
181C2×Dic53—21no—
182C2×Dic53—21no—
183C2×Dic53—22no—
184C2×Dic53—21no—
185C2×Dic53—21no—
186C2×Dic53—21no—
187C2×Dic53—21no—
188C2×Dic53—21no—
189C2×Dic53—21no—
190C2×Dic53—41no—
191C2×Dic53—41no—
192C2×Dic53—41no—
193C2×Dic53—41no—
194C2×Dic53—41no—
195C4×D53—42no—
196C4×D53—42no—
197C4×D53—41no—
198C4×D53—41no—
199C4×D53—41no—
200C4×D53—41no—
201C4×D53—41no—
202C4×D53—41no—
203C4×D53—41no—
204C4×D53—41no—
205Dic106—22no—
206Dic106—22no—
207Dic106—41no—
208Dic106—41no—
209C8×C53—42no—
210C8×C53—42no—
211#88322—42no—
212#88322—42no—
213#8941—531nomaximal, Sylow
214C22×C53#38611yescharacteristic
215C22×C53#40711yescharacteristic
216C22×C53#40811yescharacteristic
217C22×C53#41611yescharacteristic
218C22×C53#41911yescharacteristic
219C22×C53C2×#10411yescharacteristic
220C22×C53#42111yescharacteristic
221C2×D53—22no—
222C2×D53—22no—
223C2×D53—22no—
224C2×D53—41no—
225C2×D53—41no—
226C2×D53—41no—
227C2×D53—41no—
228C4×C53—21no—
229C4×C53—21no—
230C4×C53—42no—
231C4×C53—81no—
232C4×C53#43511yescharacteristic
233C4×C53#43511yescharacteristic
234C4×C53#43611yescharacteristic
235C4×C53#43611yescharacteristic
236C4×C53#45211yescharacteristic
237C4×C53#45211yescharacteristic
238C4×C53#48511yescharacteristic
239C4×C53#48511yescharacteristic
240Dic53—42no—
241Dic53—41no—
242Dic53—41no—
243Dic53—41no—
244Dic53—81no—
245C2×#432—531no—
246C4×C22⋊Q8—531no—
247C4×C22⋊C8—531no—
248C2×Q8⋊C8—531no—
249C2×Q8⋊C8—531no—
250#1279—531no—
251#1279D5311yescharacteristic, complemented
252C2×C53#148811yescharacteristic
253C2×C53#149111yescharacteristic
254C2×C53#149411yescharacteristic
255C2×C53#149711yescharacteristic
256C2×C53#187011yescharacteristic
257C2×C53#187111yescharacteristic
258C2×C53#187311yescharacteristic
259D53—42no—
260C2×C22⋊Q8—531no—
261C2×C4×Q8—531no—
262C22×C42—531no—
263#432—1062no—
264#432—1062no—
265C2×C4⋊C8—531no—
266C2×C4⋊C8C2×D5311yescharacteristic
267C2×C22⋊C8—1061no—
268C2×C22⋊C8—1061no—
269C2×#97—1061no—
270C2×C4×C8—531no—
271C2×C4×C8C2×D5311yescharacteristic
272#391—531no—
273#389—531no—
274#387—1061no—
275#385—1061no—
276#380—531no—
277C4×C4⋊C4—1061no—
278C4×C4⋊C4C2×D5311yescharacteristic
279C4×C22⋊C4—1061no—
280#336—1061no—
281Q8⋊C8—1062no—
282Q8⋊C8—1062no—
283C53#894111yescharacteristic, complemented, Sylow
284C22×Q8—531no—
285C23×C4—531no—
286C23×C4—531no—
287C23×C4—531no—
288C22×C8—21no—
289C22×C8—21no—
290C22×C8—1061no—
291C22×C8—1061no—
292C22⋊Q8—1062no—
293C22⋊Q8—1062no—
294C4×Q8—1061no—
295C4×Q8—1061no—
296C4×Q8—1061no—
297C4×Q8—1061no—
298C2×C4⋊C4—531no—
299C2×C4⋊C4—1061no—
300C2×C4⋊C4—1061no—
301C2×C4⋊C4—1061no—
302C2×C4⋊C4C4×D5311yescharacteristic
303C2×C4⋊C4C4×D5311yescharacteristic
304C2×C22⋊C4—1061no—
305C2×C22⋊C4—1061no—
306C2×C42—21no—
307C2×C42—531no—
308C2×C42—531no—
309C2×C42—531no—
310C2×C42—1061no—
311C2×C42—1061no—
312C2×C42—1061no—
313C2×C42—1061no—
314C2×C42—1061no—
315C2×C42C22×D5311yescharacteristic
316C4⋊C8—21no—
317C4⋊C8—532no—
318C4⋊C8—1061no—
319C4⋊C8D21212yes—
320C22⋊C8—1064no—
321C22⋊C8—1064no—
322#97—1064no—
323C4×C8—21no—
324C4×C8—21no—
325C4×C8—1061no—
326C4×C8—1061no—
327#95—21no—
328#95—1061no—
329#95—1061no—
330#95—1061no—
331#95—1061no—
332C24—531no—
333C2×Q8—532no—
334C2×Q8—1061no—
335C2×Q8—1061no—
336C2×Q8—1061no—
337C2×Q8—1061no—
338C2×Q8—1061no—
339C22×C4—21no—
340C22×C4—21no—
341C22×C4—531no—
342C22×C4—531no—
343C22×C4—531no—
344C22×C4—1062no—
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—1061no—
356C22×C4—1061no—
357C22×C4—1061no—
358C22×C4—1061no—
359C22×C4—1061no—
360C22×C4—1061no—
361C22×C4—2121no—
362C22×C4C2×C4×D5311yescharacteristic
363C22×C4D4×D5311yescharacteristic
364C22×C4D4×D5311yescharacteristic
365C2×C8—22no—
366C2×C8—22no—
367C2×C8—41no—
368C2×C8—41no—
369C2×C8—41no—
370C2×C8—41no—
371C2×C8—1062no—
372C2×C8—1062no—
373C2×C8—2121no—
374C2×C8—2121no—
375C2×C8—2121no—
376C2×C8—2121no—
377C4⋊C4—21no—
378C4⋊C4—21no—
379C4⋊C4—21no—
380C4⋊C4—1064no—
381C4⋊C4—1061no—
382C4⋊C4—1061no—
383C4⋊C4—2122no—
384C4⋊C4—2121no—
385C4⋊C4—2121no—
386C4⋊C4#1171991212yes—
387C22⋊C4—1064no—
388C22⋊C4—2122no—
389C42—41no—
390C42—41no—
391C42—1061no—
392C42—1061no—
393C42—1061no—
394C42—1061no—
395C42—1061no—
396C42—1061no—
397C42—2121no—
398C42—2121no—
399C42C2×D21211yescharacteristic
400C42#1171994511yescharacteristic
401C42D4×D5311yescharacteristic
402C42#1171994911yescharacteristic
403C23—532no—
404C23—1061no—
405C23—1061no—
406C23—1061no—
407C23—1061no—
408C23—1061no—
409C23—1061no—
410C23C22⋊C4×D5311yescharacteristic
411Q8—1062no—
412Q8—1062no—
413Q8—2121no—
414Q8—2121no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—22no—
418C2×C4—41no—
419C2×C4—41no—
420C2×C4—41no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—1061no—
424C2×C4—1061no—
425C2×C4—1061no—
426C2×C4—1062no—
427C2×C4—1061no—
428C2×C4—1061no—
429C2×C4—1061no—
430C2×C4—1061no—
431C2×C4—1061no—
432C2×C4—1061no—
433C2×C4—2122no—
434C2×C4—2122no—
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—2121no—
444C2×C4—2121no—
445C2×C4—2121no—
446C2×C4—2121no—
447C2×C4—2121no—
448C2×C4#1313317911yescharacteristic
449C2×C4C4×D21211yescharacteristic
450C2×C4#1313318711yescharacteristic
451C2×C4#1313319011yescharacteristic
452C2×C4#1313319211yescharacteristic
453C2×C4#1313320211yescharacteristic
454C2×C4M16×D5311yescharacteristic
455C2×C4C2×#1171991211yescharacteristic
456C2×C4#1313322311yescharacteristic
457C2×C4#1313322511yescharacteristic
458C2×C4#1313323611yescharacteristic
459C2×C4#1313323811yescharacteristic
460C2×C4#1313324611yescharacteristic
461C2×C4#1313324911yescharacteristic
462C8—42no—
463C8—42no—
464C8—2122no—
465C8—2122no—
466C22—1062no—
467C22—1062no—
468C22—1062no—
469C22—2121no—
470C22—2121no—
471C22—2121no—
472C22—2121no—
473C22#1681050111yescharacteristic
474C22#1681058511yescharacteristic
475C22#1681058811yescharacteristic
476C22#1681061611yescharacteristic
477C22#1681063011yescharacteristic
478C22#104×D5311yescharacteristic
479C22#1681063811yescharacteristic
480C4—21no—
481C4—21no—
482C4—42no—
483C4—81no—
484C4—4241no—
485C4—2121no—
486C4—2122no—
487C4—2121no—
488C4—2121no—
489C4#1681069311yescharacteristic
490C4#1681069611yescharacteristic
491C4#1681070011yescharacteristic
492C4#1681070211yescharacteristic
493C4#1681078611yescharacteristic
494C4#1681078811yescharacteristic
495C4#1681096611yescharacteristic
496C4#1681096711yescharacteristic
497C2—2122no—
498C2#2027314611yescharacteristic
499C2#2027316011yescharacteristic
500C2#2027317211yescharacteristic
501C2#2027318611yescharacteristic
502C2#2027501111yescharacteristic
503C2#2027501611yescharacteristic
504C2#2027502611yescharacteristic
subgroupG/H
centerC22×C4D4×D53
commutatorC22×C4×C53C22×C4
FrattiniC2×C42C22×D53
FittingC53×#1279C2
socleC23×C53C2×C22⋊C4
prime radicalC23×D53C22⋊C4
cosocleC2×C42×C53C23
hypercenter#1279D53
hypocenterC53#8941
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C2×C42 ◁ #1279

Complements

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

NH ≅ G/Ntype
4. C2×#16810355C2semidirect
5. C2×#16810353C2semidirect
6. #20272556C2semidirect
7. C53×#1279C2semidirect
251. #1279D53semidirect
283. C53#8941semidirect

Maximal subgroups

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

subgroupindex
C2×#168106792
C4×#131331972
C4×#131331292
C2×#168103552
C2×#168103532
#202725562
C53×#12792
#894153