Finite Group Databasebeta

← C22×#16811504 #28317995

Subgroups — C22×#16811504 #28317995

Order 13568.

2190 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
1C23×#11719940C211yesmaximal, characteristic
2C23×#11719940C211yesmaximal, characteristic
3C2×#16811504C2112yesmaximal, direct product factor
4C22×#13133173C211yesmaximal, characteristic
5C22×#11719940—21no—
6C22×#11719940—23no—
7C22×#11719940—23no—
8C22×#11719940—21no—
9C22×#11719940—21no—
10C22×#11719940—23no—
11C22×#11719940—23no—
12C22×#11719940C412yescomplemented
13C22×#11719940C416yescomplemented
14C22×#11719940C416yescomplemented
15C22×#11719940C2216yes—
16C22×#11719940C2216yes—
17C23×#88324—21no—
18C23×#88324C412yescomplemented
19#16811504C22116yesdirect product factor
20C24×Dic53C2211yescharacteristic
21C2×#13133173—23no—
22C2×#13133173—23no—
23C2×#13133173C416yes—
24C2×#13133173C2216yes—
25C22×#11719919—21no—
26C22×C22⋊Dic53C411yescharacteristic
27C22×C4×Dic53—21no—
28C22×C22⋊C4×C53C411yescharacteristic
29C2×#11719940—26no—
30C2×#11719940—26no—
31C2×#11719940—212no—
32C2×#11719940—26no—
33C2×#11719940—26no—
34C2×#11719940—212no—
35C2×#11719940—26no—
36C2×#11719940—26no—
37C2×#11719940—26no—
38C2×#11719940—26no—
39C2×#11719940—26no—
40C2×#11719940—26no—
41C2×#11719940—26no—
42C2×#11719940C2×C4112yescomplemented
43C2×#11719940C2×C4112yescomplemented
44C2×#11719940C2×C4112yescomplemented
45C2×#11719940C2314yes—
46C2×#11719940C2314yes—
47C22×#88324—26no—
48C22×#88324—22no—
49C22×#88324—26no—
50C22×#88324—22no—
51C22×#88324—26no—
52C22×#88324C2×C4112yescomplemented
53C23×Dic53—21no—
54C23×Dic53C2×C413yes—
55C23×Dic53C2×C411yescharacteristic
56C23×Dic53C2×C411yescharacteristic
57C23×Dic53C2×C413yes—
58C23×Dic53C2×C413yes—
59C23×Dic53D411yescharacteristic
60C23×Dic53D413yes—
61C23×Dic53D411yescharacteristic
62C23×Dic53D411yescharacteristic
63C23×Dic53D413yes—
64C23×Dic53D413yes—
65C23×Dic53Q811yescharacteristic
66C23×Dic53Q813yes—
67C23×Dic53C2313yes—
68#13133173—212no—
69#13133173—26no—
70#13133173—26no—
71#13133173C2×C4112yes—
72#13133173C2314yes—
73C2×#11719919—26no—
74C2×#11719919—26no—
75C2×C22⋊Dic53—23no—
76C2×C22⋊Dic53C2×C416yes—
77C2×C4×Dic53—26no—
78C2×C4×Dic53—26no—
79C25×C53C2×C411yescharacteristic
80C23×C4×C53—21no—
81C2×C22⋊C4×C53—23no—
82C2×C22⋊C4×C53C2×C416yes—
83C24×C53—44no—
84C24×C53C4211yescharacteristic
85C24×C53C4213yes—
86C24×C53C22⋊C411yescharacteristic
87C24×C53C22⋊C413yes—
88C24×C53C4⋊C411yescharacteristic
89C24×C53C4⋊C413yes—
90C24×C53C22×C413yes—
91C22×C4×C53—26no—
92C22×C4×C53—44no—
93C22×Dic53—26no—
94C22×Dic53—44no—
95C22×Dic53C4213yes—
96C22×Dic53C4213yes—
97C22×Dic53C4216yes—
98C22×Dic53C22⋊C413yes—
99C22×Dic53C22⋊C413yes—
100C22×Dic53C22⋊C416yes—
101C22×Dic53C22⋊C413yes—
102C22×Dic53C22⋊C413yes—
103C22×Dic53C22⋊C411yescharacteristic
104C22×Dic53C22⋊C411yescharacteristic
105C22×Dic53C22⋊C413yes—
106C22×Dic53C22⋊C413yes—
107C22×Dic53C22⋊C413yes—
108C22×Dic53C22⋊C413yes—
109C22×Dic53C22⋊C416yes—
110C22×Dic53C22⋊C416yes—
111C22×Dic53C4⋊C413yes—
112C22×Dic53C4⋊C413yes—
113C22×Dic53C4⋊C416yes—
114C22×Dic53C4⋊C413yes—
115C22×Dic53C4⋊C413yes—
116C22×Dic53C4⋊C413yes—
117C22×Dic53C4⋊C413yes—
118C22×Dic53C4⋊C413yes—
119C22×Dic53C4⋊C413yes—
120C22×Dic53C4⋊C416yes—
121C22×Dic53C4⋊C416yes—
122C22×Dic53C4⋊C411yescharacteristic
123C22×Dic53C4⋊C411yescharacteristic
124C22×Dic53C22×C413yes—
125C22×Dic53C22×C413yes—
126C22×Dic53C22×C413yes—
127C22×Dic53C22×C413yes—
128C22×Dic53C22×C413yes—
129C22×Dic53C2×D413yes—
130C22×Dic53C2×D413yes—
131C22×Dic53C2×D413yes—
132C22×Dic53C2×D413yes—
133C22×Dic53C2×D413yes—
134C22×Dic53C2×D413yes—
135C22×Dic53C2×Q813yes—
136C22×Dic53C2×Q813yes—
137C22×Dic53C2411yescharacteristic
138#11719940—212no—
139#11719940—212no—
140#11719940—212no—
141#11719940—212no—
142#11719940—24no—
143#11719940—24no—
144#11719940—24no—
145#11719940C22×C418yescomplemented
146C2×#88324—212no—
147C2×#88324—212no—
148C2×#88324—212no—
149C2×#88324—212no—
150C2×#88324—24no—
151C2×#88324C22×C418yescomplemented
152C22⋊C4×C53—26no—
153C22⋊C4×C53C22×C414yes—
154C22⋊Dic53—26no—
155C22⋊Dic53C22×C414yes—
156#11719919—212no—
157#11719919—24no—
158C4×Dic53—212no—
159C4×Dic53—24no—
160C23×C53—24no—
161C23×C53—24no—
162C23×C53—24no—
163C23×C53—46no—
164C23×C53—46no—
165C23×C53—46no—
166C23×C53—46no—
167C23×C53#9513yes—
168C23×C53#9513yes—
169C23×C53#9513yes—
170C23×C53#9516yes—
171C23×C53#9511yescharacteristic
172C23×C53C2×C4213yes—
173C23×C53C2×C4213yes—
174C23×C53C2×C22⋊C413yes—
175C23×C53C2×C22⋊C413yes—
176C23×C53C2×C4⋊C413yes—
177C23×C53C2×C4⋊C413yes—
178C23×C53C23×C411yescharacteristic
179C2×C4×C53—24no—
180C2×C4×C53—412no—
181C2×Dic53—24no—
182C2×Dic53—412no—
183C2×Dic53#9516yes—
184C2×Dic53#9516yes—
185C2×Dic53#9516yes—
186C2×Dic53#9516yes—
187C2×Dic53#9516yes—
188C2×Dic53#9516yes—
189C2×Dic53#9513yes—
190C2×Dic53#9513yes—
191C2×Dic53#9513yes—
192C2×Dic53#9513yes—
193C2×Dic53C2×C4213yes—
194C2×Dic53C2×C4213yes—
195C2×Dic53C2×C22⋊C413yes—
196C2×Dic53C2×C22⋊C413yes—
197C2×Dic53C2×C22⋊C413yes—
198C2×Dic53C2×C22⋊C413yes—
199C2×Dic53C2×C22⋊C413yes—
200C2×Dic53C2×C22⋊C413yes—
201C2×Dic53C2×C22⋊C413yes—
202C2×Dic53C2×C22⋊C413yes—
203C2×Dic53C2×C22⋊C413yes—
204C2×Dic53C2×C22⋊C413yes—
205C2×Dic53C2×C4⋊C413yes—
206C2×Dic53C2×C4⋊C413yes—
207C2×Dic53C2×C4⋊C413yes—
208C2×Dic53C2×C4⋊C413yes—
209C2×Dic53C2×C4⋊C413yes—
210C2×Dic53C2×C4⋊C413yes—
211C2×Dic53C2×C4⋊C413yes—
212C2×Dic53C2×C4⋊C413yes—
213C2×Dic53C2×C4⋊C413yes—
214C2×Dic53C2×C4⋊C413yes—
215C2×Dic53C23×C411yescharacteristic
216C2×Dic53C23×C411yescharacteristic
217C2×Dic53C22×D411yescharacteristic
218C2×Dic53C22×D411yescharacteristic
219C2×Dic53C22×D411yescharacteristic
220C2×Dic53C22×Q811yescharacteristic
221#88324—28no—
222#88324—28no—
223C22×#344—531nomaximal, Sylow
224C22×C53—26no—
225C22×C53—26no—
226C22×C53—26no—
227C22×C53—26no—
228C22×C53—26no—
229C22×C53—26no—
230C22×C53—46no—
231C22×C53—43no—
232C22×C53—43no—
233C22×C53—43no—
234C22×C53—41no—
235C22×C53#34414yescomplemented
236C22×C53C2×#9513yes—
237C22×C53C2×#9513yes—
238C22×C53C2×#9513yes—
239C22×C53C2×#9513yes—
240C22×C53C22×C4211yescharacteristic
241C22×C53C22×C22⋊C411yescharacteristic
242C22×C53C22×C4⋊C411yescharacteristic
243C4×C53—44no—
244Dic53—44no—
245Dic53C2×#9513yes—
246Dic53C2×#9513yes—
247Dic53C2×#9513yes—
248Dic53C2×#9513yes—
249Dic53C22×C22⋊C411yescharacteristic
250Dic53C22×C22⋊C411yescharacteristic
251Dic53C22×C4⋊C411yescharacteristic
252Dic53C22×C4⋊C411yescharacteristic
253C23×M16—531no—
254C23×M16—531no—
255C22×#117—531no—
256C2×#344—5312no—
257C2×C53—21no—
258C2×C53—21no—
259C2×C53—23no—
260C2×C53—23no—
261C2×C53—23no—
262C2×C53—21no—
263C2×C53C2×#34416yescomplemented
264C2×C53C22×#9511yescharacteristic
265C24×C4—531no—
266C22×M16—536no—
267C22×M16—536no—
268C22×M16—536no—
269C22×M16—532no—
270C22×M16—536no—
271C22×M16—1061no—
272C22×M16—1063no—
273C22×M16—1063no—
274C22×M16—1061no—
275C22×M16—1063no—
276C22×M16—1061no—
277C22×M16—1063no—
278C23×C8—532no—
279C23×C8—1061no—
280C2×#117—536no—
281C2×#117—536no—
282C2×#117—1063no—
283C2×#117—1063no—
284C22×C4⋊C4—1061no—
285C22×C22⋊C4—531no—
286C22×C22⋊C4#615711yescharacteristic
287C22×C42—1061no—
288#344—5316no—
289C53C22×#34411yescharacteristic, complemented, Sylow
290C25C2×#615711yescharacteristic
291C23×C4—21no—
292C23×C4—531no—
293C23×C4—533no—
294C23×C4—533no—
295C23×C4—531no—
296C23×C4—531no—
297C23×C4—533no—
298C23×C4—533no—
299C23×C4—531no—
300C23×C4—531no—
301C23×C4—533no—
302C23×C4—533no—
303C23×C4—531no—
304C23×C4—533no—
305C23×C4—533no—
306C23×C4—1061no—
307C2×M16—534no—
308C2×M16—5312no—
309C2×M16—534no—
310C2×M16—5312no—
311C2×M16—5312no—
312C2×M16—1066no—
313C2×M16—1066no—
314C2×M16—10612no—
315C2×M16—1066no—
316C2×M16—1066no—
317C2×M16—1066no—
318C2×M16—1066no—
319C2×M16—1066no—
320C2×M16—1066no—
321C2×M16—10612no—
322C2×M16—1066no—
323C2×M16—1066no—
324C2×M16—1066no—
325C22×C8—5312no—
326C22×C8—1066no—
327C22×C8—1062no—
328C22×C8—1066no—
329C22×C8—1062no—
330C22×C8—1066no—
331#117—534no—
332#117—5312no—
333#117—10612no—
334#117—1066no—
335#117—1066no—
336C2×C4⋊C4—1066no—
337C2×C4⋊C4—1066no—
338C2×C22⋊C4—23no—
339C2×C22⋊C4—536no—
340C2×C22⋊C4—1063no—
341C2×C22⋊C4C2×#615716yes—
342C2×C42—1066no—
343C2×C42—1066no—
344C24—44no—
345C24C4×#615711yescharacteristic
346C24C4×#615713yes—
347C24C4⋊#615711yescharacteristic
348C24C4⋊#615713yes—
349C24C22⋊#615711yescharacteristic
350C24C22⋊#615713yes—
351C24C22×#615713yes—
352C22×C4—26no—
353C22×C4—44no—
354C22×C4—533no—
355C22×C4—533no—
356C22×C4—533no—
357C22×C4—533no—
358C22×C4—533no—
359C22×C4—533no—
360C22×C4—536no—
361C22×C4—536no—
362C22×C4—533no—
363C22×C4—533no—
364C22×C4—533no—
365C22×C4—533no—
366C22×C4—533no—
367C22×C4—533no—
368C22×C4—536no—
369C22×C4—536no—
370C22×C4—531no—
371C22×C4—531no—
372C22×C4—531no—
373C22×C4—531no—
374C22×C4—533no—
375C22×C4—533no—
376C22×C4—536no—
377C22×C4—533no—
378C22×C4—533no—
379C22×C4—533no—
380C22×C4—533no—
381C22×C4—533no—
382C22×C4—533no—
383C22×C4—536no—
384C22×C4—533no—
385C22×C4—533no—
386C22×C4—533no—
387C22×C4—533no—
388C22×C4—533no—
389C22×C4—533no—
390C22×C4—536no—
391C22×C4—533no—
392C22×C4—533no—
393C22×C4—533no—
394C22×C4—533no—
395C22×C4—533no—
396C22×C4—531no—
397C22×C4—1066no—
398C22×C4—2124no—
399M16—538no—
400M16—10612no—
401M16—10612no—
402M16—1064no—
403M16—1064no—
404M16—10612no—
405M16—10612no—
406M16—1064no—
407C2×C8—538no—
408C2×C8—10612no—
409C2×C8—10612no—
410C2×C8—10612no—
411C2×C8—10612no—
412C2×C8—1064no—
413C4⋊C4—10612no—
414C4⋊C4—1064no—
415C22⋊C4—26no—
416C22⋊C4—534no—
417C22⋊C4—1066no—
418C22⋊C4C22×#615714yes—
419C42—10612no—
420C42—1064no—
421C23—24no—
422C23—24no—
423C23—24no—
424C23—46no—
425C23—46no—
426C23—46no—
427C23—46no—
428C23(C2×#6157)⋊C413yes—
429C23(C2×#6157)⋊C413yes—
430C23(C2×#6157)⋊C413yes—
431C23(C2×#6157)⋊C416yes—
432C23(C2×#6157)⋊C411yescharacteristic
433C23C2×C4×#615713yes—
434C23C2×C4×#615713yes—
435C23C2×C22⋊#615713yes—
436C23C2×C22⋊#615713yes—
437C23C2×C4⋊#615713yes—
438C23C2×C4⋊#615713yes—
439C23C23×#615711yescharacteristic
440C2×C4—24no—
441C2×C4—412no—
442C2×C4—536no—
443C2×C4—536no—
444C2×C4—536no—
445C2×C4—536no—
446C2×C4—536no—
447C2×C4—536no—
448C2×C4—533no—
449C2×C4—533no—
450C2×C4—533no—
451C2×C4—533no—
452C2×C4—533no—
453C2×C4—533no—
454C2×C4—533no—
455C2×C4—533no—
456C2×C4—533no—
457C2×C4—533no—
458C2×C4—533no—
459C2×C4—533no—
460C2×C4—533no—
461C2×C4—533no—
462C2×C4—533no—
463C2×C4—533no—
464C2×C4—533no—
465C2×C4—533no—
466C2×C4—533no—
467C2×C4—533no—
468C2×C4—533no—
469C2×C4—533no—
470C2×C4—531no—
471C2×C4—531no—
472C2×C4—533no—
473C2×C4—533no—
474C2×C4—531no—
475C2×C4—531no—
476C2×C4—533no—
477C2×C4—533no—
478C2×C4—531no—
479C2×C4—531no—
480C2×C4—1064no—
481C2×C4—21212no—
482C8—1068no—
483C8—1068no—
484C22—26no—
485C22—26no—
486C22—26no—
487C22—26no—
488C22—26no—
489C22—26no—
490C22—46no—
491C22—43no—
492C22—43no—
493C22—43no—
494C22—41no—
495C22#1681150414yesdirect product factor
496C22C2×(C2×#6157)⋊C413yes—
497C22C2×(C2×#6157)⋊C413yes—
498C22C2×(C2×#6157)⋊C413yes—
499C22C2×(C2×#6157)⋊C413yes—
500C22C22×C4×#615711yescharacteristic
501C22C22×C22⋊#615711yescharacteristic
502C22C22×C4⋊#615711yescharacteristic
503C4—44no—
504C4—531no—
505C4—531no—
506C4—531no—
507C4—531no—
508C4—533no—
509C4—533no—
510C4—533no—
511C4—533no—
512C4—2124no—
513C2—21no—
514C2—23no—
515C2—23no—
516C2—23no—
517C2—21no—
518C2—21no—
519C2C2×#1681150416yesdirect product factor
520C2C22×(C2×#6157)⋊C411yescharacteristic
subgroupG/H
centerC23(C2×#6157)⋊C4
commutatorC22×C53C22×C42
FrattiniC23C23×#6157
FittingC22×C22⋊C4×C53C4
socleC23×C53#95
prime radicalC25×C53C2×C4
cosocleC22×Dic53C24
hypercenterC22×C22⋊C4#6157
hypocenterC53C22×#344
LCSG ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C25 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×#16811504C2direct (N × H)
12. C22×#11719940C4semidirect
13. C22×#11719940C4semidirect
14. C22×#11719940C4semidirect
18. C23×#88324C4semidirect
19. #16811504C22direct (N × H)
42. C2×#11719940C2×C4semidirect
43. C2×#11719940C2×C4semidirect
44. C2×#11719940C2×C4semidirect
52. C22×#88324C2×C4semidirect
145. #11719940C22×C4semidirect
151. C2×#88324C22×C4semidirect
235. C22×C53#344semidirect
263. C2×C53C2×#344semidirect
289. C53C22×#344semidirect
495. C22#16811504direct (N × H)
519. C2C2×#16811504direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#117199402
C2×#168115042
C22×#131331732
C22×#34453