Finite Group Databasebeta

← C2×#20290575 #28319038

Subgroups — C2×#20290575 #28319038

Order 13568.

2066 conjugacy classes of subgroups.

The subgroup lattice has 580 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133275C212yesmaximal
2C2×#16811565C212yesmaximal, complemented
3#20290575C218yesmaximal, direct product factor
4C2×#16811501C212yesmaximal, complemented
5C22×C4×C22⋊D53C211yesmaximal, characteristic
6C22×C4⋊#6157—22no—
7C22×C22⋊#6157C2214yes—
8C2×#13133275—28no—
9C2×#13133275C2218yes—
10#16811565—24no—
11#16811565C2218yescomplemented
12C2×(C2×#6157)⋊C4—22no—
13#16811501C2218yescomplemented
14C23×C22⋊D53C2211yescharacteristic
15C23×C4×D53C2211yescharacteristic
16C2×C4×C22⋊D53—22no—
17C2×C4×C22⋊D53—22no—
18C2×C4×C22⋊D53—24no—
19C2×C4×C22⋊D53C414yescomplemented
20C2×C4×C22⋊D53C2214yes—
21C22×#11719919C411yescharacteristic, complemented
22C22×C22⋊Dic53C411yescharacteristic, complemented
23C22×#11719921C2211yescharacteristic
24C22×C4×Dic53C411yescharacteristic, complemented
25C24×C4×C53C411yescharacteristic, complemented
26C23×#6157—24no—
27C2×C4⋊#6157—28no—
28C2×C4⋊#6157—48no—
29C2×C22⋊#6157—28no—
30C2×C22⋊#6157—28no—
31C2×C22⋊#6157—24no—
32C2×C22⋊#6157C2318yes—
33#13133275—216no—
34(C2×#6157)⋊C4—28no—
35C24×D53C2311yescharacteristic
36C22×C22⋊D53—22no—
37C22×C22⋊D53—22no—
38C22×C22⋊D53—24no—
39C22×C22⋊D53C2×C412yes—
40C22×C22⋊D53D414yes—
41C22×C22⋊D53Q814yes—
42C22×C22⋊D53C2312yes—
43C23×Dic53C2×C411yescharacteristic
44C23×Dic53C2×C411yescharacteristic
45C23×Dic53C2×C411yescharacteristic
46C22×C4×D53—44no—
47C22×C4×D53C2×C412yes—
48C22×C4×D53D412yes—
49C22×C4×D53D412yes—
50C22×C4×D53D414yes—
51C22×C4×D53C2312yes—
52C4×C22⋊D53—28no—
53C4×C22⋊D53—28no—
54C4×C22⋊D53—28no—
55C4×C22⋊D53—28no—
56C2×#11719919—22no—
57C2×#11719919—22no—
58C2×#11719919C2×C414yescomplemented
59C2×C22⋊Dic53—22no—
60C2×C22⋊Dic53—22no—
61C2×C22⋊Dic53C2×C414yescomplemented
62C2×#11719921—22no—
63C2×#11719921—22no—
64C2×#11719921C2×C412yes—
65C2×#11719921C2312yes—
66C2×C4×Dic53—22no—
67C2×C4×Dic53—21no—
68C2×C4×Dic53—21no—
69C2×C4×Dic53C2×C414yescomplemented
70C25×C53C2×C411yescharacteristic
71C23×C4×C53—22no—
72C23×C4×C53—21no—
73C23×C4×C53—21no—
74C23×C4×C53—44no—
75C23×C4×C53C2×C411yescharacteristic
76C23×C4×C53C2×C411yescharacteristic
77C23×C4×C53C2×C414yescomplemented
78C24×C53—21no—
79C24×C53—21no—
80C24×C53—22no—
81C24×C53—42no—
82C24×C53—42no—
83C24×C53C4⋊C414yescomplemented
84C24×C53C22×C411yescharacteristic
85C24×C53C22×C412yes—
86C23×D53—42no—
87C23×D53—42no—
88C23×D53C22×C411yescharacteristic
89C23×D53C2×D412yes—
90C23×D53C2×D412yes—
91C23×D53C2×D412yes—
92C23×D53C2×Q812yes—
93C23×D53#4114yes—
94C23×D53C2411yescharacteristic
95C22×#6157—28no—
96C22×#6157—48no—
97C22×#6157—48no—
98C22×#6157—48no—
99C22×C4×C53—24no—
100C22×C4×C53—21no—
101C22×C4×C53—21no—
102C22×C4×C53—24no—
103C22×C4×C53—24no—
104C22×C4×C53—44no—
105C22×C4×C53—48no—
106C22×C4×C53—48no—
107C22×C4×C53—44no—
108C22×C4×C53—42no—
109C22×C4×C53C22⋊C411yescharacteristic
110C22×C4×C53C22⋊C412yes—
111C22×C4×C53C22⋊C411yescharacteristic
112C22×C4×C53C22×C412yes—
113C22×C4×C53C22×C412yes—
114C2×C22⋊D53—24no—
115C2×C22⋊D53—24no—
116C2×C22⋊D53—28no—
117C2×C22⋊D53—28no—
118C2×C22⋊D53—28no—
119C2×C22⋊D53—28no—
120C2×C22⋊D53—24no—
121C2×C22⋊D53—24no—
122C2×C22⋊D53C4⋊C418yescomplemented
123C2×C22⋊D53C2×D414yes—
124C2×C22⋊D53C2×Q814yes—
125C22×Dic53—21no—
126C22×Dic53—21no—
127C22×Dic53—22no—
128C22×Dic53—21no—
129C22×Dic53—21no—
130C22×Dic53—42no—
131C22×Dic53—42no—
132C22×Dic53C22⋊C412yes—
133C22×Dic53C22⋊C412yes—
134C22×Dic53C4⋊C414yescomplemented
135C22×Dic53C22×C412yes—
136C22×Dic53C22×C412yes—
137C22×Dic53C22×C412yes—
138C2×C4×D53—28no—
139C2×C4×D53—44no—
140C2×C4×D53—48no—
141C2×C4×D53—44no—
142C2×C4×D53—44no—
143C2×C4×D53C22⋊C414yes—
144C2×C4×D53C22⋊C414yes—
145C2×C4×D53C2×D414yes—
146C2×C4×D53C2×D414yes—
147C22⋊#6157—28no—
148C22⋊#6157—28no—
149C22⋊#6157—48no—
150C4⋊#6157—416no—
151C22⋊Dic53—28no—
152#11719921—24no—
153#11719921—24no—
154#11719919—28no—
155C4×Dic53—24no—
156C4×Dic53—24no—
157C23×C53—22no—
158C23×C53—22no—
159C23×C53—22no—
160C23×C53—22no—
161C23×C53—24no—
162C23×C53—24no—
163C23×C53—42no—
164C23×C53—48no—
165C23×C53—42no—
166C23×C53—42no—
167C23×C53—48no—
168C23×C53—42no—
169C23×C53—42no—
170C23×C53—42no—
171C23×C53C2×C22⋊C411yescharacteristic
172C23×C53C2×C22⋊C411yescharacteristic
173C23×C53C2×C4⋊C412yes—
174C23×C53C2×C4⋊C414yescomplemented
175C23×C53#11712yes—
176C23×C53C23×C411yescharacteristic
177C22×D53—24no—
178C22×D53—24no—
179C22×D53—41no—
180C22×D53—42no—
181C22×D53—41no—
182C22×D53—41no—
183C22×D53—44no—
184C22×D53—42no—
185C22×D53—44no—
186C22×D53—48no—
187C22×D53—41no—
188C22×D53C2×C22⋊C412yes—
189C22×D53C2×C4⋊C412yes—
190C22×D53#11712yes—
191C22×D53#12114yes—
192C22×D53#12114yes—
193C22×D53C22⋊Q814yes—
194C22×D53C22⋊Q814yes—
195C22×D53C22×D412yes—
196C22×D53C22×D411yescharacteristic
197C22×D53C22×Q811yescharacteristic
198C22×D53C2×#4112yes—
199C2×#6157—48no—
200C2×#6157—48no—
201C2×#6157—48no—
202C2×#6157—816no—
203C2×C4×C53—22no—
204C2×C4×C53—24no—
205C2×C4×C53—24no—
206C2×C4×C53—22no—
207C2×C4×C53—48no—
208C2×C4×C53—42no—
209C2×C4×C53—42no—
210C2×C4×C53—48no—
211C2×C4×C53—42no—
212C2×C4×C53#9914yescomplemented
213C2×C4×C53C2×C22⋊C412yes—
214C2×C4×C53C2×C22⋊C412yes—
215C22⋊D53—28no—
216C22⋊D53—28no—
217C22⋊D53—216no—
218C2×Dic53—22no—
219C2×Dic53—22no—
220C2×Dic53—24no—
221C2×Dic53—24no—
222C2×Dic53—24no—
223C2×Dic53—22no—
224C2×Dic53—42no—
225C2×Dic53—42no—
226C2×Dic53—42no—
227C2×Dic53—42no—
228C2×Dic53—42no—
229C2×Dic53C2×C22⋊C414yes—
230C2×Dic53C2×C4⋊C414yescomplemented
231C4×D53—28no—
232C4×D53—28no—
233C4×D53—44no—
234C4×D53—44no—
235C2×#1799—531nomaximal, Sylow
236C22×C53—24no—
237C22×C53—24no—
238C22×C53—21no—
239C22×C53—21no—
240C22×C53—21no—
241C22×C53—21no—
242C22×C53—24no—
243C22×C53—44no—
244C22×C53—44no—
245C22×C53—48no—
246C22×C53—48no—
247C22×C53—42no—
248C22×C53—42no—
249C22×C53#38012yes—
250C22×C53#38012yes—
251C22×C53C2×#9912yes—
252C22×C53#41012yes—
253C22×C53C22×C22⋊C411yescharacteristic
254C22×C53C22×C4⋊C411yescharacteristic
255C22×C53C2×#11711yescharacteristic
256C2×D53—28no—
257C2×D53—28no—
258C2×D53—22no—
259C2×D53—22no—
260C2×D53—22no—
261C2×D53—22no—
262C2×D53—42no—
263C2×D53—42no—
264C2×D53—44no—
265C2×D53—44no—
266C2×D53—41no—
267C2×D53—41no—
268C2×D53—42no—
269C2×D53#38014yes—
270C2×D53C2×#12112yes—
271C2×D53C2×C22⋊Q812yes—
272#6157—88no—
273C4×C53—22no—
274C4×C53—42no—
275C4×C53C2×#9914yescomplemented
276Dic53—24no—
277Dic53—24no—
278Dic53—42no—
279C22×C4×D4—531no—
280C22×#99—532no—
281C2×#380—532no—
282#1799—538no—
283C2×#342—532no—
284C2×C53—21no—
285C2×C53—21no—
286C2×C53—22no—
287C2×C53—44no—
288C2×C53#179914yescomplemented
289C2×C53C2×#38011yescharacteristic
290C2×C53C22×#9911yescharacteristic
291C2×C53C2×#41011yescharacteristic
292D53—22no—
293D53—22no—
294D53—24no—
295C23×D4—531no—
296C24×C4—531no—
297C24×C4#615711yescharacteristic, complemented
298C2×C4×D4—534no—
299C2×C4×D4—534no—
300C2×C4×D4—1062no—
301C2×C4×D4—1062no—
302C2×C4×D4—1064no—
303C22×C4⋊C4—531no—
304C22×C4⋊C4—1062no—
305C22×C22⋊C4—531no—
306C22×C22⋊C4—531no—
307C22×C22⋊C4—534no—
308C22×C42—531no—
309C2×#99—538no—
310C2×#99—1068no—
311#380—538no—
312#380—1064no—
313C2×#95—1062no—
314#342—538no—
315C53C2×#179911yescharacteristic, complemented, Sylow
316C25—531no—
317C25C2×#615711yescharacteristic
318C22×D4—534no—
319C22×D4—534no—
320C22×D4—532no—
321C22×D4—532no—
322C22×D4—1062no—
323C22×D4—1062no—
324C22×D4—1064no—
325C23×C4—22no—
326C23×C4—21no—
327C23×C4—21no—
328C23×C4—44no—
329C23×C4—531no—
330C23×C4—531no—
331C23×C4—532no—
332C23×C4—532no—
333C23×C4—534no—
334C23×C4—532no—
335C23×C4—532no—
336C23×C4—531no—
337C23×C4—1064no—
338C23×C4—2124no—
339C23×C4C2×#615711yescharacteristic
340C23×C4C2×#615711yescharacteristic
341C23×C4C2×#615714yescomplemented
342C4×D4—1068no—
343C4×D4—1068no—
344C4×D4—1068no—
345C4×D4—1068no—
346C2×C4⋊C4—534no—
347C2×C4⋊C4—1062no—
348C2×C4⋊C4—1062no—
349C2×C4⋊C4—1068no—
350C2×C4⋊C4—2128no—
351C2×C22⋊C4—534no—
352C2×C22⋊C4—532no—
353C2×C22⋊C4—532no—
354C2×C22⋊C4—538no—
355C2×C22⋊C4—1062no—
356C2×C22⋊C4—1062no—
357C2×C22⋊C4—1062no—
358C2×C22⋊C4—1062no—
359C2×C22⋊C4—1068no—
360C2×C22⋊C4—1068no—
361C2×C22⋊C4—1064no—
362C2×C42—534no—
363C2×C42—1062no—
364C2×C42—1061no—
365C2×C42—1061no—
366#99—10616no—
367#95—1068no—
368C24—21no—
369C24—21no—
370C24—22no—
371C24—42no—
372C24—42no—
373C24—532no—
374C24—532no—
375C24—532no—
376C24—532no—
377C24—534no—
378C24—531no—
379C24—531no—
380C24—2122no—
381C24—2122no—
382C24C4⋊#615714yescomplemented
383C24C22×#615711yescharacteristic
384C24C22×#615712yes—
385C2×D4—538no—
386C2×D4—534no—
387C2×D4—534no—
388C2×D4—1064no—
389C2×D4—1064no—
390C2×D4—1068no—
391C2×D4—1068no—
392C2×D4—1068no—
393C2×D4—1068no—
394C2×D4—1064no—
395C2×D4—1064no—
396C22×C4—24no—
397C22×C4—21no—
398C22×C4—21no—
399C22×C4—24no—
400C22×C4—24no—
401C22×C4—44no—
402C22×C4—48no—
403C22×C4—48no—
404C22×C4—44no—
405C22×C4—42no—
406C22×C4—532no—
407C22×C4—534no—
408C22×C4—532no—
409C22×C4—532no—
410C22×C4—532no—
411C22×C4—534no—
412C22×C4—534no—
413C22×C4—534no—
414C22×C4—534no—
415C22×C4—532no—
416C22×C4—1061no—
417C22×C4—1061no—
418C22×C4—1062no—
419C22×C4—1068no—
420C22×C4—1061no—
421C22×C4—1061no—
422C22×C4—1068no—
423C22×C4—2128no—
424C22×C4—2124no—
425C22×C4—2124no—
426C22×C4—2124no—
427C22×C4—2122no—
428C22×C4—2128no—
429C22×C4—2128no—
430C22×C4—2128no—
431C22×C4—2122no—
432C22×C4C22⋊#615712yes—
433C22×C4C22⋊#615711yescharacteristic
434C22×C4C22⋊#615711yescharacteristic
435C22×C4C22×#615712yes—
436C22×C4C22×#615712yes—
437C4⋊C4—1068no—
438C4⋊C4—21216no—
439C22⋊C4—1064no—
440C22⋊C4—1064no—
441C22⋊C4—1068no—
442C22⋊C4—1068no—
443C22⋊C4—1068no—
444C22⋊C4—2128no—
445C42—1064no—
446C42—1064no—
447C23—22no—
448C23—22no—
449C23—24no—
450C23—24no—
451C23—22no—
452C23—22no—
453C23—42no—
454C23—48no—
455C23—42no—
456C23—42no—
457C23—48no—
458C23—42no—
459C23—42no—
460C23—42no—
461C23—534no—
462C23—534no—
463C23—532no—
464C23—532no—
465C23—534no—
466C23—534no—
467C23—532no—
468C23—531no—
469C23—531no—
470C23—532no—
471C23—532no—
472C23—1064no—
473C23—1064no—
474C23—2122no—
475C23—2121no—
476C23—2121no—
477C23—2124no—
478C23—2124no—
479C23—2128no—
480C23—2122no—
481C23—2121no—
482C23—2121no—
483C23C2×C22⋊#615711yescharacteristic
484C23C2×C22⋊#615711yescharacteristic
485C23C2×C4⋊#615712yes—
486C23C2×C4⋊#615714yescomplemented
487C23#1313329612yes—
488C23C23×#615711yescharacteristic
489D4—1068no—
490D4—1068no—
491D4—10616no—
492C2×C4—22no—
493C2×C4—24no—
494C2×C4—24no—
495C2×C4—22no—
496C2×C4—48no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—48no—
500C2×C4—42no—
501C2×C4—534no—
502C2×C4—534no—
503C2×C4—1064no—
504C2×C4—1064no—
505C2×C4—1064no—
506C2×C4—1062no—
507C2×C4—1062no—
508C2×C4—1068no—
509C2×C4—1068no—
510C2×C4—1062no—
511C2×C4—42416no—
512C2×C4—2124no—
513C2×C4—2124no—
514C2×C4—2122no—
515C2×C4—2122no—
516C2×C4—2128no—
517C2×C4—2128no—
518C2×C4—2128no—
519C2×C4—2122no—
520C2×C4—2122no—
521C2×C4—2122no—
522C2×C4#1313327514yescomplemented
523C2×C4C2×C22⋊#615712yes—
524C2×C4C2×C22⋊#615712yes—
525C22—24no—
526C22—24no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—24no—
532C22—44no—
533C22—48no—
534C22—44no—
535C22—48no—
536C22—42no—
537C22—42no—
538C22—532no—
539C22—532no—
540C22—534no—
541C22—1068no—
542C22—1068no—
543C22—1062no—
544C22—1062no—
545C22—1062no—
546C22—1062no—
547C22—2124no—
548C22—2124no—
549C22—2121no—
550C22—2122no—
551C22—2121no—
552C22—2122no—
553C22—2122no—
554C22#1681156512yes—
555C22#1681156512yes—
556C22C2×#1313327512yes—
557C22#1681160212yes—
558C22C22×C22⋊#615711yescharacteristic
559C22C22×C4⋊#615711yescharacteristic
560C22C2×#1313329611yescharacteristic
561C4—22no—
562C4—42no—
563C4—1064no—
564C4—1064no—
565C4—4248no—
566C4—2122no—
567C4C2×#1313327514yescomplemented
568C2—22no—
569C2—21no—
570C2—21no—
571C2—44no—
572C2—1062no—
573C2—1062no—
574C2—1064no—
575C2#2029057514yesdirect product factor
576C2C2×#1681156511yescharacteristic
577C2C22×#1313327511yescharacteristic
578C2C2×#1681160211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC23×C53C23×C4
FrattiniC23C23×#6157
FittingC24×C4×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC23×C22⋊D53C22
cosocleC23×D53C24
hypercenterC24×C4#6157
hypocenterC53C2×#1799
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C24×C4

Complements

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

NH ≅ G/Ntype
2. C2×#16811565C2semidirect
3. #20290575C2direct (N × H)
4. C2×#16811501C2semidirect
11. #16811565C22semidirect
13. #16811501C22semidirect
19. C2×C4×C22⋊D53C4semidirect
21. C22×#11719919C4semidirect
22. C22×C22⋊Dic53C4semidirect
24. C22×C4×Dic53C4semidirect
25. C24×C4×C53C4semidirect
58. C2×#11719919C2×C4semidirect
61. C2×C22⋊Dic53C2×C4semidirect
69. C2×C4×Dic53C2×C4semidirect
77. C23×C4×C53C2×C4semidirect
83. C24×C53C4⋊C4semidirect
122. C2×C22⋊D53C4⋊C4semidirect
134. C22×Dic53C4⋊C4semidirect
174. C23×C53C2×C4⋊C4semidirect
212. C2×C4×C53#99semidirect
230. C2×Dic53C2×C4⋊C4semidirect
275. C4×C53C2×#99semidirect
288. C2×C53#1799semidirect
297. C24×C4#6157semidirect
315. C53C2×#1799semidirect
341. C23×C4C2×#6157semidirect
382. C24C4⋊#6157semidirect
486. C23C2×C4⋊#6157semidirect
522. C2×C4#13133275semidirect
567. C4C2×#13133275semidirect
575. C2#20290575direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131332752
C2×#168115652
#202905752
C2×#168115012
C22×C4×C22⋊D532
C2×#179953