Finite Group Databasebeta

← C2×#20290576 #28319040

Subgroups — C2×#20290576 #28319040

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×#13133276C212yesmaximal
2C2×#16811565C212yesmaximal, complemented
3#20290576C218yesmaximal, direct product factor
4C2×#16811501C212yesmaximal, complemented
5C22×C4×D212C211yesmaximal, characteristic
6C22×C4⋊#6157—22no—
7C22×C22⋊#6157C2214yes—
8C2×#13133276—28no—
9C2×#13133276C2218yes—
10#16811565—24no—
11#16811565C2218yescomplemented
12C2×(C2×#6157)⋊C4—22no—
13#16811501C2218yescomplemented
14C23×D212C2211yescharacteristic
15C23×C4×D53C411yescharacteristic, complemented
16C23×C4×D53C2211yescharacteristic
17C2×C4×D212—22no—
18C2×C4×D212—22no—
19C2×C4×D212—24no—
20C2×C4×D212C414yescomplemented
21C2×C4×D212C2214yes—
22C22×C4⋊Dic53C411yescharacteristic, complemented
23C22×#11719921C411yescharacteristic, complemented
24C22×#11719921C2211yescharacteristic
25C22×C42×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#13133276—216no—
34(C2×#6157)⋊C4—28no—
35C24×D53C2×C411yescharacteristic
36C24×D53C2311yescharacteristic
37C22×D212—22no—
38C22×D212—22no—
39C22×D212—24no—
40C22×D212C2×C412yes—
41C22×D212D414yes—
42C22×D212Q814yes—
43C22×D212C2312yes—
44C23×Dic53C2×C411yescharacteristic
45C23×Dic53C2×C411yescharacteristic
46C22×C4×D53—21no—
47C22×C4×D53—22no—
48C22×C4×D53—21no—
49C22×C4×D53—44no—
50C22×C4×D53—44no—
51C22×C4×D53C2×C412yes—
52C22×C4×D53C2×C414yescomplemented
53C22×C4×D53D412yes—
54C22×C4×D53D412yes—
55C22×C4×D53D414yes—
56C22×C4×D53C2312yes—
57C4×D212—28no—
58C4×D212—28no—
59C4×D212—28no—
60C4×D212—28no—
61C2×C4⋊Dic53—22no—
62C2×C4⋊Dic53—22no—
63C2×C4⋊Dic53C2×C414yescomplemented
64C2×#11719921—22no—
65C2×#11719921—22no—
66C2×#11719921—22no—
67C2×#11719921—22no—
68C2×#11719921C2×C412yes—
69C2×#11719921C2×C414yescomplemented
70C2×#11719921C2312yes—
71C23×C4×C53C2×C411yescharacteristic
72C23×C4×C53C2×C411yescharacteristic
73C23×C4×C53C2×C411yescharacteristic
74C2×C42×C53—22no—
75C2×C42×C53—21no—
76C2×C42×C53—21no—
77C2×C42×C53C2×C414yescomplemented
78C24×C53C22×C411yescharacteristic
79C23×D53—21no—
80C23×D53—21no—
81C23×D53—22no—
82C23×D53—42no—
83C23×D53—42no—
84C23×D53—42no—
85C23×D53—42no—
86C23×D53C4⋊C414yescomplemented
87C23×D53C22×C412yes—
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—21no—
100C22×C4×C53—21no—
101C22×C4×C53—22no—
102C22×C4×C53—24no—
103C22×C4×C53—21no—
104C22×C4×C53—21no—
105C22×C4×C53—42no—
106C22×C4×C53C22⋊C411yescharacteristic
107C22×C4×C53C22⋊C412yes—
108C22×C4×C53C22⋊C411yescharacteristic
109C22×C4×C53C4⋊C414yescomplemented
110C22×C4×C53C22×C412yes—
111C22×C4×C53C22×C412yes—
112C22×C4×C53C22×C412yes—
113C22×Dic53—21no—
114C22×Dic53—21no—
115C22×Dic53—42no—
116C22×Dic53—42no—
117C22×Dic53C22⋊C412yes—
118C22×Dic53C22⋊C412yes—
119C22×Dic53C22×C412yes—
120C22×Dic53C22×C412yes—
121C2×D212—24no—
122C2×D212—24no—
123C2×D212—28no—
124C2×D212—28no—
125C2×D212—28no—
126C2×D212—28no—
127C2×D212—24no—
128C2×D212—24no—
129C2×D212C4⋊C418yescomplemented
130C2×D212C2×D414yes—
131C2×D212C2×Q814yes—
132C2×C4×D53—28no—
133C2×C4×D53—24no—
134C2×C4×D53—24no—
135C2×C4×D53—44no—
136C2×C4×D53—48no—
137C2×C4×D53—44no—
138C2×C4×D53—44no—
139C2×C4×D53—44no—
140C2×C4×D53—48no—
141C2×C4×D53—44no—
142C2×C4×D53—48no—
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—
151C42×C53—24no—
152C42×C53—24no—
153#11719921—24no—
154#11719921—24no—
155#11719921—28no—
156C4⋊Dic53—28no—
157C23×C53—22no—
158C23×C53—22no—
159C23×C53C2×C22⋊C411yescharacteristic
160C23×C53C2×C22⋊C411yescharacteristic
161C23×C53C2×C4⋊C412yes—
162C23×C53#11712yes—
163C23×C53C23×C411yescharacteristic
164C22×D53—24no—
165C22×D53—24no—
166C22×D53—22no—
167C22×D53—22no—
168C22×D53—24no—
169C22×D53—24no—
170C22×D53—41no—
171C22×D53—42no—
172C22×D53—41no—
173C22×D53—41no—
174C22×D53—44no—
175C22×D53—42no—
176C22×D53—44no—
177C22×D53—48no—
178C22×D53—41no—
179C22×D53—42no—
180C22×D53—48no—
181C22×D53—42no—
182C22×D53—42no—
183C22×D53—42no—
184C22×D53—42no—
185C22×D53—48no—
186C22×D53—42no—
187C22×D53C2×C22⋊C412yes—
188C22×D53C2×C4⋊C412yes—
189C22×D53C2×C4⋊C414yescomplemented
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—22no—
205C2×C4×C53—24no—
206C2×C4×C53—24no—
207C2×C4×C53—22no—
208C2×C4×C53—24no—
209C2×C4×C53—24no—
210C2×C4×C53—22no—
211C2×C4×C53—42no—
212C2×C4×C53—42no—
213C2×C4×C53—42no—
214C2×C4×C53#9914yescomplemented
215C2×C4×C53C2×C22⋊C412yes—
216C2×C4×C53C2×C22⋊C412yes—
217C2×C4×C53C2×C4⋊C414yescomplemented
218C2×Dic53—24no—
219C2×Dic53—22no—
220C2×Dic53—42no—
221C2×Dic53—42no—
222C2×Dic53—42no—
223C2×Dic53—42no—
224C2×Dic53—42no—
225C2×Dic53C2×C22⋊C414yes—
226D212—28no—
227D212—28no—
228D212—216no—
229C4×D53—28no—
230C4×D53—28no—
231C4×D53—44no—
232C4×D53—44no—
233C4×D53—48no—
234C4×D53—48no—
235C2×#1799—531nomaximal, Sylow
236C22×C53—24no—
237C22×C53—24no—
238C22×C53—21no—
239C22×C53—21no—
240C22×C53—21no—
241C22×C53—21no—
242C22×C53#38012yes—
243C22×C53#38012yes—
244C22×C53C2×#9912yes—
245C22×C53#41012yes—
246C22×C53C22×C22⋊C411yescharacteristic
247C22×C53C22×C4⋊C411yescharacteristic
248C22×C53C2×#11711yescharacteristic
249C2×D53—28no—
250C2×D53—28no—
251C2×D53—22no—
252C2×D53—22no—
253C2×D53—22no—
254C2×D53—22no—
255C2×D53—24no—
256C2×D53—42no—
257C2×D53—42no—
258C2×D53—44no—
259C2×D53—44no—
260C2×D53—41no—
261C2×D53—41no—
262C2×D53—42no—
263C2×D53—44no—
264C2×D53—48no—
265C2×D53—48no—
266C2×D53—42no—
267C2×D53—42no—
268C2×D53—44no—
269C2×D53#38014yes—
270C2×D53C2×#12112yes—
271C2×D53C2×C22⋊Q812yes—
272#6157—88no—
273C4×C53—22no—
274C4×C53—24no—
275C4×C53—42no—
276C4×C53C2×#9914yescomplemented
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#179914yescomplemented
288C2×C53C2×#38011yescharacteristic
289C2×C53C22×#9911yescharacteristic
290C2×C53C2×#41011yescharacteristic
291D53—22no—
292D53—22no—
293D53—24no—
294D53—44no—
295C23×D4—531no—
296C24×C4—531no—
297C24×C4—531no—
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#615711yescharacteristic, complemented
309C2×#99—538no—
310C2×#99—1068no—
311#380—538no—
312#380—1064no—
313C2×#95—1062no—
314#342—538no—
315C53C2×#179911yescharacteristic, complemented, Sylow
316C25—531no—
317C25—531no—
318C22×D4—534no—
319C22×D4—534no—
320C22×D4—532no—
321C22×D4—532no—
322C22×D4—1062no—
323C22×D4—1062no—
324C22×D4—1064no—
325C23×C4—531no—
326C23×C4—532no—
327C23×C4—532no—
328C23×C4—534no—
329C23×C4—532no—
330C23×C4—532no—
331C23×C4—531no—
332C23×C4—534no—
333C23×C4—1062no—
334C23×C4—1061no—
335C23×C4—1061no—
336C23×C4—1064no—
337C23×C4—2124no—
338C23×C4—2124no—
339C23×C4C2×#615711yescharacteristic
340C23×C4C2×#615711yescharacteristic
341C23×C4C2×#615711yescharacteristic
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—532no—
352C2×C22⋊C4—532no—
353C2×C22⋊C4—534no—
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—22no—
363C2×C42—21no—
364C2×C42—21no—
365C2×C42C2×#615714yescomplemented
366#99—10616no—
367#95—1068no—
368C24—532no—
369C24—532no—
370C24—532no—
371C24—532no—
372C24—534no—
373C24—531no—
374C24—531no—
375C24—534no—
376C24—532no—
377C24—1061no—
378C24—1061no—
379C24—1062no—
380C24—2122no—
381C24—2122no—
382C24—2122no—
383C24—2122no—
384C24C22×#615711yescharacteristic
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—21no—
397C22×C4—21no—
398C22×C4—22no—
399C22×C4—24no—
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—42no—
403C22×C4—532no—
404C22×C4—532no—
405C22×C4—532no—
406C22×C4—534no—
407C22×C4—534no—
408C22×C4—534no—
409C22×C4—534no—
410C22×C4—532no—
411C22×C4—1068no—
412C22×C4—1061no—
413C22×C4—1061no—
414C22×C4—1064no—
415C22×C4—1064no—
416C22×C4—1068no—
417C22×C4—2128no—
418C22×C4—2124no—
419C22×C4—2124no—
420C22×C4—2124no—
421C22×C4—2124no—
422C22×C4—2128no—
423C22×C4—2128no—
424C22×C4—2122no—
425C22×C4—2128no—
426C22×C4—2128no—
427C22×C4—2128no—
428C22×C4—2124no—
429C22×C4—2122no—
430C22×C4C4⋊#615714yescomplemented
431C22×C4C22⋊#615712yes—
432C22×C4C22⋊#615711yescharacteristic
433C22×C4C22⋊#615711yescharacteristic
434C22×C4C22×#615712yes—
435C22×C4C22×#615712yes—
436C22×C4C22×#615712yes—
437C4⋊C4—1068no—
438C4⋊C4—21216no—
439C22⋊C4—1068no—
440C22⋊C4—1064no—
441C22⋊C4—1064no—
442C22⋊C4—1068no—
443C22⋊C4—1068no—
444C22⋊C4—2128no—
445C42—24no—
446C42—24no—
447C23—22no—
448C23—22no—
449C23—534no—
450C23—534no—
451C23—532no—
452C23—532no—
453C23—534no—
454C23—534no—
455C23—532no—
456C23—531no—
457C23—531no—
458C23—532no—
459C23—532no—
460C23—534no—
461C23—1064no—
462C23—1064no—
463C23—1064no—
464C23—1064no—
465C23—1062no—
466C23—1062no—
467C23—2122no—
468C23—2121no—
469C23—2121no—
470C23—2124no—
471C23—2124no—
472C23—2128no—
473C23—2122no—
474C23—2121no—
475C23—2121no—
476C23—2122no—
477C23—2128no—
478C23—2122no—
479C23—2122no—
480C23—2128no—
481C23—2122no—
482C23—2122no—
483C23—2122no—
484C23C2×C22⋊#615711yescharacteristic
485C23C2×C22⋊#615711yescharacteristic
486C23C2×C4⋊#615712yes—
487C23#1313329612yes—
488C23C23×#615711yescharacteristic
489D4—1068no—
490D4—1068no—
491D4—10616no—
492C2×C4—24no—
493C2×C4—24no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—22no—
497C2×C4—24no—
498C2×C4—24no—
499C2×C4—22no—
500C2×C4—42no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—534no—
504C2×C4—1064no—
505C2×C4—1068no—
506C2×C4—1068no—
507C2×C4—1062no—
508C2×C4—42416no—
509C2×C4—2124no—
510C2×C4—2124no—
511C2×C4—2128no—
512C2×C4—2122no—
513C2×C4—2122no—
514C2×C4—2128no—
515C2×C4—2128no—
516C2×C4—2128no—
517C2×C4—2122no—
518C2×C4—2122no—
519C2×C4—2128no—
520C2×C4—2122no—
521C2×C4#1313327614yescomplemented
522C2×C4C2×C22⋊#615712yes—
523C2×C4C2×C22⋊#615712yes—
524C2×C4C2×C4⋊#615714yescomplemented
525C22—24no—
526C22—24no—
527C22—21no—
528C22—21no—
529C22—21no—
530C22—21no—
531C22—532no—
532C22—532no—
533C22—534no—
534C22—1068no—
535C22—1068no—
536C22—1062no—
537C22—1062no—
538C22—1062no—
539C22—1062no—
540C22—1064no—
541C22—2124no—
542C22—2124no—
543C22—2121no—
544C22—2122no—
545C22—2121no—
546C22—2122no—
547C22—2122no—
548C22—2124no—
549C22—2128no—
550C22—2124no—
551C22—2128no—
552C22—2122no—
553C22—2122no—
554C22#1681156512yes—
555C22#1681156512yes—
556C22C2×#1313327612yes—
557C22#1681160212yes—
558C22C22×C22⋊#615711yescharacteristic
559C22C22×C4⋊#615711yescharacteristic
560C22C2×#1313329611yescharacteristic
561C4—24no—
562C4—22no—
563C4—42no—
564C4—1064no—
565C4—4248no—
566C4—2122no—
567C4C2×#1313327614yescomplemented
568C2—22no—
569C2—21no—
570C2—21no—
571C2—1062no—
572C2—1062no—
573C2—1064no—
574C2—2124no—
575C2#2029057614yesdirect product factor
576C2C2×#1681156511yescharacteristic
577C2C22×#1313327611yescharacteristic
578C2C2×#1681160211yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC23×C53C23×C4
FrattiniC23C23×#6157
FittingC22×C42×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC23×D212C22
cosocleC23×D53C24
hypercenterC22×C42#6157
hypocenterC53C2×#1799
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C42

Complements

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

NH ≅ G/Ntype
2. C2×#16811565C2semidirect
3. #20290576C2direct (N × H)
4. C2×#16811501C2semidirect
11. #16811565C22semidirect
13. #16811501C22semidirect
15. C23×C4×D53C4semidirect
20. C2×C4×D212C4semidirect
22. C22×C4⋊Dic53C4semidirect
23. C22×#11719921C4semidirect
25. C22×C42×C53C4semidirect
52. C22×C4×D53C2×C4semidirect
63. C2×C4⋊Dic53C2×C4semidirect
69. C2×#11719921C2×C4semidirect
77. C2×C42×C53C2×C4semidirect
86. C23×D53C4⋊C4semidirect
109. C22×C4×C53C4⋊C4semidirect
129. C2×D212C4⋊C4semidirect
189. C22×D53C2×C4⋊C4semidirect
214. C2×C4×C53#99semidirect
217. C2×C4×C53C2×C4⋊C4semidirect
276. C4×C53C2×#99semidirect
287. C2×C53#1799semidirect
308. C22×C42#6157semidirect
315. C53C2×#1799semidirect
365. C2×C42C2×#6157semidirect
430. C22×C4C4⋊#6157semidirect
521. C2×C4#13133276semidirect
524. C2×C4C2×C4⋊#6157semidirect
567. C4C2×#13133276semidirect
575. C2#20290576direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131332762
C2×#168115652
#202905762
C2×#168115012
C22×C4×D2122
C2×#179953