Finite Group Databasebeta

← C2×#20290742 #28319810

Subgroups — C2×#20290742 #28319810

Order 13568.

1854 conjugacy classes of subgroups.

The subgroup lattice has 636 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C23×#11719940C211yesmaximal, characteristic
2C22×#13133278C212yesmaximal, complemented
3C2×#16811593C211yesmaximal, characteristic
4#20290742C218yesmaximal, direct product factor
5C2×#16811503C212yesmaximal, complemented
6C22×#13133194C211yesmaximal, characteristic
7C22×#11719940—28no—
8C22×#11719940—22no—
9C22×#11719940—21no—
10C22×#11719940C2212yes—
11C22×#11719940C2214yes—
12C22×#11719940C2212yes—
13C23×#88324—21no—
14C2×#13133278—28no—
15C2×#13133278C2218yescomplemented
16#16811593—22no—
17#16811593C2214yes—
18C2×C22⋊#88324—21no—
19C2×C22⋊#88324—21no—
20#16811503C2218yescomplemented
21C23×Dic106C2211yescharacteristic
22C24×Dic53C2211yescharacteristic
23C2×#13133194—21no—
24C2×#13133194—21no—
25C2×#13133194—22no—
26C2×#13133194—21no—
27C2×#13133194—21no—
28C2×#13133194—22no—
29C2×#13133194C414yes—
30C2×#13133194C2214yes—
31C22×C4⋊Dic53C411yescharacteristic
32C22×#11719919C411yescharacteristic
33C22×#11719919C2211yescharacteristic
34C22×C22⋊Dic53C411yescharacteristic
35C22×C22⋊C4×C53C411yescharacteristic
36C2×#11719940—216no—
37C2×#11719940—28no—
38C2×#11719940—24no—
39C2×#11719940—24no—
40C2×#11719940—22no—
41C2×#11719940—216no—
42C2×#11719940—22no—
43C2×#11719940—24no—
44C2×#11719940—24no—
45C2×#11719940D418yes—
46C2×#11719940C2314yes—
47C2×#11719940C2314yes—
48C22×#88324—28no—
49C22×#88324—24no—
50C22×#88324—22no—
51C22×#88324—22no—
52#13133278—216no—
53C22⋊#88324—24no—
54C22⋊#88324—24no—
55C22×Dic106—24no—
56C22×Dic106—21no—
57C22×Dic106—21no—
58C22×Dic106—21no—
59C22×Dic106—21no—
60C22×Dic106—24no—
61C22×Dic106C2×C412yes—
62C22×Dic106C2312yes—
63C23×Dic53—24no—
64C23×Dic53—24no—
65C23×Dic53C2×C411yescharacteristic
66C23×Dic53C2×C411yescharacteristic
67C23×Dic53C2×C412yes—
68C23×Dic53C2×C411yescharacteristic
69C23×Dic53D411yescharacteristic
70C23×Dic53D411yescharacteristic
71C23×Dic53D412yes—
72C23×Dic53D411yescharacteristic
73C23×Dic53D412yes—
74C23×Dic53D411yescharacteristic
75C23×Dic53C2312yes—
76C23×Dic53C2311yescharacteristic
77#13133194—24no—
78#13133194—24no—
79#13133194—24no—
80#13133194—24no—
81#13133194—24no—
82#13133194—24no—
83#13133194—24no—
84#13133194—24no—
85C2×C4⋊Dic53—22no—
86C2×C4⋊Dic53—21no—
87C2×C4⋊Dic53—21no—
88C2×C4⋊Dic53C2×C414yes—
89C2×#11719919—22no—
90C2×#11719919—21no—
91C2×#11719919—21no—
92C2×#11719919—21no—
93C2×#11719919—21no—
94C2×#11719919—21no—
95C2×#11719919—21no—
96C2×#11719919C2×C414yes—
97C2×#11719919C2×C412yes—
98C2×#11719919C2312yes—
99C2×C22⋊Dic53—21no—
100C2×C22⋊Dic53—22no—
101C2×C22⋊Dic53—21no—
102C2×C22⋊Dic53C2×C414yes—
103C25×C53C2×C411yescharacteristic
104C23×C4×C53C2×C411yescharacteristic
105C23×C4×C53C2×C411yescharacteristic
106C2×C22⋊C4×C53—21no—
107C2×C22⋊C4×C53—22no—
108C2×C22⋊C4×C53—21no—
109C2×C22⋊C4×C53C2×C414yes—
110C24×C53—24no—
111C24×C53—24no—
112C24×C53—42no—
113C24×C53C22⋊C411yescharacteristic
114C24×C53C22⋊C412yes—
115C24×C53C22⋊C411yescharacteristic
116C24×C53C22×C412yes—
117C24×C53C22×C411yescharacteristic
118C22×C4×C53—21no—
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—24no—
123C22×C4×C53—42no—
124C22×C4×C53C22×C412yes—
125C22×C4×C53C22×C412yes—
126C22×Dic53—24no—
127C22×Dic53—28no—
128C22×Dic53—22no—
129C22×Dic53—24no—
130C22×Dic53—22no—
131C22×Dic53—24no—
132C22×Dic53—24no—
133C22×Dic53—28no—
134C22×Dic53—22no—
135C22×Dic53—24no—
136C22×Dic53—22no—
137C22×Dic53—24no—
138C22×Dic53—24no—
139C22×Dic53—21no—
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C22×Dic53—24no—
144C22×Dic53—42no—
145C22×Dic53C22⋊C412yes—
146C22×Dic53C22⋊C411yescharacteristic
147C22×Dic53C22⋊C411yescharacteristic
148C22×Dic53C22⋊C412yes—
149C22×Dic53C22⋊C412yes—
150C22×Dic53C22⋊C412yes—
151C22×Dic53C22⋊C412yes—
152C22×Dic53C22×C412yes—
153C22×Dic53C22×C412yes—
154C22×Dic53C22×C412yes—
155C22×Dic53C22×C411yescharacteristic
156C22×Dic53C2×D412yes—
157C22×Dic53C2×D412yes—
158C22×Dic53C2×D412yes—
159C22×Dic53C2×D412yes—
160C22×Dic53C2×D411yescharacteristic
161C22×Dic53C2×D411yescharacteristic
162C22×Dic53C2×D411yescharacteristic
163C22×Dic53C2×D411yescharacteristic
164C22×Dic53C2×D414yes—
165C22×Dic53#4114yes—
166C22×Dic53C2411yescharacteristic
167C2×Dic106—22no—
168C2×Dic106—22no—
169C2×Dic106—28no—
170C2×Dic106—22no—
171C2×Dic106—22no—
172C2×Dic106—24no—
173C2×Dic106—24no—
174C2×Dic106—42no—
175C2×Dic106—42no—
176C2×Dic106—44no—
177C2×Dic106—44no—
178C2×Dic106—42no—
179C2×Dic106—42no—
180#11719940—216no—
181#11719940—28no—
182#11719940—24no—
183#11719940—24no—
184#11719940—24no—
185#11719940—44no—
186#11719940C2×D418yes—
187C2×#88324—216no—
188C2×#88324—28no—
189C2×#88324—24no—
190C2×#88324—24no—
191C2×#88324—44no—
192C22⋊C4×C53—24no—
193C22⋊C4×C53—24no—
194C22⋊Dic53—24no—
195C22⋊Dic53—24no—
196C4⋊Dic53—24no—
197C4⋊Dic53—24no—
198#11719919—24no—
199#11719919—24no—
200#11719919—22no—
201#11719919—22no—
202#11719919—22no—
203#11719919—22no—
204C23×C53—22no—
205C23×C53—22no—
206C23×C53—22no—
207C23×C53—22no—
208C23×C53—24no—
209C23×C53—24no—
210C23×C53—22no—
211C23×C53—22no—
212C23×C53—24no—
213C23×C53—24no—
214C23×C53—22no—
215C23×C53—42no—
216C23×C53—42no—
217C23×C53—416no—
218C23×C53#10114yescomplemented
219C23×C53C2×C22⋊C412yes—
220C23×C53C2×C22⋊C411yescharacteristic
221C23×C53C2×C22⋊C412yes—
222C23×C53C2×C22⋊C411yescharacteristic
223C23×C53C4×D414yes—
224C23×C53C23×C411yescharacteristic
225C2×C4×C53—22no—
226C2×C4×C53—22no—
227C2×C4×C53—24no—
228C2×C4×C53—42no—
229C2×C4×C53—42no—
230C2×C4×C53—42no—
231C2×C4×C53—42no—
232C2×C4×C53—42no—
233C2×Dic53—24no—
234C2×Dic53—24no—
235C2×Dic53—22no—
236C2×Dic53—22no—
237C2×Dic53—22no—
238C2×Dic53—22no—
239C2×Dic53—24no—
240C2×Dic53—24no—
241C2×Dic53—22no—
242C2×Dic53—22no—
243C2×Dic53—24no—
244C2×Dic53—24no—
245C2×Dic53—24no—
246C2×Dic53—24no—
247C2×Dic53—22no—
248C2×Dic53—22no—
249C2×Dic53—24no—
250C2×Dic53—42no—
251C2×Dic53—42no—
252C2×Dic53—42no—
253C2×Dic53—42no—
254C2×Dic53—42no—
255C2×Dic53C2×C22⋊C412yes—
256C2×Dic53C2×C22⋊C412yes—
257C2×Dic53C2×C22⋊C411yescharacteristic
258C2×Dic53C2×C22⋊C411yescharacteristic
259C2×Dic53C4×D414yes—
260C2×Dic53#12012yes—
261C2×Dic53#12012yes—
262C2×Dic53#12114yes—
263C2×Dic53#12114yes—
264C2×Dic53#12312yes—
265C2×Dic53#12312yes—
266C2×Dic53C22×D412yes—
267C2×Dic53C22×D411yescharacteristic
268C2×Dic53C22×D411yescharacteristic
269C2×Dic53C2×#4112yes—
270Dic106—42no—
271Dic106—44no—
272Dic106—42no—
273Dic106—44no—
274Dic106—44no—
275#88324—28no—
276#88324—44no—
277C2×#1901—531nomaximal, Sylow
278C22×C53—21no—
279C22×C53—21no—
280C22×C53—21no—
281C22×C53—21no—
282C22×C53—22no—
283C22×C53—28no—
284C22×C53—28no—
285C22×C53—22no—
286C22×C53—22no—
287C22×C53—22no—
288C22×C53—22no—
289C22×C53—48no—
290C22×C53—48no—
291C22×C53—44no—
292C22×C53#38612yes—
293C22×C53#38612yes—
294C22×C53C2×#10112yes—
295C22×C53C2×#10114yescomplemented
296C22×C53#41312yes—
297C22×C53C22×C22⋊C411yescharacteristic
298C22×C53C2×C4×D412yes—
299C4×C53—42no—
300C4×C53—42no—
301Dic53—22no—
302Dic53—22no—
303Dic53—42no—
304Dic53—42no—
305Dic53#38612yes—
306Dic53#38612yes—
307Dic53C2×#12011yescharacteristic
308Dic53C2×#12112yes—
309Dic53C2×#12311yescharacteristic
310C23×M16—531no—
311C22×C22⋊Q8—531no—
312C22×#101—532no—
313C2×#407—531no—
314#1901—538no—
315C2×#343—532no—
316C2×C53—22no—
317C2×C53—22no—
318C2×C53—24no—
319C2×C53—42no—
320C2×C53#190114yescomplemented
321C2×C53C2×#38611yescharacteristic
322C2×C53C22×#10111yescharacteristic
323C2×C53C2×#41311yescharacteristic
324C23×Q8—531no—
325C24×C4—531no—
326C22×M16—532no—
327C22×M16—532no—
328C22×M16—534no—
329C22×M16—1068no—
330C22×M16—1061no—
331C22×M16—1062no—
332C23×C8—1061no—
333C2×C22⋊Q8—534no—
334C2×C22⋊Q8—534no—
335C2×C22⋊Q8—1061no—
336C2×C22⋊Q8—1062no—
337C2×C22⋊Q8—1061no—
338C2×C22⋊Q8—1061no—
339C2×C22⋊Q8—1061no—
340C2×C22⋊Q8—1062no—
341C22×C4⋊C4—531no—
342C22×C4⋊C4—531no—
343C22×C4⋊C4—531no—
344C22×C22⋊C4—531no—
345C22×C22⋊C4#615711yescharacteristic
346C2×#101—538no—
347C2×#101—1068no—
348#407—534no—
349#407—1062no—
350C2×C22⋊C8—1061no—
351C2×C22⋊C8—1061no—
352#343—538no—
353C53C2×#190111yescharacteristic, complemented, Sylow
354C25C2×#615711yescharacteristic
355C22×Q8—532no—
356C22×Q8—532no—
357C22×Q8—1061no—
358C22×Q8—1061no—
359C22×Q8—1061no—
360C22×Q8—1061no—
361C22×Q8—1064no—
362C22×Q8—1064no—
363C23×C4—531no—
364C23×C4—531no—
365C23×C4—532no—
366C23×C4—531no—
367C23×C4—531no—
368C23×C4—531no—
369C23×C4—532no—
370C23×C4—531no—
371C23×C4—532no—
372C23×C4—532no—
373C23×C4—531no—
374C23×C4—531no—
375C23×C4—1064no—
376C23×C4—1064no—
377C23×C4C2×#615711yescharacteristic
378C23×C4C2×#615711yescharacteristic
379C2×M16—534no—
380C2×M16—538no—
381C2×M16—534no—
382C2×M16—10616no—
383C2×M16—1062no—
384C2×M16—1064no—
385C2×M16—1064no—
386C2×M16—1068no—
387C2×M16—10616no—
388C2×M16—1062no—
389C2×M16—1064no—
390C2×M16—1064no—
391C22×C8—1062no—
392C22×C8—1064no—
393C22×C8—1068no—
394C22×C8—1062no—
395C22⋊Q8—1064no—
396C22⋊Q8—1064no—
397C22⋊Q8—1064no—
398C22⋊Q8—1064no—
399C22⋊Q8—1064no—
400C22⋊Q8—1064no—
401C22⋊Q8—1064no—
402C22⋊Q8—1064no—
403C2×C4⋊C4—532no—
404C2×C4⋊C4—532no—
405C2×C4⋊C4—534no—
406C2×C4⋊C4—534no—
407C2×C4⋊C4—1061no—
408C2×C4⋊C4—1061no—
409C2×C4⋊C4—1062no—
410C2×C4⋊C4—1061no—
411C2×C4⋊C4—1061no—
412C2×C4⋊C4—1061no—
413C2×C4⋊C4—1061no—
414C2×C4⋊C4—1062no—
415C2×C4⋊C4—1061no—
416C2×C4⋊C4—1061no—
417C2×C22⋊C4—22no—
418C2×C22⋊C4—21no—
419C2×C22⋊C4—21no—
420C2×C22⋊C4—534no—
421C2×C22⋊C4—1062no—
422C2×C22⋊C4—1061no—
423C2×C22⋊C4—1061no—
424C2×C22⋊C4C2×#615714yes—
425#101—10616no—
426C22⋊C8—1064no—
427C22⋊C8—1064no—
428C24—24no—
429C24—24no—
430C24—42no—
431C24C22⋊#615711yescharacteristic
432C24C22⋊#615712yes—
433C24C22⋊#615711yescharacteristic
434C24C22×#615712yes—
435C24C22×#615711yescharacteristic
436C2×Q8—1064no—
437C2×Q8—1062no—
438C2×Q8—1062no—
439C2×Q8—1062no—
440C2×Q8—1062no—
441C2×Q8—1068no—
442C2×Q8—1064no—
443C2×Q8—2122no—
444C2×Q8—2122no—
445C2×Q8—2124no—
446C2×Q8—2124no—
447C2×Q8—2122no—
448C2×Q8—2122no—
449C22×C4—21no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—24no—
454C22×C4—42no—
455C22×C4—531no—
456C22×C4—531no—
457C22×C4—531no—
458C22×C4—531no—
459C22×C4—534no—
460C22×C4—534no—
461C22×C4—531no—
462C22×C4—531no—
463C22×C4—532no—
464C22×C4—532no—
465C22×C4—532no—
466C22×C4—532no—
467C22×C4—532no—
468C22×C4—532no—
469C22×C4—532no—
470C22×C4—532no—
471C22×C4—532no—
472C22×C4—531no—
473C22×C4—531no—
474C22×C4—532no—
475C22×C4—532no—
476C22×C4—532no—
477C22×C4—1062no—
478C22×C4—1062no—
479C22×C4—1064no—
480C22×C4—1064no—
481C22×C4—1064no—
482C22×C4—1064no—
483C22×C4—1068no—
484C22×C4—1062no—
485C22×C4—1064no—
486C22×C4—1062no—
487C22×C4—1068no—
488C22×C4—1064no—
489C22×C4—1064no—
490C22×C4—1061no—
491C22×C4—1061no—
492C22×C4—1061no—
493C22×C4—1061no—
494C22×C4—1064no—
495C22×C4—2122no—
496C22×C4C22×#615712yes—
497C22×C4C22×#615712yes—
498M16—538no—
499M16—1068no—
500M16—1064no—
501M16—10616no—
502M16—1064no—
503M16—1064no—
504M16—2124no—
505C2×C8—1068no—
506C2×C8—1064no—
507C2×C8—10616no—
508C2×C8—1064no—
509C2×C8—2124no—
510C4⋊C4—1062no—
511C4⋊C4—1062no—
512C4⋊C4—1062no—
513C4⋊C4—1062no—
514C4⋊C4—1064no—
515C4⋊C4—1064no—
516C4⋊C4—1064no—
517C4⋊C4—1064no—
518C22⋊C4—24no—
519C22⋊C4—24no—
520C22⋊C4—1064no—
521C22⋊C4—1064no—
522C23—22no—
523C23—22no—
524C23—22no—
525C23—22no—
526C23—24no—
527C23—22no—
528C23—22no—
529C23—24no—
530C23—24no—
531C23—22no—
532C23—24no—
533C23—42no—
534C23—42no—
535C23—416no—
536C23#1313327814yescomplemented
537C23C2×C22⋊#615712yes—
538C23C2×C22⋊#615711yescharacteristic
539C23C2×C22⋊#615712yes—
540C23C2×C22⋊#615711yescharacteristic
541C23D4×#615714yes—
542C23C23×#615711yescharacteristic
543Q8—2122no—
544Q8—2124no—
545Q8—2122no—
546Q8—2124no—
547Q8—2124no—
548C2×C4—22no—
549C2×C4—22no—
550C2×C4—24no—
551C2×C4—42no—
552C2×C4—42no—
553C2×C4—42no—
554C2×C4—42no—
555C2×C4—42no—
556C2×C4—532no—
557C2×C4—532no—
558C2×C4—534no—
559C2×C4—534no—
560C2×C4—532no—
561C2×C4—532no—
562C2×C4—531no—
563C2×C4—531no—
564C2×C4—531no—
565C2×C4—531no—
566C2×C4—534no—
567C2×C4—532no—
568C2×C4—532no—
569C2×C4—532no—
570C2×C4—532no—
571C2×C4—1062no—
572C2×C4—1062no—
573C2×C4—1064no—
574C2×C4—1064no—
575C2×C4—1062no—
576C2×C4—1062no—
577C2×C4—1064no—
578C2×C4—1064no—
579C2×C4—1064no—
580C2×C4—1064no—
581C2×C4—1064no—
582C2×C4—1062no—
583C2×C4—1062no—
584C2×C4—1064no—
585C2×C4—1062no—
586C2×C4—1062no—
587C2×C4—1064no—
588C2×C4—2122no—
589C2×C4—2122no—
590C2×C4—2122no—
591C2×C4—2122no—
592C2×C4—2122no—
593C8—1068no—
594C8—2124no—
595C22—22no—
596C22—28no—
597C22—28no—
598C22—21no—
599C22—21no—
600C22—21no—
601C22—21no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—48no—
607C22—48no—
608C22—44no—
609C22#1681157112yes—
610C22#1681157112yes—
611C22C2×#1313327812yes—
612C22C2×#1313327814yescomplemented
613C22#1681161012yes—
614C22C22×C22⋊#615711yescharacteristic
615C22C2×D4×#615712yes—
616C4—42no—
617C4—42no—
618C4—531no—
619C4—532no—
620C4—531no—
621C4—532no—
622C4—532no—
623C4—1062no—
624C4—1062no—
625C4—2122no—
626C4—2122no—
627C2—22no—
628C2—24no—
629C2—22no—
630C2—42no—
631C2#2029074214yesdirect product factor
632C2C2×#1681157111yescharacteristic
633C2C22×#1313327811yescharacteristic
634C2C2×#1681161011yescharacteristic
subgroupG/H
centerC23C2×C22⋊#6157
commutatorC23×C53C23×C4
FrattiniC23C23×#6157
FittingC22×C22⋊C4×C53C4
socleC23×C53C2×C22⋊C4
prime radicalC25×C53C2×C4
cosocleC22×Dic53C24
hypercenterC22×C22⋊C4#6157
hypocenterC53C2×#1901
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C25 ◁ C22×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C22×#13133278C2semidirect
4. #20290742C2direct (N × H)
5. C2×#16811503C2semidirect
15. C2×#13133278C22semidirect
20. #16811503C22semidirect
218. C23×C53#101semidirect
295. C22×C53C2×#101semidirect
320. C2×C53#1901semidirect
353. C53C2×#1901semidirect
536. C23#13133278semidirect
612. C22C2×#13133278semidirect
631. C2#20290742direct (N × H)

Maximal subgroups

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

subgroupindex
C23×#117199402
C22×#131332782
C2×#168115932
#202907422
C2×#168115032
C22×#131331942
C2×#190153