Finite Group Databasebeta

← C2×#20275384 #27386673

Subgroups — C2×#20275384 #27386673

Order 13568.

1046 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
1C2×#16810671C211yesmaximal, characteristic, complemented
2C2×#16810639C211yesmaximal, characteristic
3C22×#13133145C211yesmaximal, characteristic
4C2×#16810605C211yesmaximal, characteristic
5#20275384C218yesmaximal, direct product factor
6C2×#16810395C211yesmaximal, characteristic, complemented
7C2×#16810383C211yesmaximal, characteristic, complemented
8C2×C4×M16×C53C211yesmaximal, characteristic, complemented
9C22×#11719945C2211yescharacteristic
10C22×#11719917C2211yescharacteristic
11C22×#11719917C2211yescharacteristic
12C2×#13133176C2211yescharacteristic
13#16810671—22no—
14#16810671C2214yescomplemented
15C2×#13133152—22no—
16#16810639—22no—
17#16810639C2214yes—
18C2×#13133145—21no—
19C2×#13133145—21no—
20C2×#13133145—22no—
21C2×#13133145C2214yes—
22C2×#13133143—22no—
23#16810605C414yescomplemented
24#16810605C2214yes—
25C2×#13133135C412yescomplemented
26C2×#13133133C412yescomplemented
27#16810395C2214yescomplemented
28#16810383C2214yescomplemented
29C22×M16×C53C2211yescharacteristic
30C22×M16×C53C2211yescharacteristic
31C22×C42×C53C2211yescharacteristic
32C4×M16×C53—22no—
33C4×M16×C53C2214yescomplemented
34C2×#97×C53—21no—
35C2×C4×C8×C53—21no—
36C2×#11719945—21no—
37C2×#11719945—21no—
38C2×#11719945—21no—
39C2×#11719945—21no—
40C2×#11719945—21no—
41C2×#11719945—21no—
42C2×#11719945—21no—
43C2×#11719945—21no—
44C2×#11719945—21no—
45C2×#11719945—21no—
46C2×#11719945C2×C412yes—
47C2×#11719945C2312yes—
48C22×Dic106C2×C411yescharacteristic
49C22×D212C2×C411yescharacteristic
50C22×C22⋊D53—21no—
51C22×C4×D53—21no—
52C2×#11719917—21no—
53C2×#11719917—21no—
54C2×#11719917—21no—
55C2×#11719917—21no—
56C2×#11719917—21no—
57C2×#11719917—21no—
58C2×#11719917—21no—
59C2×#11719917—21no—
60C2×#11719917C2×C412yes—
61C2×#11719917C2×C412yes—
62C2×#11719917C2312yes—
63C2×#11719917C2312yes—
64C22×#88322—21no—
65C22×#88322—21no—
66#13133176—21no—
67#13133176—21no—
68#13133176—21no—
69#13133176C2312yes—
70C2×C4⋊Dic53—21no—
71C2×C22⋊Dic53—21no—
72C2×C4×Dic53—21no—
73#13133152—28no—
74#13133145—24no—
75#13133145—24no—
76#13133143—28no—
77#13133135C2×C418yescomplemented
78#13133133C2×C418yescomplemented
79C23×C4×C53—21no—
80C23×C4×C53C2311yescharacteristic
81C2×M16×C53—21no—
82C2×M16×C53—21no—
83C2×M16×C53—21no—
84C2×M16×C53—21no—
85C2×M16×C53—21no—
86C2×M16×C53—21no—
87C2×M16×C53C2×C412yes—
88C2×M16×C53C2×C412yes—
89C2×M16×C53D412yes—
90C2×M16×C53D412yes—
91C2×M16×C53C2312yes—
92C2×M16×C53C2312yes—
93C22×C8×C53—21no—
94C22×C8×C53—21no—
95C2×C42×C53—21no—
96C2×C42×C53—21no—
97C2×C42×C53—21no—
98C2×C42×C53D411yescharacteristic
99C2×C42×C53D411yescharacteristic
100C2×C42×C53D412yes—
101C2×C42×C53C2312yes—
102#97×C53—24no—
103C4×C8×C53—24no—
104C24×C53#4111yescharacteristic
105C23×D53—21no—
106C22×C4×C53—21no—
107C22×C4×C53—22no—
108C22×C4×C53—21no—
109C22×C4×C53—22no—
110C22×C4×C53—44no—
111C22×C4×C53C22×C411yescharacteristic
112C22×C4×C53C22×C411yescharacteristic
113C22×C4×C53C2×D411yescharacteristic
114C22×C4×C53C2×D411yescharacteristic
115C22×C4×C53C2×D411yescharacteristic
116C22×C4×C53C2×D411yescharacteristic
117C22×C4×C53#4111yescharacteristic
118C22×C4×C53#4111yescharacteristic
119C22×C4×C53#4111yescharacteristic
120C22×C4×C53#4111yescharacteristic
121C22×C4×C53#4111yescharacteristic
122C22×C4×C53#4111yescharacteristic
123C22×C4×C53#4111yescharacteristic
124C22×C4×C53C2411yescharacteristic
125C2×C22⋊D53—22no—
126C2×C22⋊D53—22no—
127C2×C22⋊D53—41no—
128C2×C22⋊D53—41no—
129C2×C22⋊D53—41no—
130C2×C22⋊D53—41no—
131C22×Dic53—21no—
132C22×Dic53—41no—
133#11719945—22no—
134#11719945—22no—
135#11719945—22no—
136#11719945—22no—
137#11719945—22no—
138#11719945—22no—
139#11719945—22no—
140#11719945—22no—
141#11719945—41no—
142#11719945—41no—
143#11719945—41no—
144#11719945—41no—
145#11719945—41no—
146#11719945—41no—
147#11719945—41no—
148#11719945—41no—
149C2×D212—21no—
150C2×D212—21no—
151C2×D212—21no—
152C2×D212—21no—
153C2×D212—21no—
154C2×D212C22×C412yes—
155C2×C4×D53—22no—
156C2×C4×D53—22no—
157C2×C4×D53—41no—
158C2×C4×D53—41no—
159C2×C4×D53—41no—
160C2×C4×D53—41no—
161C2×Dic106—21no—
162C2×Dic106—21no—
163C2×Dic106—21no—
164C2×Dic106—21no—
165C2×Dic106—21no—
166C2×Dic106C22×C412yes—
167M16×C53—22no—
168M16×C53—22no—
169M16×C53—22no—
170M16×C53—22no—
171M16×C53—22no—
172M16×C53—22no—
173M16×C53C22×C414yes—
174M16×C53C2×D414yes—
175C2×C8×C53—22no—
176C2×C8×C53—22no—
177C2×C8×C53—22no—
178C2×C8×C53—22no—
179C2×C8×C53—22no—
180C2×C8×C53—22no—
181C42×C53—21no—
182C42×C53—22no—
183C42×C53—21no—
184C42×C53—22no—
185C42×C53C2×D412yes—
186C42×C53C2×D412yes—
187C22⋊Dic53—22no—
188C22⋊Dic53—41no—
189C4⋊Dic53—22no—
190C4⋊Dic53—41no—
191C4×Dic53—22no—
192C4×Dic53—41no—
193#11719917—22no—
194#11719917—22no—
195#11719917—22no—
196#11719917—22no—
197#11719917—22no—
198#11719917—41no—
199#11719917—41no—
200#11719917C22×C414yes—
201C2×#88322—22no—
202C2×#88322—22no—
203C2×#88322—22no—
204C2×#88322—22no—
205C2×#88322—41no—
206C2×#88322—41no—
207C23×C53—42no—
208C23×C53#11711yescharacteristic
209C23×C53#11711yescharacteristic
210C23×C53#12311yescharacteristic
211C23×C53#12311yescharacteristic
212C23×C53#12411yescharacteristic
213C23×C53#12411yescharacteristic
214C23×C53C2×#4111yescharacteristic
215C22×D53—22no—
216C22×D53—41no—
217C22×D53—41no—
218C22×D53—82no—
219C2×C4×C53—24no—
220C2×C4×C53—22no—
221C2×C4×C53—22no—
222C2×C4×C53—44no—
223C2×C4×C53—44no—
224C2×C4×C53—42no—
225C2×C4×C53#11711yescharacteristic
226C2×C4×C53#11711yescharacteristic
227C2×C4×C53C4×D411yescharacteristic
228C2×C4×C53C4×D411yescharacteristic
229C2×C4×C53C4×D411yescharacteristic
230C2×C4×C53C4×D411yescharacteristic
231C2×C4×C53C4×D411yescharacteristic
232C2×C4×C53C4×D411yescharacteristic
233C2×C4×C53C4×D411yescharacteristic
234C2×C4×C53C4×D411yescharacteristic
235C2×C4×C53#12111yescharacteristic
236C2×C4×C53#12111yescharacteristic
237C2×C4×C53#12111yescharacteristic
238C2×C4×C53#12111yescharacteristic
239C2×C4×C53#12311yescharacteristic
240C2×C4×C53#12311yescharacteristic
241C2×C4×C53#12411yescharacteristic
242C2×C4×C53#12411yescharacteristic
243C2×C4×C53#12611yescharacteristic
244C2×C4×C53#12611yescharacteristic
245C2×C4×C53#12611yescharacteristic
246C2×C4×C53#12611yescharacteristic
247C2×C4×C53C23×C411yescharacteristic
248C2×C4×C53C22×D411yescharacteristic
249C2×C4×C53C22×D411yescharacteristic
250C2×C4×C53C2×#4111yescharacteristic
251C2×C4×C53C2×#4111yescharacteristic
252C2×C4×C53C2×#4111yescharacteristic
253C22⋊D53—42no—
254C22⋊D53—42no—
255C22⋊D53—42no—
256C22⋊D53—42no—
257C2×Dic53—22no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—42no—
261C2×Dic53—82no—
262D212—22no—
263D212—22no—
264D212—41no—
265D212—41no—
266C4×D53—42no—
267C4×D53—42no—
268C4×D53—42no—
269C4×D53—42no—
270Dic106—22no—
271Dic106—22no—
272Dic106—41no—
273Dic106—41no—
274C8×C53—24no—
275C8×C53—24no—
276#88322—24no—
277#88322—42no—
278C2×#1932—531nomaximal, Sylow
279C22×C53—22no—
280C22×C53—22no—
281C22×C53—22no—
282C22×C53—41no—
283C22×C53—41no—
284C22×C53—41no—
285C22×C53—41no—
286C22×C53#38811yescharacteristic
287C22×C53#38811yescharacteristic
288C22×C53#38811yescharacteristic
289C22×C53#38811yescharacteristic
290C22×C53C2×#11711yescharacteristic
291C22×C53C2×#12311yescharacteristic
292C22×C53C2×#12411yescharacteristic
293C2×D53—42no—
294C2×D53—82no—
295C2×D53—82no—
296C2×D53—82no—
297C4×C53—24no—
298C4×C53—42no—
299C4×C53#38811yescharacteristic
300C4×C53#38811yescharacteristic
301C4×C53#38811yescharacteristic
302C4×C53#38811yescharacteristic
303C4×C53C2×C4×D411yescharacteristic
304C4×C53C2×C4×D411yescharacteristic
305C4×C53C2×#12111yescharacteristic
306C4×C53C2×#12611yescharacteristic
307Dic53—42no—
308Dic53—82no—
309C2×#430—531no—
310C2×#421—531no—
311C22×#104—531no—
312C2×#413—531no—
313C2×C4×M16D5311yescharacteristic, complemented
314#1932—538no—
315C2×#344—531no—
316C2×#339—531no—
317C2×C53—21no—
318C2×C53—21no—
319C2×C53—21no—
320C2×C53—21no—
321C2×C53—21no—
322C2×C53—21no—
323C2×C53#193212yescomplemented
324C2×C53C2×#38811yescharacteristic
325D53—82no—
326C22×#41—531no—
327C22×M16—531no—
328C22×M16—531no—
329C22×M16C2×D5311yescharacteristic
330C22×M16C2×D5311yescharacteristic
331C2×#117—531no—
332C22×C42C2×D5311yescharacteristic
333#430—534no—
334#430—1062no—
335C2×#108—1062no—
336#421—534no—
337#421—1062no—
338C2×#104—534no—
339C2×#104—1062no—
340C2×#104—1061no—
341C2×#104—1061no—
342C2×#104—1062no—
343#413—534no—
344#413—534no—
345C2×#101—532no—
346C2×#100—532no—
347C4×M16—22no—
348C4×M16C2×D5314yescomplemented
349C2×#97—21no—
350C2×C4×C8—21no—
351#344—534no—
352#339—534no—
353C53C2×#193211yescharacteristic, complemented, Sylow
354C2×#41—532no—
355C2×#41—532no—
356C2×#41—1061no—
357C2×#41—1061no—
358C2×#41—1061no—
359C2×#41—1061no—
360C2×#41—1061no—
361C2×#41—1061no—
362C2×#41—1061no—
363C2×#41—1061no—
364C2×#41—1061no—
365C2×#41—1061no—
366C22×Q8—531no—
367C22×D4—531no—
368C22×D4—1061no—
369C23×C4—21no—
370C23×C4—1061no—
371C23×C4C22×D5311yescharacteristic
372C2×M16—21no—
373C2×M16—21no—
374C2×M16—21no—
375C2×M16—21no—
376C2×M16—21no—
377C2×M16—21no—
378C2×M16—532no—
379C2×M16—532no—
380C2×M16—532no—
381C2×M16—532no—
382C2×M16—1061no—
383C2×M16—1061no—
384C2×M16—1061no—
385C2×M16—1061no—
386C2×M16—1061no—
387C2×M16—1061no—
388C2×M16—1061no—
389C2×M16—1061no—
390C2×M16C4×D5312yes—
391C2×M16C4×D5312yes—
392C2×M16D21212yes—
393C2×M16D21212yes—
394C2×M16C22×D5312yes—
395C2×M16C22×D5312yes—
396C22×C8—21no—
397C22×C8—21no—
398C22×C8—1061no—
399C22×C8—1061no—
400#117—532no—
401#117—1061no—
402#117—1061no—
403#117—1061no—
404C2×C4⋊C4—1061no—
405C2×C22⋊C4—1061no—
406C2×C42—21no—
407C2×C42—21no—
408C2×C42—21no—
409C2×C42—1061no—
410C2×C42C22⋊D5311yescharacteristic
411C2×C42C22⋊D5311yescharacteristic
412C2×C42C22⋊D5312yes—
413C2×C42C22×D5312yes—
414#108—1068no—
415#104—1068no—
416#104—1064no—
417#104—1064no—
418#101—538no—
419#100—538no—
420#97—24no—
421C4×C8—24no—
422C24—1061no—
423C24#1171994511yescharacteristic
424#41—1062no—
425#41—1062no—
426#41—1062no—
427#41—1062no—
428#41—1062no—
429#41—1062no—
430#41—1062no—
431#41—1062no—
432#41—2121no—
433#41—2121no—
434#41—2121no—
435#41—2121no—
436#41—2121no—
437#41—2121no—
438#41—2121no—
439#41—2121no—
440C2×Q8—532no—
441C2×Q8—1061no—
442C2×Q8—1061no—
443C2×Q8—1061no—
444C2×Q8—1061no—
445C2×Q8—1061no—
446C2×D4—532no—
447C2×D4—1062no—
448C2×D4—1062no—
449C2×D4—1061no—
450C2×D4—1061no—
451C2×D4—1061no—
452C2×D4—1061no—
453C2×D4—1061no—
454C2×D4—2121no—
455C2×D4—2121no—
456C2×D4—2121no—
457C2×D4—2121no—
458C22×C4—21no—
459C22×C4—22no—
460C22×C4—21no—
461C22×C4—22no—
462C22×C4—44no—
463C22×C4—1061no—
464C22×C4—1062no—
465C22×C4—1062no—
466C22×C4—2121no—
467C22×C4—2121no—
468C22×C4—2121no—
469C22×C4—2121no—
470C22×C4—2121no—
471C22×C4C2×C4×D5311yescharacteristic
472C22×C4C2×C4×D5311yescharacteristic
473C22×C4C2×D21211yescharacteristic
474C22×C4C2×D21211yescharacteristic
475C22×C4#1171994511yescharacteristic
476C22×C4#1171994511yescharacteristic
477C22×C4#1171994511yescharacteristic
478C22×C4#1171994511yescharacteristic
479C22×C4#1171994511yescharacteristic
480C22×C4#1171994511yescharacteristic
481C22×C4#1171994511yescharacteristic
482C22×C4C2×C22⋊D5311yescharacteristic
483C22×C4C2×C22⋊D5311yescharacteristic
484C22×C4C23×D5311yescharacteristic
485M16—22no—
486M16—22no—
487M16—22no—
488M16—22no—
489M16—22no—
490M16—22no—
491M16—534no—
492M16—1062no—
493M16—1062no—
494M16—1062no—
495M16—1062no—
496M16—1062no—
497M16—2121no—
498M16—2121no—
499M16C2×C4×D5314yes—
500M16C2×D21214yes—
501C2×C8—22no—
502C2×C8—22no—
503C2×C8—22no—
504C2×C8—22no—
505C2×C8—22no—
506C2×C8—22no—
507C2×C8—1062no—
508C2×C8—1062no—
509C2×C8—1062no—
510C2×C8—1062no—
511C2×C8—2121no—
512C2×C8—2121no—
513C4⋊C4—1062no—
514C4⋊C4—2121no—
515C22⋊C4—1062no—
516C22⋊C4—2121no—
517C42—21no—
518C42—22no—
519C42—21no—
520C42—22no—
521C42—1062no—
522C42—2121no—
523C42C2×C22⋊D5312yes—
524C42C2×C22⋊D5312yes—
525C23—42no—
526C23—1062no—
527C23—4242no—
528C23—2121no—
529C23—2121no—
530C23#1313317411yescharacteristic
531C23#1313317411yescharacteristic
532C23#1313320011yescharacteristic
533C23#1313320011yescharacteristic
534C23#1313320511yescharacteristic
535C23#1313320511yescharacteristic
536C23C2×#1171994511yescharacteristic
537Q8—1062no—
538Q8—1062no—
539Q8—2121no—
540Q8—2121no—
541D4—1062no—
542D4—1062no—
543D4—2122no—
544D4—2122no—
545D4—2122no—
546D4—2122no—
547D4—2121no—
548D4—2121no—
549C2×C4—24no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—44no—
553C2×C4—44no—
554C2×C4—42no—
555C2×C4—1062no—
556C2×C4—4242no—
557C2×C4—2122no—
558C2×C4—2121no—
559C2×C4—2121no—
560C2×C4—2122no—
561C2×C4—2122no—
562C2×C4—2122no—
563C2×C4—2122no—
564C2×C4#1313317411yescharacteristic
565C2×C4#1313317411yescharacteristic
566C2×C4C4×C22⋊D5311yescharacteristic
567C2×C4C4×C22⋊D5311yescharacteristic
568C2×C4C4×C22⋊D5311yescharacteristic
569C2×C4C4×C22⋊D5311yescharacteristic
570C2×C4C4×D21211yescharacteristic
571C2×C4C4×D21211yescharacteristic
572C2×C4C4×D21211yescharacteristic
573C2×C4C4×D21211yescharacteristic
574C2×C4#1313319111yescharacteristic
575C2×C4#1313319111yescharacteristic
576C2×C4#1313319111yescharacteristic
577C2×C4#1313319111yescharacteristic
578C2×C4#1313320011yescharacteristic
579C2×C4#1313320011yescharacteristic
580C2×C4#1313320511yescharacteristic
581C2×C4#1313320511yescharacteristic
582C2×C4#1313321111yescharacteristic
583C2×C4#1313321111yescharacteristic
584C2×C4#1313321111yescharacteristic
585C2×C4#1313321111yescharacteristic
586C2×C4C22×C4×D5311yescharacteristic
587C2×C4C22×C22⋊D5311yescharacteristic
588C2×C4C22×D21211yescharacteristic
589C2×C4C2×#1171994511yescharacteristic
590C2×C4C2×#1171994511yescharacteristic
591C2×C4C2×#1171994511yescharacteristic
592C8—24no—
593C8—24no—
594C8—1064no—
595C8—2122no—
596C22—22no—
597C22—22no—
598C22—22no—
599C22—41no—
600C22—41no—
601C22—41no—
602C22—41no—
603C22—4242no—
604C22—4242no—
605C22—4242no—
606C22—2122no—
607C22#1681051411yescharacteristic
608C22#1681051411yescharacteristic
609C22#1681051411yescharacteristic
610C22#1681051411yescharacteristic
611C22C2×#1313317411yescharacteristic
612C22C2×#1313320011yescharacteristic
613C22C2×#1313320511yescharacteristic
614C4—24no—
615C4—42no—
616C4—4242no—
617C4—2122no—
618C4#1681051411yescharacteristic
619C4#1681051411yescharacteristic
620C4#1681051411yescharacteristic
621C4#1681051411yescharacteristic
622C4C2×C4×C22⋊D5311yescharacteristic
623C4C2×C4×D21211yescharacteristic
624C4C2×#1313319111yescharacteristic
625C4C2×#1313321111yescharacteristic
626C2—21no—
627C2—21no—
628C2—21no—
629C2—21no—
630C2—21no—
631C2—21no—
632C2—4242no—
633C2#2027538412yesdirect product factor
634C2C2×#1681051411yescharacteristic
subgroupG/H
centerC2×C4#13133191
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C4×M16×C53C2
socleC22×C53#388
prime radicalC22×#11719945C22
cosocleC22×C4×C53C24
hypercenterC2×C4×M16D53
hypocenterC53C2×#1932
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×C4×M16

Complements

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

NH ≅ G/Ntype
1. C2×#16810671C2semidirect
5. #20275384C2direct (N × H)
6. C2×#16810395C2semidirect
7. C2×#16810383C2semidirect
8. C2×C4×M16×C53C2semidirect
14. #16810671C22semidirect
23. #16810605C4semidirect
25. C2×#13133135C4semidirect
26. C2×#13133133C4semidirect
27. #16810395C22semidirect
28. #16810383C22semidirect
33. C4×M16×C53C22semidirect
77. #13133135C2×C4semidirect
78. #13133133C2×C4semidirect
313. C2×C4×M16D53semidirect
323. C2×C53#1932semidirect
348. C4×M16C2×D53semidirect
353. C53C2×#1932semidirect
633. C2#20275384direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168106712
C2×#168106392
C22×#131331452
C2×#168106052
#202753842
C2×#168103952
C2×#168103832
C2×C4×M16×C532
C2×#193253