Finite Group Databasebeta

← C2×#20273256 #27351345

Subgroups — C2×#20273256 #27351345

Order 13568.

1510 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16811134C211yesmaximal, characteristic
2C2×#16810587C211yesmaximal, characteristic, complemented
3#20273256C218yesmaximal, direct product factor
4C2×#16810359C212yesmaximal, complemented
5C2×#16810356C212yesmaximal, complemented
6C2×C53×#380C211yesmaximal, characteristic, complemented
7C22×#11719945C411yescharacteristic, complemented
8C22×#11719917—21no—
9#16811134—22no—
10#16811134—22no—
11#16811134C414yescomplemented
12#16811134C2214yes—
13C2×#13133200—21no—
14C2×#13133196C411yescharacteristic, complemented
15C2×#13133196C2211yescharacteristic
16C2×#13133191C411yescharacteristic, complemented
17C2×#13133191C2211yescharacteristic
18C2×#13133186—21no—
19C2×C4×C22⋊D53—21no—
20C2×C4×C22⋊D53—21no—
21C2×#13133176C411yescharacteristic, complemented
22#16810587—22no—
23#16810587C2214yescomplemented
24C2×C22⋊#88322—21no—
25C2×C22⋊#88322C2212yes—
26#16810359C2218yescomplemented
27#16810356C2218yescomplemented
28C24×C4×C53C2211yescharacteristic
29C22×C4⋊C4×C53—21no—
30C22×C22⋊C4×C53—21no—
31C53×#380—22no—
32C53×#380C2214yescomplemented
33C2×#95×C53C2212yes—
34C2×#11719945—22no—
35C2×#11719945—44no—
36C2×#11719945C2×C414yescomplemented
37C22×Dic106C2×C411yescharacteristic
38C22×D212C2×C411yescharacteristic
39C22×C22⋊D53—21no—
40C22×C22⋊D53—21no—
41C22×C22⋊D53—21no—
42C22×C4×D53—21no—
43C2×#11719917—24no—
44C2×#11719917—44no—
45C22×#88322—22no—
46#13133200—24no—
47#13133200—24no—
48#13133196—22no—
49#13133196—21no—
50#13133196—21no—
51#13133196C2×C412yes—
52#13133196C2×C414yescomplemented
53#13133196C2312yes—
54#13133191—22no—
55#13133191—21no—
56#13133191—21no—
57#13133191C2×C412yes—
58#13133191C2×C414yescomplemented
59#13133191C2312yes—
60#13133186—24no—
61#13133186—24no—
62C4×C22⋊D53—24no—
63C4×C22⋊D53—24no—
64C4×C22⋊D53—24no—
65C4×C22⋊D53—24no—
66#13133176—22no—
67#13133176C2×C414yescomplemented
68C2×C4⋊Dic53C2×C411yescharacteristic
69C2×C4⋊Dic53C2×C411yescharacteristic
70C2×#11719919—21no—
71C2×#11719919—21no—
72C2×C22⋊Dic53—21no—
73C2×C22⋊Dic53—21no—
74C2×C22⋊Dic53—21no—
75C2×#11719921—21no—
76C2×#11719921—21no—
77C2×C4×Dic53—21no—
78C22⋊#88322—24no—
79C22⋊#88322—22no—
80C22⋊#88322C2314yes—
81C25×C53D411yescharacteristic
82C23×C4×C53—22no—
83C23×C4×C53—44no—
84C23×C4×C53C2×C411yescharacteristic
85C23×C4×C53C2×C411yescharacteristic
86C23×C4×C53C2×C412yes—
87C23×C4×C53D411yescharacteristic
88C23×C4×C53D411yescharacteristic
89C23×C4×C53D412yes—
90C23×C4×C53D411yescharacteristic
91C23×C4×C53D412yes—
92C23×C4×C53C2311yescharacteristic
93C23×C4×C53C2312yes—
94C2×C4⋊C4×C53—24no—
95C2×C4⋊C4×C53—44no—
96C2×C22⋊C4×C53—24no—
97C2×C22⋊C4×C53—44no—
98#95×C53—22no—
99#95×C53C2314yes—
100C24×C53—22no—
101C24×C53—22no—
102C24×C53—42no—
103C24×C53—42no—
104C24×C53C22⋊C411yescharacteristic
105C24×C53C22⋊C411yescharacteristic
106C24×C53C22⋊C412yes—
107C24×C53C2×D411yescharacteristic
108C24×C53C2×D412yes—
109C23×D53—21no—
110C22×C4×C53—24no—
111C22×C4×C53—24no—
112C22×C4×C53—24no—
113C22×C4×C53—24no—
114C22×C4×C53—44no—
115C22×C4×C53—44no—
116C22×C4×C53—44no—
117C22×C4×C53—44no—
118C22×C4×C53—42no—
119C22×C4×C53—42no—
120C22×C4×C53—42no—
121C22×C4×C53—44no—
122C22×C4×C53—44no—
123C22×C4×C53—44no—
124C22×C4×C53C22⋊C411yescharacteristic
125C22×C4×C53C22⋊C412yes—
126C22×C4×C53C22⋊C412yes—
127C22×C4×C53C22⋊C412yes—
128C22×C4×C53C22⋊C412yes—
129C22×C4×C53C22⋊C412yes—
130C22×C4×C53C22⋊C411yescharacteristic
131C22×C4×C53C22×C411yescharacteristic
132C22×C4×C53C22×C412yes—
133C22×C4×C53C22×C411yescharacteristic
134C22×C4×C53C22×C412yes—
135C22×C4×C53C2×D412yes—
136C22×C4×C53C2×D411yescharacteristic
137C22×C4×C53C2×D412yes—
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2×D412yes—
141C22×C4×C53C2411yescharacteristic
142C2×C22⋊D53—22no—
143C2×C22⋊D53—24no—
144C2×C22⋊D53—24no—
145C2×C22⋊D53—22no—
146C2×C22⋊D53—44no—
147C2×C22⋊D53—44no—
148C2×C22⋊D53—42no—
149C2×C22⋊D53—42no—
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—21no—
153#11719945—48no—
154#11719945—48no—
155C2×D212—21no—
156C2×D212—42no—
157C2×D212C22×C412yes—
158C2×C4×D53—24no—
159C2×C4×D53—44no—
160C2×Dic106—21no—
161C2×Dic106—42no—
162C2×Dic106C22×C412yes—
163C4⋊C4×C53—48no—
164C22⋊C4×C53—48no—
165C22⋊Dic53—24no—
166C22⋊Dic53—22no—
167C22⋊Dic53—24no—
168C22⋊Dic53—22no—
169#11719921—22no—
170#11719921—24no—
171#11719921—22no—
172C4⋊Dic53—21no—
173C4⋊Dic53—21no—
174C4⋊Dic53C22×C412yes—
175C4⋊Dic53C22×C412yes—
176#11719919—22no—
177#11719919—24no—
178#11719919—22no—
179C4×Dic53—24no—
180#11719917—48no—
181C2×#88322—24no—
182C2×#88322—44no—
183C23×C53—22no—
184C23×C53—22no—
185C23×C53—22no—
186C23×C53—22no—
187C23×C53—22no—
188C23×C53—22no—
189C23×C53—24no—
190C23×C53—22no—
191C23×C53—22no—
192C23×C53—22no—
193C23×C53—22no—
194C23×C53—42no—
195C23×C53—42no—
196C23×C53—42no—
197C23×C53—42no—
198C23×C53—48no—
199C23×C53—48no—
200C23×C53C2×C22⋊C412yes—
201C23×C53C2×C22⋊C412yes—
202C23×C53C2×C22⋊C411yescharacteristic
203C23×C53C2×C22⋊C411yescharacteristic
204C23×C53AGL(1,ℤ/8)12yes—
205C23×C53#13712yes—
206C23×C53C22×D411yescharacteristic
207C22×D53—22no—
208C22×D53—42no—
209C22×D53—42no—
210C22×D53—42no—
211C2×C4×C53—22no—
212C2×C4×C53—22no—
213C2×C4×C53—22no—
214C2×C4×C53—24no—
215C2×C4×C53—24no—
216C2×C4×C53—24no—
217C2×C4×C53—24no—
218C2×C4×C53—24no—
219C2×C4×C53—24no—
220C2×C4×C53—42no—
221C2×C4×C53—42no—
222C2×C4×C53—41no—
223C2×C4×C53—42no—
224C2×C4×C53—41no—
225C2×C4×C53—42no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—44no—
229C2×C4×C53—44no—
230C2×C4×C53—44no—
231C2×C4×C53—88no—
232C2×C4×C53#9914yes—
233C2×C4×C53C2×C22⋊C411yescharacteristic
234C2×C4×C53C2×C22⋊C412yes—
235C2×C4×C53C2×C22⋊C411yescharacteristic
236C2×C4×C53C2×C22⋊C412yes—
237C2×C4×C53C23×C411yescharacteristic
238C2×C4×C53C22×D411yescharacteristic
239C22⋊D53—48no—
240C22⋊D53—48no—
241C22⋊D53—42no—
242C22⋊D53—42no—
243C22⋊D53—44no—
244C2×Dic53—22no—
245C2×Dic53—22no—
246C2×Dic53—22no—
247C2×Dic53—42no—
248C2×Dic53—42no—
249C2×Dic53—42no—
250D212—42no—
251D212—42no—
252C4×D53—48no—
253Dic106—42no—
254Dic106—42no—
255#88322—44no—
256C2×#1513—531nomaximal, Sylow
257C22×C53—21no—
258C22×C53—21no—
259C22×C53—21no—
260C22×C53—21no—
261C22×C53—24no—
262C22×C53—24no—
263C22×C53—22no—
264C22×C53—22no—
265C22×C53—22no—
266C22×C53—22no—
267C22×C53—22no—
268C22×C53—22no—
269C22×C53—44no—
270C22×C53—44no—
271C22×C53—48no—
272C22×C53—44no—
273C22×C53—44no—
274C22×C53C2×#9912yes—
275C22×C53#41712yes—
276C22×C53#41712yes—
277C22×C53#42112yes—
278C22×C53C22×C22⋊C411yescharacteristic
279C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
280C22×C53C2×#13711yescharacteristic
281C2×D53—42no—
282C2×D53—42no—
283C2×D53—42no—
284C2×D53—84no—
285C4×C53—21no—
286C4×C53—21no—
287C4×C53—21no—
288C4×C53—22no—
289C4×C53—21no—
290C4×C53—84no—
291C4×C53C2×#9914yes—
292Dic53—42no—
293Dic53—42no—
294Dic53—42no—
295C2×#525—531no—
296C2×#407—531no—
297C2×#380D5311yescharacteristic, complemented
298#1513—538no—
299C2×#328—532no—
300C2×#327—532no—
301C2×C53—21no—
302C2×C53—21no—
303C2×C53—22no—
304C2×C53—42no—
305C2×C53—42no—
306C2×C53#151314yescomplemented
307C2×C53C22×#9911yescharacteristic
308C2×C53C2×#41711yescharacteristic
309C2×C53C2×#42111yescharacteristic
310D53—82no—
311C22×#41—531no—
312C24×C4C2×D5311yescharacteristic
313C22×M16—1061no—
314#525—534no—
315#525—534no—
316#525—1062no—
317#525—1062no—
318C2×#123—1061no—
319C2×C22⋊Q8—531no—
320C2×C22⋊Q8—531no—
321C2×#121—531no—
322C2×#121—531no—
323C2×#120—1061no—
324C2×C4×D4—1061no—
325C2×C4×D4—1061no—
326C2×#117—531no—
327C22×C4⋊C4—21no—
328C22×C22⋊C4—21no—
329#407—534no—
330#407—1062no—
331C2×C22⋊C8—532no—
332C2×C22⋊C8—1061no—
333#380—22no—
334#380C2×D5314yescomplemented
335C2×#95C2×D5312yes—
336#328—538no—
337#327—538no—
338C53C2×#151311yescharacteristic, complemented, Sylow
339C25C22⋊D5311yescharacteristic
340C2×#41—534no—
341C2×#41—1062no—
342C2×#41—2124no—
343C22×Q8—531no—
344C22×D4—531no—
345C22×D4—1061no—
346C22×D4—1061no—
347C22×D4—1061no—
348C23×C4—22no—
349C23×C4—44no—
350C23×C4—1061no—
351C23×C4C4×D5311yescharacteristic
352C23×C4C4×D5311yescharacteristic
353C23×C4C4×D5312yes—
354C23×C4D21211yescharacteristic
355C23×C4D21211yescharacteristic
356C23×C4D21212yes—
357C23×C4C22⋊D5311yescharacteristic
358C23×C4C22⋊D5312yes—
359C23×C4C22×D5311yescharacteristic
360C23×C4C22×D5312yes—
361C2×M16—1064no—
362C2×M16—2124no—
363C22×C8—1062no—
364#123—1064no—
365#123—1064no—
366C22⋊Q8—532no—
367C22⋊Q8—534no—
368C22⋊Q8—532no—
369C22⋊Q8—1061no—
370C22⋊Q8—1061no—
371C22⋊Q8—1062no—
372#121—532no—
373#121—534no—
374#121—532no—
375#121—1061no—
376#121—1061no—
377#121—1062no—
378#120—1064no—
379#120—1064no—
380C4×D4—1064no—
381C4×D4—1064no—
382C4×D4—1064no—
383C4×D4—1064no—
384#117—534no—
385#117—1062no—
386C2×C4⋊C4—24no—
387C2×C4⋊C4—44no—
388C2×C4⋊C4—531no—
389C2×C4⋊C4—531no—
390C2×C4⋊C4—1061no—
391C2×C4⋊C4—1061no—
392C2×C22⋊C4—24no—
393C2×C22⋊C4—44no—
394C2×C22⋊C4—1061no—
395C2×C22⋊C4—1061no—
396C2×C22⋊C4—1061no—
397C2×C22⋊C4—1061no—
398C2×C22⋊C4—1061no—
399C2×C42—1061no—
400C22⋊C8—534no—
401C22⋊C8—1062no—
402C22⋊C8—1064no—
403#95—22no—
404#95C22×D5314yes—
405C24—22no—
406C24—22no—
407C24—42no—
408C24—42no—
409C24—1061no—
410C24#1171992111yescharacteristic
411C24#1171992111yescharacteristic
412C24#1171992112yes—
413C24C2×C22⋊D5311yescharacteristic
414C24C2×C22⋊D5312yes—
415#41—2128no—
416#41—2128no—
417C2×Q8—532no—
418C2×Q8—1061no—
419C2×Q8—2122no—
420C2×D4—532no—
421C2×D4—1064no—
422C2×D4—1062no—
423C2×D4—1064no—
424C2×D4—1062no—
425C2×D4—1061no—
426C2×D4—2122no—
427C2×D4—2124no—
428C2×D4—2124no—
429C2×D4—2122no—
430C2×D4—2122no—
431C22×C4—24no—
432C22×C4—24no—
433C22×C4—24no—
434C22×C4—24no—
435C22×C4—44no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—44no—
439C22×C4—42no—
440C22×C4—42no—
441C22×C4—42no—
442C22×C4—44no—
443C22×C4—44no—
444C22×C4—44no—
445C22×C4—1061no—
446C22×C4—1061no—
447C22×C4—1061no—
448C22×C4—1064no—
449C22×C4—2124no—
450C22×C4#1171992112yes—
451C22×C4#1171992112yes—
452C22×C4#1171992112yes—
453C22×C4#1171992111yescharacteristic
454C22×C4#1171992112yes—
455C22×C4#1171992112yes—
456C22×C4#1171992111yescharacteristic
457C22×C4C2×C4×D5312yes—
458C22×C4C2×C4×D5311yescharacteristic
459C22×C4C2×C4×D5312yes—
460C22×C4C2×C4×D5311yescharacteristic
461C22×C4C2×D21212yes—
462C22×C4C2×D21211yescharacteristic
463C22×C4C2×D21211yescharacteristic
464C22×C4C2×D21212yes—
465C22×C4C2×C22⋊D5312yes—
466C22×C4C2×C22⋊D5311yescharacteristic
467C22×C4C23×D5311yescharacteristic
468M16—2128no—
469C2×C8—1064no—
470C2×C8—2124no—
471C4⋊C4—48no—
472C4⋊C4—532no—
473C4⋊C4—532no—
474C4⋊C4—1061no—
475C4⋊C4—1061no—
476C4⋊C4—1062no—
477C4⋊C4—1064no—
478C4⋊C4—1062no—
479C22⋊C4—48no—
480C22⋊C4—1064no—
481C22⋊C4—1064no—
482C22⋊C4—1062no—
483C22⋊C4—1062no—
484C22⋊C4—1062no—
485C22⋊C4—1064no—
486C22⋊C4—1062no—
487C42—1064no—
488C23—22no—
489C23—22no—
490C23—22no—
491C23—22no—
492C23—24no—
493C23—22no—
494C23—22no—
495C23—22no—
496C23—22no—
497C23—22no—
498C23—22no—
499C23—42no—
500C23—42no—
501C23—42no—
502C23—42no—
503C23—48no—
504C23—48no—
505C23—1062no—
506C23—2122no—
507C23—2122no—
508C23—2122no—
509C23C2×#1171992112yes—
510C23C2×#1171992112yes—
511C23C2×#1171992111yescharacteristic
512C23C2×#1171992111yescharacteristic
513C23#1313324312yes—
514C23#1313324812yes—
515C23C22×C22⋊D5311yescharacteristic
516Q8—2122no—
517Q8—2122no—
518D4—2122no—
519D4—2122no—
520D4—2128no—
521D4—2128no—
522D4—2124no—
523D4—2122no—
524D4—2122no—
525C2×C4—22no—
526C2×C4—22no—
527C2×C4—22no—
528C2×C4—24no—
529C2×C4—24no—
530C2×C4—24no—
531C2×C4—24no—
532C2×C4—24no—
533C2×C4—24no—
534C2×C4—41no—
535C2×C4—42no—
536C2×C4—41no—
537C2×C4—42no—
538C2×C4—41no—
539C2×C4—41no—
540C2×C4—42no—
541C2×C4—42no—
542C2×C4—44no—
543C2×C4—44no—
544C2×C4—44no—
545C2×C4—88no—
546C2×C4—1062no—
547C2×C4—1062no—
548C2×C4—1062no—
549C2×C4—2122no—
550C2×C4—2122no—
551C2×C4—2122no—
552C2×C4—2128no—
553C2×C4#1313313114yes—
554C2×C4C2×#1171992111yescharacteristic
555C2×C4C2×#1171992112yes—
556C2×C4C2×#1171992111yescharacteristic
557C2×C4C2×#1171992112yes—
558C2×C4C22×C4×D5311yescharacteristic
559C2×C4C22×D21211yescharacteristic
560C8—2124no—
561C22—21no—
562C22—21no—
563C22—24no—
564C22—24no—
565C22—21no—
566C22—21no—
567C22—22no—
568C22—22no—
569C22—22no—
570C22—22no—
571C22—22no—
572C22—22no—
573C22—44no—
574C22—48no—
575C22—44no—
576C22—44no—
577C22—44no—
578C22—4244no—
579C22—2122no—
580C22—2122no—
581C22—2122no—
582C22C2×#1313313112yes—
583C22#1681062212yes—
584C22#1681062212yes—
585C22#1681064112yes—
586C22C22×#1171992111yescharacteristic
587C22C2×#1313324311yescharacteristic
588C22C2×#1313324811yescharacteristic
589C4—21no—
590C4—21no—
591C4—21no—
592C4—22no—
593C4—21no—
594C4—84no—
595C4—2122no—
596C4—2122no—
597C4—2122no—
598C4C2×#1313313114yes—
599C2—21no—
600C2—22no—
601C2—21no—
602C2—42no—
603C2—42no—
604C2—4242no—
605C2#2027325614yesdirect product factor
606C2C22×#1313313111yescharacteristic
607C2C2×#1681062211yescharacteristic
608C2C2×#1681064111yescharacteristic
subgroupG/H
centerC23C2×#11719921
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#380C2
socleC23×C53C2×C22⋊C4
prime radicalC2×#16811134C2
cosocleC22×C4×C53C24
hypercenterC2×#380D53
hypocenterC53C2×#1513
LCSG ▷ C2×C4×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C24×C4 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
2. C2×#16810587C2semidirect
3. #20273256C2direct (N × H)
4. C2×#16810359C2semidirect
5. C2×#16810356C2semidirect
6. C2×C53×#380C2semidirect
7. C22×#11719945C4semidirect
11. #16811134C4semidirect
14. C2×#13133196C4semidirect
16. C2×#13133191C4semidirect
21. C2×#13133176C4semidirect
23. #16810587C22semidirect
26. #16810359C22semidirect
27. #16810356C22semidirect
32. C53×#380C22semidirect
36. C2×#11719945C2×C4semidirect
52. #13133196C2×C4semidirect
58. #13133191C2×C4semidirect
67. #13133176C2×C4semidirect
297. C2×#380D53semidirect
306. C2×C53#1513semidirect
334. #380C2×D53semidirect
338. C53C2×#1513semidirect
605. C2#20273256direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168111342
C2×#168105872
#202732562
C2×#168103592
C2×#168103562
C2×C53×#3802
C2×#151353