Finite Group Databasebeta

← C2×#20275021 #27385285

Subgroups — C2×#20275021 #27385285

Order 13568.

2270 conjugacy classes of subgroups.

The subgroup lattice has 590 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133197C211yesmaximal, characteristic
2C22×C4⋊C4×D53C211yesmaximal, characteristic
3C22×#13133142C211yesmaximal, characteristic, complemented
4C22×#13133140C211yesmaximal, characteristic, complemented
5C22×#13133129C211yesmaximal, characteristic
6#20275021C218yesmaximal, direct product factor
7C2×#16810385C211yesmaximal, characteristic, complemented
8C2×C53×#340C211yesmaximal, characteristic, complemented
9C23×Dic106C2211yescharacteristic
10C23×C4×D53—21no—
11C23×C4×D53C2211yescharacteristic
12C23×#88322C2211yescharacteristic
13C2×#13133197—24no—
14C2×#13133197—22no—
15C2×#13133197C418yescomplemented
16C2×#13133197C2214yes—
17C22×C4⋊Dic53—21no—
18C22×C4⋊Dic53C2211yescharacteristic
19C22×#11719919—21no—
20C2×C4⋊C4×D53—22no—
21C2×C4⋊C4×D53—24no—
22C2×C4⋊C4×D53—24no—
23C2×C4⋊C4×D53C2214yes—
24C22×#11719921—21no—
25C2×#13133142—24no—
26C2×#13133142C2214yescomplemented
27C2×#13133140—24no—
28C2×#13133140C2214yescomplemented
29C2×#13133129—24no—
30C2×#13133129C2214yes—
31#16810385C2214yescomplemented
32C23×C8×C53C2211yescharacteristic
33C22×C4⋊C4×C53C2211yescharacteristic
34C22×C4⋊C4×C53C2211yescharacteristic
35C53×#340C2214yescomplemented
36C24×D53D411yescharacteristic
37C22×Dic106—22no—
38C22×Dic106—22no—
39C22×Dic106—22no—
40C22×Dic106—22no—
41C22×Dic106—21no—
42C22×Dic106—22no—
43C22×Dic106C2×C414yescomplemented
44C22×Dic106C2312yes—
45C23×Dic53—21no—
46C23×Dic53—21no—
47C23×Dic53D411yescharacteristic
48C22×C4×D53—22no—
49C22×C4×D53—22no—
50C22×C4×D53—22no—
51C22×C4×D53—22no—
52C22×C4×D53—28no—
53C22×C4×D53—24no—
54C22×C4×D53—44no—
55C22×C4×D53—44no—
56C22×C4×D53C2×C414yes—
57C22×C4×D53D412yes—
58C22×C4×D53D414yes—
59C22×C4×D53C2312yes—
60C22×#88322—22no—
61C22×#88322—22no—
62C22×#88322—22no—
63C22×#88322C2312yes—
64#13133197—216no—
65#13133197—28no—
66#13133197C2×C4116yescomplemented
67C2×C4⋊Dic53—22no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—28no—
70C2×C4⋊Dic53—24no—
71C2×C4⋊Dic53D414yes—
72C2×C4⋊Dic53C2312yes—
73C2×#11719919—28no—
74C2×#11719919—24no—
75C4⋊C4×D53—216no—
76C4⋊C4×D53—28no—
77C4⋊C4×D53—28no—
78C2×#11719921—28no—
79C2×#11719921—24no—
80#13133142—28no—
81#13133140—28no—
82#13133129—28no—
83C23×C4×C53—21no—
84C23×C4×C53—21no—
85C23×C4×C53C2311yescharacteristic
86C22×C8×C53—22no—
87C22×C8×C53—22no—
88C22×C8×C53—22no—
89C22×C8×C53C2312yes—
90C2×C4⋊C4×C53—22no—
91C2×C4⋊C4×C53—21no—
92C2×C4⋊C4×C53—22no—
93C2×C4⋊C4×C53—21no—
94C2×C4⋊C4×C53C2×C414yescomplemented
95C2×C4⋊C4×C53D414yes—
96C2×C4⋊C4×C53C2312yes—
97C2×C4⋊C4×C53C2312yes—
98C24×C53C2×D411yescharacteristic
99C23×D53—22no—
100C23×D53—22no—
101C23×D53—22no—
102C23×D53—22no—
103C23×D53C22⋊C414yes—
104C23×D53SD1614yes—
105C23×D53Q1614yes—
106C23×D53C2×D412yes—
107C22×C4×C53—24no—
108C22×C4×C53—24no—
109C22×C4×C53—22no—
110C22×C4×C53—22no—
111C22×C4×C53—42no—
112C22×C4×C53—42no—
113C22×C4×C53—42no—
114C22×C4×C53—42no—
115C22×C4×C53C22×C412yes—
116C22×C4×C53C2×D411yescharacteristic
117C22×C4×C53C2×D412yes—
118C22×C4×C53C2×D412yes—
119C22×C4×C53C2×D412yes—
120C22×C4×C53#4112yes—
121C22×C4×C53#4112yes—
122C22×C4×C53C2411yescharacteristic
123C22×Dic53—22no—
124C22×Dic53—22no—
125C22×Dic53—24no—
126C22×Dic53—24no—
127C22×Dic53—22no—
128C22×Dic53—22no—
129C22×Dic53—42no—
130C22×Dic53—42no—
131C22×Dic53—42no—
132C22×Dic53—42no—
133C22×Dic53C22⋊C414yes—
134C22×Dic53C2×D412yes—
135C2×C4×D53—28no—
136C2×C4×D53—28no—
137C2×C4×D53—24no—
138C2×C4×D53—28no—
139C2×C4×D53—28no—
140C2×C4×D53—24no—
141C2×C4×D53—22no—
142C2×C4×D53—22no—
143C2×C4×D53—22no—
144C2×C4×D53—22no—
145C2×C4×D53—216no—
146C2×C4×D53—48no—
147C2×C4×D53—432no—
148C2×C4×D53—48no—
149C2×C4×D53C22⋊C418yes—
150C2×C4×D53C22×C414yes—
151C2×C4×D53C2×D414yes—
152C2×Dic106—28no—
153C2×Dic106—28no—
154C2×Dic106—22no—
155C2×Dic106—22no—
156C2×Dic106—22no—
157C2×Dic106—22no—
158C2×Dic106—24no—
159C2×Dic106—22no—
160C2×Dic106—42no—
161C2×Dic106—44no—
162C2×Dic106—44no—
163C2×Dic106—42no—
164C2×Dic106C22×C414yescomplemented
165C2×C8×C53—22no—
166C2×C8×C53—22no—
167C2×C8×C53—22no—
168C2×C8×C53—44no—
169C4⋊C4×C53—24no—
170C4⋊C4×C53—24no—
171C4⋊C4×C53—22no—
172C4⋊C4×C53—22no—
173C4⋊C4×C53C22×C414yescomplemented
174C4⋊C4×C53C2×D414yes—
175#11719921—216no—
176C4⋊Dic53—24no—
177C4⋊Dic53—22no—
178C4⋊Dic53—216no—
179C4⋊Dic53C2×D414yes—
180#11719919—216no—
181C2×#88322—22no—
182C2×#88322—22no—
183C2×#88322—22no—
184C2×#88322—44no—
185C23×C53C2×C22⋊C412yes—
186C23×C53#12012yes—
187C23×C53#12312yes—
188C23×C53C2×SD1612yes—
189C23×C53C2×Q1612yes—
190C23×C53AGL(1,ℤ/8)12yes—
191C23×C53#13712yes—
192C23×C53C22×D411yescharacteristic
193C22×D53—22no—
194C22×D53—22no—
195C22×D53—22no—
196C22×D53—22no—
197C22×D53—28no—
198C22×D53—28no—
199C22×D53—28no—
200C22×D53—28no—
201C22×D53—28no—
202C22×D53—28no—
203C22×D53#103116yescomplemented
204C22×D53C2×C22⋊C414yes—
205C22×D53C2×SD1614yes—
206C22×D53C2×Q1614yes—
207C2×C4×C53—24no—
208C2×C4×C53—24no—
209C2×C4×C53—42no—
210C2×C4×C53—42no—
211C2×C4×C53—42no—
212C2×C4×C53—42no—
213C2×C4×C53—48no—
214C2×C4×C53—48no—
215C2×C4×C53C2×C22⋊C412yes—
216C2×C4×C53C4×D414yes—
217C2×C4×C53C4×D414yes—
218C2×C4×C53#12012yes—
219C2×C4×C53#12114yes—
220C2×C4×C53#12114yes—
221C2×C4×C53#12312yes—
222C2×C4×C53C23×C411yescharacteristic
223C2×C4×C53C22×D411yescharacteristic
224C2×C4×C53C22×D411yescharacteristic
225C2×C4×C53C22×D411yescharacteristic
226C2×C4×C53C2×#4111yescharacteristic
227C2×C4×C53C2×#4111yescharacteristic
228C2×Dic53—22no—
229C2×Dic53—22no—
230C2×Dic53—28no—
231C2×Dic53—24no—
232C2×Dic53—24no—
233C2×Dic53—42no—
234C2×Dic53—42no—
235C2×Dic53—42no—
236C2×Dic53—42no—
237C2×Dic53—48no—
238C2×Dic53—48no—
239C2×Dic53C2×C22⋊C414yes—
240C4×D53—24no—
241C4×D53—24no—
242C4×D53—24no—
243C4×D53—24no—
244C4×D53—216no—
245C4×D53—432no—
246Dic106—24no—
247Dic106—24no—
248Dic106—42no—
249Dic106—42no—
250Dic106—48no—
251C8×C53—42no—
252#88322—42no—
253C2×#1872—531nomaximal, Sylow
254C22×C53#38614yes—
255C22×C53C2×#10314yes—
256C22×C53#41714yes—
257C22×C53#45014yes—
258C22×C53#45114yes—
259C22×C53#48314yes—
260C22×C53#48414yes—
261C22×C53C22×C22⋊C411yescharacteristic
262C22×C53C2×#12011yescharacteristic
263C22×C53C2×#12311yescharacteristic
264C22×C53C22×SD1611yescharacteristic
265C22×C53C22×Q1611yescharacteristic
266C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
267C22×C53C2×#13711yescharacteristic
268C2×D53—24no—
269C2×D53—24no—
270C2×D53—24no—
271C2×D53—232no—
272C2×D53—24no—
273C2×D53—24no—
274C2×D53—24no—
275C2×D53C2×#10318yescomplemented
276C4×C53—44no—
277C4×C53—44no—
278C4×C53#38614yes—
279C4×C53C2×C4×D411yescharacteristic
280C4×C53C2×C4×D411yescharacteristic
281C4×C53C2×#12111yescharacteristic
282C4×C53C2×#12111yescharacteristic
283Dic53—24no—
284Dic53—44no—
285Dic53—44no—
286C22×C22⋊Q8—531no—
287C23×C4⋊C4—531no—
288C22×#103—531no—
289C22×#103—531no—
290C22×C22⋊C8—531no—
291#1872—538no—
292C2×#340—531no—
293C2×#340D5311yescharacteristic, complemented
294C2×C53#187218yescomplemented
295C2×C53C2×#38611yescharacteristic
296C2×C53C22×#10311yescharacteristic
297C2×C53C2×#41711yescharacteristic
298C2×C53C2×#45011yescharacteristic
299C2×C53C2×#45111yescharacteristic
300C2×C53C2×#48311yescharacteristic
301C2×C53C2×#48411yescharacteristic
302D53—28no—
303C23×Q8—531no—
304C24×C4—531no—
305C24×C4—1061no—
306C23×C8—531no—
307C23×C8C2×D5311yescharacteristic
308C2×C22⋊Q8—538no—
309C2×C22⋊Q8—534no—
310C2×C22⋊Q8—1062no—
311C2×C22⋊Q8—1064no—
312C22×C4⋊C4—531no—
313C22×C4⋊C4—534no—
314C22×C4⋊C4—1064no—
315C22×C4⋊C4—1062no—
316C22×C4⋊C4—1061no—
317C22×C4⋊C4—1061no—
318C22×C4⋊C4—1064no—
319C22×C4⋊C4C2×D5311yescharacteristic
320C22×C4⋊C4C2×D5311yescharacteristic
321C22×C22⋊C4—1061no—
322C2×#103—534no—
323C2×#103—534no—
324C2×#103—1064no—
325C2×#103—1064no—
326C2×C22⋊C8—534no—
327C2×C22⋊C8—1064no—
328#340—534no—
329#340C2×D5314yescomplemented
330C53C2×#187211yescharacteristic, complemented, Sylow
331C25—531no—
332C22×Q8—534no—
333C22×Q8—532no—
334C22×Q8—1062no—
335C22×Q8—1061no—
336C22×Q8—1062no—
337C22×Q8—1062no—
338C22×Q8—1062no—
339C22×Q8—1062no—
340C23×C4—21no—
341C23×C4—21no—
342C23×C4—531no—
343C23×C4—534no—
344C23×C4—532no—
345C23×C4—534no—
346C23×C4—532no—
347C23×C4—1064no—
348C23×C4—1068no—
349C23×C4—1062no—
350C23×C4—1062no—
351C23×C4—1062no—
352C23×C4—1062no—
353C23×C4—1061no—
354C23×C4—1061no—
355C23×C4—2124no—
356C23×C4—2124no—
357C23×C4C22×D5311yescharacteristic
358C22×C8—22no—
359C22×C8—22no—
360C22×C8—22no—
361C22×C8—532no—
362C22×C8—1062no—
363C22×C8—1062no—
364C22×C8—1062no—
365C22×C8C22×D5312yes—
366C22⋊Q8—5316no—
367C22⋊Q8—10616no—
368C22⋊Q8—1068no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—22no—
373C2×C4⋊C4—534no—
374C2×C4⋊C4—532no—
375C2×C4⋊C4—1068no—
376C2×C4⋊C4—10616no—
377C2×C4⋊C4—1064no—
378C2×C4⋊C4—1068no—
379C2×C4⋊C4—1064no—
380C2×C4⋊C4—1068no—
381C2×C4⋊C4—1062no—
382C2×C4⋊C4—1061no—
383C2×C4⋊C4—1068no—
384C2×C4⋊C4C4×D5314yescomplemented
385C2×C4⋊C4D21214yes—
386C2×C4⋊C4C22×D5312yes—
387C2×C4⋊C4C22×D5312yes—
388C2×C22⋊C4—1068no—
389C2×C22⋊C4—1064no—
390#103—1068no—
391#103—1068no—
392C22⋊C8—1068no—
393C24—534no—
394C24—534no—
395C24—534no—
396C24—532no—
397C24—1062no—
398C24—1062no—
399C24—1062no—
400C24—1062no—
401C24D4×D5311yescharacteristic
402C2×Q8—534no—
403C2×Q8—1064no—
404C2×Q8—1062no—
405C2×Q8—1068no—
406C2×Q8—1068no—
407C2×Q8—1062no—
408C2×Q8—1062no—
409C2×Q8—1062no—
410C2×Q8—1062no—
411C2×Q8—2122no—
412C2×Q8—2124no—
413C2×Q8—2124no—
414C2×Q8—2122no—
415C22×C4—24no—
416C22×C4—24no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—42no—
420C22×C4—42no—
421C22×C4—42no—
422C22×C4—42no—
423C22×C4—534no—
424C22×C4—532no—
425C22×C4—538no—
426C22×C4—534no—
427C22×C4—534no—
428C22×C4—10616no—
429C22×C4—1062no—
430C22×C4—1062no—
431C22×C4—1068no—
432C22×C4—1068no—
433C22×C4—1064no—
434C22×C4—1068no—
435C22×C4—1068no—
436C22×C4—1064no—
437C22×C4—1062no—
438C22×C4—1062no—
439C22×C4—1062no—
440C22×C4—1062no—
441C22×C4—1064no—
442C22×C4—1064no—
443C22×C4—1062no—
444C22×C4—1062no—
445C22×C4—2128no—
446C22×C4—21232no—
447C22×C4—2128no—
448C22×C4—2122no—
449C22×C4—2122no—
450C22×C4—2122no—
451C22×C4—2122no—
452C22×C4C2×C4×D5312yes—
453C22×C4C2×D21212yes—
454C22×C4#1171994512yes—
455C22×C4D4×D5311yescharacteristic
456C22×C4D4×D5312yes—
457C22×C4D4×D5312yes—
458C22×C4#1171994912yes—
459C22×C4C23×D5311yescharacteristic
460C2×C8—22no—
461C2×C8—22no—
462C2×C8—22no—
463C2×C8—44no—
464C2×C8—1062no—
465C2×C8—1062no—
466C2×C8—1062no—
467C2×C8—2124no—
468C4⋊C4—24no—
469C4⋊C4—24no—
470C4⋊C4—22no—
471C4⋊C4—22no—
472C4⋊C4—534no—
473C4⋊C4—10616no—
474C4⋊C4—10616no—
475C4⋊C4—1064no—
476C4⋊C4—1062no—
477C4⋊C4C2×C4×D5314yescomplemented
478C4⋊C4C2×D21214yes—
479C22⋊C4—10616no—
480C23—534no—
481C23—5316no—
482C23—534no—
483C23—534no—
484C23—1068no—
485C23—1068no—
486C23—1068no—
487C23—1062no—
488C23—1062no—
489C23—1068no—
490C23—1062no—
491C23—1062no—
492C23—1068no—
493C23—1068no—
494C23C22⋊C4×D5312yes—
495C23#1313318712yes—
496C23#1313320212yes—
497C23SD16×D5312yes—
498C23Q16×D5312yes—
499C23#1313324112yes—
500C23#1313324612yes—
501C23C2×D4×D5311yescharacteristic
502Q8—1064no—
503Q8—1064no—
504Q8—2128no—
505Q8—2122no—
506Q8—2122no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—48no—
510C2×C4—48no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—534no—
516C2×C4—1068no—
517C2×C4—1062no—
518C2×C4—1062no—
519C2×C4—10616no—
520C2×C4—1064no—
521C2×C4—1064no—
522C2×C4—1064no—
523C2×C4—1064no—
524C2×C4—1064no—
525C2×C4—1064no—
526C2×C4—21232no—
527C2×C4—2128no—
528C2×C4—2128no—
529C2×C4—2122no—
530C2×C4—2122no—
531C2×C4—2122no—
532C2×C4—2122no—
533C2×C4C22⋊C4×D5312yes—
534C2×C4#1313317914yes—
535C2×C4C4×D21214yes—
536C2×C4#1313318712yes—
537C2×C4#1313319014yes—
538C2×C4#1313319214yes—
539C2×C4#1313320212yes—
540C2×C4C22×C4×D5311yescharacteristic
541C2×C4C2×D4×D5311yescharacteristic
542C2×C4C2×D4×D5311yescharacteristic
543C2×C4C22×D21211yescharacteristic
544C2×C4C2×#1171994511yescharacteristic
545C2×C4C2×#1171994911yescharacteristic
546C8—42no—
547C8—2122no—
548C22—538no—
549C22—1064no—
550C22—10632no—
551C22—1064no—
552C22—1064no—
553C22—1064no—
554C22—1064no—
555C22—1064no—
556C22#1681050114yes—
557C22#103×D5314yes—
558C22#1681062114yes—
559C22#1681077514yes—
560C22#1681078014yes—
561C22#1681095514yes—
562C22#1681096014yes—
563C22C2×C22⋊C4×D5311yescharacteristic
564C22C2×#1313318711yescharacteristic
565C22C2×#1313320211yescharacteristic
566C22C2×SD16×D5311yescharacteristic
567C22C2×Q16×D5311yescharacteristic
568C22C2×#1313324111yescharacteristic
569C22C2×#1313324611yescharacteristic
570C4—44no—
571C4—44no—
572C4—1064no—
573C4—2124no—
574C4—2124no—
575C4#1681050114yes—
576C4C2×#1313317911yescharacteristic
577C4C2×C4×D21211yescharacteristic
578C4C2×#1313319011yescharacteristic
579C4C2×#1313319211yescharacteristic
580C2—1068no—
581C2#2027502118yesdirect product factor
582C2C2×#1681050111yescharacteristic
583C2C2×#103×D5311yescharacteristic
584C2C2×#1681062111yescharacteristic
585C2C2×#1681077511yescharacteristic
586C2C2×#1681078011yescharacteristic
587C2C2×#1681095511yescharacteristic
588C2C2×#1681096011yescharacteristic
subgroupG/H
centerC24D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#340C2
socleC24×C53C2×D4
prime radicalC24×D53D4
cosocleC22×C4×C53C24
hypercenterC2×#340D53
hypocenterC53C2×#1872
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C24 ◁ C23×C4 ◁ C2×#340

Complements

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

NH ≅ G/Ntype
3. C22×#13133142C2semidirect
4. C22×#13133140C2semidirect
6. #20275021C2direct (N × H)
7. C2×#16810385C2semidirect
8. C2×C53×#340C2semidirect
15. C2×#13133197C4semidirect
26. C2×#13133142C22semidirect
28. C2×#13133140C22semidirect
31. #16810385C22semidirect
35. C53×#340C22semidirect
43. C22×Dic106C2×C4semidirect
66. #13133197C2×C4semidirect
94. C2×C4⋊C4×C53C2×C4semidirect
164. C2×Dic106C22×C4semidirect
173. C4⋊C4×C53C22×C4semidirect
203. C22×D53#103semidirect
275. C2×D53C2×#103semidirect
293. C2×#340D53semidirect
294. C2×C53#1872semidirect
329. #340C2×D53semidirect
330. C53C2×#1872semidirect
384. C2×C4⋊C4C4×D53semidirect
477. C4⋊C4C2×C4×D53semidirect
581. C2#20275021direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131331972
C22×C4⋊C4×D532
C22×#131331422
C22×#131331402
C22×#131331292
#202750212
C2×#168103852
C2×C53×#3402
C2×#187253