Finite Group Databasebeta

← C2×#20275029 #27385289

Subgroups — C2×#20275029 #27385289

Order 13568.

2614 conjugacy classes of subgroups.

The subgroup lattice has 646 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133188C211yesmaximal, characteristic
2C22×C4⋊C4×D53C211yesmaximal, characteristic, complemented
3C22×#13133138C211yesmaximal, characteristic, complemented
4C22×#13133129C211yesmaximal, characteristic, complemented
5#20275029C218yesmaximal, direct product factor
6C2×#16810386C211yesmaximal, characteristic, complemented
7C2×#16810385C211yesmaximal, characteristic, complemented
8C22×#102×C53C211yesmaximal, characteristic, complemented
9C23×C22⋊D53—21no—
10C23×C4×D53—21no—
11C23×C4×D53C2211yescharacteristic
12C23×#88322C2211yescharacteristic
13C2×#13133188—24no—
14C2×#13133188—22no—
15C2×#13133188C418yescomplemented
16C2×#13133188C2214yes—
17C22×C4⋊Dic53C2211yescharacteristic
18C22×C4⋊Dic53C2211yescharacteristic
19C22×#11719919—21no—
20C2×C4⋊C4×D53—22no—
21C2×C4⋊C4×D53—24no—
22C2×C4⋊C4×D53—24no—
23C2×C4⋊C4×D53C2214yescomplemented
24C22×C22⋊Dic53—21no—
25C2×#13133138—24no—
26C2×#13133138C2214yescomplemented
27C2×#13133129—24no—
28C2×#13133129C2214yescomplemented
29#16810386C2214yescomplemented
30#16810385C2214yescomplemented
31C23×D4×C53C2211yescharacteristic
32C23×C8×C53C2211yescharacteristic
33C22×C4⋊C4×C53C2211yescharacteristic
34C2×#102×C53—24no—
35C2×#102×C53C2214yescomplemented
36C24×D53D411yescharacteristic
37C22×C22⋊D53—28no—
38C22×C22⋊D53—24no—
39C22×C22⋊D53—44no—
40C22×C22⋊D53—44no—
41C23×Dic53—21no—
42C23×Dic53—21no—
43C23×Dic53D411yescharacteristic
44C22×C4×D53—22no—
45C22×C4×D53—22no—
46C22×C4×D53—22no—
47C22×C4×D53—22no—
48C22×C4×D53—28no—
49C22×C4×D53—24no—
50C22×C4×D53—44no—
51C22×C4×D53—44no—
52C22×C4×D53C2×C414yes—
53C22×C4×D53D412yes—
54C22×C4×D53D414yes—
55C22×C4×D53C2312yes—
56C22×#88322—22no—
57C22×#88322—22no—
58C22×#88322—22no—
59C22×#88322C2312yes—
60#13133188—216no—
61#13133188—28no—
62#13133188C2×C4116yescomplemented
63C2×C4⋊Dic53—22no—
64C2×C4⋊Dic53—21no—
65C2×C4⋊Dic53—22no—
66C2×C4⋊Dic53—21no—
67C2×C4⋊Dic53C2×C414yescomplemented
68C2×C4⋊Dic53D414yescomplemented
69C2×C4⋊Dic53C2312yes—
70C2×C4⋊Dic53C2312yes—
71C2×#11719919—28no—
72C2×#11719919—24no—
73C4⋊C4×D53—216no—
74C4⋊C4×D53—28no—
75C4⋊C4×D53—28no—
76C2×C22⋊Dic53—28no—
77C2×C22⋊Dic53—24no—
78#13133138—28no—
79#13133129—28no—
80C25×C53—21no—
81C22×D4×C53—22no—
82C22×D4×C53—22no—
83C22×D4×C53—21no—
84C22×D4×C53—22no—
85C22×D4×C53—22no—
86C22×D4×C53—22no—
87C22×D4×C53C2×C414yescomplemented
88C22×D4×C53C2312yes—
89C23×C4×C53—21no—
90C23×C4×C53C2311yescharacteristic
91C22×C8×C53—22no—
92C22×C8×C53—22no—
93C22×C8×C53—22no—
94C22×C8×C53C2312yes—
95C2×C4⋊C4×C53—22no—
96C2×C4⋊C4×C53—21no—
97C2×C4⋊C4×C53D414yescomplemented
98C2×C4⋊C4×C53C2312yes—
99#102×C53—28no—
100C24×C53—24no—
101C24×C53—22no—
102C24×C53—42no—
103C24×C53—42no—
104C24×C53—42no—
105C24×C53—42no—
106C24×C53—42no—
107C24×C53—42no—
108C24×C53C2×D411yescharacteristic
109C23×D53—22no—
110C23×D53—22no—
111C23×D53—22no—
112C23×D53—22no—
113C23×D53C22⋊C414yes—
114C23×D53D814yes—
115C23×D53SD1614yes—
116C23×D53C2×D412yes—
117C2×D4×C53—24no—
118C2×D4×C53—28no—
119C2×D4×C53—28no—
120C2×D4×C53—22no—
121C2×D4×C53—22no—
122C2×D4×C53—22no—
123C2×D4×C53—22no—
124C2×D4×C53—22no—
125C2×D4×C53—42no—
126C2×D4×C53—44no—
127C2×D4×C53—44no—
128C2×D4×C53—42no—
129C2×D4×C53C22×C414yescomplemented
130C22×C4×C53—24no—
131C22×C4×C53—22no—
132C22×C4×C53—42no—
133C22×C4×C53—42no—
134C22×C4×C53C22×C412yes—
135C22×C4×C53C2×D411yescharacteristic
136C22×C4×C53C2×D412yes—
137C22×C4×C53C2×D412yes—
138C22×C4×C53C2×D412yes—
139C22×C4×C53#4112yes—
140C22×C4×C53#4112yes—
141C22×C4×C53C2411yescharacteristic
142C2×C22⋊D53—216no—
143C2×C22⋊D53—432no—
144C2×C22⋊D53—48no—
145C2×C22⋊D53—48no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—24no—
149C22×Dic53—24no—
150C22×Dic53—22no—
151C22×Dic53—22no—
152C22×Dic53—42no—
153C22×Dic53—42no—
154C22×Dic53—42no—
155C22×Dic53—42no—
156C22×Dic53C22⋊C414yes—
157C22×Dic53C2×D412yes—
158C2×C4×D53—28no—
159C2×C4×D53—28no—
160C2×C4×D53—24no—
161C2×C4×D53—28no—
162C2×C4×D53—28no—
163C2×C4×D53—24no—
164C2×C4×D53—22no—
165C2×C4×D53—22no—
166C2×C4×D53—22no—
167C2×C4×D53—22no—
168C2×C4×D53—216no—
169C2×C4×D53—48no—
170C2×C4×D53—432no—
171C2×C4×D53—48no—
172C2×C4×D53C22⋊C418yes—
173C2×C4×D53C22×C414yes—
174C2×C4×D53C2×D414yes—
175C2×C8×C53—22no—
176C2×C8×C53—22no—
177C2×C8×C53—22no—
178C2×C8×C53—44no—
179C4⋊C4×C53—24no—
180C4⋊C4×C53—22no—
181C4⋊C4×C53C2×D414yescomplemented
182C22⋊Dic53—216no—
183C4⋊Dic53—24no—
184C4⋊Dic53—24no—
185C4⋊Dic53—22no—
186C4⋊Dic53—22no—
187C4⋊Dic53C22×C414yescomplemented
188C4⋊Dic53C2×D414yescomplemented
189#11719919—216no—
190C2×#88322—22no—
191C2×#88322—22no—
192C2×#88322—22no—
193C2×#88322—44no—
194C23×C53—24no—
195C23×C53—42no—
196C23×C53—42no—
197C23×C53—42no—
198C23×C53—42no—
199C23×C53—42no—
200C23×C53—42no—
201C23×C53—48no—
202C23×C53—48no—
203C23×C53—48no—
204C23×C53—84no—
205C23×C53—84no—
206C23×C53—84no—
207C23×C53—84no—
208C23×C53C2×C22⋊C412yes—
209C23×C53#12012yes—
210C23×C53#12312yes—
211C23×C53C2×D812yes—
212C23×C53C2×SD1612yes—
213C23×C53AGL(1,ℤ/8)12yes—
214C23×C53#13712yes—
215C23×C53C22×D411yescharacteristic
216C22×D53—22no—
217C22×D53—22no—
218C22×D53—22no—
219C22×D53—22no—
220C22×D53—28no—
221C22×D53—28no—
222C22×D53—28no—
223C22×D53—28no—
224C22×D53—28no—
225C22×D53—28no—
226C22×D53#102116yescomplemented
227C22×D53C2×C22⋊C414yes—
228C22×D53C2×D814yes—
229C22×D53C2×SD1614yes—
230D4×C53—24no—
231D4×C53—24no—
232D4×C53—42no—
233D4×C53—42no—
234D4×C53—48no—
235C2×C4×C53—24no—
236C2×C4×C53—42no—
237C2×C4×C53—42no—
238C2×C4×C53—48no—
239C2×C4×C53C2×C22⋊C412yes—
240C2×C4×C53C4×D414yes—
241C2×C4×C53C4×D414yes—
242C2×C4×C53#12012yes—
243C2×C4×C53#12114yes—
244C2×C4×C53#12114yes—
245C2×C4×C53#12312yes—
246C2×C4×C53C23×C411yescharacteristic
247C2×C4×C53C22×D411yescharacteristic
248C2×C4×C53C22×D411yescharacteristic
249C2×C4×C53C22×D411yescharacteristic
250C2×C4×C53C2×#4111yescharacteristic
251C2×C4×C53C2×#4111yescharacteristic
252C22⋊D53—432no—
253C2×Dic53—22no—
254C2×Dic53—22no—
255C2×Dic53—28no—
256C2×Dic53—24no—
257C2×Dic53—24no—
258C2×Dic53—42no—
259C2×Dic53—42no—
260C2×Dic53—42no—
261C2×Dic53—42no—
262C2×Dic53—48no—
263C2×Dic53—48no—
264C2×Dic53C2×C22⋊C414yes—
265C4×D53—24no—
266C4×D53—24no—
267C4×D53—24no—
268C4×D53—24no—
269C4×D53—216no—
270C4×D53—432no—
271C8×C53—42no—
272#88322—42no—
273C2×#1874—531nomaximal, Sylow
274C22×C53—44no—
275C22×C53—44no—
276C22×C53—44no—
277C22×C53—82no—
278C22×C53—82no—
279C22×C53—816no—
280C22×C53—82no—
281C22×C53—82no—
282C22×C53#38614yes—
283C22×C53C2×#10214yes—
284C22×C53#41714yes—
285C22×C53#44714yes—
286C22×C53#44814yes—
287C22×C53#48014yes—
288C22×C53#48114yes—
289C22×C53C22×C22⋊C411yescharacteristic
290C22×C53C2×#12011yescharacteristic
291C22×C53C2×#12311yescharacteristic
292C22×C53C22×D811yescharacteristic
293C22×C53C22×SD1611yescharacteristic
294C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
295C22×C53C2×#13711yescharacteristic
296C2×D53—24no—
297C2×D53—24no—
298C2×D53—24no—
299C2×D53—232no—
300C2×D53—24no—
301C2×D53—24no—
302C2×D53—24no—
303C2×D53C2×#10218yescomplemented
304C4×C53—44no—
305C4×C53#38614yes—
306C4×C53C2×C4×D411yescharacteristic
307C4×C53C2×C4×D411yescharacteristic
308C4×C53C2×#12111yescharacteristic
309C4×C53C2×#12111yescharacteristic
310Dic53—24no—
311Dic53—44no—
312Dic53—44no—
313C22×#121—531no—
314C23×C4⋊C4—531no—
315C22×#102—531no—
316C22×#102D5311yescharacteristic, complemented
317C22×C22⋊C8—531no—
318#1874—538no—
319C2×#340—531no—
320C2×#340—531no—
321C2×C53—84no—
322C2×C53#187418yescomplemented
323C2×C53C2×#38611yescharacteristic
324C2×C53C22×#10211yescharacteristic
325C2×C53C2×#41711yescharacteristic
326C2×C53C2×#44711yescharacteristic
327C2×C53C2×#44811yescharacteristic
328C2×C53C2×#48011yescharacteristic
329C2×C53C2×#48111yescharacteristic
330D53—28no—
331C23×D4—1061no—
332C23×D4C2×D5311yescharacteristic
333C24×C4—531no—
334C24×C4—1061no—
335C23×C8—531no—
336C23×C8C2×D5311yescharacteristic
337C2×#121—538no—
338C2×#121—534no—
339C2×#121—1062no—
340C2×#121—1064no—
341C22×C4⋊C4—531no—
342C22×C4⋊C4—531no—
343C22×C4⋊C4—534no—
344C22×C4⋊C4—1064no—
345C22×C4⋊C4—1062no—
346C22×C4⋊C4—1061no—
347C22×C4⋊C4—1064no—
348C22×C4⋊C4C2×D5311yescharacteristic
349C22×C22⋊C4—1061no—
350C2×#102—24no—
351C2×#102—534no—
352C2×#102—1064no—
353C2×#102C2×D5314yescomplemented
354C2×C22⋊C8—534no—
355C2×C22⋊C8—1064no—
356#340—534no—
357#340—534no—
358C53C2×#187411yescharacteristic, complemented, Sylow
359C25—21no—
360C25—531no—
361C22×D4—22no—
362C22×D4—22no—
363C22×D4—21no—
364C22×D4—22no—
365C22×D4—22no—
366C22×D4—22no—
367C22×D4—1068no—
368C22×D4—1064no—
369C22×D4—2124no—
370C22×D4—2124no—
371C22×D4C4×D5314yescomplemented
372C22×D4C22×D5312yes—
373C23×C4—21no—
374C23×C4—531no—
375C23×C4—534no—
376C23×C4—532no—
377C23×C4—534no—
378C23×C4—532no—
379C23×C4—1064no—
380C23×C4—1068no—
381C23×C4—1062no—
382C23×C4—1062no—
383C23×C4—1062no—
384C23×C4—1062no—
385C23×C4—1061no—
386C23×C4—1061no—
387C23×C4—2124no—
388C23×C4—2124no—
389C23×C4C22×D5311yescharacteristic
390C22×C8—22no—
391C22×C8—22no—
392C22×C8—22no—
393C22×C8—532no—
394C22×C8—1062no—
395C22×C8—1062no—
396C22×C8—1062no—
397C22×C8C22×D5312yes—
398#121—5316no—
399#121—10616no—
400#121—1068no—
401C2×C4⋊C4—21no—
402C2×C4⋊C4—22no—
403C2×C4⋊C4—534no—
404C2×C4⋊C4—534no—
405C2×C4⋊C4—532no—
406C2×C4⋊C4—532no—
407C2×C4⋊C4—1068no—
408C2×C4⋊C4—10616no—
409C2×C4⋊C4—1064no—
410C2×C4⋊C4—1068no—
411C2×C4⋊C4—1061no—
412C2×C4⋊C4—1061no—
413C2×C4⋊C4—1062no—
414C2×C4⋊C4—1062no—
415C2×C4⋊C4—1068no—
416C2×C4⋊C4C22⋊D5314yescomplemented
417C2×C4⋊C4C22×D5312yes—
418C2×C22⋊C4—1068no—
419C2×C22⋊C4—1064no—
420#102—28no—
421#102—1068no—
422C22⋊C8—1068no—
423C24—24no—
424C24—22no—
425C24—42no—
426C24—42no—
427C24—42no—
428C24—42no—
429C24—42no—
430C24—42no—
431C24—534no—
432C24—534no—
433C24—534no—
434C24—532no—
435C24—1062no—
436C24—1062no—
437C24—1062no—
438C24—1062no—
439C24D4×D5311yescharacteristic
440C2×D4—24no—
441C2×D4—28no—
442C2×D4—28no—
443C2×D4—22no—
444C2×D4—22no—
445C2×D4—22no—
446C2×D4—22no—
447C2×D4—22no—
448C2×D4—42no—
449C2×D4—44no—
450C2×D4—44no—
451C2×D4—42no—
452C2×D4—10616no—
453C2×D4—21232no—
454C2×D4—2128no—
455C2×D4—2128no—
456C2×D4C2×C4×D5314yescomplemented
457C22×C4—24no—
458C22×C4—22no—
459C22×C4—42no—
460C22×C4—42no—
461C22×C4—534no—
462C22×C4—532no—
463C22×C4—538no—
464C22×C4—534no—
465C22×C4—534no—
466C22×C4—10616no—
467C22×C4—1062no—
468C22×C4—1062no—
469C22×C4—1068no—
470C22×C4—1068no—
471C22×C4—1064no—
472C22×C4—1068no—
473C22×C4—1068no—
474C22×C4—1064no—
475C22×C4—1062no—
476C22×C4—1062no—
477C22×C4—1062no—
478C22×C4—1062no—
479C22×C4—1064no—
480C22×C4—1064no—
481C22×C4—1062no—
482C22×C4—1062no—
483C22×C4—2128no—
484C22×C4—21232no—
485C22×C4—2128no—
486C22×C4—2122no—
487C22×C4—2122no—
488C22×C4—2122no—
489C22×C4—2122no—
490C22×C4C2×C4×D5312yes—
491C22×C4#1171994512yes—
492C22×C4D4×D5311yescharacteristic
493C22×C4D4×D5312yes—
494C22×C4D4×D5312yes—
495C22×C4#1171994712yes—
496C22×C4C2×C22⋊D5312yes—
497C22×C4C23×D5311yescharacteristic
498C2×C8—22no—
499C2×C8—22no—
500C2×C8—22no—
501C2×C8—44no—
502C2×C8—1062no—
503C2×C8—1062no—
504C2×C8—1062no—
505C2×C8—2124no—
506C4⋊C4—24no—
507C4⋊C4—22no—
508C4⋊C4—534no—
509C4⋊C4—534no—
510C4⋊C4—10616no—
511C4⋊C4—1064no—
512C4⋊C4—1064no—
513C4⋊C4—1062no—
514C4⋊C4—1062no—
515C4⋊C4C2×C22⋊D5314yescomplemented
516C22⋊C4—10616no—
517C23—24no—
518C23—48no—
519C23—42no—
520C23—48no—
521C23—42no—
522C23—48no—
523C23—42no—
524C23—42no—
525C23—42no—
526C23—42no—
527C23—84no—
528C23—84no—
529C23—84no—
530C23—84no—
531C23—534no—
532C23—5316no—
533C23—534no—
534C23—534no—
535C23—1068no—
536C23—1068no—
537C23—1068no—
538C23—1062no—
539C23—1062no—
540C23—1068no—
541C23—1062no—
542C23—1062no—
543C23—1068no—
544C23—1068no—
545C23C22⋊C4×D5312yes—
546C23#1313318512yes—
547C23#1313319912yes—
548C23D8×D5312yes—
549C23SD16×D5312yes—
550C23#1313324112yes—
551C23#1313324612yes—
552C23C2×D4×D5311yescharacteristic
553D4—24no—
554D4—24no—
555D4—48no—
556D4—42no—
557D4—42no—
558D4—21232no—
559C2×C4—24no—
560C2×C4—48no—
561C2×C4—42no—
562C2×C4—42no—
563C2×C4—534no—
564C2×C4—1068no—
565C2×C4—1062no—
566C2×C4—1062no—
567C2×C4—10616no—
568C2×C4—1064no—
569C2×C4—1064no—
570C2×C4—1064no—
571C2×C4—1064no—
572C2×C4—1064no—
573C2×C4—1064no—
574C2×C4—21232no—
575C2×C4—2128no—
576C2×C4—2128no—
577C2×C4—2122no—
578C2×C4—2122no—
579C2×C4—2122no—
580C2×C4—2122no—
581C2×C4C22⋊C4×D5312yes—
582C2×C4#1313317714yes—
583C2×C4C4×C22⋊D5314yes—
584C2×C4#1313318512yes—
585C2×C4#1313318814yes—
586C2×C4#1313319014yes—
587C2×C4#1313319912yes—
588C2×C4C22×C4×D5311yescharacteristic
589C2×C4C2×D4×D5311yescharacteristic
590C2×C4C2×D4×D5311yescharacteristic
591C2×C4C22×C22⋊D5311yescharacteristic
592C2×C4C2×#1171994711yescharacteristic
593C2×C4C2×#1171994511yescharacteristic
594C8—42no—
595C8—2122no—
596C22—44no—
597C22—44no—
598C22—44no—
599C22—816no—
600C22—82no—
601C22—82no—
602C22—82no—
603C22—82no—
604C22—538no—
605C22—10632no—
606C22—1064no—
607C22—1064no—
608C22—1064no—
609C22—1064no—
610C22—1064no—
611C22—1064no—
612C22#1681049914yes—
613C22#102×D5314yes—
614C22#1681062114yes—
615C22#1681075514yes—
616C22#1681076014yes—
617C22#1681093514yes—
618C22#1681094014yes—
619C22C2×C22⋊C4×D5311yescharacteristic
620C22C2×#1313318511yescharacteristic
621C22C2×#1313319911yescharacteristic
622C22C2×D8×D5311yescharacteristic
623C22C2×SD16×D5311yescharacteristic
624C22C2×#1313324111yescharacteristic
625C22C2×#1313324611yescharacteristic
626C4—44no—
627C4—1064no—
628C4—2124no—
629C4—2124no—
630C4#1681049914yes—
631C4C2×#1313317711yescharacteristic
632C4C2×C4×C22⋊D5311yescharacteristic
633C4C2×#1313318811yescharacteristic
634C4C2×#1313319011yescharacteristic
635C2—84no—
636C2—1068no—
637C2#2027502918yesdirect product factor
638C2C2×#1681049911yescharacteristic
639C2C2×#102×D5311yescharacteristic
640C2C2×#1681062111yescharacteristic
641C2C2×#1681075511yescharacteristic
642C2C2×#1681076011yescharacteristic
643C2C2×#1681093511yescharacteristic
644C2C2×#1681094011yescharacteristic
subgroupG/H
centerC24D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC22×#102×C53C2
socleC24×C53C2×D4
prime radicalC22×#13133188C2
cosocleC22×C4×C53C24
hypercenterC22×#102D53
hypocenterC53C2×#1874
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C24 ◁ C23×C4 ◁ C22×#102

Complements

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

NH ≅ G/Ntype
2. C22×C4⋊C4×D53C2semidirect
3. C22×#13133138C2semidirect
4. C22×#13133129C2semidirect
5. #20275029C2direct (N × H)
6. C2×#16810386C2semidirect
7. C2×#16810385C2semidirect
8. C22×#102×C53C2semidirect
15. C2×#13133188C4semidirect
23. C2×C4⋊C4×D53C22semidirect
26. C2×#13133138C22semidirect
28. C2×#13133129C22semidirect
29. #16810386C22semidirect
30. #16810385C22semidirect
35. C2×#102×C53C22semidirect
62. #13133188C2×C4semidirect
67. C2×C4⋊Dic53C2×C4semidirect
68. C2×C4⋊Dic53D4semidirect
87. C22×D4×C53C2×C4semidirect
97. C2×C4⋊C4×C53D4semidirect
129. C2×D4×C53C22×C4semidirect
181. C4⋊C4×C53C2×D4semidirect
187. C4⋊Dic53C22×C4semidirect
188. C4⋊Dic53C2×D4semidirect
226. C22×D53#102semidirect
303. C2×D53C2×#102semidirect
316. C22×#102D53semidirect
322. C2×C53#1874semidirect
353. C2×#102C2×D53semidirect
358. C53C2×#1874semidirect
371. C22×D4C4×D53semidirect
416. C2×C4⋊C4C22⋊D53semidirect
456. C2×D4C2×C4×D53semidirect
515. C4⋊C4C2×C22⋊D53semidirect
637. C2#20275029direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131331882
C22×C4⋊C4×D532
C22×#131331382
C22×#131331292
#202750292
C2×#168103862
C2×#168103852
C22×#102×C532
C2×#187453