Finite Group Databasebeta

← C2×#20279028 #27553261

Subgroups — C2×#20279028 #27553261

Order 13568.

3374 conjugacy classes of subgroups.

The subgroup lattice has 640 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4×C22⋊D53C211yesmaximal, characteristic
2C22×C4⋊C4×D53C211yesmaximal, characteristic, complemented
3#20279028C2116yesmaximal, direct product factor
4C2×#16810564C212yesmaximal, complemented
5C2×#16810559C212yesmaximal, complemented
6C2×#16810522C211yesmaximal, characteristic, complemented
7C2×#16810519C211yesmaximal, characteristic, complemented
8C2×#16810499C212yesmaximal
9C2×#16810497C212yesmaximal, complemented
10C2×#16810483C211yesmaximal, characteristic, complemented
11C2×#16810480C211yesmaximal, characteristic, complemented
12C2×C53×#380C211yesmaximal, characteristic, complemented
13C23×C22⋊D53C2211yescharacteristic
14C23×C4×D53C2211yescharacteristic
15C23×C4×D53C2212yes—
16C2×C4×C22⋊D53—28no—
17C2×C4×C22⋊D53C2218yes—
18C22×C4⋊Dic53C2211yescharacteristic
19C22×C4⋊Dic53C2211yescharacteristic
20C22×C4⋊Dic53C2212yes—
21C22×#11719919C2212yes—
22C22×#11719919C2211yescharacteristic
23C22×#11719919C2212yes—
24C2×C4⋊C4×D53—28no—
25C2×C4⋊C4×D53C2218yescomplemented
26C22×C22⋊Dic53C2211yescharacteristic
27C22×C22⋊Dic53C2212yes—
28C22×#11719921C2212yes—
29C22×#11719921C2211yescharacteristic
30C22×C4×Dic53C2211yescharacteristic
31#16810564C22116yescomplemented
32#16810559C22116yescomplemented
33#16810522C2218yescomplemented
34#16810519C2218yescomplemented
35#16810499C22116yes—
36#16810497C22116yescomplemented
37#16810483C2218yescomplemented
38#16810480C2218yescomplemented
39C2×#13133124C2212yes—
40C2×#13133124C2212yes—
41C2×#13133124C2212yes—
42C2×#13133124C2212yes—
43C24×C4×C53C2211yescharacteristic
44C22×C4⋊C4×C53C2211yescharacteristic
45C22×C4⋊C4×C53C2211yescharacteristic
46C22×C22⋊C4×C53C2212yes—
47C53×#380C2218yescomplemented
48C2×#95×C53C2212yes—
49C24×D53C2311yescharacteristic
50C22×C22⋊D53—24no—
51C22×C22⋊D53—24no—
52C22×C22⋊D53Q818yes—
53C22×C22⋊D53C2314yes—
54C23×Dic53C2311yescharacteristic
55C23×Dic53C2312yes—
56C23×Dic53C2312yes—
57C23×Dic53C2311yescharacteristic
58C23×Dic53C2311yescharacteristic
59C22×C4×D53—24no—
60C22×C4×D53—24no—
61C22×C4×D53—28no—
62C22×C4×D53—44no—
63C22×C4×D53—44no—
64C22×C4×D53D418yes—
65C22×C4×D53C2314yes—
66C22×C4×D53C2318yes—
67C4×C22⋊D53—232no—
68C2×C4⋊Dic53—24no—
69C2×C4⋊Dic53—24no—
70C2×C4⋊Dic53—44no—
71C2×C4⋊Dic53C2314yes—
72C2×C4⋊Dic53C2314yes—
73C2×C4⋊Dic53C2318yes—
74C2×#11719919—28no—
75C2×#11719919—24no—
76C2×#11719919—44no—
77C2×#11719919C2318yes—
78C2×#11719919C2314yes—
79C2×#11719919C2318yes—
80C4⋊C4×D53—232no—
81C2×C22⋊Dic53—24no—
82C2×C22⋊Dic53—44no—
83C2×C22⋊Dic53C2314yes—
84C2×C22⋊Dic53C2318yes—
85C2×#11719921—24no—
86C2×#11719921—28no—
87C2×#11719921C2314yes—
88C2×#11719921C2318yes—
89C2×C4×Dic53—24no—
90C2×C4×Dic53C2314yes—
91#13133124C2318yes—
92#13133124C2318yes—
93#13133124C2318yes—
94#13133124C2318yes—
95C25×C53C2311yescharacteristic
96C23×C4×C53—24no—
97C23×C4×C53—24no—
98C23×C4×C53—24no—
99C23×C4×C53C2312yes—
100C23×C4×C53C2312yes—
101C23×C4×C53C2311yescharacteristic
102C23×C4×C53C2314yes—
103C23×C4×C53C2311yescharacteristic
104C2×C4⋊C4×C53—24no—
105C2×C4⋊C4×C53—24no—
106C2×C4⋊C4×C53C2314yes—
107C2×C4⋊C4×C53C2314yes—
108C2×C22⋊C4×C53—44no—
109C2×C22⋊C4×C53C2318yes—
110#95×C53C2318yes—
111C24×C53—22no—
112C24×C53—22no—
113C24×C53—22no—
114C24×C53—22no—
115C24×C53—22no—
116C24×C53—22no—
117C24×C53C2×Q814yes—
118C24×C53C2412yes—
119C24×C53C2411yescharacteristic
120C23×D53—22no—
121C23×D53—22no—
122C23×D53—22no—
123C23×D53—22no—
124C23×D53C2×D414yes—
125C23×D53C2×Q814yes—
126C23×D53#4114yes—
127C23×D53C2412yes—
128C22×C4×C53—24no—
129C22×C4×C53—22no—
130C22×C4×C53—24no—
131C22×C4×C53—22no—
132C22×C4×C53—22no—
133C22×C4×C53—24no—
134C22×C4×C53—24no—
135C22×C4×C53—24no—
136C22×C4×C53—24no—
137C22×C4×C53—28no—
138C22×C4×C53—28no—
139C22×C4×C53—28no—
140C22×C4×C53—416no—
141C22×C4×C53C2×D414yes—
142C22×C4×C53#4114yes—
143C22×C4×C53#4114yes—
144C22×C4×C53C2412yes—
145C22×C4×C53C2414yes—
146C22×C4×C53C2414yes—
147C22×C4×C53C2412yes—
148C2×C22⋊D53—232no—
149C2×C22⋊D53—28no—
150C2×C22⋊D53—28no—
151C2×C22⋊D53C2×Q8116yes—
152C22×Dic53—22no—
153C22×Dic53—22no—
154C22×Dic53—22no—
155C22×Dic53—22no—
156C22×Dic53—22no—
157C22×Dic53—24no—
158C22×Dic53—24no—
159C22×Dic53—24no—
160C22×Dic53—24no—
161C22×Dic53—24no—
162C22×Dic53—22no—
163C22×Dic53—22no—
164C22×Dic53—24no—
165C22×Dic53C2×D414yes—
166C22×Dic53C2×Q814yes—
167C22×Dic53C2414yes—
168C22×Dic53C2414yes—
169C22×Dic53C2412yes—
170C22×Dic53C2412yes—
171C22×Dic53C2412yes—
172C2×C4×D53—232no—
173C2×C4×D53—28no—
174C2×C4×D53—28no—
175C2×C4×D53—216no—
176C2×C4×D53—48no—
177C2×C4×D53—48no—
178C2×C4×D53—432no—
179C2×C4×D53C2×D4116yes—
180C4⋊C4×C53—28no—
181C4⋊C4×C53—28no—
182C22⋊C4×C53—48no—
183C22⋊Dic53—28no—
184C22⋊Dic53—48no—
185#11719921—28no—
186#11719921—216no—
187C4⋊Dic53—28no—
188C4⋊Dic53—28no—
189C4⋊Dic53—48no—
190#11719919—216no—
191#11719919—28no—
192#11719919—48no—
193C4×Dic53—28no—
194C23×C53—28no—
195C23×C53—28no—
196C23×C53—28no—
197C23×C53—22no—
198C23×C53—22no—
199C23×C53—22no—
200C23×C53—22no—
201C23×C53—22no—
202C23×C53—22no—
203C23×C53—44no—
204C23×C53—44no—
205C23×C53—44no—
206C23×C53—44no—
207C23×C53C22×D412yes—
208C23×C53C22×Q814yes—
209C23×C53C22×Q812yes—
210C23×C53C2×#4112yes—
211C23×C53C2×#4112yes—
212C23×C53C2×#4112yes—
213C23×C53H(5,𝔽2)12yes—
214C23×C53H(5,𝔽2)12yes—
215C23×C53C2511yescharacteristic
216C22×D53—28no—
217C22×D53—28no—
218C22×D53—28no—
219C22×D53—28no—
220C22×D53—28no—
221C22×D53—28no—
222C22×D53—22no—
223C22×D53—22no—
224C22×D53—22no—
225C22×D53—22no—
226C22×D53C22⋊Q8116yes—
227C22×D53C22×D414yes—
228C22×D53C22×Q814yes—
229C22×D53C2×#4114yes—
230C2×C4×C53—24no—
231C2×C4×C53—22no—
232C2×C4×C53—22no—
233C2×C4×C53—22no—
234C2×C4×C53—24no—
235C2×C4×C53—24no—
236C2×C4×C53—24no—
237C2×C4×C53—24no—
238C2×C4×C53—24no—
239C2×C4×C53—44no—
240C2×C4×C53—48no—
241C2×C4×C53—48no—
242C2×C4×C53—416no—
243C2×C4×C53#123116yes—
244C2×C4×C53C22×D414yes—
245C2×C4×C53C2×#4114yes—
246C2×C4×C53C2×#4114yes—
247C22⋊D53—232no—
248C2×Dic53—28no—
249C2×Dic53—28no—
250C2×Dic53—22no—
251C2×Dic53—22no—
252C2×Dic53—22no—
253C2×Dic53—22no—
254C2×Dic53—24no—
255C2×Dic53—22no—
256C2×Dic53—24no—
257C2×Dic53—24no—
258C2×Dic53—24no—
259C2×Dic53—24no—
260C2×Dic53—22no—
261C2×Dic53—24no—
262C2×Dic53—22no—
263C2×Dic53—48no—
264C2×Dic53—44no—
265C2×Dic53—48no—
266C2×Dic53C22×D414yes—
267C2×Dic53C22×Q814yes—
268C4×D53—232no—
269C4×D53—432no—
270C2×#2487—531nomaximal, Sylow
271C22×C53—24no—
272C22×C53—24no—
273C22×C53—24no—
274C22×C53—42no—
275C22×C53—42no—
276C22×C53—42no—
277C22×C53—42no—
278C22×C53—416no—
279C22×C53C2×C22⋊Q814yes—
280C22×C53C2×#12314yes—
281C22×C53#53514yes—
282C22×C53#55314yes—
283C22×C53#55314yes—
284C22×C53#55414yes—
285C22×C53#55414yes—
286C22×C53C23×D411yescharacteristic
287C22×C53C23×Q811yescharacteristic
288C22×C53C22×#4111yescharacteristic
289C22×C53C22×#4111yescharacteristic
290C22×C53C22×#4111yescharacteristic
291C22×C53C2×H(5,𝔽2)11yescharacteristic
292C22×C53C2×H(5,𝔽2)11yescharacteristic
293C2×D53—24no—
294C2×D53—24no—
295C2×D53—24no—
296C2×D53—24no—
297C2×D53—24no—
298C2×D53—24no—
299C2×D53—232no—
300C2×D53C2×C22⋊Q818yes—
301C4×C53—42no—
302C4×C53—44no—
303C4×C53—44no—
304C4×C53C2×#12318yes—
305Dic53—24no—
306Dic53—24no—
307Dic53—44no—
308Dic53—42no—
309Dic53—44no—
310C22×C4×D4—531no—
311C23×C4⋊C4—531no—
312#2487—5316no—
313C2×#400—532no—
314C2×#399—532no—
315C2×#389—531no—
316C2×#389—531no—
317C2×#386—532no—
318C2×#385—532no—
319C2×#380—531no—
320C2×#380—531no—
321C2×#380D5311yescharacteristic, complemented
322C2×C53—44no—
323C2×C53#248718yescomplemented
324C2×C53C22×C22⋊Q811yescharacteristic
325C2×C53C22×#12311yescharacteristic
326C2×C53C2×#53511yescharacteristic
327C2×C53C2×#55311yescharacteristic
328C2×C53C2×#55311yescharacteristic
329C2×C53C2×#55411yescharacteristic
330C2×C53C2×#55411yescharacteristic
331D53—28no—
332C23×D4—531no—
333C24×C4—531no—
334C24×C4—532no—
335C24×C4C2×D5311yescharacteristic
336C2×C4×D4—538no—
337C2×C4×D4—1068no—
338C22×C4⋊C4—532no—
339C22×C4⋊C4—531no—
340C22×C4⋊C4—531no—
341C22×C4⋊C4—531no—
342C22×C4⋊C4—532no—
343C22×C4⋊C4—532no—
344C22×C4⋊C4—538no—
345C22×C4⋊C4—1068no—
346C22×C4⋊C4C2×D5311yescharacteristic
347C22×C4⋊C4C2×D5311yescharacteristic
348C22×C22⋊C4—531no—
349C22×C22⋊C4—531no—
350C22×C22⋊C4—532no—
351C22×C22⋊C4—532no—
352C22×C22⋊C4C2×D5312yes—
353C22×C42—531no—
354#400—5316no—
355#399—5316no—
356#389—538no—
357#389—538no—
358#386—5316no—
359#385—5316no—
360#380—538no—
361#380—538no—
362#380C2×D5318yescomplemented
363C2×#95—532no—
364C2×#95—532no—
365C2×#95—532no—
366C2×#95—532no—
367C2×#95C2×D5312yes—
368C53C2×#248711yescharacteristic, complemented, Sylow
369C25—531no—
370C25C22×D5311yescharacteristic
371C22×D4—538no—
372C22×D4—534no—
373C22×D4—1064no—
374C22×D4—1064no—
375C23×C4—24no—
376C23×C4—24no—
377C23×C4—24no—
378C23×C4—531no—
379C23×C4—531no—
380C23×C4—538no—
381C23×C4—534no—
382C23×C4—531no—
383C23×C4—532no—
384C23×C4—532no—
385C23×C4—538no—
386C23×C4—1068no—
387C23×C4—1064no—
388C23×C4—1064no—
389C23×C4—2124no—
390C23×C4—2124no—
391C23×C4C22×D5314yes—
392C23×C4C22×D5312yes—
393C23×C4C22×D5311yescharacteristic
394C23×C4C22×D5312yes—
395C23×C4C22×D5311yescharacteristic
396C4×D4—10632no—
397C2×C4⋊C4—24no—
398C2×C4⋊C4—24no—
399C2×C4⋊C4—538no—
400C2×C4⋊C4—534no—
401C2×C4⋊C4—534no—
402C2×C4⋊C4—534no—
403C2×C4⋊C4—538no—
404C2×C4⋊C4—538no—
405C2×C4⋊C4—10632no—
406C2×C4⋊C4—1068no—
407C2×C4⋊C4—1064no—
408C2×C4⋊C4—1064no—
409C2×C4⋊C4—1064no—
410C2×C4⋊C4—2124no—
411C2×C4⋊C4—2124no—
412C2×C4⋊C4C22×D5314yes—
413C2×C4⋊C4C22×D5314yes—
414C2×C22⋊C4—44no—
415C2×C22⋊C4—534no—
416C2×C22⋊C4—534no—
417C2×C22⋊C4—538no—
418C2×C22⋊C4—538no—
419C2×C22⋊C4—1064no—
420C2×C22⋊C4—1068no—
421C2×C22⋊C4—1064no—
422C2×C22⋊C4—2124no—
423C2×C22⋊C4C22×D5318yes—
424C2×C42—534no—
425C2×C42—1064no—
426#95—538no—
427#95—538no—
428#95—538no—
429#95—538no—
430#95C22×D5318yes—
431C24—22no—
432C24—22no—
433C24—22no—
434C24—22no—
435C24—22no—
436C24—22no—
437C24—534no—
438C24—534no—
439C24—534no—
440C24—532no—
441C24—1062no—
442C24—1062no—
443C24—1062no—
444C24—1062no—
445C24Q8×D5314yes—
446C24C23×D5312yes—
447C24C23×D5311yescharacteristic
448C2×D4—5316no—
449C2×D4—1068no—
450C2×D4—10632no—
451C2×D4—1068no—
452C22×C4—22no—
453C22×C4—24no—
454C22×C4—24no—
455C22×C4—24no—
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—24no—
459C22×C4—24no—
460C22×C4—24no—
461C22×C4—28no—
462C22×C4—28no—
463C22×C4—28no—
464C22×C4—416no—
465C22×C4—534no—
466C22×C4—534no—
467C22×C4—532no—
468C22×C4—532no—
469C22×C4—5316no—
470C22×C4—534no—
471C22×C4—534no—
472C22×C4—532no—
473C22×C4—10616no—
474C22×C4—1062no—
475C22×C4—1062no—
476C22×C4—1062no—
477C22×C4—1062no—
478C22×C4—10632no—
479C22×C4—1068no—
480C22×C4—1068no—
481C22×C4—1062no—
482C22×C4—1064no—
483C22×C4—1064no—
484C22×C4—1064no—
485C22×C4—1062no—
486C22×C4—1064no—
487C22×C4—1064no—
488C22×C4—1064no—
489C22×C4—1062no—
490C22×C4—2128no—
491C22×C4—2128no—
492C22×C4—21232no—
493C22×C4#1171994514yes—
494C22×C4D4×D5314yes—
495C22×C4#1171994714yes—
496C22×C4C23×D5312yes—
497C22×C4C23×D5314yes—
498C22×C4C23×D5314yes—
499C22×C4C23×D5312yes—
500C4⋊C4—28no—
501C4⋊C4—28no—
502C4⋊C4—10616no—
503C4⋊C4—1068no—
504C4⋊C4—1068no—
505C4⋊C4—1068no—
506C4⋊C4—2128no—
507C4⋊C4—2128no—
508C22⋊C4—48no—
509C22⋊C4—1068no—
510C22⋊C4—10616no—
511C22⋊C4—1068no—
512C22⋊C4—2128no—
513C42—1068no—
514C23—28no—
515C23—28no—
516C23—28no—
517C23—22no—
518C23—22no—
519C23—22no—
520C23—22no—
521C23—22no—
522C23—22no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23—44no—
527C23—5316no—
528C23—534no—
529C23—534no—
530C23—534no—
531C23—1068no—
532C23—1068no—
533C23—1068no—
534C23—1068no—
535C23—1068no—
536C23—1068no—
537C23—1062no—
538C23—1062no—
539C23—1062no—
540C23—1062no—
541C23C2×D4×D5312yes—
542C23C2×Q8×D5314yes—
543C23C2×Q8×D5312yes—
544C23#41×D5312yes—
545C23C2×#1171994712yes—
546C23C2×#1171994512yes—
547C23#1313326212yes—
548C23#1313326312yes—
549C23C24×D5311yescharacteristic
550D4—10632no—
551C2×C4—22no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—24no—
555C2×C4—22no—
556C2×C4—22no—
557C2×C4—24no—
558C2×C4—24no—
559C2×C4—24no—
560C2×C4—48no—
561C2×C4—44no—
562C2×C4—48no—
563C2×C4—416no—
564C2×C4—534no—
565C2×C4—534no—
566C2×C4—1068no—
567C2×C4—1068no—
568C2×C4—1062no—
569C2×C4—1062no—
570C2×C4—1062no—
571C2×C4—1062no—
572C2×C4—10632no—
573C2×C4—1062no—
574C2×C4—1064no—
575C2×C4—1064no—
576C2×C4—1064no—
577C2×C4—1062no—
578C2×C4—1064no—
579C2×C4—1064no—
580C2×C4—1064no—
581C2×C4—1062no—
582C2×C4—21232no—
583C2×C4—2128no—
584C2×C4—2124no—
585C2×C4—2128no—
586C2×C4#13133199116yes—
587C2×C4C2×D4×D5314yes—
588C2×C4C2×#1171994714yes—
589C2×C4C2×#1171994514yes—
590C22—24no—
591C22—24no—
592C22—24no—
593C22—42no—
594C22—42no—
595C22—42no—
596C22—42no—
597C22—416no—
598C22—538no—
599C22—1064no—
600C22—1064no—
601C22—1064no—
602C22—1064no—
603C22—10632no—
604C22—1064no—
605C22—1064no—
606C22C22⋊Q8×D5314yes—
607C22C2×#1313319914yes—
608C22#1681118914yes—
609C22#1681131714yes—
610C22#1681132014yes—
611C22#1681132814yes—
612C22#1681133114yes—
613C22C22×D4×D5311yescharacteristic
614C22C22×Q8×D5311yescharacteristic
615C22C2×#41×D5311yescharacteristic
616C22C22×#1171994711yescharacteristic
617C22C22×#1171994511yescharacteristic
618C22C2×#1313326211yescharacteristic
619C22C2×#1313326311yescharacteristic
620C4—44no—
621C4—42no—
622C4—44no—
623C4—1064no—
624C4—1064no—
625C4—2124no—
626C4—2122no—
627C4—2124no—
628C4C2×#1313319918yes—
629C2—44no—
630C2—1068no—
631C2#2027902818yesdirect product factor
632C2C2×C22⋊Q8×D5311yescharacteristic
633C2C22×#1313319911yescharacteristic
634C2C2×#1681118911yescharacteristic
635C2C2×#1681131711yescharacteristic
636C2C2×#1681132011yescharacteristic
637C2C2×#1681132811yescharacteristic
638C2C2×#1681133111yescharacteristic
subgroupG/H
centerC24C23×D53
commutator, cosocleC23×C53C25
FrattiniC23C24×D53
FittingC2×C53×#380C2
socleC24×C53C24
prime radicalC23×C22⋊D53C22
hypercenterC2×#380D53
hypocenterC53C2×#2487
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
2. C22×C4⋊C4×D53C2semidirect
3. #20279028C2direct (N × H)
4. C2×#16810564C2semidirect
5. C2×#16810559C2semidirect
6. C2×#16810522C2semidirect
7. C2×#16810519C2semidirect
9. C2×#16810497C2semidirect
10. C2×#16810483C2semidirect
11. C2×#16810480C2semidirect
12. C2×C53×#380C2semidirect
25. C2×C4⋊C4×D53C22semidirect
31. #16810564C22semidirect
32. #16810559C22semidirect
33. #16810522C22semidirect
34. #16810519C22semidirect
36. #16810497C22semidirect
37. #16810483C22semidirect
38. #16810480C22semidirect
47. C53×#380C22semidirect
321. C2×#380D53semidirect
323. C2×C53#2487semidirect
362. #380C2×D53semidirect
368. C53C2×#2487semidirect
631. C2#20279028direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×C22⋊D532
C22×C4⋊C4×D532
#202790282
C2×#168105642
C2×#168105592
C2×#168105222
C2×#168105192
C2×#168104992
C2×#168104972
C2×#168104832
C2×#168104802
C2×C53×#3802
C2×#248753