Finite Group Databasebeta

← C2×#20279639 #27555865

Subgroups — C2×#20279639 #27555865

Order 13568.

3614 conjugacy classes of subgroups.

The subgroup lattice has 660 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133197C211yesmaximal, characteristic, complemented
2C22×#13133196C211yesmaximal, characteristic, complemented
3C22×#13133182C211yesmaximal, characteristic, complemented
4C22×C4×C22⋊D53C211yesmaximal, characteristic
5#20279639C2116yesmaximal, direct product factor
6C2×#16810547C212yesmaximal, complemented
7C2×#16810546C212yesmaximal, complemented
8C2×#16810523C212yesmaximal
9C2×#16810520C212yesmaximal, complemented
10C2×#16810492C211yesmaximal, characteristic, complemented
11C2×#16810480C211yesmaximal, characteristic, complemented
12C2×C53×#380C211yesmaximal, characteristic, complemented
13C23×Dic106C2211yescharacteristic
14C23×C22⋊D53C2211yescharacteristic
15C23×C4×D53C2211yescharacteristic
16C2×#13133197—28no—
17C2×#13133197C2218yescomplemented
18C2×#13133196—28no—
19C2×#13133196C2218yescomplemented
20C2×#13133182—28no—
21C2×#13133182C2218yescomplemented
22C2×C4×C22⋊D53—28no—
23C2×C4×C22⋊D53C2218yes—
24C22×C4⋊Dic53C2211yescharacteristic
25C22×C4⋊Dic53C2212yes—
26C22×#11719919C2212yes—
27C22×#11719919C2211yescharacteristic
28C22×#11719919C2212yes—
29C22×#11719919C2212yes—
30C22×#11719919C2212yes—
31C22×C22⋊Dic53C2211yescharacteristic
32C22×C22⋊Dic53C2212yes—
33C22×#11719921C2211yescharacteristic
34C22×#11719921C2212yes—
35C22×#11719921C2212yes—
36C22×C4×Dic53C2212yes—
37C22×C4×Dic53C2211yescharacteristic
38#16810547C22116yescomplemented
39#16810546C22116yescomplemented
40#16810523C22116yes—
41#16810520C22116yescomplemented
42#16810492C2218yescomplemented
43#16810480C2218yescomplemented
44C2×#13133124C2212yes—
45C24×C4×C53C2211yescharacteristic
46C22×C4⋊C4×C53C2211yescharacteristic
47C22×C4⋊C4×C53C2211yescharacteristic
48C22×C22⋊C4×C53C2212yes—
49C53×#380C2218yescomplemented
50C2×#95×C53C2212yes—
51C24×D53C2311yescharacteristic
52C22×Dic106—24no—
53C22×Dic106—24no—
54C22×Dic106D418yescomplemented
55C22×Dic106C2314yes—
56C22×C22⋊D53—24no—
57C22×C22⋊D53—24no—
58C22×C22⋊D53Q818yescomplemented
59C22×C22⋊D53C2314yes—
60C23×Dic53C2312yes—
61C23×Dic53C2311yescharacteristic
62C23×Dic53C2311yescharacteristic
63C23×Dic53C2312yes—
64C23×Dic53C2311yescharacteristic
65C22×C4×D53—24no—
66C22×C4×D53—24no—
67C22×C4×D53—24no—
68C22×C4×D53C2314yes—
69#13133197—232no—
70#13133196—232no—
71#13133182—232no—
72C4×C22⋊D53—232no—
73C2×C4⋊Dic53—24no—
74C2×C4⋊Dic53—28no—
75C2×C4⋊Dic53C2318yes—
76C2×C4⋊Dic53C2314yes—
77C2×#11719919—24no—
78C2×#11719919—28no—
79C2×#11719919—28no—
80C2×#11719919—28no—
81C2×#11719919—44no—
82C2×#11719919C2314yes—
83C2×#11719919C2318yes—
84C2×#11719919C2318yes—
85C2×#11719919C2318yes—
86C2×#11719919C2318yes—
87C2×C22⋊Dic53—24no—
88C2×C22⋊Dic53—28no—
89C2×C22⋊Dic53C2318yes—
90C2×C22⋊Dic53C2314yes—
91C2×#11719921—24no—
92C2×#11719921—28no—
93C2×#11719921—44no—
94C2×#11719921C2318yes—
95C2×#11719921C2314yes—
96C2×#11719921C2318yes—
97C2×C4×Dic53—28no—
98C2×C4×Dic53D418yes—
99C2×C4×Dic53C2318yes—
100C2×C4×Dic53C2314yes—
101#13133124C2318yes—
102C25×C53C2311yescharacteristic
103C23×C4×C53—24no—
104C23×C4×C53—24no—
105C23×C4×C53—24no—
106C23×C4×C53C2312yes—
107C23×C4×C53C2312yes—
108C23×C4×C53C2311yescharacteristic
109C23×C4×C53C2314yes—
110C23×C4×C53C2311yescharacteristic
111C2×C4⋊C4×C53—24no—
112C2×C4⋊C4×C53—24no—
113C2×C4⋊C4×C53C2314yes—
114C2×C4⋊C4×C53C2314yes—
115C2×C22⋊C4×C53—44no—
116C2×C22⋊C4×C53C2318yes—
117#95×C53C2318yes—
118C24×C53—22no—
119C24×C53—22no—
120C24×C53—22no—
121C24×C53—22no—
122C24×C53—22no—
123C24×C53—22no—
124C24×C53C2×Q814yes—
125C24×C53C2412yes—
126C24×C53C2411yescharacteristic
127C23×D53—22no—
128C23×D53—22no—
129C23×D53—22no—
130C23×D53—22no—
131C23×D53—22no—
132C23×D53—22no—
133C23×D53C2×Q814yes—
134C23×D53C2412yes—
135C22×C4×C53—24no—
136C22×C4×C53—24no—
137C22×C4×C53—22no—
138C22×C4×C53—22no—
139C22×C4×C53—22no—
140C22×C4×C53—24no—
141C22×C4×C53—24no—
142C22×C4×C53—24no—
143C22×C4×C53—24no—
144C22×C4×C53—28no—
145C22×C4×C53—28no—
146C22×C4×C53—28no—
147C22×C4×C53—416no—
148C22×C4×C53C2×D414yes—
149C22×C4×C53C2×D414yes—
150C22×C4×C53#4114yes—
151C22×C4×C53C2412yes—
152C22×C4×C53C2414yes—
153C22×C4×C53C2414yes—
154C22×C4×C53C2412yes—
155C2×C22⋊D53—232no—
156C2×C22⋊D53—28no—
157C2×C22⋊D53—28no—
158C2×C22⋊D53C2×Q8116yescomplemented
159C22×Dic53—24no—
160C22×Dic53—24no—
161C22×Dic53—22no—
162C22×Dic53—22no—
163C22×Dic53—24no—
164C22×Dic53—22no—
165C22×Dic53—22no—
166C22×Dic53—24no—
167C22×Dic53—22no—
168C22×Dic53—24no—
169C22×Dic53C2×D414yes—
170C22×Dic53C2×D418yes—
171C22×Dic53C2×D414yes—
172C22×Dic53C2×Q814yes—
173C22×Dic53#4114yes—
174C22×Dic53C2412yes—
175C22×Dic53C2414yes—
176C22×Dic53C2412yes—
177C22×Dic53C2414yes—
178C22×Dic53C2412yes—
179C2×C4×D53—28no—
180C2×C4×D53—28no—
181C2×C4×D53—28no—
182C2×C4×D53—416no—
183C2×Dic106—232no—
184C2×Dic106—28no—
185C2×Dic106—28no—
186C2×Dic106C2×D4116yescomplemented
187C4⋊C4×C53—28no—
188C4⋊C4×C53—28no—
189C22⋊C4×C53—48no—
190C22⋊Dic53—28no—
191C22⋊Dic53—216no—
192#11719921—28no—
193#11719921—216no—
194#11719921—48no—
195C4⋊Dic53—28no—
196C4⋊Dic53—216no—
197#11719919—28no—
198#11719919—216no—
199#11719919—216no—
200#11719919—216no—
201#11719919—48no—
202C4×Dic53—216no—
203C4×Dic53C2×D4116yes—
204C23×C53—28no—
205C23×C53—28no—
206C23×C53—28no—
207C23×C53—22no—
208C23×C53—22no—
209C23×C53—22no—
210C23×C53—22no—
211C23×C53—22no—
212C23×C53—22no—
213C23×C53—44no—
214C23×C53—44no—
215C23×C53—44no—
216C23×C53—44no—
217C23×C53C22×D412yes—
218C23×C53C22×D412yes—
219C23×C53C22×Q814yes—
220C23×C53C22×Q812yes—
221C23×C53C2×#4112yes—
222C23×C53C2×#4112yes—
223C23×C53H(5,𝔽2)12yes—
224C23×C532−1+412yes—
225C23×C53C2511yescharacteristic
226C22×D53—28no—
227C22×D53—28no—
228C22×D53—28no—
229C22×D53—22no—
230C22×D53—22no—
231C22×D53—22no—
232C22×D53—22no—
233C22×D53—22no—
234C22×D53—22no—
235C22×D53—44no—
236C22×D53—44no—
237C22×D53—44no—
238C22×D53—44no—
239C22×D53C22×Q814yes—
240C2×C4×C53—22no—
241C2×C4×C53—22no—
242C2×C4×C53—24no—
243C2×C4×C53—24no—
244C2×C4×C53—22no—
245C2×C4×C53—24no—
246C2×C4×C53—24no—
247C2×C4×C53—24no—
248C2×C4×C53—24no—
249C2×C4×C53—44no—
250C2×C4×C53—48no—
251C2×C4×C53—48no—
252C2×C4×C53—416no—
253C2×C4×C53#121116yes—
254C2×C4×C53C22×D414yes—
255C2×C4×C53C22×D414yes—
256C2×C4×C53C2×#4114yes—
257C22⋊D53—232no—
258C2×Dic53—216no—
259C2×Dic53—28no—
260C2×Dic53—24no—
261C2×Dic53—24no—
262C2×Dic53—22no—
263C2×Dic53—22no—
264C2×Dic53—22no—
265C2×Dic53—22no—
266C2×Dic53—24no—
267C2×Dic53—22no—
268C2×Dic53—24no—
269C2×Dic53—24no—
270C2×Dic53—44no—
271C2×Dic53—48no—
272C2×Dic53C22⋊Q8116yescomplemented
273C2×Dic53C22×D418yes—
274C2×Dic53C22×D414yes—
275C2×Dic53C22×D414yes—
276C2×Dic53C22×Q814yes—
277C2×Dic53C2×#4114yes—
278C4×D53—416no—
279Dic106—232no—
280C2×#2538—531nomaximal, Sylow
281C22×C53—24no—
282C22×C53—24no—
283C22×C53—24no—
284C22×C53—42no—
285C22×C53—42no—
286C22×C53—42no—
287C22×C53—42no—
288C22×C53—416no—
289C22×C53C2×#12114yes—
290C22×C53C2×C22⋊Q814yes—
291C22×C53#53714yes—
292C22×C53D4×Q814yes—
293C22×C53#55014yes—
294C22×C53#55414yes—
295C22×C53#55614yes—
296C22×C53C23×D411yescharacteristic
297C22×C53C23×D411yescharacteristic
298C22×C53C23×Q811yescharacteristic
299C22×C53C22×#4111yescharacteristic
300C22×C53C22×#4111yescharacteristic
301C22×C53C2×H(5,𝔽2)11yescharacteristic
302C22×C53C2×2−1+411yescharacteristic
303C2×D53—24no—
304C2×D53—24no—
305C2×D53—24no—
306C2×D53—42no—
307C2×D53—42no—
308C2×D53—42no—
309C2×D53—42no—
310C2×D53—416no—
311C4×C53—42no—
312C4×C53—44no—
313C4×C53—44no—
314C4×C53C2×#12118yes—
315Dic53—28no—
316Dic53—24no—
317Dic53—42no—
318Dic53—44no—
319Dic53C2×C22⋊Q818yescomplemented
320C22×C22⋊Q8—531no—
321C22×C22⋊Q8—531no—
322C22×C4×Q8—531no—
323C22×C4×D4—531no—
324#2538—5316no—
325C2×#396—532no—
326C2×#396—532no—
327C2×#390—532no—
328C2×#389—532no—
329C2×#384—531no—
330C2×#380—531no—
331C2×#380D5311yescharacteristic, complemented
332C2×C53—44no—
333C2×C53#253818yescomplemented
334C2×C53C22×#12111yescharacteristic
335C2×C53C22×C22⋊Q811yescharacteristic
336C2×C53C2×#53711yescharacteristic
337C2×C53C2×D4×Q811yescharacteristic
338C2×C53C2×#55011yescharacteristic
339C2×C53C2×#55411yescharacteristic
340C2×C53C2×#55611yescharacteristic
341D53—44no—
342C23×Q8—531no—
343C23×D4—531no—
344C24×C4—531no—
345C24×C4C2×D5311yescharacteristic
346C2×C22⋊Q8—538no—
347C2×C22⋊Q8—538no—
348C2×C22⋊Q8—1068no—
349C2×C22⋊Q8—1068no—
350C2×C4×Q8—538no—
351C2×C4×Q8—1068no—
352C2×C4×D4—538no—
353C2×C4×D4—1068no—
354C22×C4⋊C4—532no—
355C22×C4⋊C4—532no—
356C22×C4⋊C4—531no—
357C22×C4⋊C4—532no—
358C22×C4⋊C4—532no—
359C22×C4⋊C4—531no—
360C22×C4⋊C4—532no—
361C22×C4⋊C4C2×D5311yescharacteristic
362C22×C4⋊C4C2×D5311yescharacteristic
363C22×C22⋊C4—531no—
364C22×C22⋊C4—532no—
365C22×C22⋊C4—531no—
366C22×C22⋊C4—532no—
367C22×C22⋊C4—532no—
368C22×C22⋊C4C2×D5312yes—
369C22×C42—531no—
370C22×C42—532no—
371#396—5316no—
372#396—5316no—
373#390—5316no—
374#389—5316no—
375#384—538no—
376#380—538no—
377#380C2×D5318yescomplemented
378C2×#95—532no—
379C2×#95C2×D5312yes—
380C53C2×#253811yescharacteristic, complemented, Sylow
381C25—531no—
382C25C22×D5311yescharacteristic
383C22×Q8—538no—
384C22×Q8—534no—
385C22×Q8—1064no—
386C22×Q8—1064no—
387C22×D4—538no—
388C22×D4—534no—
389C22×D4—1064no—
390C22×D4—1064no—
391C23×C4—24no—
392C23×C4—24no—
393C23×C4—24no—
394C23×C4—531no—
395C23×C4—531no—
396C23×C4—532no—
397C23×C4—532no—
398C23×C4—531no—
399C23×C4—534no—
400C23×C4—1064no—
401C23×C4—1064no—
402C23×C4—1064no—
403C23×C4C22×D5314yes—
404C23×C4C22×D5312yes—
405C23×C4C22×D5312yes—
406C23×C4C22×D5311yescharacteristic
407C23×C4C22×D5311yescharacteristic
408C22⋊Q8—10632no—
409C22⋊Q8—10632no—
410C4×Q8—10632no—
411C4×D4—10632no—
412C2×C4⋊C4—24no—
413C2×C4⋊C4—24no—
414C2×C4⋊C4—534no—
415C2×C4⋊C4—538no—
416C2×C4⋊C4—538no—
417C2×C4⋊C4—534no—
418C2×C4⋊C4—538no—
419C2×C4⋊C4—538no—
420C2×C4⋊C4—538no—
421C2×C4⋊C4—1064no—
422C2×C4⋊C4—1068no—
423C2×C4⋊C4—1068no—
424C2×C4⋊C4—1068no—
425C2×C4⋊C4—1068no—
426C2×C4⋊C4—1064no—
427C2×C4⋊C4—2124no—
428C2×C4⋊C4C22×D5314yes—
429C2×C4⋊C4C22×D5314yes—
430C2×C22⋊C4—44no—
431C2×C22⋊C4—534no—
432C2×C22⋊C4—538no—
433C2×C22⋊C4—534no—
434C2×C22⋊C4—538no—
435C2×C22⋊C4—538no—
436C2×C22⋊C4—1064no—
437C2×C22⋊C4—1068no—
438C2×C22⋊C4—1064no—
439C2×C22⋊C4—1068no—
440C2×C22⋊C4—2124no—
441C2×C22⋊C4C22×D5318yes—
442C2×C42—538no—
443C2×C42—534no—
444C2×C42—538no—
445C2×C42—1068no—
446#95—538no—
447#95C22×D5318yes—
448C24—22no—
449C24—22no—
450C24—22no—
451C24—22no—
452C24—22no—
453C24—22no—
454C24—534no—
455C24—532no—
456C24—1062no—
457C24—1062no—
458C24—1062no—
459C24—1062no—
460C24—1062no—
461C24—1062no—
462C24Q8×D5314yes—
463C24C23×D5312yes—
464C24C23×D5311yescharacteristic
465C2×Q8—5316no—
466C2×Q8—10632no—
467C2×Q8—1068no—
468C2×Q8—1068no—
469C2×D4—5316no—
470C2×D4—1068no—
471C2×D4—10632no—
472C2×D4—1068no—
473C22×C4—24no—
474C22×C4—24no—
475C22×C4—22no—
476C22×C4—22no—
477C22×C4—24no—
478C22×C4—22no—
479C22×C4—24no—
480C22×C4—24no—
481C22×C4—24no—
482C22×C4—28no—
483C22×C4—28no—
484C22×C4—28no—
485C22×C4—416no—
486C22×C4—534no—
487C22×C4—534no—
488C22×C4—534no—
489C22×C4—532no—
490C22×C4—534no—
491C22×C4—538no—
492C22×C4—532no—
493C22×C4—534no—
494C22×C4—532no—
495C22×C4—534no—
496C22×C4—1064no—
497C22×C4—1064no—
498C22×C4—1062no—
499C22×C4—1062no—
500C22×C4—1064no—
501C22×C4—1062no—
502C22×C4—1062no—
503C22×C4—1064no—
504C22×C4—1062no—
505C22×C4—1064no—
506C22×C4—1068no—
507C22×C4—1068no—
508C22×C4—1068no—
509C22×C4—21216no—
510C22×C4#1171994514yes—
511C22×C4D4×D5314yes—
512C22×C4D4×D5314yes—
513C22×C4C23×D5314yes—
514C22×C4C23×D5312yes—
515C22×C4C23×D5314yes—
516C22×C4C23×D5312yes—
517C4⋊C4—28no—
518C4⋊C4—28no—
519C4⋊C4—1068no—
520C4⋊C4—10616no—
521C4⋊C4—10616no—
522C4⋊C4—10616no—
523C4⋊C4—10616no—
524C4⋊C4—1068no—
525C4⋊C4—2128no—
526C22⋊C4—48no—
527C22⋊C4—1068no—
528C22⋊C4—10616no—
529C22⋊C4—1068no—
530C22⋊C4—10616no—
531C22⋊C4—2128no—
532C42—5316no—
533C42—10616no—
534C23—28no—
535C23—28no—
536C23—28no—
537C23—22no—
538C23—22no—
539C23—22no—
540C23—22no—
541C23—22no—
542C23—22no—
543C23—44no—
544C23—44no—
545C23—44no—
546C23—44no—
547C23—534no—
548C23—1068no—
549C23—1068no—
550C23—1068no—
551C23—1062no—
552C23—1062no—
553C23—1062no—
554C23—1062no—
555C23—1062no—
556C23—1062no—
557C23—2124no—
558C23—2124no—
559C23—2124no—
560C23—2124no—
561C23C2×D4×D5312yes—
562C23C2×D4×D5312yes—
563C23C2×Q8×D5314yes—
564C23C2×Q8×D5312yes—
565C23#41×D5312yes—
566C23C2×#1171994512yes—
567C23#1313326212yes—
568C23#1313326412yes—
569C23C24×D5311yescharacteristic
570Q8—10632no—
571D4—10632no—
572C2×C4—24no—
573C2×C4—24no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—24no—
577C2×C4—22no—
578C2×C4—24no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—48no—
582C2×C4—44no—
583C2×C4—48no—
584C2×C4—416no—
585C2×C4—5316no—
586C2×C4—534no—
587C2×C4—534no—
588C2×C4—534no—
589C2×C4—534no—
590C2×C4—538no—
591C2×C4—10616no—
592C2×C4—1064no—
593C2×C4—1064no—
594C2×C4—1062no—
595C2×C4—1062no—
596C2×C4—1068no—
597C2×C4—1064no—
598C2×C4—1062no—
599C2×C4—1062no—
600C2×C4—1064no—
601C2×C4—1062no—
602C2×C4—1064no—
603C2×C4—2128no—
604C2×C4—2124no—
605C2×C4—21216no—
606C2×C4#13133190116yes—
607C2×C4C2×D4×D5314yes—
608C2×C4C2×D4×D5314yes—
609C2×C4C2×#1171994514yes—
610C22—24no—
611C22—24no—
612C22—24no—
613C22—42no—
614C22—42no—
615C22—416no—
616C22—42no—
617C22—42no—
618C22—1064no—
619C22—1064no—
620C22—1064no—
621C22—2122no—
622C22—2122no—
623C22—21216no—
624C22—2122no—
625C22—2122no—
626C22C2×#1313319014yes—
627C22C22⋊Q8×D5314yes—
628C22#1681120214yes—
629C22#1681128614yes—
630C22#1681129114yes—
631C22#1681132814yes—
632C22#1681134114yes—
633C22C22×D4×D5311yescharacteristic
634C22C22×D4×D5311yescharacteristic
635C22C22×Q8×D5311yescharacteristic
636C22C2×#41×D5311yescharacteristic
637C22C22×#1171994511yescharacteristic
638C22C2×#1313326211yescharacteristic
639C22C2×#1313326411yescharacteristic
640C4—44no—
641C4—42no—
642C4—44no—
643C4—538no—
644C4—1064no—
645C4—1068no—
646C4—2122no—
647C4—2124no—
648C4C2×#1313319018yes—
649C2—44no—
650C2—2124no—
651C2#2027963918yesdirect product factor
652C2C22×#1313319011yescharacteristic
653C2C2×C22⋊Q8×D5311yescharacteristic
654C2C2×#1681120211yescharacteristic
655C2C2×#1681128611yescharacteristic
656C2C2×#1681129111yescharacteristic
657C2C2×#1681132811yescharacteristic
658C2C2×#1681134111yescharacteristic
subgroupG/H
centerC24C23×D53
commutator, cosocleC23×C53C25
FrattiniC23C24×D53
FittingC2×C53×#380C2
socleC24×C53C24
prime radicalC23×C22⋊D53C22
hypercenterC2×#380D53
hypocenterC53C2×#2538
LCSG ▷ C23×C53 ▷ C53
UCSC1 ◁ C24 ◁ C2×#380

Complements

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

NH ≅ G/Ntype
1. C22×#13133197C2semidirect
2. C22×#13133196C2semidirect
3. C22×#13133182C2semidirect
5. #20279639C2direct (N × H)
6. C2×#16810547C2semidirect
7. C2×#16810546C2semidirect
9. C2×#16810520C2semidirect
10. C2×#16810492C2semidirect
11. C2×#16810480C2semidirect
12. C2×C53×#380C2semidirect
17. C2×#13133197C22semidirect
19. C2×#13133196C22semidirect
21. C2×#13133182C22semidirect
38. #16810547C22semidirect
39. #16810546C22semidirect
41. #16810520C22semidirect
42. #16810492C22semidirect
43. #16810480C22semidirect
49. C53×#380C22semidirect
54. C22×Dic106D4semidirect
58. C22×C22⋊D53Q8semidirect
158. C2×C22⋊D53C2×Q8semidirect
186. C2×Dic106C2×D4semidirect
272. C2×Dic53C22⋊Q8semidirect
319. Dic53C2×C22⋊Q8semidirect
331. C2×#380D53semidirect
333. C2×C53#2538semidirect
377. #380C2×D53semidirect
380. C53C2×#2538semidirect
651. C2#20279639direct (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×#131331962
C22×#131331822
C22×C4×C22⋊D532
#202796392
C2×#168105472
C2×#168105462
C2×#168105232
C2×#168105202
C2×#168104922
C2×#168104802
C2×C53×#3802
C2×#253853