Finite Group Databasebeta

← C2×#20285538 #28254702

Subgroups — C2×#20285538 #28254702

Order 13568.

2322 conjugacy classes of subgroups.

The subgroup lattice has 650 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133197C211yesmaximal, characteristic
2C2×C22⋊Q8×D53C211yesmaximal, characteristic
3#20285538C2116yesmaximal, direct product factor
4C2×#16810955C212yesmaximal, complemented
5C2×#16810953C211yesmaximal, characteristic, complemented
6C2×#16810882C212yesmaximal, complemented
7C2×#16810880C212yesmaximal, complemented
8C2×#16810781C211yesmaximal, characteristic, complemented
9C2×#16810780C212yesmaximal, complemented
10C2×#16810778C211yesmaximal, characteristic, complemented
11C2×#16810585C211yesmaximal, characteristic
12C2×C53×#483C211yesmaximal, characteristic, complemented
13C23×Dic106C2211yescharacteristic
14C22×Q8×D53C2211yescharacteristic
15C23×C4×D53C2211yescharacteristic
16C22×Dic212C2212yescomplemented
17C22×#11719925C2212yescomplemented
18C22×#11719912C2212yes—
19C2×#13133197—21no—
20C2×#13133197—24no—
21C2×#13133197—21no—
22C2×#13133197C2212yes—
23C2×#13133197C2218yes—
24C2×#13133197C2212yes—
25C2×#13133197C2211yescharacteristic
26C2×#13133196C2211yescharacteristic
27C2×#13133195—21no—
28C2×#13133194—21no—
29C2×#13133193C2211yescharacteristic
30C22⋊Q8×D53—24no—
31C22⋊Q8×D53C2218yes—
32C22×C4⋊Dic53—21no—
33C2×C4⋊C4×D53—21no—
34C2×C4⋊C4×D53C2211yescharacteristic
35C22×#11719921—21no—
36C2×C22⋊C4×D53—21no—
37#16810955C22116yescomplemented
38#16810953C2218yescomplemented
39#16810882C22116yescomplemented
40#16810880C22116yescomplemented
41#16810781C2218yescomplemented
42#16810780C22116yescomplemented
43#16810778C2218yescomplemented
44C2×#13133148C2212yescomplemented
45C2×#13133142C2212yescomplemented
46C2×#13133140C2212yescomplemented
47C2×#13133140C2212yescomplemented
48#16810585C2218yes—
49C2×C22⋊#88322C2211yescharacteristic
50C2×#13133129C2212yes—
51C2×C22⋊Q8×C53C2211yescharacteristic
52C22×C4⋊C4×C53C2211yescharacteristic
53C53×#483C2218yescomplemented
54C2×#106×C53C2212yescomplemented
55C2×#103×C53C2212yescomplemented
56C2×C22⋊C8×C53C2211yescharacteristic
57C24×D53D411yescharacteristic
58C22×Dic106—21no—
59C22×Dic106—22no—
60C22×Dic106—21no—
61C22×Dic106—44no—
62C22×Dic106C2311yescharacteristic
63C22×Dic106C2312yes—
64C22×Dic106C2314yes—
65C2×Q8×D53—22no—
66C2×Q8×D53—44no—
67C2×Q8×D53C2314yes—
68C23×Dic53—21no—
69C23×Dic53D411yescharacteristic
70C22×C4×D53—21no—
71C22×C4×D53—21no—
72C22×C4×D53—44no—
73C22×C4×D53D411yescharacteristic
74C22×C4×D53D414yes—
75C22×C4×D53D411yescharacteristic
76C22×C4×D53C2311yescharacteristic
77C22×C4×D53C2311yescharacteristic
78C22×C4×D53C2314yes—
79C2×Dic212—44no—
80C2×Dic212C2318yescomplemented
81C2×#11719925—44no—
82C2×#11719925C2318yescomplemented
83C2×#11719912—44no—
84C2×#11719912C2318yes—
85C22×#88322C2312yes—
86#13133197—22no—
87#13133197—24no—
88#13133197—24no—
89#13133197—28no—
90#13133197—28no—
91#13133197C2318yes—
92#13133197C2314yes—
93#13133197C2318yes—
94#13133196—22no—
95#13133196C2314yes—
96#13133195—28no—
97#13133194—28no—
98#13133193—22no—
99#13133193C2314yes—
100C2×C4⋊Dic53—21no—
101C2×C4⋊Dic53—22no—
102C2×C4⋊Dic53—22no—
103C2×C4⋊Dic53—28no—
104C2×C4⋊Dic53C2311yescharacteristic
105C2×#11719919—21no—
106C2×#11719919—21no—
107C2×#11719919—21no—
108C4⋊C4×D53—28no—
109C4⋊C4×D53—22no—
110C4⋊C4×D53C2314yes—
111C2×C22⋊Dic53—21no—
112C2×#11719921—21no—
113C2×#11719921—21no—
114C2×#11719921—22no—
115C2×#11719921—28no—
116C2×#11719921—22no—
117C22⋊C4×D53—28no—
118#13133148C2318yescomplemented
119#13133142C2318yescomplemented
120#13133140C2318yescomplemented
121#13133140C2318yescomplemented
122C22⋊#88322C2314yes—
123#13133129C2318yes—
124C22×Q8×C53C2311yescharacteristic
125C23×C4×C53—21no—
126C23×C4×C53C2311yescharacteristic
127C22×C8×C53C2312yes—
128C22⋊Q8×C53—22no—
129C22⋊Q8×C53C2314yes—
130C2×C4⋊C4×C53—21no—
131C2×C4⋊C4×C53—22no—
132C2×C4⋊C4×C53—21no—
133C2×C4⋊C4×C53C2312yes—
134C2×C4⋊C4×C53C2314yes—
135C2×C4⋊C4×C53C2311yescharacteristic
136C2×C22⋊C4×C53—21no—
137#106×C53C2318yescomplemented
138#103×C53C2318yescomplemented
139C22⋊C8×C53C2314yes—
140C24×C53C2×D411yescharacteristic
141C23×D53—24no—
142C23×D53—24no—
143C23×D53—42no—
144C23×D53C2×D411yescharacteristic
145C23×D53C2×D414yes—
146C23×D53C2×D411yescharacteristic
147C2×Q8×C53—21no—
148C2×Q8×C53—24no—
149C2×Q8×C53C2412yes—
150C22×C4×C53—24no—
151C22×C4×C53—21no—
152C22×C4×C53—21no—
153C22×C4×C53—24no—
154C22×C4×C53—22no—
155C22×C4×C53—44no—
156C22×C4×C53C2×D411yescharacteristic
157C22×C4×C53C2×D412yes—
158C22×C4×C53C2412yes—
159C22×C4×C53C2411yescharacteristic
160C22×Dic53—24no—
161C22×Dic53—22no—
162C22×Dic53—21no—
163C22×Dic53—21no—
164C22×Dic53—42no—
165C22×Dic53—44no—
166C22×Dic53C2×D411yescharacteristic
167C22×Dic53C2×D411yescharacteristic
168C22×Dic53C2×D414yes—
169Q8×D53—48no—
170Q8×D53—48no—
171C2×C4×D53—24no—
172C2×C4×D53—24no—
173C2×C4×D53—24no—
174C2×C4×D53—24no—
175C2×C4×D53—48no—
176C2×C4×D53—48no—
177C2×C4×D53—42no—
178C2×C4×D53—42no—
179C2×C4×D53—44no—
180C2×C4×D53—44no—
181C2×C4×D53C2×D412yes—
182C2×C4×D53C2×D412yes—
183C2×C4×D53C2×D414yes—
184C2×C4×D53C2×D414yes—
185C2×C4×D53C2412yes—
186C2×C4×D53C2412yes—
187C2×Dic106—24no—
188C2×Dic106—24no—
189C2×Dic106—24no—
190C2×Dic106—21no—
191C2×Dic106—24no—
192C2×Dic106—22no—
193C2×Dic106—24no—
194C2×Dic106—48no—
195C2×Dic106—44no—
196C2×Dic106—44no—
197C2×Dic106—48no—
198C2×Dic106C2412yes—
199C2×Dic106C2414yes—
200C2×C8×C53—24no—
201C2×C8×C53C2414yes—
202C4⋊C4×C53—24no—
203C4⋊C4×C53—22no—
204C4⋊C4×C53—24no—
205C4⋊C4×C53—21no—
206C4⋊C4×C53C2414yes—
207C4⋊C4×C53C2412yes—
208C22⋊C4×C53—24no—
209C22⋊Dic53—24no—
210#11719925—48no—
211#11719921—24no—
212#11719921—24no—
213#11719921—28no—
214#11719921—28no—
215C4⋊Dic53—21no—
216C4⋊Dic53—24no—
217C4⋊Dic53—28no—
218C4⋊Dic53—28no—
219C4⋊Dic53C2412yes—
220#11719919—24no—
221#11719919—24no—
222#11719919—24no—
223C2×#88322—24no—
224C2×#88322C2414yes—
225Dic212—48no—
226#11719912—48no—
227C23×C53—22no—
228C23×C53—22no—
229C23×C53#13714yes—
230C23×C53C22×D412yes—
231C23×C53C22×D411yescharacteristic
232C22×D53—22no—
233C22×D53—22no—
234C22×D53—22no—
235C22×D53—22no—
236C22×D53—28no—
237C22×D53—22no—
238C22×D53—28no—
239C22×D53—44no—
240C22×D53—416no—
241C22×D53#13714yes—
242C22×D53C22×D412yes—
243C22×D53C22×D412yes—
244Q8×C53—24no—
245Q8×C53—42no—
246C2×C4×C53—22no—
247C2×C4×C53—24no—
248C2×C4×C53—24no—
249C2×C4×C53—22no—
250C2×C4×C53—22no—
251C2×C4×C53—24no—
252C2×C4×C53—42no—
253C2×C4×C53—42no—
254C2×C4×C53—48no—
255C2×C4×C53—44no—
256C2×C4×C53C22×D412yes—
257C2×C4×C53C22×D411yescharacteristic
258C2×C4×C53H(5,𝔽2)14yes—
259C2×C4×C53C2511yescharacteristic
260C2×Dic53—22no—
261C2×Dic53—22no—
262C2×Dic53—22no—
263C2×Dic53—24no—
264C2×Dic53—22no—
265C2×Dic53—44no—
266C2×Dic53—48no—
267C2×Dic53—42no—
268C2×Dic53—44no—
269C2×Dic53—42no—
270C2×Dic53C22×D412yes—
271C2×Dic53C22×D412yes—
272C4×D53—24no—
273C4×D53—24no—
274C4×D53—24no—
275C4×D53—24no—
276C4×D53—44no—
277C4×D53—44no—
278C4×D53—48no—
279C4×D53—48no—
280Dic106—24no—
281Dic106—24no—
282Dic106—24no—
283Dic106—48no—
284Dic106—48no—
285Dic106—44no—
286Dic106—42no—
287C8×C53—24no—
288#88322—24no—
289C2×#3195—531nomaximal, Sylow
290C22×C53—22no—
291C22×C53—28no—
292C22×C53—22no—
293C22×C53#53412yes—
294C22×C53C2×#13712yes—
295C22×C53C2×#13714yes—
296C22×C53C2×#13712yes—
297C22×C53C23×D411yescharacteristic
298C2×D53—22no—
299C2×D53—28no—
300C2×D53—22no—
301C2×D53—22no—
302C2×D53—22no—
303C2×D53—22no—
304C2×D53—416no—
305C2×D53—44no—
306C2×D53C2×#13714yes—
307C4×C53—22no—
308C4×C53—42no—
309C4×C53—42no—
310C4×C53—44no—
311C4×C53#53412yes—
312C4×C53C2×H(5,𝔽2)12yes—
313Dic53—22no—
314Dic53—22no—
315Dic53—42no—
316Dic53—44no—
317Dic53—42no—
318C22×C22⋊Q8—531no—
319C22×C22⋊Q8—531no—
320#3195—5316no—
321C2×#483—532no—
322C2×#483—531no—
323C2×#483D5311yescharacteristic, complemented
324C2×#470—532no—
325C2×#470—532no—
326C2×#451—532no—
327C2×#451—531no—
328C2×#451—531no—
329C2×#407—531no—
330C2×C53—24no—
331C2×C53#319514yescomplemented
332C2×C53C2×#53411yescharacteristic
333C2×C53C22×#13711yescharacteristic
334C2×C53C22×#13711yescharacteristic
335D53—24no—
336D53—42no—
337C23×Q8—531no—
338C23×Q8—531no—
339C24×C4—531no—
340C22×Q16—532no—
341C22×Q16—532no—
342C22×M16—532no—
343C2×C22⋊Q8—531no—
344C2×C22⋊Q8—531no—
345C2×C22⋊Q8—538no—
346C2×C22⋊Q8—532no—
347C2×C22⋊Q8—538no—
348C2×C22⋊Q8—531no—
349C2×C22⋊Q8—532no—
350C2×C22⋊Q8—1061no—
351C2×C22⋊Q8—1061no—
352C2×C22⋊Q8—1061no—
353C2×C22⋊Q8—1064no—
354C2×C22⋊Q8—1061no—
355C2×C22⋊Q8—1064no—
356C2×C22⋊Q8C2×D5311yescharacteristic
357C22×C4⋊C4—531no—
358C22×C4⋊C4—1061no—
359C22×C4⋊C4—1061no—
360C22×C4⋊C4C2×D5311yescharacteristic
361C22×C22⋊C4—1061no—
362C22×C22⋊C4—1061no—
363#483—538no—
364#483—5316no—
365#483C2×D5318yescomplemented
366#470—5316no—
367#470—5316no—
368#451—538no—
369#451—5316no—
370#451—538no—
371C2×#106—532no—
372C2×#106C2×D5312yescomplemented
373C2×#103—532no—
374C2×#103—532no—
375C2×#103—532no—
376C2×#103C2×D5312yescomplemented
377#407—538no—
378C2×C22⋊C8—531no—
379C2×C22⋊C8—532no—
380C2×C22⋊C8C2×D5311yescharacteristic
381C53C2×#319511yescharacteristic, complemented, Sylow
382C25—531no—
383C22×Q8—534no—
384C22×Q8—531no—
385C22×Q8—532no—
386C22×Q8—534no—
387C22×Q8—1061no—
388C22×Q8—1062no—
389C22×Q8—1062no—
390C22×Q8—1061no—
391C22×Q8—2124no—
392C22×Q8—2124no—
393C22×Q8C22×D5311yescharacteristic
394C23×C4—21no—
395C23×C4—531no—
396C23×C4—531no—
397C23×C4—531no—
398C23×C4—531no—
399C23×C4—531no—
400C23×C4—534no—
401C23×C4—534no—
402C23×C4—1061no—
403C23×C4—1061no—
404C23×C4—1061no—
405C23×C4—2124no—
406C23×C4C22×D5311yescharacteristic
407C2×Q16—538no—
408C2×Q16—538no—
409C2×Q16—2124no—
410C2×Q16—2124no—
411C2×M16—538no—
412C2×M16—2124no—
413C22×C8—532no—
414C22×C8C22×D5312yes—
415C22⋊Q8—22no—
416C22⋊Q8—534no—
417C22⋊Q8—534no—
418C22⋊Q8—538no—
419C22⋊Q8—534no—
420C22⋊Q8—538no—
421C22⋊Q8—1068no—
422C22⋊Q8—1068no—
423C22⋊Q8—1064no—
424C22⋊Q8—1064no—
425C22⋊Q8—1068no—
426C22⋊Q8—1068no—
427C22⋊Q8—1062no—
428C22⋊Q8—1062no—
429C22⋊Q8—1062no—
430C22⋊Q8C22×D5314yes—
431C2×C4⋊C4—21no—
432C2×C4⋊C4—22no—
433C2×C4⋊C4—21no—
434C2×C4⋊C4—531no—
435C2×C4⋊C4—534no—
436C2×C4⋊C4—1062no—
437C2×C4⋊C4—1068no—
438C2×C4⋊C4—1061no—
439C2×C4⋊C4—1061no—
440C2×C4⋊C4—1068no—
441C2×C4⋊C4—1061no—
442C2×C4⋊C4—1062no—
443C2×C4⋊C4—1062no—
444C2×C4⋊C4—1061no—
445C2×C4⋊C4C22×D5311yescharacteristic
446C2×C4⋊C4C22×D5312yes—
447C2×C4⋊C4C22×D5314yes—
448C2×C22⋊C4—21no—
449C2×C22⋊C4—1068no—
450C2×C22⋊C4—1068no—
451C2×C22⋊C4—1061no—
452C2×C22⋊C4—1061no—
453C2×C22⋊C4—1062no—
454C2×C22⋊C4—1062no—
455C2×C22⋊C4—1061no—
456#106—538no—
457#106C22×D5318yescomplemented
458#103—538no—
459#103—538no—
460#103—538no—
461#103C22×D5318yescomplemented
462C22⋊C8—534no—
463C22⋊C8—538no—
464C22⋊C8C22×D5314yes—
465C24—531no—
466C24—531no—
467C24—534no—
468C24—1064no—
469C24—1064no—
470C24—2122no—
471C24D4×D5311yescharacteristic
472C2×Q8—21no—
473C2×Q8—24no—
474C2×Q8—532no—
475C2×Q8—534no—
476C2×Q8—1064no—
477C2×Q8—1064no—
478C2×Q8—1061no—
479C2×Q8—1062no—
480C2×Q8—1064no—
481C2×Q8—1064no—
482C2×Q8—1064no—
483C2×Q8—2128no—
484C2×Q8—2124no—
485C2×Q8—2128no—
486C2×Q8—2124no—
487C2×Q8—2128no—
488C2×Q8—2128no—
489C2×Q8C23×D5312yes—
490C22×C4—24no—
491C22×C4—24no—
492C22×C4—21no—
493C22×C4—21no—
494C22×C4—22no—
495C22×C4—44no—
496C22×C4—531no—
497C22×C4—531no—
498C22×C4—534no—
499C22×C4—532no—
500C22×C4—532no—
501C22×C4—532no—
502C22×C4—532no—
503C22×C4—534no—
504C22×C4—534no—
505C22×C4—1064no—
506C22×C4—1064no—
507C22×C4—1064no—
508C22×C4—1064no—
509C22×C4—1064no—
510C22×C4—1061no—
511C22×C4—1062no—
512C22×C4—1061no—
513C22×C4—2128no—
514C22×C4—2128no—
515C22×C4—2122no—
516C22×C4—2122no—
517C22×C4—2124no—
518C22×C4—2124no—
519C22×C4—2122no—
520C22×C4—2124no—
521C22×C4D4×D5312yes—
522C22×C4D4×D5311yescharacteristic
523C22×C4C23×D5312yes—
524C22×C4C23×D5311yescharacteristic
525Q16—2128no—
526Q16—2128no—
527M16—2128no—
528C2×C8—24no—
529C2×C8—534no—
530C2×C8—1064no—
531C2×C8C23×D5314yes—
532C4⋊C4—24no—
533C4⋊C4—22no—
534C4⋊C4—24no—
535C4⋊C4—21no—
536C4⋊C4—532no—
537C4⋊C4—1064no—
538C4⋊C4—1064no—
539C4⋊C4—1064no—
540C4⋊C4—1068no—
541C4⋊C4—1068no—
542C4⋊C4—1061no—
543C4⋊C4—1064no—
544C4⋊C4C23×D5312yes—
545C4⋊C4C23×D5314yes—
546C22⋊C4—24no—
547C22⋊C4—1064no—
548C22⋊C4—1064no—
549C22⋊C4—1068no—
550C22⋊C4—1068no—
551C22⋊C4—1064no—
552C23—22no—
553C23—22no—
554C23—534no—
555C23—532no—
556C23—532no—
557C23—1068no—
558C23—1062no—
559C23—1062no—
560C23—1062no—
561C23—1068no—
562C23—1062no—
563C23—1062no—
564C23—2124no—
565C23—21216no—
566C23#1313324614yes—
567C23C2×D4×D5312yes—
568C23C2×D4×D5311yescharacteristic
569Q8—24no—
570Q8—42no—
571Q8—1064no—
572Q8—1064no—
573Q8—1064no—
574Q8—2128no—
575Q8—2128no—
576Q8—2124no—
577Q8—2122no—
578C2×C4—22no—
579C2×C4—24no—
580C2×C4—24no—
581C2×C4—22no—
582C2×C4—22no—
583C2×C4—24no—
584C2×C4—42no—
585C2×C4—48no—
586C2×C4—44no—
587C2×C4—42no—
588C2×C4—532no—
589C2×C4—532no—
590C2×C4—1062no—
591C2×C4—1062no—
592C2×C4—1064no—
593C2×C4—1064no—
594C2×C4—1064no—
595C2×C4—1064no—
596C2×C4—1062no—
597C2×C4—1062no—
598C2×C4—1064no—
599C2×C4—2124no—
600C2×C4—2124no—
601C2×C4—2128no—
602C2×C4—2128no—
603C2×C4—2124no—
604C2×C4—2128no—
605C2×C4—2122no—
606C2×C4—2122no—
607C2×C4—2124no—
608C2×C4C2×D4×D5312yes—
609C2×C4C2×D4×D5311yescharacteristic
610C2×C4#1313326314yes—
611C2×C4C24×D5311yescharacteristic
612C8—24no—
613C8—1064no—
614C22—22no—
615C22—28no—
616C22—22no—
617C22—534no—
618C22—1062no—
619C22—1068no—
620C22—1062no—
621C22—1062no—
622C22—1062no—
623C22—1062no—
624C22—21216no—
625C22—2124no—
626C22#1681118412yes—
627C22#137×D5312yes—
628C22C2×#1313324614yes—
629C22C2×#1313324612yes—
630C22C22×D4×D5311yescharacteristic
631C4—22no—
632C4—42no—
633C4—42no—
634C4—44no—
635C4—1062no—
636C4—1062no—
637C4—2122no—
638C4—2124no—
639C4—2122no—
640C4#1681118412yes—
641C4C2×#1313326312yes—
642C2—24no—
643C2—1064no—
644C2—2122no—
645C2#2028553814yesdirect product factor
646C2C2×#1681118411yescharacteristic
647C2C2×#137×D5311yescharacteristic
648C2C22×#1313324611yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC2×C4×C53C25
FrattiniC2×C4C24×D53
FittingC2×C53×#483C2
socleC23×C53C22×D4
prime radicalC24×D53D4
hypercenterC2×#483D53
hypocenterC53C2×#3195
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#483

Complements

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

NH ≅ G/Ntype
3. #20285538C2direct (N × H)
4. C2×#16810955C2semidirect
5. C2×#16810953C2semidirect
6. C2×#16810882C2semidirect
7. C2×#16810880C2semidirect
8. C2×#16810781C2semidirect
9. C2×#16810780C2semidirect
10. C2×#16810778C2semidirect
12. C2×C53×#483C2semidirect
16. C22×Dic212C22semidirect
17. C22×#11719925C22semidirect
37. #16810955C22semidirect
38. #16810953C22semidirect
39. #16810882C22semidirect
40. #16810880C22semidirect
41. #16810781C22semidirect
42. #16810780C22semidirect
43. #16810778C22semidirect
44. C2×#13133148C22semidirect
45. C2×#13133142C22semidirect
46. C2×#13133140C22semidirect
47. C2×#13133140C22semidirect
53. C53×#483C22semidirect
54. C2×#106×C53C22semidirect
55. C2×#103×C53C22semidirect
80. C2×Dic212C23semidirect
82. C2×#11719925C23semidirect
118. #13133148C23semidirect
119. #13133142C23semidirect
120. #13133140C23semidirect
121. #13133140C23semidirect
137. #106×C53C23semidirect
138. #103×C53C23semidirect
323. C2×#483D53semidirect
331. C2×C53#3195semidirect
365. #483C2×D53semidirect
372. C2×#106C2×D53semidirect
376. C2×#103C2×D53semidirect
381. C53C2×#3195semidirect
457. #106C22×D53semidirect
461. #103C22×D53semidirect
645. C2#20285538direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#131331972
C2×C22⋊Q8×D532
#202855382
C2×#168109552
C2×#168109532
C2×#168108822
C2×#168108802
C2×#168107812
C2×#168107802
C2×#168107782
C2×#168105852
C2×C53×#4832
C2×#319553