Finite Group Databasebeta

← #28318140

Subgroups — #28318140

Order 13568.

1834 conjugacy classes of subgroups.

The subgroup lattice has 630 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D53⋊#99C212yesmaximal
2#20291151C212yesmaximal, complemented
3#20290462C212yesmaximal, complemented
4C4×#13133276C212yesmaximal, complemented
5#20290460C212yesmaximal, complemented
6C4×#13133275C212yesmaximal, complemented
7C2×#16811501C212yesmaximal, complemented
8C2×#16811095C211yesmaximal, characteristic
9C22×C22⋊#6157C2214yes—
10C2×#13133276C2214yes—
11D53⋊#99C4116yescomplemented
12C2×#13133275C2214yes—
13C22⋊C4×#6157—28no—
14C4×C22⋊#6157C2214yescomplemented
15C4×C22⋊#6157C2214yescomplemented
16#16811501C2214yescomplemented
17#16811501C2214yescomplemented
18C22×D4×D53C2211yescharacteristic
19#16811095—28no—
20C2×C4×D212C2211yescharacteristic
21C2×D4×Dic53C411yescharacteristic, complemented
22C2×#13133179C2211yescharacteristic
23C2×C4×C22⋊D53C411yescharacteristic, complemented
24C2×C4×C22⋊D53C2211yescharacteristic
25C2×#13133177C411yescharacteristic, complemented
26C2×#13133177C2211yescharacteristic
27C2×#13133175C411yescharacteristic, complemented
28C2×#13133174C411yescharacteristic, complemented
29C22×#11719921C411yescharacteristic, complemented
30C22×#11719921C2211yescharacteristic
31C2×C22⋊C4×D53C411yescharacteristic, complemented
32C2×C22⋊C4×D53C2211yescharacteristic
33C2×C4×D4×C53C411yescharacteristic, complemented
34C23×#6157—24no—
35C2×C22⋊#6157—24no—
36C2×C22⋊#6157—28no—
37C2×C22⋊#6157C2×C4116yescomplemented
38C2×C22⋊#6157C2318yes—
39C2×C4×#6157—24no—
40C2×C4×#6157—24no—
41#13133276C2×C4116yescomplemented
42#13133275C2×C4116yescomplemented
43(C2×#6157)⋊C4—24no—
44(C2×#6157)⋊C4—24no—
45C24×D53C2×C411yescharacteristic
46C24×D53C2311yescharacteristic
47C22×D212C2311yescharacteristic
48C22×C22⋊D53C2×C411yescharacteristic
49C22×C22⋊D53C2311yescharacteristic
50C2×D4×D53—24no—
51C2×D4×D53—24no—
52C2×D4×D53C2×C418yes—
53C23×Dic53C2×C411yescharacteristic
54C23×Dic53C2×C411yescharacteristic, complemented
55C22×C4×D53C2×C411yescharacteristic
56C22×C4×D53C2×C411yescharacteristic, complemented
57C22×C4×D53C2×C411yescharacteristic, complemented
58C22×C4×D53C2311yescharacteristic
59C22×C4×D53C2311yescharacteristic
60C4×D212—24no—
61D4×Dic53—24no—
62#13133179—24no—
63C4×C22⋊D53—24no—
64C4×C22⋊D53—24no—
65#13133177—24no—
66#13133177—24no—
67#13133175—24no—
68#13133174—24no—
69C2×C4⋊Dic53C2×C411yescharacteristic, complemented
70C2×#11719919C2×C411yescharacteristic, complemented
71C2×#11719919C2×C411yescharacteristic, complemented
72C2×C22⋊Dic53C2×C411yescharacteristic
73C2×C22⋊Dic53C2×C411yescharacteristic, complemented
74C2×#11719921—24no—
75C2×#11719921—24no—
76C2×#11719921C2×C411yescharacteristic
77C2×#11719921C2×C411yescharacteristic
78C2×#11719921C2×C411yescharacteristic
79C2×#11719921C2×C411yescharacteristic
80C2×#11719921C2×C411yescharacteristic, complemented
81C2×#11719921C2×C411yescharacteristic, complemented
82C2×#11719921C2311yescharacteristic
83C2×#11719921C2311yescharacteristic
84C22⋊C4×D53—24no—
85C22⋊C4×D53D418yes—
86C2×C4×Dic53C2×C411yescharacteristic, complemented
87C2×C4×Dic53C2×C411yescharacteristic, complemented
88C2×C4×Dic53C2×C411yescharacteristic, complemented
89C22×D4×C53C2×C411yescharacteristic
90C23×C4×C53C2×C411yescharacteristic
91C23×C4×C53C2×C411yescharacteristic, complemented
92C4×D4×C53—24no—
93C2×C4⋊C4×C53C2×C411yescharacteristic, complemented
94C2×C22⋊C4×C53C2×C411yescharacteristic
95C2×C22⋊C4×C53C2×C411yescharacteristic, complemented
96C2×C42×C53C2×C411yescharacteristic, complemented
97C24×C53C22×C411yescharacteristic
98C24×C53C22×C411yescharacteristic
99C23×D53—22no—
100C23×D53—22no—
101C23×D53—81no—
102C23×D53—81no—
103C23×D53—81no—
104C23×D53—81no—
105C23×D53C4214yescomplemented
106C23×D53C22×C411yescharacteristic
107C23×D53C22×C411yescharacteristic
108C23×D53C22×C414yes—
109C23×D53C2×D414yes—
110C23×D53#4114yes—
111C23×D53C2411yescharacteristic
112C22×#6157—28no—
113C22×#6157—216no—
114C22×#6157—84no—
115C22×#6157—84no—
116C2×D4×C53—22no—
117C2×D4×C53—22no—
118C2×D4×C53C4214yescomplemented
119C22×C4×C53—22no—
120C22×C4×C53—22no—
121C22×C4×C53—81no—
122C22×C4×C53—81no—
123C22×C4×C53C22×C411yescharacteristic
124C22×C4×C53C22×C411yescharacteristic
125C22×C4×C53C22×C411yescharacteristic
126C22×C4×C53C22×C411yescharacteristic
127C22×C4×C53C22×C411yescharacteristic
128C2×C22⋊D53—22no—
129C2×C22⋊D53—22no—
130C2×C22⋊D53—22no—
131C2×C22⋊D53—22no—
132C2×C22⋊D53C4214yescomplemented
133C2×C22⋊D53C22×C414yes—
134C22×Dic53—22no—
135C22×Dic53—22no—
136C22×Dic53—81no—
137C22×Dic53—81no—
138C22×Dic53C22×C411yescharacteristic
139C22×Dic53C22×C411yescharacteristic
140C22×Dic53C22×C411yescharacteristic
141C22×Dic53C22×C411yescharacteristic
142C22×Dic53C22×C411yescharacteristic
143D4×D53—216no—
144D4×D53—216no—
145C2×D212—22no—
146C2×D212—22no—
147C2×D212C22×C414yes—
148C2×C4×D53—22no—
149C2×C4×D53—22no—
150C2×C4×D53—22no—
151C2×C4×D53—22no—
152C2×C4×D53—81no—
153C2×C4×D53—81no—
154C2×C4×D53—81no—
155C2×C4×D53—81no—
156C2×C4×D53C4214yescomplemented
157C2×C4×D53C2×D414yes—
158C2×C4×D53C2×D414yes—
159C22⋊#6157—216no—
160C22⋊#6157—28no—
161C22⋊#6157C22×C4116yes—
162C4×#6157—84no—
163C4×#6157—84no—
164C4⋊C4×C53—22no—
165C22⋊C4×C53—22no—
166C22⋊C4×C53C22⋊C414yes—
167C42×C53—22no—
168C22⋊Dic53—22no—
169C22⋊Dic53C22⋊C414yes—
170#11719921—22no—
171#11719921—22no—
172#11719921—22no—
173#11719921—22no—
174#11719921—22no—
175#11719921—22no—
176#11719921C22⋊C414yes—
177#11719921C22⋊C414yes—
178C4⋊Dic53—22no—
179#11719919—22no—
180#11719919—22no—
181C4×Dic53—22no—
182C4×Dic53—22no—
183C4×Dic53—22no—
184C23×C53—21no—
185C23×C53—21no—
186C23×C53—41no—
187C23×C53—41no—
188C23×C53—41no—
189C23×C53—41no—
190C23×C53C2×C4212yes—
191C23×C53C2×C4212yes—
192C23×C53C2×C22⋊C412yes—
193C23×C53#11712yes—
194C23×C53C23×C411yescharacteristic
195C22×D53—21no—
196C22×D53—21no—
197C22×D53—28no—
198C22×D53—41no—
199C22×D53—41no—
200C22×D53—44no—
201C22×D53—44no—
202C22×D53—41no—
203C22×D53—41no—
204C22×D53—41no—
205C22×D53—41no—
206C22×D53—82no—
207C22×D53—82no—
208C22×D53—84no—
209C22×D53—84no—
210C22×D53—84no—
211C22×D53—84no—
212C22×D53C2×C4212yes—
213C22×D53C2×C4212yes—
214C22×D53C2×C22⋊C412yes—
215C22×D53#11712yes—
216C22×D53C4×D418yes—
217C22×D53C4×D418yes—
218C22×D53C23×C412yes—
219C22×D53C22×D412yes—
220C22×D53C2×#4112yes—
221C2×#6157—216no—
222C2×#6157—48no—
223C2×#6157—48no—
224C2×#6157—816no—
225D4×C53—28no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—21no—
229C2×C4×C53—21no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—82no—
235C2×C4×C53—82no—
236C2×C4×C53C2×C4212yes—
237C2×C4×C53C2×C22⋊C412yes—
238C2×C4×C53C2×C22⋊C412yes—
239C22⋊D53—28no—
240C22⋊D53—28no—
241C2×Dic53—21no—
242C2×Dic53—21no—
243C2×Dic53—21no—
244C2×Dic53—21no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53—82no—
250C2×Dic53—82no—
251C2×Dic53C2×C4212yes—
252C2×Dic53C2×C22⋊C412yes—
253C2×Dic53C2×C22⋊C412yes—
254D212—28no—
255C4×D53—28no—
256C4×D53—44no—
257C4×D53—44no—
258C4×D53—82no—
259C4×D53—82no—
260#20392—531nomaximal, Sylow
261C22×C53—21no—
262C22×C53—21no—
263C22×C53—22no—
264C22×C53—22no—
265C22×C53—22no—
266C22×C53—42no—
267C22×C53—42no—
268C22×C53—42no—
269C22×C53—42no—
270C22×C53—41no—
271C22×C53—41no—
272C22×C53C4×C22⋊C414yes—
273C22×C53C22×C4211yescharacteristic
274C22×C53C22×C22⋊C411yescharacteristic
275C22×C53C2×#11711yescharacteristic
276C2×D53—22no—
277C2×D53—22no—
278C2×D53—22no—
279C2×D53—22no—
280C2×D53—22no—
281C2×D53—48no—
282C2×D53—42no—
283C2×D53—42no—
284C2×D53—48no—
285C2×D53—42no—
286C2×D53—42no—
287C2×D53—42no—
288C2×D53—41no—
289C2×D53—41no—
290C2×D53—42no—
291C2×D53—88no—
292C2×D53C4×C22⋊C414yes—
293C2×D53C2×C4×D414yes—
294#6157—416no—
295C4×C53—22no—
296C4×C53—22no—
297C4×C53—22no—
298C4×C53—41no—
299C4×C53—41no—
300Dic53—22no—
301Dic53—22no—
302Dic53—22no—
303Dic53—41no—
304Dic53—41no—
305C2×#518—531no—
306C22×#99—532no—
307#2273—532no—
308C4×#99—532no—
309C4×#99—532no—
310C4×#99—532no—
311C4×#99—532no—
312C2×#342—532no—
313C2×C53—21no—
314C2×C53—21no—
315C2×C53—24no—
316C2×C53—42no—
317C2×C53C2×C4×C22⋊C411yescharacteristic
318C2×C53#288311yescharacteristic
319C2×C53#288311yescharacteristic
320D53—24no—
321D53—24no—
322D53—42no—
323C23×D4—531no—
324#518—1068no—
325C2×C4×D4—531no—
326C2×C4×D4—531no—
327C2×C4×D4—531no—
328C2×C4×D4—531no—
329C2×C4×D4—531no—
330C2×C4×D4—531no—
331C2×C4×D4—531no—
332C2×C4×D4#615711yescharacteristic, complemented
333C2×#117—531no—
334C2×#117—531no—
335C22×C22⋊C4—531no—
336C22×C22⋊C4—534no—
337C22×C22⋊C4—531no—
338C22×C22⋊C4—531no—
339C22×C22⋊C4—531no—
340C2×#99—534no—
341C2×#99—5316no—
342C2×#99—534no—
343C4×C22⋊C4—534no—
344C4×C22⋊C4—534no—
345C4×C22⋊C4—1068no—
346#342—534no—
347#342—534no—
348C53#2039211yescharacteristic, complemented, Sylow
349C25—531no—
350C25—531no—
351C22×D4—531no—
352C22×D4—531no—
353C22×D4—538no—
354C22×D4—531no—
355C22×D4—1064no—
356C22×D4—1064no—
357C22×D4C2×#615711yescharacteristic
358C23×C4—531no—
359C23×C4—531no—
360C23×C4—531no—
361C23×C4—531no—
362C23×C4—531no—
363C23×C4—531no—
364C23×C4—531no—
365C23×C4—1064no—
366C23×C4C2×#615711yescharacteristic
367C23×C4C2×#615711yescharacteristic, complemented
368C4×D4—24no—
369C4×D4—1064no—
370C4×D4—1064no—
371C4×D4—1064no—
372C4×D4—1064no—
373C4×D4—1064no—
374C4×D4—1064no—
375C4×D4—1064no—
376#117—1064no—
377#117—1064no—
378C2×C4⋊C4—531no—
379C2×C4⋊C4—531no—
380C2×C4⋊C4—531no—
381C2×C4⋊C4C2×#615711yescharacteristic, complemented
382C2×C22⋊C4—531no—
383C2×C22⋊C4—531no—
384C2×C22⋊C4—531no—
385C2×C22⋊C4—538no—
386C2×C22⋊C4—538no—
387C2×C22⋊C4—5316no—
388C2×C22⋊C4—531no—
389C2×C22⋊C4—531no—
390C2×C22⋊C4—531no—
391C2×C22⋊C4—531no—
392C2×C22⋊C4—531no—
393C2×C22⋊C4—531no—
394C2×C22⋊C4—531no—
395C2×C22⋊C4—1064no—
396C2×C22⋊C4—1064no—
397C2×C22⋊C4—1064no—
398C2×C22⋊C4—1064no—
399C2×C22⋊C4—1068no—
400C2×C22⋊C4C2×#615711yescharacteristic
401C2×C22⋊C4C2×#615711yescharacteristic, complemented
402C2×C42—531no—
403C2×C42—531no—
404C2×C42—531no—
405C2×C42—1064no—
406C2×C42—1064no—
407C2×C42C2×#615711yescharacteristic, complemented
408#99—5316no—
409#99—5316no—
410#95—1064no—
411#95—1064no—
412C24—534no—
413C24—534no—
414C24—534no—
415C24—531no—
416C24—531no—
417C24—534no—
418C24—531no—
419C24—1062no—
420C24—1062no—
421C24—4241no—
422C24—4241no—
423C24—4241no—
424C24—4241no—
425C24C22×#615711yescharacteristic
426C24C22×#615711yescharacteristic
427C2×D4—22no—
428C2×D4—22no—
429C2×D4—534no—
430C2×D4—534no—
431C2×D4—534no—
432C2×D4—1062no—
433C2×D4—1062no—
434C2×D4—10616no—
435C2×D4—10616no—
436C2×D4—1062no—
437C2×D4—1062no—
438C2×D4—1062no—
439C2×D4—1062no—
440C2×D4C4×#615714yescomplemented
441C22×C4—22no—
442C22×C4—22no—
443C22×C4—81no—
444C22×C4—81no—
445C22×C4—531no—
446C22×C4—531no—
447C22×C4—531no—
448C22×C4—534no—
449C22×C4—534no—
450C22×C4—534no—
451C22×C4—531no—
452C22×C4—531no—
453C22×C4—1062no—
454C22×C4—1062no—
455C22×C4—1062no—
456C22×C4—1062no—
457C22×C4—10616no—
458C22×C4—1062no—
459C22×C4—1068no—
460C22×C4—1062no—
461C22×C4—4241no—
462C22×C4—4241no—
463C22×C4—4241no—
464C22×C4—4241no—
465C22×C4—4244no—
466C22×C4—4244no—
467C22×C4—4241no—
468C22×C4—4241no—
469C22×C4C22×#615711yescharacteristic
470C22×C4C22×#615711yescharacteristic
471C22×C4C22×#615711yescharacteristic
472C22×C4C22×#615711yescharacteristic
473C22×C4C22×#615711yescharacteristic
474C4⋊C4—22no—
475C4⋊C4—1062no—
476C4⋊C4—1062no—
477C4⋊C4—1062no—
478C22⋊C4—22no—
479C22⋊C4—534no—
480C22⋊C4—534no—
481C22⋊C4—534no—
482C22⋊C4—5316no—
483C22⋊C4—1062no—
484C22⋊C4—1062no—
485C22⋊C4—1062no—
486C22⋊C4—1062no—
487C22⋊C4—1062no—
488C22⋊C4—1062no—
489C22⋊C4—10616no—
490C22⋊C4—1068no—
491C22⋊C4—1062no—
492C22⋊C4C22⋊#615714yes—
493C42—22no—
494C42—1062no—
495C42—1062no—
496C42—1062no—
497C42—4244no—
498C42—4244no—
499C23—21no—
500C23—21no—
501C23—41no—
502C23—41no—
503C23—41no—
504C23—41no—
505C23—532no—
506C23—532no—
507C23—538no—
508C23—538no—
509C23—532no—
510C23—532no—
511C23—532no—
512C23—532no—
513C23—532no—
514C23—1068no—
515C23—1061no—
516C23—1061no—
517C23—4242no—
518C23—4242no—
519C23—4244no—
520C23—4244no—
521C23—4244no—
522C23—4244no—
523C23—2121no—
524C23—2124no—
525C23—2121no—
526C23—2124no—
527C23—2121no—
528C23—2121no—
529C23—2121no—
530C23—2121no—
531C23C2×C4×#615712yes—
532C23C2×C4×#615712yes—
533C23C2×C22⋊#615712yes—
534C23#1313329612yes—
535C23C23×#615711yescharacteristic
536D4—28no—
537D4—1068no—
538D4—1068no—
539D4—1068no—
540C2×C4—21no—
541C2×C4—21no—
542C2×C4—21no—
543C2×C4—21no—
544C2×C4—41no—
545C2×C4—41no—
546C2×C4—41no—
547C2×C4—41no—
548C2×C4—82no—
549C2×C4—82no—
550C2×C4—532no—
551C2×C4—532no—
552C2×C4—532no—
553C2×C4—1068no—
554C2×C4—1061no—
555C2×C4—1061no—
556C2×C4—10616no—
557C2×C4—1061no—
558C2×C4—1061no—
559C2×C4—4242no—
560C2×C4—4242no—
561C2×C4—4242no—
562C2×C4—4242no—
563C2×C4—42416no—
564C2×C4—2124no—
565C2×C4—2124no—
566C2×C4—2121no—
567C2×C4—2121no—
568C2×C4—2128no—
569C2×C4—2128no—
570C2×C4—2121no—
571C2×C4—2121no—
572C2×C4C2×C4×#615712yes—
573C2×C4C2×C22⋊#615712yes—
574C2×C4C2×C22⋊#615712yes—
575C22—21no—
576C22—22no—
577C22—21no—
578C22—22no—
579C22—22no—
580C22—42no—
581C22—42no—
582C22—42no—
583C22—42no—
584C22—41no—
585C22—41no—
586C22—534no—
587C22—534no—
588C22—1062no—
589C22—1062no—
590C22—1062no—
591C22—1062no—
592C22—1062no—
593C22—4248no—
594C22—2128no—
595C22—2122no—
596C22—2122no—
597C22—2122no—
598C22—2122no—
599C22—2128no—
600C22—2122no—
601C22—2122no—
602C22—2121no—
603C22—2121no—
604C22C4×C22⋊#615714yes—
605C22C22×C4×#615711yescharacteristic
606C22C22×C22⋊#615711yescharacteristic
607C22C2×#1313329611yescharacteristic
608C4—22no—
609C4—22no—
610C4—22no—
611C4—41no—
612C4—41no—
613C4—1062no—
614C4—1062no—
615C4—1062no—
616C4—21216no—
617C4—2121no—
618C4—2121no—
619C2—21no—
620C2—21no—
621C2—24no—
622C2—42no—
623C2—1064no—
624C2—1064no—
625C2—2122no—
626C2C2×C4×C22⋊#615711yescharacteristic
627C2#2029131411yescharacteristic
628C2#2029131511yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6157
commutatorC22×C53C22×C42
FrattiniC23C23×#6157
FittingC2×C4×D4×C53C4
socleC22×C53C22×C22⋊C4
prime radicalC22×D4×D53C22
cosocleC23×D53C24
hypercenterC2×C4×D4#6157
hypocenterC53#20392
LCSG ▷ C22×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
2. #20291151C2semidirect
3. #20290462C2semidirect
4. C4×#13133276C2semidirect
5. #20290460C2semidirect
6. C4×#13133275C2semidirect
7. C2×#16811501C2semidirect
11. D53⋊#99C4semidirect
14. C4×C22⋊#6157C22semidirect
15. C4×C22⋊#6157C22semidirect
16. #16811501C22semidirect
17. #16811501C22semidirect
21. C2×D4×Dic53C4semidirect
23. C2×C4×C22⋊D53C4semidirect
25. C2×#13133177C4semidirect
27. C2×#13133175C4semidirect
28. C2×#13133174C4semidirect
29. C22×#11719921C4semidirect
31. C2×C22⋊C4×D53C4semidirect
33. C2×C4×D4×C53C4semidirect
37. C2×C22⋊#6157C2×C4semidirect
41. #13133276C2×C4semidirect
42. #13133275C2×C4semidirect
54. C23×Dic53C2×C4semidirect
56. C22×C4×D53C2×C4semidirect
57. C22×C4×D53C2×C4semidirect
69. C2×C4⋊Dic53C2×C4semidirect
70. C2×#11719919C2×C4semidirect
71. C2×#11719919C2×C4semidirect
73. C2×C22⋊Dic53C2×C4semidirect
80. C2×#11719921C2×C4semidirect
81. C2×#11719921C2×C4semidirect
86. C2×C4×Dic53C2×C4semidirect
87. C2×C4×Dic53C2×C4semidirect
88. C2×C4×Dic53C2×C4semidirect
91. C23×C4×C53C2×C4semidirect
93. C2×C4⋊C4×C53C2×C4semidirect
95. C2×C22⋊C4×C53C2×C4semidirect
96. C2×C42×C53C2×C4semidirect
105. C23×D53C42semidirect
118. C2×D4×C53C42semidirect
132. C2×C22⋊D53C42semidirect
156. C2×C4×D53C42semidirect
332. C2×C4×D4#6157semidirect
348. C53#20392semidirect
367. C23×C4C2×#6157semidirect
381. C2×C4⋊C4C2×#6157semidirect
401. C2×C22⋊C4C2×#6157semidirect
407. C2×C42C2×#6157semidirect
440. C2×D4C4×#6157semidirect

Maximal subgroups

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

subgroupindex
C2×D53⋊#992
#202911512
#202904622
C4×#131332762
#202904602
C4×#131332752
C2×#168115012
C2×#168110952
#2039253