Finite Group Databasebeta

← #28319048

Subgroups — #28319048

Order 13568.

1690 conjugacy classes of subgroups.

The subgroup lattice has 652 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D53⋊#99C212yesmaximal
2#20291194C212yesmaximal, complemented
3#20290578C212yesmaximal, complemented
4#20290577C212yesmaximal, complemented
5#20290576C212yesmaximal, complemented
6#20290575C212yesmaximal, complemented
7C2×#16811501C212yesmaximal, complemented
8C2×#16811095C211yesmaximal, characteristic
9C22×C22⋊#6157C2214yes—
10C2×#13133276C2214yes—
11D53⋊#99—28no—
12C2×#13133275C2214yes—
13#16811570—28no—
14#16811565C2214yescomplemented
15#16811565C2214yescomplemented
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×C4⋊#6157—24no—
36C2×C4⋊#6157—24no—
37C2×C22⋊#6157—28no—
38C2×C22⋊#6157—28no—
39C2×C22⋊#6157—24no—
40C2×C22⋊#6157C2318yes—
41#13133276—28no—
42#13133275—28no—
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×D53D414yes—
53C2×D4×D53Q814yes—
54C23×Dic53C2×C411yescharacteristic
55C23×Dic53C2×C411yescharacteristic, complemented
56C22×C4×D53C2×C411yescharacteristic
57C22×C4×D53C2×C411yescharacteristic, complemented
58C22×C4×D53C2×C411yescharacteristic, complemented
59C22×C4×D53C2311yescharacteristic
60C22×C4×D53C2311yescharacteristic
61C4×D212—24no—
62D4×Dic53—24no—
63#13133179—24no—
64C4×C22⋊D53—24no—
65C4×C22⋊D53—24no—
66#13133177—24no—
67#13133177—24no—
68#13133175—24no—
69#13133174—24no—
70C2×C4⋊Dic53C2×C411yescharacteristic, complemented
71C2×#11719919C2×C411yescharacteristic, complemented
72C2×#11719919C2×C411yescharacteristic, complemented
73C2×C22⋊Dic53C2×C411yescharacteristic
74C2×C22⋊Dic53C2×C411yescharacteristic, complemented
75C2×#11719921—24no—
76C2×#11719921—24no—
77C2×#11719921C2×C411yescharacteristic
78C2×#11719921C2×C411yescharacteristic
79C2×#11719921C2×C411yescharacteristic
80C2×#11719921C2×C411yescharacteristic
81C2×#11719921C2×C411yescharacteristic, complemented
82C2×#11719921C2×C411yescharacteristic, complemented
83C2×#11719921C2311yescharacteristic
84C2×#11719921C2311yescharacteristic
85C22⋊C4×D53—24no—
86C22⋊C4×D53D418yes—
87C2×C4×Dic53C2×C411yescharacteristic, complemented
88C2×C4×Dic53C2×C411yescharacteristic, complemented
89C2×C4×Dic53C2×C411yescharacteristic, complemented
90C22×D4×C53C2×C411yescharacteristic
91C23×C4×C53C2×C411yescharacteristic
92C23×C4×C53C2×C411yescharacteristic, complemented
93C4×D4×C53—24no—
94C2×C4⋊C4×C53C2×C411yescharacteristic, complemented
95C2×C22⋊C4×C53C2×C411yescharacteristic
96C2×C22⋊C4×C53C2×C411yescharacteristic, complemented
97C2×C42×C53C2×C411yescharacteristic, complemented
98C24×C53C22×C411yescharacteristic
99C24×C53C22×C411yescharacteristic
100C23×D53—22no—
101C23×D53—22no—
102C23×D53—81no—
103C23×D53—81no—
104C23×D53—81no—
105C23×D53—81no—
106C23×D53C4⋊C414yescomplemented
107C23×D53C22×C411yescharacteristic
108C23×D53C22×C411yescharacteristic
109C23×D53C2×D414yes—
110C23×D53C2×D412yes—
111C23×D53C2×Q812yes—
112C23×D53#4114yes—
113C23×D53C2411yescharacteristic
114C22×#6157—28no—
115C22×#6157—48no—
116C22×#6157—84no—
117C22×#6157—84no—
118C2×D4×C53—22no—
119C2×D4×C53—22no—
120C2×D4×C53C4⋊C414yescomplemented
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53—81no—
124C22×C4×C53—81no—
125C22×C4×C53C22×C411yescharacteristic
126C22×C4×C53C22×C411yescharacteristic
127C22×C4×C53C22×C411yescharacteristic
128C22×C4×C53C22×C411yescharacteristic
129C22×C4×C53C22×C411yescharacteristic
130C2×C22⋊D53—22no—
131C2×C22⋊D53—22no—
132C2×C22⋊D53—22no—
133C2×C22⋊D53—22no—
134C2×C22⋊D53C4⋊C414yescomplemented
135C2×C22⋊D53C2×D412yes—
136C2×C22⋊D53C2×Q812yes—
137C22×Dic53—22no—
138C22×Dic53—22no—
139C22×Dic53—81no—
140C22×Dic53—81no—
141C22×Dic53C22×C411yescharacteristic
142C22×Dic53C22×C411yescharacteristic
143C22×Dic53C22×C411yescharacteristic
144C22×Dic53C22×C411yescharacteristic
145C22×Dic53C22×C411yescharacteristic
146D4×D53—216no—
147D4×D53—216no—
148C2×D212—22no—
149C2×D212—22no—
150C2×D212C2×D412yes—
151C2×D212C2×Q812yes—
152C2×C4×D53—22no—
153C2×C4×D53—22no—
154C2×C4×D53—22no—
155C2×C4×D53—22no—
156C2×C4×D53—81no—
157C2×C4×D53—81no—
158C2×C4×D53—81no—
159C2×C4×D53—81no—
160C2×C4×D53C4⋊C414yescomplemented
161C2×C4×D53C2×D414yes—
162C2×C4×D53C2×D414yes—
163C22⋊#6157—28no—
164C22⋊#6157—28no—
165C22⋊#6157—48no—
166C4⋊#6157—84no—
167C4⋊#6157—84no—
168C4⋊C4×C53—22no—
169C22⋊C4×C53—22no—
170C22⋊C4×C53C22⋊C414yes—
171C42×C53—22no—
172C22⋊Dic53—22no—
173C22⋊Dic53C22⋊C414yes—
174#11719921—22no—
175#11719921—22no—
176#11719921—22no—
177#11719921—22no—
178#11719921—22no—
179#11719921—22no—
180#11719921C22⋊C414yes—
181#11719921C22⋊C414yes—
182C4⋊Dic53—22no—
183#11719919—22no—
184#11719919—22no—
185C4×Dic53—22no—
186C4×Dic53—22no—
187C4×Dic53—22no—
188C23×C53—21no—
189C23×C53—21no—
190C23×C53—41no—
191C23×C53—41no—
192C23×C53—41no—
193C23×C53—41no—
194C23×C53C2×C22⋊C412yes—
195C23×C53C2×C4⋊C412yes—
196C23×C53C2×C4⋊C412yes—
197C23×C53#11712yes—
198C23×C53C23×C411yescharacteristic
199C22×D53—21no—
200C22×D53—21no—
201C22×D53—28no—
202C22×D53—41no—
203C22×D53—41no—
204C22×D53—44no—
205C22×D53—44no—
206C22×D53—41no—
207C22×D53—41no—
208C22×D53—41no—
209C22×D53—41no—
210C22×D53—82no—
211C22×D53—82no—
212C22×D53—82no—
213C22×D53—82no—
214C22×D53—84no—
215C22×D53—84no—
216C22×D53—84no—
217C22×D53C2×C22⋊C412yes—
218C22×D53C2×C4⋊C412yes—
219C22×D53C2×C4⋊C412yes—
220C22×D53#11712yes—
221C22×D53#12118yes—
222C22×D53C22⋊Q818yes—
223C22×D53C22×D412yes—
224C22×D53C22×D411yescharacteristic
225C22×D53C22×Q811yescharacteristic
226C22×D53C2×#4112yes—
227C2×#6157—48no—
228C2×#6157—48no—
229C2×#6157—48no—
230C2×#6157—816no—
231D4×C53—28no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53—82no—
241C2×C4×C53—82no—
242C2×C4×C53C2×C22⋊C412yes—
243C2×C4×C53C2×C22⋊C412yes—
244C2×C4×C53C2×C4⋊C412yes—
245C22⋊D53—28no—
246C22⋊D53—24no—
247C22⋊D53—24no—
248C2×Dic53—21no—
249C2×Dic53—21no—
250C2×Dic53—21no—
251C2×Dic53—21no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—82no—
257C2×Dic53—82no—
258C2×Dic53C2×C22⋊C412yes—
259C2×Dic53C2×C22⋊C412yes—
260C2×Dic53C2×C4⋊C412yes—
261D212—24no—
262D212—24no—
263C4×D53—28no—
264C4×D53—44no—
265C4×D53—44no—
266C4×D53—82no—
267C4×D53—82no—
268#20633—531nomaximal, Sylow
269C22×C53—21no—
270C22×C53—21no—
271C22×C53—22no—
272C22×C53—22no—
273C22×C53—22no—
274C22×C53—42no—
275C22×C53—42no—
276C22×C53—42no—
277C22×C53—42no—
278C22×C53—41no—
279C22×C53—41no—
280C22×C53#38014yes—
281C22×C53C22×C22⋊C411yescharacteristic
282C22×C53C22×C4⋊C411yescharacteristic
283C22×C53C2×#11711yescharacteristic
284C2×D53—22no—
285C2×D53—22no—
286C2×D53—22no—
287C2×D53—22no—
288C2×D53—22no—
289C2×D53—48no—
290C2×D53—42no—
291C2×D53—42no—
292C2×D53—48no—
293C2×D53—41no—
294C2×D53—41no—
295C2×D53—42no—
296C2×D53—42no—
297C2×D53—41no—
298C2×D53—41no—
299C2×D53—42no—
300C2×D53—88no—
301C2×D53#38014yes—
302C2×D53C2×#12112yes—
303C2×D53C2×C22⋊Q812yes—
304#6157—88no—
305C4×C53—22no—
306C4×C53—22no—
307C4×C53—22no—
308C4×C53—41no—
309C4×C53—41no—
310Dic53—22no—
311Dic53—22no—
312Dic53—22no—
313Dic53—41no—
314Dic53—41no—
315C2×#518—531no—
316C22×#99—532no—
317#2315—532no—
318#1799—532no—
319#1799—532no—
320#1799—532no—
321#1799—532no—
322C2×#342—532no—
323C2×C53—21no—
324C2×C53—21no—
325C2×C53—24no—
326C2×C53—42no—
327C2×C53C2×#38011yescharacteristic
328C2×C53#288311yescharacteristic
329C2×C53#288311yescharacteristic
330D53—22no—
331D53—24no—
332D53—22no—
333D53—42no—
334C23×D4—531no—
335#518—1068no—
336C2×C4×D4—531no—
337C2×C4×D4—531no—
338C2×C4×D4—531no—
339C2×C4×D4—531no—
340C2×C4×D4—531no—
341C2×C4×D4—531no—
342C2×C4×D4—531no—
343C2×C4×D4#615711yescharacteristic, complemented
344C2×#117—531no—
345C2×#117—531no—
346C22×C22⋊C4—531no—
347C22×C22⋊C4—531no—
348C22×C22⋊C4—531no—
349C22×C22⋊C4—531no—
350C22×C22⋊C4—534no—
351C2×#99—534no—
352C2×#99—534no—
353C2×#99—1068no—
354#385—1068no—
355#380—534no—
356#380—534no—
357#342—534no—
358#342—534no—
359C53#2063311yescharacteristic, complemented, Sylow
360C25—531no—
361C25—531no—
362C22×D4—531no—
363C22×D4—531no—
364C22×D4—534no—
365C22×D4—534no—
366C22×D4—531no—
367C22×D4—1064no—
368C22×D4—1064no—
369C22×D4C2×#615711yescharacteristic
370C23×C4—531no—
371C23×C4—531no—
372C23×C4—531no—
373C23×C4—531no—
374C23×C4—531no—
375C23×C4—531no—
376C23×C4—531no—
377C23×C4—1064no—
378C23×C4C2×#615711yescharacteristic
379C23×C4C2×#615711yescharacteristic, complemented
380C4×D4—24no—
381C4×D4—1064no—
382C4×D4—1064no—
383C4×D4—1064no—
384C4×D4—1064no—
385C4×D4—1064no—
386C4×D4—1064no—
387C4×D4—1064no—
388#117—1064no—
389#117—1064no—
390C2×C4⋊C4—531no—
391C2×C4⋊C4—531no—
392C2×C4⋊C4—531no—
393C2×C4⋊C4—1064no—
394C2×C4⋊C4—1064no—
395C2×C4⋊C4C2×#615711yescharacteristic, complemented
396C2×C22⋊C4—531no—
397C2×C22⋊C4—531no—
398C2×C22⋊C4—531no—
399C2×C22⋊C4—538no—
400C2×C22⋊C4—531no—
401C2×C22⋊C4—531no—
402C2×C22⋊C4—531no—
403C2×C22⋊C4—531no—
404C2×C22⋊C4—531no—
405C2×C22⋊C4—531no—
406C2×C22⋊C4—531no—
407C2×C22⋊C4—538no—
408C2×C22⋊C4—1064no—
409C2×C22⋊C4—1064no—
410C2×C22⋊C4—1064no—
411C2×C22⋊C4—1068no—
412C2×C22⋊C4—1068no—
413C2×C22⋊C4—1064no—
414C2×C22⋊C4C2×#615711yescharacteristic
415C2×C22⋊C4C2×#615711yescharacteristic, complemented
416C2×C42—531no—
417C2×C42—531no—
418C2×C42—531no—
419C2×C42C2×#615711yescharacteristic, complemented
420#99—1068no—
421#99—1068no—
422#95—1064no—
423#95—1064no—
424C24—532no—
425C24—532no—
426C24—534no—
427C24—534no—
428C24—531no—
429C24—531no—
430C24—534no—
431C24—531no—
432C24—1062no—
433C24—1062no—
434C24—4241no—
435C24—4241no—
436C24—4241no—
437C24—4241no—
438C24C22×#615711yescharacteristic
439C24C22×#615711yescharacteristic
440C2×D4—22no—
441C2×D4—22no—
442C2×D4—534no—
443C2×D4—532no—
444C2×D4—532no—
445C2×D4—532no—
446C2×D4—532no—
447C2×D4—1062no—
448C2×D4—1062no—
449C2×D4—10616no—
450C2×D4—10616no—
451C2×D4—1062no—
452C2×D4—1062no—
453C2×D4—1062no—
454C2×D4—1062no—
455C2×D4C4⋊#615714yescomplemented
456C22×C4—22no—
457C22×C4—22no—
458C22×C4—81no—
459C22×C4—81no—
460C22×C4—531no—
461C22×C4—531no—
462C22×C4—531no—
463C22×C4—534no—
464C22×C4—534no—
465C22×C4—534no—
466C22×C4—531no—
467C22×C4—531no—
468C22×C4—1062no—
469C22×C4—1062no—
470C22×C4—1062no—
471C22×C4—1062no—
472C22×C4—1062no—
473C22×C4—1062no—
474C22×C4—1068no—
475C22×C4—4241no—
476C22×C4—4241no—
477C22×C4—4241no—
478C22×C4—4241no—
479C22×C4—4244no—
480C22×C4—4244no—
481C22×C4—4241no—
482C22×C4—4241no—
483C22×C4—2128no—
484C22×C4C22×#615711yescharacteristic
485C22×C4C22×#615711yescharacteristic
486C22×C4C22×#615711yescharacteristic
487C22×C4C22×#615711yescharacteristic
488C22×C4C22×#615711yescharacteristic
489C4⋊C4—22no—
490C4⋊C4—1062no—
491C4⋊C4—1062no—
492C4⋊C4—1062no—
493C4⋊C4—4244no—
494C4⋊C4—4244no—
495C22⋊C4—22no—
496C22⋊C4—534no—
497C22⋊C4—534no—
498C22⋊C4—534no—
499C22⋊C4—1062no—
500C22⋊C4—1062no—
501C22⋊C4—1062no—
502C22⋊C4—1062no—
503C22⋊C4—1062no—
504C22⋊C4—1062no—
505C22⋊C4—1062no—
506C22⋊C4—1068no—
507C22⋊C4—1068no—
508C22⋊C4—2128no—
509C22⋊C4C22⋊#615714yes—
510C42—22no—
511C42—1062no—
512C42—1062no—
513C42—1062no—
514C23—21no—
515C23—21no—
516C23—41no—
517C23—41no—
518C23—41no—
519C23—41no—
520C23—532no—
521C23—532no—
522C23—538no—
523C23—538no—
524C23—532no—
525C23—531no—
526C23—531no—
527C23—532no—
528C23—532no—
529C23—532no—
530C23—1068no—
531C23—1061no—
532C23—1061no—
533C23—4242no—
534C23—4242no—
535C23—4244no—
536C23—4244no—
537C23—4242no—
538C23—4242no—
539C23—4244no—
540C23—2121no—
541C23—2124no—
542C23—2121no—
543C23—2124no—
544C23—2121no—
545C23—2121no—
546C23—2121no—
547C23—2121no—
548C23C2×C22⋊#615712yes—
549C23C2×C4⋊#615712yes—
550C23C2×C4⋊#615712yes—
551C23#1313329612yes—
552C23C23×#615711yescharacteristic
553D4—28no—
554D4—1064no—
555D4—1064no—
556D4—1064no—
557D4—1064no—
558D4—1068no—
559C2×C4—21no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—41no—
564C2×C4—41no—
565C2×C4—41no—
566C2×C4—41no—
567C2×C4—82no—
568C2×C4—82no—
569C2×C4—532no—
570C2×C4—532no—
571C2×C4—532no—
572C2×C4—1068no—
573C2×C4—1061no—
574C2×C4—1061no—
575C2×C4—1061no—
576C2×C4—1061no—
577C2×C4—4242no—
578C2×C4—4242no—
579C2×C4—4242no—
580C2×C4—42416no—
581C2×C4—4242no—
582C2×C4—2124no—
583C2×C4—2124no—
584C2×C4—2121no—
585C2×C4—2121no—
586C2×C4—2128no—
587C2×C4—2128no—
588C2×C4—2128no—
589C2×C4—2121no—
590C2×C4—2121no—
591C2×C4C2×C22⋊#615712yes—
592C2×C4C2×C22⋊#615712yes—
593C2×C4C2×C4⋊#615712yes—
594C22—21no—
595C22—22no—
596C22—21no—
597C22—22no—
598C22—22no—
599C22—42no—
600C22—42no—
601C22—42no—
602C22—42no—
603C22—41no—
604C22—41no—
605C22—534no—
606C22—532no—
607C22—532no—
608C22—1062no—
609C22—1062no—
610C22—1062no—
611C22—1062no—
612C22—1062no—
613C22—4248no—
614C22—2128no—
615C22—2122no—
616C22—2122no—
617C22—2121no—
618C22—2121no—
619C22—2122no—
620C22—2128no—
621C22—2122no—
622C22—2122no—
623C22—2121no—
624C22—2121no—
625C22#1681156514yes—
626C22C22×C22⋊#615711yescharacteristic
627C22C22×C4⋊#615711yescharacteristic
628C22C2×#1313329611yescharacteristic
629C4—22no—
630C4—22no—
631C4—22no—
632C4—41no—
633C4—41no—
634C4—1062no—
635C4—1062no—
636C4—1062no—
637C4—4248no—
638C4—2121no—
639C4—2121no—
640C2—21no—
641C2—21no—
642C2—24no—
643C2—42no—
644C2—1062no—
645C2—1064no—
646C2—1062no—
647C2—2122no—
648C2C2×#1681156511yescharacteristic
649C2#2029131411yescharacteristic
650C2#2029131511yescharacteristic
subgroupG/H
centerC22C22×C22⋊#6157
commutatorC23×C53C23×C4
FrattiniC23C23×#6157
FittingC2×C4×D4×C53C4
socleC22×C53C22×C22⋊C4
prime radicalC22×D4×D53C22
cosocleC23×D53C24
hypercenterC2×C4×D4#6157
hypocenterC53#20633
LCSG ▷ C23×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. #20291194C2semidirect
3. #20290578C2semidirect
4. #20290577C2semidirect
5. #20290576C2semidirect
6. #20290575C2semidirect
7. C2×#16811501C2semidirect
14. #16811565C22semidirect
15. #16811565C22semidirect
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
55. C23×Dic53C2×C4semidirect
57. C22×C4×D53C2×C4semidirect
58. C22×C4×D53C2×C4semidirect
70. C2×C4⋊Dic53C2×C4semidirect
71. C2×#11719919C2×C4semidirect
72. C2×#11719919C2×C4semidirect
74. C2×C22⋊Dic53C2×C4semidirect
81. C2×#11719921C2×C4semidirect
82. C2×#11719921C2×C4semidirect
87. C2×C4×Dic53C2×C4semidirect
88. C2×C4×Dic53C2×C4semidirect
89. C2×C4×Dic53C2×C4semidirect
92. C23×C4×C53C2×C4semidirect
94. C2×C4⋊C4×C53C2×C4semidirect
96. C2×C22⋊C4×C53C2×C4semidirect
97. C2×C42×C53C2×C4semidirect
106. C23×D53C4⋊C4semidirect
120. C2×D4×C53C4⋊C4semidirect
134. C2×C22⋊D53C4⋊C4semidirect
160. C2×C4×D53C4⋊C4semidirect
343. C2×C4×D4#6157semidirect
359. C53#20633semidirect
379. C23×C4C2×#6157semidirect
395. C2×C4⋊C4C2×#6157semidirect
415. C2×C22⋊C4C2×#6157semidirect
419. C2×C42C2×#6157semidirect
455. 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
#202911942
#202905782
#202905772
#202905762
#202905752
C2×#168115012
C2×#168110952
#2063353