Finite Group Databasebeta

← #27320803

Subgroups — #27320803

Order 13568.

1934 conjugacy classes of subgroups.

The subgroup lattice has 614 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133216C211yesmaximal, characteristic
2C2×#16810544C212yesmaximal
3C2×#16810543C212yesmaximal
4C2×#16810520C211yesmaximal, characteristic
5C2×#16810492C212yesmaximal
6#20272980C211yesmaximal, characteristic
7#20272979C211yesmaximal, characteristic
8#20272978C211yesmaximal, characteristic
9#20272943C212yesmaximal
10#95×Dic53C211yesmaximal, characteristic
11C53×#1444C211yesmaximal, characteristic
12C23×Dic106C2212yes—
13C2×#13133216—28no—
14C2×#13133216—24no—
15C22×C4⋊Dic53C2211yescharacteristic
16C22×C4⋊Dic53C2211yescharacteristic
17C22×C4⋊Dic53C2212yes—
18C22×#11719919C2211yescharacteristic
19C22×#11719919C2211yescharacteristic
20C22×#11719919C2212yes—
21C22×#11719919C2212yes—
22C22×#11719919C2212yes—
23C22×#11719919C2211yescharacteristic
24C22×C4×Dic53C2211yescharacteristic
25C22×C4×Dic53C2212yes—
26#16810544—28no—
27#16810543—28no—
28#16810520—24no—
29#16810492—28no—
30C2×#13133124C2212yes—
31C2×#13133124C2212yes—
32C2×#13133124C2212yes—
33C2×#13133124C2211yescharacteristic
34C2×#13133124C2212yes—
35C2×#13133124C2211yescharacteristic
36C22×C4⋊C4×C53C2211yescharacteristic
37C22×C4⋊C4×C53C2212yes—
38C2×#95×C53C2212yes—
39C2×#95×C53C2211yescharacteristic
40C2×#95×C53C2211yescharacteristic
41C22×Dic106—24no—
42C22×Dic106—24no—
43C22×Dic106—24no—
44C22×Dic106—24no—
45C22×Dic106—24no—
46C22×Dic106—42no—
47C22×Dic106—42no—
48C23×Dic53C2311yescharacteristic
49C23×Dic53C2312yes—
50C23×Dic53C2311yescharacteristic
51C23×Dic53C2311yescharacteristic
52C23×Dic53C2312yes—
53C23×Dic53C2311yescharacteristic
54#13133216—232no—
55C2×C4⋊Dic53—22no—
56C2×C4⋊Dic53—24no—
57C2×C4⋊Dic53—24no—
58C2×C4⋊Dic53—22no—
59C2×C4⋊Dic53—24no—
60C2×C4⋊Dic53—24no—
61C2×C4⋊Dic53—42no—
62C2×#11719919—24no—
63C2×#11719919—22no—
64C2×#11719919—24no—
65C2×#11719919—24no—
66C2×#11719919—22no—
67C2×#11719919—22no—
68C2×#11719919—24no—
69C2×#11719919—24no—
70C2×#11719919—24no—
71C2×#11719919—24no—
72C2×#11719919—24no—
73C2×#11719919—22no—
74C2×#11719919—42no—
75C2×#11719919—42no—
76C2×#11719919D418yes—
77C2×#11719919Q814yescomplemented
78C2×C4×Dic53—24no—
79C2×C4×Dic53—24no—
80C2×C4×Dic53—24no—
81C2×C4×Dic53D414yes—
82C2×C4×Dic53Q818yescomplemented
83#13133124—22no—
84#13133124—24no—
85#13133124—24no—
86#13133124—24no—
87#13133124D418yes—
88#13133124Q814yescomplemented
89C23×C4×C53C2311yescharacteristic
90C23×C4×C53C2312yes—
91C23×C4×C53C2311yescharacteristic
92C23×C4×C53C2311yescharacteristic
93C23×C4×C53C2312yes—
94C2×C4⋊C4×C53—24no—
95C2×C4⋊C4×C53—22no—
96C2×C4⋊C4×C53—24no—
97C2×C4⋊C4×C53—24no—
98C2×C4⋊C4×C53—42no—
99#95×C53—22no—
100#95×C53—24no—
101#95×C53Q814yescomplemented
102C24×C53C2411yescharacteristic
103C22×C4×C53—22no—
104C22×C4×C53—22no—
105C22×C4×C53—22no—
106C22×C4×C53—22no—
107C22×C4×C53—22no—
108C22×C4×C53—22no—
109C22×C4×C53—22no—
110C22×C4×C53—21no—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—22no—
114C22×C4×C53—22no—
115C22×C4×C53—21no—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—21no—
120C22×C4×C53—22no—
121C22×C4×C53—22no—
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53C2×D414yes—
125C22×C4×C53C2×D412yes—
126C22×C4×C53C2×D414yes—
127C22×C4×C53C2×Q812yes—
128C22×C4×C53C2×Q812yes—
129C22×C4×C53C2×Q812yes—
130C22×C4×C53C2×Q814yes—
131C22×C4×C53C2×Q814yes—
132C22×Dic53—22no—
133C22×Dic53—22no—
134C22×Dic53—22no—
135C22×Dic53—22no—
136C22×Dic53—21no—
137C22×Dic53—21no—
138C22×Dic53—22no—
139C22×Dic53—22no—
140C22×Dic53—21no—
141C22×Dic53—22no—
142C22×Dic53—22no—
143C22×Dic53—22no—
144C22×Dic53—22no—
145C22×Dic53—21no—
146C22×Dic53—22no—
147C22×Dic53—22no—
148C22×Dic53—22no—
149C22×Dic53—21no—
150C22×Dic53—22no—
151C22×Dic53—22no—
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53C2×D414yes—
155C22×Dic53C2×D414yes—
156C22×Dic53C2×D412yes—
157C22×Dic53C2×D414yes—
158C22×Dic53C2×D414yes—
159C22×Dic53C2×D412yes—
160C22×Dic53C2×Q812yes—
161C22×Dic53C2×Q814yes—
162C22×Dic53C2×Q814yes—
163C22×Dic53C2×Q814yes—
164C22×Dic53C2×Q812yes—
165C22×Dic53#4112yes—
166C2×Dic106—216no—
167C2×Dic106—48no—
168C2×Dic106—48no—
169C2×Dic106—48no—
170C2×Dic106—48no—
171C2×Dic106—48no—
172C2×Dic106—48no—
173C4⋊C4×C53—28no—
174C4⋊C4×C53—48no—
175C4⋊Dic53—28no—
176C4⋊Dic53—48no—
177C4⋊Dic53—44no—
178#11719919—216no—
179#11719919—28no—
180#11719919—48no—
181#11719919—48no—
182#11719919—44no—
183#11719919—44no—
184C4×Dic53—28no—
185C4×Dic53—216no—
186C23×C53C22×D411yescharacteristic
187C23×C53C22×D412yes—
188C23×C53C22×D412yes—
189C23×C53C22×Q812yes—
190C23×C53C22×Q811yescharacteristic
191C23×C53C22×Q811yescharacteristic
192C23×C53C22×Q811yescharacteristic
193C23×C53C2×#4111yescharacteristic
194C23×C53H(5,𝔽2)11yescharacteristic
195C23×C53H(5,𝔽2)11yescharacteristic
196C23×C532−1+412yes—
197C2×C4×C53—22no—
198C2×C4×C53—24no—
199C2×C4×C53—22no—
200C2×C4×C53—24no—
201C2×C4×C53—24no—
202C2×C4×C53—24no—
203C2×C4×C53—24no—
204C2×C4×C53—24no—
205C2×C4×C53—24no—
206C2×C4×C53—24no—
207C2×C4×C53—24no—
208C2×C4×C53—22no—
209C2×C4×C53—24no—
210C2×C4×C53—24no—
211C2×C4×C53—24no—
212C2×C4×C53—42no—
213C2×C4×C53—41no—
214C2×C4×C53—42no—
215C2×C4×C53—41no—
216C2×C4×C53—42no—
217C2×C4×C53—42no—
218C2×C4×C53—42no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53—42no—
223C2×C4×C53—42no—
224C2×C4×C53C4⋊Q814yes—
225C2×C4×C53C4⋊Q818yescomplemented
226C2×Dic53—22no—
227C2×Dic53—24no—
228C2×Dic53—22no—
229C2×Dic53—24no—
230C2×Dic53—24no—
231C2×Dic53—24no—
232C2×Dic53—24no—
233C2×Dic53—24no—
234C2×Dic53—24no—
235C2×Dic53—24no—
236C2×Dic53—24no—
237C2×Dic53—22no—
238C2×Dic53—24no—
239C2×Dic53—24no—
240C2×Dic53—24no—
241C2×Dic53—42no—
242C2×Dic53—41no—
243C2×Dic53—42no—
244C2×Dic53—41no—
245C2×Dic53—42no—
246C2×Dic53—42no—
247C2×Dic53—42no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
251C2×Dic53—42no—
252C2×Dic53—42no—
253C2×Dic53#12014yes—
254C2×Dic53C22⋊Q818yes—
255C2×Dic53C22⋊Q814yes—
256C2×Dic53#12714yes—
257C2×Dic53C4⋊Q814yes—
258C2×Dic53C4⋊Q818yes—
259C2×Dic53C4⋊Q818yescomplemented
260Dic106—432no—
261#15143—531nomaximal, Sylow
262C22×C53C2×#12011yescharacteristic
263C22×C53C2×C22⋊Q811yescharacteristic
264C22×C53C2×C22⋊Q812yes—
265C22×C53C2×#12711yescharacteristic
266C22×C53C2×C4⋊Q811yescharacteristic
267C22×C53C2×C4⋊Q812yes—
268C22×C53C2×C4⋊Q812yes—
269C22×C53C2×C4⋊Q811yescharacteristic
270C22×C53C2×C4⋊Q812yes—
271C22×C53#53511yescharacteristic
272C22×C53#53611yescharacteristic
273C22×C53#54412yes—
274C22×C53#54412yes—
275C22×C53D4212yes—
276C22×C53D4212yes—
277C22×C53#54712yes—
278C22×C53D4×Q812yes—
279C22×C53D4×Q812yes—
280C22×C53D4×Q812yes—
281C22×C53#55411yescharacteristic
282C22×C53#55411yescharacteristic
283C22×C53Q8211yescharacteristic
284C22×C53Q8211yescharacteristic
285C4×C53—24no—
286C4×C53—28no—
287C4×C53—44no—
288C4×C53—42no—
289C4×C53—81no—
290Dic53—24no—
291Dic53—28no—
292Dic53—44no—
293Dic53—42no—
294Dic53—81no—
295Dic53#39518yescomplemented
296C22×C4⋊Q8—531no—
297C2×#395—532no—
298C2×#395—532no—
299C2×#389—531no—
300C2×#384—532no—
301#1444—531no—
302#1444—531no—
303#1444—531no—
304#1444D5311yescharacteristic
305C4⋊#95—532no—
306C4×#95—531no—
307C2×C53C2×#39511yescharacteristic
308C2×C53#243311yescharacteristic
309C2×C53#243411yescharacteristic
310C2×C53#244012yes—
311C2×C53#244012yes—
312C2×C53#254211yescharacteristic
313C2×C53#255112yes—
314C2×C53#255112yes—
315C2×C53#256711yescharacteristic
316C2×C53#266011yescharacteristic
317C2×C53#286811yescharacteristic
318C23×Q8—532no—
319C2×C4⋊Q8—1064no—
320C2×C4⋊Q8—1068no—
321C22×C4⋊C4—532no—
322C22×C4⋊C4—531no—
323C22×C4⋊C4—531no—
324C22×C4⋊C4—531no—
325C22×C4⋊C4—531no—
326C22×C4⋊C4—531no—
327C22×C4⋊C4—532no—
328C22×C4⋊C4—532no—
329C22×C4⋊C4—532no—
330C22×C4⋊C4C2×D5311yescharacteristic
331C22×C4⋊C4C2×D5312yes—
332C22×C42—531no—
333C22×C42—532no—
334#395—1068no—
335#395—1068no—
336#389—1064no—
337#384—1068no—
338C2×#95—532no—
339C2×#95—531no—
340C2×#95—532no—
341C2×#95—532no—
342C2×#95—532no—
343C2×#95—531no—
344C2×#95C2×D5312yes—
345C2×#95C2×D5311yescharacteristic
346C2×#95C2×D5311yescharacteristic
347C53#1514311yescharacteristic, complemented, Sylow
348C22×Q8—1064no—
349C22×Q8—1064no—
350C22×Q8—1064no—
351C22×Q8—1064no—
352C22×Q8—1064no—
353C22×Q8—2122no—
354C22×Q8—2122no—
355C23×C4—531no—
356C23×C4—531no—
357C23×C4—532no—
358C23×C4—532no—
359C23×C4—531no—
360C23×C4—531no—
361C23×C4C22×D5311yescharacteristic
362C23×C4C22×D5312yes—
363C23×C4C22×D5311yescharacteristic
364C23×C4C22×D5311yescharacteristic
365C23×C4C22×D5312yes—
366C4⋊Q8—10632no—
367C2×C4⋊C4—24no—
368C2×C4⋊C4—22no—
369C2×C4⋊C4—24no—
370C2×C4⋊C4—24no—
371C2×C4⋊C4—42no—
372C2×C4⋊C4—538no—
373C2×C4⋊C4—534no—
374C2×C4⋊C4—1064no—
375C2×C4⋊C4—1064no—
376C2×C4⋊C4—1064no—
377C2×C4⋊C4—1062no—
378C2×C4⋊C4—1062no—
379C2×C4⋊C4—1062no—
380C2×C4⋊C4—1064no—
381C2×C4⋊C4—1064no—
382C2×C4⋊C4—1062no—
383C2×C4⋊C4—1062no—
384C2×C4⋊C4—1064no—
385C2×C4⋊C4—1064no—
386C2×C4⋊C4—1064no—
387C2×C4⋊C4—1064no—
388C2×C4⋊C4—1062no—
389C2×C4⋊C4—1064no—
390C2×C4⋊C4—1064no—
391C2×C4⋊C4—1064no—
392C2×C4⋊C4—2122no—
393C2×C4⋊C4—2122no—
394C2×C4⋊C4—2122no—
395C2×C42—534no—
396C2×C42—538no—
397C2×C42—1064no—
398C2×C42—1064no—
399C2×C42—1064no—
400#95—22no—
401#95—24no—
402#95—538no—
403#95—534no—
404#95—1064no—
405#95—1064no—
406#95—1062no—
407#95—1064no—
408#95Dic10614yescomplemented
409C24C23×D5311yescharacteristic
410C2×Q8—10616no—
411C2×Q8—2128no—
412C2×Q8—2128no—
413C2×Q8—2128no—
414C2×Q8—2128no—
415C2×Q8—2128no—
416C2×Q8—2128no—
417C22×C4—22no—
418C22×C4—22no—
419C22×C4—22no—
420C22×C4—22no—
421C22×C4—22no—
422C22×C4—22no—
423C22×C4—22no—
424C22×C4—21no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—22no—
429C22×C4—21no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—21no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—21no—
437C22×C4—21no—
438C22×C4—532no—
439C22×C4—534no—
440C22×C4—534no—
441C22×C4—532no—
442C22×C4—532no—
443C22×C4—534no—
444C22×C4—534no—
445C22×C4—532no—
446C22×C4—534no—
447C22×C4—534no—
448C22×C4—534no—
449C22×C4—532no—
450C22×C4—1062no—
451C22×C4—1062no—
452C22×C4—1062no—
453C22×C4—1061no—
454C22×C4—1062no—
455C22×C4—1061no—
456C22×C4—1062no—
457C22×C4—1062no—
458C22×C4—1061no—
459C22×C4—1062no—
460C22×C4—1062no—
461C22×C4—1062no—
462C22×C4—1062no—
463C22×C4—1061no—
464C22×C4—1062no—
465C22×C4—1062no—
466C22×C4—1062no—
467C22×C4—1061no—
468C22×C4—1062no—
469C22×C4—1062no—
470C22×C4—1061no—
471C22×C4—1061no—
472C22×C4C2×Dic10612yes—
473C22×C4C2×Dic10614yes—
474C22×C4D4×D5314yes—
475C22×C4D4×D5312yes—
476C22×C4D4×D5314yes—
477C22×C4Q8×D5312yes—
478C22×C4Q8×D5312yes—
479C22×C4Q8×D5314yes—
480C4⋊C4—28no—
481C4⋊C4—48no—
482C4⋊C4—10616no—
483C4⋊C4—1068no—
484C4⋊C4—1068no—
485C4⋊C4—2128no—
486C4⋊C4—2128no—
487C4⋊C4—2124no—
488C4⋊C4—2124no—
489C4⋊C4—2124no—
490C4⋊C4—2128no—
491C42—1068no—
492C42—10616no—
493C23C2×D4×D5312yes—
494C23C2×D4×D5312yes—
495C23C2×D4×D5311yescharacteristic
496C23C2×Q8×D5311yescharacteristic
497C23C2×Q8×D5311yescharacteristic
498C23C2×Q8×D5312yes—
499C23C22×Dic10611yescharacteristic
500C23#41×D5311yescharacteristic
501C23#1313326211yescharacteristic
502C23#1313326311yescharacteristic
503C23#1313326412yes—
504Q8—21232no—
505C2×C4—24no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—24no—
514C2×C4—24no—
515C2×C4—24no—
516C2×C4—24no—
517C2×C4—22no—
518C2×C4—24no—
519C2×C4—24no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—41no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—534no—
533C2×C4—538no—
534C2×C4—534no—
535C2×C4—534no—
536C2×C4—538no—
537C2×C4—534no—
538C2×C4—538no—
539C2×C4—1064no—
540C2×C4—1064no—
541C2×C4—1064no—
542C2×C4—1062no—
543C2×C4—1064no—
544C2×C4—1062no—
545C2×C4—1064no—
546C2×C4—1064no—
547C2×C4—1064no—
548C2×C4—1064no—
549C2×C4—1064no—
550C2×C4—1064no—
551C2×C4—1062no—
552C2×C4—1064no—
553C2×C4—1064no—
554C2×C4—2121no—
555C2×C4—2121no—
556C2×C4—2121no—
557C2×C4—2122no—
558C2×C4—2122no—
559C2×C4—2122no—
560C2×C4—2121no—
561C2×C4—2122no—
562C2×C4—2122no—
563C2×C4—2121no—
564C2×C4—2122no—
565C2×C4—2122no—
566C2×C4#1313321614yes—
567C2×C4#1313321618yescomplemented
568C22#120×D5311yescharacteristic
569C22C22⋊Q8×D5311yescharacteristic
570C22C22⋊Q8×D5312yes—
571C22#127×D5311yescharacteristic
572C22C4⋊Q8×D5311yescharacteristic
573C22C4⋊Q8×D5312yes—
574C22C4⋊Q8×D5312yes—
575C22C2×#1313321611yescharacteristic
576C22C2×#1313321612yes—
577C22#1681118911yescharacteristic
578C22#1681119511yescharacteristic
579C22#1681125212yes—
580C22#1681125312yes—
581C22#1681125512yes—
582C22#1681125712yes—
583C22#1681127112yes—
584C22#1681128612yes—
585C22#1681128612yes—
586C22D4×Dic10612yes—
587C22#1681132911yescharacteristic
588C22#1681133211yescharacteristic
589C22#1681135311yescharacteristic
590C22Q8×Dic10611yescharacteristic
591C4—24no—
592C4—28no—
593C4—44no—
594C4—42no—
595C4—81no—
596C4—538no—
597C4—1064no—
598C4—1068no—
599C4—4241no—
600C4—2124no—
601C4—2122no—
602C2#395×D5311yescharacteristic
603C2#2027834711yescharacteristic
604C2#2027835711yescharacteristic
605C2#2027841212yes—
606C2#2027841612yes—
607C2#2027966311yescharacteristic
608C2#2027974012yes—
609C2#2027974312yes—
610C2#2027992211yescharacteristic
611C2#2028083811yescharacteristic
612C2#2028311211yescharacteristic
subgroupG/H
center, FrattiniC24C23×D53
commutator, socle, prime radical, cosocleC24×C53C24
FittingC53×#1444C2
hypercenter#1444D53
hypocenterC53#15143
LCSG ▷ C24×C53 ▷ C53
UCSC1 ◁ C24 ◁ #1444

Complements

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

NH ≅ G/Ntype
77. C2×#11719919Q8semidirect
82. C2×C4×Dic53Q8semidirect
88. #13133124Q8semidirect
101. #95×C53Q8semidirect
225. C2×C4×C53C4⋊Q8semidirect
259. C2×Dic53C4⋊Q8semidirect
295. Dic53#395semidirect
347. C53#15143semidirect
408. #95Dic106semidirect
567. C2×C4#13133216semidirect

Maximal subgroups

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

subgroupindex
C22×#131332162
C2×#168105442
C2×#168105432
C2×#168105202
C2×#168104922
#202729802
#202729792
#202729782
#202729432
#95×Dic532
C53×#14442
#1514353