Finite Group Databasebeta

← #27289336

Subgroups — #27289336

Order 13568.

1414 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13133174C211yesmaximal, characteristic
2C2×#16810586C211yesmaximal, characteristic
3#20277094C211yesmaximal, characteristic
4C2×#16810384C211yesmaximal, characteristic, complemented
5C2×#16810383C211yesmaximal, characteristic, complemented
6C2×#16810382C211yesmaximal, characteristic, complemented
7C2×C53×#339C211yesmaximal, characteristic, complemented
8C23×C4×D53C2211yescharacteristic
9C22×#11719917C2211yescharacteristic
10C2×#13133176C2211yescharacteristic
11C2×#13133174—22no—
12C2×#13133174—22no—
13C2×#13133174—22no—
14C2×#13133174—22no—
15C2×#13133174—22no—
16C2×#13133174C414yescomplemented
17C22×#11719919—21no—
18C22×#11719921—21no—
19C22×C4×Dic53C2211yescharacteristic
20#16810586—22no—
21#16810586C414yescomplemented
22C2×#13133129—22no—
23#16810484—21no—
24#16810480C414yescomplemented
25#16810478—21no—
26C4×#11719921—22no—
27#16810384—22no—
28#16810383—22no—
29#16810382—22no—
30C22×M16×C53C2211yescharacteristic
31C2×#117×C53C2211yescharacteristic
32C22×C42×C53C2211yescharacteristic
33C53×#339—22no—
34C24×D53D411yescharacteristic
35C23×Dic53—21no—
36C23×Dic53D411yescharacteristic
37C22×C4×D53—21no—
38C22×C4×D53—21no—
39C22×C4×D53—21no—
40C22×C4×D53—21no—
41C22×C4×D53—21no—
42C22×C4×D53—21no—
43C22×C4×D53—21no—
44C22×C4×D53—21no—
45C22×C4×D53C2×C412yes—
46C22×C4×D53C2×C412yes—
47C22×C4×D53C2×C412yes—
48C22×C4×D53D412yes—
49C22×C4×D53D412yes—
50C22×C4×D53Q812yes—
51C2×#11719917—21no—
52C2×#11719917—21no—
53C2×#11719917—21no—
54C2×#11719917—21no—
55C2×#11719917—21no—
56C2×#11719917C2×C412yes—
57C22×#88322—21no—
58#13133176—21no—
59#13133176—21no—
60#13133176—21no—
61#13133176—21no—
62#13133174—28no—
63#13133174—28no—
64#13133174—28no—
65#13133174—28no—
66C2×C4⋊Dic53C2×C412yes—
67C2×#11719919—24no—
68C2×#11719919—24no—
69C2×#11719919—24no—
70C2×C22⋊Dic53—21no—
71C2×#11719921—24no—
72C2×#11719921—24no—
73C2×#11719921—24no—
74C2×#11719921—24no—
75C2×C4×Dic53—21no—
76C2×C4×Dic53—21no—
77C2×C4×Dic53—21no—
78C2×C4×Dic53—21no—
79C2×C4×Dic53C2×C412yes—
80C2×C4×Dic53D412yes—
81C2×C4×Dic53D412yes—
82#13133129—28no—
83#13133124—24no—
84C23×C4×C53—21no—
85C23×C4×C53C2311yescharacteristic
86C2×M16×C53—21no—
87C2×M16×C53—21no—
88C2×M16×C53—21no—
89C2×M16×C53—21no—
90C2×M16×C53—21no—
91C2×M16×C53C2×C412yes—
92C22×C8×C53—21no—
93#117×C53—21no—
94#117×C53—21no—
95#117×C53—21no—
96#117×C53—21no—
97C2×C4⋊C4×C53C2×C412yes—
98C2×C22⋊C4×C53—21no—
99C2×C42×C53—21no—
100C2×C42×C53—21no—
101C2×C42×C53—21no—
102C2×C42×C53—21no—
103C2×C42×C53C2×C412yes—
104C2×C42×C53D412yes—
105C2×C42×C53D412yes—
106C24×C53C2×D411yescharacteristic
107C23×D53—21no—
108C23×D53—21no—
109C23×D53—21no—
110C23×D53—21no—
111C23×D53—44no—
112C23×D53C22⋊C412yes—
113C23×D53C22⋊C412yes—
114C23×D53C4⋊C412yes—
115C22×C4×C53—22no—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—44no—
120C22×C4×C53C22×C411yescharacteristic
121C22×C4×C53C22×C411yescharacteristic
122C22×C4×C53C22×C411yescharacteristic
123C22×C4×C53C2×D411yescharacteristic
124C22×C4×C53C2×D411yescharacteristic
125C22×C4×C53C2×D411yescharacteristic
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53C2×Q811yescharacteristic
130C22×C4×C53#4111yescharacteristic
131C22×C4×C53#4111yescharacteristic
132C22×C4×C53#4111yescharacteristic
133C22×C4×C53#4111yescharacteristic
134C22×Dic53—21no—
135C22×Dic53—21no—
136C22×Dic53—21no—
137C22×Dic53—21no—
138C22×Dic53—22no—
139C22×Dic53—22no—
140C22×Dic53—22no—
141C22×Dic53—22no—
142C22×Dic53—44no—
143C22×Dic53C22⋊C412yes—
144C22×Dic53C22⋊C412yes—
145C22×Dic53C4⋊C412yes—
146C2×C4×D53—22no—
147C2×C4×D53—22no—
148C2×C4×D53—22no—
149C2×C4×D53—22no—
150C2×C4×D53—22no—
151C2×C4×D53—22no—
152C2×C4×D53—22no—
153C2×C4×D53—22no—
154C2×C4×D53—22no—
155C2×C4×D53—22no—
156C2×C4×D53—22no—
157C2×C4×D53—22no—
158C2×C4×D53—22no—
159C2×C4×D53—22no—
160C2×C4×D53—22no—
161C2×C4×D53—22no—
162C2×C4×D53—22no—
163C2×C4×D53—22no—
164C2×C4×D53—22no—
165C2×C4×D53—22no—
166C2×C4×D53—22no—
167C2×C4×D53—22no—
168C2×C4×D53—22no—
169C2×C4×D53—22no—
170C2×C4×D53C4214yes—
171C2×C4×D53C22⋊C414yes—
172C2×C4×D53C4⋊C414yes—
173C2×C4×D53C4⋊C414yes—
174M16×C53—22no—
175M16×C53—22no—
176M16×C53—41no—
177M16×C53—41no—
178C2×C8×C53—22no—
179C2×C8×C53—41no—
180C2×C8×C53—41no—
181C4⋊C4×C53—22no—
182C4⋊C4×C53—22no—
183C22⋊C4×C53—41no—
184C22⋊C4×C53—41no—
185C42×C53—22no—
186C42×C53—22no—
187C42×C53—22no—
188C42×C53—41no—
189C42×C53—41no—
190C42×C53C22⋊C414yescomplemented
191C22⋊Dic53—41no—
192C22⋊Dic53—41no—
193#11719921—216no—
194#11719921—48no—
195C4⋊Dic53—22no—
196C4⋊Dic53—22no—
197#11719919—216no—
198C4×Dic53—22no—
199C4×Dic53—22no—
200C4×Dic53—22no—
201C4×Dic53—41no—
202C4×Dic53—41no—
203C4×Dic53C22⋊C414yescomplemented
204#11719917—22no—
205#11719917—22no—
206#11719917—41no—
207#11719917—41no—
208C2×#88322—22no—
209C2×#88322—41no—
210C2×#88322—41no—
211C23×C53—24no—
212C23×C53C2×C22⋊C411yescharacteristic
213C23×C53C2×C22⋊C411yescharacteristic
214C23×C53C2×C4⋊C411yescharacteristic
215C23×C53#12011yescharacteristic
216C23×C53#12011yescharacteristic
217C23×C53#12311yescharacteristic
218C23×C53#12311yescharacteristic
219C22×D53—22no—
220C22×D53—22no—
221C22×D53—22no—
222C22×D53—22no—
223C22×D53—22no—
224C22×D53—22no—
225C22×D53—24no—
226C22×D53—24no—
227C22×D53—44no—
228C22×D53—44no—
229C22×D53—44no—
230C22×D53—44no—
231C22×D53—44no—
232C22×D53—44no—
233C22×D53#9514yes—
234C2×C4×C53—24no—
235C2×C4×C53—44no—
236C2×C4×C53—44no—
237C2×C4×C53—44no—
238C2×C4×C53—42no—
239C2×C4×C53—44no—
240C2×C4×C53C2×C4211yescharacteristic
241C2×C4×C53C2×C22⋊C411yescharacteristic
242C2×C4×C53C2×C22⋊C411yescharacteristic
243C2×C4×C53C2×C22⋊C411yescharacteristic
244C2×C4×C53C2×C4⋊C411yescharacteristic
245C2×C4×C53C2×C4⋊C411yescharacteristic
246C2×C4×C53#11711yescharacteristic
247C2×C4×C53#11711yescharacteristic
248C2×C4×C53C4×D411yescharacteristic
249C2×C4×C53C4×D411yescharacteristic
250C2×C4×C53C4×D411yescharacteristic
251C2×C4×C53C4×D411yescharacteristic
252C2×C4×C53C4×D411yescharacteristic
253C2×C4×C53C4×D411yescharacteristic
254C2×C4×C53C4×D411yescharacteristic
255C2×C4×C53C4×D411yescharacteristic
256C2×C4×C53#12011yescharacteristic
257C2×C4×C53#12011yescharacteristic
258C2×C4×C53#12111yescharacteristic
259C2×C4×C53#12111yescharacteristic
260C2×C4×C53#12111yescharacteristic
261C2×C4×C53#12111yescharacteristic
262C2×C4×C53C22⋊Q811yescharacteristic
263C2×C4×C53C22⋊Q811yescharacteristic
264C2×C4×C53C22⋊Q811yescharacteristic
265C2×C4×C53C22⋊Q811yescharacteristic
266C2×C4×C53#12311yescharacteristic
267C2×C4×C53#12311yescharacteristic
268C2×Dic53—22no—
269C2×Dic53—22no—
270C2×Dic53—22no—
271C2×Dic53—22no—
272C2×Dic53—22no—
273C2×Dic53—22no—
274C2×Dic53—24no—
275C2×Dic53—44no—
276C2×Dic53—44no—
277C2×Dic53—44no—
278C2×Dic53—42no—
279C2×Dic53—44no—
280C2×Dic53#9514yes—
281C4×D53—24no—
282C4×D53—24no—
283C4×D53—24no—
284C4×D53—24no—
285C4×D53—24no—
286C4×D53—24no—
287C4×D53—24no—
288C4×D53—24no—
289C8×C53—42no—
290#88322—42no—
291#10720—531nomaximal, Sylow
292C22×C53—22no—
293C22×C53—22no—
294C22×C53—22no—
295C22×C53—22no—
296C22×C53—22no—
297C22×C53—22no—
298C22×C53C2×#9511yescharacteristic
299C22×C53#37911yescharacteristic
300C22×C53#38111yescharacteristic
301C22×C53#38511yescharacteristic
302C22×C53#38511yescharacteristic
303C22×C53#38611yescharacteristic
304C22×C53#38611yescharacteristic
305C22×C53#42114yes—
306C2×D53—24no—
307C2×D53—24no—
308C2×D53—24no—
309C2×D53—44no—
310C2×D53—44no—
311C2×D53—44no—
312C2×D53—44no—
313C2×D53—44no—
314C2×D53—44no—
315C4×C53—44no—
316C4×C53—82no—
317C4×C53C4×C22⋊C411yescharacteristic
318C4×C53C4×C22⋊C411yescharacteristic
319C4×C53#38011yescharacteristic
320C4×C53#38011yescharacteristic
321C4×C53#38511yescharacteristic
322C4×C53#38511yescharacteristic
323C4×C53#38611yescharacteristic
324C4×C53#38611yescharacteristic
325Dic53—24no—
326Dic53—44no—
327Dic53—82no—
328C22×#117—531no—
329C2×#407—531no—
330#2280—531no—
331C2×#339—531no—
332C2×#339—531no—
333C2×#339—531no—
334C2×#339D5311yescharacteristic, complemented
335C2×C53—21no—
336C2×C53—21no—
337C2×C53—21no—
338C2×C53—21no—
339C2×C53C22⋊#9511yescharacteristic
340C2×C53#173312yes—
341C2×C53#189712yes—
342C2×C53#189712yes—
343D53—44no—
344C24×C4—531no—
345C22×M16—531no—
346C22×M16C2×D5311yescharacteristic
347C2×#117—534no—
348C2×#117—531no—
349C2×#117—1062no—
350C2×#117—1062no—
351C2×#117—1062no—
352C2×#117—1062no—
353C2×#117—1062no—
354C2×#117C2×D5311yescharacteristic
355C22×C4⋊C4—1061no—
356C22×C22⋊C4—1061no—
357C22×C42—531no—
358C22×C42C2×D5311yescharacteristic
359#407—534no—
360#407—1062no—
361C2×C22⋊C8—1062no—
362#381—1061no—
363#380—534no—
364#379—1061no—
365C4×C22⋊C4—1062no—
366#339—22no—
367#339—1062no—
368#339—1062no—
369#339—1062no—
370C53#1072011yescharacteristic, complemented, Sylow
371C25—531no—
372C23×C4—21no—
373C23×C4—531no—
374C23×C4—532no—
375C23×C4—532no—
376C23×C4—532no—
377C23×C4—532no—
378C23×C4—532no—
379C23×C4—532no—
380C23×C4—1061no—
381C23×C4—1061no—
382C23×C4—1061no—
383C23×C4—1061no—
384C23×C4—1061no—
385C23×C4—1061no—
386C23×C4—1061no—
387C23×C4—1061no—
388C23×C4—1061no—
389C23×C4C22×D5311yescharacteristic
390C2×M16—21no—
391C2×M16—21no—
392C2×M16—21no—
393C2×M16—21no—
394C2×M16—21no—
395C2×M16—532no—
396C2×M16—1061no—
397C2×M16—1061no—
398C2×M16—1061no—
399C2×M16—1061no—
400C2×M16—1061no—
401C2×M16C4×D5312yes—
402C22×C8—21no—
403C22×C8—1061no—
404#117—21no—
405#117—21no—
406#117—21no—
407#117—21no—
408#117—1068no—
409#117—1068no—
410#117—1068no—
411#117—1068no—
412#117—1061no—
413#117—1061no—
414#117—1061no—
415#117—1061no—
416C2×C4⋊C4—532no—
417C2×C4⋊C4—1064no—
418C2×C4⋊C4—1064no—
419C2×C4⋊C4—1064no—
420C2×C4⋊C4C4×D5312yes—
421C2×C22⋊C4—21no—
422C2×C22⋊C4—1064no—
423C2×C22⋊C4—1064no—
424C2×C22⋊C4—1064no—
425C2×C22⋊C4—1064no—
426C2×C22⋊C4—1061no—
427C2×C42—21no—
428C2×C42—21no—
429C2×C42—21no—
430C2×C42—21no—
431C2×C42—532no—
432C2×C42—532no—
433C2×C42—532no—
434C2×C42—1061no—
435C2×C42—1061no—
436C2×C42—1061no—
437C2×C42—1061no—
438C2×C42C4×D5312yes—
439C2×C42D21212yes—
440C2×C42C22⋊D5312yes—
441C22⋊C8—1068no—
442#95—1064no—
443C24—532no—
444C24—532no—
445C24—532no—
446C24—1061no—
447C24—1061no—
448C24—1061no—
449C24—1061no—
450C24—2124no—
451C24D4×D5311yescharacteristic
452C22×C4—22no—
453C22×C4—22no—
454C22×C4—22no—
455C22×C4—22no—
456C22×C4—44no—
457C22×C4—532no—
458C22×C4—532no—
459C22×C4—532no—
460C22×C4—534no—
461C22×C4—534no—
462C22×C4—534no—
463C22×C4—534no—
464C22×C4—1061no—
465C22×C4—1061no—
466C22×C4—1061no—
467C22×C4—1061no—
468C22×C4—1062no—
469C22×C4—1062no—
470C22×C4—1062no—
471C22×C4—1062no—
472C22×C4—1062no—
473C22×C4—1062no—
474C22×C4—1062no—
475C22×C4—1062no—
476C22×C4—1062no—
477C22×C4—1062no—
478C22×C4—1062no—
479C22×C4—1062no—
480C22×C4—1062no—
481C22×C4—1062no—
482C22×C4—1062no—
483C22×C4—1062no—
484C22×C4—1062no—
485C22×C4—1062no—
486C22×C4—1062no—
487C22×C4—1062no—
488C22×C4—1062no—
489C22×C4—1062no—
490C22×C4—1062no—
491C22×C4—1062no—
492C22×C4—1062no—
493C22×C4—1062no—
494C22×C4—1062no—
495C22×C4—1062no—
496C22×C4—2124no—
497C22×C4C2×C4×D5311yescharacteristic
498C22×C4C2×C4×D5311yescharacteristic
499C22×C4C2×C4×D5311yescharacteristic
500C22×C4C2×D21211yescharacteristic
501C22×C4#1171994511yescharacteristic
502C22×C4#1171994511yescharacteristic
503C22×C4D4×D5311yescharacteristic
504C22×C4D4×D5311yescharacteristic
505C22×C4D4×D5311yescharacteristic
506C22×C4D4×D5311yescharacteristic
507C22×C4#1171994711yescharacteristic
508C22×C4Q8×D5311yescharacteristic
509C22×C4#1171994911yescharacteristic
510C22×C4C2×C22⋊D5311yescharacteristic
511M16—22no—
512M16—22no—
513M16—41no—
514M16—41no—
515M16—1062no—
516M16—1062no—
517M16—2121no—
518M16—2121no—
519C2×C8—22no—
520C2×C8—41no—
521C2×C8—41no—
522C2×C8—1062no—
523C2×C8—2121no—
524C2×C8—2121no—
525C4⋊C4—22no—
526C4⋊C4—22no—
527C4⋊C4—10616no—
528C4⋊C4—1062no—
529C4⋊C4—1062no—
530C22⋊C4—41no—
531C22⋊C4—41no—
532C22⋊C4—10616no—
533C22⋊C4—2128no—
534C22⋊C4—2121no—
535C22⋊C4—2121no—
536C42—22no—
537C42—22no—
538C42—22no—
539C42—41no—
540C42—41no—
541C42—534no—
542C42—1062no—
543C42—1062no—
544C42—1062no—
545C42—2121no—
546C42—2121no—
547C42#1171992114yescomplemented
548C23—24no—
549C23—534no—
550C23—1062no—
551C23—1062no—
552C23—1062no—
553C23—1064no—
554C23—1064no—
555C23—1062no—
556C23—1062no—
557C23—1062no—
558C23—2124no—
559C23—2124no—
560C23—2124no—
561C23—2124no—
562C23—2124no—
563C23—2124no—
564C23C22⋊C4×D5311yescharacteristic
565C23C22⋊C4×D5311yescharacteristic
566C23C4⋊C4×D5311yescharacteristic
567C23#1313318511yescharacteristic
568C23#1313318711yescharacteristic
569C23#1313319911yescharacteristic
570C23#1313320211yescharacteristic
571C2×C4—24no—
572C2×C4—44no—
573C2×C4—44no—
574C2×C4—44no—
575C2×C4—42no—
576C2×C4—44no—
577C2×C4—534no—
578C2×C4—1062no—
579C2×C4—1062no—
580C2×C4—1062no—
581C2×C4—1062no—
582C2×C4—1062no—
583C2×C4—1062no—
584C2×C4—1064no—
585C2×C4—1064no—
586C2×C4—1064no—
587C2×C4—1064no—
588C2×C4—1064no—
589C2×C4—1064no—
590C2×C4—1064no—
591C2×C4—1064no—
592C2×C4—1064no—
593C2×C4—2122no—
594C2×C4—2124no—
595C2×C4—2124no—
596C2×C4—2124no—
597C2×C4—2124no—
598C2×C4C42×D5311yescharacteristic
599C2×C4C22⋊C4×D5311yescharacteristic
600C2×C4C22⋊C4×D5311yescharacteristic
601C2×C4C2×#1171992111yescharacteristic
602C2×C4C4⋊C4×D5311yescharacteristic
603C2×C4C4⋊C4×D5311yescharacteristic
604C2×C4#1313317411yescharacteristic
605C2×C4#1313317511yescharacteristic
606C2×C4#1313317711yescharacteristic
607C2×C4#1313317711yescharacteristic
608C2×C4C4×C22⋊D5311yescharacteristic
609C2×C4C4×C22⋊D5311yescharacteristic
610C2×C4#1313317911yescharacteristic
611C2×C4#1313317911yescharacteristic
612C2×C4C4×D21211yescharacteristic
613C2×C4C4×D21211yescharacteristic
614C2×C4#1313318511yescharacteristic
615C2×C4#1313318711yescharacteristic
616C2×C4#1313318811yescharacteristic
617C2×C4#1313319011yescharacteristic
618C2×C4#1313319011yescharacteristic
619C2×C4#1313319211yescharacteristic
620C2×C4#1313319311yescharacteristic
621C2×C4#1313319511yescharacteristic
622C2×C4#1313319511yescharacteristic
623C2×C4#1313319711yescharacteristic
624C2×C4#1313319911yescharacteristic
625C2×C4#1313320211yescharacteristic
626C8—42no—
627C8—2122no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—22no—
632C22—22no—
633C22—22no—
634C22—1064no—
635C22—1064no—
636C22—1064no—
637C22—2124no—
638C22—2124no—
639C22—2124no—
640C22—2124no—
641C22—2124no—
642C22—2124no—
643C22#95×D5311yescharacteristic
644C22#1681047811yescharacteristic
645C22#1681048511yescharacteristic
646C22#1681049411yescharacteristic
647C22#1681049711yescharacteristic
648C22#1681049911yescharacteristic
649C22#1681050111yescharacteristic
650C22#1681063814yes—
651C4—44no—
652C4—82no—
653C4—1064no—
654C4—4242no—
655C4—2124no—
656C4#1681047011yescharacteristic
657C4C4×#1171992111yescharacteristic
658C4#1681048011yescharacteristic
659C4#1681048211yescharacteristic
660C4#1681049411yescharacteristic
661C4#1681049711yescharacteristic
662C4#1681049911yescharacteristic
663C4#1681050111yescharacteristic
664C2—21no—
665C2—21no—
666C2—21no—
667C2—21no—
668C2—2124no—
669C2#2027293611yescharacteristic
670C2#2027442312yes—
671C2#2027516012yes—
672C2#2027516212yes—
subgroupG/H
centerC22×C4D4×D53
commutatorC2×C4×C53C2×C42
FrattiniC23×C4C22×D53
FittingC2×C53×#339C2
socleC23×C53C2×C22⋊C4
prime radicalC24×D53D4
cosocleC23×C4×C53C23
hypercenterC2×#339D53
hypocenterC53#10720
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C22×C4 ◁ C23×C4 ◁ C2×#339

Complements

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

NH ≅ G/Ntype
4. C2×#16810384C2semidirect
5. C2×#16810383C2semidirect
6. C2×#16810382C2semidirect
7. C2×C53×#339C2semidirect
16. C2×#13133174C4semidirect
21. #16810586C4semidirect
24. #16810480C4semidirect
190. C42×C53C22⋊C4semidirect
203. C4×Dic53C22⋊C4semidirect
334. C2×#339D53semidirect
370. C53#10720semidirect
547. C42#11719921semidirect

Maximal subgroups

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

subgroupindex
C22×#131331742
C2×#168105862
#202770942
C2×#168103842
C2×#168103832
C2×#168103822
C2×C53×#3392
#1072053