Finite Group Databasebeta

← #27290417

Subgroups — #27290417

Order 13568.

1066 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810586C211yesmaximal, characteristic
2C2×#16810586C211yesmaximal, characteristic
3#20283133C211yesmaximal, characteristic, complemented
4#20277094C211yesmaximal, characteristic
5C2×#16810393C211yesmaximal, characteristic, complemented
6#20274453C211yesmaximal, characteristic, complemented
7C2×C53×#344C211yesmaximal, characteristic, complemented
8C23×C4×D53C2211yescharacteristic
9C22×#11719917C2211yescharacteristic
10C22×#11719917C2211yescharacteristic
11C2×#13133176C2211yescharacteristic
12#16810586—22no—
13#16810586—22no—
14#16810586C414yescomplemented
15#16810586C414yescomplemented
16C2×#13133129—22no—
17C2×#13133129—22no—
18#16810581—22no—
19#16810578—22no—
20M16×Dic53—22no—
21#16810484C412yes—
22#16810480—22no—
23#16810478C412yes—
24C4×#11719921—22no—
25#16810393—22no—
26#16810391—22no—
27C22×M16×C53C2211yescharacteristic
28C22×M16×C53C2211yescharacteristic
29C2×#117×C53C2211yescharacteristic
30C53×#344—22no—
31C24×D53C2×C411yescharacteristic
32C23×Dic53C2×C411yescharacteristic
33C22×C4×D53—21no—
34C22×C4×D53—21no—
35C22×C4×D53—21no—
36C22×C4×D53—21no—
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—
47C2×#11719917—21no—
48C2×#11719917—21no—
49C2×#11719917—21no—
50C2×#11719917—21no—
51C2×#11719917—21no—
52C2×#11719917—21no—
53C2×#11719917—21no—
54C2×#11719917—21no—
55C2×#11719917—21no—
56C2×#11719917C2×C412yes—
57C2×#11719917C2×C412yes—
58C2×#11719917D412yes—
59C22×#88322—21no—
60C22×#88322D412yes—
61#13133176—21no—
62#13133176—21no—
63#13133176—21no—
64#13133176—21no—
65C2×C4⋊Dic53D412yes—
66C2×C22⋊Dic53C2×C412yes—
67C2×#11719921—24no—
68C2×C4×Dic53D412yes—
69#13133129—28no—
70#13133129—28no—
71#13133127—24no—
72C8×Dic53—24no—
73#13133124—24no—
74C23×C4×C53C2311yescharacteristic
75C2×M16×C53—21no—
76C2×M16×C53—21no—
77C2×M16×C53—21no—
78C2×M16×C53—21no—
79C2×M16×C53—21no—
80C2×M16×C53—21no—
81C2×M16×C53—21no—
82C2×M16×C53—21no—
83C2×M16×C53—21no—
84C2×M16×C53C2×C412yes—
85C2×M16×C53C2×C412yes—
86C2×M16×C53D412yes—
87C22×C8×C53—21no—
88C22×C8×C53D412yes—
89#117×C53—21no—
90#117×C53—21no—
91#117×C53—21no—
92#117×C53—21no—
93C2×C4⋊C4×C53—21no—
94C2×C22⋊C4×C53C2×C412yes—
95C2×C42×C53—21no—
96C24×C53C22×C411yescharacteristic
97C23×D53—21no—
98C23×D53—21no—
99C23×D53—21no—
100C23×D53—21no—
101C23×D53—21no—
102C23×D53—21no—
103C23×D53—44no—
104C23×D53C4212yes—
105C22×C4×C53—22no—
106C22×C4×C53C22×C411yescharacteristic
107C22×C4×C53C22×C411yescharacteristic
108C22×C4×C53C2×D411yescharacteristic
109C22×C4×C53C2×D411yescharacteristic
110C22×C4×C53C2×D411yescharacteristic
111C22×C4×C53C2×D411yescharacteristic
112C22×C4×C53C2×D411yescharacteristic
113C22×C4×C53C2×D411yescharacteristic
114C22×C4×C53#4111yescharacteristic
115C22×C4×C53#4111yescharacteristic
116C22×C4×C53#4111yescharacteristic
117C22×C4×C53#4111yescharacteristic
118C22×C4×C53#4111yescharacteristic
119C22×C4×C53#4111yescharacteristic
120C22×Dic53—21no—
121C22×Dic53—21no—
122C22×Dic53—21no—
123C22×Dic53—21no—
124C22×Dic53—21no—
125C22×Dic53—21no—
126C22×Dic53C4212yes—
127C22×Dic53C22⋊C414yes—
128C2×C4×D53—22no—
129C2×C4×D53—22no—
130C2×C4×D53—22no—
131C2×C4×D53—22no—
132C2×C4×D53—22no—
133C2×C4×D53—22no—
134C2×C4×D53—22no—
135C2×C4×D53—22no—
136C2×C4×D53—22no—
137C2×C4×D53—22no—
138C2×C4×D53—22no—
139C2×C4×D53—22no—
140C2×C4×D53—41no—
141C2×C4×D53—41no—
142C2×C4×D53—41no—
143C2×C4×D53—41no—
144C2×C4×D53—41no—
145C2×C4×D53—41no—
146C2×C4×D53—41no—
147C2×C4×D53—41no—
148C2×C4×D53—41no—
149C2×C4×D53—41no—
150C2×C4×D53—41no—
151C2×C4×D53—41no—
152C2×C4×D53—41no—
153C2×C4×D53—41no—
154C2×C4×D53—41no—
155C2×C4×D53—41no—
156M16×C53—22no—
157M16×C53—22no—
158M16×C53—22no—
159M16×C53—22no—
160M16×C53—22no—
161M16×C53—22no—
162M16×C53—41no—
163M16×C53—41no—
164C2×C8×C53—22no—
165C2×C8×C53—22no—
166C2×C8×C53—22no—
167C2×C8×C53—41no—
168C2×C8×C53—41no—
169C2×C8×C53C22⋊C414yescomplemented
170C4⋊C4×C53—41no—
171C4⋊C4×C53—41no—
172C22⋊C4×C53—22no—
173C22⋊C4×C53—22no—
174C42×C53—41no—
175C42×C53—41no—
176C22⋊Dic53—22no—
177C22⋊Dic53—22no—
178#11719921—48no—
179C4⋊Dic53—22no—
180C4⋊Dic53—22no—
181C4×Dic53—22no—
182C4×Dic53—22no—
183#11719917—22no—
184#11719917—22no—
185#11719917—22no—
186#11719917—22no—
187#11719917—22no—
188#11719917—22no—
189#11719917—41no—
190#11719917—41no—
191C2×#88322—22no—
192C2×#88322—22no—
193C2×#88322—22no—
194C2×#88322—41no—
195C2×#88322—41no—
196C2×#88322C22⋊C414yescomplemented
197C23×C53—42no—
198C23×C53C2×C4211yescharacteristic
199C23×C53C2×C22⋊C411yescharacteristic
200C23×C53#11711yescharacteristic
201C23×C53C4×D411yescharacteristic
202C23×C53C4×D411yescharacteristic
203C23×C53C4×D411yescharacteristic
204C23×C53C4×D411yescharacteristic
205C22×D53—24no—
206C22×D53—24no—
207C22×D53—24no—
208C22×D53—22no—
209C22×D53—22no—
210C22×D53—22no—
211C22×D53—44no—
212C22×D53—44no—
213C22×D53—44no—
214C22×D53—44no—
215C22×D53—42no—
216C22×D53—42no—
217C22×D53—42no—
218C22×D53—41no—
219C22×D53—41no—
220C22×D53—41no—
221C22×D53—41no—
222C2×C4×C53—44no—
223C2×C4×C53—42no—
224C2×C4×C53C2×C22⋊C411yescharacteristic
225C2×C4×C53C2×C22⋊C411yescharacteristic
226C2×C4×C53#11711yescharacteristic
227C2×C4×C53#11711yescharacteristic
228C2×C4×C53C4×D411yescharacteristic
229C2×C4×C53C4×D411yescharacteristic
230C2×C4×C53C4×D411yescharacteristic
231C2×C4×C53C4×D411yescharacteristic
232C2×C4×C53C4×D411yescharacteristic
233C2×C4×C53C4×D411yescharacteristic
234C2×C4×C53C4×D411yescharacteristic
235C2×C4×C53C4×D411yescharacteristic
236C2×C4×C53#12011yescharacteristic
237C2×C4×C53#12111yescharacteristic
238C2×C4×C53#12111yescharacteristic
239C2×C4×C53#12111yescharacteristic
240C2×C4×C53#12111yescharacteristic
241C2×C4×C53#12111yescharacteristic
242C2×C4×C53#12111yescharacteristic
243C2×C4×C53#12311yescharacteristic
244C2×C4×C53#12311yescharacteristic
245C2×C4×C53#12311yescharacteristic
246C2×C4×C53#12411yescharacteristic
247C2×C4×C53#12411yescharacteristic
248C2×C4×C53#12411yescharacteristic
249C2×C4×C53#12611yescharacteristic
250C2×C4×C53#12611yescharacteristic
251C2×C4×C53#12711yescharacteristic
252C2×Dic53—22no—
253C2×Dic53—22no—
254C2×Dic53—22no—
255C2×Dic53—24no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—44no—
261C4×D53—42no—
262C4×D53—42no—
263C4×D53—42no—
264C4×D53—42no—
265C4×D53—42no—
266C4×D53—42no—
267C4×D53—42no—
268C4×D53—42no—
269C8×C53—24no—
270C8×C53—42no—
271#88322—24no—
272#88322—42no—
273#10937—531nomaximal, Sylow
274C22×C53—22no—
275C22×C53—22no—
276C22×C53—22no—
277C22×C53—41no—
278C22×C53—41no—
279C22×C53—41no—
280C22×C53—41no—
281C22×C53C4×C22⋊C411yescharacteristic
282C22×C53C4×C22⋊C411yescharacteristic
283C22×C53C4×C22⋊C411yescharacteristic
284C22×C53#38611yescharacteristic
285C22×C53#38811yescharacteristic
286C22×C53#38811yescharacteristic
287C22×C53#39011yescharacteristic
288C2×D53—24no—
289C2×D53—24no—
290C2×D53—24no—
291C2×D53—44no—
292C2×D53—42no—
293C2×D53—42no—
294C2×D53—42no—
295C2×D53—44no—
296C2×D53—44no—
297C2×D53—44no—
298C2×D53—42no—
299C4×C53—82no—
300C4×C53#38611yescharacteristic
301C4×C53#38611yescharacteristic
302C4×C53#38811yescharacteristic
303C4×C53#38811yescharacteristic
304C4×C53#38811yescharacteristic
305C4×C53#38811yescharacteristic
306C4×C53#39011yescharacteristic
307C4×C53#39011yescharacteristic
308Dic53—44no—
309Dic53—42no—
310C2×#407—531no—
311C2×#407—531no—
312#2873—531no—
313#2280—531no—
314C2×#344—531no—
315C2×#344D5311yescharacteristic, complemented
316#1742—531no—
317C2×C53—21no—
318C2×C53—21no—
319C2×C53—21no—
320C2×C53—21no—
321C2×C53—21no—
322C2×C53—21no—
323C2×C53#143711yescharacteristic
324C2×C53#189112yes—
325D53—44no—
326C24×C4—531no—
327C22×M16—531no—
328C22×M16—531no—
329C22×M16C2×D5311yescharacteristic
330C22×M16C2×D5311yescharacteristic
331C2×#117—531no—
332C2×#117C2×D5311yescharacteristic
333#407—534no—
334#407—534no—
335#407—1062no—
336#407—1062no—
337C2×C22⋊C8—1062no—
338C2×C22⋊C8—1062no—
339#405—1062no—
340#405—1062no—
341C4×M16—1062no—
342#381—532no—
343#380—1062no—
344#379—532no—
345C4×C22⋊C4—1062no—
346#344—22no—
347#344—1062no—
348#343—1062no—
349C53#1093711yescharacteristic, complemented, Sylow
350C25—531no—
351C23×C4—531no—
352C23×C4—532no—
353C23×C4—532no—
354C23×C4—1061no—
355C23×C4—1061no—
356C23×C4—1061no—
357C23×C4—1061no—
358C23×C4—1061no—
359C23×C4—1061no—
360C23×C4—1061no—
361C23×C4—1061no—
362C23×C4—1061no—
363C23×C4—1061no—
364C23×C4—1061no—
365C23×C4—1061no—
366C23×C4C22×D5311yescharacteristic
367C2×M16—21no—
368C2×M16—21no—
369C2×M16—21no—
370C2×M16—21no—
371C2×M16—21no—
372C2×M16—21no—
373C2×M16—21no—
374C2×M16—21no—
375C2×M16—21no—
376C2×M16—532no—
377C2×M16—532no—
378C2×M16—532no—
379C2×M16—1061no—
380C2×M16—1061no—
381C2×M16—1061no—
382C2×M16—1061no—
383C2×M16—1061no—
384C2×M16—1061no—
385C2×M16—1061no—
386C2×M16—1061no—
387C2×M16—1061no—
388C2×M16C4×D5312yes—
389C2×M16C4×D5312yes—
390C2×M16C22⋊D5312yes—
391C22×C8—21no—
392C22×C8—532no—
393C22×C8—1061no—
394C22×C8D21212yes—
395#117—21no—
396#117—21no—
397#117—21no—
398#117—21no—
399#117—1061no—
400#117—1061no—
401#117—1061no—
402#117—1061no—
403C2×C4⋊C4—21no—
404C2×C4⋊C4—532no—
405C2×C22⋊C4—532no—
406C2×C22⋊C4—1064no—
407C2×C22⋊C4C4×D5312yes—
408C2×C42—21no—
409C2×C42—532no—
410C22⋊C8—1068no—
411C22⋊C8—1068no—
412#97—1064no—
413C4×C8—1064no—
414#95—1064no—
415C24—532no—
416C24—1061no—
417C24—1061no—
418C24—1061no—
419C24—1061no—
420C24—1061no—
421C24—1061no—
422C24—2124no—
423C24C2×C4×D5311yescharacteristic
424C22×C4—22no—
425C22×C4—532no—
426C22×C4—534no—
427C22×C4—1061no—
428C22×C4—1061no—
429C22×C4—1061no—
430C22×C4—1061no—
431C22×C4—1061no—
432C22×C4—1061no—
433C22×C4—1062no—
434C22×C4—1062no—
435C22×C4—1062no—
436C22×C4—1062no—
437C22×C4—1062no—
438C22×C4—1062no—
439C22×C4—1062no—
440C22×C4—1062no—
441C22×C4—1062no—
442C22×C4—1062no—
443C22×C4—1062no—
444C22×C4—1062no—
445C22×C4—2121no—
446C22×C4—2121no—
447C22×C4—2121no—
448C22×C4—2121no—
449C22×C4—2121no—
450C22×C4—2121no—
451C22×C4—2121no—
452C22×C4—2121no—
453C22×C4—2121no—
454C22×C4—2121no—
455C22×C4—2121no—
456C22×C4—2121no—
457C22×C4—2121no—
458C22×C4—2121no—
459C22×C4—2121no—
460C22×C4—2121no—
461C22×C4C2×C4×D5311yescharacteristic
462C22×C4C2×C4×D5311yescharacteristic
463C22×C4C2×D21211yescharacteristic
464C22×C4#1171994511yescharacteristic
465C22×C4#1171994511yescharacteristic
466C22×C4D4×D5311yescharacteristic
467C22×C4D4×D5311yescharacteristic
468C22×C4D4×D5311yescharacteristic
469C22×C4D4×D5311yescharacteristic
470C22×C4#1171994711yescharacteristic
471C22×C4#1171994711yescharacteristic
472C22×C4#1171994911yescharacteristic
473C22×C4#1171994911yescharacteristic
474C22×C4C2×C22⋊D5311yescharacteristic
475M16—22no—
476M16—22no—
477M16—22no—
478M16—22no—
479M16—22no—
480M16—22no—
481M16—41no—
482M16—41no—
483M16—1062no—
484M16—1062no—
485M16—1062no—
486M16—1062no—
487M16—1062no—
488M16—1062no—
489M16—2121no—
490M16—2121no—
491C2×C8—22no—
492C2×C8—22no—
493C2×C8—22no—
494C2×C8—41no—
495C2×C8—41no—
496C2×C8—534no—
497C2×C8—1062no—
498C2×C8—1062no—
499C2×C8—1062no—
500C2×C8—2121no—
501C2×C8—2121no—
502C2×C8#1171992114yescomplemented
503C4⋊C4—41no—
504C4⋊C4—41no—
505C4⋊C4—1062no—
506C4⋊C4—1062no—
507C22⋊C4—22no—
508C22⋊C4—22no—
509C22⋊C4—1062no—
510C22⋊C4—1062no—
511C22⋊C4—2128no—
512C42—41no—
513C42—41no—
514C42—1062no—
515C42—1062no—
516C23—42no—
517C23—1064no—
518C23—1064no—
519C23—1064no—
520C23—1062no—
521C23—1062no—
522C23—1062no—
523C23—2122no—
524C23—2122no—
525C23—2122no—
526C23—2124no—
527C23—2124no—
528C23—2124no—
529C23—2124no—
530C23—2121no—
531C23—2121no—
532C23—2121no—
533C23—2121no—
534C23C42×D5311yescharacteristic
535C23C22⋊C4×D5311yescharacteristic
536C23#1313317511yescharacteristic
537C23#1313317711yescharacteristic
538C23C4×C22⋊D5311yescharacteristic
539C23#1313317911yescharacteristic
540C23C4×D21211yescharacteristic
541C2×C4—44no—
542C2×C4—42no—
543C2×C4—1064no—
544C2×C4—1062no—
545C2×C4—1062no—
546C2×C4—1062no—
547C2×C4—2121no—
548C2×C4—2121no—
549C2×C4—2121no—
550C2×C4—2121no—
551C2×C4—2122no—
552C2×C4—2122no—
553C2×C4—2122no—
554C2×C4—2122no—
555C2×C4—2122no—
556C2×C4—2122no—
557C2×C4—2122no—
558C2×C4—2122no—
559C2×C4—2124no—
560C2×C4C22⋊C4×D5311yescharacteristic
561C2×C4C2×#1171992111yescharacteristic
562C2×C4#1313317411yescharacteristic
563C2×C4#1313317511yescharacteristic
564C2×C4#1313317711yescharacteristic
565C2×C4#1313317711yescharacteristic
566C2×C4#1313317711yescharacteristic
567C2×C4C4×C22⋊D5311yescharacteristic
568C2×C4#1313317911yescharacteristic
569C2×C4#1313317911yescharacteristic
570C2×C4#1313317911yescharacteristic
571C2×C4C4×D21211yescharacteristic
572C2×C4#1313318711yescharacteristic
573C2×C4#1313318911yescharacteristic
574C2×C4#1313318911yescharacteristic
575C2×C4#1313319011yescharacteristic
576C2×C4#1313319011yescharacteristic
577C2×C4#1313319211yescharacteristic
578C2×C4#1313319211yescharacteristic
579C2×C4#1313320111yescharacteristic
580C2×C4#1313320211yescharacteristic
581C2×C4#1313320211yescharacteristic
582C2×C4#1313320311yescharacteristic
583C2×C4#1313320311yescharacteristic
584C2×C4#1313320611yescharacteristic
585C2×C4#1313321211yescharacteristic
586C2×C4#1313321211yescharacteristic
587C2×C4#1313321311yescharacteristic
588C8—24no—
589C8—42no—
590C8—1064no—
591C8—2122no—
592C22—22no—
593C22—22no—
594C22—22no—
595C22—41no—
596C22—41no—
597C22—41no—
598C22—41no—
599C22—1064no—
600C22—1064no—
601C22—1064no—
602C22—2124no—
603C22—2122no—
604C22—2122no—
605C22—2122no—
606C22—2124no—
607C22—2124no—
608C22—2122no—
609C22—2124no—
610C22#1681047011yescharacteristic
611C22#1681047011yescharacteristic
612C22C4×#1171992111yescharacteristic
613C22#1681050111yescharacteristic
614C22#1681051311yescharacteristic
615C22#1681051511yescharacteristic
616C22#1681052311yescharacteristic
617C4—82no—
618C4—2122no—
619C4—2124no—
620C4#1681050011yescharacteristic
621C4#1681050111yescharacteristic
622C4#1681051211yescharacteristic
623C4#1681051211yescharacteristic
624C4#1681051311yescharacteristic
625C4#1681051511yescharacteristic
626C4#1681052311yescharacteristic
627C4#1681052411yescharacteristic
628C2—21no—
629C2—21no—
630C2—21no—
631C2—21no—
632C2—21no—
633C2—21no—
634C2—2124no—
635C2#2027295111yescharacteristic
636C2#2027513212yes—
subgroupG/H
centerC2×C4#13133187
commutatorC23×C53C2×C42
FrattiniC23×C4C22×D53
FittingC2×C53×#344C2
socleC22×C53#386
prime radicalC24×D53C2×C4
cosocleC23×C4×C53C23
hypercenterC2×#344D53
hypocenterC53#10937
LCSG ▷ C23×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C2×#344

Complements

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

NH ≅ G/Ntype
3. #20283133C2semidirect
5. C2×#16810393C2semidirect
6. #20274453C2semidirect
7. C2×C53×#344C2semidirect
14. #16810586C4semidirect
15. #16810586C4semidirect
169. C2×C8×C53C22⋊C4semidirect
196. C2×#88322C22⋊C4semidirect
315. C2×#344D53semidirect
349. C53#10937semidirect
502. C2×C8#11719921semidirect

Maximal subgroups

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

subgroupindex
C2×#168105862
#202831332
#202770942
C2×#168103932
#202744532
C2×C53×#3442
#1093753