Finite Group Databasebeta

← #27428263

Subgroups — #27428263

Order 13568.

926 conjugacy classes of subgroups.

The subgroup lattice has 686 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810730C211yesmaximal, characteristic, complemented
2#20278214C211yesmaximal, characteristic
3Q8×C22⋊Dic53C211yesmaximal, characteristic
4#20276265C211yesmaximal, characteristic, complemented
5#20276148C211yesmaximal, characteristic
6#20276035C211yesmaximal, characteristic, complemented
7#20275915C211yesmaximal, characteristic
8#20275402C211yesmaximal, characteristic, complemented
9#20275325C211yesmaximal, characteristic
10#20275058C211yesmaximal, characteristic, complemented
11#20275058C211yesmaximal, characteristic, complemented
12#20274967C211yesmaximal, characteristic, complemented
13#20274728C211yesmaximal, characteristic, complemented
14#20274650C211yesmaximal, characteristic
15C2×C53×#451C211yesmaximal, characteristic
16C2×Q8×Dic53C2211yescharacteristic
17C2×Q8×Dic53C2211yescharacteristic
18C2×Q8×Dic53C2211yescharacteristic
19#16810730—24no—
20C2×#13133150C2211yescharacteristic
21C2×#13133149C2211yescharacteristic
22C2×#13133146C2211yescharacteristic
23C2×#13133141C2211yescharacteristic
24C2×#13133141C2211yescharacteristic
25C2×#13133141C2211yescharacteristic
26C2×#13133141C2211yescharacteristic
27C2×#13133141C2211yescharacteristic
28C2×#13133141C2211yescharacteristic
29C2×C22⋊#88322C2211yescharacteristic
30C2×#13133127C2211yescharacteristic
31#16810562—21no—
32#16810548—21no—
33#16810530—21no—
34#16810530—21no—
35#16810530C2211yescharacteristic
36#16810530C2211yescharacteristic
37#16810530C2211yescharacteristic
38#16810519C2211yescharacteristic
39#16810511—21no—
40#16810504—21no—
41#16810488C2211yescharacteristic
42#16810483—21no—
43#16810483C2211yescharacteristic
44#16810483C2211yescharacteristic
45C4×C22⋊Dic53—21no—
46C4×C22⋊Dic53C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810377C2211yescharacteristic
52#16810377C2211yescharacteristic
53C23×Q8×C53C2211yescharacteristic
54C22×Q16×C53C2211yescharacteristic
55C22×Q16×C53C2211yescharacteristic
56C2×C22⋊Q8×C53C2211yescharacteristic
57C53×#451—24no—
58C2×#103×C53C2211yescharacteristic
59C2×#103×C53C2211yescharacteristic
60C2×C22⋊C8×C53C2211yescharacteristic
61C22×#88322C2311yescharacteristic
62C22×#88322C2311yescharacteristic
63Q8×Dic53—22no—
64Q8×Dic53—22no—
65Q8×Dic53—22no—
66Q8×Dic53—41no—
67Q8×Dic53—41no—
68Q8×Dic53—41no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53—21no—
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×#11719919—21no—
77C2×C22⋊Dic53—21no—
78C2×C22⋊Dic53—21no—
79C2×C22⋊Dic53D411yescharacteristic
80C2×C22⋊Dic53D411yescharacteristic
81C2×C4×Dic53—21no—
82C2×C4×Dic53—21no—
83C2×C4×Dic53—21no—
84C2×C4×Dic53C2311yescharacteristic
85C2×C4×Dic53C2311yescharacteristic
86#13133150—41no—
87#13133149—22no—
88#13133146—41no—
89#13133141—22no—
90#13133141—22no—
91#13133141—22no—
92#13133141—22no—
93#13133141—41no—
94#13133141—41no—
95C22⋊#88322—41no—
96#13133127—22no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124—21no—
107#13133124—21no—
108#13133124D411yescharacteristic
109#13133124D411yescharacteristic
110C22×Q8×C53—22no—
111C22×Q8×C53—22no—
112C22×Q8×C53—21no—
113C22×Q8×C53—22no—
114C22×Q8×C53—22no—
115C22×Q8×C53—41no—
116C22×Q8×C53—41no—
117C22×Q8×C53C2311yescharacteristic
118C22×Q8×C53C2311yescharacteristic
119C22×Q8×C53C2311yescharacteristic
120C23×C4×C53—21no—
121C23×C4×C53C2311yescharacteristic
122C2×Q16×C53—22no—
123C2×Q16×C53—22no—
124C2×Q16×C53—22no—
125C2×Q16×C53—22no—
126C2×Q16×C53—41no—
127C2×Q16×C53D414yes—
128C22×C8×C53C2311yescharacteristic
129C22×C8×C53C2311yescharacteristic
130C22⋊Q8×C53—22no—
131C22⋊Q8×C53—41no—
132C2×C4⋊C4×C53—21no—
133C2×C4⋊C4×C53C2311yescharacteristic
134C2×C22⋊C4×C53—21no—
135#103×C53—22no—
136#103×C53—22no—
137C22⋊C8×C53—22no—
138C24×C53C2×D411yescharacteristic
139C2×Q8×C53—24no—
140C2×Q8×C53—21no—
141C2×Q8×C53—21no—
142C2×Q8×C53—21no—
143C2×Q8×C53—21no—
144C2×Q8×C53—21no—
145C2×Q8×C53—21no—
146C2×Q8×C53—21no—
147C2×Q8×C53—21no—
148C2×Q8×C53—21no—
149C2×Q8×C53—21no—
150C2×Q8×C53—21no—
151C2×Q8×C53—21no—
152C2×Q8×C53—21no—
153C2×Q8×C53—42no—
154C2×Q8×C53—42no—
155C2×Q8×C53—44no—
156C2×Q8×C53—44no—
157C2×Q8×C53—44no—
158C2×Q8×C53—44no—
159C2×Q8×C53C2×D412yes—
160C2×Q8×C53C2×D412yes—
161C2×Q8×C53#4112yes—
162C2×Q8×C53#4112yes—
163C2×Q8×C53#4112yes—
164C22×C4×C53—21no—
165C22×C4×C53—24no—
166C22×C4×C53—21no—
167C22×C4×C53—21no—
168C22×C4×C53—21no—
169C22×C4×C53—21no—
170C22×C4×C53—21no—
171C22×C4×C53—21no—
172C22×C4×C53—21no—
173C22×C4×C53—21no—
174C22×C4×C53—21no—
175C22×C4×C53—42no—
176C22×C4×C53—42no—
177C22×C4×C53C2×D411yescharacteristic
178C22×C4×C53C2411yescharacteristic
179C22×Dic53—21no—
180C22×Dic53—21no—
181C22×Dic53—21no—
182C22×Dic53—21no—
183C22×Dic53C2×D411yescharacteristic
184C22×Dic53C2×D411yescharacteristic
185C22×Dic53C2×D411yescharacteristic
186C22×Dic53C2×D411yescharacteristic
187Q16×C53—28no—
188Q16×C53—44no—
189C2×C8×C53—21no—
190C2×C8×C53—21no—
191C2×C8×C53—21no—
192C2×C8×C53—21no—
193C2×C8×C53—21no—
194C2×C8×C53C2×D412yes—
195C4⋊C4×C53—21no—
196C4⋊C4×C53—21no—
197C4⋊C4×C53—42no—
198C22⋊C4×C53—42no—
199C22⋊Dic53—22no—
200C22⋊Dic53—41no—
201C22⋊Dic53—42no—
202C22⋊Dic53—42no—
203C4⋊Dic53—21no—
204C4⋊Dic53—21no—
205C4⋊Dic53—21no—
206C4⋊Dic53—21no—
207C4⋊Dic53—21no—
208C4⋊Dic53—21no—
209C4⋊Dic53—21no—
210C4⋊Dic53—21no—
211C4⋊Dic53—42no—
212C4⋊Dic53—42no—
213C4⋊Dic53—42no—
214#11719919—42no—
215C4×Dic53—21no—
216C4×Dic53—21no—
217C4×Dic53—21no—
218C4×Dic53—21no—
219C4×Dic53—42no—
220C4×Dic53—42no—
221C4×Dic53—42no—
222C2×#88322—21no—
223C2×#88322—21no—
224C2×#88322—21no—
225C2×#88322—21no—
226C2×#88322—21no—
227C2×#88322—21no—
228C23×C53—21no—
229C23×C53—21no—
230C23×C53—21no—
231C23×C53—21no—
232C23×C53—21no—
233C23×C53—21no—
234C23×C53#13712yes—
235C23×C53C22×D411yescharacteristic
236Q8×C53—22no—
237Q8×C53—22no—
238Q8×C53—22no—
239Q8×C53—22no—
240Q8×C53—22no—
241Q8×C53—22no—
242Q8×C53—22no—
243Q8×C53—48no—
244Q8×C53—41no—
245Q8×C53—41no—
246Q8×C53—41no—
247Q8×C53—41no—
248Q8×C53#12314yes—
249C2×C4×C53—22no—
250C2×C4×C53—22no—
251C2×C4×C53—22no—
252C2×C4×C53—22no—
253C2×C4×C53—21no—
254C2×C4×C53—21no—
255C2×C4×C53—22no—
256C2×C4×C53—21no—
257C2×C4×C53—48no—
258C2×C4×C53—41no—
259C2×C4×C53—41no—
260C2×C4×C53—41no—
261C2×C4×C53—41no—
262C2×C4×C53—41no—
263C2×C4×C53—41no—
264C2×C4×C53—41no—
265C2×C4×C53—41no—
266C2×C4×C53—41no—
267C2×C4×C53—41no—
268C2×C4×C53—41no—
269C2×C4×C53C22×D411yescharacteristic
270C2×C4×C53C2×#4111yescharacteristic
271C2×C4×C53C2×#4111yescharacteristic
272C2×C4×C53C2×#4111yescharacteristic
273C2×C4×C53H(5,𝔽2)11yescharacteristic
274C2×C4×C53H(5,𝔽2)11yescharacteristic
275C2×Dic53—21no—
276C2×Dic53—21no—
277C2×Dic53—21no—
278C2×Dic53—21no—
279C2×Dic53—21no—
280C2×Dic53—21no—
281C2×Dic53—21no—
282C2×Dic53—21no—
283C2×Dic53—21no—
284C2×Dic53—21no—
285C2×Dic53—21no—
286C2×Dic53—21no—
287C2×Dic53—41no—
288C2×Dic53—41no—
289C2×Dic53—41no—
290C2×Dic53—41no—
291C2×Dic53—41no—
292C2×Dic53—41no—
293C2×Dic53—41no—
294C2×Dic53—41no—
295C2×Dic53—41no—
296C2×Dic53—41no—
297C2×Dic53—41no—
298C2×Dic53—41no—
299C8×C53—22no—
300C8×C53—41no—
301#88322—41no—
302#88322—41no—
303#24922—531nomaximal, Sylow
304C22×C53—22no—
305C22×C53—22no—
306C22×C53—22no—
307C22×C53—41no—
308C22×C53—41no—
309C22×C53—41no—
310C22×C53—41no—
311C22×C53#52511yescharacteristic
312C22×C53D4211yescharacteristic
313C22×C53#54611yescharacteristic
314C22×C53C2×#13711yescharacteristic
315C22×C53#57511yescharacteristic
316C22×C53#57811yescharacteristic
317C22×C53#57811yescharacteristic
318C4×C53—24no—
319C4×C53—41no—
320C4×C53—42no—
321C4×C53—42no—
322C4×C53—81no—
323C4×C53C2×#12311yescharacteristic
324C4×C53#53811yescharacteristic
325C4×C53#54611yescharacteristic
326C4×C53#55111yescharacteristic
327Dic53—41no—
328Dic53—41no—
329Dic53—41no—
330Dic53—41no—
331Dic53—81no—
332Dic53—81no—
333Dic53—81no—
334Dic53—81no—
335C2×#451D5311yescharacteristic
336C2×#441—531no—
337#2421—531no—
338Q8×C22⋊C4—531no—
339#2097—531no—
340#2077—531no—
341#2056—531no—
342#2036—531no—
343#1935—531no—
344#1924—531no—
345#1878—531no—
346#1878—531no—
347#1863—531no—
348#1805—531no—
349#1787—531no—
350C2×C53—42no—
351C2×C53#241811yescharacteristic
352C2×C53#311711yescharacteristic
353C2×C53#312711yescharacteristic
354C2×C53#327011yescharacteristic
355C2×C53#327711yescharacteristic
356C2×C53#331211yescharacteristic
357C2×C53#331311yescharacteristic
358C23×Q8C2×D5311yescharacteristic
359C22×Q16C2×D5311yescharacteristic
360C22×Q16C2×D5311yescharacteristic
361C2×C22⋊Q8C2×D5311yescharacteristic
362C2×C4×Q8—531no—
363C2×C4×Q8—531no—
364C2×C4×Q8—531no—
365#451—24no—
366#441—1064no—
367C2×#107—531no—
368C2×#106—531no—
369C2×C4⋊C8—531no—
370C2×#103—531no—
371C2×#103—531no—
372C2×#103—531no—
373C2×#103—531no—
374C2×#103—531no—
375C2×#103—531no—
376C2×#103C2×D5311yescharacteristic
377C2×#103C2×D5311yescharacteristic
378C2×C22⋊C8—531no—
379C2×C22⋊C8C2×D5311yescharacteristic
380C2×#97—531no—
381#400—1061no—
382#397—1061no—
383#391—531no—
384#391—531no—
385#391—531no—
386#391—1061no—
387#391—1061no—
388#389—531no—
389#388—1061no—
390#387—1061no—
391#382—531no—
392#380—531no—
393#380—531no—
394#380—1061no—
395C4×C22⋊C4—531no—
396C4×C22⋊C4—1061no—
397#340—531no—
398#340—531no—
399#340—531no—
400#340—531no—
401#336—531no—
402#336—531no—
403C53#2492211yescharacteristic, complemented, Sylow
404C22×Q8—22no—
405C22×Q8—22no—
406C22×Q8—21no—
407C22×Q8—22no—
408C22×Q8—22no—
409C22×Q8—41no—
410C22×Q8—41no—
411C22×Q8C22×D5311yescharacteristic
412C22×Q8C22×D5311yescharacteristic
413C22×Q8C22×D5311yescharacteristic
414C23×C4—21no—
415C23×C4C22×D5311yescharacteristic
416C2×Q16—22no—
417C2×Q16—22no—
418C2×Q16—22no—
419C2×Q16—22no—
420C2×Q16—41no—
421C2×Q16C22⋊D5314yes—
422C22×C8—531no—
423C22×C8—531no—
424C22×C8C22×D5311yescharacteristic
425C22×C8C22×D5311yescharacteristic
426C22⋊Q8—22no—
427C22⋊Q8—41no—
428C4×Q8—1062no—
429C4×Q8—1062no—
430C4×Q8—1062no—
431C4×Q8—2121no—
432C4×Q8—2121no—
433C4×Q8—2121no—
434C2×C4⋊C4—21no—
435C2×C4⋊C4—531no—
436C2×C4⋊C4—531no—
437C2×C4⋊C4—531no—
438C2×C4⋊C4—531no—
439C2×C4⋊C4—1061no—
440C2×C4⋊C4—1061no—
441C2×C4⋊C4—1061no—
442C2×C4⋊C4—1061no—
443C2×C4⋊C4C22×D5311yescharacteristic
444C2×C22⋊C4—21no—
445C2×C22⋊C4—531no—
446C2×C22⋊C4—531no—
447C2×C22⋊C4—1061no—
448C2×C22⋊C4—1061no—
449C2×C42—531no—
450C2×C42—531no—
451C2×C42—1061no—
452C2×C42—1061no—
453C2×C42—1061no—
454#107—2121no—
455#106—1062no—
456C4⋊C8—2121no—
457#103—22no—
458#103—22no—
459#103—1062no—
460#103—1062no—
461#103—1062no—
462#103—1062no—
463#103—2121no—
464#103—2121no—
465C22⋊C8—22no—
466C22⋊C8—2121no—
467#97—1062no—
468#95—531no—
469#95—531no—
470#95—1061no—
471#95—1061no—
472#95—1061no—
473#95—1061no—
474#95—1061no—
475#95—1061no—
476#95—1061no—
477#95—1061no—
478#95—1061no—
479#95—1061no—
480#95—1061no—
481C24D4×D5311yescharacteristic
482C2×Q8—24no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—21no—
486C2×Q8—21no—
487C2×Q8—21no—
488C2×Q8—21no—
489C2×Q8—21no—
490C2×Q8—21no—
491C2×Q8—21no—
492C2×Q8—21no—
493C2×Q8—21no—
494C2×Q8—21no—
495C2×Q8—21no—
496C2×Q8—42no—
497C2×Q8—42no—
498C2×Q8—44no—
499C2×Q8—44no—
500C2×Q8—44no—
501C2×Q8—44no—
502C2×Q8#1171994712yes—
503C2×Q8#1171994712yes—
504C2×Q8#1171994712yes—
505C2×Q8C2×C22⋊D5312yes—
506C2×Q8C2×C22⋊D5312yes—
507C22×C4—21no—
508C22×C4—24no—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—21no—
514C22×C4—21no—
515C22×C4—21no—
516C22×C4—21no—
517C22×C4—21no—
518C22×C4—42no—
519C22×C4—42no—
520C22×C4—531no—
521C22×C4—531no—
522C22×C4—531no—
523C22×C4—531no—
524C22×C4—1061no—
525C22×C4—1061no—
526C22×C4—1061no—
527C22×C4—1061no—
528C22×C4D4×D5311yescharacteristic
529C22×C4C23×D5311yescharacteristic
530Q16—28no—
531Q16—44no—
532C2×C8—21no—
533C2×C8—21no—
534C2×C8—21no—
535C2×C8—21no—
536C2×C8—21no—
537C2×C8—1061no—
538C2×C8—1061no—
539C2×C8—1061no—
540C2×C8—1061no—
541C2×C8—1061no—
542C2×C8—1061no—
543C2×C8C2×C22⋊D5312yes—
544C4⋊C4—21no—
545C4⋊C4—21no—
546C4⋊C4—42no—
547C4⋊C4—1061no—
548C4⋊C4—1061no—
549C4⋊C4—1061no—
550C4⋊C4—1061no—
551C4⋊C4—1061no—
552C4⋊C4—1061no—
553C4⋊C4—1061no—
554C4⋊C4—1061no—
555C4⋊C4—2122no—
556C4⋊C4—2122no—
557C4⋊C4—2122no—
558C4⋊C4—2122no—
559C22⋊C4—42no—
560C22⋊C4—1062no—
561C22⋊C4—2121no—
562C22⋊C4—2122no—
563C22⋊C4—2122no—
564C42—1061no—
565C42—1061no—
566C42—1061no—
567C42—1061no—
568C42—2122no—
569C42—2122no—
570C42—2122no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23#1313324912yes—
578C23C2×D4×D5311yescharacteristic
579Q8—22no—
580Q8—22no—
581Q8—22no—
582Q8—22no—
583Q8—22no—
584Q8—22no—
585Q8—22no—
586Q8—48no—
587Q8—41no—
588Q8—41no—
589Q8—41no—
590Q8—41no—
591Q8#1313319814yes—
592C2×C4—22no—
593C2×C4—22no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—21no—
597C2×C4—22no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—48no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—1061no—
613C2×C4—1061no—
614C2×C4—1061no—
615C2×C4—1061no—
616C2×C4—1061no—
617C2×C4—1061no—
618C2×C4—1061no—
619C2×C4—1061no—
620C2×C4—1061no—
621C2×C4—1061no—
622C2×C4—1061no—
623C2×C4—1061no—
624C2×C4—2121no—
625C2×C4—2121no—
626C2×C4—2121no—
627C2×C4—2121no—
628C2×C4—2121no—
629C2×C4—2121no—
630C2×C4—2121no—
631C2×C4—2121no—
632C2×C4—2121no—
633C2×C4—2121no—
634C2×C4—2121no—
635C2×C4—2121no—
636C2×C4C22×C22⋊D5311yescharacteristic
637C2×C4C2×#1171994711yescharacteristic
638C2×C4C2×#1171994711yescharacteristic
639C2×C4C2×#1171994711yescharacteristic
640C2×C4#1313326211yescharacteristic
641C2×C4#1313326211yescharacteristic
642C8—22no—
643C8—41no—
644C8—2121no—
645C8—2121no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—41no—
650C22—41no—
651C22—41no—
652C22—41no—
653C22#1681112911yescharacteristic
654C22D4×C22⋊D5311yescharacteristic
655C22#1681126011yescharacteristic
656C22C2×#1313324911yescharacteristic
657C22#1681142211yescharacteristic
658C22#1681143611yescharacteristic
659C22#1681143911yescharacteristic
660C4—24no—
661C4—41no—
662C4—42no—
663C4—42no—
664C4—81no—
665C4—4241no—
666C4—4241no—
667C4—4241no—
668C4—4241no—
669C4—2121no—
670C4—2121no—
671C4—2121no—
672C4—2121no—
673C4C2×#1313319811yescharacteristic
674C4#1681120811yescharacteristic
675C4#1681126711yescharacteristic
676C4#1681130411yescharacteristic
677C2—42no—
678C2#2027817311yescharacteristic
679C2#2028488711yescharacteristic
680C2#2028500911yescharacteristic
681C2#2028628511yescharacteristic
682C2#2028636611yescharacteristic
683C2#2028677011yescharacteristic
684C2#2028677411yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×C53×#451C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenterC2×#451D53
hypocenterC53#24922
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#451

Complements

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

NH ≅ G/Ntype
1. C2×#16810730C2semidirect
4. #20276265C2semidirect
6. #20276035C2semidirect
8. #20275402C2semidirect
10. #20275058C2semidirect
11. #20275058C2semidirect
12. #20274967C2semidirect
13. #20274728C2semidirect
403. C53#24922semidirect

Maximal subgroups

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

subgroupindex
C2×#168107302
#202782142
Q8×C22⋊Dic532
#202762652
#202761482
#202760352
#202759152
#202754022
#202753252
#202750582
#202749672
#202747282
#202746502
C2×C53×#4512
#2492253