Finite Group Databasebeta

← #27428023

Subgroups — #27428023

Order 13568.

940 conjugacy classes of subgroups.

The subgroup lattice has 690 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×Q16×Dic53C211yesmaximal, characteristic, complemented
2#20278236C211yesmaximal, characteristic
3Q8×C22⋊Dic53C211yesmaximal, characteristic
4#20276251C211yesmaximal, characteristic, complemented
5#20276143C211yesmaximal, characteristic
6#20276035C211yesmaximal, characteristic, complemented
7#20275915C211yesmaximal, characteristic
8#20275388C211yesmaximal, characteristic, complemented
9#20275318C211yesmaximal, characteristic
10#20275058C211yesmaximal, characteristic, complemented
11#20275058C211yesmaximal, characteristic, complemented
12#20274967C211yesmaximal, characteristic, complemented
13#20274718C211yesmaximal, characteristic, complemented
14#20274650C211yesmaximal, characteristic
15C2×C53×#451C211yesmaximal, characteristic
16C2×Q8×Dic53C2211yescharacteristic
17C2×Q8×Dic53C2211yescharacteristic
18C2×Q8×Dic53C2211yescharacteristic
19Q16×Dic53—24no—
20C2×#13133151C2211yescharacteristic
21C2×#13133150C2211yescharacteristic
22C2×#13133146C2211yescharacteristic
23C2×#13133141C2211yescharacteristic
24C2×#13133141C2211yescharacteristic
25C2×#13133141C2211yescharacteristic
26C2×#13133141C2211yescharacteristic
27C2×#13133141C2211yescharacteristic
28C2×#13133141C2211yescharacteristic
29C2×C22⋊#88322C2211yescharacteristic
30C2×C8×Dic53C2211yescharacteristic
31#16810559—21no—
32#16810553—21no—
33#16810530—21no—
34#16810530—21no—
35#16810530C2211yescharacteristic
36#16810530C2211yescharacteristic
37#16810530C2211yescharacteristic
38#16810519C2211yescharacteristic
39#16810511—21no—
40#16810504—21no—
41#16810492C2211yescharacteristic
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×#11719919—21no—
78C2×#11719919—21no—
79C2×C22⋊Dic53—21no—
80C2×C22⋊Dic53—21no—
81C2×C22⋊Dic53D411yescharacteristic
82C2×C22⋊Dic53D411yescharacteristic
83C2×C4×Dic53—21no—
84C2×C4×Dic53—21no—
85C2×C4×Dic53—21no—
86C2×C4×Dic53C2311yescharacteristic
87C2×C4×Dic53C2311yescharacteristic
88#13133151—22no—
89#13133150—41no—
90#13133146—41no—
91#13133141—22no—
92#13133141—22no—
93#13133141—22no—
94#13133141—22no—
95#13133141—41no—
96#13133141—41no—
97C22⋊#88322—41no—
98C8×Dic53—22no—
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—41no—
200C22⋊Dic53—42no—
201C22⋊Dic53—81no—
202C22⋊Dic53Q1614yescomplemented
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—
215#11719919—81no—
216#11719919—81no—
217C4×Dic53—21no—
218C4×Dic53—21no—
219C4×Dic53—21no—
220C4×Dic53—21no—
221C4×Dic53—42no—
222C4×Dic53—42no—
223C4×Dic53—42no—
224C2×#88322—21no—
225C2×#88322—21no—
226C2×#88322—21no—
227C2×#88322—21no—
228C2×#88322—21no—
229C2×#88322—21no—
230C23×C53—21no—
231C23×C53—21no—
232C23×C53—21no—
233C23×C53—21no—
234C23×C53—21no—
235C23×C53—21no—
236C23×C53C2×Q1612yes—
237C23×C53C22×D411yescharacteristic
238Q8×C53—22no—
239Q8×C53—22no—
240Q8×C53—22no—
241Q8×C53—22no—
242Q8×C53—22no—
243Q8×C53—22no—
244Q8×C53—22no—
245Q8×C53—48no—
246Q8×C53—41no—
247Q8×C53—41no—
248Q8×C53—41no—
249Q8×C53—41no—
250Q8×C53#12314yes—
251C2×C4×C53—22no—
252C2×C4×C53—22no—
253C2×C4×C53—22no—
254C2×C4×C53—22no—
255C2×C4×C53—21no—
256C2×C4×C53—21no—
257C2×C4×C53—22no—
258C2×C4×C53—21no—
259C2×C4×C53—48no—
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×C53—41no—
270C2×C4×C53—41no—
271C2×C4×C53C22×D411yescharacteristic
272C2×C4×C53C2×#4111yescharacteristic
273C2×C4×C53C2×#4111yescharacteristic
274C2×C4×C53C2×#4111yescharacteristic
275C2×C4×C53H(5,𝔽2)11yescharacteristic
276C2×C4×C53H(5,𝔽2)11yescharacteristic
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—
299C2×Dic53C2×Q1612yes—
300C2×Dic53C2×Q1612yes—
301C8×C53—22no—
302C8×C53—41no—
303#88322—41no—
304#88322—41no—
305#24906—531nomaximal, Sylow
306C22×C53—22no—
307C22×C53—22no—
308C22×C53—22no—
309C22×C53—41no—
310C22×C53—41no—
311C22×C53—41no—
312C22×C53—41no—
313C22×C53#52511yescharacteristic
314C22×C53D4211yescharacteristic
315C22×C53#54611yescharacteristic
316C22×C53C22×Q1611yescharacteristic
317C22×C53#57511yescharacteristic
318C22×C53#57711yescharacteristic
319C22×C53#57811yescharacteristic
320C4×C53—24no—
321C4×C53—41no—
322C4×C53—42no—
323C4×C53—42no—
324C4×C53—81no—
325C4×C53C2×#12311yescharacteristic
326C4×C53#53811yescharacteristic
327C4×C53#54611yescharacteristic
328C4×C53#55111yescharacteristic
329Dic53—22no—
330Dic53—22no—
331Dic53—41no—
332Dic53—41no—
333Dic53—81no—
334Dic53—81no—
335Dic53—81no—
336Dic53—81no—
337C2×#451D5311yescharacteristic
338C2×C4×Q16—531no—
339#2423—531no—
340Q8×C22⋊C4—531no—
341#2095—531no—
342#2076—531no—
343#2056—531no—
344#2036—531no—
345#1933—531no—
346#1923—531no—
347#1878—531no—
348#1878—531no—
349#1863—531no—
350#1803—531no—
351#1787—531no—
352C2×C53—42no—
353C2×C53#241811yescharacteristic
354C2×C53#310411yescharacteristic
355C2×C53#312111yescharacteristic
356C2×C53#327111yescharacteristic
357C2×C53D4×Q1611yescharacteristic
358C2×C53#329911yescharacteristic
359C2×C53#332711yescharacteristic
360C23×Q8C2×D5311yescharacteristic
361C22×Q16C2×D5311yescharacteristic
362C22×Q16C2×D5311yescharacteristic
363C2×C22⋊Q8C2×D5311yescharacteristic
364C2×C4×Q8—531no—
365C2×C4×Q8—531no—
366C2×C4×Q8—531no—
367#451—24no—
368C4×Q16—1064no—
369C2×#107—531no—
370C2×#107—531no—
371C2×C4⋊C8—531no—
372C2×#103—531no—
373C2×#103—531no—
374C2×#103—531no—
375C2×#103—531no—
376C2×#103—531no—
377C2×#103—531no—
378C2×#103C2×D5311yescharacteristic
379C2×#103C2×D5311yescharacteristic
380C2×C22⋊C8—531no—
381C2×C22⋊C8C2×D5311yescharacteristic
382C2×C4×C8—531no—
383#399—1061no—
384#398—1061no—
385#391—531no—
386#391—531no—
387#391—531no—
388#391—1061no—
389#391—1061no—
390#389—531no—
391#388—1061no—
392#387—1061no—
393#384—531no—
394#380—531no—
395#380—531no—
396#380—1061no—
397C4×C22⋊C4—531no—
398C4×C22⋊C4—1061no—
399#340—531no—
400#340—531no—
401#340—531no—
402#340—531no—
403#336—531no—
404#336—531no—
405C53#2490611yescharacteristic, complemented, Sylow
406C22×Q8—22no—
407C22×Q8—22no—
408C22×Q8—21no—
409C22×Q8—22no—
410C22×Q8—22no—
411C22×Q8—41no—
412C22×Q8—41no—
413C22×Q8C22×D5311yescharacteristic
414C22×Q8C22×D5311yescharacteristic
415C22×Q8C22×D5311yescharacteristic
416C23×C4—21no—
417C23×C4C22×D5311yescharacteristic
418C2×Q16—22no—
419C2×Q16—22no—
420C2×Q16—22no—
421C2×Q16—22no—
422C2×Q16—41no—
423C2×Q16C22⋊D5314yes—
424C22×C8—531no—
425C22×C8—531no—
426C22×C8C22×D5311yescharacteristic
427C22×C8C22×D5311yescharacteristic
428C22⋊Q8—22no—
429C22⋊Q8—41no—
430C4×Q8—1062no—
431C4×Q8—1062no—
432C4×Q8—1062no—
433C4×Q8—2121no—
434C4×Q8—2121no—
435C4×Q8—2121no—
436C2×C4⋊C4—21no—
437C2×C4⋊C4—531no—
438C2×C4⋊C4—531no—
439C2×C4⋊C4—531no—
440C2×C4⋊C4—531no—
441C2×C4⋊C4—1061no—
442C2×C4⋊C4—1061no—
443C2×C4⋊C4—1061no—
444C2×C4⋊C4—1061no—
445C2×C4⋊C4—1061no—
446C2×C4⋊C4—1061no—
447C2×C4⋊C4C22×D5311yescharacteristic
448C2×C22⋊C4—21no—
449C2×C22⋊C4—531no—
450C2×C22⋊C4—531no—
451C2×C22⋊C4—1061no—
452C2×C22⋊C4—1061no—
453C2×C42—531no—
454C2×C42—531no—
455C2×C42—1061no—
456C2×C42—1061no—
457C2×C42—1061no—
458#107—1062no—
459#107—2121no—
460C4⋊C8—2121no—
461#103—22no—
462#103—22no—
463#103—1062no—
464#103—1062no—
465#103—1062no—
466#103—1062no—
467#103—2121no—
468#103—2121no—
469C22⋊C8—22no—
470C22⋊C8—2121no—
471C4×C8—1062no—
472#95—531no—
473#95—531no—
474#95—1061no—
475#95—1061no—
476#95—1061no—
477#95—1061no—
478#95—1061no—
479#95—1061no—
480#95—1061no—
481#95—1061no—
482#95—1061no—
483C24D4×D5311yescharacteristic
484C2×Q8—24no—
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—21no—
497C2×Q8—21no—
498C2×Q8—42no—
499C2×Q8—42no—
500C2×Q8—44no—
501C2×Q8—44no—
502C2×Q8—44no—
503C2×Q8—44no—
504C2×Q8#1171994712yes—
505C2×Q8#1171994712yes—
506C2×Q8#1171994712yes—
507C2×Q8C2×C22⋊D5312yes—
508C2×Q8C2×C22⋊D5312yes—
509C22×C4—21no—
510C22×C4—24no—
511C22×C4—21no—
512C22×C4—21no—
513C22×C4—21no—
514C22×C4—21no—
515C22×C4—21no—
516C22×C4—21no—
517C22×C4—21no—
518C22×C4—21no—
519C22×C4—21no—
520C22×C4—42no—
521C22×C4—42no—
522C22×C4—531no—
523C22×C4—531no—
524C22×C4—531no—
525C22×C4—531no—
526C22×C4—1061no—
527C22×C4—1061no—
528C22×C4—1061no—
529C22×C4—1061no—
530C22×C4D4×D5311yescharacteristic
531C22×C4C23×D5311yescharacteristic
532Q16—28no—
533Q16—44no—
534C2×C8—21no—
535C2×C8—21no—
536C2×C8—21no—
537C2×C8—21no—
538C2×C8—21no—
539C2×C8—1061no—
540C2×C8—1061no—
541C2×C8—1061no—
542C2×C8—1061no—
543C2×C8—1061no—
544C2×C8—1061no—
545C2×C8C2×C22⋊D5312yes—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—42no—
549C4⋊C4—1061no—
550C4⋊C4—1061no—
551C4⋊C4—1061no—
552C4⋊C4—1061no—
553C4⋊C4—1061no—
554C4⋊C4—1061no—
555C4⋊C4—1061no—
556C4⋊C4—1061no—
557C4⋊C4—4241no—
558C4⋊C4—4241no—
559C4⋊C4—2122no—
560C4⋊C4—2122no—
561C4⋊C4—2122no—
562C4⋊C4—2122no—
563C22⋊C4—42no—
564C22⋊C4—534no—
565C22⋊C4—4241no—
566C22⋊C4—2121no—
567C22⋊C4—2122no—
568C42—1061no—
569C42—1061no—
570C42—1061no—
571C42—1061no—
572C42—2122no—
573C42—2122no—
574C42—2122no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—21no—
581C23Q16×D5312yes—
582C23C2×D4×D5311yescharacteristic
583Q8—22no—
584Q8—22no—
585Q8—22no—
586Q8—22no—
587Q8—22no—
588Q8—22no—
589Q8—22no—
590Q8—48no—
591Q8—41no—
592Q8—41no—
593Q8—41no—
594Q8—41no—
595Q8#1313319814yes—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—22no—
600C2×C4—21no—
601C2×C4—22no—
602C2×C4—21no—
603C2×C4—21no—
604C2×C4—48no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—41no—
609C2×C4—41no—
610C2×C4—41no—
611C2×C4—41no—
612C2×C4—41no—
613C2×C4—41no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—532no—
617C2×C4—532no—
618C2×C4—1061no—
619C2×C4—1061no—
620C2×C4—1061no—
621C2×C4—1061no—
622C2×C4—1061no—
623C2×C4—1061no—
624C2×C4—1061no—
625C2×C4—1061no—
626C2×C4—1061no—
627C2×C4—1061no—
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×C4—2121no—
637C2×C4—2121no—
638C2×C4—2121no—
639C2×C4—2121no—
640C2×C4C22×C22⋊D5311yescharacteristic
641C2×C4C2×#1171994711yescharacteristic
642C2×C4C2×#1171994711yescharacteristic
643C2×C4C2×#1171994711yescharacteristic
644C2×C4#1313326211yescharacteristic
645C2×C4#1313326211yescharacteristic
646C8—22no—
647C8—41no—
648C8—2121no—
649C8—2121no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—41no—
654C22—41no—
655C22—41no—
656C22—41no—
657C22#1681112911yescharacteristic
658C22D4×C22⋊D5311yescharacteristic
659C22#1681126011yescharacteristic
660C22C2×Q16×D5311yescharacteristic
661C22#1681142211yescharacteristic
662C22#1681143211yescharacteristic
663C22#1681143611yescharacteristic
664C4—24no—
665C4—41no—
666C4—42no—
667C4—42no—
668C4—81no—
669C4—1062no—
670C4—1062no—
671C4—4241no—
672C4—4241no—
673C4—4241no—
674C4—4241no—
675C4—2121no—
676C4—2121no—
677C4C2×#1313319811yescharacteristic
678C4#1681120811yescharacteristic
679C4#1681126711yescharacteristic
680C4#1681130411yescharacteristic
681C2—42no—
682C2#2027817311yescharacteristic
683C2#2028474911yescharacteristic
684C2#2028494511yescharacteristic
685C2#2028629411yescharacteristic
686C2Q16×C22⋊D5311yescharacteristic
687C2#2028658411yescharacteristic
688C2#2028698711yescharacteristic
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#24906
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×Q16×Dic53C2semidirect
4. #20276251C2semidirect
6. #20276035C2semidirect
8. #20275388C2semidirect
10. #20275058C2semidirect
11. #20275058C2semidirect
12. #20274967C2semidirect
13. #20274718C2semidirect
202. C22⋊Dic53Q16semidirect
405. C53#24906semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×Dic532
#202782362
Q8×C22⋊Dic532
#202762512
#202761432
#202760352
#202759152
#202753882
#202753182
#202750582
#202749672
#202747182
#202746502
C2×C53×#4512
#2490653