Finite Group Databasebeta

← #27441226

Subgroups — #27441226

Order 13568.

990 conjugacy classes of subgroups.

The subgroup lattice has 684 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810906C211yesmaximal, characteristic, complemented
2C22×#13133141C211yesmaximal, characteristic
3#20278418C211yesmaximal, characteristic
4Q8×C22⋊Dic53C211yesmaximal, characteristic
5#20276256C211yesmaximal, characteristic, complemented
6#20276094C211yesmaximal, characteristic, complemented
7#20276092C211yesmaximal, characteristic, complemented
8#20275854C211yesmaximal, characteristic, complemented
9#20274985C211yesmaximal, characteristic, complemented
10#20274981C211yesmaximal, characteristic, complemented
11#20274910C211yesmaximal, characteristic
12#20274892C211yesmaximal, characteristic
13#20274738C211yesmaximal, characteristic
14#20274718C211yesmaximal, characteristic, complemented
15C53×#1787C211yesmaximal, characteristic
16C23×#88322C2211yescharacteristic
17C2×#13133216C2211yescharacteristic
18C2×#13133194—21no—
19C2×Q8×Dic53C2211yescharacteristic
20C2×Q8×Dic53C2211yescharacteristic
21C22×C4⋊Dic53C2211yescharacteristic
22#16810906—24no—
23C2×#13133149C2211yescharacteristic
24C2×#13133149C2211yescharacteristic
25C2×#13133148C2211yescharacteristic
26C2×#13133148C2211yescharacteristic
27C2×#13133146C2211yescharacteristic
28C2×#13133146C2211yescharacteristic
29C2×#13133141—24no—
30C2×#13133141C2211yescharacteristic
31C2×#13133141C2211yescharacteristic
32C2×#13133141C2211yescharacteristic
33C2×#13133141C2211yescharacteristic
34#16810557—21no—
35#16810544—21no—
36#16810530—21no—
37#16810530—21no—
38#16810530C2211yescharacteristic
39#16810530C2211yescharacteristic
40#16810522C2211yescharacteristic
41#16810519C2211yescharacteristic
42#16810495—21no—
43#16810492C2211yescharacteristic
44#16810483—21no—
45#16810483C2211yescharacteristic
46C4×C22⋊Dic53—21no—
47C4×C22⋊Dic53C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810385C2211yescharacteristic
52#16810377C2211yescharacteristic
53#16810377C2211yescharacteristic
54C23×Q8×C53C2211yescharacteristic
55C2×#103×C53C2211yescharacteristic
56C2×#103×C53C2211yescharacteristic
57C2×#103×C53C2211yescharacteristic
58C2×#103×C53C2211yescharacteristic
59C2×C22⋊C8×C53C2211yescharacteristic
60C53×#380C2211yescharacteristic
61C22×Dic106—21no—
62C23×Dic53—21no—
63C22×#88322—22no—
64C22×#88322—22no—
65C22×#88322—41no—
66C22×#88322C2311yescharacteristic
67C22×#88322C2311yescharacteristic
68#13133216—22no—
69#13133216—41no—
70#13133194—44no—
71Q8×Dic53—22no—
72Q8×Dic53—22no—
73Q8×Dic53—41no—
74Q8×Dic53—41no—
75C2×C4⋊Dic53—21no—
76C2×C4⋊Dic53—21no—
77C2×C4⋊Dic53—21no—
78C2×C4⋊Dic53—21no—
79C2×C4⋊Dic53—22no—
80C2×C4⋊Dic53—41no—
81C2×C4⋊Dic53C2311yescharacteristic
82C2×C4⋊Dic53C2311yescharacteristic
83C2×C4⋊Dic53C2311yescharacteristic
84C2×C4⋊Dic53C2311yescharacteristic
85C2×#11719919—21no—
86C2×#11719919—21no—
87C2×#11719919—21no—
88C2×#11719919—21no—
89C2×C22⋊Dic53—21no—
90C2×C22⋊Dic53D411yescharacteristic
91C2×C22⋊Dic53D411yescharacteristic
92C2×C4×Dic53—21no—
93C2×C4×Dic53—21no—
94C2×C4×Dic53C2311yescharacteristic
95C2×C4×Dic53C2311yescharacteristic
96#13133149—22no—
97#13133149—41no—
98#13133148—22no—
99#13133148—41no—
100#13133146—22no—
101#13133146—41no—
102#13133141—22no—
103#13133141—22no—
104#13133141—22no—
105#13133141D414yes—
106#13133124—21no—
107#13133124—21no—
108#13133124—21no—
109#13133124—21no—
110#13133124—21no—
111#13133124—21no—
112#13133124—21no—
113#13133124D411yescharacteristic
114#13133124D411yescharacteristic
115C22×Q8×C53—21no—
116C22×Q8×C53—22no—
117C22×Q8×C53—22no—
118C22×Q8×C53—22no—
119C22×Q8×C53—41no—
120C22×Q8×C53—41no—
121C22×Q8×C53Q814yescomplemented
122C22×Q8×C53C2311yescharacteristic
123C22×Q8×C53C2311yescharacteristic
124C23×C4×C53—21no—
125C23×C4×C53C2311yescharacteristic
126C22×C8×C53C2311yescharacteristic
127C22×C8×C53C2311yescharacteristic
128C2×C4⋊C4×C53C2311yescharacteristic
129C2×C4⋊C4×C53C2311yescharacteristic
130C2×C22⋊C4×C53—21no—
131#103×C53—22no—
132#103×C53—22no—
133#103×C53—41no—
134#103×C53—41no—
135C22⋊C8×C53—22no—
136#95×C53—21no—
137C24×C53C2×D411yescharacteristic
138C2×Q8×C53—24no—
139C2×Q8×C53—21no—
140C2×Q8×C53—21no—
141C2×Q8×C53—21no—
142C2×Q8×C53—21no—
143C2×Q8×C53—28no—
144C2×Q8×C53—21no—
145C2×Q8×C53—21no—
146C2×Q8×C53—21no—
147C2×Q8×C53—21no—
148C2×Q8×C53—42no—
149C2×Q8×C53—42no—
150C2×Q8×C53—44no—
151C2×Q8×C53—44no—
152C2×Q8×C53—44no—
153C2×Q8×C53C2×D412yes—
154C2×Q8×C53C2×Q812yes—
155C2×Q8×C53C2×Q812yes—
156C2×Q8×C53#4112yes—
157C22×C4×C53—21no—
158C22×C4×C53—21no—
159C22×C4×C53—21no—
160C22×C4×C53—21no—
161C22×C4×C53—21no—
162C22×C4×C53—24no—
163C22×C4×C53—21no—
164C22×C4×C53—21no—
165C22×C4×C53—21no—
166C22×C4×C53—42no—
167C22×C4×C53—42no—
168C22×C4×C53C2×D411yescharacteristic
169C22×C4×C53C2×Q812yes—
170C22×C4×C53#4112yes—
171C22×C4×C53C2411yescharacteristic
172C22×Dic53—21no—
173C22×Dic53—21no—
174C22×Dic53—21no—
175C22×Dic53—21no—
176C22×Dic53—42no—
177C22×Dic53—42no—
178C22×Dic53—81no—
179C22×Dic53C2×D411yescharacteristic
180C22×Dic53C2×D411yescharacteristic
181C22×Dic53C2×D411yescharacteristic
182C22×Dic53C2×D411yescharacteristic
183C2×Dic106—42no—
184C2×Dic106—42no—
185C2×Dic106—81no—
186C2×C8×C53—21no—
187C2×C8×C53—21no—
188C2×C8×C53—21no—
189C2×C8×C53—21no—
190C2×C8×C53—21no—
191C2×C8×C53—21no—
192C4⋊C4×C53—21no—
193C4⋊C4×C53—21no—
194C4⋊C4×C53—21no—
195C4⋊C4×C53—21no—
196C22⋊C4×C53—42no—
197C22⋊Dic53—22no—
198C22⋊Dic53—42no—
199C22⋊Dic53SD1614yes—
200C4⋊Dic53—21no—
201C4⋊Dic53—21no—
202C4⋊Dic53—21no—
203C4⋊Dic53—21no—
204C4⋊Dic53—21no—
205C4⋊Dic53—21no—
206C4⋊Dic53—21no—
207C4⋊Dic53—42no—
208C4⋊Dic53—42no—
209C4⋊Dic53—42no—
210C4⋊Dic53—42no—
211C4⋊Dic53C2×D412yes—
212#11719919—42no—
213#11719919—42no—
214#11719919—42no—
215#11719919—81no—
216C4×Dic53—21no—
217C4×Dic53—21no—
218C4×Dic53—21no—
219C4×Dic53—21no—
220C4×Dic53—42no—
221C4×Dic53—42no—
222C2×#88322—21no—
223C2×#88322—21no—
224C2×#88322—21no—
225C2×#88322—21no—
226C2×#88322—21no—
227C2×#88322—44no—
228C2×#88322C2×D412yes—
229C23×C53—21no—
230C23×C53—21no—
231C23×C53—21no—
232C23×C53—21no—
233C23×C53C22⋊Q812yes—
234C23×C53C2×SD1612yes—
235C23×C53#13712yes—
236C23×C53C22×D411yescharacteristic
237Q8×C53—22no—
238Q8×C53—22no—
239Q8×C53—22no—
240Q8×C53—22no—
241Q8×C53—22no—
242Q8×C53—48no—
243Q8×C53—41no—
244Q8×C53—41no—
245Q8×C53C22⋊Q814yescomplemented
246C2×C4×C53—22no—
247C2×C4×C53—21no—
248C2×C4×C53—22no—
249C2×C4×C53—21no—
250C2×C4×C53—22no—
251C2×C4×C53—22no—
252C2×C4×C53—22no—
253C2×C4×C53—22no—
254C2×C4×C53—22no—
255C2×C4×C53—22no—
256C2×C4×C53—48no—
257C2×C4×C53—41no—
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×C53C22⋊Q812yes—
266C2×C4×C53C22×D411yescharacteristic
267C2×C4×C53C22×Q811yescharacteristic
268C2×C4×C53C2×#4111yescharacteristic
269C2×C4×C53C2×#4111yescharacteristic
270C2×C4×C53H(5,𝔽2)11yescharacteristic
271C2×C4×C53H(5,𝔽2)11yescharacteristic
272C2×Dic53—21no—
273C2×Dic53—21no—
274C2×Dic53—21no—
275C2×Dic53—21no—
276C2×Dic53—22no—
277C2×Dic53—21no—
278C2×Dic53—21no—
279C2×Dic53—21no—
280C2×Dic53—21no—
281C2×Dic53—21no—
282C2×Dic53—21no—
283C2×Dic53—41no—
284C2×Dic53—41no—
285C2×Dic53—41no—
286C2×Dic53—41no—
287C2×Dic53—41no—
288C2×Dic53—41no—
289C2×Dic53—41no—
290C2×Dic53—41no—
291C2×Dic53—41no—
292C2×Dic53—41no—
293C2×Dic53—41no—
294C2×Dic53—84no—
295C2×Dic53C2×SD1612yes—
296C2×Dic53C2×SD1612yes—
297Dic106—84no—
298C8×C53—41no—
299C8×C53—41no—
300#88322—22no—
301#88322—41no—
302#25955—531nomaximal, Sylow
303C22×C53—22no—
304C22×C53—22no—
305C22×C53—22no—
306C22×C53—22no—
307C22×C53—22no—
308C22×C53—22no—
309C22×C53#47514yescomplemented
310C22×C53C2×C22⋊Q811yescharacteristic
311C22×C53D4211yescharacteristic
312C22×C53#54611yescharacteristic
313C22×C53C22×SD1611yescharacteristic
314C22×C53C2×#13711yescharacteristic
315C22×C53#57711yescharacteristic
316C22×C53#57811yescharacteristic
317C4×C53—22no—
318C4×C53—24no—
319C4×C53—42no—
320C4×C53—81no—
321C4×C53—81no—
322C4×C53C2×C22⋊Q811yescharacteristic
323C4×C53#54311yescharacteristic
324C4×C53#54611yescharacteristic
325C4×C53#55111yescharacteristic
326Dic53—22no—
327Dic53—22no—
328Dic53—41no—
329Dic53—42no—
330Dic53—41no—
331Dic53—81no—
332Dic53—81no—
333Dic53—81no—
334C2×#475—531no—
335C22×#103—531no—
336#2440—531no—
337Q8×C22⋊C4—531no—
338#2096—531no—
339#2066—531no—
340#2066—531no—
341#2024—531no—
342#1865—531no—
343#1865—531no—
344#1853—531no—
345#1850—531no—
346#1807—531no—
347#1803—531no—
348#1787D5311yescharacteristic
349C2×C53—24no—
350C2×C53#243711yescharacteristic
351C2×C53C2×#47511yescharacteristic
352C2×C53#308211yescharacteristic
353C2×C53#327011yescharacteristic
354C2×C53D4×SD1611yescharacteristic
355C2×C53#329811yescharacteristic
356C2×C53#331611yescharacteristic
357C23×Q8C2×D5311yescharacteristic
358C23×C8—531no—
359C2×C4⋊Q8—531no—
360C2×C22⋊Q8—1061no—
361C2×C4×Q8—531no—
362C2×C4×Q8—531no—
363C22×C4⋊C4—531no—
364#475—1064no—
365C2×#106—531no—
366C2×#106—531no—
367C2×#106—531no—
368C2×#106—531no—
369C2×C4⋊C8—531no—
370C2×C4⋊C8—531no—
371C2×#103—531no—
372C2×#103—531no—
373C2×#103—531no—
374C2×#103—531no—
375C2×#103—1064no—
376C2×#103C2×D5311yescharacteristic
377C2×#103C2×D5311yescharacteristic
378C2×#103C2×D5311yescharacteristic
379C2×#103C2×D5311yescharacteristic
380C2×C22⋊C8C2×D5311yescharacteristic
381#398—1061no—
382#395—1061no—
383#391—531no—
384#391—531no—
385#391—1061no—
386#391—1061no—
387#389—531no—
388#389—531no—
389#385—1061no—
390#384—531no—
391#380—531no—
392#380—1061no—
393#380C2×D5311yescharacteristic
394C4×C22⋊C4—531no—
395C4×C22⋊C4—1061no—
396#340—531no—
397#340—531no—
398#340—531no—
399#340—531no—
400#336—531no—
401#336—531no—
402C53#2595511yescharacteristic, complemented, Sylow
403C22×Q8—21no—
404C22×Q8—22no—
405C22×Q8—22no—
406C22×Q8—22no—
407C22×Q8—41no—
408C22×Q8—41no—
409C22×Q8—1061no—
410C22×Q8Dic10614yescomplemented
411C22×Q8C22×D5311yescharacteristic
412C22×Q8C22×D5311yescharacteristic
413C23×C4—21no—
414C23×C4—1061no—
415C23×C4C22×D5311yescharacteristic
416C22×C8—531no—
417C22×C8—531no—
418C22×C8—1062no—
419C22×C8—1062no—
420C22×C8—2121no—
421C22×C8C22×D5311yescharacteristic
422C22×C8C22×D5311yescharacteristic
423C4⋊Q8—1062no—
424C4⋊Q8—2121no—
425C22⋊Q8—2124no—
426C4×Q8—1062no—
427C4×Q8—1062no—
428C4×Q8—2121no—
429C4×Q8—2121no—
430C2×C4⋊C4—531no—
431C2×C4⋊C4—531no—
432C2×C4⋊C4—531no—
433C2×C4⋊C4—531no—
434C2×C4⋊C4—1062no—
435C2×C4⋊C4—1061no—
436C2×C4⋊C4—1061no—
437C2×C4⋊C4—1061no—
438C2×C4⋊C4—1061no—
439C2×C4⋊C4—1061no—
440C2×C4⋊C4—1061no—
441C2×C4⋊C4—1061no—
442C2×C4⋊C4—1061no—
443C2×C4⋊C4—2121no—
444C2×C4⋊C4C22×D5311yescharacteristic
445C2×C4⋊C4C22×D5311yescharacteristic
446C2×C22⋊C4—21no—
447C2×C22⋊C4—531no—
448C2×C22⋊C4—531no—
449C2×C22⋊C4—1061no—
450C2×C42—531no—
451C2×C42—531no—
452C2×C42—1061no—
453C2×C42—1061no—
454#106—1062no—
455#106—1062no—
456#106—2121no—
457#106—2121no—
458C4⋊C8—1062no—
459C4⋊C8—2121no—
460#103—22no—
461#103—22no—
462#103—41no—
463#103—41no—
464#103—534no—
465#103—1062no—
466#103—1062no—
467#103—1062no—
468C22⋊C8—22no—
469#95—21no—
470#95—531no—
471#95—531no—
472#95—1061no—
473#95—1061no—
474#95—1061no—
475#95—1061no—
476#95—1061no—
477#95—1061no—
478#95—1061no—
479C24D4×D5311yescharacteristic
480C2×Q8—24no—
481C2×Q8—21no—
482C2×Q8—21no—
483C2×Q8—21no—
484C2×Q8—21no—
485C2×Q8—28no—
486C2×Q8—21no—
487C2×Q8—21no—
488C2×Q8—21no—
489C2×Q8—21no—
490C2×Q8—42no—
491C2×Q8—42no—
492C2×Q8—44no—
493C2×Q8—44no—
494C2×Q8—44no—
495C2×Q8—4241no—
496C2×Q8—2122no—
497C2×Q8—2122no—
498C2×Q8C2×Dic10612yes—
499C2×Q8C2×Dic10612yes—
500C2×Q8D4×D5312yes—
501C2×Q8#1171994712yes—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—24no—
508C22×C4—21no—
509C22×C4—21no—
510C22×C4—21no—
511C22×C4—42no—
512C22×C4—42no—
513C22×C4—531no—
514C22×C4—531no—
515C22×C4—531no—
516C22×C4—531no—
517C22×C4—1061no—
518C22×C4—1061no—
519C22×C4—1061no—
520C22×C4—1061no—
521C22×C4—4241no—
522C22×C4—2122no—
523C22×C4—2122no—
524C22×C4C2×Dic10612yes—
525C22×C4D4×D5311yescharacteristic
526C22×C4#1171994712yes—
527C22×C4C23×D5311yescharacteristic
528C2×C8—21no—
529C2×C8—21no—
530C2×C8—21no—
531C2×C8—21no—
532C2×C8—21no—
533C2×C8—21no—
534C2×C8—532no—
535C2×C8—1061no—
536C2×C8—1061no—
537C2×C8—1061no—
538C2×C8—1061no—
539C2×C8—1061no—
540C2×C8—2124no—
541C4⋊C4—21no—
542C4⋊C4—21no—
543C4⋊C4—21no—
544C4⋊C4—21no—
545C4⋊C4—532no—
546C4⋊C4—1061no—
547C4⋊C4—1061no—
548C4⋊C4—1061no—
549C4⋊C4—1061no—
550C4⋊C4—1061no—
551C4⋊C4—1061no—
552C4⋊C4—1061no—
553C4⋊C4—4241no—
554C4⋊C4—2122no—
555C4⋊C4—2122no—
556C4⋊C4—2122no—
557C4⋊C4—2122no—
558C4⋊C4—2122no—
559C4⋊C4—2122no—
560C4⋊C4—2122no—
561C22⋊C4—42no—
562C22⋊C4—534no—
563C22⋊C4—1062no—
564C22⋊C4—2122no—
565C42—1061no—
566C42—1061no—
567C42—1061no—
568C42—1061no—
569C42—2122no—
570C42—2122no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23#1313319412yes—
576C23SD16×D5312yes—
577C23#1313324912yes—
578C23C2×D4×D5311yescharacteristic
579Q8—22no—
580Q8—22no—
581Q8—22no—
582Q8—22no—
583Q8—22no—
584Q8—48no—
585Q8—41no—
586Q8—41no—
587Q8—4244no—
588Q8#1313319414yescomplemented
589C2×C4—22no—
590C2×C4—22no—
591C2×C4—21no—
592C2×C4—22no—
593C2×C4—21no—
594C2×C4—22no—
595C2×C4—22no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—22no—
599C2×C4—48no—
600C2×C4—41no—
601C2×C4—41no—
602C2×C4—41no—
603C2×C4—41no—
604C2×C4—41no—
605C2×C4—41no—
606C2×C4—41no—
607C2×C4—41no—
608C2×C4—532no—
609C2×C4—532no—
610C2×C4—1061no—
611C2×C4—1061no—
612C2×C4—1061no—
613C2×C4—1061no—
614C2×C4—1061no—
615C2×C4—1061no—
616C2×C4—1061no—
617C2×C4—1061no—
618C2×C4—1061no—
619C2×C4—1062no—
620C2×C4—1061no—
621C2×C4—4244no—
622C2×C4—2121no—
623C2×C4—2121no—
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#1313319412yes—
634C2×C4C2×D4×D5311yescharacteristic
635C2×C4C22×Dic10611yescharacteristic
636C2×C4C2×#1171994711yescharacteristic
637C2×C4C2×#1171994711yescharacteristic
638C2×C4#1313326211yescharacteristic
639C2×C4#1313326211yescharacteristic
640C8—41no—
641C8—41no—
642C8—1062no—
643C8—2121no—
644C22—22no—
645C22—22no—
646C22—22no—
647C22—22no—
648C22—22no—
649C22—22no—
650C22#1681090614yescomplemented
651C22C2×#1313319411yescharacteristic
652C22#1681125511yescharacteristic
653C22#1681126011yescharacteristic
654C22C2×SD16×D5311yescharacteristic
655C22C2×#1313324911yescharacteristic
656C22#1681143211yescharacteristic
657C22#1681143911yescharacteristic
658C4—22no—
659C4—24no—
660C4—42no—
661C4—81no—
662C4—81no—
663C4—1062no—
664C4—1062no—
665C4—4241no—
666C4—4241no—
667C4—4241no—
668C4—2121no—
669C4—2121no—
670C4—2122no—
671C4C2×#1313319411yescharacteristic
672C4#1681124911yescharacteristic
673C4#1681126011yescharacteristic
674C4#1681130411yescharacteristic
675C2—24no—
676C2#2027838111yescharacteristic
677C2C2×#1681090611yescharacteristic
678C2#2028455911yescharacteristic
679C2#2028628911yescharacteristic
680C2#2028640411yescharacteristic
681C2#2028657311yescharacteristic
682C2#2028683011yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1787C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
hypercenter#1787D53
hypocenterC53#25955
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1787

Complements

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

NH ≅ G/Ntype
1. C2×#16810906C2semidirect
5. #20276256C2semidirect
6. #20276094C2semidirect
7. #20276092C2semidirect
8. #20275854C2semidirect
9. #20274985C2semidirect
10. #20274981C2semidirect
14. #20274718C2semidirect
121. C22×Q8×C53Q8semidirect
245. Q8×C53C22⋊Q8semidirect
309. C22×C53#475semidirect
402. C53#25955semidirect
410. C22×Q8Dic106semidirect
588. Q8#13133194semidirect
650. C22#16810906semidirect

Maximal subgroups

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

subgroupindex
C2×#168109062
C22×#131331412
#202784182
Q8×C22⋊Dic532
#202762562
#202760942
#202760922
#202758542
#202749852
#202749812
#202749102
#202748922
#202747382
#202747182
C53×#17872
#2595553