Finite Group Databasebeta

← #27444680

Subgroups — #27444680

Order 13568.

844 conjugacy classes of subgroups.

The subgroup lattice has 704 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20277644C211yesmaximal, characteristic, complemented
2#20276283C211yesmaximal, characteristic, complemented
3#20276152C211yesmaximal, characteristic, complemented
4#20276128C211yesmaximal, characteristic, complemented
5#20275894C211yesmaximal, characteristic, complemented
6#20275339C211yesmaximal, characteristic, complemented
7#20275283C211yesmaximal, characteristic, complemented
8#20275144C211yesmaximal, characteristic, complemented
9#20275038C211yesmaximal, characteristic, complemented
10#20275033C211yesmaximal, characteristic, complemented
11#20275008C211yesmaximal, characteristic, complemented
12#20274914C211yesmaximal, characteristic, complemented
13#20274803C211yesmaximal, characteristic, complemented
14#20274431C211yesmaximal, characteristic, complemented
15C53×#2414C211yesmaximal, characteristic
16C23×#88322C2211yescharacteristic
17C2×#13133176C2211yescharacteristic
18C22×C4⋊Dic53C2211yescharacteristic
19C2×#13133150C2211yescharacteristic
20C2×#13133148C2211yescharacteristic
21C2×#13133138C2211yescharacteristic
22C2×#13133138C2211yescharacteristic
23C2×#13133138C2211yescharacteristic
24C2×#13133138C2211yescharacteristic
25C2×C22⋊#88322C2211yescharacteristic
26C2×C22⋊#88322C2211yescharacteristic
27C2×C22⋊#88322C2211yescharacteristic
28#16810516—21no—
29#16810509—21no—
30#16810503—21no—
31#16810489C2211yescharacteristic
32#16810483C2211yescharacteristic
33#16810483C2211yescharacteristic
34C4×C4⋊Dic53C2211yescharacteristic
35C22⋊C4×Dic53—21no—
36#16810385C2211yescharacteristic
37#16810385C2211yescharacteristic
38#16810385C2211yescharacteristic
39#16810385C2211yescharacteristic
40#16810385C2211yescharacteristic
41#16810385C2211yescharacteristic
42#16810385C2211yescharacteristic
43#16810385C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810378C2211yescharacteristic
47#16810378C2211yescharacteristic
48C2×#121×C53C2211yescharacteristic
49C2×#121×C53C2211yescharacteristic
50C2×C4×D4×C53C2211yescharacteristic
51C2×#117×C53C2211yescharacteristic
52C53×#394—21no—
53C53×#392—21no—
54C53×#390—21no—
55C53×#390C2211yescharacteristic
56C53×#389C2211yescharacteristic
57C53×#388—21no—
58C53×#380C2211yescharacteristic
59C23×Dic53—21no—
60C22×#88322—22no—
61C22×#88322—41no—
62C22×#88322—41no—
63C22×#88322C2311yescharacteristic
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67#13133176—22no—
68#13133176—41no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53—22no—
71C2×C4⋊Dic53—41no—
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C22⋊Dic53—21no—
77C2×C22⋊Dic53—21no—
78C2×C22⋊Dic53—21no—
79C2×C4×Dic53—21no—
80C2×C4×Dic53—21no—
81#13133150—22no—
82#13133148—41no—
83#13133138—22no—
84#13133138—41no—
85#13133138—41no—
86#13133138—41no—
87C22⋊#88322—22no—
88C22⋊#88322—41no—
89C22⋊#88322—41no—
90#13133124—21no—
91#13133124—21no—
92#13133124—21no—
93#13133124—21no—
94#13133124—21no—
95C22×D4×C53—21no—
96C22×D4×C53—21no—
97C22×D4×C53D411yescharacteristic
98C22×D4×C53C2311yescharacteristic
99C23×C4×C53C2311yescharacteristic
100C23×C4×C53C2311yescharacteristic
101#121×C53—22no—
102#121×C53—22no—
103#121×C53—22no—
104#121×C53—41no—
105C4×D4×C53—22no—
106C4×D4×C53—22no—
107#117×C53—22no—
108#117×C53—41no—
109C2×C4⋊C4×C53—21no—
110C2×C4⋊C4×C53D411yescharacteristic
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C22⋊C4×C53—21no—
115C2×C22⋊C4×C53—21no—
116C2×C22⋊C4×C53—21no—
117C2×C22⋊C4×C53—21no—
118C2×C22⋊C4×C53—21no—
119C2×C22⋊C4×C53—21no—
120C2×C22⋊C4×C53D411yescharacteristic
121C2×C22⋊C4×C53D411yescharacteristic
122C2×C42×C53—21no—
123C2×C42×C53C2311yescharacteristic
124#95×C53—21no—
125#95×C53—21no—
126C24×C53—21no—
127C24×C53C2×D411yescharacteristic
128C24×C53C2×D411yescharacteristic
129C2×D4×C53—22no—
130C2×D4×C53—22no—
131C2×D4×C53—21no—
132C2×D4×C53—21no—
133C2×D4×C53—21no—
134C2×D4×C53—21no—
135C2×D4×C53—21no—
136C2×D4×C53—21no—
137C2×D4×C53—41no—
138C2×D4×C53—42no—
139C2×D4×C53—42no—
140C2×D4×C53—42no—
141C2×D4×C53—81no—
142C2×D4×C53—81no—
143C2×D4×C53—81no—
144C22×C4×C53—21no—
145C22×C4×C53—21no—
146C22×C4×C53—21no—
147C22×C4×C53—21no—
148C22×C4×C53—21no—
149C22×C4×C53—21no—
150C22×C4×C53—21no—
151C22×C4×C53—21no—
152C22×C4×C53—21no—
153C22×C4×C53—21no—
154C22×C4×C53—21no—
155C22×C4×C53—21no—
156C22×C4×C53—21no—
157C22×C4×C53C2×D411yescharacteristic
158C22×C4×C53C2×D411yescharacteristic
159C22×C4×C53C2×D411yescharacteristic
160C22×C4×C53C2×D411yescharacteristic
161C22×C4×C53#4112yes—
162C22×C4×C53#4112yes—
163C22×C4×C53C2411yescharacteristic
164C22×Dic53—21no—
165C22×Dic53—21no—
166C22×Dic53—21no—
167C22×Dic53—21no—
168C22×Dic53—42no—
169C22×Dic53—42no—
170C22×Dic53—81no—
171C4⋊C4×C53—22no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—21no—
175C4⋊C4×C53—21no—
176C4⋊C4×C53—21no—
177C4⋊C4×C53—81no—
178C4⋊C4×C53#4112yes—
179C22⋊C4×C53—22no—
180C22⋊C4×C53—22no—
181C22⋊C4×C53—42no—
182C22⋊C4×C53—42no—
183C22⋊C4×C53—42no—
184C22⋊C4×C53—42no—
185C22⋊C4×C53—81no—
186C22⋊C4×C53—81no—
187C42×C53—21no—
188C42×C53—21no—
189C42×C53—42no—
190C22⋊Dic53—42no—
191C22⋊Dic53—42no—
192C22⋊Dic53—81no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198C4⋊Dic53—21no—
199C4⋊Dic53—21no—
200C4⋊Dic53—42no—
201C4⋊Dic53#4112yes—
202C4×Dic53—42no—
203C4×Dic53—81no—
204C2×#88322—21no—
205C2×#88322—21no—
206C2×#88322—21no—
207C2×#88322—21no—
208C2×#88322—21no—
209C2×#88322—21no—
210C2×#88322—21no—
211C2×#88322—21no—
212C2×#88322—21no—
213C2×#88322—21no—
214C2×#88322—21no—
215C2×#88322—21no—
216C2×#88322—44no—
217C23×C53—21no—
218C23×C53—21no—
219C23×C53—21no—
220C23×C53—21no—
221C23×C53—21no—
222C23×C53—21no—
223C23×C53—21no—
224C23×C53—21no—
225C23×C53—21no—
226C23×C53—21no—
227C23×C53—21no—
228C23×C53—21no—
229C23×C53—41no—
230C23×C53—41no—
231C23×C53—41no—
232C23×C53—41no—
233C23×C53—41no—
234C23×C53—41no—
235C23×C53—41no—
236C23×C53#12312yes—
237C23×C53#13512yes—
238C23×C53C22×D411yescharacteristic
239D4×C53—44no—
240D4×C53—41no—
241D4×C53—41no—
242D4×C53—41no—
243D4×C53—41no—
244D4×C53—84no—
245D4×C53—84no—
246C2×C4×C53—21no—
247C2×C4×C53—21no—
248C2×C4×C53—22no—
249C2×C4×C53—22no—
250C2×C4×C53—22no—
251C2×C4×C53—22no—
252C2×C4×C53—21no—
253C2×C4×C53—21no—
254C2×C4×C53—21no—
255C2×C4×C53—21no—
256C2×C4×C53—21no—
257C2×C4×C53—22no—
258C2×C4×C53—21no—
259C2×C4×C53—21no—
260C2×C4×C53—21no—
261C2×C4×C53—21no—
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×C53—41no—
272C2×C4×C53—41no—
273C2×C4×C53—41no—
274C2×C4×C53#12312yes—
275C2×C4×C53C2×#4111yescharacteristic
276C2×C4×C53C2×#4111yescharacteristic
277C2×C4×C53C2×#4111yescharacteristic
278C2×C4×C53C2×#4111yescharacteristic
279C2×C4×C53H(5,𝔽2)11yescharacteristic
280C2×C4×C53H(5,𝔽2)11yescharacteristic
281C2×Dic53—22no—
282C2×Dic53—41no—
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—84no—
294#88322—41no—
295#88322—41no—
296#88322—41no—
297#88322—41no—
298#26199—531nomaximal, Sylow
299C22×C53—22no—
300C22×C53—22no—
301C22×C53—22no—
302C22×C53—22no—
303C22×C53—22no—
304C22×C53—22no—
305C22×C53—41no—
306C22×C53—41no—
307C22×C53—41no—
308C22×C53—41no—
309C22×C53—41no—
310C22×C53—41no—
311C22×C53—41no—
312C22×C53—41no—
313C22×C53—81no—
314C22×C53—81no—
315C22×C53—81no—
316C22×C53—81no—
317C22×C53—81no—
318C22×C53—81no—
319C22×C53—81no—
320C22×C53C2×#12311yescharacteristic
321C22×C53#54611yescharacteristic
322C22×C53#54611yescharacteristic
323C22×C53C2×#13511yescharacteristic
324C22×C53#57511yescharacteristic
325C22×C53#57611yescharacteristic
326C22×C53#57711yescharacteristic
327C4×C53—22no—
328C4×C53—41no—
329C4×C53—41no—
330C4×C53—41no—
331C4×C53—42no—
332C4×C53—81no—
333C4×C53—81no—
334C4×C53#52911yescharacteristic
335C4×C53#53811yescharacteristic
336C4×C53#55311yescharacteristic
337C4×C53#55311yescharacteristic
338Dic53—42no—
339Dic53—81no—
340Dic53—81no—
341Dic53—81no—
342#2414D5311yescharacteristic
343#2352—531no—
344#2101—531no—
345#2077—531no—
346#2073—531no—
347#2033—531no—
348#1926—531no—
349#1918—531no—
350#1893—531no—
351#1875—531no—
352#1874—531no—
353#1869—531no—
354#1854—531no—
355#1824—531no—
356#1735—531no—
357C2×C53—42no—
358C2×C53—42no—
359C2×C53—161no—
360C2×C53#244411yescharacteristic
361C2×C53#309111yescharacteristic
362C2×C53#309611yescharacteristic
363C2×C53#329711yescharacteristic
364C2×C53#330111yescharacteristic
365C2×C53#331711yescharacteristic
366C2×C53#331911yescharacteristic
367C23×C8—531no—
368C2×#121C2×D5311yescharacteristic
369C2×#121C2×D5311yescharacteristic
370C2×C4×D4C2×D5311yescharacteristic
371C2×#117—531no—
372C2×#117C2×D5311yescharacteristic
373C22×C4⋊C4—531no—
374C2×#107—531no—
375C2×#106—531no—
376C2×#102—531no—
377C2×#102—531no—
378C2×#102—531no—
379C2×#102—531no—
380C2×C22⋊C8—531no—
381C2×C22⋊C8—531no—
382C2×C22⋊C8—531no—
383#394—21no—
384#392—21no—
385#390—21no—
386#390C2×D5311yescharacteristic
387#389C2×D5311yescharacteristic
388#388—21no—
389#388—1061no—
390#387—1061no—
391#386—1061no—
392#382—531no—
393#380—531no—
394#380—531no—
395#380C2×D5311yescharacteristic
396C4×C4⋊C4—531no—
397C4×C22⋊C4—1061no—
398#340—531no—
399#340—531no—
400#340—531no—
401#340—531no—
402#340—531no—
403#340—531no—
404#340—531no—
405#340—531no—
406#340—531no—
407#340—531no—
408#336—531no—
409#336—531no—
410C53#2619911yescharacteristic, complemented, Sylow
411C22×D4—21no—
412C22×D4—21no—
413C22×D4C22⋊D5311yescharacteristic
414C22×D4C22×D5311yescharacteristic
415C23×C4—1061no—
416C23×C4C22×D5311yescharacteristic
417C23×C4C22×D5311yescharacteristic
418C22×C8—531no—
419C22×C8—531no—
420C22×C8—531no—
421C22×C8—531no—
422C22×C8—1062no—
423C22×C8—2121no—
424C22×C8—2121no—
425#121—22no—
426#121—22no—
427#121—22no—
428#121—41no—
429C4×D4—22no—
430C4×D4—22no—
431#117—22no—
432#117—41no—
433#117—1062no—
434#117—2121no—
435C2×C4⋊C4—21no—
436C2×C4⋊C4—531no—
437C2×C4⋊C4—531no—
438C2×C4⋊C4—531no—
439C2×C4⋊C4—531no—
440C2×C4⋊C4—1062no—
441C2×C4⋊C4—1061no—
442C2×C4⋊C4—2121no—
443C2×C4⋊C4C22⋊D5311yescharacteristic
444C2×C4⋊C4C22×D5311yescharacteristic
445C2×C4⋊C4C22×D5311yescharacteristic
446C2×C4⋊C4C22×D5311yescharacteristic
447C2×C22⋊C4—21no—
448C2×C22⋊C4—21no—
449C2×C22⋊C4—21no—
450C2×C22⋊C4—21no—
451C2×C22⋊C4—21no—
452C2×C22⋊C4—21no—
453C2×C22⋊C4—1061no—
454C2×C22⋊C4—1061no—
455C2×C22⋊C4—1061no—
456C2×C22⋊C4C22⋊D5311yescharacteristic
457C2×C22⋊C4C22⋊D5311yescharacteristic
458C2×C42—21no—
459C2×C42—1061no—
460C2×C42—1061no—
461C2×C42C22×D5311yescharacteristic
462#107—1062no—
463#106—2121no—
464#102—1062no—
465#102—2121no—
466#102—2121no—
467#102—2121no—
468C22⋊C8—1062no—
469C22⋊C8—2121no—
470C22⋊C8—2121no—
471#95—21no—
472#95—21no—
473#95—1061no—
474#95—1061no—
475#95—1061no—
476#95—1061no—
477#95—1061no—
478C24—21no—
479C24C2×C22⋊D5311yescharacteristic
480C24C2×C22⋊D5311yescharacteristic
481C2×D4—22no—
482C2×D4—22no—
483C2×D4—21no—
484C2×D4—21no—
485C2×D4—21no—
486C2×D4—21no—
487C2×D4—21no—
488C2×D4—21no—
489C2×D4—41no—
490C2×D4—42no—
491C2×D4—42no—
492C2×D4—42no—
493C2×D4—81no—
494C2×D4—81no—
495C2×D4—81no—
496C22×C4—21no—
497C22×C4—21no—
498C22×C4—21no—
499C22×C4—21no—
500C22×C4—21no—
501C22×C4—21no—
502C22×C4—21no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—21no—
506C22×C4—21no—
507C22×C4—21no—
508C22×C4—21no—
509C22×C4—1061no—
510C22×C4—1061no—
511C22×C4—1061no—
512C22×C4—1061no—
513C22×C4—4241no—
514C22×C4—2122no—
515C22×C4—2122no—
516C22×C4#1171994712yes—
517C22×C4#1171994712yes—
518C22×C4C2×C22⋊D5311yescharacteristic
519C22×C4C2×C22⋊D5311yescharacteristic
520C22×C4C2×C22⋊D5311yescharacteristic
521C22×C4C2×C22⋊D5311yescharacteristic
522C22×C4C23×D5311yescharacteristic
523C2×C8—1061no—
524C2×C8—1061no—
525C2×C8—1061no—
526C2×C8—1061no—
527C2×C8—1061no—
528C2×C8—1061no—
529C2×C8—1061no—
530C2×C8—1061no—
531C2×C8—1061no—
532C2×C8—1061no—
533C2×C8—1061no—
534C2×C8—1061no—
535C2×C8—2124no—
536C4⋊C4—22no—
537C4⋊C4—21no—
538C4⋊C4—21no—
539C4⋊C4—21no—
540C4⋊C4—21no—
541C4⋊C4—21no—
542C4⋊C4—81no—
543C4⋊C4—532no—
544C4⋊C4—1061no—
545C4⋊C4—1061no—
546C4⋊C4—1061no—
547C4⋊C4—1061no—
548C4⋊C4—1061no—
549C4⋊C4—1061no—
550C4⋊C4—1061no—
551C4⋊C4—2122no—
552C4⋊C4#1171994712yes—
553C22⋊C4—22no—
554C22⋊C4—22no—
555C22⋊C4—42no—
556C22⋊C4—42no—
557C22⋊C4—42no—
558C22⋊C4—42no—
559C22⋊C4—81no—
560C22⋊C4—81no—
561C22⋊C4—4241no—
562C22⋊C4—2122no—
563C22⋊C4—2122no—
564C42—21no—
565C42—21no—
566C42—42no—
567C42—4241no—
568C42—2122no—
569C23—21no—
570C23—21no—
571C23—21no—
572C23—21no—
573C23—21no—
574C23—21no—
575C23—21no—
576C23—21no—
577C23—21no—
578C23—21no—
579C23—21no—
580C23—21no—
581C23—41no—
582C23—41no—
583C23—41no—
584C23—41no—
585C23—41no—
586C23—41no—
587C23—41no—
588C23#1313319812yes—
589C23#1313323712yes—
590C23C22×C22⋊D5311yescharacteristic
591D4—44no—
592D4—41no—
593D4—41no—
594D4—41no—
595D4—41no—
596D4—84no—
597D4—84no—
598C2×C4—21no—
599C2×C4—21no—
600C2×C4—22no—
601C2×C4—22no—
602C2×C4—22no—
603C2×C4—22no—
604C2×C4—22no—
605C2×C4—21no—
606C2×C4—21no—
607C2×C4—21no—
608C2×C4—21no—
609C2×C4—21no—
610C2×C4—21no—
611C2×C4—21no—
612C2×C4—21no—
613C2×C4—21no—
614C2×C4—41no—
615C2×C4—41no—
616C2×C4—41no—
617C2×C4—41no—
618C2×C4—41no—
619C2×C4—41no—
620C2×C4—41no—
621C2×C4—41no—
622C2×C4—41no—
623C2×C4—41no—
624C2×C4—41no—
625C2×C4—41no—
626C2×C4—1062no—
627C2×C4—4244no—
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#1313319812yes—
640C2×C4#41×D5311yescharacteristic
641C2×C4C2×#1171994711yescharacteristic
642C2×C4C2×#1171994711yescharacteristic
643C2×C4C2×#1171994711yescharacteristic
644C2×C4#1313326211yescharacteristic
645C2×C4#1313326311yescharacteristic
646C8—2121no—
647C8—2121no—
648C8—2121no—
649C8—2121no—
650C22—22no—
651C22—22no—
652C22—22no—
653C22—22no—
654C22—22no—
655C22—22no—
656C22—41no—
657C22—41no—
658C22—41no—
659C22—41no—
660C22—41no—
661C22—41no—
662C22—41no—
663C22—41no—
664C22—81no—
665C22—81no—
666C22—81no—
667C22—81no—
668C22—81no—
669C22—81no—
670C22—81no—
671C22C2×#1313319811yescharacteristic
672C22#1681126111yescharacteristic
673C22#1681126711yescharacteristic
674C22C2×#1313323711yescharacteristic
675C22#1681142111yescharacteristic
676C22#1681142611yescharacteristic
677C22#1681143411yescharacteristic
678C4—22no—
679C4—42no—
680C4—41no—
681C4—41no—
682C4—41no—
683C4—81no—
684C4—81no—
685C4—4241no—
686C4—4241no—
687C4—4241no—
688C4—2122no—
689C4#1681115411yescharacteristic
690C4#1681121011yescharacteristic
691C4#1681131611yescharacteristic
692C4#1681132511yescharacteristic
693C2—42no—
694C2—42no—
695C2—161no—
696C2#2027846111yescharacteristic
697C2#2028462411yescharacteristic
698C2#2028466711yescharacteristic
699C2#2028656511yescharacteristic
700C2#2028661111yescharacteristic
701C2#2028684111yescharacteristic
702C2#2028686511yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
Fitting, prime radicalC53×#2414C2
socleC23×C53C22×D4
hypercenter#2414D53
hypocenterC53#26199
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #2414

Complements

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

NH ≅ G/Ntype
1. #20277644C2semidirect
2. #20276283C2semidirect
3. #20276152C2semidirect
4. #20276128C2semidirect
5. #20275894C2semidirect
6. #20275339C2semidirect
7. #20275283C2semidirect
8. #20275144C2semidirect
9. #20275038C2semidirect
10. #20275033C2semidirect
11. #20275008C2semidirect
12. #20274914C2semidirect
13. #20274803C2semidirect
14. #20274431C2semidirect
410. C53#26199semidirect

Maximal subgroups

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

subgroupindex
#202776442
#202762832
#202761522
#202761282
#202758942
#202753392
#202752832
#202751442
#202750382
#202750332
#202750082
#202749142
#202748032
#202744312
C53×#24142
#2619953