Finite Group Databasebeta

← #27447647

Subgroups — #27447647

Order 13568.

902 conjugacy classes of subgroups.

The subgroup lattice has 672 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20278411C211yesmaximal, characteristic, complemented
2#20277735C211yesmaximal, characteristic
3#20276276C211yesmaximal, characteristic, complemented
4#20276205C211yesmaximal, characteristic, complemented
5#20276101C211yesmaximal, characteristic, complemented
6#20275849C211yesmaximal, characteristic, complemented
7#20275665C211yesmaximal, characteristic, complemented
8#20275285C211yesmaximal, characteristic, complemented
9#20275144C211yesmaximal, characteristic
10#20274964C211yesmaximal, characteristic, complemented
11#20274905C211yesmaximal, characteristic, complemented
12#20274892C211yesmaximal, characteristic, complemented
13#20274713C211yesmaximal, characteristic, complemented
14C2×#16810385C211yesmaximal, characteristic, complemented
15C53×#1874C211yesmaximal, characteristic
16C23×#88322C2211yescharacteristic
17C2×#13133216C2211yescharacteristic
18C2×#13133196—21no—
19C2×D4×Dic53—21no—
20C2×D4×Dic53C2211yescharacteristic
21C22×C4⋊Dic53C2211yescharacteristic
22C2×#13133151C2211yescharacteristic
23C2×#13133151C2211yescharacteristic
24C2×#13133146C2211yescharacteristic
25C2×#13133146C2211yescharacteristic
26C2×#13133138C2211yescharacteristic
27C2×#13133138C2211yescharacteristic
28#16810556—21no—
29#16810543—21no—
30#16810526C2211yescharacteristic
31#16810522C2211yescharacteristic
32#16810522C2211yescharacteristic
33#16810519C2211yescharacteristic
34#16810516—21no—
35#16810503—21no—
36#16810498—21no—
37#16810496—21no—
38#16810492C2211yescharacteristic
39#16810483C2211yescharacteristic
40C4⋊C4×Dic53C2211yescharacteristic
41C4×C22⋊Dic53C2211yescharacteristic
42#16810386C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810385C2211yescharacteristic
45#16810385C2211yescharacteristic
46#16810385C2211yescharacteristic
47#16810385C2211yescharacteristic
48#16810385C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810377C2211yescharacteristic
51#16810377C2211yescharacteristic
52C2×#121×C53C2211yescharacteristic
53C22×C4⋊C4×C53C2211yescharacteristic
54C2×#102×C53C2211yescharacteristic
55C2×#102×C53C2211yescharacteristic
56C2×C22⋊C8×C53C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C22×Dic106—21no—
60C23×Dic53—21no—
61C23×Dic53—21no—
62C22×#88322—22no—
63C22×#88322—41no—
64C22×#88322—41no—
65C22×#88322C2311yescharacteristic
66C22×#88322C2311yescharacteristic
67#13133216—22no—
68#13133216—41no—
69#13133196—44no—
70D4×Dic53—22no—
71D4×Dic53—41no—
72D4×Dic53—44no—
73C2×C4⋊Dic53—21no—
74C2×C4⋊Dic53—21no—
75C2×C4⋊Dic53—21no—
76C2×C4⋊Dic53—22no—
77C2×C4⋊Dic53—41no—
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×C4⋊Dic53C2311yescharacteristic
82C2×#11719919—21no—
83C2×#11719919—21no—
84C2×#11719919—21no—
85C2×#11719919—21no—
86C2×C22⋊Dic53—21no—
87C2×C22⋊Dic53—21no—
88C2×C22⋊Dic53—21no—
89C2×C22⋊Dic53—21no—
90C2×C22⋊Dic53D411yescharacteristic
91C2×C22⋊Dic53D411yescharacteristic
92C2×C4×Dic53—21no—
93C2×C4×Dic53C2311yescharacteristic
94C2×C4×Dic53C2311yescharacteristic
95#13133151—22no—
96#13133151—41no—
97#13133146—41no—
98#13133146—41no—
99#13133138—41no—
100#13133138—41no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124—21no—
105#13133124—21no—
106#13133124D411yescharacteristic
107#13133124D411yescharacteristic
108C22×D4×C53—21no—
109C22×D4×C53C2311yescharacteristic
110C23×C4×C53—21no—
111C23×C4×C53C2311yescharacteristic
112C22×C8×C53C2311yescharacteristic
113C22×C8×C53C2311yescharacteristic
114#121×C53—22no—
115#121×C53Q814yescomplemented
116C2×C4⋊C4×C53—21no—
117C2×C4⋊C4×C53—22no—
118C2×C4⋊C4×C53—41no—
119C2×C4⋊C4×C53C2311yescharacteristic
120C2×C4⋊C4×C53C2311yescharacteristic
121C2×C4⋊C4×C53C2311yescharacteristic
122C2×C22⋊C4×C53—21no—
123#102×C53—22no—
124#102×C53—41no—
125C22⋊C8×C53—22no—
126C24×C53—21no—
127C24×C53C2×D411yescharacteristic
128C2×D4×C53—24no—
129C2×D4×C53—21no—
130C2×D4×C53—21no—
131C2×D4×C53—21no—
132C2×D4×C53—21no—
133C2×D4×C53—21no—
134C2×D4×C53—42no—
135C2×D4×C53—42no—
136C2×D4×C53C2×Q812yes—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—21no—
141C22×C4×C53—21no—
142C22×C4×C53—21no—
143C22×C4×C53—21no—
144C22×C4×C53—42no—
145C22×C4×C53—42no—
146C22×C4×C53—81no—
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2×Q812yes—
149C22×C4×C53#4112yes—
150C22×C4×C53C2411yescharacteristic
151C22×Dic53—21no—
152C22×Dic53—21no—
153C22×Dic53—21no—
154C22×Dic53—21no—
155C22×Dic53—42no—
156C22×Dic53—42no—
157C22×Dic53—42no—
158C22×Dic53—42no—
159C22×Dic53—81no—
160C22×Dic53—81no—
161C22×Dic53C2×D411yescharacteristic
162C22×Dic53C2×D411yescharacteristic
163C22×Dic53C2×D411yescharacteristic
164C22×Dic53C2×D411yescharacteristic
165C2×Dic106—42no—
166C2×Dic106—42no—
167C2×Dic106—81no—
168C2×C8×C53—21no—
169C2×C8×C53—21no—
170C2×C8×C53—21no—
171C2×C8×C53—21no—
172C2×C8×C53—21no—
173C2×C8×C53—21no—
174C4⋊C4×C53—21no—
175C4⋊C4×C53—21no—
176C4⋊C4×C53—21no—
177C4⋊C4×C53—21no—
178C4⋊C4×C53—42no—
179C4⋊C4×C53C2×Q812yes—
180C4⋊C4×C53#4112yes—
181C22⋊C4×C53—24no—
182C22⋊Dic53—22no—
183C22⋊Dic53—42no—
184C22⋊Dic53—42no—
185C22⋊Dic53—42no—
186C22⋊Dic53—81no—
187C22⋊Dic53D814yes—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—42no—
196C4⋊Dic53—42no—
197C4⋊Dic53—42no—
198C4⋊Dic53#4112yes—
199#11719919—42no—
200#11719919—42no—
201#11719919—42no—
202#11719919—81no—
203C4×Dic53—21no—
204C4×Dic53—21no—
205C4×Dic53—21no—
206C4×Dic53—21no—
207C4×Dic53—42no—
208C2×#88322—21no—
209C2×#88322—21no—
210C2×#88322—21no—
211C2×#88322—21no—
212C2×#88322—21no—
213C2×#88322—21no—
214C2×#88322—44no—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53—21no—
218C23×C53—21no—
219C23×C53—22no—
220C23×C53—41no—
221C23×C53—41no—
222C23×C53—41no—
223C23×C53—41no—
224C23×C53—41no—
225C23×C53—41no—
226C23×C53C22⋊Q812yes—
227C23×C53C2×D812yes—
228C23×C53#13712yes—
229C23×C53C22×D411yescharacteristic
230D4×C53—22no—
231D4×C53—22no—
232D4×C53—48no—
233D4×C53—41no—
234D4×C53—41no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—22no—
238C2×C4×C53—22no—
239C2×C4×C53—22no—
240C2×C4×C53—22no—
241C2×C4×C53—22no—
242C2×C4×C53—22no—
243C2×C4×C53—41no—
244C2×C4×C53—41no—
245C2×C4×C53—41no—
246C2×C4×C53—41no—
247C2×C4×C53—41no—
248C2×C4×C53—41no—
249C2×C4×C53—41no—
250C2×C4×C53—84no—
251C2×C4×C53C22⋊Q812yes—
252C2×C4×C53C22×Q811yescharacteristic
253C2×C4×C53C2×#4111yescharacteristic
254C2×C4×C53C2×#4111yescharacteristic
255C2×C4×C53C2×#4111yescharacteristic
256C2×C4×C53H(5,𝔽2)11yescharacteristic
257C2×C4×C53H(5,𝔽2)11yescharacteristic
258C2×Dic53—21no—
259C2×Dic53—21no—
260C2×Dic53—21no—
261C2×Dic53—21no—
262C2×Dic53—22no—
263C2×Dic53—21no—
264C2×Dic53—21no—
265C2×Dic53—21no—
266C2×Dic53—21no—
267C2×Dic53—21no—
268C2×Dic53—21no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—41no—
275C2×Dic53—41no—
276C2×Dic53—41no—
277C2×Dic53—41no—
278C2×Dic53—41no—
279C2×Dic53—41no—
280C2×Dic53—84no—
281C2×Dic53—84no—
282C2×Dic53C2×D812yes—
283C2×Dic53C2×D812yes—
284Dic106—84no—
285C8×C53—41no—
286C8×C53—41no—
287#88322—41no—
288#88322—41no—
289#26408—531nomaximal, Sylow
290C22×C53—22no—
291C22×C53—22no—
292C22×C53—22no—
293C22×C53—22no—
294C22×C53—22no—
295C22×C53—22no—
296C22×C53—42no—
297C22×C53—42no—
298C22×C53—42no—
299C22×C53—81no—
300C22×C53—81no—
301C22×C53—81no—
302C22×C53—81no—
303C22×C53#47414yes—
304C22×C53C2×C22⋊Q811yescharacteristic
305C22×C53#54611yescharacteristic
306C22×C53#54611yescharacteristic
307C22×C53C22×D811yescharacteristic
308C22×C53C2×#13711yescharacteristic
309C22×C53#57711yescharacteristic
310C22×C53#57711yescharacteristic
311C4×C53—22no—
312C4×C53—42no—
313C4×C53—42no—
314C4×C53—81no—
315C4×C53#52911yescharacteristic
316C4×C53#53811yescharacteristic
317C4×C53#55411yescharacteristic
318C4×C53#55411yescharacteristic
319Dic53—22no—
320Dic53—22no—
321Dic53—42no—
322Dic53—41no—
323Dic53—41no—
324Dic53—81no—
325Dic53—81no—
326Dic53—81no—
327#2440—531no—
328#2361—531no—
329#2099—531no—
330#2086—531no—
331#2067—531no—
332#2023—531no—
333#1988—531no—
334#1918—531no—
335#1893—531no—
336#1874D5311yescharacteristic
337#1862—531no—
338#1852—531no—
339#1850—531no—
340#1802—531no—
341C2×#340—531no—
342C2×C53—24no—
343C2×C53—82no—
344C2×C53#245011yescharacteristic
345C2×C53C2×#47411yescharacteristic
346C2×C53#308411yescharacteristic
347C2×C53#329411yescharacteristic
348C2×C53#329411yescharacteristic
349C2×C53#331611yescharacteristic
350C2×C53#332011yescharacteristic
351C23×C8—531no—
352C2×C4⋊Q8—531no—
353C2×C22⋊Q8—1061no—
354C2×#121C2×D5311yescharacteristic
355C2×C4×D4—531no—
356C2×C4×D4—1061no—
357C22×C4⋊C4—531no—
358C22×C4⋊C4C2×D5311yescharacteristic
359C2×#107—531no—
360C2×#107—531no—
361C2×C4⋊C8—531no—
362C2×C4⋊C8—531no—
363C2×#102—531no—
364C2×#102—531no—
365C2×#102C2×D5311yescharacteristic
366C2×#102C2×D5311yescharacteristic
367C2×C22⋊C8C2×D5311yescharacteristic
368#398—1061no—
369#395—1061no—
370#390—531no—
371#389—531no—
372#389—531no—
373#389—531no—
374#388—1061no—
375#386—1061no—
376#385—1061no—
377#385—1061no—
378#384—531no—
379#380—531no—
380C4×C4⋊C4—531no—
381C4×C22⋊C4—531no—
382#340—531no—
383#340—531no—
384#340—531no—
385#340—531no—
386#340—531no—
387#340—531no—
388#340—531no—
389#340—531no—
390#340C2×D5311yescharacteristic
391#340C2×D5311yescharacteristic
392#336—531no—
393#336—531no—
394C53#2640811yescharacteristic, complemented, Sylow
395C22×Q8—1061no—
396C22×D4—21no—
397C22×D4C22×D5311yescharacteristic
398C23×C4—21no—
399C23×C4—1061no—
400C23×C4—1061no—
401C23×C4C22×D5311yescharacteristic
402C22×C8—531no—
403C22×C8—531no—
404C22×C8—1062no—
405C22×C8—2121no—
406C22×C8—2121no—
407C22×C8C22×D5311yescharacteristic
408C22×C8C22×D5311yescharacteristic
409C4⋊Q8—1062no—
410C4⋊Q8—2121no—
411C22⋊Q8—2124no—
412#121—22no—
413#121Dic10614yescomplemented
414C4×D4—1062no—
415C4×D4—2121no—
416C4×D4—2124no—
417C2×C4⋊C4—22no—
418C2×C4⋊C4—21no—
419C2×C4⋊C4—41no—
420C2×C4⋊C4—531no—
421C2×C4⋊C4—531no—
422C2×C4⋊C4—531no—
423C2×C4⋊C4—531no—
424C2×C4⋊C4—1062no—
425C2×C4⋊C4—1061no—
426C2×C4⋊C4—1061no—
427C2×C4⋊C4—1061no—
428C2×C4⋊C4—1061no—
429C2×C4⋊C4—1061no—
430C2×C4⋊C4—1061no—
431C2×C4⋊C4—1061no—
432C2×C4⋊C4—2121no—
433C2×C4⋊C4C22×D5311yescharacteristic
434C2×C4⋊C4C22×D5311yescharacteristic
435C2×C4⋊C4C22×D5311yescharacteristic
436C2×C22⋊C4—21no—
437C2×C22⋊C4—531no—
438C2×C22⋊C4—531no—
439C2×C22⋊C4—1061no—
440C2×C22⋊C4—1061no—
441C2×C22⋊C4—1061no—
442C2×C22⋊C4—1061no—
443C2×C42—531no—
444C2×C42—531no—
445C2×C42—1061no—
446#107—1062no—
447#107—2121no—
448C4⋊C8—2121no—
449C4⋊C8—2121no—
450#102—22no—
451#102—41no—
452#102—2121no—
453#102—2121no—
454C22⋊C8—22no—
455#95—531no—
456#95—531no—
457#95—1061no—
458#95—1061no—
459#95—1061no—
460#95—1061no—
461#95—1061no—
462C24—21no—
463C24D4×D5311yescharacteristic
464C2×Q8—4241no—
465C2×Q8—2122no—
466C2×Q8—2122no—
467C2×D4—24no—
468C2×D4—21no—
469C2×D4—21no—
470C2×D4—21no—
471C2×D4—21no—
472C2×D4—21no—
473C2×D4—42no—
474C2×D4—42no—
475C2×D4C2×Dic10612yes—
476C22×C4—21no—
477C22×C4—21no—
478C22×C4—21no—
479C22×C4—21no—
480C22×C4—21no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—42no—
484C22×C4—42no—
485C22×C4—81no—
486C22×C4—531no—
487C22×C4—531no—
488C22×C4—531no—
489C22×C4—531no—
490C22×C4—1061no—
491C22×C4—1061no—
492C22×C4—1061no—
493C22×C4—1061no—
494C22×C4—4241no—
495C22×C4—4241no—
496C22×C4—2122no—
497C22×C4—2122no—
498C22×C4—2122no—
499C22×C4—2122no—
500C22×C4C2×Dic10612yes—
501C22×C4D4×D5311yescharacteristic
502C22×C4#1171994712yes—
503C22×C4C23×D5311yescharacteristic
504C2×C8—21no—
505C2×C8—21no—
506C2×C8—21no—
507C2×C8—21no—
508C2×C8—21no—
509C2×C8—21no—
510C2×C8—1061no—
511C2×C8—1061no—
512C2×C8—1061no—
513C2×C8—1061no—
514C2×C8—1061no—
515C2×C8—1061no—
516C2×C8—2124no—
517C4⋊C4—21no—
518C4⋊C4—21no—
519C4⋊C4—21no—
520C4⋊C4—21no—
521C4⋊C4—42no—
522C4⋊C4—532no—
523C4⋊C4—1061no—
524C4⋊C4—1061no—
525C4⋊C4—1061no—
526C4⋊C4—1061no—
527C4⋊C4—1061no—
528C4⋊C4—1061no—
529C4⋊C4—1061no—
530C4⋊C4—4241no—
531C4⋊C4—2122no—
532C4⋊C4—2122no—
533C4⋊C4—2122no—
534C4⋊C4—2122no—
535C4⋊C4—2122no—
536C4⋊C4—2122no—
537C4⋊C4C2×Dic10612yes—
538C4⋊C4#1171994712yes—
539C22⋊C4—24no—
540C22⋊C4—534no—
541C22⋊C4—1062no—
542C22⋊C4—4241no—
543C22⋊C4—2122no—
544C22⋊C4—2122no—
545C22⋊C4—2122no—
546C42—1061no—
547C42—1061no—
548C42—1061no—
549C42—1061no—
550C42—2122no—
551C23—21no—
552C23—21no—
553C23—21no—
554C23—21no—
555C23—22no—
556C23—41no—
557C23—41no—
558C23—41no—
559C23—41no—
560C23—41no—
561C23—41no—
562C23#1313319412yes—
563C23D8×D5312yes—
564C23#1313324612yes—
565C23C2×D4×D5311yescharacteristic
566Q8—4244no—
567D4—22no—
568D4—22no—
569D4—48no—
570D4—41no—
571D4—41no—
572C2×C4—21no—
573C2×C4—21no—
574C2×C4—22no—
575C2×C4—22no—
576C2×C4—22no—
577C2×C4—22no—
578C2×C4—22no—
579C2×C4—22no—
580C2×C4—41no—
581C2×C4—41no—
582C2×C4—41no—
583C2×C4—41no—
584C2×C4—41no—
585C2×C4—41no—
586C2×C4—41no—
587C2×C4—84no—
588C2×C4—532no—
589C2×C4—532no—
590C2×C4—1061no—
591C2×C4—1061no—
592C2×C4—1061no—
593C2×C4—1061no—
594C2×C4—1061no—
595C2×C4—1062no—
596C2×C4—1061no—
597C2×C4—1061no—
598C2×C4—1061no—
599C2×C4—1061no—
600C2×C4—1061no—
601C2×C4—4244no—
602C2×C4—4244no—
603C2×C4—2121no—
604C2×C4—2121no—
605C2×C4—2121no—
606C2×C4—2121no—
607C2×C4—2121no—
608C2×C4—2121no—
609C2×C4—2121no—
610C2×C4—2121no—
611C2×C4—2121no—
612C2×C4—2121no—
613C2×C4—2121no—
614C2×C4#1313319412yes—
615C2×C4C22×Dic10611yescharacteristic
616C2×C4#41×D5311yescharacteristic
617C2×C4C2×#1171994711yescharacteristic
618C2×C4C2×#1171994711yescharacteristic
619C2×C4#1313326211yescharacteristic
620C2×C4#1313326311yescharacteristic
621C8—41no—
622C8—41no—
623C8—2121no—
624C8—2121no—
625C22—22no—
626C22—22no—
627C22—22no—
628C22—22no—
629C22—22no—
630C22—22no—
631C22—42no—
632C22—42no—
633C22—42no—
634C22—81no—
635C22—81no—
636C22—81no—
637C22—81no—
638C22#1681090114yes—
639C22C2×#1313319411yescharacteristic
640C22#1681126011yescharacteristic
641C22#1681126511yescharacteristic
642C22C2×D8×D5311yescharacteristic
643C22C2×#1313324611yescharacteristic
644C22#1681143211yescharacteristic
645C22#1681143311yescharacteristic
646C4—22no—
647C4—42no—
648C4—42no—
649C4—81no—
650C4—1062no—
651C4—1062no—
652C4—4241no—
653C4—4241no—
654C4—4241no—
655C4—2122no—
656C4—2121no—
657C4—2121no—
658C4#1681115411yescharacteristic
659C4#1681121011yescharacteristic
660C4#1681133011yescharacteristic
661C4#1681133211yescharacteristic
662C2—24no—
663C2—82no—
664C2#2027854311yescharacteristic
665C2C2×#1681090111yescharacteristic
666C2#2028458111yescharacteristic
667C2#2028652111yescharacteristic
668C2#2028652611yescharacteristic
669C2#2028682411yescharacteristic
670C2#2028689011yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1874C2
socleC23×C53C22×D4
prime radicalC2×#121×C53C22
hypercenter#1874D53
hypocenterC53#26408
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1874

Complements

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

NH ≅ G/Ntype
1. #20278411C2semidirect
3. #20276276C2semidirect
4. #20276205C2semidirect
5. #20276101C2semidirect
6. #20275849C2semidirect
7. #20275665C2semidirect
8. #20275285C2semidirect
10. #20274964C2semidirect
11. #20274905C2semidirect
12. #20274892C2semidirect
13. #20274713C2semidirect
14. C2×#16810385C2semidirect
115. #121×C53Q8semidirect
394. C53#26408semidirect
413. #121Dic106semidirect

Maximal subgroups

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

subgroupindex
#202784112
#202777352
#202762762
#202762052
#202761012
#202758492
#202756652
#202752852
#202751442
#202749642
#202749052
#202748922
#202747132
C2×#168103852
C53×#18742
#2640853