Finite Group Databasebeta

← #27428430

Subgroups — #27428430

Order 13568.

854 conjugacy classes of subgroups.

The subgroup lattice has 656 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279023C211yesmaximal, characteristic
2D4×C4⋊Dic53C211yesmaximal, characteristic
3#20276274C211yesmaximal, characteristic, complemented
4#20276188C211yesmaximal, characteristic, complemented
5#20276138C211yesmaximal, characteristic, complemented
6#20276138C211yesmaximal, characteristic, complemented
7#20275287C211yesmaximal, characteristic, complemented
8#20275287C211yesmaximal, characteristic, complemented
9#20274905C211yesmaximal, characteristic, complemented
10#20274905C211yesmaximal, characteristic, complemented
11#20274802C211yesmaximal, characteristic, complemented
12#20274802C211yesmaximal, characteristic, complemented
13#20274744C211yesmaximal, characteristic, complemented
14C2×#16810386C211yesmaximal, characteristic, complemented
15C53×#1869C211yesmaximal, characteristic
16C2×#13133198—21no—
17C2×D4×Dic53—21no—
18C22×C4⋊Dic53C2211yescharacteristic
19C22×C4⋊Dic53C2211yescharacteristic
20C22×C4⋊Dic53C2211yescharacteristic
21C2×#13133151C2211yescharacteristic
22C2×#13133151C2211yescharacteristic
23C2×#13133151C2211yescharacteristic
24C2×#13133151C2211yescharacteristic
25#16810553—21no—
26#16810548—21no—
27#16810522—21no—
28#16810521C2211yescharacteristic
29#16810521C2211yescharacteristic
30#16810498—21no—
31#16810498—21no—
32#16810493C2211yescharacteristic
33#16810484—21no—
34#16810483C2211yescharacteristic
35#16810483C2211yescharacteristic
36#16810483C2211yescharacteristic
37#16810483C2211yescharacteristic
38#16810483C2211yescharacteristic
39C4×C4⋊Dic53C2211yescharacteristic
40#16810386C2211yescharacteristic
41#16810386C2211yescharacteristic
42#16810386C2211yescharacteristic
43#16810386C2211yescharacteristic
44#16810386C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810386C2211yescharacteristic
50#16810386C2211yescharacteristic
51#16810386C2211yescharacteristic
52C23×C8×C53C2211yescharacteristic
53C2×C4×D4×C53C2211yescharacteristic
54C2×C22⋊C8×C53C2211yescharacteristic
55C2×C22⋊C8×C53C2211yescharacteristic
56C2×C22⋊C8×C53C2211yescharacteristic
57C53×#336C2211yescharacteristic
58C53×#336C2211yescharacteristic
59C23×Dic53—21no—
60C23×Dic53—21no—
61C23×Dic53—21no—
62#13133198—44no—
63D4×Dic53—44no—
64C2×C4⋊Dic53—21no—
65C2×C4⋊Dic53—21no—
66C2×C4⋊Dic53—21no—
67C2×C4⋊Dic53—21no—
68C2×C4⋊Dic53—21no—
69C2×C4⋊Dic53—22no—
70C2×C4⋊Dic53—22no—
71C2×C4⋊Dic53—22no—
72C2×C4⋊Dic53—22no—
73C2×C4⋊Dic53—41no—
74C2×C4⋊Dic53—41no—
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×C4⋊Dic53C2311yescharacteristic
81C2×C4⋊Dic53C2311yescharacteristic
82C2×C4⋊Dic53C2311yescharacteristic
83C2×#11719919—21no—
84C2×#11719919—21no—
85C2×C22⋊Dic53—21no—
86C2×C22⋊Dic53—21no—
87C2×C22⋊Dic53—21no—
88C2×C22⋊Dic53—21no—
89C2×C22⋊Dic53—21no—
90C2×C4×Dic53—21no—
91#13133151—22no—
92#13133151—22no—
93#13133151—41no—
94#13133151—41no—
95#13133124—21no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103C22×D4×C53D411yescharacteristic
104C23×C4×C53C2311yescharacteristic
105C23×C4×C53C2311yescharacteristic
106C22×C8×C53—22no—
107C22×C8×C53—22no—
108C22×C8×C53—41no—
109C22×C8×C53C2311yescharacteristic
110C22×C8×C53C2311yescharacteristic
111C22×C8×C53C2311yescharacteristic
112C22×C8×C53C2311yescharacteristic
113C4×D4×C53—22no—
114C4×D4×C53—22no—
115C2×C4⋊C4×C53D411yescharacteristic
116C2×C22⋊C4×C53D411yescharacteristic
117C2×C22⋊C4×C53D411yescharacteristic
118C2×C42×C53C2311yescharacteristic
119C22⋊C8×C53—22no—
120C22⋊C8×C53—22no—
121C22⋊C8×C53—41no—
122C24×C53C2×D411yescharacteristic
123C24×C53C2×D411yescharacteristic
124C2×D4×C53—22no—
125C2×D4×C53—22no—
126C2×D4×C53D814yes—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53—21no—
130C22×C4×C53—21no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—21no—
136C22×C4×C53—21no—
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53C2×D411yescharacteristic
141C22×C4×C53#4112yes—
142C22×C4×C53#4112yes—
143C22×C4×C53C2411yescharacteristic
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—21no—
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53—21no—
152C22×Dic53—42no—
153C22×Dic53—42no—
154C22×Dic53—42no—
155C22×Dic53—42no—
156C22×Dic53—42no—
157C22×Dic53—42no—
158C22×Dic53—42no—
159C22×Dic53—81no—
160C22×Dic53—81no—
161C2×C8×C53—21no—
162C2×C8×C53—21no—
163C2×C8×C53—21no—
164C2×C8×C53—21no—
165C2×C8×C53—21no—
166C2×C8×C53—21no—
167C2×C8×C53—21no—
168C2×C8×C53—21no—
169C2×C8×C53—21no—
170C2×C8×C53—21no—
171C2×C8×C53—21no—
172C2×C8×C53—44no—
173C2×C8×C53#4112yes—
174C4⋊C4×C53—22no—
175C22⋊C4×C53—22no—
176C22⋊C4×C53—22no—
177C42×C53—21no—
178C42×C53—21no—
179C22⋊Dic53—42no—
180C22⋊Dic53—42no—
181C22⋊Dic53—42no—
182C22⋊Dic53—42no—
183C22⋊Dic53—42no—
184C4⋊Dic53—21no—
185C4⋊Dic53—21no—
186C4⋊Dic53—21no—
187C4⋊Dic53—21no—
188C4⋊Dic53—21no—
189C4⋊Dic53—21no—
190C4⋊Dic53—21no—
191C4⋊Dic53—21no—
192C4⋊Dic53—21no—
193C4⋊Dic53—21no—
194C4⋊Dic53—21no—
195C4⋊Dic53—21no—
196C4⋊Dic53—21no—
197C4⋊Dic53—21no—
198C4⋊Dic53—21no—
199C4⋊Dic53—42no—
200C4⋊Dic53—42no—
201C4⋊Dic53—42no—
202C4⋊Dic53—42no—
203C4⋊Dic53—81no—
204C4⋊Dic53#4112yes—
205#11719919—42no—
206#11719919—42no—
207C4×Dic53—42no—
208C23×C53—21no—
209C23×C53—21no—
210C23×C53—21no—
211C23×C53—21no—
212C23×C53—21no—
213C23×C53—21no—
214C23×C53—21no—
215C23×C53—21no—
216C23×C53—21no—
217C23×C53—21no—
218C23×C53#12312yes—
219C23×C53C2×D812yes—
220C23×C53C2×D812yes—
221C23×C53#13712yes—
222C23×C53C22×D411yescharacteristic
223D4×C53—28no—
224C2×C4×C53—21no—
225C2×C4×C53—21no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—22no—
229C2×C4×C53—22no—
230C2×C4×C53—22no—
231C2×C4×C53—22no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
235C2×C4×C53—21no—
236C2×C4×C53—21no—
237C2×C4×C53—21no—
238C2×C4×C53—41no—
239C2×C4×C53—41no—
240C2×C4×C53—41no—
241C2×C4×C53—41no—
242C2×C4×C53#12312yes—
243C2×C4×C53C2×D812yes—
244C2×C4×C53C2×#4111yescharacteristic
245C2×C4×C53C2×#4111yescharacteristic
246C2×C4×C53C2×#4111yescharacteristic
247C2×C4×C53C2×#4111yescharacteristic
248C2×C4×C53H(5,𝔽2)11yescharacteristic
249C2×C4×C532−1+411yescharacteristic
250C2×Dic53—22no—
251C2×Dic53—41no—
252C2×Dic53—41no—
253C2×Dic53—41no—
254C2×Dic53—41no—
255C2×Dic53—41no—
256C2×Dic53—41no—
257C2×Dic53—41no—
258C2×Dic53—41no—
259C2×Dic53—41no—
260C2×Dic53—41no—
261C2×Dic53—41no—
262C2×Dic53—41no—
263C2×Dic53—41no—
264C2×Dic53—41no—
265C2×Dic53—41no—
266C2×Dic53—41no—
267C2×Dic53—41no—
268C2×Dic53—41no—
269C2×Dic53—41no—
270C2×Dic53—41no—
271C2×Dic53—41no—
272C2×Dic53—41no—
273C2×Dic53—41no—
274C2×Dic53—84no—
275C2×Dic53—84no—
276C2×Dic53—84no—
277C8×C53—22no—
278C8×C53—41no—
279C8×C53—41no—
280C8×C53—41no—
281#24933—531nomaximal, Sylow
282C22×C53—22no—
283C22×C53—22no—
284C22×C53—22no—
285C22×C53—22no—
286C22×C53—22no—
287C22×C53—22no—
288C22×C53—22no—
289C22×C53—22no—
290C22×C53—22no—
291C22×C53—41no—
292C22×C53—41no—
293C22×C53—41no—
294C22×C53—41no—
295C22×C53#48014yes—
296C22×C53C2×#12311yescharacteristic
297C22×C53#54611yescharacteristic
298C22×C53#54711yescharacteristic
299C22×C53C22×D811yescharacteristic
300C22×C53C2×#13711yescharacteristic
301C22×C53#57711yescharacteristic
302C22×C53#57811yescharacteristic
303C4×C53—22no—
304C4×C53—22no—
305C4×C53—41no—
306C4×C53—41no—
307C4×C53#52911yescharacteristic
308C4×C53#53911yescharacteristic
309C4×C53#55111yescharacteristic
310C4×C53#55211yescharacteristic
311Dic53—42no—
312Dic53—81no—
313Dic53—81no—
314Dic53—81no—
315Dic53—81no—
316Dic53—81no—
317Dic53—81no—
318Dic53—81no—
319#2486—531no—
320D4×C4⋊C4—531no—
321#2099—531no—
322#2083—531no—
323#2075—531no—
324#2075—531no—
325#1918—531no—
326#1918—531no—
327#1869D5311yescharacteristic
328#1852—531no—
329#1852—531no—
330#1823—531no—
331#1823—531no—
332#1808—531no—
333C2×#340—531no—
334C2×C53—24no—
335C2×C53—42no—
336C2×C53#241911yescharacteristic
337C2×C53C2×#48011yescharacteristic
338C2×C53#309711yescharacteristic
339C2×C53#329411yescharacteristic
340C2×C53#331211yescharacteristic
341C2×C53#333411yescharacteristic
342C2×C53#334411yescharacteristic
343C23×C8C2×D5311yescharacteristic
344C2×#123—1061no—
345C2×C4×D4—1061no—
346C2×C4×D4C2×D5311yescharacteristic
347C22×C4⋊C4—531no—
348C22×C4⋊C4—531no—
349C22×C4⋊C4—531no—
350C2×#107—531no—
351C2×#107—531no—
352C2×#107—531no—
353C2×#107—531no—
354C2×C22⋊C8C2×D5311yescharacteristic
355C2×C22⋊C8C2×D5311yescharacteristic
356C2×C22⋊C8C2×D5311yescharacteristic
357#398—1061no—
358#397—1061no—
359#389—531no—
360#389—531no—
361#389—1061no—
362#385—1061no—
363#385—1061no—
364#384—531no—
365#381—1061no—
366#380—531no—
367#380—531no—
368#380—531no—
369#380—531no—
370#380—531no—
371C4×C4⋊C4—531no—
372#340—531no—
373#340—531no—
374#340—531no—
375#340—531no—
376#340—531no—
377#340—531no—
378#340—531no—
379#340—531no—
380#340—531no—
381#340—531no—
382#340—531no—
383#340—531no—
384#336C2×D5311yescharacteristic
385#336C2×D5311yescharacteristic
386C53#2493311yescharacteristic, complemented, Sylow
387C22×D4D21211yescharacteristic
388C23×C4—1061no—
389C23×C4—1061no—
390C23×C4—1061no—
391C23×C4C22×D5311yescharacteristic
392C23×C4C22×D5311yescharacteristic
393C22×C8—22no—
394C22×C8—22no—
395C22×C8—41no—
396C22×C8C22×D5311yescharacteristic
397C22×C8C22×D5311yescharacteristic
398C22×C8C22×D5311yescharacteristic
399C22×C8C22×D5311yescharacteristic
400#123—2124no—
401C4×D4—22no—
402C4×D4—22no—
403C4×D4—2124no—
404C2×C4⋊C4—531no—
405C2×C4⋊C4—531no—
406C2×C4⋊C4—531no—
407C2×C4⋊C4—531no—
408C2×C4⋊C4—531no—
409C2×C4⋊C4—531no—
410C2×C4⋊C4—531no—
411C2×C4⋊C4—531no—
412C2×C4⋊C4—1062no—
413C2×C4⋊C4—1062no—
414C2×C4⋊C4—1062no—
415C2×C4⋊C4—1062no—
416C2×C4⋊C4—1061no—
417C2×C4⋊C4—1061no—
418C2×C4⋊C4—1061no—
419C2×C4⋊C4—1061no—
420C2×C4⋊C4—1061no—
421C2×C4⋊C4—1061no—
422C2×C4⋊C4—1061no—
423C2×C4⋊C4—2121no—
424C2×C4⋊C4—2121no—
425C2×C4⋊C4D21211yescharacteristic
426C2×C22⋊C4—1061no—
427C2×C22⋊C4—1061no—
428C2×C22⋊C4—1061no—
429C2×C22⋊C4—1061no—
430C2×C22⋊C4—1061no—
431C2×C22⋊C4D21211yescharacteristic
432C2×C22⋊C4D21211yescharacteristic
433C2×C42—1061no—
434C2×C42C22×D5311yescharacteristic
435#107—1062no—
436#107—1062no—
437#107—2121no—
438#107—2121no—
439C22⋊C8—22no—
440C22⋊C8—22no—
441C22⋊C8—41no—
442#95—1061no—
443#95—1061no—
444#95—1061no—
445#95—1061no—
446#95—1061no—
447#95—1061no—
448#95—1061no—
449#95—1061no—
450C24C2×D21211yescharacteristic
451C24C2×D21211yescharacteristic
452C2×D4—22no—
453C2×D4—22no—
454C2×D4D42414yes—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—21no—
461C22×C4—21no—
462C22×C4—21no—
463C22×C4—21no—
464C22×C4—21no—
465C22×C4—1061no—
466C22×C4—1061no—
467C22×C4—1061no—
468C22×C4—1061no—
469C22×C4—1061no—
470C22×C4—1061no—
471C22×C4—1061no—
472C22×C4—1061no—
473C22×C4—4241no—
474C22×C4—4241no—
475C22×C4—2122no—
476C22×C4—2122no—
477C22×C4—2122no—
478C22×C4—2122no—
479C22×C4—2122no—
480C22×C4—2122no—
481C22×C4—2122no—
482C22×C4C2×D21211yescharacteristic
483C22×C4C2×D21211yescharacteristic
484C22×C4C2×D21211yescharacteristic
485C22×C4C2×D21211yescharacteristic
486C22×C4#1171994712yes—
487C22×C4#1171994712yes—
488C22×C4C23×D5311yescharacteristic
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8—21no—
494C2×C8—21no—
495C2×C8—21no—
496C2×C8—21no—
497C2×C8—21no—
498C2×C8—21no—
499C2×C8—21no—
500C2×C8—44no—
501C2×C8#1171994712yes—
502C4⋊C4—22no—
503C4⋊C4—532no—
504C4⋊C4—1061no—
505C4⋊C4—1061no—
506C4⋊C4—1061no—
507C4⋊C4—1061no—
508C4⋊C4—1061no—
509C4⋊C4—1061no—
510C4⋊C4—1061no—
511C4⋊C4—1061no—
512C4⋊C4—1061no—
513C4⋊C4—1061no—
514C4⋊C4—1061no—
515C4⋊C4—1061no—
516C4⋊C4—1061no—
517C4⋊C4—1061no—
518C4⋊C4—1061no—
519C4⋊C4—4241no—
520C4⋊C4—2122no—
521C4⋊C4—2122no—
522C4⋊C4—2122no—
523C4⋊C4—2122no—
524C4⋊C4—2122no—
525C4⋊C4—2122no—
526C22⋊C4—22no—
527C22⋊C4—22no—
528C22⋊C4—2122no—
529C22⋊C4—2122no—
530C22⋊C4—2122no—
531C22⋊C4—2122no—
532C22⋊C4—2122no—
533C42—21no—
534C42—21no—
535C42—2122no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23—21no—
542C23—21no—
543C23—21no—
544C23—21no—
545C23—21no—
546C23#1313320112yes—
547C23C2×D42412yes—
548C23C2×D42412yes—
549C23#1313325012yes—
550C23C22×D21211yescharacteristic
551D4—28no—
552C2×C4—21no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—22no—
557C2×C4—22no—
558C2×C4—22no—
559C2×C4—22no—
560C2×C4—21no—
561C2×C4—21no—
562C2×C4—21no—
563C2×C4—21no—
564C2×C4—21no—
565C2×C4—21no—
566C2×C4—41no—
567C2×C4—41no—
568C2×C4—41no—
569C2×C4—41no—
570C2×C4—1062no—
571C2×C4—4244no—
572C2×C4—4244no—
573C2×C4—4244no—
574C2×C4—2121no—
575C2×C4—2121no—
576C2×C4—2121no—
577C2×C4—2121no—
578C2×C4—2121no—
579C2×C4—2121no—
580C2×C4—2121no—
581C2×C4—2121no—
582C2×C4—2121no—
583C2×C4—2121no—
584C2×C4—2121no—
585C2×C4—2121no—
586C2×C4—2121no—
587C2×C4—2121no—
588C2×C4—2121no—
589C2×C4—2121no—
590C2×C4—2121no—
591C2×C4—2121no—
592C2×C4—2121no—
593C2×C4—2121no—
594C2×C4—2121no—
595C2×C4—2121no—
596C2×C4—2121no—
597C2×C4#1313320112yes—
598C2×C4C2×D42412yes—
599C2×C4#41×D5311yescharacteristic
600C2×C4C2×#1171994711yescharacteristic
601C2×C4C2×#1171994711yescharacteristic
602C2×C4C2×#1171994711yescharacteristic
603C2×C4#1313326211yescharacteristic
604C2×C4#1313326411yescharacteristic
605C8—22no—
606C8—41no—
607C8—41no—
608C8—41no—
609C22—22no—
610C22—22no—
611C22—22no—
612C22—22no—
613C22—22no—
614C22—22no—
615C22—22no—
616C22—22no—
617C22—22no—
618C22—41no—
619C22—41no—
620C22—41no—
621C22—41no—
622C22#1681093714yes—
623C22C2×#1313320111yescharacteristic
624C22#1681126811yescharacteristic
625C22#1681127711yescharacteristic
626C22C22×D42411yescharacteristic
627C22C2×#1313325011yescharacteristic
628C22#1681143511yescharacteristic
629C22#1681144011yescharacteristic
630C4—22no—
631C4—22no—
632C4—41no—
633C4—41no—
634C4—4241no—
635C4—4241no—
636C4—4241no—
637C4—4241no—
638C4—4241no—
639C4—4241no—
640C4—4241no—
641C4—2122no—
642C4#1681115411yescharacteristic
643C4#1681121811yescharacteristic
644C4#1681130411yescharacteristic
645C4#1681131011yescharacteristic
646C2—24no—
647C2—42no—
648C2#2027819011yescharacteristic
649C2C2×#1681093711yescharacteristic
650C2#2028468511yescharacteristic
651C2#2028652911yescharacteristic
652C2#2028676811yescharacteristic
653C2#2028706311yescharacteristic
654C2#2028717211yescharacteristic
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1869C2
socleC23×C53C22×D4
prime radicalC22×D4×C53D4
hypercenter#1869D53
hypocenterC53#24933
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #1869

Complements

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

NH ≅ G/Ntype
3. #20276274C2semidirect
4. #20276188C2semidirect
5. #20276138C2semidirect
6. #20276138C2semidirect
7. #20275287C2semidirect
8. #20275287C2semidirect
9. #20274905C2semidirect
10. #20274905C2semidirect
11. #20274802C2semidirect
12. #20274802C2semidirect
13. #20274744C2semidirect
14. C2×#16810386C2semidirect
386. C53#24933semidirect

Maximal subgroups

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

subgroupindex
#202790232
D4×C4⋊Dic532
#202762742
#202761882
#202761382
#202752872
#202749052
#202748022
#202747442
C2×#168103862
C53×#18692
#2493353