Finite Group Databasebeta

← #27428670

Subgroups — #27428670

Order 13568.

800 conjugacy classes of subgroups.

The subgroup lattice has 648 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20279012C211yesmaximal, characteristic
2D4×C4⋊Dic53C211yesmaximal, characteristic
3#20276279C211yesmaximal, characteristic, complemented
4#20276193C211yesmaximal, characteristic, complemented
5#20276148C211yesmaximal, characteristic, complemented
6#20276138C211yesmaximal, characteristic, complemented
7#20275308C211yesmaximal, characteristic, complemented
8#20275294C211yesmaximal, characteristic, complemented
9#20274910C211yesmaximal, characteristic, complemented
10#20274910C211yesmaximal, characteristic, complemented
11#20274811C211yesmaximal, characteristic, complemented
12#20274802C211yesmaximal, characteristic, complemented
13#20274748C211yesmaximal, characteristic, complemented
14#20274435C211yesmaximal, characteristic, complemented
15C53×#1870C211yesmaximal, characteristic
16C2×D4×Dic53—21no—
17C2×#13133176C2211yescharacteristic
18C22×C4⋊Dic53C2211yescharacteristic
19C22×C4⋊Dic53C2211yescharacteristic
20C2×#13133151C2211yescharacteristic
21C2×#13133149C2211yescharacteristic
22C2×#13133149C2211yescharacteristic
23C2×#13133149C2211yescharacteristic
24#16810551—21no—
25#16810537—21no—
26#16810522—21no—
27#16810521C2211yescharacteristic
28#16810521C2211yescharacteristic
29#16810511—21no—
30#16810504—21no—
31#16810498—21no—
32#16810498—21no—
33#16810493C2211yescharacteristic
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
52C22×M16×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—
61D4×Dic53—44no—
62#13133176—22no—
63#13133176—22no—
64C2×C4⋊Dic53—21no—
65C2×C4⋊Dic53—21no—
66C2×C4⋊Dic53—21no—
67C2×C4⋊Dic53—21no—
68C2×C4⋊Dic53—22no—
69C2×C4⋊Dic53—22no—
70C2×C4⋊Dic53—41no—
71C2×C4⋊Dic53—41no—
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×C4⋊Dic53C2311yescharacteristic
77C2×C4⋊Dic53C2311yescharacteristic
78C2×C4⋊Dic53C2311yescharacteristic
79C2×C4⋊Dic53C2311yescharacteristic
80C2×#11719919—21no—
81C2×#11719919—21no—
82C2×C22⋊Dic53—21no—
83C2×C22⋊Dic53—21no—
84C2×C22⋊Dic53—21no—
85C2×C22⋊Dic53—21no—
86C2×C22⋊Dic53—21no—
87C2×C22⋊Dic53—21no—
88C2×C4×Dic53—21no—
89C2×C4×Dic53—21no—
90#13133151—22no—
91#13133149—22no—
92#13133149—41no—
93#13133149—41no—
94#13133124—21no—
95#13133124—21no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102C22×D4×C53D411yescharacteristic
103C23×C4×C53C2311yescharacteristic
104C23×C4×C53C2311yescharacteristic
105C2×M16×C53—22no—
106C2×M16×C53—22no—
107C2×M16×C53—41no—
108C22×C8×C53C2311yescharacteristic
109C22×C8×C53C2311yescharacteristic
110C22×C8×C53C2311yescharacteristic
111C22×C8×C53C2311yescharacteristic
112C4×D4×C53—22no—
113C4×D4×C53—22no—
114C2×C4⋊C4×C53D411yescharacteristic
115C2×C22⋊C4×C53D411yescharacteristic
116C2×C22⋊C4×C53D411yescharacteristic
117C2×C42×C53C2311yescharacteristic
118C22⋊C8×C53—22no—
119C22⋊C8×C53—22no—
120C22⋊C8×C53—41no—
121C24×C53C2×D411yescharacteristic
122C24×C53C2×D411yescharacteristic
123C2×D4×C53—22no—
124C2×D4×C53—22no—
125C2×D4×C53—22no—
126C22×C4×C53—21no—
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×C53C2×D411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139C22×C4×C53C2×D411yescharacteristic
140C22×C4×C53#4112yes—
141C22×C4×C53#4112yes—
142C22×C4×C53C2411yescharacteristic
143C22×Dic53—21no—
144C22×Dic53—21no—
145C22×Dic53—21no—
146C22×Dic53—21no—
147C22×Dic53—21no—
148C22×Dic53—21no—
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53—42no—
152C22×Dic53—42no—
153C22×Dic53—42no—
154C22×Dic53—42no—
155C22×Dic53—81no—
156C22×Dic53—81no—
157M16×C53—44no—
158C2×C8×C53—21no—
159C2×C8×C53—21no—
160C2×C8×C53—21no—
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#4112yes—
170C4⋊C4×C53—22no—
171C22⋊C4×C53—22no—
172C22⋊C4×C53—22no—
173C42×C53—21no—
174C42×C53—21no—
175C22⋊Dic53—42no—
176C22⋊Dic53—42no—
177C22⋊Dic53—42no—
178C22⋊Dic53—42no—
179C22⋊Dic53—81no—
180C22⋊Dic53—81no—
181C4⋊Dic53—21no—
182C4⋊Dic53—21no—
183C4⋊Dic53—21no—
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—42no—
197C4⋊Dic53—42no—
198C4⋊Dic53—42no—
199C4⋊Dic53—81no—
200C4⋊Dic53#4112yes—
201#11719919—42no—
202#11719919—81no—
203C4×Dic53—42no—
204C4×Dic53—42no—
205C23×C53—21no—
206C23×C53—21no—
207C23×C53—21no—
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#12312yes—
218C23×C53AGL(1,ℤ/8)12yes—
219C23×C53C22×D411yescharacteristic
220D4×C53—28no—
221C2×C4×C53—21no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—22no—
226C2×C4×C53—22no—
227C2×C4×C53—22no—
228C2×C4×C53—22no—
229C2×C4×C53—21no—
230C2×C4×C53—21no—
231C2×C4×C53—21no—
232C2×C4×C53—21no—
233C2×C4×C53—21no—
234C2×C4×C53—21no—
235C2×C4×C53—41no—
236C2×C4×C53—41no—
237C2×C4×C53—41no—
238C2×C4×C53—41no—
239C2×C4×C53#12312yes—
240C2×C4×C53#13712yes—
241C2×C4×C53C2×#4111yescharacteristic
242C2×C4×C53C2×#4111yescharacteristic
243C2×C4×C53C2×#4111yescharacteristic
244C2×C4×C53C2×#4111yescharacteristic
245C2×C4×C53H(5,𝔽2)11yescharacteristic
246C2×C4×C532−1+411yescharacteristic
247C2×Dic53—22no—
248C2×Dic53—41no—
249C2×Dic53—41no—
250C2×Dic53—41no—
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—84no—
272C2×Dic53—84no—
273C8×C53—22no—
274C8×C53—41no—
275C8×C53—41no—
276C8×C53—41no—
277#24949—531nomaximal, Sylow
278C22×C53—22no—
279C22×C53—22no—
280C22×C53—22no—
281C22×C53—22no—
282C22×C53—22no—
283C22×C53—22no—
284C22×C53—41no—
285C22×C53—41no—
286C22×C53—41no—
287C22×C53—41no—
288C22×C53—41no—
289C22×C53—41no—
290C22×C53—41no—
291C22×C53—41no—
292C22×C53C2×#12311yescharacteristic
293C22×C53#54611yescharacteristic
294C22×C53#54711yescharacteristic
295C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
296C22×C53C2×#13711yescharacteristic
297C22×C53#57711yescharacteristic
298C22×C53#57711yescharacteristic
299C4×C53—22no—
300C4×C53—22no—
301C4×C53—41no—
302C4×C53—41no—
303C4×C53#52911yescharacteristic
304C4×C53#53911yescharacteristic
305C4×C53#55111yescharacteristic
306C4×C53#55211yescharacteristic
307Dic53—42no—
308Dic53—81no—
309Dic53—81no—
310Dic53—81no—
311Dic53—81no—
312Dic53—81no—
313Dic53—81no—
314Dic53—81no—
315#2485—531no—
316D4×C4⋊C4—531no—
317#2100—531no—
318#2084—531no—
319#2077—531no—
320#2075—531no—
321#1921—531no—
322#1919—531no—
323#1870D5311yescharacteristic
324#1853—531no—
325#1853—531no—
326#1825—531no—
327#1823—531no—
328#1809—531no—
329#1736—531no—
330C2×C53—42no—
331C2×C53—42no—
332C2×C53#241911yescharacteristic
333C2×C53#309211yescharacteristic
334C2×C53#309911yescharacteristic
335C2×C53#331011yescharacteristic
336C2×C53#332911yescharacteristic
337C2×C53#334511yescharacteristic
338C2×C53#335611yescharacteristic
339C22×M16C2×D5311yescharacteristic
340C2×C4×D4—1061no—
341C2×C4×D4C2×D5311yescharacteristic
342C2×#117—531no—
343C22×C4⋊C4—531no—
344C22×C4⋊C4—531no—
345C2×#107—531no—
346C2×#106—531no—
347C2×#106—531no—
348C2×#106—531no—
349C2×C22⋊C8C2×D5311yescharacteristic
350C2×C22⋊C8C2×D5311yescharacteristic
351C2×C22⋊C8C2×D5311yescharacteristic
352#397—1061no—
353#394—1061no—
354#389—531no—
355#389—531no—
356#389—1061no—
357#388—1061no—
358#387—1061no—
359#385—1061no—
360#385—1061no—
361#384—531no—
362#380—531no—
363#380—531no—
364#380—531no—
365#380—531no—
366#380—531no—
367C4×C4⋊C4—531no—
368#340—531no—
369#340—531no—
370#340—531no—
371#340—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#336C2×D5311yescharacteristic
381#336C2×D5311yescharacteristic
382C53#2494911yescharacteristic, complemented, Sylow
383C22×D4D21211yescharacteristic
384C23×C4—1061no—
385C23×C4—1061no—
386C23×C4C22×D5311yescharacteristic
387C23×C4C22×D5311yescharacteristic
388C2×M16—22no—
389C2×M16—22no—
390C2×M16—41no—
391C22×C8C22×D5311yescharacteristic
392C22×C8C22×D5311yescharacteristic
393C22×C8C22×D5311yescharacteristic
394C22×C8C22×D5311yescharacteristic
395C4×D4—22no—
396C4×D4—22no—
397C4×D4—2124no—
398#117—1062no—
399#117—1062no—
400C2×C4⋊C4—531no—
401C2×C4⋊C4—531no—
402C2×C4⋊C4—531no—
403C2×C4⋊C4—531no—
404C2×C4⋊C4—531no—
405C2×C4⋊C4—531no—
406C2×C4⋊C4—531no—
407C2×C4⋊C4—531no—
408C2×C4⋊C4—1062no—
409C2×C4⋊C4—1062no—
410C2×C4⋊C4—1061no—
411C2×C4⋊C4—1061no—
412C2×C4⋊C4—1061no—
413C2×C4⋊C4—1061no—
414C2×C4⋊C4—1061no—
415C2×C4⋊C4—1061no—
416C2×C4⋊C4—2121no—
417C2×C4⋊C4—2121no—
418C2×C4⋊C4D21211yescharacteristic
419C2×C22⋊C4—1061no—
420C2×C22⋊C4—1061no—
421C2×C22⋊C4—1061no—
422C2×C22⋊C4—1061no—
423C2×C22⋊C4—1061no—
424C2×C22⋊C4—1061no—
425C2×C22⋊C4D21211yescharacteristic
426C2×C22⋊C4D21211yescharacteristic
427C2×C42—1061no—
428C2×C42—1061no—
429C2×C42C22×D5311yescharacteristic
430#107—1062no—
431#106—1062no—
432#106—2121no—
433#106—2121no—
434C22⋊C8—22no—
435C22⋊C8—22no—
436C22⋊C8—41no—
437#95—1061no—
438#95—1061no—
439#95—1061no—
440#95—1061no—
441#95—1061no—
442#95—1061no—
443#95—1061no—
444#95—1061no—
445C24C2×D21211yescharacteristic
446C24C2×D21211yescharacteristic
447C2×D4—22no—
448C2×D4—22no—
449C2×D4—22no—
450C22×C4—21no—
451C22×C4—21no—
452C22×C4—21no—
453C22×C4—21no—
454C22×C4—21no—
455C22×C4—21no—
456C22×C4—21no—
457C22×C4—21no—
458C22×C4—21no—
459C22×C4—21no—
460C22×C4—1061no—
461C22×C4—1061no—
462C22×C4—1061no—
463C22×C4—1061no—
464C22×C4—1061no—
465C22×C4—1061no—
466C22×C4—1061no—
467C22×C4—1061no—
468C22×C4—4241no—
469C22×C4—4241no—
470C22×C4—2122no—
471C22×C4—2122no—
472C22×C4—2122no—
473C22×C4—2122no—
474C22×C4C2×D21211yescharacteristic
475C22×C4C2×D21211yescharacteristic
476C22×C4C2×D21211yescharacteristic
477C22×C4C2×D21211yescharacteristic
478C22×C4#1171994712yes—
479C22×C4#1171994712yes—
480C22×C4C23×D5311yescharacteristic
481M16—44no—
482C2×C8—21no—
483C2×C8—21no—
484C2×C8—21no—
485C2×C8—21no—
486C2×C8—21no—
487C2×C8—21no—
488C2×C8—21no—
489C2×C8—21no—
490C2×C8—21no—
491C2×C8—21no—
492C2×C8—21no—
493C2×C8#1171994712yes—
494C4⋊C4—22no—
495C4⋊C4—532no—
496C4⋊C4—1061no—
497C4⋊C4—1061no—
498C4⋊C4—1061no—
499C4⋊C4—1061no—
500C4⋊C4—1061no—
501C4⋊C4—1061no—
502C4⋊C4—1061no—
503C4⋊C4—1061no—
504C4⋊C4—1061no—
505C4⋊C4—1061no—
506C4⋊C4—1061no—
507C4⋊C4—1061no—
508C4⋊C4—1061no—
509C4⋊C4—1061no—
510C4⋊C4—1061no—
511C4⋊C4—4241no—
512C4⋊C4—4241no—
513C4⋊C4—2122no—
514C4⋊C4—2122no—
515C4⋊C4—2122no—
516C4⋊C4—2122no—
517C22⋊C4—22no—
518C22⋊C4—22no—
519C22⋊C4—4241no—
520C22⋊C4—4241no—
521C22⋊C4—2122no—
522C22⋊C4—2122no—
523C22⋊C4—2122no—
524C22⋊C4—2122no—
525C42—21no—
526C42—21no—
527C42—2122no—
528C42—2122no—
529C23—21no—
530C23—21no—
531C23—21no—
532C23—21no—
533C23—21no—
534C23—21no—
535C23—21no—
536C23—21no—
537C23—21no—
538C23—21no—
539C23—21no—
540C23—21no—
541C23#1313320112yes—
542C23#1313324512yes—
543C23C22×D21211yescharacteristic
544D4—28no—
545C2×C4—21no—
546C2×C4—21no—
547C2×C4—21no—
548C2×C4—21no—
549C2×C4—22no—
550C2×C4—22no—
551C2×C4—22no—
552C2×C4—22no—
553C2×C4—21no—
554C2×C4—21no—
555C2×C4—21no—
556C2×C4—21no—
557C2×C4—21no—
558C2×C4—21no—
559C2×C4—41no—
560C2×C4—41no—
561C2×C4—41no—
562C2×C4—41no—
563C2×C4—1062no—
564C2×C4—4244no—
565C2×C4—4244no—
566C2×C4—2121no—
567C2×C4—2121no—
568C2×C4—2121no—
569C2×C4—2121no—
570C2×C4—2121no—
571C2×C4—2121no—
572C2×C4—2121no—
573C2×C4—2121no—
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#1313320112yes—
590C2×C4#1313325012yes—
591C2×C4#41×D5311yescharacteristic
592C2×C4C2×#1171994711yescharacteristic
593C2×C4C2×#1171994711yescharacteristic
594C2×C4C2×#1171994711yescharacteristic
595C2×C4#1313326211yescharacteristic
596C2×C4#1313326411yescharacteristic
597C8—22no—
598C8—41no—
599C8—41no—
600C8—41no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—22no—
606C22—22no—
607C22—41no—
608C22—41no—
609C22—41no—
610C22—41no—
611C22—41no—
612C22—41no—
613C22—41no—
614C22—41no—
615C22C2×#1313320111yescharacteristic
616C22#1681126811yescharacteristic
617C22#1681127711yescharacteristic
618C22C2×#1313324511yescharacteristic
619C22C2×#1313325011yescharacteristic
620C22#1681143511yescharacteristic
621C22#1681143511yescharacteristic
622C4—22no—
623C4—22no—
624C4—41no—
625C4—41no—
626C4—4241no—
627C4—4241no—
628C4—4241no—
629C4—4241no—
630C4—4241no—
631C4—4241no—
632C4—4241no—
633C4—2122no—
634C4#1681115411yescharacteristic
635C4#1681121811yescharacteristic
636C4#1681130411yescharacteristic
637C4#1681131011yescharacteristic
638C2—42no—
639C2—42no—
640C2#2027819011yescharacteristic
641C2#2028464011yescharacteristic
642C2#2028470611yescharacteristic
643C2#2028673811yescharacteristic
644C2#2028701011yescharacteristic
645C2#2028718711yescharacteristic
646C2#2028733111yescharacteristic
subgroupG/H
centerC23C22×D212
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1870C2
socleC23×C53C22×D4
prime radicalC22×D4×C53D4
hypercenter#1870D53
hypocenterC53#24949
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ #1870

Complements

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

NH ≅ G/Ntype
3. #20276279C2semidirect
4. #20276193C2semidirect
5. #20276148C2semidirect
6. #20276138C2semidirect
7. #20275308C2semidirect
8. #20275294C2semidirect
9. #20274910C2semidirect
10. #20274910C2semidirect
11. #20274811C2semidirect
12. #20274802C2semidirect
13. #20274748C2semidirect
14. #20274435C2semidirect
382. C53#24949semidirect

Maximal subgroups

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

subgroupindex
#202790122
D4×C4⋊Dic532
#202762792
#202761932
#202761482
#202761382
#202753082
#202752942
#202749102
#202748112
#202748022
#202747482
#202744352
C53×#18702
#2494953