Finite Group Databasebeta

← #27401836

Subgroups — #27401836

Order 13568.

810 conjugacy classes of subgroups.

The subgroup lattice has 588 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810666C211yesmaximal, characteristic
2#20277608C211yesmaximal, characteristic
3#20277586C211yesmaximal, characteristic
4#20275452C211yesmaximal, characteristic, complemented
5#20275447C211yesmaximal, characteristic, complemented
6#20275310C211yesmaximal, characteristic, complemented
7#20275298C211yesmaximal, characteristic, complemented
8#20275291C211yesmaximal, characteristic, complemented
9#20275289C211yesmaximal, characteristic, complemented
10#20274806C211yesmaximal, characteristic
11#20274746C211yesmaximal, characteristic
12#20274611C211yesmaximal, characteristic, complemented
13#20274603C211yesmaximal, characteristic, complemented
14#20274501C211yesmaximal, characteristic
15C53×#1825C211yesmaximal, characteristic
16C22×#11719917C2211yescharacteristic
17C2×#13133173—21no—
18C22×C4⋊Dic53C2211yescharacteristic
19#16810666C418yescomplemented
20C2×#13133150C2211yescharacteristic
21C2×#13133150C2211yescharacteristic
22C2×#13133150C2211yescharacteristic
23C2×#13133148C2211yescharacteristic
24C2×#13133148C2211yescharacteristic
25C2×#13133148C2211yescharacteristic
26C2×#13133127C2211yescharacteristic
27C2×C8×Dic53C2211yescharacteristic
28#16810522C2211yescharacteristic
29#16810522C2211yescharacteristic
30#16810519C2211yescharacteristic
31#16810519C2211yescharacteristic
32#16810516—21no—
33#16810516—21no—
34#16810506—21no—
35#16810506—21no—
36#16810491—21no—
37#16810484—21no—
38#16810483C2211yescharacteristic
39C4⋊C4×Dic53C2211yescharacteristic
40C4⋊C4×Dic53C2211yescharacteristic
41C4⋊C4×Dic53C2211yescharacteristic
42C4⋊C4×Dic53C2211yescharacteristic
43C4×C4⋊Dic53—21no—
44C4×C22⋊Dic53C2211yescharacteristic
45#16810386C2211yescharacteristic
46#16810386C2211yescharacteristic
47#16810386C2211yescharacteristic
48#16810386C2211yescharacteristic
49#16810385C2211yescharacteristic
50#16810385C2211yescharacteristic
51#16810377C2211yescharacteristic
52#16810377C2211yescharacteristic
53C2×#117×C53C2211yescharacteristic
54C2×C22⋊C8×C53C2211yescharacteristic
55C53×#380C2211yescharacteristic
56C53×#340C2211yescharacteristic
57C53×#340C2211yescharacteristic
58C53×#340C2211yescharacteristic
59C53×#340C2211yescharacteristic
60C23×Dic53—21no—
61C2×#11719917—22no—
62C2×#11719917—22no—
63C2×#11719917C2×C414yes—
64C22×#88322C2311yescharacteristic
65C22×#88322C2311yescharacteristic
66#13133173—44no—
67C2×C4⋊Dic53—21no—
68C2×C4⋊Dic53—22no—
69C2×C4⋊Dic53—21no—
70C2×C4⋊Dic53—21no—
71C2×C4⋊Dic53C2×C414yes—
72C2×C4⋊Dic53C2311yescharacteristic
73C2×C4⋊Dic53C2311yescharacteristic
74C2×C4⋊Dic53C2311yescharacteristic
75C2×C4⋊Dic53C2311yescharacteristic
76C2×#11719919—21no—
77C2×#11719919—21no—
78C2×C22⋊Dic53—21no—
79C2×C22⋊Dic53D411yescharacteristic
80C2×C22⋊Dic53D411yescharacteristic
81C2×C4×Dic53—21no—
82C2×C4×Dic53—21no—
83C2×C4×Dic53—21no—
84C2×C4×Dic53—21no—
85C2×C4×Dic53C2311yescharacteristic
86C2×C4×Dic53C2311yescharacteristic
87#13133150—22no—
88#13133150C2×C414yes—
89#13133150C2×C414yes—
90#13133148—22no—
91#13133148C2×C414yes—
92#13133148C2×C414yes—
93#13133127—22no—
94C8×Dic53—22no—
95#13133124—21no—
96#13133124—21no—
97#13133124—21no—
98#13133124—21no—
99#13133124—21no—
100#13133124—21no—
101#13133124—21no—
102#13133124—21no—
103#13133124—21no—
104#13133124D411yescharacteristic
105#13133124D411yescharacteristic
106C23×C4×C53C2311yescharacteristic
107C22×C8×C53C2311yescharacteristic
108C22×C8×C53C2311yescharacteristic
109#117×C53—22no—
110#117×C53C2×C414yes—
111C2×C4⋊C4×C53C2311yescharacteristic
112C2×C4⋊C4×C53C2311yescharacteristic
113C2×C4⋊C4×C53C2311yescharacteristic
114C2×C4⋊C4×C53C2311yescharacteristic
115C2×C22⋊C4×C53—21no—
116C2×C22⋊C4×C53—21no—
117C2×C42×C53—21no—
118C22⋊C8×C53—22no—
119#95×C53—21no—
120C24×C53C2×D411yescharacteristic
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53C22×C412yes—
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53#4112yes—
132C22×C4×C53C2411yescharacteristic
133C22×Dic53—24no—
134C22×Dic53—21no—
135C22×Dic53—21no—
136C22×Dic53—21no—
137C22×Dic53—21no—
138C22×Dic53—42no—
139C22×Dic53—42no—
140C22×Dic53C2×D411yescharacteristic
141C22×Dic53C2×D411yescharacteristic
142C22×Dic53C2×D411yescharacteristic
143C22×Dic53C2×D411yescharacteristic
144C2×C8×C53—21no—
145C2×C8×C53—21no—
146C2×C8×C53—21no—
147C2×C8×C53—21no—
148C2×C8×C53—21no—
149C2×C8×C53—21no—
150C4⋊C4×C53—21no—
151C4⋊C4×C53—21no—
152C4⋊C4×C53—21no—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53C22×C412yes—
157C4⋊C4×C53C22×C412yes—
158C22⋊C4×C53—24no—
159C22⋊C4×C53—42no—
160C42×C53—24no—
161C22⋊Dic53—22no—
162C22⋊Dic53—41no—
163C22⋊Dic53—81no—
164C4⋊Dic53—24no—
165C4⋊Dic53—21no—
166C4⋊Dic53—21no—
167C4⋊Dic53—21no—
168C4⋊Dic53—21no—
169C4⋊Dic53—21no—
170C4⋊Dic53—21no—
171C4⋊Dic53—42no—
172C4⋊Dic53—42no—
173C4⋊Dic53C22×C412yes—
174C4⋊Dic53C22×C412yes—
175#11719919—42no—
176#11719919—42no—
177C4×Dic53—21no—
178C4×Dic53—21no—
179C4×Dic53—21no—
180C4×Dic53—21no—
181C4×Dic53—42no—
182C4×Dic53—42no—
183C4×Dic53—81no—
184C4×Dic53—81no—
185#11719917—28no—
186C2×#88322—21no—
187C2×#88322—21no—
188C2×#88322C22×C412yes—
189C2×#88322C22×C412yes—
190C2×#88322#4112yes—
191C2×#88322#4112yes—
192C23×C53—21no—
193C23×C53—21no—
194C23×C53—21no—
195C23×C53—21no—
196C23×C53—21no—
197C23×C53—21no—
198C23×C53C4×D412yes—
199C23×C53C22×D411yescharacteristic
200C2×C4×C53—21no—
201C2×C4×C53—21no—
202C2×C4×C53—22no—
203C2×C4×C53—22no—
204C2×C4×C53—22no—
205C2×C4×C53—22no—
206C2×C4×C53—22no—
207C2×C4×C53—22no—
208C2×C4×C53—41no—
209C2×C4×C53—41no—
210C2×C4×C53—41no—
211C2×C4×C53—41no—
212C2×C4×C53—41no—
213C2×C4×C53—41no—
214C2×C4×C53—41no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53—41no—
218C2×C4×C53C4×D412yes—
219C2×C4×C53C23×C411yescharacteristic
220C2×C4×C53C2×#4111yescharacteristic
221C2×C4×C53C2×#4111yescharacteristic
222C2×C4×C53C2×#4111yescharacteristic
223C2×C4×C532−1+411yescharacteristic
224C2×C4×C532−1+411yescharacteristic
225C2×Dic53—21no—
226C2×Dic53—21no—
227C2×Dic53—22no—
228C2×Dic53—22no—
229C2×Dic53—21no—
230C2×Dic53—21no—
231C2×Dic53—21no—
232C2×Dic53—21no—
233C2×Dic53—21no—
234C2×Dic53—21no—
235C2×Dic53—21no—
236C2×Dic53—21no—
237C2×Dic53—21no—
238C2×Dic53—48no—
239C2×Dic53—41no—
240C2×Dic53—41no—
241C2×Dic53—41no—
242C2×Dic53—41no—
243C2×Dic53—41no—
244C2×Dic53—41no—
245C2×Dic53—41no—
246C2×Dic53—41no—
247C2×Dic53—41no—
248C2×Dic53—41no—
249C2×Dic53AGL(1,ℤ/8)12yes—
250C8×C53—41no—
251C8×C53—41no—
252#88322—22no—
253#88322#11714yescomplemented
254#22622—531nomaximal, Sylow
255C22×C53—22no—
256C22×C53—22no—
257C22×C53—22no—
258C22×C53—41no—
259C22×C53—41no—
260C22×C53—41no—
261C22×C53—41no—
262C22×C53C2×C4×D411yescharacteristic
263C22×C53#54711yescharacteristic
264C22×C53#54711yescharacteristic
265C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
266C22×C53#57511yescharacteristic
267C22×C53#57711yescharacteristic
268C22×C53#57811yescharacteristic
269C4×C53—22no—
270C4×C53—42no—
271C4×C53—42no—
272C4×C53—81no—
273C4×C53—81no—
274C4×C53C2×#11711yescharacteristic
275C4×C53#51911yescharacteristic
276C4×C53#55211yescharacteristic
277C4×C53#55211yescharacteristic
278Dic53—22no—
279Dic53—41no—
280Dic53—41no—
281Dic53—41no—
282Dic53—42no—
283Dic53—42no—
284Dic53—81no—
285Dic53—81no—
286C2×#428—531no—
287#2347—531no—
288#2344—531no—
289#1943—531no—
290#1942—531no—
291#1921—531no—
292#1920—531no—
293#1919—531no—
294#1918—531no—
295#1825D5311yescharacteristic
296#1824—531no—
297#1809—531no—
298#1776—531no—
299C4×#106—531no—
300#1753—531no—
301C2×C53—42no—
302C2×C53#234411yescharacteristic
303C2×C53C4×AGL(1,ℤ/8)11yescharacteristic
304C2×C53#294711yescharacteristic
305C2×C53#334511yescharacteristic
306C2×C53#334911yescharacteristic
307C2×C53#335111yescharacteristic
308C2×C53#335311yescharacteristic
309C22×M16—531no—
310C2×#117—1061no—
311C2×#117C2×D5311yescharacteristic
312C22×C4⋊C4—531no—
313#428—538no—
314C2×#107—531no—
315C2×#107—531no—
316C2×#107—531no—
317C2×#106—531no—
318C2×#106—531no—
319C2×#106—531no—
320C2×C22⋊C8C2×D5311yescharacteristic
321C2×#97—531no—
322C2×C4×C8—531no—
323#389—531no—
324#389—531no—
325#389—531no—
326#389—531no—
327#388—1061no—
328#388—1061no—
329#387—1061no—
330#387—1061no—
331#383—1061no—
332#381—1061no—
333#380—531no—
334#380C2×D5311yescharacteristic
335C4×C4⋊C4—531no—
336C4×C4⋊C4—531no—
337C4×C4⋊C4—531no—
338C4×C4⋊C4—531no—
339C4×C4⋊C4—1061no—
340C4×C22⋊C4—531no—
341#340—531no—
342#340—531no—
343#340—531no—
344#340—531no—
345#340—531no—
346#340—531no—
347#340C2×D5311yescharacteristic
348#340C2×D5311yescharacteristic
349#340C2×D5311yescharacteristic
350#340C2×D5311yescharacteristic
351#336—531no—
352#336—531no—
353C53#2262211yescharacteristic, complemented, Sylow
354C23×C4—1061no—
355C23×C4C22×D5311yescharacteristic
356C2×M16—534no—
357C2×M16—1062no—
358C2×M16—1062no—
359C22×C8—531no—
360C22×C8—531no—
361C22×C8C22×D5311yescharacteristic
362C22×C8C22×D5311yescharacteristic
363#117—22no—
364#117—2124no—
365#117C4×D5314yes—
366C2×C4⋊C4—534no—
367C2×C4⋊C4—531no—
368C2×C4⋊C4—531no—
369C2×C4⋊C4—531no—
370C2×C4⋊C4—531no—
371C2×C4⋊C4—1061no—
372C2×C4⋊C4—1062no—
373C2×C4⋊C4—1061no—
374C2×C4⋊C4—1061no—
375C2×C4⋊C4—1061no—
376C2×C4⋊C4—1061no—
377C2×C4⋊C4C22×D5311yescharacteristic
378C2×C4⋊C4C22×D5311yescharacteristic
379C2×C4⋊C4C22×D5311yescharacteristic
380C2×C4⋊C4C22×D5311yescharacteristic
381C2×C22⋊C4—21no—
382C2×C22⋊C4—21no—
383C2×C22⋊C4—531no—
384C2×C22⋊C4—531no—
385C2×C22⋊C4—1061no—
386C2×C42—21no—
387C2×C42—531no—
388C2×C42—531no—
389C2×C42—1061no—
390C2×C42—1061no—
391C2×C42—1061no—
392C2×C42—1061no—
393#107—534no—
394#107—534no—
395#107—1062no—
396#106—534no—
397#106—534no—
398#106—1062no—
399C22⋊C8—22no—
400#97—1062no—
401C4×C8—1062no—
402#95—21no—
403#95—531no—
404#95—531no—
405#95—1061no—
406#95—1061no—
407#95—1061no—
408#95—1061no—
409#95—1061no—
410#95—1061no—
411#95—1061no—
412#95—1061no—
413#95—1061no—
414C24D4×D5311yescharacteristic
415C22×C4—21no—
416C22×C4—21no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—21no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—531no—
424C22×C4—531no—
425C22×C4—531no—
426C22×C4—531no—
427C22×C4—1064no—
428C22×C4—1061no—
429C22×C4—1061no—
430C22×C4—1061no—
431C22×C4—1061no—
432C22×C4—2122no—
433C22×C4—2122no—
434C22×C4C2×C4×D5312yes—
435C22×C4D4×D5311yescharacteristic
436C22×C4#1171994712yes—
437C22×C4C23×D5311yescharacteristic
438M16—1068no—
439C2×C8—21no—
440C2×C8—21no—
441C2×C8—21no—
442C2×C8—21no—
443C2×C8—21no—
444C2×C8—21no—
445C2×C8—532no—
446C2×C8—532no—
447C2×C8—532no—
448C2×C8—532no—
449C2×C8—1061no—
450C2×C8—1061no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—21no—
454C4⋊C4—21no—
455C4⋊C4—21no—
456C4⋊C4—21no—
457C4⋊C4—532no—
458C4⋊C4—532no—
459C4⋊C4—1064no—
460C4⋊C4—1061no—
461C4⋊C4—1061no—
462C4⋊C4—1061no—
463C4⋊C4—1061no—
464C4⋊C4—1061no—
465C4⋊C4—1061no—
466C4⋊C4—2122no—
467C4⋊C4—2122no—
468C4⋊C4—2122no—
469C4⋊C4—2122no—
470C4⋊C4C2×C4×D5312yes—
471C4⋊C4C2×C4×D5312yes—
472C22⋊C4—24no—
473C22⋊C4—42no—
474C22⋊C4—1062no—
475C22⋊C4—4241no—
476C22⋊C4—2121no—
477C42—24no—
478C42—1061no—
479C42—1061no—
480C42—1061no—
481C42—1061no—
482C42—4241no—
483C42—4241no—
484C42—2122no—
485C42—2122no—
486C23—21no—
487C23—21no—
488C23—21no—
489C23—21no—
490C23—21no—
491C23—21no—
492C23#1313317712yes—
493C23C2×D4×D5311yescharacteristic
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—22no—
497C2×C4—22no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—41no—
503C2×C4—41no—
504C2×C4—41no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—532no—
513C2×C4—1061no—
514C2×C4—1061no—
515C2×C4—1061no—
516C2×C4—1061no—
517C2×C4—1061no—
518C2×C4—1062no—
519C2×C4—1062no—
520C2×C4—1061no—
521C2×C4—1061no—
522C2×C4—1061no—
523C2×C4—1061no—
524C2×C4—1061no—
525C2×C4—1061no—
526C2×C4—2121no—
527C2×C4—2121no—
528C2×C4—2121no—
529C2×C4—2121no—
530C2×C4—2121no—
531C2×C4—2121no—
532C2×C4—2128no—
533C2×C4—2121no—
534C2×C4—2121no—
535C2×C4—2121no—
536C2×C4—2121no—
537C2×C4#1313317712yes—
538C2×C4C22×C4×D5311yescharacteristic
539C2×C4#41×D5311yescharacteristic
540C2×C4#41×D5311yescharacteristic
541C2×C4C2×#1171994711yescharacteristic
542C2×C4#1313326411yescharacteristic
543C2×C4#1313326411yescharacteristic
544C8—41no—
545C8—41no—
546C8—534no—
547C8—1062no—
548C22—22no—
549C22—22no—
550C22—22no—
551C22—41no—
552C22—41no—
553C22—41no—
554C22—41no—
555C22C2×#1313317711yescharacteristic
556C22#1681127111yescharacteristic
557C22#1681127111yescharacteristic
558C22AGL(1,ℤ/8)×D5311yescharacteristic
559C22#1681142211yescharacteristic
560C22#1681143011yescharacteristic
561C22#1681143711yescharacteristic
562C4—22no—
563C4—42no—
564C4—42no—
565C4—81no—
566C4—81no—
567C4—1062no—
568C4—4241no—
569C4—4241no—
570C4—2121no—
571C4—2121no—
572C4—2121no—
573C4—2122no—
574C4—2122no—
575C4#117×D5311yescharacteristic
576C4#1681110011yescharacteristic
577C4#1681131011yescharacteristic
578C4#1681131011yescharacteristic
579C2—42no—
580C2#2027757911yescharacteristic
581C2#2028352811yescharacteristic
582C2#2028356111yescharacteristic
583C2#2028718811yescharacteristic
584C2#2028724811yescharacteristic
585C2#2028727311yescharacteristic
586C2#2028729711yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC53×#1825C2
socleC23×C53C22×D4
prime radicalC24×C53C2×D4
cosocleC22×C4×C53C24
hypercenter#1825D53
hypocenterC53#22622
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C23×C4 ◁ #1825

Complements

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

NH ≅ G/Ntype
4. #20275452C2semidirect
5. #20275447C2semidirect
6. #20275310C2semidirect
7. #20275298C2semidirect
8. #20275291C2semidirect
9. #20275289C2semidirect
12. #20274611C2semidirect
13. #20274603C2semidirect
19. #16810666C4semidirect
253. #88322#117semidirect
353. C53#22622semidirect

Maximal subgroups

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

subgroupindex
C2×#168106662
#202776082
#202775862
#202754522
#202754472
#202753102
#202752982
#202752912
#202752892
#202748062
#202747462
#202746112
#202746032
#202745012
C53×#18252
#2262253