Finite Group Databasebeta

← #27395308

Subgroups — #27395308

Order 13568.

1080 conjugacy classes of subgroups.

The subgroup lattice has 568 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810762C211yesmaximal, characteristic
2C4×#13133191C211yesmaximal, characteristic
3#20275599C212yesmaximal, complemented
4#20275408C212yesmaximal, complemented
5#20275353C212yesmaximal, complemented
6#20275052C212yesmaximal, complemented
7#20275032C211yesmaximal, characteristic
8#20274903C211yesmaximal, characteristic, complemented
9#20274650C211yesmaximal, characteristic, complemented
10#20274477C211yesmaximal, characteristic, complemented
11C2×C53×#391C211yesmaximal, characteristic, complemented
12C22×#11719924C2212yes—
13C2×#13133191C2211yescharacteristic
14C2×C4×D212C2211yescharacteristic
15C2×C4×C22⋊D53—21no—
16#16810762C418yescomplemented
17C2×#13133150C2212yes—
18C2×#13133141C2212yes—
19C2×#13133141C2212yes—
20C2×#13133141C2212yes—
21C2×#13133137C2212yes—
22C2×C22⋊#88322C2211yescharacteristic
23C2×C22⋊#88322C2211yescharacteristic
24C2×#13133128C2212yes—
25#16810525C2211yescharacteristic
26#16810521C2211yescharacteristic
27#16810514—21no—
28#16810502—21no—
29#16810483C2211yescharacteristic
30C4×C4⋊Dic53C2211yescharacteristic
31C4×#11719921—21no—
32#16810385C2212yes—
33#16810385C2212yes—
34#16810378C2212yes—
35C23×Q8×C53C2211yescharacteristic
36C22×C4⋊C4×C53C2211yescharacteristic
37C22×C42×C53C2211yescharacteristic
38C53×#391—21no—
39C53×#391—21no—
40C53×#391C2212yes—
41C53×#391C2212yes—
42C2×#95×C53—21no—
43C2×#95×C53—21no—
44C22×D212C2311yescharacteristic
45C22×C22⋊D53—21no—
46C22×C4×D53—21no—
47C2×#11719924—42no—
48C2×#11719924—42no—
49C2×#11719924C2×C418yescomplemented
50C22×#88322C2312yes—
51C22×#88322C2312yes—
52#13133191—22no—
53#13133191C2×C414yes—
54C4×D212—42no—
55C4×C22⋊D53—44no—
56C2×C4⋊Dic53C2311yescharacteristic
57C2×C4⋊Dic53C2311yescharacteristic
58C2×C4⋊Dic53C2311yescharacteristic
59C2×#11719919—21no—
60C2×C22⋊Dic53—21no—
61C2×#11719921—21no—
62C2×#11719921—21no—
63C2×#11719921—21no—
64C2×#11719921—21no—
65C2×C4×Dic53—21no—
66#13133150—42no—
67#13133141—42no—
68#13133141—42no—
69#13133141C2×C418yescomplemented
70#13133137—42no—
71C22⋊#88322—22no—
72C22⋊#88322C2×C414yescomplemented
73#13133128—42no—
74#13133124—21no—
75#13133124—21no—
76#13133124—21no—
77C22×Q8×C53—21no—
78C22×Q8×C53—22no—
79C22×Q8×C53—42no—
80C22×Q8×C53—42no—
81C22×Q8×C53C2×C414yescomplemented
82C22×Q8×C53C2312yes—
83C23×C4×C53—21no—
84C23×C4×C53—21no—
85C23×C4×C53D411yescharacteristic
86C23×C4×C53D411yescharacteristic
87C23×C4×C53C2311yescharacteristic
88C2×C4⋊C4×C53—21no—
89C2×C4⋊C4×C53—42no—
90C2×C4⋊C4×C53C2312yes—
91C2×C42×C53—42no—
92C2×C42×C53D411yescharacteristic
93C2×C42×C53D411yescharacteristic
94C2×C42×C53C2311yescharacteristic
95C2×C42×C53C2311yescharacteristic
96#95×C53—22no—
97#95×C53—22no—
98#95×C53—22no—
99#95×C53—22no—
100C24×C53C2×D411yescharacteristic
101C23×D53—21no—
102C2×Q8×C53—24no—
103C2×Q8×C53—22no—
104C2×Q8×C53—22no—
105C2×Q8×C53—22no—
106C2×Q8×C53—22no—
107C2×Q8×C53—22no—
108C2×Q8×C53—48no—
109C2×Q8×C53—42no—
110C2×Q8×C53—42no—
111C2×Q8×C53—48no—
112C2×Q8×C53C22×C414yes—
113C22×C4×C53—22no—
114C22×C4×C53—22no—
115C22×C4×C53—24no—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—22no—
120C22×C4×C53—22no—
121C22×C4×C53—41no—
122C22×C4×C53—41no—
123C22×C4×C53—41no—
124C22×C4×C53—42no—
125C22×C4×C53—42no—
126C22×C4×C53—42no—
127C22×C4×C53—42no—
128C22×C4×C53—42no—
129C22×C4×C53C22×C412yes—
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53C2×D411yescharacteristic
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53#4112yes—
136C22×C4×C53C2411yescharacteristic
137C2×C22⋊D53—24no—
138C2×C22⋊D53—42no—
139C2×C22⋊D53—42no—
140C22×Dic53—21no—
141C22×Dic53—21no—
142C22×Dic53—21no—
143C2×D212—22no—
144C2×D212—22no—
145C2×D212—21no—
146C2×D212C22×C412yes—
147C2×C4×D53—42no—
148C2×C4×D53—42no—
149C2×C4×D53—42no—
150C4⋊C4×C53—22no—
151C4⋊C4×C53—22no—
152C4⋊C4×C53—42no—
153C42×C53—22no—
154C42×C53—22no—
155C42×C53—41no—
156C42×C53—41no—
157C22⋊Dic53—42no—
158#11719924—48no—
159#11719921—24no—
160#11719921—42no—
161#11719921—42no—
162#11719921—42no—
163C4⋊Dic53—21no—
164C4⋊Dic53—22no—
165C4⋊Dic53—22no—
166C4⋊Dic53C22×C412yes—
167#11719919—42no—
168C4×Dic53—42no—
169C2×#88322—22no—
170C2×#88322—22no—
171C2×#88322—22no—
172C2×#88322—22no—
173C2×#88322—22no—
174C2×#88322C22×C414yes—
175C23×C53—21no—
176C23×C53—21no—
177C23×C53—21no—
178C23×C53—21no—
179C23×C53C4×D412yes—
180C23×C53AGL(1,ℤ/8)12yes—
181C23×C53#13712yes—
182C23×C53C22×D411yescharacteristic
183C22×D53—22no—
184C22×D53—41no—
185C22×D53—41no—
186C22×D53—41no—
187C22×D53—41no—
188C22×D53—41no—
189C22×D53—41no—
190Q8×C53—24no—
191Q8×C53—24no—
192Q8×C53—48no—
193Q8×C53—42no—
194Q8×C53—42no—
195C2×C4×C53—21no—
196C2×C4×C53—21no—
197C2×C4×C53—21no—
198C2×C4×C53—21no—
199C2×C4×C53—24no—
200C2×C4×C53—24no—
201C2×C4×C53—24no—
202C2×C4×C53—21no—
203C2×C4×C53—21no—
204C2×C4×C53—21no—
205C2×C4×C53—21no—
206C2×C4×C53—21no—
207C2×C4×C53—21no—
208C2×C4×C53—21no—
209C2×C4×C53—21no—
210C2×C4×C53—21no—
211C2×C4×C53—44no—
212C2×C4×C53—44no—
213C2×C4×C53—48no—
214C2×C4×C53—42no—
215C2×C4×C53—42no—
216C2×C4×C53—42no—
217C2×C4×C53—42no—
218C2×C4×C53—42no—
219C2×C4×C53—84no—
220C2×C4×C53C4×D412yes—
221C2×C4×C53#13712yes—
222C2×C4×C53C23×C411yescharacteristic
223C2×C4×C53C2×#4111yescharacteristic
224C2×C4×C53H(5,𝔽2)12yes—
225C2×C4×C53H(5,𝔽2)12yes—
226C22⋊D53—48no—
227C2×Dic53—22no—
228C2×Dic53—41no—
229C2×Dic53—41no—
230C2×Dic53—41no—
231C2×Dic53—41no—
232C2×Dic53—41no—
233C2×Dic53—41no—
234C2×Dic53—41no—
235C2×Dic53—41no—
236D212—24no—
237D212—42no—
238C4×D53—84no—
239#88322—24no—
240#88322—42no—
241#22047—531nomaximal, Sylow
242C22×C53—22no—
243C22×C53—22no—
244C22×C53—22no—
245C22×C53—22no—
246C22×C53—22no—
247C22×C53—22no—
248C22×C53#44014yes—
249C22×C53C2×C4×D411yescharacteristic
250C22×C53#53411yescharacteristic
251C22×C53#53511yescharacteristic
252C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
253C22×C53C2×#13711yescharacteristic
254C22×C53C2×#13711yescharacteristic
255C22×C53#57511yescharacteristic
256C2×D53—42no—
257C2×D53—42no—
258C2×D53—42no—
259C2×D53—81no—
260C2×D53—81no—
261C2×D53—81no—
262C2×D53—81no—
263C4×C53—22no—
264C4×C53—22no—
265C4×C53—44no—
266C4×C53—41no—
267C4×C53—41no—
268C4×C53—41no—
269C4×C53—82no—
270C4×C53#51812yes—
271C4×C53#53812yes—
272Dic53—42no—
273Dic53—81no—
274Dic53—81no—
275C2×#448—531no—
276C4×#121—531no—
277C2×#391D5311yescharacteristic, complemented
278#1973—532no—
279#1935—532no—
280#1928—532no—
281#1877—532no—
282#1874—531no—
283#1852—531no—
284#1787—531no—
285#1749—531no—
286C2×C53—24no—
287C2×C53#232111yescharacteristic
288C2×C53C2×#44011yescharacteristic
289C2×C53C4×#13711yescharacteristic
290C2×C53#319311yescharacteristic
291C2×C53#319611yescharacteristic
292C2×C53#321311yescharacteristic
293C2×C53#321811yescharacteristic
294D53—82no—
295C23×Q8C2×D5311yescharacteristic
296C22×SD16—532no—
297C2×#121—531no—
298C2×C4×D4—531no—
299C2×C4×D4—1061no—
300C22×C4⋊C4C2×D5311yescharacteristic
301C22×C42C2×D5311yescharacteristic
302#448—538no—
303C2×#107—532no—
304C2×#103—532no—
305C2×#103—532no—
306C2×#103—532no—
307C2×#102—532no—
308C2×C22⋊C8—531no—
309C2×C22⋊C8—531no—
310C2×#97—532no—
311#391—21no—
312#391—21no—
313#391C2×D5312yes—
314#391C2×D5312yes—
315#390—531no—
316#389—531no—
317#388—1061no—
318#386—1061no—
319#380—531no—
320C4×C4⋊C4—531no—
321C4×C22⋊C4—1061no—
322C2×#95—21no—
323C2×#95—21no—
324#340—532no—
325#340—532no—
326#336—532no—
327C53#2204711yescharacteristic, complemented, Sylow
328C22×Q8—22no—
329C22×Q8—21no—
330C22×Q8—42no—
331C22×Q8—42no—
332C22×Q8C4×D5314yescomplemented
333C22×Q8C22×D5312yes—
334C22×D4—531no—
335C22×D4—1061no—
336C23×C4—21no—
337C23×C4—21no—
338C23×C4—1061no—
339C23×C4C22⋊D5311yescharacteristic
340C23×C4C22⋊D5311yescharacteristic
341C23×C4C22×D5311yescharacteristic
342C2×SD16—538no—
343C2×SD16—2122no—
344C2×SD16—2122no—
345C22×C8—532no—
346C22×C8—532no—
347#121—534no—
348#121—1062no—
349C4×D4—2122no—
350C4×D4—2124no—
351C2×C4⋊C4—21no—
352C2×C4⋊C4—42no—
353C2×C4⋊C4—531no—
354C2×C4⋊C4—531no—
355C2×C4⋊C4—531no—
356C2×C4⋊C4—1061no—
357C2×C4⋊C4C22×D5312yes—
358C2×C22⋊C4—1061no—
359C2×C22⋊C4—1061no—
360C2×C22⋊C4—1061no—
361C2×C22⋊C4—1061no—
362C2×C22⋊C4—1061no—
363C2×C42—42no—
364C2×C42—1061no—
365C2×C42C22⋊D5311yescharacteristic
366C2×C42C22⋊D5311yescharacteristic
367C2×C42C22×D5311yescharacteristic
368C2×C42C22×D5311yescharacteristic
369#107—2122no—
370#103—538no—
371#103—2122no—
372#103—2122no—
373#102—2122no—
374C22⋊C8—534no—
375C22⋊C8—1062no—
376#97—2122no—
377#95—22no—
378#95—22no—
379#95—22no—
380#95—22no—
381#95—1061no—
382#95—1061no—
383#95—1061no—
384C24—1061no—
385C24C2×C22⋊D5311yescharacteristic
386C2×Q8—24no—
387C2×Q8—22no—
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—48no—
393C2×Q8—42no—
394C2×Q8—42no—
395C2×Q8—48no—
396C2×Q8C2×C4×D5314yes—
397C2×D4—532no—
398C2×D4—1064no—
399C2×D4—1061no—
400C2×D4—1062no—
401C2×D4—1062no—
402C2×D4—2122no—
403C2×D4—2122no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—24no—
407C22×C4—22no—
408C22×C4—22no—
409C22×C4—22no—
410C22×C4—22no—
411C22×C4—22no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—42no—
416C22×C4—42no—
417C22×C4—42no—
418C22×C4—42no—
419C22×C4—42no—
420C22×C4—1061no—
421C22×C4—1061no—
422C22×C4—1061no—
423C22×C4—2122no—
424C22×C4—2122no—
425C22×C4—2122no—
426C22×C4C2×C4×D5312yes—
427C22×C4#1171994512yes—
428C22×C4C2×C22⋊D5311yescharacteristic
429C22×C4C2×C22⋊D5311yescharacteristic
430C22×C4C2×C22⋊D5311yescharacteristic
431C22×C4C2×C22⋊D5311yescharacteristic
432C22×C4C2×C22⋊D5311yescharacteristic
433C22×C4C23×D5311yescharacteristic
434SD16—2128no—
435C2×C8—534no—
436C2×C8—1062no—
437C2×C8—1062no—
438C2×C8—1062no—
439C2×C8—1062no—
440C2×C8—1062no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—42no—
444C4⋊C4—532no—
445C4⋊C4—1061no—
446C4⋊C4—1062no—
447C4⋊C4—1062no—
448C4⋊C4—2122no—
449C22⋊C4—1064no—
450C22⋊C4—2122no—
451C22⋊C4—2122no—
452C22⋊C4—2122no—
453C22⋊C4—2122no—
454C42—22no—
455C42—22no—
456C42—41no—
457C42—41no—
458C42—2122no—
459C23—21no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—1062no—
464C23—2121no—
465C23—2121no—
466C23—2121no—
467C23—2121no—
468C23—2121no—
469C23—2121no—
470C23C4×C22⋊D5312yes—
471C23#1313324212yes—
472C23#1313324812yes—
473C23C22×C22⋊D5311yescharacteristic
474Q8—24no—
475Q8—24no—
476Q8—48no—
477Q8—42no—
478Q8—42no—
479D4—1064no—
480D4—2128no—
481D4—2122no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—24no—
487C2×C4—24no—
488C2×C4—24no—
489C2×C4—21no—
490C2×C4—21no—
491C2×C4—21no—
492C2×C4—21no—
493C2×C4—21no—
494C2×C4—21no—
495C2×C4—21no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—44no—
499C2×C4—44no—
500C2×C4—48no—
501C2×C4—42no—
502C2×C4—42no—
503C2×C4—42no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—84no—
507C2×C4—1062no—
508C2×C4—4244no—
509C2×C4—2121no—
510C2×C4—2121no—
511C2×C4—2121no—
512C2×C4—2121no—
513C2×C4—2121no—
514C2×C4—2121no—
515C2×C4—2121no—
516C2×C4—2121no—
517C2×C4C4×C22⋊D5312yes—
518C2×C4#1313324812yes—
519C2×C4C22×C4×D5311yescharacteristic
520C2×C4C2×#1171994511yescharacteristic
521C2×C4#1313326212yes—
522C2×C4#1313326212yes—
523C8—1064no—
524C8—2122no—
525C22—22no—
526C22—22no—
527C22—22no—
528C22—22no—
529C22—22no—
530C22—22no—
531C22—4241no—
532C22—4241no—
533C22—4241no—
534C22—4241no—
535C22—2122no—
536C22—2122no—
537C22—2122no—
538C22#1681072114yes—
539C22C2×C4×C22⋊D5311yescharacteristic
540C22#1681118211yescharacteristic
541C22#1681119011yescharacteristic
542C22C2×#1313324211yescharacteristic
543C22C2×#1313324811yescharacteristic
544C22C2×#1313324811yescharacteristic
545C22#1681142111yescharacteristic
546C4—22no—
547C4—22no—
548C4—41no—
549C4—41no—
550C4—41no—
551C4—44no—
552C4—82no—
553C4—4241no—
554C4—4241no—
555C4—2122no—
556C4#1681109612yes—
557C4#1681121112yes—
558C2—24no—
559C2—4242no—
560C2#2027737311yescharacteristic
561C2C2×#1681072111yescharacteristic
562C2C4×#1313324811yescharacteristic
563C2#2028551711yescharacteristic
564C2#2028554511yescharacteristic
565C2#2028570211yescharacteristic
566C2#2028574211yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
FittingC2×C53×#391C2
socleC23×C53C22×D4
prime radicalC2×#13133191C22
cosocleC22×C4×C53C24
hypercenterC2×#391D53
hypocenterC53#22047
LCSG ▷ C2×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C22×C42 ◁ C2×#391

Complements

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

NH ≅ G/Ntype
3. #20275599C2semidirect
4. #20275408C2semidirect
5. #20275353C2semidirect
6. #20275052C2semidirect
8. #20274903C2semidirect
9. #20274650C2semidirect
10. #20274477C2semidirect
11. C2×C53×#391C2semidirect
16. #16810762C4semidirect
49. C2×#11719924C2×C4semidirect
69. #13133141C2×C4semidirect
72. C22⋊#88322C2×C4semidirect
81. C22×Q8×C53C2×C4semidirect
277. C2×#391D53semidirect
327. C53#22047semidirect
332. C22×Q8C4×D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168107622
C4×#131331912
#202755992
#202754082
#202753532
#202750522
#202750322
#202749032
#202746502
#202744772
C2×C53×#3912
#2204753