Finite Group Databasebeta

← #27466127

Subgroups — #27466127

Order 13568.

886 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#13133197C211yesmaximal, characteristic
2#384×D53C211yesmaximal, characteristic
3#20276276C212yesmaximal, complemented
4#20276255C212yesmaximal, complemented
5#20275478C211yesmaximal, characteristic, complemented
6#20275021C212yesmaximal
7#20274906C212yesmaximal
8#20274721C211yesmaximal, characteristic, complemented
9#20274717C211yesmaximal, characteristic, complemented
10#20274692C211yesmaximal, characteristic
11C53×#1948C211yesmaximal, characteristic, complemented
12C2×#13133197C2211yescharacteristic
13C2×C4×Dic106C2211yescharacteristic
14C2×C4⋊C4×D53—21no—
15C2×C4⋊C4×D53—21no—
16C2×C4⋊C4×D53C2212yes—
17C2×C42×D53C2211yescharacteristic
18C2×#13133148C2212yes—
19C2×C4⋊#88322C2211yescharacteristic
20C2×#13133142C2212yes—
21C2×#13133140C2212yes—
22C2×#13133129C2212yes—
23#16810528C2211yescharacteristic
24#16810522C2211yescharacteristic
25#16810508—21no—
26#16810497—21no—
27#16810493C2211yescharacteristic
28#16810492—21no—
29#16810492—21no—
30#16810492C2212yes—
31#16810480C2211yescharacteristic
32C4×C4⋊Dic53—21no—
33C4×#11719921—21no—
34#16810386C2212yes—
35#16810385C2212yes—
36#16810385C2212yes—
37#16810377C2212yes—
38C2×#106×C53C2212yes—
39C2×C4⋊C8×C53C2211yescharacteristic
40C53×#384C2211yescharacteristic
41C4×C4⋊C4×C53C2211yescharacteristic
42C53×#340C2212yes—
43C22×Dic106C2311yescharacteristic
44C22×C4×D53—22no—
45C22×C4×D53D411yescharacteristic
46C22×C4×D53D411yescharacteristic
47C22×C4×D53C2311yescharacteristic
48C22×#88322C2312yes—
49#13133197—22no—
50#13133197Q814yes—
51C4×Dic106—22no—
52C4×Dic106—22no—
53C2×C4⋊Dic53—21no—
54C2×C4⋊Dic53—21no—
55C2×C4⋊Dic53—21no—
56C2×C4⋊Dic53—21no—
57C2×C4⋊Dic53C2311yescharacteristic
58C2×C4⋊Dic53C2312yes—
59C2×#11719919—21no—
60C2×#11719919—22no—
61C2×#11719919—22no—
62C2×#11719919—22no—
63C4⋊C4×D53—42no—
64C4⋊C4×D53—42no—
65C4⋊C4×D53—44no—
66C4⋊C4×D53—44no—
67C2×#11719921—21no—
68C2×#11719921—21no—
69C2×C4×Dic53—21no—
70C2×C4×Dic53D411yescharacteristic
71C2×C4×Dic53D411yescharacteristic
72C2×C4×Dic53C2311yescharacteristic
73C42×D53—22no—
74C42×D53—22no—
75#13133148—42no—
76C4⋊#88322—22no—
77#13133142—42no—
78#13133140—42no—
79#13133129—24no—
80#13133124—21no—
81#13133124—21no—
82#13133124—21no—
83#13133124—21no—
84C22×C8×C53C2312yes—
85C2×C4⋊C4×C53—21no—
86C2×C4⋊C4×C53—21no—
87C2×C4⋊C4×C53C2311yescharacteristic
88C2×C4⋊C4×C53C2312yes—
89C2×C4⋊C4×C53C2311yescharacteristic
90C2×C42×C53—21no—
91C2×C42×C53C2311yescharacteristic
92#106×C53—42no—
93C4⋊C8×C53—22no—
94#95×C53—21no—
95C23×D53C2×D411yescharacteristic
96C22×C4×C53—21no—
97C22×C4×C53—21no—
98C22×C4×C53—22no—
99C22×C4×C53C2×D411yescharacteristic
100C22×C4×C53C2×D411yescharacteristic
101C22×C4×C53C2411yescharacteristic
102C22×Dic53—21no—
103C22×Dic53—21no—
104C22×Dic53—22no—
105C22×Dic53C2×D411yescharacteristic
106C22×Dic53C2×D411yescharacteristic
107C22×Dic53C2×D411yescharacteristic
108C2×C4×D53—22no—
109C2×C4×D53—22no—
110C2×C4×D53—22no—
111C2×C4×D53—22no—
112C2×C4×D53—21no—
113C2×C4×D53—21no—
114C2×C4×D53—21no—
115C2×C4×D53—21no—
116C2×C4×D53—22no—
117C2×C4×D53—44no—
118C2×C4×D53—44no—
119C2×C4×D53—44no—
120C2×C4×D53SD1614yes—
121C2×C4×D53C2×Q812yes—
122C2×C4×D53#4112yes—
123C2×Dic106—21no—
124C2×Dic106—21no—
125C2×Dic106—21no—
126C2×Dic106—21no—
127C2×Dic106—21no—
128C2×Dic106C2×Q812yes—
129C2×C8×C53—22no—
130C2×C8×C53—22no—
131C2×C8×C53—22no—
132C4⋊C4×C53—21no—
133C4⋊C4×C53—21no—
134C4⋊C4×C53—22no—
135C4⋊C4×C53—22no—
136C4⋊C4×C53—21no—
137C4⋊C4×C53—42no—
138C4⋊C4×C53—42no—
139C4⋊C4×C53C2×Q812yes—
140C42×C53—42no—
141C42×C53#4112yes—
142C42×C53#4112yes—
143#11719921—24no—
144#11719921—42no—
145C4⋊Dic53—24no—
146C4⋊Dic53—21no—
147C4⋊Dic53—21no—
148C4⋊Dic53—22no—
149C4⋊Dic53—22no—
150C4⋊Dic53—42no—
151C4⋊Dic53—42no—
152C4⋊Dic53—42no—
153#11719919—44no—
154#11719919—42no—
155#11719919—44no—
156#11719919—44no—
157C4×Dic53—22no—
158C4×Dic53—22no—
159C4×Dic53—21no—
160C4×Dic53—21no—
161C4×Dic53—42no—
162C2×#88322—22no—
163C2×#88322—22no—
164C2×#88322—22no—
165C23×C53C22×D411yescharacteristic
166C22×D53—21no—
167C22×D53—21no—
168C22×D53—21no—
169C22×D53—21no—
170C22×D53C22⋊Q812yes—
171C22×D53C2×SD1612yes—
172C22×D53#13712yes—
173C2×C4×C53—21no—
174C2×C4×C53—21no—
175C2×C4×C53—22no—
176C2×C4×C53—41no—
177C2×C4×C53—41no—
178C2×C4×C53—42no—
179C2×C4×C53—41no—
180C2×C4×C53—42no—
181C2×C4×C53—42no—
182C2×C4×C53—41no—
183C2×C4×C53—41no—
184C2×C4×C53#12312yes—
185C2×C4×C53#12312yes—
186C2×C4×C53C2×SD1612yes—
187C2×C4×C53#13712yes—
188C2×C4×C53C22×Q811yescharacteristic
189C2×C4×C53C2×#4111yescharacteristic
190C2×C4×C53C2×#4111yescharacteristic
191C2×C4×C53C2×#4111yescharacteristic
192C2×C4×C53H(5,𝔽2)11yescharacteristic
193C2×C4×C53H(5,𝔽2)11yescharacteristic
194C2×Dic53—21no—
195C2×Dic53—21no—
196C2×Dic53—21no—
197C2×Dic53—21no—
198C2×Dic53—21no—
199C2×Dic53—21no—
200C2×Dic53—22no—
201C2×Dic53—41no—
202C2×Dic53—41no—
203C2×Dic53—41no—
204C2×Dic53—42no—
205C2×Dic53—42no—
206C2×Dic53—42no—
207C2×Dic53—41no—
208C2×Dic53—41no—
209C2×Dic53C22⋊Q812yes—
210C2×Dic53C2×SD1612yes—
211C2×Dic53AGL(1,ℤ/8)12yes—
212C4×D53—28no—
213C4×D53—22no—
214C4×D53—22no—
215C4×D53—22no—
216C4×D53—22no—
217C4×D53—28no—
218C4×D53—88no—
219Dic106—22no—
220Dic106—22no—
221Dic106—41no—
222Dic106—41no—
223C8×C53—42no—
224#88322—42no—
225#27680—531nomaximal, Sylow
226C22×C53C2×C22⋊Q811yescharacteristic
227C22×C53C2×#12311yescharacteristic
228C22×C53#53511yescharacteristic
229C22×C53C22×SD1611yescharacteristic
230C22×C53C2×AGL(1,ℤ/8)11yescharacteristic
231C22×C53C2×#13711yescharacteristic
232C22×C53C2×#13711yescharacteristic
233C2×D53—22no—
234C2×D53—22no—
235C2×D53—22no—
236C2×D53—22no—
237C2×D53—22no—
238C2×D53—22no—
239C2×D53#47514yes—
240C4×C53—22no—
241C4×C53—42no—
242C4×C53—82no—
243C4×C53—81no—
244C4×C53#48314yes—
245C4×C53#55311yescharacteristic
246C4×C53#55311yescharacteristic
247C4×C53#55411yescharacteristic
248C4×C53#55411yescharacteristic
249Dic53—22no—
250Dic53—22no—
251Dic53—22no—
252Dic53—42no—
253Dic53—81no—
254Dic53—82no—
255C4×C22⋊Q8—531no—
256C2×#384—531no—
257#2099—532no—
258#2096—532no—
259#1948—531no—
260#1948D5311yescharacteristic, complemented
261#1872—532no—
262#1853—532no—
263#1803—531no—
264#1803—531no—
265#1797—531no—
266C2×C53#248711yescharacteristic
267C2×C53C2×#47511yescharacteristic
268C2×C53#307911yescharacteristic
269C2×C53C2×#48311yescharacteristic
270C2×C53#309211yescharacteristic
271C2×C53#320811yescharacteristic
272C2×C53#321811yescharacteristic
273D53—24no—
274C2×C22⋊Q8—531no—
275C2×C4×Q8—531no—
276C22×C4⋊C4—532no—
277C22×C4⋊C4—1061no—
278C22×C4⋊C4—1061no—
279C22×C42—531no—
280C2×#106—532no—
281C2×#106C2×D5312yes—
282C2×C4⋊C8—531no—
283C2×C4⋊C8C2×D5311yescharacteristic
284C2×#103—532no—
285C2×#103—532no—
286C2×C22⋊C8—532no—
287#391—531no—
288#389—531no—
289#387—1061no—
290#385—1061no—
291#384—532no—
292#384—531no—
293#384—1061no—
294#384—1061no—
295#384C2×D5311yescharacteristic
296#380—531no—
297C4×C4⋊C4—1061no—
298C4×C4⋊C4C2×D5311yescharacteristic
299C4×C22⋊C4—1061no—
300#340—532no—
301#340—532no—
302#340—532no—
303#340C2×D5312yes—
304#336—532no—
305C53#2768011yescharacteristic, complemented, Sylow
306C22×Q8—531no—
307C23×C4—531no—
308C23×C4—531no—
309C23×C4—531no—
310C23×C4—1062no—
311C22×C8—532no—
312C22×C8C22×D5312yes—
313C22⋊Q8—534no—
314C22⋊Q8—1062no—
315C4×Q8—1062no—
316C4×Q8—1062no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—532no—
320C2×C4⋊C4—531no—
321C2×C4⋊C4—1061no—
322C2×C4⋊C4—1062no—
323C2×C4⋊C4—1062no—
324C2×C4⋊C4—1062no—
325C2×C4⋊C4—1061no—
326C2×C4⋊C4—1061no—
327C2×C4⋊C4—1061no—
328C2×C4⋊C4—1061no—
329C2×C4⋊C4—2122no—
330C2×C4⋊C4—2122no—
331C2×C4⋊C4—2124no—
332C2×C4⋊C4—2124no—
333C2×C4⋊C4C22×D5312yes—
334C2×C4⋊C4C22×D5311yescharacteristic
335C2×C4⋊C4C22×D5311yescharacteristic
336C2×C22⋊C4—1061no—
337C2×C22⋊C4—1061no—
338C2×C42—21no—
339C2×C42—531no—
340C2×C42—531no—
341C2×C42—531no—
342C2×C42—1062no—
343C2×C42—1062no—
344C2×C42—1061no—
345C2×C42C22×D5311yescharacteristic
346#106—42no—
347#106—2122no—
348C4⋊C8—22no—
349C4⋊C8—1062no—
350#103—2122no—
351#103—2122no—
352C22⋊C8—1064no—
353#95—21no—
354#95—1061no—
355#95—1061no—
356#95—1061no—
357#95—1061no—
358C24—531no—
359C2×Q8—532no—
360C2×Q8—1061no—
361C2×Q8—1061no—
362C2×Q8—1061no—
363C2×Q8—1061no—
364C2×Q8—1061no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—531no—
369C22×C4—531no—
370C22×C4—531no—
371C22×C4—534no—
372C22×C4—532no—
373C22×C4—532no—
374C22×C4—1062no—
375C22×C4—1062no—
376C22×C4—1062no—
377C22×C4—1062no—
378C22×C4—1062no—
379C22×C4—1061no—
380C22×C4—1061no—
381C22×C4—1061no—
382C22×C4—1061no—
383C22×C4—1061no—
384C22×C4—1061no—
385C22×C4—1062no—
386C22×C4—2124no—
387C22×C4—2124no—
388C22×C4—2124no—
389C22×C4D4×D5311yescharacteristic
390C22×C4D4×D5311yescharacteristic
391C22×C4C23×D5311yescharacteristic
392C2×C8—22no—
393C2×C8—22no—
394C2×C8—22no—
395C2×C8—1062no—
396C2×C8—1062no—
397C2×C8—1062no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4—22no—
401C4⋊C4—22no—
402C4⋊C4—21no—
403C4⋊C4—42no—
404C4⋊C4—42no—
405C4⋊C4—1064no—
406C4⋊C4—1061no—
407C4⋊C4—1061no—
408C4⋊C4—1062no—
409C4⋊C4—1062no—
410C4⋊C4—2122no—
411C4⋊C4—2124no—
412C4⋊C4—2124no—
413C4⋊C4—2122no—
414C4⋊C4—2124no—
415C4⋊C4—2122no—
416C4⋊C4—2122no—
417C4⋊C4Q8×D5312yes—
418C22⋊C4—1064no—
419C22⋊C4—2122no—
420C42—42no—
421C42—1062no—
422C42—1062no—
423C42—1061no—
424C42—1061no—
425C42—2122no—
426C42#1171994512yes—
427C42#1171994912yes—
428C23—532no—
429C23—532no—
430C23—532no—
431C23—1061no—
432C23—1061no—
433C23—1061no—
434C23—1061no—
435C23C2×D4×D5311yescharacteristic
436Q8—1062no—
437Q8—1062no—
438Q8—2121no—
439Q8—2121no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—22no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—532no—
452C2×C4—532no—
453C2×C4—532no—
454C2×C4—1068no—
455C2×C4—1061no—
456C2×C4—1061no—
457C2×C4—1061no—
458C2×C4—1061no—
459C2×C4—1061no—
460C2×C4—1061no—
461C2×C4—1062no—
462C2×C4—1062no—
463C2×C4—1062no—
464C2×C4—1062no—
465C2×C4—1068no—
466C2×C4—1062no—
467C2×C4—4248no—
468C2×C4—2121no—
469C2×C4—2122no—
470C2×C4—2122no—
471C2×C4—2121no—
472C2×C4—2121no—
473C2×C4—2122no—
474C2×C4—2121no—
475C2×C4—2121no—
476C2×C4#1313320212yes—
477C2×C4#1313320212yes—
478C2×C4SD16×D5312yes—
479C2×C4#1313324612yes—
480C2×C4C2×Q8×D5311yescharacteristic
481C2×C4#41×D5311yescharacteristic
482C2×C4C2×#1171994511yescharacteristic
483C2×C4C2×#1171994911yescharacteristic
484C2×C4#1313326211yescharacteristic
485C2×C4#1313326311yescharacteristic
486C8—42no—
487C8—2122no—
488C22—534no—
489C22—1062no—
490C22—1062no—
491C22—1062no—
492C22—1062no—
493C22—1062no—
494C22—1062no—
495C22C22⋊Q8×D5311yescharacteristic
496C22C2×#1313320211yescharacteristic
497C22#1681118911yescharacteristic
498C22C2×SD16×D5311yescharacteristic
499C22AGL(1,ℤ/8)×D5311yescharacteristic
500C22#137×D5311yescharacteristic
501C22C2×#1313324611yescharacteristic
502C4—22no—
503C4—42no—
504C4—81no—
505C4—82no—
506C4—1062no—
507C4—1062no—
508C4—1062no—
509C4—4242no—
510C4—4241no—
511C4—2122no—
512C4#1681095514yes—
513C4#1681132311yescharacteristic
514C4#1681132611yescharacteristic
515C4#1681133311yescharacteristic
516C4#1681133411yescharacteristic
517C2—1064no—
518C2#2027903011yescharacteristic
519C2#475×D5311yescharacteristic
520C2#2028452711yescharacteristic
521C2C2×#1681095511yescharacteristic
522C2#2028464111yescharacteristic
523C2#2028565511yescharacteristic
524C2#2028574111yescharacteristic
subgroupG/H
centerC23C2×D4×D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC53×#1948C2
socleC23×C53C22×D4
prime radicalC23×D53C2×D4
hypercenter#1948D53
hypocenterC53#27680
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1948

Complements

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

NH ≅ G/Ntype
3. #20276276C2semidirect
4. #20276255C2semidirect
5. #20275478C2semidirect
8. #20274721C2semidirect
9. #20274717C2semidirect
11. C53×#1948C2semidirect
260. #1948D53semidirect
305. C53#27680semidirect

Maximal subgroups

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

subgroupindex
C4×#131331972
#384×D532
#202762762
#202762552
#202754782
#202750212
#202749062
#202747212
#202747172
#202746922
C53×#19482
#2768053