Finite Group Databasebeta

← #27440597

Subgroups — #27440597

Order 13568.

1292 conjugacy classes of subgroups.

The subgroup lattice has 538 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810991C211yesmaximal, characteristic, complemented
2C2×#16810923C211yesmaximal, characteristic, complemented
3#20279881C211yesmaximal, characteristic, complemented
4#20275915C212yesmaximal, complemented
5#20275908C212yesmaximal, complemented
6#20275639C211yesmaximal, characteristic, complemented
7#20275514C211yesmaximal, characteristic, complemented
8#20274655C212yesmaximal, complemented
9#20274655C212yesmaximal, complemented
10#20274540C211yesmaximal, characteristic, complemented
11C2×C53×#547C211yesmaximal, characteristic
12C2×#13133209C2211yescharacteristic
13#16810991—24no—
14#16810923—24no—
15C2×#13133150C2211yescharacteristic
16C2×#13133148C2211yescharacteristic
17C2×C4⋊#88322C2211yescharacteristic
18C2×#13133141C2212yes—
19C2×#13133141C2212yes—
20C2×#13133141C2212yes—
21C2×#13133141C2212yes—
22C2×#13133138C2211yescharacteristic
23C2×#13133138C2211yescharacteristic
24C2×#13133138C2212yes—
25C2×C22⋊#88322C2212yes—
26C2×C22⋊#88322C2212yes—
27C2×#13133128C2211yescharacteristic
28#16810561—21no—
29#16810537—21no—
30#16810517—22no—
31#16810483C2212yes—
32#16810483C2212yes—
33C4×C4⋊Dic53C2211yescharacteristic
34#16810385C2212yes—
35C22×#41×C53C2212yes—
36C53×#547—24no—
37C53×#547—24no—
38C2×C4⋊Q8×C53C2211yescharacteristic
39C2×#123×C53—22no—
40C2×C22⋊Q8×C53C2212yes—
41C2×#121×C53—21no—
42C2×C4×D4×C53C2211yescharacteristic
43C2×C4×D4×C53C2211yescharacteristic
44C22×C4⋊C4×C53—21no—
45C22×#88322C2312yes—
46C22×#88322C2311yescharacteristic
47C22×#88322C2311yescharacteristic
48#13133209—22no—
49#13133209—22no—
50C2×C4⋊Dic53C2311yescharacteristic
51C2×C4⋊Dic53C2311yescharacteristic
52C2×C4⋊Dic53C2312yes—
53C2×#11719919—21no—
54C2×#11719919—21no—
55C2×C22⋊Dic53—22no—
56C2×C22⋊Dic53—22no—
57C2×C4×Dic53—21no—
58#13133150—22no—
59#13133148—22no—
60C4⋊#88322—22no—
61#13133141—24no—
62#13133141—24no—
63#13133141—42no—
64#13133141—42no—
65#13133138—22no—
66#13133138—22no—
67#13133138—24no—
68C22⋊#88322—42no—
69C22⋊#88322—42no—
70#13133128—22no—
71#13133124—22no—
72#13133124—22no—
73#13133124—21no—
74C2×#41×C53—24no—
75C2×#41×C53—24no—
76C2×#41×C53—24no—
77C2×#41×C53—24no—
78C2×#41×C53—42no—
79C2×#41×C53—42no—
80C22×Q8×C53C2312yes—
81C22×D4×C53—22no—
82C22×D4×C53D411yescharacteristic
83C22×D4×C53C2311yescharacteristic
84C23×C4×C53—21no—
85C23×C4×C53—22no—
86C23×C4×C53C2312yes—
87C4⋊Q8×C53—22no—
88C4⋊Q8×C53D414yes—
89#123×C53—216no—
90C22⋊Q8×C53—24no—
91C22⋊Q8×C53—24no—
92#121×C53—28no—
93C4×D4×C53—22no—
94C4×D4×C53—22no—
95C4×D4×C53—22no—
96C4×D4×C53D414yes—
97C2×C4⋊C4×C53—28no—
98C2×C4⋊C4×C53—21no—
99C2×C4⋊C4×C53—22no—
100C2×C4⋊C4×C53—21no—
101C2×C4⋊C4×C53D411yescharacteristic
102C2×C4⋊C4×C53C2311yescharacteristic
103C2×C22⋊C4×C53—22no—
104C2×C22⋊C4×C53—21no—
105C2×C22⋊C4×C53D412yes—
106C2×C42×C53C2311yescharacteristic
107C24×C53—21no—
108C24×C53C2×D412yes—
109#41×C53—48no—
110#41×C53—48no—
111#41×C53—48no—
112#41×C53—48no—
113C2×Q8×C53—22no—
114C2×Q8×C53—22no—
115C2×Q8×C53—22no—
116C2×Q8×C53—22no—
117C2×Q8×C53—22no—
118C2×Q8×C53C2×D414yes—
119C2×D4×C53—28no—
120C2×D4×C53—22no—
121C2×D4×C53—21no—
122C2×D4×C53—22no—
123C2×D4×C53—42no—
124C2×D4×C53—44no—
125C2×D4×C53—44no—
126C2×D4×C53C2×D412yes—
127C2×D4×C53#4114yes—
128C22×C4×C53—28no—
129C22×C4×C53—21no—
130C22×C4×C53—22no—
131C22×C4×C53—24no—
132C22×C4×C53—22no—
133C22×C4×C53—22no—
134C22×C4×C53—22no—
135C22×C4×C53—22no—
136C22×C4×C53—22no—
137C22×C4×C53—22no—
138C22×C4×C53—44no—
139C22×C4×C53—44no—
140C22×C4×C53—44no—
141C22×C4×C53C2×D412yes—
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2411yescharacteristic
145C22×Dic53—22no—
146C22×Dic53—21no—
147C22×Dic53—21no—
148C4⋊C4×C53—28no—
149C4⋊C4×C53—22no—
150C4⋊C4×C53—24no—
151C4⋊C4×C53—24no—
152C4⋊C4×C53—21no—
153C4⋊C4×C53C2×D412yes—
154C22⋊C4×C53—28no—
155C22⋊C4×C53—24no—
156C22⋊C4×C53—24no—
157C42×C53C2×D412yes—
158C42×C53C2×D412yes—
159C22⋊Dic53—44no—
160C22⋊Dic53—82no—
161C4⋊Dic53—21no—
162C4⋊Dic53—21no—
163C4⋊Dic53—21no—
164C4⋊Dic53—21no—
165C4⋊Dic53—22no—
166C4⋊Dic53—22no—
167#11719919—42no—
168#11719919—42no—
169C4×Dic53—81no—
170C2×#88322—22no—
171C2×#88322—21no—
172C2×#88322—21no—
173C2×#88322—22no—
174C2×#88322—22no—
175C2×#88322—22no—
176C2×#88322—22no—
177C23×C53—22no—
178C23×C53—24no—
179C23×C53—22no—
180C23×C53—22no—
181C23×C53—22no—
182C23×C53—22no—
183C23×C53—22no—
184C23×C53—22no—
185C23×C53—22no—
186C23×C53—42no—
187C23×C53—41no—
188C23×C53—41no—
189C23×C53C22×D411yescharacteristic
190Q8×C53—24no—
191Q8×C53—24no—
192Q8×C53—42no—
193Q8×C53—42no—
194D4×C53—22no—
195D4×C53—24no—
196D4×C53—44no—
197D4×C53—416no—
198D4×C53#12414yes—
199C2×C4×C53—22no—
200C2×C4×C53—21no—
201C2×C4×C53—21no—
202C2×C4×C53—22no—
203C2×C4×C53—24no—
204C2×C4×C53—22no—
205C2×C4×C53—22no—
206C2×C4×C53—22no—
207C2×C4×C53—48no—
208C2×C4×C53—416no—
209C2×C4×C53—42no—
210C2×C4×C53—42no—
211C2×C4×C53—42no—
212C2×C4×C53—42no—
213C2×C4×C53—42no—
214C2×C4×C53—42no—
215C2×C4×C53—42no—
216C2×C4×C53#12012yes—
217C2×C4×C53#12012yes—
218C2×C4×C53C22×D411yescharacteristic
219C2×C4×C53C22×D411yescharacteristic
220C2×C4×C53C2×#4112yes—
221C2×C4×C53H(5,𝔽2)12yes—
222C2×Dic53—41no—
223C2×Dic53—41no—
224C2×Dic53—42no—
225C2×Dic53—42no—
226C2×Dic53—42no—
227C2×Dic53—42no—
228C2×Dic53—41no—
229C2×Dic53—41no—
230#88322—41no—
231#88322—41no—
232#88322—42no—
233#25892—531nomaximal, Sylow
234C22×C53—24no—
235C22×C53—42no—
236C22×C53—42no—
237C22×C53—44no—
238C22×C53—44no—
239C22×C53—42no—
240C22×C53—42no—
241C22×C53—42no—
242C22×C53—42no—
243C22×C53—42no—
244C22×C53—42no—
245C22×C53—42no—
246C22×C53C2×#12011yescharacteristic
247C22×C53#54612yes—
248C22×C53#57512yes—
249C22×C53#57711yescharacteristic
250C22×C53#57811yescharacteristic
251C4×C53—22no—
252C4×C53—22no—
253C4×C53—42no—
254C4×C53—44no—
255C4×C53—42no—
256C4×C53C2×#12411yescharacteristic
257C4×C53#53511yescharacteristic
258C4×C53#54612yes—
259Dic53—81no—
260Dic53—81no—
261Dic53—82no—
262C2×#547D5311yescharacteristic
263C2×#491—531no—
264C2×#478—531no—
265#2562—531no—
266#2036—532no—
267#2035—532no—
268#1981—531no—
269#1955—531no—
270#1788—532no—
271#1788—532no—
272#1762—531no—
273C2×C53—44no—
274C2×C53—82no—
275C2×C53#243511yescharacteristic
276C2×C53#300511yescharacteristic
277C2×C53#301311yescharacteristic
278C2×C53#330712yes—
279C2×C53#331512yes—
280C22×#41C2×D5312yes—
281#547—24no—
282#547—24no—
283C2×C4⋊Q8C2×D5311yescharacteristic
284C2×#125—531no—
285C2×#123—22no—
286C2×C22⋊Q8C2×D5312yes—
287C2×#121—21no—
288C2×C4×D4C2×D5311yescharacteristic
289C2×C4×D4C2×D5311yescharacteristic
290C22×C4⋊C4—21no—
291#491—1064no—
292#478—1064no—
293C2×#107—531no—
294C2×#106—531no—
295C2×C4⋊C8—531no—
296C2×#103—532no—
297C2×#103—532no—
298C2×#103—532no—
299C2×#103—532no—
300C2×#102—531no—
301C2×#102—531no—
302C2×#102—532no—
303C2×C22⋊C8—532no—
304C2×C22⋊C8—532no—
305C2×#97—531no—
306#400—1061no—
307#394—1061no—
308#388—1062no—
309#380—532no—
310#380—532no—
311C4×C4⋊C4—531no—
312#340—532no—
313C53#2589211yescharacteristic, complemented, Sylow
314C2×#41—24no—
315C2×#41—24no—
316C2×#41—24no—
317C2×#41—24no—
318C2×#41—42no—
319C2×#41—42no—
320C22×Q8C22×D5312yes—
321C22×D4—22no—
322C22×D4C22⋊D5311yescharacteristic
323C22×D4C22×D5311yescharacteristic
324C23×C4—21no—
325C23×C4—22no—
326C23×C4C22×D5312yes—
327C22×C8—532no—
328C22×C8—531no—
329C22×C8—531no—
330C4⋊Q8—22no—
331C4⋊Q8C22⋊D5314yes—
332#125—1062no—
333#125—1062no—
334#123—216no—
335C22⋊Q8—24no—
336C22⋊Q8—24no—
337#121—28no—
338C4×D4—22no—
339C4×D4—22no—
340C4×D4—22no—
341C4×D4C22⋊D5314yes—
342C2×C4⋊C4—28no—
343C2×C4⋊C4—21no—
344C2×C4⋊C4—22no—
345C2×C4⋊C4—21no—
346C2×C4⋊C4—531no—
347C2×C4⋊C4—531no—
348C2×C4⋊C4—532no—
349C2×C4⋊C4—1061no—
350C2×C4⋊C4—1061no—
351C2×C4⋊C4C22⋊D5311yescharacteristic
352C2×C4⋊C4C22×D5311yescharacteristic
353C2×C22⋊C4—22no—
354C2×C22⋊C4—21no—
355C2×C22⋊C4—1062no—
356C2×C22⋊C4—1062no—
357C2×C22⋊C4C22⋊D5312yes—
358C2×C42—1061no—
359C2×C42C22×D5311yescharacteristic
360#107—1062no—
361#106—1062no—
362C4⋊C8—1062no—
363#103—1064no—
364#103—1064no—
365#103—2122no—
366#103—2122no—
367#102—1064no—
368#102—1062no—
369#102—1062no—
370C22⋊C8—2122no—
371C22⋊C8—2122no—
372#97—1062no—
373#95—1061no—
374#95—1062no—
375#95—1062no—
376C24—21no—
377C24C2×C22⋊D5312yes—
378#41—48no—
379#41—48no—
380#41—48no—
381#41—48no—
382C2×Q8—22no—
383C2×Q8—22no—
384C2×Q8—22no—
385C2×Q8—22no—
386C2×Q8—22no—
387C2×Q8C2×C22⋊D5314yes—
388C2×D4—28no—
389C2×D4—22no—
390C2×D4—21no—
391C2×D4—22no—
392C2×D4—42no—
393C2×D4—44no—
394C2×D4—44no—
395C2×D4#1171994714yes—
396C2×D4C2×C22⋊D5312yes—
397C22×C4—28no—
398C22×C4—21no—
399C22×C4—22no—
400C22×C4—24no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—44no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—1062no—
411C22×C4—1061no—
412C22×C4—1061no—
413C22×C4C2×C22⋊D5312yes—
414C22×C4C2×C22⋊D5311yescharacteristic
415C22×C4C2×C22⋊D5311yescharacteristic
416C22×C4C23×D5311yescharacteristic
417C2×C8—1062no—
418C2×C8—1061no—
419C2×C8—1061no—
420C2×C8—1062no—
421C2×C8—1062no—
422C2×C8—1062no—
423C2×C8—1062no—
424C4⋊C4—28no—
425C4⋊C4—22no—
426C4⋊C4—24no—
427C4⋊C4—24no—
428C4⋊C4—21no—
429C4⋊C4—1061no—
430C4⋊C4—1061no—
431C4⋊C4—1061no—
432C4⋊C4—1061no—
433C4⋊C4—1062no—
434C4⋊C4—1062no—
435C4⋊C4—2122no—
436C4⋊C4—2122no—
437C4⋊C4C2×C22⋊D5312yes—
438C22⋊C4—28no—
439C22⋊C4—24no—
440C22⋊C4—24no—
441C22⋊C4—4242no—
442C22⋊C4—2124no—
443C42—4241no—
444C42C2×C22⋊D5312yes—
445C42C2×C22⋊D5312yes—
446C23—24no—
447C23—22no—
448C23—22no—
449C23—22no—
450C23—22no—
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—42no—
456C23—41no—
457C23—41no—
458C23C22×C22⋊D5311yescharacteristic
459Q8—24no—
460Q8—24no—
461Q8—42no—
462Q8—42no—
463D4—24no—
464D4—22no—
465D4—44no—
466D4—416no—
467D4#1313320414yes—
468C2×C4—22no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—22no—
472C2×C4—22no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—24no—
476C2×C4—48no—
477C2×C4—416no—
478C2×C4—42no—
479C2×C4—42no—
480C2×C4—42no—
481C2×C4—42no—
482C2×C4—42no—
483C2×C4—42no—
484C2×C4—42no—
485C2×C4—2122no—
486C2×C4—2121no—
487C2×C4—2121no—
488C2×C4—2122no—
489C2×C4—2122no—
490C2×C4—2122no—
491C2×C4—2121no—
492C2×C4—2121no—
493C2×C4#1313318612yes—
494C2×C4#1313318612yes—
495C2×C4C22×C22⋊D5311yescharacteristic
496C2×C4C22×C22⋊D5311yescharacteristic
497C2×C4C2×#1171994712yes—
498C2×C4#1313326212yes—
499C8—2122no—
500C8—2121no—
501C8—2121no—
502C22—24no—
503C22—42no—
504C22—44no—
505C22—42no—
506C22—44no—
507C22—42no—
508C22—42no—
509C22—42no—
510C22—42no—
511C22—42no—
512C22—42no—
513C22—42no—
514C22C2×#1313318611yescharacteristic
515C22#1681126712yes—
516C22#1681142112yes—
517C22#1681143111yescharacteristic
518C22#1681143811yescharacteristic
519C4—22no—
520C4—22no—
521C4—42no—
522C4—42no—
523C4—44no—
524C4—4241no—
525C4—4241no—
526C4—4242no—
527C4C2×#1313320411yescharacteristic
528C4#1681119011yescharacteristic
529C4#1681126712yes—
530C2—44no—
531C2—82no—
532C2#2027836211yescharacteristic
533C2#2028395711yescharacteristic
534C2#2028403011yescharacteristic
535C2#2028669112yes—
536C2#2028681112yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
Fitting, prime radicalC2×C53×#547C2
socleC23×C53C22×D4
hypercenterC2×#547D53
hypocenterC53#25892
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. C2×#16810991C2semidirect
2. C2×#16810923C2semidirect
3. #20279881C2semidirect
4. #20275915C2semidirect
5. #20275908C2semidirect
6. #20275639C2semidirect
7. #20275514C2semidirect
8. #20274655C2semidirect
9. #20274655C2semidirect
10. #20274540C2semidirect
313. C53#25892semidirect

Maximal subgroups

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

subgroupindex
C2×#168109912
C2×#168109232
#202798812
#202759152
#202759082
#202756392
#202755142
#202746552
#202745402
C2×C53×#5472
#2589253