Finite Group Databasebeta

← #27419652

Subgroups — #27419652

Order 13568.

1392 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×#16810830C211yesmaximal, characteristic, complemented
2C2×#16810779C212yesmaximal, complemented
3C2×#16810728C211yesmaximal, characteristic, complemented
4#20279624C211yesmaximal, characteristic
5#20275908C212yesmaximal, complemented
6#20275723C212yesmaximal, complemented
7#20275536C211yesmaximal, characteristic, complemented
8#20275514C211yesmaximal, characteristic
9#20274730C211yesmaximal, characteristic, complemented
10#20274650C212yesmaximal, complemented
11C2×D4×Q8×C53C211yesmaximal, characteristic
12C22×#11719925C2212yes—
13C22×#11719925C2211yescharacteristic
14C2×#13133196C2212yes—
15C2×C4×Dic106C2211yescharacteristic
16#16810830—24no—
17#16810779—28no—
18#16810728—24no—
19C2×#13133148C2211yescharacteristic
20C2×C4⋊#88322C2211yescharacteristic
21C2×#13133141C2212yes—
22C2×#13133141C2212yes—
23C2×#13133141C2211yescharacteristic
24C2×#13133141C2212yes—
25C2×#13133141C2212yes—
26C2×#13133140C2211yescharacteristic
27C2×C22⋊#88322C2212yes—
28C2×C22⋊#88322C2212yes—
29C2×#13133128C2211yescharacteristic
30#16810554—21no—
31#16810537—21no—
32#16810511—22no—
33#16810489C2211yescharacteristic
34#16810483C2212yes—
35#16810385C2212yes—
36C23×Q8×C53C2212yes—
37D4×Q8×C53—24no—
38D4×Q8×C53—24no—
39C2×C4⋊Q8×C53—21no—
40C2×C4⋊Q8×C53C2211yescharacteristic
41C2×C22⋊Q8×C53—22no—
42C2×C22⋊Q8×C53C2212yes—
43C2×C4×Q8×C53C2211yescharacteristic
44C2×C4×D4×C53—21no—
45C2×C4×D4×C53C2211yescharacteristic
46C22×Dic106C2311yescharacteristic
47C2×#11719925—22no—
48C2×#11719925—24no—
49C2×#11719925—22no—
50C2×#11719925—24no—
51C2×#11719925—42no—
52C2×#11719925D414yescomplemented
53C22×#88322C2311yescharacteristic
54C22×#88322C2312yes—
55C22×#88322C2311yescharacteristic
56#13133196—24no—
57#13133196—42no—
58C4×Dic106—22no—
59C4×Dic106—22no—
60C2×C4⋊Dic53C2311yescharacteristic
61C2×C4⋊Dic53C2312yes—
62C2×#11719919—22no—
63C2×#11719919—21no—
64C2×C22⋊Dic53—22no—
65C2×C22⋊Dic53—22no—
66C2×C4×Dic53—21no—
67#13133148—22no—
68C4⋊#88322—22no—
69#13133141—24no—
70#13133141—24no—
71#13133141—24no—
72#13133141—22no—
73#13133141—42no—
74#13133140—22no—
75C22⋊#88322—24no—
76C22⋊#88322—42no—
77#13133128—22no—
78#13133124—21no—
79#13133124—22no—
80#13133124—21no—
81C22×Q8×C53—24no—
82C22×Q8×C53—24no—
83C22×Q8×C53—22no—
84C22×Q8×C53—24no—
85C22×Q8×C53—24no—
86C22×Q8×C53—42no—
87C22×Q8×C53—42no—
88C22×Q8×C53C2312yes—
89C22×Q8×C53C2311yescharacteristic
90C22×D4×C53D411yescharacteristic
91C23×C4×C53—22no—
92C23×C4×C53C2312yes—
93C4⋊Q8×C53—28no—
94C4⋊Q8×C53—22no—
95C4⋊Q8×C53D414yes—
96C22⋊Q8×C53—216no—
97C22⋊Q8×C53—24no—
98C22⋊Q8×C53—24no—
99C4×Q8×C53—22no—
100C4×Q8×C53D414yes—
101C4×D4×C53—28no—
102C4×D4×C53—22no—
103C4×D4×C53—22no—
104C2×C4⋊C4×C53—21no—
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53—21no—
107C2×C4⋊C4×C53—22no—
108C2×C4⋊C4×C53D411yescharacteristic
109C2×C4⋊C4×C53C2311yescharacteristic
110C2×C22⋊C4×C53—22no—
111C2×C22⋊C4×C53D412yes—
112C2×C42×C53—21no—
113C2×C42×C53C2311yescharacteristic
114C24×C53C2×D412yes—
115C2×Q8×C53—28no—
116C2×Q8×C53—21no—
117C2×Q8×C53—22no—
118C2×Q8×C53—21no—
119C2×Q8×C53—21no—
120C2×Q8×C53—22no—
121C2×Q8×C53—22no—
122C2×Q8×C53—22no—
123C2×Q8×C53—22no—
124C2×Q8×C53—44no—
125C2×Q8×C53—44no—
126C2×Q8×C53—48no—
127C2×Q8×C53—48no—
128C2×Q8×C53—48no—
129C2×Q8×C53—48no—
130C2×Q8×C53C2×D414yes—
131C2×Q8×C53C2×D412yes—
132C2×Q8×C53C2×D412yes—
133C2×Q8×C53#4112yes—
134C2×D4×C53—24no—
135C2×D4×C53—22no—
136C22×C4×C53—21no—
137C22×C4×C53—28no—
138C22×C4×C53—21no—
139C22×C4×C53—22no—
140C22×C4×C53—22no—
141C22×C4×C53—22no—
142C22×C4×C53—22no—
143C22×C4×C53—22no—
144C22×C4×C53—22no—
145C22×C4×C53—22no—
146C22×C4×C53—44no—
147C22×C4×C53—44no—
148C22×C4×C53C2×D412yes—
149C22×C4×C53C2×D411yescharacteristic
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2411yescharacteristic
152C22×Dic53—21no—
153C22×Dic53—22no—
154C22×Dic53—21no—
155C2×Dic106—21no—
156C2×Dic106—21no—
157C2×Dic106—21no—
158C2×Dic106—21no—
159C2×Dic106—21no—
160C2×Dic106C2×D412yes—
161C4⋊C4×C53—28no—
162C4⋊C4×C53—24no—
163C4⋊C4×C53—24no—
164C4⋊C4×C53—24no—
165C4⋊C4×C53—22no—
166C4⋊C4×C53—21no—
167C4⋊C4×C53C2×D412yes—
168C22⋊C4×C53—28no—
169C22⋊C4×C53—24no—
170C42×C53—24no—
171C42×C53C2×D412yes—
172C42×C53C2×D412yes—
173C22⋊Dic53—44no—
174C22⋊Dic53—44no—
175#11719925—28no—
176#11719925—48no—
177C4⋊Dic53—21no—
178C4⋊Dic53—22no—
179C4⋊Dic53—21no—
180C4⋊Dic53—22no—
181#11719919—44no—
182#11719919—42no—
183C4×Dic53—42no—
184C2×#88322—21no—
185C2×#88322—21no—
186C2×#88322—21no—
187C2×#88322—22no—
188C2×#88322—22no—
189C2×#88322—21no—
190C2×#88322—22no—
191C2×#88322—21no—
192C2×#88322C2×D412yes—
193C23×C53—22no—
194C23×C53—22no—
195C23×C53—22no—
196C23×C53—22no—
197C23×C53—22no—
198C23×C53—22no—
199C23×C53#13714yes—
200C23×C53C22×D411yescharacteristic
201Q8×C53—22no—
202Q8×C53—22no—
203Q8×C53—22no—
204Q8×C53—24no—
205Q8×C53—24no—
206Q8×C53—416no—
207Q8×C53—42no—
208Q8×C53—42no—
209Q8×C53#12114yes—
210D4×C53—28no—
211C2×C4×C53—22no—
212C2×C4×C53—22no—
213C2×C4×C53—22no—
214C2×C4×C53—22no—
215C2×C4×C53—22no—
216C2×C4×C53—24no—
217C2×C4×C53—22no—
218C2×C4×C53—22no—
219C2×C4×C53—22no—
220C2×C4×C53—22no—
221C2×C4×C53—416no—
222C2×C4×C53—42no—
223C2×C4×C53—42no—
224C2×C4×C53—42no—
225C2×C4×C53—42no—
226C2×C4×C53—42no—
227C2×C4×C53—42no—
228C2×C4×C53—42no—
229C2×C4×C53#12012yes—
230C2×C4×C53#12012yes—
231C2×C4×C53C22×D411yescharacteristic
232C2×C4×C53C22×D411yescharacteristic
233C2×C4×C53C22×D411yescharacteristic
234C2×C4×C53C2×#4111yescharacteristic
235C2×C4×C53H(5,𝔽2)11yescharacteristic
236C2×C4×C53H(5,𝔽2)11yescharacteristic
237C2×Dic53—22no—
238C2×Dic53—41no—
239C2×Dic53—41no—
240C2×Dic53—41no—
241C2×Dic53—42no—
242C2×Dic53—42no—
243C2×Dic53—42no—
244C2×Dic53—42no—
245Dic106—22no—
246Dic106—22no—
247Dic106—41no—
248Dic106—41no—
249#88322—22no—
250#88322—41no—
251#88322—42no—
252#24255—531nomaximal, Sylow
253C22×C53—24no—
254C22×C53—24no—
255C22×C53—24no—
256C22×C53—42no—
257C22×C53—42no—
258C22×C53—42no—
259C22×C53—42no—
260C22×C53C2×#12011yescharacteristic
261C22×C53D4211yescharacteristic
262C22×C53#54611yescharacteristic
263C22×C53C2×#13712yes—
264C22×C53#57812yes—
265C4×C53—24no—
266C4×C53—22no—
267C4×C53—22no—
268C4×C53—42no—
269C4×C53—44no—
270C4×C53—42no—
271C4×C53C2×#12111yescharacteristic
272C4×C53#53511yescharacteristic
273C4×C53D4211yescharacteristic
274C4×C53#54611yescharacteristic
275Dic53—42no—
276Dic53—81no—
277Dic53—82no—
278C2×D4×Q8D5311yescharacteristic
279C2×#462—531no—
280C2×#451—532no—
281C2×#441—531no—
282#2536—531no—
283#2035—532no—
284#2001—532no—
285#1960—531no—
286#1955—531no—
287#1805—531no—
288#1787—532no—
289C2×C53—44no—
290C2×C53#240811yescharacteristic
291C2×C53#300411yescharacteristic
292C2×C53#301111yescharacteristic
293C2×C53#327012yes—
294C2×C53#331212yes—
295C23×Q8C2×D5312yes—
296C22×Q16—531no—
297C22×Q16—532no—
298D4×Q8—24no—
299D4×Q8—24no—
300C2×C4⋊Q8—21no—
301C2×C4⋊Q8C2×D5311yescharacteristic
302C2×C22⋊Q8—22no—
303C2×C22⋊Q8—532no—
304C2×C22⋊Q8C2×D5312yes—
305C2×C4×Q8—531no—
306C2×C4×Q8C2×D5311yescharacteristic
307C2×C4×D4—21no—
308C2×C4×D4C2×D5311yescharacteristic
309#462—1064no—
310#451—1068no—
311#441—1064no—
312C2×#106—531no—
313C2×C4⋊C8—531no—
314C2×#103—532no—
315C2×#103—531no—
316C2×#103—532no—
317C2×#103—532no—
318C2×#103—531no—
319C2×#103—532no—
320C2×C22⋊C8—532no—
321C2×C22⋊C8—532no—
322C2×#97—531no—
323#398—1061no—
324#394—1061no—
325#388—1062no—
326#382—531no—
327#380—532no—
328#340—532no—
329C53#2425511yescharacteristic, complemented, Sylow
330C22×Q8—24no—
331C22×Q8—24no—
332C22×Q8—22no—
333C22×Q8—24no—
334C22×Q8—24no—
335C22×Q8—42no—
336C22×Q8—42no—
337C22×Q8—531no—
338C22×Q8C22×D5312yes—
339C22×Q8C22×D5311yescharacteristic
340C22×D4C22⋊D5311yescharacteristic
341C23×C4—22no—
342C23×C4C22×D5312yes—
343C2×Q16—534no—
344C2×Q16—1062no—
345C2×Q16—1062no—
346C2×Q16—1064no—
347C2×Q16—1064no—
348C2×Q16—2122no—
349C22×C8—531no—
350C22×C8—532no—
351C22×C8—531no—
352C4⋊Q8—28no—
353C4⋊Q8—22no—
354C4⋊Q8C22⋊D5314yes—
355C22⋊Q8—216no—
356C22⋊Q8—24no—
357C22⋊Q8—24no—
358C22⋊Q8—1064no—
359C22⋊Q8—2122no—
360C4×Q8—22no—
361C4×Q8—1062no—
362C4×Q8—1062no—
363C4×Q8C22⋊D5314yes—
364C4×D4—28no—
365C4×D4—22no—
366C4×D4—22no—
367C2×C4⋊C4—21no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—21no—
370C2×C4⋊C4—22no—
371C2×C4⋊C4—531no—
372C2×C4⋊C4—532no—
373C2×C4⋊C4—1061no—
374C2×C4⋊C4—1062no—
375C2×C4⋊C4C22⋊D5311yescharacteristic
376C2×C4⋊C4C22×D5311yescharacteristic
377C2×C22⋊C4—22no—
378C2×C22⋊C4—1062no—
379C2×C22⋊C4—1062no—
380C2×C22⋊C4C22⋊D5312yes—
381C2×C42—21no—
382C2×C42—1061no—
383C2×C42C22×D5311yescharacteristic
384#106—1062no—
385C4⋊C8—1062no—
386#103—1064no—
387#103—1064no—
388#103—1062no—
389#103—1064no—
390#103—1062no—
391#103—2122no—
392C22⋊C8—1064no—
393C22⋊C8—2122no—
394#97—1062no—
395#95—1062no—
396#95—1061no—
397#95—1061no—
398C24C2×C22⋊D5312yes—
399C2×Q8—28no—
400C2×Q8—21no—
401C2×Q8—22no—
402C2×Q8—21no—
403C2×Q8—21no—
404C2×Q8—22no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—44no—
409C2×Q8—44no—
410C2×Q8—48no—
411C2×Q8—48no—
412C2×Q8—48no—
413C2×Q8—48no—
414C2×Q8—532no—
415C2×Q8—1061no—
416C2×Q8—1061no—
417C2×Q8—1061no—
418C2×Q8—1061no—
419C2×Q8—1061no—
420C2×Q8D4×D5312yes—
421C2×Q8#1171994712yes—
422C2×Q8C2×C22⋊D5314yes—
423C2×Q8C2×C22⋊D5312yes—
424C2×D4—24no—
425C2×D4—22no—
426C22×C4—21no—
427C22×C4—28no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—22no—
434C22×C4—22no—
435C22×C4—22no—
436C22×C4—44no—
437C22×C4—44no—
438C22×C4—1061no—
439C22×C4—1062no—
440C22×C4—1061no—
441C22×C4C2×C22⋊D5312yes—
442C22×C4C2×C22⋊D5311yescharacteristic
443C22×C4C2×C22⋊D5311yescharacteristic
444C22×C4C23×D5311yescharacteristic
445Q16—1068no—
446Q16—2128no—
447C2×C8—532no—
448C2×C8—1061no—
449C2×C8—1061no—
450C2×C8—1062no—
451C2×C8—1061no—
452C2×C8—1062no—
453C2×C8—1061no—
454C2×C8—1062no—
455C2×C8—1061no—
456C4⋊C4—28no—
457C4⋊C4—24no—
458C4⋊C4—24no—
459C4⋊C4—24no—
460C4⋊C4—22no—
461C4⋊C4—21no—
462C4⋊C4—1062no—
463C4⋊C4—1061no—
464C4⋊C4—1061no—
465C4⋊C4—1062no—
466C4⋊C4—2124no—
467C4⋊C4—2122no—
468C4⋊C4C2×C22⋊D5312yes—
469C22⋊C4—28no—
470C22⋊C4—24no—
471C22⋊C4—2124no—
472C22⋊C4—2124no—
473C42—24no—
474C42—2122no—
475C42C2×C22⋊D5312yes—
476C42C2×C22⋊D5312yes—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23#1313324814yes—
484C23C22×C22⋊D5311yescharacteristic
485Q8—22no—
486Q8—22no—
487Q8—22no—
488Q8—24no—
489Q8—24no—
490Q8—416no—
491Q8—42no—
492Q8—42no—
493Q8—1062no—
494Q8—1062no—
495Q8—2121no—
496Q8—2121no—
497Q8#1313318814yes—
498D4—28no—
499C2×C4—22no—
500C2×C4—22no—
501C2×C4—22no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—22no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—416no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—42no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—1062no—
518C2×C4—2121no—
519C2×C4—2122no—
520C2×C4—2122no—
521C2×C4—2122no—
522C2×C4—2122no—
523C2×C4—2121no—
524C2×C4—2121no—
525C2×C4#1313318612yes—
526C2×C4#1313318612yes—
527C2×C4C2×D4×D5311yescharacteristic
528C2×C4C22×C22⋊D5311yescharacteristic
529C2×C4C22×C22⋊D5311yescharacteristic
530C2×C4C2×#1171994711yescharacteristic
531C2×C4#1313326211yescharacteristic
532C2×C4#1313326211yescharacteristic
533C8—1062no—
534C8—2121no—
535C8—2122no—
536C22—24no—
537C22—24no—
538C22—24no—
539C22—42no—
540C22—42no—
541C22—42no—
542C22—42no—
543C22C2×#1313318611yescharacteristic
544C22D4×C22⋊D5311yescharacteristic
545C22#1681126711yescharacteristic
546C22C2×#1313324812yes—
547C22#1681143812yes—
548C4—22no—
549C4—24no—
550C4—22no—
551C4—42no—
552C4—44no—
553C4—42no—
554C4—4241no—
555C4—4242no—
556C4—2122no—
557C4C2×#1313318811yescharacteristic
558C4#1681119011yescharacteristic
559C4D4×C22⋊D5311yescharacteristic
560C4#1681126711yescharacteristic
561C2—44no—
562C2#2027806511yescharacteristic
563C2#2028394911yescharacteristic
564C2#2028401011yescharacteristic
565C2#2028628612yes—
566C2#2028676612yes—
subgroupG/H
centerC23C22×C22⋊D53
commutator, cosocleC22×C4×C53C24
FrattiniC22×C4C23×D53
FittingC2×D4×Q8×C53C2
socleC23×C53C22×D4
prime radicalC22×D4×C53D4
hypercenterC2×D4×Q8D53
hypocenterC53#24255
LCSG ▷ C22×C4×C53 ▷ C2×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ C2×D4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#16810830C2semidirect
2. C2×#16810779C2semidirect
3. C2×#16810728C2semidirect
5. #20275908C2semidirect
6. #20275723C2semidirect
7. #20275536C2semidirect
9. #20274730C2semidirect
10. #20274650C2semidirect
52. C2×#11719925D4semidirect
329. C53#24255semidirect

Maximal subgroups

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

subgroupindex
C2×#168108302
C2×#168107792
C2×#168107282
#202796242
#202759082
#202757232
#202755362
#202755142
#202747302
#202746502
C2×D4×Q8×C532
#2425553