Finite Group Databasebeta

← #27399550

Subgroups — #27399550

Order 13568.

1490 conjugacy classes of subgroups.

The subgroup lattice has 584 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×D4×#88322C211yesmaximal, characteristic, complemented
2C2×#16810619C212yesmaximal, complemented
3#20277664C211yesmaximal, characteristic, complemented
4#20275639C211yesmaximal, characteristic, complemented
5#20275518C211yesmaximal, characteristic, complemented
6#20275139C212yesmaximal, complemented
7#20275028C212yesmaximal, complemented
8#20274715C211yesmaximal, characteristic, complemented
9#20274655C212yesmaximal, complemented
10C4×#13133138C211yesmaximal, characteristic, complemented
11C2×C53×#547C211yesmaximal, characteristic
12C23×#88322C2212yes—
13C2×#13133176C2212yes—
14D4×#88322—24no—
15C2×C4⋊#88322C2211yescharacteristic
16#16810619—28no—
17C2×#13133141C2212yes—
18C2×#13133141C2212yes—
19C2×#13133141C2212yes—
20C2×#13133141C2212yes—
21C2×#13133138C2212yes—
22C2×#13133138C2211yescharacteristic
23C2×#13133138C2211yescharacteristic
24C2×C22⋊#88322C2212yes—
25C2×C4×#88322C2211yescharacteristic
26#16810526—21no—
27#16810516—22no—
28#16810493C2211yescharacteristic
29#16810483C2212yes—
30C4⋊C4×Dic53—21no—
31C4×C4⋊Dic53C2211yescharacteristic
32#16810385C2212yes—
33#16810385C2211yescharacteristic
34#16810385C2211yescharacteristic
35C22×#41×C53C2212yes—
36C53×#547—24no—
37C53×#547C418yescomplemented
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×#88322—24no—
46C22×#88322—24no—
47C22×#88322—42no—
48C22×#88322C2311yescharacteristic
49C22×#88322C2312yes—
50C22×#88322C2311yescharacteristic
51#13133176—24no—
52#13133176—42no—
53C2×C4⋊Dic53—21no—
54C2×C4⋊Dic53—21no—
55C2×C4⋊Dic53C2312yes—
56C2×C4⋊Dic53C2311yescharacteristic
57C2×C4⋊Dic53C2311yescharacteristic
58C2×C22⋊Dic53—22no—
59C2×C22⋊Dic53—22no—
60C2×C4×Dic53—22no—
61C2×C4×Dic53—21no—
62C4⋊#88322—22no—
63#13133141—24no—
64#13133141—24no—
65#13133141—42no—
66#13133141—42no—
67#13133138—24no—
68#13133138—22no—
69#13133138D414yescomplemented
70C22⋊#88322—24no—
71C4×#88322—22no—
72#13133124—21no—
73#13133124—22no—
74C2×#41×C53—24no—
75C2×#41×C53—24no—
76C2×#41×C53—24no—
77C2×#41×C53—42no—
78C2×#41×C53—42no—
79C2×#41×C53C2×C418yes—
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×C53C2×C414yes—
89#123×C53—216no—
90C22⋊Q8×C53—24no—
91C22⋊Q8×C53C2×C418yes—
92#121×C53—28no—
93C4×D4×C53—22no—
94C4×D4×C53C2×C414yes—
95C4×D4×C53C2×C414yes—
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—216no—
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×C53C22×C414yes—
119C2×D4×C53—28no—
120C2×D4×C53—22no—
121C2×D4×C53—22no—
122C2×D4×C53—21no—
123C2×D4×C53—21no—
124C2×D4×C53—21no—
125C2×D4×C53—44no—
126C2×D4×C53—44no—
127C2×D4×C53C22⋊C414yes—
128C2×D4×C53C22×C412yes—
129C2×D4×C53C2×D412yes—
130C2×D4×C53#4112yes—
131C22×C4×C53—22no—
132C22×C4×C53—22no—
133C22×C4×C53—22no—
134C22×C4×C53—22no—
135C22×C4×C53—21no—
136C22×C4×C53—22no—
137C22×C4×C53—24no—
138C22×C4×C53—28no—
139C22×C4×C53—44no—
140C22×C4×C53—44no—
141C22×C4×C53—42no—
142C22×C4×C53—42no—
143C22×C4×C53C22×C414yes—
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2×D412yes—
146C22×C4×C53C2×D414yes—
147C22×C4×C53C2×D411yescharacteristic
148C22×C4×C53C2411yescharacteristic
149C22×Dic53—21no—
150C22×Dic53—21no—
151C22×Dic53—22no—
152C4⋊C4×C53—24no—
153C4⋊C4×C53—28no—
154C4⋊C4×C53—24no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53C22⋊C414yes—
157C4⋊C4×C53C22×C412yes—
158C22⋊C4×C53—28no—
159C22⋊C4×C53—24no—
160C22⋊C4×C53C22⋊C418yes—
161C42×C53C22×C412yes—
162C42×C53C2×D412yes—
163C22⋊Dic53—44no—
164C22⋊Dic53—82no—
165C4⋊Dic53—21no—
166C4⋊Dic53—22no—
167C4⋊Dic53—21no—
168C4⋊Dic53—21no—
169C4⋊Dic53—22no—
170C4⋊Dic53—42no—
171C4⋊Dic53—81no—
172C4⋊Dic53C2×D412yes—
173C4×Dic53—44no—
174C4×Dic53—42no—
175C2×#88322—21no—
176C2×#88322—21no—
177C2×#88322—21no—
178C2×#88322—22no—
179C2×#88322—22no—
180C2×#88322—21no—
181C2×#88322—21no—
182C2×#88322—22no—
183C2×#88322—48no—
184C2×#88322C2×D412yes—
185C23×C53—22no—
186C23×C53—22no—
187C23×C53—22no—
188C23×C53—22no—
189C23×C53—22no—
190C23×C53—22no—
191C23×C53—22no—
192C23×C53—22no—
193C23×C53—22no—
194C23×C53—41no—
195C23×C53—41no—
196C23×C53—41no—
197C23×C53—41no—
198C23×C53C2×C22⋊C414yes—
199C23×C53C22×D411yescharacteristic
200Q8×C53—24no—
201Q8×C53—24no—
202Q8×C53—42no—
203Q8×C53—42no—
204D4×C53—22no—
205D4×C53—22no—
206D4×C53—22no—
207D4×C53—28no—
208D4×C53—416no—
209D4×C53C4×D414yes—
210C2×C4×C53—22no—
211C2×C4×C53—22no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—24no—
215C2×C4×C53—24no—
216C2×C4×C53—24no—
217C2×C4×C53—24no—
218C2×C4×C53—24no—
219C2×C4×C53—22no—
220C2×C4×C53—416no—
221C2×C4×C53—42no—
222C2×C4×C53—42no—
223C2×C4×C53—41no—
224C2×C4×C53—41no—
225C2×C4×C53—48no—
226C2×C4×C53C2×C22⋊C414yes—
227C2×C4×C53C2×C22⋊C412yes—
228C2×C4×C53C2×C22⋊C412yes—
229C2×C4×C53#13512yes—
230C2×C4×C53#13512yes—
231C2×C4×C53C23×C411yescharacteristic
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—42no—
241C2×Dic53—41no—
242C2×Dic53—41no—
243C2×Dic53—42no—
244C2×Dic53—42no—
245C2×Dic53—41no—
246#88322—22no—
247#88322—42no—
248#88322—41no—
249#22417—531nomaximal, Sylow
250C22×C53—24no—
251C22×C53—24no—
252C22×C53—24no—
253C22×C53—24no—
254C22×C53—42no—
255C22×C53—42no—
256C22×C53—42no—
257C22×C53—42no—
258C22×C53—42no—
259C22×C53—42no—
260C22×C53—42no—
261C22×C53—42no—
262C22×C53—42no—
263C22×C53—42no—
264C22×C53C22×C22⋊C411yescharacteristic
265C22×C53D4211yescharacteristic
266C22×C53#54611yescharacteristic
267C22×C53C2×#13511yescharacteristic
268C22×C53C2×#13511yescharacteristic
269C22×C53#57711yescharacteristic
270C22×C53#57811yescharacteristic
271C4×C53—24no—
272C4×C53—22no—
273C4×C53—44no—
274C4×C53—42no—
275C4×C53#41614yescomplemented
276C4×C53C2×C4×D411yescharacteristic
277C4×C53#51811yescharacteristic
278C4×C53D4211yescharacteristic
279C4×C53#54611yescharacteristic
280Dic53—42no—
281Dic53—82no—
282Dic53—81no—
283C2×#547D5311yescharacteristic
284C2×C8×D4—531no—
285C2×#416—532no—
286#2354—531no—
287#1981—531no—
288#1956—531no—
289#1892—532no—
290#1873—532no—
291#1802—531no—
292#1788—532no—
293C4×#102—531no—
294C2×C53—44no—
295C2×C53—44no—
296C2×C53D4×C22⋊C411yescharacteristic
297C2×C53C2×#41611yescharacteristic
298C2×C53#289811yescharacteristic
299C2×C53#328311yescharacteristic
300C2×C53#328411yescharacteristic
301C2×C53#330211yescharacteristic
302C2×C53#330311yescharacteristic
303C22×#41C2×D5312yes—
304C23×C8—532no—
305#547—24no—
306#547Dic5318yescomplemented
307C2×C4⋊Q8C2×D5311yescharacteristic
308C2×#123—22no—
309C2×C22⋊Q8C2×D5312yes—
310C2×#121—21no—
311C2×C4×D4C2×D5311yescharacteristic
312C2×C4×D4C2×D5311yescharacteristic
313C2×#117—532no—
314C22×C4⋊C4—21no—
315C8×D4—1064no—
316C2×C4⋊C8—531no—
317#416—1068no—
318C2×#103—532no—
319C2×#103—532no—
320C2×#103—532no—
321C2×#103—532no—
322C2×#102—531no—
323C2×#102—532no—
324C2×#102—531no—
325C2×C22⋊C8—532no—
326C2×C4×C8—531no—
327#390—1061no—
328#388—1062no—
329#384—531no—
330#380—532no—
331C4×C4⋊C4—531no—
332C4×C4⋊C4—1061no—
333#340—532no—
334#340—531no—
335#340—531no—
336C53#2241711yescharacteristic, complemented, Sylow
337C2×#41—24no—
338C2×#41—24no—
339C2×#41—24no—
340C2×#41—42no—
341C2×#41—42no—
342C2×#41C2×Dic5318yes—
343C22×Q8C22×D5312yes—
344C22×D4—22no—
345C22×D4C22⋊D5311yescharacteristic
346C22×D4C22×D5311yescharacteristic
347C23×C4—21no—
348C23×C4—22no—
349C23×C4C22×D5312yes—
350C22×C8—531no—
351C22×C8—532no—
352C22×C8—531no—
353C22×C8—1064no—
354C22×C8—1064no—
355C22×C8—2122no—
356C4⋊Q8—22no—
357C4⋊Q8C2×Dic5314yes—
358#123—216no—
359C22⋊Q8—24no—
360C22⋊Q8C2×Dic5318yes—
361#121—28no—
362C4×D4—22no—
363C4×D4C2×Dic5314yes—
364C4×D4C2×Dic5314yes—
365C4×D4C22⋊D5314yes—
366#117—1064no—
367#117—2122no—
368C2×C4⋊C4—21no—
369C2×C4⋊C4—22no—
370C2×C4⋊C4—21no—
371C2×C4⋊C4—28no—
372C2×C4⋊C4—531no—
373C2×C4⋊C4—532no—
374C2×C4⋊C4—531no—
375C2×C4⋊C4—1061no—
376C2×C4⋊C4—1061no—
377C2×C4⋊C4C22⋊D5311yescharacteristic
378C2×C4⋊C4C22×D5311yescharacteristic
379C2×C22⋊C4—22no—
380C2×C22⋊C4—21no—
381C2×C22⋊C4—1062no—
382C2×C22⋊C4—1062no—
383C2×C22⋊C4C22⋊D5312yes—
384C2×C42—1062no—
385C2×C42—1061no—
386C2×C42C22×D5311yescharacteristic
387C4⋊C8—1062no—
388#103—1064no—
389#103—1064no—
390#103—2122no—
391#103—2122no—
392#102—534no—
393#102—1064no—
394#102—1062no—
395C22⋊C8—1064no—
396C4×C8—1062no—
397#95—1061no—
398#95—1062no—
399C24—21no—
400C24C2×C22⋊D5312yes—
401#41—216no—
402#41—48no—
403#41—48no—
404#41—48no—
405C2×Q8—22no—
406C2×Q8—22no—
407C2×Q8—22no—
408C2×Q8—22no—
409C2×Q8—22no—
410C2×Q8C22×Dic5314yes—
411C2×D4—28no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—21no—
417C2×D4—44no—
418C2×D4—44no—
419C2×D4C22⋊Dic5314yes—
420C2×D4D4×D5312yes—
421C2×D4#1171994712yes—
422C2×D4C22×Dic5312yes—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—21no—
428C22×C4—22no—
429C22×C4—24no—
430C22×C4—28no—
431C22×C4—44no—
432C22×C4—44no—
433C22×C4—42no—
434C22×C4—42no—
435C22×C4—1061no—
436C22×C4—1061no—
437C22×C4—1062no—
438C22×C4C22×Dic5314yes—
439C22×C4C2×C22⋊D5312yes—
440C22×C4C2×C22⋊D5311yescharacteristic
441C22×C4C2×C22⋊D5314yes—
442C22×C4C2×C22⋊D5311yescharacteristic
443C22×C4C23×D5311yescharacteristic
444C2×C8—532no—
445C2×C8—1061no—
446C2×C8—1061no—
447C2×C8—1062no—
448C2×C8—1061no—
449C2×C8—1062no—
450C2×C8—1061no—
451C2×C8—1061no—
452C2×C8—1062no—
453C2×C8—2128no—
454C4⋊C4—24no—
455C4⋊C4—28no—
456C4⋊C4—24no—
457C4⋊C4—21no—
458C4⋊C4—532no—
459C4⋊C4—1061no—
460C4⋊C4—1062no—
461C4⋊C4—1061no—
462C4⋊C4—1061no—
463C4⋊C4—1062no—
464C4⋊C4—4241no—
465C4⋊C4—2122no—
466C4⋊C4C22⋊Dic5314yes—
467C4⋊C4C22×Dic5312yes—
468C22⋊C4—28no—
469C22⋊C4—24no—
470C22⋊C4—4242no—
471C22⋊C4—2124no—
472C22⋊C4C22⋊Dic5318yes—
473C42—2124no—
474C42—2122no—
475C42C22×Dic5312yes—
476C42C2×C22⋊D5312yes—
477C23—22no—
478C23—22no—
479C23—22no—
480C23—22no—
481C23—22no—
482C23—22no—
483C23—22no—
484C23—22no—
485C23—22no—
486C23—41no—
487C23—41no—
488C23—41no—
489C23—41no—
490C23C2×C22⋊Dic5314yes—
491C23C22×C22⋊D5311yescharacteristic
492Q8—24no—
493Q8—24no—
494Q8—42no—
495Q8—42no—
496D4—22no—
497D4—22no—
498D4—28no—
499D4—22no—
500D4—416no—
501D4D4×Dic5314yes—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—24no—
508C2×C4—24no—
509C2×C4—24no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—416no—
513C2×C4—42no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—42no—
517C2×C4—48no—
518C2×C4—1062no—
519C2×C4—2121no—
520C2×C4—2121no—
521C2×C4—2122no—
522C2×C4—2121no—
523C2×C4—2122no—
524C2×C4—2122no—
525C2×C4—2121no—
526C2×C4—2121no—
527C2×C4C2×C22⋊Dic5312yes—
528C2×C4C2×C22⋊Dic5314yes—
529C2×C4C2×C22⋊Dic5312yes—
530C2×C4#1313323712yes—
531C2×C4#1313323712yes—
532C2×C4C23×Dic5311yescharacteristic
533C2×C4C2×D4×D5311yescharacteristic
534C2×C4C22×C22⋊D5311yescharacteristic
535C2×C4C2×#1171994711yescharacteristic
536C2×C4#1313326211yescharacteristic
537C2×C4#1313326211yescharacteristic
538C8—1062no—
539C8—2121no—
540C8—2122no—
541C22—24no—
542C22—24no—
543C22—24no—
544C22—24no—
545C22—42no—
546C22—42no—
547C22—42no—
548C22—42no—
549C22—42no—
550C22—42no—
551C22—42no—
552C22—42no—
553C22—42no—
554C22—42no—
555C22C22×C22⋊Dic5311yescharacteristic
556C22D4×C22⋊D5311yescharacteristic
557C22#1681126711yescharacteristic
558C22C2×#1313323711yescharacteristic
559C22C2×#1313323711yescharacteristic
560C22#1681143111yescharacteristic
561C22#1681143811yescharacteristic
562C4—24no—
563C4—22no—
564C4—44no—
565C4—42no—
566C4—4241no—
567C4—4242no—
568C4—2122no—
569C4#1681061914yescomplemented
570C4C2×D4×Dic5311yescharacteristic
571C4#1681109811yescharacteristic
572C4D4×C22⋊D5311yescharacteristic
573C4#1681126711yescharacteristic
574C2—44no—
575C2—44no—
576C2D4×C22⋊Dic5311yescharacteristic
577C2C2×#1681061911yescharacteristic
578C2#2028324811yescharacteristic
579C2#2028643311yescharacteristic
580C2#2028644111yescharacteristic
581C2#2028663211yescharacteristic
582C2#2028664311yescharacteristic
subgroupG/H
centerC23C22×C22⋊D53
commutatorC2×C4×C53C23×C4
FrattiniC22×C4C23×D53
Fitting, prime radicalC2×C53×#547C2
socleC23×C53C22×D4
cosocleC22×C4×C53C24
hypercenterC2×#547D53
hypocenterC53#22417
LCSG ▷ C2×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×D4×#88322C2semidirect
2. C2×#16810619C2semidirect
3. #20277664C2semidirect
4. #20275639C2semidirect
5. #20275518C2semidirect
6. #20275139C2semidirect
7. #20275028C2semidirect
8. #20274715C2semidirect
9. #20274655C2semidirect
10. C4×#13133138C2semidirect
37. C53×#547C4semidirect
69. #13133138D4semidirect
275. C4×C53#416semidirect
306. #547Dic53semidirect
336. C53#22417semidirect
569. C4#16810619semidirect

Maximal subgroups

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

subgroupindex
C2×D4×#883222
C2×#168106192
#202776642
#202756392
#202755182
#202751392
#202750282
#202747152
#202746552
C4×#131331382
C2×C53×#5472
#2241753