Finite Group Databasebeta

← #27276176

Subgroups — #27276176

Order 13568.

1048 conjugacy classes of subgroups.

The subgroup lattice has 564 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20275677C211yesmaximal, characteristic, complemented
2#20274715C211yesmaximal, characteristic, complemented
3#20274715C211yesmaximal, characteristic, complemented
4C2×#16810369C211yesmaximal, characteristic, complemented
5C2×D4⋊#88322C211yesmaximal, characteristic, complemented
6C2×D4⋊#88322C211yesmaximal, characteristic, complemented
7C2×C53×#547C211yesmaximal, characteristic
8C2×C4⋊#88322C2211yescharacteristic
9C2×C4⋊#88322C2211yescharacteristic
10C2×#13133138—21no—
11C2×#13133138—21no—
12C2×C4×#88322C2211yescharacteristic
13#16810493C2211yescharacteristic
14#16810385—21no—
15#16810385—21no—
16#16810385—22no—
17#16810369—22no—
18D4⋊#88322—22no—
19D4⋊#88322—22no—
20C22×#41×C53—21no—
21C53×#547—22no—
22C53×#547—22no—
23C53×#547—22no—
24C53×#547C414yescomplemented
25C2×C4⋊Q8×C53C2211yescharacteristic
26C2×#123×C53—22no—
27C2×C22⋊Q8×C53—21no—
28C2×#121×C53—21no—
29C2×C4×D4×C53C2211yescharacteristic
30C2×C4×D4×C53C2211yescharacteristic
31C22×C4⋊C4×C53—21no—
32C22×#88322—21no—
33C22×#88322—21no—
34C22×#88322—21no—
35C2×C4⋊Dic53—21no—
36C2×C4⋊Dic53—21no—
37C2×C4⋊Dic53—21no—
38C4⋊#88322—21no—
39C4⋊#88322—21no—
40C4⋊#88322—21no—
41C4⋊#88322—21no—
42#13133138—42no—
43#13133138—42no—
44C4×#88322—21no—
45C4×#88322—21no—
46C2×#41×C53—24no—
47C2×#41×C53—24no—
48C2×#41×C53—42no—
49C2×#41×C53—42no—
50C2×#41×C53—42no—
51C2×#41×C53—42no—
52C22×Q8×C53—21no—
53C22×D4×C53—22no—
54C22×D4×C53D411yescharacteristic
55C22×D4×C53D411yescharacteristic
56C23×C4×C53—22no—
57C23×C4×C53—21no—
58C23×C4×C53—21no—
59C4⋊Q8×C53—21no—
60C4⋊Q8×C53—21no—
61C4⋊Q8×C53C2×C412yes—
62C4⋊Q8×C53D412yes—
63#123×C53—28no—
64#123×C53—28no—
65C22⋊Q8×C53—24no—
66C22⋊Q8×C53—24no—
67#121×C53—24no—
68#121×C53—24no—
69C4×D4×C53—21no—
70C4×D4×C53—21no—
71C4×D4×C53—21no—
72C4×D4×C53—21no—
73C4×D4×C53C2×C412yes—
74C4×D4×C53C2×C412yes—
75C4×D4×C53D412yes—
76C4×D4×C53D412yes—
77C2×C4⋊C4×C53—22no—
78C2×C4⋊C4×C53—24no—
79C2×C4⋊C4×C53—24no—
80C2×C4⋊C4×C53—21no—
81C2×C4⋊C4×C53D411yescharacteristic
82C2×C4⋊C4×C53D411yescharacteristic
83C2×C4⋊C4×C53D412yes—
84C2×C22⋊C4×C53—22no—
85C2×C22⋊C4×C53—21no—
86C2×C22⋊C4×C53—21no—
87C2×C42×C53C2311yescharacteristic
88C24×C53—21no—
89C24×C53—21no—
90#41×C53—48no—
91#41×C53—48no—
92#41×C53—48no—
93#41×C53—48no—
94C2×Q8×C53—22no—
95C2×Q8×C53—22no—
96C2×Q8×C53—41no—
97C2×Q8×C53—41no—
98C2×Q8×C53—41no—
99C2×Q8×C53—41no—
100C2×D4×C53—28no—
101C2×D4×C53—21no—
102C2×D4×C53—21no—
103C2×D4×C53—21no—
104C2×D4×C53—21no—
105C2×D4×C53—21no—
106C2×D4×C53—21no—
107C2×D4×C53—21no—
108C2×D4×C53—21no—
109C2×D4×C53—44no—
110C2×D4×C53—44no—
111C2×D4×C53C22⋊C412yes—
112C2×D4×C53C22⋊C412yes—
113C2×D4×C53D812yes—
114C2×D4×C53SD1612yes—
115C22×C4×C53—28no—
116C22×C4×C53—21no—
117C22×C4×C53—21no—
118C22×C4×C53—22no—
119C22×C4×C53—22no—
120C22×C4×C53—22no—
121C22×C4×C53—22no—
122C22×C4×C53—22no—
123C22×C4×C53—44no—
124C22×C4×C53—44no—
125C22×C4×C53—41no—
126C22×C4×C53—41no—
127C22×C4×C53—41no—
128C22×C4×C53—41no—
129C22×C4×C53—41no—
130C22×C4×C53—41no—
131C22×C4×C53—41no—
132C22×C4×C53—41no—
133C22×C4×C53C2×D411yescharacteristic
134C22×C4×C53C2×D411yescharacteristic
135C22×C4×C53C2×D411yescharacteristic
136C22×Dic53—41no—
137C4⋊C4×C53—28no—
138C4⋊C4×C53—22no—
139C4⋊C4×C53—21no—
140C4⋊C4×C53—22no—
141C4⋊C4×C53—21no—
142C4⋊C4×C53—22no—
143C4⋊C4×C53C22⋊C412yes—
144C4⋊C4×C53C22⋊C412yes—
145C4⋊C4×C53C22⋊C414yes—
146C22⋊C4×C53—28no—
147C22⋊C4×C53—22no—
148C22⋊C4×C53—22no—
149C22⋊C4×C53—22no—
150C22⋊C4×C53—22no—
151C42×C53C22×C411yescharacteristic
152C42×C53C2×D411yescharacteristic
153C42×C53C2×D411yescharacteristic
154C42×C53C2×D411yescharacteristic
155C4⋊Dic53—42no—
156C4⋊Dic53—41no—
157C4⋊Dic53—41no—
158C4⋊Dic53—41no—
159C4⋊Dic53—41no—
160C2×#88322—22no—
161C2×#88322—41no—
162C2×#88322—41no—
163C2×#88322—41no—
164C2×#88322—41no—
165C2×#88322—41no—
166C2×#88322—41no—
167C2×#88322—41no—
168C2×#88322—41no—
169C23×C53—22no—
170C23×C53—22no—
171C23×C53—22no—
172C23×C53—22no—
173C23×C53—41no—
174C23×C53—41no—
175C23×C53—41no—
176C23×C53—41no—
177C23×C53—41no—
178C23×C53—41no—
179C23×C53—41no—
180C23×C53—41no—
181C23×C53—41no—
182C23×C53—41no—
183C23×C53#12011yescharacteristic
184Q8×C53—42no—
185Q8×C53—42no—
186Q8×C53—42no—
187Q8×C53—42no—
188D4×C53—22no—
189D4×C53—22no—
190D4×C53—22no—
191D4×C53—22no—
192D4×C53—22no—
193D4×C53—416no—
194D4×C53—41no—
195D4×C53—41no—
196D4×C53#10214yescomplemented
197C2×C4×C53—21no—
198C2×C4×C53—21no—
199C2×C4×C53—22no—
200C2×C4×C53—24no—
201C2×C4×C53—22no—
202C2×C4×C53—416no—
203C2×C4×C53—41no—
204C2×C4×C53—42no—
205C2×C4×C53—41no—
206C2×C4×C53—41no—
207C2×C4×C53—42no—
208C2×C4×C53—41no—
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—42no—
217C2×C4×C53C2×C22⋊C411yescharacteristic
218C2×C4×C53C2×C22⋊C411yescharacteristic
219C2×C4×C53C2×C22⋊C411yescharacteristic
220C2×C4×C53#12011yescharacteristic
221C2×C4×C53#12011yescharacteristic
222C2×C4×C53#12011yescharacteristic
223C2×C4×C53C2×D811yescharacteristic
224C2×C4×C53C2×SD1611yescharacteristic
225C2×C4×C53#13511yescharacteristic
226C2×C4×C53#13511yescharacteristic
227C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
228C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
229C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
230C2×C4×C53#13711yescharacteristic
231C2×Dic53—81no—
232C2×Dic53—81no—
233C2×Dic53—81no—
234#88322—42no—
235#88322—81no—
236#88322—81no—
237#8446—531nomaximal, Sylow
238C22×C53—24no—
239C22×C53—42no—
240C22×C53—42no—
241C22×C53—42no—
242C22×C53—42no—
243C22×C53—42no—
244C22×C53—42no—
245C22×C53—42no—
246C22×C53—42no—
247C22×C53—42no—
248C22×C53—81no—
249C22×C53—81no—
250C22×C53—81no—
251C22×C53—81no—
252C22×C53#37911yescharacteristic
253C22×C53#44711yescharacteristic
254C22×C53#44911yescharacteristic
255C22×C53#45011yescharacteristic
256C22×C53#45211yescharacteristic
257C22×C53#45411yescharacteristic
258C22×C53#45611yescharacteristic
259C4×C53—21no—
260C4×C53—21no—
261C4×C53—42no—
262C4×C53—44no—
263C4×C53—42no—
264C4×C53C2×#10211yescharacteristic
265C4×C53#41611yescharacteristic
266C4×C53#41711yescharacteristic
267C4×C53#41811yescharacteristic
268C4×C53#44711yescharacteristic
269C4×C53#44911yescharacteristic
270C4×C53#45011yescharacteristic
271C4×C53#45211yescharacteristic
272Dic53—161no—
273C2×#547D5311yescharacteristic
274#1990—531no—
275#1802—531no—
276#1802—531no—
277C2×#332—531no—
278C2×D4⋊C8—531no—
279C2×D4⋊C8—531no—
280C2×C53—44no—
281C2×C53—82no—
282C2×C53#162511yescharacteristic
283C2×C53#163511yescharacteristic
284C2×C53#164711yescharacteristic
285C2×C53#165411yescharacteristic
286C2×C53#178611yescharacteristic
287C2×C53#178811yescharacteristic
288C2×C53#179311yescharacteristic
289C22×#41—21no—
290#547—22no—
291#547—22no—
292#547—22no—
293#547Dic5314yescomplemented
294C2×C4⋊Q8C2×D5311yescharacteristic
295C2×#123—22no—
296C2×C22⋊Q8—21no—
297C2×#121—21no—
298C2×C4×D4C2×D5311yescharacteristic
299C2×C4×D4C2×D5311yescharacteristic
300C22×C4⋊C4—21no—
301C2×C4⋊C8—531no—
302C2×C4⋊C8—531no—
303C2×#102—1061no—
304C2×#102—1061no—
305C2×C4×C8—531no—
306#384—531no—
307#340—1061no—
308#340—1061no—
309#340—1062no—
310#332—1062no—
311D4⋊C8—1062no—
312D4⋊C8—1062no—
313C53#844611yescharacteristic, complemented, Sylow
314C2×#41—24no—
315C2×#41—24no—
316C2×#41—42no—
317C2×#41—42no—
318C2×#41—42no—
319C2×#41—42no—
320C22×Q8—21no—
321C22×D4—22no—
322C22×D4C22⋊D5311yescharacteristic
323C22×D4C22⋊D5311yescharacteristic
324C23×C4—22no—
325C23×C4—21no—
326C23×C4—21no—
327C22×C8—1061no—
328C22×C8—1061no—
329C22×C8—1061no—
330C4⋊Q8—21no—
331C4⋊Q8—21no—
332C4⋊Q8C2×Dic5312yes—
333C4⋊Q8C22⋊D5312yes—
334#123—28no—
335#123—28no—
336C22⋊Q8—24no—
337C22⋊Q8—24no—
338#121—24no—
339#121—24no—
340C4×D4—21no—
341C4×D4—21no—
342C4×D4—21no—
343C4×D4—21no—
344C4×D4C2×Dic5312yes—
345C4×D4C2×Dic5312yes—
346C4×D4C22⋊D5312yes—
347C4×D4C22⋊D5312yes—
348C2×C4⋊C4—22no—
349C2×C4⋊C4—24no—
350C2×C4⋊C4—21no—
351C2×C4⋊C4—24no—
352C2×C4⋊C4—1061no—
353C2×C4⋊C4—1061no—
354C2×C4⋊C4—1061no—
355C2×C4⋊C4C22⋊D5311yescharacteristic
356C2×C4⋊C4C22⋊D5311yescharacteristic
357C2×C4⋊C4C22⋊D5312yes—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—21no—
360C2×C22⋊C4—21no—
361C2×C42C22×D5311yescharacteristic
362C4⋊C8—1061no—
363C4⋊C8—1061no—
364C4⋊C8—1061no—
365C4⋊C8—1061no—
366#102—2122no—
367#102—2122no—
368C4×C8—1061no—
369C4×C8—1061no—
370C24—21no—
371C24—21no—
372#41—48no—
373#41—48no—
374#41—48no—
375#41—48no—
376C2×Q8—22no—
377C2×Q8—22no—
378C2×Q8—41no—
379C2×Q8—41no—
380C2×Q8—41no—
381C2×Q8—41no—
382C2×D4—28no—
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—21no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—44no—
392C2×D4—44no—
393C2×D4#1171992212yes—
394C2×D4#1171992312yes—
395C2×D4C22⋊Dic5312yes—
396C2×D4C22⋊Dic5312yes—
397C22×C4—28no—
398C22×C4—21no—
399C22×C4—21no—
400C22×C4—22no—
401C22×C4—22no—
402C22×C4—22no—
403C22×C4—22no—
404C22×C4—22no—
405C22×C4—44no—
406C22×C4—44no—
407C22×C4—41no—
408C22×C4—41no—
409C22×C4—41no—
410C22×C4—41no—
411C22×C4—41no—
412C22×C4—41no—
413C22×C4—41no—
414C22×C4—41no—
415C22×C4—2121no—
416C22×C4C2×C22⋊D5311yescharacteristic
417C22×C4C2×C22⋊D5311yescharacteristic
418C22×C4C2×C22⋊D5311yescharacteristic
419C2×C8—1062no—
420C2×C8—2121no—
421C2×C8—2121no—
422C2×C8—2121no—
423C2×C8—2121no—
424C2×C8—2121no—
425C2×C8—2121no—
426C2×C8—2121no—
427C2×C8—2121no—
428C4⋊C4—28no—
429C4⋊C4—22no—
430C4⋊C4—21no—
431C4⋊C4—22no—
432C4⋊C4—21no—
433C4⋊C4—22no—
434C4⋊C4—2122no—
435C4⋊C4—2121no—
436C4⋊C4—2121no—
437C4⋊C4—2121no—
438C4⋊C4—2121no—
439C4⋊C4C22⋊Dic5312yes—
440C4⋊C4C22⋊Dic5312yes—
441C4⋊C4C22⋊Dic5314yes—
442C22⋊C4—28no—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—22no—
447C42C22×Dic5311yescharacteristic
448C42C2×C22⋊D5311yescharacteristic
449C42C2×C22⋊D5311yescharacteristic
450C42C2×C22⋊D5311yescharacteristic
451C23—22no—
452C23—22no—
453C23—22no—
454C23—22no—
455C23—41no—
456C23—41no—
457C23—41no—
458C23—41no—
459C23—41no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—41no—
464C23—41no—
465C23#1313318611yescharacteristic
466Q8—42no—
467Q8—42no—
468Q8—42no—
469Q8—42no—
470D4—22no—
471D4—22no—
472D4—22no—
473D4—22no—
474D4—22no—
475D4—416no—
476D4—41no—
477D4—41no—
478D4#1313313814yescomplemented
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—22no—
482C2×C4—22no—
483C2×C4—24no—
484C2×C4—416no—
485C2×C4—41no—
486C2×C4—42no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—41no—
491C2×C4—42no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—42no—
499C2×C4—4241no—
500C2×C4—4241no—
501C2×C4—4241no—
502C2×C4C2×C22⋊Dic5311yescharacteristic
503C2×C4C2×C22⋊Dic5311yescharacteristic
504C2×C4C2×C22⋊Dic5311yescharacteristic
505C2×C4#1313318611yescharacteristic
506C2×C4#1313318611yescharacteristic
507C2×C4#1313318611yescharacteristic
508C2×C4C2×#1171992211yescharacteristic
509C2×C4C2×#1171992311yescharacteristic
510C2×C4#1313323711yescharacteristic
511C2×C4#1313323711yescharacteristic
512C2×C4#1313324211yescharacteristic
513C2×C4#1313324311yescharacteristic
514C2×C4#1313324311yescharacteristic
515C2×C4#1313324711yescharacteristic
516C8—4241no—
517C8—4241no—
518C8—2122no—
519C22—24no—
520C22—42no—
521C22—42no—
522C22—42no—
523C22—42no—
524C22—42no—
525C22—42no—
526C22—42no—
527C22—42no—
528C22—42no—
529C22—81no—
530C22—81no—
531C22—81no—
532C22—81no—
533C22#1681047911yescharacteristic
534C22#1681075711yescharacteristic
535C22#1681076811yescharacteristic
536C22#1681077411yescharacteristic
537C22#1681078511yescharacteristic
538C22#1681079611yescharacteristic
539C22#1681080611yescharacteristic
540C4—21no—
541C4—21no—
542C4—42no—
543C4—44no—
544C4—42no—
545C4—8481no—
546C4C2×#1313313811yescharacteristic
547C4#1681061911yescharacteristic
548C4#1681062411yescharacteristic
549C4#1681062811yescharacteristic
550C4#1681075711yescharacteristic
551C4#1681076811yescharacteristic
552C4#1681077411yescharacteristic
553C4#1681078511yescharacteristic
554C2—44no—
555C2—82no—
556C2#2027380411yescharacteristic
557C2#2027386611yescharacteristic
558C2#2027395011yescharacteristic
559C2#2027399911yescharacteristic
560C2#2027464611yescharacteristic
561C2#2027465511yescharacteristic
562C2#2027467411yescharacteristic
subgroupG/H
centerC23#13133186
commutatorC42×C53C22×C4
FrattiniC2×C42C22×D53
Fitting, prime radicalC2×C53×#547C2
socleC23×C53#120
cosocleC2×C42×C53C23
hypercenterC2×#547D53
hypocenterC53#8446
LCSG ▷ C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#547

Complements

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

NH ≅ G/Ntype
1. #20275677C2semidirect
2. #20274715C2semidirect
3. #20274715C2semidirect
4. C2×#16810369C2semidirect
5. C2×D4⋊#88322C2semidirect
6. C2×D4⋊#88322C2semidirect
24. C53×#547C4semidirect
196. D4×C53#102semidirect
293. #547Dic53semidirect
313. C53#8446semidirect
478. D4#13133138semidirect

Maximal subgroups

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

subgroupindex
#202756772
#202747152
C2×#168103692
C2×D4⋊#883222
C2×C53×#5472
#844653