Finite Group Databasebeta

← #27286432

Subgroups — #27286432

Order 13568.

712 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16810910C211yesmaximal, characteristic, complemented
2#20279628C211yesmaximal, characteristic
3#20275518C211yesmaximal, characteristic, complemented
4C2×D4⋊#88322C211yesmaximal, characteristic, complemented
5C2×#16810350C211yesmaximal, characteristic, complemented
6#20272599C211yesmaximal, characteristic, complemented
7C2×C53×#476C211yesmaximal, characteristic, complemented
8C2×#13133191—21no—
9C2×#13133188—21no—
10C2×C4×D212C2211yescharacteristic
11#16810910—22no—
12#16810910—22no—
13C2×#13133148—21no—
14C2×C4⋊#88322C2211yescharacteristic
15C2×#13133141—22no—
16C2×#13133137—21no—
17C2×C4×#88322C2211yescharacteristic
18#16810539—22no—
19#16810493C2211yescharacteristic
20#16810483—21no—
21#16810480—21no—
22D4⋊#88322—22no—
23#16810350—22no—
24C2×C4⋊Q8×C53C2211yescharacteristic
25C2×C4×D4×C53C2211yescharacteristic
26C53×#476—22no—
27C53×#476—22no—
28C2×#106×C53—21no—
29C2×C4⋊C8×C53C2211yescharacteristic
30C2×#102×C53—21no—
31C22×D212D411yescharacteristic
32C22×C22⋊D53—22no—
33C22×C4×D53—21no—
34C22×#88322—21no—
35C22×#88322—21no—
36#13133191—42no—
37#13133191—42no—
38#13133188—42no—
39#13133188—42no—
40C4×D212—21no—
41C4×D212—21no—
42C4×D212—21no—
43C4×D212—21no—
44C2×C4⋊Dic53—21no—
45C2×C4⋊Dic53—21no—
46C2×C4⋊Dic53D411yescharacteristic
47C2×C4⋊Dic53D412yes—
48C2×C22⋊Dic53—22no—
49C2×#11719921—22no—
50C2×#11719921—21no—
51#13133148—24no—
52C4⋊#88322—21no—
53C4⋊#88322D412yescomplemented
54#13133141—44no—
55#13133137—24no—
56C4×#88322—21no—
57C4×#88322—21no—
58#13133124—22no—
59C22×Q8×C53—21no—
60C22×D4×C53D411yescharacteristic
61C23×C4×C53—21no—
62C22×C8×C53—21no—
63C4⋊Q8×C53—21no—
64C4⋊Q8×C53—21no—
65C4⋊Q8×C53—21no—
66C4⋊Q8×C53D412yescomplemented
67C4×D4×C53—21no—
68C4×D4×C53—21no—
69C4×D4×C53—21no—
70C4×D4×C53—21no—
71C2×C4⋊C4×C53—21no—
72C2×C4⋊C4×C53—21no—
73C2×C4⋊C4×C53D411yescharacteristic
74C2×C22⋊C4×C53—21no—
75C2×C42×C53C2311yescharacteristic
76#106×C53—24no—
77C4⋊C8×C53—21no—
78C4⋊C8×C53D412yescomplemented
79#102×C53—24no—
80C24×C53—21no—
81C23×D53—21no—
82C2×Q8×C53—22no—
83C2×Q8×C53—42no—
84C2×Q8×C53—42no—
85C2×Q8×C53—41no—
86C2×D4×C53—21no—
87C2×D4×C53—21no—
88C2×D4×C53—21no—
89C2×D4×C53—21no—
90C2×D4×C53—21no—
91C2×D4×C53SD1612yes—
92C22×C4×C53—21no—
93C22×C4×C53—41no—
94C22×C4×C53—41no—
95C22×C4×C53—41no—
96C22×C4×C53—41no—
97C22×C4×C53—41no—
98C22×C4×C53—41no—
99C22×C4×C53—41no—
100C22×C4×C53C2×D411yescharacteristic
101C22×C4×C53C2×D411yescharacteristic
102C22×C4×C53C2×D411yescharacteristic
103C2×C22⋊D53—44no—
104C2×C22⋊D53—44no—
105C2×C22⋊D53—82no—
106C22×Dic53—21no—
107C22×Dic53—22no—
108C2×D212—21no—
109C2×D212—21no—
110C2×D212—21no—
111C2×D212—21no—
112C2×D212—21no—
113C2×D212SD1612yes—
114C2×C4×D53—41no—
115C2×C4×D53—41no—
116C2×C4×D53—41no—
117C2×C4×D53—41no—
118C2×C4×D53—41no—
119C2×C4×D53—41no—
120C2×C8×C53—22no—
121C2×C8×C53—41no—
122C2×C8×C53—41no—
123C4⋊C4×C53—22no—
124C4⋊C4×C53—21no—
125C4⋊C4×C53—22no—
126C4⋊C4×C53—21no—
127C4⋊C4×C53—41no—
128C4⋊C4×C53—41no—
129C22⋊C4×C53—41no—
130C22⋊C4×C53—41no—
131C42×C53C2×D411yescharacteristic
132C42×C53C2×D411yescharacteristic
133C42×C53C2×D411yescharacteristic
134C42×C53#4111yescharacteristic
135C22⋊Dic53—44no—
136#11719921—44no—
137#11719921—41no—
138#11719921—41no—
139C4⋊Dic53—21no—
140C4⋊Dic53—21no—
141C4⋊Dic53—41no—
142C4⋊Dic53—41no—
143C4⋊Dic53—41no—
144C4⋊Dic53—41no—
145C4⋊Dic53—42no—
146C2×#88322—22no—
147C2×#88322—22no—
148C2×#88322—41no—
149C2×#88322—41no—
150C2×#88322—41no—
151C2×#88322—41no—
152C23×C53—22no—
153C23×C53—41no—
154C23×C53—41no—
155C23×C53—41no—
156C23×C53—41no—
157C23×C53—41no—
158C23×C53—41no—
159C23×C53#12011yescharacteristic
160C22×D53—22no—
161C22×D53—41no—
162C22×D53—41no—
163C22×D53—41no—
164C22×D53—41no—
165C22×D53—41no—
166C22×D53—41no—
167Q8×C53—44no—
168Q8×C53—82no—
169D4×C53—22no—
170D4×C53—22no—
171D4×C53—41no—
172D4×C53—41no—
173C2×C4×C53—21no—
174C2×C4×C53—21no—
175C2×C4×C53—41no—
176C2×C4×C53—42no—
177C2×C4×C53—41no—
178C2×C4×C53—41no—
179C2×C4×C53—81no—
180C2×C4×C53—81no—
181C2×C4×C53—81no—
182C2×C4×C53—81no—
183C2×C4×C53—81no—
184C2×C4×C53—81no—
185C2×C4×C53#12011yescharacteristic
186C2×C4×C53#12011yescharacteristic
187C2×C4×C53#12111yescharacteristic
188C2×C4×C53#12111yescharacteristic
189C2×C4×C53#12111yescharacteristic
190C2×C4×C53#12711yescharacteristic
191C2×C4×C53C2×SD1611yescharacteristic
192C2×C4×C53C2×SD1611yescharacteristic
193C2×C4×C53#13511yescharacteristic
194C2×C4×C53#13511yescharacteristic
195C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
196C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
197C2×C4×C53#13711yescharacteristic
198C2×C4×C53#13711yescharacteristic
199C22⋊D53—88no—
200C2×Dic53—41no—
201C2×Dic53—41no—
202C2×Dic53—42no—
203C2×Dic53—42no—
204C2×Dic53—41no—
205C2×Dic53—42no—
206D212—22no—
207D212—22no—
208D212—41no—
209D212—41no—
210C4×D53—81no—
211C4×D53—81no—
212C4×D53—81no—
213C4×D53—81no—
214C8×C53—42no—
215#88322—42no—
216#88322—42no—
217#10235—531nomaximal, Sylow
218C22×C53—42no—
219C22×C53—42no—
220C22×C53—42no—
221C22×C53—81no—
222C22×C53—81no—
223C22×C53—81no—
224C22×C53—81no—
225C22×C53#39211yescharacteristic
226C22×C53#45011yescharacteristic
227C22×C53#45011yescharacteristic
228C22×C53#45211yescharacteristic
229C22×C53#45211yescharacteristic
230C22×C53#45311yescharacteristic
231C22×C53#45411yescharacteristic
232C2×D53—42no—
233C2×D53—42no—
234C2×D53—42no—
235C2×D53—81no—
236C2×D53—81no—
237C2×D53—81no—
238C2×D53—81no—
239C4×C53—21no—
240C4×C53—21no—
241C4×C53—81no—
242C4×C53—82no—
243C4×C53#44811yescharacteristic
244C4×C53#44811yescharacteristic
245C4×C53#44911yescharacteristic
246C4×C53#44911yescharacteristic
247C4×C53#46511yescharacteristic
248C4×C53#46511yescharacteristic
249C4×C53#46811yescharacteristic
250C4×C53#46811yescharacteristic
251Dic53—82no—
252Dic53—81no—
253C2×#476—531no—
254C2×#476D5311yescharacteristic, complemented
255#2537—531no—
256#1956—531no—
257C2×D4⋊C8—531no—
258C2×D4⋊C8—531no—
259#1296—531no—
260C2×C53—82no—
261C2×C53#162911yescharacteristic
262C2×C53#163411yescharacteristic
263C2×C53#163511yescharacteristic
264C2×C53#163511yescharacteristic
265C2×C53#200011yescharacteristic
266C2×C53#200011yescharacteristic
267C2×C53#200611yescharacteristic
268D53—82no—
269C2×C4⋊Q8C2×D5311yescharacteristic
270C2×#121—1061no—
271C2×#121—1061no—
272C2×C4×D4—531no—
273C2×C4×D4C2×D5311yescharacteristic
274#476—22no—
275#476—22no—
276#476—1062no—
277#476—1062no—
278C2×#106—21no—
279C2×#106—1061no—
280C2×C4⋊C8—531no—
281C2×C4⋊C8C2×D5311yescharacteristic
282C2×#103—1062no—
283C2×#102—21no—
284C2×#102—1061no—
285C2×C4×C8—531no—
286#394—1062no—
287#384—531no—
288#380—1061no—
289#380—1061no—
290D4⋊C8—1062no—
291D4⋊C8—1062no—
292C53#1023511yescharacteristic, complemented, Sylow
293C22×Q8—21no—
294C22×D4—531no—
295C22×D4—1062no—
296C22×D4C22⋊D5311yescharacteristic
297C23×C4—21no—
298C23×C4—1061no—
299C22×C8—21no—
300C22×C8—1061no—
301C22×C8—1061no—
302C4⋊Q8—21no—
303C4⋊Q8—21no—
304C4⋊Q8—21no—
305C4⋊Q8C22⋊D5312yescomplemented
306#121—2122no—
307#121—2122no—
308#121—2122no—
309#121—2122no—
310C4×D4—21no—
311C4×D4—21no—
312C4×D4—21no—
313C4×D4—21no—
314C4×D4—1061no—
315C4×D4—1061no—
316C4×D4—1061no—
317C4×D4—1061no—
318C2×C4⋊C4—21no—
319C2×C4⋊C4—21no—
320C2×C4⋊C4—531no—
321C2×C4⋊C4—532no—
322C2×C4⋊C4—1061no—
323C2×C4⋊C4—1061no—
324C2×C4⋊C4C22⋊D5311yescharacteristic
325C2×C22⋊C4—21no—
326C2×C22⋊C4—1062no—
327C2×C22⋊C4—1062no—
328C2×C22⋊C4—1061no—
329C2×C42C22×D5311yescharacteristic
330#106—24no—
331#106—1064no—
332C4⋊C8—21no—
333C4⋊C8—532no—
334C4⋊C8—1061no—
335C4⋊C8C22⋊D5312yescomplemented
336#103—2124no—
337#102—24no—
338#102—1064no—
339C4×C8—1061no—
340C4×C8—1061no—
341#95—1062no—
342C24—21no—
343C24—1061no—
344C2×Q8—22no—
345C2×Q8—42no—
346C2×Q8—42no—
347C2×Q8—41no—
348C2×D4—21no—
349C2×D4—21no—
350C2×D4—21no—
351C2×D4—21no—
352C2×D4—21no—
353C2×D4—532no—
354C2×D4—1061no—
355C2×D4—1061no—
356C2×D4—1061no—
357C2×D4—1061no—
358C2×D4—1061no—
359C2×D4—4242no—
360C2×D4—2124no—
361C2×D4—2124no—
362C2×D4#1171992412yes—
363C22×C4—21no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4—41no—
367C22×C4—41no—
368C22×C4—41no—
369C22×C4—41no—
370C22×C4—41no—
371C22×C4—1061no—
372C22×C4—1062no—
373C22×C4—2121no—
374C22×C4—2121no—
375C22×C4—2121no—
376C22×C4—2121no—
377C22×C4—2121no—
378C22×C4—2121no—
379C22×C4D4×D5311yescharacteristic
380C22×C4D4×D5311yescharacteristic
381C22×C4C2×C22⋊D5311yescharacteristic
382C2×C8—22no—
383C2×C8—41no—
384C2×C8—41no—
385C2×C8—1062no—
386C2×C8—1062no—
387C2×C8—2121no—
388C2×C8—2121no—
389C2×C8—2121no—
390C2×C8—2121no—
391C4⋊C4—22no—
392C4⋊C4—22no—
393C4⋊C4—21no—
394C4⋊C4—21no—
395C4⋊C4—41no—
396C4⋊C4—41no—
397C4⋊C4—1061no—
398C4⋊C4—1061no—
399C4⋊C4—2122no—
400C4⋊C4—2121no—
401C4⋊C4—2121no—
402C4⋊C4—2121no—
403C4⋊C4—2121no—
404C22⋊C4—41no—
405C22⋊C4—41no—
406C22⋊C4—2124no—
407C22⋊C4—2124no—
408C22⋊C4—2121no—
409C22⋊C4—2121no—
410C42D4×D5311yescharacteristic
411C42#1171994711yescharacteristic
412C42C2×C22⋊D5311yescharacteristic
413C42C2×C22⋊D5311yescharacteristic
414C23—22no—
415C23—41no—
416C23—41no—
417C23—41no—
418C23—41no—
419C23—41no—
420C23—41no—
421C23—1062no—
422C23—2121no—
423C23—2121no—
424C23—2121no—
425C23—2121no—
426C23—2121no—
427C23—2121no—
428C23#1313318511yescharacteristic
429Q8—44no—
430Q8—82no—
431D4—22no—
432D4—22no—
433D4—41no—
434D4—41no—
435D4—1062no—
436D4—1062no—
437D4—4248no—
438D4—2121no—
439D4—2121no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—41no—
445C2×C4—42no—
446C2×C4—81no—
447C2×C4—81no—
448C2×C4—81no—
449C2×C4—81no—
450C2×C4—81no—
451C2×C4—81no—
452C2×C4—4241no—
453C2×C4—4241no—
454C2×C4—4241no—
455C2×C4—4241no—
456C2×C4—2122no—
457C2×C4—2121no—
458C2×C4—2122no—
459C2×C4—2121no—
460C2×C4—2121no—
461C2×C4—2122no—
462C2×C4#1313318511yescharacteristic
463C2×C4#1313318611yescharacteristic
464C2×C4#1313318811yescharacteristic
465C2×C4#1313318911yescharacteristic
466C2×C4#1313318911yescharacteristic
467C2×C4#1313321311yescharacteristic
468C2×C4SD16×D5311yescharacteristic
469C2×C4C2×#1171992411yescharacteristic
470C2×C4#1313323711yescharacteristic
471C2×C4#1313323811yescharacteristic
472C2×C4#1313324211yescharacteristic
473C2×C4#1313324411yescharacteristic
474C2×C4#1313324611yescharacteristic
475C2×C4#1313324811yescharacteristic
476C8—42no—
477C8—2122no—
478C8—2122no—
479C22—42no—
480C22—42no—
481C22—42no—
482C22—81no—
483C22—81no—
484C22—81no—
485C22—81no—
486C22—4241no—
487C22—4241no—
488C22—4241no—
489C22—4241no—
490C22—2122no—
491C22—2122no—
492C22—2122no—
493C22#1681053111yescharacteristic
494C22#1681077211yescharacteristic
495C22#1681077311yescharacteristic
496C22#1681078211yescharacteristic
497C22#1681078311yescharacteristic
498C22#1681078911yescharacteristic
499C22#1681079411yescharacteristic
500C4—21no—
501C4—21no—
502C4—81no—
503C4—82no—
504C4—4241no—
505C4—4242no—
506C4#1681076011yescharacteristic
507C4#1681076211yescharacteristic
508C4#1681076711yescharacteristic
509C4#1681076811yescharacteristic
510C4#1681084911yescharacteristic
511C4#1681085411yescharacteristic
512C4#1681086611yescharacteristic
513C4#1681087111yescharacteristic
514C2—82no—
515C2—4242no—
516C2#2027383011yescharacteristic
517C2#2027385911yescharacteristic
518C2#2027386411yescharacteristic
519C2#2027386511yescharacteristic
520C2#2027571511yescharacteristic
521C2#2027571611yescharacteristic
522C2#2027575511yescharacteristic
subgroupG/H
centerC23#13133185
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC2×C53×#476C2
socleC23×C53#120
prime radical#20279628C2
hypercenterC2×#476D53
hypocenterC53#10235
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×#476

Complements

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

NH ≅ G/Ntype
1. C2×#16810910C2semidirect
3. #20275518C2semidirect
4. C2×D4⋊#88322C2semidirect
5. C2×#16810350C2semidirect
6. #20272599C2semidirect
7. C2×C53×#476C2semidirect
53. C4⋊#88322D4semidirect
66. C4⋊Q8×C53D4semidirect
78. C4⋊C8×C53D4semidirect
254. C2×#476D53semidirect
292. C53#10235semidirect
305. C4⋊Q8C22⋊D53semidirect
335. C4⋊C8C22⋊D53semidirect

Maximal subgroups

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

subgroupindex
C2×#168109102
#202796282
#202755182
C2×D4⋊#883222
C2×#168103502
#202725992
C2×C53×#4762
#1023553