Finite Group Databasebeta

← C53×#8421 #27215933

Subgroups — C53×#8421 #27215933

Order 13568.

942 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3499C211yesmaximal, characteristic
2C53×#1988C211yesmaximal, characteristic, complemented
3C53×#1806C211yesmaximal, characteristic, complemented
4C53×#1547C211yesmaximal, characteristic, complemented
5C53×#1546C211yesmaximal, characteristic, complemented
6C53×#1502C211yesmaximal, characteristic, complemented
7C53×#1477C211yesmaximal, characteristic, complemented
8C2×2−1+4×C53—21no—
9D4×Q8×C53—21no—
10D4×Q8×C53C412yescomplemented
11C53×#547—22no—
12C53×#547—21no—
13C53×#547—22no—
14C53×#547C412yescomplemented
15C53×#536—22no—
16C53×#536—21no—
17C53×#532—21no—
18C53×#532C2211yescharacteristic
19C2×C4⋊Q8×C53—21no—
20C2×C4⋊Q8×C53C2211yescharacteristic
21C4×#41×C53C2211yescharacteristic
22C53×#431C2211yescharacteristic
23C53×#423C2211yescharacteristic
24C2×C4⋊C8×C53C2211yescharacteristic
25C2×#104×C53—21no—
26C4×C4⋊C4×C53C2211yescharacteristic
27C53×#340—22no—
28C53×#339—21no—
29C53×#332—22no—
30C53×#332—21no—
31C53×#331—21no—
32C53×#327—22no—
33C53×Q8⋊C8—21no—
34C53×D4⋊C8—21no—
352−1+4×C53—42no—
362−1+4×C53—42no—
372−1+4×C53—42no—
382−1+4×C53—42no—
39C2×#41×C53—22no—
40C2×#41×C53—24no—
41C2×#41×C53—22no—
42C2×#41×C53—21no—
43C2×#41×C53D411yescharacteristic
44C22×Q8×C53—22no—
45C22×Q8×C53—22no—
46C22×Q8×C53—21no—
47C2×M16×C53—21no—
48C22×C8×C53—21no—
49C4⋊Q8×C53—22no—
50C4⋊Q8×C53—22no—
51C4⋊Q8×C53—22no—
52C4⋊Q8×C53—21no—
53C4⋊Q8×C53—21no—
54C4⋊Q8×C53C2×C412yes—
55C4⋊Q8×C53D411yescharacteristic
56C4⋊Q8×C53D412yes—
57#127×C53D411yescharacteristic
58#124×C53—22no—
59#124×C53—21no—
60#123×C53—24no—
61#123×C53—24no—
62#123×C53—24no—
63C22⋊Q8×C53—24no—
64C22⋊Q8×C53—24no—
65C22⋊Q8×C53—22no—
66C22⋊Q8×C53—22no—
67#121×C53—22no—
68#121×C53—22no—
69#121×C53—21no—
70#121×C53D412yes—
71C4×Q8×C53C2×C411yescharacteristic
72C4×Q8×C53D411yescharacteristic
73C4×D4×C53—22no—
74C4×D4×C53—21no—
75C4×D4×C53—22no—
76C4×D4×C53—21no—
77C4×D4×C53—21no—
78C4×D4×C53C2×C411yescharacteristic
79C4×D4×C53C2×C412yes—
80C4×D4×C53D411yescharacteristic
81#117×C53—21no—
82#117×C53D411yescharacteristic
83C2×C4⋊C4×C53—22no—
84C2×C4⋊C4×C53—22no—
85C2×C4⋊C4×C53—21no—
86C2×C4⋊C4×C53—21no—
87C2×C4⋊C4×C53D412yes—
88C2×C42×C53—21no—
89C2×C42×C53—21no—
90C2×C42×C53C2311yescharacteristic
91C4⋊C8×C53—21no—
92C4⋊C8×C53—21no—
93C4⋊C8×C53—21no—
94C4⋊C8×C53—21no—
95C4⋊C8×C53—21no—
96#104×C53—42no—
97C22⋊C8×C53—21no—
98C4×C8×C53—21no—
99#95×C53—21no—
100#41×C53—21no—
101#41×C53—21no—
102#41×C53—44no—
103#41×C53—44no—
104#41×C53—48no—
105#41×C53—48no—
106#41×C53—44no—
107#41×C53—44no—
108#41×C53—41no—
109#41×C53—41no—
110#41×C53—41no—
111#41×C53—41no—
112#41×C53D812yes—
113#41×C53SD1612yes—
114C2×Q8×C53—24no—
115C2×Q8×C53—22no—
116C2×Q8×C53—22no—
117C2×Q8×C53—21no—
118C2×Q8×C53—44no—
119C2×Q8×C53—44no—
120C2×Q8×C53—44no—
121C2×Q8×C53—44no—
122C2×Q8×C53—41no—
123C2×Q8×C53—41no—
124C2×Q8×C53—41no—
125C2×Q8×C53—41no—
126C2×Q8×C53C22⋊C411yescharacteristic
127C2×D4×C53—21no—
128C2×D4×C53—24no—
129C2×D4×C53—22no—
130C2×D4×C53—21no—
131C2×D4×C53—21no—
132C2×D4×C53C22⋊C411yescharacteristic
133C2×D4×C53C22⋊C412yes—
134C22×C4×C53—21no—
135C22×C4×C53—24no—
136C22×C4×C53—24no—
137C22×C4×C53—22no—
138C22×C4×C53—21no—
139C22×C4×C53—22no—
140C22×C4×C53—21no—
141C22×C4×C53—41no—
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2×D411yescharacteristic
145M16×C53—41no—
146M16×C53—41no—
147C2×C8×C53—42no—
148C2×C8×C53—41no—
149C2×C8×C53—41no—
150C2×C8×C53—41no—
151C4⋊C4×C53—24no—
152C4⋊C4×C53—24no—
153C4⋊C4×C53—22no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—22no—
157C4⋊C4×C53—22no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—22no—
160C4⋊C4×C53—41no—
161C4⋊C4×C53C22⋊C412yes—
162C4⋊C4×C53C22⋊C414yes—
163C4⋊C4×C53C22⋊C411yescharacteristic
164C4⋊C4×C53C22⋊C411yescharacteristic
165C22⋊C4×C53—24no—
166C22⋊C4×C53—24no—
167C22⋊C4×C53—21no—
168C22⋊C4×C53—21no—
169C22⋊C4×C53—22no—
170C22⋊C4×C53—21no—
171C22⋊C4×C53—22no—
172C42×C53—21no—
173C42×C53—21no—
174C42×C53—21no—
175C42×C53—41no—
176C42×C53—41no—
177C42×C53C22×C411yescharacteristic
178C42×C53C2×D411yescharacteristic
179C42×C53C2×D411yescharacteristic
180C42×C53C2×D411yescharacteristic
181C23×C53—21no—
182C23×C53—22no—
183C23×C53#12011yescharacteristic
184Q8×C53—21no—
185Q8×C53—48no—
186Q8×C53—42no—
187Q8×C53—42no—
188Q8×C53—41no—
189Q8×C53—42no—
190Q8×C53—41no—
191Q8×C53—42no—
192Q8×C53#10212yes—
193D4×C53—22no—
194D4×C53—22no—
195D4×C53—21no—
196D4×C53—42no—
197D4×C53—48no—
198D4×C53—42no—
199D4×C53—42no—
200D4×C53—41no—
201D4×C53—41no—
202D4×C53—41no—
203D4×C53—41no—
204D4×C53#10212yes—
205C2×C4×C53—21no—
206C2×C4×C53—22no—
207C2×C4×C53—22no—
208C2×C4×C53—21no—
209C2×C4×C53—21no—
210C2×C4×C53—22no—
211C2×C4×C53—24no—
212C2×C4×C53—21no—
213C2×C4×C53—21no—
214C2×C4×C53—41no—
215C2×C4×C53—42no—
216C2×C4×C53—48no—
217C2×C4×C53—48no—
218C2×C4×C53—42no—
219C2×C4×C53—42no—
220C2×C4×C53—42no—
221C2×C4×C53—42no—
222C2×C4×C53—41no—
223C2×C4×C53—41no—
224C2×C4×C53—42no—
225C2×C4×C53—81no—
226C2×C4×C53—81no—
227C2×C4×C53C2×C22⋊C411yescharacteristic
228C2×C4×C53C2×C22⋊C411yescharacteristic
229C2×C4×C53C2×C22⋊C411yescharacteristic
230C2×C4×C53#12011yescharacteristic
231C2×C4×C53#12011yescharacteristic
232C2×C4×C53#12011yescharacteristic
233C2×C4×C53C2×D811yescharacteristic
234C2×C4×C53C2×SD1611yescharacteristic
235C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
236C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
237C8×C53—81no—
238C8×C53—81no—
239C8×C53—81no—
240#8421C5311yesmaximal, characteristic, direct product factor, Sylow
241C22×C53—21no—
242C22×C53—21no—
243C22×C53—22no—
244C22×C53—41no—
245C22×C53—41no—
246C22×C53—42no—
247C22×C53—42no—
248C22×C53—42no—
249C22×C53#37911yescharacteristic
250C22×C53#44711yescharacteristic
251C22×C53#45011yescharacteristic
252C4×C53—21no—
253C4×C53—21no—
254C4×C53—21no—
255C4×C53—22no—
256C4×C53—41no—
257C4×C53—41no—
258C4×C53—44no—
259C4×C53—42no—
260C4×C53—82no—
261C4×C53C2×#10211yescharacteristic
262C4×C53#41811yescharacteristic
263C4×C53#44711yescharacteristic
264C4×C53#45011yescharacteristic
265C4×C53#45411yescharacteristic
266C4×C53#45611yescharacteristic
267#3499C2×C5311yescharacteristic
268#1988C2×C5311yescharacteristic, complemented
269#1806C2×C5311yescharacteristic, complemented
270#1547C2×C5311yescharacteristic, complemented
271#1546C2×C5311yescharacteristic, complemented
272#1502C2×C5311yescharacteristic, complemented
273#1477C2×C5311yescharacteristic, complemented
274C2×C53—21no—
275C2×C53—21no—
276C2×C53—42no—
277C2×C53—82no—
278C2×C53#178611yescharacteristic
279C2×C53#179311yescharacteristic
280C2×C53#179611yescharacteristic
281C2×2−1+4—21no—
282D4×Q8—21no—
283D4×Q8C4×C5312yescomplemented
284#547—22no—
285#547—21no—
286#547—22no—
287#547C4×C5312yescomplemented
288#536—22no—
289#536—21no—
290#532—21no—
291#532C22×C5311yescharacteristic
292C2×C4⋊Q8—21no—
293C2×C4⋊Q8C22×C5311yescharacteristic
294C4×#41C22×C5311yescharacteristic
295#431C22×C5311yescharacteristic
296#423C22×C5311yescharacteristic
297C2×C4⋊C8C22×C5311yescharacteristic
298C2×#104—21no—
299C4×C4⋊C4C22×C5311yescharacteristic
300#340—22no—
301#339—21no—
302#332—22no—
303#332—21no—
304#331—21no—
305#327—22no—
306Q8⋊C8—21no—
307D4⋊C8—21no—
308C53#842111yescharacteristic, direct product factor, Sylow
3092−1+4—42no—
3102−1+4—42no—
3112−1+4—42no—
3122−1+4—42no—
313C2×#41—22no—
314C2×#41—24no—
315C2×#41—22no—
316C2×#41—21no—
317C2×#41D4×C5311yescharacteristic
318C22×Q8—22no—
319C22×Q8—22no—
320C22×Q8—21no—
321C2×M16—21no—
322C22×C8—21no—
323C4⋊Q8—22no—
324C4⋊Q8—22no—
325C4⋊Q8—22no—
326C4⋊Q8—21no—
327C4⋊Q8—21no—
328C4⋊Q8C2×C4×C5312yes—
329C4⋊Q8D4×C5311yescharacteristic
330C4⋊Q8D4×C5312yes—
331#127D4×C5311yescharacteristic
332#124—22no—
333#124—21no—
334#123—24no—
335#123—24no—
336#123—24no—
337C22⋊Q8—24no—
338C22⋊Q8—22no—
339C22⋊Q8—24no—
340C22⋊Q8—22no—
341#121—22no—
342#121—22no—
343#121—21no—
344#121D4×C5312yes—
345C4×Q8C2×C4×C5311yescharacteristic
346C4×Q8D4×C5311yescharacteristic
347C4×D4—22no—
348C4×D4—22no—
349C4×D4—21no—
350C4×D4—21no—
351C4×D4—21no—
352C4×D4C2×C4×C5311yescharacteristic
353C4×D4C2×C4×C5312yes—
354C4×D4D4×C5311yescharacteristic
355#117—21no—
356#117D4×C5311yescharacteristic
357C2×C4⋊C4—22no—
358C2×C4⋊C4—21no—
359C2×C4⋊C4—22no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4D4×C5312yes—
362C2×C42—21no—
363C2×C42—21no—
364C2×C42C23×C5311yescharacteristic
365C4⋊C8—21no—
366C4⋊C8—21no—
367C4⋊C8—21no—
368C4⋊C8—21no—
369C4⋊C8—21no—
370#104—42no—
371C22⋊C8—21no—
372C4×C8—21no—
373#95—21no—
374#41—21no—
375#41—21no—
376#41—44no—
377#41—44no—
378#41—48no—
379#41—48no—
380#41—44no—
381#41—44no—
382#41—41no—
383#41—41no—
384#41—41no—
385#41—41no—
386#41D8×C5312yes—
387#41SD16×C5312yes—
388C2×Q8—24no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—21no—
392C2×Q8—44no—
393C2×Q8—44no—
394C2×Q8—44no—
395C2×Q8—44no—
396C2×Q8—41no—
397C2×Q8—41no—
398C2×Q8—41no—
399C2×Q8—41no—
400C2×Q8C22⋊C4×C5311yescharacteristic
401C2×D4—21no—
402C2×D4—24no—
403C2×D4—22no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4C22⋊C4×C5311yescharacteristic
407C2×D4C22⋊C4×C5312yes—
408C22×C4—21no—
409C22×C4—24no—
410C22×C4—24no—
411C22×C4—22no—
412C22×C4—21no—
413C22×C4—22no—
414C22×C4—21no—
415C22×C4—41no—
416C22×C4C2×D4×C5311yescharacteristic
417C22×C4C2×D4×C5311yescharacteristic
418C22×C4C2×D4×C5311yescharacteristic
419M16—41no—
420M16—41no—
421C2×C8—42no—
422C2×C8—41no—
423C2×C8—41no—
424C2×C8—41no—
425C4⋊C4—24no—
426C4⋊C4—24no—
427C4⋊C4—22no—
428C4⋊C4—21no—
429C4⋊C4—21no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—21no—
433C4⋊C4—22no—
434C4⋊C4—41no—
435C4⋊C4C22⋊C4×C5312yes—
436C4⋊C4C22⋊C4×C5314yes—
437C4⋊C4C22⋊C4×C5311yescharacteristic
438C4⋊C4C22⋊C4×C5311yescharacteristic
439C22⋊C4—24no—
440C22⋊C4—24no—
441C22⋊C4—21no—
442C22⋊C4—21no—
443C22⋊C4—22no—
444C22⋊C4—21no—
445C22⋊C4—22no—
446C42—21no—
447C42—21no—
448C42—21no—
449C42—41no—
450C42—41no—
451C42C22×C4×C5311yescharacteristic
452C42C2×D4×C5311yescharacteristic
453C42C2×D4×C5311yescharacteristic
454C42C2×D4×C5311yescharacteristic
455C23—21no—
456C23—22no—
457C23#120×C5311yescharacteristic
458Q8—21no—
459Q8—48no—
460Q8—42no—
461Q8—42no—
462Q8—41no—
463Q8—42no—
464Q8—41no—
465Q8—42no—
466Q8#102×C5312yes—
467D4—22no—
468D4—22no—
469D4—21no—
470D4—42no—
471D4—48no—
472D4—42no—
473D4—42no—
474D4—41no—
475D4—41no—
476D4—41no—
477D4—41no—
478D4#102×C5312yes—
479C2×C4—21no—
480C2×C4—22no—
481C2×C4—22no—
482C2×C4—21no—
483C2×C4—21no—
484C2×C4—21no—
485C2×C4—21no—
486C2×C4—24no—
487C2×C4—22no—
488C2×C4—41no—
489C2×C4—42no—
490C2×C4—48no—
491C2×C4—48no—
492C2×C4—42no—
493C2×C4—42no—
494C2×C4—42no—
495C2×C4—42no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—42no—
499C2×C4—81no—
500C2×C4—81no—
501C2×C4C2×C22⋊C4×C5311yescharacteristic
502C2×C4C2×C22⋊C4×C5311yescharacteristic
503C2×C4C2×C22⋊C4×C5311yescharacteristic
504C2×C4#120×C5311yescharacteristic
505C2×C4#120×C5311yescharacteristic
506C2×C4#120×C5311yescharacteristic
507C2×C4C2×D8×C5311yescharacteristic
508C2×C4C2×SD16×C5311yescharacteristic
509C2×C4AGL(1,ℤ/8)×C5311yescharacteristic
510C2×C4AGL(1,ℤ/8)×C5311yescharacteristic
511C8—81no—
512C8—81no—
513C8—81no—
514C22—21no—
515C22—21no—
516C22—22no—
517C22—41no—
518C22—41no—
519C22—42no—
520C22—42no—
521C22—42no—
522C22C53×#37911yescharacteristic
523C22C53×#44711yescharacteristic
524C22C53×#45011yescharacteristic
525C4—22no—
526C4—21no—
527C4—21no—
528C4—21no—
529C4—44no—
530C4—42no—
531C4—41no—
532C4—41no—
533C4—82no—
534C4C2×#102×C5311yescharacteristic
535C4C53×#41811yescharacteristic
536C4C53×#44711yescharacteristic
537C4C53×#45011yescharacteristic
538C4C53×#45411yescharacteristic
539C4C53×#45611yescharacteristic
540C2—21no—
541C2—21no—
542C2—42no—
543C2—82no—
544C2C53×#178611yescharacteristic
545C2C53×#179311yescharacteristic
546C2C53×#179611yescharacteristic
subgroupG/H
center, socleC22×C53#379
commutatorC42C22×C4×C53
Frattini, cosocleC2×C42C23×C53
prime radicalC53×#3499C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#1988C2semidirect
3. C53×#1806C2semidirect
4. C53×#1547C2semidirect
5. C53×#1546C2semidirect
6. C53×#1502C2semidirect
7. C53×#1477C2semidirect
10. D4×Q8×C53C4semidirect
14. C53×#547C4semidirect
240. #8421C53direct (N × H)
268. #1988C2×C53semidirect
269. #1806C2×C53semidirect
270. #1547C2×C53semidirect
271. #1546C2×C53semidirect
272. #1502C2×C53semidirect
273. #1477C2×C53semidirect
283. D4×Q8C4×C53semidirect
287. #547C4×C53semidirect
308. C53#8421direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#34992
C53×#19882
C53×#18062
C53×#15472
C53×#15462
C53×#15022
C53×#14772
#842153