Finite Group Databasebeta

← C53×#8500 #27216012

Subgroups — C53×#8500 #27216012

Order 13568.

736 conjugacy classes of subgroups.

The subgroup lattice has 570 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3546C211yesmaximal, characteristic
2C53×#1553C211yesmaximal, characteristic, complemented
3C53×#1535C211yesmaximal, characteristic, complemented
4C53×#1521C211yesmaximal, characteristic, complemented
5C53×#1514C211yesmaximal, characteristic, complemented
6C53×#1514C211yesmaximal, characteristic, complemented
7C53×#1496C211yesmaximal, characteristic, complemented
8C53×#560—22no—
9C53×#556C411yescharacteristic, complemented
10C53×#553—21no—
11C53×#550—21no—
12C53×#550C411yescharacteristic, complemented
13C53×#547C411yescharacteristic, complemented
14C53×#546—21no—
15C53×#546—21no—
16D42×C53C411yescharacteristic, complemented
17C53×#542—21no—
18C53×#538—22no—
19C53×#537—21no—
20C53×#535—21no—
21C53×#535—21no—
22C53×#532C2211yescharacteristic
23C53×#532C2211yescharacteristic
24C53×#529C2211yescharacteristic
25C53×#432C2211yescharacteristic, complemented
26C53×#432C2211yescharacteristic, complemented
27C53×#431C2211yescharacteristic, complemented
28C53×#383C2211yescharacteristic, complemented
29C53×#332—21no—
30C53×#331—21no—
31C53×#330—21no—
32C53×#329—21no—
33C53×#328—21no—
34C53×#327—21no—
35C53×#327—21no—
36C53×#327—22no—
37C53×#327—21no—
38C53×D4⋊C8—22no—
39C2×#41×C53—21no—
40C2×#41×C53—21no—
41C2×#41×C53—21no—
42C2×#41×C53—21no—
43C22×D4×C53—21no—
44C22×D4×C53—21no—
45C4⋊Q8×C53C2×C411yescharacteristic
46C4⋊Q8×C53D411yescharacteristic
47#127×C53—22no—
48#127×C53—21no—
49#127×C53C2×C411yescharacteristic
50#127×C53D411yescharacteristic
51#126×C53—22no—
52#126×C53—22no—
53#126×C53—21no—
54#125×C53D411yescharacteristic
55#124×C53—22no—
56#124×C53—21no—
57#124×C53—21no—
58#124×C53—21no—
59#124×C53D411yescharacteristic
60#123×C53—22no—
61#123×C53—22no—
62#123×C53—22no—
63#123×C53—21no—
64C22⋊Q8×C53—22no—
65C22⋊Q8×C53—21no—
66C22⋊Q8×C53—21no—
67C22⋊Q8×C53D411yescharacteristic
68#121×C53—22no—
69#121×C53—22no—
70#121×C53—22no—
71#121×C53—22no—
72#121×C53—22no—
73#121×C53—21no—
74#121×C53—21no—
75#121×C53—21no—
76#121×C53—21no—
77#121×C53—22no—
78#121×C53—21no—
79#121×C53—21no—
80#121×C53D411yescharacteristic
81#121×C53D411yescharacteristic
82#121×C53D411yescharacteristic
83#121×C53D412yes—
84#120×C53—22no—
85#120×C53—22no—
86#120×C53—22no—
87#120×C53—22no—
88C4×Q8×C53—21no—
89C4×Q8×C53C2×C411yescharacteristic
90C4×D4×C53—22no—
91C4×D4×C53—21no—
92C4×D4×C53—21no—
93C4×D4×C53—21no—
94C4×D4×C53—21no—
95C4×D4×C53C2×C411yescharacteristic
96C4×D4×C53C2×C411yescharacteristic
97C4×D4×C53C2×C411yescharacteristic
98C4×D4×C53D412yes—
99#117×C53—21no—
100#117×C53—21no—
101C2×C4⋊C4×C53—21no—
102C2×C22⋊C4×C53—22no—
103C2×C42×C53C2311yescharacteristic
104C4⋊C8×C53—21no—
105C4⋊C8×C53—21no—
106C4⋊C8×C53—21no—
107C22⋊C8×C53—21no—
108C22⋊C8×C53—21no—
109C22⋊C8×C53—21no—
110#97×C53—21no—
111#97×C53—21no—
112C4×C8×C53—21no—
113#95×C53—21no—
114#95×C53—21no—
115#95×C53—21no—
116C24×C53—22no—
117#41×C53—82no—
118#41×C53—82no—
119#41×C53—82no—
120#41×C53—82no—
121C2×Q8×C53—21no—
122C2×Q8×C53—21no—
123C2×Q8×C53C22⋊C411yescharacteristic
124C2×D4×C53—22no—
125C2×D4×C53—22no—
126C2×D4×C53—22no—
127C2×D4×C53—22no—
128C2×D4×C53—22no—
129C2×D4×C53—21no—
130C2×D4×C53—21no—
131C2×D4×C53—22no—
132C2×D4×C53—21no—
133C2×D4×C53—21no—
134C2×D4×C53—21no—
135C2×D4×C53—21no—
136C2×D4×C53—21no—
137C2×D4×C53—82no—
138C2×D4×C53—82no—
139C2×D4×C53C22⋊C411yescharacteristic
140C2×D4×C53C22⋊C411yescharacteristic
141C2×D4×C53C22⋊C411yescharacteristic
142C2×D4×C53C22⋊C412yes—
143C22×C4×C53—22no—
144C22×C4×C53—22no—
145C22×C4×C53—21no—
146C22×C4×C53—21no—
147C22×C4×C53—21no—
148C22×C4×C53—22no—
149C22×C4×C53—41no—
150C22×C4×C53C2×D411yescharacteristic
151C22×C4×C53C2×D411yescharacteristic
152C22×C4×C53C2×D411yescharacteristic
153C2×C8×C53—41no—
154C2×C8×C53—41no—
155C2×C8×C53—41no—
156C4⋊C4×C53—22no—
157C4⋊C4×C53—22no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—21no—
160C4⋊C4×C53—21no—
161C4⋊C4×C53—21no—
162C4⋊C4×C53—21no—
163C4⋊C4×C53—21no—
164C4⋊C4×C53C22⋊C411yescharacteristic
165C4⋊C4×C53C22⋊C411yescharacteristic
166C4⋊C4×C53C22⋊C411yescharacteristic
167C4⋊C4×C53C22⋊C411yescharacteristic
168C4⋊C4×C53C22⋊C412yes—
169C22⋊C4×C53—22no—
170C22⋊C4×C53—22no—
171C22⋊C4×C53—22no—
172C22⋊C4×C53—22no—
173C22⋊C4×C53—22no—
174C22⋊C4×C53—21no—
175C22⋊C4×C53—21no—
176C22⋊C4×C53—21no—
177C22⋊C4×C53—21no—
178C22⋊C4×C53—21no—
179C22⋊C4×C53—21no—
180C22⋊C4×C53—21no—
181C22⋊C4×C53—21no—
182C22⋊C4×C53—21no—
183C22⋊C4×C53—22no—
184C22⋊C4×C53—22no—
185C42×C53—22no—
186C42×C53—21no—
187C42×C53C22×C411yescharacteristic
188C42×C53C2×D411yescharacteristic
189C42×C53C2×D411yescharacteristic
190C42×C53C2×D411yescharacteristic
191C23×C53—21no—
192C23×C53—22no—
193C23×C53—21no—
194C23×C53—21no—
195C23×C53—82no—
196C23×C53—82no—
197C23×C53—82no—
198C23×C53#12011yescharacteristic
199Q8×C53—42no—
200Q8×C53—41no—
201Q8×C53—81no—
202D4×C53—41no—
203D4×C53—41no—
204D4×C53—42no—
205D4×C53—41no—
206D4×C53—42no—
207D4×C53—42no—
208D4×C53—81no—
209D4×C53—81no—
210D4×C53—81no—
211D4×C53—81no—
212D4×C53—81no—
213D4×C53—82no—
214D4×C53—82no—
215D4×C53—82no—
216D4×C53—82no—
217D4×C53—82no—
218D4×C53—82no—
219C2×C4×C53—21no—
220C2×C4×C53—21no—
221C2×C4×C53—22no—
222C2×C4×C53—21no—
223C2×C4×C53—21no—
224C2×C4×C53—21no—
225C2×C4×C53—41no—
226C2×C4×C53—41no—
227C2×C4×C53—41no—
228C2×C4×C53—42no—
229C2×C4×C53—42no—
230C2×C4×C53—42no—
231C2×C4×C53—82no—
232C2×C4×C53—82no—
233C2×C4×C53—82no—
234C2×C4×C53—81no—
235C2×C4×C53—81no—
236C2×C4×C53—81no—
237C2×C4×C53C2×C22⋊C411yescharacteristic
238C2×C4×C53C2×C22⋊C411yescharacteristic
239C2×C4×C53C2×C22⋊C411yescharacteristic
240C2×C4×C53#12011yescharacteristic
241C2×C4×C53#12011yescharacteristic
242C2×C4×C53#12011yescharacteristic
243C8×C53—81no—
244C8×C53—81no—
245C8×C53—81no—
246#8500C5311yesmaximal, characteristic, direct product factor, Sylow
247C22×C53—21no—
248C22×C53—21no—
249C22×C53—21no—
250C22×C53—42no—
251C22×C53—41no—
252C22×C53—41no—
253C22×C53—41no—
254C22×C53—41no—
255C22×C53—41no—
256C22×C53—41no—
257C22×C53—41no—
258C22×C53—42no—
259C22×C53—42no—
260C22×C53—41no—
261C22×C53—41no—
262C22×C53—164no—
263C22×C53#37911yescharacteristic
264C4×C53—21no—
265C4×C53—21no—
266C4×C53—21no—
267C4×C53—21no—
268C4×C53—21no—
269C4×C53—21no—
270C4×C53—41no—
271C4×C53—41no—
272C4×C53—41no—
273C4×C53—41no—
274C4×C53—41no—
275C4×C53—42no—
276C4×C53—161no—
277#3546C2×C5311yescharacteristic
278#1553C2×C5311yescharacteristic, complemented
279#1535C2×C5311yescharacteristic, complemented
280#1521C2×C5311yescharacteristic, complemented
281#1514C2×C5311yescharacteristic, complemented
282#1514C2×C5311yescharacteristic, complemented
283#1496C2×C5311yescharacteristic, complemented
284C2×C53—41no—
285C2×C53—81no—
286C2×C53—82no—
287C2×C53—81no—
288C2×C53—81no—
289C2×C53#179411yescharacteristic
290C2×C53#179511yescharacteristic
291C2×C53#179511yescharacteristic
292#560—22no—
293#556C4×C5311yescharacteristic, complemented
294#553—21no—
295#550—21no—
296#550C4×C5311yescharacteristic, complemented
297#547C4×C5311yescharacteristic, complemented
298#546—21no—
299#546—21no—
300D42C4×C5311yescharacteristic, complemented
301#542—21no—
302#538—22no—
303#537—21no—
304#535—21no—
305#535—21no—
306#532C22×C5311yescharacteristic
307#532C22×C5311yescharacteristic
308#529C22×C5311yescharacteristic
309#432C22×C5311yescharacteristic, complemented
310#432C22×C5311yescharacteristic, complemented
311#431C22×C5311yescharacteristic, complemented
312#383C22×C5311yescharacteristic, complemented
313#332—21no—
314#331—21no—
315#330—21no—
316#329—21no—
317#328—21no—
318#327—21no—
319#327—21no—
320#327—22no—
321#327—21no—
322D4⋊C8—22no—
323C53#850011yescharacteristic, direct product factor, Sylow
324C2×#41—21no—
325C2×#41—21no—
326C2×#41—21no—
327C2×#41—21no—
328C22×D4—21no—
329C22×D4—21no—
330C4⋊Q8C2×C4×C5311yescharacteristic
331C4⋊Q8D4×C5311yescharacteristic
332#127—22no—
333#127—21no—
334#127C2×C4×C5311yescharacteristic
335#127D4×C5311yescharacteristic
336#126—22no—
337#126—22no—
338#126—21no—
339#125D4×C5311yescharacteristic
340#124—22no—
341#124—21no—
342#124—21no—
343#124—21no—
344#124D4×C5311yescharacteristic
345#123—22no—
346#123—22no—
347#123—21no—
348#123—22no—
349C22⋊Q8—22no—
350C22⋊Q8—21no—
351C22⋊Q8—21no—
352C22⋊Q8D4×C5311yescharacteristic
353#121—22no—
354#121—22no—
355#121—22no—
356#121—22no—
357#121—22no—
358#121—21no—
359#121—22no—
360#121—21no—
361#121—21no—
362#121—21no—
363#121—21no—
364#121—21no—
365#121D4×C5311yescharacteristic
366#121D4×C5311yescharacteristic
367#121D4×C5311yescharacteristic
368#121D4×C5312yes—
369#120—22no—
370#120—22no—
371#120—22no—
372#120—22no—
373C4×Q8—21no—
374C4×Q8C2×C4×C5311yescharacteristic
375C4×D4—22no—
376C4×D4—21no—
377C4×D4—21no—
378C4×D4—21no—
379C4×D4—21no—
380C4×D4C2×C4×C5311yescharacteristic
381C4×D4C2×C4×C5311yescharacteristic
382C4×D4C2×C4×C5311yescharacteristic
383C4×D4D4×C5312yes—
384#117—21no—
385#117—21no—
386C2×C4⋊C4—21no—
387C2×C22⋊C4—22no—
388C2×C42C23×C5311yescharacteristic
389C4⋊C8—21no—
390C4⋊C8—21no—
391C4⋊C8—21no—
392C22⋊C8—21no—
393C22⋊C8—21no—
394C22⋊C8—21no—
395#97—21no—
396#97—21no—
397C4×C8—21no—
398#95—21no—
399#95—21no—
400#95—21no—
401C24—22no—
402#41—82no—
403#41—82no—
404#41—82no—
405#41—82no—
406C2×Q8—21no—
407C2×Q8—21no—
408C2×Q8C22⋊C4×C5311yescharacteristic
409C2×D4—22no—
410C2×D4—22no—
411C2×D4—22no—
412C2×D4—22no—
413C2×D4—22no—
414C2×D4—21no—
415C2×D4—21no—
416C2×D4—22no—
417C2×D4—21no—
418C2×D4—21no—
419C2×D4—21no—
420C2×D4—21no—
421C2×D4—21no—
422C2×D4—82no—
423C2×D4—82no—
424C2×D4C22⋊C4×C5311yescharacteristic
425C2×D4C22⋊C4×C5311yescharacteristic
426C2×D4C22⋊C4×C5311yescharacteristic
427C2×D4C22⋊C4×C5312yes—
428C22×C4—22no—
429C22×C4—22no—
430C22×C4—21no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—41no—
435C22×C4C2×D4×C5311yescharacteristic
436C22×C4C2×D4×C5311yescharacteristic
437C22×C4C2×D4×C5311yescharacteristic
438C2×C8—41no—
439C2×C8—41no—
440C2×C8—41no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—21no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4C22⋊C4×C5311yescharacteristic
450C4⋊C4C22⋊C4×C5311yescharacteristic
451C4⋊C4C22⋊C4×C5311yescharacteristic
452C4⋊C4C22⋊C4×C5311yescharacteristic
453C4⋊C4C22⋊C4×C5312yes—
454C22⋊C4—22no—
455C22⋊C4—22no—
456C22⋊C4—22no—
457C22⋊C4—22no—
458C22⋊C4—22no—
459C22⋊C4—21no—
460C22⋊C4—21no—
461C22⋊C4—21no—
462C22⋊C4—21no—
463C22⋊C4—21no—
464C22⋊C4—21no—
465C22⋊C4—21no—
466C22⋊C4—21no—
467C22⋊C4—21no—
468C22⋊C4—22no—
469C22⋊C4—22no—
470C42—22no—
471C42—21no—
472C42C22×C4×C5311yescharacteristic
473C42C2×D4×C5311yescharacteristic
474C42C2×D4×C5311yescharacteristic
475C42C2×D4×C5311yescharacteristic
476C23—21no—
477C23—21no—
478C23—22no—
479C23—21no—
480C23—82no—
481C23—82no—
482C23—82no—
483C23#120×C5311yescharacteristic
484Q8—41no—
485Q8—42no—
486Q8—81no—
487D4—41no—
488D4—41no—
489D4—42no—
490D4—41no—
491D4—42no—
492D4—42no—
493D4—81no—
494D4—81no—
495D4—81no—
496D4—81no—
497D4—81no—
498D4—82no—
499D4—82no—
500D4—82no—
501D4—82no—
502D4—82no—
503D4—82no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—21no—
509C2×C4—21no—
510C2×C4—41no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—42no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4—81no—
520C2×C4—81no—
521C2×C4—81no—
522C2×C4C2×C22⋊C4×C5311yescharacteristic
523C2×C4C2×C22⋊C4×C5311yescharacteristic
524C2×C4C2×C22⋊C4×C5311yescharacteristic
525C2×C4#120×C5311yescharacteristic
526C2×C4#120×C5311yescharacteristic
527C2×C4#120×C5311yescharacteristic
528C8—81no—
529C8—81no—
530C8—81no—
531C22—21no—
532C22—21no—
533C22—21no—
534C22—42no—
535C22—41no—
536C22—41no—
537C22—41no—
538C22—41no—
539C22—41no—
540C22—41no—
541C22—41no—
542C22—42no—
543C22—42no—
544C22—41no—
545C22—41no—
546C22—164no—
547C22C53×#37911yescharacteristic
548C4—21no—
549C4—21no—
550C4—21no—
551C4—21no—
552C4—21no—
553C4—21no—
554C4—41no—
555C4—41no—
556C4—41no—
557C4—41no—
558C4—41no—
559C4—42no—
560C4—161no—
561C2—41no—
562C2—81no—
563C2—81no—
564C2—82no—
565C2—81no—
566C2C53×#179411yescharacteristic
567C2C53×#179511yescharacteristic
568C2C53×#179511yescharacteristic
subgroupG/H
center, socleC22×C53#379
commutatorC42C22×C4×C53
Frattini, cosocleC2×C42C23×C53
prime radicalC53×#3546C2
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×#1553C2semidirect
3. C53×#1535C2semidirect
4. C53×#1521C2semidirect
5. C53×#1514C2semidirect
6. C53×#1514C2semidirect
7. C53×#1496C2semidirect
9. C53×#556C4semidirect
12. C53×#550C4semidirect
13. C53×#547C4semidirect
16. D42×C53C4semidirect
25. C53×#432C22semidirect
26. C53×#432C22semidirect
27. C53×#431C22semidirect
28. C53×#383C22semidirect
246. #8500C53direct (N × H)
278. #1553C2×C53semidirect
279. #1535C2×C53semidirect
280. #1521C2×C53semidirect
281. #1514C2×C53semidirect
282. #1514C2×C53semidirect
283. #1496C2×C53semidirect
293. #556C4×C53semidirect
296. #550C4×C53semidirect
297. #547C4×C53semidirect
300. D42C4×C53semidirect
309. #432C22×C53semidirect
310. #432C22×C53semidirect
311. #431C22×C53semidirect
312. #383C22×C53semidirect
323. C53#8500direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#35462
C53×#15532
C53×#15352
C53×#15212
C53×#15142
C53×#14962
#850053