Finite Group Databasebeta

← C53×#18441 #27225953

Subgroups — C53×#18441 #27225953

Order 13568.

888 conjugacy classes of subgroups.

The subgroup lattice has 558 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3559C211yesmaximal, characteristic
2C53×#3239C211yesmaximal, characteristic, complemented
3C53×#3227C211yesmaximal, characteristic, complemented
4C53×#2642C211yesmaximal, characteristic, complemented
5C53×#1616C214yesmaximal, complemented
6C53×#1615C214yesmaximal, complemented
7C53×#1505C211yesmaximal, characteristic, complemented
8C53×#1500C211yesmaximal, characteristic, complemented
9C53×#1470C211yesmaximal, characteristic, complemented
10C53×#563—21no—
11C53×#562—21no—
12C53×#554—21no—
13C53×#553—21no—
14C53×#553—21no—
15C53×#547—22no—
16C53×#547—21no—
17C53×#544—21no—
18C53×#540—21no—
19C53×#539—22no—
20C53×#539C2212yes—
21C53×#537C2212yes—
22C53×#532C2211yescharacteristic
23C2×C4⋊Q8×C53C2211yescharacteristic
24C53×#529C2211yescharacteristic
25C2×#125×C53C2211yescharacteristic
26C53×#501—21no—
27C53×#498—21no—
28C53×#489—21no—
29C53×#486—21no—
30C53×#484C2212yes—
31C53×#481C2214yes—
32C53×#481C2212yes—
33C53×#476—22no—
34C53×#475—21no—
35C53×#474—21no—
36C53×#432C2211yescharacteristic
37C53×#431C2211yescharacteristic
38C53×#400—22no—
39C53×#400C2214yes—
40C53×#391—21no—
41C53×#391—21no—
42C53×#376C2211yescharacteristic
43C53×#328C2212yes—
44C53×#327C2212yes—
45C53×#327C2214yes—
46C53×#324C2214yes—
47C2×#41×C53—21no—
48C2×#41×C53—22no—
49C22×Q8×C53—21no—
50C4⋊Q8×C53—21no—
51C4⋊Q8×C53—22no—
52C4⋊Q8×C53—21no—
53C4⋊Q8×C53—21no—
54C4⋊Q8×C53—21no—
55C4⋊Q8×C53—21no—
56C4⋊Q8×C53D411yescharacteristic
57#127×C53D411yescharacteristic
58#126×C53—22no—
59#126×C53—22no—
60#126×C53—21no—
61#125×C53—21no—
62#125×C53—22no—
63#125×C53—21no—
64#125×C53—21no—
65#125×C53—21no—
66#125×C53—21no—
67#125×C53D411yescharacteristic
68#124×C53—21no—
69#124×C53D411yescharacteristic
70#123×C53—22no—
71#123×C53—22no—
72#123×C53—22no—
73#123×C53—22no—
74#123×C53—22no—
75#123×C53—21no—
76#123×C53—22no—
77C22⋊Q8×C53—22no—
78C22⋊Q8×C53—22no—
79C22⋊Q8×C53—22no—
80C22⋊Q8×C53—22no—
81C22⋊Q8×C53—22no—
82C22⋊Q8×C53—22no—
83C22⋊Q8×C53C2311yescharacteristic
84#121×C53—21no—
85#121×C53—22no—
86#121×C53—21no—
87#121×C53—22no—
88#121×C53—22no—
89#121×C53—21no—
90#121×C53—22no—
91#121×C53C2311yescharacteristic
92#121×C53C2312yes—
93C4×Q8×C53D411yescharacteristic
94C4×D4×C53—22no—
95C4×D4×C53—21no—
96C4×D4×C53—21no—
97C4×D4×C53—21no—
98C4×D4×C53—21no—
99C4×D4×C53—22no—
100C4×D4×C53D411yescharacteristic
101C4×D4×C53D412yes—
102#117×C53—21no—
103#117×C53—21no—
104C2×C4⋊C4×C53—22no—
105C2×C4⋊C4×C53—21no—
106C2×C4⋊C4×C53—21no—
107C2×C4⋊C4×C53D411yescharacteristic
108C2×C4⋊C4×C53D412yes—
109C2×C4⋊C4×C53D411yescharacteristic
110C2×C4⋊C4×C53C2312yes—
111C2×C4⋊C4×C53C2312yes—
112C2×C42×C53—21no—
113C2×C42×C53C2311yescharacteristic
114#107×C53—22no—
115#106×C53—24no—
116#106×C53—22no—
117C4⋊C8×C53—21no—
118C4⋊C8×C53—21no—
119#103×C53—22no—
120#102×C53—24no—
121#102×C53—22no—
122C22⋊C8×C53C2312yes—
123C22⋊C8×C53C2312yes—
124#97×C53—21no—
125C4×C8×C53—21no—
126#95×C53—21no—
127#95×C53—24no—
128#95×C53—24no—
129#95×C53C2312yes—
130#41×C53—82no—
131#41×C53—84no—
132C2×Q8×C53—22no—
133C2×Q8×C53—21no—
134C2×Q8×C53—44no—
135C2×Q8×C53—41no—
136C2×Q8×C53—41no—
137C2×Q8×C53C2×D411yescharacteristic
138C2×D4×C53—21no—
139C2×D4×C53—22no—
140C2×D4×C53—22no—
141C2×D4×C53—22no—
142C2×D4×C53—21no—
143C2×D4×C53—21no—
144C2×D4×C53C2×D411yescharacteristic
145C2×D4×C53C2×D412yes—
146C22×C4×C53—22no—
147C22×C4×C53—22no—
148C22×C4×C53—22no—
149C22×C4×C53—21no—
150C22×C4×C53—22no—
151C22×C4×C53—22no—
152C22×C4×C53—22no—
153C22×C4×C53C2×D412yes—
154C22×C4×C53C2×D411yescharacteristic
155C22×C4×C53C2×D411yescharacteristic
156C22×C4×C53C2×D412yes—
157C22×C4×C53C2411yescharacteristic
158C2×C8×C53—22no—
159C2×C8×C53—22no—
160C4⋊C4×C53—22no—
161C4⋊C4×C53—22no—
162C4⋊C4×C53—22no—
163C4⋊C4×C53—22no—
164C4⋊C4×C53—21no—
165C4⋊C4×C53—21no—
166C4⋊C4×C53—22no—
167C4⋊C4×C53—21no—
168C4⋊C4×C53—21no—
169C4⋊C4×C53—22no—
170C4⋊C4×C53—21no—
171C4⋊C4×C53—21no—
172C4⋊C4×C53—21no—
173C4⋊C4×C53—21no—
174C4⋊C4×C53—22no—
175C4⋊C4×C53—22no—
176C4⋊C4×C53—21no—
177C4⋊C4×C53—22no—
178C4⋊C4×C53—22no—
179C4⋊C4×C53—41no—
180C4⋊C4×C53—41no—
181C4⋊C4×C53C2×D412yes—
182C4⋊C4×C53C2×D411yescharacteristic
183C4⋊C4×C53C2×D411yescharacteristic
184C22⋊C4×C53—22no—
185C22⋊C4×C53—22no—
186C22⋊C4×C53—22no—
187C22⋊C4×C53—22no—
188C22⋊C4×C53—22no—
189C22⋊C4×C53—21no—
190C22⋊C4×C53—21no—
191C22⋊C4×C53—21no—
192C22⋊C4×C53—21no—
193C22⋊C4×C53—22no—
194C22⋊C4×C53—21no—
195C22⋊C4×C53—22no—
196C42×C53—22no—
197C42×C53—21no—
198C42×C53—42no—
199C42×C53C2×D411yescharacteristic
200C42×C53C2×D411yescharacteristic
201C42×C53C2×D411yescharacteristic
202C42×C53C2×D411yescharacteristic
203C23×C53—21no—
204C23×C53—22no—
205C23×C53C22×D411yescharacteristic
206Q8×C53—41no—
207Q8×C53—81no—
208Q8×C53—82no—
209Q8×C53—82no—
210Q8×C53—82no—
211D4×C53—42no—
212D4×C53—41no—
213D4×C53—44no—
214D4×C53—42no—
215D4×C53—42no—
216D4×C53—81no—
217D4×C53—82no—
218D4×C53—82no—
219C2×C4×C53—22no—
220C2×C4×C53—22no—
221C2×C4×C53—22no—
222C2×C4×C53—21no—
223C2×C4×C53—22no—
224C2×C4×C53—21no—
225C2×C4×C53—22no—
226C2×C4×C53—21no—
227C2×C4×C53—24no—
228C2×C4×C53—41no—
229C2×C4×C53—42no—
230C2×C4×C53—42no—
231C2×C4×C53—42no—
232C2×C4×C53—44no—
233C2×C4×C53—42no—
234C2×C4×C53—42no—
235C2×C4×C53—42no—
236C2×C4×C53—82no—
237C2×C4×C53—82no—
238C2×C4×C53—82no—
239C2×C4×C53—82no—
240C2×C4×C53#12011yescharacteristic
241C2×C4×C53#12011yescharacteristic
242C2×C4×C53#12011yescharacteristic
243C2×C4×C53#12011yescharacteristic
244C2×C4×C53C22×D411yescharacteristic
245C2×C4×C53C22×D411yescharacteristic
246C8×C53—42no—
247C8×C53—42no—
248#18441C5311yesmaximal, characteristic, direct product factor, Sylow
249C22×C53—21no—
250C22×C53—21no—
251C22×C53—42no—
252C22×C53—41no—
253C22×C53—41no—
254C22×C53—42no—
255C22×C53—42no—
256C22×C53—41no—
257C22×C53C2×#12011yescharacteristic
258C22×C53#57712yes—
259C4×C53—21no—
260C4×C53—21no—
261C4×C53—21no—
262C4×C53—21no—
263C4×C53—21no—
264C4×C53—42no—
265C4×C53—42no—
266C4×C53—41no—
267C4×C53—41no—
268C4×C53—42no—
269C4×C53—41no—
270C4×C53—82no—
271C4×C53—82no—
272C4×C53#45412yes—
273#3559C2×C5311yescharacteristic
274#3239C2×C5311yescharacteristic, complemented
275#3227C2×C5311yescharacteristic, complemented
276#2642C2×C5311yescharacteristic, complemented
277#1616C2×C5314yescomplemented
278#1615C2×C5314yescomplemented
279#1505C2×C5311yescharacteristic, complemented
280#1500C2×C5311yescharacteristic, complemented
281#1470C2×C5311yescharacteristic, complemented
282C2×C53—22no—
283C2×C53—81no—
284C2×C53—82no—
285C2×C53#300911yescharacteristic
286C2×C53C2×#45411yescharacteristic
287C2×C53#302811yescharacteristic
288#563—21no—
289#562—21no—
290#554—21no—
291#553—21no—
292#553—21no—
293#547—22no—
294#547—21no—
295#544—21no—
296#540—21no—
297#539—22no—
298#539C22×C5312yes—
299#537C22×C5312yes—
300#532C22×C5311yescharacteristic
301C2×C4⋊Q8C22×C5311yescharacteristic
302#529C22×C5311yescharacteristic
303C2×#125C22×C5311yescharacteristic
304#501—21no—
305#498—21no—
306#489—21no—
307#486—21no—
308#484C22×C5312yes—
309#481C22×C5314yes—
310#481C22×C5312yes—
311#476—22no—
312#475—21no—
313#474—21no—
314#432C22×C5311yescharacteristic
315#431C22×C5311yescharacteristic
316#400—22no—
317#400C22×C5314yes—
318#391—21no—
319#391—21no—
320#376C22×C5311yescharacteristic
321#328C22×C5312yes—
322#327C22×C5314yes—
323#327C22×C5312yes—
324#324C22×C5314yes—
325C53#1844111yescharacteristic, direct product factor, Sylow
326C2×#41—21no—
327C2×#41—22no—
328C22×Q8—21no—
329C4⋊Q8—21no—
330C4⋊Q8—22no—
331C4⋊Q8—21no—
332C4⋊Q8—21no—
333C4⋊Q8—21no—
334C4⋊Q8—21no—
335C4⋊Q8D4×C5311yescharacteristic
336#127D4×C5311yescharacteristic
337#126—22no—
338#126—22no—
339#126—21no—
340#125—21no—
341#125—22no—
342#125—21no—
343#125—21no—
344#125—21no—
345#125—21no—
346#125D4×C5311yescharacteristic
347#124—21no—
348#124D4×C5311yescharacteristic
349#123—22no—
350#123—22no—
351#123—22no—
352#123—22no—
353#123—22no—
354#123—21no—
355#123—22no—
356C22⋊Q8—22no—
357C22⋊Q8—22no—
358C22⋊Q8—22no—
359C22⋊Q8—22no—
360C22⋊Q8—22no—
361C22⋊Q8—22no—
362C22⋊Q8C23×C5311yescharacteristic
363#121—21no—
364#121—22no—
365#121—21no—
366#121—22no—
367#121—22no—
368#121—21no—
369#121—22no—
370#121C23×C5311yescharacteristic
371#121C23×C5312yes—
372C4×Q8D4×C5311yescharacteristic
373C4×D4—22no—
374C4×D4—21no—
375C4×D4—21no—
376C4×D4—21no—
377C4×D4—22no—
378C4×D4—21no—
379C4×D4D4×C5311yescharacteristic
380C4×D4D4×C5312yes—
381#117—21no—
382#117—21no—
383C2×C4⋊C4—22no—
384C2×C4⋊C4—21no—
385C2×C4⋊C4—21no—
386C2×C4⋊C4D4×C5311yescharacteristic
387C2×C4⋊C4D4×C5312yes—
388C2×C4⋊C4D4×C5311yescharacteristic
389C2×C4⋊C4C23×C5312yes—
390C2×C4⋊C4C23×C5312yes—
391C2×C42—21no—
392C2×C42C23×C5311yescharacteristic
393#107—22no—
394#106—22no—
395#106—24no—
396C4⋊C8—21no—
397C4⋊C8—21no—
398#103—22no—
399#102—22no—
400#102—24no—
401C22⋊C8C23×C5312yes—
402C22⋊C8C23×C5312yes—
403#97—21no—
404C4×C8—21no—
405#95—21no—
406#95—24no—
407#95—24no—
408#95C23×C5312yes—
409#41—82no—
410#41—84no—
411C2×Q8—22no—
412C2×Q8—21no—
413C2×Q8—44no—
414C2×Q8—41no—
415C2×Q8—41no—
416C2×Q8C2×D4×C5311yescharacteristic
417C2×D4—21no—
418C2×D4—22no—
419C2×D4—22no—
420C2×D4—22no—
421C2×D4—21no—
422C2×D4—21no—
423C2×D4C2×D4×C5311yescharacteristic
424C2×D4C2×D4×C5312yes—
425C22×C4—22no—
426C22×C4—22no—
427C22×C4—22no—
428C22×C4—21no—
429C22×C4—22no—
430C22×C4—22no—
431C22×C4—22no—
432C22×C4C2×D4×C5312yes—
433C22×C4C2×D4×C5311yescharacteristic
434C22×C4C2×D4×C5311yescharacteristic
435C22×C4C2×D4×C5312yes—
436C22×C4C24×C5311yescharacteristic
437C2×C8—22no—
438C2×C8—22no—
439C4⋊C4—22no—
440C4⋊C4—22no—
441C4⋊C4—22no—
442C4⋊C4—22no—
443C4⋊C4—21no—
444C4⋊C4—21no—
445C4⋊C4—22no—
446C4⋊C4—21no—
447C4⋊C4—21no—
448C4⋊C4—22no—
449C4⋊C4—21no—
450C4⋊C4—21no—
451C4⋊C4—21no—
452C4⋊C4—21no—
453C4⋊C4—22no—
454C4⋊C4—22no—
455C4⋊C4—21no—
456C4⋊C4—22no—
457C4⋊C4—22no—
458C4⋊C4—41no—
459C4⋊C4—41no—
460C4⋊C4C2×D4×C5312yes—
461C4⋊C4C2×D4×C5311yescharacteristic
462C4⋊C4C2×D4×C5311yescharacteristic
463C22⋊C4—22no—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—22no—
467C22⋊C4—22no—
468C22⋊C4—21no—
469C22⋊C4—21no—
470C22⋊C4—21no—
471C22⋊C4—21no—
472C22⋊C4—22no—
473C22⋊C4—21no—
474C22⋊C4—22no—
475C42—22no—
476C42—21no—
477C42—42no—
478C42C2×D4×C5311yescharacteristic
479C42C2×D4×C5311yescharacteristic
480C42C2×D4×C5311yescharacteristic
481C42C2×D4×C5311yescharacteristic
482C23—21no—
483C23—22no—
484C23C22×D4×C5311yescharacteristic
485Q8—41no—
486Q8—81no—
487Q8—82no—
488Q8—82no—
489Q8—82no—
490D4—42no—
491D4—41no—
492D4—44no—
493D4—42no—
494D4—42no—
495D4—81no—
496D4—82no—
497D4—82no—
498C2×C4—22no—
499C2×C4—22no—
500C2×C4—21no—
501C2×C4—21no—
502C2×C4—22no—
503C2×C4—22no—
504C2×C4—21no—
505C2×C4—22no—
506C2×C4—24no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—42no—
510C2×C4—42no—
511C2×C4—42no—
512C2×C4—44no—
513C2×C4—42no—
514C2×C4—42no—
515C2×C4—82no—
516C2×C4—82no—
517C2×C4—82no—
518C2×C4—82no—
519C2×C4#120×C5311yescharacteristic
520C2×C4#120×C5311yescharacteristic
521C2×C4#120×C5311yescharacteristic
522C2×C4#120×C5311yescharacteristic
523C2×C4C22×D4×C5311yescharacteristic
524C2×C4C22×D4×C5311yescharacteristic
525C8—42no—
526C8—42no—
527C22—21no—
528C22—21no—
529C22—42no—
530C22—41no—
531C22—41no—
532C22—42no—
533C22—42no—
534C22—41no—
535C22C2×#120×C5311yescharacteristic
536C22C53×#57712yes—
537C4—21no—
538C4—21no—
539C4—21no—
540C4—21no—
541C4—21no—
542C4—42no—
543C4—42no—
544C4—41no—
545C4—41no—
546C4—42no—
547C4—41no—
548C4—82no—
549C4—82no—
550C4C53×#45412yes—
551C2—22no—
552C2—81no—
553C2—82no—
554C2C53×#300911yescharacteristic
555C2C2×C53×#45411yescharacteristic
556C2C53×#302811yescharacteristic
subgroupG/H
center, socleC22×C53C2×#120
commutator, Frattini, cosocleC22×C4C24×C53
prime radicalC53×#3559C2
LCSG ▷ C22×C4 ▷ 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×#3239C2semidirect
3. C53×#3227C2semidirect
4. C53×#2642C2semidirect
5. C53×#1616C2semidirect
6. C53×#1615C2semidirect
7. C53×#1505C2semidirect
8. C53×#1500C2semidirect
9. C53×#1470C2semidirect
248. #18441C53direct (N × H)
274. #3239C2×C53semidirect
275. #3227C2×C53semidirect
276. #2642C2×C53semidirect
277. #1616C2×C53semidirect
278. #1615C2×C53semidirect
279. #1505C2×C53semidirect
280. #1500C2×C53semidirect
281. #1470C2×C53semidirect
325. C53#18441direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#35592
C53×#32392
C53×#32272
C53×#26422
C53×#16162
C53×#16152
C53×#15052
C53×#15002
C53×#14702
#1844153