Finite Group Databasebeta

← C53×#19050 #27226562

Subgroups — C53×#19050 #27226562

Order 13568.

1026 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3545C211yesmaximal, characteristic, complemented
2C53×#3217C212yesmaximal, complemented
3C2×C53×#496C211yesmaximal, characteristic, complemented
4C53×#1657C212yesmaximal, complemented
5C53×#1656C214yesmaximal, complemented
6C53×#1655C212yesmaximal, complemented
7C53×#1533C211yesmaximal, characteristic, complemented
8C53×#1532C212yesmaximal, complemented
9C22×SD16×C53—21no—
10C22×D8×C53—21no—
11C53×#563C2211yescharacteristic, complemented
12C53×#562C2212yescomplemented
13C53×H(3,𝔽4)C2211yescharacteristic, complemented
14C53×#555—21no—
15C53×#551—21no—
16C53×#551—22no—
17C53×#546—22no—
18C53×#542—22no—
19C53×#538—22no—
20C53×#536—21no—
21C53×#534—21no—
22C2×#127×C53C2211yescharacteristic, complemented
23C53×#529C2212yes—
24C2×#124×C53C2211yescharacteristic
25C53×#496—21no—
26C53×#496—21no—
27C53×#496C2212yes—
28C53×#496C2214yescomplemented
29C53×#496C2212yescomplemented
30C53×#490C2214yescomplemented
31C53×#460—22no—
32C53×#459—22no—
33C53×#450—22no—
34C53×#447—22no—
35C53×#432C2212yescomplemented
36C2×#97×C53C2211yescharacteristic, complemented
37C53×#330C2214yescomplemented
38C53×#329C2214yescomplemented
39C53×#329C2212yescomplemented
40C53×#329C2212yescomplemented
41C2×#41×C53—21no—
42C22×Q8×C53D411yescharacteristic
43C22×D4×C53—21no—
44C22×D4×C53—21no—
45C22×D4×C53—22no—
46C22×D4×C53D411yescharacteristic
47C2×SD16×C53—22no—
48C2×SD16×C53—22no—
49C2×SD16×C53—24no—
50C2×SD16×C53—44no—
51C2×D8×C53—22no—
52C2×D8×C53—22no—
53C2×D8×C53—24no—
54C2×D8×C53—44no—
55C22×C8×C53—21no—
56C4⋊Q8×C53—21no—
57C4⋊Q8×C53—21no—
58#127×C53—21no—
59#127×C53—22no—
60#127×C53C2312yes—
61#126×C53—22no—
62#126×C53—22no—
63#126×C53—22no—
64#125×C53C2312yes—
65#124×C53—21no—
66#124×C53—22no—
67#124×C53—22no—
68#124×C53—21no—
69#124×C53—22no—
70#124×C53D411yescharacteristic
71#124×C53D411yescharacteristic
72#124×C53C2311yescharacteristic
73#124×C53C2311yescharacteristic
74#124×C53C2312yes—
75#123×C53—21no—
76#123×C53—22no—
77#123×C53—22no—
78#123×C53—22no—
79#123×C53—22no—
80#123×C53—21no—
81#123×C53—22no—
82C22⋊Q8×C53—22no—
83C22⋊Q8×C53—22no—
84C22⋊Q8×C53—22no—
85C22⋊Q8×C53—22no—
86C22⋊Q8×C53D412yes—
87#121×C53—22no—
88#121×C53—22no—
89#121×C53—22no—
90#121×C53—22no—
91#121×C53D412yes—
92#120×C53—21no—
93#120×C53—22no—
94#120×C53—21no—
95#120×C53—22no—
96C4×Q8×C53D412yes—
97C4×D4×C53—22no—
98C4×D4×C53—22no—
99C4×D4×C53—22no—
100C4×D4×C53D412yes—
101#117×C53—21no—
102#117×C53—22no—
103#117×C53—22no—
104C2×C22⋊C4×C53—21no—
105C2×C22⋊C4×C53—22no—
106C2×C22⋊C4×C53—22no—
107C2×C42×C53C2311yescharacteristic
108C4⋊C8×C53—22no—
109#102×C53—24no—
110#102×C53—24no—
111C22⋊C8×C53—22no—
112#97×C53—21no—
113#97×C53C2314yes—
114#97×C53C2312yes—
115C24×C53—21no—
116C24×C53—21no—
117C24×C53—41no—
118#41×C53—82no—
119C2×Q8×C53—21no—
120C2×Q8×C53—21no—
121C2×Q8×C53—42no—
122C2×Q8×C53C2×D412yes—
123C2×Q8×C53C2×D411yescharacteristic
124C2×Q8×C53C2×D411yescharacteristic
125C2×D4×C53—22no—
126C2×D4×C53—22no—
127C2×D4×C53—22no—
128C2×D4×C53—21no—
129C2×D4×C53—22no—
130C2×D4×C53—21no—
131C2×D4×C53—24no—
132C2×D4×C53—22no—
133C2×D4×C53—42no—
134C2×D4×C53—42no—
135C2×D4×C53—42no—
136C2×D4×C53—42no—
137C2×D4×C53—41no—
138C2×D4×C53—84no—
139C2×D4×C53—82no—
140C2×D4×C53C2×D412yes—
141C2×D4×C53C2×D411yescharacteristic
142C2×D4×C53C2×D411yescharacteristic
143C22×C4×C53—21no—
144C22×C4×C53—22no—
145C22×C4×C53—22no—
146C22×C4×C53—21no—
147C22×C4×C53—22no—
148C22×C4×C53C2×D411yescharacteristic
149C22×C4×C53C2×D412yes—
150SD16×C53—44no—
151SD16×C53—44no—
152SD16×C53—84no—
153D8×C53—44no—
154D8×C53—44no—
155D8×C53—84no—
156C2×C8×C53—24no—
157C2×C8×C53—22no—
158C2×C8×C53—24no—
159C4⋊C4×C53—22no—
160C4⋊C4×C53—22no—
161C4⋊C4×C53—22no—
162C4⋊C4×C53—22no—
163C4⋊C4×C53—22no—
164C4⋊C4×C53—22no—
165C4⋊C4×C53—22no—
166C4⋊C4×C53C2×D412yes—
167C4⋊C4×C53C2×D412yes—
168C22⋊C4×C53—22no—
169C22⋊C4×C53—21no—
170C22⋊C4×C53—21no—
171C22⋊C4×C53—22no—
172C22⋊C4×C53—22no—
173C22⋊C4×C53—22no—
174C22⋊C4×C53—22no—
175C22⋊C4×C53—22no—
176C22⋊C4×C53—22no—
177C22⋊C4×C53—22no—
178C22⋊C4×C53—22no—
179C22⋊C4×C53—22no—
180C22⋊C4×C53—22no—
181C22⋊C4×C53—22no—
182C22⋊C4×C53—22no—
183C42×C53—21no—
184C42×C53—21no—
185C42×C53—22no—
186C42×C53C2×D411yescharacteristic
187C42×C53C2×D412yes—
188C42×C53C2411yescharacteristic
189C23×C53—22no—
190C23×C53—21no—
191C23×C53—21no—
192C23×C53—42no—
193C23×C53—42no—
194C23×C53—82no—
195C23×C53—84no—
196C23×C53—81no—
197C23×C53—82no—
198C23×C53—82no—
199C23×C53#12011yescharacteristic
200Q8×C53—41no—
201Q8×C53—41no—
202Q8×C53—42no—
203Q8×C53—81no—
204Q8×C53—81no—
205D4×C53—41no—
206D4×C53—41no—
207D4×C53—42no—
208D4×C53—42no—
209D4×C53—42no—
210D4×C53—44no—
211D4×C53—44no—
212D4×C53—44no—
213D4×C53—81no—
214D4×C53—82no—
215D4×C53—84no—
216D4×C53—82no—
217D4×C53—82no—
218D4×C53—82no—
219D4×C53—81no—
220C2×C4×C53—22no—
221C2×C4×C53—21no—
222C2×C4×C53—22no—
223C2×C4×C53—22no—
224C2×C4×C53—21no—
225C2×C4×C53—22no—
226C2×C4×C53—42no—
227C2×C4×C53—42no—
228C2×C4×C53—82no—
229C2×C4×C53—82no—
230C2×C4×C53—81no—
231C2×C4×C53—82no—
232C2×C4×C53#12012yes—
233C2×C4×C53#12011yescharacteristic
234C2×C4×C53C22×D411yescharacteristic
235C2×C4×C53C22×D412yes—
236C8×C53—24no—
237C8×C53—44no—
238#19050C5311yesmaximal, characteristic, direct product factor, Sylow
239C22×C53—22no—
240C22×C53—42no—
241C22×C53—41no—
242C22×C53—41no—
243C22×C53—42no—
244C22×C53—42no—
245C22×C53—42no—
246C22×C53—42no—
247C22×C53—82no—
248C22×C53—84no—
249C22×C53—162no—
250C22×C53—164no—
251C22×C53—162no—
252C22×C53#45312yes—
253C22×C53C2×#12011yescharacteristic
254C4×C53—21no—
255C4×C53—21no—
256C4×C53—22no—
257C4×C53—22no—
258C4×C53—41no—
259C4×C53—41no—
260C4×C53—42no—
261C4×C53—42no—
262C4×C53—42no—
263#3545C2×C5311yescharacteristic, complemented
264#3217C2×C5312yescomplemented
265C2×#496C2×C5311yescharacteristic, complemented
266#1657C2×C5312yescomplemented
267#1656C2×C5314yescomplemented
268#1655C2×C5312yescomplemented
269#1533C2×C5311yescharacteristic, complemented
270#1532C2×C5312yescomplemented
271C2×C53—22no—
272C2×C53—82no—
273C2×C53—81no—
274C2×C53—81no—
275C2×C53—162no—
276C2×C53C2×#45311yescharacteristic
277C2×C53#302112yes—
278C22×SD16—21no—
279C22×D8—21no—
280#563C22×C5311yescharacteristic, complemented
281#562C22×C5312yescomplemented
282H(3,𝔽4)C22×C5311yescharacteristic, complemented
283#555—21no—
284#551—21no—
285#551—22no—
286#546—22no—
287#542—22no—
288#538—22no—
289#536—21no—
290#534—21no—
291C2×#127C22×C5311yescharacteristic, complemented
292#529C22×C5312yes—
293C2×#124C22×C5311yescharacteristic
294#496—21no—
295#496—21no—
296#496C22×C5312yes—
297#496C22×C5312yescomplemented
298#496C22×C5314yescomplemented
299#490C22×C5314yescomplemented
300#460—22no—
301#459—22no—
302#450—22no—
303#447—22no—
304#432C22×C5312yescomplemented
305C2×#97C22×C5311yescharacteristic, complemented
306#330C22×C5314yescomplemented
307#329C22×C5314yescomplemented
308#329C22×C5312yescomplemented
309#329C22×C5312yescomplemented
310C53#1905011yescharacteristic, direct product factor, Sylow
311C2×#41—21no—
312C22×Q8D4×C5311yescharacteristic
313C22×D4—21no—
314C22×D4—21no—
315C22×D4—22no—
316C22×D4D4×C5311yescharacteristic
317C2×SD16—22no—
318C2×SD16—22no—
319C2×SD16—24no—
320C2×SD16—44no—
321C2×D8—22no—
322C2×D8—22no—
323C2×D8—24no—
324C2×D8—44no—
325C22×C8—21no—
326C4⋊Q8—21no—
327C4⋊Q8—21no—
328#127—22no—
329#127—21no—
330#127C23×C5312yes—
331#126—22no—
332#126—22no—
333#126—22no—
334#125C23×C5312yes—
335#124—21no—
336#124—22no—
337#124—21no—
338#124—22no—
339#124—22no—
340#124D4×C5311yescharacteristic
341#124D4×C5311yescharacteristic
342#124C23×C5311yescharacteristic
343#124C23×C5311yescharacteristic
344#124C23×C5312yes—
345#123—21no—
346#123—22no—
347#123—22no—
348#123—21no—
349#123—22no—
350#123—22no—
351#123—22no—
352C22⋊Q8—22no—
353C22⋊Q8—22no—
354C22⋊Q8—22no—
355C22⋊Q8—22no—
356C22⋊Q8D4×C5312yes—
357#121—22no—
358#121—22no—
359#121—22no—
360#121—22no—
361#121D4×C5312yes—
362#120—21no—
363#120—22no—
364#120—21no—
365#120—22no—
366C4×Q8D4×C5312yes—
367C4×D4—22no—
368C4×D4—22no—
369C4×D4—22no—
370C4×D4D4×C5312yes—
371#117—21no—
372#117—22no—
373#117—22no—
374C2×C22⋊C4—21no—
375C2×C22⋊C4—22no—
376C2×C22⋊C4—22no—
377C2×C42C23×C5311yescharacteristic
378C4⋊C8—22no—
379#102—24no—
380#102—24no—
381C22⋊C8—22no—
382#97—21no—
383#97C23×C5314yes—
384#97C23×C5312yes—
385C24—21no—
386C24—21no—
387C24—41no—
388#41—82no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—42no—
392C2×Q8C2×D4×C5312yes—
393C2×Q8C2×D4×C5311yescharacteristic
394C2×Q8C2×D4×C5311yescharacteristic
395C2×D4—22no—
396C2×D4—22no—
397C2×D4—22no—
398C2×D4—21no—
399C2×D4—22no—
400C2×D4—21no—
401C2×D4—24no—
402C2×D4—22no—
403C2×D4—42no—
404C2×D4—42no—
405C2×D4—42no—
406C2×D4—42no—
407C2×D4—41no—
408C2×D4—84no—
409C2×D4—82no—
410C2×D4C2×D4×C5312yes—
411C2×D4C2×D4×C5311yescharacteristic
412C2×D4C2×D4×C5311yescharacteristic
413C22×C4—21no—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—21no—
417C22×C4—22no—
418C22×C4C2×D4×C5311yescharacteristic
419C22×C4C2×D4×C5312yes—
420SD16—44no—
421SD16—44no—
422SD16—84no—
423D8—44no—
424D8—44no—
425D8—84no—
426C2×C8—24no—
427C2×C8—22no—
428C2×C8—24no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4—22no—
432C4⋊C4—22no—
433C4⋊C4—22no—
434C4⋊C4—22no—
435C4⋊C4—22no—
436C4⋊C4C2×D4×C5312yes—
437C4⋊C4C2×D4×C5312yes—
438C22⋊C4—22no—
439C22⋊C4—21no—
440C22⋊C4—21no—
441C22⋊C4—22no—
442C22⋊C4—22no—
443C22⋊C4—22no—
444C22⋊C4—22no—
445C22⋊C4—22no—
446C22⋊C4—22no—
447C22⋊C4—22no—
448C22⋊C4—22no—
449C22⋊C4—22no—
450C22⋊C4—22no—
451C22⋊C4—22no—
452C22⋊C4—22no—
453C42—21no—
454C42—21no—
455C42—22no—
456C42C2×D4×C5311yescharacteristic
457C42C2×D4×C5312yes—
458C42C24×C5311yescharacteristic
459C23—21no—
460C23—22no—
461C23—21no—
462C23—42no—
463C23—42no—
464C23—82no—
465C23—84no—
466C23—81no—
467C23—82no—
468C23—82no—
469C23#120×C5311yescharacteristic
470Q8—41no—
471Q8—41no—
472Q8—42no—
473Q8—81no—
474Q8—81no—
475D4—41no—
476D4—41no—
477D4—42no—
478D4—42no—
479D4—44no—
480D4—42no—
481D4—44no—
482D4—44no—
483D4—81no—
484D4—82no—
485D4—84no—
486D4—82no—
487D4—82no—
488D4—82no—
489D4—81no—
490C2×C4—22no—
491C2×C4—22no—
492C2×C4—21no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—21no—
496C2×C4—42no—
497C2×C4—42no—
498C2×C4—82no—
499C2×C4—82no—
500C2×C4—81no—
501C2×C4—82no—
502C2×C4#120×C5312yes—
503C2×C4#120×C5311yescharacteristic
504C2×C4C22×D4×C5312yes—
505C2×C4C22×D4×C5311yescharacteristic
506C8—24no—
507C8—44no—
508C22—22no—
509C22—42no—
510C22—41no—
511C22—41no—
512C22—42no—
513C22—42no—
514C22—42no—
515C22—42no—
516C22—82no—
517C22—84no—
518C22—162no—
519C22—164no—
520C22—162no—
521C22C53×#45312yes—
522C22C2×#120×C5311yescharacteristic
523C4—21no—
524C4—21no—
525C4—22no—
526C4—22no—
527C4—41no—
528C4—41no—
529C4—42no—
530C4—42no—
531C4—42no—
532C2—22no—
533C2—81no—
534C2—82no—
535C2—81no—
536C2—162no—
537C2C2×C53×#45311yescharacteristic
538C2C53×#302112yes—
subgroupG/H
center, socleC22×C53C2×#120
commutator, Frattini, cosocleC42C24×C53
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
1. C53×#3545C2semidirect
2. C53×#3217C2semidirect
3. C2×C53×#496C2semidirect
4. C53×#1657C2semidirect
5. C53×#1656C2semidirect
6. C53×#1655C2semidirect
7. C53×#1533C2semidirect
8. C53×#1532C2semidirect
11. C53×#563C22semidirect
12. C53×#562C22semidirect
13. C53×H(3,𝔽4)C22semidirect
22. C2×#127×C53C22semidirect
28. C53×#496C22semidirect
29. C53×#496C22semidirect
30. C53×#490C22semidirect
35. C53×#432C22semidirect
36. C2×#97×C53C22semidirect
37. C53×#330C22semidirect
38. C53×#329C22semidirect
39. C53×#329C22semidirect
40. C53×#329C22semidirect
238. #19050C53direct (N × H)
263. #3545C2×C53semidirect
264. #3217C2×C53semidirect
265. C2×#496C2×C53semidirect
266. #1657C2×C53semidirect
267. #1656C2×C53semidirect
268. #1655C2×C53semidirect
269. #1533C2×C53semidirect
270. #1532C2×C53semidirect
280. #563C22×C53semidirect
281. #562C22×C53semidirect
282. H(3,𝔽4)C22×C53semidirect
291. C2×#127C22×C53semidirect
297. #496C22×C53semidirect
298. #496C22×C53semidirect
299. #490C22×C53semidirect
304. #432C22×C53semidirect
305. C2×#97C22×C53semidirect
306. #330C22×C53semidirect
307. #329C22×C53semidirect
308. #329C22×C53semidirect
309. #329C22×C53semidirect
310. C53#19050direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#35452
C53×#32172
C2×C53×#4962
C53×#16572
C53×#16562
C53×#16552
C53×#15332
C53×#15322
#1905053