Finite Group Databasebeta

← C53×#33559 #27241071

Subgroups — C53×#33559 #27241071

Order 13568.

1248 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×#3549C211yesmaximal, characteristic, complemented
2C53×#3027C212yesmaximal, complemented
3C53×#3018C211yesmaximal, characteristic, complemented
4C53×#2199C214yesmaximal, complemented
5C53×#2196C214yesmaximal, complemented
6C53×#2126C211yesmaximal, characteristic, complemented
7C53×#2122C212yesmaximal, complemented
8C53×#585C2211yescharacteristic, complemented
9C53×#575—21no—
10C53×H(3,𝔽4)—21no—
11C53×H(3,𝔽4)C2212yescomplemented
12C53×#555—21no—
13C53×#555—22no—
14D42×C53—22no—
15D42×C53C2212yescomplemented
16C53×#535—21no—
17C53×#535—22no—
18C53×#535—24no—
19C53×#535C2212yes—
20C53×#532—21no—
21C53×#532C2211yescharacteristic
22C53×B(4,𝔽2)—24no—
23C53×B(4,𝔽2)C2218yescomplemented
24C53×#456—21no—
25C53×#455C2212yescomplemented
26C53×#454—21no—
27C53×#453C2212yes—
28C2×#104×C53—21no—
29C53×#413C2211yescharacteristic, complemented
30C53×#410—22no—
31C53×#410C2212yescomplemented
32C53×#354C2212yescomplemented
33C53×#353C2212yescomplemented
34C53×AΣL(1,𝔽16)C2218yescomplemented
352−1+4×C53—21no—
362−1+4×C53—41no—
37H(5,𝔽2)×C53—22no—
38H(5,𝔽2)×C53—41no—
39H(5,𝔽2)×C53C2312yes—
40C2×#41×C53—22no—
41C2×#41×C53—22no—
42C2×#41×C53—21no—
43C2×#41×C53—42no—
44C2×#41×C53D411yescharacteristic, complemented
45C2×#41×C53C2311yescharacteristic
46C22×D4×C53—21no—
47C22×D4×C53—24no—
48C22×D4×C53D412yescomplemented
49#137×C53—22no—
50#137×C53—41no—
51AGL(1,ℤ/8)×C53—22no—
52AGL(1,ℤ/8)×C53—41no—
53#135×C53—42no—
54C2×M16×C53—21no—
55C4⋊Q8×C53—21no—
56C4⋊Q8×C53D411yescharacteristic, complemented
57#127×C53—21no—
58#127×C53—22no—
59#127×C53—22no—
60#127×C53—42no—
61#127×C53C2311yescharacteristic
62#124×C53—24no—
63#124×C53—24no—
64#124×C53—22no—
65#124×C53—22no—
66#124×C53—22no—
67#124×C53—42no—
68#124×C53D411yescharacteristic, complemented
69#124×C53C2311yescharacteristic
70#121×C53—22no—
71#121×C53—24no—
72#121×C53—22no—
73#121×C53—24no—
74#121×C53—22no—
75#121×C53—22no—
76#121×C53—22no—
77#121×C53—44no—
78#120×C53—22no—
79#120×C53—24no—
80#120×C53—24no—
81#120×C53—22no—
82#120×C53—28no—
83#120×C53D414yescomplemented
84#120×C53C2314yes—
85C4×D4×C53—22no—
86C4×D4×C53—22no—
87C4×D4×C53—22no—
88#117×C53—24no—
89#117×C53—21no—
90#117×C53—22no—
91#117×C53D412yescomplemented
92C2×C42×C53D411yescharacteristic, complemented
93#104×C53—22no—
94#104×C53—22no—
95#101×C53—21no—
96#100×C53C2312yes—
97#99×C53—28no—
98#99×C53—22no—
99#99×C53C2314yes—
100C24×C53—22no—
101C24×C53C2×D414yes—
102#41×C53—22no—
103#41×C53—22no—
104#41×C53—41no—
105#41×C53—41no—
106#41×C53—42no—
107#41×C53—42no—
108#41×C53—42no—
109#41×C53—41no—
110#41×C53—44no—
111#41×C53—41no—
112#41×C53—41no—
113#41×C53—42no—
114#41×C53—42no—
115#41×C53—41no—
116#41×C53—84no—
117C2×Q8×C53—22no—
118C2×Q8×C53—22no—
119C2×Q8×C53—21no—
120C2×Q8×C53—42no—
121C2×Q8×C53C2×D411yescharacteristic
122C2×Q8×C53C2×D411yescharacteristic
123C2×D4×C53—21no—
124C2×D4×C53—22no—
125C2×D4×C53—22no—
126C2×D4×C53—24no—
127C2×D4×C53—22no—
128C2×D4×C53—22no—
129C2×D4×C53—24no—
130C2×D4×C53—22no—
131C2×D4×C53—21no—
132C2×D4×C53—24no—
133C2×D4×C53—24no—
134C2×D4×C53—22no—
135C2×D4×C53—22no—
136C2×D4×C53—42no—
137C2×D4×C53—42no—
138C2×D4×C53—42no—
139C2×D4×C53—44no—
140C2×D4×C53—41no—
141C2×D4×C53—48no—
142C2×D4×C53—88no—
143C2×D4×C53—81no—
144C2×D4×C53—81no—
145C2×D4×C53C2×D412yes—
146C2×D4×C53C2×D411yescharacteristic
147C2×D4×C53C2411yescharacteristic
148C22×C4×C53—24no—
149C22×C4×C53—24no—
150C22×C4×C53—21no—
151C22×C4×C53—21no—
152C22×C4×C53—41no—
153C22×C4×C53C2×D412yes—
154C22×C4×C53C2×D411yescharacteristic
155Q16×C53—42no—
156SD16×C53—42no—
157SD16×C53—42no—
158D8×C53—42no—
159M16×C53—22no—
160M16×C53—41no—
161C2×C8×C53—41no—
162C4⋊C4×C53—22no—
163C4⋊C4×C53—22no—
164C4⋊C4×C53—22no—
165C4⋊C4×C53—42no—
166C22⋊C4×C53—22no—
167C22⋊C4×C53—24no—
168C22⋊C4×C53—24no—
169C22⋊C4×C53—22no—
170C22⋊C4×C53—24no—
171C22⋊C4×C53—22no—
172C22⋊C4×C53—24no—
173C22⋊C4×C53—24no—
174C22⋊C4×C53—48no—
175C22⋊C4×C53C2×D414yes—
176C42×C53—22no—
177C42×C53—22no—
178C42×C53—21no—
179C42×C53—42no—
180C42×C53C2×D411yescharacteristic
181C42×C53C2×D411yescharacteristic
182C23×C53—24no—
183C23×C53—24no—
184C23×C53—21no—
185C23×C53—41no—
186C23×C53—44no—
187C23×C53—84no—
188C23×C53—84no—
189C23×C53—82no—
190C23×C53#12011yescharacteristic
191C23×C53C22×D412yes—
192Q8×C53—22no—
193Q8×C53—41no—
194Q8×C53—42no—
195Q8×C53—41no—
196Q8×C53—41no—
197Q8×C53—82no—
198D4×C53—22no—
199D4×C53—42no—
200D4×C53—41no—
201D4×C53—41no—
202D4×C53—42no—
203D4×C53—44no—
204D4×C53—44no—
205D4×C53—44no—
206D4×C53—42no—
207D4×C53—42no—
208D4×C53—41no—
209D4×C53—42no—
210D4×C53—42no—
211D4×C53—82no—
212D4×C53—81no—
213D4×C53—84no—
214D4×C53—84no—
215D4×C53—82no—
216D4×C53—84no—
217D4×C53—82no—
218D4×C53—168no—
219C2×C4×C53—21no—
220C2×C4×C53—22no—
221C2×C4×C53—21no—
222C2×C4×C53—22no—
223C2×C4×C53—21no—
224C2×C4×C53—24no—
225C2×C4×C53—24no—
226C2×C4×C53—21no—
227C2×C4×C53—21no—
228C2×C4×C53—44no—
229C2×C4×C53—48no—
230C2×C4×C53—42no—
231C2×C4×C53—42no—
232C2×C4×C53—41no—
233C2×C4×C53—42no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—44no—
237C2×C4×C53#12012yes—
238C2×C4×C53#12011yescharacteristic
239C2×C4×C53C22×D411yescharacteristic
240C8×C53—42no—
241#33559C5311yesmaximal, characteristic, direct product factor, Sylow
242C22×C53—22no—
243C22×C53—21no—
244C22×C53—41no—
245C22×C53—42no—
246C22×C53—44no—
247C22×C53—44no—
248C22×C53—42no—
249C22×C53—42no—
250C22×C53—41no—
251C22×C53—44no—
252C22×C53—84no—
253C22×C53—88no—
254C22×C53—161no—
255C22×C53—164no—
256C22×C53C2×#12011yescharacteristic
257C4×C53—22no—
258C4×C53—21no—
259C4×C53—22no—
260C4×C53—44no—
261C4×C53—41no—
262C4×C53—42no—
263C4×C53—41no—
264C4×C53—84no—
265C4×C53B(4,𝔽2)11yescharacteristic
266C4×C53B(4,𝔽2)11yescharacteristic
267#3549C2×C5311yescharacteristic, complemented
268#3027C2×C5312yescomplemented
269#3018C2×C5311yescharacteristic, complemented
270#2199C2×C5314yescomplemented
271#2196C2×C5314yescomplemented
272#2126C2×C5311yescharacteristic, complemented
273#2122C2×C5312yescomplemented
274C2×C53—21no—
275C2×C53—42no—
276C2×C53—41no—
277C2×C53—84no—
278C2×C53—81no—
279C2×C53—82no—
280C2×C53C2×B(4,𝔽2)11yescharacteristic
281#585C22×C5311yescharacteristic, complemented
282#575—21no—
283H(3,𝔽4)—21no—
284H(3,𝔽4)C22×C5312yescomplemented
285#555—21no—
286#555—22no—
287D42—22no—
288D42C22×C5312yescomplemented
289#535—21no—
290#535—22no—
291#535—24no—
292#535C22×C5312yes—
293#532—21no—
294#532C22×C5311yescharacteristic
295B(4,𝔽2)—24no—
296B(4,𝔽2)C22×C5318yescomplemented
297#456—21no—
298#455C22×C5312yescomplemented
299#454—21no—
300#453C22×C5312yes—
301C2×#104—21no—
302#413C22×C5311yescharacteristic, complemented
303#410—22no—
304#410C22×C5312yescomplemented
305#354C22×C5312yescomplemented
306#353C22×C5312yescomplemented
307AΣL(1,𝔽16)C22×C5318yescomplemented
308C53#3355911yescharacteristic, direct product factor, Sylow
3092−1+4—21no—
3102−1+4—41no—
311H(5,𝔽2)—22no—
312H(5,𝔽2)—41no—
313H(5,𝔽2)C23×C5312yes—
314C2×#41—22no—
315C2×#41—22no—
316C2×#41—21no—
317C2×#41—42no—
318C2×#41D4×C5311yescharacteristic, complemented
319C2×#41C23×C5311yescharacteristic
320C22×D4—21no—
321C22×D4—24no—
322C22×D4D4×C5312yescomplemented
323#137—22no—
324#137—41no—
325AGL(1,ℤ/8)—22no—
326AGL(1,ℤ/8)—41no—
327#135—42no—
328C2×M16—21no—
329C4⋊Q8—21no—
330C4⋊Q8D4×C5311yescharacteristic, complemented
331#127—21no—
332#127—22no—
333#127—22no—
334#127—42no—
335#127C23×C5311yescharacteristic
336#124—24no—
337#124—22no—
338#124—22no—
339#124—24no—
340#124—22no—
341#124—42no—
342#124D4×C5311yescharacteristic, complemented
343#124C23×C5311yescharacteristic
344#121—22no—
345#121—22no—
346#121—24no—
347#121—22no—
348#121—22no—
349#121—22no—
350#121—24no—
351#121—44no—
352#120—22no—
353#120—24no—
354#120—24no—
355#120—22no—
356#120—28no—
357#120D4×C5314yescomplemented
358#120C23×C5314yes—
359C4×D4—22no—
360C4×D4—22no—
361C4×D4—22no—
362#117—21no—
363#117—24no—
364#117—22no—
365#117D4×C5312yescomplemented
366C2×C42D4×C5311yescharacteristic, complemented
367#104—22no—
368#104—22no—
369#101—21no—
370#100C23×C5312yes—
371#99—22no—
372#99—28no—
373#99C23×C5314yes—
374C24—22no—
375C24C2×D4×C5314yes—
376#41—22no—
377#41—22no—
378#41—41no—
379#41—41no—
380#41—42no—
381#41—42no—
382#41—42no—
383#41—41no—
384#41—44no—
385#41—41no—
386#41—41no—
387#41—42no—
388#41—41no—
389#41—42no—
390#41—84no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—21no—
394C2×Q8—42no—
395C2×Q8C2×D4×C5311yescharacteristic
396C2×Q8C2×D4×C5311yescharacteristic
397C2×D4—21no—
398C2×D4—22no—
399C2×D4—22no—
400C2×D4—24no—
401C2×D4—22no—
402C2×D4—22no—
403C2×D4—22no—
404C2×D4—24no—
405C2×D4—21no—
406C2×D4—24no—
407C2×D4—24no—
408C2×D4—22no—
409C2×D4—22no—
410C2×D4—42no—
411C2×D4—42no—
412C2×D4—42no—
413C2×D4—44no—
414C2×D4—41no—
415C2×D4—48no—
416C2×D4—88no—
417C2×D4—81no—
418C2×D4—81no—
419C2×D4C2×D4×C5312yes—
420C2×D4C2×D4×C5311yescharacteristic
421C2×D4C24×C5311yescharacteristic
422C22×C4—24no—
423C22×C4—24no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—41no—
427C22×C4C2×D4×C5312yes—
428C22×C4C2×D4×C5311yescharacteristic
429Q16—42no—
430SD16—42no—
431SD16—42no—
432D8—42no—
433M16—22no—
434M16—41no—
435C2×C8—41no—
436C4⋊C4—22no—
437C4⋊C4—22no—
438C4⋊C4—22no—
439C4⋊C4—42no—
440C22⋊C4—24no—
441C22⋊C4—22no—
442C22⋊C4—24no—
443C22⋊C4—22no—
444C22⋊C4—24no—
445C22⋊C4—22no—
446C22⋊C4—24no—
447C22⋊C4—24no—
448C22⋊C4—48no—
449C22⋊C4C2×D4×C5314yes—
450C42—22no—
451C42—21no—
452C42—22no—
453C42—42no—
454C42C2×D4×C5311yescharacteristic
455C42C2×D4×C5311yescharacteristic
456C23—24no—
457C23—24no—
458C23—21no—
459C23—41no—
460C23—44no—
461C23—84no—
462C23—84no—
463C23—82no—
464C23#120×C5311yescharacteristic
465C23C22×D4×C5312yes—
466Q8—22no—
467Q8—41no—
468Q8—41no—
469Q8—42no—
470Q8—41no—
471Q8—82no—
472D4—22no—
473D4—42no—
474D4—41no—
475D4—41no—
476D4—42no—
477D4—44no—
478D4—44no—
479D4—44no—
480D4—42no—
481D4—42no—
482D4—41no—
483D4—42no—
484D4—42no—
485D4—82no—
486D4—81no—
487D4—84no—
488D4—84no—
489D4—82no—
490D4—84no—
491D4—82no—
492D4—168no—
493C2×C4—21no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—21no—
497C2×C4—21no—
498C2×C4—24no—
499C2×C4—21no—
500C2×C4—24no—
501C2×C4—21no—
502C2×C4—44no—
503C2×C4—48no—
504C2×C4—42no—
505C2×C4—42no—
506C2×C4—41no—
507C2×C4—42no—
508C2×C4—41no—
509C2×C4—41no—
510C2×C4—44no—
511C2×C4#120×C5312yes—
512C2×C4#120×C5311yescharacteristic
513C2×C4C22×D4×C5311yescharacteristic
514C8—42no—
515C22—22no—
516C22—21no—
517C22—41no—
518C22—42no—
519C22—44no—
520C22—44no—
521C22—42no—
522C22—42no—
523C22—41no—
524C22—44no—
525C22—84no—
526C22—88no—
527C22—161no—
528C22—164no—
529C22C2×#120×C5311yescharacteristic
530C4—22no—
531C4—21no—
532C4—22no—
533C4—44no—
534C4—41no—
535C4—42no—
536C4—41no—
537C4—84no—
538C4C53×B(4,𝔽2)11yescharacteristic
539C4C53×B(4,𝔽2)11yescharacteristic
540C2—21no—
541C2—42no—
542C2—41no—
543C2—84no—
544C2—81no—
545C2—82no—
546C2C2×C53×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC2×C53C2×B(4,𝔽2)
commutator, Frattini, cosocleC2×D4C24×C53
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C53 ◁ C2×C4×C53 ◁ C2×#41×C53 ◁ G

Complements

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

NH ≅ G/Ntype
1. C53×#3549C2semidirect
2. C53×#3027C2semidirect
3. C53×#3018C2semidirect
4. C53×#2199C2semidirect
5. C53×#2196C2semidirect
6. C53×#2126C2semidirect
7. C53×#2122C2semidirect
8. C53×#585C22semidirect
11. C53×H(3,𝔽4)C22semidirect
15. D42×C53C22semidirect
23. C53×B(4,𝔽2)C22semidirect
25. C53×#455C22semidirect
29. C53×#413C22semidirect
31. C53×#410C22semidirect
32. C53×#354C22semidirect
33. C53×#353C22semidirect
34. C53×AΣL(1,𝔽16)C22semidirect
44. C2×#41×C53D4semidirect
48. C22×D4×C53D4semidirect
56. C4⋊Q8×C53D4semidirect
68. #124×C53D4semidirect
83. #120×C53D4semidirect
91. #117×C53D4semidirect
92. C2×C42×C53D4semidirect
241. #33559C53direct (N × H)
267. #3549C2×C53semidirect
268. #3027C2×C53semidirect
269. #3018C2×C53semidirect
270. #2199C2×C53semidirect
271. #2196C2×C53semidirect
272. #2126C2×C53semidirect
273. #2122C2×C53semidirect
281. #585C22×C53semidirect
284. H(3,𝔽4)C22×C53semidirect
288. D42C22×C53semidirect
296. B(4,𝔽2)C22×C53semidirect
298. #455C22×C53semidirect
302. #413C22×C53semidirect
304. #410C22×C53semidirect
305. #354C22×C53semidirect
306. #353C22×C53semidirect
307. AΣL(1,𝔽16)C22×C53semidirect
308. C53#33559direct (N × H)
318. C2×#41D4×C53semidirect
322. C22×D4D4×C53semidirect
330. C4⋊Q8D4×C53semidirect
342. #124D4×C53semidirect
357. #120D4×C53semidirect
365. #117D4×C53semidirect
366. C2×C42D4×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#35492
C53×#30272
C53×#30182
C53×#21992
C53×#21962
C53×#21262
C53×#21222
#3355953