Finite Group Databasebeta

← C53×#12765 #27220277

Subgroups — C53×#12765 #27220277

Order 13568.

852 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3481C211yesmaximal, characteristic
2C53×#2132C211yesmaximal, characteristic, complemented
3C53×#2120C212yesmaximal, complemented
4C53×#1891C212yesmaximal, complemented
5C53×#1884C211yesmaximal, characteristic, complemented
6C53×#568C2211yescharacteristic, complemented
7C53×#559C411yescharacteristic, complemented
8C53×#554—21no—
9C53×#553—22no—
10C53×#551—21no—
11C53×#551C412yescomplemented
12C53×#546—22no—
13D42×C53C411yescharacteristic, complemented
14C53×#532C2211yescharacteristic
15C53×#529—22no—
16C53×#525—21no—
17C53×#525C2212yes—
18C4×#41×C53—21no—
19C2×C4×D4×C53—22no—
20C53×#421—21no—
21C53×#413C2212yescomplemented
22C53×#410C2211yescharacteristic, complemented
23C53×#407—22no—
24C53×#356—21no—
25C53×#355—21no—
26C53×#352—22no—
27C53×AΣL(1,𝔽16)—22no—
28C53×#344—22no—
29C53×#339—21no—
30C2×#41×C53—21no—
31C2×#41×C53D411yescharacteristic
32C2×#41×C53C2311yescharacteristic
33C22×D4×C53—22no—
34C23×C4×C53—21no—
35C23×C4×C53D412yes—
36#131×C53—42no—
37C2×M16×C53—22no—
38C2×M16×C53—21no—
39C2×M16×C53—22no—
40C4⋊Q8×C53D411yescharacteristic
41#127×C53C2×C411yescharacteristic
42#126×C53—22no—
43#126×C53—22no—
44#125×C53—22no—
45#124×C53—22no—
46#124×C53C2×C411yescharacteristic
47#124×C53D411yescharacteristic
48#123×C53—21no—
49#123×C53—22no—
50#123×C53—22no—
51#123×C53—21no—
52#123×C53C2×C412yes—
53#123×C53D412yes—
54C22⋊Q8×C53—22no—
55C22⋊Q8×C53—22no—
56C22⋊Q8×C53—22no—
57#121×C53—22no—
58#121×C53—22no—
59#121×C53—22no—
60#121×C53—22no—
61#120×C53—21no—
62#120×C53—21no—
63#120×C53C2×C412yes—
64#120×C53D412yes—
65C4×Q8×C53—21no—
66C4×Q8×C53—21no—
67C4×D4×C53—21no—
68C4×D4×C53—22no—
69C4×D4×C53—22no—
70C4×D4×C53—22no—
71C4×D4×C53—22no—
72C4×D4×C53—21no—
73C4×D4×C53—22no—
74C4×D4×C53—22no—
75C4×D4×C53—22no—
76C4×D4×C53—22no—
77C4×D4×C53—22no—
78#117×C53—21no—
79#117×C53—21no—
80#117×C53—22no—
81#117×C53—21no—
82#117×C53D412yes—
83C2×C4⋊C4×C53—22no—
84C2×C22⋊C4×C53—22no—
85C2×C22⋊C4×C53—22no—
86C2×C42×C53—22no—
87C2×C42×C53D411yescharacteristic
88#104×C53—42no—
89#101×C53—22no—
90#100×C53—22no—
91#99×C53—22no—
92C22⋊C8×C53—44no—
93C24×C53—21no—
94C24×C53C22⋊C412yes—
95#41×C53—41no—
96#41×C53—41no—
97#41×C53—41no—
98#41×C53—41no—
99#41×C53—44no—
100C2×Q8×C53—21no—
101C2×Q8×C53C22⋊C411yescharacteristic
102C2×Q8×C53C2×D411yescharacteristic
103C2×D4×C53—21no—
104C2×D4×C53—22no—
105C2×D4×C53—22no—
106C2×D4×C53—22no—
107C2×D4×C53—22no—
108C2×D4×C53—21no—
109C2×D4×C53—48no—
110C2×D4×C53C22⋊C411yescharacteristic
111C2×D4×C53C22×C411yescharacteristic
112C2×D4×C53C2×D412yes—
113C22×C4×C53—22no—
114C22×C4×C53—22no—
115C22×C4×C53—21no—
116C22×C4×C53—22no—
117C22×C4×C53—21no—
118C22×C4×C53—22no—
119C22×C4×C53—21no—
120C22×C4×C53—22no—
121C22×C4×C53—42no—
122C22×C4×C53—42no—
123C22×C4×C53—81no—
124C22×C4×C53—81no—
125C22×C4×C53C22⋊C412yes—
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53C2×D412yes—
128M16×C53—22no—
129M16×C53—42no—
130M16×C53—41no—
131M16×C53—42no—
132M16×C53—42no—
133C2×C8×C53—24no—
134C2×C8×C53—41no—
135C2×C8×C53—42no—
136C4⋊C4×C53—21no—
137C4⋊C4×C53—22no—
138C4⋊C4×C53—21no—
139C4⋊C4×C53—22no—
140C4⋊C4×C53—22no—
141C4⋊C4×C53—22no—
142C4⋊C4×C53—22no—
143C4⋊C4×C53—22no—
144C4⋊C4×C53—22no—
145C4⋊C4×C53—41no—
146C22⋊C4×C53—22no—
147C22⋊C4×C53—22no—
148C22⋊C4×C53—22no—
149C22⋊C4×C53—21no—
150C22⋊C4×C53—21no—
151C22⋊C4×C53—22no—
152C22⋊C4×C53—22no—
153C22⋊C4×C53—22no—
154C22⋊C4×C53—22no—
155C22⋊C4×C53—22no—
156C22⋊C4×C53—22no—
157C22⋊C4×C53—41no—
158C22⋊C4×C53C22⋊C412yes—
159C22⋊C4×C53C22⋊C412yes—
160C42×C53—22no—
161C42×C53—22no—
162C42×C53—21no—
163C42×C53—22no—
164C42×C53—41no—
165C42×C53C22⋊C411yescharacteristic
166C42×C53C22⋊C411yescharacteristic
167C23×C53—22no—
168C23×C53—21no—
169C23×C53—42no—
170C23×C53—44no—
171C23×C53—44no—
172C23×C53—81no—
173C23×C53C2×C22⋊C412yes—
174C23×C53#12011yescharacteristic
175Q8×C53—41no—
176Q8×C53—42no—
177Q8×C53—41no—
178D4×C53—24no—
179D4×C53—41no—
180D4×C53—42no—
181D4×C53—41no—
182D4×C53—44no—
183D4×C53—44no—
184D4×C53—44no—
185D4×C53—44no—
186D4×C53—81no—
187C2×C4×C53—21no—
188C2×C4×C53—22no—
189C2×C4×C53—21no—
190C2×C4×C53—22no—
191C2×C4×C53—21no—
192C2×C4×C53—22no—
193C2×C4×C53—21no—
194C2×C4×C53—21no—
195C2×C4×C53—22no—
196C2×C4×C53—22no—
197C2×C4×C53—44no—
198C2×C4×C53—42no—
199C2×C4×C53—44no—
200C2×C4×C53—42no—
201C2×C4×C53—41no—
202C2×C4×C53—42no—
203C2×C4×C53—41no—
204C2×C4×C53—42no—
205C2×C4×C53—82no—
206C2×C4×C53—81no—
207C2×C4×C53—82no—
208C2×C4×C53—81no—
209C2×C4×C53C2×C22⋊C411yescharacteristic
210C2×C4×C53#12012yes—
211C2×C4×C53#12011yescharacteristic
212C8×C53—42no—
213C8×C53—82no—
214#12765C5311yesmaximal, characteristic, direct product factor, Sylow
215C22×C53—24no—
216C22×C53—22no—
217C22×C53—21no—
218C22×C53—41no—
219C22×C53—42no—
220C22×C53—42no—
221C22×C53—42no—
222C22×C53—44no—
223C22×C53—42no—
224C22×C53—41no—
225C22×C53—82no—
226C22×C53—84no—
227C22×C53#37911yescharacteristic
228C4×C53—22no—
229C4×C53—22no—
230C4×C53—21no—
231C4×C53—41no—
232C4×C53—42no—
233C4×C53—42no—
234C4×C53—42no—
235C4×C53—41no—
236C4×C53—161no—
237C4×C53B(4,𝔽2)11yescharacteristic
238C4×C53#45811yescharacteristic
239#3481C2×C5311yescharacteristic
240#2132C2×C5311yescharacteristic, complemented
241#2120C2×C5312yescomplemented
242#1891C2×C5312yescomplemented
243#1884C2×C5311yescharacteristic, complemented
244C2×C53—21no—
245C2×C53—41no—
246C2×C53—42no—
247C2×C53—44no—
248C2×C53—81no—
249C2×C53#179011yescharacteristic
250#568C22×C5311yescharacteristic, complemented
251#559C4×C5311yescharacteristic, complemented
252#554—21no—
253#553—22no—
254#551—21no—
255#551C4×C5312yescomplemented
256#546—22no—
257D42C4×C5311yescharacteristic, complemented
258#532C22×C5311yescharacteristic
259#529—22no—
260#525—21no—
261#525C22×C5312yes—
262C4×#41—21no—
263C2×C4×D4—22no—
264#421—21no—
265#413C22×C5312yescomplemented
266#410C22×C5311yescharacteristic, complemented
267#407—22no—
268#356—21no—
269#355—21no—
270#352—22no—
271AΣL(1,𝔽16)—22no—
272#344—22no—
273#339—21no—
274C53#1276511yescharacteristic, direct product factor, Sylow
275C2×#41—21no—
276C2×#41D4×C5311yescharacteristic
277C2×#41C23×C5311yescharacteristic
278C22×D4—22no—
279C23×C4—21no—
280C23×C4D4×C5312yes—
281#131—42no—
282C2×M16—22no—
283C2×M16—21no—
284C2×M16—22no—
285C4⋊Q8D4×C5311yescharacteristic
286#127C2×C4×C5311yescharacteristic
287#126—22no—
288#126—22no—
289#125—22no—
290#124—22no—
291#124C2×C4×C5311yescharacteristic
292#124D4×C5311yescharacteristic
293#123—21no—
294#123—22no—
295#123—21no—
296#123—22no—
297#123C2×C4×C5312yes—
298#123D4×C5312yes—
299C22⋊Q8—22no—
300C22⋊Q8—22no—
301C22⋊Q8—22no—
302#121—22no—
303#121—22no—
304#121—22no—
305#121—22no—
306#120—21no—
307#120—21no—
308#120C2×C4×C5312yes—
309#120D4×C5312yes—
310C4×Q8—21no—
311C4×Q8—21no—
312C4×D4—21no—
313C4×D4—22no—
314C4×D4—22no—
315C4×D4—22no—
316C4×D4—22no—
317C4×D4—22no—
318C4×D4—22no—
319C4×D4—22no—
320C4×D4—22no—
321C4×D4—22no—
322C4×D4—21no—
323#117—21no—
324#117—21no—
325#117—22no—
326#117—21no—
327#117D4×C5312yes—
328C2×C4⋊C4—22no—
329C2×C22⋊C4—22no—
330C2×C22⋊C4—22no—
331C2×C42—22no—
332C2×C42D4×C5311yescharacteristic
333#104—42no—
334#101—22no—
335#100—22no—
336#99—22no—
337C22⋊C8—44no—
338C24—21no—
339C24C22⋊C4×C5312yes—
340#41—41no—
341#41—41no—
342#41—41no—
343#41—41no—
344#41—44no—
345C2×Q8—21no—
346C2×Q8C22⋊C4×C5311yescharacteristic
347C2×Q8C2×D4×C5311yescharacteristic
348C2×D4—21no—
349C2×D4—22no—
350C2×D4—22no—
351C2×D4—22no—
352C2×D4—22no—
353C2×D4—21no—
354C2×D4—48no—
355C2×D4C22⋊C4×C5311yescharacteristic
356C2×D4C22×C4×C5311yescharacteristic
357C2×D4C2×D4×C5312yes—
358C22×C4—22no—
359C22×C4—22no—
360C22×C4—21no—
361C22×C4—22no—
362C22×C4—21no—
363C22×C4—22no—
364C22×C4—21no—
365C22×C4—22no—
366C22×C4—42no—
367C22×C4—42no—
368C22×C4—81no—
369C22×C4—81no—
370C22×C4C22⋊C4×C5312yes—
371C22×C4C2×D4×C5311yescharacteristic
372C22×C4C2×D4×C5312yes—
373M16—22no—
374M16—42no—
375M16—41no—
376M16—42no—
377M16—42no—
378C2×C8—24no—
379C2×C8—41no—
380C2×C8—42no—
381C4⋊C4—21no—
382C4⋊C4—22no—
383C4⋊C4—21no—
384C4⋊C4—22no—
385C4⋊C4—22no—
386C4⋊C4—22no—
387C4⋊C4—22no—
388C4⋊C4—22no—
389C4⋊C4—22no—
390C4⋊C4—41no—
391C22⋊C4—22no—
392C22⋊C4—22no—
393C22⋊C4—22no—
394C22⋊C4—21no—
395C22⋊C4—21no—
396C22⋊C4—22no—
397C22⋊C4—22no—
398C22⋊C4—22no—
399C22⋊C4—22no—
400C22⋊C4—22no—
401C22⋊C4—22no—
402C22⋊C4—41no—
403C22⋊C4C22⋊C4×C5312yes—
404C22⋊C4C22⋊C4×C5312yes—
405C42—22no—
406C42—22no—
407C42—21no—
408C42—22no—
409C42—41no—
410C42C22⋊C4×C5311yescharacteristic
411C42C22⋊C4×C5311yescharacteristic
412C23—22no—
413C23—21no—
414C23—42no—
415C23—44no—
416C23—44no—
417C23—81no—
418C23C2×C22⋊C4×C5312yes—
419C23#120×C5311yescharacteristic
420Q8—41no—
421Q8—41no—
422Q8—42no—
423D4—24no—
424D4—41no—
425D4—42no—
426D4—41no—
427D4—44no—
428D4—44no—
429D4—44no—
430D4—44no—
431D4—81no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—22no—
435C2×C4—21no—
436C2×C4—22no—
437C2×C4—21no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—22no—
441C2×C4—21no—
442C2×C4—44no—
443C2×C4—42no—
444C2×C4—44no—
445C2×C4—42no—
446C2×C4—41no—
447C2×C4—42no—
448C2×C4—41no—
449C2×C4—42no—
450C2×C4—82no—
451C2×C4—81no—
452C2×C4—82no—
453C2×C4—81no—
454C2×C4C2×C22⋊C4×C5311yescharacteristic
455C2×C4#120×C5312yes—
456C2×C4#120×C5311yescharacteristic
457C8—42no—
458C8—82no—
459C22—24no—
460C22—22no—
461C22—21no—
462C22—41no—
463C22—42no—
464C22—42no—
465C22—42no—
466C22—44no—
467C22—42no—
468C22—41no—
469C22—82no—
470C22—84no—
471C22C53×#37911yescharacteristic
472C4—22no—
473C4—22no—
474C4—21no—
475C4—41no—
476C4—42no—
477C4—42no—
478C4—42no—
479C4—41no—
480C4—161no—
481C4C53×B(4,𝔽2)11yescharacteristic
482C4C53×#45811yescharacteristic
483C2—21no—
484C2—41no—
485C2—44no—
486C2—42no—
487C2—81no—
488C2C53×#179011yescharacteristic
subgroupG/H
centerC4×C53B(4,𝔽2)
commutatorC2×D4C22×C4×C53
Frattini, cosocleC2×#41C23×C53
socleC2×C53#1790
prime radicalC53×#3481C2
LCSG ▷ C2×D4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C4×C53 ◁ C2×C4×C53 ◁ C2×#41×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#2132C2semidirect
3. C53×#2120C2semidirect
4. C53×#1891C2semidirect
5. C53×#1884C2semidirect
6. C53×#568C22semidirect
7. C53×#559C4semidirect
11. C53×#551C4semidirect
13. D42×C53C4semidirect
21. C53×#413C22semidirect
22. C53×#410C22semidirect
214. #12765C53direct (N × H)
240. #2132C2×C53semidirect
241. #2120C2×C53semidirect
242. #1891C2×C53semidirect
243. #1884C2×C53semidirect
250. #568C22×C53semidirect
251. #559C4×C53semidirect
255. #551C4×C53semidirect
257. D42C4×C53semidirect
265. #413C22×C53semidirect
266. #410C22×C53semidirect
274. C53#12765direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#34812
C53×#21322
C53×#21202
C53×#18912
C53×#18842
#1276553