Finite Group Databasebeta

← C53×#13101 #27220613

Subgroups — C53×#13101 #27220613

Order 13568.

630 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3011C211yesmaximal, characteristic
2C53×#2884C211yesmaximal, characteristic
3C53×#2129C211yesmaximal, characteristic, complemented
4C53×#2123C211yesmaximal, characteristic, complemented
5C53×#1796C211yesmaximal, characteristic
6C53×#1395C211yesmaximal, characteristic, complemented
7C53×#1391C211yesmaximal, characteristic, complemented
8C2×2−1+4×C53C2211yescharacteristic
9C2×#137×C53C411yescharacteristic, complemented
10C22×Q16×C53C411yescharacteristic, complemented
11C53×#536C2211yescharacteristic
12C53×#519—21no—
13C53×#452—21no—
14C53×#452—21no—
15C53×#451—21no—
16C53×#451—21no—
17C53×#421—21no—
18C53×#419—21no—
19C53×#419C411yescharacteristic, complemented
20C53×#416C411yescharacteristic, complemented
21C53×#413C2211yescharacteristic
22C53×#410—21no—
23C53×#410—22no—
24C53×#410—22no—
25C53×#410C2211yescharacteristic
26C53×#410C2211yescharacteristic
27C53×#410C2211yescharacteristic
28C53×#408C2211yescharacteristic
29C53×#354—22no—
30C53×#352—22no—
312−1+4×C53—42no—
322−1+4×C53—42no—
33C2×#41×C53—21no—
34C2×#41×C53—21no—
35C2×#41×C53—21no—
36C2×#41×C53C2×C411yescharacteristic
37C2×#41×C53D411yescharacteristic
38C2×#41×C53D411yescharacteristic
39C2×#41×C53C2311yescharacteristic
40C22×Q8×C53—21no—
41C22×Q8×C53C2×C411yescharacteristic
42C22×Q8×C53C2×C411yescharacteristic
43C22×Q8×C53D411yescharacteristic
44C22×Q8×C53D411yescharacteristic
45#137×C53—42no—
46C2×Q16×C53—21no—
47C2×Q16×C53—21no—
48C2×Q16×C53—21no—
49C2×Q16×C53—42no—
50C2×SD16×C53—21no—
51C2×M16×C53—21no—
52C2×M16×C53C2×C411yescharacteristic
53C22×C8×C53C2×C411yescharacteristic
54C4⋊Q8×C53—21no—
55#124×C53—21no—
56#123×C53—21no—
57#123×C53—21no—
58C22⋊Q8×C53—21no—
59C22⋊Q8×C53—21no—
60C4×Q8×C53—42no—
61C4×Q8×C53—42no—
62#117×C53—24no—
63#117×C53—21no—
64#117×C53—21no—
65#117×C53—21no—
66#117×C53C2×C411yescharacteristic
67#104×C53—42no—
68#103×C53—21no—
69#103×C53—21no—
70#103×C53—21no—
71#103×C53—42no—
72#102×C53—21no—
73#101×C53—21no—
74#100×C53—21no—
75#99×C53—24no—
76#99×C53—21no—
77#99×C53—21no—
78#99×C53—21no—
79#99×C53—21no—
80#99×C53D412yes—
81#99×C53D412yes—
82C22⋊C8×C53—21no—
83C22⋊C8×C53—21no—
84#41×C53—22no—
85#41×C53—22no—
86#41×C53—41no—
87#41×C53—41no—
88#41×C53—41no—
89#41×C53—41no—
90#41×C53—41no—
91#41×C53—41no—
92#41×C53—44no—
93#41×C53—44no—
94#41×C53—44no—
95C2×Q8×C53—22no—
96C2×Q8×C53—22no—
97C2×Q8×C53—21no—
98C2×Q8×C53—21no—
99C2×Q8×C53—21no—
100C2×Q8×C53—22no—
101C2×Q8×C53—22no—
102C2×Q8×C53—21no—
103C2×Q8×C53—21no—
104C2×Q8×C53—41no—
105C2×Q8×C53—41no—
106C2×Q8×C53—41no—
107C2×Q8×C53—41no—
108C2×Q8×C53—44no—
109C2×Q8×C53C22⋊C411yescharacteristic
110C2×Q8×C53C22⋊C411yescharacteristic
111C2×Q8×C53C22⋊C411yescharacteristic
112C2×Q8×C53#4111yescharacteristic
113C2×D4×C53—21no—
114C2×D4×C53—21no—
115C2×D4×C53—21no—
116C2×D4×C53C22⋊C411yescharacteristic
117C2×D4×C53C2×D411yescharacteristic
118C2×D4×C53C2×D411yescharacteristic
119C2×D4×C53#4111yescharacteristic
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53C22×C411yescharacteristic
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53C2×D411yescharacteristic
130Q16×C53—42no—
131Q16×C53—42no—
132Q16×C53—42no—
133SD16×C53—42no—
134M16×C53—41no—
135M16×C53—41no—
136M16×C53—41no—
137C2×C8×C53—21no—
138C2×C8×C53—21no—
139C2×C8×C53—41no—
140C2×C8×C53—41no—
141C4⋊C4×C53—21no—
142C4⋊C4×C53—44no—
143C4⋊C4×C53—41no—
144C4⋊C4×C53—41no—
145C4⋊C4×C53—41no—
146C4⋊C4×C53—41no—
147C4⋊C4×C53—41no—
148C22⋊C4×C53—21no—
149C22⋊C4×C53—22no—
150C22⋊C4×C53—22no—
151C22⋊C4×C53—41no—
152C22⋊C4×C53—41no—
153C22⋊C4×C53—41no—
154C42×C53—21no—
155C42×C53—44no—
156C42×C53—41no—
157C42×C53—41no—
158C42×C53—41no—
159C23×C53—21no—
160C23×C53C4×D411yescharacteristic
161C23×C53#12011yescharacteristic
162C23×C53#12111yescharacteristic
163Q8×C53—24no—
164Q8×C53—21no—
165Q8×C53—21no—
166Q8×C53—21no—
167Q8×C53—21no—
168Q8×C53—41no—
169Q8×C53—42no—
170Q8×C53—42no—
171Q8×C53—41no—
172Q8×C53—41no—
173Q8×C53—81no—
174Q8×C53#9912yes—
175Q8×C53#9912yes—
176D4×C53—21no—
177D4×C53—21no—
178D4×C53—41no—
179D4×C53—41no—
180D4×C53—41no—
181D4×C53—42no—
182D4×C53—42no—
183D4×C53—42no—
184C2×C4×C53—24no—
185C2×C4×C53—21no—
186C2×C4×C53—21no—
187C2×C4×C53—21no—
188C2×C4×C53—21no—
189C2×C4×C53—21no—
190C2×C4×C53—21no—
191C2×C4×C53—21no—
192C2×C4×C53—21no—
193C2×C4×C53—21no—
194C2×C4×C53—42no—
195C2×C4×C53—42no—
196C2×C4×C53—42no—
197C2×C4×C53—42no—
198C2×C4×C53—42no—
199C2×C4×C53—42no—
200C2×C4×C53—42no—
201C2×C4×C53—41no—
202C2×C4×C53—41no—
203C2×C4×C53—81no—
204C2×C4×C53—81no—
205C2×C4×C53C2×C22⋊C411yescharacteristic
206C2×C4×C53C4×D411yescharacteristic
207C2×C4×C53#12111yescharacteristic
208C2×C4×C53#12311yescharacteristic
209C8×C53—41no—
210C8×C53—41no—
211C8×C53—81no—
212#13101C5311yesmaximal, characteristic, direct product factor, Sylow
213C22×C53—21no—
214C22×C53—21no—
215C22×C53—21no—
216C22×C53—41no—
217C22×C53—41no—
218C22×C53—41no—
219C22×C53—41no—
220C22×C53—81no—
221C22×C53#38611yescharacteristic
222C4×C53—21no—
223C4×C53—22no—
224C4×C53—22no—
225C4×C53—21no—
226C4×C53—21no—
227C4×C53—41no—
228C4×C53—81no—
229C4×C53—82no—
230C4×C53—82no—
231C4×C53—81no—
232C4×C53—161no—
233C4×C53C2×#9911yescharacteristic
234C4×C53C2×#9911yescharacteristic
235#3011C2×C5311yescharacteristic
236#2884C2×C5311yescharacteristic
237#2129C2×C5311yescharacteristic, complemented
238#2123C2×C5311yescharacteristic, complemented
239#1796C2×C5311yescharacteristic
240#1395C2×C5311yescharacteristic, complemented
241#1391C2×C5311yescharacteristic, complemented
242C2×C53—21no—
243C2×C53—41no—
244C2×C53—41no—
245C2×C53—41no—
246C2×C53—81no—
247C2×C53#189811yescharacteristic
248C2×2−1+4C22×C5311yescharacteristic
249C2×#137C4×C5311yescharacteristic, complemented
250C22×Q16C4×C5311yescharacteristic, complemented
251#536C22×C5311yescharacteristic
252#519—21no—
253#452—21no—
254#452—21no—
255#451—21no—
256#451—21no—
257#421—21no—
258#419—21no—
259#419C4×C5311yescharacteristic, complemented
260#416C4×C5311yescharacteristic, complemented
261#413C22×C5311yescharacteristic
262#410—21no—
263#410—22no—
264#410—22no—
265#410C22×C5311yescharacteristic
266#410C22×C5311yescharacteristic
267#410C22×C5311yescharacteristic
268#408C22×C5311yescharacteristic
269#354—22no—
270#352—22no—
271C53#1310111yescharacteristic, direct product factor, Sylow
2722−1+4—42no—
2732−1+4—42no—
274C2×#41—21no—
275C2×#41—21no—
276C2×#41—21no—
277C2×#41C2×C4×C5311yescharacteristic
278C2×#41D4×C5311yescharacteristic
279C2×#41D4×C5311yescharacteristic
280C2×#41C23×C5311yescharacteristic
281C22×Q8—21no—
282C22×Q8C2×C4×C5311yescharacteristic
283C22×Q8C2×C4×C5311yescharacteristic
284C22×Q8D4×C5311yescharacteristic
285C22×Q8D4×C5311yescharacteristic
286#137—42no—
287C2×Q16—21no—
288C2×Q16—21no—
289C2×Q16—21no—
290C2×Q16—42no—
291C2×SD16—21no—
292C2×M16—21no—
293C2×M16C2×C4×C5311yescharacteristic
294C22×C8C2×C4×C5311yescharacteristic
295C4⋊Q8—21no—
296#124—21no—
297#123—21no—
298#123—21no—
299C22⋊Q8—21no—
300C22⋊Q8—21no—
301C4×Q8—42no—
302C4×Q8—42no—
303#117—24no—
304#117—21no—
305#117—21no—
306#117—21no—
307#117C2×C4×C5311yescharacteristic
308#104—42no—
309#103—21no—
310#103—21no—
311#103—21no—
312#103—42no—
313#102—21no—
314#101—21no—
315#100—21no—
316#99—24no—
317#99—21no—
318#99—21no—
319#99—21no—
320#99—21no—
321#99D4×C5312yes—
322#99D4×C5312yes—
323C22⋊C8—21no—
324C22⋊C8—21no—
325#41—22no—
326#41—22no—
327#41—41no—
328#41—41no—
329#41—41no—
330#41—41no—
331#41—41no—
332#41—41no—
333#41—44no—
334#41—44no—
335#41—44no—
336C2×Q8—22no—
337C2×Q8—22no—
338C2×Q8—21no—
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8—22no—
342C2×Q8—22no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8—41no—
346C2×Q8—41no—
347C2×Q8—41no—
348C2×Q8—41no—
349C2×Q8—44no—
350C2×Q8C22⋊C4×C5311yescharacteristic
351C2×Q8C22⋊C4×C5311yescharacteristic
352C2×Q8C22⋊C4×C5311yescharacteristic
353C2×Q8#41×C5311yescharacteristic
354C2×D4—21no—
355C2×D4—21no—
356C2×D4—21no—
357C2×D4C22⋊C4×C5311yescharacteristic
358C2×D4C2×D4×C5311yescharacteristic
359C2×D4C2×D4×C5311yescharacteristic
360C2×D4#41×C5311yescharacteristic
361C22×C4—21no—
362C22×C4—21no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—21no—
368C22×C4C22×C4×C5311yescharacteristic
369C22×C4C2×D4×C5311yescharacteristic
370C22×C4C2×D4×C5311yescharacteristic
371Q16—42no—
372Q16—42no—
373Q16—42no—
374SD16—42no—
375M16—41no—
376M16—41no—
377M16—41no—
378C2×C8—21no—
379C2×C8—21no—
380C2×C8—41no—
381C2×C8—41no—
382C4⋊C4—21no—
383C4⋊C4—44no—
384C4⋊C4—41no—
385C4⋊C4—41no—
386C4⋊C4—41no—
387C4⋊C4—41no—
388C4⋊C4—41no—
389C22⋊C4—21no—
390C22⋊C4—22no—
391C22⋊C4—22no—
392C22⋊C4—41no—
393C22⋊C4—41no—
394C22⋊C4—41no—
395C42—21no—
396C42—44no—
397C42—41no—
398C42—41no—
399C42—41no—
400C23—21no—
401C23C4×D4×C5311yescharacteristic
402C23#120×C5311yescharacteristic
403C23#121×C5311yescharacteristic
404Q8—24no—
405Q8—21no—
406Q8—21no—
407Q8—21no—
408Q8—21no—
409Q8—41no—
410Q8—42no—
411Q8—42no—
412Q8—41no—
413Q8—41no—
414Q8—81no—
415Q8#99×C5312yes—
416Q8#99×C5312yes—
417D4—21no—
418D4—21no—
419D4—41no—
420D4—41no—
421D4—41no—
422D4—42no—
423D4—42no—
424D4—42no—
425C2×C4—24no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—21no—
431C2×C4—21no—
432C2×C4—21no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—42no—
436C2×C4—42no—
437C2×C4—42no—
438C2×C4—42no—
439C2×C4—42no—
440C2×C4—42no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—81no—
445C2×C4—81no—
446C2×C4C2×C22⋊C4×C5311yescharacteristic
447C2×C4C4×D4×C5311yescharacteristic
448C2×C4#121×C5311yescharacteristic
449C2×C4#123×C5311yescharacteristic
450C8—41no—
451C8—41no—
452C8—81no—
453C22—21no—
454C22—21no—
455C22—21no—
456C22—41no—
457C22—41no—
458C22—41no—
459C22—41no—
460C22—81no—
461C22C53×#38611yescharacteristic
462C4—21no—
463C4—22no—
464C4—22no—
465C4—21no—
466C4—21no—
467C4—41no—
468C4—81no—
469C4—82no—
470C4—82no—
471C4—81no—
472C4—161no—
473C4C2×#99×C5311yescharacteristic
474C4C2×#99×C5311yescharacteristic
475C2—21no—
476C2—41no—
477C2—41no—
478C2—41no—
479C2—81no—
480C2C53×#189811yescharacteristic
subgroupG/H
center, socleC2×C53#1898
commutatorC22×C4C22×C4×C53
Frattini, cosocleC2×#41C23×C53
prime radicalC2×2−1+4×C53C22
LCSG ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C53 ◁ C2×C4×C53 ◁ C2×2−1+4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
3. C53×#2129C2semidirect
4. C53×#2123C2semidirect
6. C53×#1395C2semidirect
7. C53×#1391C2semidirect
9. C2×#137×C53C4semidirect
10. C22×Q16×C53C4semidirect
19. C53×#419C4semidirect
20. C53×#416C4semidirect
212. #13101C53direct (N × H)
237. #2129C2×C53semidirect
238. #2123C2×C53semidirect
240. #1395C2×C53semidirect
241. #1391C2×C53semidirect
249. C2×#137C4×C53semidirect
250. C22×Q16C4×C53semidirect
259. #419C4×C53semidirect
260. #416C4×C53semidirect
271. C53#13101direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#30112
C53×#28842
C53×#21292
C53×#21232
C53×#17962
C53×#13952
C53×#13912
#1310153