Finite Group Databasebeta

← C53×#13098 #27220610

Subgroups — C53×#13098 #27220610

Order 13568.

640 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3010C211yesmaximal, characteristic
2C53×#2889C211yesmaximal, characteristic, complemented
3C53×#2134C211yesmaximal, characteristic, complemented
4C53×#2122C211yesmaximal, characteristic, complemented
5C53×#1796C211yesmaximal, characteristic, complemented
6C53×#1395C211yesmaximal, characteristic, complemented
7C53×#1392C211yesmaximal, characteristic, complemented
8C2×2−1+4×C53C2211yescharacteristic
9C2×AGL(1,ℤ/8)×C53C411yescharacteristic, complemented
10C22×SD16×C53C411yescharacteristic, complemented
11C53×#568—21no—
12C53×#535C2211yescharacteristic
13C53×#449—21no—
14C53×#449—21no—
15C53×#448—21no—
16C53×#448—21no—
17C53×#421—21no—
18C53×#419—21no—
19C53×#419C411yescharacteristic, complemented
20C53×#416C411yescharacteristic, complemented
21C53×#413—22no—
22C53×#413C2211yescharacteristic
23C53×#413C2211yescharacteristic
24C53×#413C2211yescharacteristic
25C2×#101×C53—21no—
26C2×#101×C53—22no—
27C53×#410C2211yescharacteristic
28C53×#408C2211yescharacteristic
29C53×#355—22no—
30C53×AΣL(1,𝔽16)—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×C53D411yescharacteristic
43C22×Q8×C53D411yescharacteristic
44C22×D4×C53C2×C411yescharacteristic
45AGL(1,ℤ/8)×C53—42no—
46C2×SD16×C53—21no—
47C2×SD16×C53—21no—
48C2×SD16×C53—21no—
49C2×SD16×C53—42no—
50C2×D8×C53—21no—
51#131×C53—42no—
52#131×C53—42no—
53C2×M16×C53—24no—
54C2×M16×C53—21no—
55C2×M16×C53—21no—
56C2×M16×C53—21no—
57C2×M16×C53C2×C411yescharacteristic
58C22×C8×C53C2×C411yescharacteristic
59#127×C53—21no—
60#124×C53—21no—
61#121×C53—21no—
62#121×C53—21no—
63#120×C53—21no—
64#120×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—24no—
74#101×C53—21no—
75#101×C53—21no—
76#101×C53D412yes—
77#100×C53—21no—
78#100×C53—21no—
79#100×C53D412yes—
80#99×C53—21no—
81#99×C53—21no—
82C22⋊C8×C53—21no—
83C22⋊C8×C53—21no—
84C24×C53—21no—
85#41×C53—22no—
86#41×C53—22no—
87#41×C53—22no—
88#41×C53—22no—
89#41×C53—41no—
90#41×C53—41no—
91#41×C53—41no—
92#41×C53—41no—
93#41×C53—44no—
94#41×C53—44no—
95#41×C53—44no—
96C2×Q8×C53—21no—
97C2×Q8×C53—21no—
98C2×Q8×C53—21no—
99C2×Q8×C53—22no—
100C2×Q8×C53—22no—
101C2×Q8×C53—41no—
102C2×Q8×C53—41no—
103C2×Q8×C53—41no—
104C2×Q8×C53—41no—
105C2×Q8×C53—44no—
106C2×Q8×C53C22⋊C411yescharacteristic
107C2×Q8×C53C22⋊C411yescharacteristic
108C2×Q8×C53C22⋊C411yescharacteristic
109C2×Q8×C53C2×D411yescharacteristic
110C2×D4×C53—21no—
111C2×D4×C53—21no—
112C2×D4×C53—21no—
113C2×D4×C53—21no—
114C2×D4×C53—21no—
115C2×D4×C53—41no—
116C2×D4×C53—41no—
117C2×D4×C53—41no—
118C2×D4×C53—41no—
119C2×D4×C53C22⋊C411yescharacteristic
120C2×D4×C53C2×D411yescharacteristic
121C2×D4×C53#4111yescharacteristic
122C2×D4×C53#4111yescharacteristic
123C22×C4×C53—21no—
124C22×C4×C53—21no—
125C22×C4×C53—21no—
126C22×C4×C53—21no—
127C22×C4×C53—21no—
128C22×C4×C53—21no—
129C22×C4×C53C22×C411yescharacteristic
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2×D411yescharacteristic
132SD16×C53—42no—
133SD16×C53—42no—
134SD16×C53—42no—
135D8×C53—42no—
136M16×C53—22no—
137M16×C53—22no—
138M16×C53—44no—
139M16×C53—41no—
140M16×C53—41no—
141M16×C53—41no—
142M16×C53—41no—
143M16×C53—41no—
144C2×C8×C53—21no—
145C2×C8×C53—21no—
146C2×C8×C53—44no—
147C2×C8×C53—41no—
148C2×C8×C53—41no—
149C2×C8×C53—41no—
150C2×C8×C53—41no—
151C4⋊C4×C53—21no—
152C4⋊C4×C53—41no—
153C22⋊C4×C53—21no—
154C22⋊C4×C53—41no—
155C22⋊C4×C53—41no—
156C22⋊C4×C53—41no—
157C42×C53—21no—
158C42×C53—41no—
159C23×C53—21no—
160C23×C53—41no—
161C23×C53—81no—
162C23×C53—161no—
163C23×C53C4×D411yescharacteristic
164C23×C53#12111yescharacteristic
165C23×C53#12311yescharacteristic
166Q8×C53—21no—
167Q8×C53—21no—
168Q8×C53—21no—
169Q8×C53—21no—
170Q8×C53—21no—
171Q8×C53—41no—
172Q8×C53—42no—
173Q8×C53—42no—
174Q8×C53—42no—
175Q8×C53#9912yes—
176D4×C53—24no—
177D4×C53—21no—
178D4×C53—41no—
179D4×C53—41no—
180D4×C53—41no—
181D4×C53—42no—
182D4×C53—42no—
183D4×C53—41no—
184D4×C53—41no—
185D4×C53—81no—
186D4×C53—161no—
187D4×C53—161no—
188D4×C53#9912yes—
189C2×C4×C53—24no—
190C2×C4×C53—21no—
191C2×C4×C53—21no—
192C2×C4×C53—21no—
193C2×C4×C53—21no—
194C2×C4×C53—21no—
195C2×C4×C53—21no—
196C2×C4×C53—21no—
197C2×C4×C53—21no—
198C2×C4×C53—21no—
199C2×C4×C53—42no—
200C2×C4×C53—42no—
201C2×C4×C53—42no—
202C2×C4×C53—42no—
203C2×C4×C53—42no—
204C2×C4×C53—41no—
205C2×C4×C53—81no—
206C2×C4×C53C2×C22⋊C411yescharacteristic
207C2×C4×C53C4×D411yescharacteristic
208C2×C4×C53#12011yescharacteristic
209C2×C4×C53#12111yescharacteristic
210C8×C53—42no—
211C8×C53—42no—
212C8×C53—41no—
213C8×C53—41no—
214C8×C53—81no—
215#13098C5311yesmaximal, characteristic, direct product factor, Sylow
216C22×C53—21no—
217C22×C53—21no—
218C22×C53—22no—
219C22×C53—21no—
220C22×C53—41no—
221C22×C53—41no—
222C22×C53—41no—
223C22×C53—81no—
224C22×C53—81no—
225C22×C53—81no—
226C22×C53—81no—
227C22×C53—162no—
228C22×C53#38611yescharacteristic
229C4×C53—21no—
230C4×C53—21no—
231C4×C53—22no—
232C4×C53—21no—
233C4×C53—41no—
234C4×C53—41no—
235C4×C53—81no—
236C4×C53—161no—
237C4×C53C2×#9911yescharacteristic
238C4×C53C2×#9911yescharacteristic
239#3010C2×C5311yescharacteristic
240#2889C2×C5311yescharacteristic, complemented
241#2134C2×C5311yescharacteristic, complemented
242#2122C2×C5311yescharacteristic, complemented
243#1796C2×C5311yescharacteristic, complemented
244#1395C2×C5311yescharacteristic, complemented
245#1392C2×C5311yescharacteristic, complemented
246C2×C53—21no—
247C2×C53—41no—
248C2×C53—41no—
249C2×C53—41no—
250C2×C53—42no—
251C2×C53—161no—
252C2×C53#189811yescharacteristic
253C2×2−1+4C22×C5311yescharacteristic
254C2×AGL(1,ℤ/8)C4×C5311yescharacteristic, complemented
255C22×SD16C4×C5311yescharacteristic, complemented
256#568—21no—
257#535C22×C5311yescharacteristic
258#449—21no—
259#449—21no—
260#448—21no—
261#448—21no—
262#421—21no—
263#419—21no—
264#419C4×C5311yescharacteristic, complemented
265#416C4×C5311yescharacteristic, complemented
266#413—22no—
267#413C22×C5311yescharacteristic
268#413C22×C5311yescharacteristic
269#413C22×C5311yescharacteristic
270C2×#101—21no—
271C2×#101—22no—
272#410C22×C5311yescharacteristic
273#408C22×C5311yescharacteristic
274#355—22no—
275AΣL(1,𝔽16)—22no—
276C53#1309811yescharacteristic, direct product factor, Sylow
2772−1+4—42no—
2782−1+4—42no—
279C2×#41—21no—
280C2×#41—21no—
281C2×#41—21no—
282C2×#41C2×C4×C5311yescharacteristic
283C2×#41D4×C5311yescharacteristic
284C2×#41D4×C5311yescharacteristic
285C2×#41C23×C5311yescharacteristic
286C22×Q8—21no—
287C22×Q8C2×C4×C5311yescharacteristic
288C22×Q8D4×C5311yescharacteristic
289C22×Q8D4×C5311yescharacteristic
290C22×D4C2×C4×C5311yescharacteristic
291AGL(1,ℤ/8)—42no—
292C2×SD16—21no—
293C2×SD16—21no—
294C2×SD16—21no—
295C2×SD16—42no—
296C2×D8—21no—
297#131—42no—
298#131—42no—
299C2×M16—24no—
300C2×M16—21no—
301C2×M16—21no—
302C2×M16—21no—
303C2×M16C2×C4×C5311yescharacteristic
304C22×C8C2×C4×C5311yescharacteristic
305#127—21no—
306#124—21no—
307#121—21no—
308#121—21no—
309#120—21no—
310#120—21no—
311#117—21no—
312#117C2×C4×C5311yescharacteristic
313#104—42no—
314#103—21no—
315#103—21no—
316#103—21no—
317#103—42no—
318#102—21no—
319#101—24no—
320#101—21no—
321#101—21no—
322#101D4×C5312yes—
323#100—21no—
324#100—21no—
325#100D4×C5312yes—
326#99—21no—
327#99—21no—
328C22⋊C8—21no—
329C22⋊C8—21no—
330C24—21no—
331#41—22no—
332#41—22no—
333#41—22no—
334#41—22no—
335#41—41no—
336#41—41no—
337#41—41no—
338#41—41no—
339#41—44no—
340#41—44no—
341#41—44no—
342C2×Q8—21no—
343C2×Q8—21no—
344C2×Q8—21no—
345C2×Q8—22no—
346C2×Q8—22no—
347C2×Q8—41no—
348C2×Q8—41no—
349C2×Q8—41no—
350C2×Q8—41no—
351C2×Q8—44no—
352C2×Q8C22⋊C4×C5311yescharacteristic
353C2×Q8C22⋊C4×C5311yescharacteristic
354C2×Q8C22⋊C4×C5311yescharacteristic
355C2×Q8C2×D4×C5311yescharacteristic
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—21no—
359C2×D4—21no—
360C2×D4—21no—
361C2×D4—41no—
362C2×D4—41no—
363C2×D4—41no—
364C2×D4—41no—
365C2×D4C22⋊C4×C5311yescharacteristic
366C2×D4C2×D4×C5311yescharacteristic
367C2×D4#41×C5311yescharacteristic
368C2×D4#41×C5311yescharacteristic
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—21no—
374C22×C4—21no—
375C22×C4C22×C4×C5311yescharacteristic
376C22×C4C2×D4×C5311yescharacteristic
377C22×C4C2×D4×C5311yescharacteristic
378SD16—42no—
379SD16—42no—
380SD16—42no—
381D8—42no—
382M16—22no—
383M16—22no—
384M16—44no—
385M16—41no—
386M16—41no—
387M16—41no—
388M16—41no—
389M16—41no—
390C2×C8—21no—
391C2×C8—21no—
392C2×C8—44no—
393C2×C8—41no—
394C2×C8—41no—
395C2×C8—41no—
396C2×C8—41no—
397C4⋊C4—21no—
398C4⋊C4—41no—
399C22⋊C4—21no—
400C22⋊C4—41no—
401C22⋊C4—41no—
402C22⋊C4—41no—
403C42—21no—
404C42—41no—
405C23—21no—
406C23—41no—
407C23—81no—
408C23—161no—
409C23C4×D4×C5311yescharacteristic
410C23#121×C5311yescharacteristic
411C23#123×C5311yescharacteristic
412Q8—21no—
413Q8—21no—
414Q8—21no—
415Q8—21no—
416Q8—21no—
417Q8—41no—
418Q8—42no—
419Q8—42no—
420Q8—42no—
421Q8#99×C5312yes—
422D4—24no—
423D4—21no—
424D4—41no—
425D4—41no—
426D4—41no—
427D4—42no—
428D4—42no—
429D4—41no—
430D4—41no—
431D4—81no—
432D4—161no—
433D4—161no—
434D4#99×C5312yes—
435C2×C4—24no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—21no—
443C2×C4—21no—
444C2×C4—21no—
445C2×C4—42no—
446C2×C4—42no—
447C2×C4—42no—
448C2×C4—42no—
449C2×C4—42no—
450C2×C4—41no—
451C2×C4—81no—
452C2×C4C2×C22⋊C4×C5311yescharacteristic
453C2×C4C4×D4×C5311yescharacteristic
454C2×C4#120×C5311yescharacteristic
455C2×C4#121×C5311yescharacteristic
456C8—42no—
457C8—42no—
458C8—41no—
459C8—41no—
460C8—81no—
461C22—21no—
462C22—21no—
463C22—22no—
464C22—21no—
465C22—41no—
466C22—41no—
467C22—41no—
468C22—81no—
469C22—81no—
470C22—81no—
471C22—81no—
472C22—162no—
473C22C53×#38611yescharacteristic
474C4—21no—
475C4—21no—
476C4—22no—
477C4—21no—
478C4—41no—
479C4—41no—
480C4—81no—
481C4—161no—
482C4C2×#99×C5311yescharacteristic
483C4C2×#99×C5311yescharacteristic
484C2—21no—
485C2—41no—
486C2—41no—
487C2—41no—
488C2—42no—
489C2—161no—
490C2C53×#189811yescharacteristic
subgroupG/H
center, socleC2×C53#1898
commutatorC22×C4C22×C4×C53
Frattini, cosocleC2×#41C23×C53
prime radicalC53×#3010C2
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
2. C53×#2889C2semidirect
3. C53×#2134C2semidirect
4. C53×#2122C2semidirect
5. C53×#1796C2semidirect
6. C53×#1395C2semidirect
7. C53×#1392C2semidirect
9. C2×AGL(1,ℤ/8)×C53C4semidirect
10. C22×SD16×C53C4semidirect
19. C53×#419C4semidirect
20. C53×#416C4semidirect
215. #13098C53direct (N × H)
240. #2889C2×C53semidirect
241. #2134C2×C53semidirect
242. #2122C2×C53semidirect
243. #1796C2×C53semidirect
244. #1395C2×C53semidirect
245. #1392C2×C53semidirect
254. C2×AGL(1,ℤ/8)C4×C53semidirect
255. C22×SD16C4×C53semidirect
264. #419C4×C53semidirect
265. #416C4×C53semidirect
276. C53#13098direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#30102
C53×#28892
C53×#21342
C53×#21222
C53×#17962
C53×#13952
C53×#13922
#1309853