Finite Group Databasebeta

← C53×#8484 #27215996

Subgroups — C53×#8484 #27215996

Order 13568.

706 conjugacy classes of subgroups.

The subgroup lattice has 526 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3561C211yesmaximal, characteristic
2C53×#1967C211yesmaximal, characteristic, complemented
3C53×#1559C211yesmaximal, characteristic, complemented
4C53×#1529C211yesmaximal, characteristic, complemented
5C53×#1495C211yesmaximal, characteristic, complemented
6C53×#1485C211yesmaximal, characteristic, complemented
7C53×#1469C211yesmaximal, characteristic, complemented
8C53×#563—21no—
9C53×#562—21no—
10C53×#556—21no—
11C53×#551—22no—
12D4×Q8×C53—21no—
13C53×#548—21no—
14C53×#546—21no—
15C53×#543C412yes—
16C53×#542—21no—
17C53×#542—22no—
18C53×#538C412yes—
19C53×#536—21no—
20C53×#533C2211yescharacteristic
21C53×#529C2211yescharacteristic
22C2×#124×C53C2211yescharacteristic
23C53×#432C2211yescharacteristic
24C53×#431C2211yescharacteristic
25C53×#423C2211yescharacteristic
26C2×#101×C53—21no—
27C2×#100×C53—21no—
28C4×M16×C53C2211yescharacteristic
29C53×#333—21no—
30C53×#332—21no—
31C53×#330—21no—
32C53×#329—21no—
33C53×Q8⋊C8—22no—
34C53×Q8⋊C8—21no—
35C53×D4⋊C8—21no—
36C53×#323—22no—
37C2×#41×C53—21no—
38C22×Q8×C53—21no—
39C22×Q8×C53D411yescharacteristic
40C22×D4×C53D411yescharacteristic
41C2×M16×C53—21no—
42C2×M16×C53—21no—
43C4⋊Q8×C53—21no—
44C4⋊Q8×C53—22no—
45C4⋊Q8×C53D411yescharacteristic
46C4⋊Q8×C53D411yescharacteristic
47#126×C53—22no—
48#126×C53—22no—
49#126×C53—22no—
50#126×C53—21no—
51#125×C53—21no—
52#125×C53D411yescharacteristic
53#124×C53—21no—
54#124×C53—22no—
55#124×C53—21no—
56#124×C53—21no—
57#124×C53—21no—
58#124×C53—21no—
59#124×C53—22no—
60#124×C53D411yescharacteristic
61#123×C53—22no—
62#123×C53—22no—
63#123×C53—22no—
64#123×C53—22no—
65#123×C53—21no—
66C22⋊Q8×C53—22no—
67C22⋊Q8×C53—21no—
68C22⋊Q8×C53—22no—
69C22⋊Q8×C53—22no—
70C22⋊Q8×C53—22no—
71C22⋊Q8×C53—22no—
72C22⋊Q8×C53—22no—
73C22⋊Q8×C53—22no—
74C22⋊Q8×C53—21no—
75C22⋊Q8×C53C2×C411yescharacteristic
76C22⋊Q8×C53C2×C412yes—
77#121×C53—21no—
78#121×C53—22no—
79#121×C53—22no—
80#121×C53C2×C411yescharacteristic
81#120×C53—22no—
82C4×Q8×C53—21no—
83C4×Q8×C53—21no—
84C4×Q8×C53D411yescharacteristic
85C4×Q8×C53D412yes—
86C4×D4×C53—21no—
87C4×D4×C53—22no—
88C4×D4×C53—22no—
89C4×D4×C53—21no—
90C4×D4×C53D411yescharacteristic
91#117×C53—22no—
92#117×C53—21no—
93#117×C53—21no—
94C2×C22⋊C4×C53—22no—
95C2×C22⋊C4×C53C2×C412yes—
96C2×C22⋊C4×C53D412yes—
97C2×C42×C53C2311yescharacteristic
98C4⋊C8×C53—21no—
99C4⋊C8×C53—21no—
100C4⋊C8×C53—21no—
101C4⋊C8×C53—21no—
102#101×C53—42no—
103#100×C53—42no—
104C22⋊C8×C53—21no—
105C22⋊C8×C53—21no—
106#97×C53—21no—
107#97×C53—21no—
108C4×C8×C53—21no—
109C4×C8×C53—21no—
110C24×C53C22⋊C412yes—
111#41×C53—81no—
112#41×C53—81no—
113C2×Q8×C53—22no—
114C2×Q8×C53—21no—
115C2×Q8×C53—21no—
116C2×Q8×C53—21no—
117C2×Q8×C53—41no—
118C2×Q8×C53—41no—
119C2×Q8×C53—81no—
120C2×Q8×C53—81no—
121C2×Q8×C53C22⋊C411yescharacteristic
122C2×Q8×C53C22⋊C412yes—
123C2×D4×C53—21no—
124C2×D4×C53—22no—
125C2×D4×C53—22no—
126C2×D4×C53—21no—
127C2×D4×C53—21no—
128C2×D4×C53—41no—
129C2×D4×C53—41no—
130C2×D4×C53C22⋊C411yescharacteristic
131C22×C4×C53—22no—
132C22×C4×C53—22no—
133C22×C4×C53—22no—
134C22×C4×C53—21no—
135C22×C4×C53C22⋊C412yes—
136C22×C4×C53C22×C411yescharacteristic
137C22×C4×C53C2×D411yescharacteristic
138C22×C4×C53C2×D411yescharacteristic
139M16×C53—22no—
140M16×C53—41no—
141M16×C53—41no—
142M16×C53—41no—
143C2×C8×C53—41no—
144C2×C8×C53—41no—
145C2×C8×C53—41no—
146C2×C8×C53—41no—
147C4⋊C4×C53—22no—
148C4⋊C4×C53—22no—
149C4⋊C4×C53—22no—
150C4⋊C4×C53—22no—
151C4⋊C4×C53—21no—
152C4⋊C4×C53—22no—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—21no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—21no—
158C4⋊C4×C53—21no—
159C4⋊C4×C53—22no—
160C4⋊C4×C53C22⋊C411yescharacteristic
161C4⋊C4×C53C22⋊C411yescharacteristic
162C4⋊C4×C53C22⋊C412yes—
163C22⋊C4×C53—22no—
164C22⋊C4×C53—22no—
165C22⋊C4×C53—22no—
166C22⋊C4×C53—22no—
167C22⋊C4×C53—22no—
168C22⋊C4×C53—21no—
169C22⋊C4×C53—22no—
170C22⋊C4×C53—22no—
171C22⋊C4×C53—22no—
172C22⋊C4×C53—21no—
173C22⋊C4×C53—22no—
174C22⋊C4×C53—22no—
175C22⋊C4×C53—21no—
176C22⋊C4×C53—21no—
177C22⋊C4×C53—21no—
178C42×C53—22no—
179C42×C53—22no—
180C42×C53—21no—
181C42×C53C2×D411yescharacteristic
182C42×C53C2×D411yescharacteristic
183C42×C53C2×D411yescharacteristic
184C42×C53C2×D411yescharacteristic
185C23×C53—21no—
186C23×C53—22no—
187C23×C53—42no—
188C23×C53—42no—
189C23×C53—42no—
190C23×C53C2×C22⋊C411yescharacteristic
191Q8×C53—41no—
192Q8×C53—42no—
193Q8×C53—41no—
194Q8×C53—41no—
195Q8×C53—41no—
196Q8×C53—41no—
197Q8×C53—81no—
198Q8×C53—82no—
199D4×C53—41no—
200D4×C53—42no—
201D4×C53—41no—
202D4×C53—41no—
203D4×C53—81no—
204D4×C53—82no—
205D4×C53—82no—
206C2×C4×C53—21no—
207C2×C4×C53—21no—
208C2×C4×C53—21no—
209C2×C4×C53—22no—
210C2×C4×C53—21no—
211C2×C4×C53—22no—
212C2×C4×C53—22no—
213C2×C4×C53—21no—
214C2×C4×C53—41no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53—41no—
218C2×C4×C53—42no—
219C2×C4×C53—82no—
220C2×C4×C53—82no—
221C2×C4×C53—82no—
222C2×C4×C53C2×C22⋊C411yescharacteristic
223C2×C4×C53C2×C22⋊C411yescharacteristic
224C2×C4×C53#12011yescharacteristic
225C2×C4×C53#12011yescharacteristic
226C2×C4×C53#12011yescharacteristic
227C2×C4×C53#12011yescharacteristic
228C8×C53—42no—
229C8×C53—81no—
230C8×C53—81no—
231C8×C53—81no—
232#8484C5311yesmaximal, characteristic, direct product factor, Sylow
233C22×C53—21no—
234C22×C53—21no—
235C22×C53—21no—
236C22×C53—41no—
237C22×C53—42no—
238C22×C53—41no—
239C22×C53—42no—
240C22×C53—42no—
241C22×C53—41no—
242C22×C53—162no—
243C22×C53#37911yescharacteristic
244C4×C53—21no—
245C4×C53—21no—
246C4×C53—21no—
247C4×C53—21no—
248C4×C53—21no—
249C4×C53—41no—
250C4×C53—42no—
251C4×C53—42no—
252C4×C53—41no—
253C4×C53—41no—
254C4×C53—42no—
255C4×C53#45311yescharacteristic
256C4×C53#45511yescharacteristic
257#3561C2×C5311yescharacteristic
258#1967C2×C5311yescharacteristic, complemented
259#1559C2×C5311yescharacteristic, complemented
260#1529C2×C5311yescharacteristic, complemented
261#1495C2×C5311yescharacteristic, complemented
262#1485C2×C5311yescharacteristic, complemented
263#1469C2×C5311yescharacteristic, complemented
264C2×C53—41no—
265C2×C53—81no—
266C2×C53—82no—
267C2×C53#179111yescharacteristic
268C2×C53#179211yescharacteristic
269C2×C53#179611yescharacteristic
270#563—21no—
271#562—21no—
272#556—21no—
273#551—22no—
274D4×Q8—21no—
275#548—21no—
276#546—21no—
277#543C4×C5312yes—
278#542—21no—
279#542—22no—
280#538C4×C5312yes—
281#536—21no—
282#533C22×C5311yescharacteristic
283#529C22×C5311yescharacteristic
284C2×#124C22×C5311yescharacteristic
285#432C22×C5311yescharacteristic
286#431C22×C5311yescharacteristic
287#423C22×C5311yescharacteristic
288C2×#101—21no—
289C2×#100—21no—
290C4×M16C22×C5311yescharacteristic
291#333—21no—
292#332—21no—
293#330—21no—
294#329—21no—
295Q8⋊C8—22no—
296Q8⋊C8—21no—
297D4⋊C8—21no—
298#323—22no—
299C53#848411yescharacteristic, direct product factor, Sylow
300C2×#41—21no—
301C22×Q8—21no—
302C22×Q8D4×C5311yescharacteristic
303C22×D4D4×C5311yescharacteristic
304C2×M16—21no—
305C2×M16—21no—
306C4⋊Q8—21no—
307C4⋊Q8—22no—
308C4⋊Q8D4×C5311yescharacteristic
309C4⋊Q8D4×C5311yescharacteristic
310#126—22no—
311#126—22no—
312#126—22no—
313#126—21no—
314#125—21no—
315#125D4×C5311yescharacteristic
316#124—21no—
317#124—22no—
318#124—22no—
319#124—21no—
320#124—21no—
321#124—21no—
322#124—21no—
323#124D4×C5311yescharacteristic
324#123—22no—
325#123—22no—
326#123—22no—
327#123—21no—
328#123—22no—
329C22⋊Q8—22no—
330C22⋊Q8—21no—
331C22⋊Q8—22no—
332C22⋊Q8—22no—
333C22⋊Q8—22no—
334C22⋊Q8—21no—
335C22⋊Q8—22no—
336C22⋊Q8—22no—
337C22⋊Q8—22no—
338C22⋊Q8C2×C4×C5311yescharacteristic
339C22⋊Q8C2×C4×C5312yes—
340#121—21no—
341#121—22no—
342#121—22no—
343#121C2×C4×C5311yescharacteristic
344#120—22no—
345C4×Q8—21no—
346C4×Q8—21no—
347C4×Q8D4×C5311yescharacteristic
348C4×Q8D4×C5312yes—
349C4×D4—21no—
350C4×D4—22no—
351C4×D4—21no—
352C4×D4—22no—
353C4×D4D4×C5311yescharacteristic
354#117—22no—
355#117—21no—
356#117—21no—
357C2×C22⋊C4—22no—
358C2×C22⋊C4C2×C4×C5312yes—
359C2×C22⋊C4D4×C5312yes—
360C2×C42C23×C5311yescharacteristic
361C4⋊C8—21no—
362C4⋊C8—21no—
363C4⋊C8—21no—
364C4⋊C8—21no—
365#101—42no—
366#100—42no—
367C22⋊C8—21no—
368C22⋊C8—21no—
369#97—21no—
370#97—21no—
371C4×C8—21no—
372C4×C8—21no—
373C24C22⋊C4×C5312yes—
374#41—81no—
375#41—81no—
376C2×Q8—22no—
377C2×Q8—21no—
378C2×Q8—21no—
379C2×Q8—21no—
380C2×Q8—41no—
381C2×Q8—41no—
382C2×Q8—81no—
383C2×Q8—81no—
384C2×Q8C22⋊C4×C5311yescharacteristic
385C2×Q8C22⋊C4×C5312yes—
386C2×D4—21no—
387C2×D4—22no—
388C2×D4—22no—
389C2×D4—21no—
390C2×D4—21no—
391C2×D4—41no—
392C2×D4—41no—
393C2×D4C22⋊C4×C5311yescharacteristic
394C22×C4—22no—
395C22×C4—22no—
396C22×C4—22no—
397C22×C4—21no—
398C22×C4C22⋊C4×C5312yes—
399C22×C4C22×C4×C5311yescharacteristic
400C22×C4C2×D4×C5311yescharacteristic
401C22×C4C2×D4×C5311yescharacteristic
402M16—22no—
403M16—41no—
404M16—41no—
405M16—41no—
406C2×C8—41no—
407C2×C8—41no—
408C2×C8—41no—
409C2×C8—41no—
410C4⋊C4—22no—
411C4⋊C4—22no—
412C4⋊C4—22no—
413C4⋊C4—22no—
414C4⋊C4—21no—
415C4⋊C4—22no—
416C4⋊C4—21no—
417C4⋊C4—21no—
418C4⋊C4—21no—
419C4⋊C4—21no—
420C4⋊C4—21no—
421C4⋊C4—21no—
422C4⋊C4—22no—
423C4⋊C4C22⋊C4×C5311yescharacteristic
424C4⋊C4C22⋊C4×C5311yescharacteristic
425C4⋊C4C22⋊C4×C5312yes—
426C22⋊C4—22no—
427C22⋊C4—22no—
428C22⋊C4—22no—
429C22⋊C4—22no—
430C22⋊C4—22no—
431C22⋊C4—21no—
432C22⋊C4—22no—
433C22⋊C4—22no—
434C22⋊C4—22no—
435C22⋊C4—21no—
436C22⋊C4—22no—
437C22⋊C4—22no—
438C22⋊C4—21no—
439C22⋊C4—21no—
440C22⋊C4—21no—
441C42—22no—
442C42—22no—
443C42—21no—
444C42C2×D4×C5311yescharacteristic
445C42C2×D4×C5311yescharacteristic
446C42C2×D4×C5311yescharacteristic
447C42C2×D4×C5311yescharacteristic
448C23—22no—
449C23—21no—
450C23—42no—
451C23—42no—
452C23—42no—
453C23C2×C22⋊C4×C5311yescharacteristic
454Q8—41no—
455Q8—42no—
456Q8—41no—
457Q8—41no—
458Q8—41no—
459Q8—41no—
460Q8—81no—
461Q8—82no—
462D4—41no—
463D4—42no—
464D4—41no—
465D4—41no—
466D4—81no—
467D4—82no—
468D4—82no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—22no—
473C2×C4—21no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—21no—
477C2×C4—41no—
478C2×C4—41no—
479C2×C4—41no—
480C2×C4—41no—
481C2×C4—42no—
482C2×C4—82no—
483C2×C4—82no—
484C2×C4—82no—
485C2×C4C2×C22⋊C4×C5311yescharacteristic
486C2×C4C2×C22⋊C4×C5311yescharacteristic
487C2×C4#120×C5311yescharacteristic
488C2×C4#120×C5311yescharacteristic
489C2×C4#120×C5311yescharacteristic
490C2×C4#120×C5311yescharacteristic
491C8—42no—
492C8—81no—
493C8—81no—
494C8—81no—
495C22—21no—
496C22—21no—
497C22—21no—
498C22—42no—
499C22—41no—
500C22—41no—
501C22—42no—
502C22—41no—
503C22—42no—
504C22—162no—
505C22C53×#37911yescharacteristic
506C4—21no—
507C4—21no—
508C4—21no—
509C4—21no—
510C4—21no—
511C4—41no—
512C4—42no—
513C4—41no—
514C4—42no—
515C4—42no—
516C4—41no—
517C4C53×#45311yescharacteristic
518C4C53×#45511yescharacteristic
519C2—41no—
520C2—82no—
521C2—81no—
522C2C53×#179111yescharacteristic
523C2C53×#179211yescharacteristic
524C2C53×#179611yescharacteristic
subgroupG/H
center, socleC22×C53#379
commutatorC22×C4C22×C4×C53
Frattini, cosocleC2×C42C23×C53
prime radicalC53×#3561C2
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#1967C2semidirect
3. C53×#1559C2semidirect
4. C53×#1529C2semidirect
5. C53×#1495C2semidirect
6. C53×#1485C2semidirect
7. C53×#1469C2semidirect
232. #8484C53direct (N × H)
258. #1967C2×C53semidirect
259. #1559C2×C53semidirect
260. #1529C2×C53semidirect
261. #1495C2×C53semidirect
262. #1485C2×C53semidirect
263. #1469C2×C53semidirect
299. C53#8484direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#35612
C53×#19672
C53×#15592
C53×#15292
C53×#14952
C53×#14852
C53×#14692
#848453