Finite Group Databasebeta

← C53×#9943 #27217455

Subgroups — C53×#9943 #27217455

Order 13568.

798 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 798 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3007C211yesmaximal, characteristic
2C53×#2583C211yesmaximal, characteristic, complemented
3C53×#2406C211yesmaximal, characteristic
4C53×#1789C211yesmaximal, characteristic
5C53×#1511C211yesmaximal, characteristic, complemented
6C53×#1463C211yesmaximal, characteristic, complemented
7C53×#1284C211yesmaximal, characteristic, complemented
8C22×#41×C53C2211yescharacteristic
9C2×#137×C53—21no—
10C2×#123×C53—21no—
11C2×C22⋊Q8×C53C2211yescharacteristic
12C22×C4⋊C4×C53C2211yescharacteristic
13C53×#452—22no—
14C53×#451—21no—
15C53×#450—21no—
16C2×#104×C53—22no—
17C53×#417—21no—
18C53×#407C2211yescharacteristic
19C53×#407C2211yescharacteristic
20C53×#398—21no—
21C53×#395—21no—
22C53×#395—21no—
23C53×#394—21no—
24C53×#389—22no—
25C53×#386—21no—
26C53×#385—21no—
27C53×#385—21no—
28C53×#385—22no—
29C53×#381C2211yescharacteristic
30C4×C22⋊C4×C53C2211yescharacteristic
31C53×#328—22no—
32C53×#324—22no—
33C2×#41×C53—21no—
34C2×#41×C53—21no—
35C2×#41×C53—21no—
36C2×#41×C53—21no—
37C2×#41×C53—44no—
38C22×Q8×C53D411yescharacteristic
39C22×Q8×C53D411yescharacteristic
40C22×D4×C53—21no—
41C22×D4×C53D411yescharacteristic
42C23×C4×C53—21no—
43C23×C4×C53D411yescharacteristic
44C23×C4×C53D411yescharacteristic
45C23×C4×C53C2311yescharacteristic
46#137×C53—44no—
47C2×Q16×C53—22no—
48C2×SD16×C53—22no—
49C2×M16×C53—21no—
50C2×M16×C53—21no—
51#123×C53—44no—
52C22⋊Q8×C53—21no—
53C22⋊Q8×C53—21no—
54C22⋊Q8×C53—21no—
55C22⋊Q8×C53—21no—
56C2×C4⋊C4×C53—22no—
57C2×C4⋊C4×C53—22no—
58C2×C4⋊C4×C53—22no—
59C2×C4⋊C4×C53—21no—
60C2×C4⋊C4×C53—21no—
61C2×C4⋊C4×C53—21no—
62C2×C4⋊C4×C53—21no—
63C2×C4⋊C4×C53—21no—
64C2×C4⋊C4×C53—21no—
65C2×C4⋊C4×C53—21no—
66C2×C4⋊C4×C53D411yescharacteristic
67C2×C22⋊C4×C53—22no—
68C2×C22⋊C4×C53—22no—
69C2×C22⋊C4×C53—21no—
70C2×C22⋊C4×C53—21no—
71C2×C22⋊C4×C53—21no—
72C2×C42×C53D412yes—
73#104×C53—44no—
74#103×C53—22no—
75#102×C53—22no—
76C22⋊C8×C53—21no—
77C22⋊C8×C53—21no—
78C22⋊C8×C53—21no—
79C22⋊C8×C53D412yes—
80#95×C53—22no—
81#95×C53—22no—
82#95×C53—22no—
83#95×C53—21no—
84#95×C53—21no—
85#95×C53D412yes—
86C24×C53—21no—
87C24×C53C2×D411yescharacteristic
88#41×C53—24no—
89#41×C53—24no—
90#41×C53—42no—
91#41×C53—42no—
92#41×C53—42no—
93#41×C53—42no—
94#41×C53—42no—
95#41×C53—42no—
96C2×Q8×C53—21no—
97C2×Q8×C53—21no—
98C2×Q8×C53—21no—
99C2×Q8×C53—21no—
100C2×Q8×C53—42no—
101C2×Q8×C53—42no—
102C2×D4×C53—24no—
103C2×D4×C53—21no—
104C2×D4×C53—21no—
105C2×D4×C53—42no—
106C2×D4×C53—42no—
107C2×D4×C53—41no—
108C2×D4×C53—41no—
109C22×C4×C53—22no—
110C22×C4×C53—21no—
111C22×C4×C53—21no—
112C22×C4×C53—21no—
113C22×C4×C53—21no—
114C22×C4×C53—22no—
115C22×C4×C53—21no—
116C22×C4×C53—21no—
117C22×C4×C53—21no—
118C22×C4×C53—21no—
119C22×C4×C53—21no—
120C22×C4×C53—42no—
121C22×C4×C53—42no—
122C22×C4×C53—42no—
123C22×C4×C53—42no—
124C22×C4×C53—41no—
125C22×C4×C53—41no—
126C22×C4×C53—42no—
127C22×C4×C53C2×D411yescharacteristic
128C22×C4×C53C2×D411yescharacteristic
129C22×C4×C53C2×D411yescharacteristic
130C22×C4×C53C2×D411yescharacteristic
131C22×C4×C53C2×D411yescharacteristic
132C22×C4×C53#4111yescharacteristic
133Q16×C53—44no—
134SD16×C53—44no—
135M16×C53—42no—
136M16×C53—42no—
137C2×C8×C53—22no—
138C2×C8×C53—41no—
139C4⋊C4×C53—21no—
140C4⋊C4×C53—21no—
141C4⋊C4×C53—44no—
142C4⋊C4×C53—44no—
143C4⋊C4×C53—41no—
144C4⋊C4×C53—41no—
145C4⋊C4×C53—41no—
146C4⋊C4×C53—41no—
147C4⋊C4×C53—41no—
148C4⋊C4×C53—41no—
149C4⋊C4×C53—82no—
150C22⋊C4×C53—24no—
151C22⋊C4×C53—44no—
152C22⋊C4×C53—44no—
153C22⋊C4×C53—42no—
154C22⋊C4×C53—41no—
155C22⋊C4×C53—41no—
156C42×C53—42no—
157C23×C53—22no—
158C23×C53—22no—
159C23×C53—22no—
160C23×C53—41no—
161C23×C53—41no—
162C23×C53—42no—
163C23×C53—41no—
164C23×C53—42no—
165C23×C53#12011yescharacteristic
166C23×C53#12011yescharacteristic
167C23×C53#12111yescharacteristic
168Q8×C53—22no—
169Q8×C53—22no—
170Q8×C53—41no—
171Q8×C53—41no—
172Q8×C53—42no—
173Q8×C53—81no—
174D4×C53—22no—
175D4×C53—22no—
176D4×C53—41no—
177D4×C53—41no—
178D4×C53—44no—
179D4×C53—44no—
180C2×C4×C53—22no—
181C2×C4×C53—22no—
182C2×C4×C53—22no—
183C2×C4×C53—22no—
184C2×C4×C53—22no—
185C2×C4×C53—22no—
186C2×C4×C53—24no—
187C2×C4×C53—41no—
188C2×C4×C53—41no—
189C2×C4×C53—41no—
190C2×C4×C53—41no—
191C2×C4×C53—44no—
192C2×C4×C53—44no—
193C2×C4×C53—41no—
194C2×C4×C53—41no—
195C2×C4×C53—41no—
196C2×C4×C53—41no—
197C2×C4×C53—42no—
198C2×C4×C53—41no—
199C2×C4×C53—41no—
200C2×C4×C53—42no—
201C2×C4×C53—42no—
202C2×C4×C53—42no—
203C2×C4×C53—42no—
204C2×C4×C53—42no—
205C2×C4×C53—41no—
206C2×C4×C53—41no—
207C2×C4×C53—41no—
208C2×C4×C53—42no—
209C2×C4×C53—42no—
210C2×C4×C53—42no—
211C2×C4×C53—81no—
212C2×C4×C53#12011yescharacteristic
213C2×C4×C53#12111yescharacteristic
214C2×C4×C53#12111yescharacteristic
215C2×C4×C53#12711yescharacteristic
216C8×C53—42no—
217C8×C53—81no—
218#9943C5311yesmaximal, characteristic, direct product factor, Sylow
219C22×C53—21no—
220C22×C53—21no—
221C22×C53—21no—
222C22×C53—21no—
223C22×C53—21no—
224C22×C53—21no—
225C22×C53—21no—
226C22×C53—44no—
227C22×C53—44no—
228C22×C53—42no—
229C22×C53—82no—
230C22×C53—81no—
231C22×C53—81no—
232C22×C53#39211yescharacteristic
233C22×C53#45612yes—
234C22×C53B(4,𝔽2)12yes—
235C4×C53—21no—
236C4×C53—21no—
237C4×C53—21no—
238C4×C53—21no—
239C4×C53—44no—
240C4×C53—42no—
241C4×C53—42no—
242C4×C53—82no—
243C4×C53—81no—
244C4×C53—81no—
245C4×C53—81no—
246#3007C2×C5311yescharacteristic
247#2583C2×C5311yescharacteristic, complemented
248#2406C2×C5311yescharacteristic
249#1789C2×C5311yescharacteristic
250#1511C2×C5311yescharacteristic, complemented
251#1463C2×C5311yescharacteristic, complemented
252#1284C2×C5311yescharacteristic, complemented
253C2×C53—22no—
254C2×C53—22no—
255C2×C53—41no—
256C2×C53—82no—
257C2×C53#200511yescharacteristic
258C2×C53#201011yescharacteristic
259C2×C53#202111yescharacteristic
260C22×#41C22×C5311yescharacteristic
261C2×#137—21no—
262C2×#123—21no—
263C2×C22⋊Q8C22×C5311yescharacteristic
264C22×C4⋊C4C22×C5311yescharacteristic
265#452—22no—
266#451—21no—
267#450—21no—
268C2×#104—22no—
269#417—21no—
270#407C22×C5311yescharacteristic
271#407C22×C5311yescharacteristic
272#398—21no—
273#395—21no—
274#395—21no—
275#394—21no—
276#389—22no—
277#386—21no—
278#385—21no—
279#385—21no—
280#385—22no—
281#381C22×C5311yescharacteristic
282C4×C22⋊C4C22×C5311yescharacteristic
283#328—22no—
284#324—22no—
285C53#994311yescharacteristic, direct product factor, Sylow
286C2×#41—21no—
287C2×#41—21no—
288C2×#41—21no—
289C2×#41—21no—
290C2×#41—44no—
291C22×Q8D4×C5311yescharacteristic
292C22×Q8D4×C5311yescharacteristic
293C22×D4—21no—
294C22×D4D4×C5311yescharacteristic
295C23×C4—21no—
296C23×C4D4×C5311yescharacteristic
297C23×C4D4×C5311yescharacteristic
298C23×C4C23×C5311yescharacteristic
299#137—44no—
300C2×Q16—22no—
301C2×SD16—22no—
302C2×M16—21no—
303C2×M16—21no—
304#123—44no—
305C22⋊Q8—21no—
306C22⋊Q8—21no—
307C22⋊Q8—21no—
308C22⋊Q8—21no—
309C2×C4⋊C4—22no—
310C2×C4⋊C4—22no—
311C2×C4⋊C4—21no—
312C2×C4⋊C4—21no—
313C2×C4⋊C4—21no—
314C2×C4⋊C4—21no—
315C2×C4⋊C4—21no—
316C2×C4⋊C4—21no—
317C2×C4⋊C4—21no—
318C2×C4⋊C4—22no—
319C2×C4⋊C4D4×C5311yescharacteristic
320C2×C22⋊C4—21no—
321C2×C22⋊C4—22no—
322C2×C22⋊C4—22no—
323C2×C22⋊C4—21no—
324C2×C22⋊C4—21no—
325C2×C42D4×C5312yes—
326#104—44no—
327#103—22no—
328#102—22no—
329C22⋊C8—21no—
330C22⋊C8—21no—
331C22⋊C8—21no—
332C22⋊C8D4×C5312yes—
333#95—22no—
334#95—22no—
335#95—22no—
336#95—21no—
337#95—21no—
338#95D4×C5312yes—
339C24—21no—
340C24C2×D4×C5311yescharacteristic
341#41—24no—
342#41—24no—
343#41—42no—
344#41—42no—
345#41—42no—
346#41—42no—
347#41—42no—
348#41—42no—
349C2×Q8—21no—
350C2×Q8—21no—
351C2×Q8—21no—
352C2×Q8—21no—
353C2×Q8—42no—
354C2×Q8—42no—
355C2×D4—24no—
356C2×D4—21no—
357C2×D4—21no—
358C2×D4—42no—
359C2×D4—42no—
360C2×D4—41no—
361C2×D4—41no—
362C22×C4—22no—
363C22×C4—21no—
364C22×C4—21no—
365C22×C4—21no—
366C22×C4—21no—
367C22×C4—22no—
368C22×C4—21no—
369C22×C4—21no—
370C22×C4—21no—
371C22×C4—21no—
372C22×C4—21no—
373C22×C4—42no—
374C22×C4—42no—
375C22×C4—42no—
376C22×C4—42no—
377C22×C4—41no—
378C22×C4—41no—
379C22×C4—42no—
380C22×C4C2×D4×C5311yescharacteristic
381C22×C4C2×D4×C5311yescharacteristic
382C22×C4C2×D4×C5311yescharacteristic
383C22×C4C2×D4×C5311yescharacteristic
384C22×C4C2×D4×C5311yescharacteristic
385C22×C4#41×C5311yescharacteristic
386Q16—44no—
387SD16—44no—
388M16—42no—
389M16—42no—
390C2×C8—22no—
391C2×C8—41no—
392C4⋊C4—21no—
393C4⋊C4—21no—
394C4⋊C4—44no—
395C4⋊C4—44no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C4⋊C4—41no—
400C4⋊C4—41no—
401C4⋊C4—41no—
402C4⋊C4—82no—
403C22⋊C4—24no—
404C22⋊C4—44no—
405C22⋊C4—42no—
406C22⋊C4—44no—
407C22⋊C4—41no—
408C22⋊C4—41no—
409C42—42no—
410C23—22no—
411C23—22no—
412C23—22no—
413C23—41no—
414C23—42no—
415C23—41no—
416C23—41no—
417C23—42no—
418C23#120×C5311yescharacteristic
419C23#120×C5311yescharacteristic
420C23#121×C5311yescharacteristic
421Q8—22no—
422Q8—22no—
423Q8—41no—
424Q8—41no—
425Q8—42no—
426Q8—81no—
427D4—22no—
428D4—22no—
429D4—41no—
430D4—41no—
431D4—44no—
432D4—44no—
433C2×C4—22no—
434C2×C4—22no—
435C2×C4—24no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—22no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—41no—
444C2×C4—44no—
445C2×C4—44no—
446C2×C4—42no—
447C2×C4—41no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—41no—
451C2×C4—41no—
452C2×C4—41no—
453C2×C4—42no—
454C2×C4—42no—
455C2×C4—42no—
456C2×C4—42no—
457C2×C4—42no—
458C2×C4—41no—
459C2×C4—41no—
460C2×C4—41no—
461C2×C4—42no—
462C2×C4—42no—
463C2×C4—42no—
464C2×C4—81no—
465C2×C4#120×C5311yescharacteristic
466C2×C4#121×C5311yescharacteristic
467C2×C4#121×C5311yescharacteristic
468C2×C4#127×C5311yescharacteristic
469C8—42no—
470C8—81no—
471C22—21no—
472C22—21no—
473C22—21no—
474C22—21no—
475C22—21no—
476C22—21no—
477C22—21no—
478C22—44no—
479C22—44no—
480C22—42no—
481C22—82no—
482C22—81no—
483C22—81no—
484C22C53×#39211yescharacteristic
485C22C53×#45612yes—
486C22C53×B(4,𝔽2)12yes—
487C4—21no—
488C4—21no—
489C4—21no—
490C4—21no—
491C4—44no—
492C4—42no—
493C4—42no—
494C4—82no—
495C4—81no—
496C4—81no—
497C4—81no—
498C2—22no—
499C2—22no—
500C2—41no—
501C2—82no—
502C2C53×#200511yescharacteristic
503C2C53×#201011yescharacteristic
504C2C53×#202111yescharacteristic
subgroupG/H
center, socleC22×C53#392
commutator, Frattini, cosocleC23×C4C23×C53
prime radicalC22×#41×C53C22
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C53 ◁ C23×C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#2583C2semidirect
5. C53×#1511C2semidirect
6. C53×#1463C2semidirect
7. C53×#1284C2semidirect
218. #9943C53direct (N × H)
247. #2583C2×C53semidirect
250. #1511C2×C53semidirect
251. #1463C2×C53semidirect
252. #1284C2×C53semidirect
285. C53#9943direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#30072
C53×#25832
C53×#24062
C53×#17892
C53×#15112
C53×#14632
C53×#12842
#994353