Finite Group Databasebeta

← C53×#9781 #27217293

Subgroups — C53×#9781 #27217293

Order 13568.

564 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C53×#465C211yesmaximal, characteristic
2C53×#1996C211yesmaximal, characteristic, complemented
3C53×#1964C211yesmaximal, characteristic
4C53×#1807C211yesmaximal, characteristic, complemented
5C8×C22⋊C4×C53C211yesmaximal, characteristic, complemented
6C53×#1520C211yesmaximal, characteristic, complemented
7C53×#1519C211yesmaximal, characteristic
8C22×SD16×C53C411yescharacteristic, complemented
9C23×C8×C53C2211yescharacteristic
10C2×C22⋊Q8×C53C2211yescharacteristic
11C2×#121×C53C2211yescharacteristic
12C53×#465—21no—
13C53×#465—21no—
14C53×#465—21no—
15C53×#465—21no—
16C2×#106×C53—21no—
17C2×#106×C53C411yescharacteristic, complemented
18C2×#103×C53C411yescharacteristic, complemented
19C2×#102×C53C411yescharacteristic, complemented
20C2×#99×C53—22no—
21C2×C22⋊C8×C53C2211yescharacteristic
22C2×C4×C8×C53—21no—
23C53×#380C2211yescharacteristic
24C53×#380C2211yescharacteristic
25C4×C22⋊C4×C53C2211yescharacteristic
26C53×#342—22no—
27C53×#340—21no—
28C53×#336—21no—
29C53×#328—22no—
30C53×#327—22no—
31C22×Q8×C53C2×C411yescharacteristic
32C22×D4×C53—21no—
33C22×D4×C53C2×C411yescharacteristic
34C23×C4×C53C2311yescharacteristic
35C2×SD16×C53—21no—
36C2×SD16×C53—21no—
37C2×SD16×C53—42no—
38C22×C8×C53—21no—
39C22×C8×C53—42no—
40C22×C8×C53C2×C411yescharacteristic
41C22×C8×C53C2×C411yescharacteristic
42C22×C8×C53D411yescharacteristic
43C22×C8×C53D411yescharacteristic
44C22×C8×C53D411yescharacteristic
45C22×C8×C53D411yescharacteristic
46C22⋊Q8×C53—21no—
47C22⋊Q8×C53—21no—
48C22⋊Q8×C53—21no—
49C22⋊Q8×C53—21no—
50#121×C53—21no—
51#121×C53—21no—
52#121×C53—21no—
53#121×C53—21no—
54C2×C4⋊C4×C53—21no—
55C2×C4⋊C4×C53—21no—
56C2×C4⋊C4×C53—21no—
57C2×C4⋊C4×C53C2×C411yescharacteristic
58C2×C4⋊C4×C53C2×C411yescharacteristic
59C2×C22⋊C4×C53—21no—
60C2×C22⋊C4×C53—21no—
61C2×C22⋊C4×C53—21no—
62C2×C22⋊C4×C53—21no—
63C2×C22⋊C4×C53D412yes—
64C2×C42×C53—21no—
65#106×C53—21no—
66#106×C53—21no—
67#106×C53—42no—
68#103×C53—21no—
69#103×C53—21no—
70#102×C53—21no—
71#102×C53—21no—
72#99×C53—44no—
73C22⋊C8×C53—21no—
74C22⋊C8×C53D412yes—
75C4×C8×C53—42no—
76#95×C53—21no—
77#95×C53—21no—
78#95×C53—21no—
79C24×C53—21no—
80C24×C53C2×D411yescharacteristic
81C2×Q8×C53—21no—
82C2×Q8×C53—21no—
83C2×Q8×C53—41no—
84C2×Q8×C53—41no—
85C2×D4×C53—21no—
86C2×D4×C53—21no—
87C2×D4×C53—41no—
88C2×D4×C53—41no—
89C2×D4×C53—41no—
90C2×D4×C53—41no—
91C2×D4×C53—42no—
92C2×D4×C53—42no—
93C22×C4×C53—22no—
94C22×C4×C53—21no—
95C22×C4×C53—21no—
96C22×C4×C53—21no—
97C22×C4×C53—41no—
98C22×C4×C53—41no—
99C22×C4×C53—41no—
100C22×C4×C53—41no—
101C22×C4×C53C22×C411yescharacteristic
102C22×C4×C53C2×D411yescharacteristic
103C22×C4×C53C2×D411yescharacteristic
104C22×C4×C53C2×D411yescharacteristic
105C22×C4×C53#4111yescharacteristic
106C22×C4×C53#4111yescharacteristic
107SD16×C53—42no—
108SD16×C53—42no—
109C2×C8×C53—22no—
110C2×C8×C53—22no—
111C2×C8×C53—22no—
112C2×C8×C53—22no—
113C2×C8×C53—41no—
114C2×C8×C53—41no—
115C2×C8×C53—41no—
116C2×C8×C53—41no—
117C2×C8×C53—41no—
118C2×C8×C53C22⋊C411yescharacteristic
119C2×C8×C53C22⋊C411yescharacteristic
120C2×C8×C53C22⋊C411yescharacteristic
121C2×C8×C53C22⋊C411yescharacteristic
122C4⋊C4×C53—21no—
123C4⋊C4×C53—21no—
124C4⋊C4×C53—21no—
125C4⋊C4×C53—21no—
126C4⋊C4×C53—41no—
127C4⋊C4×C53—41no—
128C4⋊C4×C53—41no—
129C4⋊C4×C53—41no—
130C4⋊C4×C53—41no—
131C4⋊C4×C53—41no—
132C22⋊C4×C53—22no—
133C22⋊C4×C53—41no—
134C22⋊C4×C53—41no—
135C22⋊C4×C53—41no—
136C22⋊C4×C53—41no—
137C22⋊C4×C53—42no—
138C22⋊C4×C53—42no—
139C22⋊C4×C53SD1614yescomplemented
140C42×C53—41no—
141C42×C53—41no—
142C23×C53—41no—
143C23×C53—41no—
144C23×C53—41no—
145C23×C53—42no—
146C23×C53—81no—
147C23×C53—81no—
148C23×C53C4×D411yescharacteristic
149C23×C53#12111yescharacteristic
150C23×C53#12711yescharacteristic
151C23×C53C2×SD1612yes—
152C23×C53#13512yes—
153Q8×C53—41no—
154Q8×C53—41no—
155Q8×C53—81no—
156D4×C53—41no—
157D4×C53—41no—
158D4×C53—81no—
159D4×C53—82no—
160D4×C53—81no—
161D4×C53—81no—
162C2×C4×C53—21no—
163C2×C4×C53—21no—
164C2×C4×C53—21no—
165C2×C4×C53—21no—
166C2×C4×C53—21no—
167C2×C4×C53—21no—
168C2×C4×C53—24no—
169C2×C4×C53—41no—
170C2×C4×C53—41no—
171C2×C4×C53—41no—
172C2×C4×C53—42no—
173C2×C4×C53—42no—
174C2×C4×C53—41no—
175C2×C4×C53—41no—
176C2×C4×C53—41no—
177C2×C4×C53—42no—
178C2×C4×C53—81no—
179C2×C4×C53—81no—
180C2×C4×C53—81no—
181C2×C4×C53—81no—
182C2×C4×C53—81no—
183C2×C4×C53—81no—
184C2×C4×C53—81no—
185C2×C4×C53—81no—
186C2×C4×C53C2×C22⋊C411yescharacteristic
187C2×C4×C53C4×D411yescharacteristic
188C2×C4×C53#12111yescharacteristic
189C2×C4×C53#12411yescharacteristic
190C8×C53—21no—
191C8×C53—21no—
192C8×C53—21no—
193C8×C53—42no—
194C8×C53#9912yescomplemented
195#9781C5311yesmaximal, characteristic, direct product factor, Sylow
196C22×C53—21no—
197C22×C53—21no—
198C22×C53—21no—
199C22×C53—21no—
200C22×C53—21no—
201C22×C53—21no—
202C22×C53—42no—
203C22×C53—42no—
204C22×C53—81no—
205C22×C53—81no—
206C22×C53—81no—
207C22×C53—162no—
208C22×C53#39011yescharacteristic
209C22×C53C4×SD1612yes—
210C22×C53#46512yes—
211C22×C53#49211yescharacteristic
212C22×C53#49511yescharacteristic
213C4×C53—21no—
214C4×C53—21no—
215C4×C53—21no—
216C4×C53—44no—
217C4×C53—81no—
218C4×C53—81no—
219C4×C53—81no—
220C4×C53—161no—
221C4×C53—161no—
222C4×C53C2×#9911yescharacteristic
223C4×C53#41011yescharacteristic
224C2×#465C2×C5311yescharacteristic
225#1996C2×C5311yescharacteristic, complemented
226#1964C2×C5311yescharacteristic
227#1807C2×C5311yescharacteristic, complemented
228C8×C22⋊C4C2×C5311yescharacteristic, complemented
229#1520C2×C5311yescharacteristic, complemented
230#1519C2×C5311yescharacteristic
231C2×C53—22no—
232C2×C53—21no—
233C2×C53—21no—
234C2×C53—22no—
235C2×C53—161no—
236C2×C53#196411yescharacteristic
237C2×C53#196811yescharacteristic
238C2×C53#197611yescharacteristic
239C22×SD16C4×C5311yescharacteristic, complemented
240C23×C8C22×C5311yescharacteristic
241C2×C22⋊Q8C22×C5311yescharacteristic
242C2×#121C22×C5311yescharacteristic
243#465—21no—
244#465—21no—
245#465—21no—
246#465—21no—
247C2×#106—21no—
248C2×#106C4×C5311yescharacteristic, complemented
249C2×#103C4×C5311yescharacteristic, complemented
250C2×#102C4×C5311yescharacteristic, complemented
251C2×#99—22no—
252C2×C22⋊C8C22×C5311yescharacteristic
253C2×C4×C8—21no—
254#380C22×C5311yescharacteristic
255#380C22×C5311yescharacteristic
256C4×C22⋊C4C22×C5311yescharacteristic
257#342—22no—
258#340—21no—
259#336—21no—
260#328—22no—
261#327—22no—
262C53#978111yescharacteristic, direct product factor, Sylow
263C22×Q8C2×C4×C5311yescharacteristic
264C22×D4—21no—
265C22×D4C2×C4×C5311yescharacteristic
266C23×C4C23×C5311yescharacteristic
267C2×SD16—21no—
268C2×SD16—21no—
269C2×SD16—42no—
270C22×C8—21no—
271C22×C8—42no—
272C22×C8C2×C4×C5311yescharacteristic
273C22×C8C2×C4×C5311yescharacteristic
274C22×C8D4×C5311yescharacteristic
275C22×C8D4×C5311yescharacteristic
276C22×C8D4×C5311yescharacteristic
277C22×C8D4×C5311yescharacteristic
278C22⋊Q8—21no—
279C22⋊Q8—21no—
280C22⋊Q8—21no—
281C22⋊Q8—21no—
282#121—21no—
283#121—21no—
284#121—21no—
285#121—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4C2×C4×C5311yescharacteristic
290C2×C4⋊C4C2×C4×C5311yescharacteristic
291C2×C22⋊C4—21no—
292C2×C22⋊C4—21no—
293C2×C22⋊C4—21no—
294C2×C22⋊C4—21no—
295C2×C22⋊C4D4×C5312yes—
296C2×C42—21no—
297#106—21no—
298#106—21no—
299#106—42no—
300#103—21no—
301#103—21no—
302#102—21no—
303#102—21no—
304#99—44no—
305C22⋊C8—21no—
306C22⋊C8D4×C5312yes—
307C4×C8—42no—
308#95—21no—
309#95—21no—
310#95—21no—
311C24—21no—
312C24C2×D4×C5311yescharacteristic
313C2×Q8—21no—
314C2×Q8—21no—
315C2×Q8—41no—
316C2×Q8—41no—
317C2×D4—21no—
318C2×D4—21no—
319C2×D4—41no—
320C2×D4—41no—
321C2×D4—41no—
322C2×D4—41no—
323C2×D4—42no—
324C2×D4—42no—
325C22×C4—22no—
326C22×C4—21no—
327C22×C4—21no—
328C22×C4—21no—
329C22×C4—41no—
330C22×C4—41no—
331C22×C4—41no—
332C22×C4—41no—
333C22×C4C22×C4×C5311yescharacteristic
334C22×C4C2×D4×C5311yescharacteristic
335C22×C4C2×D4×C5311yescharacteristic
336C22×C4C2×D4×C5311yescharacteristic
337C22×C4#41×C5311yescharacteristic
338C22×C4#41×C5311yescharacteristic
339SD16—42no—
340SD16—42no—
341C2×C8—22no—
342C2×C8—22no—
343C2×C8—22no—
344C2×C8—22no—
345C2×C8—41no—
346C2×C8—41no—
347C2×C8—41no—
348C2×C8—41no—
349C2×C8—41no—
350C2×C8C22⋊C4×C5311yescharacteristic
351C2×C8C22⋊C4×C5311yescharacteristic
352C2×C8C22⋊C4×C5311yescharacteristic
353C2×C8C22⋊C4×C5311yescharacteristic
354C4⋊C4—21no—
355C4⋊C4—21no—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—41no—
359C4⋊C4—41no—
360C4⋊C4—41no—
361C4⋊C4—41no—
362C4⋊C4—41no—
363C4⋊C4—41no—
364C22⋊C4—22no—
365C22⋊C4—41no—
366C22⋊C4—41no—
367C22⋊C4—41no—
368C22⋊C4—41no—
369C22⋊C4—42no—
370C22⋊C4—42no—
371C22⋊C4SD16×C5314yescomplemented
372C42—41no—
373C42—41no—
374C23—41no—
375C23—41no—
376C23—41no—
377C23—42no—
378C23—81no—
379C23—81no—
380C23C4×D4×C5311yescharacteristic
381C23#121×C5311yescharacteristic
382C23#127×C5311yescharacteristic
383C23C2×SD16×C5312yes—
384C23#135×C5312yes—
385Q8—41no—
386Q8—41no—
387Q8—81no—
388D4—41no—
389D4—41no—
390D4—81no—
391D4—81no—
392D4—81no—
393D4—82no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—24no—
401C2×C4—41no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—42no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—42no—
410C2×C4—81no—
411C2×C4—81no—
412C2×C4—81no—
413C2×C4—81no—
414C2×C4—81no—
415C2×C4—81no—
416C2×C4—81no—
417C2×C4—81no—
418C2×C4C2×C22⋊C4×C5311yescharacteristic
419C2×C4C4×D4×C5311yescharacteristic
420C2×C4#121×C5311yescharacteristic
421C2×C4#124×C5311yescharacteristic
422C8—21no—
423C8—21no—
424C8—21no—
425C8—42no—
426C8#99×C5312yescomplemented
427C22—21no—
428C22—21no—
429C22—21no—
430C22—21no—
431C22—21no—
432C22—21no—
433C22—42no—
434C22—42no—
435C22—81no—
436C22—81no—
437C22—81no—
438C22—162no—
439C22C53×#39011yescharacteristic
440C22C4×SD16×C5312yes—
441C22C53×#46512yes—
442C22C53×#49211yescharacteristic
443C22C53×#49511yescharacteristic
444C4—21no—
445C4—21no—
446C4—21no—
447C4—44no—
448C4—81no—
449C4—81no—
450C4—81no—
451C4—161no—
452C4—161no—
453C4C2×#99×C5311yescharacteristic
454C4C53×#41011yescharacteristic
455C2—22no—
456C2—21no—
457C2—21no—
458C2—22no—
459C2—161no—
460C2C53×#196411yescharacteristic
461C2C53×#196811yescharacteristic
462C2C53×#197611yescharacteristic
subgroupG/H
center, socleC22×C53#390
commutatorC22×C4C22×C4×C53
Frattini, cosocleC23×C4C23×C53
prime radicalC2×#121×C53C22
LCSG ▷ C22×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×#1996C2semidirect
4. C53×#1807C2semidirect
5. C8×C22⋊C4×C53C2semidirect
6. C53×#1520C2semidirect
8. C22×SD16×C53C4semidirect
17. C2×#106×C53C4semidirect
18. C2×#103×C53C4semidirect
19. C2×#102×C53C4semidirect
139. C22⋊C4×C53SD16semidirect
194. C8×C53#99semidirect
195. #9781C53direct (N × H)
225. #1996C2×C53semidirect
227. #1807C2×C53semidirect
228. C8×C22⋊C4C2×C53semidirect
229. #1520C2×C53semidirect
239. C22×SD16C4×C53semidirect
248. C2×#106C4×C53semidirect
249. C2×#103C4×C53semidirect
250. C2×#102C4×C53semidirect
262. C53#9781direct (N × H)
371. C22⋊C4SD16×C53semidirect
426. C8#99×C53semidirect

Maximal subgroups

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

subgroupindex
C2×C53×#4652
C53×#19962
C53×#19642
C53×#18072
C8×C22⋊C4×C532
C53×#15202
C53×#15192
#978153