Finite Group Databasebeta

← C53×#19564 #27227076

Subgroups — C53×#19564 #27227076

Order 13568.

626 conjugacy classes of subgroups.

The subgroup lattice has 444 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3255C211yesmaximal, characteristic, complemented
2C53×#3208C211yesmaximal, characteristic
3C53×#3208C211yesmaximal, characteristic
4C53×#2986C211yesmaximal, characteristic, complemented
5C53×#1702C212yesmaximal, complemented
6C53×#1686C212yesmaximal, complemented
7C53×#1682C212yesmaximal, complemented
8C53×#1666C212yesmaximal, complemented
9C53×#1574C211yesmaximal, characteristic, complemented
10C53×#1545C211yesmaximal, characteristic
11C53×#1544C211yesmaximal, characteristic, complemented
12C53×#533C2211yescharacteristic
13C53×#533C2211yescharacteristic
14C2×#127×C53C2211yescharacteristic
15C53×#492—21no—
16C53×#492C2212yes—
17C53×#486—21no—
18C53×#486C2212yes—
19C53×#475—21no—
20C53×#475C2212yescomplemented
21C53×#470—22no—
22C53×#462—21no—
23C53×#462C2212yescomplemented
24C53×#460—21no—
25C53×#460—21no—
26C53×#460C2212yes—
27C53×#460C2212yes—
28C53×#450—22no—
29C53×#450—22no—
30C53×#446—21no—
31C53×#446C2212yescomplemented
32C8×Q8×C53—21no—
33C8×Q8×C53C2212yescomplemented
34C53×#433—22no—
35C53×#431C2211yescharacteristic
36C53×#431C2211yescharacteristic
37C53×#431C2211yescharacteristic
38C53×#423C2211yescharacteristic
39C53×#335C2212yescomplemented
40C53×#334C2212yescomplemented
41C53×#332C2211yescharacteristic, complemented
42C53×#332C2211yescharacteristic, complemented
43C53×#332C2211yescharacteristic, complemented
44C53×#332C2211yescharacteristic, complemented
45C53×#331C2212yes—
46C53×#331C2212yes—
47C22×D4×C53—21no—
48C22×D4×C53—21no—
49C22×D4×C53—21no—
50C2×Q16×C53—22no—
51C2×SD16×C53—22no—
52C2×SD16×C53—22no—
53C2×SD16×C53—22no—
54C2×SD16×C53—22no—
55C2×M16×C53—21no—
56C4⋊Q8×C53D411yescharacteristic
57C4⋊Q8×C53D411yescharacteristic
58C4⋊Q8×C53C2311yescharacteristic
59C4⋊Q8×C53C2311yescharacteristic
60#127×C53—21no—
61#127×C53—21no—
62#127×C53—21no—
63#127×C53C2312yes—
64#125×C53—21no—
65#125×C53—21no—
66C22⋊Q8×C53—21no—
67C22⋊Q8×C53—21no—
68C22⋊Q8×C53—21no—
69C22⋊Q8×C53D412yes—
70C4×Q8×C53—21no—
71C4×Q8×C53—21no—
72C4×Q8×C53—21no—
73C4×Q8×C53C2312yes—
74#117×C53—21no—
75#117×C53—21no—
76C2×C42×C53C2311yescharacteristic
77#106×C53—22no—
78C4⋊C8×C53—22no—
79C4⋊C8×C53—22no—
80C4⋊C8×C53—21no—
81C4⋊C8×C53D411yescharacteristic
82C4⋊C8×C53D411yescharacteristic
83C4⋊C8×C53C2311yescharacteristic
84C4⋊C8×C53C2311yescharacteristic
85C4⋊C8×C53C2312yes—
86C4⋊C8×C53C2311yescharacteristic
87C4⋊C8×C53C2311yescharacteristic
88#103×C53—22no—
89#103×C53—22no—
90#102×C53—22no—
91#102×C53—22no—
92#102×C53—22no—
93#102×C53—22no—
94C22⋊C8×C53—21no—
95C22⋊C8×C53—21no—
96C22⋊C8×C53D412yes—
97#97×C53—22no—
98C4×C8×C53—22no—
99C4×C8×C53—21no—
100C4×C8×C53—21no—
101C4×C8×C53C2312yes—
102C24×C53—41no—
103C2×Q8×C53—21no—
104C2×Q8×C53—21no—
105C2×Q8×C53—21no—
106C2×Q8×C53C2×D412yes—
107C2×D4×C53—22no—
108C2×D4×C53—22no—
109C2×D4×C53—22no—
110C2×D4×C53—41no—
111C2×D4×C53—41no—
112C2×D4×C53—41no—
113C2×D4×C53—81no—
114C2×D4×C53—81no—
115C2×D4×C53—81no—
116C2×D4×C53—81no—
117C2×D4×C53—82no—
118C22×C4×C53C2×D411yescharacteristic
119C22×C4×C53C2×D411yescharacteristic
120C22×C4×C53#4111yescharacteristic
121Q16×C53—82no—
122SD16×C53—82no—
123SD16×C53—82no—
124SD16×C53—82no—
125SD16×C53—82no—
126M16×C53—81no—
127C2×C8×C53—21no—
128C2×C8×C53—21no—
129C2×C8×C53—21no—
130C2×C8×C53—21no—
131C2×C8×C53—22no—
132C2×C8×C53C2×D412yes—
133C2×C8×C53C2×D412yes—
134C4⋊C4×C53—21no—
135C4⋊C4×C53—21no—
136C4⋊C4×C53—21no—
137C4⋊C4×C53—22no—
138C4⋊C4×C53—22no—
139C4⋊C4×C53—21no—
140C4⋊C4×C53—21no—
141C4⋊C4×C53—21no—
142C4⋊C4×C53—21no—
143C4⋊C4×C53—41no—
144C4⋊C4×C53—41no—
145C4⋊C4×C53C2×D412yes—
146C22⋊C4×C53—22no—
147C22⋊C4×C53—41no—
148C42×C53—22no—
149C42×C53—41no—
150C42×C53C2×D411yescharacteristic
151C42×C53C2×D411yescharacteristic
152C42×C53#4111yescharacteristic
153C42×C53C2411yescharacteristic
154C23×C53—42no—
155C23×C53—82no—
156C23×C53—82no—
157C23×C53—82no—
158C23×C53#12111yescharacteristic
159Q8×C53—22no—
160Q8×C53—22no—
161Q8×C53—42no—
162Q8×C53—41no—
163Q8×C53—41no—
164Q8×C53—41no—
165Q8×C53—41no—
166D4×C53—42no—
167D4×C53—42no—
168D4×C53—42no—
169D4×C53—42no—
170D4×C53—82no—
171D4×C53—82no—
172D4×C53—82no—
173D4×C53—82no—
174C2×C4×C53—21no—
175C2×C4×C53—21no—
176C2×C4×C53—21no—
177C2×C4×C53—21no—
178C2×C4×C53—22no—
179C2×C4×C53—22no—
180C2×C4×C53—22no—
181C2×C4×C53—41no—
182C2×C4×C53—41no—
183C2×C4×C53#12111yescharacteristic
184C2×C4×C53#12111yescharacteristic
185C2×C4×C53#12111yescharacteristic
186C2×C4×C53C22×D411yescharacteristic
187C2×C4×C53C22×D411yescharacteristic
188C2×C4×C53C2×#4111yescharacteristic
189C8×C53—22no—
190C8×C53—22no—
191C8×C53—42no—
192C8×C53—41no—
193C8×C53—41no—
194C8×C53—41no—
195C8×C53—41no—
196#19564C5311yesmaximal, characteristic, direct product factor, Sylow
197C22×C53—21no—
198C22×C53—21no—
199C22×C53—21no—
200C22×C53—82no—
201C22×C53—82no—
202C22×C53—82no—
203C22×C53—164no—
204C22×C53C2×#12111yescharacteristic
205C4×C53—21no—
206C4×C53—21no—
207C4×C53—21no—
208C4×C53—42no—
209C4×C53—42no—
210C4×C53—81no—
211C4×C53—81no—
212C4×C53#47112yes—
213C4×C53#57611yescharacteristic
214C4×C53#57711yescharacteristic
215C4×C53#57711yescharacteristic
216C4×C53#57711yescharacteristic
217#3255C2×C5311yescharacteristic, complemented
218#3208C2×C5311yescharacteristic
219#3208C2×C5311yescharacteristic
220#2986C2×C5311yescharacteristic, complemented
221#1702C2×C5312yescomplemented
222#1686C2×C5312yescomplemented
223#1682C2×C5312yescomplemented
224#1666C2×C5312yescomplemented
225#1574C2×C5311yescharacteristic, complemented
226#1545C2×C5311yescharacteristic
227#1544C2×C5311yescharacteristic, complemented
228C2×C53—41no—
229C2×C53—162no—
230C2×C53#304311yescharacteristic
231C2×C53#306211yescharacteristic
232C2×C53C2×#47111yescharacteristic
233#533C22×C5311yescharacteristic
234#533C22×C5311yescharacteristic
235C2×#127C22×C5311yescharacteristic
236#492—21no—
237#492C22×C5312yes—
238#486—21no—
239#486C22×C5312yes—
240#475—21no—
241#475C22×C5312yescomplemented
242#470—22no—
243#462—21no—
244#462C22×C5312yescomplemented
245#460—21no—
246#460—21no—
247#460C22×C5312yes—
248#460C22×C5312yes—
249#450—22no—
250#450—22no—
251#446—21no—
252#446C22×C5312yescomplemented
253C8×Q8—21no—
254C8×Q8C22×C5312yescomplemented
255#433—22no—
256#431C22×C5311yescharacteristic
257#431C22×C5311yescharacteristic
258#431C22×C5311yescharacteristic
259#423C22×C5311yescharacteristic
260#335C22×C5312yescomplemented
261#334C22×C5312yescomplemented
262#332C22×C5311yescharacteristic, complemented
263#332C22×C5311yescharacteristic, complemented
264#332C22×C5311yescharacteristic, complemented
265#332C22×C5311yescharacteristic, complemented
266#331C22×C5312yes—
267#331C22×C5312yes—
268C53#1956411yescharacteristic, direct product factor, Sylow
269C22×D4—21no—
270C22×D4—21no—
271C22×D4—21no—
272C2×Q16—22no—
273C2×SD16—22no—
274C2×SD16—22no—
275C2×SD16—22no—
276C2×SD16—22no—
277C2×M16—21no—
278C4⋊Q8D4×C5311yescharacteristic
279C4⋊Q8D4×C5311yescharacteristic
280C4⋊Q8C23×C5311yescharacteristic
281C4⋊Q8C23×C5311yescharacteristic
282#127—21no—
283#127—21no—
284#127—21no—
285#127C23×C5312yes—
286#125—21no—
287#125—21no—
288C22⋊Q8—21no—
289C22⋊Q8—21no—
290C22⋊Q8—21no—
291C22⋊Q8D4×C5312yes—
292C4×Q8—21no—
293C4×Q8—21no—
294C4×Q8—21no—
295C4×Q8C23×C5312yes—
296#117—21no—
297#117—21no—
298C2×C42C23×C5311yescharacteristic
299#106—22no—
300C4⋊C8—22no—
301C4⋊C8—22no—
302C4⋊C8—21no—
303C4⋊C8D4×C5311yescharacteristic
304C4⋊C8D4×C5311yescharacteristic
305C4⋊C8C23×C5311yescharacteristic
306C4⋊C8C23×C5311yescharacteristic
307C4⋊C8C23×C5312yes—
308C4⋊C8C23×C5311yescharacteristic
309C4⋊C8C23×C5311yescharacteristic
310#103—22no—
311#103—22no—
312#102—22no—
313#102—22no—
314#102—22no—
315#102—22no—
316C22⋊C8—21no—
317C22⋊C8—21no—
318C22⋊C8D4×C5312yes—
319#97—22no—
320C4×C8—22no—
321C4×C8—21no—
322C4×C8—21no—
323C4×C8C23×C5312yes—
324C24—41no—
325C2×Q8—21no—
326C2×Q8—21no—
327C2×Q8—21no—
328C2×Q8C2×D4×C5312yes—
329C2×D4—22no—
330C2×D4—22no—
331C2×D4—22no—
332C2×D4—41no—
333C2×D4—41no—
334C2×D4—41no—
335C2×D4—81no—
336C2×D4—81no—
337C2×D4—81no—
338C2×D4—81no—
339C2×D4—82no—
340C22×C4C2×D4×C5311yescharacteristic
341C22×C4C2×D4×C5311yescharacteristic
342C22×C4#41×C5311yescharacteristic
343Q16—82no—
344SD16—82no—
345SD16—82no—
346SD16—82no—
347SD16—82no—
348M16—81no—
349C2×C8—21no—
350C2×C8—21no—
351C2×C8—21no—
352C2×C8—21no—
353C2×C8—22no—
354C2×C8C2×D4×C5312yes—
355C2×C8C2×D4×C5312yes—
356C4⋊C4—21no—
357C4⋊C4—21no—
358C4⋊C4—21no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—21no—
364C4⋊C4—21no—
365C4⋊C4—41no—
366C4⋊C4—41no—
367C4⋊C4C2×D4×C5312yes—
368C22⋊C4—22no—
369C22⋊C4—41no—
370C42—22no—
371C42—41no—
372C42C2×D4×C5311yescharacteristic
373C42C2×D4×C5311yescharacteristic
374C42#41×C5311yescharacteristic
375C42C24×C5311yescharacteristic
376C23—42no—
377C23—82no—
378C23—82no—
379C23—82no—
380C23#121×C5311yescharacteristic
381Q8—22no—
382Q8—22no—
383Q8—42no—
384Q8—41no—
385Q8—41no—
386Q8—41no—
387Q8—41no—
388D4—42no—
389D4—42no—
390D4—42no—
391D4—42no—
392D4—82no—
393D4—82no—
394D4—82no—
395D4—82no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—22no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4#121×C5311yescharacteristic
406C2×C4#121×C5311yescharacteristic
407C2×C4#121×C5311yescharacteristic
408C2×C4C22×D4×C5311yescharacteristic
409C2×C4C22×D4×C5311yescharacteristic
410C2×C4C2×#41×C5311yescharacteristic
411C8—22no—
412C8—22no—
413C8—42no—
414C8—41no—
415C8—41no—
416C8—41no—
417C8—41no—
418C22—21no—
419C22—21no—
420C22—21no—
421C22—82no—
422C22—82no—
423C22—82no—
424C22—164no—
425C22C2×#121×C5311yescharacteristic
426C4—21no—
427C4—21no—
428C4—21no—
429C4—42no—
430C4—42no—
431C4—81no—
432C4—81no—
433C4C53×#47112yes—
434C4C53×#57611yescharacteristic
435C4C53×#57711yescharacteristic
436C4C53×#57711yescharacteristic
437C4C53×#57711yescharacteristic
438C2—41no—
439C2—162no—
440C2C53×#304311yescharacteristic
441C2C53×#306211yescharacteristic
442C2C2×C53×#47111yescharacteristic
subgroupG/H
center, socleC22×C53C2×#121
commutator, Frattini, cosocleC42C24×C53
prime radicalC2×#127×C53C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G

Complements

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

NH ≅ G/Ntype
1. C53×#3255C2semidirect
4. C53×#2986C2semidirect
5. C53×#1702C2semidirect
6. C53×#1686C2semidirect
7. C53×#1682C2semidirect
8. C53×#1666C2semidirect
9. C53×#1574C2semidirect
11. C53×#1544C2semidirect
20. C53×#475C22semidirect
23. C53×#462C22semidirect
31. C53×#446C22semidirect
33. C8×Q8×C53C22semidirect
39. C53×#335C22semidirect
40. C53×#334C22semidirect
41. C53×#332C22semidirect
42. C53×#332C22semidirect
43. C53×#332C22semidirect
44. C53×#332C22semidirect
196. #19564C53direct (N × H)
217. #3255C2×C53semidirect
220. #2986C2×C53semidirect
221. #1702C2×C53semidirect
222. #1686C2×C53semidirect
223. #1682C2×C53semidirect
224. #1666C2×C53semidirect
225. #1574C2×C53semidirect
227. #1544C2×C53semidirect
241. #475C22×C53semidirect
244. #462C22×C53semidirect
252. #446C22×C53semidirect
254. C8×Q8C22×C53semidirect
260. #335C22×C53semidirect
261. #334C22×C53semidirect
262. #332C22×C53semidirect
263. #332C22×C53semidirect
264. #332C22×C53semidirect
265. #332C22×C53semidirect
268. C53#19564direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#32552
C53×#32082
C53×#29862
C53×#17022
C53×#16862
C53×#16822
C53×#16662
C53×#15742
C53×#15452
C53×#15442
#1956453