Finite Group Databasebeta

← C53×#33605 #27241117

Subgroups — C53×#33605 #27241117

Order 13568.

738 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
1C2×C53×#509C211yesmaximal, characteristic, complemented
2C22×M32×C53C211yesmaximal, characteristic, complemented
3C53×#3050C211yesmaximal, characteristic, complemented
4C2×C53×#466C211yesmaximal, characteristic, complemented
5C53×#2220C212yesmaximal, complemented
6C53×#2218C212yesmaximal, complemented
7C53×#2217C212yesmaximal, complemented
8C53×#2215C212yesmaximal, complemented
9C53×#2159C211yesmaximal, characteristic, complemented
10C53×#2140C211yesmaximal, characteristic, complemented
11C53×#2139C211yesmaximal, characteristic, complemented
12C2×#135×C53C2211yescharacteristic
13C22×D8×C53C2211yescharacteristic
14C23×C8×C53C2211yescharacteristic
15C53×#525—21no—
16C2×#121×C53—21no—
17C53×#509—24no—
18C2×SD32×C53C2212yescomplemented
19C2×D16×C53C2212yescomplemented
20C2×M32×C53—24no—
21C2×M32×C53C2211yescharacteristic
22C2×M32×C53C2211yescharacteristic
23C2×M32×C53C2212yescomplemented
24C22×C16×C53C2212yescomplemented
25C53×#467C2211yescharacteristic, complemented
26C53×#467C2211yescharacteristic, complemented
27C53×#466—21no—
28C53×#466—21no—
29C53×#466C2212yes—
30C53×#466C2212yes—
31C53×#466C2211yescharacteristic, complemented
32C53×#466C2211yescharacteristic, complemented
33C53×#465—21no—
34C53×#465—21no—
35C53×#427C2211yescharacteristic
36C2×#107×C53C2211yescharacteristic
37C53×#416—21no—
38C2×#102×C53—21no—
39C53×#367C2212yescomplemented
40C53×#366C2212yescomplemented
41C53×#358C2212yescomplemented
42C53×#357C2212yescomplemented
43C53×#357C2212yescomplemented
44C53×#357C2212yescomplemented
45C2×#41×C53—21no—
46C22×D4×C53—21no—
47C22×D4×C53—41no—
48C23×C4×C53D411yescharacteristic
49#135×C53—22no—
50#135×C53—22no—
51C2×Q16×C53C2311yescharacteristic
52C2×SD16×C53—21no—
53C2×D8×C53—21no—
54C2×D8×C53—22no—
55C2×D8×C53—22no—
56C2×D8×C53C2311yescharacteristic
57C2×D8×C53C2312yes—
58C22×C8×C53—22no—
59C22×C8×C53—22no—
60C22×C8×C53D411yescharacteristic
61C22×C8×C53D411yescharacteristic
62C22×C8×C53D411yescharacteristic
63C22×C8×C53C2311yescharacteristic
64C22×C8×C53C2311yescharacteristic
65C22×C8×C53C2311yescharacteristic
66#123×C53—41no—
67C22⋊Q8×C53—21no—
68C22⋊Q8×C53—21no—
69#121×C53—22no—
70#121×C53—22no—
71#121×C53—21no—
72#121×C53—21no—
73#121×C53—41no—
74#121×C53—41no—
75#120×C53—41no—
76C4×D4×C53—41no—
77C4×D4×C53—41no—
78#117×C53—21no—
79C2×C4⋊C4×C53—21no—
80C2×C22⋊C4×C53—41no—
81SD32×C53—24no—
82D16×C53—24no—
83M32×C53—22no—
84M32×C53—24no—
85M32×C53D414yescomplemented
86C2×C16×C53—24no—
87C2×C16×C53C2312yes—
88C2×C16×C53C2312yes—
89#107×C53—21no—
90#107×C53C2311yescharacteristic
91#107×C53C2311yescharacteristic
92#107×C53C2312yes—
93#106×C53—21no—
94#103×C53—21no—
95#103×C53—21no—
96#102×C53—22no—
97#102×C53—22no—
98#102×C53—21no—
99#102×C53—21no—
100C24×C53—41no—
101C24×C53D811yescharacteristic
102#41×C53—42no—
103#41×C53—42no—
104C2×Q8×C53—21no—
105C2×D4×C53—22no—
106C2×D4×C53—21no—
107C2×D4×C53—41no—
108C2×D4×C53—41no—
109C2×D4×C53—41no—
110C2×D4×C53—42no—
111C2×D4×C53—42no—
112C2×D4×C53—42no—
113C2×D4×C53—41no—
114C2×D4×C53—42no—
115C2×D4×C53—84no—
116C22×C4×C53—22no—
117C22×C4×C53—22no—
118C22×C4×C53—41no—
119C22×C4×C53—41no—
120C22×C4×C53D811yescharacteristic
121C22×C4×C53D811yescharacteristic
122C22×C4×C53D811yescharacteristic
123C22×C4×C53C2×D411yescharacteristic
124C22×C4×C53C2×D411yescharacteristic
125C22×C4×C53C2×D411yescharacteristic
126Q16×C53—21no—
127Q16×C53—21no—
128SD16×C53—42no—
129D8×C53—21no—
130D8×C53—22no—
131D8×C53—22no—
132D8×C53—21no—
133D8×C53—42no—
134C2×C8×C53—22no—
135C2×C8×C53—22no—
136C2×C8×C53—22no—
137C2×C8×C53—21no—
138C2×C8×C53—21no—
139C2×C8×C53—21no—
140C2×C8×C53—21no—
141C2×C8×C53C2×D411yescharacteristic
142C2×C8×C53C2×D411yescharacteristic
143C2×C8×C53C2×D411yescharacteristic
144C2×C8×C53C2×D412yes—
145C2×C8×C53#4112yes—
146C2×C8×C53C2411yescharacteristic
147C4⋊C4×C53—22no—
148C4⋊C4×C53—21no—
149C4⋊C4×C53—21no—
150C4⋊C4×C53—41no—
151C4⋊C4×C53—41no—
152C4⋊C4×C53—41no—
153C22⋊C4×C53—41no—
154C22⋊C4×C53—41no—
155C22⋊C4×C53—41no—
156C22⋊C4×C53—42no—
157C22⋊C4×C53—42no—
158C22⋊C4×C53—41no—
159C22⋊C4×C53—41no—
160C42×C53—41no—
161C16×C53—22no—
162C16×C53C2×D414yescomplemented
163C23×C53—22no—
164C23×C53—22no—
165C23×C53—41no—
166C23×C53—41no—
167C23×C53—42no—
168C23×C53—82no—
169C23×C53—82no—
170C23×C53—82no—
171C23×C53C2×D811yescharacteristic
172C23×C53C2×D811yescharacteristic
173C23×C53C2×D811yescharacteristic
174Q8×C53—41no—
175Q8×C53—41no—
176D4×C53—42no—
177D4×C53—41no—
178D4×C53—42no—
179D4×C53—41no—
180D4×C53—82no—
181D4×C53—84no—
182D4×C53—82no—
183D4×C53—82no—
184D4×C53—84no—
185D4×C53—82no—
186C2×C4×C53—22no—
187C2×C4×C53—21no—
188C2×C4×C53—21no—
189C2×C4×C53—21no—
190C2×C4×C53—21no—
191C2×C4×C53—22no—
192C2×C4×C53—22no—
193C2×C4×C53—42no—
194C2×C4×C53—41no—
195C2×C4×C53—41no—
196C2×C4×C53—41no—
197C2×C4×C53—82no—
198C2×C4×C53—82no—
199C2×C4×C53#12112yes—
200C2×C4×C53C2×D811yescharacteristic
201C2×C4×C53C2×D811yescharacteristic
202C2×C4×C53C2×D811yescharacteristic
203C2×C4×C53#13512yes—
204C2×C4×C53C22×D411yescharacteristic
205C8×C53—21no—
206C8×C53—21no—
207C8×C53#12112yes—
208C8×C53C22×D411yescharacteristic
209C8×C53C2×#4111yescharacteristic
210#33605C5311yesmaximal, characteristic, direct product factor, Sylow
211C22×C53—21no—
212C22×C53—21no—
213C22×C53—21no—
214C22×C53—21no—
215C22×C53—21no—
216C22×C53—21no—
217C22×C53—21no—
218C22×C53—44no—
219C22×C53—82no—
220C22×C53—81no—
221C22×C53—81no—
222C22×C53—82no—
223C22×C53—81no—
224C22×C53—82no—
225C22×C53—164no—
226C22×C53#46612yes—
227C22×C53#50912yescomplemented
228C22×C53C22×D811yescharacteristic
229C4×C53—21no—
230C4×C53—21no—
231C4×C53—82no—
232C4×C53—81no—
233C4×C53—81no—
234C4×C53—81no—
235C4×C53#46612yes—
236C4×C53C2×#12111yescharacteristic
237C4×C53C2×#13511yescharacteristic
238C2×#509C2×C5311yescharacteristic, complemented
239C22×M32C2×C5311yescharacteristic, complemented
240#3050C2×C5311yescharacteristic, complemented
241C2×#466C2×C5311yescharacteristic, complemented
242#2220C2×C5312yescomplemented
243#2218C2×C5312yescomplemented
244#2217C2×C5312yescomplemented
245#2215C2×C5312yescomplemented
246#2159C2×C5311yescharacteristic, complemented
247#2140C2×C5311yescharacteristic, complemented
248#2139C2×C5311yescharacteristic, complemented
249C2×C53—22no—
250C2×C53—22no—
251C2×C53—41no—
252C2×C53—161no—
253C2×C53—162no—
254C2×C53C2×#46611yescharacteristic
255C2×C53C2×#50911yescharacteristic
256C2×C53#341411yescharacteristic
257C2×#135C22×C5311yescharacteristic
258C22×D8C22×C5311yescharacteristic
259C23×C8C22×C5311yescharacteristic
260#525—21no—
261C2×#121—21no—
262#509—24no—
263C2×SD32C22×C5312yescomplemented
264C2×D16C22×C5312yescomplemented
265C2×M32—24no—
266C2×M32C22×C5311yescharacteristic
267C2×M32C22×C5311yescharacteristic
268C2×M32C22×C5312yescomplemented
269C22×C16C22×C5312yescomplemented
270#467C22×C5311yescharacteristic, complemented
271#467C22×C5311yescharacteristic, complemented
272#466—21no—
273#466—21no—
274#466C22×C5312yes—
275#466C22×C5312yes—
276#466C22×C5311yescharacteristic, complemented
277#466C22×C5311yescharacteristic, complemented
278#465—21no—
279#465—21no—
280#427C22×C5311yescharacteristic
281C2×#107C22×C5311yescharacteristic
282#416—21no—
283C2×#102—21no—
284#367C22×C5312yescomplemented
285#366C22×C5312yescomplemented
286#358C22×C5312yescomplemented
287#357C22×C5312yescomplemented
288#357C22×C5312yescomplemented
289#357C22×C5312yescomplemented
290C53#3360511yescharacteristic, direct product factor, Sylow
291C2×#41—21no—
292C22×D4—21no—
293C22×D4—41no—
294C23×C4D4×C5311yescharacteristic
295#135—22no—
296#135—22no—
297C2×Q16C23×C5311yescharacteristic
298C2×SD16—21no—
299C2×D8—22no—
300C2×D8—21no—
301C2×D8—22no—
302C2×D8C23×C5312yes—
303C2×D8C23×C5311yescharacteristic
304C22×C8—22no—
305C22×C8—22no—
306C22×C8D4×C5311yescharacteristic
307C22×C8D4×C5311yescharacteristic
308C22×C8D4×C5311yescharacteristic
309C22×C8C23×C5311yescharacteristic
310C22×C8C23×C5311yescharacteristic
311C22×C8C23×C5311yescharacteristic
312#123—41no—
313C22⋊Q8—21no—
314C22⋊Q8—21no—
315#121—22no—
316#121—22no—
317#121—21no—
318#121—21no—
319#121—41no—
320#121—41no—
321#120—41no—
322C4×D4—41no—
323C4×D4—41no—
324#117—21no—
325C2×C4⋊C4—21no—
326C2×C22⋊C4—41no—
327SD32—24no—
328D16—24no—
329M32—22no—
330M32—24no—
331M32D4×C5314yescomplemented
332C2×C16—24no—
333C2×C16C23×C5312yes—
334C2×C16C23×C5312yes—
335#107—21no—
336#107C23×C5311yescharacteristic
337#107C23×C5311yescharacteristic
338#107C23×C5312yes—
339#106—21no—
340#103—21no—
341#103—21no—
342#102—22no—
343#102—22no—
344#102—21no—
345#102—21no—
346C24—41no—
347C24D8×C5311yescharacteristic
348#41—42no—
349#41—42no—
350C2×Q8—21no—
351C2×D4—22no—
352C2×D4—21no—
353C2×D4—41no—
354C2×D4—41no—
355C2×D4—41no—
356C2×D4—42no—
357C2×D4—42no—
358C2×D4—42no—
359C2×D4—41no—
360C2×D4—42no—
361C2×D4—84no—
362C22×C4—22no—
363C22×C4—22no—
364C22×C4—41no—
365C22×C4—41no—
366C22×C4D8×C5311yescharacteristic
367C22×C4D8×C5311yescharacteristic
368C22×C4D8×C5311yescharacteristic
369C22×C4C2×D4×C5311yescharacteristic
370C22×C4C2×D4×C5311yescharacteristic
371C22×C4C2×D4×C5311yescharacteristic
372Q16—21no—
373Q16—21no—
374SD16—42no—
375D8—21no—
376D8—22no—
377D8—22no—
378D8—21no—
379D8—42no—
380C2×C8—22no—
381C2×C8—22no—
382C2×C8—22no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—21no—
386C2×C8—21no—
387C2×C8C2×D4×C5311yescharacteristic
388C2×C8C2×D4×C5311yescharacteristic
389C2×C8C2×D4×C5311yescharacteristic
390C2×C8C2×D4×C5312yes—
391C2×C8#41×C5312yes—
392C2×C8C24×C5311yescharacteristic
393C4⋊C4—22no—
394C4⋊C4—21no—
395C4⋊C4—21no—
396C4⋊C4—41no—
397C4⋊C4—41no—
398C4⋊C4—41no—
399C22⋊C4—41no—
400C22⋊C4—41no—
401C22⋊C4—41no—
402C22⋊C4—42no—
403C22⋊C4—42no—
404C22⋊C4—41no—
405C22⋊C4—41no—
406C42—41no—
407C16—22no—
408C16C2×D4×C5314yescomplemented
409C23—22no—
410C23—22no—
411C23—41no—
412C23—41no—
413C23—42no—
414C23—82no—
415C23—82no—
416C23—82no—
417C23C2×D8×C5311yescharacteristic
418C23C2×D8×C5311yescharacteristic
419C23C2×D8×C5311yescharacteristic
420Q8—41no—
421Q8—41no—
422D4—42no—
423D4—41no—
424D4—42no—
425D4—41no—
426D4—82no—
427D4—84no—
428D4—82no—
429D4—84no—
430D4—82no—
431D4—82no—
432C2×C4—22no—
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—22no—
438C2×C4—22no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—42no—
442C2×C4—41no—
443C2×C4—82no—
444C2×C4—82no—
445C2×C4#121×C5312yes—
446C2×C4C2×D8×C5311yescharacteristic
447C2×C4C2×D8×C5311yescharacteristic
448C2×C4C2×D8×C5311yescharacteristic
449C2×C4#135×C5312yes—
450C2×C4C22×D4×C5311yescharacteristic
451C8—21no—
452C8—21no—
453C8#121×C5312yes—
454C8C22×D4×C5311yescharacteristic
455C8C2×#41×C5311yescharacteristic
456C22—21no—
457C22—21no—
458C22—21no—
459C22—21no—
460C22—21no—
461C22—21no—
462C22—21no—
463C22—44no—
464C22—81no—
465C22—82no—
466C22—82no—
467C22—81no—
468C22—81no—
469C22—82no—
470C22—164no—
471C22C53×#46612yes—
472C22C53×#50912yescomplemented
473C22C22×D8×C5311yescharacteristic
474C4—21no—
475C4—21no—
476C4—81no—
477C4—82no—
478C4—81no—
479C4—81no—
480C4C53×#46612yes—
481C4C2×#121×C5311yescharacteristic
482C4C2×#135×C5311yescharacteristic
483C2—22no—
484C2—22no—
485C2—41no—
486C2—162no—
487C2—161no—
488C2C2×C53×#46611yescharacteristic
489C2C2×C53×#50911yescharacteristic
490C2C53×#341411yescharacteristic
subgroupG/H
center, socleC22×C53C22×D8
commutator, Frattini, cosocleC2×C8C24×C53
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ C23×C4×C53 ◁ C23×C8×C53 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C53×#509C2semidirect
2. C22×M32×C53C2semidirect
3. C53×#3050C2semidirect
4. C2×C53×#466C2semidirect
5. C53×#2220C2semidirect
6. C53×#2218C2semidirect
7. C53×#2217C2semidirect
8. C53×#2215C2semidirect
9. C53×#2159C2semidirect
10. C53×#2140C2semidirect
11. C53×#2139C2semidirect
18. C2×SD32×C53C22semidirect
19. C2×D16×C53C22semidirect
23. C2×M32×C53C22semidirect
24. C22×C16×C53C22semidirect
25. C53×#467C22semidirect
26. C53×#467C22semidirect
31. C53×#466C22semidirect
32. C53×#466C22semidirect
39. C53×#367C22semidirect
40. C53×#366C22semidirect
41. C53×#358C22semidirect
42. C53×#357C22semidirect
43. C53×#357C22semidirect
44. C53×#357C22semidirect
85. M32×C53D4semidirect
162. C16×C53C2×D4semidirect
210. #33605C53direct (N × H)
227. C22×C53#509semidirect
238. C2×#509C2×C53semidirect
239. C22×M32C2×C53semidirect
240. #3050C2×C53semidirect
241. C2×#466C2×C53semidirect
242. #2220C2×C53semidirect
243. #2218C2×C53semidirect
244. #2217C2×C53semidirect
245. #2215C2×C53semidirect
246. #2159C2×C53semidirect
247. #2140C2×C53semidirect
248. #2139C2×C53semidirect
263. C2×SD32C22×C53semidirect
264. C2×D16C22×C53semidirect
268. C2×M32C22×C53semidirect
269. C22×C16C22×C53semidirect
270. #467C22×C53semidirect
271. #467C22×C53semidirect
276. #466C22×C53semidirect
277. #466C22×C53semidirect
284. #367C22×C53semidirect
285. #366C22×C53semidirect
286. #358C22×C53semidirect
287. #357C22×C53semidirect
288. #357C22×C53semidirect
289. #357C22×C53semidirect
290. C53#33605direct (N × H)
331. M32D4×C53semidirect
408. C16C2×D4×C53semidirect
472. C22C53×#509semidirect

Maximal subgroups

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

subgroupindex
C2×C53×#5092
C22×M32×C532
C53×#30502
C2×C53×#4662
C53×#22202
C53×#22182
C53×#22172
C53×#22152
C53×#21592
C53×#21402
C53×#21392
#3360553