Finite Group Databasebeta

← C53×#8502 #27216014

Subgroups — C53×#8502 #27216014

Order 13568.

672 conjugacy classes of subgroups.

The subgroup lattice has 540 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3547C211yesmaximal, characteristic
2C53×#1553C211yesmaximal, characteristic, complemented
3C53×#1535C211yesmaximal, characteristic, complemented
4C53×#1521C211yesmaximal, characteristic, complemented
5C53×#1515C211yesmaximal, characteristic, complemented
6C53×#1514C211yesmaximal, characteristic, complemented
7C53×#1497C211yesmaximal, characteristic, complemented
8C53×#562—22no—
9C53×#555—21no—
10C53×#555C411yescharacteristic, complemented
11C53×#554C411yescharacteristic, complemented
12C53×#553—21no—
13C53×#552—21no—
14D4×Q8×C53C411yescharacteristic, complemented
15C53×#548—21no—
16C53×#547C411yescharacteristic, complemented
17C53×#544—21no—
18C53×#542—22no—
19C53×#540—21no—
20C53×#537—21no—
21C53×#536—21no—
22C53×#533C2211yescharacteristic
23C53×#532C2211yescharacteristic
24C53×#529C2211yescharacteristic
25C53×#432C2211yescharacteristic
26C53×#432C2211yescharacteristic
27C53×#431C2211yescharacteristic
28C53×#383C2211yescharacteristic
29C53×#332—21no—
30C53×#331—21no—
31C53×#330—21no—
32C53×#329—21no—
33C53×#328—22no—
34C53×#328—21no—
35C53×#327—21no—
36C53×#327—21no—
37C53×#327—21no—
38C53×Q8⋊C8—22no—
39C2×#41×C53—21no—
40C2×#41×C53—21no—
41C22×Q8×C53—21no—
42C4⋊Q8×C53—22no—
43C4⋊Q8×C53—21no—
44C4⋊Q8×C53C2×C411yescharacteristic
45C4⋊Q8×C53C2×C411yescharacteristic
46C4⋊Q8×C53D411yescharacteristic
47#127×C53D411yescharacteristic
48#126×C53—22no—
49#126×C53—21no—
50#125×C53—22no—
51#125×C53—21no—
52#125×C53—21no—
53#125×C53D411yescharacteristic
54#124×C53—22no—
55#124×C53—22no—
56#124×C53—21no—
57#124×C53—22no—
58#124×C53D411yescharacteristic
59#123×C53—22no—
60#123×C53—22no—
61#123×C53—22no—
62#123×C53—22no—
63#123×C53—21no—
64C22⋊Q8×C53—22no—
65C22⋊Q8×C53—22no—
66C22⋊Q8×C53—21no—
67C22⋊Q8×C53—21no—
68C22⋊Q8×C53—22no—
69C22⋊Q8×C53—21no—
70C22⋊Q8×C53—22no—
71C22⋊Q8×C53—21no—
72C22⋊Q8×C53D411yescharacteristic
73C22⋊Q8×C53D412yes—
74#121×C53—22no—
75#121×C53—22no—
76#121×C53—22no—
77#121×C53—22no—
78#121×C53—21no—
79#121×C53—21no—
80#121×C53—21no—
81#121×C53—21no—
82#121×C53D411yescharacteristic
83#121×C53D411yescharacteristic
84#121×C53D411yescharacteristic
85C4×Q8×C53—21no—
86C4×Q8×C53C2×C411yescharacteristic
87C4×Q8×C53D412yes—
88C4×D4×C53—22no—
89C4×D4×C53—21no—
90C4×D4×C53—21no—
91C4×D4×C53—21no—
92C4×D4×C53—21no—
93C4×D4×C53C2×C411yescharacteristic
94C4×D4×C53C2×C411yescharacteristic
95C4×D4×C53C2×C411yescharacteristic
96#117×C53—22no—
97#117×C53—21no—
98#117×C53—21no—
99#117×C53—21no—
100#117×C53—21no—
101C2×C4⋊C4×C53—21no—
102C2×C4⋊C4×C53—21no—
103C2×C42×C53C2311yescharacteristic
104C4⋊C8×C53—21no—
105C4⋊C8×C53—21no—
106C4⋊C8×C53—21no—
107C22⋊C8×C53—21no—
108C22⋊C8×C53—21no—
109C22⋊C8×C53—21no—
110#97×C53—21no—
111#97×C53—21no—
112C4×C8×C53—21no—
113#95×C53—21no—
114#95×C53—21no—
115#95×C53—21no—
116#41×C53—82no—
117#41×C53—82no—
118C2×Q8×C53—22no—
119C2×Q8×C53—21no—
120C2×Q8×C53—21no—
121C2×Q8×C53—82no—
122C2×Q8×C53C22⋊C411yescharacteristic
123C2×Q8×C53C22⋊C412yes—
124C2×D4×C53—22no—
125C2×D4×C53—22no—
126C2×D4×C53—21no—
127C2×D4×C53—21no—
128C2×D4×C53—21no—
129C2×D4×C53—21no—
130C2×D4×C53—21no—
131C2×D4×C53C22⋊C411yescharacteristic
132C2×D4×C53C22⋊C411yescharacteristic
133C2×D4×C53C22⋊C411yescharacteristic
134C22×C4×C53—22no—
135C22×C4×C53—22no—
136C22×C4×C53—22no—
137C22×C4×C53—21no—
138C22×C4×C53—21no—
139C22×C4×C53—21no—
140C22×C4×C53—41no—
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D411yescharacteristic
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—21no—
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×C53—22no—
161C4⋊C4×C53C22⋊C411yescharacteristic
162C4⋊C4×C53C22⋊C411yescharacteristic
163C4⋊C4×C53C22⋊C411yescharacteristic
164C4⋊C4×C53C22⋊C411yescharacteristic
165C4⋊C4×C53C22⋊C412yes—
166C22⋊C4×C53—22no—
167C22⋊C4×C53—22no—
168C22⋊C4×C53—22no—
169C22⋊C4×C53—22no—
170C22⋊C4×C53—22no—
171C22⋊C4×C53—22no—
172C22⋊C4×C53—21no—
173C22⋊C4×C53—21no—
174C22⋊C4×C53—21no—
175C22⋊C4×C53—21no—
176C22⋊C4×C53—21no—
177C22⋊C4×C53—21no—
178C22⋊C4×C53—21no—
179C22⋊C4×C53—21no—
180C22⋊C4×C53—21no—
181C42×C53—22no—
182C42×C53—21no—
183C42×C53—22no—
184C42×C53C22×C411yescharacteristic
185C42×C53C2×D411yescharacteristic
186C42×C53C2×D411yescharacteristic
187C42×C53C2×D411yescharacteristic
188C23×C53—21no—
189C23×C53—21no—
190C23×C53—21no—
191C23×C53#12011yescharacteristic
192Q8×C53—42no—
193Q8×C53—41no—
194Q8×C53—42no—
195Q8×C53—81no—
196Q8×C53—82no—
197D4×C53—41no—
198D4×C53—41no—
199D4×C53—41no—
200D4×C53—81no—
201D4×C53—81no—
202D4×C53—81no—
203D4×C53—81no—
204D4×C53—81no—
205D4×C53—82no—
206D4×C53—82no—
207C2×C4×C53—21no—
208C2×C4×C53—22no—
209C2×C4×C53—21no—
210C2×C4×C53—21no—
211C2×C4×C53—21no—
212C2×C4×C53—22no—
213C2×C4×C53—21no—
214C2×C4×C53—41no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53—42no—
218C2×C4×C53—42no—
219C2×C4×C53—42no—
220C2×C4×C53—82no—
221C2×C4×C53—82no—
222C2×C4×C53—82no—
223C2×C4×C53—81no—
224C2×C4×C53—81no—
225C2×C4×C53—81no—
226C2×C4×C53C2×C22⋊C411yescharacteristic
227C2×C4×C53C2×C22⋊C411yescharacteristic
228C2×C4×C53C2×C22⋊C411yescharacteristic
229C2×C4×C53#12011yescharacteristic
230C2×C4×C53#12011yescharacteristic
231C2×C4×C53#12011yescharacteristic
232C8×C53—81no—
233C8×C53—81no—
234C8×C53—81no—
235#8502C5311yesmaximal, characteristic, direct product factor, Sylow
236C22×C53—21no—
237C22×C53—21no—
238C22×C53—21no—
239C22×C53—41no—
240C22×C53—41no—
241C22×C53—41no—
242C22×C53—41no—
243C22×C53—41no—
244C22×C53—41no—
245C22×C53—41no—
246C22×C53—41no—
247C22×C53—41no—
248C22×C53#37911yescharacteristic
249C4×C53—21no—
250C4×C53—21no—
251C4×C53—21no—
252C4×C53—21no—
253C4×C53—21no—
254C4×C53—21no—
255C4×C53—41no—
256C4×C53—42no—
257C4×C53—41no—
258C4×C53—41no—
259C4×C53—42no—
260C4×C53—41no—
261C4×C53—41no—
262C4×C53—161no—
263#3547C2×C5311yescharacteristic
264#1553C2×C5311yescharacteristic, complemented
265#1535C2×C5311yescharacteristic, complemented
266#1521C2×C5311yescharacteristic, complemented
267#1515C2×C5311yescharacteristic, complemented
268#1514C2×C5311yescharacteristic, complemented
269#1497C2×C5311yescharacteristic, complemented
270C2×C53—41no—
271C2×C53—81no—
272C2×C53—81no—
273C2×C53—81no—
274C2×C53#179411yescharacteristic
275C2×C53#179511yescharacteristic
276C2×C53#179611yescharacteristic
277#562—22no—
278#555—21no—
279#555C4×C5311yescharacteristic, complemented
280#554C4×C5311yescharacteristic, complemented
281#553—21no—
282#552—21no—
283D4×Q8C4×C5311yescharacteristic, complemented
284#548—21no—
285#547C4×C5311yescharacteristic, complemented
286#544—21no—
287#542—22no—
288#540—21no—
289#537—21no—
290#536—21no—
291#533C22×C5311yescharacteristic
292#532C22×C5311yescharacteristic
293#529C22×C5311yescharacteristic
294#432C22×C5311yescharacteristic
295#432C22×C5311yescharacteristic
296#431C22×C5311yescharacteristic
297#383C22×C5311yescharacteristic
298#332—21no—
299#331—21no—
300#330—21no—
301#329—21no—
302#328—22no—
303#328—21no—
304#327—21no—
305#327—21no—
306#327—21no—
307Q8⋊C8—22no—
308C53#850211yescharacteristic, direct product factor, Sylow
309C2×#41—21no—
310C2×#41—21no—
311C22×Q8—21no—
312C4⋊Q8—22no—
313C4⋊Q8—21no—
314C4⋊Q8C2×C4×C5311yescharacteristic
315C4⋊Q8C2×C4×C5311yescharacteristic
316C4⋊Q8D4×C5311yescharacteristic
317#127D4×C5311yescharacteristic
318#126—22no—
319#126—21no—
320#125—22no—
321#125—21no—
322#125—21no—
323#125D4×C5311yescharacteristic
324#124—22no—
325#124—22no—
326#124—22no—
327#124—21no—
328#124D4×C5311yescharacteristic
329#123—22no—
330#123—22no—
331#123—22no—
332#123—21no—
333#123—22no—
334C22⋊Q8—22no—
335C22⋊Q8—22no—
336C22⋊Q8—21no—
337C22⋊Q8—22no—
338C22⋊Q8—21no—
339C22⋊Q8—21no—
340C22⋊Q8—21no—
341C22⋊Q8—22no—
342C22⋊Q8D4×C5311yescharacteristic
343C22⋊Q8D4×C5312yes—
344#121—22no—
345#121—22no—
346#121—22no—
347#121—22no—
348#121—21no—
349#121—21no—
350#121—21no—
351#121—21no—
352#121D4×C5311yescharacteristic
353#121D4×C5311yescharacteristic
354#121D4×C5311yescharacteristic
355C4×Q8—21no—
356C4×Q8C2×C4×C5311yescharacteristic
357C4×Q8D4×C5312yes—
358C4×D4—22no—
359C4×D4—21no—
360C4×D4—21no—
361C4×D4—21no—
362C4×D4—21no—
363C4×D4C2×C4×C5311yescharacteristic
364C4×D4C2×C4×C5311yescharacteristic
365C4×D4C2×C4×C5311yescharacteristic
366#117—22no—
367#117—21no—
368#117—21no—
369#117—21no—
370#117—21no—
371C2×C4⋊C4—21no—
372C2×C4⋊C4—21no—
373C2×C42C23×C5311yescharacteristic
374C4⋊C8—21no—
375C4⋊C8—21no—
376C4⋊C8—21no—
377C22⋊C8—21no—
378C22⋊C8—21no—
379C22⋊C8—21no—
380#97—21no—
381#97—21no—
382C4×C8—21no—
383#95—21no—
384#95—21no—
385#95—21no—
386#41—82no—
387#41—82no—
388C2×Q8—22no—
389C2×Q8—21no—
390C2×Q8—21no—
391C2×Q8—82no—
392C2×Q8C22⋊C4×C5311yescharacteristic
393C2×Q8C22⋊C4×C5312yes—
394C2×D4—22no—
395C2×D4—22no—
396C2×D4—21no—
397C2×D4—21no—
398C2×D4—21no—
399C2×D4—21no—
400C2×D4—21no—
401C2×D4C22⋊C4×C5311yescharacteristic
402C2×D4C22⋊C4×C5311yescharacteristic
403C2×D4C22⋊C4×C5311yescharacteristic
404C22×C4—22no—
405C22×C4—22no—
406C22×C4—22no—
407C22×C4—21no—
408C22×C4—21no—
409C22×C4—21no—
410C22×C4—41no—
411C22×C4C2×D4×C5311yescharacteristic
412C22×C4C2×D4×C5311yescharacteristic
413C22×C4C2×D4×C5311yescharacteristic
414C2×C8—41no—
415C2×C8—41no—
416C2×C8—41no—
417C4⋊C4—22no—
418C4⋊C4—22no—
419C4⋊C4—22no—
420C4⋊C4—22no—
421C4⋊C4—21no—
422C4⋊C4—21no—
423C4⋊C4—21no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—21no—
427C4⋊C4—21no—
428C4⋊C4—21no—
429C4⋊C4—22no—
430C4⋊C4—22no—
431C4⋊C4C22⋊C4×C5311yescharacteristic
432C4⋊C4C22⋊C4×C5311yescharacteristic
433C4⋊C4C22⋊C4×C5311yescharacteristic
434C4⋊C4C22⋊C4×C5311yescharacteristic
435C4⋊C4C22⋊C4×C5312yes—
436C22⋊C4—22no—
437C22⋊C4—22no—
438C22⋊C4—22no—
439C22⋊C4—22no—
440C22⋊C4—22no—
441C22⋊C4—22no—
442C22⋊C4—21no—
443C22⋊C4—21no—
444C22⋊C4—21no—
445C22⋊C4—21no—
446C22⋊C4—21no—
447C22⋊C4—21no—
448C22⋊C4—21no—
449C22⋊C4—21no—
450C22⋊C4—21no—
451C42—22no—
452C42—21no—
453C42—22no—
454C42C22×C4×C5311yescharacteristic
455C42C2×D4×C5311yescharacteristic
456C42C2×D4×C5311yescharacteristic
457C42C2×D4×C5311yescharacteristic
458C23—21no—
459C23—21no—
460C23—21no—
461C23#120×C5311yescharacteristic
462Q8—41no—
463Q8—42no—
464Q8—42no—
465Q8—81no—
466Q8—82no—
467D4—41no—
468D4—41no—
469D4—41no—
470D4—81no—
471D4—81no—
472D4—81no—
473D4—81no—
474D4—81no—
475D4—82no—
476D4—82no—
477C2×C4—21no—
478C2×C4—22no—
479C2×C4—21no—
480C2×C4—21no—
481C2×C4—21no—
482C2×C4—22no—
483C2×C4—21no—
484C2×C4—41no—
485C2×C4—41no—
486C2×C4—41no—
487C2×C4—42no—
488C2×C4—42no—
489C2×C4—42no—
490C2×C4—82no—
491C2×C4—82no—
492C2×C4—82no—
493C2×C4—81no—
494C2×C4—81no—
495C2×C4—81no—
496C2×C4C2×C22⋊C4×C5311yescharacteristic
497C2×C4C2×C22⋊C4×C5311yescharacteristic
498C2×C4C2×C22⋊C4×C5311yescharacteristic
499C2×C4#120×C5311yescharacteristic
500C2×C4#120×C5311yescharacteristic
501C2×C4#120×C5311yescharacteristic
502C8—81no—
503C8—81no—
504C8—81no—
505C22—21no—
506C22—21no—
507C22—21no—
508C22—41no—
509C22—41no—
510C22—41no—
511C22—41no—
512C22—41no—
513C22—41no—
514C22—41no—
515C22—41no—
516C22—41no—
517C22C53×#37911yescharacteristic
518C4—21no—
519C4—21no—
520C4—21no—
521C4—21no—
522C4—21no—
523C4—21no—
524C4—41no—
525C4—42no—
526C4—41no—
527C4—41no—
528C4—42no—
529C4—41no—
530C4—41no—
531C4—161no—
532C2—41no—
533C2—81no—
534C2—81no—
535C2—81no—
536C2C53×#179411yescharacteristic
537C2C53×#179511yescharacteristic
538C2C53×#179611yescharacteristic
subgroupG/H
center, socleC22×C53#379
commutatorC42C22×C4×C53
Frattini, cosocleC2×C42C23×C53
prime radicalC53×#3547C2
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
2. C53×#1553C2semidirect
3. C53×#1535C2semidirect
4. C53×#1521C2semidirect
5. C53×#1515C2semidirect
6. C53×#1514C2semidirect
7. C53×#1497C2semidirect
10. C53×#555C4semidirect
11. C53×#554C4semidirect
14. D4×Q8×C53C4semidirect
16. C53×#547C4semidirect
235. #8502C53direct (N × H)
264. #1553C2×C53semidirect
265. #1535C2×C53semidirect
266. #1521C2×C53semidirect
267. #1515C2×C53semidirect
268. #1514C2×C53semidirect
269. #1497C2×C53semidirect
279. #555C4×C53semidirect
280. #554C4×C53semidirect
283. D4×Q8C4×C53semidirect
285. #547C4×C53semidirect
308. C53#8502direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#35472
C53×#15532
C53×#15352
C53×#15212
C53×#15152
C53×#15142
C53×#14972
#850253