Finite Group Databasebeta

← C53×#9259 #27216771

Subgroups — C53×#9259 #27216771

Order 13568.

966 conjugacy classes of subgroups.

The subgroup lattice has 578 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C53×#447C211yesmaximal, characteristic
2D4×C22⋊C4×C53C211yesmaximal, characteristic, complemented
3C53×#1964C211yesmaximal, characteristic, complemented
4C53×#1786C211yesmaximal, characteristic, complemented
5C53×#1519C211yesmaximal, characteristic, complemented
6C53×#1289C211yesmaximal, characteristic, complemented
7C53×#1282C211yesmaximal, characteristic, complemented
8C23×D4×C53C2211yescharacteristic
9C22×D8×C53C412yescomplemented
10C2×#121×C53C2211yescharacteristic
11C2×C4×D4×C53—21no—
12C22×C22⋊C4×C53—21no—
13C53×#447—21no—
14C53×#447—22no—
15C53×#447—21no—
16C2×#102×C53—22no—
17C2×#102×C53C412yescomplemented
18C2×#99×C53—21no—
19C2×#99×C53—21no—
20C2×C22⋊C8×C53C2211yescharacteristic
21C2×C22⋊C8×C53C2211yescharacteristic
22C53×#390—21no—
23C53×#386—22no—
24C53×#380C2211yescharacteristic
25C53×#380C2211yescharacteristic
26C53×#379—21no—
27C4×C22⋊C4×C53C2211yescharacteristic
28C53×#327—22no—
29C25×C53—21no—
30C22×D4×C53—21no—
31C22×D4×C53—21no—
32C22×D4×C53—21no—
33C22×D4×C53—21no—
34C22×D4×C53—44no—
35C22×D4×C53C2×C412yes—
36C22×D4×C53C2×C411yescharacteristic, complemented
37C22×D4×C53D411yescharacteristic
38C22×D4×C53D412yes—
39C22×D4×C53D411yescharacteristic
40C23×C4×C53—22no—
41C23×C4×C53C2311yescharacteristic
42C2×D8×C53—22no—
43C2×D8×C53—22no—
44C2×D8×C53—44no—
45C22×C8×C53—21no—
46C22×C8×C53C2×C412yescomplemented
47#121×C53—21no—
48#121×C53—21no—
49#121×C53—21no—
50#121×C53—21no—
51C4×D4×C53—44no—
52C2×C4⋊C4×C53—21no—
53C2×C4⋊C4×C53—21no—
54C2×C4⋊C4×C53C2×C411yescharacteristic, complemented
55C2×C22⋊C4×C53—22no—
56C2×C22⋊C4×C53—22no—
57C2×C22⋊C4×C53—22no—
58C2×C22⋊C4×C53—21no—
59C2×C22⋊C4×C53—21no—
60C2×C22⋊C4×C53—21no—
61C2×C22⋊C4×C53—42no—
62C2×C22⋊C4×C53—42no—
63C2×C22⋊C4×C53D411yescharacteristic
64C2×C22⋊C4×C53D411yescharacteristic
65C2×C42×C53—21no—
66#102×C53—22no—
67#102×C53—22no—
68#102×C53—44no—
69#99×C53—42no—
70#99×C53—42no—
71C22⋊C8×C53—21no—
72C22⋊C8×C53—21no—
73C22⋊C8×C53—21no—
74C22⋊C8×C53—21no—
75#95×C53—21no—
76#95×C53—21no—
77#95×C53D412yes—
78C24×C53—21no—
79C24×C53—21no—
80C24×C53—22no—
81C24×C53—22no—
82C24×C53—21no—
83C24×C53—44no—
84C24×C53—41no—
85C24×C53—41no—
86C24×C53—41no—
87C24×C53—41no—
88C24×C53—82no—
89C24×C53C2×D411yescharacteristic
90C2×D4×C53—24no—
91C2×D4×C53—24no—
92C2×D4×C53—24no—
93C2×D4×C53—21no—
94C2×D4×C53—22no—
95C2×D4×C53—21no—
96C2×D4×C53—22no—
97C2×D4×C53—21no—
98C2×D4×C53—21no—
99C2×D4×C53—42no—
100C2×D4×C53—42no—
101C2×D4×C53—42no—
102C2×D4×C53—42no—
103C2×D4×C53—44no—
104C2×D4×C53—44no—
105C2×D4×C53—44no—
106C2×D4×C53—41no—
107C2×D4×C53—41no—
108C2×D4×C53—41no—
109C2×D4×C53—41no—
110C2×D4×C53—41no—
111C2×D4×C53—41no—
112C2×D4×C53C22⋊C412yes—
113C2×D4×C53C22⋊C412yes—
114C22×C4×C53—21no—
115C22×C4×C53—21no—
116C22×C4×C53—21no—
117C22×C4×C53—22no—
118C22×C4×C53—42no—
119C22×C4×C53—44no—
120C22×C4×C53—44no—
121C22×C4×C53—41no—
122C22×C4×C53—82no—
123C22×C4×C53C22×C411yescharacteristic
124C22×C4×C53C2×D411yescharacteristic
125C22×C4×C53C2×D411yescharacteristic
126C22×C4×C53C2×D411yescharacteristic
127C22×C4×C53#4111yescharacteristic
128C22×C4×C53#4111yescharacteristic
129D8×C53—44no—
130D8×C53—44no—
131C2×C8×C53—22no—
132C2×C8×C53—42no—
133C2×C8×C53—42no—
134C2×C8×C53—41no—
135C4⋊C4×C53—21no—
136C4⋊C4×C53—21no—
137C4⋊C4×C53—42no—
138C4⋊C4×C53—42no—
139C22⋊C4×C53—21no—
140C22⋊C4×C53—21no—
141C22⋊C4×C53—21no—
142C22⋊C4×C53—44no—
143C22⋊C4×C53—44no—
144C22⋊C4×C53—41no—
145C22⋊C4×C53—41no—
146C22⋊C4×C53—41no—
147C22⋊C4×C53—41no—
148C22⋊C4×C53—41no—
149C22⋊C4×C53—41no—
150C22⋊C4×C53—82no—
151C22⋊C4×C53D812yescomplemented
152C42×C53—42no—
153C23×C53—24no—
154C23×C53—22no—
155C23×C53—22no—
156C23×C53—41no—
157C23×C53—41no—
158C23×C53—44no—
159C23×C53—44no—
160C23×C53—42no—
161C23×C53—42no—
162C23×C53—42no—
163C23×C53—41no—
164C23×C53—41no—
165C23×C53—42no—
166C23×C53—42no—
167C23×C53—41no—
168C23×C53—41no—
169C23×C53—42no—
170C23×C53—42no—
171C23×C53—41no—
172C23×C53—41no—
173C23×C53—41no—
174C23×C53—82no—
175C23×C53—82no—
176C23×C53—81no—
177C23×C53—81no—
178C23×C53—84no—
179C23×C53—84no—
180C23×C53—84no—
181C23×C53—84no—
182C23×C53C4×D411yescharacteristic
183C23×C53#12011yescharacteristic
184C23×C53#12111yescharacteristic
185C23×C53C2×D811yescharacteristic
186C23×C53#13511yescharacteristic
187C23×C53AGL(1,ℤ/8)11yescharacteristic
188C23×C53AGL(1,ℤ/8)11yescharacteristic
189D4×C53—24no—
190D4×C53—22no—
191D4×C53—22no—
192D4×C53—22no—
193D4×C53—44no—
194D4×C53—44no—
195D4×C53—42no—
196D4×C53—42no—
197D4×C53—81no—
198D4×C53—81no—
199D4×C53—81no—
200D4×C53—81no—
201D4×C53—81no—
202D4×C53#9914yescomplemented
203C2×C4×C53—22no—
204C2×C4×C53—22no—
205C2×C4×C53—22no—
206C2×C4×C53—22no—
207C2×C4×C53—41no—
208C2×C4×C53—41no—
209C2×C4×C53—41no—
210C2×C4×C53—41no—
211C2×C4×C53—41no—
212C2×C4×C53—42no—
213C2×C4×C53—42no—
214C2×C4×C53—41no—
215C2×C4×C53—41no—
216C2×C4×C53—41no—
217C2×C4×C53—41no—
218C2×C4×C53—41no—
219C2×C4×C53—42no—
220C2×C4×C53—41no—
221C2×C4×C53—88no—
222C2×C4×C53—81no—
223C2×C4×C53—81no—
224C2×C4×C53—81no—
225C2×C4×C53C2×C22⋊C411yescharacteristic
226C2×C4×C53C4×D411yescharacteristic
227C2×C4×C53#12111yescharacteristic
228C2×C4×C53#12311yescharacteristic
229C8×C53—42no—
230C8×C53—81no—
231#9259C5311yesmaximal, characteristic, direct product factor, Sylow
232C22×C53—21no—
233C22×C53—21no—
234C22×C53—21no—
235C22×C53—21no—
236C22×C53—21no—
237C22×C53—21no—
238C22×C53—24no—
239C22×C53—41no—
240C22×C53—41no—
241C22×C53—41no—
242C22×C53—41no—
243C22×C53—44no—
244C22×C53—44no—
245C22×C53—42no—
246C22×C53—42no—
247C22×C53—42no—
248C22×C53—42no—
249C22×C53—42no—
250C22×C53—84no—
251C22×C53—84no—
252C22×C53—84no—
253C22×C53—84no—
254C22×C53—82no—
255C22×C53—82no—
256C22×C53—81no—
257C22×C53—81no—
258C22×C53—81no—
259C22×C53—161no—
260C22×C53—161no—
261C22×C53#38611yescharacteristic
262C22×C53C4×D811yescharacteristic
263C22×C53#44211yescharacteristic
264C22×C53#44711yescharacteristic
265C22×C53#44911yescharacteristic
266C22×C53#45911yescharacteristic
267C22×C53#46311yescharacteristic
268C4×C53—21no—
269C4×C53—21no—
270C4×C53—21no—
271C4×C53—42no—
272C4×C53—81no—
273C4×C53—82no—
274C4×C53—81no—
275C4×C53—161no—
276C4×C53C2×#9912yes—
277C2×#447C2×C5311yescharacteristic
278D4×C22⋊C4C2×C5311yescharacteristic, complemented
279#1964C2×C5311yescharacteristic, complemented
280#1786C2×C5311yescharacteristic, complemented
281#1519C2×C5311yescharacteristic, complemented
282#1289C2×C5311yescharacteristic, complemented
283#1282C2×C5311yescharacteristic, complemented
284C2×C53—21no—
285C2×C53—21no—
286C2×C53—21no—
287C2×C53—21no—
288C2×C53—21no—
289C2×C53—21no—
290C2×C53—44no—
291C2×C53—82no—
292C2×C53—161no—
293C2×C53#187911yescharacteristic
294C2×C53#189811yescharacteristic
295C2×C53#190411yescharacteristic
296C23×D4C22×C5311yescharacteristic
297C22×D8C4×C5312yescomplemented
298C2×#121C22×C5311yescharacteristic
299C2×C4×D4—21no—
300C22×C22⋊C4—21no—
301#447—21no—
302#447—22no—
303#447—21no—
304C2×#102—22no—
305C2×#102C4×C5312yescomplemented
306C2×#99—21no—
307C2×#99—21no—
308C2×C22⋊C8C22×C5311yescharacteristic
309C2×C22⋊C8C22×C5311yescharacteristic
310#390—21no—
311#386—22no—
312#380C22×C5311yescharacteristic
313#380C22×C5311yescharacteristic
314#379—21no—
315C4×C22⋊C4C22×C5311yescharacteristic
316#327—22no—
317C53#925911yescharacteristic, direct product factor, Sylow
318C25—21no—
319C22×D4—21no—
320C22×D4—21no—
321C22×D4—21no—
322C22×D4—21no—
323C22×D4—44no—
324C22×D4C2×C4×C5312yes—
325C22×D4C2×C4×C5311yescharacteristic, complemented
326C22×D4D4×C5311yescharacteristic
327C22×D4D4×C5312yes—
328C22×D4D4×C5311yescharacteristic
329C23×C4—22no—
330C23×C4C23×C5311yescharacteristic
331C2×D8—22no—
332C2×D8—22no—
333C2×D8—44no—
334C22×C8—21no—
335C22×C8C2×C4×C5312yescomplemented
336#121—21no—
337#121—21no—
338#121—21no—
339#121—21no—
340C4×D4—44no—
341C2×C4⋊C4—21no—
342C2×C4⋊C4—21no—
343C2×C4⋊C4C2×C4×C5311yescharacteristic, complemented
344C2×C22⋊C4—22no—
345C2×C22⋊C4—22no—
346C2×C22⋊C4—21no—
347C2×C22⋊C4—21no—
348C2×C22⋊C4—21no—
349C2×C22⋊C4—22no—
350C2×C22⋊C4—42no—
351C2×C22⋊C4—42no—
352C2×C22⋊C4D4×C5311yescharacteristic
353C2×C22⋊C4D4×C5311yescharacteristic
354C2×C42—21no—
355#102—22no—
356#102—22no—
357#102—44no—
358#99—42no—
359#99—42no—
360C22⋊C8—21no—
361C22⋊C8—21no—
362C22⋊C8—21no—
363C22⋊C8—21no—
364#95—21no—
365#95—21no—
366#95D4×C5312yes—
367C24—21no—
368C24—21no—
369C24—22no—
370C24—22no—
371C24—21no—
372C24—44no—
373C24—41no—
374C24—41no—
375C24—41no—
376C24—41no—
377C24—82no—
378C24C2×D4×C5311yescharacteristic
379C2×D4—24no—
380C2×D4—24no—
381C2×D4—24no—
382C2×D4—21no—
383C2×D4—22no—
384C2×D4—21no—
385C2×D4—22no—
386C2×D4—21no—
387C2×D4—21no—
388C2×D4—42no—
389C2×D4—42no—
390C2×D4—42no—
391C2×D4—42no—
392C2×D4—44no—
393C2×D4—44no—
394C2×D4—44no—
395C2×D4—41no—
396C2×D4—41no—
397C2×D4—41no—
398C2×D4—41no—
399C2×D4—41no—
400C2×D4—41no—
401C2×D4C22⋊C4×C5312yes—
402C2×D4C22⋊C4×C5312yes—
403C22×C4—21no—
404C22×C4—21no—
405C22×C4—21no—
406C22×C4—22no—
407C22×C4—42no—
408C22×C4—44no—
409C22×C4—44no—
410C22×C4—41no—
411C22×C4—82no—
412C22×C4C22×C4×C5311yescharacteristic
413C22×C4C2×D4×C5311yescharacteristic
414C22×C4C2×D4×C5311yescharacteristic
415C22×C4C2×D4×C5311yescharacteristic
416C22×C4#41×C5311yescharacteristic
417C22×C4#41×C5311yescharacteristic
418D8—44no—
419D8—44no—
420C2×C8—22no—
421C2×C8—42no—
422C2×C8—42no—
423C2×C8—41no—
424C4⋊C4—21no—
425C4⋊C4—21no—
426C4⋊C4—42no—
427C4⋊C4—42no—
428C22⋊C4—21no—
429C22⋊C4—21no—
430C22⋊C4—21no—
431C22⋊C4—44no—
432C22⋊C4—44no—
433C22⋊C4—41no—
434C22⋊C4—41no—
435C22⋊C4—41no—
436C22⋊C4—41no—
437C22⋊C4—41no—
438C22⋊C4—41no—
439C22⋊C4—82no—
440C22⋊C4D8×C5312yescomplemented
441C42—42no—
442C23—24no—
443C23—22no—
444C23—22no—
445C23—41no—
446C23—41no—
447C23—44no—
448C23—44no—
449C23—42no—
450C23—42no—
451C23—42no—
452C23—41no—
453C23—41no—
454C23—42no—
455C23—42no—
456C23—41no—
457C23—41no—
458C23—42no—
459C23—42no—
460C23—41no—
461C23—41no—
462C23—41no—
463C23—82no—
464C23—82no—
465C23—81no—
466C23—81no—
467C23—84no—
468C23—84no—
469C23—84no—
470C23—84no—
471C23C4×D4×C5311yescharacteristic
472C23#120×C5311yescharacteristic
473C23#121×C5311yescharacteristic
474C23C2×D8×C5311yescharacteristic
475C23#135×C5311yescharacteristic
476C23AGL(1,ℤ/8)×C5311yescharacteristic
477C23AGL(1,ℤ/8)×C5311yescharacteristic
478D4—24no—
479D4—22no—
480D4—22no—
481D4—22no—
482D4—44no—
483D4—44no—
484D4—42no—
485D4—42no—
486D4—81no—
487D4—81no—
488D4—81no—
489D4—81no—
490D4—81no—
491D4#99×C5314yescomplemented
492C2×C4—22no—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—22no—
496C2×C4—41no—
497C2×C4—41no—
498C2×C4—41no—
499C2×C4—41no—
500C2×C4—41no—
501C2×C4—41no—
502C2×C4—42no—
503C2×C4—41no—
504C2×C4—42no—
505C2×C4—41no—
506C2×C4—41no—
507C2×C4—41no—
508C2×C4—42no—
509C2×C4—41no—
510C2×C4—88no—
511C2×C4—81no—
512C2×C4—81no—
513C2×C4—81no—
514C2×C4C2×C22⋊C4×C5311yescharacteristic
515C2×C4C4×D4×C5311yescharacteristic
516C2×C4#121×C5311yescharacteristic
517C2×C4#123×C5311yescharacteristic
518C8—42no—
519C8—81no—
520C22—21no—
521C22—21no—
522C22—21no—
523C22—21no—
524C22—21no—
525C22—21no—
526C22—24no—
527C22—41no—
528C22—41no—
529C22—41no—
530C22—41no—
531C22—44no—
532C22—44no—
533C22—42no—
534C22—42no—
535C22—42no—
536C22—42no—
537C22—42no—
538C22—84no—
539C22—84no—
540C22—84no—
541C22—84no—
542C22—82no—
543C22—82no—
544C22—81no—
545C22—81no—
546C22—81no—
547C22—161no—
548C22—161no—
549C22C53×#38611yescharacteristic
550C22C4×D8×C5311yescharacteristic
551C22C53×#44211yescharacteristic
552C22C53×#44711yescharacteristic
553C22C53×#44911yescharacteristic
554C22C53×#45911yescharacteristic
555C22C53×#46311yescharacteristic
556C4—21no—
557C4—21no—
558C4—21no—
559C4—42no—
560C4—82no—
561C4—81no—
562C4—81no—
563C4—161no—
564C4C2×#99×C5312yes—
565C2—21no—
566C2—21no—
567C2—21no—
568C2—21no—
569C2—21no—
570C2—21no—
571C2—44no—
572C2—82no—
573C2—161no—
574C2C53×#187911yescharacteristic
575C2C53×#189811yescharacteristic
576C2C53×#190411yescharacteristic
subgroupG/H
center, socleC22×C53#386
commutatorC22×C4C22×C4×C53
Frattini, cosocleC23×C4C23×C53
prime radicalC2×C53×#447C2
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. D4×C22⋊C4×C53C2semidirect
3. C53×#1964C2semidirect
4. C53×#1786C2semidirect
5. C53×#1519C2semidirect
6. C53×#1289C2semidirect
7. C53×#1282C2semidirect
9. C22×D8×C53C4semidirect
17. C2×#102×C53C4semidirect
36. C22×D4×C53C2×C4semidirect
46. C22×C8×C53C2×C4semidirect
54. C2×C4⋊C4×C53C2×C4semidirect
151. C22⋊C4×C53D8semidirect
202. D4×C53#99semidirect
231. #9259C53direct (N × H)
278. D4×C22⋊C4C2×C53semidirect
279. #1964C2×C53semidirect
280. #1786C2×C53semidirect
281. #1519C2×C53semidirect
282. #1289C2×C53semidirect
283. #1282C2×C53semidirect
297. C22×D8C4×C53semidirect
305. C2×#102C4×C53semidirect
317. C53#9259direct (N × H)
325. C22×D4C2×C4×C53semidirect
335. C22×C8C2×C4×C53semidirect
343. C2×C4⋊C4C2×C4×C53semidirect
440. C22⋊C4D8×C53semidirect
491. D4#99×C53semidirect

Maximal subgroups

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

subgroupindex
C2×C53×#4472
D4×C22⋊C4×C532
C53×#19642
C53×#17862
C53×#15192
C53×#12892
C53×#12822
#925953