Finite Group Databasebeta

← C53×#9039 #27216551

Subgroups — C53×#9039 #27216551

Order 13568.

834 conjugacy classes of subgroups.

The subgroup lattice has 606 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C53×#448C211yesmaximal, characteristic
2C53×#2325C211yesmaximal, characteristic
3C53×#1787C211yesmaximal, characteristic, complemented
4C2×C53×#327C211yesmaximal, characteristic
5C53×#1446C211yesmaximal, characteristic, complemented
6C53×#1294C211yesmaximal, characteristic, complemented
7C53×#1280C211yesmaximal, characteristic, complemented
8C23×Q8×C53C2211yescharacteristic
9C22×SD16×C53—21no—
10C2×#121×C53C2211yescharacteristic
11C2×C4×D4×C53—21no—
12C53×#448—22no—
13C53×#448C414yescomplemented
14C2×#103×C53—21no—
15C2×#103×C53—21no—
16C2×#103×C53—21no—
17C2×C22⋊C8×C53C2211yescharacteristic
18C2×C22⋊C8×C53C2211yescharacteristic
19C53×#388—21no—
20C53×#387—21no—
21C53×#386—21no—
22C53×#386—21no—
23C53×#385—21no—
24C53×#380C2211yescharacteristic
25C2×#95×C53—21no—
26C2×#95×C53—21no—
27C2×#95×C53C2211yescharacteristic
28C2×#95×C53C2211yescharacteristic
29C53×#327—22no—
30C22×Q8×C53—21no—
31C22×Q8×C53—21no—
32C22×Q8×C53—21no—
33C22×Q8×C53—21no—
34C22×Q8×C53—21no—
35C22×Q8×C53—42no—
36C22×Q8×C53—42no—
37C22×Q8×C53C2×C412yescomplemented
38C22×D4×C53—21no—
39C22×D4×C53D411yescharacteristic
40C23×C4×C53—21no—
41C23×C4×C53—21no—
42C23×C4×C53—21no—
43C23×C4×C53D411yescharacteristic
44C23×C4×C53D411yescharacteristic
45C23×C4×C53C2311yescharacteristic
46C2×SD16×C53—24no—
47C2×SD16×C53—42no—
48C2×SD16×C53—42no—
49C22×C8×C53—21no—
50C22×C8×C53—21no—
51#121×C53—21no—
52#121×C53—21no—
53#121×C53C2×C412yes—
54#121×C53D412yes—
55C4×D4×C53—42no—
56C4×D4×C53—42no—
57C2×C4⋊C4×C53—21no—
58C2×C4⋊C4×C53D411yescharacteristic
59C2×C4⋊C4×C53D411yescharacteristic
60C2×C4⋊C4×C53D411yescharacteristic
61C2×C22⋊C4×C53—21no—
62C2×C22⋊C4×C53—21no—
63C2×C22⋊C4×C53—21no—
64C2×C22⋊C4×C53—21no—
65C2×C22⋊C4×C53—21no—
66C2×C42×C53—21no—
67#103×C53—24no—
68#103×C53—42no—
69#103×C53—42no—
70C22⋊C8×C53—21no—
71C22⋊C8×C53—21no—
72C22⋊C8×C53—21no—
73C22⋊C8×C53C2×C412yescomplemented
74#95×C53—21no—
75#95×C53—21no—
76#95×C53—21no—
77#95×C53—21no—
78#95×C53—21no—
79#95×C53—21no—
80#95×C53—21no—
81#95×C53—42no—
82#95×C53—42no—
83C24×C53—21no—
84C24×C53C2×D411yescharacteristic
85C2×Q8×C53—22no—
86C2×Q8×C53—22no—
87C2×Q8×C53—22no—
88C2×Q8×C53—22no—
89C2×Q8×C53—22no—
90C2×Q8×C53—22no—
91C2×Q8×C53—44no—
92C2×Q8×C53—41no—
93C2×Q8×C53—41no—
94C2×Q8×C53—41no—
95C2×Q8×C53—41no—
96C2×Q8×C53—44no—
97C2×Q8×C53—44no—
98C2×Q8×C53—44no—
99C2×Q8×C53—41no—
100C2×Q8×C53—41no—
101C2×D4×C53—22no—
102C2×D4×C53—22no—
103C2×D4×C53—21no—
104C2×D4×C53—41no—
105C2×D4×C53—41no—
106C2×D4×C53—41no—
107C2×D4×C53—41no—
108C2×D4×C53—41no—
109C2×D4×C53—41no—
110C2×D4×C53C22⋊C412yes—
111C22×C4×C53—22no—
112C22×C4×C53—22no—
113C22×C4×C53—22no—
114C22×C4×C53—21no—
115C22×C4×C53—21no—
116C22×C4×C53—21no—
117C22×C4×C53—22no—
118C22×C4×C53—21no—
119C22×C4×C53—21no—
120C22×C4×C53—21no—
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—41no—
124C22×C4×C53—41no—
125C22×C4×C53—42no—
126C22×C4×C53—41no—
127C22×C4×C53—41no—
128C22×C4×C53—41no—
129C22×C4×C53—41no—
130C22×C4×C53—41no—
131C22×C4×C53—41no—
132C22×C4×C53—41no—
133C22×C4×C53—41no—
134C22×C4×C53—41no—
135C22×C4×C53—41no—
136C22×C4×C53—41no—
137C22×C4×C53—41no—
138C22×C4×C53—41no—
139C22×C4×C53—41no—
140C22×C4×C53—81no—
141C22×C4×C53—81no—
142C22×C4×C53SD1612yescomplemented
143C22×C4×C53C22×C411yescharacteristic
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53C2×D411yescharacteristic
147C22×C4×C53#4111yescharacteristic
148C22×C4×C53#4111yescharacteristic
149SD16×C53—48no—
150C2×C8×C53—22no—
151C2×C8×C53—41no—
152C2×C8×C53—41no—
153C2×C8×C53—41no—
154C2×C8×C53—41no—
155C2×C8×C53—41no—
156C4⋊C4×C53—21no—
157C4⋊C4×C53—42no—
158C4⋊C4×C53—41no—
159C4⋊C4×C53—41no—
160C4⋊C4×C53C22⋊C412yes—
161C22⋊C4×C53—22no—
162C22⋊C4×C53—22no—
163C22⋊C4×C53—42no—
164C22⋊C4×C53—41no—
165C22⋊C4×C53—41no—
166C22⋊C4×C53—81no—
167C22⋊C4×C53—81no—
168C42×C53—42no—
169C23×C53—21no—
170C23×C53—21no—
171C23×C53—21no—
172C23×C53—21no—
173C23×C53—22no—
174C23×C53—41no—
175C23×C53—41no—
176C23×C53—41no—
177C23×C53—41no—
178C23×C53—41no—
179C23×C53—41no—
180C23×C53C4×D411yescharacteristic
181C23×C53#12011yescharacteristic
182C23×C53#12111yescharacteristic
183C23×C53C2×SD1611yescharacteristic
184C23×C53#13511yescharacteristic
185C23×C53AGL(1,ℤ/8)11yescharacteristic
186C23×C53#13711yescharacteristic
187Q8×C53—24no—
188Q8×C53—24no—
189Q8×C53—42no—
190Q8×C53—42no—
191Q8×C53—42no—
192Q8×C53—42no—
193Q8×C53—42no—
194Q8×C53—42no—
195D4×C53—42no—
196D4×C53—42no—
197D4×C53—42no—
198D4×C53—42no—
199D4×C53—42no—
200D4×C53—81no—
201C2×C4×C53—21no—
202C2×C4×C53—21no—
203C2×C4×C53—21no—
204C2×C4×C53—21no—
205C2×C4×C53—24no—
206C2×C4×C53—22no—
207C2×C4×C53—22no—
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—42no—
215C2×C4×C53—42no—
216C2×C4×C53—41no—
217C2×C4×C53—41no—
218C2×C4×C53—41no—
219C2×C4×C53—42no—
220C2×C4×C53—42no—
221C2×C4×C53—41no—
222C2×C4×C53—42no—
223C2×C4×C53—42no—
224C2×C4×C53—42no—
225C2×C4×C53—42no—
226C2×C4×C53—42no—
227C2×C4×C53—41no—
228C2×C4×C53—41no—
229C2×C4×C53—41no—
230C2×C4×C53—41no—
231C2×C4×C53—41no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—81no—
237C2×C4×C53—84no—
238C2×C4×C53—81no—
239C2×C4×C53—81no—
240C2×C4×C53—81no—
241C2×C4×C53—81no—
242C2×C4×C53—81no—
243C2×C4×C53C2×C22⋊C411yescharacteristic
244C2×C4×C53C4×D411yescharacteristic
245C2×C4×C53#12111yescharacteristic
246C2×C4×C53#12311yescharacteristic
247C8×C53—42no—
248C8×C53—81no—
249#9039C5311yesmaximal, characteristic, direct product factor, Sylow
250C22×C53—21no—
251C22×C53—21no—
252C22×C53—21no—
253C22×C53—21no—
254C22×C53—21no—
255C22×C53—21no—
256C22×C53—21no—
257C22×C53—21no—
258C22×C53—21no—
259C22×C53—21no—
260C22×C53—21no—
261C22×C53—21no—
262C22×C53—42no—
263C22×C53—42no—
264C22×C53—42no—
265C22×C53—81no—
266C22×C53—81no—
267C22×C53—81no—
268C22×C53—81no—
269C22×C53#38611yescharacteristic
270C22×C53C4×SD1611yescharacteristic
271C22×C53#44011yescharacteristic
272C22×C53#45011yescharacteristic
273C22×C53#45211yescharacteristic
274C22×C53#45511yescharacteristic
275C22×C53#45611yescharacteristic
276C22×C53B(4,𝔽2)11yescharacteristic
277C22×C53B(4,𝔽2)11yescharacteristic
278C22×C53#46111yescharacteristic
279C22×C53#46311yescharacteristic
280C4×C53—21no—
281C4×C53—21no—
282C4×C53—21no—
283C4×C53—21no—
284C4×C53—42no—
285C4×C53—42no—
286C4×C53—44no—
287C4×C53—82no—
288C4×C53—81no—
289C4×C53—81no—
290C4×C53—81no—
291C2×#448C2×C5311yescharacteristic
292#2325C2×C5311yescharacteristic
293#1787C2×C5311yescharacteristic, complemented
294C2×#327C2×C5311yescharacteristic
295#1446C2×C5311yescharacteristic, complemented
296#1294C2×C5311yescharacteristic, complemented
297#1280C2×C5311yescharacteristic, complemented
298C2×C53—21no—
299C2×C53—21no—
300C2×C53—21no—
301C2×C53—21no—
302C2×C53—82no—
303C2×C53#160111yescharacteristic
304C2×C53#160311yescharacteristic
305C2×C53#160411yescharacteristic
306C2×C53#160611yescharacteristic
307C2×C53#187711yescharacteristic
308C2×C53#188211yescharacteristic
309C2×C53#188911yescharacteristic
310C23×Q8C22×C5311yescharacteristic
311C22×SD16—21no—
312C2×#121C22×C5311yescharacteristic
313C2×C4×D4—21no—
314#448—22no—
315#448C4×C5314yescomplemented
316C2×#103—21no—
317C2×#103—21no—
318C2×#103—21no—
319C2×C22⋊C8C22×C5311yescharacteristic
320C2×C22⋊C8C22×C5311yescharacteristic
321#388—21no—
322#387—21no—
323#386—21no—
324#386—21no—
325#385—21no—
326#380C22×C5311yescharacteristic
327C2×#95—21no—
328C2×#95—21no—
329C2×#95C22×C5311yescharacteristic
330C2×#95C22×C5311yescharacteristic
331#327—22no—
332C53#903911yescharacteristic, direct product factor, Sylow
333C22×Q8—21no—
334C22×Q8—21no—
335C22×Q8—21no—
336C22×Q8—21no—
337C22×Q8—21no—
338C22×Q8—42no—
339C22×Q8—42no—
340C22×Q8C2×C4×C5312yescomplemented
341C22×D4—21no—
342C22×D4D4×C5311yescharacteristic
343C23×C4—21no—
344C23×C4—21no—
345C23×C4—21no—
346C23×C4D4×C5311yescharacteristic
347C23×C4D4×C5311yescharacteristic
348C23×C4C23×C5311yescharacteristic
349C2×SD16—24no—
350C2×SD16—42no—
351C2×SD16—42no—
352C22×C8—21no—
353C22×C8—21no—
354#121—21no—
355#121—21no—
356#121C2×C4×C5312yes—
357#121D4×C5312yes—
358C4×D4—42no—
359C4×D4—42no—
360C2×C4⋊C4—21no—
361C2×C4⋊C4D4×C5311yescharacteristic
362C2×C4⋊C4D4×C5311yescharacteristic
363C2×C4⋊C4D4×C5311yescharacteristic
364C2×C22⋊C4—21no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—21no—
369C2×C42—21no—
370#103—24no—
371#103—42no—
372#103—42no—
373C22⋊C8—21no—
374C22⋊C8—21no—
375C22⋊C8—21no—
376C22⋊C8C2×C4×C5312yescomplemented
377#95—21no—
378#95—21no—
379#95—21no—
380#95—21no—
381#95—21no—
382#95—21no—
383#95—21no—
384#95—42no—
385#95—42no—
386C24—21no—
387C24C2×D4×C5311yescharacteristic
388C2×Q8—22no—
389C2×Q8—22no—
390C2×Q8—22no—
391C2×Q8—22no—
392C2×Q8—22no—
393C2×Q8—22no—
394C2×Q8—44no—
395C2×Q8—41no—
396C2×Q8—41no—
397C2×Q8—41no—
398C2×Q8—41no—
399C2×Q8—44no—
400C2×Q8—44no—
401C2×Q8—44no—
402C2×Q8—41no—
403C2×Q8—41no—
404C2×D4—22no—
405C2×D4—22no—
406C2×D4—21no—
407C2×D4—41no—
408C2×D4—41no—
409C2×D4—41no—
410C2×D4—41no—
411C2×D4—41no—
412C2×D4—41no—
413C2×D4C22⋊C4×C5312yes—
414C22×C4—22no—
415C22×C4—22no—
416C22×C4—22no—
417C22×C4—21no—
418C22×C4—21no—
419C22×C4—21no—
420C22×C4—22no—
421C22×C4—21no—
422C22×C4—21no—
423C22×C4—21no—
424C22×C4—21no—
425C22×C4—21no—
426C22×C4—41no—
427C22×C4—41no—
428C22×C4—42no—
429C22×C4—41no—
430C22×C4—41no—
431C22×C4—41no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—41no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—41no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—41no—
441C22×C4—41no—
442C22×C4—41no—
443C22×C4—81no—
444C22×C4—81no—
445C22×C4SD16×C5312yescomplemented
446C22×C4C22×C4×C5311yescharacteristic
447C22×C4C2×D4×C5311yescharacteristic
448C22×C4C2×D4×C5311yescharacteristic
449C22×C4C2×D4×C5311yescharacteristic
450C22×C4#41×C5311yescharacteristic
451C22×C4#41×C5311yescharacteristic
452SD16—48no—
453C2×C8—22no—
454C2×C8—41no—
455C2×C8—41no—
456C2×C8—41no—
457C2×C8—41no—
458C2×C8—41no—
459C4⋊C4—21no—
460C4⋊C4—41no—
461C4⋊C4—41no—
462C4⋊C4—42no—
463C4⋊C4C22⋊C4×C5312yes—
464C22⋊C4—22no—
465C22⋊C4—22no—
466C22⋊C4—42no—
467C22⋊C4—41no—
468C22⋊C4—41no—
469C22⋊C4—81no—
470C22⋊C4—81no—
471C42—42no—
472C23—21no—
473C23—21no—
474C23—21no—
475C23—21no—
476C23—22no—
477C23—41no—
478C23—41no—
479C23—41no—
480C23—41no—
481C23—41no—
482C23—41no—
483C23C4×D4×C5311yescharacteristic
484C23#120×C5311yescharacteristic
485C23#121×C5311yescharacteristic
486C23C2×SD16×C5311yescharacteristic
487C23#135×C5311yescharacteristic
488C23AGL(1,ℤ/8)×C5311yescharacteristic
489C23#137×C5311yescharacteristic
490Q8—24no—
491Q8—24no—
492Q8—42no—
493Q8—42no—
494Q8—42no—
495Q8—42no—
496Q8—42no—
497Q8—42no—
498D4—42no—
499D4—42no—
500D4—42no—
501D4—42no—
502D4—42no—
503D4—81no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—24no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—41no—
512C2×C4—41no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—42no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—42no—
519C2×C4—41no—
520C2×C4—41no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—42no—
527C2×C4—42no—
528C2×C4—42no—
529C2×C4—42no—
530C2×C4—41no—
531C2×C4—41no—
532C2×C4—41no—
533C2×C4—41no—
534C2×C4—41no—
535C2×C4—41no—
536C2×C4—41no—
537C2×C4—41no—
538C2×C4—41no—
539C2×C4—81no—
540C2×C4—84no—
541C2×C4—81no—
542C2×C4—81no—
543C2×C4—81no—
544C2×C4—81no—
545C2×C4—81no—
546C2×C4C2×C22⋊C4×C5311yescharacteristic
547C2×C4C4×D4×C5311yescharacteristic
548C2×C4#121×C5311yescharacteristic
549C2×C4#123×C5311yescharacteristic
550C8—42no—
551C8—81no—
552C22—21no—
553C22—21no—
554C22—21no—
555C22—21no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—21no—
561C22—21no—
562C22—21no—
563C22—21no—
564C22—42no—
565C22—42no—
566C22—42no—
567C22—81no—
568C22—81no—
569C22—81no—
570C22—81no—
571C22C53×#38611yescharacteristic
572C22C4×SD16×C5311yescharacteristic
573C22C53×#44011yescharacteristic
574C22C53×#45011yescharacteristic
575C22C53×#45211yescharacteristic
576C22C53×#45511yescharacteristic
577C22C53×#45611yescharacteristic
578C22C53×B(4,𝔽2)11yescharacteristic
579C22C53×B(4,𝔽2)11yescharacteristic
580C22C53×#46111yescharacteristic
581C22C53×#46311yescharacteristic
582C4—21no—
583C4—21no—
584C4—21no—
585C4—21no—
586C4—42no—
587C4—42no—
588C4—44no—
589C4—82no—
590C4—81no—
591C4—81no—
592C4—81no—
593C2—21no—
594C2—21no—
595C2—21no—
596C2—21no—
597C2—82no—
598C2C53×#160111yescharacteristic
599C2C53×#160311yescharacteristic
600C2C53×#160411yescharacteristic
601C2C53×#160611yescharacteristic
602C2C53×#187711yescharacteristic
603C2C53×#188211yescharacteristic
604C2C53×#188911yescharacteristic
subgroupG/H
center, socleC23×C53#120
commutatorC22×C4C22×C4×C53
Frattini, cosocleC23×C4C23×C53
prime radicalC2×#121×C53C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C53 ◁ C23×C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
3. C53×#1787C2semidirect
5. C53×#1446C2semidirect
6. C53×#1294C2semidirect
7. C53×#1280C2semidirect
13. C53×#448C4semidirect
37. C22×Q8×C53C2×C4semidirect
73. C22⋊C8×C53C2×C4semidirect
142. C22×C4×C53SD16semidirect
249. #9039C53direct (N × H)
293. #1787C2×C53semidirect
295. #1446C2×C53semidirect
296. #1294C2×C53semidirect
297. #1280C2×C53semidirect
315. #448C4×C53semidirect
332. C53#9039direct (N × H)
340. C22×Q8C2×C4×C53semidirect
376. C22⋊C8C2×C4×C53semidirect
445. C22×C4SD16×C53semidirect

Maximal subgroups

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

subgroupindex
C2×C53×#4482
C53×#23252
C53×#17872
C2×C53×#3272
C53×#14462
C53×#12942
C53×#12802
#903953