Finite Group Databasebeta

← C53×#9938 #27217450

Subgroups — C53×#9938 #27217450

Order 13568.

1008 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 1008 conjugacy classes of subgroups — too many to draw. Only its 102 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3006C211yesmaximal, characteristic, complemented
2C53×#2539C211yesmaximal, characteristic, complemented
3C53×#2404C211yesmaximal, characteristic, complemented
4C53×#1789C211yesmaximal, characteristic, complemented
5C53×#1510C211yesmaximal, characteristic, complemented
6C53×#1462C211yesmaximal, characteristic, complemented
7C53×#1284C211yesmaximal, characteristic, complemented
8C22×#41×C53C2211yescharacteristic
9C2×AGL(1,ℤ/8)×C53—21no—
10C2×#123×C53—21no—
11C2×#121×C53C2211yescharacteristic
12C2×#120×C53—21no—
13C2×#120×C53—21no—
14C22×C22⋊C4×C53C2211yescharacteristic
15C53×#449—22no—
16C53×#448—21no—
17C53×#447—21no—
18C2×#104×C53—22no—
19C53×#417—21no—
20C53×#407C2211yescharacteristic
21C53×#407C2211yescharacteristic
22C53×#399—21no—
23C53×#393—21no—
24C53×#392—21no—
25C53×#392—21no—
26C53×#390—22no—
27C53×#386—21no—
28C53×#386—22no—
29C53×#381C2211yescharacteristic
30C4×C22⋊C4×C53C2211yescharacteristic
31C53×#327—22no—
32C53×#323—22no—
33C25×C53D411yescharacteristic
34C2×#41×C53—21no—
35C2×#41×C53—21no—
36C2×#41×C53—21no—
37C2×#41×C53—21no—
38C2×#41×C53—44no—
39C22×Q8×C53D411yescharacteristic, complemented
40C22×D4×C53—21no—
41C22×D4×C53—21no—
42C22×D4×C53—22no—
43C22×D4×C53—22no—
44C22×D4×C53D411yescharacteristic, complemented
45C22×D4×C53D411yescharacteristic, complemented
46C23×C4×C53—21no—
47C23×C4×C53D411yescharacteristic
48C23×C4×C53C2311yescharacteristic
49AGL(1,ℤ/8)×C53—44no—
50C2×SD16×C53—22no—
51C2×D8×C53—22no—
52C2×M16×C53—21no—
53C2×M16×C53—21no—
54#123×C53—44no—
55#121×C53—21no—
56#121×C53—21no—
57#121×C53—21no—
58#121×C53—21no—
59#120×C53—44no—
60#120×C53—44no—
61C2×C4⋊C4×C53—22no—
62C2×C4⋊C4×C53D411yescharacteristic, complemented
63C2×C22⋊C4×C53—22no—
64C2×C22⋊C4×C53—22no—
65C2×C22⋊C4×C53—22no—
66C2×C22⋊C4×C53—22no—
67C2×C22⋊C4×C53—21no—
68C2×C22⋊C4×C53—21no—
69C2×C22⋊C4×C53—21no—
70C2×C22⋊C4×C53—21no—
71C2×C22⋊C4×C53—21no—
72C2×C22⋊C4×C53—21no—
73C2×C22⋊C4×C53—21no—
74C2×C22⋊C4×C53—21no—
75C2×C22⋊C4×C53—21no—
76C2×C42×C53D412yescomplemented
77#104×C53—44no—
78#103×C53—22no—
79#102×C53—22no—
80C22⋊C8×C53—21no—
81C22⋊C8×C53—21no—
82C22⋊C8×C53—21no—
83C22⋊C8×C53D412yescomplemented
84#95×C53—22no—
85#95×C53—21no—
86#95×C53—21no—
87#95×C53D412yescomplemented
88C24×C53—21no—
89C24×C53—21no—
90C24×C53—21no—
91C24×C53—21no—
92C24×C53—21no—
93C24×C53—42no—
94C24×C53—42no—
95C24×C53—42no—
96C24×C53C2×D411yescharacteristic
97#41×C53—24no—
98#41×C53—24no—
99#41×C53—42no—
100#41×C53—42no—
101#41×C53—42no—
102#41×C53—42no—
103#41×C53—42no—
104#41×C53—42no—
105C2×Q8×C53—21no—
106C2×Q8×C53—21no—
107C2×Q8×C53—42no—
108C2×D4×C53—24no—
109C2×D4×C53—21no—
110C2×D4×C53—21no—
111C2×D4×C53—21no—
112C2×D4×C53—21no—
113C2×D4×C53—42no—
114C2×D4×C53—44no—
115C2×D4×C53—42no—
116C2×D4×C53—42no—
117C2×D4×C53—41no—
118C2×D4×C53—41no—
119C2×D4×C53—42no—
120C2×D4×C53—44no—
121C2×D4×C53—42no—
122C2×D4×C53—44no—
123C2×D4×C53—44no—
124C2×D4×C53—41no—
125C2×D4×C53—41no—
126C2×D4×C53—82no—
127C2×D4×C53—82no—
128C22×C4×C53—21no—
129C22×C4×C53—22no—
130C22×C4×C53—22no—
131C22×C4×C53—21no—
132C22×C4×C53—21no—
133C22×C4×C53—21no—
134C22×C4×C53—21no—
135C22×C4×C53—42no—
136C22×C4×C53—42no—
137C22×C4×C53—42no—
138C22×C4×C53—41no—
139C22×C4×C53—41no—
140C22×C4×C53—42no—
141C22×C4×C53C2×D411yescharacteristic
142C22×C4×C53C2×D411yescharacteristic
143C22×C4×C53C2×D411yescharacteristic
144C22×C4×C53C2×D411yescharacteristic
145C22×C4×C53C2×D411yescharacteristic
146C22×C4×C53#4111yescharacteristic
147SD16×C53—44no—
148D8×C53—44no—
149M16×C53—42no—
150M16×C53—42no—
151C2×C8×C53—22no—
152C2×C8×C53—41no—
153C4⋊C4×C53—21no—
154C4⋊C4×C53—21no—
155C4⋊C4×C53—44no—
156C22⋊C4×C53—24no—
157C22⋊C4×C53—44no—
158C22⋊C4×C53—44no—
159C22⋊C4×C53—44no—
160C22⋊C4×C53—42no—
161C22⋊C4×C53—41no—
162C22⋊C4×C53—41no—
163C22⋊C4×C53—41no—
164C22⋊C4×C53—41no—
165C22⋊C4×C53—41no—
166C22⋊C4×C53—41no—
167C22⋊C4×C53—82no—
168C42×C53—42no—
169C23×C53—22no—
170C23×C53—22no—
171C23×C53—22no—
172C23×C53—22no—
173C23×C53—22no—
174C23×C53—22no—
175C23×C53—41no—
176C23×C53—41no—
177C23×C53—42no—
178C23×C53—42no—
179C23×C53—42no—
180C23×C53—41no—
181C23×C53—41no—
182C23×C53—41no—
183C23×C53—41no—
184C23×C53—42no—
185C23×C53—41no—
186C23×C53—41no—
187C23×C53—41no—
188C23×C53—42no—
189C23×C53—42no—
190C23×C53—42no—
191C23×C53—41no—
192C23×C53—42no—
193C23×C53—81no—
194C23×C53—81no—
195C23×C53—88no—
196C23×C53#12011yescharacteristic
197C23×C53#12011yescharacteristic
198C23×C53#12111yescharacteristic
199Q8×C53—22no—
200Q8×C53—22no—
201Q8×C53—41no—
202Q8×C53—41no—
203D4×C53—22no—
204D4×C53—22no—
205D4×C53—41no—
206D4×C53—41no—
207D4×C53—42no—
208D4×C53—44no—
209D4×C53—44no—
210D4×C53—81no—
211D4×C53—88no—
212D4×C53—82no—
213D4×C53—88no—
214D4×C53—82no—
215C2×C4×C53—22no—
216C2×C4×C53—22no—
217C2×C4×C53—22no—
218C2×C4×C53—24no—
219C2×C4×C53—41no—
220C2×C4×C53—41no—
221C2×C4×C53—41no—
222C2×C4×C53—41no—
223C2×C4×C53—44no—
224C2×C4×C53—44no—
225C2×C4×C53—41no—
226C2×C4×C53—42no—
227C2×C4×C53—41no—
228C2×C4×C53—42no—
229C2×C4×C53—42no—
230C2×C4×C53—42no—
231C2×C4×C53—42no—
232C2×C4×C53—41no—
233C2×C4×C53—41no—
234C2×C4×C53—41no—
235C2×C4×C53—41no—
236C2×C4×C53—42no—
237C2×C4×C53—42no—
238C2×C4×C53—81no—
239C2×C4×C53#12011yescharacteristic
240C2×C4×C53#12111yescharacteristic
241C2×C4×C53#12111yescharacteristic
242C2×C4×C53#12711yescharacteristic
243C8×C53—42no—
244C8×C53—81no—
245#9938C5311yesmaximal, characteristic, direct product factor, Sylow
246C22×C53—21no—
247C22×C53—21no—
248C22×C53—21no—
249C22×C53—21no—
250C22×C53—21no—
251C22×C53—21no—
252C22×C53—21no—
253C22×C53—44no—
254C22×C53—44no—
255C22×C53—44no—
256C22×C53—44no—
257C22×C53—42no—
258C22×C53—42no—
259C22×C53—82no—
260C22×C53—82no—
261C22×C53—82no—
262C22×C53—82no—
263C22×C53—82no—
264C22×C53—82no—
265C22×C53—82no—
266C22×C53—81no—
267C22×C53—81no—
268C22×C53—81no—
269C22×C53—81no—
270C22×C53—81no—
271C22×C53—81no—
272C22×C53—81no—
273C22×C53#39211yescharacteristic
274C22×C53#45312yes—
275C22×C53B(4,𝔽2)12yes—
276C4×C53—21no—
277C4×C53—21no—
278C4×C53—21no—
279C4×C53—21no—
280C4×C53—44no—
281C4×C53—42no—
282C4×C53—82no—
283C4×C53—81no—
284C4×C53—81no—
285#3006C2×C5311yescharacteristic, complemented
286#2539C2×C5311yescharacteristic, complemented
287#2404C2×C5311yescharacteristic, complemented
288#1789C2×C5311yescharacteristic, complemented
289#1510C2×C5311yescharacteristic, complemented
290#1462C2×C5311yescharacteristic, complemented
291#1284C2×C5311yescharacteristic, complemented
292C2×C53—22no—
293C2×C53—22no—
294C2×C53—41no—
295C2×C53—82no—
296C2×C53—82no—
297C2×C53—161no—
298C2×C53#200211yescharacteristic
299C2×C53#201011yescharacteristic
300C2×C53#202111yescharacteristic
301C22×#41C22×C5311yescharacteristic
302C2×AGL(1,ℤ/8)—21no—
303C2×#123—21no—
304C2×#121C22×C5311yescharacteristic
305C2×#120—21no—
306C2×#120—21no—
307C22×C22⋊C4C22×C5311yescharacteristic
308#449—22no—
309#448—21no—
310#447—21no—
311C2×#104—22no—
312#417—21no—
313#407C22×C5311yescharacteristic
314#407C22×C5311yescharacteristic
315#399—21no—
316#393—21no—
317#392—21no—
318#392—21no—
319#390—22no—
320#386—21no—
321#386—22no—
322#381C22×C5311yescharacteristic
323C4×C22⋊C4C22×C5311yescharacteristic
324#327—22no—
325#323—22no—
326C53#993811yescharacteristic, direct product factor, Sylow
327C25D4×C5311yescharacteristic
328C2×#41—21no—
329C2×#41—21no—
330C2×#41—21no—
331C2×#41—21no—
332C2×#41—44no—
333C22×Q8D4×C5311yescharacteristic, complemented
334C22×D4—21no—
335C22×D4—21no—
336C22×D4—22no—
337C22×D4—22no—
338C22×D4D4×C5311yescharacteristic, complemented
339C22×D4D4×C5311yescharacteristic, complemented
340C23×C4—21no—
341C23×C4D4×C5311yescharacteristic
342C23×C4C23×C5311yescharacteristic
343AGL(1,ℤ/8)—44no—
344C2×SD16—22no—
345C2×D8—22no—
346C2×M16—21no—
347C2×M16—21no—
348#123—44no—
349#121—21no—
350#121—21no—
351#121—21no—
352#121—21no—
353#120—44no—
354#120—44no—
355C2×C4⋊C4—22no—
356C2×C4⋊C4D4×C5311yescharacteristic, complemented
357C2×C22⋊C4—21no—
358C2×C22⋊C4—22no—
359C2×C22⋊C4—22no—
360C2×C22⋊C4—22no—
361C2×C22⋊C4—21no—
362C2×C22⋊C4—21no—
363C2×C22⋊C4—21no—
364C2×C22⋊C4—21no—
365C2×C22⋊C4—21no—
366C2×C22⋊C4—21no—
367C2×C22⋊C4—21no—
368C2×C22⋊C4—21no—
369C2×C22⋊C4—22no—
370C2×C42D4×C5312yescomplemented
371#104—44no—
372#103—22no—
373#102—22no—
374C22⋊C8—21no—
375C22⋊C8—21no—
376C22⋊C8—21no—
377C22⋊C8D4×C5312yescomplemented
378#95—22no—
379#95—21no—
380#95—21no—
381#95D4×C5312yescomplemented
382C24—21no—
383C24—21no—
384C24—21no—
385C24—21no—
386C24—21no—
387C24—42no—
388C24—42no—
389C24—42no—
390C24C2×D4×C5311yescharacteristic
391#41—24no—
392#41—24no—
393#41—42no—
394#41—42no—
395#41—42no—
396#41—42no—
397#41—42no—
398#41—42no—
399C2×Q8—21no—
400C2×Q8—21no—
401C2×Q8—42no—
402C2×D4—24no—
403C2×D4—21no—
404C2×D4—21no—
405C2×D4—21no—
406C2×D4—21no—
407C2×D4—42no—
408C2×D4—44no—
409C2×D4—42no—
410C2×D4—42no—
411C2×D4—41no—
412C2×D4—41no—
413C2×D4—42no—
414C2×D4—44no—
415C2×D4—42no—
416C2×D4—44no—
417C2×D4—44no—
418C2×D4—41no—
419C2×D4—41no—
420C2×D4—82no—
421C2×D4—82no—
422C22×C4—21no—
423C22×C4—22no—
424C22×C4—22no—
425C22×C4—21no—
426C22×C4—21no—
427C22×C4—21no—
428C22×C4—21no—
429C22×C4—42no—
430C22×C4—42no—
431C22×C4—42no—
432C22×C4—41no—
433C22×C4—41no—
434C22×C4—42no—
435C22×C4C2×D4×C5311yescharacteristic
436C22×C4C2×D4×C5311yescharacteristic
437C22×C4C2×D4×C5311yescharacteristic
438C22×C4C2×D4×C5311yescharacteristic
439C22×C4C2×D4×C5311yescharacteristic
440C22×C4#41×C5311yescharacteristic
441SD16—44no—
442D8—44no—
443M16—42no—
444M16—42no—
445C2×C8—22no—
446C2×C8—41no—
447C4⋊C4—21no—
448C4⋊C4—21no—
449C4⋊C4—44no—
450C22⋊C4—24no—
451C22⋊C4—44no—
452C22⋊C4—44no—
453C22⋊C4—44no—
454C22⋊C4—42no—
455C22⋊C4—41no—
456C22⋊C4—41no—
457C22⋊C4—41no—
458C22⋊C4—41no—
459C22⋊C4—41no—
460C22⋊C4—41no—
461C22⋊C4—82no—
462C42—42no—
463C23—22no—
464C23—22no—
465C23—22no—
466C23—22no—
467C23—22no—
468C23—22no—
469C23—41no—
470C23—42no—
471C23—42no—
472C23—41no—
473C23—42no—
474C23—41no—
475C23—41no—
476C23—41no—
477C23—41no—
478C23—41no—
479C23—42no—
480C23—41no—
481C23—41no—
482C23—42no—
483C23—42no—
484C23—42no—
485C23—41no—
486C23—42no—
487C23—81no—
488C23—81no—
489C23—88no—
490C23#120×C5311yescharacteristic
491C23#120×C5311yescharacteristic
492C23#121×C5311yescharacteristic
493Q8—22no—
494Q8—22no—
495Q8—41no—
496Q8—41no—
497D4—22no—
498D4—22no—
499D4—41no—
500D4—41no—
501D4—42no—
502D4—44no—
503D4—44no—
504D4—81no—
505D4—88no—
506D4—82no—
507D4—88no—
508D4—82no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—24no—
512C2×C4—22no—
513C2×C4—41no—
514C2×C4—41no—
515C2×C4—41no—
516C2×C4—41no—
517C2×C4—44no—
518C2×C4—44no—
519C2×C4—41no—
520C2×C4—42no—
521C2×C4—41no—
522C2×C4—42no—
523C2×C4—42no—
524C2×C4—41no—
525C2×C4—42no—
526C2×C4—41no—
527C2×C4—42no—
528C2×C4—41no—
529C2×C4—41no—
530C2×C4—42no—
531C2×C4—42no—
532C2×C4—81no—
533C2×C4#120×C5311yescharacteristic
534C2×C4#121×C5311yescharacteristic
535C2×C4#121×C5311yescharacteristic
536C2×C4#127×C5311yescharacteristic
537C8—42no—
538C8—81no—
539C22—21no—
540C22—21no—
541C22—21no—
542C22—21no—
543C22—21no—
544C22—21no—
545C22—21no—
546C22—44no—
547C22—44no—
548C22—44no—
549C22—44no—
550C22—42no—
551C22—42no—
552C22—82no—
553C22—82no—
554C22—82no—
555C22—82no—
556C22—82no—
557C22—82no—
558C22—82no—
559C22—81no—
560C22—81no—
561C22—81no—
562C22—81no—
563C22—81no—
564C22—81no—
565C22—81no—
566C22C53×#39211yescharacteristic
567C22C53×#45312yes—
568C22C53×B(4,𝔽2)12yes—
569C4—21no—
570C4—21no—
571C4—21no—
572C4—21no—
573C4—44no—
574C4—42no—
575C4—82no—
576C4—81no—
577C4—81no—
578C2—22no—
579C2—22no—
580C2—41no—
581C2—82no—
582C2—82no—
583C2—161no—
584C2C53×#200211yescharacteristic
585C2C53×#201011yescharacteristic
586C2C53×#202111yescharacteristic
subgroupG/H
center, socleC22×C53#392
commutator, Frattini, cosocleC23×C4C23×C53
LCSG ▷ C23×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C53 ◁ C23×C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
1. C53×#3006C2semidirect
2. C53×#2539C2semidirect
3. C53×#2404C2semidirect
4. C53×#1789C2semidirect
5. C53×#1510C2semidirect
6. C53×#1462C2semidirect
7. C53×#1284C2semidirect
39. C22×Q8×C53D4semidirect
44. C22×D4×C53D4semidirect
45. C22×D4×C53D4semidirect
62. C2×C4⋊C4×C53D4semidirect
76. C2×C42×C53D4semidirect
83. C22⋊C8×C53D4semidirect
87. #95×C53D4semidirect
245. #9938C53direct (N × H)
285. #3006C2×C53semidirect
286. #2539C2×C53semidirect
287. #2404C2×C53semidirect
288. #1789C2×C53semidirect
289. #1510C2×C53semidirect
290. #1462C2×C53semidirect
291. #1284C2×C53semidirect
326. C53#9938direct (N × H)
333. C22×Q8D4×C53semidirect
338. C22×D4D4×C53semidirect
339. C22×D4D4×C53semidirect
356. C2×C4⋊C4D4×C53semidirect
370. C2×C42D4×C53semidirect
377. C22⋊C8D4×C53semidirect
381. #95D4×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#30062
C53×#25392
C53×#24042
C53×#17892
C53×#15102
C53×#14622
C53×#12842
#993853