Finite Group Databasebeta

← C53×#12705 #27220217

Subgroups — C53×#12705 #27220217

Order 13568.

1186 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1186 conjugacy classes of subgroups — too many to draw. Only its 146 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×#3484C211yesmaximal, characteristic
2C53×#2121C214yesmaximal, complemented
3C53×#1783C211yesmaximal, characteristic, complemented
4C53×#1781C211yesmaximal, characteristic, complemented
5C2×H(5,𝔽2)×C53C411yescharacteristic, complemented
6C23×D4×C53C2211yescharacteristic
7C53×#546—24no—
8D42×C53—24no—
9C53×#535—21no—
10C53×#534—21no—
11C53×#534C2212yes—
12C53×#525—21no—
13C2×#120×C53—21no—
14C2×#120×C53C412yescomplemented
15C53×#518C411yescharacteristic, complemented
16C2×#100×C53—21no—
17C2×#100×C53C2212yescomplemented
18C2×#99×C53—21no—
19C2×#99×C53C2212yescomplemented
20C53×#407—21no—
21C53×#379—21no—
22C53×#352—24no—
23C53×AΣL(1,𝔽16)—24no—
24C25×C53C2×C411yescharacteristic
25C25×C53D411yescharacteristic
26H(5,𝔽2)×C53—42no—
27H(5,𝔽2)×C53—42no—
28C2×#41×C53—21no—
29C2×#41×C53—22no—
30C22×D4×C53—21no—
31C22×D4×C53—22no—
32C22×D4×C53—22no—
33C22×D4×C53—22no—
34C22×D4×C53—24no—
35C22×D4×C53—42no—
36C22×D4×C53—42no—
37C22×D4×C53C2×C411yescharacteristic
38C22×D4×C53C2×C412yes—
39C22×D4×C53D411yescharacteristic
40C22×D4×C53D411yescharacteristic
41C22×D4×C53C2311yescharacteristic
42C23×C4×C53D411yescharacteristic
43C2×M16×C53—22no—
44#127×C53—22no—
45#124×C53—22no—
46#123×C53—22no—
47#123×C53—22no—
48#123×C53D414yes—
49C22⋊Q8×C53—22no—
50#121×C53—22no—
51#121×C53—24no—
52#121×C53—24no—
53#121×C53—24no—
54#120×C53—24no—
55#120×C53—22no—
56#120×C53—24no—
57#120×C53—22no—
58#120×C53—22no—
59#120×C53—22no—
60#120×C53D414yes—
61C4×D4×C53—24no—
62C4×D4×C53—24no—
63#117×C53—21no—
64C2×C22⋊C4×C53—22no—
65C2×C22⋊C4×C53—21no—
66C2×C22⋊C4×C53—22no—
67C2×C22⋊C4×C53—22no—
68C2×C22⋊C4×C53—42no—
69C2×C22⋊C4×C53C2×C412yes—
70#100×C53—24no—
71#100×C53—42no—
72#99×C53—24no—
73#99×C53—42no—
74C22⋊C8×C53—42no—
75C24×C53—21no—
76C24×C53—22no—
77C24×C53—22no—
78C24×C53—22no—
79C24×C53—41no—
80C24×C53—41no—
81C24×C53—41no—
82C24×C53—82no—
83C24×C53—82no—
84C24×C53C22⋊C411yescharacteristic
85C24×C53C22⋊C411yescharacteristic
86C24×C53C22⋊C411yescharacteristic
87C24×C53C22⋊C412yescomplemented
88C24×C53C22×C411yescharacteristic
89C24×C53C2×D411yescharacteristic
90#41×C53—42no—
91#41×C53—48no—
92#41×C53—42no—
93C2×Q8×C53—21no—
94C2×D4×C53—22no—
95C2×D4×C53—22no—
96C2×D4×C53—24no—
97C2×D4×C53—22no—
98C2×D4×C53—21no—
99C2×D4×C53—22no—
100C2×D4×C53—22no—
101C2×D4×C53—22no—
102C2×D4×C53—22no—
103C2×D4×C53—24no—
104C2×D4×C53—24no—
105C2×D4×C53—22no—
106C2×D4×C53—22no—
107C2×D4×C53—22no—
108C2×D4×C53—41no—
109C2×D4×C53—41no—
110C2×D4×C53—41no—
111C2×D4×C53—41no—
112C2×D4×C53—416no—
113C2×D4×C53—48no—
114C2×D4×C53—42no—
115C2×D4×C53—42no—
116C2×D4×C53—42no—
117C2×D4×C53—42no—
118C2×D4×C53—84no—
119C2×D4×C53—84no—
120C2×D4×C53C2×D414yes—
121C22×C4×C53—21no—
122C22×C4×C53—22no—
123C22×C4×C53—22no—
124C22×C4×C53—22no—
125C22×C4×C53—22no—
126C22×C4×C53—22no—
127C22×C4×C53—41no—
128C22×C4×C53—42no—
129C22×C4×C53C22⋊C411yescharacteristic
130C22×C4×C53C22⋊C412yescomplemented
131C22×C4×C53C2×D411yescharacteristic
132M16×C53—42no—
133M16×C53—42no—
134C2×C8×C53—42no—
135C4⋊C4×C53—22no—
136C4⋊C4×C53—24no—
137C22⋊C4×C53—24no—
138C22⋊C4×C53—22no—
139C22⋊C4×C53—24no—
140C22⋊C4×C53—22no—
141C22⋊C4×C53—24no—
142C22⋊C4×C53—24no—
143C22⋊C4×C53—42no—
144C22⋊C4×C53—42no—
145C22⋊C4×C53—84no—
146C22⋊C4×C53C22⋊C414yescomplemented
147C42×C53—22no—
148C23×C53—21no—
149C23×C53—21no—
150C23×C53—21no—
151C23×C53—21no—
152C23×C53—21no—
153C23×C53—21no—
154C23×C53—21no—
155C23×C53—21no—
156C23×C53—21no—
157C23×C53—22no—
158C23×C53—21no—
159C23×C53—21no—
160C23×C53—21no—
161C23×C53—42no—
162C23×C53—44no—
163C23×C53—44no—
164C23×C53—44no—
165C23×C53—44no—
166C23×C53—44no—
167C23×C53—42no—
168C23×C53—42no—
169C23×C53—82no—
170C23×C53—82no—
171C23×C53—82no—
172C23×C53—81no—
173C23×C53—81no—
174C23×C53—81no—
175C23×C53—81no—
176C23×C53—82no—
177C23×C53—82no—
178C23×C53—82no—
179C23×C53—82no—
180C23×C53—82no—
181C23×C53—82no—
182C23×C53—82no—
183C23×C53—82no—
184C23×C53#9912yes—
185C23×C53C2×C22⋊C412yes—
186C23×C53C2×C22⋊C411yescharacteristic
187C23×C53#12012yes—
188Q8×C53—42no—
189D4×C53—28no—
190D4×C53—44no—
191D4×C53—44no—
192D4×C53—44no—
193D4×C53—42no—
194D4×C53—44no—
195D4×C53—44no—
196D4×C53—48no—
197D4×C53—42no—
198D4×C53—42no—
199D4×C53—84no—
200D4×C53—84no—
201C2×C4×C53—21no—
202C2×C4×C53—21no—
203C2×C4×C53—21no—
204C2×C4×C53—21no—
205C2×C4×C53—22no—
206C2×C4×C53—24no—
207C2×C4×C53—44no—
208C2×C4×C53—44no—
209C2×C4×C53—42no—
210C2×C4×C53—42no—
211C2×C4×C53—82no—
212C2×C4×C53—82no—
213C2×C4×C53—82no—
214C2×C4×C53—84no—
215C2×C4×C53#9912yes—
216C2×C4×C53#12012yes—
217C8×C53—82no—
218#12705C5311yesmaximal, characteristic, direct product factor, Sylow
219C22×C53—22no—
220C22×C53—22no—
221C22×C53—22no—
222C22×C53—22no—
223C22×C53—22no—
224C22×C53—22no—
225C22×C53—41no—
226C22×C53—42no—
227C22×C53—42no—
228C22×C53—42no—
229C22×C53—42no—
230C22×C53—42no—
231C22×C53—41no—
232C22×C53—41no—
233C22×C53—41no—
234C22×C53—44no—
235C22×C53—42no—
236C22×C53—41no—
237C22×C53—42no—
238C22×C53—41no—
239C22×C53—81no—
240C22×C53—81no—
241C22×C53—81no—
242C22×C53—81no—
243C22×C53—81no—
244C22×C53—81no—
245C22×C53—88no—
246C22×C53—88no—
247C22×C53—162no—
248C22×C53—162no—
249C22×C53#37911yescharacteristic
250C22×C53C2×#9911yescharacteristic
251C22×C53C2×#9911yescharacteristic
252C4×C53—22no—
253C4×C53—22no—
254C4×C53—44no—
255C4×C53—42no—
256C4×C53—162no—
257#3484C2×C5311yescharacteristic
258#2121C2×C5314yescomplemented
259#1783C2×C5311yescharacteristic, complemented
260#1781C2×C5311yescharacteristic, complemented
261C2×C53—21no—
262C2×C53—21no—
263C2×C53—21no—
264C2×C53—42no—
265C2×C53—42no—
266C2×C53—42no—
267C2×C53—42no—
268C2×C53—81no—
269C2×C53—82no—
270C2×C53#178111yescharacteristic
271C2×H(5,𝔽2)C4×C5311yescharacteristic, complemented
272C23×D4C22×C5311yescharacteristic
273#546—24no—
274D42—24no—
275#535—21no—
276#534—21no—
277#534C22×C5312yes—
278#525—21no—
279C2×#120—21no—
280C2×#120C4×C5312yescomplemented
281#518C4×C5311yescharacteristic, complemented
282C2×#100—21no—
283C2×#100C22×C5312yescomplemented
284C2×#99—21no—
285C2×#99C22×C5312yescomplemented
286#407—21no—
287#379—21no—
288#352—24no—
289AΣL(1,𝔽16)—24no—
290C53#1270511yescharacteristic, direct product factor, Sylow
291C25C2×C4×C5311yescharacteristic
292C25D4×C5311yescharacteristic
293H(5,𝔽2)—42no—
294H(5,𝔽2)—42no—
295C2×#41—21no—
296C2×#41—22no—
297C22×D4—21no—
298C22×D4—22no—
299C22×D4—22no—
300C22×D4—22no—
301C22×D4—24no—
302C22×D4—42no—
303C22×D4—42no—
304C22×D4C2×C4×C5311yescharacteristic
305C22×D4C2×C4×C5312yes—
306C22×D4D4×C5311yescharacteristic
307C22×D4D4×C5311yescharacteristic
308C22×D4C23×C5311yescharacteristic
309C23×C4D4×C5311yescharacteristic
310C2×M16—22no—
311#127—22no—
312#124—22no—
313#123—22no—
314#123—22no—
315#123D4×C5314yes—
316C22⋊Q8—22no—
317#121—22no—
318#121—24no—
319#121—24no—
320#121—24no—
321#120—24no—
322#120—22no—
323#120—22no—
324#120—22no—
325#120—22no—
326#120—24no—
327#120D4×C5314yes—
328C4×D4—24no—
329C4×D4—24no—
330#117—21no—
331C2×C22⋊C4—22no—
332C2×C22⋊C4—21no—
333C2×C22⋊C4—22no—
334C2×C22⋊C4—22no—
335C2×C22⋊C4—42no—
336C2×C22⋊C4C2×C4×C5312yes—
337#100—24no—
338#100—42no—
339#99—24no—
340#99—42no—
341C22⋊C8—42no—
342C24—21no—
343C24—22no—
344C24—22no—
345C24—22no—
346C24—41no—
347C24—41no—
348C24—41no—
349C24—82no—
350C24—82no—
351C24C22⋊C4×C5311yescharacteristic
352C24C22⋊C4×C5311yescharacteristic
353C24C22⋊C4×C5311yescharacteristic
354C24C22⋊C4×C5312yescomplemented
355C24C22×C4×C5311yescharacteristic
356C24C2×D4×C5311yescharacteristic
357#41—42no—
358#41—48no—
359#41—42no—
360C2×Q8—21no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—24no—
364C2×D4—22no—
365C2×D4—21no—
366C2×D4—22no—
367C2×D4—22no—
368C2×D4—22no—
369C2×D4—22no—
370C2×D4—24no—
371C2×D4—24no—
372C2×D4—22no—
373C2×D4—22no—
374C2×D4—22no—
375C2×D4—41no—
376C2×D4—41no—
377C2×D4—41no—
378C2×D4—41no—
379C2×D4—416no—
380C2×D4—48no—
381C2×D4—42no—
382C2×D4—42no—
383C2×D4—42no—
384C2×D4—42no—
385C2×D4—84no—
386C2×D4—84no—
387C2×D4C2×D4×C5314yes—
388C22×C4—21no—
389C22×C4—22no—
390C22×C4—22no—
391C22×C4—22no—
392C22×C4—22no—
393C22×C4—22no—
394C22×C4—41no—
395C22×C4—42no—
396C22×C4C22⋊C4×C5311yescharacteristic
397C22×C4C22⋊C4×C5312yescomplemented
398C22×C4C2×D4×C5311yescharacteristic
399M16—42no—
400M16—42no—
401C2×C8—42no—
402C4⋊C4—22no—
403C4⋊C4—24no—
404C22⋊C4—24no—
405C22⋊C4—22no—
406C22⋊C4—24no—
407C22⋊C4—22no—
408C22⋊C4—24no—
409C22⋊C4—24no—
410C22⋊C4—42no—
411C22⋊C4—42no—
412C22⋊C4—84no—
413C22⋊C4C22⋊C4×C5314yescomplemented
414C42—22no—
415C23—21no—
416C23—21no—
417C23—21no—
418C23—21no—
419C23—21no—
420C23—21no—
421C23—21no—
422C23—21no—
423C23—21no—
424C23—21no—
425C23—22no—
426C23—21no—
427C23—21no—
428C23—42no—
429C23—44no—
430C23—44no—
431C23—44no—
432C23—44no—
433C23—44no—
434C23—42no—
435C23—42no—
436C23—82no—
437C23—82no—
438C23—82no—
439C23—81no—
440C23—81no—
441C23—81no—
442C23—81no—
443C23—82no—
444C23—82no—
445C23—82no—
446C23—82no—
447C23—82no—
448C23—82no—
449C23—82no—
450C23—82no—
451C23#99×C5312yes—
452C23C2×C22⋊C4×C5312yes—
453C23C2×C22⋊C4×C5311yescharacteristic
454C23#120×C5312yes—
455Q8—42no—
456D4—28no—
457D4—44no—
458D4—44no—
459D4—44no—
460D4—42no—
461D4—44no—
462D4—44no—
463D4—48no—
464D4—42no—
465D4—42no—
466D4—84no—
467D4—84no—
468C2×C4—21no—
469C2×C4—21no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—24no—
473C2×C4—22no—
474C2×C4—44no—
475C2×C4—44no—
476C2×C4—42no—
477C2×C4—42no—
478C2×C4—82no—
479C2×C4—82no—
480C2×C4—82no—
481C2×C4—84no—
482C2×C4#99×C5312yes—
483C2×C4#120×C5312yes—
484C8—82no—
485C22—22no—
486C22—22no—
487C22—22no—
488C22—22no—
489C22—22no—
490C22—22no—
491C22—41no—
492C22—42no—
493C22—42no—
494C22—42no—
495C22—42no—
496C22—42no—
497C22—41no—
498C22—41no—
499C22—41no—
500C22—44no—
501C22—42no—
502C22—41no—
503C22—42no—
504C22—41no—
505C22—81no—
506C22—81no—
507C22—81no—
508C22—81no—
509C22—81no—
510C22—81no—
511C22—88no—
512C22—88no—
513C22—162no—
514C22—162no—
515C22C53×#37911yescharacteristic
516C22C2×#99×C5311yescharacteristic
517C22C2×#99×C5311yescharacteristic
518C4—22no—
519C4—22no—
520C4—44no—
521C4—42no—
522C4—162no—
523C2—21no—
524C2—21no—
525C2—21no—
526C2—42no—
527C2—42no—
528C2—42no—
529C2—42no—
530C2—81no—
531C2—82no—
532C2C53×#178111yescharacteristic
subgroupG/H
center, socleC2×C53#1781
commutatorC24C22×C4×C53
Frattini, cosocleC22×D4C23×C53
prime radicalC53×#3484C2
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C2×C53 ◁ C23×C53 ◁ C23×D4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#2121C2semidirect
3. C53×#1783C2semidirect
4. C53×#1781C2semidirect
5. C2×H(5,𝔽2)×C53C4semidirect
14. C2×#120×C53C4semidirect
15. C53×#518C4semidirect
17. C2×#100×C53C22semidirect
19. C2×#99×C53C22semidirect
87. C24×C53C22⋊C4semidirect
130. C22×C4×C53C22⋊C4semidirect
146. C22⋊C4×C53C22⋊C4semidirect
218. #12705C53direct (N × H)
258. #2121C2×C53semidirect
259. #1783C2×C53semidirect
260. #1781C2×C53semidirect
271. C2×H(5,𝔽2)C4×C53semidirect
280. C2×#120C4×C53semidirect
281. #518C4×C53semidirect
283. C2×#100C22×C53semidirect
285. C2×#99C22×C53semidirect
290. C53#12705direct (N × H)
354. C24C22⋊C4×C53semidirect
397. C22×C4C22⋊C4×C53semidirect
413. C22⋊C4C22⋊C4×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#34842
C53×#21212
C53×#17832
C53×#17812
#1270553