Finite Group Databasebeta

← C53×#12687 #27220199

Subgroups — C53×#12687 #27220199

Order 13568.

2508 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 2508 conjugacy classes of subgroups — too many to draw. Only its 158 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
1C22×#120×C53C211yesmaximal, characteristic
2C2×C53×AΣL(1,𝔽16)C214yesmaximal, complemented
3C53×#1783C211yesmaximal, characteristic, complemented
4C53×#1781C211yesmaximal, characteristic, complemented
5C26×C53C411yescharacteristic, complemented
6C23×D4×C53—21no—
7C23×D4×C53C2211yescharacteristic
8C2×#120×C53—21no—
9C2×#120×C53—21no—
10C2×#120×C53—28no—
11C2×#120×C53C412yescomplemented
12C2×#120×C53C2212yes—
13C22×C22⋊C4×C53—21no—
14C22×C22⋊C4×C53C411yescharacteristic, complemented
15C2×#100×C53—21no—
16C2×#100×C53C2212yescomplemented
17C2×#99×C53—21no—
18C2×#99×C53C2212yescomplemented
19C53×#407—21no—
20C53×#379—21no—
21C53×AΣL(1,𝔽16)—28no—
22C25×C53—21no—
23C25×C53—21no—
24C25×C53—24no—
25C25×C53—48no—
26C25×C53—42no—
27C25×C53—42no—
28C25×C53C2×C411yescharacteristic
29C25×C53C2×C412yes—
30C25×C53D411yescharacteristic
31C22×D4×C53—28no—
32C22×D4×C53—22no—
33C22×D4×C53—22no—
34C22×D4×C53—24no—
35C22×D4×C53—42no—
36C22×D4×C53—42no—
37C22×D4×C53—48no—
38C22×D4×C53C2×C411yescharacteristic
39C22×D4×C53D411yescharacteristic
40C22×D4×C53D411yescharacteristic
41C22×D4×C53C2311yescharacteristic
42C23×C4×C53—21no—
43C23×C4×C53D411yescharacteristic
44C2×M16×C53—22no—
45#120×C53—28no—
46#120×C53—28no—
47#120×C53—24no—
48#120×C53—24no—
49#120×C53—24no—
50#120×C53D418yes—
51C2×C22⋊C4×C53—28no—
52C2×C22⋊C4×C53—24no—
53C2×C22⋊C4×C53—21no—
54C2×C22⋊C4×C53—22no—
55C2×C22⋊C4×C53—22no—
56C2×C22⋊C4×C53—22no—
57C2×C22⋊C4×C53—42no—
58C2×C22⋊C4×C53C2×C412yes—
59#100×C53—24no—
60#100×C53—42no—
61#99×C53—24no—
62#99×C53—42no—
63C22⋊C8×C53—42no—
64C24×C53—24no—
65C24×C53—28no—
66C24×C53—22no—
67C24×C53—22no—
68C24×C53—22no—
69C24×C53—22no—
70C24×C53—24no—
71C24×C53—22no—
72C24×C53—22no—
73C24×C53—22no—
74C24×C53—41no—
75C24×C53—44no—
76C24×C53—42no—
77C24×C53—48no—
78C24×C53—432no—
79C24×C53—416no—
80C24×C53—42no—
81C24×C53—44no—
82C24×C53—48no—
83C24×C53—42no—
84C24×C53—42no—
85C24×C53—44no—
86C24×C53—416no—
87C24×C53—432no—
88C24×C53—48no—
89C24×C53—42no—
90C24×C53—42no—
91C24×C53—48no—
92C24×C53C22⋊C411yescharacteristic
93C24×C53C22⋊C411yescharacteristic
94C24×C53C22⋊C411yescharacteristic
95C24×C53C22⋊C414yescomplemented
96C24×C53C22×C411yescharacteristic
97C24×C53C2×D411yescharacteristic
98C2×D4×C53—28no—
99C2×D4×C53—22no—
100C2×D4×C53—22no—
101C2×D4×C53—22no—
102C2×D4×C53—22no—
103C2×D4×C53—28no—
104C2×D4×C53—22no—
105C2×D4×C53—41no—
106C2×D4×C53—41no—
107C2×D4×C53—41no—
108C2×D4×C53—41no—
109C2×D4×C53—41no—
110C2×D4×C53—41no—
111C2×D4×C53—416no—
112C2×D4×C53—48no—
113C2×D4×C53—48no—
114C2×D4×C53—432no—
115C2×D4×C53C2×D414yes—
116C22×C4×C53—24no—
117C22×C4×C53—22no—
118C22×C4×C53—22no—
119C22×C4×C53—22no—
120C22×C4×C53—41no—
121C22×C4×C53—44no—
122C22×C4×C53—42no—
123C22×C4×C53C22⋊C411yescharacteristic
124C22×C4×C53C2×D411yescharacteristic
125M16×C53—42no—
126M16×C53—42no—
127C2×C8×C53—42no—
128C22⋊C4×C53—24no—
129C22⋊C4×C53—28no—
130C22⋊C4×C53—22no—
131C22⋊C4×C53—28no—
132C22⋊C4×C53—42no—
133C22⋊C4×C53—42no—
134C22⋊C4×C53—84no—
135C22⋊C4×C53C22⋊C414yescomplemented
136C23×C53—24no—
137C23×C53—21no—
138C23×C53—21no—
139C23×C53—216no—
140C23×C53—28no—
141C23×C53—28no—
142C23×C53—21no—
143C23×C53—24no—
144C23×C53—21no—
145C23×C53—21no—
146C23×C53—21no—
147C23×C53—21no—
148C23×C53—21no—
149C23×C53—21no—
150C23×C53—21no—
151C23×C53—21no—
152C23×C53—44no—
153C23×C53—42no—
154C23×C53—48no—
155C23×C53—44no—
156C23×C53—42no—
157C23×C53—48no—
158C23×C53—48no—
159C23×C53—48no—
160C23×C53—416no—
161C23×C53—44no—
162C23×C53—48no—
163C23×C53—44no—
164C23×C53—48no—
165C23×C53—48no—
166C23×C53—416no—
167C23×C53—48no—
168C23×C53—44no—
169C23×C53—42no—
170C23×C53—42no—
171C23×C53—48no—
172C23×C53—48no—
173C23×C53—416no—
174C23×C53—464no—
175C23×C53—416no—
176C23×C53—464no—
177C23×C53—416no—
178C23×C53—48no—
179C23×C53—48no—
180C23×C53—44no—
181C23×C53—42no—
182C23×C53—42no—
183C23×C53—44no—
184C23×C53—82no—
185C23×C53—84no—
186C23×C53#9912yes—
187C23×C53#9912yes—
188C23×C53C2×C22⋊C411yescharacteristic
189C23×C53C2×C22⋊C412yes—
190C23×C53#12012yes—
191D4×C53—44no—
192D4×C53—44no—
193D4×C53—44no—
194D4×C53—416no—
195D4×C53—416no—
196D4×C53—44no—
197C2×C4×C53—21no—
198C2×C4×C53—21no—
199C2×C4×C53—24no—
200C2×C4×C53—21no—
201C2×C4×C53—21no—
202C2×C4×C53—21no—
203C2×C4×C53—44no—
204C2×C4×C53—48no—
205C2×C4×C53—44no—
206C2×C4×C53—82no—
207C2×C4×C53—84no—
208C2×C4×C53#12012yes—
209C8×C53—82no—
210#12687C5311yesmaximal, characteristic, direct product factor, Sylow
211C22×C53—24no—
212C22×C53—24no—
213C22×C53—22no—
214C22×C53—22no—
215C22×C53—22no—
216C22×C53—28no—
217C22×C53—22no—
218C22×C53—22no—
219C22×C53—22no—
220C22×C53—42no—
221C22×C53—42no—
222C22×C53—42no—
223C22×C53—42no—
224C22×C53—44no—
225C22×C53—432no—
226C22×C53—416no—
227C22×C53—48no—
228C22×C53—42no—
229C22×C53—42no—
230C22×C53—48no—
231C22×C53—48no—
232C22×C53—416no—
233C22×C53—432no—
234C22×C53—42no—
235C22×C53—48no—
236C22×C53—42no—
237C22×C53—42no—
238C22×C53—44no—
239C22×C53—42no—
240C22×C53—41no—
241C22×C53—81no—
242C22×C53—88no—
243C22×C53—81no—
244C22×C53AΣL(1,𝔽16)14yescomplemented
245C22×C53#37911yescharacteristic
246C22×C53C2×#9911yescharacteristic
247C22×C53C2×#9911yescharacteristic
248C4×C53—22no—
249C4×C53—44no—
250C4×C53—41no—
251C4×C53—162no—
252C22×#120C2×C5311yescharacteristic
253C2×AΣL(1,𝔽16)C2×C5314yescomplemented
254#1783C2×C5311yescharacteristic, complemented
255#1781C2×C5311yescharacteristic, complemented
256C2×C53—21no—
257C2×C53—24no—
258C2×C53—21no—
259C2×C53—42no—
260C2×C53—48no—
261C2×C53—42no—
262C2×C53—42no—
263C2×C53—81no—
264C2×C53#178111yescharacteristic
265C2×C53C2×AΣL(1,𝔽16)12yes—
266C26C4×C5311yescharacteristic, complemented
267C23×D4—21no—
268C23×D4C22×C5311yescharacteristic
269C2×#120—21no—
270C2×#120—21no—
271C2×#120—28no—
272C2×#120C4×C5312yescomplemented
273C2×#120C22×C5312yes—
274C22×C22⋊C4—21no—
275C22×C22⋊C4C4×C5311yescharacteristic, complemented
276C2×#100—21no—
277C2×#100C22×C5312yescomplemented
278C2×#99—21no—
279C2×#99C22×C5312yescomplemented
280#407—21no—
281#379—21no—
282AΣL(1,𝔽16)—28no—
283C53#1268711yescharacteristic, direct product factor, Sylow
284C25—21no—
285C25—21no—
286C25—24no—
287C25—48no—
288C25—42no—
289C25—42no—
290C25C2×C4×C5311yescharacteristic
291C25C2×C4×C5312yes—
292C25D4×C5311yescharacteristic
293C22×D4—28no—
294C22×D4—22no—
295C22×D4—22no—
296C22×D4—24no—
297C22×D4—42no—
298C22×D4—42no—
299C22×D4—48no—
300C22×D4C2×C4×C5311yescharacteristic
301C22×D4D4×C5311yescharacteristic
302C22×D4D4×C5311yescharacteristic
303C22×D4C23×C5311yescharacteristic
304C23×C4—21no—
305C23×C4D4×C5311yescharacteristic
306C2×M16—22no—
307#120—28no—
308#120—28no—
309#120—24no—
310#120—24no—
311#120—24no—
312#120D4×C5318yes—
313C2×C22⋊C4—28no—
314C2×C22⋊C4—21no—
315C2×C22⋊C4—22no—
316C2×C22⋊C4—22no—
317C2×C22⋊C4—24no—
318C2×C22⋊C4—22no—
319C2×C22⋊C4—42no—
320C2×C22⋊C4C2×C4×C5312yes—
321#100—24no—
322#100—42no—
323#99—24no—
324#99—42no—
325C22⋊C8—42no—
326C24—24no—
327C24—28no—
328C24—22no—
329C24—22no—
330C24—22no—
331C24—22no—
332C24—24no—
333C24—22no—
334C24—22no—
335C24—22no—
336C24—41no—
337C24—44no—
338C24—48no—
339C24—42no—
340C24—42no—
341C24—44no—
342C24—42no—
343C24—48no—
344C24—432no—
345C24—416no—
346C24—42no—
347C24—44no—
348C24—48no—
349C24—416no—
350C24—432no—
351C24—48no—
352C24—42no—
353C24—42no—
354C24C22⋊C4×C5311yescharacteristic
355C24C22⋊C4×C5311yescharacteristic
356C24C22⋊C4×C5311yescharacteristic
357C24C22⋊C4×C5314yescomplemented
358C24C22×C4×C5311yescharacteristic
359C24C2×D4×C5311yescharacteristic
360C2×D4—28no—
361C2×D4—22no—
362C2×D4—22no—
363C2×D4—22no—
364C2×D4—22no—
365C2×D4—28no—
366C2×D4—22no—
367C2×D4—41no—
368C2×D4—41no—
369C2×D4—41no—
370C2×D4—41no—
371C2×D4—41no—
372C2×D4—41no—
373C2×D4—416no—
374C2×D4—48no—
375C2×D4—48no—
376C2×D4—432no—
377C2×D4C2×D4×C5314yes—
378C22×C4—24no—
379C22×C4—22no—
380C22×C4—22no—
381C22×C4—22no—
382C22×C4—41no—
383C22×C4—44no—
384C22×C4—42no—
385C22×C4C22⋊C4×C5311yescharacteristic
386C22×C4C2×D4×C5311yescharacteristic
387M16—42no—
388M16—42no—
389C2×C8—42no—
390C22⋊C4—28no—
391C22⋊C4—24no—
392C22⋊C4—22no—
393C22⋊C4—28no—
394C22⋊C4—42no—
395C22⋊C4—42no—
396C22⋊C4—84no—
397C22⋊C4C22⋊C4×C5314yescomplemented
398C23—24no—
399C23—21no—
400C23—21no—
401C23—216no—
402C23—28no—
403C23—28no—
404C23—21no—
405C23—24no—
406C23—21no—
407C23—21no—
408C23—21no—
409C23—21no—
410C23—21no—
411C23—21no—
412C23—21no—
413C23—21no—
414C23—44no—
415C23—42no—
416C23—42no—
417C23—48no—
418C23—44no—
419C23—48no—
420C23—48no—
421C23—48no—
422C23—416no—
423C23—48no—
424C23—44no—
425C23—48no—
426C23—44no—
427C23—48no—
428C23—416no—
429C23—48no—
430C23—44no—
431C23—42no—
432C23—42no—
433C23—48no—
434C23—48no—
435C23—416no—
436C23—416no—
437C23—464no—
438C23—464no—
439C23—48no—
440C23—416no—
441C23—48no—
442C23—44no—
443C23—42no—
444C23—42no—
445C23—44no—
446C23—82no—
447C23—84no—
448C23#99×C5312yes—
449C23#99×C5312yes—
450C23C2×C22⋊C4×C5311yescharacteristic
451C23C2×C22⋊C4×C5312yes—
452C23#120×C5312yes—
453D4—44no—
454D4—44no—
455D4—44no—
456D4—416no—
457D4—416no—
458D4—44no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—24no—
463C2×C4—21no—
464C2×C4—21no—
465C2×C4—44no—
466C2×C4—48no—
467C2×C4—44no—
468C2×C4—82no—
469C2×C4—84no—
470C2×C4#120×C5312yes—
471C8—82no—
472C22—24no—
473C22—24no—
474C22—22no—
475C22—22no—
476C22—22no—
477C22—28no—
478C22—22no—
479C22—22no—
480C22—22no—
481C22—42no—
482C22—42no—
483C22—42no—
484C22—42no—
485C22—44no—
486C22—432no—
487C22—416no—
488C22—48no—
489C22—42no—
490C22—42no—
491C22—48no—
492C22—48no—
493C22—416no—
494C22—432no—
495C22—44no—
496C22—42no—
497C22—48no—
498C22—42no—
499C22—42no—
500C22—41no—
501C22—42no—
502C22—81no—
503C22—88no—
504C22—81no—
505C22C53×AΣL(1,𝔽16)14yescomplemented
506C22C53×#37911yescharacteristic
507C22C2×#99×C5311yescharacteristic
508C22C2×#99×C5311yescharacteristic
509C4—22no—
510C4—44no—
511C4—41no—
512C4—162no—
513C2—21no—
514C2—24no—
515C2—21no—
516C2—42no—
517C2—48no—
518C2—42no—
519C2—42no—
520C2—81no—
521C2C53×#178111yescharacteristic
522C2C2×C53×AΣL(1,𝔽16)12yes—
subgroupG/H
center, socleC22×C53C2×#99
commutatorC24C22×C4×C53
Frattini, cosocleC22×D4C23×C53
prime radicalC22×#120×C53C2
LCSG ▷ C24 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ C24×C53 ◁ C23×D4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C53×AΣL(1,𝔽16)C2semidirect
3. C53×#1783C2semidirect
4. C53×#1781C2semidirect
5. C26×C53C4semidirect
11. C2×#120×C53C4semidirect
14. C22×C22⋊C4×C53C4semidirect
16. C2×#100×C53C22semidirect
18. C2×#99×C53C22semidirect
95. C24×C53C22⋊C4semidirect
135. C22⋊C4×C53C22⋊C4semidirect
210. #12687C53direct (N × H)
244. C22×C53AΣL(1,𝔽16)semidirect
253. C2×AΣL(1,𝔽16)C2×C53semidirect
254. #1783C2×C53semidirect
255. #1781C2×C53semidirect
266. C26C4×C53semidirect
272. C2×#120C4×C53semidirect
275. C22×C22⋊C4C4×C53semidirect
277. C2×#100C22×C53semidirect
279. C2×#99C22×C53semidirect
283. C53#12687direct (N × H)
357. C24C22⋊C4×C53semidirect
397. C22⋊C4C22⋊C4×C53semidirect
505. C22C53×AΣL(1,𝔽16)semidirect

Maximal subgroups

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

subgroupindex
C22×#120×C532
C2×C53×AΣL(1,𝔽16)2
C53×#17832
C53×#17812
#1268753