Finite Group Databasebeta

← C53×#8315 #27215827

Subgroups — C53×#8315 #27215827

Order 13568.

868 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C53×#529C211yesmaximal, characteristic
2C53×#1797C211yesmaximal, characteristic, complemented
3C4×C22⋊C8×C53C211yesmaximal, characteristic, complemented
4C2×C53×Q8⋊C8C212yesmaximal, complemented
5C2×C53×D4⋊C8C212yesmaximal, complemented
6C53×#529—24no—
7C53×#529—24no—
8C2×#126×C53—21no—
9C2×#125×C53C411yescharacteristic
10C2×#124×C53C411yescharacteristic
11C2×#123×C53—21no—
12C2×C22⋊Q8×C53C411yescharacteristic
13C2×#121×C53C411yescharacteristic
14C2×C4×Q8×C53C2211yescharacteristic
15C2×C4×D4×C53—21no—
16C2×C4×D4×C53C2211yescharacteristic
17C2×#117×C53—21no—
18C22×C42×C53C2211yescharacteristic
19C2×C4⋊C8×C53C2212yes—
20C2×C22⋊C8×C53—21no—
21C2×C22⋊C8×C53—21no—
22C2×C4×C8×C53C2212yes—
23C53×#336—21no—
24C53×#336—21no—
25C53×Q8⋊C8—24no—
26C53×D4⋊C8—24no—
27C22×Q8×C53C2×C411yescharacteristic
28C22×D4×C53—21no—
29C22×D4×C53C2×C411yescharacteristic
30C23×C4×C53—21no—
31C23×C4×C53C2×C411yescharacteristic
32C23×C4×C53D411yescharacteristic
33C23×C4×C53D411yescharacteristic
34C22×C8×C53—22no—
35C22×C8×C53—22no—
36#126×C53—28no—
37#125×C53—22no—
38#125×C53—22no—
39#124×C53—22no—
40#124×C53—22no—
41#123×C53—28no—
42C22⋊Q8×C53—22no—
43C22⋊Q8×C53—22no—
44#121×C53—22no—
45#121×C53—22no—
46C4×Q8×C53—22no—
47C4×Q8×C53D414yes—
48C4×D4×C53—28no—
49C4×D4×C53—22no—
50C4×D4×C53D414yes—
51#117×C53—28no—
52C2×C4⋊C4×C53—21no—
53C2×C4⋊C4×C53—21no—
54C2×C4⋊C4×C53—21no—
55C2×C4⋊C4×C53—21no—
56C2×C4⋊C4×C53C2×C411yescharacteristic
57C2×C4⋊C4×C53C2×C411yescharacteristic
58C2×C22⋊C4×C53—21no—
59C2×C22⋊C4×C53—21no—
60C2×C22⋊C4×C53—21no—
61C2×C22⋊C4×C53—21no—
62C2×C22⋊C4×C53—21no—
63C2×C42×C53—22no—
64C2×C42×C53—22no—
65C2×C42×C53—21no—
66C2×C42×C53C2×C411yescharacteristic
67C2×C42×C53D411yescharacteristic
68C2×C42×C53D411yescharacteristic
69C2×C42×C53C2311yescharacteristic
70C4⋊C8×C53—22no—
71C4⋊C8×C53—22no—
72C22⋊C8×C53—24no—
73C22⋊C8×C53—42no—
74C4×C8×C53—22no—
75C4×C8×C53—22no—
76C24×C53—21no—
77C24×C53C22⋊C411yescharacteristic
78C2×Q8×C53—21no—
79C2×Q8×C53—22no—
80C2×Q8×C53—22no—
81C2×Q8×C53C22⋊C412yes—
82C2×D4×C53—24no—
83C2×D4×C53—21no—
84C2×D4×C53—22no—
85C2×D4×C53—22no—
86C2×D4×C53—42no—
87C2×D4×C53—42no—
88C2×D4×C53C22⋊C412yes—
89C22×C4×C53—22no—
90C22×C4×C53—21no—
91C22×C4×C53—21no—
92C22×C4×C53—21no—
93C22×C4×C53—21no—
94C22×C4×C53—21no—
95C22×C4×C53—22no—
96C22×C4×C53—22no—
97C22×C4×C53—24no—
98C22×C4×C53—22no—
99C22×C4×C53—22no—
100C22×C4×C53—22no—
101C22×C4×C53—21no—
102C22×C4×C53—21no—
103C22×C4×C53—41no—
104C22×C4×C53—42no—
105C22×C4×C53—41no—
106C22×C4×C53—41no—
107C22×C4×C53C22⋊C411yescharacteristic
108C22×C4×C53C22⋊C411yescharacteristic
109C22×C4×C53C22⋊C411yescharacteristic
110C22×C4×C53C22×C411yescharacteristic
111C22×C4×C53C2×D411yescharacteristic
112C22×C4×C53C2×D411yescharacteristic
113C2×C8×C53—24no—
114C2×C8×C53—42no—
115C2×C8×C53—42no—
116C2×C8×C53—42no—
117C2×C8×C53—42no—
118C2×C8×C53—42no—
119C4⋊C4×C53—24no—
120C4⋊C4×C53—24no—
121C4⋊C4×C53—24no—
122C4⋊C4×C53—21no—
123C4⋊C4×C53—21no—
124C4⋊C4×C53—24no—
125C4⋊C4×C53C22⋊C412yes—
126C4⋊C4×C53C22⋊C412yes—
127C22⋊C4×C53—24no—
128C22⋊C4×C53—24no—
129C22⋊C4×C53—24no—
130C22⋊C4×C53—24no—
131C22⋊C4×C53—24no—
132C42×C53—21no—
133C42×C53—21no—
134C42×C53—22no—
135C42×C53—22no—
136C42×C53—24no—
137C42×C53C2×D412yes—
138C42×C53C2×D412yes—
139C23×C53—21no—
140C23×C53—21no—
141C23×C53—21no—
142C23×C53—21no—
143C23×C53—21no—
144C23×C53—21no—
145C23×C53—22no—
146C23×C53—41no—
147C23×C53—41no—
148C23×C53—41no—
149C23×C53—41no—
150C23×C53—41no—
151C23×C53—41no—
152C23×C53#10412yes—
153C23×C53C2×C22⋊C411yescharacteristic
154Q8×C53—24no—
155Q8×C53—42no—
156D4×C53—24no—
157D4×C53—48no—
158D4×C53—42no—
159C2×C4×C53—21no—
160C2×C4×C53—21no—
161C2×C4×C53—22no—
162C2×C4×C53—21no—
163C2×C4×C53—21no—
164C2×C4×C53—21no—
165C2×C4×C53—22no—
166C2×C4×C53—22no—
167C2×C4×C53—21no—
168C2×C4×C53—21no—
169C2×C4×C53—21no—
170C2×C4×C53—21no—
171C2×C4×C53—21no—
172C2×C4×C53—21no—
173C2×C4×C53—44no—
174C2×C4×C53—41no—
175C2×C4×C53—41no—
176C2×C4×C53—41no—
177C2×C4×C53—41no—
178C2×C4×C53—41no—
179C2×C4×C53—41no—
180C2×C4×C53—42no—
181C2×C4×C53—42no—
182C2×C4×C53—42no—
183C2×C4×C53—44no—
184C2×C4×C53—42no—
185C2×C4×C53—44no—
186C2×C4×C53C2×C22⋊C411yescharacteristic
187C2×C4×C53C2×C22⋊C411yescharacteristic
188C2×C4×C53#12012yes—
189C2×C4×C53#12012yes—
190C2×C4×C53#13112yes—
191C2×C4×C53#13112yes—
192C2×C4×C53#13512yes—
193C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
194C2×C4×C53#13711yescharacteristic
195C8×C53—44no—
196C8×C53—82no—
197#8315C5311yesmaximal, characteristic, direct product factor, Sylow
198C22×C53—22no—
199C22×C53—22no—
200C22×C53—22no—
201C22×C53—41no—
202C22×C53—41no—
203C22×C53—41no—
204C22×C53—41no—
205C22×C53—42no—
206C22×C53—42no—
207C22×C53—42no—
208C22×C53—81no—
209C22×C53—81no—
210C22×C53—81no—
211C22×C53—81no—
212C22×C53#37911yescharacteristic
213C22×C53#40811yescharacteristic
214C22×C53#40811yescharacteristic
215C22×C53#41611yescharacteristic
216C22×C53#41711yescharacteristic
217C22×C53C2×#10411yescharacteristic
218C22×C53#42111yescharacteristic
219C4×C53—22no—
220C4×C53—41no—
221C4×C53—42no—
222C4×C53—41no—
223C4×C53—42no—
224C4×C53—41no—
225C4×C53—42no—
226C4×C53#40812yes—
227C4×C53#40812yes—
228C4×C53#44912yes—
229C4×C53#45212yes—
230C2×#529C2×C5311yescharacteristic
231#1797C2×C5311yescharacteristic, complemented
232C4×C22⋊C8C2×C5311yescharacteristic, complemented
233C2×Q8⋊C8C2×C5312yescomplemented
234C2×D4⋊C8C2×C5312yescomplemented
235C2×C53—42no—
236C2×C53—82no—
237C2×C53#148211yescharacteristic
238C2×C53#148311yescharacteristic
239C2×C53#149411yescharacteristic
240C2×C53#149511yescharacteristic
241C2×C53#178011yescharacteristic
242C2×C53#178811yescharacteristic
243C2×C53#178911yescharacteristic
244#529—24no—
245#529—24no—
246C2×#126—21no—
247C2×#125C4×C5311yescharacteristic
248C2×#124C4×C5311yescharacteristic
249C2×#123—21no—
250C2×C22⋊Q8C4×C5311yescharacteristic
251C2×#121C4×C5311yescharacteristic
252C2×C4×Q8C22×C5311yescharacteristic
253C2×C4×D4—21no—
254C2×C4×D4C22×C5311yescharacteristic
255C2×#117—21no—
256C22×C42C22×C5311yescharacteristic
257C2×C4⋊C8C22×C5312yes—
258C2×C22⋊C8—21no—
259C2×C22⋊C8—21no—
260C2×C4×C8C22×C5312yes—
261#336—21no—
262#336—21no—
263Q8⋊C8—24no—
264D4⋊C8—24no—
265C53#831511yescharacteristic, direct product factor, Sylow
266C22×Q8C2×C4×C5311yescharacteristic
267C22×D4—21no—
268C22×D4C2×C4×C5311yescharacteristic
269C23×C4—21no—
270C23×C4C2×C4×C5311yescharacteristic
271C23×C4D4×C5311yescharacteristic
272C23×C4D4×C5311yescharacteristic
273C22×C8—22no—
274C22×C8—22no—
275#126—28no—
276#125—22no—
277#125—22no—
278#124—22no—
279#124—22no—
280#123—28no—
281C22⋊Q8—22no—
282C22⋊Q8—22no—
283#121—22no—
284#121—22no—
285C4×Q8—22no—
286C4×Q8D4×C5314yes—
287C4×D4—28no—
288C4×D4—22no—
289C4×D4D4×C5314yes—
290#117—28no—
291C2×C4⋊C4—21no—
292C2×C4⋊C4—21no—
293C2×C4⋊C4—21no—
294C2×C4⋊C4—21no—
295C2×C4⋊C4C2×C4×C5311yescharacteristic
296C2×C4⋊C4C2×C4×C5311yescharacteristic
297C2×C22⋊C4—21no—
298C2×C22⋊C4—21no—
299C2×C22⋊C4—21no—
300C2×C22⋊C4—21no—
301C2×C22⋊C4—21no—
302C2×C42—21no—
303C2×C42—22no—
304C2×C42—22no—
305C2×C42C2×C4×C5311yescharacteristic
306C2×C42D4×C5311yescharacteristic
307C2×C42D4×C5311yescharacteristic
308C2×C42C23×C5311yescharacteristic
309C4⋊C8—22no—
310C4⋊C8—22no—
311C22⋊C8—24no—
312C22⋊C8—42no—
313C4×C8—22no—
314C4×C8—22no—
315C24—21no—
316C24C22⋊C4×C5311yescharacteristic
317C2×Q8—21no—
318C2×Q8—22no—
319C2×Q8—22no—
320C2×Q8C22⋊C4×C5312yes—
321C2×D4—24no—
322C2×D4—21no—
323C2×D4—22no—
324C2×D4—22no—
325C2×D4—42no—
326C2×D4—42no—
327C2×D4C22⋊C4×C5312yes—
328C22×C4—22no—
329C22×C4—21no—
330C22×C4—21no—
331C22×C4—21no—
332C22×C4—21no—
333C22×C4—21no—
334C22×C4—22no—
335C22×C4—22no—
336C22×C4—24no—
337C22×C4—22no—
338C22×C4—22no—
339C22×C4—22no—
340C22×C4—21no—
341C22×C4—21no—
342C22×C4—41no—
343C22×C4—42no—
344C22×C4—41no—
345C22×C4—41no—
346C22×C4C22⋊C4×C5311yescharacteristic
347C22×C4C22⋊C4×C5311yescharacteristic
348C22×C4C22⋊C4×C5311yescharacteristic
349C22×C4C22×C4×C5311yescharacteristic
350C22×C4C2×D4×C5311yescharacteristic
351C22×C4C2×D4×C5311yescharacteristic
352C2×C8—24no—
353C2×C8—42no—
354C2×C8—42no—
355C2×C8—42no—
356C2×C8—42no—
357C2×C8—42no—
358C4⋊C4—24no—
359C4⋊C4—24no—
360C4⋊C4—24no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—24no—
364C4⋊C4C22⋊C4×C5312yes—
365C4⋊C4C22⋊C4×C5312yes—
366C22⋊C4—24no—
367C22⋊C4—24no—
368C22⋊C4—24no—
369C22⋊C4—24no—
370C22⋊C4—24no—
371C42—21no—
372C42—21no—
373C42—22no—
374C42—22no—
375C42—24no—
376C42C2×D4×C5312yes—
377C42C2×D4×C5312yes—
378C23—21no—
379C23—21no—
380C23—21no—
381C23—21no—
382C23—21no—
383C23—21no—
384C23—22no—
385C23—41no—
386C23—41no—
387C23—41no—
388C23—41no—
389C23—41no—
390C23—41no—
391C23#104×C5312yes—
392C23C2×C22⋊C4×C5311yescharacteristic
393Q8—24no—
394Q8—42no—
395D4—24no—
396D4—48no—
397D4—42no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—22no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—21no—
411C2×C4—21no—
412C2×C4—44no—
413C2×C4—41no—
414C2×C4—41no—
415C2×C4—41no—
416C2×C4—41no—
417C2×C4—41no—
418C2×C4—41no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—44no—
423C2×C4—42no—
424C2×C4—44no—
425C2×C4C2×C22⋊C4×C5311yescharacteristic
426C2×C4C2×C22⋊C4×C5311yescharacteristic
427C2×C4#120×C5312yes—
428C2×C4#120×C5312yes—
429C2×C4#131×C5312yes—
430C2×C4#131×C5312yes—
431C2×C4#135×C5312yes—
432C2×C4AGL(1,ℤ/8)×C5311yescharacteristic
433C2×C4#137×C5311yescharacteristic
434C8—44no—
435C8—82no—
436C22—22no—
437C22—22no—
438C22—22no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—42no—
444C22—42no—
445C22—42no—
446C22—81no—
447C22—81no—
448C22—81no—
449C22—81no—
450C22C53×#37911yescharacteristic
451C22C53×#40811yescharacteristic
452C22C53×#40811yescharacteristic
453C22C53×#41611yescharacteristic
454C22C53×#41711yescharacteristic
455C22C2×#104×C5311yescharacteristic
456C22C53×#42111yescharacteristic
457C4—22no—
458C4—41no—
459C4—42no—
460C4—41no—
461C4—42no—
462C4—41no—
463C4—42no—
464C4C53×#40812yes—
465C4C53×#40812yes—
466C4C53×#44912yes—
467C4C53×#45212yes—
468C2—42no—
469C2—82no—
470C2C53×#148211yescharacteristic
471C2C53×#148311yescharacteristic
472C2C53×#149411yescharacteristic
473C2C53×#149511yescharacteristic
474C2C53×#178011yescharacteristic
475C2C53×#178811yescharacteristic
476C2C53×#178911yescharacteristic
subgroupG/H
centerC22×C4×C53C2×D4
commutatorC22×C4C22×C4×C53
Frattini, cosocleC2×C42C23×C53
socleC23×C53C2×C22⋊C4
prime radicalC2×#121×C53C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C4×C53 ◁ C22×C42×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#1797C2semidirect
3. C4×C22⋊C8×C53C2semidirect
4. C2×C53×Q8⋊C8C2semidirect
5. C2×C53×D4⋊C8C2semidirect
197. #8315C53direct (N × H)
231. #1797C2×C53semidirect
232. C4×C22⋊C8C2×C53semidirect
233. C2×Q8⋊C8C2×C53semidirect
234. C2×D4⋊C8C2×C53semidirect
265. C53#8315direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C53×#5292
C53×#17972
C4×C22⋊C8×C532
C2×C53×Q8⋊C82
C2×C53×D4⋊C82
#831553