Finite Group Databasebeta

← C53×#36567 #27244079

Subgroups — C53×#36567 #27244079

Order 13568.

1574 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#2864C212yesmaximal
2C53×#2859C211yesmaximal, characteristic
3C53×#2590C214yesmaximal
4C53×#2585C214yesmaximal
5C53×#2584C212yesmaximal
6C53×#2507C212yesmaximal
7C53×#2506C212yesmaximal
8C53×#2462C214yesmaximal
9C53×#2370C214yesmaximal
10C53×#2370C212yesmaximal
11C4×#125×C53C211yesmaximal, characteristic
12C4×#125×C53C212yesmaximal
13C42×Q8×C53C211yesmaximal, characteristic
14C2×#125×C53C2214yes—
15C2×#125×C53C2212yes—
16C2×#125×C53C2211yescharacteristic
17C2×C4×Q8×C53C2212yes—
18C2×C4×Q8×C53C2211yescharacteristic
19C53×#400C2214yes—
20C53×#398C2214yes—
21C53×#398C2214yes—
22C53×#398C2214yes—
23C53×#398C2214yes—
24C53×#395C2214yes—
25C53×#389C2214yes—
26C53×#389C2214yes—
27C53×#389C2212yes—
28C53×#389C2212yes—
29C53×#389C2212yes—
30C53×#389C2212yes—
31C53×#389C2212yes—
32C53×#389C2212yes—
33C53×#389C2214yes—
34C53×#389C2212yes—
35C53×#389C2214yes—
36C53×#389C2214yes—
37C53×#389C2214yes—
38C53×#387C2214yes—
39C53×#387C2214yes—
40C53×#387C2214yes—
41C53×#387C2214yes—
42C53×#387C2214yes—
43C53×#387C2214yes—
44C53×#387C2214yes—
45C53×#387C2214yes—
46C53×#387C2214yes—
47C53×#387C2214yes—
48C53×#387C2214yes—
49C53×#384C2211yescharacteristic
50C53×#384C2211yescharacteristic
51C53×#382C2212yes—
52C53×#382C2211yescharacteristic
53C4×C4⋊C4×C53C2214yes—
54C4×C4⋊C4×C53C2212yes—
55C4×C4⋊C4×C53C2212yes—
56C4×C4⋊C4×C53C2212yes—
57C4×C4⋊C4×C53C2212yes—
58C4×C4⋊C4×C53C2212yes—
59C4×C4⋊C4×C53C2211yescharacteristic
60C4×C4⋊C4×C53C2212yes—
61C4×C4⋊C4×C53C2212yes—
62C4×C4⋊C4×C53C2214yes—
63C4×C4⋊C4×C53C2214yes—
64C4×C4⋊C4×C53C2214yes—
65C43×C53C2211yescharacteristic
66C43×C53C2212yes—
67C22×Q8×C53C2311yescharacteristic
68#125×C53—216no—
69#125×C53—28no—
70#125×C53Q818yescomplemented
71C4×Q8×C53—28no—
72C4×Q8×C53—24no—
73C2×C4⋊C4×C53C2312yes—
74C2×C4⋊C4×C53C2312yes—
75C2×C4⋊C4×C53C2312yes—
76C2×C4⋊C4×C53C2311yescharacteristic
77C2×C4⋊C4×C53C2312yes—
78C2×C4⋊C4×C53C2312yes—
79C2×C4⋊C4×C53C2314yes—
80C2×C4⋊C4×C53C2314yes—
81C2×C4⋊C4×C53C2314yes—
82C2×C4⋊C4×C53C2314yes—
83C2×C4⋊C4×C53C2314yes—
84C2×C4⋊C4×C53C2314yes—
85C2×C4⋊C4×C53C2314yes—
86C2×C4⋊C4×C53C2312yes—
87C2×C4⋊C4×C53C2314yes—
88C2×C4⋊C4×C53C2314yes—
89C2×C4⋊C4×C53C2314yes—
90C2×C4⋊C4×C53C2314yes—
91C2×C4⋊C4×C53C2314yes—
92C2×C4⋊C4×C53C2312yes—
93C2×C4⋊C4×C53C2312yes—
94C2×C42×C53C2312yes—
95C2×C42×C53C2312yes—
96C2×C42×C53C2312yes—
97C2×C42×C53C2312yes—
98C2×C42×C53C2314yes—
99C2×C42×C53C2312yes—
100C2×C42×C53C2311yescharacteristic
101C2×C42×C53C2312yes—
102C2×C42×C53C2312yes—
103C2×C42×C53C2312yes—
104C2×C42×C53C2314yes—
105C2×C42×C53C2314yes—
106C2×C42×C53C2311yescharacteristic
107C2×C42×C53C2311yescharacteristic
108#95×C53C2311yescharacteristic
109#95×C53C2311yescharacteristic
110#95×C53C2312yes—
111#95×C53C2312yes—
112#95×C53C2314yes—
113#95×C53C2314yes—
114#95×C53C2314yes—
115#95×C53C2314yes—
116#95×C53C2314yes—
117#95×C53C2314yes—
118#95×C53C2314yes—
119#95×C53C2314yes—
120#95×C53C2314yes—
121#95×C53C2314yes—
122#95×C53C2314yes—
123#95×C53C2314yes—
124#95×C53C2314yes—
125C2×Q8×C53—24no—
126C2×Q8×C53—22no—
127C22×C4×C53C2412yes—
128C22×C4×C53C2414yes—
129C22×C4×C53C2412yes—
130C22×C4×C53C2414yes—
131C22×C4×C53C2414yes—
132C22×C4×C53C2412yes—
133C22×C4×C53C2414yes—
134C22×C4×C53C2412yes—
135C22×C4×C53C2412yes—
136C22×C4×C53C2412yes—
137C22×C4×C53C2411yescharacteristic
138C22×C4×C53C2411yescharacteristic
139C22×C4×C53C2411yescharacteristic
140C4⋊C4×C53—24no—
141C4⋊C4×C53—24no—
142C4⋊C4×C53—24no—
143C4⋊C4×C53—22no—
144C4⋊C4×C53—28no—
145C4⋊C4×C53—28no—
146C4⋊C4×C53—28no—
147C4⋊C4×C53—28no—
148C4⋊C4×C53—28no—
149C4⋊C4×C53—28no—
150C4⋊C4×C53—28no—
151C4⋊C4×C53—24no—
152C4⋊C4×C53—24no—
153C4⋊C4×C53—24no—
154C4⋊C4×C53—44no—
155C4⋊C4×C53—44no—
156C4⋊C4×C53—44no—
157C4⋊C4×C53—44no—
158C4⋊C4×C53C2×Q8116yes—
159C4⋊C4×C53C2×Q818yes—
160C4⋊C4×C53#4118yes—
161C42×C53—24no—
162C42×C53—24no—
163C42×C53—24no—
164C42×C53—24no—
165C42×C53—28no—
166C42×C53—22no—
167C42×C53—24no—
168C42×C53—42no—
169C42×C53—42no—
170C42×C53—44no—
171C42×C53—44no—
172C42×C53—42no—
173C42×C53C2×Q814yes—
174C42×C53#4114yes—
175C23×C53C2511yescharacteristic
176Q8×C53—28no—
177C2×C4×C53—22no—
178C2×C4×C53—22no—
179C2×C4×C53—22no—
180C2×C4×C53—22no—
181C2×C4×C53—24no—
182C2×C4×C53—22no—
183C2×C4×C53—22no—
184C2×C4×C53—22no—
185C2×C4×C53—22no—
186C2×C4×C53—22no—
187C2×C4×C53—24no—
188C2×C4×C53—22no—
189C2×C4×C53—24no—
190C2×C4×C53—22no—
191C2×C4×C53—22no—
192C2×C4×C53—24no—
193C2×C4×C53—24no—
194C2×C4×C53—24no—
195C2×C4×C53—24no—
196C2×C4×C53—22no—
197C2×C4×C53—24no—
198C2×C4×C53—22no—
199C2×C4×C53—22no—
200C2×C4×C53—24no—
201C2×C4×C53—22no—
202C2×C4×C53—24no—
203C2×C4×C53—24no—
204C2×C4×C53C22×Q814yes—
205C2×C4×C53C22×Q818yes—
206C2×C4×C53C22×Q812yes—
207C2×C4×C53C2×#4114yes—
208C2×C4×C53C2×#4112yes—
209C2×C4×C53C2×#4114yes—
210C2×C4×C53C2×#4114yes—
211C2×C4×C53C2×#4112yes—
212C2×C4×C53C2×#4112yes—
213C2×C4×C532−1+412yes—
214#36567C5311yesmaximal, characteristic, direct product factor, Sylow
215C22×C53C23×Q811yescharacteristic
216C22×C53C22×#4111yescharacteristic
217C22×C53C22×#4112yes—
218C22×C53C2×2−1+411yescharacteristic
219C22×C53#58512yes—
220C4×C53—24no—
221C4×C53—22no—
222C4×C53—24no—
223C4×C53—28no—
224C4×C53—22no—
225C4×C53—24no—
226C4×C53—24no—
227C4×C53—44no—
228C4×C53—42no—
229C4×C53—44no—
230C4×C53—42no—
231C4×C53—44no—
232C4×C53#55714yes—
233#2864C2×C5312yes—
234#2859C2×C5311yescharacteristic
235#2590C2×C5314yes—
236#2585C2×C5314yes—
237#2584C2×C5312yes—
238#2507C2×C5312yes—
239#2506C2×C5312yes—
240#2462C2×C5314yes—
241#2370C2×C5312yes—
242#2370C2×C5314yes—
243C4×#125C2×C5311yescharacteristic
244C4×#125C2×C5312yes—
245C42×Q8C2×C5311yescharacteristic
246C2×C53#344411yescharacteristic
247C2×C53C2×#55711yescharacteristic
248C2×C53Q8×#4112yes—
249C2×C53#351211yescharacteristic
250C2×C53#351512yes—
251C2×#125C22×C5312yes—
252C2×#125C22×C5314yes—
253C2×#125C22×C5311yescharacteristic
254C2×C4×Q8C22×C5311yescharacteristic
255C2×C4×Q8C22×C5312yes—
256#400C22×C5314yes—
257#398C22×C5314yes—
258#398C22×C5314yes—
259#398C22×C5314yes—
260#398C22×C5314yes—
261#395C22×C5314yes—
262#389C22×C5312yes—
263#389C22×C5312yes—
264#389C22×C5312yes—
265#389C22×C5314yes—
266#389C22×C5314yes—
267#389C22×C5312yes—
268#389C22×C5312yes—
269#389C22×C5314yes—
270#389C22×C5314yes—
271#389C22×C5314yes—
272#389C22×C5314yes—
273#389C22×C5312yes—
274#389C22×C5312yes—
275#387C22×C5314yes—
276#387C22×C5314yes—
277#387C22×C5314yes—
278#387C22×C5314yes—
279#387C22×C5314yes—
280#387C22×C5314yes—
281#387C22×C5314yes—
282#387C22×C5314yes—
283#387C22×C5314yes—
284#387C22×C5314yes—
285#387C22×C5314yes—
286#384C22×C5311yescharacteristic
287#384C22×C5311yescharacteristic
288#382C22×C5312yes—
289#382C22×C5311yescharacteristic
290C4×C4⋊C4C22×C5312yes—
291C4×C4⋊C4C22×C5314yes—
292C4×C4⋊C4C22×C5314yes—
293C4×C4⋊C4C22×C5314yes—
294C4×C4⋊C4C22×C5312yes—
295C4×C4⋊C4C22×C5312yes—
296C4×C4⋊C4C22×C5312yes—
297C4×C4⋊C4C22×C5312yes—
298C4×C4⋊C4C22×C5314yes—
299C4×C4⋊C4C22×C5311yescharacteristic
300C4×C4⋊C4C22×C5312yes—
301C4×C4⋊C4C22×C5312yes—
302C43C22×C5311yescharacteristic
303C43C22×C5312yes—
304C53#3656711yescharacteristic, direct product factor, Sylow
305C22×Q8C23×C5311yescharacteristic
306#125—216no—
307#125—28no—
308#125Q8×C5318yescomplemented
309C4×Q8—28no—
310C4×Q8—24no—
311C2×C4⋊C4C23×C5312yes—
312C2×C4⋊C4C23×C5312yes—
313C2×C4⋊C4C23×C5312yes—
314C2×C4⋊C4C23×C5311yescharacteristic
315C2×C4⋊C4C23×C5312yes—
316C2×C4⋊C4C23×C5312yes—
317C2×C4⋊C4C23×C5314yes—
318C2×C4⋊C4C23×C5314yes—
319C2×C4⋊C4C23×C5314yes—
320C2×C4⋊C4C23×C5314yes—
321C2×C4⋊C4C23×C5314yes—
322C2×C4⋊C4C23×C5314yes—
323C2×C4⋊C4C23×C5314yes—
324C2×C4⋊C4C23×C5312yes—
325C2×C4⋊C4C23×C5314yes—
326C2×C4⋊C4C23×C5314yes—
327C2×C4⋊C4C23×C5314yes—
328C2×C4⋊C4C23×C5314yes—
329C2×C4⋊C4C23×C5314yes—
330C2×C4⋊C4C23×C5312yes—
331C2×C4⋊C4C23×C5312yes—
332C2×C42C23×C5312yes—
333C2×C42C23×C5312yes—
334C2×C42C23×C5312yes—
335C2×C42C23×C5312yes—
336C2×C42C23×C5314yes—
337C2×C42C23×C5312yes—
338C2×C42C23×C5311yescharacteristic
339C2×C42C23×C5312yes—
340C2×C42C23×C5312yes—
341C2×C42C23×C5312yes—
342C2×C42C23×C5314yes—
343C2×C42C23×C5314yes—
344C2×C42C23×C5311yescharacteristic
345C2×C42C23×C5311yescharacteristic
346#95C23×C5311yescharacteristic
347#95C23×C5311yescharacteristic
348#95C23×C5312yes—
349#95C23×C5312yes—
350#95C23×C5314yes—
351#95C23×C5314yes—
352#95C23×C5314yes—
353#95C23×C5314yes—
354#95C23×C5314yes—
355#95C23×C5314yes—
356#95C23×C5314yes—
357#95C23×C5314yes—
358#95C23×C5314yes—
359#95C23×C5314yes—
360#95C23×C5314yes—
361#95C23×C5314yes—
362#95C23×C5314yes—
363C2×Q8—24no—
364C2×Q8—22no—
365C22×C4C24×C5312yes—
366C22×C4C24×C5314yes—
367C22×C4C24×C5312yes—
368C22×C4C24×C5314yes—
369C22×C4C24×C5314yes—
370C22×C4C24×C5312yes—
371C22×C4C24×C5314yes—
372C22×C4C24×C5312yes—
373C22×C4C24×C5312yes—
374C22×C4C24×C5312yes—
375C22×C4C24×C5311yescharacteristic
376C22×C4C24×C5311yescharacteristic
377C22×C4C24×C5311yescharacteristic
378C4⋊C4—24no—
379C4⋊C4—24no—
380C4⋊C4—24no—
381C4⋊C4—22no—
382C4⋊C4—28no—
383C4⋊C4—28no—
384C4⋊C4—28no—
385C4⋊C4—28no—
386C4⋊C4—28no—
387C4⋊C4—28no—
388C4⋊C4—28no—
389C4⋊C4—24no—
390C4⋊C4—24no—
391C4⋊C4—24no—
392C4⋊C4—44no—
393C4⋊C4—44no—
394C4⋊C4—44no—
395C4⋊C4—44no—
396C4⋊C4C2×Q8×C53116yes—
397C4⋊C4C2×Q8×C5318yes—
398C4⋊C4#41×C5318yes—
399C42—24no—
400C42—24no—
401C42—24no—
402C42—24no—
403C42—28no—
404C42—22no—
405C42—24no—
406C42—42no—
407C42—42no—
408C42—44no—
409C42—44no—
410C42—42no—
411C42C2×Q8×C5314yes—
412C42#41×C5314yes—
413C23C25×C5311yescharacteristic
414Q8—28no—
415C2×C4—22no—
416C2×C4—24no—
417C2×C4—22no—
418C2×C4—22no—
419C2×C4—22no—
420C2×C4—22no—
421C2×C4—22no—
422C2×C4—22no—
423C2×C4—22no—
424C2×C4—22no—
425C2×C4—24no—
426C2×C4—22no—
427C2×C4—22no—
428C2×C4—22no—
429C2×C4—24no—
430C2×C4—24no—
431C2×C4—24no—
432C2×C4—24no—
433C2×C4—24no—
434C2×C4—22no—
435C2×C4—24no—
436C2×C4—22no—
437C2×C4—22no—
438C2×C4—24no—
439C2×C4—22no—
440C2×C4—24no—
441C2×C4—24no—
442C2×C4C22×Q8×C5314yes—
443C2×C4C22×Q8×C5318yes—
444C2×C4C22×Q8×C5312yes—
445C2×C4C2×#41×C5314yes—
446C2×C4C2×#41×C5314yes—
447C2×C4C2×#41×C5314yes—
448C2×C4C2×#41×C5312yes—
449C2×C4C2×#41×C5312yes—
450C2×C4C2×#41×C5312yes—
451C2×C42−1+4×C5312yes—
452C22C23×Q8×C5311yescharacteristic
453C22C22×#41×C5311yescharacteristic
454C22C22×#41×C5312yes—
455C22C2×2−1+4×C5311yescharacteristic
456C22C53×#58512yes—
457C4—24no—
458C4—28no—
459C4—22no—
460C4—24no—
461C4—24no—
462C4—24no—
463C4—22no—
464C4—42no—
465C4—44no—
466C4—42no—
467C4—44no—
468C4—44no—
469C4C53×#55714yes—
470C2C53×#344411yescharacteristic
471C2C2×C53×#55711yescharacteristic
472C2Q8×#41×C5312yes—
473C2C53×#351211yescharacteristic
474C2C53×#351512yes—
subgroupG/H
center, socle, prime radicalC23×C53C25
commutator, Frattini, cosocleC23C25×C53
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C53 ◁ G

Complements

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

NH ≅ G/Ntype
70. #125×C53Q8semidirect
214. #36567C53direct (N × H)
304. C53#36567direct (N × H)
308. #125Q8×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#28642
C53×#28592
C53×#25902
C53×#25852
C53×#25842
C53×#25072
C53×#25062
C53×#24622
C53×#23702
C4×#125×C532
C42×Q8×C532
#3656753