Finite Group Databasebeta

← C53×#8221 #27215733

Subgroups — C53×#8221 #27215733

Order 13568.

640 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×C22⋊Q8×C53C211yesmaximal, characteristic
2C4×C22⋊C8×C53C211yesmaximal, characteristic
3C2×C53×#328C212yesmaximal
4C53×#1434C211yesmaximal, characteristic
5C53×#1294C212yesmaximal
6C2×C22⋊Q8×C53C2211yescharacteristic
7C2×C4×Q8×C53C411yescharacteristic, complemented
8C22×C42×C53C2211yescharacteristic
9C2×C22⋊C8×C53C2212yes—
10C2×C4×C8×C53—21no—
11C53×#391C411yescharacteristic, complemented
12C53×#389C411yescharacteristic, complemented
13C53×#387—21no—
14C53×#385—21no—
15C53×#380C2211yescharacteristic
16C4×C4⋊C4×C53—21no—
17C4×C4⋊C4×C53C411yescharacteristic, complemented
18C4×C22⋊C4×C53—21no—
19C2×#95×C53—21no—
20C2×#95×C53—21no—
21C2×#95×C53C2212yes—
22C53×#336—21no—
23C53×#328C418yescomplemented
24C22×Q8×C53C2×C411yescharacteristic
25C23×C4×C53—22no—
26C23×C4×C53D411yescharacteristic
27C23×C4×C53D411yescharacteristic
28C23×C4×C53C2311yescharacteristic
29C22×C8×C53—22no—
30C22⋊Q8×C53—22no—
31C22⋊Q8×C53C2×C414yes—
32C4×Q8×C53—42no—
33C2×C4⋊C4×C53—21no—
34C2×C4⋊C4×C53—21no—
35C2×C4⋊C4×C53—21no—
36C2×C4⋊C4×C53C2×C411yescharacteristic
37C2×C4⋊C4×C53C2×C411yescharacteristic
38C2×C4⋊C4×C53C2×C411yescharacteristic
39C2×C22⋊C4×C53—21no—
40C2×C22⋊C4×C53—21no—
41C2×C42×C53—21no—
42C2×C42×C53—21no—
43C2×C42×C53—42no—
44C2×C42×C53C2×C411yescharacteristic
45C2×C42×C53C2×C411yescharacteristic
46C2×C42×C53D411yescharacteristic
47C2×C42×C53D411yescharacteristic
48C22⋊C8×C53—22no—
49C22⋊C8×C53C2×C414yes—
50C4×C8×C53—42no—
51#95×C53—21no—
52#95×C53—21no—
53#95×C53—21no—
54#95×C53—21no—
55#95×C53—21no—
56#95×C53—22no—
57#95×C53—42no—
58#95×C53—42no—
59#95×C53C2×C414yes—
60C24×C53C2×D411yescharacteristic
61C2×Q8×C53—21no—
62C2×Q8×C53—41no—
63C2×Q8×C53—41no—
64C2×Q8×C53C4212yes—
65C22×C4×C53—21no—
66C22×C4×C53—21no—
67C22×C4×C53—22no—
68C22×C4×C53—22no—
69C22×C4×C53—21no—
70C22×C4×C53—21no—
71C22×C4×C53—24no—
72C22×C4×C53—41no—
73C22×C4×C53—41no—
74C22×C4×C53—41no—
75C22×C4×C53—41no—
76C22×C4×C53—41no—
77C22×C4×C53—42no—
78C22×C4×C53—42no—
79C22×C4×C53—42no—
80C22×C4×C53—42no—
81C22×C4×C53—42no—
82C22×C4×C53—42no—
83C22×C4×C53C22⋊C411yescharacteristic
84C22×C4×C53C22⋊C411yescharacteristic
85C22×C4×C53C22⋊C411yescharacteristic
86C22×C4×C53C22⋊C411yescharacteristic
87C22×C4×C53SD1612yes—
88C22×C4×C53Q1612yes—
89C22×C4×C53C22×C411yescharacteristic
90C22×C4×C53C22×C412yes—
91C22×C4×C53C2×D411yescharacteristic
92C22×C4×C53#4112yes—
93C2×C8×C53—24no—
94C2×C8×C53—42no—
95C2×C8×C53—42no—
96C4⋊C4×C53—21no—
97C4⋊C4×C53—24no—
98C4⋊C4×C53—42no—
99C4⋊C4×C53—41no—
100C4⋊C4×C53—41no—
101C4⋊C4×C53—42no—
102C4⋊C4×C53C4212yes—
103C22⋊C4×C53—24no—
104C22⋊C4×C53—42no—
105C42×C53—22no—
106C42×C53—22no—
107C42×C53—41no—
108C42×C53—41no—
109C42×C53—42no—
110C42×C53—42no—
111C23×C53—21no—
112C23×C53—21no—
113C23×C53—21no—
114C23×C53—21no—
115C23×C53C2×C22⋊C411yescharacteristic
116C23×C53C4×D412yes—
117C23×C53C2×SD1611yescharacteristic
118C23×C53C2×Q1611yescharacteristic
119C23×C53#13512yes—
120Q8×C53—42no—
121Q8×C53—42no—
122C2×C4×C53—21no—
123C2×C4×C53—21no—
124C2×C4×C53—21no—
125C2×C4×C53—21no—
126C2×C4×C53—21no—
127C2×C4×C53—21no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—21no—
131C2×C4×C53—21no—
132C2×C4×C53—22no—
133C2×C4×C53—22no—
134C2×C4×C53—21no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—42no—
138C2×C4×C53—42no—
139C2×C4×C53—42no—
140C2×C4×C53—42no—
141C2×C4×C53—44no—
142C2×C4×C53—41no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53—44no—
147C2×C4×C53—44no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53—41no—
152C2×C4×C53—41no—
153C2×C4×C53—41no—
154C2×C4×C53—44no—
155C2×C4×C53—82no—
156C2×C4×C53—82no—
157C2×C4×C53—82no—
158C2×C4×C53—82no—
159C2×C4×C53#9912yes—
160C2×C4×C53#10312yes—
161C2×C4×C53#10312yes—
162C2×C4×C53C2×C4211yescharacteristic
163C2×C4×C53#11711yescharacteristic
164C2×C4×C53C4×D412yes—
165C8×C53—44no—
166#8221C5311yesmaximal, characteristic, direct product factor, Sylow
167C22×C53—22no—
168C22×C53—22no—
169C22×C53—22no—
170C22×C53—21no—
171C22×C53—21no—
172C22×C53—21no—
173C22×C53—21no—
174C22×C53—22no—
175C22×C53C4×C22⋊C411yescharacteristic
176C22×C53C2×#9911yescharacteristic
177C22×C53#41011yescharacteristic
178C22×C53C2×#10311yescharacteristic
179C22×C53#41611yescharacteristic
180C22×C53#42111yescharacteristic
181C22×C53#42111yescharacteristic
182C22×C53C4×SD1612yes—
183C22×C53C4×Q1612yes—
184C4×C53—22no—
185C4×C53—21no—
186C4×C53—21no—
187C4×C53—22no—
188C4×C53—22no—
189C4×C53—22no—
190C4×C53—42no—
191C4×C53—42no—
192C4×C53—41no—
193C4×C53—84no—
194C4×C53—81no—
195C4×C53—81no—
196C4×C22⋊Q8C2×C5311yescharacteristic
197C4×C22⋊C8C2×C5311yescharacteristic
198C2×#328C2×C5312yes—
199#1434C2×C5311yescharacteristic
200#1294C2×C5312yes—
201C2×C53—22no—
202C2×C53—22no—
203C2×C53#151611yescharacteristic
204C2×C53#151811yescharacteristic
205C2×C53#152011yescharacteristic
206C2×C53#152111yescharacteristic
207C2×C53C4×#9911yescharacteristic
208C2×C53#175911yescharacteristic
209C2×C53C4×#10311yescharacteristic
210C2×C22⋊Q8C22×C5311yescharacteristic
211C2×C4×Q8C4×C5311yescharacteristic, complemented
212C22×C42C22×C5311yescharacteristic
213C2×C22⋊C8C22×C5312yes—
214C2×C4×C8—21no—
215#391C4×C5311yescharacteristic, complemented
216#389C4×C5311yescharacteristic, complemented
217#387—21no—
218#385—21no—
219#380C22×C5311yescharacteristic
220C4×C4⋊C4—21no—
221C4×C4⋊C4C4×C5311yescharacteristic, complemented
222C4×C22⋊C4—21no—
223C2×#95—21no—
224C2×#95—21no—
225C2×#95C22×C5312yes—
226#336—21no—
227#328C4×C5318yescomplemented
228C53#822111yescharacteristic, direct product factor, Sylow
229C22×Q8C2×C4×C5311yescharacteristic
230C23×C4—22no—
231C23×C4D4×C5311yescharacteristic
232C23×C4D4×C5311yescharacteristic
233C23×C4C23×C5311yescharacteristic
234C22×C8—22no—
235C22⋊Q8—22no—
236C22⋊Q8C2×C4×C5314yes—
237C4×Q8—42no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4—21no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4C2×C4×C5311yescharacteristic
242C2×C4⋊C4C2×C4×C5311yescharacteristic
243C2×C4⋊C4C2×C4×C5311yescharacteristic
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C42—21no—
247C2×C42—21no—
248C2×C42—42no—
249C2×C42C2×C4×C5311yescharacteristic
250C2×C42C2×C4×C5311yescharacteristic
251C2×C42D4×C5311yescharacteristic
252C2×C42D4×C5311yescharacteristic
253C22⋊C8—22no—
254C22⋊C8C2×C4×C5314yes—
255C4×C8—42no—
256#95—21no—
257#95—22no—
258#95—21no—
259#95—21no—
260#95—21no—
261#95—21no—
262#95—42no—
263#95—42no—
264#95C2×C4×C5314yes—
265C24C2×D4×C5311yescharacteristic
266C2×Q8—21no—
267C2×Q8—41no—
268C2×Q8—41no—
269C2×Q8C42×C5312yes—
270C22×C4—21no—
271C22×C4—21no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—21no—
275C22×C4—21no—
276C22×C4—24no—
277C22×C4—41no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4—42no—
286C22×C4—42no—
287C22×C4—42no—
288C22×C4C22⋊C4×C5311yescharacteristic
289C22×C4C22⋊C4×C5311yescharacteristic
290C22×C4C22⋊C4×C5311yescharacteristic
291C22×C4C22⋊C4×C5311yescharacteristic
292C22×C4SD16×C5312yes—
293C22×C4Q16×C5312yes—
294C22×C4C22×C4×C5311yescharacteristic
295C22×C4C22×C4×C5312yes—
296C22×C4C2×D4×C5311yescharacteristic
297C22×C4#41×C5312yes—
298C2×C8—24no—
299C2×C8—42no—
300C2×C8—42no—
301C4⋊C4—21no—
302C4⋊C4—24no—
303C4⋊C4—41no—
304C4⋊C4—41no—
305C4⋊C4—42no—
306C4⋊C4—42no—
307C4⋊C4C42×C5312yes—
308C22⋊C4—24no—
309C22⋊C4—42no—
310C42—22no—
311C42—22no—
312C42—41no—
313C42—41no—
314C42—42no—
315C42—42no—
316C23—21no—
317C23—21no—
318C23—21no—
319C23—21no—
320C23C2×C22⋊C4×C5311yescharacteristic
321C23C4×D4×C5312yes—
322C23C2×SD16×C5311yescharacteristic
323C23C2×Q16×C5311yescharacteristic
324C23#135×C5312yes—
325Q8—42no—
326Q8—42no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—22no—
338C2×C4—22no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—44no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—44no—
352C2×C4—44no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—41no—
358C2×C4—41no—
359C2×C4—44no—
360C2×C4—82no—
361C2×C4—82no—
362C2×C4—82no—
363C2×C4—82no—
364C2×C4#99×C5312yes—
365C2×C4#103×C5312yes—
366C2×C4#103×C5312yes—
367C2×C4C2×C42×C5311yescharacteristic
368C2×C4#117×C5311yescharacteristic
369C2×C4C4×D4×C5312yes—
370C8—44no—
371C22—22no—
372C22—22no—
373C22—22no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—21no—
378C22—22no—
379C22C4×C22⋊C4×C5311yescharacteristic
380C22C2×#99×C5311yescharacteristic
381C22C53×#41011yescharacteristic
382C22C2×#103×C5311yescharacteristic
383C22C53×#41611yescharacteristic
384C22C53×#42111yescharacteristic
385C22C53×#42111yescharacteristic
386C22C4×SD16×C5312yes—
387C22C4×Q16×C5312yes—
388C4—22no—
389C4—21no—
390C4—21no—
391C4—22no—
392C4—22no—
393C4—22no—
394C4—42no—
395C4—42no—
396C4—41no—
397C4—84no—
398C4—81no—
399C4—81no—
400C2—22no—
401C2—22no—
402C2C53×#151611yescharacteristic
403C2C53×#151811yescharacteristic
404C2C53×#152011yescharacteristic
405C2C53×#152111yescharacteristic
406C2C4×#99×C5311yescharacteristic
407C2C53×#175911yescharacteristic
408C2C4×#103×C5311yescharacteristic
subgroupG/H
center, socleC23×C53C2×C22⋊C4
commutatorC2×C4C2×C42×C53
Frattini, cosocleC23×C4C23×C53
prime radicalC24×C53C2×D4
LCSG ▷ C2×C4 ▷ C22 ▷ C1
UCSC1 ◁ C23×C53 ◁ C22×C42×C53 ◁ G

Complements

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

NH ≅ G/Ntype
7. C2×C4×Q8×C53C4semidirect
11. C53×#391C4semidirect
12. C53×#389C4semidirect
17. C4×C4⋊C4×C53C4semidirect
23. C53×#328C4semidirect
166. #8221C53direct (N × H)
211. C2×C4×Q8C4×C53semidirect
215. #391C4×C53semidirect
216. #389C4×C53semidirect
221. C4×C4⋊C4C4×C53semidirect
227. #328C4×C53semidirect
228. C53#8221direct (N × H)

Maximal subgroups

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

subgroupindex
C4×C22⋊Q8×C532
C4×C22⋊C8×C532
C2×C53×#3282
C53×#14342
C53×#12942
#822153