Finite Group Databasebeta

← C53×#8363 #27215875

Subgroups — C53×#8363 #27215875

Order 13568.

788 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C53×#552C211yesmaximal, characteristic
2C53×#1826C212yesmaximal, complemented
3C4×#99×C53C211yesmaximal, characteristic
4C53×#1508C212yesmaximal, complemented
5C53×#1465C211yesmaximal, characteristic
6C53×#552—24no—
7C53×#552—24no—
8C2×#126×C53—21no—
9C2×#125×C53—21no—
10C2×#125×C53C411yescharacteristic, complemented
11C2×#123×C53—21no—
12C2×C22⋊Q8×C53C411yescharacteristic, complemented
13C2×C22⋊Q8×C53C2211yescharacteristic
14C2×C4×Q8×C53C411yescharacteristic, complemented
15C2×C4×D4×C53C2211yescharacteristic
16C2×#117×C53—21no—
17C2×#117×C53C2211yescharacteristic
18C22×C4⋊C4×C53C411yescharacteristic, complemented
19C2×#99×C53—21no—
20C53×#407C2212yes—
21C4×C22⋊C4×C53C2212yes—
22C53×#342—21no—
23C53×#339—24no—
24C53×#328—24no—
25C53×#324—21no—
26C53×#323—21no—
27C22×Q8×C53C2×C411yescharacteristic
28C22×D4×C53D411yescharacteristic
29C23×C4×C53—21no—
30C23×C4×C53C2×C411yescharacteristic
31C23×C4×C53C2311yescharacteristic
32C2×M16×C53—22no—
33#126×C53—28no—
34#125×C53—28no—
35#125×C53—22no—
36#125×C53—22no—
37#123×C53—28no—
38C22⋊Q8×C53—22no—
39C22⋊Q8×C53—22no—
40C22⋊Q8×C53—22no—
41C22⋊Q8×C53D414yes—
42C4×Q8×C53—22no—
43C4×Q8×C53—22no—
44C4×D4×C53—22no—
45C4×D4×C53—22no—
46#117×C53—28no—
47#117×C53—22no—
48#117×C53D414yes—
49C2×C4⋊C4×C53—22no—
50C2×C4⋊C4×C53—22no—
51C2×C4⋊C4×C53—21no—
52C2×C4⋊C4×C53—21no—
53C2×C4⋊C4×C53—21no—
54C2×C4⋊C4×C53—21no—
55C2×C4⋊C4×C53—21no—
56C2×C4⋊C4×C53—21no—
57C2×C4⋊C4×C53C2×C411yescharacteristic
58C2×C4⋊C4×C53C2×C411yescharacteristic
59C2×C4⋊C4×C53C2×C411yescharacteristic
60C2×C4⋊C4×C53D411yescharacteristic
61C2×C22⋊C4×C53—21no—
62C2×C22⋊C4×C53—21no—
63C2×C22⋊C4×C53—21no—
64C2×C22⋊C4×C53—22no—
65C2×C22⋊C4×C53D411yescharacteristic
66C2×C22⋊C4×C53D411yescharacteristic
67C2×C42×C53—21no—
68C2×C42×C53—21no—
69C2×C42×C53—22no—
70C2×C42×C53C2×C411yescharacteristic
71#99×C53—24no—
72C22⋊C8×C53—22no—
73C22⋊C8×C53—22no—
74#95×C53—22no—
75C24×C53C22⋊C411yescharacteristic
76C24×C53C2×D411yescharacteristic
77C2×Q8×C53—21no—
78C2×Q8×C53—42no—
79C2×Q8×C53C22⋊C412yes—
80C2×D4×C53—22no—
81C2×D4×C53—22no—
82C2×D4×C53—41no—
83C22×C4×C53—24no—
84C22×C4×C53—21no—
85C22×C4×C53—21no—
86C22×C4×C53—21no—
87C22×C4×C53—21no—
88C22×C4×C53—21no—
89C22×C4×C53—21no—
90C22×C4×C53—42no—
91C22×C4×C53—42no—
92C22×C4×C53—42no—
93C22×C4×C53—42no—
94C22×C4×C53—42no—
95C22×C4×C53C22⋊C411yescharacteristic
96C22×C4×C53C22⋊C411yescharacteristic
97C22×C4×C53C22⋊C411yescharacteristic
98C22×C4×C53C22×C411yescharacteristic
99C22×C4×C53C2×D412yes—
100C22×C4×C53C2×D412yes—
101C22×C4×C53C2×D411yescharacteristic
102M16×C53—44no—
103C2×C8×C53—42no—
104C4⋊C4×C53—24no—
105C4⋊C4×C53—24no—
106C4⋊C4×C53—24no—
107C4⋊C4×C53—24no—
108C4⋊C4×C53—24no—
109C4⋊C4×C53—21no—
110C4⋊C4×C53—21no—
111C4⋊C4×C53—21no—
112C4⋊C4×C53—24no—
113C4⋊C4×C53—22no—
114C4⋊C4×C53C22⋊C412yes—
115C4⋊C4×C53C22⋊C412yes—
116C4⋊C4×C53C22⋊C412yes—
117C22⋊C4×C53—24no—
118C22⋊C4×C53—24no—
119C22⋊C4×C53—24no—
120C22⋊C4×C53—22no—
121C22⋊C4×C53—22no—
122C22⋊C4×C53—24no—
123C22⋊C4×C53—42no—
124C42×C53—24no—
125C42×C53—21no—
126C42×C53—21no—
127C42×C53—24no—
128C42×C53—44no—
129C23×C53—21no—
130C23×C53—21no—
131C23×C53—21no—
132C23×C53—21no—
133C23×C53—41no—
134C23×C53—41no—
135C23×C53—41no—
136C23×C53—41no—
137C23×C53#10412yes—
138C23×C53C2×C22⋊C411yescharacteristic
139C23×C53#12012yes—
140Q8×C53—42no—
141Q8×C53—42no—
142D4×C53—44no—
143C2×C4×C53—22no—
144C2×C4×C53—24no—
145C2×C4×C53—22no—
146C2×C4×C53—22no—
147C2×C4×C53—21no—
148C2×C4×C53—22no—
149C2×C4×C53—22no—
150C2×C4×C53—21no—
151C2×C4×C53—21no—
152C2×C4×C53—21no—
153C2×C4×C53—21no—
154C2×C4×C53—41no—
155C2×C4×C53—42no—
156C2×C4×C53—42no—
157C2×C4×C53—42no—
158C2×C4×C53—48no—
159C2×C4×C53—42no—
160C2×C4×C53—41no—
161C2×C4×C53—41no—
162C2×C4×C53—41no—
163C2×C4×C53—41no—
164C2×C4×C53—42no—
165C2×C4×C53—41no—
166C2×C4×C53—41no—
167C2×C4×C53—44no—
168C2×C4×C53—82no—
169C2×C4×C53—82no—
170C2×C4×C53C2×C22⋊C411yescharacteristic
171C2×C4×C53C2×C22⋊C411yescharacteristic
172C2×C4×C53#12012yes—
173C8×C53—82no—
174#8363C5311yesmaximal, characteristic, direct product factor, Sylow
175C22×C53—21no—
176C22×C53—21no—
177C22×C53—22no—
178C22×C53—22no—
179C22×C53—22no—
180C22×C53—22no—
181C22×C53—42no—
182C22×C53—42no—
183C22×C53—41no—
184C22×C53—41no—
185C22×C53—81no—
186C22×C53—81no—
187C22×C53#37911yescharacteristic
188C22×C53C2×#10411yescharacteristic
189C22×C53#42111yescharacteristic
190C4×C53—22no—
191C4×C53—21no—
192C4×C53—41no—
193C4×C53—42no—
194C4×C53—41no—
195C4×C53—42no—
196C4×C53—41no—
197C4×C53—42no—
198C4×C53—42no—
199C4×C53—84no—
200C4×C53#41012yes—
201C2×#552C2×C5311yescharacteristic
202#1826C2×C5312yescomplemented
203C4×#99C2×C5311yescharacteristic
204#1508C2×C5312yescomplemented
205#1465C2×C5311yescharacteristic
206C2×C53—21no—
207C2×C53—21no—
208C2×C53—42no—
209C2×C53—42no—
210C2×C53#178211yescharacteristic
211C2×C53#178911yescharacteristic
212C2×C53#179611yescharacteristic
213#552—24no—
214#552—24no—
215C2×#126—21no—
216C2×#125—21no—
217C2×#125C4×C5311yescharacteristic, complemented
218C2×#123—21no—
219C2×C22⋊Q8C4×C5311yescharacteristic, complemented
220C2×C22⋊Q8C22×C5311yescharacteristic
221C2×C4×Q8C4×C5311yescharacteristic, complemented
222C2×C4×D4C22×C5311yescharacteristic
223C2×#117—21no—
224C2×#117C22×C5311yescharacteristic
225C22×C4⋊C4C4×C5311yescharacteristic, complemented
226C2×#99—21no—
227#407C22×C5312yes—
228C4×C22⋊C4C22×C5312yes—
229#342—21no—
230#339—24no—
231#328—24no—
232#324—21no—
233#323—21no—
234C53#836311yescharacteristic, direct product factor, Sylow
235C22×Q8C2×C4×C5311yescharacteristic
236C22×D4D4×C5311yescharacteristic
237C23×C4—21no—
238C23×C4C2×C4×C5311yescharacteristic
239C23×C4C23×C5311yescharacteristic
240C2×M16—22no—
241#126—28no—
242#125—28no—
243#125—22no—
244#125—22no—
245#123—28no—
246C22⋊Q8—22no—
247C22⋊Q8—22no—
248C22⋊Q8—22no—
249C22⋊Q8D4×C5314yes—
250C4×Q8—22no—
251C4×Q8—22no—
252C4×D4—22no—
253C4×D4—22no—
254#117—28no—
255#117—22no—
256#117D4×C5314yes—
257C2×C4⋊C4—21no—
258C2×C4⋊C4—21no—
259C2×C4⋊C4—21no—
260C2×C4⋊C4—21no—
261C2×C4⋊C4—21no—
262C2×C4⋊C4—21no—
263C2×C4⋊C4—22no—
264C2×C4⋊C4—22no—
265C2×C4⋊C4C2×C4×C5311yescharacteristic
266C2×C4⋊C4C2×C4×C5311yescharacteristic
267C2×C4⋊C4C2×C4×C5311yescharacteristic
268C2×C4⋊C4D4×C5311yescharacteristic
269C2×C22⋊C4—21no—
270C2×C22⋊C4—21no—
271C2×C22⋊C4—21no—
272C2×C22⋊C4—22no—
273C2×C22⋊C4D4×C5311yescharacteristic
274C2×C22⋊C4D4×C5311yescharacteristic
275C2×C42—21no—
276C2×C42—21no—
277C2×C42—22no—
278C2×C42C2×C4×C5311yescharacteristic
279#99—24no—
280C22⋊C8—22no—
281C22⋊C8—22no—
282#95—22no—
283C24C22⋊C4×C5311yescharacteristic
284C24C2×D4×C5311yescharacteristic
285C2×Q8—21no—
286C2×Q8—42no—
287C2×Q8C22⋊C4×C5312yes—
288C2×D4—22no—
289C2×D4—22no—
290C2×D4—41no—
291C22×C4—24no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—21no—
296C22×C4—21no—
297C22×C4—21no—
298C22×C4—42no—
299C22×C4—42no—
300C22×C4—42no—
301C22×C4—42no—
302C22×C4—42no—
303C22×C4C22⋊C4×C5311yescharacteristic
304C22×C4C22⋊C4×C5311yescharacteristic
305C22×C4C22⋊C4×C5311yescharacteristic
306C22×C4C22×C4×C5311yescharacteristic
307C22×C4C2×D4×C5312yes—
308C22×C4C2×D4×C5312yes—
309C22×C4C2×D4×C5311yescharacteristic
310M16—44no—
311C2×C8—42no—
312C4⋊C4—24no—
313C4⋊C4—24no—
314C4⋊C4—24no—
315C4⋊C4—24no—
316C4⋊C4—24no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—21no—
320C4⋊C4—24no—
321C4⋊C4—22no—
322C4⋊C4C22⋊C4×C5312yes—
323C4⋊C4C22⋊C4×C5312yes—
324C4⋊C4C22⋊C4×C5312yes—
325C22⋊C4—24no—
326C22⋊C4—24no—
327C22⋊C4—24no—
328C22⋊C4—22no—
329C22⋊C4—22no—
330C22⋊C4—24no—
331C22⋊C4—42no—
332C42—24no—
333C42—21no—
334C42—21no—
335C42—24no—
336C42—44no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23#104×C5312yes—
346C23C2×C22⋊C4×C5311yescharacteristic
347C23#120×C5312yes—
348Q8—42no—
349Q8—42no—
350D4—44no—
351C2×C4—24no—
352C2×C4—22no—
353C2×C4—21no—
354C2×C4—22no—
355C2×C4—22no—
356C2×C4—22no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—22no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—41no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—48no—
367C2×C4—42no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—42no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—44no—
376C2×C4—82no—
377C2×C4—82no—
378C2×C4C2×C22⋊C4×C5311yescharacteristic
379C2×C4C2×C22⋊C4×C5311yescharacteristic
380C2×C4#120×C5312yes—
381C8—82no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—22no—
388C22—42no—
389C22—42no—
390C22—41no—
391C22—41no—
392C22—81no—
393C22—81no—
394C22C53×#37911yescharacteristic
395C22C2×#104×C5311yescharacteristic
396C22C53×#42111yescharacteristic
397C4—22no—
398C4—21no—
399C4—41no—
400C4—42no—
401C4—41no—
402C4—42no—
403C4—42no—
404C4—42no—
405C4—41no—
406C4—84no—
407C4C53×#41012yes—
408C2—21no—
409C2—21no—
410C2—42no—
411C2—42no—
412C2C53×#178211yescharacteristic
413C2C53×#178911yescharacteristic
414C2C53×#179611yescharacteristic
subgroupG/H
center, socleC22×C53#379
commutatorC22×C4C22×C4×C53
Frattini, cosocleC23×C4C23×C53
prime radicalC22×D4×C53D4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C53 ◁ C23×C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#1826C2semidirect
4. C53×#1508C2semidirect
10. C2×#125×C53C4semidirect
12. C2×C22⋊Q8×C53C4semidirect
14. C2×C4×Q8×C53C4semidirect
18. C22×C4⋊C4×C53C4semidirect
174. #8363C53direct (N × H)
202. #1826C2×C53semidirect
204. #1508C2×C53semidirect
217. C2×#125C4×C53semidirect
219. C2×C22⋊Q8C4×C53semidirect
221. C2×C4×Q8C4×C53semidirect
225. C22×C4⋊C4C4×C53semidirect
234. C53#8363direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C53×#5522
C53×#18262
C4×#99×C532
C53×#15082
C53×#14652
#836353