Finite Group Databasebeta

← C53×#11716 #27219228

Subgroups — C53×#11716 #27219228

Order 13568.

592 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#2626C211yesmaximal, characteristic
2C2×C53×#342C211yesmaximal, characteristic
3C2×C53×#323C211yesmaximal, characteristic
4C53×#1352C212yesmaximal, complemented
5C53×#1351C212yesmaximal, complemented
6C2×C22⋊Q8×C53C411yescharacteristic, complemented
7C22×C22⋊C4×C53—21no—
8C22×C22⋊C4×C53C2211yescharacteristic
9C2×C22⋊C8×C53—21no—
10C53×#399C411yescharacteristic, complemented
11C53×#399C2211yescharacteristic
12C53×#396—21no—
13C53×#396—21no—
14C53×#393C411yescharacteristic, complemented
15C53×#393C2211yescharacteristic
16C53×#385—21no—
17C53×#380C411yescharacteristic, complemented
18C53×#379—21no—
19C53×#342—21no—
20C53×#342C2212yes—
21C53×#323—21no—
22C53×#323C2212yes—
23C25×C53D411yescharacteristic
24C22×Q8×C53C2×C411yescharacteristic, complemented
25C23×C4×C53—41no—
26C23×C4×C53C2×C411yescharacteristic
27C23×C4×C53D411yescharacteristic
28C22×C8×C53—41no—
29C22⋊Q8×C53—41no—
30C22⋊Q8×C53—41no—
31C2×C4⋊C4×C53—21no—
32C2×C4⋊C4×C53—21no—
33C2×C4⋊C4×C53—21no—
34C2×C4⋊C4×C53—21no—
35C2×C4⋊C4×C53C2×C411yescharacteristic, complemented
36C2×C22⋊C4×C53—21no—
37C2×C22⋊C4×C53—21no—
38C2×C22⋊C4×C53—21no—
39C2×C22⋊C4×C53—21no—
40C2×C22⋊C4×C53—21no—
41C2×C22⋊C4×C53—21no—
42C2×C22⋊C4×C53—24no—
43C2×C22⋊C4×C53—22no—
44C2×C22⋊C4×C53—41no—
45C2×C22⋊C4×C53—41no—
46C2×C22⋊C4×C53—41no—
47C2×C22⋊C4×C53—42no—
48C2×C22⋊C4×C53C2×C411yescharacteristic
49C2×C22⋊C4×C53D411yescharacteristic
50C2×C22⋊C4×C53D411yescharacteristic
51C2×C22⋊C4×C53C2311yescharacteristic
52C22⋊C8×C53—22no—
53C22⋊C8×C53—41no—
54#95×C53—21no—
55#95×C53—21no—
56#95×C53C2×C411yescharacteristic, complemented
57#95×C53C2×C411yescharacteristic, complemented
58C24×C53—21no—
59C24×C53—21no—
60C24×C53—22no—
61C24×C53—81no—
62C24×C53—81no—
63C24×C53C22⋊C411yescharacteristic
64C24×C53C22⋊C411yescharacteristic
65C24×C53SD1612yes—
66C24×C53Q1612yes—
67C24×C53C2×D411yescharacteristic
68C2×Q8×C53—41no—
69C2×Q8×C53—81no—
70C22×C4×C53—21no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53—21no—
75C22×C4×C53—21no—
76C22×C4×C53—21no—
77C22×C4×C53—21no—
78C22×C4×C53—44no—
79C22×C4×C53—42no—
80C22×C4×C53—81no—
81C22×C4×C53—81no—
82C22×C4×C53—82no—
83C22×C4×C53—82no—
84C22×C4×C53C22⋊C411yescharacteristic
85C22×C4×C53C22⋊C411yescharacteristic
86C22×C4×C53C22×C411yescharacteristic
87C22×C4×C53C2×D411yescharacteristic
88C2×C8×C53—42no—
89C2×C8×C53—82no—
90C4⋊C4×C53—42no—
91C4⋊C4×C53—41no—
92C4⋊C4×C53—81no—
93C4⋊C4×C53—81no—
94C4⋊C4×C53—81no—
95C22⋊C4×C53—24no—
96C22⋊C4×C53—41no—
97C22⋊C4×C53—41no—
98C22⋊C4×C53—41no—
99C22⋊C4×C53—41no—
100C22⋊C4×C53—44no—
101C22⋊C4×C53—41no—
102C22⋊C4×C53—41no—
103C22⋊C4×C53—42no—
104C22⋊C4×C53—42no—
105C22⋊C4×C53—81no—
106C22⋊C4×C53—81no—
107C22⋊C4×C53—81no—
108C22⋊C4×C53—81no—
109C23×C53—22no—
110C23×C53—22no—
111C23×C53—21no—
112C23×C53—21no—
113C23×C53—22no—
114C23×C53—22no—
115C23×C53—21no—
116C23×C53—21no—
117C23×C53—41no—
118C23×C53—41no—
119C23×C53—42no—
120C23×C53—42no—
121C23×C53—41no—
122C23×C53—41no—
123C23×C53—41no—
124C23×C53—41no—
125C23×C53—81no—
126C23×C53—81no—
127C23×C53—82no—
128C23×C53—82no—
129C23×C53—84no—
130C23×C53#9911yescharacteristic
131C23×C53#9911yescharacteristic
132C23×C53#10312yes—
133C23×C53#10312yes—
134C23×C53#10412yes—
135C23×C53C2×C22⋊C411yescharacteristic
136C23×C53C2×SD1611yescharacteristic
137C23×C53C2×Q1611yescharacteristic
138Q8×C53—82no—
139C2×C4×C53—21no—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—42no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53—41no—
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—41no—
155C2×C4×C53—41no—
156C2×C4×C53—44no—
157C2×C4×C53—41no—
158C2×C4×C53—41no—
159C2×C4×C53—41no—
160C2×C4×C53—41no—
161C2×C4×C53—81no—
162C2×C4×C53—81no—
163C2×C4×C53—81no—
164C2×C4×C53—88no—
165C2×C4×C53—82no—
166C2×C4×C53—82no—
167C8×C53—82no—
168#11716C5311yesmaximal, characteristic, direct product factor, Sylow
169C22×C53—21no—
170C22×C53—21no—
171C22×C53—21no—
172C22×C53—21no—
173C22×C53—21no—
174C22×C53—21no—
175C22×C53—21no—
176C22×C53—22no—
177C22×C53—21no—
178C22×C53—42no—
179C22×C53—42no—
180C22×C53—42no—
181C22×C53—41no—
182C22×C53—41no—
183C22×C53—44no—
184C22×C53—44no—
185C22×C53—41no—
186C22×C53—41no—
187C22×C53—41no—
188C22×C53—41no—
189C22×C53—81no—
190C22×C53—81no—
191C22×C53—82no—
192C22×C53—82no—
193C22×C53#32812yes—
194C22×C53#32812yes—
195C22×C53C2×#9911yescharacteristic
196C22×C53C2×#10311yescharacteristic
197C22×C53C2×#10411yescharacteristic
198C4×C53—41no—
199C4×C53—41no—
200C4×C53—41no—
201C4×C53—41no—
202C4×C53—81no—
203C4×C53—81no—
204C4×C53—84no—
205C4×C53—81no—
206C4×C53—81no—
207#2626C2×C5311yescharacteristic
208C2×#342C2×C5311yescharacteristic
209C2×#323C2×C5311yescharacteristic
210#1352C2×C5312yescomplemented
211#1351C2×C5312yescomplemented
212C2×C53—21no—
213C2×C53—22no—
214C2×C53—21no—
215C2×C53—22no—
216C2×C53—41no—
217C2×C53—42no—
218C2×C53—41no—
219C2×C53C2×#32811yescharacteristic
220C2×C53#212111yescharacteristic
221C2×C53#212711yescharacteristic
222C2×C22⋊Q8C4×C5311yescharacteristic, complemented
223C22×C22⋊C4—21no—
224C22×C22⋊C4C22×C5311yescharacteristic
225C2×C22⋊C8—21no—
226#399C4×C5311yescharacteristic, complemented
227#399C22×C5311yescharacteristic
228#396—21no—
229#396—21no—
230#393C4×C5311yescharacteristic, complemented
231#393C22×C5311yescharacteristic
232#385—21no—
233#380C4×C5311yescharacteristic, complemented
234#379—21no—
235#342—21no—
236#342C22×C5312yes—
237#323—21no—
238#323C22×C5312yes—
239C53#1171611yescharacteristic, direct product factor, Sylow
240C25D4×C5311yescharacteristic
241C22×Q8C2×C4×C5311yescharacteristic, complemented
242C23×C4—41no—
243C23×C4C2×C4×C5311yescharacteristic
244C23×C4D4×C5311yescharacteristic
245C22×C8—41no—
246C22⋊Q8—41no—
247C22⋊Q8—41no—
248C2×C4⋊C4—21no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4C2×C4×C5311yescharacteristic, complemented
253C2×C22⋊C4—24no—
254C2×C22⋊C4—22no—
255C2×C22⋊C4—21no—
256C2×C22⋊C4—21no—
257C2×C22⋊C4—21no—
258C2×C22⋊C4—21no—
259C2×C22⋊C4—21no—
260C2×C22⋊C4—21no—
261C2×C22⋊C4—41no—
262C2×C22⋊C4—41no—
263C2×C22⋊C4—41no—
264C2×C22⋊C4—42no—
265C2×C22⋊C4C2×C4×C5311yescharacteristic
266C2×C22⋊C4D4×C5311yescharacteristic
267C2×C22⋊C4D4×C5311yescharacteristic
268C2×C22⋊C4C23×C5311yescharacteristic
269C22⋊C8—22no—
270C22⋊C8—41no—
271#95—21no—
272#95—21no—
273#95C2×C4×C5311yescharacteristic, complemented
274#95C2×C4×C5311yescharacteristic, complemented
275C24—21no—
276C24—21no—
277C24—22no—
278C24—81no—
279C24—81no—
280C24C22⋊C4×C5311yescharacteristic
281C24C22⋊C4×C5311yescharacteristic
282C24SD16×C5312yes—
283C24Q16×C5312yes—
284C24C2×D4×C5311yescharacteristic
285C2×Q8—41no—
286C2×Q8—81no—
287C22×C4—21no—
288C22×C4—21no—
289C22×C4—21no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—21no—
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—44no—
296C22×C4—42no—
297C22×C4—81no—
298C22×C4—81no—
299C22×C4—82no—
300C22×C4—82no—
301C22×C4C22⋊C4×C5311yescharacteristic
302C22×C4C22⋊C4×C5311yescharacteristic
303C22×C4C22×C4×C5311yescharacteristic
304C22×C4C2×D4×C5311yescharacteristic
305C2×C8—42no—
306C2×C8—82no—
307C4⋊C4—41no—
308C4⋊C4—42no—
309C4⋊C4—81no—
310C4⋊C4—81no—
311C4⋊C4—81no—
312C22⋊C4—24no—
313C22⋊C4—41no—
314C22⋊C4—41no—
315C22⋊C4—41no—
316C22⋊C4—41no—
317C22⋊C4—44no—
318C22⋊C4—41no—
319C22⋊C4—41no—
320C22⋊C4—42no—
321C22⋊C4—42no—
322C22⋊C4—81no—
323C22⋊C4—81no—
324C22⋊C4—81no—
325C22⋊C4—81no—
326C23—22no—
327C23—22no—
328C23—21no—
329C23—21no—
330C23—22no—
331C23—22no—
332C23—21no—
333C23—21no—
334C23—41no—
335C23—41no—
336C23—42no—
337C23—42no—
338C23—41no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—81no—
343C23—81no—
344C23—82no—
345C23—82no—
346C23—84no—
347C23#99×C5311yescharacteristic
348C23#99×C5311yescharacteristic
349C23#103×C5312yes—
350C23#103×C5312yes—
351C23#104×C5312yes—
352C23C2×C22⋊C4×C5311yescharacteristic
353C23C2×SD16×C5311yescharacteristic
354C23C2×Q16×C5311yescharacteristic
355Q8—82no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—42no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—44no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—81no—
379C2×C4—81no—
380C2×C4—81no—
381C2×C4—88no—
382C2×C4—82no—
383C2×C4—82no—
384C8—82no—
385C22—21no—
386C22—21no—
387C22—21no—
388C22—21no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—22no—
393C22—21no—
394C22—42no—
395C22—42no—
396C22—42no—
397C22—41no—
398C22—41no—
399C22—44no—
400C22—44no—
401C22—41no—
402C22—41no—
403C22—41no—
404C22—41no—
405C22—81no—
406C22—81no—
407C22—82no—
408C22—82no—
409C22C53×#32812yes—
410C22C53×#32812yes—
411C22C2×#99×C5311yescharacteristic
412C22C2×#103×C5311yescharacteristic
413C22C2×#104×C5311yescharacteristic
414C4—41no—
415C4—41no—
416C4—41no—
417C4—41no—
418C4—81no—
419C4—81no—
420C4—84no—
421C4—81no—
422C4—81no—
423C2—21no—
424C2—22no—
425C2—21no—
426C2—22no—
427C2—41no—
428C2—42no—
429C2—41no—
430C2C2×C53×#32811yescharacteristic
431C2C53×#212111yescharacteristic
432C2C53×#212711yescharacteristic
subgroupG/H
center, socleC22×C53C2×#99
commutatorC22×C4C22×C4×C53
Frattini, cosocleC2×C22⋊C4C23×C53
prime radicalC25×C53D4
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ C24×C53 ◁ C22×C22⋊C4×C53 ◁ G

Complements

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

NH ≅ G/Ntype
4. C53×#1352C2semidirect
5. C53×#1351C2semidirect
6. C2×C22⋊Q8×C53C4semidirect
10. C53×#399C4semidirect
14. C53×#393C4semidirect
17. C53×#380C4semidirect
24. C22×Q8×C53C2×C4semidirect
35. C2×C4⋊C4×C53C2×C4semidirect
56. #95×C53C2×C4semidirect
57. #95×C53C2×C4semidirect
168. #11716C53direct (N × H)
210. #1352C2×C53semidirect
211. #1351C2×C53semidirect
222. C2×C22⋊Q8C4×C53semidirect
226. #399C4×C53semidirect
230. #393C4×C53semidirect
233. #380C4×C53semidirect
239. C53#11716direct (N × H)
241. C22×Q8C2×C4×C53semidirect
252. C2×C4⋊C4C2×C4×C53semidirect
273. #95C2×C4×C53semidirect
274. #95C2×C4×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#26262
C2×C53×#3422
C2×C53×#3232
C53×#13522
C53×#13512
#1171653