Finite Group Databasebeta

← C53×#19123 #27226635

Subgroups — C53×#19123 #27226635

Order 13568.

628 conjugacy classes of subgroups.

The subgroup lattice has 450 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3131C211yesmaximal, characteristic
2C53×#3104C211yesmaximal, characteristic, complemented
3C53×#3103C211yesmaximal, characteristic
4C2×C8×Q8×C53C211yesmaximal, characteristic, complemented
5C53×#1684C212yesmaximal, complemented
6C53×#1682C212yesmaximal, complemented
7C53×#1680C212yesmaximal, complemented
8C53×#1678C212yesmaximal, complemented
9C53×#1564C211yesmaximal, characteristic, complemented
10C53×#1542C211yesmaximal, characteristic, complemented
11C53×#1540C211yesmaximal, characteristic
12C53×#533C2211yescharacteristic
13C53×#532C2211yescharacteristic
14C2×C4×Q8×C53C2211yescharacteristic
15C53×#488—21no—
16C53×#488—21no—
17C53×#487C2211yescharacteristic
18C53×#487C2211yescharacteristic
19C53×#486C2211yescharacteristic
20C53×#486C2211yescharacteristic
21C53×#484—21no—
22C53×#481—21no—
23C53×#477C2212yes—
24C53×#475C2212yes—
25C53×#462C2212yes—
26C53×#460C2212yes—
27C53×#451—21no—
28C53×#448—21no—
29C8×Q8×C53—21no—
30C8×Q8×C53—21no—
31C8×Q8×C53C2212yes—
32C8×Q8×C53C2212yes—
33C4×Q16×C53—22no—
34C4×SD16×C53—22no—
35C53×#431C2211yescharacteristic
36C53×#431C2211yescharacteristic
37C2×C4⋊C8×C53—21no—
38C2×C4⋊C8×C53C2211yescharacteristic
39C53×#416—21no—
40C53×#416—21no—
41C2×C4×C8×C53—21no—
42C2×C4×C8×C53C2211yescharacteristic
43C53×#335C2212yes—
44C53×#334C2212yes—
45C53×#333C2212yes—
46C53×#332C2212yes—
47C53×#332C2212yes—
48C53×#331C2212yes—
49C2×#41×C53—21no—
50C22×Q8×C53D411yescharacteristic
51C2×Q16×C53—22no—
52C2×SD16×C53—22no—
53C22×C8×C53—21no—
54C22×C8×C53D411yescharacteristic
55C22×C8×C53D411yescharacteristic
56C4⋊Q8×C53C2311yescharacteristic
57C4⋊Q8×C53C2311yescharacteristic
58C4⋊Q8×C53C2311yescharacteristic
59#127×C53C2311yescharacteristic
60#125×C53—21no—
61#124×C53—21no—
62C22⋊Q8×C53—21no—
63C22⋊Q8×C53—21no—
64#121×C53—21no—
65#121×C53—21no—
66C4×Q8×C53—21no—
67C4×Q8×C53—21no—
68C4×Q8×C53—21no—
69C4×Q8×C53—21no—
70C4×Q8×C53D412yes—
71C4×Q8×C53C2312yes—
72C4×D4×C53—21no—
73C4×D4×C53—21no—
74#117×C53—21no—
75C2×C4⋊C4×C53—21no—
76C2×C4⋊C4×C53D411yescharacteristic
77C2×C42×C53—21no—
78C2×C42×C53C2311yescharacteristic
79#107×C53—22no—
80#106×C53—22no—
81C4⋊C8×C53—22no—
82C4⋊C8×C53—22no—
83C4⋊C8×C53—21no—
84C4⋊C8×C53C2312yes—
85C4⋊C8×C53C2312yes—
86C4⋊C8×C53C2312yes—
87#103×C53—21no—
88#103×C53—21no—
89#103×C53—21no—
90#103×C53—21no—
91#103×C53—22no—
92#103×C53—22no—
93#103×C53—22no—
94#102×C53—21no—
95#102×C53—21no—
96#102×C53—21no—
97#102×C53—21no—
98#102×C53—22no—
99C22⋊C8×C53—21no—
100C22⋊C8×C53—21no—
101C4×C8×C53—22no—
102C4×C8×C53—22no—
103C4×C8×C53—21no—
104C4×C8×C53—21no—
105C4×C8×C53D411yescharacteristic
106C4×C8×C53D411yescharacteristic
107C4×C8×C53C2311yescharacteristic
108C4×C8×C53C2311yescharacteristic
109#41×C53—42no—
110#41×C53—42no—
111C2×Q8×C53—24no—
112C2×Q8×C53—21no—
113C2×Q8×C53—21no—
114C2×Q8×C53—21no—
115C2×Q8×C53—21no—
116C2×Q8×C53C2×D412yes—
117C2×D4×C53—21no—
118C2×D4×C53—21no—
119C2×D4×C53—21no—
120C2×D4×C53—41no—
121C22×C4×C53—21no—
122C22×C4×C53—21no—
123C22×C4×C53—41no—
124C22×C4×C53C2×D411yescharacteristic
125C22×C4×C53C2×D411yescharacteristic
126C22×C4×C53#4111yescharacteristic
127Q16×C53—44no—
128SD16×C53—44no—
129C2×C8×C53—22no—
130C2×C8×C53—22no—
131C2×C8×C53—22no—
132C2×C8×C53—22no—
133C2×C8×C53—22no—
134C2×C8×C53—42no—
135C2×C8×C53C2×D411yescharacteristic
136C2×C8×C53C2×D411yescharacteristic
137C2×C8×C53C2×D411yescharacteristic
138C2×C8×C53C2×D411yescharacteristic
139C4⋊C4×C53—22no—
140C4⋊C4×C53—21no—
141C4⋊C4×C53—22no—
142C4⋊C4×C53—21no—
143C4⋊C4×C53—21no—
144C4⋊C4×C53—21no—
145C4⋊C4×C53—21no—
146C4⋊C4×C53—21no—
147C4⋊C4×C53—21no—
148C4⋊C4×C53—41no—
149C4⋊C4×C53—41no—
150C4⋊C4×C53C2×D412yes—
151C22⋊C4×C53—41no—
152C22⋊C4×C53—41no—
153C22⋊C4×C53—41no—
154C42×C53—22no—
155C42×C53—22no—
156C42×C53—41no—
157C42×C53C2×D411yescharacteristic
158C42×C53C2×D411yescharacteristic
159C42×C53#4111yescharacteristic
160C42×C53C2411yescharacteristic
161C23×C53—41no—
162C23×C53#12111yescharacteristic
163Q8×C53—24no—
164Q8×C53—22no—
165Q8×C53—41no—
166Q8×C53—42no—
167Q8×C53—41no—
168Q8×C53—81no—
169Q8×C53#13514yes—
170D4×C53—41no—
171D4×C53—42no—
172D4×C53—41no—
173D4×C53—81no—
174D4×C53—82no—
175C2×C4×C53—24no—
176C2×C4×C53—22no—
177C2×C4×C53—22no—
178C2×C4×C53—22no—
179C2×C4×C53—41no—
180C2×C4×C53—42no—
181C2×C4×C53—41no—
182C2×C4×C53—41no—
183C2×C4×C53—41no—
184C2×C4×C53—82no—
185C2×C4×C53#12111yescharacteristic
186C2×C4×C53#12111yescharacteristic
187C2×C4×C53#12111yescharacteristic
188C2×C4×C53#13512yes—
189C2×C4×C53#13712yes—
190C2×C4×C53C22×D411yescharacteristic
191C2×C4×C53C22×D411yescharacteristic
192C2×C4×C53C2×#4111yescharacteristic
193C8×C53—21no—
194C8×C53—21no—
195C8×C53—21no—
196C8×C53—21no—
197C8×C53—42no—
198C8×C53—42no—
199C8×C53—42no—
200#19123C5311yesmaximal, characteristic, direct product factor, Sylow
201C22×C53—21no—
202C22×C53—21no—
203C22×C53—81no—
204C22×C53—81no—
205C22×C53—81no—
206C22×C53#46412yes—
207C22×C53C2×#12111yescharacteristic
208C4×C53—24no—
209C4×C53—21no—
210C4×C53—21no—
211C4×C53—21no—
212C4×C53—42no—
213C4×C53—81no—
214C4×C53—81no—
215C4×C53—81no—
216C4×C53#46412yes—
217C4×C53C2×#13511yescharacteristic
218C4×C53C2×#13512yes—
219C4×C53C2×#13711yescharacteristic
220#3131C2×C5311yescharacteristic
221#3104C2×C5311yescharacteristic, complemented
222#3103C2×C5311yescharacteristic
223C2×C8×Q8C2×C5311yescharacteristic, complemented
224#1684C2×C5312yescomplemented
225#1682C2×C5312yescomplemented
226#1680C2×C5312yescomplemented
227#1678C2×C5312yescomplemented
228#1564C2×C5311yescharacteristic, complemented
229#1542C2×C5311yescharacteristic, complemented
230#1540C2×C5311yescharacteristic
231C2×C53—22no—
232C2×C53—161no—
233C2×C53C2×#46411yescharacteristic
234C2×C53#305011yescharacteristic
235C2×C53#306711yescharacteristic
236#533C22×C5311yescharacteristic
237#532C22×C5311yescharacteristic
238C2×C4×Q8C22×C5311yescharacteristic
239#488—21no—
240#488—21no—
241#487C22×C5311yescharacteristic
242#487C22×C5311yescharacteristic
243#486C22×C5311yescharacteristic
244#486C22×C5311yescharacteristic
245#484—21no—
246#481—21no—
247#477C22×C5312yes—
248#475C22×C5312yes—
249#462C22×C5312yes—
250#460C22×C5312yes—
251#451—21no—
252#448—21no—
253C8×Q8—21no—
254C8×Q8—21no—
255C8×Q8C22×C5312yes—
256C8×Q8C22×C5312yes—
257C4×Q16—22no—
258C4×SD16—22no—
259#431C22×C5311yescharacteristic
260#431C22×C5311yescharacteristic
261C2×C4⋊C8—21no—
262C2×C4⋊C8C22×C5311yescharacteristic
263#416—21no—
264#416—21no—
265C2×C4×C8—21no—
266C2×C4×C8C22×C5311yescharacteristic
267#335C22×C5312yes—
268#334C22×C5312yes—
269#333C22×C5312yes—
270#332C22×C5312yes—
271#332C22×C5312yes—
272#331C22×C5312yes—
273C53#1912311yescharacteristic, direct product factor, Sylow
274C2×#41—21no—
275C22×Q8D4×C5311yescharacteristic
276C2×Q16—22no—
277C2×SD16—22no—
278C22×C8—21no—
279C22×C8D4×C5311yescharacteristic
280C22×C8D4×C5311yescharacteristic
281C4⋊Q8C23×C5311yescharacteristic
282C4⋊Q8C23×C5311yescharacteristic
283C4⋊Q8C23×C5311yescharacteristic
284#127C23×C5311yescharacteristic
285#125—21no—
286#124—21no—
287C22⋊Q8—21no—
288C22⋊Q8—21no—
289#121—21no—
290#121—21no—
291C4×Q8—21no—
292C4×Q8—21no—
293C4×Q8—21no—
294C4×Q8—21no—
295C4×Q8D4×C5312yes—
296C4×Q8C23×C5312yes—
297C4×D4—21no—
298C4×D4—21no—
299#117—21no—
300C2×C4⋊C4—21no—
301C2×C4⋊C4D4×C5311yescharacteristic
302C2×C42—21no—
303C2×C42C23×C5311yescharacteristic
304#107—22no—
305#106—22no—
306C4⋊C8—22no—
307C4⋊C8—22no—
308C4⋊C8—21no—
309C4⋊C8C23×C5312yes—
310C4⋊C8C23×C5312yes—
311C4⋊C8C23×C5312yes—
312#103—22no—
313#103—22no—
314#103—21no—
315#103—21no—
316#103—21no—
317#103—21no—
318#103—22no—
319#102—21no—
320#102—21no—
321#102—21no—
322#102—21no—
323#102—22no—
324C22⋊C8—21no—
325C22⋊C8—21no—
326C4×C8—22no—
327C4×C8—22no—
328C4×C8—21no—
329C4×C8—21no—
330C4×C8D4×C5311yescharacteristic
331C4×C8D4×C5311yescharacteristic
332C4×C8C23×C5311yescharacteristic
333C4×C8C23×C5311yescharacteristic
334#41—42no—
335#41—42no—
336C2×Q8—24no—
337C2×Q8—21no—
338C2×Q8—21no—
339C2×Q8—21no—
340C2×Q8—21no—
341C2×Q8C2×D4×C5312yes—
342C2×D4—21no—
343C2×D4—21no—
344C2×D4—21no—
345C2×D4—41no—
346C22×C4—21no—
347C22×C4—21no—
348C22×C4—41no—
349C22×C4C2×D4×C5311yescharacteristic
350C22×C4C2×D4×C5311yescharacteristic
351C22×C4#41×C5311yescharacteristic
352Q16—44no—
353SD16—44no—
354C2×C8—22no—
355C2×C8—22no—
356C2×C8—22no—
357C2×C8—22no—
358C2×C8—22no—
359C2×C8—42no—
360C2×C8C2×D4×C5311yescharacteristic
361C2×C8C2×D4×C5311yescharacteristic
362C2×C8C2×D4×C5311yescharacteristic
363C2×C8C2×D4×C5311yescharacteristic
364C4⋊C4—22no—
365C4⋊C4—21no—
366C4⋊C4—22no—
367C4⋊C4—21no—
368C4⋊C4—21no—
369C4⋊C4—21no—
370C4⋊C4—21no—
371C4⋊C4—21no—
372C4⋊C4—21no—
373C4⋊C4—41no—
374C4⋊C4—41no—
375C4⋊C4C2×D4×C5312yes—
376C22⋊C4—41no—
377C22⋊C4—41no—
378C22⋊C4—41no—
379C42—22no—
380C42—22no—
381C42—41no—
382C42C2×D4×C5311yescharacteristic
383C42C2×D4×C5311yescharacteristic
384C42#41×C5311yescharacteristic
385C42C24×C5311yescharacteristic
386C23—41no—
387C23#121×C5311yescharacteristic
388Q8—24no—
389Q8—22no—
390Q8—42no—
391Q8—41no—
392Q8—41no—
393Q8—81no—
394Q8#135×C5314yes—
395D4—42no—
396D4—41no—
397D4—41no—
398D4—81no—
399D4—82no—
400C2×C4—24no—
401C2×C4—22no—
402C2×C4—22no—
403C2×C4—22no—
404C2×C4—41no—
405C2×C4—42no—
406C2×C4—41no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—82no—
410C2×C4#121×C5311yescharacteristic
411C2×C4#121×C5311yescharacteristic
412C2×C4#121×C5311yescharacteristic
413C2×C4#135×C5312yes—
414C2×C4#137×C5312yes—
415C2×C4C22×D4×C5311yescharacteristic
416C2×C4C22×D4×C5311yescharacteristic
417C2×C4C2×#41×C5311yescharacteristic
418C8—21no—
419C8—21no—
420C8—21no—
421C8—21no—
422C8—42no—
423C8—42no—
424C8—42no—
425C22—21no—
426C22—21no—
427C22—81no—
428C22—81no—
429C22—81no—
430C22C53×#46412yes—
431C22C2×#121×C5311yescharacteristic
432C4—24no—
433C4—21no—
434C4—21no—
435C4—21no—
436C4—42no—
437C4—81no—
438C4—81no—
439C4—81no—
440C4C53×#46412yes—
441C4C2×#135×C5311yescharacteristic
442C4C2×#135×C5312yes—
443C4C2×#137×C5311yescharacteristic
444C2—22no—
445C2—161no—
446C2C2×C53×#46411yescharacteristic
447C2C53×#305011yescharacteristic
448C2C53×#306711yescharacteristic
subgroupG/H
centerC2×C4×C53#121
commutator, Frattini, cosocleC42C24×C53
socleC22×C53C2×#121
prime radicalC53×#532C22
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C53 ◁ C2×C42×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#3104C2semidirect
4. C2×C8×Q8×C53C2semidirect
5. C53×#1684C2semidirect
6. C53×#1682C2semidirect
7. C53×#1680C2semidirect
8. C53×#1678C2semidirect
9. C53×#1564C2semidirect
10. C53×#1542C2semidirect
200. #19123C53direct (N × H)
221. #3104C2×C53semidirect
223. C2×C8×Q8C2×C53semidirect
224. #1684C2×C53semidirect
225. #1682C2×C53semidirect
226. #1680C2×C53semidirect
227. #1678C2×C53semidirect
228. #1564C2×C53semidirect
229. #1542C2×C53semidirect
273. C53#19123direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#31312
C53×#31042
C53×#31032
C2×C8×Q8×C532
C53×#16842
C53×#16822
C53×#16802
C53×#16782
C53×#15642
C53×#15422
C53×#15402
#1912353