Finite Group Databasebeta

← C53×#19431 #27226943

Subgroups — C53×#19431 #27226943

Order 13568.

604 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3231C211yesmaximal, characteristic, complemented
2C53×#3110C211yesmaximal, characteristic
3C53×#3101C211yesmaximal, characteristic
4C53×#2975C211yesmaximal, characteristic
5C53×#1703C212yesmaximal, complemented
6C53×#1699C212yesmaximal, complemented
7C53×#1693C212yesmaximal, complemented
8C53×#1691C212yesmaximal, complemented
9C53×#1572C211yesmaximal, characteristic, complemented
10C53×#1554C211yesmaximal, characteristic, complemented
11C53×#1544C211yesmaximal, characteristic, complemented
12C53×#533C2211yescharacteristic
13C53×#533C2211yescharacteristic
14C2×C4×D4×C53C2211yescharacteristic
15C53×#500C2211yescharacteristic, complemented
16C53×#498C2211yescharacteristic, complemented
17C53×#487C2211yescharacteristic, complemented
18C53×#487C2211yescharacteristic, complemented
19C53×#485—22no—
20C53×#484—21no—
21C53×#483—21no—
22C53×#479—22no—
23C53×#476C2212yes—
24C53×#474C2212yes—
25C53×#461C2212yes—
26C53×#461C2212yes—
27C53×#450—21no—
28C53×#450—21no—
29C53×#440—22no—
30C4×SD16×C53—22no—
31C53×#436—22no—
32C53×#436C2212yes—
33C53×#436C2212yes—
34C53×#432C2211yescharacteristic
35C53×#431C2211yescharacteristic
36C53×#431C2211yescharacteristic
37C53×#431C2211yescharacteristic
38C53×#407—22no—
39C53×#335C2212yescomplemented
40C53×#334C2212yescomplemented
41C53×#333C2212yescomplemented
42C53×#332C2212yescomplemented
43C53×#332C2212yescomplemented
44C53×#332C2212yescomplemented
45C22×D4×C53D411yescharacteristic
46C23×C4×C53—21no—
47C2×SD16×C53—22no—
48C2×SD16×C53—22no—
49C2×M16×C53—22no—
50C2×M16×C53—22no—
51C4⋊Q8×C53C2311yescharacteristic
52C4⋊Q8×C53C2311yescharacteristic
53C4⋊Q8×C53C2311yescharacteristic
54C4⋊Q8×C53C2311yescharacteristic
55#125×C53—21no—
56#125×C53—21no—
57C22⋊Q8×C53—21no—
58C22⋊Q8×C53—21no—
59C22⋊Q8×C53—21no—
60C22⋊Q8×C53—21no—
61C4×Q8×C53—21no—
62C4×Q8×C53—21no—
63C4×Q8×C53—21no—
64C4×Q8×C53—21no—
65C4×D4×C53—21no—
66C4×D4×C53—21no—
67C4×D4×C53D412yes—
68C4×D4×C53C2312yes—
69#117×C53—21no—
70#117×C53—21no—
71C2×C4⋊C4×C53D411yescharacteristic
72C2×C22⋊C4×C53—21no—
73C2×C42×C53C2311yescharacteristic
74#107×C53—22no—
75#107×C53—22no—
76#106×C53—22no—
77#106×C53—22no—
78C4⋊C8×C53D412yes—
79C4⋊C8×C53C2312yes—
80C4⋊C8×C53C2312yes—
81C4⋊C8×C53C2312yes—
82#103×C53—22no—
83#103×C53—22no—
84#103×C53—22no—
85#103×C53—22no—
86#102×C53—22no—
87#102×C53—22no—
88C22⋊C8×C53—22no—
89C22⋊C8×C53—22no—
90C22⋊C8×C53—21no—
91C22⋊C8×C53—21no—
92C22⋊C8×C53—21no—
93C22⋊C8×C53D412yes—
94#97×C53—21no—
95C4×C8×C53—21no—
96C4×C8×C53—21no—
97C4×C8×C53C2311yescharacteristic
98C4×C8×C53C2311yescharacteristic
99C24×C53—21no—
100C2×Q8×C53—21no—
101C2×Q8×C53—21no—
102C2×Q8×C53—21no—
103C2×Q8×C53—21no—
104C2×D4×C53—21no—
105C2×D4×C53—42no—
106C2×D4×C53C2×D412yes—
107C22×C4×C53—21no—
108C22×C4×C53—22no—
109C22×C4×C53—22no—
110C22×C4×C53—82no—
111C22×C4×C53C2×D411yescharacteristic
112C22×C4×C53C2×D411yescharacteristic
113C22×C4×C53#4111yescharacteristic
114SD16×C53—44no—
115SD16×C53—44no—
116M16×C53—82no—
117M16×C53—82no—
118C2×C8×C53—22no—
119C2×C8×C53—22no—
120C2×C8×C53—22no—
121C2×C8×C53C2×D412yes—
122C2×C8×C53C2×D412yes—
123C4⋊C4×C53—21no—
124C4⋊C4×C53—21no—
125C4⋊C4×C53—21no—
126C4⋊C4×C53—21no—
127C4⋊C4×C53—21no—
128C4⋊C4×C53—21no—
129C4⋊C4×C53—21no—
130C4⋊C4×C53—21no—
131C4⋊C4×C53—21no—
132C4⋊C4×C53—41no—
133C4⋊C4×C53—41no—
134C4⋊C4×C53—41no—
135C4⋊C4×C53—41no—
136C4⋊C4×C53C2×D412yes—
137C22⋊C4×C53—22no—
138C22⋊C4×C53—22no—
139C22⋊C4×C53—41no—
140C22⋊C4×C53—41no—
141C42×C53—41no—
142C42×C53—41no—
143C42×C53C2×D411yescharacteristic
144C42×C53C2×D411yescharacteristic
145C42×C53#4111yescharacteristic
146C42×C53C2411yescharacteristic
147C23×C53—22no—
148C23×C53—42no—
149C23×C53—81no—
150C23×C53—81no—
151C23×C53#12111yescharacteristic
152Q8×C53—42no—
153Q8×C53—42no—
154Q8×C53—42no—
155Q8×C53—81no—
156D4×C53—22no—
157D4×C53—22no—
158D4×C53—42no—
159C2×C4×C53—22no—
160C2×C4×C53—22no—
161C2×C4×C53—22no—
162C2×C4×C53—41no—
163C2×C4×C53—42no—
164C2×C4×C53—42no—
165C2×C4×C53—41no—
166C2×C4×C53—41no—
167C2×C4×C53—41no—
168C2×C4×C53—41no—
169C2×C4×C53—82no—
170C2×C4×C53—81no—
171C2×C4×C53#12111yescharacteristic
172C2×C4×C53#12111yescharacteristic
173C2×C4×C53#12111yescharacteristic
174C2×C4×C53C22×D411yescharacteristic
175C2×C4×C53C22×D411yescharacteristic
176C2×C4×C53C2×#4111yescharacteristic
177C8×C53—22no—
178C8×C53—22no—
179C8×C53—42no—
180C8×C53—42no—
181C8×C53—42no—
182#19431C5311yesmaximal, characteristic, direct product factor, Sylow
183C22×C53—21no—
184C22×C53—21no—
185C22×C53—21no—
186C22×C53—42no—
187C22×C53—42no—
188C22×C53—42no—
189C22×C53—84no—
190C22×C53C2×#12111yescharacteristic
191C4×C53—21no—
192C4×C53—21no—
193C4×C53—21no—
194C4×C53—21no—
195C4×C53—42no—
196C4×C53—81no—
197C4×C53—81no—
198C4×C53—81no—
199C4×C53—81no—
200C4×C53#57512yes—
201C4×C53#57712yes—
202#3231C2×C5311yescharacteristic, complemented
203#3110C2×C5311yescharacteristic
204#3101C2×C5311yescharacteristic
205#2975C2×C5311yescharacteristic
206#1703C2×C5312yescomplemented
207#1699C2×C5312yescomplemented
208#1693C2×C5312yescomplemented
209#1691C2×C5312yescomplemented
210#1572C2×C5311yescharacteristic, complemented
211#1554C2×C5311yescharacteristic, complemented
212#1544C2×C5311yescharacteristic, complemented
213C2×C53—41no—
214C2×C53—82no—
215C2×C53#304211yescharacteristic
216C2×C53#305411yescharacteristic
217C2×C53#306911yescharacteristic
218#533C22×C5311yescharacteristic
219#533C22×C5311yescharacteristic
220C2×C4×D4C22×C5311yescharacteristic
221#500C22×C5311yescharacteristic, complemented
222#498C22×C5311yescharacteristic, complemented
223#487C22×C5311yescharacteristic, complemented
224#487C22×C5311yescharacteristic, complemented
225#485—22no—
226#484—21no—
227#483—21no—
228#479—22no—
229#476C22×C5312yes—
230#474C22×C5312yes—
231#461C22×C5312yes—
232#461C22×C5312yes—
233#450—21no—
234#450—21no—
235#440—22no—
236C4×SD16—22no—
237#436—22no—
238#436C22×C5312yes—
239#436C22×C5312yes—
240#432C22×C5311yescharacteristic
241#431C22×C5311yescharacteristic
242#431C22×C5311yescharacteristic
243#431C22×C5311yescharacteristic
244#407—22no—
245#335C22×C5312yescomplemented
246#334C22×C5312yescomplemented
247#333C22×C5312yescomplemented
248#332C22×C5312yescomplemented
249#332C22×C5312yescomplemented
250#332C22×C5312yescomplemented
251C53#1943111yescharacteristic, direct product factor, Sylow
252C22×D4D4×C5311yescharacteristic
253C23×C4—21no—
254C2×SD16—22no—
255C2×SD16—22no—
256C2×M16—22no—
257C2×M16—22no—
258C4⋊Q8C23×C5311yescharacteristic
259C4⋊Q8C23×C5311yescharacteristic
260C4⋊Q8C23×C5311yescharacteristic
261C4⋊Q8C23×C5311yescharacteristic
262#125—21no—
263#125—21no—
264C22⋊Q8—21no—
265C22⋊Q8—21no—
266C22⋊Q8—21no—
267C22⋊Q8—21no—
268C4×Q8—21no—
269C4×Q8—21no—
270C4×Q8—21no—
271C4×Q8—21no—
272C4×D4—21no—
273C4×D4—21no—
274C4×D4D4×C5312yes—
275C4×D4C23×C5312yes—
276#117—21no—
277#117—21no—
278C2×C4⋊C4D4×C5311yescharacteristic
279C2×C22⋊C4—21no—
280C2×C42C23×C5311yescharacteristic
281#107—22no—
282#107—22no—
283#106—22no—
284#106—22no—
285C4⋊C8D4×C5312yes—
286C4⋊C8C23×C5312yes—
287C4⋊C8C23×C5312yes—
288C4⋊C8C23×C5312yes—
289#103—22no—
290#103—22no—
291#103—22no—
292#103—22no—
293#102—22no—
294#102—22no—
295C22⋊C8—21no—
296C22⋊C8—21no—
297C22⋊C8—21no—
298C22⋊C8—22no—
299C22⋊C8—22no—
300C22⋊C8D4×C5312yes—
301#97—21no—
302C4×C8—21no—
303C4×C8—21no—
304C4×C8C23×C5311yescharacteristic
305C4×C8C23×C5311yescharacteristic
306C24—21no—
307C2×Q8—21no—
308C2×Q8—21no—
309C2×Q8—21no—
310C2×Q8—21no—
311C2×D4—21no—
312C2×D4—42no—
313C2×D4C2×D4×C5312yes—
314C22×C4—21no—
315C22×C4—22no—
316C22×C4—22no—
317C22×C4—82no—
318C22×C4C2×D4×C5311yescharacteristic
319C22×C4C2×D4×C5311yescharacteristic
320C22×C4#41×C5311yescharacteristic
321SD16—44no—
322SD16—44no—
323M16—82no—
324M16—82no—
325C2×C8—22no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8C2×D4×C5312yes—
329C2×C8C2×D4×C5312yes—
330C4⋊C4—21no—
331C4⋊C4—21no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—21no—
336C4⋊C4—21no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—41no—
340C4⋊C4—41no—
341C4⋊C4—41no—
342C4⋊C4—41no—
343C4⋊C4C2×D4×C5312yes—
344C22⋊C4—22no—
345C22⋊C4—22no—
346C22⋊C4—41no—
347C22⋊C4—41no—
348C42—41no—
349C42—41no—
350C42C2×D4×C5311yescharacteristic
351C42C2×D4×C5311yescharacteristic
352C42#41×C5311yescharacteristic
353C42C24×C5311yescharacteristic
354C23—22no—
355C23—42no—
356C23—81no—
357C23—81no—
358C23#121×C5311yescharacteristic
359Q8—42no—
360Q8—42no—
361Q8—42no—
362Q8—81no—
363D4—22no—
364D4—22no—
365D4—42no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—22no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—82no—
377C2×C4—81no—
378C2×C4#121×C5311yescharacteristic
379C2×C4#121×C5311yescharacteristic
380C2×C4#121×C5311yescharacteristic
381C2×C4C22×D4×C5311yescharacteristic
382C2×C4C22×D4×C5311yescharacteristic
383C2×C4C2×#41×C5311yescharacteristic
384C8—22no—
385C8—22no—
386C8—42no—
387C8—42no—
388C8—42no—
389C22—21no—
390C22—21no—
391C22—21no—
392C22—42no—
393C22—42no—
394C22—42no—
395C22—84no—
396C22C2×#121×C5311yescharacteristic
397C4—21no—
398C4—21no—
399C4—21no—
400C4—21no—
401C4—42no—
402C4—81no—
403C4—81no—
404C4—81no—
405C4—81no—
406C4C53×#57512yes—
407C4C53×#57712yes—
408C2—41no—
409C2—82no—
410C2C53×#304211yescharacteristic
411C2C53×#305411yescharacteristic
412C2C53×#306911yescharacteristic
subgroupG/H
center, socleC22×C53C2×#121
commutator, Frattini, cosocleC42C24×C53
prime radicalC22×D4×C53D4
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G

Complements

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

NH ≅ G/Ntype
1. C53×#3231C2semidirect
5. C53×#1703C2semidirect
6. C53×#1699C2semidirect
7. C53×#1693C2semidirect
8. C53×#1691C2semidirect
9. C53×#1572C2semidirect
10. C53×#1554C2semidirect
11. C53×#1544C2semidirect
15. C53×#500C22semidirect
16. C53×#498C22semidirect
17. C53×#487C22semidirect
18. C53×#487C22semidirect
39. C53×#335C22semidirect
40. C53×#334C22semidirect
41. C53×#333C22semidirect
42. C53×#332C22semidirect
43. C53×#332C22semidirect
44. C53×#332C22semidirect
182. #19431C53direct (N × H)
202. #3231C2×C53semidirect
206. #1703C2×C53semidirect
207. #1699C2×C53semidirect
208. #1693C2×C53semidirect
209. #1691C2×C53semidirect
210. #1572C2×C53semidirect
211. #1554C2×C53semidirect
212. #1544C2×C53semidirect
221. #500C22×C53semidirect
222. #498C22×C53semidirect
223. #487C22×C53semidirect
224. #487C22×C53semidirect
245. #335C22×C53semidirect
246. #334C22×C53semidirect
247. #333C22×C53semidirect
248. #332C22×C53semidirect
249. #332C22×C53semidirect
250. #332C22×C53semidirect
251. C53#19431direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#32312
C53×#31102
C53×#31012
C53×#29752
C53×#17032
C53×#16992
C53×#16932
C53×#16912
C53×#15722
C53×#15542
C53×#15442
#1943153