Finite Group Databasebeta

← C53×#11691 #27219203

Subgroups — C53×#11691 #27219203

Order 13568.

552 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#2392C211yesmaximal, characteristic
2C53×#1740C211yesmaximal, characteristic, complemented
3C53×#1458C211yesmaximal, characteristic, complemented
4C53×#1344C212yesmaximal, complemented
5C53×#1343C212yesmaximal
6C2×C4×D4×C53—21no—
7C2×C4×D4×C53C411yescharacteristic, complemented
8C2×#117×C53—21no—
9C2×#117×C53C2211yescharacteristic
10C53×#431—21no—
11C53×#398C411yescharacteristic, complemented
12C53×#397C2211yescharacteristic
13C53×#396—21no—
14C53×#394—21no—
15C53×#394C411yescharacteristic, complemented
16C53×#392C2211yescharacteristic
17C4×C4⋊C4×C53C411yescharacteristic, complemented
18C4×C22⋊C4×C53—21no—
19C53×#342—21no—
20C53×#342C2212yes—
21C53×#324—21no—
22C53×#323C2212yes—
23C22×D4×C53—21no—
24C22×D4×C53C2×C411yescharacteristic
25C23×C4×C53—21no—
26C23×C4×C53D411yescharacteristic
27C4×D4×C53—42no—
28C4×D4×C53—42no—
29C4×D4×C53—41no—
30C4×D4×C53—41no—
31#117×C53—24no—
32#117×C53—41no—
33#117×C53—41no—
34#117×C53—42no—
35C2×C4⋊C4×C53—21no—
36C2×C4⋊C4×C53—21no—
37C2×C4⋊C4×C53—41no—
38C2×C4⋊C4×C53C2×C411yescharacteristic
39C2×C4⋊C4×C53C2×C411yescharacteristic
40C2×C4⋊C4×C53D411yescharacteristic
41C2×C22⋊C4×C53—21no—
42C2×C22⋊C4×C53—21no—
43C2×C22⋊C4×C53—21no—
44C2×C22⋊C4×C53—21no—
45C2×C22⋊C4×C53—21no—
46C2×C22⋊C4×C53—22no—
47C2×C22⋊C4×C53D411yescharacteristic
48C2×C22⋊C4×C53C2311yescharacteristic
49C2×C42×C53—21no—
50C2×C42×C53—41no—
51C2×C42×C53C2×C411yescharacteristic
52C2×C42×C53D411yescharacteristic
53C4⋊C8×C53—41no—
54C22⋊C8×C53—22no—
55C4×C8×C53—41no—
56#95×C53—21no—
57#95×C53—21no—
58#95×C53C2×C411yescharacteristic
59#95×C53C2×C411yescharacteristic
60C24×C53—21no—
61C24×C53C2×D411yescharacteristic
62C2×D4×C53—42no—
63C2×D4×C53—41no—
64C2×D4×C53—42no—
65C2×D4×C53—41no—
66C2×D4×C53—41no—
67C2×D4×C53—42no—
68C22×C4×C53—21no—
69C22×C4×C53—21no—
70C22×C4×C53—22no—
71C22×C4×C53—21no—
72C22×C4×C53—21no—
73C22×C4×C53—21no—
74C22×C4×C53—42no—
75C22×C4×C53—41no—
76C22×C4×C53—41no—
77C22×C4×C53—42no—
78C22×C4×C53—42no—
79C22×C4×C53C22⋊C411yescharacteristic
80C22×C4×C53C22⋊C411yescharacteristic
81C22×C4×C53C22⋊C411yescharacteristic
82C22×C4×C53C22⋊C411yescharacteristic
83C22×C4×C53D812yes—
84C22×C4×C53SD1612yes—
85C22×C4×C53C22×C411yescharacteristic
86C22×C4×C53C2×D411yescharacteristic
87C2×C8×C53—42no—
88C4⋊C4×C53—41no—
89C4⋊C4×C53—41no—
90C4⋊C4×C53—42no—
91C4⋊C4×C53—44no—
92C4⋊C4×C53—41no—
93C4⋊C4×C53—42no—
94C22⋊C4×C53—24no—
95C22⋊C4×C53—41no—
96C22⋊C4×C53—41no—
97C22⋊C4×C53—42no—
98C22⋊C4×C53—42no—
99C22⋊C4×C53—42no—
100C22⋊C4×C53—42no—
101C22⋊C4×C53—41no—
102C22⋊C4×C53—41no—
103C22⋊C4×C53—42no—
104C42×C53—21no—
105C42×C53—21no—
106C42×C53—21no—
107C42×C53—21no—
108C42×C53—44no—
109C42×C53—42no—
110C23×C53—21no—
111C23×C53—21no—
112C23×C53—22no—
113C23×C53—41no—
114C23×C53—41no—
115C23×C53—41no—
116C23×C53—41no—
117C23×C53—41no—
118C23×C53—41no—
119C23×C53—41no—
120C23×C53—41no—
121C23×C53C2×C22⋊C411yescharacteristic
122C23×C53C2×D811yescharacteristic
123C23×C53C2×SD1611yescharacteristic
124D4×C53—42no—
125D4×C53—44no—
126D4×C53—44no—
127D4×C53—42no—
128C2×C4×C53—21no—
129C2×C4×C53—21no—
130C2×C4×C53—22no—
131C2×C4×C53—22no—
132C2×C4×C53—22no—
133C2×C4×C53—22no—
134C2×C4×C53—22no—
135C2×C4×C53—21no—
136C2×C4×C53—21no—
137C2×C4×C53—21no—
138C2×C4×C53—21no—
139C2×C4×C53—41no—
140C2×C4×C53—41no—
141C2×C4×C53—41no—
142C2×C4×C53—41no—
143C2×C4×C53—41no—
144C2×C4×C53—41no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53—44no—
148C2×C4×C53—41no—
149C2×C4×C53—41no—
150C2×C4×C53—41no—
151C2×C4×C53—41no—
152C2×C4×C53—82no—
153C2×C4×C53—82no—
154C2×C4×C53—82no—
155C2×C4×C53—82no—
156C2×C4×C53#9911yescharacteristic
157C2×C4×C53#9911yescharacteristic
158C2×C4×C53#10212yes—
159C2×C4×C53#10212yes—
160C2×C4×C53#10412yes—
161C8×C53—82no—
162#11691C5311yesmaximal, characteristic, direct product factor, Sylow
163C22×C53—21no—
164C22×C53—21no—
165C22×C53—21no—
166C22×C53—21no—
167C22×C53—41no—
168C22×C53—41no—
169C22×C53—42no—
170C22×C53—42no—
171C22×C53—41no—
172C22×C53—41no—
173C22×C53—42no—
174C22×C53—81no—
175C22×C53—81no—
176C22×C53—81no—
177C22×C53—81no—
178C22×C53—81no—
179C22×C53C2×#9911yescharacteristic
180C22×C53C2×#10211yescharacteristic
181C22×C53C2×#10411yescharacteristic
182C4×C53—22no—
183C4×C53—42no—
184C4×C53—41no—
185C4×C53—41no—
186C4×C53—41no—
187C4×C53—41no—
188C4×C53—81no—
189C4×C53—84no—
190C4×C53—81no—
191C4×C53#32712yes—
192C4×C53#32712yes—
193#2392C2×C5311yescharacteristic
194#1740C2×C5311yescharacteristic, complemented
195#1458C2×C5311yescharacteristic, complemented
196#1344C2×C5312yescomplemented
197#1343C2×C5312yes—
198C2×C53—21no—
199C2×C53—21no—
200C2×C53—41no—
201C2×C53—41no—
202C2×C53—82no—
203C2×C53C2×#32711yescharacteristic
204C2×C53#212011yescharacteristic
205C2×C53#212611yescharacteristic
206C2×C4×D4—21no—
207C2×C4×D4C4×C5311yescharacteristic, complemented
208C2×#117—21no—
209C2×#117C22×C5311yescharacteristic
210#431—21no—
211#398C4×C5311yescharacteristic, complemented
212#397C22×C5311yescharacteristic
213#396—21no—
214#394—21no—
215#394C4×C5311yescharacteristic, complemented
216#392C22×C5311yescharacteristic
217C4×C4⋊C4C4×C5311yescharacteristic, complemented
218C4×C22⋊C4—21no—
219#342—21no—
220#342C22×C5312yes—
221#324—21no—
222#323C22×C5312yes—
223C53#1169111yescharacteristic, direct product factor, Sylow
224C22×D4—21no—
225C22×D4C2×C4×C5311yescharacteristic
226C23×C4—21no—
227C23×C4D4×C5311yescharacteristic
228C4×D4—41no—
229C4×D4—41no—
230C4×D4—42no—
231C4×D4—42no—
232#117—24no—
233#117—41no—
234#117—41no—
235#117—42no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—41no—
239C2×C4⋊C4C2×C4×C5311yescharacteristic
240C2×C4⋊C4C2×C4×C5311yescharacteristic
241C2×C4⋊C4D4×C5311yescharacteristic
242C2×C22⋊C4—22no—
243C2×C22⋊C4—21no—
244C2×C22⋊C4—21no—
245C2×C22⋊C4—21no—
246C2×C22⋊C4—21no—
247C2×C22⋊C4—21no—
248C2×C22⋊C4D4×C5311yescharacteristic
249C2×C22⋊C4C23×C5311yescharacteristic
250C2×C42—21no—
251C2×C42—41no—
252C2×C42C2×C4×C5311yescharacteristic
253C2×C42D4×C5311yescharacteristic
254C4⋊C8—41no—
255C22⋊C8—22no—
256C4×C8—41no—
257#95—21no—
258#95—21no—
259#95C2×C4×C5311yescharacteristic
260#95C2×C4×C5311yescharacteristic
261C24—21no—
262C24C2×D4×C5311yescharacteristic
263C2×D4—41no—
264C2×D4—42no—
265C2×D4—42no—
266C2×D4—41no—
267C2×D4—41no—
268C2×D4—42no—
269C22×C4—21no—
270C22×C4—21no—
271C22×C4—22no—
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—42no—
276C22×C4—41no—
277C22×C4—41no—
278C22×C4—42no—
279C22×C4—42no—
280C22×C4C22⋊C4×C5311yescharacteristic
281C22×C4C22⋊C4×C5311yescharacteristic
282C22×C4C22⋊C4×C5311yescharacteristic
283C22×C4C22⋊C4×C5311yescharacteristic
284C22×C4D8×C5312yes—
285C22×C4SD16×C5312yes—
286C22×C4C22×C4×C5311yescharacteristic
287C22×C4C2×D4×C5311yescharacteristic
288C2×C8—42no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—41no—
292C4⋊C4—42no—
293C4⋊C4—44no—
294C4⋊C4—42no—
295C22⋊C4—24no—
296C22⋊C4—41no—
297C22⋊C4—41no—
298C22⋊C4—42no—
299C22⋊C4—42no—
300C22⋊C4—42no—
301C22⋊C4—42no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—42no—
305C42—21no—
306C42—21no—
307C42—21no—
308C42—21no—
309C42—44no—
310C42—42no—
311C23—21no—
312C23—21no—
313C23—22no—
314C23—41no—
315C23—41no—
316C23—41no—
317C23—41no—
318C23—41no—
319C23—41no—
320C23—41no—
321C23—41no—
322C23C2×C22⋊C4×C5311yescharacteristic
323C23C2×D8×C5311yescharacteristic
324C23C2×SD16×C5311yescharacteristic
325D4—42no—
326D4—44no—
327D4—44no—
328D4—42no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—22no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—41no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—41no—
346C2×C4—44no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—82no—
354C2×C4—82no—
355C2×C4—82no—
356C2×C4—82no—
357C2×C4#99×C5311yescharacteristic
358C2×C4#99×C5311yescharacteristic
359C2×C4#102×C5312yes—
360C2×C4#102×C5312yes—
361C2×C4#104×C5312yes—
362C8—82no—
363C22—21no—
364C22—21no—
365C22—21no—
366C22—21no—
367C22—41no—
368C22—41no—
369C22—42no—
370C22—42no—
371C22—41no—
372C22—41no—
373C22—42no—
374C22—81no—
375C22—81no—
376C22—81no—
377C22—81no—
378C22—81no—
379C22C2×#99×C5311yescharacteristic
380C22C2×#102×C5311yescharacteristic
381C22C2×#104×C5311yescharacteristic
382C4—22no—
383C4—42no—
384C4—41no—
385C4—41no—
386C4—41no—
387C4—41no—
388C4—81no—
389C4—84no—
390C4—81no—
391C4C53×#32712yes—
392C4C53×#32712yes—
393C2—21no—
394C2—21no—
395C2—41no—
396C2—41no—
397C2—82no—
398C2C2×C53×#32711yescharacteristic
399C2C53×#212011yescharacteristic
400C2C53×#212611yescharacteristic
subgroupG/H
centerC2×C4×C53#99
commutatorC22×C4C22×C4×C53
Frattini, cosocleC2×C22⋊C4C23×C53
socleC22×C53C2×#99
prime radicalC53×#392C22
LCSG ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C53 ◁ C22×C4×C53 ◁ C2×#117×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#1740C2semidirect
3. C53×#1458C2semidirect
4. C53×#1344C2semidirect
7. C2×C4×D4×C53C4semidirect
11. C53×#398C4semidirect
15. C53×#394C4semidirect
17. C4×C4⋊C4×C53C4semidirect
162. #11691C53direct (N × H)
194. #1740C2×C53semidirect
195. #1458C2×C53semidirect
196. #1344C2×C53semidirect
207. C2×C4×D4C4×C53semidirect
211. #398C4×C53semidirect
215. #394C4×C53semidirect
217. C4×C4⋊C4C4×C53semidirect
223. C53#11691direct (N × H)

Maximal subgroups

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

subgroupindex
C53×#23922
C53×#17402
C53×#14582
C53×#13442
C53×#13432
#1169153