Finite Group Databasebeta

← C53×#19397 #27226909

Subgroups — C53×#19397 #27226909

Order 13568.

656 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C53×#3206C211yesmaximal, characteristic
2C2×C53×#475C211yesmaximal, characteristic, complemented
3C53×#3037C211yesmaximal, characteristic
4C53×#2982C211yesmaximal, characteristic, complemented
5C53×#1680C214yesmaximal, complemented
6C53×#1666C214yesmaximal, complemented
7C53×#1570C211yesmaximal, characteristic, complemented
8C53×#1549C211yesmaximal, characteristic, complemented
9C53×#1540C211yesmaximal, characteristic
10C2×#137×C53—21no—
11C53×#532C2211yescharacteristic
12C2×C4⋊Q8×C53C2211yescharacteristic
13C2×C4×Q8×C53C2211yescharacteristic
14C53×#492C2212yes—
15C53×#487C2212yes—
16C53×#475—22no—
17C53×#475C2214yes—
18C53×#464—22no—
19C53×#462C2212yes—
20C53×#461—22no—
21C53×#460C2212yes—
22C53×#448—22no—
23C8×Q8×C53—22no—
24C8×Q8×C53C2214yes—
25C53×#431—22no—
26C53×#431C2211yescharacteristic
27C53×#431C2211yescharacteristic
28C2×#106×C53—21no—
29C53×#423C2211yescharacteristic
30C2×C4⋊C8×C53C2211yescharacteristic
31C53×#417—21no—
32C2×#103×C53—21no—
33C53×#334C2214yes—
34C53×#332C2214yes—
35C53×#332C2212yes—
36C53×#331C2212yes—
37C2×#41×C53—21no—
38C22×Q8×C53—21no—
39C22×Q8×C53D411yescharacteristic
40#137×C53—44no—
41C2×Q16×C53—22no—
42C2×SD16×C53—24no—
43C2×SD16×C53—22no—
44C2×M16×C53—21no—
45C22×C8×C53—21no—
46C4⋊Q8×C53—21no—
47C4⋊Q8×C53—21no—
48C4⋊Q8×C53—21no—
49C4⋊Q8×C53C2311yescharacteristic
50C4⋊Q8×C53C2312yes—
51#127×C53C2311yescharacteristic
52#124×C53—21no—
53#121×C53—21no—
54#121×C53—21no—
55C4×Q8×C53—21no—
56C4×Q8×C53—21no—
57C4×Q8×C53D412yes—
58C4×Q8×C53C2312yes—
59C4×D4×C53—21no—
60C4×D4×C53—21no—
61C2×C4⋊C4×C53—21no—
62C2×C4⋊C4×C53—21no—
63C2×C4⋊C4×C53—21no—
64C2×C4⋊C4×C53D411yescharacteristic
65C2×C42×C53—21no—
66C2×C42×C53C2311yescharacteristic
67#106×C53—24no—
68C4⋊C8×C53—24no—
69C4⋊C8×C53—21no—
70C4⋊C8×C53—21no—
71C4⋊C8×C53D412yes—
72C4⋊C8×C53C2312yes—
73C4⋊C8×C53C2312yes—
74C4⋊C8×C53C2312yes—
75#103×C53—24no—
76#103×C53—22no—
77#103×C53—24no—
78#102×C53—22no—
79C22⋊C8×C53—21no—
80C22⋊C8×C53D412yes—
81C4×C8×C53—24no—
82C4×C8×C53—21no—
83C4×C8×C53C2312yes—
84#41×C53—42no—
85#41×C53—42no—
86C2×Q8×C53—21no—
87C2×Q8×C53—22no—
88C2×Q8×C53—21no—
89C2×Q8×C53—22no—
90C2×Q8×C53—41no—
91C2×Q8×C53—82no—
92C2×Q8×C53SD1614yescomplemented
93C2×Q8×C53C2×D412yes—
94C2×D4×C53—21no—
95C2×D4×C53—21no—
96C2×D4×C53—21no—
97C2×D4×C53—41no—
98C22×C4×C53—21no—
99C22×C4×C53—21no—
100C22×C4×C53—41no—
101C22×C4×C53—41no—
102C22×C4×C53C2×D411yescharacteristic
103C22×C4×C53C2×D411yescharacteristic
104C22×C4×C53#4111yescharacteristic
105Q16×C53—44no—
106SD16×C53—44no—
107SD16×C53—84no—
108M16×C53—42no—
109C2×C8×C53—22no—
110C2×C8×C53—22no—
111C2×C8×C53—22no—
112C2×C8×C53—42no—
113C2×C8×C53C2×D414yes—
114C4⋊C4×C53—22no—
115C4⋊C4×C53—21no—
116C4⋊C4×C53—22no—
117C4⋊C4×C53—21no—
118C4⋊C4×C53—22no—
119C4⋊C4×C53—22no—
120C4⋊C4×C53—21no—
121C4⋊C4×C53—41no—
122C4⋊C4×C53—41no—
123C4⋊C4×C53C2×D412yes—
124C22⋊C4×C53—41no—
125C22⋊C4×C53—41no—
126C42×C53—22no—
127C42×C53—22no—
128C42×C53C2×D411yescharacteristic
129C42×C53C2×D411yescharacteristic
130C42×C53#4111yescharacteristic
131C42×C53C2411yescharacteristic
132C23×C53—41no—
133C23×C53#12111yescharacteristic
134Q8×C53—22no—
135Q8×C53—24no—
136Q8×C53—41no—
137Q8×C53—41no—
138Q8×C53—44no—
139Q8×C53—82no—
140Q8×C53C2×SD1614yes—
141D4×C53—41no—
142D4×C53—41no—
143D4×C53—42no—
144D4×C53—81no—
145D4×C53—82no—
146C2×C4×C53—22no—
147C2×C4×C53—24no—
148C2×C4×C53—22no—
149C2×C4×C53—22no—
150C2×C4×C53—41no—
151C2×C4×C53—41no—
152C2×C4×C53—42no—
153C2×C4×C53—81no—
154C2×C4×C53—82no—
155C2×C4×C53—82no—
156C2×C4×C53#12111yescharacteristic
157C2×C4×C53#12111yescharacteristic
158C2×C4×C53#12111yescharacteristic
159C2×C4×C53C2×SD1612yes—
160C2×C4×C53#13512yes—
161C2×C4×C53C22×D411yescharacteristic
162C2×C4×C53C22×D411yescharacteristic
163C2×C4×C53C2×#4111yescharacteristic
164C8×C53—24no—
165C8×C53—42no—
166C8×C53—42no—
167C8×C53—42no—
168#19397C5311yesmaximal, characteristic, direct product factor, Sylow
169C22×C53—21no—
170C22×C53—21no—
171C22×C53—81no—
172C22×C53—81no—
173C22×C53—81no—
174C22×C53#46512yes—
175C22×C53C2×#12111yescharacteristic
176C4×C53—21no—
177C4×C53—24no—
178C4×C53—21no—
179C4×C53—21no—
180C4×C53—42no—
181C4×C53—82no—
182C4×C53—81no—
183C4×C53#46512yes—
184C4×C53C22×SD1611yescharacteristic
185C4×C53C2×#13511yescharacteristic
186C4×C53#57712yes—
187#3206C2×C5311yescharacteristic
188C2×#475C2×C5311yescharacteristic, complemented
189#3037C2×C5311yescharacteristic
190#2982C2×C5311yescharacteristic, complemented
191#1680C2×C5314yescomplemented
192#1666C2×C5314yescomplemented
193#1570C2×C5311yescharacteristic, complemented
194#1549C2×C5311yescharacteristic, complemented
195#1540C2×C5311yescharacteristic
196C2×C53—22no—
197C2×C53—161no—
198C2×C53#304311yescharacteristic
199C2×C53C2×#46511yescharacteristic
200C2×C53#306911yescharacteristic
201C2×#137—21no—
202#532C22×C5311yescharacteristic
203C2×C4⋊Q8C22×C5311yescharacteristic
204C2×C4×Q8C22×C5311yescharacteristic
205#492C22×C5312yes—
206#487C22×C5312yes—
207#475—22no—
208#475C22×C5314yes—
209#464—22no—
210#462C22×C5312yes—
211#461—22no—
212#460C22×C5312yes—
213#448—22no—
214C8×Q8—22no—
215C8×Q8C22×C5314yes—
216#431—22no—
217#431C22×C5311yescharacteristic
218#431C22×C5311yescharacteristic
219C2×#106—21no—
220#423C22×C5311yescharacteristic
221C2×C4⋊C8C22×C5311yescharacteristic
222#417—21no—
223C2×#103—21no—
224#334C22×C5314yes—
225#332C22×C5312yes—
226#332C22×C5314yes—
227#331C22×C5312yes—
228C53#1939711yescharacteristic, direct product factor, Sylow
229C2×#41—21no—
230C22×Q8—21no—
231C22×Q8D4×C5311yescharacteristic
232#137—44no—
233C2×Q16—22no—
234C2×SD16—22no—
235C2×SD16—24no—
236C2×M16—21no—
237C22×C8—21no—
238C4⋊Q8—21no—
239C4⋊Q8—21no—
240C4⋊Q8—21no—
241C4⋊Q8C23×C5312yes—
242C4⋊Q8C23×C5311yescharacteristic
243#127C23×C5311yescharacteristic
244#124—21no—
245#121—21no—
246#121—21no—
247C4×Q8—21no—
248C4×Q8—21no—
249C4×Q8D4×C5312yes—
250C4×Q8C23×C5312yes—
251C4×D4—21no—
252C4×D4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4—21no—
255C2×C4⋊C4—21no—
256C2×C4⋊C4D4×C5311yescharacteristic
257C2×C42—21no—
258C2×C42C23×C5311yescharacteristic
259#106—24no—
260C4⋊C8—24no—
261C4⋊C8—21no—
262C4⋊C8—21no—
263C4⋊C8D4×C5312yes—
264C4⋊C8C23×C5312yes—
265C4⋊C8C23×C5312yes—
266C4⋊C8C23×C5312yes—
267#103—24no—
268#103—24no—
269#103—22no—
270#102—22no—
271C22⋊C8—21no—
272C22⋊C8D4×C5312yes—
273C4×C8—24no—
274C4×C8—21no—
275C4×C8C23×C5312yes—
276#41—42no—
277#41—42no—
278C2×Q8—21no—
279C2×Q8—22no—
280C2×Q8—21no—
281C2×Q8—22no—
282C2×Q8—41no—
283C2×Q8—82no—
284C2×Q8SD16×C5314yescomplemented
285C2×Q8C2×D4×C5312yes—
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—41no—
290C22×C4—21no—
291C22×C4—21no—
292C22×C4—41no—
293C22×C4—41no—
294C22×C4C2×D4×C5311yescharacteristic
295C22×C4C2×D4×C5311yescharacteristic
296C22×C4#41×C5311yescharacteristic
297Q16—44no—
298SD16—44no—
299SD16—84no—
300M16—42no—
301C2×C8—22no—
302C2×C8—22no—
303C2×C8—22no—
304C2×C8—42no—
305C2×C8C2×D4×C5314yes—
306C4⋊C4—22no—
307C4⋊C4—21no—
308C4⋊C4—22no—
309C4⋊C4—21no—
310C4⋊C4—22no—
311C4⋊C4—22no—
312C4⋊C4—21no—
313C4⋊C4—41no—
314C4⋊C4—41no—
315C4⋊C4C2×D4×C5312yes—
316C22⋊C4—41no—
317C22⋊C4—41no—
318C42—22no—
319C42—22no—
320C42C2×D4×C5311yescharacteristic
321C42C2×D4×C5311yescharacteristic
322C42#41×C5311yescharacteristic
323C42C24×C5311yescharacteristic
324C23—41no—
325C23#121×C5311yescharacteristic
326Q8—22no—
327Q8—24no—
328Q8—41no—
329Q8—41no—
330Q8—44no—
331Q8—82no—
332Q8C2×SD16×C5314yes—
333D4—41no—
334D4—41no—
335D4—42no—
336D4—81no—
337D4—82no—
338C2×C4—22no—
339C2×C4—24no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—41no—
343C2×C4—41no—
344C2×C4—42no—
345C2×C4—81no—
346C2×C4—82no—
347C2×C4—82no—
348C2×C4#121×C5311yescharacteristic
349C2×C4#121×C5311yescharacteristic
350C2×C4#121×C5311yescharacteristic
351C2×C4C2×SD16×C5312yes—
352C2×C4#135×C5312yes—
353C2×C4C22×D4×C5311yescharacteristic
354C2×C4C22×D4×C5311yescharacteristic
355C2×C4C2×#41×C5311yescharacteristic
356C8—24no—
357C8—42no—
358C8—42no—
359C8—42no—
360C22—21no—
361C22—21no—
362C22—81no—
363C22—81no—
364C22—81no—
365C22C53×#46512yes—
366C22C2×#121×C5311yescharacteristic
367C4—21no—
368C4—24no—
369C4—21no—
370C4—21no—
371C4—42no—
372C4—82no—
373C4—81no—
374C4C53×#46512yes—
375C4C22×SD16×C5311yescharacteristic
376C4C2×#135×C5311yescharacteristic
377C4C53×#57712yes—
378C2—22no—
379C2—161no—
380C2C53×#304311yescharacteristic
381C2C2×C53×#46511yescharacteristic
382C2C53×#306911yescharacteristic
subgroupG/H
center, socleC22×C53C2×#121
commutator, Frattini, cosocleC42C24×C53
prime radicalC53×#532C22
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
2. C2×C53×#475C2semidirect
4. C53×#2982C2semidirect
5. C53×#1680C2semidirect
6. C53×#1666C2semidirect
7. C53×#1570C2semidirect
8. C53×#1549C2semidirect
92. C2×Q8×C53SD16semidirect
168. #19397C53direct (N × H)
188. C2×#475C2×C53semidirect
190. #2982C2×C53semidirect
191. #1680C2×C53semidirect
192. #1666C2×C53semidirect
193. #1570C2×C53semidirect
194. #1549C2×C53semidirect
228. C53#19397direct (N × H)
284. C2×Q8SD16×C53semidirect

Maximal subgroups

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

subgroupindex
C53×#32062
C2×C53×#4752
C53×#30372
C53×#29822
C53×#16802
C53×#16662
C53×#15702
C53×#15492
C53×#15402
#1939753