Finite Group Databasebeta

← C53×#8325 #27215837

Subgroups — C53×#8325 #27215837

Order 13568.

748 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C53×#542C211yesmaximal, characteristic
2C53×#1479C212yesmaximal, complemented
3C53×#1478C212yesmaximal, complemented
4C53×#1458C211yesmaximal, characteristic, complemented
5C53×#1458C211yesmaximal, characteristic, complemented
6C53×#542—24no—
7C53×#542—24no—
8C2×C4⋊Q8×C53C411yescharacteristic
9C2×#126×C53—21no—
10C2×#124×C53—21no—
11C2×#124×C53C411yescharacteristic
12C2×#123×C53—21no—
13C2×C22⋊Q8×C53—21no—
14C2×C22⋊Q8×C53C411yescharacteristic
15C2×#121×C53C411yescharacteristic
16C2×C4×Q8×C53C2211yescharacteristic
17C2×C4×D4×C53C2211yescharacteristic
18C2×#117×C53C2211yescharacteristic
19C53×#431C2212yes—
20C53×#431C2212yes—
21C53×Q8⋊C8—24no—
22C53×D4⋊C8—24no—
23C53×#324—21no—
24C53×#324—21no—
25C53×#323—21no—
26C53×#323—21no—
27C22×Q8×C53—21no—
28C22×Q8×C53C2×C411yescharacteristic
29C22×D4×C53—21no—
30C22×D4×C53C2×C411yescharacteristic
31C23×C4×C53—21no—
32C23×C4×C53C2×C411yescharacteristic
33C4⋊Q8×C53—22no—
34C4⋊Q8×C53—22no—
35#126×C53—28no—
36#124×C53—28no—
37#124×C53—22no—
38#124×C53—22no—
39#123×C53—28no—
40C22⋊Q8×C53—28no—
41C22⋊Q8×C53—22no—
42C22⋊Q8×C53—22no—
43#121×C53—22no—
44#121×C53—22no—
45C4×Q8×C53—22no—
46C4×Q8×C53D414yes—
47C4×D4×C53—22no—
48C4×D4×C53D414yes—
49#117×C53—22no—
50#117×C53—22no—
51C2×C4⋊C4×C53—21no—
52C2×C4⋊C4×C53—21no—
53C2×C4⋊C4×C53—21no—
54C2×C4⋊C4×C53C2×C411yescharacteristic
55C2×C4⋊C4×C53C2×C411yescharacteristic
56C2×C4⋊C4×C53D411yescharacteristic
57C2×C4⋊C4×C53D411yescharacteristic
58C2×C22⋊C4×C53—21no—
59C2×C22⋊C4×C53—21no—
60C2×C22⋊C4×C53—21no—
61C2×C22⋊C4×C53—21no—
62C2×C22⋊C4×C53—21no—
63C2×C22⋊C4×C53D411yescharacteristic
64C2×C22⋊C4×C53D411yescharacteristic
65C2×C42×C53—21no—
66C2×C42×C53C2×C411yescharacteristic
67C2×C42×C53C2311yescharacteristic
68C4⋊C8×C53—22no—
69C4⋊C8×C53—22no—
70C22⋊C8×C53—22no—
71C22⋊C8×C53—22no—
72C4×C8×C53—22no—
73C4×C8×C53—22no—
74C24×C53—21no—
75C24×C53C22⋊C411yescharacteristic
76C2×Q8×C53—24no—
77C2×Q8×C53—21no—
78C2×Q8×C53—24no—
79C2×Q8×C53—82no—
80C2×Q8×C53C22⋊C412yes—
81C2×D4×C53—24no—
82C2×D4×C53—21no—
83C2×D4×C53—24no—
84C2×D4×C53—82no—
85C2×D4×C53C22⋊C412yes—
86C22×C4×C53—21no—
87C22×C4×C53—21no—
88C22×C4×C53—21no—
89C22×C4×C53—21no—
90C22×C4×C53—21no—
91C22×C4×C53—24no—
92C22×C4×C53—24no—
93C22×C4×C53—82no—
94C22×C4×C53C22⋊C411yescharacteristic
95C22×C4×C53C22⋊C411yescharacteristic
96C22×C4×C53C22⋊C411yescharacteristic
97C22×C4×C53C22×C411yescharacteristic
98C22×C4×C53C2×D411yescharacteristic
99C22×C4×C53C2×D411yescharacteristic
100C2×C8×C53—42no—
101C2×C8×C53—42no—
102C4⋊C4×C53—24no—
103C4⋊C4×C53—24no—
104C4⋊C4×C53—21no—
105C4⋊C4×C53—21no—
106C4⋊C4×C53—24no—
107C4⋊C4×C53—22no—
108C4⋊C4×C53—22no—
109C4⋊C4×C53C22⋊C412yes—
110C4⋊C4×C53C22⋊C412yes—
111C22⋊C4×C53—24no—
112C22⋊C4×C53—24no—
113C22⋊C4×C53—24no—
114C22⋊C4×C53—24no—
115C22⋊C4×C53—24no—
116C22⋊C4×C53—22no—
117C22⋊C4×C53—22no—
118C42×C53—21no—
119C42×C53—21no—
120C42×C53—24no—
121C42×C53C2×D412yes—
122C42×C53C2×D412yes—
123C23×C53—21no—
124C23×C53—22no—
125C23×C53—22no—
126C23×C53—42no—
127C23×C53—42no—
128C23×C53—41no—
129C23×C53—41no—
130C23×C53—42no—
131C23×C53C2×C22⋊C411yescharacteristic
132Q8×C53—24no—
133Q8×C53—42no—
134Q8×C53—84no—
135D4×C53—24no—
136D4×C53—42no—
137D4×C53—84no—
138C2×C4×C53—22no—
139C2×C4×C53—21no—
140C2×C4×C53—22no—
141C2×C4×C53—22no—
142C2×C4×C53—21no—
143C2×C4×C53—21no—
144C2×C4×C53—22no—
145C2×C4×C53—41no—
146C2×C4×C53—41no—
147C2×C4×C53—42no—
148C2×C4×C53—42no—
149C2×C4×C53—42no—
150C2×C4×C53—42no—
151C2×C4×C53—42no—
152C2×C4×C53—41no—
153C2×C4×C53—41no—
154C2×C4×C53—84no—
155C2×C4×C53C2×C22⋊C411yescharacteristic
156C2×C4×C53C2×C22⋊C411yescharacteristic
157C2×C4×C53#12012yes—
158C2×C4×C53#12012yes—
159C2×C4×C53#13114yes—
160C8×C53—82no—
161C8×C53—82no—
162#8325C5311yesmaximal, characteristic, direct product factor, Sylow
163C22×C53—21no—
164C22×C53—21no—
165C22×C53—42no—
166C22×C53—41no—
167C22×C53—41no—
168C22×C53—42no—
169C22×C53—41no—
170C22×C53—42no—
171C22×C53—82no—
172C22×C53—84no—
173C22×C53#37911yescharacteristic
174C22×C53#40812yes—
175C4×C53—22no—
176C4×C53—22no—
177C4×C53—42no—
178C4×C53—42no—
179C4×C53—41no—
180C4×C53—41no—
181C4×C53—41no—
182C4×C53—42no—
183C4×C53#40812yes—
184C4×C53#40812yes—
185C2×#542C2×C5311yescharacteristic
186#1479C2×C5312yescomplemented
187#1478C2×C5312yescomplemented
188#1458C2×C5311yescharacteristic, complemented
189#1458C2×C5311yescharacteristic, complemented
190C2×C53—22no—
191C2×C53—81no—
192C2×C53—82no—
193C2×C53#178011yescharacteristic
194C2×C53#179511yescharacteristic
195C2×C53#179611yescharacteristic
196#542—24no—
197#542—24no—
198C2×C4⋊Q8C4×C5311yescharacteristic
199C2×#126—21no—
200C2×#124—21no—
201C2×#124C4×C5311yescharacteristic
202C2×#123—21no—
203C2×C22⋊Q8—21no—
204C2×C22⋊Q8C4×C5311yescharacteristic
205C2×#121C4×C5311yescharacteristic
206C2×C4×Q8C22×C5311yescharacteristic
207C2×C4×D4C22×C5311yescharacteristic
208C2×#117C22×C5311yescharacteristic
209#431C22×C5312yes—
210#431C22×C5312yes—
211Q8⋊C8—24no—
212D4⋊C8—24no—
213#324—21no—
214#324—21no—
215#323—21no—
216#323—21no—
217C53#832511yescharacteristic, direct product factor, Sylow
218C22×Q8—21no—
219C22×Q8C2×C4×C5311yescharacteristic
220C22×D4—21no—
221C22×D4C2×C4×C5311yescharacteristic
222C23×C4—21no—
223C23×C4C2×C4×C5311yescharacteristic
224C4⋊Q8—22no—
225C4⋊Q8—22no—
226#126—28no—
227#124—28no—
228#124—22no—
229#124—22no—
230#123—28no—
231C22⋊Q8—28no—
232C22⋊Q8—22no—
233C22⋊Q8—22no—
234#121—22no—
235#121—22no—
236C4×Q8—22no—
237C4×Q8D4×C5314yes—
238C4×D4—22no—
239C4×D4D4×C5314yes—
240#117—22no—
241#117—22no—
242C2×C4⋊C4—21no—
243C2×C4⋊C4—21no—
244C2×C4⋊C4—21no—
245C2×C4⋊C4C2×C4×C5311yescharacteristic
246C2×C4⋊C4C2×C4×C5311yescharacteristic
247C2×C4⋊C4D4×C5311yescharacteristic
248C2×C4⋊C4D4×C5311yescharacteristic
249C2×C22⋊C4—21no—
250C2×C22⋊C4—21no—
251C2×C22⋊C4—21no—
252C2×C22⋊C4—21no—
253C2×C22⋊C4—21no—
254C2×C22⋊C4D4×C5311yescharacteristic
255C2×C22⋊C4D4×C5311yescharacteristic
256C2×C42—21no—
257C2×C42C2×C4×C5311yescharacteristic
258C2×C42C23×C5311yescharacteristic
259C4⋊C8—22no—
260C4⋊C8—22no—
261C22⋊C8—22no—
262C22⋊C8—22no—
263C4×C8—22no—
264C4×C8—22no—
265C24—21no—
266C24C22⋊C4×C5311yescharacteristic
267C2×Q8—24no—
268C2×Q8—21no—
269C2×Q8—24no—
270C2×Q8—82no—
271C2×Q8C22⋊C4×C5312yes—
272C2×D4—24no—
273C2×D4—21no—
274C2×D4—24no—
275C2×D4—82no—
276C2×D4C22⋊C4×C5312yes—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—24no—
283C22×C4—24no—
284C22×C4—82no—
285C22×C4C22⋊C4×C5311yescharacteristic
286C22×C4C22⋊C4×C5311yescharacteristic
287C22×C4C22⋊C4×C5311yescharacteristic
288C22×C4C22×C4×C5311yescharacteristic
289C22×C4C2×D4×C5311yescharacteristic
290C22×C4C2×D4×C5311yescharacteristic
291C2×C8—42no—
292C2×C8—42no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—21no—
296C4⋊C4—21no—
297C4⋊C4—24no—
298C4⋊C4—22no—
299C4⋊C4—22no—
300C4⋊C4C22⋊C4×C5312yes—
301C4⋊C4C22⋊C4×C5312yes—
302C22⋊C4—24no—
303C22⋊C4—24no—
304C22⋊C4—24no—
305C22⋊C4—24no—
306C22⋊C4—24no—
307C22⋊C4—22no—
308C22⋊C4—22no—
309C42—21no—
310C42—21no—
311C42—24no—
312C42C2×D4×C5312yes—
313C42C2×D4×C5312yes—
314C23—21no—
315C23—22no—
316C23—22no—
317C23—42no—
318C23—42no—
319C23—41no—
320C23—41no—
321C23—42no—
322C23C2×C22⋊C4×C5311yescharacteristic
323Q8—24no—
324Q8—42no—
325Q8—84no—
326D4—24no—
327D4—42no—
328D4—84no—
329C2×C4—21no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—21no—
334C2×C4—22no—
335C2×C4—22no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—42no—
342C2×C4—42no—
343C2×C4—41no—
344C2×C4—41no—
345C2×C4—84no—
346C2×C4C2×C22⋊C4×C5311yescharacteristic
347C2×C4C2×C22⋊C4×C5311yescharacteristic
348C2×C4#120×C5312yes—
349C2×C4#120×C5312yes—
350C2×C4#131×C5314yes—
351C8—82no—
352C8—82no—
353C22—21no—
354C22—21no—
355C22—41no—
356C22—42no—
357C22—42no—
358C22—41no—
359C22—42no—
360C22—41no—
361C22—82no—
362C22—84no—
363C22C53×#37911yescharacteristic
364C22C53×#40812yes—
365C4—22no—
366C4—22no—
367C4—41no—
368C4—42no—
369C4—42no—
370C4—42no—
371C4—41no—
372C4—41no—
373C4C53×#40812yes—
374C4C53×#40812yes—
375C2—22no—
376C2—82no—
377C2—81no—
378C2C53×#178011yescharacteristic
379C2C53×#179511yescharacteristic
380C2C53×#179611yescharacteristic
subgroupG/H
center, socleC22×C53#379
commutatorC22×C4C22×C4×C53
Frattini, cosocleC2×C42C23×C53
prime radicalC2×#121×C53C4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C53×#1479C2semidirect
3. C53×#1478C2semidirect
4. C53×#1458C2semidirect
5. C53×#1458C2semidirect
162. #8325C53direct (N × H)
186. #1479C2×C53semidirect
187. #1478C2×C53semidirect
188. #1458C2×C53semidirect
189. #1458C2×C53semidirect
217. C53#8325direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C53×#5422
C53×#14792
C53×#14782
C53×#14582
#832553