Finite Group Databasebeta

← C53×#8372 #27215884

Subgroups — C53×#8372 #27215884

Order 13568.

788 conjugacy classes of subgroups.

The subgroup lattice has 394 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C53×#554C211yesmaximal, characteristic
2C4×#101×C53C211yesmaximal, characteristic, complemented
3C53×#1528C212yesmaximal, complemented
4C53×#1490C212yesmaximal
5C53×#1467C211yesmaximal, characteristic, complemented
6C53×#554—24no—
7C53×#554—24no—
8C2×C4⋊Q8×C53C411yescharacteristic
9C2×#125×C53C411yescharacteristic
10C2×#125×C53C2211yescharacteristic
11C2×C22⋊Q8×C53—21no—
12C2×C22⋊Q8×C53—21no—
13C2×C22⋊Q8×C53—21no—
14C2×C4×Q8×C53C2211yescharacteristic
15C2×C4×D4×C53C411yescharacteristic
16C2×C4×D4×C53C411yescharacteristic
17C2×C4×D4×C53C2211yescharacteristic
18C22×C4⋊C4×C53—21no—
19C53×#432C2212yes—
20C2×#101×C53—21no—
21C4×M16×C53C2212yes—
22C53×#343—21no—
23C53×#330—24no—
24C53×D4⋊C8—24no—
25C53×#324—21no—
26C53×#324—21no—
27C22×Q8×C53—21no—
28C22×Q8×C53D411yescharacteristic
29C22×D4×C53C2×C411yescharacteristic
30C23×C4×C53—21no—
31C23×C4×C53—21no—
32C23×C4×C53—21no—
33C2×M16×C53—22no—
34C4⋊Q8×C53—22no—
35C4⋊Q8×C53—22no—
36#125×C53—22no—
37#125×C53—22no—
38#125×C53—22no—
39#125×C53D414yes—
40C22⋊Q8×C53—28no—
41C22⋊Q8×C53—28no—
42C22⋊Q8×C53—28no—
43C4×Q8×C53—22no—
44C4×Q8×C53—22no—
45C4×D4×C53—22no—
46C4×D4×C53—22no—
47C4×D4×C53—22no—
48C4×D4×C53—22no—
49C4×D4×C53—22no—
50C4×D4×C53D414yes—
51C2×C4⋊C4×C53—28no—
52C2×C4⋊C4×C53—21no—
53C2×C4⋊C4×C53—21no—
54C2×C4⋊C4×C53—21no—
55C2×C4⋊C4×C53—21no—
56C2×C4⋊C4×C53—21no—
57C2×C4⋊C4×C53C2×C411yescharacteristic
58C2×C4⋊C4×C53C2×C411yescharacteristic
59C2×C4⋊C4×C53C2×C411yescharacteristic
60C2×C4⋊C4×C53D411yescharacteristic
61C2×C4⋊C4×C53D411yescharacteristic
62C2×C4⋊C4×C53D411yescharacteristic
63C2×C22⋊C4×C53—21no—
64C2×C22⋊C4×C53—21no—
65C2×C22⋊C4×C53—21no—
66C2×C42×C53C2×C411yescharacteristic
67C2×C42×C53C2×C411yescharacteristic
68C2×C42×C53C2311yescharacteristic
69C4⋊C8×C53—22no—
70#101×C53—24no—
71C22⋊C8×C53—22no—
72#97×C53—22no—
73#97×C53—22no—
74C4×C8×C53—22no—
75C24×C53—21no—
76C2×Q8×C53—24no—
77C2×Q8×C53—22no—
78C2×Q8×C53—22no—
79C2×Q8×C53—41no—
80C2×Q8×C53—82no—
81C2×D4×C53—24no—
82C2×D4×C53—21no—
83C2×D4×C53C22⋊C412yes—
84C22×C4×C53—24no—
85C22×C4×C53—21no—
86C22×C4×C53—21no—
87C22×C4×C53—21no—
88C22×C4×C53—24no—
89C22×C4×C53—24no—
90C22×C4×C53—42no—
91C22×C4×C53—42no—
92C22×C4×C53—82no—
93C22×C4×C53—82no—
94C22×C4×C53C22⋊C411yescharacteristic
95C22×C4×C53C22⋊C411yescharacteristic
96C22×C4×C53C22⋊C411yescharacteristic
97C22×C4×C53C22⋊C411yescharacteristic
98C22×C4×C53C22×C411yescharacteristic
99C22×C4×C53C2×D411yescharacteristic
100C22×C4×C53C2×D411yescharacteristic
101M16×C53—24no—
102M16×C53—42no—
103C2×C8×C53—42no—
104C2×C8×C53—42no—
105C4⋊C4×C53—24no—
106C4⋊C4×C53—24no—
107C4⋊C4×C53—24no—
108C4⋊C4×C53—21no—
109C4⋊C4×C53—21no—
110C4⋊C4×C53—21no—
111C4⋊C4×C53—22no—
112C4⋊C4×C53—24no—
113C4⋊C4×C53—24no—
114C4⋊C4×C53—22no—
115C4⋊C4×C53—22no—
116C4⋊C4×C53C22⋊C412yes—
117C4⋊C4×C53C22⋊C412yes—
118C4⋊C4×C53C22⋊C412yes—
119C22⋊C4×C53—24no—
120C22⋊C4×C53—24no—
121C22⋊C4×C53—24no—
122C42×C53—21no—
123C42×C53—21no—
124C42×C53—21no—
125C42×C53—21no—
126C42×C53C2×D412yes—
127C42×C53C2×D412yes—
128C23×C53—24no—
129C23×C53—22no—
130C23×C53—42no—
131C23×C53—41no—
132C23×C53—41no—
133C23×C53C2×C22⋊C411yescharacteristic
134Q8×C53—44no—
135Q8×C53—84no—
136D4×C53—24no—
137D4×C53—22no—
138D4×C53#10114yescomplemented
139C2×C4×C53—21no—
140C2×C4×C53—21no—
141C2×C4×C53—21no—
142C2×C4×C53—21no—
143C2×C4×C53—22no—
144C2×C4×C53—21no—
145C2×C4×C53—22no—
146C2×C4×C53—22no—
147C2×C4×C53—22no—
148C2×C4×C53—21no—
149C2×C4×C53—21no—
150C2×C4×C53—21no—
151C2×C4×C53—21no—
152C2×C4×C53—41no—
153C2×C4×C53—48no—
154C2×C4×C53—42no—
155C2×C4×C53—42no—
156C2×C4×C53—41no—
157C2×C4×C53—41no—
158C2×C4×C53—41no—
159C2×C4×C53—41no—
160C2×C4×C53—84no—
161C2×C4×C53—84no—
162C2×C4×C53#10412yes—
163C2×C4×C53C2×C22⋊C411yescharacteristic
164C2×C4×C53C2×C22⋊C411yescharacteristic
165C2×C4×C53#12012yes—
166C2×C4×C53#12012yes—
167C8×C53—44no—
168C8×C53—82no—
169#8372C5311yesmaximal, characteristic, direct product factor, Sylow
170C22×C53—21no—
171C22×C53—21no—
172C22×C53—24no—
173C22×C53—44no—
174C22×C53—44no—
175C22×C53—42no—
176C22×C53—42no—
177C22×C53#37911yescharacteristic
178C22×C53C2×#10411yescharacteristic
179C22×C53#42111yescharacteristic
180C4×C53—22no—
181C4×C53—21no—
182C4×C53—22no—
183C4×C53—22no—
184C4×C53—42no—
185C4×C53—42no—
186C4×C53—41no—
187C4×C53—41no—
188C4×C53—41no—
189C4×C53—42no—
190C4×C53C2×#10112yes—
191C2×#554C2×C5311yescharacteristic
192C4×#101C2×C5311yescharacteristic, complemented
193#1528C2×C5312yescomplemented
194#1490C2×C5312yes—
195#1467C2×C5311yescharacteristic, complemented
196C2×C53—21no—
197C2×C53—21no—
198C2×C53—44no—
199C2×C53#178411yescharacteristic
200C2×C53#178911yescharacteristic
201C2×C53#179511yescharacteristic
202#554—24no—
203#554—24no—
204C2×C4⋊Q8C4×C5311yescharacteristic
205C2×#125C4×C5311yescharacteristic
206C2×#125C22×C5311yescharacteristic
207C2×C22⋊Q8—21no—
208C2×C22⋊Q8—21no—
209C2×C22⋊Q8—21no—
210C2×C4×Q8C22×C5311yescharacteristic
211C2×C4×D4C4×C5311yescharacteristic
212C2×C4×D4C4×C5311yescharacteristic
213C2×C4×D4C22×C5311yescharacteristic
214C22×C4⋊C4—21no—
215#432C22×C5312yes—
216C2×#101—21no—
217C4×M16C22×C5312yes—
218#343—21no—
219#330—24no—
220D4⋊C8—24no—
221#324—21no—
222#324—21no—
223C53#837211yescharacteristic, direct product factor, Sylow
224C22×Q8—21no—
225C22×Q8D4×C5311yescharacteristic
226C22×D4C2×C4×C5311yescharacteristic
227C23×C4—21no—
228C23×C4—21no—
229C23×C4—21no—
230C2×M16—22no—
231C4⋊Q8—22no—
232C4⋊Q8—22no—
233#125—22no—
234#125—22no—
235#125—22no—
236#125D4×C5314yes—
237C22⋊Q8—28no—
238C22⋊Q8—28no—
239C22⋊Q8—28no—
240C4×Q8—22no—
241C4×Q8—22no—
242C4×D4—22no—
243C4×D4—22no—
244C4×D4—22no—
245C4×D4—22no—
246C4×D4—22no—
247C4×D4D4×C5314yes—
248C2×C4⋊C4—28no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4—21no—
252C2×C4⋊C4—21no—
253C2×C4⋊C4—21no—
254C2×C4⋊C4C2×C4×C5311yescharacteristic
255C2×C4⋊C4C2×C4×C5311yescharacteristic
256C2×C4⋊C4C2×C4×C5311yescharacteristic
257C2×C4⋊C4D4×C5311yescharacteristic
258C2×C4⋊C4D4×C5311yescharacteristic
259C2×C4⋊C4D4×C5311yescharacteristic
260C2×C22⋊C4—21no—
261C2×C22⋊C4—21no—
262C2×C22⋊C4—21no—
263C2×C42C2×C4×C5311yescharacteristic
264C2×C42C2×C4×C5311yescharacteristic
265C2×C42C23×C5311yescharacteristic
266C4⋊C8—22no—
267#101—24no—
268C22⋊C8—22no—
269#97—22no—
270#97—22no—
271C4×C8—22no—
272C24—21no—
273C2×Q8—24no—
274C2×Q8—22no—
275C2×Q8—22no—
276C2×Q8—41no—
277C2×Q8—82no—
278C2×D4—24no—
279C2×D4—21no—
280C2×D4C22⋊C4×C5312yes—
281C22×C4—24no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—24no—
286C22×C4—24no—
287C22×C4—42no—
288C22×C4—42no—
289C22×C4—82no—
290C22×C4—82no—
291C22×C4C22⋊C4×C5311yescharacteristic
292C22×C4C22⋊C4×C5311yescharacteristic
293C22×C4C22⋊C4×C5311yescharacteristic
294C22×C4C22⋊C4×C5311yescharacteristic
295C22×C4C22×C4×C5311yescharacteristic
296C22×C4C2×D4×C5311yescharacteristic
297C22×C4C2×D4×C5311yescharacteristic
298M16—24no—
299M16—42no—
300C2×C8—42no—
301C2×C8—42no—
302C4⋊C4—24no—
303C4⋊C4—24no—
304C4⋊C4—24no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—22no—
309C4⋊C4—24no—
310C4⋊C4—24no—
311C4⋊C4—22no—
312C4⋊C4—22no—
313C4⋊C4C22⋊C4×C5312yes—
314C4⋊C4C22⋊C4×C5312yes—
315C4⋊C4C22⋊C4×C5312yes—
316C22⋊C4—24no—
317C22⋊C4—24no—
318C22⋊C4—24no—
319C42—21no—
320C42—21no—
321C42—21no—
322C42—21no—
323C42C2×D4×C5312yes—
324C42C2×D4×C5312yes—
325C23—24no—
326C23—22no—
327C23—42no—
328C23—41no—
329C23—41no—
330C23C2×C22⋊C4×C5311yescharacteristic
331Q8—44no—
332Q8—84no—
333D4—24no—
334D4—22no—
335D4#101×C5314yescomplemented
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—22no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—22no—
345C2×C4—22no—
346C2×C4—22no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—41no—
350C2×C4—48no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4—84no—
358C2×C4—84no—
359C2×C4#104×C5312yes—
360C2×C4C2×C22⋊C4×C5311yescharacteristic
361C2×C4C2×C22⋊C4×C5311yescharacteristic
362C2×C4#120×C5312yes—
363C2×C4#120×C5312yes—
364C8—44no—
365C8—82no—
366C22—21no—
367C22—21no—
368C22—24no—
369C22—44no—
370C22—44no—
371C22—42no—
372C22—42no—
373C22C53×#37911yescharacteristic
374C22C2×#104×C5311yescharacteristic
375C22C53×#42111yescharacteristic
376C4—21no—
377C4—22no—
378C4—22no—
379C4—22no—
380C4—41no—
381C4—42no—
382C4—42no—
383C4—42no—
384C4—41no—
385C4—41no—
386C4C2×#101×C5312yes—
387C2—21no—
388C2—21no—
389C2—44no—
390C2C53×#178411yescharacteristic
391C2C53×#178911yescharacteristic
392C2C53×#179511yescharacteristic
subgroupG/H
center, socleC22×C53#379
commutatorC22×C4C22×C4×C53
Frattini, cosocleC2×C42C23×C53
prime radicalC22×D4×C53C2×C4
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C53 ◁ C2×C42×C53 ◁ G

Complements

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

NH ≅ G/Ntype
2. C4×#101×C53C2semidirect
3. C53×#1528C2semidirect
5. C53×#1467C2semidirect
138. D4×C53#101semidirect
169. #8372C53direct (N × H)
192. C4×#101C2×C53semidirect
193. #1528C2×C53semidirect
195. #1467C2×C53semidirect
223. C53#8372direct (N × H)
335. D4#101×C53semidirect

Maximal subgroups

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

subgroupindex
C2×C53×#5542
C4×#101×C532
C53×#15282
C53×#14902
C53×#14672
#837253