Finite Group Databasebeta

← #11756841

Subgroups — #11756841

Order 960.

516 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D3×#711C211yesmaximal, characteristic
2#91990C211yesmaximal, characteristic, complemented
3C3×#4131C211yesmaximal, characteristic, complemented
4C2×#6739C212yesmaximal, complemented
5#91929C212yesmaximal, complemented
6D3×#704C212yesmaximal, complemented
7#91927C212yesmaximal, complemented
8D5⋊ΓL(1,𝔽25)C211yesmaximal, characteristic, complemented
9#91915C211yesmaximal, characteristic, complemented
10#91913C211yesmaximal, characteristic, complemented
11#91911C211yesmaximal, characteristic, complemented
12#66256—31nomaximal, Hall
13#6819C411yescharacteristic
14C5×D3×D4C411yescharacteristic
15C3×#711C2211yescharacteristic
16D3×C22⋊D5C412yes—
17#6782C2212yes—
18C2×D3×Dic5C2212yes—
19#6780C412yes—
20C4×D3×D5C2211yescharacteristic
21#6772C2211yescharacteristic
22#6768C411yescharacteristic
23D3×Dic10C411yescharacteristic
24#6762C2212yescomplemented
25C2×C3⋊#170C2212yescomplemented
26#6758C2211yescharacteristic, complemented
27D5⋊(C3⋊C8)C2211yescharacteristic, complemented
28C3×#704C2212yescomplemented
29C2×C3×#170C2212yescomplemented
30C3×#700C2211yescharacteristic, complemented
31C3×D5⋊C8C2211yescharacteristic, complemented
32#6740C2211yescharacteristic, complemented
33#6740C2211yescharacteristic, complemented
34#6739C2212yes—
35#6739C2211yescharacteristic
36#6739C2211yescharacteristic
37#6739C2212yescomplemented
38#6739C2212yescomplemented
39#6738C2212yescomplemented
40D3×#170C2212yescomplemented
41D3×#170C2212yescomplemented
42#5750—51nomaximal, Hall
43C2×#711—31no—
44C2×#704—32no—
45C2×#704—32no—
46#4131—31no—
47#4131—32no—
48#4131—32no—
49#4131—31no—
50#4131—31no—
51#4131D311yescharacteristic, complemented
52D5⋊M16—31no—
53D5⋊M16—31no—
54C22×C5×D3C2×C412yes—
55C2×D3×D5C2×C411yescharacteristic
56C3×C5×D4C2×C411yescharacteristic
57#1220C2×C412yes—
58C2×Dic15C2×C412yes—
59C4×D15C2×C411yescharacteristic
60Dic30C2×C411yescharacteristic
61C5×#67C2×C412yes—
62C5×D12C2×C411yescharacteristic
63C4×C5×D3C2×C411yescharacteristic
64C3×C22⋊D5C2×C412yes—
65C2×C3×Dic5C2312yes—
66C3×C4×D5C2311yescharacteristic
67C3×Dic10C2×C411yescharacteristic
68#1204C2×C412yes—
69#1203C2×C412yes—
70#1201C2×C412yes—
71#1200C2311yescharacteristic
72D3×Dic5C2×C412yes—
73D3×Dic5C2×C412yes—
74D3×Dic5C2312yes—
75D3×Dic5C2311yescharacteristic
76D5×Dic3C2×C411yescharacteristic
77C3⋊#170C2311yescharacteristic
78C3⋊#170C2312yes—
79C3⋊#170C2311yescharacteristic
80C3×#170C2311yescharacteristic
81C3×#170C2312yes—
82C3×#170C2311yescharacteristic
83D3×#41—51no—
84C3×#131—51no—
85#948—51no—
86#907—51no—
87#907—52no—
88D3×M16—51no—
89D3×M16—52no—
90#901—51no—
91#901—52no—
92C2×ΓL(1,𝔽25)—51no—
93C2×ΓL(1,𝔽25)—52no—
94C2×C5×D4—31no—
95C2×C22⋊D5—32no—
96C22×Dic5—32no—
97#711—31no—
98#711—32no—
99#711—32no—
100#711—31no—
101#711—31no—
102#711C2×D311yescharacteristic
103C2×C4×D5—31no—
104C2×Dic10—31no—
105#704—32no—
106#704—32no—
107#704—31no—
108#704—31no—
109#704—32no—
110#704—32no—
111#704C2×D312yescomplemented
112C2×#170—32no—
113C2×#170—32no—
114C2×#170—32no—
115C2×#170C2×D312yescomplemented
116#700—31no—
117#700—31no—
118#700—31no—
119#700C2×D311yescharacteristic, complemented
120D5⋊C8—31no—
121D5⋊C8—32no—
122D5⋊C8C2×D311yescharacteristic, complemented
123#568—151noSylow
124C22×C3×C5C22×C412yes—
125C2×D15C22×C411yescharacteristic
126C2×C5×D3—42no—
127C2×C5×D3C22×C411yescharacteristic
128C2×C5×D3C22×C412yes—
129C2×C3×D5C22×C411yescharacteristic
130D3×D5—41no—
131C3×C4×C5C22×C411yescharacteristic
132Dic15C22×C411yescharacteristic
133Dic15C22×C412yes—
134C3×Dic5C22×C412yes—
135C3×Dic5C2411yescharacteristic
136C5×Dic3C22×C411yescharacteristic
137C3×#41—51no—
138#239—51no—
139D3×Q8—51no—
140#237—51no—
141#237—52no—
142D3×D4—52no—
143D3×D4AGL(1,𝔽5)11yescharacteristic
144#235—52no—
145#235—51no—
146C2×C4×D3—52no—
147C2×C4×D3—51no—
148C3×M16—51no—
149C3×M16—52no—
150C2×C3×C8—51no—
151C2×C3×C8—52no—
152#208—51no—
153#208—52no—
154C2×C3⋊C8—51no—
155C2×C3⋊C8—52no—
156ΓL(1,𝔽25)—51no—
157ΓL(1,𝔽25)—52no—
158ΓL(1,𝔽25)—51no—
159ΓL(1,𝔽25)—52no—
160ΓL(1,𝔽25)—52no—
161ΓL(1,𝔽25)—51no—
162ΓL(1,𝔽25)—51no—
163D3×C8—52no—
164D3×C8—52no—
165D3×C8—52no—
166C23×C5—32no—
167C22×D5—31no—
168C5×D4—32no—
169C5×D4—31no—
170C5×D4C4×D311yescharacteristic
171C2×C4×C5—31no—
172C22⋊D5—32no—
173C22⋊D5—32no—
174C22⋊D5—32no—
175C22⋊D5C4×D312yes—
176C2×Dic5—32no—
177C2×Dic5—32no—
178C2×Dic5—32no—
179C2×Dic5—32no—
180C2×Dic5—31no—
181C2×Dic5C22×D312yes—
182C4×D5—31no—
183C4×D5—31no—
184C4×D5—31no—
185C4×D5C22×D311yescharacteristic
186Dic10—31no—
187Dic10—32no—
188Dic10C4×D311yescharacteristic
189#170—31no—
190#170—32no—
191#170—31no—
192#170C22×D312yes—
193#170C22×D311yescharacteristic
194#170C22×D311yescharacteristic
195C2×#41—151no—
196#131—51no—
197#131—151no—
198#131—152no—
199#131—152no—
200#131—151no—
201#131—151no—
202C2×M16—151no—
203C2×M16—152no—
204C2×M16—151no—
205C2×M16—152no—
206C2×C3×C5—22no—
207C2×C3×C5C23×C411yescharacteristic
208D15—21no—
209C3×D5—21no—
210C5×D3—21no—
211C5×D3—22no—
212C22×D3—51no—
213C22×D3C2×AGL(1,𝔽5)12yes—
214C3×Q8—51no—
215C3×D4—52no—
216C3×D4C2×AGL(1,𝔽5)11yescharacteristic
217C2×C3×C4—52no—
218C2×C3×C4—51no—
219#67—52no—
220#67—52no—
221#67C2×AGL(1,𝔽5)12yes—
222C2×Dic3—52no—
223C2×Dic3—51no—
224D12—52no—
225D12C2×AGL(1,𝔽5)11yescharacteristic
226C4×D3—52no—
227C4×D3—52no—
228C4×D3—52no—
229C4×D3—51no—
230C4×D3—51no—
231C4×D3—51no—
232C4×D3C2×AGL(1,𝔽5)11yescharacteristic
233Dic6—51no—
234Dic6—52no—
235C3×C8—51no—
236C3×C8—52no—
237C3×C8—51no—
238C3⋊C8—51no—
239C3⋊C8—52no—
240C3⋊C8—51no—
241C22×C5—31no—
242C22×C5—32no—
243C22×C5—122no—
244C22×C5C2×C4×D312yes—
245C2×D5—31no—
246C2×D5—121no—
247C2×D5C2×C4×D311yescharacteristic
248C4×C5—31no—
249C4×C5C2×C4×D311yescharacteristic
250Dic5—32no—
251Dic5—31no—
252Dic5C2×C4×D312yes—
253Dic5C23×D311yescharacteristic
254#41—51no—
255#41—152no—
256#41—151no—
257#41—152no—
258#41—151no—
259#41—151no—
260C2×Q8—151no—
261C2×D4—31no—
262C2×D4—152no—
263C22×C4—151no—
264C22×C4—152no—
265M16—51no—
266M16—52no—
267M16—151no—
268M16—151no—
269M16—152no—
270M16—151no—
271M16—151no—
272M16—152no—
273M16—152no—
274M16—151no—
275M16—152no—
276C2×C8—51no—
277C2×C8—52no—
278C2×C8—152no—
279C2×C8—152no—
280C2×C8—152no—
281C2×C8—151no—
282C2×C8—152no—
283C3×C5#56811yescharacteristic, complemented, Hall
284C22×C3—51no—
285C22×C3C22×AGL(1,𝔽5)12yes—
286C2×D3—42no—
287C2×D3—51no—
288C2×D3—201no—
289C2×D3C22×AGL(1,𝔽5)11yescharacteristic
290C2×D3C22×AGL(1,𝔽5)12yes—
291C3×C4—52no—
292C3×C4—51no—
293C3×C4C22×AGL(1,𝔽5)11yescharacteristic
294Dic3—52no—
295Dic3—51no—
296Dic3C22×AGL(1,𝔽5)11yescharacteristic
297C2×C5—22no—
298C2×C5—61no—
299C2×C5—62no—
300C2×C5C22×C4×D311yescharacteristic
301D5—21no—
302D5—61no—
303C23—32no—
304C23—151no—
305Q8—51no—
306Q8—152no—
307Q8—151no—
308D4—32no—
309D4—31no—
310D4—52no—
311D4—152no—
312D4—152no—
313D4—152no—
314D4D3×AGL(1,𝔽5)11yescharacteristic
315C2×C4—31no—
316C2×C4—52no—
317C2×C4—51no—
318C2×C4—151no—
319C2×C4—151no—
320C2×C4—152no—
321C2×C4—151no—
322C2×C4—152no—
323C2×C4—152no—
324C2×C4—152no—
325C2×C4—151no—
326C8—51no—
327C8—51no—
328C8—52no—
329C8—152no—
330C8—151no—
331C8—151no—
332C2×C3—22no—
333C2×C3—101no—
334C2×C3C23×AGL(1,𝔽5)11yescharacteristic
335D3—21no—
336D3—22no—
337D3—101no—
338C5#575011yescharacteristic, complemented, Sylow
339C22—31no—
340C22—32no—
341C22—51no—
342C22—122no—
343C22—151no—
344C22—601no—
345C22C2×D3×AGL(1,𝔽5)12yes—
346C4—31no—
347C4—51no—
348C4—52no—
349C4—151no—
350C4—152no—
351C4C2×D3×AGL(1,𝔽5)11yescharacteristic
352C3#6625611yescharacteristic, complemented, Sylow
353C2—22no—
354C2—61no—
355C2—62no—
356C2—101no—
357C2—301no—
358C2C22×D3×AGL(1,𝔽5)11yescharacteristic
subgroupG/H
center, FrattiniC2C22×D3×AGL(1,𝔽5)
commutator, socleC2×C3×C5C23×C4
FittingC3×C5×D4C2×C4
prime radicalD3×#711C2
cosocleC3×Dic5C24
hypercenterD4D3×AGL(1,𝔽5)
hypocenterC3×C5#568
LCSG ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ D4

Complements

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

NH ≅ G/Ntype
2. #91990C2semidirect
3. C3×#4131C2semidirect
4. C2×#6739C2semidirect
5. #91929C2semidirect
6. D3×#704C2semidirect
7. #91927C2semidirect
8. D5⋊ΓL(1,𝔽25)C2semidirect
9. #91915C2semidirect
10. #91913C2semidirect
11. #91911C2semidirect
24. #6762C22semidirect
25. C2×C3⋊#170C22semidirect
26. #6758C22semidirect
27. D5⋊(C3⋊C8)C22semidirect
28. C3×#704C22semidirect
29. C2×C3×#170C22semidirect
30. C3×#700C22semidirect
31. C3×D5⋊C8C22semidirect
32. #6740C22semidirect
33. #6740C22semidirect
37. #6739C22semidirect
38. #6739C22semidirect
39. #6738C22semidirect
40. D3×#170C22semidirect
41. D3×#170C22semidirect
51. #4131D3semidirect
111. #704C2×D3semidirect
115. C2×#170C2×D3semidirect
119. #700C2×D3semidirect
122. D5⋊C8C2×D3semidirect
283. C3×C5#568semidirect
338. C5#5750semidirect
352. C3#66256semidirect

Maximal subgroups

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

subgroupindex
D3×#7112
#919902
C3×#41312
C2×#67392
#919292
D3×#7042
#919272
D5⋊ΓL(1,𝔽25)2
#919152
#919132
#919112
#662563
#57505