Finite Group Databasebeta

← #11755325

Subgroups — #11755325

Order 960.

536 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#1009C211yesmaximal, characteristic, complemented
2#92019C211yesmaximal, characteristic, complemented
3#91838C211yesmaximal, characteristic, complemented
4C3×#4096C211yesmaximal, characteristic, complemented
5#91519C212yesmaximal, complemented
6#91504C212yesmaximal, complemented
7#91496C212yesmaximal, complemented
8D3×#686C212yesmaximal, complemented
9C2×#6654C211yesmaximal, characteristic, complemented
10#91299C211yesmaximal, characteristic, complemented
11#91286C211yesmaximal, characteristic, complemented
12#66169—31nomaximal, Hall
13C3×C5×#41C2211yescharacteristic, complemented
14C2×D60C2211yescharacteristic, complemented
15C5×#239C2212yescomplemented
16C5×D3×D4—22no—
17C5×D3×D4C2212yescomplemented
18C5×#235—21no—
19C5×#235C2211yescharacteristic
20C2×C5×D12—21no—
21C2×C5×D12C2211yescharacteristic, complemented
22C3×#709C2211yescharacteristic, complemented
23#6790—22no—
24D3×D20C2212yescomplemented
25D5×D12—21no—
26#6773—21no—
27#6769C2212yescomplemented
28#6717C2212yescomplemented
29#6715C2212yescomplemented
30#6710C2211yescharacteristic, complemented
31C3×#689C2211yescharacteristic, complemented
32C3×#688C2211yescharacteristic, complemented
33C3×#687C2211yescharacteristic, complemented
34C3×#686C2211yescharacteristic, complemented
35C2×C3×#168C2211yescharacteristic, complemented
36#6656C2212yescomplemented
37#6654C2212yescomplemented
38#6654C2212yescomplemented
39#6652C2212yescomplemented
40#6650C2212yescomplemented
41D3×#168C2212yescomplemented
42#5769—51nomaximal, Hall
43C5×H(5,𝔽2)—31no—
44#4149—31no—
45#4096—32no—
46#4096D311yescharacteristic, complemented
47#4095—31no—
48#4095—32no—
49#4094—31no—
50#4078—31no—
51#4078—32no—
52C2×#686—32no—
53C2×#686—31no—
54C22×D15—21no—
55C22×C5×D3—21no—
56C22×C5×D3—22no—
57C2×D3×D5—22no—
58C3×C5×Q8C2311yescharacteristic
59C3×C5×D4—21no—
60C3×C5×D4C2311yescharacteristic
61C2×C3×C4×C5—21no—
62C2×C3×C4×C5C2311yescharacteristic
63D60—21no—
64D60C2312yes—
65C5×#67—22no—
66C5×#67D412yes—
67C5×D12—22no—
68C5×D12—21no—
69C5×D12D411yescharacteristic
70C5×D12C2312yes—
71C4×C5×D3—22no—
72C4×C5×D3C2312yes—
73C5×Dic6D411yescharacteristic
74C3×C22⋊D5—21no—
75C3×D20C2311yescharacteristic
76C3×C4×D5—21no—
77C3×Dic10C2311yescharacteristic
78#1203—22no—
79#1202—22no—
80#1200—22no—
81C3⋊#168C2312yes—
82C3×#168C2311yescharacteristic
83C3×#168C2311yescharacteristic
84#1009—51no—
85#1009D511yescharacteristic, complemented
86C3×#135—51no—
87#949—51no—
88#949—51no—
89#919—52no—
90#914—52no—
91#914—52no—
92D3×D8—52no—
93C2×D24—51no—
94#901—51no—
95C5×#41—31no—
96C5×#41—32no—
97C5×#41—61no—
98C5×#41C2×D311yescharacteristic, complemented
99C2×C5×D4—32no—
100C2×C5×D4—31no—
101C2×C5×D4—61no—
102C2×C5×D4—62no—
103#713—61no—
104D4×D5—61no—
105D4×D5—62no—
106#709—32no—
107#709C2×D311yescharacteristic, complemented
108C2×D20—31no—
109C2×D20—32no—
110#689C2×D311yescharacteristic, complemented
111#688—32no—
112#688C2×D311yescharacteristic, complemented
113#687—32no—
114#687C2×D311yescharacteristic, complemented
115#686—32no—
116#686—32no—
117#686—32no—
118#686—32no—
119#686C2×D311yescharacteristic, complemented
120#681—31no—
121#681—32no—
122C2×#168—32no—
123C2×#168C2×D311yescharacteristic, complemented
124#576—151noSylow
125C22×C3×C5—21no—
126C22×C3×C5C2×D411yescharacteristic
127C2×D15—22no—
128C2×D15—81no—
129C2×C5×D3—22no—
130C2×C5×D3—82no—
131C2×C5×D3—81no—
132C2×C5×D3C2×D412yes—
133C2×C3×D5—21no—
134D3×D5—82no—
135C3×C4×C5—21no—
136C3×C4×C5C2×D411yescharacteristic
137C3×C4×C5C2411yescharacteristic
138C3×Dic5—21no—
139C5×Dic3C2×D412yes—
140C3×#41—51no—
141C3×#41C2×D511yescharacteristic, complemented
142#239—52no—
143#239C2×D512yescomplemented
144D3×D4—22no—
145D3×D4—52no—
146D3×D4—102no—
147D3×D4C2×D512yescomplemented
148#235—21no—
149#235—101no—
150#235C2×D511yescharacteristic
151C2×D12—21no—
152C2×D12—51no—
153C2×D12—101no—
154C2×D12C2×D511yescharacteristic, complemented
155C3×Q16—51no—
156C3×SD16—51no—
157C3×SD16—51no—
158C3×D8—51no—
159C2×C3×C8—51no—
160#215—52no—
161#215—52no—
162#213—52no—
163#213—52no—
164#208—51no—
165D24—52no—
166D24—52no—
167ΓL(1,𝔽25)—52no—
168D3×C8—52no—
169C23×C5—62no—
170C23×C5—61no—
171C22×D5—62no—
172C22×D5—61no—
173C5×Q8—31no—
174C5×Q8C22×D311yescharacteristic
175C5×D4—21no—
176C5×D4—32no—
177C5×D4—31no—
178C5×D4—32no—
179C5×D4—62no—
180C5×D4—62no—
181C5×D4—61no—
182C5×D4C22×D311yescharacteristic
183C2×C4×C5—21no—
184C2×C4×C5—32no—
185C2×C4×C5—62no—
186C2×C4×C5C22×D311yescharacteristic
187C22⋊D5—21no—
188C22⋊D5—62no—
189D20—32no—
190D20—62no—
191D20—61no—
192D20C22×D311yescharacteristic
193C4×D5—21no—
194C4×D5—62no—
195Dic10C22×D311yescharacteristic
196#168—32no—
197#168C22×D311yescharacteristic
198#168C22×D311yescharacteristic
199H(5,𝔽2)—31no—
200H(5,𝔽2)—151no—
201AGL(1,ℤ/8)—152no—
202AGL(1,ℤ/8)—152no—
203AGL(1,ℤ/8)—151no—
204AGL(1,ℤ/8)—151no—
205#135—51no—
206#135—152no—
207C2×D8—151no—
208C2×D8—152no—
209#131—151no—
210C2×C3×C5—21no—
211C2×C3×C5—41no—
212C2×C3×C5C22×D411yescharacteristic
213D15—42no—
214C3×D5—41no—
215C5×D3—22no—
216C5×D3—42no—
217C22×D3—21no—
218C22×D3—22no—
219C22×D3—101no—
220C22×D3—102no—
221C3×Q8—51no—
222C3×Q8C22×D511yescharacteristic
223C3×D4—21no—
224C3×D4—51no—
225C3×D4—101no—
226C3×D4C22×D511yescharacteristic
227C2×C3×C4—21no—
228C2×C3×C4—101no—
229C2×C3×C4C22×D511yescharacteristic
230#67—22no—
231#67—102no—
232#67C22⋊D512yes—
233D12—22no—
234D12—21no—
235D12—52no—
236D12—102no—
237D12—101no—
238D12C22⋊D511yescharacteristic
239D12C22×D512yes—
240C4×D3—22no—
241C4×D3—102no—
242C4×D3C22×D512yes—
243Dic6C22⋊D511yescharacteristic
244C3×C8—51no—
245C3×C8—51no—
246C3⋊C8—52no—
247C22×C5—21no—
248C22×C5—32no—
249C22×C5—62no—
250C22×C5—241no—
251C22×C5—242no—
252C22×C5D3×D411yescharacteristic
253C2×D5—21no—
254C2×D5—62no—
255C2×D5—242no—
256C2×D5—241no—
257C4×C5—21no—
258C4×C5—32no—
259C4×C5D3×D411yescharacteristic
260C4×C5C23×D311yescharacteristic
261Dic5—21no—
262#41—31no—
263#41—32no—
264#41—51no—
265#41—61no—
266#41—152no—
267#41—301no—
268#41D3×D511yescharacteristic, complemented
269C2×D4—32no—
270C2×D4—31no—
271C2×D4—61no—
272C2×D4—62no—
273C2×D4—152no—
274C2×D4—151no—
275C2×D4—301no—
276C2×D4—302no—
277Q16—51no—
278SD16—51no—
279SD16—51no—
280SD16—152no—
281SD16—152no—
282D8—51no—
283D8—152no—
284D8—152no—
285D8—152no—
286D8—152no—
287M16—152no—
288M16—151no—
289C2×C8—51no—
290C2×C8—152no—
291C3×C5#57611yescharacteristic, complemented, Hall
292C22×C3—21no—
293C22×C3—101no—
294C22×C3C2×C22⋊D511yescharacteristic
295C2×D3—22no—
296C2×D3—82no—
297C2×D3—81no—
298C2×D3—102no—
299C2×D3—402no—
300C2×D3—401no—
301C2×D3C2×C22⋊D512yes—
302C3×C4—21no—
303C3×C4—101no—
304C3×C4C2×C22⋊D511yescharacteristic
305C3×C4C23×D511yescharacteristic
306Dic3C2×C22⋊D512yes—
307C2×C5—21no—
308C2×C5—41no—
309C2×C5—62no—
310C2×C5—122no—
311C2×C5C2×D3×D411yescharacteristic
312D5—41no—
313D5—122no—
314C23—62no—
315C23—61no—
316C23—302no—
317C23—301no—
318Q8—31no—
319Q8—51no—
320Q8C2×D3×D511yescharacteristic
321D4—21no—
322D4—32no—
323D4—31no—
324D4—32no—
325D4—51no—
326D4—62no—
327D4—62no—
328D4—61no—
329D4—101no—
330D4—152no—
331D4—302no—
332D4—302no—
333D4—301no—
334D4C2×D3×D511yescharacteristic
335C2×C4—21no—
336C2×C4—32no—
337C2×C4—62no—
338C2×C4—101no—
339C2×C4—302no—
340C2×C4C2×D3×D511yescharacteristic
341C8—51no—
342C8—51no—
343C8—152no—
344C2×C3—21no—
345C2×C3—41no—
346C2×C3—201no—
347C2×C3C22×C22⋊D511yescharacteristic
348D3—22no—
349D3—42no—
350D3—202no—
351C5#576911yescharacteristic, complemented, Sylow
352C22—21no—
353C22—32no—
354C22—62no—
355C22—101no—
356C22—241no—
357C22—242no—
358C22—302no—
359C22—1201no—
360C22—1202no—
361C22D3×C22⋊D511yescharacteristic
362C4—21no—
363C4—32no—
364C4—101no—
365C4D3×C22⋊D511yescharacteristic
366C4C22×D3×D511yescharacteristic
367C3#6616911yescharacteristic, complemented, Sylow
368C2—21no—
369C2—41no—
370C2—62no—
371C2—122no—
372C2—201no—
373C2—602no—
374C2C2×D3×C22⋊D511yescharacteristic
subgroupG/H
centerC2C2×D3×C22⋊D5
commutator, cosocleC3×C4×C5C24
FrattiniC4C22×D3×D5
FittingC3×C5×#41C22
socleC2×C3×C5C22×D4
hypercenter#41D3×D5
hypocenterC3×C5#576
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C2×C4 ◁ #41

Complements

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

NH ≅ G/Ntype
1. C5×#1009C2semidirect
2. #92019C2semidirect
3. #91838C2semidirect
4. C3×#4096C2semidirect
5. #91519C2semidirect
6. #91504C2semidirect
7. #91496C2semidirect
8. D3×#686C2semidirect
9. C2×#6654C2semidirect
10. #91299C2semidirect
11. #91286C2semidirect
13. C3×C5×#41C22semidirect
14. C2×D60C22semidirect
15. C5×#239C22semidirect
17. C5×D3×D4C22semidirect
21. C2×C5×D12C22semidirect
22. C3×#709C22semidirect
24. D3×D20C22semidirect
27. #6769C22semidirect
28. #6717C22semidirect
29. #6715C22semidirect
30. #6710C22semidirect
31. C3×#689C22semidirect
32. C3×#688C22semidirect
33. C3×#687C22semidirect
34. C3×#686C22semidirect
35. C2×C3×#168C22semidirect
36. #6656C22semidirect
37. #6654C22semidirect
38. #6654C22semidirect
39. #6652C22semidirect
40. #6650C22semidirect
41. D3×#168C22semidirect
46. #4096D3semidirect
85. #1009D5semidirect
98. C5×#41C2×D3semidirect
107. #709C2×D3semidirect
110. #689C2×D3semidirect
112. #688C2×D3semidirect
114. #687C2×D3semidirect
119. #686C2×D3semidirect
123. C2×#168C2×D3semidirect
141. C3×#41C2×D5semidirect
143. #239C2×D5semidirect
147. D3×D4C2×D5semidirect
154. C2×D12C2×D5semidirect
268. #41D3×D5semidirect
291. C3×C5#576semidirect
351. C5#5769semidirect
367. C3#66169semidirect

Maximal subgroups

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

subgroupindex
C5×#10092
#920192
#918382
C3×#40962
#915192
#915042
#914962
D3×#6862
C2×#66542
#912992
#912862
#661693
#57695