Finite Group Databasebeta

← #11754029

Subgroups — #11754029

Order 960.

524 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1D3×#709C211yesmaximal, characteristic, complemented
2C2×#6765C211yesmaximal, characteristic, complemented
3#91798C211yesmaximal, characteristic, complemented
4C5×D3×M16C211yesmaximal, characteristic, complemented
5C3×#4054C211yesmaximal, characteristic, complemented
6#91313C211yesmaximal, characteristic, complemented
7#91312C211yesmaximal, characteristic, complemented
8#91274C212yesmaximal, complemented
9#91273C212yesmaximal, complemented
10#91272C212yesmaximal, complemented
11#91254C212yesmaximal, complemented
12#66082—31nomaximal, Hall
13#6817C2211yescharacteristic
14C2×Dic30C2211yescharacteristic, complemented
15C2×C4×C5×D3C2211yescharacteristic
16C3×#709C2211yescharacteristic
17C2×C3×D20C2211yescharacteristic, complemented
18D3×C22⋊D5—21no—
19C2×#1201—21no—
20#6779—21no—
21C2×D5×Dic3—21no—
22D3×D20C2211yescharacteristic, complemented
23C4×D3×D5—21no—
24#6771—21no—
25#6769C2211yescharacteristic, complemented
26D3×Dic10C2211yescharacteristic, complemented
27#6765—21no—
28#6765—21no—
29#6765C2212yes—
30#6765C2212yescomplemented
31#6765C2211yescharacteristic, complemented
32C3×C5×M16C2211yescharacteristic, complemented
33Dic60C2212yescomplemented
34#6706C2212yescomplemented
35C5×#208C2211yescharacteristic, complemented
36C5×ΓL(1,𝔽25)C2212yescomplemented
37C5×D3×C8C2212yescomplemented
38C3×D40C2212yescomplemented
39C3×#677C2212yescomplemented
40#6662C2212yescomplemented
41#6657C2212yescomplemented
42#6657C2212yescomplemented
43#6653C2212yescomplemented
44#5773—51nomaximal, Hall
45C2×#709—31no—
46C2×#709—31no—
47C2×C5×M16—31no—
48#4055—31no—
49#4055—31no—
50#4055—32no—
51#4054—31no—
52#4054—32no—
53#4054D311yescharacteristic, complemented
54#4050—32no—
55#4050—32no—
56C22×C5×D3D411yescharacteristic
57C22×C3×D5—21no—
58C2×D3×D5—21no—
59C2×C3×C4×C5C2311yescharacteristic
60#1220—21no—
61C2×Dic15—21no—
62D60C2311yescharacteristic
63C4×D15—21no—
64Dic30—21no—
65Dic30C2311yescharacteristic
66Dic30C2312yes—
67C2×C5×Dic3D411yescharacteristic
68C4×C5×D3D411yescharacteristic
69C4×C5×D3D411yescharacteristic
70C4×C5×D3C2311yescharacteristic
71C4×C5×D3C2311yescharacteristic
72C3×C22⋊D5—21no—
73C3×D20—21no—
74C3×D20C2311yescharacteristic
75C3×D20C2312yes—
76C3×C4×D5—21no—
77C3×Dic10C2311yescharacteristic
78#1204—21no—
79#1203—21no—
80#1202—21no—
81#1201—22no—
82#1201—22no—
83#1201—21no—
84#1200—21no—
85D3×Dic5—21no—
86D5×Dic3—22no—
87D5×Dic3—22no—
88D5×Dic3—21no—
89C3×C5×C8C2312yes—
90C5×C3⋊C8C2312yes—
91D3×#41—51no—
92C2×#237—51no—
93C3×AGL(1,ℤ/8)—51no—
94#951—51no—
95#932—51no—
96#916—52no—
97#915—52no—
98#912—52no—
99#911—52no—
100#909—51no—
101D3×M16D511yescharacteristic, complemented
102C22×C4×C5—31no—
103C2×C22⋊D5—61no—
104C2×C22⋊D5—61no—
105#709—31no—
106#709—32no—
107#709—31no—
108#709—32no—
109#709—31no—
110#709—61no—
111#709—61no—
112#709—61no—
113#709—61no—
114#709C2×D311yescharacteristic
115C2×D20—31no—
116C2×D20C2×D311yescharacteristic, complemented
117C2×C4×D5—61no—
118C2×C4×D5—61no—
119C2×Dic10—31no—
120C2×Dic10—31no—
121C5×M16—31no—
122C5×M16—32no—
123C5×M16C2×D311yescharacteristic, complemented
124C2×C5×C8—32no—
125Dic20—32no—
126Dic20—32no—
127D40—32no—
128D40C2×D312yescomplemented
129#677—32no—
130#677—32no—
131#677—32no—
132#677C2×D312yescomplemented
133#575—151noSylow
134C22×C3×C5C2×D411yescharacteristic
135C2×D15—21no—
136C2×C5×D3—41no—
137C2×C5×D3C2×D411yescharacteristic
138C2×C5×D3C2×D411yescharacteristic
139C2×C3×D5—22no—
140C2×C3×D5—21no—
141C2×C3×D5—42no—
142D3×D5—42no—
143C3×C4×C5C2×D411yescharacteristic
144C3×C4×C5C2411yescharacteristic
145Dic15—22no—
146Dic15—21no—
147C3×Dic5—21no—
148C5×Dic3C2×D411yescharacteristic
149C5×Dic3C2×D411yescharacteristic
150C3×#41—51no—
151C2×C3×D4—51no—
152C2×#67—101no—
153C22×Dic3—101no—
154#239—51no—
155D3×Q8—51no—
156#237—52no—
157#237—51no—
158#237—52no—
159#237—101no—
160#237—101no—
161#237—101no—
162D3×D4—51no—
163D3×D4—101no—
164#235—51no—
165#235—101no—
166C2×C4×D3—101no—
167C2×C4×D3C2×D511yescharacteristic
168C2×Dic6—51no—
169C3×SD16—52no—
170C3×D8—52no—
171C3×M16C2×D511yescharacteristic, complemented
172#216—52no—
173#214—52no—
174#214—52no—
175#213—52no—
176#208C2×D511yescharacteristic, complemented
177Dic12—52no—
178#204—52no—
179ΓL(1,𝔽25)C2×D512yescomplemented
180D3×C8C2×D512yescomplemented
181C23×C5—31no—
182C22×D5—21no—
183C22×D5—61no—
184C2×C4×C5—31no—
185C2×C4×C5—31no—
186C2×C4×C5—31no—
187C2×C4×C5—31no—
188C2×C4×C5—31no—
189C2×C4×C5C22×D311yescharacteristic
190C22⋊D5—21no—
191C22⋊D5—61no—
192C22⋊D5—61no—
193C22⋊D5—62no—
194C22⋊D5—62no—
195C22⋊D5—61no—
196C2×Dic5—61no—
197C2×Dic5—61no—
198D20—21no—
199D20—31no—
200D20—61no—
201D20C22×D311yescharacteristic
202D20C22×D312yes—
203C4×D5—21no—
204C4×D5—61no—
205C4×D5—62no—
206C4×D5—61no—
207C4×D5—62no—
208C4×D5—61no—
209Dic10—31no—
210Dic10—32no—
211Dic10—61no—
212Dic10—61no—
213Dic10C22×D311yescharacteristic
214C5×C8—32no—
215C5×C8C22×D312yes—
216C2×#41—151no—
217C2×#41—151no—
218#137—152no—
219#137—151no—
220#137—151no—
221AGL(1,ℤ/8)—51no—
222AGL(1,ℤ/8)—152no—
223AGL(1,ℤ/8)—151no—
224#135—152no—
225#135—152no—
226C2×M16—31no—
227C2×C3×C5—21no—
228C2×C3×C5C22×D411yescharacteristic
229D15—41no—
230C3×D5—41no—
231C3×D5—42no—
232C5×D3—21no—
233C5×D3—21no—
234C23×C3—101no—
235C22×D3—101no—
236C22×D3D2011yescharacteristic
237C3×Q8—51no—
238C3×D4—52no—
239C3×D4—51no—
240C3×D4—101no—
241C3×D4—101no—
242C2×C3×C4—101no—
243C2×C3×C4C22×D511yescharacteristic
244#67—101no—
245#67—101no—
246#67—102no—
247#67—102no—
248#67—101no—
249C2×Dic3—101no—
250C2×Dic3—101no—
251C2×Dic3—102no—
252C2×Dic3—102no—
253C2×Dic3D2011yescharacteristic
254D12—51no—
255D12—101no—
256C4×D3—101no—
257C4×D3—101no—
258C4×D3—101no—
259C4×D3D2011yescharacteristic
260C4×D3D2011yescharacteristic
261C4×D3C22×D511yescharacteristic
262C4×D3C22×D511yescharacteristic
263Dic6—52no—
264Dic6—51no—
265Dic6—101no—
266Dic6—101no—
267C3×C8C22×D512yes—
268C3⋊C8C22×D512yes—
269C22×C5—31no—
270C22×C5—31no—
271C22×C5—121no—
272C22×C5D3×D411yescharacteristic
273C2×D5—21no—
274C2×D5—22no—
275C2×D5—42no—
276C2×D5—61no—
277C2×D5—122no—
278C4×C5—31no—
279C4×C5—31no—
280C4×C5D3×D411yescharacteristic
281C4×C5C23×D311yescharacteristic
282Dic5—21no—
283Dic5—62no—
284Dic5—61no—
285#41—51no—
286#41—151no—
287#41—152no—
288#41—152no—
289#41—151no—
290#41—151no—
291#41—301no—
292#41—301no—
293#41—301no—
294#41—301no—
295C2×Q8—151no—
296C2×Q8—151no—
297C2×D4—51no—
298C2×D4—151no—
299C2×D4—301no—
300C2×D4—301no—
301C22×C4—31no—
302C22×C4—301no—
303C22×C4—301no—
304Q16—152no—
305Q16—152no—
306SD16—52no—
307SD16—152no—
308SD16—152no—
309SD16—152no—
310D8—52no—
311D8—152no—
312M16—31no—
313M16—32no—
314M16D3×D511yescharacteristic, complemented
315C2×C8—32no—
316C3×C5#57511yescharacteristic, complemented, Hall
317C22×C3—102no—
318C22×C3—101no—
319C22×C3—202no—
320C22×C3C2×D2011yescharacteristic
321C2×D3—41no—
322C2×D3—101no—
323C2×D3—202no—
324C2×D3C2×D2011yescharacteristic
325C2×D3C2×D2011yescharacteristic
326C3×C4—101no—
327C3×C4C2×D2011yescharacteristic
328C3×C4C23×D511yescharacteristic
329Dic3—101no—
330Dic3—102no—
331Dic3C2×D2011yescharacteristic
332Dic3C2×D2011yescharacteristic
333C2×C5—21no—
334C2×C5—61no—
335C2×C5—61no—
336C2×C5C2×D3×D411yescharacteristic
337D5—42no—
338D5—41no—
339D5—121no—
340C23—31no—
341C23—101no—
342C23—301no—
343Q8—51no—
344Q8—151no—
345Q8—152no—
346Q8—301no—
347Q8—301no—
348D4—51no—
349D4—52no—
350D4—101no—
351D4—101no—
352D4—151no—
353D4—301no—
354D4—301no—
355D4—302no—
356D4—301no—
357D4—302no—
358D4—301no—
359C2×C4—31no—
360C2×C4—31no—
361C2×C4—31no—
362C2×C4—31no—
363C2×C4—31no—
364C2×C4—101no—
365C2×C4—301no—
366C2×C4—301no—
367C2×C4—302no—
368C2×C4—302no—
369C2×C4—301no—
370C2×C4—301no—
371C2×C4—301no—
372C2×C4C2×D3×D511yescharacteristic
373C8—32no—
374C8C2×D3×D512yes—
375C2×C3—21no—
376C2×C3—201no—
377C2×C3—202no—
378C2×C3C22×D2011yescharacteristic
379D3—21no—
380D3—21no—
381D3—201no—
382C5#577311yescharacteristic, complemented, Sylow
383C22—31no—
384C22—31no—
385C22—101no—
386C22—102no—
387C22—121no—
388C22—202no—
389C22—301no—
390C22—602no—
391C22D3×D2011yescharacteristic
392C4—31no—
393C4—31no—
394C4—101no—
395C4—302no—
396C4—301no—
397C4D3×D2011yescharacteristic
398C4C22×D3×D511yescharacteristic
399C3#6608211yescharacteristic, complemented, Sylow
400C2—21no—
401C2—61no—
402C2—61no—
403C2—202no—
404C2—201no—
405C2—601no—
406C2C2×D3×D2011yescharacteristic
subgroupG/H
centerC2C2×D3×D20
commutator, cosocleC3×C4×C5C24
FrattiniC4C22×D3×D5
FittingC3×C5×M16C22
socleC2×C3×C5C22×D4
hypercenterM16D3×D5
hypocenterC3×C5#575
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C2×C4 ◁ M16

Complements

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

NH ≅ G/Ntype
1. D3×#709C2semidirect
2. C2×#6765C2semidirect
3. #91798C2semidirect
4. C5×D3×M16C2semidirect
5. C3×#4054C2semidirect
6. #91313C2semidirect
7. #91312C2semidirect
8. #91274C2semidirect
9. #91273C2semidirect
10. #91272C2semidirect
11. #91254C2semidirect
14. C2×Dic30C22semidirect
17. C2×C3×D20C22semidirect
22. D3×D20C22semidirect
25. #6769C22semidirect
26. D3×Dic10C22semidirect
30. #6765C22semidirect
31. #6765C22semidirect
32. C3×C5×M16C22semidirect
33. Dic60C22semidirect
34. #6706C22semidirect
35. C5×#208C22semidirect
36. C5×ΓL(1,𝔽25)C22semidirect
37. C5×D3×C8C22semidirect
38. C3×D40C22semidirect
39. C3×#677C22semidirect
40. #6662C22semidirect
41. #6657C22semidirect
42. #6657C22semidirect
43. #6653C22semidirect
53. #4054D3semidirect
101. D3×M16D5semidirect
116. C2×D20C2×D3semidirect
123. C5×M16C2×D3semidirect
128. D40C2×D3semidirect
132. #677C2×D3semidirect
171. C3×M16C2×D5semidirect
176. #208C2×D5semidirect
179. ΓL(1,𝔽25)C2×D5semidirect
180. D3×C8C2×D5semidirect
314. M16D3×D5semidirect
316. C3×C5#575semidirect
382. C5#5773semidirect
399. C3#66082semidirect

Maximal subgroups

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

subgroupindex
D3×#7092
C2×#67652
#917982
C5×D3×M162
C3×#40542
#913132
#913122
#912742
#912732
#912722
#912542
#660823
#57735