Finite Group Databasebeta

← #11754035

Subgroups — #11754035

Order 960.

524 conjugacy classes of subgroups; only the 128 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 128 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#92016C211yesmaximal, characteristic, complemented
2C2×#6768C211yesmaximal, characteristic, complemented
3M16×D15C211yesmaximal, characteristic, complemented
4C5×#908C211yesmaximal, characteristic, complemented
5C3×#4055C211yesmaximal, characteristic, complemented
6#91309C211yesmaximal, characteristic, complemented
7#91299C211yesmaximal, characteristic, complemented
8#91280C212yesmaximal, complemented
9#91279C212yesmaximal, complemented
10#91276C212yesmaximal, complemented
11#91257C212yesmaximal, complemented
12#66107—31nomaximal, Hall
13C2×C4×D15C2211yescharacteristic
14C5×#235C2211yescharacteristic
15C2×C5×D12C2211yescharacteristic, complemented
16C3×#709C2211yescharacteristic
17C2×C3×Dic10C2211yescharacteristic, complemented
18#6790—21no—
19C2×#1203—21no—
20C2×D3×Dic5—21no—
21#6780—21no—
22#6777C2211yescharacteristic, complemented
23C4×D3×D5—21no—
24#6771—21no—
25D15⋊Q8C2211yescharacteristic, complemented
26#6768—21no—
27#6768—21no—
28#6768C2212yes—
29#6768C2212yescomplemented
30#6768C2211yescharacteristic, complemented
31#6766C2211yescharacteristic, complemented
32C3×C5×M16C2211yescharacteristic, complemented
33#6710C2211yescharacteristic, complemented
34#6705C2212yescomplemented
35C8×D15C2212yescomplemented
36C5×D24C2212yescomplemented
37C5×#204C2212yescomplemented
38C3×Dic20C2212yescomplemented
39C3×#677C2212yescomplemented
40#6661C2212yescomplemented
41#6656C2212yescomplemented
42#6656C2212yescomplemented
43#6652C2212yescomplemented
44#5777—51nomaximal, Hall
45D5×#41—31no—
46C2×#711—31no—
47C5×AGL(1,ℤ/8)—31no—
48#4097—31no—
49#4078—31no—
50#4062—32no—
51#4061—32no—
52#4058—32no—
53#4057—32no—
54#4055D311yescharacteristic, complemented
55D5×M16—31no—
56C22×D15D411yescharacteristic
57C22×C5×D3—21no—
58C2×D3×D5—21no—
59C2×C3×C4×C5C2311yescharacteristic
60C2×Dic15D411yescharacteristic
61C4×D15D411yescharacteristic
62C4×D15D411yescharacteristic
63C4×D15C2311yescharacteristic
64C4×D15C2311yescharacteristic
65C5×#67—21no—
66C5×D12—21no—
67C5×D12C2311yescharacteristic
68C5×D12C2312yes—
69C4×C5×D3—21no—
70C5×Dic6C2311yescharacteristic
71C3×C22⋊D5—21no—
72C2×C3×Dic5—21no—
73C3×D20C2311yescharacteristic
74C3×C4×D5—21no—
75C3×Dic10—21no—
76C3×Dic10C2311yescharacteristic
77C3×Dic10C2312yes—
78#1204—21no—
79#1203—22no—
80#1203—22no—
81#1203—21no—
82#1202—21no—
83#1201—21no—
84#1200—21no—
85D3×Dic5—22no—
86D3×Dic5—22no—
87D3×Dic5—21no—
88D5×Dic3—21no—
89C3×C5×C8C2312yes—
90C3⋊#168C2312yes—
91D3×#41—51no—
92C2×#239—51no—
93C3×#137—51no—
94#949—51no—
95#942—51no—
96#919—52no—
97#918—52no—
98#916—52no—
99#914—52no—
100#908D511yescharacteristic, complemented
101D3×M16—51no—
102C5×#41—31no—
103C2×C5×D4—31no—
104C2×C22⋊D5—61no—
105C22×Dic5—61no—
106#713—31no—
107Q8×D5—31no—
108#711—32no—
109#711—32no—
110#711—31no—
111#711—61no—
112#711—61no—
113#711—61no—
114D4×D5—31no—
115D4×D5—61no—
116#709—61no—
117#709C2×D311yescharacteristic
118C2×C4×D5—31no—
119C2×C4×D5—61no—
120C2×Dic10C2×D311yescharacteristic, complemented
121C5×SD16—32no—
122C5×D8—32no—
123C5×M16C2×D311yescharacteristic, complemented
124#689—32no—
125#687—32no—
126#687—32no—
127#686—32no—
128#681—31no—
129Dic20C2×D312yescomplemented
130#677C2×D312yescomplemented
131#676—32no—
132C8×D5—32no—
133#575—151noSylow
134C22×C3×C5C2×D411yescharacteristic
135C2×D15—41no—
136C2×D15C2×D411yescharacteristic
137C2×D15C2×D411yescharacteristic
138C2×C5×D3—22no—
139C2×C5×D3—21no—
140C2×C5×D3—42no—
141C2×C3×D5—21no—
142D3×D5—42no—
143C3×C4×C5C2×D411yescharacteristic
144C3×C4×C5C2411yescharacteristic
145Dic15C2×D411yescharacteristic
146Dic15C2×D411yescharacteristic
147C3×Dic5—22no—
148C3×Dic5—21no—
149C5×Dic3—21no—
150C3×#41—51no—
151C2×C3×Q8—51no—
152#239—52no—
153#239—52no—
154#239—51no—
155#239—101no—
156#239—101no—
157D3×Q8—51no—
158#237—51no—
159#237—101no—
160D3×D4—51no—
161D3×D4—101no—
162#235—101no—
163#235C2×D511yescharacteristic
164C2×D12—101no—
165C2×D12C2×D511yescharacteristic, complemented
166C2×C4×D3—51no—
167C2×C4×D3—101no—
168C2×C4×D3—101no—
169C3×Q16—52no—
170C3×SD16—52no—
171C3×M16C2×D511yescharacteristic, complemented
172#216—52no—
173#215—52no—
174#215—52no—
175#213—52no—
176#208—51no—
177D24C2×D512yescomplemented
178#204C2×D512yescomplemented
179ΓL(1,𝔽25)—52no—
180D3×C8—52no—
181C23×C5—61no—
182C22×D5—31no—
183C22×D5—61no—
184C5×Q8—31no—
185C5×D4—32no—
186C5×D4—31no—
187C5×D4—61no—
188C5×D4—61no—
189C2×C4×C5—61no—
190C2×C4×C5C22×D311yescharacteristic
191C22⋊D5—21no—
192C22⋊D5—61no—
193C22⋊D5—62no—
194C22⋊D5—62no—
195C22⋊D5—61no—
196C2×Dic5—21no—
197C2×Dic5—31no—
198C2×Dic5—61no—
199C2×Dic5—62no—
200C2×Dic5—62no—
201D20—61no—
202D20C22×D311yescharacteristic
203C4×D5—21no—
204C4×D5—31no—
205C4×D5—31no—
206C4×D5—31no—
207C4×D5—31no—
208C4×D5—61no—
209C4×D5—61no—
210Dic10—21no—
211Dic10—61no—
212Dic10C22×D312yes—
213Dic10C22×D311yescharacteristic
214C5×C8C22×D312yes—
215#168—32no—
216C2×#41—151no—
217C2×#41—151no—
218#137—51no—
219#137—152no—
220#137—151no—
221AGL(1,ℤ/8)—31no—
222AGL(1,ℤ/8)—152no—
223AGL(1,ℤ/8)—151no—
224#135—152no—
225#135—152no—
226C2×M16—151no—
227C2×C3×C5—21no—
228C2×C3×C5C22×D411yescharacteristic
229D15—21no—
230D15—21no—
231C3×D5—41no—
232C5×D3—41no—
233C5×D3—42no—
234C22×D3—21no—
235C22×D3—51no—
236C22×D3—101no—
237C3×Q8—51no—
238C3×Q8—52no—
239C3×Q8—101no—
240C3×D4—51no—
241C3×D4—101no—
242C2×C3×C4—101no—
243C2×C3×C4—101no—
244C2×C3×C4C22×D511yescharacteristic
245#67—21no—
246#67—101no—
247#67—101no—
248C2×Dic3—51no—
249C2×Dic3—101no—
250D12—21no—
251D12—102no—
252D12—102no—
253D12—101no—
254D12C22×D512yes—
255D12C22×D511yescharacteristic
256C4×D3—21no—
257C4×D3—51no—
258C4×D3—51no—
259C4×D3—51no—
260C4×D3—51no—
261C4×D3—101no—
262C4×D3—101no—
263C4×D3—102no—
264C4×D3—102no—
265Dic6—101no—
266Dic6C22×D511yescharacteristic
267C3×C8C22×D512yes—
268C3⋊C8—52no—
269C22×C5—62no—
270C22×C5—61no—
271C22×C5—122no—
272C22×C5D3×D411yescharacteristic
273C2×D5—21no—
274C2×D5—31no—
275C2×D5—31no—
276C2×D5—122no—
277C2×D5—121no—
278C4×C5—61no—
279C4×C5D3×D411yescharacteristic
280C4×C5C23×D311yescharacteristic
281Dic5—22no—
282Dic5—21no—
283Dic5—31no—
284Dic5—31no—
285#41—31no—
286#41—51no—
287#41—151no—
288#41—152no—
289#41—152no—
290#41—151no—
291#41—301no—
292#41—301no—
293#41—301no—
294#41—301no—
295C2×Q8—51no—
296C2×Q8—151no—
297C2×D4—31no—
298C2×D4—151no—
299C2×D4—301no—
300C2×D4—301no—
301C22×C4—151no—
302C22×C4—301no—
303C22×C4—301no—
304Q16—52no—
305Q16—152no—
306SD16—32no—
307SD16—52no—
308SD16—152no—
309SD16—152no—
310D8—32no—
311D8—152no—
312M16—151no—
313M16—152no—
314M16D3×D511yescharacteristic, complemented
315C2×C8—152no—
316C3×C5#57511yescharacteristic, complemented, Hall
317C22×C3—101no—
318C22×C3D4×D511yescharacteristic
319C2×D3—21no—
320C2×D3—22no—
321C2×D3—42no—
322C2×D3—51no—
323C2×D3—51no—
324C2×D3—202no—
325C2×D3—201no—
326C3×C4—101no—
327C3×C4—102no—
328C3×C4D4×D511yescharacteristic
329C3×C4C23×D511yescharacteristic
330Dic3—21no—
331Dic3—51no—
332Dic3—51no—
333C2×C5—21no—
334C2×C5—121no—
335C2×C5—122no—
336C2×C5C2×D3×D411yescharacteristic
337D5—41no—
338D5—61no—
339D5—61no—
340C23—61no—
341C23—151no—
342C23—301no—
343Q8—31no—
344Q8—52no—
345Q8—51no—
346Q8—101no—
347Q8—301no—
348D4—31no—
349D4—32no—
350D4—51no—
351D4—61no—
352D4—61no—
353D4—101no—
354D4—301no—
355D4—301no—
356D4—302no—
357D4—302no—
358D4—301no—
359C2×C4—61no—
360C2×C4—101no—
361C2×C4—101no—
362C2×C4—151no—
363C2×C4—151no—
364C2×C4—151no—
365C2×C4—151no—
366C2×C4—151no—
367C2×C4—301no—
368C2×C4—301no—
369C2×C4—301no—
370C2×C4—302no—
371C2×C4—302no—
372C2×C4C2×D3×D511yescharacteristic
373C8—152no—
374C8C2×D3×D512yes—
375C2×C3—21no—
376C2×C3—201no—
377C2×C3C2×D4×D511yescharacteristic
378D3—42no—
379D3—41no—
380D3—101no—
381D3—101no—
382C5#577711yescharacteristic, complemented, Sylow
383C22—61no—
384C22—62no—
385C22—101no—
386C22—122no—
387C22—151no—
388C22—151no—
389C22—601no—
390C22—602no—
391C22#677711yescharacteristic
392C4—61no—
393C4—101no—
394C4—102no—
395C4—151no—
396C4—151no—
397C4#677711yescharacteristic
398C4C22×D3×D511yescharacteristic
399C3#6610711yescharacteristic, complemented, Sylow
400C2—21no—
401C2—122no—
402C2—121no—
403C2—201no—
404C2—301no—
405C2—301no—
406C2C2×#677711yescharacteristic
subgroupG/H
centerC2C2×#6777
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. #92016C2semidirect
2. C2×#6768C2semidirect
3. M16×D15C2semidirect
4. C5×#908C2semidirect
5. C3×#4055C2semidirect
6. #91309C2semidirect
7. #91299C2semidirect
8. #91280C2semidirect
9. #91279C2semidirect
10. #91276C2semidirect
11. #91257C2semidirect
15. C2×C5×D12C22semidirect
17. C2×C3×Dic10C22semidirect
22. #6777C22semidirect
25. D15⋊Q8C22semidirect
29. #6768C22semidirect
30. #6768C22semidirect
31. #6766C22semidirect
32. C3×C5×M16C22semidirect
33. #6710C22semidirect
34. #6705C22semidirect
35. C8×D15C22semidirect
36. C5×D24C22semidirect
37. C5×#204C22semidirect
38. C3×Dic20C22semidirect
39. C3×#677C22semidirect
40. #6661C22semidirect
41. #6656C22semidirect
42. #6656C22semidirect
43. #6652C22semidirect
54. #4055D3semidirect
100. #908D5semidirect
120. C2×Dic10C2×D3semidirect
123. C5×M16C2×D3semidirect
129. Dic20C2×D3semidirect
130. #677C2×D3semidirect
165. C2×D12C2×D5semidirect
171. C3×M16C2×D5semidirect
177. D24C2×D5semidirect
178. #204C2×D5semidirect
314. M16D3×D5semidirect
316. C3×C5#575semidirect
382. C5#5777semidirect
399. C3#66107semidirect

Maximal subgroups

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

subgroupindex
#920162
C2×#67682
M16×D152
C5×#9082
C3×#40552
#913092
#912992
#912802
#912792
#912762
#912572
#661073
#57775