Finite Group Databasebeta

← #11754037

Subgroups — #11754037

Order 960.

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

normal subgroups only

This group has 516 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#92017C211yesmaximal, characteristic, complemented
2#92000C211yesmaximal, characteristic, complemented
3#91796C211yesmaximal, characteristic, complemented
4C5×#909C211yesmaximal, characteristic, complemented
5C3×#4054C211yesmaximal, characteristic, complemented
6C2×#6655C211yesmaximal, characteristic, complemented
7#91297C211yesmaximal, characteristic, complemented
8#91278C212yesmaximal, complemented
9#91277C212yesmaximal, complemented
10#91258C212yesmaximal, complemented
11#91255C212yesmaximal, complemented
12#66112—31nomaximal, Hall
13#6817C2211yescharacteristic
14C5×#235C2211yescharacteristic
15C2×C5×Dic6C2211yescharacteristic, complemented
16C3×#709C2211yescharacteristic, complemented
17C2×C3×D20C2211yescharacteristic
18D5×#67—22no—
19#6779—22no—
20#6777C2212yes—
21D3×D20—21no—
22#6773—21no—
23D15⋊Q8C2212yescomplemented
24#6766C2212yes—
25#6766C2212yescomplemented
26#6765—21no—
27D5×Dic6—21no—
28C3×C5×M16C2211yescharacteristic, complemented
29C2×C3⋊#168C2211yescharacteristic, complemented
30#6705C2212yescomplemented
31C8×D15C2212yescomplemented
32C5×Dic12C2212yescomplemented
33C5×#204C2212yescomplemented
34C3×D40C2212yescomplemented
35C3×#677C2212yescomplemented
36#6661C2211yescharacteristic, complemented
37#6656C2211yescharacteristic, complemented
38#6655C2212yescomplemented
39#6655C2212yescomplemented
40#6655C2211yescharacteristic, complemented
41#6652C2211yescharacteristic, complemented
42#5775—51nomaximal, Hall
43#4149—31no—
44#4147—31no—
45C5×#137—31no—
46#4096—31no—
47C2×#688—31no—
48#4065—32no—
49#4064—32no—
50#4060—32no—
51D5×SD16—32no—
52#4054D311yescharacteristic, complemented
53#4053—31no—
54C22×C3×D5—21no—
55C2×D3×D5—22no—
56C2×C3×C4×C5C2311yescharacteristic
57#1220D412yes—
58D60D411yescharacteristic
59C4×D15C2312yes—
60Dic30D411yescharacteristic
61C5×#67—21no—
62C2×C5×Dic3—21no—
63C5×D12C2311yescharacteristic
64C4×C5×D3—21no—
65C5×Dic6—21no—
66C5×Dic6C2311yescharacteristic
67C5×Dic6C2312yes—
68C3×C22⋊D5—21no—
69C3×D20—21no—
70C3×D20C2312yes—
71C3×D20C2311yescharacteristic
72C3×C4×D5—21no—
73C3×Dic10C2311yescharacteristic
74#1204—22no—
75#1202—22no—
76#1202—22no—
77#1201—22no—
78#1200—22no—
79D5×Dic3—22no—
80D5×Dic3—22no—
81C3×C5×C8C2312yes—
82C3⋊#168C2311yescharacteristic
83C3⋊#168C2311yescharacteristic
84#1010—51no—
85#1004—51no—
86C3×AGL(1,ℤ/8)—51no—
87#950—51no—
88C2×#214—51no—
89#915—52no—
90D3×SD16—52no—
91#912—52no—
92#911—52no—
93#909D511yescharacteristic, complemented
94#907—51no—
95C5×#41—31no—
96C2×C5×Q8—31no—
97#713—32no—
98#713—32no—
99#713—61no—
100Q8×D5—32no—
101Q8×D5—61no—
102D4×D5—32no—
103D4×D5—62no—
104#709—31no—
105#709—61no—
106#709—62no—
107#709C2×D311yescharacteristic, complemented
108C2×D20—61no—
109C2×D20C2×D311yescharacteristic
110C5×Q16—32no—
111C5×SD16—32no—
112C5×M16C2×D311yescharacteristic, complemented
113#689—31no—
114#688—31no—
115#688—32no—
116#688—32no—
117#687—31no—
118#686—31no—
119C2×#168—31no—
120D40C2×D312yescomplemented
121#677C2×D312yescomplemented
122#676—32no—
123C8×D5—32no—
124#577—151noSylow
125C22×C3×C5C2×D411yescharacteristic
126C2×D15C2×D412yes—
127C2×C5×D3—21no—
128C2×C3×D5—21no—
129C2×C3×D5—22no—
130C2×C3×D5—81no—
131D3×D5—82no—
132C3×C4×C5C2×D411yescharacteristic
133C3×C4×C5C2411yescharacteristic
134Dic15C2×D412yes—
135C3×Dic5—21no—
136C5×Dic3—21no—
137C5×Dic3—22no—
138C3×#41—51no—
139C2×C3×D4—51no—
140C2×#67—102no—
141D3×Q8—52no—
142#237—52no—
143#237—52no—
144#237—101no—
145#237—102no—
146D3×D4—52no—
147D3×D4—101no—
148#235—51no—
149#235—101no—
150#235C2×D511yescharacteristic
151C2×Dic6—101no—
152C2×Dic6C2×D511yescharacteristic, complemented
153C3×SD16—52no—
154C3×D8—52no—
155C3×M16C2×D511yescharacteristic, complemented
156#216—51no—
157#215—51no—
158#214—52no—
159#214—51no—
160#214—52no—
161#213—51no—
162C2×C3⋊C8—51no—
163Dic12C2×D512yescomplemented
164#204C2×D512yescomplemented
165ΓL(1,𝔽25)—52no—
166D3×C8—52no—
167C22×D5—21no—
168C22×D5—62no—
169C5×Q8—32no—
170C5×Q8—31no—
171C5×Q8—61no—
172C5×D4—31no—
173C5×D4—61no—
174C2×C4×C5—61no—
175C2×C4×C5—61no—
176C2×C4×C5C22×D311yescharacteristic
177C22⋊D5—21no—
178C22⋊D5—32no—
179C22⋊D5—62no—
180D20—21no—
181D20—31no—
182D20—62no—
183D20—62no—
184D20C22×D312yes—
185D20C22×D311yescharacteristic
186C4×D5—21no—
187C4×D5—32no—
188C4×D5—62no—
189C4×D5—62no—
190C4×D5—62no—
191Dic10—31no—
192Dic10—62no—
193Dic10C22×D311yescharacteristic
194C5×C8C22×D312yes—
195#168—31no—
196#168—31no—
1972−1+4—151no—
198H(5,𝔽2)—151no—
199#137—31no—
200#137—152no—
201AGL(1,ℤ/8)—51no—
202AGL(1,ℤ/8)—152no—
203#135—152no—
204#135—151no—
205C2×SD16—152no—
206C2×SD16—151no—
207#131—151no—
208C2×C3×C5—21no—
209C2×C3×C5C22×D411yescharacteristic
210D15—22no—
211C3×D5—42no—
212C3×D5—41no—
213C5×D3—41no—
214C23×C3—101no—
215C22×D3—102no—
216C3×Q8—51no—
217C3×D4—51no—
218C3×D4—52no—
219C3×D4—101no—
220C3×D4—101no—
221C2×C3×C4—101no—
222C2×C3×C4C22×D511yescharacteristic
223#67—21no—
224#67—52no—
225#67—102no—
226#67—102no—
227#67—102no—
228C2×Dic3—21no—
229C2×Dic3—102no—
230C2×Dic3—102no—
231D12—51no—
232D12C22×D511yescharacteristic
233C4×D3—21no—
234C4×D3—52no—
235C4×D3—102no—
236Dic6—21no—
237Dic6—51no—
238Dic6—102no—
239Dic6C22×D512yes—
240Dic6C22×D511yescharacteristic
241C3×C8C22×D512yes—
242C3⋊C8—51no—
243C3⋊C8—51no—
244C22×C5—61no—
245C22×C5D3×D411yescharacteristic
246C2×D5—21no—
247C2×D5—22no—
248C2×D5—32no—
249C2×D5—81no—
250C2×D5—242no—
251C4×C5—62no—
252C4×C5—61no—
253C4×C5D3×D411yescharacteristic
254C4×C5C23×D311yescharacteristic
255Dic5—21no—
256Dic5—32no—
257#41—31no—
258#41—51no—
259#41—151no—
260#41—152no—
261#41—152no—
262#41—302no—
263#41—301no—
264#41—301no—
265C2×Q8—31no—
266C2×Q8—152no—
267C2×Q8—301no—
268C2×D4—51no—
269C2×D4—152no—
270C2×D4—301no—
271C2×D4—302no—
272Q16—32no—
273Q16—151no—
274SD16—32no—
275SD16—52no—
276SD16—151no—
277SD16—152no—
278SD16—152no—
279SD16—151no—
280D8—52no—
281D8—151no—
282M16—152no—
283M16D3×D511yescharacteristic, complemented
284C2×C8—151no—
285C2×C8—152no—
286C3×C5#57711yescharacteristic, complemented, Hall
287C22×C3—101no—
288C22×C3—102no—
289C22×C3—401no—
290C22×C3D4×D511yescharacteristic
291C2×D3—21no—
292C2×D3—52no—
293C2×D3—402no—
294C3×C4—101no—
295C3×C4D4×D511yescharacteristic
296C3×C4C23×D511yescharacteristic
297Dic3—22no—
298Dic3—21no—
299Dic3—52no—
300C2×C5—21no—
301C2×C5—121no—
302C2×C5C2×D3×D411yescharacteristic
303D5—42no—
304D5—41no—
305D5—62no—
306C23—101no—
307C23—302no—
308Q8—32no—
309Q8—31no—
310Q8—51no—
311Q8—61no—
312Q8—151no—
313Q8—302no—
314D4—31no—
315D4—51no—
316D4—52no—
317D4—61no—
318D4—101no—
319D4—101no—
320D4—152no—
321D4—151no—
322D4—302no—
323D4—302no—
324D4—302no—
325C2×C4—61no—
326C2×C4—61no—
327C2×C4—101no—
328C2×C4—152no—
329C2×C4—302no—
330C2×C4—302no—
331C2×C4—302no—
332C2×C4C2×D3×D511yescharacteristic
333C8—151no—
334C8—151no—
335C8C2×D3×D512yes—
336C2×C3—21no—
337C2×C3—202no—
338C2×C3—201no—
339C2×C3C2×D4×D511yescharacteristic
340D3—41no—
341D3—102no—
342C5#577511yescharacteristic, complemented, Sylow
343C22—61no—
344C22—101no—
345C22—102no—
346C22—152no—
347C22—401no—
348C22—1202no—
349C22#677711yescharacteristic
350C4—61no—
351C4—62no—
352C4—101no—
353C4—152no—
354C4#677711yescharacteristic
355C4C22×D3×D511yescharacteristic
356C3#6611211yescharacteristic, complemented, Sylow
357C2—21no—
358C2—121no—
359C2—202no—
360C2—201no—
361C2—302no—
362C2C2×#677711yescharacteristic
subgroupG/H
centerC2C2×#6777
commutator, cosocleC3×C4×C5C24
FrattiniC4C22×D3×D5
FittingC3×C5×M16C22
socleC2×C3×C5C22×D4
hypercenterM16D3×D5
hypocenterC3×C5#577
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. #92017C2semidirect
2. #92000C2semidirect
3. #91796C2semidirect
4. C5×#909C2semidirect
5. C3×#4054C2semidirect
6. C2×#6655C2semidirect
7. #91297C2semidirect
8. #91278C2semidirect
9. #91277C2semidirect
10. #91258C2semidirect
11. #91255C2semidirect
15. C2×C5×Dic6C22semidirect
16. C3×#709C22semidirect
23. D15⋊Q8C22semidirect
25. #6766C22semidirect
28. C3×C5×M16C22semidirect
29. C2×C3⋊#168C22semidirect
30. #6705C22semidirect
31. C8×D15C22semidirect
32. C5×Dic12C22semidirect
33. C5×#204C22semidirect
34. C3×D40C22semidirect
35. C3×#677C22semidirect
36. #6661C22semidirect
37. #6656C22semidirect
38. #6655C22semidirect
39. #6655C22semidirect
40. #6655C22semidirect
41. #6652C22semidirect
52. #4054D3semidirect
93. #909D5semidirect
107. #709C2×D3semidirect
112. C5×M16C2×D3semidirect
120. D40C2×D3semidirect
121. #677C2×D3semidirect
152. C2×Dic6C2×D5semidirect
155. C3×M16C2×D5semidirect
163. Dic12C2×D5semidirect
164. #204C2×D5semidirect
283. M16D3×D5semidirect
286. C3×C5#577semidirect
342. C5#5775semidirect
356. C3#66112semidirect

Maximal subgroups

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

subgroupindex
#920172
#920002
#917962
C5×#9092
C3×#40542
C2×#66552
#912972
#912782
#912772
#912582
#912552
#661123
#57755