Finite Group Databasebeta

← #11754635

Subgroups — #11754635

Order 960.

668 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×#709C211yesmaximal, characteristic, complemented
2C2×D5×D12C211yesmaximal, characteristic, complemented
3#91819C211yesmaximal, characteristic, complemented
4C5×#930C211yesmaximal, characteristic, complemented
5#91577C212yesmaximal, complemented
6#91567C212yesmaximal, complemented
7#91474C212yesmaximal, complemented
8#91450C211yesmaximal, characteristic, complemented
9#91449C211yesmaximal, characteristic, complemented
10#91358C211yesmaximal, characteristic, complemented
11#91326C212yesmaximal, complemented
12#65941—31nomaximal, Hall
13C22×D3×D5C2212yes—
14C22×C3×C4×C5C2211yescharacteristic, complemented
15C2×#1220C2212yescomplemented
16C2×D60C2211yescharacteristic
17C2×C5×#67C2212yescomplemented
18C2×C5×D12C2211yescharacteristic
19C2×C3×C22⋊D5C2211yescharacteristic
20C2×C3×C22⋊D5C2211yescharacteristic, complemented
21C3×#709—24no—
22C2×C3×D20C2211yescharacteristic
23C2×C3×C4×D5C2211yescharacteristic
24C2×C3×C4×D5C2211yescharacteristic, complemented
25C2×C3×Dic10C2211yescharacteristic, complemented
26C2×#1203C2212yes—
27C2×#1202C2212yescomplemented
28C2×#1201C2212yescomplemented
29D5×D12—24no—
30#6714C2212yescomplemented
31C4⋊Dic15C2211yescharacteristic, complemented
32C5×(C2×D3)⋊C4C2212yescomplemented
33C5×C4⋊Dic3C2211yescharacteristic, complemented
34#6667C2212yescomplemented
35(C2×D3)⋊Dic5C2212yescomplemented
36#6665C2212yes—
37Dic3×Dic5C2212yescomplemented
38#5830—51nomaximal, Hall
39C2×D4×D5—31no—
40C2×#709D311yescharacteristic, complemented
41C5×#121—31no—
42#4086—32no—
43#4084—31no—
44#4083—32no—
45#4076—31no—
46#4041—31no—
47#4038—31no—
48#4031—32no—
49#4028—32no—
50C23×C3×C5C2311yescharacteristic
51C22×D15C2312yes—
52C22×C5×D3C2312yes—
53C22×C3×D5C2311yescharacteristic
54C22×C3×D5C2311yescharacteristic
55C2×D3×D5—24no—
56C2×D3×D5—42no—
57C2×D3×D5—42no—
58C2×C3×C4×C5—22no—
59C2×C3×C4×C5C2311yescharacteristic
60C2×C3×C4×C5C2311yescharacteristic
61#1220—42no—
62C2×Dic15C2312yes—
63D60—22no—
64C5×#67—42no—
65C2×C5×Dic3C2312yes—
66C5×D12—22no—
67C3×C22⋊D5—22no—
68C3×C22⋊D5—22no—
69C2×C3×Dic5C2311yescharacteristic
70C2×C3×Dic5C2311yescharacteristic
71C3×D20—22no—
72C3×C4×D5—22no—
73C3×C4×D5D414yescomplemented
74C3×Dic10—22no—
75#1203—24no—
76#1202—42no—
77#1201—42no—
78C2×C3×#41—51no—
79C22×D12—51no—
80#947—52no—
81#940—52no—
82#937—52no—
83#937—52no—
84#930—51no—
85#930—51no—
86#930—51no—
87#930D511yescharacteristic, complemented
88#926—51no—
89C23×D5—32no—
90C2×C5×D4—32no—
91C2×C5×D4—31no—
92C22×C4×C5C2×D311yescharacteristic, complemented
93C2×C22⋊D5—32no—
94C2×C22⋊D5—32no—
95C2×C22⋊D5—32no—
96C2×C22⋊D5C2×D311yescharacteristic
97C2×C22⋊D5C2×D311yescharacteristic, complemented
98D4×D5—64no—
99#709—24no—
100C2×D20—31no—
101C2×D20—32no—
102C2×D20C2×D311yescharacteristic
103C2×C4×D5C2×D311yescharacteristic
104C2×C4×D5C2×D311yescharacteristic, complemented
105C2×Dic10C2×D311yescharacteristic, complemented
106C5×C4⋊C4—31no—
107C5×C22⋊C4—32no—
108C22⋊Dic5—32no—
109#685—32no—
110#685—32no—
111#685—32no—
112C4⋊Dic5—31no—
113C4×Dic5—32no—
114#535—151noSylow
115C22×C3×C5—21no—
116C22×C3×C5—21no—
117C22×C3×C5—21no—
118C22×C3×C5C2411yescharacteristic
119C2×D15—22no—
120C2×D15—22no—
121C2×D15—22no—
122C2×C5×D3—22no—
123C2×C5×D3—22no—
124C2×C5×D3—22no—
125C2×C3×D5—21no—
126C2×C3×D5—21no—
127C2×C3×D5—21no—
128C2×C3×D5—21no—
129C2×C3×D5—21no—
130C2×C3×D5C2×D412yes—
131D3×D5—48no—
132C3×C4×C5—21no—
133C3×C4×C5C2×D412yes—
134Dic15—22no—
135C3×Dic5—21no—
136C3×Dic5C2×D412yes—
137C5×Dic3—22no—
138C23×D3—52no—
139C3×#41—104no—
140C2×C3×Q8—51no—
141C2×C3×D4—51no—
142C2×C3×D4—51no—
143C2×C3×D4—51no—
144C22×C3×C4—51no—
145C22×C3×C4—51no—
146C22×C3×C4C2×D511yescharacteristic, complemented
147C2×#67—52no—
148C2×#67—52no—
149C2×#67—52no—
150C2×#67C2×D512yescomplemented
151C2×D12—52no—
152C2×D12—51no—
153C2×D12—104no—
154C2×D12C2×D511yescharacteristic
155#217—52no—
156(C2×D3)⋊C4—52no—
157(C2×D3)⋊C4—52no—
158(C2×D3)⋊C4—52no—
159(C2×D3)⋊C4C2×D512yescomplemented
160C4⋊Dic3—51no—
161C4⋊Dic3C2×D511yescharacteristic, complemented
162C4×Dic3—52no—
163C23×C5—32no—
164C23×C5C22×D311yescharacteristic
165C22×D5—32no—
166C22×D5—64no—
167C22×D5—122no—
168C22×D5—122no—
169C22×D5C22×D311yescharacteristic
170C22×D5C22×D311yescharacteristic
171C5×D4—62no—
172C5×D4—122no—
173C2×C4×C5—22no—
174C2×C4×C5—32no—
175C2×C4×C5C22×D311yescharacteristic
176C2×C4×C5C22×D311yescharacteristic
177C22⋊D5—22no—
178C22⋊D5—22no—
179C22⋊D5—64no—
180C22⋊D5—122no—
181C22⋊D5—122no—
182C2×Dic5—32no—
183C2×Dic5C22×D311yescharacteristic
184C2×Dic5C22×D311yescharacteristic
185D20—22no—
186D20—62no—
187D20—122no—
188C4×D5—22no—
189C4×D5#6714yescomplemented
190Dic10—22no—
191C2×#41—51no—
192C22×D4—151no—
193#127—152no—
194#124—152no—
195#121—31no—
196#121—151no—
197#121—151no—
198#121—151no—
199#120—152no—
200#120—152no—
201#117—151no—
202C2×C3×C5—41no—
203C2×C3×C5C22×D411yescharacteristic
204C2×C3×C5H(5,𝔽2)11yescharacteristic
205C2×C3×C5H(5,𝔽2)11yescharacteristic
206D15—42no—
207C3×D5—22no—
208C3×D5—41no—
209C5×D3—42no—
210C23×C3—51no—
211C23×C3—51no—
212C23×C3C22×D511yescharacteristic
213C22×D3—52no—
214C22×D3—104no—
215C22×D3—202no—
216C22×D3—202no—
217C22×D3C22×D512yes—
218C3×Q8—102no—
219C3×D4—102no—
220C3×D4—102no—
221C3×D4—102no—
222C2×C3×C4—22no—
223C2×C3×C4—51no—
224C2×C3×C4—54no—
225C2×C3×C4—51no—
226C2×C3×C4—102no—
227C2×C3×C4C22×D511yescharacteristic
228C2×C3×C4C22×D511yescharacteristic
229#67—42no—
230#67—202no—
231#67—202no—
232#67—202no—
233C2×Dic3—52no—
234C2×Dic3C22×D512yes—
235D12—22no—
236D12—104no—
237D12—102no—
238C22×C5—21no—
239C22×C5—21no—
240C22×C5—21no—
241C22×C5—62no—
242C22×C5—62no—
243C22×C5—62no—
244C22×C5C23×D311yescharacteristic
245C2×D5—21no—
246C2×D5—21no—
247C2×D5—21no—
248C2×D5—21no—
249C2×D5—21no—
250C2×D5—62no—
251C2×D5—62no—
252C2×D5—62no—
253C2×D5—128no—
254C2×D5C2×#6712yes—
255C4×C5—21no—
256C4×C5—62no—
257C4×C5C2×#6712yes—
258Dic5—21no—
259Dic5—62no—
260Dic5C2×#6712yes—
261C24—152no—
262#41—104no—
263C2×Q8—51no—
264C2×D4—32no—
265C2×D4—31no—
266C2×D4—51no—
267C2×D4—51no—
268C2×D4—51no—
269C2×D4—152no—
270C2×D4—152no—
271C2×D4—152no—
272C2×D4—152no—
273C2×D4—151no—
274C2×D4—304no—
275C22×C4—51no—
276C22×C4—51no—
277C22×C4D3×D511yescharacteristic, complemented
278C4⋊C4—31no—
279C4⋊C4—151no—
280C22⋊C4—32no—
281C22⋊C4—152no—
282C22⋊C4—152no—
283C22⋊C4—152no—
284C22⋊C4—152no—
285C42—152no—
286C3×C5#53511yescharacteristic, complemented, Hall
287C22×C3—21no—
288C22×C3—21no—
289C22×C3—21no—
290C22×C3—52no—
291C22×C3—101no—
292C22×C3—101no—
293C22×C3—101no—
294C22×C3—101no—
295C22×C3—101no—
296C22×C3C23×D511yescharacteristic
297C2×D3—22no—
298C2×D3—22no—
299C2×D3—22no—
300C2×D3—102no—
301C2×D3—102no—
302C2×D3—102no—
303C2×D3—208no—
304C3×C4—21no—
305C3×C4—52no—
306C3×C4—101no—
307C3×C4D4×D512yes—
308Dic3—22no—
309Dic3—102no—
310C2×C5—41no—
311C2×C5—122no—
312C2×C5#100911yescharacteristic
313C2×C5#100911yescharacteristic
314C2×C5C22×#6711yescharacteristic
315D5—22no—
316D5—41no—
317D5—122no—
318C23—32no—
319C23—51no—
320C23—51no—
321C23—152no—
322C23—304no—
323C23—602no—
324C23—602no—
325C23C2×D3×D511yescharacteristic
326Q8—102no—
327D4—62no—
328D4—102no—
329D4—102no—
330D4—102no—
331D4—122no—
332D4—304no—
333D4—302no—
334D4—602no—
335D4—602no—
336D4—602no—
337C2×C4—22no—
338C2×C4—32no—
339C2×C4—51no—
340C2×C4—54no—
341C2×C4—51no—
342C2×C4—102no—
343C2×C4—152no—
344C2×C4C2×D3×D511yescharacteristic
345C2×C4C2×D3×D511yescharacteristic
346C2×C3—41no—
347C2×C3—102no—
348C2×C3—201no—
349C2×C3C2×D4×D511yescharacteristic
350C2×C3#414411yescharacteristic
351C2×C3#414911yescharacteristic
352D3—42no—
353D3—202no—
354C5#583011yescharacteristic, complemented, Sylow
355C22—21no—
356C22—21no—
357C22—21no—
358C22—52no—
359C22—62no—
360C22—62no—
361C22—62no—
362C22—101no—
363C22—101no—
364C22—101no—
365C22—101no—
366C22—101no—
367C22—302no—
368C22—302no—
369C22—302no—
370C22—608no—
371C22C22×D3×D511yescharacteristic
372C4—21no—
373C4—52no—
374C4—62no—
375C4—101no—
376C4—302no—
377C4D5×#6712yes—
378C3#6594111yescharacteristic, complemented, Sylow
379C2—41no—
380C2—102no—
381C2—122no—
382C2—201no—
383C2—602no—
384C2#9201811yescharacteristic
385C2#9201911yescharacteristic
386C2C2×D5×#6711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×D5
commutator, socle, cosocleC22×C3×C5C24
FittingC22×C3×C4×C5C22
hypercenterC22×C4D3×D5
hypocenterC3×C5#535
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×C3×#709C2semidirect
2. C2×D5×D12C2semidirect
3. #91819C2semidirect
4. C5×#930C2semidirect
5. #91577C2semidirect
6. #91567C2semidirect
7. #91474C2semidirect
8. #91450C2semidirect
9. #91449C2semidirect
10. #91358C2semidirect
11. #91326C2semidirect
14. C22×C3×C4×C5C22semidirect
15. C2×#1220C22semidirect
17. C2×C5×#67C22semidirect
20. C2×C3×C22⋊D5C22semidirect
24. C2×C3×C4×D5C22semidirect
25. C2×C3×Dic10C22semidirect
27. C2×#1202C22semidirect
28. C2×#1201C22semidirect
30. #6714C22semidirect
31. C4⋊Dic15C22semidirect
32. C5×(C2×D3)⋊C4C22semidirect
33. C5×C4⋊Dic3C22semidirect
34. #6667C22semidirect
35. (C2×D3)⋊Dic5C22semidirect
37. Dic3×Dic5C22semidirect
40. C2×#709D3semidirect
73. C3×C4×D5D4semidirect
87. #930D5semidirect
92. C22×C4×C5C2×D3semidirect
97. C2×C22⋊D5C2×D3semidirect
104. C2×C4×D5C2×D3semidirect
105. C2×Dic10C2×D3semidirect
146. C22×C3×C4C2×D5semidirect
150. C2×#67C2×D5semidirect
159. (C2×D3)⋊C4C2×D5semidirect
161. C4⋊Dic3C2×D5semidirect
189. C4×D5#67semidirect
277. C22×C4D3×D5semidirect
286. C3×C5#535semidirect
354. C5#5830semidirect
378. C3#65941semidirect

Maximal subgroups

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

subgroupindex
C2×C3×#7092
C2×D5×D122
#918192
C5×#9302
#915772
#915672
#914742
#914502
#914492
#913582
#913262
#659413
#58305