Finite Group Databasebeta

← #14328007

Subgroups — #14328007

Order 1920.

1120 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×D60C211yesmaximal, characteristic
2C3×C5×#382C211yesmaximal, characteristic, complemented
3#11751089C212yesmaximal
4C4⋊C4×Dic15C212yesmaximal, complemented
5C4×C4⋊Dic15C211yesmaximal, characteristic, complemented
6#11750771C214yesmaximal, complemented
7#11750767C212yesmaximal, complemented
8#11750765C212yesmaximal, complemented
9#10605891—31nomaximal, Hall
10C22×D60C2211yescharacteristic
11C22×C4×D15C2212yes—
12C2×C3×C5×C4⋊C4C2212yes—
13C2×C3×C42×C5C2211yescharacteristic
14C2×#6714C2214yes—
15C2×#6714C2214yes—
16C2×#6714C2212yes—
17C2×C4⋊Dic15C2212yes—
18C2×C4⋊Dic15C2211yescharacteristic
19C2×C4×Dic15C2212yes—
20C2×C4×Dic15C2214yes—
21C4×D60C418yescomplemented
22C3×C5×#95C2212yes—
23C3×C5×#95C2212yes—
24#91115C2211yescharacteristic
25#91115C2214yes—
26#91115C2211yescharacteristic
27#74160—51nomaximal, Hall
28C2×C4×D20—31no—
29C5×#382D311yescharacteristic, complemented
30#65277—32no—
31C4⋊C4×Dic5—32no—
32C4×C4⋊Dic5—31no—
33#64959—34no—
34#64955—32no—
35#64953—32no—
36C23×D15C2312yes—
37C22×C3×C4×C5C2314yes—
38C22×C3×C4×C5C2311yescharacteristic
39C22×C3×C4×C5C2311yescharacteristic
40C22×C3×C4×C5C2311yescharacteristic
41C22×Dic15C2312yes—
42C22×Dic15C2312yes—
43C22×Dic15C2312yes—
44C2×D60—22no—
45C2×D60—22no—
46C2×D60C2×C414yes—
47C2×C4×D15—44no—
48C2×C4×D15C2×C418yes—
49C3×C5×C4⋊C4—24no—
50C3×C42×C5C2×C414yes—
51#6714—28no—
52#6714—44no—
53#6714C2×C418yes—
54C4⋊Dic15—24no—
55C4⋊Dic15C2×C414yes—
56C4×Dic15—24no—
57C4×Dic15—44no—
58C2×C4×D12—51no—
59C3×#382D511yescharacteristic, complemented
60#4988—52no—
61Dic3×C4⋊C4—52no—
62C4×C4⋊Dic3—51no—
63#4670—54no—
64#4666—52no—
65#4664—52no—
66C22×D20—31no—
67C22×C4×D5—32no—
68C2×C5×C4⋊C4C2×D312yes—
69C2×C42×C5C2×D311yescharacteristic
70C2×#685—32no—
71C2×#685—34no—
72C2×#685—34no—
73C2×C4⋊Dic5—31no—
74C2×C4⋊Dic5—32no—
75C2×C4×Dic5—32no—
76C2×C4×Dic5—34no—
77C4×D20—38no—
78C5×#95C2×D312yes—
79C5×#95C2×D312yes—
80#3963—31no—
81#3963—34no—
82#3963—31no—
83#2357—151noSylow
84C23×C3×C5C2411yescharacteristic
85C22×D15—24no—
86C22×D15—22no—
87C22×D15—22no—
88C22×D15—24no—
89C22×D15C22×C414yes—
90C2×C3×C4×C5—22no—
91C2×C3×C4×C5—22no—
92C2×C3×C4×C5—24no—
93C2×C3×C4×C5—24no—
94C2×C3×C4×C5—24no—
95C2×C3×C4×C5C22×C412yes—
96C2×C3×C4×C5C22×C412yes—
97C2×C3×C4×C5C22×C412yes—
98C2×C3×C4×C5#4112yes—
99C2×C3×C4×C5#4112yes—
100C2×Dic15—22no—
101C2×Dic15—22no—
102C2×Dic15—24no—
103C2×Dic15—22no—
104C2×Dic15—22no—
105C2×Dic15—22no—
106C2×Dic15C22×C414yes—
107C2×Dic15#4114yes—
108D60—28no—
109C4×D15—48no—
110C22×D12—51no—
111C22×C4×D3—52no—
112C2×C3×C4⋊C4C2×D512yes—
113C2×C3×C42C2×D511yescharacteristic
114C2×(C2×D3)⋊C4—52no—
115C2×(C2×D3)⋊C4—54no—
116C2×(C2×D3)⋊C4—54no—
117C2×C4⋊Dic3—51no—
118C2×C4⋊Dic3—52no—
119C2×C4×Dic3—52no—
120C2×C4×Dic3—54no—
121C4×D12—58no—
122C3×#95C2×D512yes—
123C3×#95C2×D512yes—
124C3⋊#95—51no—
125C3⋊#95—54no—
126C3⋊#95—51no—
127C23×D5—32no—
128C22×C4×C5C22×D314yes—
129C22×C4×C5C22×D311yescharacteristic
130C22×C4×C5C22×D311yescharacteristic
131C22×C4×C5C22×D311yescharacteristic
132C22×Dic5—32no—
133C22×Dic5—32no—
134C22×Dic5—32no—
135C2×D20—34no—
136C2×D20—62no—
137C2×D20—62no—
138C2×C4×D5—38no—
139C2×C4×D5—124no—
140C5×C4⋊C4—24no—
141C42×C5C4×D314yes—
142#685—38no—
143#685—68no—
144#685—124no—
145C4⋊Dic5—34no—
146C4⋊Dic5—64no—
147C4×Dic5—64no—
148C4×Dic5—124no—
149C2×C4×D4—151no—
150#390—152no—
151#388—154no—
152#382D1511yescharacteristic, complemented
153#381—152no—
154C4×C4⋊C4—152no—
155C4×C4⋊C4—151no—
156C4×C22⋊C4—152no—
157C22×C3×C5C23×C411yescharacteristic
158C22×C3×C5C2×#4111yescharacteristic
159C22×C3×C5C2×#4111yescharacteristic
160C22×C3×C5C2×#4112yes—
161C22×C3×C5H(5,𝔽2)12yes—
162C2×D15—24no—
163C2×D15—24no—
164C2×D15—24no—
165C2×D15—44no—
166C2×D15—44no—
167C3×C4×C5—22no—
168C3×C4×C5—22no—
169C3×C4×C5—44no—
170C3×C4×C5#11714yes—
171Dic15—24no—
172Dic15—24no—
173Dic15—42no—
174C23×D3—52no—
175C22×C3×C4C22×D514yes—
176C22×C3×C4C22×D511yescharacteristic
177C22×C3×C4C22×D511yescharacteristic
178C22×C3×C4C22×D511yescharacteristic
179C22×Dic3—52no—
180C22×Dic3—52no—
181C22×Dic3—52no—
182C2×D12—54no—
183C2×D12—102no—
184C2×D12—102no—
185C2×C4×D3—58no—
186C2×C4×D3—204no—
187C3×C4⋊C4—24no—
188C3×C42C4×D514yes—
189(C2×D3)⋊C4—58no—
190(C2×D3)⋊C4—108no—
191(C2×D3)⋊C4—204no—
192C4⋊Dic3—54no—
193C4⋊Dic3—104no—
194C4×Dic3—104no—
195C4×Dic3—204no—
196C23×C5C23×D311yescharacteristic
197C22×D5—34no—
198C22×D5—62no—
199C22×D5—62no—
200C22×D5—64no—
201C22×D5—64no—
202C2×C4×C5—22no—
203C2×C4×C5—22no—
204C2×C4×C5—24no—
205C2×C4×C5—24no—
206C2×C4×C5—24no—
207C2×C4×C5C2×C4×D312yes—
208C2×C4×C5C2×C4×D312yes—
209C2×C4×C5C2×C4×D312yes—
210C2×C4×C5#23712yes—
211C2×C4×C5#23912yes—
212C2×Dic5—34no—
213C2×Dic5—34no—
214C2×Dic5—62no—
215C2×Dic5—62no—
216C2×Dic5—64no—
217C2×Dic5—62no—
218C2×Dic5—62no—
219C2×Dic5—62no—
220D20—68no—
221C4×D5—128no—
222C22×D4—151no—
223C23×C4—152no—
224C4×D4—158no—
225C2×C4⋊C4—151no—
226C2×C4⋊C4—152no—
227C2×C4⋊C4C2×D1512yes—
228C2×C22⋊C4—152no—
229C2×C22⋊C4—154no—
230C2×C22⋊C4—154no—
231C2×C42—152no—
232C2×C42—154no—
233C2×C42C2×D1511yescharacteristic
234#95—151no—
235#95—151no—
236#95—154no—
237#95C2×D1512yes—
238#95C2×D1512yes—
239C2×C3×C5C2×#11711yescharacteristic
240C2×C3×C5C4×#4111yescharacteristic
241C2×C3×C5#51811yescharacteristic
242C2×C3×C5#55312yes—
243C2×C3×C5#55912yes—
244D15—44no—
245C23×C3C23×D511yescharacteristic
246C22×D3—54no—
247C22×D3—102no—
248C22×D3—102no—
249C22×D3—104no—
250C22×D3—104no—
251C2×C3×C4—22no—
252C2×C3×C4—22no—
253C2×C3×C4—24no—
254C2×C3×C4—24no—
255C2×C3×C4—24no—
256C2×C3×C4C2×C4×D512yes—
257C2×C3×C4C2×C4×D512yes—
258C2×C3×C4C2×C4×D512yes—
259C2×C3×C4#71112yes—
260C2×C3×C4#71312yes—
261C2×Dic3—54no—
262C2×Dic3—54no—
263C2×Dic3—102no—
264C2×Dic3—102no—
265C2×Dic3—104no—
266C2×Dic3—102no—
267C2×Dic3—102no—
268C2×Dic3—102no—
269D12—108no—
270C4×D3—208no—
271C22×C5C22×C4×D311yescharacteristic
272C22×C5C2×#23711yescharacteristic
273C22×C5C2×#23911yescharacteristic
274C22×C5D3×#4112yes—
275C22×C5#100912yes—
276C2×D5—64no—
277C2×D5—64no—
278C2×D5—64no—
279C2×D5—124no—
280C2×D5—124no—
281C4×C5—22no—
282C4×C5—22no—
283C4×C5—44no—
284C4×C5#89214yes—
285Dic5—64no—
286Dic5—64no—
287Dic5—122no—
288C24—152no—
289C2×D4—154no—
290C2×D4—302no—
291C2×D4—302no—
292C22×C4—152no—
293C22×C4—152no—
294C22×C4—158no—
295C22×C4—152no—
296C22×C4—604no—
297C22×C4C22×D1514yes—
298C22×C4C22×D1511yescharacteristic
299C22×C4C22×D1511yescharacteristic
300C22×C4C22×D1511yescharacteristic
301C4⋊C4—24no—
302C4⋊C4—154no—
303C4⋊C4—304no—
304C22⋊C4—158no—
305C22⋊C4—308no—
306C22⋊C4—604no—
307C42—304no—
308C42—604no—
309C42C4×D1514yes—
310C3×C5#235711yescharacteristic, complemented, Hall
311C22×C3C22×C4×D511yescharacteristic
312C22×C3C2×#71111yescharacteristic
313C22×C3C2×#71311yescharacteristic
314C22×C3D5×#4112yes—
315C22×C3#414912yes—
316C2×D3—104no—
317C2×D3—104no—
318C2×D3—104no—
319C2×D3—204no—
320C2×D3—204no—
321C3×C4—22no—
322C3×C4—22no—
323C3×C4—44no—
324C3×C4#403814yes—
325Dic3—104no—
326Dic3—104no—
327Dic3—202no—
328C2×C5C2×#89211yescharacteristic
329C2×C5#552111yescharacteristic
330C2×C5#552211yescharacteristic
331C2×C5#568112yes—
332C2×C5#569512yes—
333D5—124no—
334C23—154no—
335C23—302no—
336C23—302no—
337C23—304no—
338C23—304no—
339C23C23×D1511yescharacteristic
340D4—308no—
341C2×C4—22no—
342C2×C4—22no—
343C2×C4—24no—
344C2×C4—24no—
345C2×C4—24no—
346C2×C4—154no—
347C2×C4—154no—
348C2×C4—302no—
349C2×C4—302no—
350C2×C4—304no—
351C2×C4—302no—
352C2×C4—302no—
353C2×C4—302no—
354C2×C4—608no—
355C2×C4C2×C4×D1512yes—
356C2×C4C2×C4×D1512yes—
357C2×C4C2×C4×D1512yes—
358C2×C4#681912yes—
359C2×C4#682112yes—
360C2×C3C2×#403811yescharacteristic
361C2×C3#6585511yescharacteristic
362C2×C3#6585611yescharacteristic
363C2×C3#6601512yes—
364C2×C3#6602912yes—
365D3—204no—
366C5#7416011yescharacteristic, complemented, Sylow
367C22—304no—
368C22—304no—
369C22—304no—
370C22—604no—
371C22—604no—
372C22C22×C4×D1511yescharacteristic
373C22C2×#681911yescharacteristic
374C22C2×#682111yescharacteristic
375C22#41×D1512yes—
376C22#9210012yes—
377C4—22no—
378C4—22no—
379C4—44no—
380C4—304no—
381C4—304no—
382C4—602no—
383C4#9178114yes—
384C3#1060589111yescharacteristic, complemented, Sylow
385C2—604no—
386C2C2×#9178111yescharacteristic
387C2#1175628011yescharacteristic
388C2#1175628111yescharacteristic
389C2#1175644012yes—
390C2#1175645412yes—
subgroupG/H
center, FrattiniC23C23×D15
commutatorC22×C3×C5C23×C4
FittingC3×C5×#382C2
socle, cosocleC23×C3×C5C24
prime radicalC22×D60C22
hypercenter#382D15
hypocenterC3×C5#2357
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ #382

Complements

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

NH ≅ G/Ntype
2. C3×C5×#382C2semidirect
4. C4⋊C4×Dic15C2semidirect
5. C4×C4⋊Dic15C2semidirect
6. #11750771C2semidirect
7. #11750767C2semidirect
8. #11750765C2semidirect
21. C4×D60C4semidirect
29. C5×#382D3semidirect
59. C3×#382D5semidirect
152. #382D15semidirect
310. C3×C5#2357semidirect
366. C5#74160semidirect
384. C3#10605891semidirect

Maximal subgroups

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

subgroupindex
C2×C4×D602
C3×C5×#3822
#117510892
C4⋊C4×Dic152
C4×C4⋊Dic152
#117507712
#117507672
#117507652
#106058913
#741605