Finite Group Databasebeta

← #14322935

Subgroups — #14322935

Order 1920.

1200 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91761C211yesmaximal, characteristic
2C2×#91437C211yesmaximal, characteristic
3C2×#91347C211yesmaximal, characteristic
4C5×#4936C211yesmaximal, characteristic, complemented
5C3×#65224C211yesmaximal, characteristic, complemented
6#11748898C214yesmaximal
7#11748866C211yesmaximal, characteristic, complemented
8#11748676C214yesmaximal, complemented
9#11748623C211yesmaximal, characteristic, complemented
10#10605956—31nomaximal, Hall
11C22×C4×D15C2211yescharacteristic
12C22×#1200C2212yes—
13C2×C3×C42×C5C2211yescharacteristic
14C2×#6714C2212yes—
15C2×#6712C2212yes—
16C2×C4×Dic15C2211yescharacteristic
17#91761C418yescomplemented
18C2×C5×C4⋊Dic3C2211yescharacteristic
19C2×C5×C4⋊Dic3C2211yescharacteristic
20C2×C3×C4⋊Dic5C2211yescharacteristic
21C2×C3×C4⋊Dic5C2211yescharacteristic
22C2×C3×C4⋊Dic5C2214yes—
23C2×#6669C2212yes—
24C2×#6669C2214yes—
25C2×#6667C2212yes—
26C2×Dic3×Dic5C2212yes—
27#91437—24no—
28#91347—24no—
29C5×C3⋊#95C2214yes—
30#90994C2214yes—
31#74359—51nomaximal, Hall
32C2×#4038—31no—
33C2×D5×C4⋊C4—31no—
34C2×#4018—31no—
35C5×#382—31no—
36#65266—31no—
37#65265—31no—
38#65224D311yescharacteristic, complemented
39#64957—34no—
40#64936—34no—
41C23×D15C2311yescharacteristic
42C22×C3×C4×C5C2312yes—
43C22×C3×C4×C5C2311yescharacteristic
44C22×Dic15C2312yes—
45C22×Dic15C2311yescharacteristic
46C2×C4×D15C2×C414yes—
47C2×C4×D15D414yes—
48C2×C4×D15Q814yes—
49C22×C5×Dic3C2312yes—
50C22×C5×Dic3C2312yes—
51C22×C3×Dic5C2312yes—
52C22×C3×Dic5C2312yes—
53C2×#1200—24no—
54C2×#1200—44no—
55C3×C42×C5C2×C414yescomplemented
56#6714C2×C418yes—
57#6712C2×C418yescomplemented
58C4×Dic15C2×C414yescomplemented
59C5×C4⋊Dic3—22no—
60C5×C4⋊Dic3—22no—
61C3×C4⋊Dic5—22no—
62C3×C4⋊Dic5—22no—
63C3×C4⋊Dic5—44no—
64#6669—24no—
65#6669—44no—
66#6667—24no—
67Dic3×Dic5—24no—
68C2×#892—51no—
69C2×D3×C4⋊C4—51no—
70C2×#872—51no—
71C3×#384—51no—
72#4989—54no—
73#4977—51no—
74#4976—54no—
75#4975—51no—
76#4936D511yescharacteristic, complemented
77C22×C4×D5—31no—
78C22×C4×D5—32no—
79C2×C5×C4⋊C4—31no—
80C2×C5×C4⋊C4—31no—
81C2×C42×C5C2×D311yescharacteristic
82C2×#685—32no—
83C2×#685—32no—
84C2×C4⋊Dic5C2×D311yescharacteristic
85C2×C4⋊Dic5C2×D311yescharacteristic
86C2×C4⋊Dic5C2×D314yes—
87C2×#683—32no—
88C2×#683—32no—
89C2×#683—34no—
90C2×C4×Dic5—31no—
91C2×C4×Dic5—32no—
92#4038—64no—
93D5×C4⋊C4—64no—
94#4018—38no—
95C5×#95—34no—
96#3963—34no—
97#2317—151noSylow
98C23×C3×C5C2411yescharacteristic
99C22×D15—22no—
100C22×D15—22no—
101C22×D15C22×C412yes—
102C22×D15C2×D412yes—
103C22×D15C2×Q812yes—
104C2×C3×C4×C5—22no—
105C2×C3×C4×C5—22no—
106C2×C3×C4×C5C22×C414yes—
107C2×C3×C4×C5C22×C412yes—
108C2×C3×C4×C5C2×D412yes—
109C2×C3×C4×C5C2×Q812yes—
110C2×Dic15—22no—
111C2×Dic15—22no—
112C2×Dic15C22×C414yes—
113C2×Dic15C22×C412yes—
114C2×Dic15C2×D412yes—
115C2×Dic15C2×Q812yes—
116C4×D15C4⋊C4116yescomplemented
117C2×C5×Dic3—22no—
118C2×C5×Dic3—22no—
119C2×C5×Dic3—24no—
120C2×C5×Dic3—24no—
121C2×C3×Dic5—22no—
122C2×C3×Dic5—22no—
123C2×C3×Dic5—24no—
124C2×C3×Dic5—24no—
125#1200—48no—
126C22×C4×D3—51no—
127C22×C4×D3—52no—
128C2×C3×C4⋊C4—51no—
129C2×C3×C4⋊C4—51no—
130C2×C3×C4⋊C4—54no—
131C2×C3×C42C2×D511yescharacteristic
132C2×(C2×D3)⋊C4—52no—
133C2×(C2×D3)⋊C4—52no—
134C2×C4⋊Dic3C2×D511yescharacteristic
135C2×C4⋊Dic3C2×D511yescharacteristic
136C2×Dic3⋊C4—52no—
137C2×Dic3⋊C4—52no—
138C2×Dic3⋊C4—54no—
139C2×C4×Dic3—51no—
140C2×C4×Dic3—52no—
141#892—104no—
142D3×C4⋊C4—104no—
143#872—58no—
144C3⋊#95—54no—
145C3⋊#95C2×D514yes—
146C23×D5—31no—
147C22×C4×C5—32no—
148C22×C4×C5—32no—
149C22×C4×C5C22×D312yes—
150C22×C4×C5C22×D311yescharacteristic
151C22×Dic5—31no—
152C22×Dic5—32no—
153C22×Dic5C22×D312yes—
154C22×Dic5C22×D312yes—
155C2×C4×D5—34no—
156C2×C4×D5—34no—
157C2×C4×D5—34no—
158C2×C4×D5—64no—
159C2×C4×D5—124no—
160C5×C4⋊C4—62no—
161C5×C4⋊C4—62no—
162C42×C5C4×D314yescomplemented
163#685—38no—
164#685—64no—
165C4⋊Dic5—22no—
166C4⋊Dic5—22no—
167C4⋊Dic5—44no—
168#683—38no—
169#683—64no—
170#683—124no—
171C4×Dic5—34no—
172C4×Dic5—64no—
173C2×#117—151no—
174C2×#117—151no—
175C22×C4⋊C4—151no—
176#389—154no—
177#385—154no—
178#384—51no—
179#384—151no—
180#382—31no—
181#382—151no—
182C22×C3×C5C23×C411yescharacteristic
183C22×C3×C5C22×D411yescharacteristic
184C22×C3×C5C22×Q811yescharacteristic
185C22×C3×C5H(5,𝔽2)12yes—
186C22×C3×C52−1+412yes—
187C2×D15—24no—
188C2×D15—22no—
189C2×D15—22no—
190C2×D15—24no—
191C2×D15C2×C4⋊C414yes—
192C3×C4×C5—24no—
193C3×C4×C5C2×C4⋊C414yes—
194Dic15—24no—
195Dic15C2×C4⋊C414yes—
196C3×Dic5—42no—
197C3×Dic5—42no—
198C5×Dic3—42no—
199C5×Dic3—42no—
200C23×D3—51no—
201C22×C3×C4—52no—
202C22×C3×C4—52no—
203C22×C3×C4C22×D512yes—
204C22×C3×C4C22×D511yescharacteristic
205C22×Dic3—51no—
206C22×Dic3—52no—
207C22×Dic3C22×D512yes—
208C22×Dic3C22×D512yes—
209C2×C4×D3—54no—
210C2×C4×D3—54no—
211C2×C4×D3—54no—
212C2×C4×D3—104no—
213C2×C4×D3—204no—
214C3×C4⋊C4—102no—
215C3×C4⋊C4—102no—
216C3×C4⋊C4—204no—
217C3×C42C4×D514yescomplemented
218(C2×D3)⋊C4—58no—
219(C2×D3)⋊C4—104no—
220C4⋊Dic3—22no—
221C4⋊Dic3—22no—
222Dic3⋊C4—58no—
223Dic3⋊C4—104no—
224Dic3⋊C4—204no—
225C4×Dic3—54no—
226C4×Dic3—104no—
227C23×C5C23×D311yescharacteristic
228C22×D5—32no—
229C22×D5—32no—
230C22×D5—32no—
231C22×D5—62no—
232C22×D5—62no—
233C2×C4×C5—22no—
234C2×C4×C5—22no—
235C2×C4×C5—62no—
236C2×C4×C5—62no—
237C2×C4×C5—64no—
238C2×C4×C5—64no—
239C2×C4×C5C2×C4×D314yes—
240C2×C4×C5C2×C4×D312yes—
241C2×C4×C5D3×D412yes—
242C2×C4×C5D3×Q812yes—
243C2×Dic5—22no—
244C2×Dic5—22no—
245C2×Dic5—24no—
246C2×Dic5—24no—
247C2×Dic5—32no—
248C2×Dic5—32no—
249C2×Dic5—32no—
250C2×Dic5—34no—
251C2×Dic5—62no—
252C2×Dic5—62no—
253C4×D5—316no—
254C4×D5—128no—
255C23×C4—152no—
256C23×C4—151no—
257#117—158no—
258#117—304no—
259C2×C4⋊C4—31no—
260C2×C4⋊C4—31no—
261C2×C4⋊C4—51no—
262C2×C4⋊C4—51no—
263C2×C4⋊C4—54no—
264C2×C4⋊C4—152no—
265C2×C4⋊C4—152no—
266C2×C4⋊C4—154no—
267C2×C4⋊C4—304no—
268C2×C22⋊C4—152no—
269C2×C22⋊C4—152no—
270C2×C42—151no—
271C2×C42—152no—
272C2×C42D3×D511yescharacteristic
273#95—34no—
274#95—154no—
275C2×C3×C5C22×C4⋊C411yescharacteristic
276C2×C3×C5#52012yes—
277C2×C3×C5#53511yescharacteristic
278C2×C3×C5#53611yescharacteristic
279C2×C3×C5#54412yes—
280D15—24no—
281C23×C3C23×D511yescharacteristic
282C22×D3—52no—
283C22×D3—52no—
284C22×D3—52no—
285C22×D3—102no—
286C22×D3—102no—
287C2×C3×C4—22no—
288C2×C3×C4—22no—
289C2×C3×C4—102no—
290C2×C3×C4—102no—
291C2×C3×C4—104no—
292C2×C3×C4—104no—
293C2×C3×C4C2×C4×D514yes—
294C2×C3×C4C2×C4×D512yes—
295C2×C3×C4D4×D512yes—
296C2×C3×C4Q8×D512yes—
297C2×Dic3—22no—
298C2×Dic3—22no—
299C2×Dic3—24no—
300C2×Dic3—24no—
301C2×Dic3—52no—
302C2×Dic3—52no—
303C2×Dic3—52no—
304C2×Dic3—54no—
305C2×Dic3—102no—
306C2×Dic3—102no—
307C4×D3—516no—
308C4×D3—208no—
309C22×C5C22×C4×D311yescharacteristic
310C22×C5C2×D3×D411yescharacteristic
311C22×C5#100412yes—
312C22×C5C2×D3×Q811yescharacteristic
313C22×C5#100712yes—
314C2×D5—34no—
315C2×D5—62no—
316C2×D5—62no—
317C2×D5—64no—
318C2×D5—64no—
319C4×C5—24no—
320C4×C5—122no—
321C4×C5—122no—
322C4×C5D3×C4⋊C414yes—
323Dic5—34no—
324Dic5—42no—
325Dic5—42no—
326Dic5—64no—
327C24—151no—
328C22×C4—32no—
329C22×C4—32no—
330C22×C4—52no—
331C22×C4—52no—
332C22×C4—151no—
333C22×C4—154no—
334C22×C4—154no—
335C22×C4—154no—
336C22×C4—152no—
337C22×C4—304no—
338C22×C4—604no—
339C22×C4C2×D3×D512yes—
340C22×C4C2×D3×D511yescharacteristic
341C4⋊C4—62no—
342C4⋊C4—62no—
343C4⋊C4—102no—
344C4⋊C4—102no—
345C4⋊C4—158no—
346C4⋊C4—204no—
347C4⋊C4—304no—
348C4⋊C4—604no—
349C22⋊C4—158no—
350C22⋊C4—304no—
351C42—154no—
352C42—304no—
353C42#120014yescomplemented
354C3×C5#231711yescharacteristic, complemented, Hall
355C22×C3C22×C4×D511yescharacteristic
356C22×C3C2×D4×D511yescharacteristic
357C22×C3C2×Q8×D511yescharacteristic
358C22×C3#414912yes—
359C22×C3#415012yes—
360C2×D3—54no—
361C2×D3—104no—
362C2×D3—102no—
363C2×D3—102no—
364C2×D3—104no—
365C3×C4—24no—
366C3×C4—202no—
367C3×C4—202no—
368C3×C4D5×C4⋊C414yes—
369Dic3—42no—
370Dic3—42no—
371Dic3—54no—
372Dic3—104no—
373C2×C5C2×D3×C4⋊C411yescharacteristic
374C2×C5#550412yes—
375C2×C5#571511yescharacteristic
376C2×C5#572411yescharacteristic
377C2×C5#572912yes—
378D5—64no—
379C23—152no—
380C23—152no—
381C23—152no—
382C23—302no—
383C23—302no—
384C23C22×D3×D511yescharacteristic
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—64no—
388C2×C4—62no—
389C2×C4—62no—
390C2×C4—64no—
391C2×C4—102no—
392C2×C4—102no—
393C2×C4—104no—
394C2×C4—104no—
395C2×C4—152no—
396C2×C4—152no—
397C2×C4—152no—
398C2×C4—1516no—
399C2×C4—154no—
400C2×C4—302no—
401C2×C4—302no—
402C2×C4—608no—
403C2×C4D15⋊Q812yes—
404C2×C4#677712yes—
405C2×C4C2×#120014yes—
406C2×C4C2×#120012yes—
407C2×C3C2×D5×C4⋊C411yescharacteristic
408C2×C3#6585712yes—
409C2×C3#6600811yescharacteristic
410C2×C3#6600911yescharacteristic
411C2×C3#6602312yes—
412D3—104no—
413C5#7435911yescharacteristic, complemented, Sylow
414C22—154no—
415C22—304no—
416C22—302no—
417C22—302no—
418C22—304no—
419C22#9200412yes—
420C22C2×D15⋊Q811yescharacteristic
421C22C2×#677711yescharacteristic
422C22#9201712yes—
423C22C22×#120011yescharacteristic
424C4—24no—
425C4—122no—
426C4—122no—
427C4—154no—
428C4—202no—
429C4—202no—
430C4—304no—
431C4#9143714yes—
432C3#1060595611yescharacteristic, complemented, Sylow
433C2—304no—
434C2#1175256011yescharacteristic
435C2#1175264412yes—
436C2#1175275211yescharacteristic
437C2C2×#9143711yescharacteristic
438C2#1175449612yes—
subgroupG/H
center, FrattiniC23C22×D3×D5
commutatorC22×C3×C5C23×C4
FittingC2×C3×C42×C5C22
socle, cosocleC23×C3×C5C24
prime radicalC23×D15C23
hypercenterC2×C42D3×D5
hypocenterC3×C5#2317
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
4. C5×#4936C2semidirect
5. C3×#65224C2semidirect
7. #11748866C2semidirect
8. #11748676C2semidirect
9. #11748623C2semidirect
17. #91761C4semidirect
38. #65224D3semidirect
55. C3×C42×C5C2×C4semidirect
57. #6712C2×C4semidirect
58. C4×Dic15C2×C4semidirect
76. #4936D5semidirect
116. C4×D15C4⋊C4semidirect
162. C42×C5C4×D3semidirect
217. C3×C42C4×D5semidirect
353. C42#1200semidirect
354. C3×C5#2317semidirect
413. C5#74359semidirect
432. C3#10605956semidirect

Maximal subgroups

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

subgroupindex
C2×#917612
C2×#914372
C2×#913472
C5×#49362
C3×#652242
#117488982
#117488662
#117486762
#117486232
#106059563
#743595