Finite Group Databasebeta

← #14307122

Subgroups — #14307122

Order 1920.

520 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11756276C211yesmaximal, characteristic
2M16×Dic15C211yesmaximal, characteristic
3C4×#6712C211yesmaximal, characteristic
4C3×C5×#339C211yesmaximal, characteristic
5#11746376C211yesmaximal, characteristic
6#11746353C211yesmaximal, characteristic
7#11746344C211yesmaximal, characteristic
8#10599523—31nomaximal, Hall
9C2×C3×C5×M16C2211yescharacteristic
10C3×C5×#117C2211yescharacteristic
11C2×C3×C42×C5C2211yescharacteristic
12#91815C2211yescharacteristic
13C2×#6712—21no—
14C2×C4×Dic15C2211yescharacteristic
15C2×C4×Dic15C2211yescharacteristic
16C2×#6710C2211yescharacteristic
17#91779C411yescharacteristic, complemented
18#91779C411yescharacteristic, complemented
19#91777C411yescharacteristic, complemented
20#91777C411yescharacteristic, complemented
21#91767—21no—
22#91767—21no—
23C4×Dic30—21no—
24C4×Dic30—21no—
25#91115—21no—
26#91099—21no—
27C8×Dic15—21no—
28#68730—51nomaximal, Hall
29#65851—31no—
30M16×Dic5—31no—
31C4×#683—31no—
32C5×#339D311yescharacteristic
33#64775—31no—
34#64752—31no—
35#64743—31no—
36C22×C3×C4×C5—21no—
37C22×C3×C4×C5C2311yescharacteristic
38C22×Dic15—21no—
39C22×Dic15D411yescharacteristic
40C22×Dic15D411yescharacteristic
41C2×Dic30—21no—
42C2×Dic30—21no—
43C3×C5×M16—21no—
44C3×C5×M16Q812yescomplemented
45C2×C3×C5×C8—21no—
46C3×C5×C4⋊C4C2×C411yescharacteristic
47C3×C5×C4⋊C4C2×C411yescharacteristic
48C3×C5×C22⋊C4—21no—
49C3×C42×C5—21no—
50C3×C42×C5—21no—
51C3×C42×C5—21no—
52#6719—21no—
53C4⋊Dic15—21no—
54C4⋊Dic15—21no—
55C4⋊Dic15C2×C411yescharacteristic
56C4⋊Dic15C2×C411yescharacteristic
57#6712—24no—
58#6712—21no—
59#6712—21no—
60#6712—21no—
61#6712—21no—
62C4×Dic15—21no—
63C4×Dic15—21no—
64C4×Dic15—21no—
65C4×Dic15C2×C411yescharacteristic
66C4×Dic15C2×C411yescharacteristic
67C4×Dic15D411yescharacteristic
68C4×Dic15D411yescharacteristic
69#6710—21no—
70#6710Q812yescomplemented
71C2×C3⋊#168—21no—
72#5517—51no—
73Dic3×M16—51no—
74C4×Dic3⋊C4—51no—
75C3×#339D511yescharacteristic
76#4555—51no—
77#4532—51no—
78#4523—51no—
79C2×C5×M16C2×D311yescharacteristic
80C5×#117C2×D311yescharacteristic
81C2×C42×C5C2×D311yescharacteristic
82#4072—31no—
83C2×#683—61no—
84C2×C4×Dic5—31no—
85C2×C4×Dic5—31no—
86C2×#681—31no—
87#4036—31no—
88#4036—31no—
89#4034—31no—
90#4034—31no—
91#4024—61no—
92#4024—61no—
93C4×Dic10—61no—
94C4×Dic10—61no—
95#3963—61no—
96#3947—61no—
97C8×Dic5—61no—
98#1986—151noSylow
99C23×C3×C5C2×D411yescharacteristic
100C2×C3×C4×C5—22no—
101C2×C3×C4×C5—21no—
102C2×C3×C4×C5—21no—
103C2×C3×C4×C5—41no—
104C2×C3×C4×C5—41no—
105C2×C3×C4×C5C22×C411yescharacteristic
106C2×C3×C4×C5C2×D411yescharacteristic
107C2×C3×C4×C5C2×Q811yescharacteristic
108C2×C3×C4×C5C2×Q811yescharacteristic
109C2×C3×C4×C5#4111yescharacteristic
110C2×C3×C4×C5#4111yescharacteristic
111C2×Dic15—22no—
112C2×Dic15—22no—
113C2×Dic15—21no—
114C2×Dic15—21no—
115C2×Dic15—41no—
116C2×Dic15—41no—
117C2×Dic15—41no—
118C2×Dic15C22⋊C411yescharacteristic
119C2×Dic15C22⋊C411yescharacteristic
120C2×Dic15C22⋊C411yescharacteristic
121C2×Dic15C22⋊C411yescharacteristic
122Dic30—42no—
123Dic30—42no—
124C3×C5×C8—22no—
125C3⋊#168—22no—
126C2×C3×M16C2×D511yescharacteristic
127C3×#117C2×D511yescharacteristic
128C2×C3×C42C2×D511yescharacteristic
129#926—51no—
130C2×Dic3⋊C4—101no—
131C2×C4×Dic3—51no—
132C2×C4×Dic3—51no—
133C2×#208—51no—
134#890—51no—
135#890—51no—
136#888—51no—
137#888—51no—
138#878—101no—
139#878—101no—
140C4×Dic6—101no—
141C4×Dic6—101no—
142C3⋊#95—101no—
143#815—101no—
144C8×Dic3—101no—
145C22×C4×C5—21no—
146C22×C4×C5C22×D311yescharacteristic
147C22×Dic5—31no—
148C22×Dic5—31no—
149C22×Dic5—61no—
150C2×Dic10—61no—
151C2×Dic10—61no—
152C5×M16—21no—
153C5×M16Dic612yescomplemented
154C2×C5×C8—21no—
155C5×C4⋊C4C4×D311yescharacteristic
156C5×C4⋊C4C4×D311yescharacteristic
157C5×C22⋊C4—21no—
158C42×C5—21no—
159C42×C5—21no—
160C42×C5—21no—
161C22⋊Dic5—61no—
162C4⋊Dic5—31no—
163C4⋊Dic5—31no—
164C4⋊Dic5—61no—
165C4⋊Dic5—61no—
166#683—61no—
167#683—61no—
168#683—61no—
169#683—61no—
170#683—64no—
171C4×Dic5—31no—
172C4×Dic5—31no—
173C4×Dic5—31no—
174C4×Dic5—31no—
175C4×Dic5—61no—
176C4×Dic5—61no—
177C4×Dic5—61no—
178#681—32no—
179#681—61no—
180C2×#168—61no—
181#533—151no—
182C4×M16—151no—
183C4×C4⋊C4—151no—
184#339—151no—
185#339—151no—
186#339—151no—
187#339D1511yescharacteristic
188C22×C3×C5—21no—
189C22×C3×C5—21no—
190C22×C3×C5C2×C22⋊C411yescharacteristic
191C22×C3×C5#12411yescharacteristic
192C22×C3×C5C4⋊Q811yescharacteristic
193C3×C4×C5—42no—
194C3×C4×C5—41no—
195C3×C4×C5—41no—
196C3×C4×C5C4×Q811yescharacteristic
197C3×C4×C5C4×Q811yescharacteristic
198C3×C4×C5C22⋊Q811yescharacteristic
199C3×C4×C5C22⋊Q811yescharacteristic
200Dic15—21no—
201Dic15—21no—
202Dic15—21no—
203Dic15—42no—
204Dic15—41no—
205Dic15—41no—
206Dic15#10412yes—
207C22×C3×C4—21no—
208C22×C3×C4C22×D511yescharacteristic
209C22×Dic3—51no—
210C22×Dic3—51no—
211C22×Dic3—101no—
212C2×Dic6—101no—
213C2×Dic6—101no—
214C3×M16—21no—
215C3×M16Dic1012yescomplemented
216C2×C3×C8—21no—
217C3×C4⋊C4C4×D511yescharacteristic
218C3×C4⋊C4C4×D511yescharacteristic
219C3×C22⋊C4—21no—
220C3×C42—21no—
221C3×C42—21no—
222C3×C42—21no—
223#217—101no—
224C4⋊Dic3—51no—
225C4⋊Dic3—51no—
226C4⋊Dic3—101no—
227C4⋊Dic3—101no—
228Dic3⋊C4—101no—
229Dic3⋊C4—101no—
230Dic3⋊C4—101no—
231Dic3⋊C4—101no—
232Dic3⋊C4—104no—
233C4×Dic3—51no—
234C4×Dic3—51no—
235C4×Dic3—51no—
236C4×Dic3—51no—
237C4×Dic3—101no—
238C4×Dic3—101no—
239C4×Dic3—101no—
240#208—52no—
241#208—101no—
242C2×C3⋊C8—101no—
243C23×C5D3×D411yescharacteristic
244C2×C4×C5—22no—
245C2×C4×C5—21no—
246C2×C4×C5—21no—
247C2×C4×C5—41no—
248C2×C4×C5—41no—
249C2×C4×C5C2×Dic611yescharacteristic
250C2×C4×C5C2×C4×D311yescharacteristic
251C2×C4×C5#23511yescharacteristic
252C2×C4×C5D3×D411yescharacteristic
253C2×C4×C5#23711yescharacteristic
254C2×C4×C5D3×Q811yescharacteristic
255C2×Dic5—31no—
256C2×Dic5—31no—
257C2×Dic5—31no—
258C2×Dic5—31no—
259C2×Dic5—62no—
260C2×Dic5—62no—
261C2×Dic5—61no—
262C2×Dic5—61no—
263C2×Dic5—121no—
264C2×Dic5—121no—
265C2×Dic5—121no—
266Dic10—122no—
267Dic10—122no—
268C5×C8—22no—
269#168—62no—
270C2×M16—151no—
271C2×M16C2×D1511yescharacteristic
272C4⋊Q8—151no—
273C4⋊Q8—151no—
274#125—151no—
275#125—151no—
276C22⋊Q8—301no—
277C22⋊Q8—301no—
278C4×Q8—301no—
279C4×Q8—301no—
280#117—151no—
281#117C2×D1511yescharacteristic
282C2×C4⋊C4—301no—
283C2×C42—151no—
284C2×C42—151no—
285C2×C42C2×D1511yescharacteristic
286#97—301no—
287C4×C8—301no—
288#95—301no—
289C2×C3×C5—21no—
290C2×C3×C5—21no—
291C2×C3×C5#39111yescharacteristic
292C2×C3×C5C2×#10411yescharacteristic
293C2×C3×C5#42111yescharacteristic
294C23×C3D4×D511yescharacteristic
295C2×C3×C4—22no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—41no—
299C2×C3×C4—41no—
300C2×C3×C4C2×Dic1011yescharacteristic
301C2×C3×C4C2×C4×D511yescharacteristic
302C2×C3×C4#70911yescharacteristic
303C2×C3×C4D4×D511yescharacteristic
304C2×C3×C4#71111yescharacteristic
305C2×C3×C4Q8×D511yescharacteristic
306C2×Dic3—51no—
307C2×Dic3—51no—
308C2×Dic3—51no—
309C2×Dic3—51no—
310C2×Dic3—102no—
311C2×Dic3—102no—
312C2×Dic3—101no—
313C2×Dic3—101no—
314C2×Dic3—201no—
315C2×Dic3—201no—
316C2×Dic3—201no—
317Dic6—202no—
318Dic6—202no—
319C3×C8—22no—
320C3⋊C8—102no—
321C22×C5—21no—
322C22×C5—21no—
323C22×C5D3×C22⋊C411yescharacteristic
324C22×C5#88511yescharacteristic
325C22×C5#88811yescharacteristic
326C4×C5—42no—
327C4×C5—41no—
328C4×C5—41no—
329C4×C5C4×Dic611yescharacteristic
330C4×C5#87811yescharacteristic
331C4×C5#88711yescharacteristic
332C4×C5#89611yescharacteristic
333Dic5—32no—
334Dic5—61no—
335Dic5—61no—
336Dic5—61no—
337Dic5—122no—
338Dic5—121no—
339Dic5—121no—
340C2×Q8—301no—
341C2×Q8—301no—
342C22×C4—21no—
343C22×C4—151no—
344C22×C4—151no—
345C22×C4—301no—
346C22×C4C22×D1511yescharacteristic
347M16—21no—
348M16—152no—
349M16—301no—
350M16Dic3012yescomplemented
351C2×C8—21no—
352C2×C8—301no—
353C4⋊C4—151no—
354C4⋊C4—151no—
355C4⋊C4—301no—
356C4⋊C4—301no—
357C4⋊C4—301no—
358C4⋊C4—301no—
359C4⋊C4—301no—
360C4⋊C4—301no—
361C4⋊C4—304no—
362C4⋊C4C4×D1511yescharacteristic
363C4⋊C4C4×D1511yescharacteristic
364C22⋊C4—21no—
365C22⋊C4—301no—
366C42—21no—
367C42—21no—
368C42—21no—
369C42—151no—
370C42—151no—
371C42—151no—
372C42—151no—
373C42—301no—
374C42—301no—
375C42—301no—
376C3×C5#198611yescharacteristic, complemented, Hall
377C22×C3—21no—
378C22×C3—21no—
379C22×C3D5×C22⋊C411yescharacteristic
380C22×C3#403111yescharacteristic
381C22×C3#403411yescharacteristic
382C3×C4—42no—
383C3×C4—41no—
384C3×C4—41no—
385C3×C4C4×Dic1011yescharacteristic
386C3×C4#402411yescharacteristic
387C3×C4#403311yescharacteristic
388C3×C4#404211yescharacteristic
389Dic3—52no—
390Dic3—101no—
391Dic3—101no—
392Dic3—101no—
393Dic3—202no—
394Dic3—201no—
395Dic3—201no—
396C2×C5—21no—
397C2×C5—21no—
398C2×C5#464611yescharacteristic
399C2×C5D3×#10411yescharacteristic
400C2×C5#482111yescharacteristic
401C23D4×D1511yescharacteristic
402Q8—602no—
403Q8—602no—
404C2×C4—22no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—151no—
410C2×C4—151no—
411C2×C4—151no—
412C2×C4—151no—
413C2×C4—302no—
414C2×C4—302no—
415C2×C4—301no—
416C2×C4—301no—
417C2×C4—601no—
418C2×C4—601no—
419C2×C4—601no—
420C2×C4C2×Dic3011yescharacteristic
421C2×C4C2×C4×D1511yescharacteristic
422C2×C4#681711yescharacteristic
423C2×C4D4×D1511yescharacteristic
424C2×C4#681911yescharacteristic
425C2×C4Q8×D1511yescharacteristic
426C8—22no—
427C8—302no—
428C2×C3—21no—
429C2×C3—21no—
430C2×C3#6493511yescharacteristic
431C2×C3D5×#10411yescharacteristic
432C2×C3#6511011yescharacteristic
433C5#6873011yescharacteristic, complemented, Sylow
434C22—21no—
435C22—21no—
436C22C22⋊C4×D1511yescharacteristic
437C22#9177411yescharacteristic
438C22#9177711yescharacteristic
439C4—42no—
440C4—41no—
441C4—41no—
442C4—152no—
443C4—301no—
444C4—301no—
445C4—301no—
446C4—602no—
447C4—601no—
448C4—601no—
449C4C4×Dic3011yescharacteristic
450C4#9176711yescharacteristic
451C4#9177611yescharacteristic
452C4#9178511yescharacteristic
453C3#1059952311yescharacteristic, complemented, Sylow
454C2—21no—
455C2—21no—
456C2#1175074711yescharacteristic
457C2D15×#10411yescharacteristic
458C2#1175092211yescharacteristic
subgroupG/H
centerC2×C4D4×D15
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C22×D15
FittingC3×C5×#339C2
socleC22×C3×C5C2×C22⋊C4
prime radicalC23×C3×C5C2×D4
cosocleC22×C3×C4×C5C23
hypercenter#339D15
hypocenterC3×C5#1986
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
17. #91779C4semidirect
18. #91779C4semidirect
19. #91777C4semidirect
20. #91777C4semidirect
44. C3×C5×M16Q8semidirect
70. #6710Q8semidirect
153. C5×M16Dic6semidirect
215. C3×M16Dic10semidirect
350. M16Dic30semidirect
376. C3×C5#1986semidirect
433. C5#68730semidirect
453. C3#10599523semidirect

Maximal subgroups

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

subgroupindex
#117562762
M16×Dic152
C4×#67122
C3×C5×#3392
#117463762
#117463532
#117463442
#105995233
#687305