Finite Group Databasebeta

← #14307142

Subgroups — #14307142

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#11754617C211yesmaximal, characteristic
2M16×Dic15C211yesmaximal, characteristic
3C4×#6669C211yesmaximal, characteristic
4C5×#4555C211yesmaximal, characteristic
5C3×#64775C211yesmaximal, characteristic
6#11745977C211yesmaximal, characteristic
7#11745974C211yesmaximal, characteristic
8#10599523—31nomaximal, Hall
9C2×C3×C5×M16C2211yescharacteristic
10C2×C4×Dic15C2211yescharacteristic
11C2×#6710C2211yescharacteristic
12C5×#926C2211yescharacteristic
13C2×C4×C5×Dic3C2211yescharacteristic
14C3×#4072C2211yescharacteristic
15C2×C3×C4×Dic5C2211yescharacteristic
16#91576—21no—
17#91576—21no—
18C2×#6669—21no—
19#91470C411yescharacteristic, complemented
20#91470C411yescharacteristic, complemented
21C4×#1204—21no—
22C4×#1204—21no—
23#91384C411yescharacteristic, complemented
24#91384C411yescharacteristic, complemented
25#91099—21no—
26C8×Dic15—21no—
27#90994—21no—
28#68730—51nomaximal, Hall
29#65851—31no—
30M16×Dic5—31no—
31C4×#683—31no—
32C5×#339—31no—
33#64775D311yescharacteristic
34#64752—31no—
35#64743—31no—
36C22×C3×C4×C5C2311yescharacteristic
37C22×Dic15D411yescharacteristic
38C22×Dic15D411yescharacteristic
39C22×C5×Dic3—21no—
40C22×C3×Dic5—21no—
41C2×#1204—21no—
42C2×#1204—21no—
43C3×C5×M16—21no—
44C3×C5×M16Q812yescomplemented
45C2×C3×C5×C8—21no—
46C4×Dic15C2×C411yescharacteristic
47C4×Dic15C2×C411yescharacteristic
48C4×Dic15D411yescharacteristic
49C4×Dic15D411yescharacteristic
50#6710—21no—
51#6710Q812yescomplemented
52C2×C3⋊#168—21no—
53C5×#217—21no—
54C5×C4⋊Dic3C2×C411yescharacteristic
55C5×C4⋊Dic3C2×C411yescharacteristic
56C4×C5×Dic3—21no—
57C4×C5×Dic3—21no—
58C4×C5×Dic3—21no—
59C3×C22⋊Dic5—21no—
60C3×C4⋊Dic5C2×C411yescharacteristic
61C3×C4⋊Dic5C2×C411yescharacteristic
62C3×C4×Dic5—21no—
63C3×C4×Dic5—21no—
64C3×C4×Dic5—21no—
65Dic3⋊Dic5—21no—
66Dic3⋊Dic5—21no—
67#6669—24no—
68#6669—21no—
69#6669—21no—
70#6668—21no—
71#6668—21no—
72#5517—51no—
73Dic3×M16—51no—
74C4×Dic3⋊C4—51no—
75C3×#339—51no—
76#4555D511yescharacteristic
77#4532—51no—
78#4523—51no—
79C2×C5×M16C2×D311yescharacteristic
80C5×#117—31no—
81C2×C42×C5—31no—
82#4072C2×D311yescharacteristic
83C2×#683—61no—
84C2×C4×Dic5—31no—
85C2×C4×Dic5C2×D311yescharacteristic
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×C5C22×C411yescharacteristic
101C2×C3×C4×C5C2×D411yescharacteristic
102C2×C3×C4×C5C2×Q811yescharacteristic
103C2×C3×C4×C5C2×Q811yescharacteristic
104C2×C3×C4×C5#4111yescharacteristic
105C2×C3×C4×C5#4111yescharacteristic
106C2×Dic15—22no—
107C2×Dic15—41no—
108C2×Dic15C22⋊C411yescharacteristic
109C2×Dic15C22⋊C411yescharacteristic
110C2×Dic15C22⋊C411yescharacteristic
111C2×Dic15C22⋊C411yescharacteristic
112C2×C5×Dic3—22no—
113C2×C5×Dic3—21no—
114C2×C5×Dic3—21no—
115C2×C5×Dic3—41no—
116C2×C5×Dic3—41no—
117C2×C3×Dic5—22no—
118C2×C3×Dic5—21no—
119C2×C3×Dic5—21no—
120C2×C3×Dic5—41no—
121C2×C3×Dic5—41no—
122#1204—42no—
123#1204—42no—
124C3×C5×C8—22no—
125C3⋊#168—22no—
126C2×C3×M16C2×D511yescharacteristic
127C3×#117—51no—
128C2×C3×C42—51no—
129#926C2×D511yescharacteristic
130C2×Dic3⋊C4—101no—
131C2×C4×Dic3—51no—
132C2×C4×Dic3C2×D511yescharacteristic
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—61no—
146C22×C4×C5C22×D311yescharacteristic
147C22×Dic5—21no—
148C22×Dic5—31no—
149C22×Dic5—31no—
150C2×Dic10—61no—
151C2×Dic10—61no—
152C5×M16—21no—
153C5×M16Dic612yescomplemented
154C2×C5×C8—21no—
155C5×C4⋊C4—31no—
156C5×C4⋊C4—31no—
157C5×C22⋊C4—61no—
158C42×C5—61no—
159C42×C5—61no—
160C42×C5—61no—
161C22⋊Dic5—21no—
162C4⋊Dic5—61no—
163C4⋊Dic5—61no—
164C4⋊Dic5C4×D311yescharacteristic
165C4⋊Dic5C4×D311yescharacteristic
166#683—61no—
167#683—61no—
168#683—61no—
169#683—61no—
170#683—64no—
171C4×Dic5—21no—
172C4×Dic5—21no—
173C4×Dic5—21no—
174C4×Dic5—31no—
175C4×Dic5—31no—
176C4×Dic5—31no—
177C4×Dic5—31no—
178#681—32no—
179#681—61no—
180C2×#168—61no—
181#533—151no—
182C4×M16—151no—
183C4×C4⋊C4—151no—
184#339—31no—
185#339—51no—
186#339—151no—
187#339—151no—
188C22×C3×C5—21no—
189C22×C3×C5—21no—
190C22×C3×C5C2×C22⋊C411yescharacteristic
191C22×C3×C5#12411yescharacteristic
192C22×C3×C5C4⋊Q811yescharacteristic
193C3×C4×C5C4×Q811yescharacteristic
194C3×C4×C5C4×Q811yescharacteristic
195C3×C4×C5C22⋊Q811yescharacteristic
196C3×C4×C5C22⋊Q811yescharacteristic
197Dic15—21no—
198Dic15—21no—
199Dic15—21no—
200Dic15#10412yes—
201C3×Dic5—42no—
202C3×Dic5—41no—
203C3×Dic5—41no—
204C5×Dic3—42no—
205C5×Dic3—41no—
206C5×Dic3—41no—
207C22×C3×C4—101no—
208C22×C3×C4C22×D511yescharacteristic
209C22×Dic3—21no—
210C22×Dic3—51no—
211C22×Dic3—51no—
212C2×Dic6—101no—
213C2×Dic6—101no—
214C3×M16—21no—
215C3×M16Dic1012yescomplemented
216C2×C3×C8—21no—
217C3×C4⋊C4—51no—
218C3×C4⋊C4—51no—
219C3×C22⋊C4—101no—
220C3×C42—101no—
221C3×C42—101no—
222C3×C42—101no—
223#217—21no—
224C4⋊Dic3—101no—
225C4⋊Dic3—101no—
226C4⋊Dic3C4×D511yescharacteristic
227C4⋊Dic3C4×D511yescharacteristic
228Dic3⋊C4—101no—
229Dic3⋊C4—101no—
230Dic3⋊C4—101no—
231Dic3⋊C4—101no—
232Dic3⋊C4—104no—
233C4×Dic3—21no—
234C4×Dic3—21no—
235C4×Dic3—21no—
236C4×Dic3—51no—
237C4×Dic3—51no—
238C4×Dic3—51no—
239C4×Dic3—51no—
240#208—52no—
241#208—101no—
242C2×C3⋊C8—101no—
243C23×C5D3×D411yescharacteristic
244C2×C4×C5—62no—
245C2×C4×C5—61no—
246C2×C4×C5—61no—
247C2×C4×C5—121no—
248C2×C4×C5—121no—
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—22no—
256C2×Dic5—21no—
257C2×Dic5—21no—
258C2×Dic5—31no—
259C2×Dic5—31no—
260C2×Dic5—31no—
261C2×Dic5—31no—
262C2×Dic5—41no—
263C2×Dic5—41no—
264C2×Dic5—62no—
265C2×Dic5—121no—
266Dic10—122no—
267Dic10—122no—
268C5×C8—22no—
269#168—62no—
270C2×M16—151no—
271C2×M16D3×D511yescharacteristic
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—31no—
281#117—51no—
282C2×C4⋊C4—301no—
283C2×C42—31no—
284C2×C42—51no—
285C2×C42—151no—
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—102no—
296C2×C3×C4—101no—
297C2×C3×C4—101no—
298C2×C3×C4—201no—
299C2×C3×C4—201no—
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—22no—
307C2×Dic3—21no—
308C2×Dic3—21no—
309C2×Dic3—41no—
310C2×Dic3—41no—
311C2×Dic3—51no—
312C2×Dic3—51no—
313C2×Dic3—51no—
314C2×Dic3—51no—
315C2×Dic3—102no—
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—122no—
327C4×C5—121no—
328C4×C5—121no—
329C4×C5C4×Dic611yescharacteristic
330C4×C5#87811yescharacteristic
331C4×C5#88711yescharacteristic
332C4×C5#89611yescharacteristic
333Dic5—32no—
334Dic5—42no—
335Dic5—41no—
336Dic5—41no—
337Dic5—61no—
338Dic5—61no—
339Dic5—61no—
340C2×Q8—301no—
341C2×Q8—301no—
342C22×C4—61no—
343C22×C4—101no—
344C22×C4—151no—
345C22×C4—151no—
346C22×C4C2×D3×D511yescharacteristic
347M16—21no—
348M16—152no—
349M16—301no—
350M16#120412yescomplemented
351C2×C8—21no—
352C2×C8—301no—
353C4⋊C4—31no—
354C4⋊C4—31no—
355C4⋊C4—51no—
356C4⋊C4—51no—
357C4⋊C4—301no—
358C4⋊C4—301no—
359C4⋊C4—301no—
360C4⋊C4—301no—
361C4⋊C4—301no—
362C4⋊C4—301no—
363C4⋊C4—304no—
364C22⋊C4—61no—
365C22⋊C4—101no—
366C42—61no—
367C42—61no—
368C42—61no—
369C42—101no—
370C42—101no—
371C42—101no—
372C42—151no—
373C42—151no—
374C42—151no—
375C42—151no—
376C3×C5#198611yescharacteristic, complemented, Hall
377C22×C3—21no—
378C22×C3—21no—
379C22×C3D5×C22⋊C411yescharacteristic
380C22×C3#403111yescharacteristic
381C22×C3#403411yescharacteristic
382C3×C4—202no—
383C3×C4—201no—
384C3×C4—201no—
385C3×C4C4×Dic1011yescharacteristic
386C3×C4#402411yescharacteristic
387C3×C4#403311yescharacteristic
388C3×C4#404211yescharacteristic
389Dic3—42no—
390Dic3—41no—
391Dic3—41no—
392Dic3—52no—
393Dic3—101no—
394Dic3—101no—
395Dic3—101no—
396C2×C5—21no—
397C2×C5—21no—
398C2×C5#464611yescharacteristic
399C2×C5D3×#10411yescharacteristic
400C2×C5#482111yescharacteristic
401C23#677711yescharacteristic
402Q8—602no—
403Q8—602no—
404C2×C4—62no—
405C2×C4—61no—
406C2×C4—61no—
407C2×C4—102no—
408C2×C4—101no—
409C2×C4—101no—
410C2×C4—121no—
411C2×C4—121no—
412C2×C4—151no—
413C2×C4—151no—
414C2×C4—151no—
415C2×C4—151no—
416C2×C4—201no—
417C2×C4—201no—
418C2×C4—302no—
419C2×C4—601no—
420C2×C4D15⋊Q811yescharacteristic
421C2×C4#677111yescharacteristic
422C2×C4C4×D3×D511yescharacteristic
423C2×C4#678011yescharacteristic
424C2×C4C2×#120411yescharacteristic
425C2×C4#679011yescharacteristic
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—
436C22#9146411yescharacteristic
437C22#9147011yescharacteristic
438C22#9147311yescharacteristic
439C4—122no—
440C4—121no—
441C4—121no—
442C4—152no—
443C4—202no—
444C4—201no—
445C4—201no—
446C4—301no—
447C4—301no—
448C4—301no—
449C4#9132511yescharacteristic
450C4#9137711yescharacteristic
451C4C4×#120411yescharacteristic
452C4#9157611yescharacteristic
453C3#1059952311yescharacteristic, complemented, Sylow
454C2—21no—
455C2—21no—
456C2#1174823611yescharacteristic
457C2#1174823711yescharacteristic
458C2#1174893511yescharacteristic
subgroupG/H
centerC2×C4#6790
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC2×C3×C5×M16C22
socleC22×C3×C5C2×C22⋊C4
prime radicalC23×C3×C5C2×D4
cosocleC22×C3×C4×C5C23
hypercenterC2×M16D3×D5
hypocenterC3×C5#1986
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
19. #91470C4semidirect
20. #91470C4semidirect
23. #91384C4semidirect
24. #91384C4semidirect
44. C3×C5×M16Q8semidirect
51. #6710Q8semidirect
153. C5×M16Dic6semidirect
215. C3×M16Dic10semidirect
350. M16#1204semidirect
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
#117546172
M16×Dic152
C4×#66692
C5×#45552
C3×#647752
#117459772
#117459742
#105995233
#687305