Finite Group Databasebeta

← #14302912

Subgroups — #14302912

Order 1920.

724 conjugacy classes of subgroups.

The subgroup lattice has 468 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#66164C211yesmaximal, characteristic
2#11754488C211yesmaximal, characteristic, complemented
3#11751337C211yesmaximal, characteristic, complemented
4C5×#5236C211yesmaximal, characteristic, complemented
5C2×#90958C211yesmaximal, characteristic, complemented
6#11745970C211yesmaximal, characteristic, complemented
7#11745969C211yesmaximal, characteristic, complemented
8#10601201—31nomaximal, Hall
9C2×C3×C5×#41C2211yescharacteristic
10C22×C3×D20C2211yescharacteristic
11#91815C2211yescharacteristic
12C2×#6710C2211yescharacteristic
13C5×#926C2211yescharacteristic
14C2×C5×#208C2211yescharacteristic
15C3×#4093C412yescomplemented
16C3×#4086C412yescomplemented
17C3×#4083—22no—
18C3×#4076—22no—
19C3×#4072C2211yescharacteristic
20D5×#217—22no—
21#91434—22no—
22Dic3×D20—22no—
23#91121—22no—
24C5×#837—22no—
25#90958—22no—
26#71853—51nomaximal, Hall
27#66164D311yescharacteristic
28#65855—31no—
29C5×#421—31no—
30#65525—31no—
31C2×#3955—31no—
32#64777—31no—
33#64751—31no—
34C23×C3×D5D412yes—
35C3×C5×#41—21no—
36C3×C5×#41—21no—
37C3×C5×#41—21no—
38C3×C5×#41—21no—
39C2×C3×C5×Q8C2×C411yescharacteristic, complemented
40C2×C3×C5×D4—21no—
41C2×C3×C5×D4C2×C411yescharacteristic, complemented
42C22×C3×C4×C5—21no—
43C22×C3×C4×C5C2311yescharacteristic
44C2×C3×C22⋊D5—24no—
45C2×C3×D20—21no—
46C2×C3×D20—21no—
47C2×C3×D20—21no—
48C2×C3×D20—21no—
49C2×C3×D20C2×C412yes—
50C2×C3×D20D412yes—
51C2×D5×Dic3—24no—
52#6719—21no—
53C4⋊Dic15—21no—
54C4×Dic15D412yescomplemented
55#6710—21no—
56#6710—21no—
57C2×C3⋊#168—21no—
58C5×#217—21no—
59C5×C4⋊Dic3—21no—
60C4×C5×Dic3D412yescomplemented
61C5×#208—21no—
62C5×#208—21no—
63C2×C5×C3⋊C8—21no—
64C3×C22⋊Dic5—21no—
65C3×#685—24no—
66C3×C4⋊Dic5—21no—
67C3×C4×Dic5C2×C412yescomplemented
68#6665—24no—
69C3×#535—51no—
70#5798—51no—
71#5236—51no—
72#5236D511yescharacteristic, complemented
73C2×#833—51no—
74#4531—51no—
75#4531—51no—
76C2×C5×#41C2×D311yescharacteristic
77C22×D20C2×D311yescharacteristic
78C2×C5×M16—31no—
79C5×#117—31no—
80#4093Dic312yescomplemented
81#4086Dic312yescomplemented
82#4083—22no—
83#4076—22no—
84#4072—31no—
85#4072C2×D311yescharacteristic
86C2×#681—31no—
87#4039—62no—
88D5×C22⋊C4—62no—
89C4×D20—62no—
90C5×#104—62no—
91#3969—62no—
92#3955—62no—
93#1911—151noSylow
94C23×C3×C5—21no—
95C23×C3×C5C2×D411yescharacteristic
96C22×C3×D5—22no—
97C22×C3×D5—22no—
98C22×C3×D5—82no—
99C22×C3×D5C22⋊C414yes—
100C3×C5×Q8—21no—
101C3×C5×Q8—21no—
102C3×C5×D4—21no—
103C3×C5×D4—21no—
104C3×C5×D4—41no—
105C3×C5×D4—41no—
106C2×C3×C4×C5—21no—
107C2×C3×C4×C5—41no—
108C2×C3×C4×C5—41no—
109C2×C3×C4×C5C22×C411yescharacteristic
110C2×C3×C4×C5C2×D411yescharacteristic
111C2×C3×C4×C5C2×D411yescharacteristic
112C2×C3×C4×C5C2×D411yescharacteristic
113C2×C3×C4×C5#4111yescharacteristic
114C2×C3×C4×C5#4111yescharacteristic
115C2×Dic15—22no—
116C2×C5×Dic3—22no—
117C3×C22⋊D5—84no—
118C2×C3×Dic5—22no—
119C3×D20—22no—
120C3×D20—22no—
121C3×D20—22no—
122C3×D20—22no—
123D5×Dic3—84no—
124C3⋊#168—41no—
125C5×C3⋊C8—41no—
126C2×C3×#41C2×D511yescharacteristic
127C22×C3×D4—51no—
128C3×#127—52no—
129C3×#124—52no—
130C3×#121—102no—
131C3×#120—102no—
132C3×#117—51no—
133C2×#217—102no—
134D4×Dic3—102no—
135D4×Dic3—102no—
136#926—51no—
137#926C2×D511yescharacteristic
138C2×#208—51no—
139C2×#208C2×D511yescharacteristic
140#837—22no—
141#837—102no—
142#833—102no—
143C23×D5#6712yes—
144C5×#41—21no—
145C5×#41—21no—
146C5×#41—21no—
147C5×#41—21no—
148C2×C5×Q8C2×Dic311yescharacteristic, complemented
149C2×C5×D4—21no—
150C2×C5×D4C2×Dic311yescharacteristic, complemented
151C22×C4×C5—21no—
152C22×C4×C5C22×D311yescharacteristic
153C2×C22⋊D5—24no—
154C2×D20—21no—
155C2×D20—21no—
156C2×D20—21no—
157C2×D20—21no—
158C2×D20C2×Dic312yes—
159C2×D20#6712yes—
160C2×C4×D5—64no—
161C5×M16—61no—
162C5×M16—61no—
163C2×C5×C8—61no—
164C5×C4⋊C4—61no—
165C5×C22⋊C4—61no—
166C42×C5—32no—
167C22⋊Dic5—21no—
168C22⋊Dic5—61no—
169#685—24no—
170#685—64no—
171C4⋊Dic5—21no—
172C4⋊Dic5—61no—
173C4×Dic5—32no—
174C4×Dic5C2×Dic312yescomplemented
175#681—61no—
176#681—61no—
177C2×#168—61no—
178#535—51no—
179#518—151no—
180#421—31no—
181#421—151no—
182C2×#100—151no—
183H(3,ℤ/4)—151no—
184H(3,ℤ/4)—151no—
185C22×C3×C5—41no—
186C22×C3×C5—42no—
187C22×C3×C5—41no—
188C22×C3×C5C2×C22⋊C411yescharacteristic
189C22×C3×C5#12011yescharacteristic
190C22×C3×C5#12311yescharacteristic
191C2×C3×D5—24no—
192C2×C3×D5—44no—
193C2×C3×D5—44no—
194C2×C3×D5—82no—
195C3×C4×C5—41no—
196C3×C4×C5C4×D411yescharacteristic
197C3×C4×C5C4×D411yescharacteristic
198C3×C4×C5#12111yescharacteristic
199C3×C4×C5#12111yescharacteristic
200Dic15—42no—
201C3×Dic5—42no—
202C5×Dic3—42no—
203C24×C3—52no—
204C3×#41—21no—
205C3×#41—21no—
206C3×#41—21no—
207C3×#41—21no—
208C2×C3×Q8C4×D511yescharacteristic, complemented
209C2×C3×D4—21no—
210C2×C3×D4—52no—
211C2×C3×D4—52no—
212C2×C3×D4—104no—
213C2×C3×D4—101no—
214C2×C3×D4—101no—
215C2×C3×D4—101no—
216C2×C3×D4—101no—
217C2×C3×D4C4×D511yescharacteristic, complemented
218C22×C3×C4—21no—
219C22×C3×C4C22×D511yescharacteristic
220C22×Dic3—104no—
221C3×C4⋊C4—101no—
222C3×C22⋊C4—104no—
223C3×C22⋊C4—101no—
224C3×C42—52no—
225#217—21no—
226#217—104no—
227#217—101no—
228C4⋊Dic3—21no—
229C4⋊Dic3—101no—
230C4×Dic3—52no—
231C4×Dic3C22⋊D512yescomplemented
232#208—21no—
233#208—21no—
234#208—101no—
235#208—101no—
236C2×C3⋊C8—21no—
237C2×C3⋊C8—101no—
238C23×C5—21no—
239C23×C5C2×#6711yescharacteristic
240C22×D5—22no—
241C22×D5—22no—
242C22×D5—82no—
243C22×D5#21714yes—
244C5×Q8—21no—
245C5×Q8—21no—
246C5×D4—21no—
247C5×D4—21no—
248C5×D4—41no—
249C5×D4—41no—
250C2×C4×C5—21no—
251C2×C4×C5—41no—
252C2×C4×C5—41no—
253C2×C4×C5—62no—
254C2×C4×C5D3×D411yescharacteristic
255C2×C4×C5D3×D411yescharacteristic
256C2×C4×C5#23711yescharacteristic
257C2×C4×C5#23711yescharacteristic
258C2×C4×C5C22×Dic311yescharacteristic
259C2×C4×C5C2×#6711yescharacteristic
260C22⋊D5—84no—
261C2×Dic5—22no—
262C2×Dic5—62no—
263D20—22no—
264D20—22no—
265D20—22no—
266D20—22no—
267C4×D5—244no—
268C5×C8—121no—
269#168—121no—
270C2×#41D3×D511yescharacteristic
271C22×D4—51no—
272C2×M16—31no—
273C2×M16—151no—
274#127—52no—
275#124—52no—
276#121—102no—
277#120—102no—
278C4×D4—302no—
279C4×D4—302no—
280#117—31no—
281#117—51no—
282#117—151no—
283C2×C22⋊C4—302no—
284#104—62no—
285#104—302no—
286#100—302no—
287C2×C3×C5—21no—
288C2×C3×C5—21no—
289C2×C3×C5—21no—
290C2×C3×C5—81no—
291C2×C3×C5#38611yescharacteristic
292C3×D5—44no—
293C23×C3—21no—
294C23×C3—54no—
295C23×C3—102no—
296C23×C3—102no—
297C23×C3—402no—
298C23×C3D4×D511yescharacteristic
299C3×Q8—21no—
300C3×Q8—21no—
301C3×D4—21no—
302C3×D4—21no—
303C3×D4—41no—
304C3×D4—41no—
305C3×D4—102no—
306C3×D4—102no—
307C3×D4—102no—
308C3×D4—102no—
309C3×D4—404no—
310C2×C3×C4—21no—
311C2×C3×C4—41no—
312C2×C3×C4—41no—
313C2×C3×C4—102no—
314C2×C3×C4C2×C4×D511yescharacteristic
315C2×C3×C4#70911yescharacteristic
316C2×C3×C4D4×D511yescharacteristic
317C2×C3×C4D4×D511yescharacteristic
318C2×C3×C4#71111yescharacteristic
319C2×C3×C4C2×C22⋊D511yescharacteristic
320C2×Dic3—22no—
321C2×Dic3—102no—
322C2×Dic3—404no—
323C3⋊C8—41no—
324C3⋊C8—201no—
325C22×C5—41no—
326C22×C5—42no—
327C22×C5—41no—
328C22×C5#93511yescharacteristic
329C22×C5#93711yescharacteristic
330C22×C5C2×#21711yescharacteristic
331C2×D5—24no—
332C2×D5—44no—
333C2×D5—44no—
334C2×D5—82no—
335C4×C5—41no—
336C4×C5—122no—
337C4×C5D4×Dic311yescharacteristic
338C4×C5D4×Dic311yescharacteristic
339C4×C5#93911yescharacteristic
340C4×C5#93911yescharacteristic
341Dic5—42no—
342Dic5—122no—
343C24—52no—
344#41—21no—
345#41—21no—
346#41—21no—
347#41—21no—
348C2×Q8D5×Dic311yescharacteristic, complemented
349C2×D4—21no—
350C2×D4—52no—
351C2×D4—52no—
352C2×D4—104no—
353C2×D4—101no—
354C2×D4—101no—
355C2×D4—101no—
356C2×D4—101no—
357C2×D4D5×Dic311yescharacteristic, complemented
358C22×C4—21no—
359C22×C4—304no—
360C22×C4C2×D3×D511yescharacteristic
361M16—61no—
362M16—61no—
363M16—301no—
364M16—301no—
365C2×C8—61no—
366C2×C8—301no—
367C4⋊C4—61no—
368C4⋊C4—101no—
369C4⋊C4—301no—
370C22⋊C4—61no—
371C22⋊C4—104no—
372C22⋊C4—101no—
373C22⋊C4—304no—
374C22⋊C4—301no—
375C42—32no—
376C42—52no—
377C42—152no—
378C3×C5#191111yescharacteristic, complemented, Hall
379C22×C3—41no—
380C22×C3—42no—
381C22×C3—41no—
382C22×C3—104no—
383C22×C3—204no—
384C22×C3—204no—
385C22×C3—402no—
386C22×C3D5×C22⋊C411yescharacteristic
387C22×C3#402911yescharacteristic
388C22×C3#408311yescharacteristic
389C3×C4—41no—
390C3×C4—202no—
391C3×C4#402711yescharacteristic
392C3×C4#403011yescharacteristic
393C3×C4C4×C22⋊D511yescharacteristic
394C3×C4#408411yescharacteristic
395Dic3—42no—
396Dic3—202no—
397C2×C5—21no—
398C2×C5—21no—
399C2×C5—21no—
400C2×C5—81no—
401C2×C5#522011yescharacteristic
402D5—44no—
403C23—21no—
404C23—54no—
405C23—102no—
406C23—102no—
407C23—402no—
408C23D5×#6711yescharacteristic
409Q8—21no—
410Q8—21no—
411D4—21no—
412D4—21no—
413D4—41no—
414D4—41no—
415D4—102no—
416D4—102no—
417D4—102no—
418D4—102no—
419D4—404no—
420C2×C4—21no—
421C2×C4—41no—
422C2×C4—41no—
423C2×C4—62no—
424C2×C4—102no—
425C2×C4—302no—
426C2×C4—1204no—
427C2×C4C2×D5×Dic311yescharacteristic
428C2×C4#677911yescharacteristic
429C2×C4#678011yescharacteristic
430C2×C4D5×#6711yescharacteristic
431C2×C4D3×C22⋊D511yescharacteristic
432C2×C4#679011yescharacteristic
433C8—121no—
434C8—601no—
435C2×C3—21no—
436C2×C3—21no—
437C2×C3—21no—
438C2×C3—81no—
439C2×C3—204no—
440C2×C3#6524511yescharacteristic
441C5#7185311yescharacteristic, complemented, Sylow
442C22—41no—
443C22—42no—
444C22—41no—
445C22—104no—
446C22—204no—
447C22—204no—
448C22—402no—
449C22D5×#21711yescharacteristic
450C22#9154811yescharacteristic
451C22#9157711yescharacteristic
452C4—41no—
453C4—122no—
454C4—202no—
455C4—602no—
456C4#9154911yescharacteristic
457C4#9155011yescharacteristic
458C4Dic3×C22⋊D511yescharacteristic
459C4#9155911yescharacteristic
460C3#1060120111yescharacteristic, complemented, Sylow
461C2—21no—
462C2—21no—
463C2—21no—
464C2—81no—
465C2—204no—
466C2#1174951811yescharacteristic
subgroupG/H
centerC2#11749518
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC2×C3×C5×#41C22
socleC2×C3×C5#386
prime radicalC3×#66164C2
cosocleC22×C3×C4×C5C23
hypercenterC2×#41D3×D5
hypocenterC3×C5#1911
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#41

Complements

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

NH ≅ G/Ntype
2. #11754488C2semidirect
3. #11751337C2semidirect
4. C5×#5236C2semidirect
5. C2×#90958C2semidirect
6. #11745970C2semidirect
7. #11745969C2semidirect
15. C3×#4093C4semidirect
16. C3×#4086C4semidirect
39. C2×C3×C5×Q8C2×C4semidirect
41. C2×C3×C5×D4C2×C4semidirect
54. C4×Dic15D4semidirect
60. C4×C5×Dic3D4semidirect
67. C3×C4×Dic5C2×C4semidirect
72. #5236D5semidirect
80. #4093Dic3semidirect
81. #4086Dic3semidirect
148. C2×C5×Q8C2×Dic3semidirect
150. C2×C5×D4C2×Dic3semidirect
174. C4×Dic5C2×Dic3semidirect
208. C2×C3×Q8C4×D5semidirect
217. C2×C3×D4C4×D5semidirect
231. C4×Dic3C22⋊D5semidirect
348. C2×Q8D5×Dic3semidirect
357. C2×D4D5×Dic3semidirect
378. C3×C5#1911semidirect
441. C5#71853semidirect
460. C3#10601201semidirect

Maximal subgroups

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

subgroupindex
C3×#661642
#117544882
#117513372
C5×#52362
C2×#909582
#117459702
#117459692
#106012013
#718535