Finite Group Databasebeta

← #14302915

Subgroups — #14302915

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×#65859C211yesmaximal, characteristic
2#11754488C211yesmaximal, characteristic, complemented
3#11748421C211yesmaximal, characteristic, complemented
4#11748420C211yesmaximal, characteristic, complemented
5C2×#90958C211yesmaximal, characteristic, complemented
6#11746352C211yesmaximal, characteristic, complemented
7C5×#4531C211yesmaximal, characteristic, complemented
8#10599498—31nomaximal, Hall
9C2×C3×#709C2211yescharacteristic
10C22×C3×D20C2211yescharacteristic
11C3×C5×#117C2211yescharacteristic
12#91815C2211yescharacteristic
13C2×#6710C2211yescharacteristic
14C5×#926C2211yescharacteristic
15C2×C5×#208C2211yescharacteristic
16C3×#4041—22no—
17C3×#4028—22no—
18C3×#4021C412yescomplemented
19C3×#4020C412yescomplemented
20D5×#217—22no—
21#91434—22no—
22Dic3×D20—22no—
23#90977—22no—
24#90976—22no—
25#90958—22no—
26#71853—51nomaximal, Hall
27#65859D311yescharacteristic
28#65855—31no—
29#65428—31no—
30C2×#3955—31no—
31#65294—31no—
32C5×H(3,ℤ/4)—31no—
33#64751—31no—
34C23×C3×D5D412yes—
35C22×C3×C4×C5C2311yescharacteristic
36C2×C3×C22⋊D5—21no—
37C3×#709—21no—
38C3×#709—21no—
39C3×#709—21no—
40C3×#709—21no—
41C2×C3×D20—21no—
42C2×C3×D20—21no—
43C2×C3×D20—21no—
44C2×C3×D20—21no—
45C2×C3×D20—24no—
46C2×C3×D20C2×C412yes—
47C2×C3×D20C2×C411yescharacteristic, complemented
48C2×C3×D20D412yes—
49C2×C3×C4×D5—21no—
50C2×C3×Dic10C2×C411yescharacteristic, complemented
51C2×D5×Dic3—24no—
52C3×C5×C4⋊C4—21no—
53C3×C5×C22⋊C4—21no—
54C3×C42×C5C2×C412yescomplemented
55#6719—21no—
56C4⋊Dic15—21no—
57C4×Dic15D412yescomplemented
58#6710—21no—
59#6710—21no—
60C2×C3⋊#168—21no—
61C5×#217—21no—
62C5×C4⋊Dic3—21no—
63C4×C5×Dic3D412yescomplemented
64C5×#208—21no—
65C5×#208—21no—
66C2×C5×C3⋊C8—21no—
67C3×#685—24no—
68#6665—24no—
69C3×#535—51no—
70#5798—51no—
71#5236—51no—
72#5236—51no—
73C2×#833—51no—
74#4531—51no—
75#4531D511yescharacteristic, complemented
76C2×#709C2×D311yescharacteristic
77C22×D20C2×D311yescharacteristic
78C2×C5×M16—31no—
79C5×#117—31no—
80C5×#117C2×D311yescharacteristic
81#4072—31no—
82C2×#681—31no—
83#4041—22no—
84#4039—62no—
85#4028—22no—
86D5×C22⋊C4—62no—
87#4021Dic312yescomplemented
88#4020Dic312yescomplemented
89C4×D20—62no—
90#3957—62no—
91#3955—62no—
92#3937—62no—
93#1911—151noSylow
94C23×C3×C5C2×D411yescharacteristic
95C22×C3×D5—22no—
96C22×C3×D5—22no—
97C22×C3×D5—21no—
98C22×C3×D5—82no—
99C22×C3×D5C22⋊C414yes—
100C2×C3×C4×C5—22no—
101C2×C3×C4×C5C22×C411yescharacteristic
102C2×C3×C4×C5C2×D411yescharacteristic
103C2×C3×C4×C5C2×D411yescharacteristic
104C2×C3×C4×C5C2×D411yescharacteristic
105C2×C3×C4×C5#4111yescharacteristic
106C2×C3×C4×C5#4111yescharacteristic
107C2×Dic15—22no—
108C2×C5×Dic3—22no—
109C3×C22⋊D5—41no—
110C3×C22⋊D5—41no—
111C2×C3×Dic5—21no—
112C3×D20—22no—
113C3×D20—22no—
114C3×D20—21no—
115C3×D20—21no—
116C3×D20—22no—
117C3×D20—22no—
118C3×D20—84no—
119C3×C4×D5—41no—
120C3×C4×D5—41no—
121C3×Dic10—21no—
122C3×Dic10—21no—
123D5×Dic3—84no—
124C3⋊#168—41no—
125C5×C3⋊C8—41no—
126C2×C3×#41—51no—
127C22×C3×D4—51no—
128C3×#127—52no—
129C3×#124—52no—
130C3×#121—102no—
131C3×#120—102no—
132C3×#117C2×D511yescharacteristic
133C2×#217—102no—
134D4×Dic3—102no—
135D4×Dic3—102no—
136#926—51no—
137#926C2×D511yescharacteristic
138C2×#208—51no—
139C2×#208C2×D511yescharacteristic
140#837—102no—
141#837—102no—
142#833—102no—
143C23×D5#6712yes—
144C22×C4×C5C22×D311yescharacteristic
145C2×C22⋊D5—21no—
146#709—21no—
147#709—21no—
148#709—21no—
149#709—21no—
150C2×D20—21no—
151C2×D20—21no—
152C2×D20—21no—
153C2×D20—21no—
154C2×D20—24no—
155C2×D20C2×Dic312yes—
156C2×D20C2×Dic311yescharacteristic, complemented
157C2×D20#6712yes—
158C2×C4×D5—21no—
159C2×C4×D5—64no—
160C2×Dic10C2×Dic311yescharacteristic, complemented
161C5×M16—61no—
162C5×M16—61no—
163C2×C5×C8—61no—
164C5×C4⋊C4—21no—
165C5×C4⋊C4—61no—
166C5×C22⋊C4—21no—
167C5×C22⋊C4—61no—
168C42×C5—32no—
169C42×C5C2×Dic312yescomplemented
170C22⋊Dic5—61no—
171#685—24no—
172#685—64no—
173C4⋊Dic5—61no—
174C4×Dic5—32no—
175#681—61no—
176#681—61no—
177C2×#168—61no—
178#535—51no—
179#518—151no—
180#421—151no—
181#421—151no—
182C2×#100—151no—
183H(3,ℤ/4)—31no—
184H(3,ℤ/4)—151no—
185C22×C3×C5—41no—
186C22×C3×C5C2×C22⋊C411yescharacteristic
187C22×C3×C5#12011yescharacteristic
188C22×C3×C5#12311yescharacteristic
189C2×C3×D5—24no—
190C2×C3×D5—44no—
191C2×C3×D5—44no—
192C2×C3×D5—42no—
193C2×C3×D5—41no—
194C2×C3×D5—82no—
195C3×C4×C5—42no—
196C3×C4×C5C4×D411yescharacteristic
197C3×C4×C5C4×D411yescharacteristic
198C3×C4×C5#12111yescharacteristic
199C3×C4×C5#12111yescharacteristic
200Dic15—42no—
201C3×Dic5—41no—
202C5×Dic3—42no—
203C24×C3—52no—
204C3×#41—101no—
205C3×#41—101no—
206C3×#41—101no—
207C3×#41—101no—
208C2×C3×Q8—51no—
209C2×C3×D4—52no—
210C2×C3×D4—51no—
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×D4—101no—
218C22×C3×C4—101no—
219C22×C3×C4C22×D511yescharacteristic
220C22×Dic3—104no—
221C3×C4⋊C4—21no—
222C3×C22⋊C4—21no—
223C3×C22⋊C4—104no—
224C3×C42C4×D512yescomplemented
225#217—21no—
226#217—104no—
227#217—101no—
228C4⋊Dic3—21no—
229C4⋊Dic3—101no—
230C4×Dic3—52no—
231C4×Dic3D2012yescomplemented
232#208—21no—
233#208—21no—
234#208—101no—
235#208—101no—
236C2×C3⋊C8—21no—
237C2×C3⋊C8—101no—
238C23×C5C2×#6711yescharacteristic
239C22×D5—22no—
240C22×D5—22no—
241C22×D5—21no—
242C22×D5—82no—
243C22×D5#21714yes—
244C2×C4×C5—22no—
245C2×C4×C5—62no—
246C2×C4×C5D3×D411yescharacteristic
247C2×C4×C5D3×D411yescharacteristic
248C2×C4×C5#23711yescharacteristic
249C2×C4×C5#23711yescharacteristic
250C2×C4×C5C22×Dic311yescharacteristic
251C2×C4×C5C2×#6711yescharacteristic
252C22⋊D5—41no—
253C22⋊D5—41no—
254C2×Dic5—21no—
255C2×Dic5—62no—
256D20—22no—
257D20—22no—
258D20—21no—
259D20—21no—
260D20—22no—
261D20—22no—
262D20—84no—
263C4×D5—41no—
264C4×D5—41no—
265C4×D5—244no—
266Dic10—21no—
267Dic10—21no—
268C5×C8—121no—
269#168—121no—
270C2×#41—51no—
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—151no—
282#117D3×D511yescharacteristic
283C2×C22⋊C4—302no—
284#104—302no—
285#104—302no—
286#100—302no—
287C2×C3×C5—21no—
288C2×C3×C5—21no—
289C2×C3×C5—21no—
290C2×C3×C5#38611yescharacteristic
291C3×D5—44no—
292C3×D5—81no—
293C23×C3—54no—
294C23×C3—102no—
295C23×C3—102no—
296C23×C3—101no—
297C23×C3—402no—
298C23×C3D4×D511yescharacteristic
299C3×Q8—101no—
300C3×Q8—101no—
301C3×D4—102no—
302C3×D4—101no—
303C3×D4—102no—
304C3×D4—102no—
305C3×D4—101no—
306C3×D4—102no—
307C3×D4—201no—
308C3×D4—201no—
309C3×D4—404no—
310C2×C3×C4—22no—
311C2×C3×C4—101no—
312C2×C3×C4—201no—
313C2×C3×C4—201no—
314C2×C3×C4C2×C4×D511yescharacteristic
315C2×C3×C4C2×D2011yescharacteristic
316C2×C3×C4#70911yescharacteristic
317C2×C3×C4D4×D511yescharacteristic
318C2×C3×C4D4×D511yescharacteristic
319C2×C3×C4#71311yescharacteristic
320C2×Dic3—22no—
321C2×Dic3—102no—
322C2×Dic3—404no—
323C3⋊C8—41no—
324C3⋊C8—201no—
325C22×C5—41no—
326C22×C5#93511yescharacteristic
327C22×C5#93711yescharacteristic
328C22×C5C2×#21711yescharacteristic
329C2×D5—24no—
330C2×D5—44no—
331C2×D5—44no—
332C2×D5—42no—
333C2×D5—41no—
334C2×D5—82no—
335C4×C5—42no—
336C4×C5—122no—
337C4×C5D4×Dic311yescharacteristic
338C4×C5D4×Dic311yescharacteristic
339C4×C5#93911yescharacteristic
340C4×C5#93911yescharacteristic
341Dic5—41no—
342Dic5—122no—
343C24—52no—
344#41—101no—
345#41—101no—
346#41—101no—
347#41—101no—
348C2×Q8—51no—
349C2×D4—52no—
350C2×D4—51no—
351C2×D4—52no—
352C2×D4—104no—
353C2×D4—101no—
354C2×D4—101no—
355C2×D4—101no—
356C2×D4—101no—
357C2×D4—101no—
358C22×C4—101no—
359C22×C4—304no—
360C22×C4C2×D3×D511yescharacteristic
361M16—61no—
362M16—61no—
363M16—301no—
364M16—301no—
365C2×C8—61no—
366C2×C8—301no—
367C4⋊C4—21no—
368C4⋊C4—61no—
369C4⋊C4—301no—
370C22⋊C4—21no—
371C22⋊C4—61no—
372C22⋊C4—104no—
373C22⋊C4—304no—
374C22⋊C4—301no—
375C42—32no—
376C42—152no—
377C42D5×Dic312yescomplemented
378C3×C5#191111yescharacteristic, complemented, Hall
379C22×C3—41no—
380C22×C3—104no—
381C22×C3—204no—
382C22×C3—204no—
383C22×C3—202no—
384C22×C3—201no—
385C22×C3—402no—
386C22×C3D5×C22⋊C411yescharacteristic
387C22×C3#402811yescharacteristic
388C22×C3#404011yescharacteristic
389C3×C4—42no—
390C3×C4—201no—
391C3×C4C4×D2011yescharacteristic
392C3×C4#403011yescharacteristic
393C3×C4#403911yescharacteristic
394C3×C4#404111yescharacteristic
395Dic3—42no—
396Dic3—202no—
397C2×C5—21no—
398C2×C5—21no—
399C2×C5—21no—
400C2×C5#522011yescharacteristic
401D5—44no—
402D5—81no—
403C23—54no—
404C23—102no—
405C23—102no—
406C23—101no—
407C23—402no—
408C23D5×#6711yescharacteristic
409Q8—101no—
410Q8—101no—
411D4—102no—
412D4—102no—
413D4—102no—
414D4—101no—
415D4—102no—
416D4—101no—
417D4—201no—
418D4—201no—
419D4—404no—
420C2×C4—22no—
421C2×C4—62no—
422C2×C4—101no—
423C2×C4—201no—
424C2×C4—201no—
425C2×C4—302no—
426C2×C4—1204no—
427C2×C4#676511yescharacteristic
428C2×C4#676611yescharacteristic
429C2×C4D3×D2011yescharacteristic
430C2×C4#677711yescharacteristic
431C2×C4C2×D5×Dic311yescharacteristic
432C2×C4D5×#6711yescharacteristic
433C8—121no—
434C8—601no—
435C2×C3—21no—
436C2×C3—21no—
437C2×C3—21no—
438C2×C3—204no—
439C2×C3—401no—
440C2×C3#6495411yescharacteristic
441C5#7185311yescharacteristic, complemented, Sylow
442C22—41no—
443C22—104no—
444C22—204no—
445C22—204no—
446C22—202no—
447C22—201no—
448C22—402no—
449C22#9140711yescharacteristic
450C22#9147411yescharacteristic
451C22D5×#21711yescharacteristic
452C4—42no—
453C4—122no—
454C4—201no—
455C4—602no—
456C4#9140811yescharacteristic
457C4#9140911yescharacteristic
458C4Dic3×D2011yescharacteristic
459C4#9143411yescharacteristic
460C3#1059949811yescharacteristic, complemented, Sylow
461C2—21no—
462C2—21no—
463C2—21no—
464C2—204no—
465C2—401no—
466C2#1174889611yescharacteristic
subgroupG/H
centerC2#11748896
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC3×C5×#117C22
socleC2×C3×C5#386
prime radicalC3×#65859C2
cosocleC22×C3×C4×C5C23
hypercenter#117D3×D5
hypocenterC3×C5#1911
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
2. #11754488C2semidirect
3. #11748421C2semidirect
4. #11748420C2semidirect
5. C2×#90958C2semidirect
6. #11746352C2semidirect
7. C5×#4531C2semidirect
18. C3×#4021C4semidirect
19. C3×#4020C4semidirect
47. C2×C3×D20C2×C4semidirect
50. C2×C3×Dic10C2×C4semidirect
54. C3×C42×C5C2×C4semidirect
57. C4×Dic15D4semidirect
63. C4×C5×Dic3D4semidirect
75. #4531D5semidirect
87. #4021Dic3semidirect
88. #4020Dic3semidirect
156. C2×D20C2×Dic3semidirect
160. C2×Dic10C2×Dic3semidirect
169. C42×C5C2×Dic3semidirect
224. C3×C42C4×D5semidirect
231. C4×Dic3D20semidirect
377. C42D5×Dic3semidirect
378. C3×C5#1911semidirect
441. C5#71853semidirect
460. C3#10599498semidirect

Maximal subgroups

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

subgroupindex
C3×#658592
#117544882
#117484212
#117484202
C2×#909582
#117463522
C5×#45312
#105994983
#718535