Finite Group Databasebeta

← #14302916

Subgroups — #14302916

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
1#11756284C211yesmaximal, characteristic
2#11754515C211yesmaximal, characteristic, complemented
3#11748429C211yesmaximal, characteristic, complemented
4#11748428C211yesmaximal, characteristic, complemented
5C2×#90960C211yesmaximal, characteristic, complemented
6C5×#4531C211yesmaximal, characteristic, complemented
7C3×#64751C211yesmaximal, characteristic, complemented
8#10599498—31nomaximal, Hall
9C2×#6817C2211yescharacteristic
10C22×D60C2211yescharacteristic
11C3×C5×#117C2211yescharacteristic
12#91784—22no—
13#91771—22no—
14#91764C412yescomplemented
15#91763C412yescomplemented
16C5×#926C2211yescharacteristic
17C2×C5×#208C2211yescharacteristic
18C3×#4072C2211yescharacteristic
19C2×C3×#681C2211yescharacteristic
20#91561—22no—
21#91428—22no—
22#91418—22no—
23#90981—22no—
24#90980—22no—
25#90960—22no—
26#68705—51nomaximal, Hall
27#65859—31no—
28#65855—31no—
29#65428—31no—
30C2×#3955—31no—
31#65294—31no—
32C5×H(3,ℤ/4)—31no—
33#64751D311yescharacteristic, complemented
34C23×D15D412yes—
35C22×C3×C4×C5C2311yescharacteristic
36C2×#1220—21no—
37#6817—21no—
38#6817—21no—
39#6817—21no—
40#6817—21no—
41C2×D60—21no—
42C2×D60—21no—
43C2×D60—21no—
44C2×D60—21no—
45C2×D60—24no—
46C2×D60C2×C412yes—
47C2×D60C2×C411yescharacteristic, complemented
48C2×D60D412yes—
49C2×C4×D15—21no—
50C2×Dic30C2×C411yescharacteristic, complemented
51C2×#1200—24no—
52C3×C5×C4⋊C4—21no—
53C3×C5×C22⋊C4—21no—
54C3×C42×C5C2×C412yescomplemented
55#6714—24no—
56C5×#217—21no—
57C5×C4⋊Dic3—21no—
58C4×C5×Dic3D412yescomplemented
59C5×#208—21no—
60C5×#208—21no—
61C2×C5×C3⋊C8—21no—
62C3×C22⋊Dic5—21no—
63C3×C4⋊Dic5—21no—
64C3×C4×Dic5D412yescomplemented
65C3×#681—21no—
66C3×#681—21no—
67C2×C3×#168—21no—
68#6667—24no—
69#5525—51no—
70#5521—51no—
71#5139—51no—
72C2×#823—51no—
73#5005—51no—
74C3×H(3,ℤ/4)—51no—
75#4531D511yescharacteristic, complemented
76C2×#709—31no—
77C22×D20—31no—
78C2×C5×M16—31no—
79C5×#117—31no—
80C5×#117C2×D311yescharacteristic
81#4072C2×D311yescharacteristic
82C2×#681C2×D311yescharacteristic
83#4041—62no—
84#4039—62no—
85#4028—62no—
86D5×C22⋊C4—62no—
87#4021—32no—
88#4020—32no—
89C4×D20—62no—
90#3957—62no—
91#3955—62no—
92#3937—62no—
93#1911—151noSylow
94C23×C3×C5C2×D411yescharacteristic
95C22×D15—22no—
96C22×D15—22no—
97C22×D15—21no—
98C22×D15—82no—
99C22×D15C22⋊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
107#1220—41no—
108#1220—41no—
109C2×Dic15—21no—
110D60—22no—
111D60—22no—
112D60—21no—
113D60—21no—
114D60—22no—
115D60—22no—
116D60—84no—
117C4×D15—41no—
118C4×D15—41no—
119Dic30—21no—
120Dic30—21no—
121C2×C5×Dic3—22no—
122C2×C3×Dic5—22no—
123#1200—84no—
124C3×#168—41no—
125C5×C3⋊C8—41no—
126C2×#235—51no—
127C22×D12—51no—
128C2×C3×M16—51no—
129C3×#117—51no—
130C3×#117C2×D511yescharacteristic
131#926C2×D511yescharacteristic
132C2×#208C2×D511yescharacteristic
133#895—102no—
134#893—102no—
135#882—102no—
136D3×C22⋊C4—102no—
137#875—52no—
138#874—52no—
139C4×D12—102no—
140#825—102no—
141#823—102no—
142#805—102no—
143C23×D5—32no—
144C22×C4×C5C22×D311yescharacteristic
145C2×C22⋊D5—61no—
146#709—61no—
147#709—61no—
148#709—61no—
149#709—61no—
150C2×D20—32no—
151C2×D20—31no—
152C2×D20—32no—
153C2×D20—61no—
154C2×D20—61no—
155C2×D20—61no—
156C2×D20—61no—
157C2×D20—64no—
158C2×C4×D5—61no—
159C2×C4×D5—64no—
160C2×Dic10—31no—
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×C5C4×D312yescomplemented
170C22⋊Dic5—21no—
171#685—64no—
172#685—64no—
173C4⋊Dic5—21no—
174C4×Dic5D1212yescomplemented
175#681—21no—
176#681—21no—
177C2×#168—21no—
178#535—151no—
179#518—151no—
180#421—151no—
181#421—151no—
182C2×#100—151no—
183H(3,ℤ/4)—31no—
184H(3,ℤ/4)—51no—
185C22×C3×C5—41no—
186C22×C3×C5C2×C22⋊C411yescharacteristic
187C22×C3×C5#12011yescharacteristic
188C22×C3×C5#12311yescharacteristic
189C2×D15—24no—
190C2×D15—44no—
191C2×D15—44no—
192C2×D15—42no—
193C2×D15—41no—
194C2×D15—82no—
195C3×C4×C5—42no—
196C3×C4×C5C4×D411yescharacteristic
197C3×C4×C5C4×D411yescharacteristic
198C3×C4×C5#12111yescharacteristic
199C3×C4×C5#12111yescharacteristic
200Dic15—41no—
201C3×Dic5—42no—
202C5×Dic3—42no—
203C23×D3—52no—
204C22×C3×C4C22×D511yescharacteristic
205C2×#67—101no—
206#235—101no—
207#235—101no—
208#235—101no—
209#235—101no—
210C2×D12—52no—
211C2×D12—51no—
212C2×D12—52no—
213C2×D12—101no—
214C2×D12—101no—
215C2×D12—101no—
216C2×D12—101no—
217C2×D12—104no—
218C2×C4×D3—101no—
219C2×C4×D3—104no—
220C2×Dic6—51no—
221C3×M16—101no—
222C3×M16—101no—
223C2×C3×C8—101no—
224C3×C4⋊C4—21no—
225C3×C4⋊C4—101no—
226C3×C22⋊C4—21no—
227C3×C22⋊C4—101no—
228C3×C42—52no—
229C3×C42C4×D512yescomplemented
230#217—21no—
231(C2×D3)⋊C4—104no—
232(C2×D3)⋊C4—104no—
233C4⋊Dic3—21no—
234C4×Dic3D2012yescomplemented
235#208—21no—
236#208—21no—
237C2×C3⋊C8—21no—
238C23×C5D3×D411yescharacteristic
239C22×D5—34no—
240C22×D5—62no—
241C22×D5—62no—
242C22×D5—61no—
243C22×D5—242no—
244C2×C4×C5—22no—
245C2×C4×C5—62no—
246C2×C4×C5C2×C4×D311yescharacteristic
247C2×C4×C5C2×D1211yescharacteristic
248C2×C4×C5#23511yescharacteristic
249C2×C4×C5D3×D411yescharacteristic
250C2×C4×C5D3×D411yescharacteristic
251C2×C4×C5#23911yescharacteristic
252C22⋊D5—121no—
253C22⋊D5—121no—
254C2×Dic5—22no—
255C2×Dic5—61no—
256D20—62no—
257D20—61no—
258D20—62no—
259D20—62no—
260D20—61no—
261D20—62no—
262D20—244no—
263C4×D5—121no—
264C4×D5—121no—
265C4×D5—244no—
266Dic10—61no—
267Dic10—61no—
268C5×C8—121no—
269#168—41no—
270C2×#41—151no—
271C22×D4—151no—
272C2×M16—31no—
273C2×M16—51no—
274#127—152no—
275#124—152no—
276#121—302no—
277#120—302no—
278C4×D4—302no—
279C4×D4—302no—
280#117—31no—
281#117—51no—
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
291D15—44no—
292D15—81no—
293C23×C3D4×D511yescharacteristic
294C22×D3—54no—
295C22×D3—102no—
296C22×D3—102no—
297C22×D3—101no—
298C22×D3—402no—
299C2×C3×C4—22no—
300C2×C3×C4—102no—
301C2×C3×C4C2×C4×D511yescharacteristic
302C2×C3×C4C2×D2011yescharacteristic
303C2×C3×C4#70911yescharacteristic
304C2×C3×C4D4×D511yescharacteristic
305C2×C3×C4D4×D511yescharacteristic
306C2×C3×C4#71311yescharacteristic
307#67—201no—
308#67—201no—
309C2×Dic3—22no—
310C2×Dic3—101no—
311D12—102no—
312D12—101no—
313D12—102no—
314D12—102no—
315D12—101no—
316D12—102no—
317D12—404no—
318C4×D3—201no—
319C4×D3—201no—
320C4×D3—404no—
321Dic6—101no—
322Dic6—101no—
323C3×C8—201no—
324C3⋊C8—41no—
325C22×C5—41no—
326C22×C5D3×C22⋊C411yescharacteristic
327C22×C5#88211yescharacteristic
328C22×C5#89411yescharacteristic
329C2×D5—64no—
330C2×D5—124no—
331C2×D5—124no—
332C2×D5—122no—
333C2×D5—121no—
334C2×D5—242no—
335C4×C5—42no—
336C4×C5—122no—
337C4×C5C4×D1211yescharacteristic
338C4×C5#88411yescharacteristic
339C4×C5#89311yescharacteristic
340C4×C5#89511yescharacteristic
341Dic5—42no—
342Dic5—121no—
343C24—152no—
344#41—301no—
345#41—301no—
346#41—301no—
347#41—301no—
348C2×Q8—151no—
349C2×D4—152no—
350C2×D4—151no—
351C2×D4—152no—
352C2×D4—304no—
353C2×D4—301no—
354C2×D4—301no—
355C2×D4—301no—
356C2×D4—301no—
357C2×D4—301no—
358C22×C4—304no—
359C22×C4—301no—
360C22×C4C2×D3×D511yescharacteristic
361M16—61no—
362M16—61no—
363M16—101no—
364M16—101no—
365C2×C8—61no—
366C2×C8—101no—
367C4⋊C4—21no—
368C4⋊C4—61no—
369C4⋊C4—101no—
370C22⋊C4—21no—
371C22⋊C4—61no—
372C22⋊C4—101no—
373C22⋊C4—304no—
374C22⋊C4—304no—
375C42—32no—
376C42—52no—
377C42#120012yescomplemented
378C3×C5#191111yescharacteristic, complemented, Hall
379C22×C3—41no—
380C22×C3D5×C22⋊C411yescharacteristic
381C22×C3#402811yescharacteristic
382C22×C3#404011yescharacteristic
383C2×D3—104no—
384C2×D3—204no—
385C2×D3—204no—
386C2×D3—202no—
387C2×D3—201no—
388C2×D3—402no—
389C3×C4—42no—
390C3×C4—202no—
391C3×C4C4×D2011yescharacteristic
392C3×C4#403011yescharacteristic
393C3×C4#403911yescharacteristic
394C3×C4#404111yescharacteristic
395Dic3—42no—
396Dic3—201no—
397C2×C5—21no—
398C2×C5—21no—
399C2×C5—21no—
400C2×C5#466511yescharacteristic
401D5—124no—
402D5—241no—
403C23—154no—
404C23—302no—
405C23—302no—
406C23—301no—
407C23—1202no—
408C23#679011yescharacteristic
409Q8—301no—
410Q8—301no—
411D4—302no—
412D4—301no—
413D4—302no—
414D4—302no—
415D4—301no—
416D4—302no—
417D4—601no—
418D4—601no—
419D4—1204no—
420C2×C4—22no—
421C2×C4—62no—
422C2×C4—102no—
423C2×C4—301no—
424C2×C4—601no—
425C2×C4—601no—
426C2×C4—1204no—
427C2×C4#676911yescharacteristic
428C2×C4#677311yescharacteristic
429C2×C4D5×D1211yescharacteristic
430C2×C4D3×D2011yescharacteristic
431C2×C4C2×#120011yescharacteristic
432C2×C4#679011yescharacteristic
433C8—121no—
434C8—201no—
435C2×C3—21no—
436C2×C3—21no—
437C2×C3—21no—
438C2×C3#6495411yescharacteristic
439D3—204no—
440D3—401no—
441C5#6870511yescharacteristic, complemented, Sylow
442C22—41no—
443C22—304no—
444C22—604no—
445C22—604no—
446C22—602no—
447C22—601no—
448C22—1202no—
449C22#9143311yescharacteristic
450C22#9147611yescharacteristic
451C22#9156111yescharacteristic
452C4—42no—
453C4—122no—
454C4—202no—
455C4—601no—
456C4#9141811yescharacteristic
457C4#9142011yescharacteristic
458C4#9142811yescharacteristic
459C4#9145911yescharacteristic
460C3#1059949811yescharacteristic, complemented, Sylow
461C2—21no—
462C2—21no—
463C2—21no—
464C2—604no—
465C2—1201no—
466C2#1174890411yescharacteristic
subgroupG/H
centerC2#11748904
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC3×C5×#117C22
socleC2×C3×C5#386
prime radical#11756284C2
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. #11754515C2semidirect
3. #11748429C2semidirect
4. #11748428C2semidirect
5. C2×#90960C2semidirect
6. C5×#4531C2semidirect
7. C3×#64751C2semidirect
14. #91764C4semidirect
15. #91763C4semidirect
33. #64751D3semidirect
47. C2×D60C2×C4semidirect
50. C2×Dic30C2×C4semidirect
54. C3×C42×C5C2×C4semidirect
58. C4×C5×Dic3D4semidirect
64. C3×C4×Dic5D4semidirect
75. #4531D5semidirect
169. C42×C5C4×D3semidirect
174. C4×Dic5D12semidirect
229. C3×C42C4×D5semidirect
234. C4×Dic3D20semidirect
377. C42#1200semidirect
378. C3×C5#1911semidirect
441. C5#68705semidirect
460. C3#10599498semidirect

Maximal subgroups

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

subgroupindex
#117562842
#117545152
#117484292
#117484282
C2×#909602
C5×#45312
C3×#647512
#105994983
#687055