Finite Group Databasebeta

← #14302846

Subgroups — #14302846

Order 1920.

688 conjugacy classes of subgroups.

The subgroup lattice has 464 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11754610C211yesmaximal, characteristic
2#11754125C211yesmaximal, characteristic, complemented
3#11751102C211yesmaximal, characteristic, complemented
4C5×#5006C211yesmaximal, characteristic, complemented
5#11748396C211yesmaximal, characteristic, complemented
6C3×#64751C211yesmaximal, characteristic, complemented
7#11745969C211yesmaximal, characteristic, complemented
8#10600840—31nomaximal, Hall
9C22×D60C2211yescharacteristic
10C2×C5×#235C2211yescharacteristic
11C3×C5×#117C2211yescharacteristic
12C2×#6710C2211yescharacteristic
13C5×#926C2211yescharacteristic
14C3×#4072C2211yescharacteristic
15C2×C3×#681C2211yescharacteristic
16#91573—22no—
17#91460—22no—
18#91451C412yescomplemented
19#91375C412yescomplemented
20#91288—22no—
21D3×#681—22no—
22#91286—22no—
23#91093—22no—
24C5×#806—22no—
25#90969—22no—
26#68713—51nomaximal, Hall
27#66164—31no—
28#66153—31no—
29C5×#410—31no—
30#65290—31no—
31#65290—31no—
32#64751—31no—
33#64751D311yescharacteristic, complemented
34C23×D15—21no—
35C22×C3×C4×C5C2311yescharacteristic
36C2×D60—21no—
37C2×D60—21no—
38C2×D60—21no—
39C2×D60—21no—
40C2×D60—21no—
41C2×D60C2×C412yes—
42C2×C5×#67D412yes—
43C5×#235—21no—
44C5×#235—21no—
45C5×#235—21no—
46C5×#235—21no—
47C2×C5×D12C2×C411yescharacteristic
48C2×C4×C5×D3D412yes—
49C2×C5×Dic6C2×C411yescharacteristic
50C2×#1203—24no—
51C3×C5×C4⋊C4—21no—
52C3×C5×C22⋊C4—21no—
53C3×C42×C5—21no—
54#6710—21no—
55#6710—21no—
56C2×C3⋊#168D412yescomplemented
57C5×#217—21no—
58C5×C4⋊Dic3—21no—
59C4×C5×Dic3—21no—
60C3×C22⋊Dic5—21no—
61C3×C4⋊Dic5—21no—
62C3×C4×Dic5C2×C412yes—
63C3×#681—21no—
64C3×#681—21no—
65C2×C3×#168D412yescomplemented
66#6667—24no—
67#6650—24no—
68D3×#168—24no—
69#5745—51no—
70#5525—51no—
71#5131—51no—
72#5006D511yescharacteristic, complemented
73#5001—51no—
74C3×H(3,ℤ/4)—51no—
75#4531—51no—
76C2×C5×#41—31no—
77C22×D20—31no—
78C5×#117—31no—
79C5×#117C2×D311yescharacteristic
80#4094—62no—
81#4094—62no—
82#4093—32no—
83#4086—32no—
84#4083—62no—
85#4076—62no—
86#4072C2×D311yescharacteristic
87C2×#681—31no—
88C2×#681—62no—
89C2×#681C2×D311yescharacteristic
90C5×#99—62no—
91#3941—62no—
92#3941—62no—
93#1910—151noSylow
94C23×C3×C5C2×D411yescharacteristic
95C22×D15—22no—
96C22×D15—41no—
97C22×D15—41no—
98C22×D15—82no—
99C22×C5×D3C22⋊C412yes—
100C2×C3×C4×C5—41no—
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
107D60—22no—
108D60—22no—
109D60—41no—
110D60—41no—
111C5×#67—22no—
112C5×#67—22no—
113C2×C5×Dic3—41no—
114C2×C5×Dic3C22⋊C412yes—
115C5×D12—21no—
116C5×D12—21no—
117C4×C5×D3—22no—
118C4×C5×D3—22no—
119C5×Dic6—21no—
120C5×Dic6—21no—
121C2×C3×Dic5—22no—
122#1203—84no—
123C3⋊#168—22no—
124C3×#168—22no—
125C2×#235C2×D511yescharacteristic
126C22×D12—51no—
127C2×C3×M16—51no—
128C3×#117—51no—
129C3×#117C2×D511yescharacteristic
130#926C2×D511yescharacteristic
131C2×#208—51no—
132#907—102no—
133D3×M16—102no—
134#901—102no—
135#895—102no—
136#882—102no—
137#875—52no—
138#874—52no—
139#821—102no—
140#809—102no—
141#806—22no—
142C23×D5—61no—
143C5×#41—61no—
144C5×#41—61no—
145C5×#41—61no—
146C5×#41—61no—
147C2×C5×Q8—31no—
148C2×C5×D4—32no—
149C2×C5×D4—31no—
150C22×C4×C5—32no—
151C22×C4×C5C22×D311yescharacteristic
152C2×C22⋊D5—64no—
153C2×D20—32no—
154C2×D20—61no—
155C2×D20—61no—
156C2×D20—61no—
157C2×D20—61no—
158C2×D20—61no—
159C5×C4⋊C4—21no—
160C5×C4⋊C4—61no—
161C5×C22⋊C4—21no—
162C5×C22⋊C4—61no—
163C42×C5—21no—
164C42×C5—61no—
165C22⋊Dic5—21no—
166#685—64no—
167C4⋊Dic5—21no—
168C4×Dic5C4×D312yes—
169#681—21no—
170#681—21no—
171#681—64no—
172#681—61no—
173#681—61no—
174C2×#168—32no—
175C2×#168—64no—
176C2×#168D1212yescomplemented
177#568—151no—
178#535—151no—
179#418—151no—
180#418—151no—
181#410—31no—
182H(3,ℤ/4)—51no—
183H(3,ℤ/4)—151no—
184C22×C3×C5—41no—
185C22×C3×C5C2×C22⋊C411yescharacteristic
186C22×C3×C5#12111yescharacteristic
187C22×C3×C5#12111yescharacteristic
188C2×D15—44no—
189C2×D15—42no—
190C2×D15—82no—
191C2×D15—82no—
192C2×C5×D3—22no—
193C2×C5×D3—42no—
194C3×C4×C5—81no—
195C3×C4×C5C4×D411yescharacteristic
196C3×C4×C5C4×D411yescharacteristic
197C3×C4×C5#12011yescharacteristic
198C3×C4×C5#12311yescharacteristic
199C3×Dic5—42no—
200C5×Dic3—22no—
201C5×Dic3—81no—
202C23×D3—101no—
203C22×C3×C4C22×D511yescharacteristic
204C2×#67C22⋊D512yes—
205#235—21no—
206#235—21no—
207#235—21no—
208#235—21no—
209C2×D12—52no—
210C2×D12—101no—
211C2×D12—101no—
212C2×D12—101no—
213C2×D12—101no—
214C2×D12—101no—
215C2×D12—104no—
216C2×D12C4×D511yescharacteristic
217C2×C4×D3D2012yes—
218C2×Dic6C4×D511yescharacteristic
219C3×M16—101no—
220C3×M16—101no—
221C2×C3×C8—52no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—101no—
224C3×C22⋊C4—21no—
225C3×C22⋊C4—101no—
226C3×C42—21no—
227C3×C42—52no—
228#217—21no—
229(C2×D3)⋊C4—104no—
230C4⋊Dic3—21no—
231C4×Dic3—21no—
232#208—101no—
233#208—101no—
234C2×C3⋊C8—52no—
235ΓL(1,𝔽25)—104no—
236D3×C8—104no—
237C23×C5—32no—
238C23×C5D3×D411yescharacteristic
239C22×D5—62no—
240C22×D5—121no—
241C22×D5—121no—
242C22×D5—242no—
243C5×Q8—61no—
244C5×Q8—61no—
245C5×D4—62no—
246C5×D4—62no—
247C5×D4—61no—
248C5×D4—61no—
249C2×C4×C5—32no—
250C2×C4×C5—41no—
251C2×C4×C5—62no—
252C2×C4×C5—62no—
253C2×C4×C5—121no—
254C2×C4×C5C2×C4×D311yescharacteristic
255C2×C4×C5C2×D1211yescharacteristic
256C2×C4×C5#23511yescharacteristic
257C2×C4×C5D3×D411yescharacteristic
258C2×C4×C5D3×D411yescharacteristic
259C2×C4×C5#23911yescharacteristic
260C22⋊D5—244no—
261C2×Dic5—22no—
262D20—62no—
263D20—62no—
264D20—121no—
265D20—121no—
266#168—22no—
267#168—62no—
268C2×#41—31no—
269C22×D4—151no—
270#131—302no—
271#131—302no—
272C2×M16—51no—
273C2×M16—151no—
274C2×M16—302no—
275#127—152no—
276#124—152no—
277#121—302no—
278#120—302no—
279#117—31no—
280#117—51no—
281#117D3×D511yescharacteristic
282#102—302no—
283#102—302no—
284#99—62no—
285C2×C3×C5—21no—
286C2×C3×C5—21no—
287C2×C3×C5—21no—
288C2×C3×C5#38611yescharacteristic
289D15—82no—
290C5×D3—42no—
291C23×C3C2×C22⋊D511yescharacteristic
292C22×D3—102no—
293C22×D3—201no—
294C22×D3—201no—
295C22×D3—402no—
296C22×D3#68512yes—
297C2×C3×C4—41no—
298C2×C3×C4—102no—
299C2×C3×C4C2×C4×D511yescharacteristic
300C2×C3×C4C2×D2011yescharacteristic
301C2×C3×C4D4×D511yescharacteristic
302C2×C3×C4D4×D511yescharacteristic
303C2×C3×C4#71111yescharacteristic
304C2×C3×C4#71111yescharacteristic
305#67—22no—
306#67—22no—
307C2×Dic3—41no—
308C2×Dic3#68512yes—
309D12—21no—
310D12—21no—
311D12—102no—
312D12—102no—
313D12—201no—
314D12—201no—
315D12—404no—
316C4×D3—22no—
317C4×D3—22no—
318Dic6—21no—
319Dic6—21no—
320C3×C8—102no—
321C3⋊C8—102no—
322C22×C5—41no—
323C22×C5—62no—
324C22×C5—122no—
325C22×C5D3×C22⋊C411yescharacteristic
326C22×C5#88411yescharacteristic
327C22×C5#89511yescharacteristic
328C2×D5—124no—
329C2×D5—122no—
330C2×D5—242no—
331C2×D5—242no—
332C4×C5—62no—
333C4×C5—81no—
334C4×C5—241no—
335C4×C5C4×D1211yescharacteristic
336C4×C5#88211yescharacteristic
337C4×C5#89311yescharacteristic
338C4×C5#89411yescharacteristic
339Dic5—42no—
340C24—301no—
341#41—61no—
342#41—61no—
343#41—61no—
344#41—61no—
345C2×Q8—31no—
346C2×D4—32no—
347C2×D4—31no—
348C2×D4—152no—
349C2×D4—304no—
350C2×D4—301no—
351C2×D4—301no—
352C2×D4—301no—
353C2×D4—301no—
354C2×D4—301no—
355C22×C4—32no—
356C22×C4C2×D3×D511yescharacteristic
357M16—101no—
358M16—101no—
359M16—304no—
360M16—301no—
361M16—301no—
362C2×C8—52no—
363C2×C8—152no—
364C2×C8—304no—
365C4⋊C4—21no—
366C4⋊C4—61no—
367C4⋊C4—101no—
368C22⋊C4—21no—
369C22⋊C4—61no—
370C22⋊C4—101no—
371C22⋊C4—304no—
372C42—21no—
373C42—52no—
374C42—61no—
375C3×C5#191011yescharacteristic, complemented, Hall
376C22×C3—41no—
377C22×C3C2×#68511yescharacteristic
378C22×C3#408511yescharacteristic
379C22×C3#408511yescharacteristic
380C2×D3—22no—
381C2×D3—42no—
382C2×D3—204no—
383C2×D3—202no—
384C2×D3—402no—
385C2×D3—402no—
386C3×C4—81no—
387C3×C4—202no—
388C3×C4#402711yescharacteristic
389C3×C4#402711yescharacteristic
390C3×C4#402811yescharacteristic
391C3×C4#403211yescharacteristic
392Dic3—22no—
393Dic3—81no—
394C2×C5—21no—
395C2×C5—21no—
396C2×C5—21no—
397C2×C5—122no—
398C2×C5#466511yescharacteristic
399D5—242no—
400C23—32no—
401C23—302no—
402C23—601no—
403C23—601no—
404C23—1202no—
405C23D3×C22⋊D511yescharacteristic
406Q8—61no—
407Q8—61no—
408D4—62no—
409D4—62no—
410D4—61no—
411D4—61no—
412D4—302no—
413D4—302no—
414D4—601no—
415D4—601no—
416D4—1204no—
417C2×C4—32no—
418C2×C4—41no—
419C2×C4—62no—
420C2×C4—62no—
421C2×C4—102no—
422C2×C4—121no—
423C2×C4#676811yescharacteristic
424C2×C4C4×D3×D511yescharacteristic
425C2×C4D5×D1211yescharacteristic
426C2×C4D3×D2011yescharacteristic
427C2×C4#678211yescharacteristic
428C2×C4#679011yescharacteristic
429C8—102no—
430C8—302no—
431C2×C3—21no—
432C2×C3—21no—
433C2×C3—21no—
434C2×C3#6524711yescharacteristic
435D3—42no—
436D3—402no—
437C5#6871311yescharacteristic, complemented, Sylow
438C22—41no—
439C22—62no—
440C22—122no—
441C22—604no—
442C22—602no—
443C22—1202no—
444C22—1202no—
445C22#9141611yescharacteristic
446C22D3×#68511yescharacteristic
447C22#9155811yescharacteristic
448C4—62no—
449C4—81no—
450C4—202no—
451C4—241no—
452C4#9140511yescharacteristic
453C4#9140611yescharacteristic
454C4#9142711yescharacteristic
455C4#9147611yescharacteristic
456C3#1060084011yescharacteristic, complemented, Sylow
457C2—21no—
458C2—21no—
459C2—21no—
460C2—122no—
461C2—1202no—
462C2#1174888311yescharacteristic
subgroupG/H
centerC2#11748883
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC3×C5×#117C22
socleC2×C3×C5#386
prime radical#11754610C2
cosocleC22×C3×C4×C5C23
hypercenter#117D3×D5
hypocenterC3×C5#1910
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. #11754125C2semidirect
3. #11751102C2semidirect
4. C5×#5006C2semidirect
5. #11748396C2semidirect
6. C3×#64751C2semidirect
7. #11745969C2semidirect
18. #91451C4semidirect
19. #91375C4semidirect
33. #64751D3semidirect
56. C2×C3⋊#168D4semidirect
65. C2×C3×#168D4semidirect
72. #5006D5semidirect
176. C2×#168D12semidirect
375. C3×C5#1910semidirect
437. C5#68713semidirect
456. C3#10600840semidirect

Maximal subgroups

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

subgroupindex
#117546102
#117541252
#117511022
C5×#50062
#117483962
C3×#647512
#117459692
#106008403
#687135