Finite Group Databasebeta

← #14302848

Subgroups — #14302848

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#11754609C211yesmaximal, characteristic
2#11754120C211yesmaximal, characteristic, complemented
3#11751102C211yesmaximal, characteristic, complemented
4C3×#65295C211yesmaximal, characteristic, complemented
5#11748392C211yesmaximal, characteristic, complemented
6C5×#4531C211yesmaximal, characteristic, complemented
7#11745969C211yesmaximal, characteristic, complemented
8#10599506—31nomaximal, Hall
9C22×D60C2211yescharacteristic
10C2×C3×#709C2211yescharacteristic
11C3×C5×#117C2211yescharacteristic
12C2×#6710C2211yescharacteristic
13C5×#926C2211yescharacteristic
14C2×C5×#208C2211yescharacteristic
15C3×#4072C2211yescharacteristic
16#91572—22no—
17#91458C412yescomplemented
18#91450—22no—
19#91374C412yescomplemented
20#91284—22no—
21#91283—22no—
22D5×#208—22no—
23#91093—22no—
24C3×#3938—22no—
25#90968—22no—
26#70047—51nomaximal, Hall
27#66079—31no—
28#65859—31no—
29#65420—31no—
30#65295D311yescharacteristic, complemented
31#65290—31no—
32C5×H(3,ℤ/4)—31no—
33#64751—31no—
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×C3×C22⋊D5D412yes—
43C3×#709—21no—
44C3×#709—21no—
45C3×#709—21no—
46C3×#709—21no—
47C2×C3×D20C2×C411yescharacteristic
48C2×C3×C4×D5D412yes—
49C2×C3×Dic10C2×C411yescharacteristic
50C2×#1202—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×Dic3C2×C412yes—
60C5×#208—21no—
61C5×#208—21no—
62C2×C5×C3⋊C8D412yescomplemented
63C3×C22⋊Dic5—21no—
64C3×C4⋊Dic5—21no—
65C3×C4×Dic5—21no—
66#6667—24no—
67#6649—24no—
68D5×C3⋊C8—24no—
69#5830—51no—
70#5819—51no—
71C3×#410—51no—
72#5001—51no—
73#5001—51no—
74#4531—51no—
75#4531D511yescharacteristic, complemented
76C2×#709C2×D311yescharacteristic
77C22×D20—31no—
78C2×C5×M16—31no—
79C5×#117—31no—
80C5×#117C2×D311yescharacteristic
81#4072C2×D311yescharacteristic
82C2×#681—31no—
83#4053—62no—
84D5×M16—62no—
85#4047—62no—
86#4041—62no—
87#4028—62no—
88#4021—32no—
89#4020—32no—
90#3953—62no—
91#3941—62no—
92#3938—22no—
93#1910—151noSylow
94C23×C3×C5C2×D411yescharacteristic
95C22×D15—22no—
96C22×D15—41no—
97C22×D15—41no—
98C22×D15—82no—
99C22×C3×D5C22⋊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—
111C2×C5×Dic3—22no—
112C3×C22⋊D5—22no—
113C3×C22⋊D5—22no—
114C2×C3×Dic5—41no—
115C2×C3×Dic5C22⋊C412yes—
116C3×D20—21no—
117C3×D20—21no—
118C3×C4×D5—22no—
119C3×C4×D5—22no—
120C3×Dic10—21no—
121C3×Dic10—21no—
122#1202—84no—
123C3⋊#168—22no—
124C5×C3⋊C8—22no—
125C2×C3×#41—51no—
126C22×D12—51no—
127C3×#117—51no—
128C3×#117C2×D511yescharacteristic
129#948—102no—
130#948—102no—
131#947—52no—
132#940—52no—
133#937—102no—
134#930—102no—
135#926C2×D511yescharacteristic
136C2×#208—51no—
137C2×#208—102no—
138C2×#208C2×D511yescharacteristic
139C3×#99—102no—
140#809—102no—
141#809—102no—
142C23×D5—61no—
143C22×C4×C5C22×D311yescharacteristic
144C2×C22⋊D5#6712yes—
145#709—21no—
146#709—21no—
147#709—21no—
148#709—21no—
149C2×D20—32no—
150C2×D20—61no—
151C2×D20—61no—
152C2×D20—61no—
153C2×D20—61no—
154C2×D20—61no—
155C2×D20—64no—
156C2×D20C4×D311yescharacteristic
157C2×C4×D5D1212yes—
158C2×Dic10C4×D311yescharacteristic
159C5×M16—61no—
160C5×M16—61no—
161C2×C5×C8—32no—
162C5×C4⋊C4—21no—
163C5×C4⋊C4—61no—
164C5×C22⋊C4—21no—
165C5×C22⋊C4—61no—
166C42×C5—21no—
167C42×C5—32no—
168C22⋊Dic5—21no—
169#685—64no—
170C4⋊Dic5—21no—
171C4×Dic5—21no—
172#681—61no—
173#681—61no—
174C2×#168—32no—
175#676—64no—
176C8×D5—64no—
177#568—151no—
178#535—151no—
179#418—151no—
180#418—151no—
181#410—51no—
182H(3,ℤ/4)—31no—
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×C3×D5—22no—
193C2×C3×D5—42no—
194C3×C4×C5—81no—
195C3×C4×C5C4×D411yescharacteristic
196C3×C4×C5C4×D411yescharacteristic
197C3×C4×C5#12011yescharacteristic
198C3×C4×C5#12311yescharacteristic
199C3×Dic5—22no—
200C3×Dic5—81no—
201C5×Dic3—42no—
202C23×D3—101no—
203C3×#41—101no—
204C3×#41—101no—
205C3×#41—101no—
206C3×#41—101no—
207C2×C3×Q8—51no—
208C2×C3×D4—52no—
209C2×C3×D4—51no—
210C22×C3×C4—52no—
211C22×C3×C4C22×D511yescharacteristic
212C2×#67—104no—
213C2×D12—52no—
214C2×D12—101no—
215C2×D12—101no—
216C2×D12—101no—
217C2×D12—101no—
218C2×D12—101no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—101no—
221C3×C22⋊C4—21no—
222C3×C22⋊C4—101no—
223C3×C42—21no—
224C3×C42—101no—
225#217—21no—
226(C2×D3)⋊C4—104no—
227C4⋊Dic3—21no—
228C4×Dic3C4×D512yes—
229#208—21no—
230#208—21no—
231#208—104no—
232#208—101no—
233#208—101no—
234C2×C3⋊C8—52no—
235C2×C3⋊C8—104no—
236C2×C3⋊C8D2012yescomplemented
237C23×C5C2×#6711yescharacteristic
238C22×D5—62no—
239C22×D5—121no—
240C22×D5—121no—
241C22×D5—242no—
242C22×D5(C2×D3)⋊C412yes—
243C2×C4×C5—41no—
244C2×C4×C5—62no—
245C2×C4×C5C2×C4×D311yescharacteristic
246C2×C4×C5C2×D1211yescharacteristic
247C2×C4×C5D3×D411yescharacteristic
248C2×C4×C5D3×D411yescharacteristic
249C2×C4×C5#23711yescharacteristic
250C2×C4×C5#23711yescharacteristic
251C22⋊D5—22no—
252C22⋊D5—22no—
253C2×Dic5—41no—
254C2×Dic5(C2×D3)⋊C412yes—
255D20—21no—
256D20—21no—
257D20—62no—
258D20—62no—
259D20—121no—
260D20—121no—
261D20—244no—
262C4×D5—22no—
263C4×D5—22no—
264Dic10—21no—
265Dic10—21no—
266C5×C8—62no—
267#168—62no—
268C2×#41—51no—
269C22×D4—151no—
270#131—302no—
271#131—302no—
272C2×M16—31no—
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—102no—
285C2×C3×C5—21no—
286C2×C3×C5—21no—
287C2×C3×C5—21no—
288C2×C3×C5#38611yescharacteristic
289D15—82no—
290C3×D5—42no—
291C23×C3—52no—
292C23×C3D4×D511yescharacteristic
293C22×D3—102no—
294C22×D3—201no—
295C22×D3—201no—
296C22×D3—402no—
297C3×Q8—101no—
298C3×Q8—101no—
299C3×D4—102no—
300C3×D4—102no—
301C3×D4—101no—
302C3×D4—101no—
303C2×C3×C4—41no—
304C2×C3×C4—52no—
305C2×C3×C4—102no—
306C2×C3×C4—102no—
307C2×C3×C4—201no—
308C2×C3×C4C2×C4×D511yescharacteristic
309C2×C3×C4C2×D2011yescharacteristic
310C2×C3×C4#70911yescharacteristic
311C2×C3×C4D4×D511yescharacteristic
312C2×C3×C4D4×D511yescharacteristic
313C2×C3×C4#71311yescharacteristic
314#67—404no—
315C2×Dic3—22no—
316D12—102no—
317D12—102no—
318D12—201no—
319D12—201no—
320C3⋊C8—22no—
321C3⋊C8—102no—
322C22×C5—41no—
323C22×C5C2×(C2×D3)⋊C411yescharacteristic
324C22×C5#93911yescharacteristic
325C22×C5#93911yescharacteristic
326C2×D5—22no—
327C2×D5—42no—
328C2×D5—124no—
329C2×D5—122no—
330C2×D5—242no—
331C2×D5—242no—
332C4×C5—81no—
333C4×C5—122no—
334C4×C5#67⋊C411yescharacteristic
335C4×C5#67⋊C411yescharacteristic
336C4×C5#88211yescharacteristic
337C4×C5#88611yescharacteristic
338Dic5—22no—
339Dic5—81no—
340C24—301no—
341#41—101no—
342#41—101no—
343#41—101no—
344#41—101no—
345C2×Q8—51no—
346C2×D4—52no—
347C2×D4—51no—
348C2×D4—152no—
349C2×D4—304no—
350C2×D4—301no—
351C2×D4—301no—
352C2×D4—301no—
353C2×D4—301no—
354C2×D4—301no—
355C22×C4—52no—
356C22×C4C2×D3×D511yescharacteristic
357M16—61no—
358M16—61no—
359M16—304no—
360M16—301no—
361M16—301no—
362C2×C8—32no—
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—32no—
374C42—101no—
375C3×C5#191011yescharacteristic, complemented, Hall
376C22×C3—41no—
377C22×C3—102no—
378C22×C3—202no—
379C22×C3D5×C22⋊C411yescharacteristic
380C22×C3#403011yescharacteristic
381C22×C3#404111yescharacteristic
382C2×D3—204no—
383C2×D3—202no—
384C2×D3—402no—
385C2×D3—402no—
386C3×C4—81no—
387C3×C4—102no—
388C3×C4—401no—
389C3×C4C4×D2011yescharacteristic
390C3×C4#402811yescharacteristic
391C3×C4#403911yescharacteristic
392C3×C4#404011yescharacteristic
393Dic3—42no—
394C2×C5—21no—
395C2×C5—21no—
396C2×C5—21no—
397C2×C5#495811yescharacteristic
398D5—42no—
399D5—242no—
400C23—52no—
401C23—302no—
402C23—601no—
403C23—601no—
404C23—1202no—
405C23D5×#6711yescharacteristic
406Q8—101no—
407Q8—101no—
408D4—102no—
409D4—102no—
410D4—101no—
411D4—101no—
412D4—302no—
413D4—302no—
414D4—601no—
415D4—601no—
416D4—1204no—
417C2×C4—41no—
418C2×C4—52no—
419C2×C4—62no—
420C2×C4—102no—
421C2×C4—102no—
422C2×C4—201no—
423C2×C4#676611yescharacteristic
424C2×C4C4×D3×D511yescharacteristic
425C2×C4D5×D1211yescharacteristic
426C2×C4D3×D2011yescharacteristic
427C2×C4#677911yescharacteristic
428C2×C4#679011yescharacteristic
429C8—62no—
430C8—302no—
431C2×C3—21no—
432C2×C3—21no—
433C2×C3—21no—
434C2×C3—202no—
435C2×C3#6495411yescharacteristic
436D3—402no—
437C5#7004711yescharacteristic, complemented, Sylow
438C22—41no—
439C22—102no—
440C22—202no—
441C22—604no—
442C22—602no—
443C22—1202no—
444C22—1202no—
445C22#9140911yescharacteristic
446C22D5×(C2×D3)⋊C411yescharacteristic
447C22#9155011yescharacteristic
448C4—81no—
449C4—102no—
450C4—122no—
451C4—401no—
452C4#9139511yescharacteristic
453C4#9139611yescharacteristic
454C4#9141311yescharacteristic
455C4#9147611yescharacteristic
456C3#1059950611yescharacteristic, complemented, Sylow
457C2—21no—
458C2—21no—
459C2—21no—
460C2—202no—
461C2—1202no—
462C2#1174887211yescharacteristic
subgroupG/H
centerC2#11748872
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC3×C5×#117C22
socleC2×C3×C5#386
prime radical#11754609C2
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. #11754120C2semidirect
3. #11751102C2semidirect
4. C3×#65295C2semidirect
5. #11748392C2semidirect
6. C5×#4531C2semidirect
7. #11745969C2semidirect
17. #91458C4semidirect
19. #91374C4semidirect
30. #65295D3semidirect
56. C2×C3⋊#168D4semidirect
62. C2×C5×C3⋊C8D4semidirect
75. #4531D5semidirect
236. C2×C3⋊C8D20semidirect
375. C3×C5#1910semidirect
437. C5#70047semidirect
456. C3#10599506semidirect

Maximal subgroups

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

subgroupindex
#117546092
#117541202
#117511022
C3×#652952
#117483922
C5×#45312
#117459692
#105995063
#700475