Finite Group Databasebeta

← #14302844

Subgroups — #14302844

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#11754606C211yesmaximal, characteristic
2#11754130C211yesmaximal, characteristic, complemented
3#11751107C211yesmaximal, characteristic, complemented
4C5×#5001C211yesmaximal, characteristic, complemented
5#11748389C211yesmaximal, characteristic, complemented
6C3×#64751C211yesmaximal, characteristic, complemented
7#11745970C211yesmaximal, characteristic, complemented
8#10601123—31nomaximal, Hall
9C2×#6817C2211yescharacteristic
10C22×C5×D12C2211yescharacteristic
11C3×C5×#117C2211yescharacteristic
12#91815C2211yescharacteristic
13C2×C5×#208C2211yescharacteristic
14C3×#4072C2211yescharacteristic
15C2×C3×#681C2211yescharacteristic
16#91570—22no—
17#91466—22no—
18#91451C412yescomplemented
19#91373C412yescomplemented
20#91292—22no—
21#91291—22no—
22#91290—22no—
23#91090—22no—
24C5×#809—22no—
25#90967—22no—
26#68713—51nomaximal, Hall
27#66139—31no—
28#66079—31no—
29C5×#418—31no—
30#65504—31no—
31#65295—31no—
32#64777—31no—
33#64751D311yescharacteristic, complemented
34C23×C5×D3—21no—
35C22×C3×C4×C5C2311yescharacteristic
36C2×#1220D412yes—
37#6817—21no—
38#6817—21no—
39#6817—21no—
40#6817—21no—
41C2×D60C2×C411yescharacteristic
42C2×C4×D15D412yes—
43C2×Dic30C2×C411yescharacteristic
44C2×C5×D12—21no—
45C2×C5×D12—21no—
46C2×C5×D12—21no—
47C2×C5×D12—21no—
48C2×C5×D12—21no—
49C2×C5×D12C2×C412yes—
50C2×#1203—24no—
51C3×C5×C4⋊C4—21no—
52C3×C5×C22⋊C4—21no—
53C3×C42×C5—21no—
54#6719—21no—
55C4⋊Dic15—21no—
56C4×Dic15—21no—
57C5×#208—21no—
58C5×#208—21no—
59C2×C5×C3⋊C8D412yescomplemented
60C3×C22⋊Dic5—21no—
61C3×C4⋊Dic5—21no—
62C3×C4×Dic5C2×C412yes—
63C3×#681—21no—
64C3×#681—21no—
65C2×C3×#168D412yescomplemented
66(C2×D3)⋊Dic5—24no—
67#6651—24no—
68#6648—24no—
69#5745—51no—
70#5525—51no—
71#5131—51no—
72#5006—51no—
73#5001D511yescharacteristic, complemented
74C3×H(3,ℤ/4)—51no—
75#4531—51no—
76C22×C5×D4—31no—
77C2×#709—31no—
78C2×C5×M16—31no—
79C5×#117C2×D311yescharacteristic
80#4099—62no—
81#4086—32no—
82#4084—62no—
83#4082—32no—
84#4072—31no—
85#4072C2×D311yescharacteristic
86C2×#681C2×D311yescharacteristic
87#4053—62no—
88D5×M16—62no—
89#4047—62no—
90C5×#102—62no—
91#3964—62no—
92#3938—62no—
93#1910—151noSylow
94C23×C3×C5C2×D411yescharacteristic
95C22×D15C22⋊C412yes—
96C22×C5×D3—22no—
97C22×C5×D3—41no—
98C22×C5×D3—41no—
99C22×C5×D3—82no—
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
107#1220—22no—
108#1220—22no—
109C2×Dic15—41no—
110C2×Dic15C22⋊C412yes—
111D60—21no—
112D60—21no—
113C4×D15—22no—
114C4×D15—22no—
115Dic30—21no—
116Dic30—21no—
117C5×D12—22no—
118C5×D12—22no—
119C5×D12—41no—
120C5×D12—41no—
121C2×C3×Dic5—22no—
122#1203—84no—
123C3×#168—22no—
124C5×C3⋊C8—22no—
125C2×#235—51no—
126C22×D12C2×D511yescharacteristic
127C2×C3×M16—51no—
128C3×#117—51no—
129C3×#117C2×D511yescharacteristic
130#926—51no—
131C2×#208C2×D511yescharacteristic
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—22no—
141#806—102no—
142C24×C5—61no—
143C2×C5×D4—32no—
144C2×C5×D4—61no—
145C2×C5×D4—61no—
146C2×C5×D4—61no—
147C2×C5×D4—61no—
148C2×C5×D4—61no—
149C22×C4×C5C22×D311yescharacteristic
150C2×C22⋊D5—32no—
151C2×C22⋊D5—64no—
152#709—61no—
153#709—61no—
154#709—61no—
155#709—61no—
156C2×D20—31no—
157C2×C4×D5—32no—
158C2×Dic10—31no—
159C5×M16—61no—
160C5×M16—61no—
161C2×C5×C8—32no—
162C5×C4⋊C4—21no—
163C5×C22⋊C4—21no—
164C42×C5—21no—
165C22⋊Dic5—21no—
166C22⋊Dic5—64no—
167C22⋊Dic5—61no—
168C4⋊Dic5—21no—
169C4⋊Dic5—61no—
170C4×Dic5—61no—
171C4×Dic5C4×D312yes—
172#681—21no—
173#681—21no—
174C2×#168D1212yescomplemented
175#676—64no—
176C8×D5—64no—
177#568—151no—
178#535—151no—
179#418—31no—
180#418—151no—
181#410—151no—
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—22no—
189C2×D15—42no—
190C2×C5×D3—44no—
191C2×C5×D3—42no—
192C2×C5×D3—82no—
193C2×C5×D3—82no—
194C3×C4×C5—81no—
195C3×C4×C5C4×D411yescharacteristic
196C3×C4×C5C4×D411yescharacteristic
197C3×C4×C5#12011yescharacteristic
198C3×C4×C5#12311yescharacteristic
199Dic15—22no—
200Dic15—81no—
201C3×Dic5—42no—
202C23×D3—21no—
203C22×C3×C4C22×D511yescharacteristic
204C2×#67—52no—
205#235—101no—
206#235—101no—
207#235—101no—
208#235—101no—
209C2×D12—21no—
210C2×D12—21no—
211C2×D12—21no—
212C2×D12—21no—
213C2×D12—21no—
214C2×D12—51no—
215C2×D12—104no—
216C2×D12C4×D512yes—
217C2×C4×D3—52no—
218C2×Dic6—51no—
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—101no—
229(C2×D3)⋊C4—104no—
230C4⋊Dic3—101no—
231C4×Dic3—101no—
232#208—21no—
233#208—21no—
234C2×C3⋊C8C22⋊D512yescomplemented
235ΓL(1,𝔽25)—104no—
236D3×C8—104no—
237C23×C5—62no—
238C23×C5—121no—
239C23×C5—121no—
240C23×C5—242no—
241C23×C5D3×D411yescharacteristic
242C22×D5—32no—
243C5×D4—62no—
244C5×D4—62no—
245C5×D4—121no—
246C5×D4—121no—
247C2×C4×C5—41no—
248C2×C4×C5C2×C4×D311yescharacteristic
249C2×C4×C5C2×D1211yescharacteristic
250C2×C4×C5#23511yescharacteristic
251C2×C4×C5D3×D411yescharacteristic
252C2×C4×C5D3×D411yescharacteristic
253C2×C4×C5#23911yescharacteristic
254C22⋊D5—62no—
255C22⋊D5—62no—
256C22⋊D5—244no—
257C2×Dic5—22no—
258C2×Dic5—32no—
259C2×Dic5—121no—
260D20—61no—
261D20—61no—
262C4×D5—62no—
263C4×D5—62no—
264Dic10—61no—
265Dic10—61no—
266C5×C8—62no—
267#168—22no—
268C2×#41—151no—
269C22×D4—31no—
270#131—302no—
271#131—302no—
272C2×M16—31no—
273C2×M16—51no—
274C2×M16—302no—
275#127—152no—
276#124—152no—
277#121—302no—
278#120—302no—
279#117—51no—
280#117—151no—
281#117D3×D511yescharacteristic
282#102—62no—
283#102—302no—
284#99—302no—
285C2×C3×C5—21no—
286C2×C3×C5—21no—
287C2×C3×C5—21no—
288C2×C3×C5#38611yescharacteristic
289D15—42no—
290C5×D3—82no—
291C23×C3D4×D511yescharacteristic
292C22×D3—22no—
293C22×D3—41no—
294C22×D3—41no—
295C22×D3—52no—
296C22×D3—82no—
297C2×C3×C4—41no—
298C2×C3×C4—102no—
299C2×C3×C4C2×C4×D511yescharacteristic
300C2×C3×C4#70911yescharacteristic
301C2×C3×C4D4×D511yescharacteristic
302C2×C3×C4D4×D511yescharacteristic
303C2×C3×C4#71111yescharacteristic
304C2×C3×C4C2×C22⋊D511yescharacteristic
305#67—102no—
306#67—102no—
307C2×Dic3—52no—
308C2×Dic3—201no—
309D12—22no—
310D12—22no—
311D12—41no—
312D12—41no—
313D12—101no—
314D12—101no—
315D12—404no—
316C4×D3—102no—
317C4×D3—102no—
318Dic6—101no—
319Dic6—101no—
320C3×C8—102no—
321C3⋊C8—22no—
322C22×C5—41no—
323C22×C5—124no—
324C22×C5—122no—
325C22×C5—242no—
326C22×C5—242no—
327C22×C5D3×C22⋊C411yescharacteristic
328C22×C5#88411yescharacteristic
329C22×C5#89511yescharacteristic
330C2×D5—62no—
331C2×D5—122no—
332C4×C5—81no—
333C4×C5C4×D1211yescharacteristic
334C4×C5#88211yescharacteristic
335C4×C5#89311yescharacteristic
336C4×C5#89411yescharacteristic
337Dic5—42no—
338Dic5—62no—
339Dic5—241no—
340C24—61no—
341#41—301no—
342#41—301no—
343#41—301no—
344#41—301no—
345C2×Q8—151no—
346C2×D4—32no—
347C2×D4—61no—
348C2×D4—61no—
349C2×D4—61no—
350C2×D4—61no—
351C2×D4—61no—
352C2×D4—152no—
353C2×D4—151no—
354C2×D4—304no—
355C22×C4—152no—
356C22×C4C2×D3×D511yescharacteristic
357M16—61no—
358M16—61no—
359M16—101no—
360M16—101no—
361M16—304no—
362C2×C8—32no—
363C2×C8—52no—
364C2×C8—304no—
365C4⋊C4—21no—
366C4⋊C4—101no—
367C4⋊C4—301no—
368C22⋊C4—21no—
369C22⋊C4—101no—
370C22⋊C4—304no—
371C22⋊C4—301no—
372C42—21no—
373C42—52no—
374C42—301no—
375C3×C5#191011yescharacteristic, complemented, Hall
376C22×C3—41no—
377C22×C3D5×C22⋊C411yescharacteristic
378C22×C3#403011yescharacteristic
379C22×C3#408411yescharacteristic
380C2×D3—44no—
381C2×D3—42no—
382C2×D3—82no—
383C2×D3—82no—
384C2×D3—102no—
385C2×D3—202no—
386C3×C4—81no—
387C3×C4—202no—
388C3×C4#402711yescharacteristic
389C3×C4#402911yescharacteristic
390C3×C4C4×C22⋊D511yescharacteristic
391C3×C4#408311yescharacteristic
392Dic3—102no—
393Dic3—401no—
394C2×C5—21no—
395C2×C5—21no—
396C2×C5—21no—
397C2×C5—242no—
398C2×C5#466511yescharacteristic
399D5—122no—
400C23—62no—
401C23—121no—
402C23—121no—
403C23—152no—
404C23—242no—
405C23#679011yescharacteristic
406Q8—301no—
407Q8—301no—
408D4—62no—
409D4—62no—
410D4—121no—
411D4—121no—
412D4—302no—
413D4—302no—
414D4—301no—
415D4—301no—
416D4—1204no—
417C2×C4—41no—
418C2×C4—102no—
419C2×C4—152no—
420C2×C4—302no—
421C2×C4—302no—
422C2×C4—601no—
423C2×C4#676911yescharacteristic
424C2×C4C4×D3×D511yescharacteristic
425C2×C4D5×D1211yescharacteristic
426C2×C4#677711yescharacteristic
427C2×C4#678211yescharacteristic
428C2×C4D3×C22⋊D511yescharacteristic
429C8—62no—
430C8—102no—
431C2×C3—21no—
432C2×C3—21no—
433C2×C3—21no—
434C2×C3#6524511yescharacteristic
435D3—82no—
436D3—202no—
437C5#6871311yescharacteristic, complemented, Sylow
438C22—41no—
439C22—124no—
440C22—122no—
441C22—242no—
442C22—242no—
443C22—302no—
444C22—602no—
445C22#9142011yescharacteristic
446C22#9147311yescharacteristic
447C22#9155711yescharacteristic
448C4—81no—
449C4—202no—
450C4—302no—
451C4—1201no—
452C4#9140511yescharacteristic
453C4#9143811yescharacteristic
454C4#9144611yescharacteristic
455C4#9147511yescharacteristic
456C3#1060112311yescharacteristic, complemented, Sylow
457C2—21no—
458C2—21no—
459C2—21no—
460C2—242no—
461C2—602no—
462C2#1174891911yescharacteristic
subgroupG/H
centerC2#11748919
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC3×C5×#117C22
socleC2×C3×C5#386
prime radical#11754606C2
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. #11754130C2semidirect
3. #11751107C2semidirect
4. C5×#5001C2semidirect
5. #11748389C2semidirect
6. C3×#64751C2semidirect
7. #11745970C2semidirect
18. #91451C4semidirect
19. #91373C4semidirect
33. #64751D3semidirect
59. C2×C5×C3⋊C8D4semidirect
65. C2×C3×#168D4semidirect
73. #5001D5semidirect
174. C2×#168D12semidirect
234. C2×C3⋊C8C22⋊D5semidirect
375. C3×C5#1910semidirect
437. C5#68713semidirect
456. C3#10601123semidirect

Maximal subgroups

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

subgroupindex
#117546062
#117541302
#117511072
C5×#50012
#117483892
C3×#647512
#117459702
#106011233
#687135