Finite Group Databasebeta

← #14301165

Subgroups — #14301165

Order 1920.

620 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11756581C211yesmaximal, characteristic
2#11756504C211yesmaximal, characteristic, complemented
3C3×C5×#408C211yesmaximal, characteristic, complemented
4#11751337C211yesmaximal, characteristic, complemented
5#11751214C211yesmaximal, characteristic, complemented
6#11746377C211yesmaximal, characteristic, complemented
7#11746373C211yesmaximal, characteristic, complemented
8#10600805—31nomaximal, Hall
9C2×C3×C5×#41C2211yescharacteristic
10C2×#6817C2211yescharacteristic
11C2×C3×C5×M16C2211yescharacteristic
12C22×C3×C5×C8C2211yescharacteristic
13#91840—21no—
14#91839—22no—
15#91838—21no—
16#91831C412yescomplemented
17#91821C412yescomplemented
18#91815C2211yescharacteristic
19C2×#6710C2211yescharacteristic
20C2×#6710C2211yescharacteristic
21#91796—22no—
22M16×D15—22no—
23#91790—22no—
24C3×C5×C22⋊C8—22no—
25#91121—22no—
26#91105—21no—
27#91100—21no—
28#70012—51nomaximal, Hall
29#66156—31no—
30#66079—31no—
31C5×#408D311yescharacteristic, complemented
32#65525—31no—
33#65402—31no—
34#64776—31no—
35#64772—31no—
36C3×C5×#41—21no—
37C3×C5×#41—21no—
38C3×C5×#41—21no—
39C3×C5×#41—21no—
40C2×C3×C5×Q8C2×C411yescharacteristic
41C2×C3×C5×D4—21no—
42C2×C3×C5×D4C2×C411yescharacteristic
43C22×C3×C4×C5—21no—
44C22×C3×C4×C5C2311yescharacteristic
45C2×#1220—21no—
46#6817—21no—
47#6817—21no—
48#6817C2×C412yes—
49#6817D412yes—
50C2×D60D411yescharacteristic
51C2×C4×D15—21no—
52C2×Dic30D411yescharacteristic
53C3×C5×M16—21no—
54C3×C5×M16—21no—
55C2×C3×C5×C8—21no—
56C2×C3×C5×C8—21no—
57C2×C3×C5×C8—21no—
58C2×C3×C5×C8D412yes—
59#6719—21no—
60#6718—22no—
61#6717—22no—
62#6716—22no—
63#6715—22no—
64C4⋊Dic15—21no—
65C4×Dic15—21no—
66#6710—21no—
67#6710—21no—
68#6710C2×C412yes—
69#6710D412yes—
70C2×C3⋊#168—21no—
71C2×C3⋊#168—21no—
72#6705—24no—
73C8×D15—24no—
74#5822—51no—
75#5745—51no—
76C3×#408D511yescharacteristic, complemented
77#5236—51no—
78#5113—51no—
79#4556—51no—
80#4552—51no—
81C2×C5×#41C2×D311yescharacteristic
82C2×#709—31no—
83C2×C5×M16C2×D311yescharacteristic
84C22×C5×C8C2×D311yescharacteristic
85#4097—61no—
86#4096—62no—
87#4095—61no—
88#4088—32no—
89#4078—32no—
90#4072—31no—
91C2×#681—31no—
92C2×#681—31no—
93#4053—62no—
94D5×M16—62no—
95#4047—62no—
96C5×C22⋊C8—22no—
97#3969—62no—
98#3953—61no—
99#3948—61no—
100#1881—151noSylow
101C23×C3×C5—21no—
102C23×C3×C5#4111yescharacteristic
103C22×D15—21no—
104C3×C5×Q8—21no—
105C3×C5×Q8—21no—
106C3×C5×D4—21no—
107C3×C5×D4—21no—
108C3×C5×D4—41no—
109C3×C5×D4—41no—
110C2×C3×C4×C5—21no—
111C2×C3×C4×C5—41no—
112C2×C3×C4×C5—41no—
113C2×C3×C4×C5C22×C411yescharacteristic
114C2×C3×C4×C5C2×D411yescharacteristic
115C2×C3×C4×C5C2×D411yescharacteristic
116C2×C3×C4×C5C2×D411yescharacteristic
117C2×C3×C4×C5C2×D411yescharacteristic
118C2×C3×C4×C5#4111yescharacteristic
119#1220—22no—
120#1220—22no—
121C2×Dic15—21no—
122C2×Dic15—41no—
123D60—21no—
124D60C22⋊C412yes—
125C4×D15—22no—
126C4×D15—22no—
127Dic30—21no—
128Dic30C22⋊C412yes—
129C3×C5×C8—22no—
130C3×C5×C8—41no—
131C3⋊#168—22no—
132C3⋊#168—22no—
133C2×C3×#41C2×D511yescharacteristic
134C2×#235—51no—
135C2×C3×M16C2×D511yescharacteristic
136C22×C3×C8C2×D511yescharacteristic
137#951—101no—
138#950—102no—
139#949—101no—
140#942—52no—
141#932—52no—
142#926—51no—
143C2×#208—51no—
144C2×#208—51no—
145#907—102no—
146D3×M16—102no—
147#901—102no—
148C3×C22⋊C8—22no—
149#837—102no—
150#821—101no—
151#816—101no—
152C5×#41—21no—
153C5×#41—21no—
154C5×#41—21no—
155C5×#41—21no—
156C2×C5×Q8C4×D311yescharacteristic
157C2×C5×D4—21no—
158C2×C5×D4C4×D311yescharacteristic
159C22×C4×C5—21no—
160C22×C4×C5C22×D311yescharacteristic
161C2×C22⋊D5—61no—
162#709—32no—
163#709—32no—
164#709—61no—
165#709—61no—
166C2×D20—31no—
167C2×C4×D5—61no—
168C2×Dic10—31no—
169C5×M16—21no—
170C5×M16—21no—
171C2×C5×C8—21no—
172C2×C5×C8—21no—
173C2×C5×C8—21no—
174C2×C5×C8#6712yes—
175C22⋊Dic5—61no—
176#689—62no—
177#688—62no—
178#687—62no—
179#686—62no—
180C4⋊Dic5—61no—
181C4×Dic5—61no—
182#681—32no—
183#681—32no—
184#681—61no—
185#681—61no—
186C2×#168—61no—
187C2×#168—61no—
188#676—64no—
189C8×D5—64no—
190#575—151no—
191#568—151no—
192#421—151no—
193#416—151no—
194#408D1511yescharacteristic, complemented
195#344—151no—
196#341—151no—
197C22×C3×C5—41no—
198C22×C3×C5—42no—
199C22×C3×C5—41no—
200C22×C3×C5C4×D411yescharacteristic
201C22×C3×C5#12111yescharacteristic
202C22×C3×C5#12311yescharacteristic
203C2×D15—22no—
204C2×D15—81no—
205C3×C4×C5—41no—
206C3×C4×C5C2×C22⋊C411yescharacteristic
207C3×C4×C5C4×D411yescharacteristic
208C3×C4×C5#12011yescharacteristic
209C3×C4×C5#12111yescharacteristic
210Dic15—22no—
211Dic15—81no—
212C3×#41—21no—
213C3×#41—21no—
214C3×#41—21no—
215C3×#41—21no—
216C2×C3×Q8C4×D511yescharacteristic
217C2×C3×D4—21no—
218C2×C3×D4C4×D511yescharacteristic
219C22×C3×C4—21no—
220C22×C3×C4C22×D511yescharacteristic
221C2×#67—101no—
222#235—52no—
223#235—52no—
224#235—101no—
225#235—101no—
226C2×D12—51no—
227C2×C4×D3—101no—
228C2×Dic6—51no—
229C3×M16—21no—
230C3×M16—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8C22⋊D512yes—
235#217—101no—
236#216—102no—
237#215—102no—
238#214—102no—
239#213—102no—
240C4⋊Dic3—101no—
241C4×Dic3—101no—
242#208—52no—
243#208—52no—
244#208—101no—
245#208—101no—
246C2×C3⋊C8—101no—
247C2×C3⋊C8—101no—
248ΓL(1,𝔽25)—104no—
249D3×C8—104no—
250C23×C5—21no—
251C23×C5#23511yescharacteristic
252C22×D5—61no—
253C5×Q8—21no—
254C5×Q8—21no—
255C5×D4—21no—
256C5×D4—21no—
257C5×D4—41no—
258C5×D4—41no—
259C2×C4×C5—21no—
260C2×C4×C5—41no—
261C2×C4×C5—41no—
262C2×C4×C5C2×C4×D311yescharacteristic
263C2×C4×C5D3×D411yescharacteristic
264C2×C4×C5D3×D411yescharacteristic
265C2×C4×C5D3×D411yescharacteristic
266C2×C4×C5#23711yescharacteristic
267C2×C4×C5C2×#6711yescharacteristic
268C22⋊D5—62no—
269C22⋊D5—62no—
270C2×Dic5—61no—
271C2×Dic5—121no—
272D20—32no—
273D20—61no—
274C4×D5—62no—
275C4×D5—62no—
276Dic10—32no—
277Dic10—61no—
278C5×C8—22no—
279C5×C8—41no—
280#168—62no—
281#168—62no—
282C2×#41—151no—
283C2×#41C2×D1511yescharacteristic
284#137—152no—
285#137—301no—
286AGL(1,ℤ/8)—152no—
287AGL(1,ℤ/8)—301no—
288#135—302no—
289#131—302no—
290#131—302no—
291C2×M16—151no—
292C2×M16—151no—
293C2×M16—302no—
294C2×M16C2×D1511yescharacteristic
295C22×C8C2×D1511yescharacteristic
296#117—151no—
297#104—302no—
298#103—301no—
299#102—301no—
300C22⋊C8—22no—
301C2×C3×C5—21no—
302C2×C3×C5—21no—
303C2×C3×C5—21no—
304C2×C3×C5—81no—
305C2×C3×C5#38611yescharacteristic
306D15—42no—
307C23×C3—21no—
308C23×C3#70911yescharacteristic
309C22×D3—101no—
310C3×Q8—21no—
311C3×Q8—21no—
312C3×D4—21no—
313C3×D4—21no—
314C3×D4—41no—
315C3×D4—41no—
316C2×C3×C4—21no—
317C2×C3×C4—41no—
318C2×C3×C4—41no—
319C2×C3×C4C2×C4×D511yescharacteristic
320C2×C3×C4D4×D511yescharacteristic
321C2×C3×C4D4×D511yescharacteristic
322C2×C3×C4D4×D511yescharacteristic
323C2×C3×C4#71111yescharacteristic
324C2×C3×C4C2×C22⋊D511yescharacteristic
325#67—102no—
326#67—102no—
327C2×Dic3—101no—
328C2×Dic3—201no—
329D12—52no—
330D12—101no—
331C4×D3—102no—
332C4×D3—102no—
333Dic6—52no—
334Dic6—101no—
335C3×C8—22no—
336C3×C8—41no—
337C3⋊C8—102no—
338C3⋊C8—102no—
339C22×C5—41no—
340C22×C5—42no—
341C22×C5—41no—
342C22×C5#88311yescharacteristic
343C22×C5#88411yescharacteristic
344C22×C5C4×#6711yescharacteristic
345C2×D5—62no—
346C2×D5—241no—
347C4×C5—41no—
348C4×C5D3×C22⋊C411yescharacteristic
349C4×C5#67⋊C411yescharacteristic
350C4×C5#93711yescharacteristic
351C4×C5#93811yescharacteristic
352Dic5—62no—
353Dic5—241no—
354#41—21no—
355#41—21no—
356#41—21no—
357#41—21no—
358#41—152no—
359#41—152no—
360#41—301no—
361#41—301no—
362C2×Q8—151no—
363C2×Q8C4×D1511yescharacteristic
364C2×D4—21no—
365C2×D4—151no—
366C2×D4—301no—
367C2×D4C4×D1511yescharacteristic
368C22×C4—21no—
369C22×C4—301no—
370C22×C4C22×D1511yescharacteristic
371Q16—302no—
372SD16—302no—
373SD16—302no—
374D8—302no—
375M16—21no—
376M16—21no—
377M16—152no—
378M16—152no—
379M16—304no—
380M16—301no—
381M16—301no—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—304no—
386C2×C8—301no—
387C2×C8—301no—
388C2×C8#122012yes—
389C4⋊C4—301no—
390C22⋊C4—301no—
391C42—301no—
392C3×C5#188111yescharacteristic, complemented, Hall
393C22×C3—41no—
394C22×C3—42no—
395C22×C3—41no—
396C22×C3#402911yescharacteristic
397C22×C3#403011yescharacteristic
398C22×C3C4×C22⋊D511yescharacteristic
399C2×D3—102no—
400C2×D3—401no—
401C3×C4—41no—
402C3×C4D5×C22⋊C411yescharacteristic
403C3×C4#402711yescharacteristic
404C3×C4#408311yescharacteristic
405C3×C4#408411yescharacteristic
406Dic3—102no—
407Dic3—401no—
408C2×C5—21no—
409C2×C5—21no—
410C2×C5—21no—
411C2×C5—81no—
412C2×C5#495611yescharacteristic
413D5—122no—
414C23—21no—
415C23—301no—
416C23#681711yescharacteristic
417Q8—21no—
418Q8—21no—
419Q8—152no—
420Q8—301no—
421D4—21no—
422D4—21no—
423D4—41no—
424D4—41no—
425D4—152no—
426D4—302no—
427D4—302no—
428D4—301no—
429C2×C4—21no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—301no—
433C2×C4—302no—
434C2×C4—302no—
435C2×C4—601no—
436C2×C4C2×C4×D1511yescharacteristic
437C2×C4D4×D1511yescharacteristic
438C2×C4D4×D1511yescharacteristic
439C2×C4D4×D1511yescharacteristic
440C2×C4#681911yescharacteristic
441C2×C4C2×#122011yescharacteristic
442C8—22no—
443C8—41no—
444C8—302no—
445C8—302no—
446C2×C3—21no—
447C2×C3—21no—
448C2×C3—21no—
449C2×C3—81no—
450C2×C3#6524511yescharacteristic
451D3—202no—
452C5#7001211yescharacteristic, complemented, Sylow
453C22—41no—
454C22—42no—
455C22—41no—
456C22—302no—
457C22—1201no—
458C22#9177211yescharacteristic
459C22#9177311yescharacteristic
460C22C4×#122011yescharacteristic
461C4—41no—
462C4—302no—
463C4—1201no—
464C4C22⋊C4×D1511yescharacteristic
465C4#9177011yescharacteristic
466C4#9182611yescharacteristic
467C4#9182711yescharacteristic
468C3#1060080511yescharacteristic, complemented, Sylow
469C2—21no—
470C2—21no—
471C2—21no—
472C2—81no—
473C2—602no—
474C2#1175105711yescharacteristic
subgroupG/H
centerC4#91826
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C22×D15
FittingC3×C5×#408C2
socleC2×C3×C5#386
prime radical#11756581C2
cosocleC22×C3×C4×C5C23
hypercenter#408D15
hypocenterC3×C5#1881
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C4 ◁ C22×C4 ◁ #408

Complements

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

NH ≅ G/Ntype
2. #11756504C2semidirect
3. C3×C5×#408C2semidirect
4. #11751337C2semidirect
5. #11751214C2semidirect
6. #11746377C2semidirect
7. #11746373C2semidirect
16. #91831C4semidirect
17. #91821C4semidirect
31. C5×#408D3semidirect
76. C3×#408D5semidirect
194. #408D15semidirect
392. C3×C5#1881semidirect
452. C5#70012semidirect
468. C3#10600805semidirect

Maximal subgroups

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

subgroupindex
#117565812
#117565042
C3×C5×#4082
#117513372
#117512142
#117463772
#117463732
#106008053
#700125