Finite Group Databasebeta

← #14301202

Subgroups — #14301202

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
1C5×#5745C211yesmaximal, characteristic, complemented
2#11754145C211yesmaximal, characteristic
3#11751230C211yesmaximal, characteristic, complemented
4#11748429C211yesmaximal, characteristic, complemented
5#11748382C211yesmaximal, characteristic, complemented
6C3×#64776C211yesmaximal, characteristic, complemented
7#11745982C211yesmaximal, characteristic, complemented
8#10602813—31nomaximal, Hall
9C2×#6817C2211yescharacteristic
10C2×C5×#235C2211yescharacteristic
11C2×C3×C5×M16C2211yescharacteristic
12C22×C3⋊#168C2211yescharacteristic
13C2×C5×#208C2211yescharacteristic
14C5×#907—22no—
15C5×D3×M16—22no—
16C5×#901—22no—
17C3×#4072C2211yescharacteristic
18C2×C3×#681C2211yescharacteristic
19#91322C412yescomplemented
20#91321—21no—
21#91305C412yescomplemented
22#91304—21no—
23#91303—22no—
24#91104—22no—
25#90981—22no—
26#90972—21no—
27#90966—21no—
28#69035—51nomaximal, Hall
29C5×#568—31no—
30#66156—31no—
31#65522—31no—
32#65428—31no—
33#65418—31no—
34#64778—31no—
35#64776D311yescharacteristic, complemented
36C22×C3×C4×C5C2311yescharacteristic
37C2×#1220—21no—
38#6817—21no—
39#6817—21no—
40#6817—21no—
41#6817—21no—
42C2×D60C2×C411yescharacteristic
43C2×C4×D15—21no—
44C2×Dic30C2×C411yescharacteristic
45C2×C5×#67—21no—
46C5×#235—21no—
47C5×#235—21no—
48C5×#235C2×C412yes—
49C5×#235D412yes—
50C2×C5×D12D411yescharacteristic
51C2×C4×C5×D3—21no—
52C2×C5×Dic6D411yescharacteristic
53C3×C5×M16—21no—
54C3×C5×M16—21no—
55C2×C3×C5×C8D412yes—
56C2×C3⋊#168—21no—
57C2×C3⋊#168—21no—
58C2×C3⋊#168—21no—
59C5×#208—21no—
60C5×#208D412yes—
61C2×C5×C3⋊C8—21no—
62C5×ΓL(1,𝔽25)—24no—
63C5×D3×C8—24no—
64C3×C22⋊Dic5—21no—
65C3×C4⋊Dic5—21no—
66C3×C4×Dic5—21no—
67C3×#681—21no—
68C3×#681C2×C412yes—
69C2×C3×#168—21no—
70#6663—22no—
71#6660—22no—
72#6659—22no—
73#6654—22no—
74#5748—51no—
75#5745D511yescharacteristic, complemented
76#5129—51no—
77#5005—51no—
78#5002—51no—
79C3×#344—51no—
80#4558—51no—
81C2×C5×#41—31no—
82C2×#709—31no—
83C5×#131—62no—
84C5×#131—62no—
85C2×C5×M16—31no—
86C2×C5×M16—62no—
87C2×C5×M16C2×D311yescharacteristic
88#4097—32no—
89#4096—62no—
90#4095—32no—
91#4088—61no—
92#4078—61no—
93#4072C2×D311yescharacteristic
94C2×#681C2×D311yescharacteristic
95C22×#168—31no—
96#3967—61no—
97#3964—61no—
98#3957—62no—
99#3952—62no—
100#1881—151noSylow
101C23×C3×C5#4111yescharacteristic
102C22×D15—21no—
103C22×C5×D3—21no—
104C2×C3×C4×C5C22×C411yescharacteristic
105C2×C3×C4×C5C2×D411yescharacteristic
106C2×C3×C4×C5C2×D411yescharacteristic
107C2×C3×C4×C5C2×D411yescharacteristic
108C2×C3×C4×C5C2×D411yescharacteristic
109C2×C3×C4×C5#4111yescharacteristic
110#1220—41no—
111#1220—41no—
112C2×Dic15—21no—
113D60—21no—
114D60—21no—
115C4×D15—41no—
116C4×D15—41no—
117Dic30—21no—
118Dic30—21no—
119C5×#67—22no—
120C5×#67—22no—
121C2×C5×Dic3—21no—
122C5×D12—21no—
123C5×D12C22⋊C412yes—
124C4×C5×D3—22no—
125C4×C5×D3—22no—
126C5×Dic6—21no—
127C5×Dic6C22⋊C412yes—
128C2×C3×Dic5—41no—
129C3×C5×C8—22no—
130C3⋊#168—41no—
131C3×#168—22no—
132C5×C3⋊C8—22no—
133C2×#235—51no—
134C2×#235C2×D511yescharacteristic
135C2×C3×M16—51no—
136C2×C3×M16C2×D511yescharacteristic
137C3×#117—51no—
138C2×#208C2×D511yescharacteristic
139C22×C3⋊C8—51no—
140#909—52no—
141#909—101no—
142#908—52no—
143#908—101no—
144#907—22no—
145D3×M16—22no—
146#904—102no—
147#901—22no—
148(C2×D3)⋊C8—102no—
149#810—101no—
150#809—101no—
151#805—102no—
152C5×#41—32no—
153C5×#41—32no—
154C5×#41—61no—
155C5×#41—61no—
156C2×C5×Q8—31no—
157C2×C5×D4—31no—
158C2×C5×D4—61no—
159C22×C4×C5—61no—
160C22×C4×C5C22×D311yescharacteristic
161C2×C22⋊D5—61no—
162#709—61no—
163#709—61no—
164#709—61no—
165#709—61no—
166C2×D20—31no—
167C2×C4×D5—61no—
168C2×Dic10—31no—
169C5×M16—21no—
170C5×M16—21no—
171C5×M16—32no—
172C5×M16—64no—
173C5×M16—61no—
174C2×C5×C8—64no—
175C2×C5×C8—61no—
176C2×C5×C8D1212yes—
177C22⋊Dic5—21no—
178#689—62no—
179#688—62no—
180#687—62no—
181#686—62no—
182C4⋊Dic5—21no—
183C4×Dic5—21no—
184#681—21no—
185#681C4×D312yes—
186C2×#168—21no—
187C2×#168—61no—
188C2×#168—61no—
189C2×#168—61no—
190#575—151no—
191#568—31no—
192#421—151no—
193#416—151no—
194#408—151no—
195#344—51no—
196#341—151no—
197C22×C3×C5—41no—
198C22×C3×C5C4×D411yescharacteristic
199C22×C3×C5#12111yescharacteristic
200C22×C3×C5#12311yescharacteristic
201C2×D15—42no—
202C2×D15—41no—
203C2×C5×D3—22no—
204C2×C5×D3—81no—
205C3×C4×C5C2×C22⋊C411yescharacteristic
206C3×C4×C5C4×D411yescharacteristic
207C3×C4×C5#12011yescharacteristic
208C3×C4×C5#12111yescharacteristic
209Dic15—41no—
210C3×Dic5—81no—
211C5×Dic3—22no—
212C22×C3×C4C22×D511yescharacteristic
213C2×#67—21no—
214C2×#67—101no—
215#235—21no—
216#235—21no—
217#235—101no—
218#235—101no—
219#235—101no—
220#235—101no—
221#235C4×D512yes—
222#235C22⋊D512yes—
223C2×D12—51no—
224C2×D12D2011yescharacteristic
225C2×C4×D3—21no—
226C2×C4×D3—101no—
227C2×Dic6—51no—
228C2×Dic6D2011yescharacteristic
229C3×M16—21no—
230C3×M16—21no—
231C3×M16—52no—
232C3×M16—101no—
233C2×C3×C8—101no—
234C2×C3×C8C22⋊D512yes—
235C3×C4⋊C4—101no—
236C3×C22⋊C4—101no—
237C3×C42—101no—
238#208—21no—
239#208C22⋊D512yes—
240C2×C3⋊C8—21no—
241C2×C3⋊C8—101no—
242C2×C3⋊C8—101no—
243C2×C3⋊C8—101no—
244Dic12—102no—
245D24—102no—
246#204—102no—
247#204—102no—
248ΓL(1,𝔽25)—24no—
249D3×C8—24no—
250C23×C5—61no—
251C23×C5#23511yescharacteristic
252C22×D5—61no—
253C5×Q8—32no—
254C5×Q8—61no—
255C5×D4—32no—
256C5×D4—62no—
257C5×D4—62no—
258C5×D4—61no—
259C2×C4×C5—61no—
260C2×C4×C5—62no—
261C2×C4×C5—62no—
262C2×C4×C5C2×C4×D311yescharacteristic
263C2×C4×C5C2×D1211yescharacteristic
264C2×C4×C5D3×D411yescharacteristic
265C2×C4×C5D3×D411yescharacteristic
266C2×C4×C5D3×D411yescharacteristic
267C2×C4×C5#23911yescharacteristic
268C22⋊D5—121no—
269C22⋊D5—121no—
270C2×Dic5—41no—
271C2×Dic5—61no—
272D20—61no—
273D20—61no—
274C4×D5—121no—
275C4×D5—121no—
276Dic10—61no—
277Dic10—61no—
278C5×C8—22no—
279C5×C8—62no—
280#168—22no—
281#168—121no—
282C2×#41—31no—
283C2×#41—151no—
284#137—152no—
285#137—301no—
286AGL(1,ℤ/8)—152no—
287AGL(1,ℤ/8)—301no—
288#135—302no—
289#131—62no—
290#131—62no—
291C2×M16—31no—
292C2×M16—51no—
293C2×M16—62no—
294C2×M16D3×D511yescharacteristic
295C22×C8—151no—
296#117—51no—
297#104—302no—
298#103—301no—
299#102—301no—
300C22⋊C8—302no—
301C2×C3×C5—21no—
302C2×C3×C5—21no—
303C2×C3×C5—21no—
304C2×C3×C5#38611yescharacteristic
305D15—81no—
306C5×D3—42no—
307C23×C3#70911yescharacteristic
308C22×D3—21no—
309C22×D3—101no—
310C2×C3×C4—201no—
311C2×C3×C4C2×C4×D511yescharacteristic
312C2×C3×C4C2×D2011yescharacteristic
313C2×C3×C4#70911yescharacteristic
314C2×C3×C4C2×C22⋊D511yescharacteristic
315C2×C3×C4C2×C22⋊D511yescharacteristic
316C2×C3×C4C2×C22⋊D511yescharacteristic
317#67—22no—
318#67—22no—
319#67—201no—
320#67—201no—
321C2×Dic3—21no—
322C2×Dic3—101no—
323D12—21no—
324D12—101no—
325D12—101no—
326D12#68512yes—
327C4×D3—22no—
328C4×D3—22no—
329C4×D3—201no—
330C4×D3—201no—
331Dic6—21no—
332Dic6—101no—
333Dic6—101no—
334Dic6#68512yes—
335C3×C8—22no—
336C3×C8—102no—
337C3⋊C8—22no—
338C3⋊C8—201no—
339C22×C5—41no—
340C22×C5—62no—
341C22×C5—241no—
342C22×C5C4×D1211yescharacteristic
343C22×C5#88411yescharacteristic
344C22×C5#89411yescharacteristic
345C2×D5—122no—
346C2×D5—121no—
347C4×C5—62no—
348C4×C5D3×C22⋊C411yescharacteristic
349C4×C5#88211yescharacteristic
350C4×C5#89311yescharacteristic
351C4×C5#89511yescharacteristic
352Dic5—81no—
353Dic5—121no—
354#41—32no—
355#41—32no—
356#41—61no—
357#41—61no—
358#41—301no—
359#41—301no—
360#41—301no—
361#41—301no—
362C2×Q8—31no—
363C2×Q8—151no—
364C2×D4—31no—
365C2×D4—61no—
366C2×D4—151no—
367C2×D4—301no—
368C22×C4—61no—
369C22×C4—301no—
370C22×C4C2×D3×D511yescharacteristic
371Q16—302no—
372SD16—302no—
373SD16—302no—
374D8—302no—
375M16—21no—
376M16—21no—
377M16—32no—
378M16—52no—
379M16—64no—
380M16—61no—
381M16—101no—
382C2×C8—64no—
383C2×C8—61no—
384C2×C8—101no—
385C2×C8—301no—
386C2×C8—301no—
387C2×C8—301no—
388C2×C8#120312yes—
389C4⋊C4—101no—
390C22⋊C4—101no—
391C42—101no—
392C3×C5#188111yescharacteristic, complemented, Hall
393C22×C3—41no—
394C22×C3C4×C22⋊D511yescharacteristic
395C22×C3#407511yescharacteristic
396C22×C3#407611yescharacteristic
397C2×D3—22no—
398C2×D3—81no—
399C2×D3—202no—
400C2×D3—201no—
401C3×C4—401no—
402C3×C4C2×#68511yescharacteristic
403C3×C4C4×C22⋊D511yescharacteristic
404C3×C4#407611yescharacteristic
405C3×C4#409911yescharacteristic
406Dic3—22no—
407Dic3—201no—
408C2×C5—21no—
409C2×C5—21no—
410C2×C5—21no—
411C2×C5—122no—
412C2×C5#466511yescharacteristic
413D5—241no—
414C23—61no—
415C23—301no—
416C23#677111yescharacteristic
417Q8—32no—
418Q8—61no—
419Q8—301no—
420Q8—301no—
421D4—32no—
422D4—62no—
423D4—62no—
424D4—61no—
425D4—301no—
426D4—301no—
427D4—601no—
428D4—601no—
429C2×C4—61no—
430C2×C4—62no—
431C2×C4—62no—
432C2×C4—201no—
433C2×C4—301no—
434C2×C4—601no—
435C2×C4—601no—
436C2×C4#676911yescharacteristic
437C2×C4C4×D3×D511yescharacteristic
438C2×C4D3×D2011yescharacteristic
439C2×C4C2×#120311yescharacteristic
440C2×C4D3×C22⋊D511yescharacteristic
441C2×C4D3×C22⋊D511yescharacteristic
442C8—22no—
443C8—62no—
444C8—102no—
445C8—601no—
446C2×C3—21no—
447C2×C3—21no—
448C2×C3—21no—
449C2×C3#6550211yescharacteristic
450D3—42no—
451D3—401no—
452C5#6903511yescharacteristic, complemented, Sylow
453C22—41no—
454C22—62no—
455C22—241no—
456C22—602no—
457C22—601no—
458C22#9135111yescharacteristic
459C22C4×#120311yescharacteristic
460C22#9145411yescharacteristic
461C4—62no—
462C4—401no—
463C4—601no—
464C4#9144611yescharacteristic
465C4#9146011yescharacteristic
466C4D3×#68511yescharacteristic
467C4#9157011yescharacteristic
468C3#1060281311yescharacteristic, complemented, Sylow
469C2—21no—
470C2—21no—
471C2—21no—
472C2—122no—
473C2—1201no—
474C2#1174891411yescharacteristic
subgroupG/H
centerC4#91570
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC2×C3×C5×M16C22
socleC2×C3×C5#386
prime radical#11754145C2
cosocleC22×C3×C4×C5C23
hypercenterC2×M16D3×D5
hypocenterC3×C5#1881
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
1. C5×#5745C2semidirect
3. #11751230C2semidirect
4. #11748429C2semidirect
5. #11748382C2semidirect
6. C3×#64776C2semidirect
7. #11745982C2semidirect
19. #91322C4semidirect
21. #91305C4semidirect
35. #64776D3semidirect
75. #5745D5semidirect
392. C3×C5#1881semidirect
452. C5#69035semidirect
468. C3#10602813semidirect

Maximal subgroups

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

subgroupindex
C5×#57452
#117541452
#117512302
#117484292
#117483822
C3×#647762
#117459822
#106028133
#690355