Finite Group Databasebeta

← #14301207

Subgroups — #14301207

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#11756507C211yesmaximal, characteristic
2#11754130C211yesmaximal, characteristic, complemented
3#11748429C211yesmaximal, characteristic, complemented
4#11748411C211yesmaximal, characteristic, complemented
5#11748350C211yesmaximal, characteristic, complemented
6C5×#4558C211yesmaximal, characteristic, complemented
7C3×#64779C211yesmaximal, characteristic, complemented
8#10599740—31nomaximal, Hall
9C2×#6817C2211yescharacteristic
10C2×#6817C2211yescharacteristic
11C2×C3×C5×M16C2211yescharacteristic
12#91798—21no—
13#91798C412yescomplemented
14#91797—21no—
15#91797C412yescomplemented
16#91793—22no—
17C2×C5×#208C2211yescharacteristic
18C22×C5×C3⋊C8C2211yescharacteristic
19C3×#4072C2211yescharacteristic
20C2×C3×#681C2211yescharacteristic
21#91292—22no—
22#91291—22no—
23#91290—22no—
24#90981—22no—
25#90974—21no—
26#90968—21no—
27#90957—22no—
28#69035—51nomaximal, Hall
29#66082—31no—
30#66079—31no—
31#65428—31no—
32#65402—31no—
33#65399—31no—
34C5×#341—31no—
35#64779D311yescharacteristic, complemented
36C22×C3×C4×C5C2311yescharacteristic
37C2×#1220—21no—
38C2×#1220—21no—
39#6817—21no—
40#6817—21no—
41#6817—21no—
42#6817—21no—
43#6817—21no—
44#6817—21no—
45#6817C2×C412yes—
46#6817D412yes—
47C2×D60C2×C411yescharacteristic
48C2×D60D411yescharacteristic
49C2×C4×D15—21no—
50C2×C4×D15—21no—
51C2×Dic30C2×C411yescharacteristic
52C2×Dic30D411yescharacteristic
53C3×C5×M16—21no—
54C3×C5×M16C2×C412yes—
55C2×C3×C5×C8—21no—
56Dic60—22no—
57D120—22no—
58#6706—22no—
59#6706—22no—
60C5×#208—21no—
61C5×#208D412yes—
62C2×C5×C3⋊C8—21no—
63C2×C5×C3⋊C8—21no—
64C2×C5×C3⋊C8—21no—
65C2×C5×C3⋊C8—21no—
66C3×C22⋊Dic5—21no—
67C3×C4⋊Dic5—21no—
68C3×C4×Dic5—21no—
69C3×#681—21no—
70C3×#681—21no—
71C2×C3×#168D412yes—
72#6651—24no—
73#6648—24no—
74#5748—51no—
75#5745—51no—
76#5129—51no—
77#5005—51no—
78#5002—51no—
79C3×#344—51no—
80#4558D511yescharacteristic, complemented
81C2×#709—31no—
82C2×#709—31no—
83C2×C5×M16—31no—
84C2×C5×M16C2×D311yescharacteristic
85C22×C5×C8—31no—
86#4072C2×D311yescharacteristic
87C2×#681C2×D311yescharacteristic
88#4055—32no—
89#4055—61no—
90#4054—32no—
91#4054—61no—
92#4053—62no—
93D5×M16—62no—
94#4050—62no—
95#4047—62no—
96#3957—62no—
97#3953—61no—
98#3952—62no—
99#3948—61no—
100#1881—151noSylow
101C23×C3×C5#4111yescharacteristic
102C22×D15—21no—
103C22×D15—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—22no—
111#1220—22no—
112#1220—41no—
113#1220—41no—
114C2×Dic15—21no—
115C2×Dic15—21no—
116D60—21no—
117D60—21no—
118D60—21no—
119D60C22⋊C412yes—
120C4×D15—22no—
121C4×D15—22no—
122C4×D15—41no—
123C4×D15—41no—
124Dic30—21no—
125Dic30—21no—
126Dic30—21no—
127Dic30C22⋊C412yes—
128C2×C3×Dic5—41no—
129C3×C5×C8—22no—
130C3×#168—22no—
131C5×C3⋊C8—22no—
132C5×C3⋊C8—41no—
133C2×#235—51no—
134C2×#235—51no—
135C2×C3×M16—51no—
136C2×C3×M16C2×D511yescharacteristic
137C3×#117—51no—
138C2×#208C2×D511yescharacteristic
139C22×C3⋊C8C2×D511yescharacteristic
140#909—52no—
141#909—101no—
142#908—52no—
143#908—101no—
144#907—102no—
145D3×M16—102no—
146#904—102no—
147#901—102no—
148(C2×D3)⋊C8—102no—
149#810—101no—
150#809—101no—
151#805—102no—
152C22×C4×C5C22×D311yescharacteristic
153C2×C22⋊D5—61no—
154C2×C22⋊D5—61no—
155#709—32no—
156#709—32no—
157#709—61no—
158#709—61no—
159#709—61no—
160#709—61no—
161#709—61no—
162#709—61no—
163C2×D20—31no—
164C2×D20—31no—
165C2×C4×D5—61no—
166C2×C4×D5—61no—
167C2×Dic10—31no—
168C2×Dic10—31no—
169C5×M16—21no—
170C5×M16—32no—
171C5×M16—61no—
172C5×M16C4×D312yes—
173C2×C5×C8—21no—
174C2×C5×C8—61no—
175C2×C5×C8—61no—
176C2×C5×C8—61no—
177C2×C5×C8—61no—
178C22⋊Dic5—21no—
179C4⋊Dic5—21no—
180C4×Dic5—21no—
181#681—21no—
182#681—21no—
183C2×#168D1212yes—
184Dic20—62no—
185D40—62no—
186#677—62no—
187#677—62no—
188#676—64no—
189C8×D5—64no—
190#575—151no—
191#568—151no—
192#421—151no—
193#416—151no—
194#408—151no—
195#344—51no—
196#341—31no—
197C22×C3×C5—41no—
198C22×C3×C5C4×D411yescharacteristic
199C22×C3×C5#12111yescharacteristic
200C22×C3×C5#12311yescharacteristic
201C2×D15—22no—
202C2×D15—42no—
203C2×D15—41no—
204C2×D15—81no—
205C3×C4×C5C2×C22⋊C411yescharacteristic
206C3×C4×C5C4×D411yescharacteristic
207C3×C4×C5#12011yescharacteristic
208C3×C4×C5#12111yescharacteristic
209Dic15—22no—
210Dic15—41no—
211C3×Dic5—81no—
212C22×C3×C4C22×D511yescharacteristic
213C2×#67—101no—
214C2×#67—101no—
215#235—52no—
216#235—52no—
217#235—101no—
218#235—101no—
219#235—101no—
220#235—101no—
221#235—101no—
222#235—101no—
223C2×D12—51no—
224C2×D12—51no—
225C2×C4×D3—101no—
226C2×C4×D3—101no—
227C2×Dic6—51no—
228C2×Dic6—51no—
229C3×M16—21no—
230C3×M16—101no—
231C3×M16—101no—
232C3×M16C4×D512yes—
233C2×C3×C8—21no—
234C2×C3×C8—52no—
235C3×C4⋊C4—101no—
236C3×C22⋊C4—101no—
237C3×C42—101no—
238#208—21no—
239#208D2012yes—
240C2×C3⋊C8—21no—
241C2×C3⋊C8—21no—
242C2×C3⋊C8—21no—
243C2×C3⋊C8—21no—
244Dic12—102no—
245D24—102no—
246#204—102no—
247#204—102no—
248ΓL(1,𝔽25)—104no—
249D3×C8—104no—
250C23×C5#23511yescharacteristic
251C22×D5—61no—
252C22×D5—61no—
253C2×C4×C5C2×C4×D311yescharacteristic
254C2×C4×C5C2×D1211yescharacteristic
255C2×C4×C5D3×D411yescharacteristic
256C2×C4×C5D3×D411yescharacteristic
257C2×C4×C5D3×D411yescharacteristic
258C2×C4×C5#23911yescharacteristic
259C22⋊D5—62no—
260C22⋊D5—62no—
261C22⋊D5—121no—
262C22⋊D5—121no—
263C2×Dic5—41no—
264C2×Dic5—61no—
265C2×Dic5—61no—
266D20—32no—
267D20—61no—
268D20—61no—
269D20—61no—
270C4×D5—62no—
271C4×D5—62no—
272C4×D5—121no—
273C4×D5—121no—
274Dic10—32no—
275Dic10—61no—
276Dic10—61no—
277Dic10—61no—
278C5×C8—22no—
279C5×C8—62no—
280C5×C8—121no—
281#168—22no—
282C2×#41—151no—
283C2×#41—151no—
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—31no—
292C2×M16—51no—
293C2×M16—302no—
294C2×M16D3×D511yescharacteristic
295C22×C8—31no—
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—42no—
306D15—81no—
307C23×C3#71311yescharacteristic
308C22×D3—101no—
309C22×D3—101no—
310C2×C3×C4—201no—
311C2×C3×C4C2×C4×D511yescharacteristic
312C2×C3×C4C2×D2011yescharacteristic
313C2×C3×C4#70911yescharacteristic
314C2×C3×C4D4×D511yescharacteristic
315C2×C3×C4D4×D511yescharacteristic
316C2×C3×C4D4×D511yescharacteristic
317#67—102no—
318#67—102no—
319#67—201no—
320#67—201no—
321C2×Dic3—101no—
322C2×Dic3—101no—
323D12—52no—
324D12—101no—
325D12—101no—
326D12—101no—
327C4×D3—102no—
328C4×D3—102no—
329C4×D3—201no—
330C4×D3—201no—
331Dic6—52no—
332Dic6—101no—
333Dic6—101no—
334Dic6—101no—
335C3×C8—22no—
336C3×C8—102no—
337C3⋊C8—22no—
338C3⋊C8—41no—
339C22×C5—41no—
340C22×C5C4×D1211yescharacteristic
341C22×C5#88411yescharacteristic
342C22×C5#89411yescharacteristic
343C2×D5—62no—
344C2×D5—122no—
345C2×D5—121no—
346C2×D5—241no—
347C4×C5D3×C22⋊C411yescharacteristic
348C4×C5#88211yescharacteristic
349C4×C5#89311yescharacteristic
350C4×C5#89511yescharacteristic
351Dic5—62no—
352Dic5—81no—
353Dic5—121no—
354#41—152no—
355#41—152no—
356#41—301no—
357#41—301no—
358#41—301no—
359#41—301no—
360#41—301no—
361#41—301no—
362C2×Q8—151no—
363C2×Q8—151no—
364C2×D4—151no—
365C2×D4—151no—
366C2×D4—301no—
367C2×D4—301no—
368C22×C4—301no—
369C22×C4—301no—
370C22×C4C2×D3×D511yescharacteristic
371Q16—302no—
372SD16—302no—
373SD16—302no—
374D8—302no—
375M16—21no—
376M16—32no—
377M16—61no—
378M16—101no—
379M16—101no—
380M16—304no—
381M16#120012yes—
382C2×C8—21no—
383C2×C8—52no—
384C2×C8—61no—
385C2×C8—61no—
386C2×C8—61no—
387C2×C8—61no—
388C2×C8—304no—
389C4⋊C4—101no—
390C22⋊C4—101no—
391C42—101no—
392C3×C5#188111yescharacteristic, complemented, Hall
393C22×C3—41no—
394C22×C3#403911yescharacteristic
395C22×C3#404011yescharacteristic
396C22×C3#404111yescharacteristic
397C2×D3—102no—
398C2×D3—202no—
399C2×D3—201no—
400C2×D3—401no—
401C3×C4—401no—
402C3×C4C4×D2011yescharacteristic
403C3×C4D5×C22⋊C411yescharacteristic
404C3×C4#402811yescharacteristic
405C3×C4#403011yescharacteristic
406Dic3—102no—
407Dic3—201no—
408C2×C5—21no—
409C2×C5—21no—
410C2×C5—21no—
411C2×C5#466511yescharacteristic
412D5—122no—
413D5—241no—
414C23—301no—
415C23—301no—
416C23#677311yescharacteristic
417Q8—152no—
418Q8—301no—
419Q8—301no—
420Q8—301no—
421D4—152no—
422D4—302no—
423D4—302no—
424D4—301no—
425D4—301no—
426D4—301no—
427D4—601no—
428D4—601no—
429C2×C4—201no—
430C2×C4—301no—
431C2×C4—302no—
432C2×C4—302no—
433C2×C4—301no—
434C2×C4—601no—
435C2×C4—601no—
436C2×C4#676911yescharacteristic
437C2×C4D5×D1211yescharacteristic
438C2×C4D3×D2011yescharacteristic
439C2×C4C2×#120011yescharacteristic
440C2×C4#679011yescharacteristic
441C2×C4#679011yescharacteristic
442C8—22no—
443C8—62no—
444C8—102no—
445C8—121no—
446C2×C3—21no—
447C2×C3—21no—
448C2×C3—21no—
449C2×C3#6495411yescharacteristic
450D3—202no—
451D3—401no—
452C5#6903511yescharacteristic, complemented, Sylow
453C22—41no—
454C22—302no—
455C22—602no—
456C22—601no—
457C22—1201no—
458C22#9141811yescharacteristic
459C22#9143311yescharacteristic
460C22#9145911yescharacteristic
461C4—302no—
462C4—401no—
463C4—601no—
464C4#9142011yescharacteristic
465C4#9142811yescharacteristic
466C4#9147611yescharacteristic
467C4#9156111yescharacteristic
468C3#1059974011yescharacteristic, complemented, Sylow
469C2—21no—
470C2—21no—
471C2—21no—
472C2—602no—
473C2—1201no—
474C2#1174890411yescharacteristic
subgroupG/H
centerC4#91476
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC2×C3×C5×M16C22
socleC2×C3×C5#386
prime radical#11756507C2
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
2. #11754130C2semidirect
3. #11748429C2semidirect
4. #11748411C2semidirect
5. #11748350C2semidirect
6. C5×#4558C2semidirect
7. C3×#64779C2semidirect
13. #91798C4semidirect
15. #91797C4semidirect
35. #64779D3semidirect
80. #4558D5semidirect
392. C3×C5#1881semidirect
452. C5#69035semidirect
468. C3#10599740semidirect

Maximal subgroups

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

subgroupindex
#117565072
#117541302
#117484292
#117484112
#117483502
C5×#45582
C3×#647792
#105997403
#690355