Finite Group Databasebeta

← #14301286

Subgroups — #14301286

Order 1920.

668 conjugacy classes of subgroups.

The subgroup lattice has 500 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#909C211yesmaximal, characteristic
2#11754492C211yesmaximal, characteristic
3#11748607C211yesmaximal, characteristic, complemented
4C2×#90978C211yesmaximal, characteristic
5#11748409C211yesmaximal, characteristic, complemented
6#11746376C211yesmaximal, characteristic, complemented
7C3×#64774C211yesmaximal, characteristic, complemented
8#10602893—31nomaximal, Hall
9C2×C5×#235C2211yescharacteristic
10C22×C5×Dic6C2211yescharacteristic
11C2×C3×C5×M16C2211yescharacteristic
12#91815C2211yescharacteristic
13C2×C4×Dic15C2211yescharacteristic
14C5×#909—22no—
15C5×#909C414yescomplemented
16C2×C5×Dic12—21no—
17C2×C5×#204—21no—
18C2×C3×C4⋊Dic5C2211yescharacteristic
19C2×C3×#681C2211yescharacteristic
20Dic5×#67—22no—
21#91435—21no—
22D3×C4⋊Dic5—21no—
23#91344—21no—
24#91338—21no—
25#90994—22no—
26#90978—22no—
27#90973—21no—
28#90973—21no—
29#69004—51nomaximal, Hall
30C2×C5×#137—31no—
31#66161—31no—
32C2×#3969—31no—
33#65519—31no—
34#65514—31no—
35#64775—31no—
36#64774D311yescharacteristic, complemented
37C22×C3×C4×C5C2311yescharacteristic
38C22×Dic15—21no—
39C2×C5×#67—21no—
40C22×C5×Dic3—21no—
41C5×#235—21no—
42C5×#235—21no—
43C5×#235C2×C412yes—
44C5×#235D412yes—
45C2×C5×D12D411yescharacteristic
46C2×C4×C5×D3—21no—
47C2×C5×Dic6—21no—
48C2×C5×Dic6—21no—
49C2×C5×Dic6—21no—
50C2×C5×Dic6—21no—
51C2×C5×Dic6—21no—
52C2×C5×Dic6C2×C412yescomplemented
53C2×C5×Dic6D411yescharacteristic
54C22×C3×Dic5D412yes—
55C2×D3×Dic5—22no—
56C3×C5×M16—21no—
57C3×C5×M16C2×C412yescomplemented
58C2×C3×C5×C8—21no—
59#6719—21no—
60C4⋊Dic15D411yescharacteristic
61C4⋊Dic15D411yescharacteristic
62C4×Dic15—21no—
63C4×Dic15—21no—
64C4×Dic15—21no—
65C5×Dic12—24no—
66C5×#204—24no—
67C3×C4⋊Dic5—21no—
68C3×C4⋊Dic5—21no—
69C3×#681—21no—
70C3×#681—21no—
71C2×C3×#168—21no—
72Dic3⋊Dic5—22no—
73(C2×D3)⋊Dic5—22no—
74Dic3×Dic5—22no—
75C2×#909D511yescharacteristic
76#5504—51no—
77C2×#825—51no—
78#5120—51no—
79#4973—51no—
80C3×#343—51no—
81#4555—51no—
82C2×C5×#41—31no—
83C22×C5×Q8—31no—
84C5×#137—34no—
85C5×#137—62no—
86C2×C5×Q16—61no—
87C2×C5×SD16—61no—
88C2×C5×M16C2×D311yescharacteristic
89Q8×Dic5—61no—
90D4×Dic5—61no—
91D4×Dic5—62no—
92#4072—31no—
93#4072—61no—
94C2×C4⋊Dic5—61no—
95C2×C4⋊Dic5C2×D311yescharacteristic
96C2×C4×Dic5—31no—
97C2×#681C2×D311yescharacteristic
98#3969—62no—
99#3967—61no—
100#3967—61no—
101#3963—62no—
102#1882—151noSylow
103C23×C3×C5C2×D411yescharacteristic
104C22×C5×D3—21no—
105C2×C3×C4×C5C22×C411yescharacteristic
106C2×C3×C4×C5C2×D411yescharacteristic
107C2×C3×C4×C5C2×D411yescharacteristic
108C2×C3×C4×C5C2×D411yescharacteristic
109C2×C3×C4×C5#4111yescharacteristic
110C2×C3×C4×C5#4111yescharacteristic
111C2×Dic15—22no—
112C2×Dic15—21no—
113C2×Dic15—21no—
114C2×Dic15—41no—
115C2×Dic15—41no—
116C5×#67—22no—
117C5×#67—22no—
118C2×C5×Dic3—21no—
119C2×C5×Dic3—22no—
120C2×C5×Dic3—41no—
121C2×C5×Dic3—41no—
122C5×D12—21no—
123C5×D12C22⋊C412yes—
124C4×C5×D3—22no—
125C4×C5×D3—22no—
126C5×Dic6—21no—
127C5×Dic6—22no—
128C5×Dic6—22no—
129C5×Dic6—41no—
130C5×Dic6—41no—
131C5×Dic6C22⋊C412yes—
132C2×C3×Dic5—22no—
133C2×C3×Dic5—42no—
134D3×Dic5—82no—
135C3×C5×C8—22no—
136C3×#168—41no—
137C2×#235C2×D511yescharacteristic
138C22×Dic6C2×D511yescharacteristic
139C2×C3×M16—51no—
140C2×C3×M16C2×D511yescharacteristic
141C2×C3×C4⋊C4—51no—
142C4×#67—102no—
143#926—51no—
144C2×C4×Dic3—51no—
145#909—22no—
146#909Dic514yescomplemented
147C2×Dic12—21no—
148C2×#204—21no—
149#893—101no—
150#892—101no—
151D3×C4⋊C4—101no—
152#887—101no—
153C3⋊#95—102no—
154#825—102no—
155#816—101no—
156#816—101no—
157C5×#41—32no—
158C5×#41—32no—
159C5×#41—61no—
160C5×#41—61no—
161C2×C5×Q8—32no—
162C2×C5×Q8—31no—
163C2×C5×Q8—61no—
164C2×C5×Q8—61no—
165C2×C5×Q8—61no—
166C2×C5×Q8—61no—
167C2×C5×Q8—61no—
168C2×C5×D4—31no—
169C2×C5×D4—61no—
170C22×C4×C5—61no—
171C22×C4×C5—61no—
172C22×C4×C5C22×D311yescharacteristic
173C22×Dic5—62no—
174C22×Dic5—61no—
175C22×Dic5D1212yes—
176C5×Q16—64no—
177C5×SD16—64no—
178C5×M16—21no—
179C5×M16C4×D312yescomplemented
180C2×C5×C8—21no—
181C22⋊Dic5—62no—
182C22⋊Dic5—61no—
183C4⋊Dic5—21no—
184C4⋊Dic5—21no—
185C4⋊Dic5—31no—
186C4⋊Dic5—31no—
187C4⋊Dic5—62no—
188C4×Dic5—62no—
189C4×Dic5—61no—
190C4×Dic5—61no—
191C4×Dic5—61no—
192#681—21no—
193#681—21no—
194C2×#168—21no—
195C2×#137—31no—
196#520—151no—
197C2×#104—151no—
198#419—151no—
199#391—151no—
200#343—51no—
201#339—151no—
202C22×C3×C5—21no—
203C22×C3×C5—21no—
204C22×C3×C5C4×D411yescharacteristic
205C22×C3×C5#12011yescharacteristic
206C22×C3×C5#12111yescharacteristic
207C2×C5×D3—22no—
208C2×C5×D3—41no—
209C2×C5×D3—41no—
210C3×C4×C5C2×C22⋊C411yescharacteristic
211C3×C4×C5C4×D411yescharacteristic
212C3×C4×C5#12111yescharacteristic
213C3×C4×C5#12311yescharacteristic
214Dic15—42no—
215Dic15—41no—
216Dic15—41no—
217C3×Dic5—42no—
218C5×Dic3—22no—
219C5×Dic3—42no—
220C22×C3×C4—52no—
221C22×C3×C4C22×D511yescharacteristic
222C2×#67—21no—
223C22×Dic3—21no—
224C22×Dic3—101no—
225#235—21no—
226#235—21no—
227#235C2×Dic512yes—
228#235C22⋊D512yes—
229C2×D12C22⋊D511yescharacteristic
230C2×C4×D3—21no—
231C2×C4×D3—102no—
232C2×Dic6—21no—
233C2×Dic6—21no—
234C2×Dic6—21no—
235C2×Dic6—21no—
236C2×Dic6—21no—
237C2×Dic6C2×Dic512yescomplemented
238C2×Dic6C22⋊D511yescharacteristic
239C3×M16—21no—
240C3×M16—101no—
241C3×M16—101no—
242C3×M16C2×Dic512yescomplemented
243C2×C3×C8—21no—
244C2×C3×C8—101no—
245C3×C4⋊C4—101no—
246C3×C4⋊C4—101no—
247#217—101no—
248(C2×D3)⋊C4—102no—
249C4⋊Dic3—51no—
250C4⋊Dic3—51no—
251Dic3⋊C4—102no—
252C4×Dic3—102no—
253C4×Dic3—101no—
254C4×Dic3—101no—
255C4×Dic3—101no—
256Dic12—24no—
257#204—24no—
258C23×C5—61no—
259C23×C5C2×D1211yescharacteristic
260C5×Q8—32no—
261C5×Q8—61no—
262C5×Q8—62no—
263C5×Q8—62no—
264C5×Q8—121no—
265C5×Q8—121no—
266C5×D4—32no—
267C5×D4—62no—
268C5×D4—62no—
269C5×D4—61no—
270C2×C4×C5—61no—
271C2×C4×C5—62no—
272C2×C4×C5—62no—
273C2×C4×C5—62no—
274C2×C4×C5—121no—
275C2×C4×C5—121no—
276C2×C4×C5C2×C4×D311yescharacteristic
277C2×C4×C5#23511yescharacteristic
278C2×C4×C5D3×D411yescharacteristic
279C2×C4×C5D3×D411yescharacteristic
280C2×C4×C5D3×D411yescharacteristic
281C2×C4×C5#23911yescharacteristic
282C2×Dic5—22no—
283C2×Dic5—42no—
284C2×Dic5—62no—
285C2×Dic5—61no—
286C2×Dic5—61no—
287C2×Dic5—121no—
288C2×Dic5—121no—
289C2×Dic5—242no—
290C5×C8—22no—
291#168—41no—
292C2×#41—31no—
293C22×Q8—31no—
294#137—34no—
295#137—62no—
296C2×Q16—61no—
297C2×SD16—61no—
298C2×M16—51no—
299C2×M16D3×D511yescharacteristic
300C4×Q8—301no—
301C4×D4—301no—
302C4×D4—302no—
303#117—151no—
304#117—301no—
305C2×C4⋊C4—51no—
306C2×C4⋊C4—301no—
307C2×C42—151no—
308#104—302no—
309#103—301no—
310#103—301no—
311#95—302no—
312C2×C3×C5—21no—
313C2×C3×C5—21no—
314C2×C3×C5#38611yescharacteristic
315C2×C3×C5#45511yescharacteristic
316C2×C3×C5#45611yescharacteristic
317C5×D3—42no—
318C23×C3D4×D511yescharacteristic
319C22×D3—21no—
320C2×C3×C4—102no—
321C2×C3×C4—202no—
322C2×C3×C4D4×D511yescharacteristic
323C2×C3×C4#71111yescharacteristic
324C2×C3×C4#71111yescharacteristic
325C2×C3×C4C22×Dic511yescharacteristic
326C2×C3×C4C2×C22⋊D511yescharacteristic
327C2×C3×C4C2×C22⋊D511yescharacteristic
328#67—22no—
329#67—22no—
330C2×Dic3—21no—
331C2×Dic3—22no—
332C2×Dic3—41no—
333C2×Dic3—41no—
334C2×Dic3—102no—
335C2×Dic3—101no—
336C2×Dic3—101no—
337C2×Dic3—201no—
338C2×Dic3—201no—
339D12—21no—
340D12C22⋊Dic512yes—
341C4×D3—22no—
342C4×D3—22no—
343C4×D3—402no—
344Dic6—21no—
345Dic6—22no—
346Dic6—22no—
347Dic6—41no—
348Dic6—41no—
349Dic6C22⋊Dic512yes—
350C3×C8—22no—
351C3×C8—201no—
352C22×C5—21no—
353C22×C5—21no—
354C22×C5—62no—
355C22×C5—121no—
356C22×C5—121no—
357C22×C5C4×D1211yescharacteristic
358C22×C5#88211yescharacteristic
359C22×C5#89511yescharacteristic
360C4×C5—62no—
361C4×C5—122no—
362C4×C5D3×C22⋊C411yescharacteristic
363C4×C5#88411yescharacteristic
364C4×C5#89311yescharacteristic
365C4×C5#89411yescharacteristic
366Dic5—42no—
367Dic5—122no—
368Dic5—121no—
369Dic5—121no—
370#41—32no—
371#41—32no—
372#41—61no—
373#41—61no—
374C2×Q8—32no—
375C2×Q8—31no—
376C2×Q8—61no—
377C2×Q8—61no—
378C2×Q8—61no—
379C2×Q8—61no—
380C2×Q8—61no—
381C2×D4—31no—
382C2×D4—61no—
383C22×C4—52no—
384C22×C4—61no—
385C22×C4—61no—
386C22×C4—302no—
387C22×C4—301no—
388C22×C4C2×D3×D511yescharacteristic
389Q16—64no—
390SD16—64no—
391M16—21no—
392M16—101no—
393M16—101no—
394M16D3×Dic512yescomplemented
395C2×C8—21no—
396C2×C8—101no—
397C4⋊C4—101no—
398C4⋊C4—101no—
399C4⋊C4—151no—
400C4⋊C4—151no—
401C4⋊C4—302no—
402C22⋊C4—302no—
403C22⋊C4—301no—
404C42—302no—
405C42—301no—
406C42—301no—
407C42—301no—
408C3×C5#188211yescharacteristic, complemented, Hall
409C22×C3—21no—
410C22×C3—21no—
411C22×C3D4×Dic511yescharacteristic
412C22×C3#408311yescharacteristic
413C22×C3#408511yescharacteristic
414C2×D3—22no—
415C2×D3—41no—
416C2×D3—41no—
417C3×C4—202no—
418C3×C4D4×Dic511yescharacteristic
419C3×C4#408111yescharacteristic
420C3×C4#408511yescharacteristic
421C3×C4C2×C22⋊Dic511yescharacteristic
422Dic3—22no—
423Dic3—42no—
424Dic3—202no—
425Dic3—201no—
426Dic3—201no—
427C2×C5—21no—
428C2×C5—21no—
429C2×C5—122no—
430C2×C5#466511yescharacteristic
431C2×C5#474611yescharacteristic
432C2×C5#475211yescharacteristic
433C23—61no—
434C23D5×D1211yescharacteristic
435Q8—32no—
436Q8—61no—
437Q8—62no—
438Q8—62no—
439Q8—121no—
440Q8—121no—
441D4—32no—
442D4—62no—
443D4—62no—
444D4—61no—
445C2×C4—61no—
446C2×C4—62no—
447C2×C4—62no—
448C2×C4—62no—
449C2×C4—102no—
450C2×C4—121no—
451C2×C4—121no—
452C2×C4—202no—
453C2×C4—302no—
454C2×C4—301no—
455C2×C4—301no—
456C2×C4—601no—
457C2×C4—601no—
458C2×C4—1202no—
459C2×C4#676811yescharacteristic
460C2×C4#677211yescharacteristic
461C2×C4#677711yescharacteristic
462C2×C4C2×D3×Dic511yescharacteristic
463C2×C4D3×C22⋊D511yescharacteristic
464C2×C4D3×C22⋊D511yescharacteristic
465C8—22no—
466C8—201no—
467C2×C3—21no—
468C2×C3—21no—
469C2×C3#6549111yescharacteristic
470C2×C3#6549411yescharacteristic
471C2×C3#6550911yescharacteristic
472D3—42no—
473C5#6900411yescharacteristic, complemented, Sylow
474C22—21no—
475C22—21no—
476C22—62no—
477C22—121no—
478C22—121no—
479C22Dic5×D1211yescharacteristic
480C22#9141611yescharacteristic
481C22#9147511yescharacteristic
482C4—62no—
483C4—122no—
484C4—202no—
485C4—602no—
486C4—601no—
487C4—601no—
488C4#9142311yescharacteristic
489C4#9143511yescharacteristic
490C4#9145311yescharacteristic
491C4D3×C22⋊Dic511yescharacteristic
492C3#1060289311yescharacteristic, complemented, Sylow
493C2—21no—
494C2—21no—
495C2—122no—
496C2#1174824411yescharacteristic
497C2#1174825811yescharacteristic
498C2#1174890111yescharacteristic
subgroupG/H
centerC22#91475
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC2×C3×C5×M16C22
socleC22×C3×C5#120
prime radicalC2×C5×#235C22
cosocleC22×C3×C4×C5C23
hypercenterC2×M16D3×D5
hypocenterC3×C5#1882
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×M16

Complements

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

NH ≅ G/Ntype
3. #11748607C2semidirect
5. #11748409C2semidirect
6. #11746376C2semidirect
7. C3×#64774C2semidirect
15. C5×#909C4semidirect
36. #64774D3semidirect
52. C2×C5×Dic6C2×C4semidirect
57. C3×C5×M16C2×C4semidirect
146. #909Dic5semidirect
179. C5×M16C4×D3semidirect
237. C2×Dic6C2×Dic5semidirect
242. C3×M16C2×Dic5semidirect
394. M16D3×Dic5semidirect
408. C3×C5#1882semidirect
473. C5#69004semidirect
492. C3#10602893semidirect

Maximal subgroups

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

subgroupindex
C2×C5×#9092
#117544922
#117486072
C2×#909782
#117484092
#117463762
C3×#647742
#106028933
#690045