Finite Group Databasebeta

← #14301284

Subgroups — #14301284

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×#91798C211yesmaximal, characteristic
2#11754496C211yesmaximal, characteristic
3#11748599C211yesmaximal, characteristic, complemented
4C2×#90980C211yesmaximal, characteristic
5#11748417C211yesmaximal, characteristic, complemented
6C5×#4555C211yesmaximal, characteristic, complemented
7C3×#64774C211yesmaximal, characteristic, complemented
8#10599568—31nomaximal, Hall
9C2×#6817C2211yescharacteristic
10C22×Dic30C2211yescharacteristic
11C2×C3×C5×M16C2211yescharacteristic
12#91798—22no—
13#91798C414yescomplemented
14C2×Dic60—21no—
15C2×#6706—21no—
16C5×#926C2211yescharacteristic
17C2×C4×C5×Dic3C2211yescharacteristic
18C2×C3×C4⋊Dic5C2211yescharacteristic
19C2×C3×#681C2211yescharacteristic
20#91552—22no—
21#91437—21no—
22#91428—21no—
23#91347—21no—
24#91340—21no—
25#90994—22no—
26#90980—22no—
27#90975—21no—
28#90975—21no—
29#69004—51nomaximal, Hall
30C2×#4055—31no—
31#65857—31no—
32#65293—31no—
33#65218—31no—
34C2×#3937—31no—
35C5×#339—31no—
36#64774D311yescharacteristic, complemented
37C22×C3×C4×C5C2311yescharacteristic
38C2×#1220—21no—
39C22×Dic15—21no—
40#6817—21no—
41#6817—21no—
42#6817C2×C412yes—
43#6817D412yes—
44C2×D60D411yescharacteristic
45C2×C4×D15—21no—
46C2×Dic30—21no—
47C2×Dic30—21no—
48C2×Dic30—21no—
49C2×Dic30—21no—
50C2×Dic30—21no—
51C2×Dic30C2×C412yescomplemented
52C2×Dic30D411yescharacteristic
53C22×C5×Dic3—21no—
54C22×C3×Dic5D412yes—
55C2×#1200—22no—
56C3×C5×M16—21no—
57C3×C5×M16C2×C412yescomplemented
58C2×C3×C5×C8—21no—
59Dic60—24no—
60#6706—24no—
61C5×#217—21no—
62C5×C4⋊Dic3D411yescharacteristic
63C5×C4⋊Dic3D411yescharacteristic
64C4×C5×Dic3—21no—
65C4×C5×Dic3—21no—
66C4×C5×Dic3—21no—
67C3×C4⋊Dic5—21no—
68C3×C4⋊Dic5—21no—
69C3×#681—21no—
70C3×#681—21no—
71C2×C3×#168—21no—
72#6669—22no—
73#6667—22no—
74Dic3×Dic5—22no—
75C2×#909—51no—
76#5504—51no—
77C2×#825—51no—
78#5120—51no—
79#4973—51no—
80C3×#343—51no—
81#4555D511yescharacteristic, complemented
82C2×#709—31no—
83C22×Dic10—31no—
84C2×C5×M16C2×D311yescharacteristic
85C5×#117—31no—
86C2×C42×C5—31no—
87C2×C4⋊Dic5C2×D311yescharacteristic
88C2×#681C2×D311yescharacteristic
89#4055—34no—
90#4055—62no—
91C2×Dic20—61no—
92C2×#677—61no—
93#4038—61no—
94D5×C4⋊C4—61no—
95#4027—62no—
96C4×D20—61no—
97C4×Dic10—61no—
98#3963—62no—
99#3942—61no—
100#3942—61no—
101#3937—62no—
102#1882—151noSylow
103C23×C3×C5C2×D411yescharacteristic
104C22×D15—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
111#1220—22no—
112#1220—22no—
113C2×Dic15—21no—
114C2×Dic15—22no—
115C2×Dic15—41no—
116C2×Dic15—41no—
117D60—21no—
118D60C22⋊C412yes—
119C4×D15—22no—
120C4×D15—22no—
121Dic30—21no—
122Dic30—22no—
123Dic30—22no—
124Dic30—41no—
125Dic30—41no—
126Dic30C22⋊C412yes—
127C2×C5×Dic3—22no—
128C2×C5×Dic3—21no—
129C2×C5×Dic3—21no—
130C2×C5×Dic3—41no—
131C2×C5×Dic3—41no—
132C2×C3×Dic5—22no—
133C2×C3×Dic5—42no—
134#1200—82no—
135C3×C5×C8—22no—
136C3×#168—41no—
137C2×#235—51no—
138C22×Dic6—51no—
139C2×C3×M16—51no—
140C2×C3×M16C2×D511yescharacteristic
141C2×C3×C4⋊C4—51no—
142C4×#67—102no—
143#926C2×D511yescharacteristic
144C2×C4×Dic3C2×D511yescharacteristic
145#909—54no—
146#909—102no—
147C2×Dic12—101no—
148C2×#204—101no—
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—
157C22×C4×C5—61no—
158C22×C4×C5C22×D311yescharacteristic
159C2×C22⋊D5—61no—
160C22×Dic5—61no—
161C22×Dic5D1212yes—
162#709—32no—
163#709—32no—
164#709—61no—
165#709—61no—
166C2×D20—31no—
167C2×C4×D5—61no—
168C2×C4×D5—62no—
169C2×Dic10—31no—
170C2×Dic10—32no—
171C2×Dic10—61no—
172C2×Dic10—61no—
173C2×Dic10—61no—
174C2×Dic10—61no—
175C2×Dic10—61no—
176C5×M16—21no—
177C5×M16C4×D312yescomplemented
178C2×C5×C8—21no—
179C5×C4⋊C4—31no—
180C5×C4⋊C4—31no—
181C5×C22⋊C4—61no—
182C42×C5—61no—
183C42×C5—61no—
184C42×C5—61no—
185#685—62no—
186C4⋊Dic5—21no—
187C4⋊Dic5—21no—
188#683—62no—
189C4×Dic5—62no—
190#681—21no—
191#681—21no—
192C2×#168—21no—
193Dic20—64no—
194#677—64no—
195C2×#137—151no—
196#520—151no—
197C2×#104—151no—
198#419—151no—
199#391—151no—
200#343—51no—
201#339—31no—
202C22×C3×C5—21no—
203C22×C3×C5—21no—
204C22×C3×C5C4×D411yescharacteristic
205C22×C3×C5#12011yescharacteristic
206C22×C3×C5#12111yescharacteristic
207C2×D15—22no—
208C2×D15—41no—
209C2×D15—41no—
210C3×C4×C5C2×C22⋊C411yescharacteristic
211C3×C4×C5C4×D411yescharacteristic
212C3×C4×C5#12111yescharacteristic
213C3×C4×C5#12311yescharacteristic
214Dic15—22no—
215Dic15—42no—
216C3×Dic5—42no—
217C5×Dic3—42no—
218C5×Dic3—41no—
219C5×Dic3—41no—
220C22×C3×C4—52no—
221C22×C3×C4C22×D511yescharacteristic
222C2×#67—101no—
223C22×Dic3—21no—
224C22×Dic3—101no—
225#235—52no—
226#235—52no—
227#235—101no—
228#235—101no—
229C2×D12—51no—
230C2×C4×D3—101no—
231C2×C4×D3—102no—
232C2×Dic6—51no—
233C2×Dic6—52no—
234C2×Dic6—101no—
235C2×Dic6—101no—
236C2×Dic6—101no—
237C2×Dic6—101no—
238C2×Dic6—101no—
239C3×M16—21no—
240C3×M16—101no—
241C3×M16—101no—
242C3×M16C4×D512yescomplemented
243C2×C3×C8—21no—
244C2×C3×C8—101no—
245C3×C4⋊C4—101no—
246C3×C4⋊C4—101no—
247#217—21no—
248(C2×D3)⋊C4—102no—
249C4⋊Dic3D2011yescharacteristic
250C4⋊Dic3D2011yescharacteristic
251Dic3⋊C4—102no—
252C4×Dic3—21no—
253C4×Dic3—21no—
254C4×Dic3—21no—
255C4×Dic3—102no—
256Dic12—104no—
257#204—104no—
258C23×C5C2×D1211yescharacteristic
259C22×D5—61no—
260C2×C4×C5—62no—
261C2×C4×C5—61no—
262C2×C4×C5—61no—
263C2×C4×C5—121no—
264C2×C4×C5—121no—
265C2×C4×C5C2×C4×D311yescharacteristic
266C2×C4×C5#23511yescharacteristic
267C2×C4×C5D3×D411yescharacteristic
268C2×C4×C5D3×D411yescharacteristic
269C2×C4×C5D3×D411yescharacteristic
270C2×C4×C5#23911yescharacteristic
271C22⋊D5—62no—
272C22⋊D5—62no—
273C2×Dic5—22no—
274C2×Dic5—42no—
275C2×Dic5—61no—
276C2×Dic5—62no—
277C2×Dic5—121no—
278C2×Dic5—121no—
279D20—32no—
280D20—61no—
281C4×D5—62no—
282C4×D5—62no—
283C4×D5—242no—
284Dic10—32no—
285Dic10—61no—
286Dic10—62no—
287Dic10—62no—
288Dic10—121no—
289Dic10—121no—
290C5×C8—22no—
291#168—41no—
292C2×#41—151no—
293C22×Q8—151no—
294#137—154no—
295#137—302no—
296C2×Q16—301no—
297C2×SD16—301no—
298C2×M16—51no—
299C2×M16D3×D511yescharacteristic
300C4×Q8—301no—
301C4×D4—301no—
302C4×D4—302no—
303#117—31no—
304#117—301no—
305C2×C4⋊C4—51no—
306C2×C4⋊C4—301no—
307C2×C42—31no—
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
317D15—42no—
318C23×C3D4×D511yescharacteristic
319C22×D3—101no—
320C2×C3×C4—102no—
321C2×C3×C4—202no—
322C2×C3×C4C2×C4×D511yescharacteristic
323C2×C3×C4C2×D2011yescharacteristic
324C2×C3×C4#70911yescharacteristic
325C2×C3×C4D4×D511yescharacteristic
326C2×C3×C4D4×D511yescharacteristic
327C2×C3×C4#71311yescharacteristic
328#67—102no—
329#67—102no—
330C2×Dic3—22no—
331C2×Dic3—21no—
332C2×Dic3—21no—
333C2×Dic3—41no—
334C2×Dic3—41no—
335C2×Dic3—101no—
336C2×Dic3—102no—
337C2×Dic3—201no—
338C2×Dic3—201no—
339D12—52no—
340D12—101no—
341C4×D3—102no—
342C4×D3—102no—
343C4×D3—402no—
344Dic6—52no—
345Dic6—101no—
346Dic6—102no—
347Dic6—102no—
348Dic6—201no—
349Dic6—201no—
350C3×C8—22no—
351C3×C8—201no—
352C22×C5—21no—
353C22×C5—21no—
354C22×C5C4×D1211yescharacteristic
355C22×C5#88211yescharacteristic
356C22×C5#89511yescharacteristic
357C2×D5—62no—
358C2×D5—121no—
359C2×D5—121no—
360C4×C5—122no—
361C4×C5—121no—
362C4×C5—121no—
363C4×C5D3×C22⋊C411yescharacteristic
364C4×C5#88411yescharacteristic
365C4×C5#89311yescharacteristic
366C4×C5#89411yescharacteristic
367Dic5—42no—
368Dic5—62no—
369Dic5—122no—
370#41—152no—
371#41—152no—
372#41—301no—
373#41—301no—
374C2×Q8—152no—
375C2×Q8—151no—
376C2×Q8—301no—
377C2×Q8—301no—
378C2×Q8—301no—
379C2×Q8—301no—
380C2×Q8—301no—
381C2×D4—151no—
382C2×D4—301no—
383C22×C4—52no—
384C22×C4—61no—
385C22×C4—302no—
386C22×C4—301no—
387C22×C4—301no—
388C22×C4C2×D3×D511yescharacteristic
389Q16—304no—
390SD16—304no—
391M16—21no—
392M16—101no—
393M16—101no—
394M16#120012yescomplemented
395C2×C8—21no—
396C2×C8—101no—
397C4⋊C4—31no—
398C4⋊C4—31no—
399C4⋊C4—101no—
400C4⋊C4—101no—
401C4⋊C4—302no—
402C22⋊C4—61no—
403C22⋊C4—302no—
404C42—61no—
405C42—61no—
406C42—61no—
407C42—302no—
408C3×C5#188211yescharacteristic, complemented, Hall
409C22×C3—21no—
410C22×C3—21no—
411C22×C3#402811yescharacteristic
412C22×C3#403011yescharacteristic
413C22×C3#403911yescharacteristic
414C2×D3—102no—
415C2×D3—201no—
416C2×D3—201no—
417C3×C4—202no—
418C3×C4C4×D2011yescharacteristic
419C3×C4D5×C22⋊C411yescharacteristic
420C3×C4#404011yescharacteristic
421C3×C4#404111yescharacteristic
422Dic3—42no—
423Dic3—41no—
424Dic3—41no—
425Dic3—102no—
426Dic3—202no—
427C2×C5—21no—
428C2×C5—21no—
429C2×C5#466511yescharacteristic
430C2×C5#474611yescharacteristic
431C2×C5#475211yescharacteristic
432D5—122no—
433C23—301no—
434C23D5×D1211yescharacteristic
435Q8—152no—
436Q8—301no—
437Q8—302no—
438Q8—302no—
439Q8—601no—
440Q8—601no—
441D4—152no—
442D4—302no—
443D4—302no—
444D4—301no—
445C2×C4—62no—
446C2×C4—61no—
447C2×C4—61no—
448C2×C4—102no—
449C2×C4—121no—
450C2×C4—121no—
451C2×C4—202no—
452C2×C4—301no—
453C2×C4—302no—
454C2×C4—302no—
455C2×C4—302no—
456C2×C4—601no—
457C2×C4—601no—
458C2×C4—1202no—
459C2×C4#676911yescharacteristic
460C2×C4#677311yescharacteristic
461C2×C4D3×D2011yescharacteristic
462C2×C4C2×#120011yescharacteristic
463C2×C4#679011yescharacteristic
464C2×C4#679011yescharacteristic
465C8—22no—
466C8—201no—
467C2×C3—21no—
468C2×C3—21no—
469C2×C3#6495411yescharacteristic
470C2×C3#6511411yescharacteristic
471C2×C3#6511511yescharacteristic
472D3—202no—
473C5#6900411yescharacteristic, complemented, Sylow
474C22—21no—
475C22—21no—
476C22—302no—
477C22—601no—
478C22—601no—
479C22#9141811yescharacteristic
480C22#9142011yescharacteristic
481C22#9147611yescharacteristic
482C4—122no—
483C4—121no—
484C4—121no—
485C4—202no—
486C4—302no—
487C4—602no—
488C4#9142811yescharacteristic
489C4#9143311yescharacteristic
490C4#9145911yescharacteristic
491C4#9156111yescharacteristic
492C3#1059956811yescharacteristic, complemented, Sylow
493C2—21no—
494C2—21no—
495C2—602no—
496C2#1174825011yescharacteristic
497C2#1174826211yescharacteristic
498C2#1174890411yescharacteristic
subgroupG/H
centerC22#91476
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C2×D3×D5
FittingC2×C3×C5×M16C22
socleC22×C3×C5#120
prime radicalC2×#6817C22
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. #11748599C2semidirect
5. #11748417C2semidirect
6. C5×#4555C2semidirect
7. C3×#64774C2semidirect
13. #91798C4semidirect
36. #64774D3semidirect
51. C2×Dic30C2×C4semidirect
57. C3×C5×M16C2×C4semidirect
81. #4555D5semidirect
177. C5×M16C4×D3semidirect
242. C3×M16C4×D5semidirect
394. M16#1200semidirect
408. C3×C5#1882semidirect
473. C5#69004semidirect
492. C3#10599568semidirect

Maximal subgroups

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

subgroupindex
C2×#917982
#117544962
#117485992
C2×#909802
#117484172
C5×#45552
C3×#647742
#105995683
#690045