Finite Group Databasebeta

← #14301238

Subgroups — #14301238

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#11756282C211yesmaximal, characteristic
3#11751105C211yesmaximal, characteristic, complemented
4#11751030C211yesmaximal, characteristic, complemented
5C2×#91089C211yesmaximal, characteristic
6C3×C5×#339C211yesmaximal, characteristic, complemented
7#11746375C211yesmaximal, characteristic, complemented
8#10599568—31nomaximal, Hall
9C2×#6817C2211yescharacteristic
10C22×Dic30C2211yescharacteristic
11C2×C3×C5×M16C2211yescharacteristic
12C3×C5×#117C2211yescharacteristic
13C2×C3×C42×C5C2211yescharacteristic
14C2×C4⋊Dic15C2211yescharacteristic
15C2×#6710C2211yescharacteristic
16#91798—22no—
17#91798C414yescomplemented
18C2×Dic60—21no—
19C2×#6706—21no—
20#91781—21no—
21C4⋊C4×D15—21no—
22#91770—22no—
23C4×D60—21no—
24C4×Dic30—21no—
25#91115—22no—
26#91094—21no—
27#91094—21no—
28#91089—22no—
29#68775—51nomaximal, Hall
30C2×#4055—31no—
31#65857—31no—
32#65293—31no—
33#65218—31no—
34C2×#3937—31no—
35C5×#339D311yescharacteristic, complemented
36#64774—31no—
37C22×C3×C4×C5—21no—
38C22×C3×C4×C5C2311yescharacteristic
39C2×#1220—21no—
40C22×Dic15—21no—
41C22×Dic15D412yes—
42#6817—21no—
43#6817—21no—
44#6817C2×C412yes—
45#6817D412yes—
46C2×D60D411yescharacteristic
47C2×C4×D15—21no—
48C2×C4×D15—22no—
49C2×Dic30—21no—
50C2×Dic30—21no—
51C2×Dic30—21no—
52C2×Dic30—21no—
53C2×Dic30—21no—
54C2×Dic30C2×C412yescomplemented
55C2×Dic30D411yescharacteristic
56C3×C5×M16—21no—
57C3×C5×M16C2×C412yescomplemented
58C2×C3×C5×C8—21no—
59C3×C5×C4⋊C4D411yescharacteristic
60C3×C5×C4⋊C4D411yescharacteristic
61C3×C5×C22⋊C4—21no—
62C3×C42×C5—21no—
63C3×C42×C5—21no—
64C3×C42×C5—21no—
65#6714—22no—
66C4⋊Dic15—21no—
67C4⋊Dic15—21no—
68#6712—22no—
69C4×Dic15—22no—
70#6710—21no—
71#6710—21no—
72C2×C3⋊#168—21no—
73Dic60—24no—
74#6706—24no—
75C2×#909—51no—
76#5523—51no—
77#5004—51no—
78#4929—51no—
79C2×#805—51no—
80C3×#339D511yescharacteristic, complemented
81#4554—51no—
82C2×#709—31no—
83C22×Dic10—31no—
84C2×C5×M16C2×D311yescharacteristic
85C5×#117C2×D311yescharacteristic
86C2×C42×C5C2×D311yescharacteristic
87C2×C4⋊Dic5—31no—
88C2×#681—31no—
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×C5—22no—
106C2×C3×C4×C5—21no—
107C2×C3×C4×C5—21no—
108C2×C3×C4×C5—41no—
109C2×C3×C4×C5—41no—
110C2×C3×C4×C5C22×C411yescharacteristic
111C2×C3×C4×C5C2×D411yescharacteristic
112C2×C3×C4×C5C2×D411yescharacteristic
113C2×C3×C4×C5C2×D411yescharacteristic
114C2×C3×C4×C5#4111yescharacteristic
115C2×C3×C4×C5#4111yescharacteristic
116#1220—22no—
117#1220—22no—
118C2×Dic15—21no—
119C2×Dic15—22no—
120C2×Dic15—22no—
121C2×Dic15—41no—
122C2×Dic15—41no—
123C2×Dic15—42no—
124D60—21no—
125D60C22⋊C412yes—
126C4×D15—22no—
127C4×D15—22no—
128C4×D15—82no—
129Dic30—21no—
130Dic30—22no—
131Dic30—22no—
132Dic30—41no—
133Dic30—41no—
134Dic30C22⋊C412yes—
135C3×C5×C8—22no—
136C3⋊#168—41no—
137C2×#235—51no—
138C22×Dic6—51no—
139C2×C3×M16C2×D511yescharacteristic
140C3×#117C2×D511yescharacteristic
141C2×C3×C42C2×D511yescharacteristic
142C2×C4⋊Dic3—51no—
143C2×#208—51no—
144#909—54no—
145#909—102no—
146C2×Dic12—101no—
147C2×#204—101no—
148#892—101no—
149D3×C4⋊C4—101no—
150#67⋊C4—102no—
151C4×D12—101no—
152C4×Dic6—101no—
153C3⋊#95—102no—
154#810—101no—
155#810—101no—
156#805—102no—
157C22×C4×C5—21no—
158C22×C4×C5C22×D311yescharacteristic
159C2×C22⋊D5—61no—
160C22×Dic5—32no—
161C22×Dic5—61no—
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⋊C4D1211yescharacteristic
180C5×C4⋊C4D1211yescharacteristic
181C5×C22⋊C4—21no—
182C42×C5—21no—
183C42×C5—21no—
184C42×C5—21no—
185#685—62no—
186C4⋊Dic5—61no—
187C4⋊Dic5—61no—
188#683—62no—
189C4×Dic5—62no—
190#681—61no—
191#681—61no—
192C2×#168—61no—
193Dic20—64no—
194#677—64no—
195C2×#137—151no—
196#520—151no—
197C2×#104—151no—
198#419—151no—
199#391—151no—
200#343—151no—
201#339D1511yescharacteristic, complemented
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×C5—42no—
211C3×C4×C5—41no—
212C3×C4×C5—41no—
213C3×C4×C5C2×C22⋊C411yescharacteristic
214C3×C4×C5C4×D411yescharacteristic
215C3×C4×C5#12111yescharacteristic
216C3×C4×C5#12311yescharacteristic
217Dic15—22no—
218Dic15—42no—
219Dic15—42no—
220C22×C3×C4—21no—
221C22×C3×C4C22×D511yescharacteristic
222C2×#67—101no—
223C22×Dic3—52no—
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×M16C4×D512yescomplemented
241C2×C3×C8—21no—
242C3×C4⋊C4D2011yescharacteristic
243C3×C4⋊C4D2011yescharacteristic
244C3×C22⋊C4—21no—
245C3×C42—21no—
246C3×C42—21no—
247C3×C42—21no—
248(C2×D3)⋊C4—102no—
249C4⋊Dic3—101no—
250C4⋊Dic3—101no—
251Dic3⋊C4—102no—
252C4×Dic3—102no—
253#208—101no—
254#208—101no—
255C2×C3⋊C8—101no—
256Dic12—104no—
257#204—104no—
258C23×C5D3×D411yescharacteristic
259C22×D5—61no—
260C2×C4×C5—22no—
261C2×C4×C5—21no—
262C2×C4×C5—21no—
263C2×C4×C5—41no—
264C2×C4×C5—41no—
265C2×C4×C5C2×C4×D311yescharacteristic
266C2×C4×C5C2×D1211yescharacteristic
267C2×C4×C5#23511yescharacteristic
268C2×C4×C5D3×D411yescharacteristic
269C2×C4×C5D3×D411yescharacteristic
270C2×C4×C5#23911yescharacteristic
271C22⋊D5—62no—
272C22⋊D5—62no—
273C2×Dic5—61no—
274C2×Dic5—62no—
275C2×Dic5—62no—
276C2×Dic5—121no—
277C2×Dic5—121no—
278C2×Dic5—122no—
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—121no—
292C2×#41—151no—
293C22×Q8—151no—
294#137—154no—
295#137—302no—
296C2×Q16—301no—
297C2×SD16—301no—
298C2×M16—151no—
299C2×M16C2×D1511yescharacteristic
300C4×Q8—301no—
301C4×D4—301no—
302C4×D4—302no—
303#117—301no—
304#117C2×D1511yescharacteristic
305C2×C4⋊C4—151no—
306C2×C4⋊C4—301no—
307C2×C42C2×D1511yescharacteristic
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—22no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—41no—
324C2×C3×C4—41no—
325C2×C3×C4C2×C4×D511yescharacteristic
326C2×C3×C4C2×D2011yescharacteristic
327C2×C3×C4#70911yescharacteristic
328C2×C3×C4D4×D511yescharacteristic
329C2×C3×C4D4×D511yescharacteristic
330C2×C3×C4#71311yescharacteristic
331#67—102no—
332#67—102no—
333C2×Dic3—101no—
334C2×Dic3—102no—
335C2×Dic3—102no—
336C2×Dic3—201no—
337C2×Dic3—201no—
338C2×Dic3—202no—
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×C5#88211yescharacteristic
355C22×C5#88411yescharacteristic
356C22×C5#89311yescharacteristic
357C2×D5—62no—
358C2×D5—121no—
359C2×D5—121no—
360C4×C5—42no—
361C4×C5—41no—
362C4×C5—41no—
363C4×C5C4×D1211yescharacteristic
364C4×C5D3×C22⋊C411yescharacteristic
365C4×C5#89411yescharacteristic
366C4×C5#89511yescharacteristic
367Dic5—62no—
368Dic5—122no—
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—21no—
384C22×C4—152no—
385C22×C4—302no—
386C22×C4—301no—
387C22×C4—301no—
388C22×C4C22×D1511yescharacteristic
389Q16—304no—
390SD16—304no—
391M16—21no—
392M16—301no—
393M16—301no—
394M16C4×D1512yescomplemented
395C2×C8—21no—
396C2×C8—301no—
397C4⋊C4—302no—
398C4⋊C4—301no—
399C4⋊C4—301no—
400C4⋊C4D6011yescharacteristic
401C4⋊C4D6011yescharacteristic
402C22⋊C4—21no—
403C22⋊C4—302no—
404C42—21no—
405C42—21no—
406C42—21no—
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—42no—
418C3×C4—41no—
419C3×C4—41no—
420C3×C4C4×D2011yescharacteristic
421C3×C4D5×C22⋊C411yescharacteristic
422C3×C4#404011yescharacteristic
423C3×C4#404111yescharacteristic
424Dic3—102no—
425Dic3—202no—
426Dic3—202no—
427C2×C5—21no—
428C2×C5—21no—
429C2×C5#466511yescharacteristic
430C2×C5#482511yescharacteristic
431C2×C5#482611yescharacteristic
432D5—122no—
433C23—301no—
434C23D4×D1511yescharacteristic
435Q8—152no—
436Q8—301no—
437Q8—302no—
438Q8—302no—
439Q8—601no—
440Q8—601no—
441D4—152no—
442D4—302no—
443D4—302no—
444D4—301no—
445C2×C4—22no—
446C2×C4—21no—
447C2×C4—21no—
448C2×C4—41no—
449C2×C4—41no—
450C2×C4—301no—
451C2×C4—302no—
452C2×C4—302no—
453C2×C4—302no—
454C2×C4—302no—
455C2×C4—601no—
456C2×C4—601no—
457C2×C4—602no—
458C2×C4—1202no—
459C2×C4C2×C4×D1511yescharacteristic
460C2×C4C2×D6011yescharacteristic
461C2×C4#681711yescharacteristic
462C2×C4D4×D1511yescharacteristic
463C2×C4D4×D1511yescharacteristic
464C2×C4#682111yescharacteristic
465C8—22no—
466C8—601no—
467C2×C3—21no—
468C2×C3—21no—
469C2×C3#6495411yescharacteristic
470C2×C3#6511411yescharacteristic
471C2×C3#6511511yescharacteristic
472D3—202no—
473C5#6877511yescharacteristic, complemented, Sylow
474C22—21no—
475C22—21no—
476C22—302no—
477C22—601no—
478C22—601no—
479C22#9177111yescharacteristic
480C22#9177311yescharacteristic
481C22#9178211yescharacteristic
482C4—42no—
483C4—41no—
484C4—41no—
485C4—302no—
486C4—602no—
487C4—602no—
488C4C4×D6011yescharacteristic
489C4C22⋊C4×D1511yescharacteristic
490C4#9178311yescharacteristic
491C4#9178411yescharacteristic
492C3#1059956811yescharacteristic, complemented, Sylow
493C2—21no—
494C2—21no—
495C2—602no—
496C2#1175076611yescharacteristic
497C2#1175092611yescharacteristic
498C2#1175092711yescharacteristic
subgroupG/H
centerC22#91771
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C22×D15
FittingC3×C5×#339C2
socleC22×C3×C5#120
prime radicalC2×#6817C22
cosocleC22×C3×C4×C5C23
hypercenter#339D15
hypocenterC3×C5#1882
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
3. #11751105C2semidirect
4. #11751030C2semidirect
6. C3×C5×#339C2semidirect
7. #11746375C2semidirect
17. #91798C4semidirect
35. C5×#339D3semidirect
54. C2×Dic30C2×C4semidirect
57. C3×C5×M16C2×C4semidirect
80. C3×#339D5semidirect
177. C5×M16C4×D3semidirect
201. #339D15semidirect
240. C3×M16C4×D5semidirect
394. M16C4×D15semidirect
408. C3×C5#1882semidirect
473. C5#68775semidirect
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
#117562822
#117511052
#117510302
C2×#910892
C3×C5×#3392
#117463752
#105995683
#687755