Finite Group Databasebeta

← #14301235

Subgroups — #14301235

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
1C3×C5×#520C211yesmaximal, characteristic
2C2×#91840C211yesmaximal, characteristic
3C2×#91121C211yesmaximal, characteristic
4#11751105C211yesmaximal, characteristic, complemented
5#11751074C211yesmaximal, characteristic, complemented
6#11746353C211yesmaximal, characteristic, complemented
7#11746351C211yesmaximal, characteristic, complemented
8#10602459—31nomaximal, Hall
9C2×C3×C5×#41C2211yescharacteristic
10C22×Dic30C2211yescharacteristic
11C3×C4×C5×Q8—21no—
12C3×C4×C5×D4—21no—
13C3×C4×C5×D4—22no—
14C3×C5×#117—21no—
15C3×C5×#117C2211yescharacteristic
16C2×C3×C5×C4⋊C4—21no—
17C2×C3×C5×C4⋊C4C2211yescharacteristic
18#91840—22no—
19#91840C414yescomplemented
20C2×#6718—21no—
21C2×#6716—21no—
22C2×C4×Dic15C2211yescharacteristic
23C2×#6710C2211yescharacteristic
24C2×#6710C2211yescharacteristic
25#91121—22no—
26#91115—22no—
27#91094—21no—
28#91094—21no—
29#71666—51nomaximal, Hall
30C5×#520D311yescharacteristic
31C2×#4097—31no—
32C2×#3969—31no—
33#65293—31no—
34#65263—31no—
35#64752—31no—
36#64750—31no—
37C3×C5×#41—21no—
38C3×C5×#41—21no—
39C3×C5×#41C2×C412yes—
40C3×C5×#41D412yes—
41C2×C3×C5×Q8D411yescharacteristic
42C2×C3×C5×D4—21no—
43C2×C3×C5×D4D411yescharacteristic
44C22×C3×C4×C5—21no—
45C22×C3×C4×C5—22no—
46C22×C3×C4×C5D412yes—
47C22×C3×C4×C5C2311yescharacteristic
48C22×Dic15—21no—
49C22×Dic15—21no—
50C2×Dic30—21no—
51C2×Dic30—21no—
52C2×Dic30—21no—
53C2×Dic30—21no—
54C2×Dic30—21no—
55C2×Dic30C2×C412yescomplemented
56C3×C5×C4⋊C4—22no—
57C3×C5×C4⋊C4—21no—
58C3×C5×C4⋊C4—21no—
59C3×C5×C4⋊C4D411yescharacteristic
60C3×C5×C4⋊C4D411yescharacteristic
61C3×C5×C22⋊C4—22no—
62C3×C5×C22⋊C4—21no—
63C3×C42×C5—22no—
64C3×C42×C5—21no—
65#6718—24no—
66#6716—24no—
67C4×Dic15—21no—
68C4×Dic15—21no—
69#6710—21no—
70#6710—21no—
71#6710—21no—
72#6710C2×C412yescomplemented
73C2×C3⋊#168—21no—
74C2×C3⋊#168—21no—
75C3×#520D511yescharacteristic
76C2×#951—51no—
77C2×#837—51no—
78#5004—51no—
79#4973—51no—
80#4532—51no—
81#4530—51no—
82C2×C5×#41C2×D311yescharacteristic
83C22×Dic10—31no—
84C4×C5×Q8—21no—
85C4×C5×D4—21no—
86C4×C5×D4—22no—
87C5×#117—21no—
88C5×#117C2×D311yescharacteristic
89C2×C5×C4⋊C4—21no—
90C2×C5×C4⋊C4C2×D311yescharacteristic
91#4097—34no—
92#4097—62no—
93C2×#689—61no—
94C2×#687—61no—
95C2×C4×Dic5—31no—
96C2×#681—31no—
97C2×#681—31no—
98#3969—62no—
99#3963—62no—
100#3942—61no—
101#3942—61no—
102#1882—151noSylow
103C23×C3×C5—21no—
104C23×C3×C5C2×D411yescharacteristic
105C3×C5×Q8—21no—
106C3×C5×Q8C22⋊C412yes—
107C3×C5×D4—22no—
108C3×C5×D4—22no—
109C3×C5×D4—21no—
110C3×C5×D4C22⋊C412yes—
111C2×C3×C4×C5—21no—
112C2×C3×C4×C5—22no—
113C2×C3×C4×C5—22no—
114C2×C3×C4×C5—21no—
115C2×C3×C4×C5—22no—
116C2×C3×C4×C5—21no—
117C2×C3×C4×C5—42no—
118C2×C3×C4×C5—82no—
119C2×C3×C4×C5C22×C411yescharacteristic
120C2×C3×C4×C5C2×D411yescharacteristic
121C2×C3×C4×C5C2×D411yescharacteristic
122C2×C3×C4×C5C2×D411yescharacteristic
123C2×C3×C4×C5#4111yescharacteristic
124C2×C3×C4×C5#4111yescharacteristic
125C2×Dic15—22no—
126C2×Dic15—22no—
127C2×Dic15—41no—
128C2×Dic15—41no—
129C2×Dic15—41no—
130C2×Dic15—41no—
131Dic30—22no—
132Dic30—22no—
133Dic30—41no—
134Dic30—41no—
135C3⋊#168—22no—
136C3⋊#168—41no—
137C2×C3×#41C2×D511yescharacteristic
138C22×Dic6—51no—
139C3×C4×Q8—21no—
140C3×C4×D4—21no—
141C3×C4×D4—22no—
142C3×#117—21no—
143C3×#117C2×D511yescharacteristic
144C2×C3×C4⋊C4—21no—
145C2×C3×C4⋊C4C2×D511yescharacteristic
146#951—54no—
147#951—102no—
148C2×#216—101no—
149C2×#214—101no—
150C2×C4×Dic3—51no—
151C2×#208—51no—
152C2×#208—51no—
153#837—102no—
154C3⋊#95—102no—
155#810—101no—
156#810—101no—
157C5×#41—21no—
158C5×#41—21no—
159C5×#41C4×D312yes—
160C5×#41D1212yes—
161C2×C5×Q8#6711yescharacteristic
162C2×C5×D4—21no—
163C2×C5×D4#6711yescharacteristic
164C22×C4×C5—21no—
165C22×C4×C5—22no—
166C22×C4×C5#6712yes—
167C22×C4×C5C22×D311yescharacteristic
168C22×Dic5—61no—
169C22×Dic5—61no—
170C2×Dic10—32no—
171C2×Dic10—61no—
172C2×Dic10—61no—
173C2×Dic10—61no—
174C2×Dic10—61no—
175C2×Dic10—61no—
176C5×C4⋊C4—22no—
177C5×C4⋊C4—21no—
178C5×C4⋊C4—21no—
179C5×C4⋊C4#6711yescharacteristic
180C5×C4⋊C4#6711yescharacteristic
181C5×C22⋊C4—22no—
182C5×C22⋊C4—21no—
183C42×C5—22no—
184C42×C5—21no—
185#689—64no—
186#687—64no—
187C4×Dic5—61no—
188C4×Dic5—61no—
189#681—32no—
190#681—61no—
191#681—61no—
192#681—61no—
193C2×#168—61no—
194C2×#168—61no—
195C2×#137—151no—
196#520D1511yescharacteristic
197C2×#104—151no—
198#419—151no—
199#391—151no—
200#343—151no—
201#339—151no—
202C22×C3×C5—21no—
203C22×C3×C5—21no—
204C22×C3×C5—22no—
205C22×C3×C5—41no—
206C22×C3×C5—41no—
207C22×C3×C5C4×D411yescharacteristic
208C22×C3×C5#12011yescharacteristic
209C22×C3×C5#12111yescharacteristic
210C3×C4×C5—22no—
211C3×C4×C5—42no—
212C3×C4×C5—41no—
213C3×C4×C5—41no—
214C3×C4×C5C2×C22⋊C411yescharacteristic
215C3×C4×C5C4×D411yescharacteristic
216C3×C4×C5#12111yescharacteristic
217C3×C4×C5#12311yescharacteristic
218Dic15—42no—
219Dic15—42no—
220C3×#41—21no—
221C3×#41—21no—
222C3×#41C4×D512yes—
223C3×#41D2012yes—
224C2×C3×Q8C22⋊D511yescharacteristic
225C2×C3×D4—21no—
226C2×C3×D4C22⋊D511yescharacteristic
227C22×C3×C4—21no—
228C22×C3×C4—22no—
229C22×C3×C4C22⋊D512yes—
230C22×C3×C4C22×D511yescharacteristic
231C22×Dic3—101no—
232C22×Dic3—101no—
233C2×Dic6—52no—
234C2×Dic6—101no—
235C2×Dic6—101no—
236C2×Dic6—101no—
237C2×Dic6—101no—
238C2×Dic6—101no—
239C3×C4⋊C4—22no—
240C3×C4⋊C4—21no—
241C3×C4⋊C4—21no—
242C3×C4⋊C4C22⋊D511yescharacteristic
243C3×C4⋊C4C22⋊D511yescharacteristic
244C3×C22⋊C4—22no—
245C3×C22⋊C4—21no—
246C3×C42—22no—
247C3×C42—21no—
248#216—104no—
249#214—104no—
250C4×Dic3—101no—
251C4×Dic3—101no—
252#208—52no—
253#208—101no—
254#208—101no—
255#208—101no—
256C2×C3⋊C8—101no—
257C2×C3⋊C8—101no—
258C23×C5—21no—
259C23×C5C2×#6711yescharacteristic
260C5×Q8—21no—
261C5×Q8(C2×D3)⋊C412yes—
262C5×D4—22no—
263C5×D4—22no—
264C5×D4—21no—
265C5×D4(C2×D3)⋊C412yes—
266C2×C4×C5—21no—
267C2×C4×C5—22no—
268C2×C4×C5—22no—
269C2×C4×C5—21no—
270C2×C4×C5—22no—
271C2×C4×C5—21no—
272C2×C4×C5—42no—
273C2×C4×C5—82no—
274C2×C4×C5C2×C4×D311yescharacteristic
275C2×C4×C5C2×D1211yescharacteristic
276C2×C4×C5#23511yescharacteristic
277C2×C4×C5#23511yescharacteristic
278C2×C4×C5C2×#6711yescharacteristic
279C2×C4×C5C2×#6711yescharacteristic
280C2×Dic5—62no—
281C2×Dic5—62no—
282C2×Dic5—121no—
283C2×Dic5—121no—
284C2×Dic5—121no—
285C2×Dic5—121no—
286Dic10—62no—
287Dic10—62no—
288Dic10—121no—
289Dic10—121no—
290#168—62no—
291#168—121no—
292C2×#41C2×D1511yescharacteristic
293C22×Q8—151no—
294#137—154no—
295#137—302no—
296C2×Q16—301no—
297C2×SD16—301no—
298C2×M16—151no—
299C2×M16—151no—
300C4×Q8—21no—
301C4×D4—21no—
302C4×D4—22no—
303#117—21no—
304#117C2×D1511yescharacteristic
305C2×C4⋊C4—21no—
306C2×C4⋊C4C2×D1511yescharacteristic
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—42no—
315C2×C3×C5#38611yescharacteristic
316C2×C3×C5#45511yescharacteristic
317C2×C3×C5#45611yescharacteristic
318C23×C3—21no—
319C23×C3C2×C22⋊D511yescharacteristic
320C3×Q8—21no—
321C3×Q8#68512yes—
322C3×D4—22no—
323C3×D4—22no—
324C3×D4—21no—
325C3×D4#68512yes—
326C2×C3×C4—21no—
327C2×C3×C4—22no—
328C2×C3×C4—22no—
329C2×C3×C4—21no—
330C2×C3×C4—22no—
331C2×C3×C4—21no—
332C2×C3×C4—42no—
333C2×C3×C4—82no—
334C2×C3×C4C2×C4×D511yescharacteristic
335C2×C3×C4C2×D2011yescharacteristic
336C2×C3×C4#70911yescharacteristic
337C2×C3×C4#70911yescharacteristic
338C2×C3×C4C2×C22⋊D511yescharacteristic
339C2×C3×C4C2×C22⋊D511yescharacteristic
340C2×Dic3—102no—
341C2×Dic3—102no—
342C2×Dic3—201no—
343C2×Dic3—201no—
344C2×Dic3—201no—
345C2×Dic3—201no—
346Dic6—102no—
347Dic6—102no—
348Dic6—201no—
349Dic6—201no—
350C3⋊C8—102no—
351C3⋊C8—201no—
352C22×C5—21no—
353C22×C5—21no—
354C22×C5—22no—
355C22×C5—41no—
356C22×C5—41no—
357C22×C5C4×#6711yescharacteristic
358C22×C5#93011yescharacteristic
359C22×C5#95311yescharacteristic
360C4×C5—22no—
361C4×C5—42no—
362C4×C5—41no—
363C4×C5—41no—
364C4×C5C2×(C2×D3)⋊C411yescharacteristic
365C4×C5C4×#6711yescharacteristic
366C4×C5#92911yescharacteristic
367C4×C5#93011yescharacteristic
368Dic5—122no—
369Dic5—122no—
370#41—21no—
371#41—21no—
372#41C4×D1512yes—
373#41D6012yes—
374C2×Q8—152no—
375C2×Q8—301no—
376C2×Q8—301no—
377C2×Q8—301no—
378C2×Q8—301no—
379C2×Q8—301no—
380C2×Q8#122011yescharacteristic
381C2×D4—21no—
382C2×D4#122011yescharacteristic
383C22×C4—21no—
384C22×C4—22no—
385C22×C4—301no—
386C22×C4—301no—
387C22×C4#122012yes—
388C22×C4C22×D1511yescharacteristic
389Q16—304no—
390SD16—304no—
391M16—152no—
392M16—301no—
393M16—301no—
394M16—301no—
395C2×C8—301no—
396C2×C8—301no—
397C4⋊C4—22no—
398C4⋊C4—21no—
399C4⋊C4—21no—
400C4⋊C4#122011yescharacteristic
401C4⋊C4#122011yescharacteristic
402C22⋊C4—22no—
403C22⋊C4—21no—
404C42—22no—
405C42—21no—
406C42—301no—
407C42—301no—
408C3×C5#188211yescharacteristic, complemented, Hall
409C22×C3—21no—
410C22×C3—21no—
411C22×C3—22no—
412C22×C3—41no—
413C22×C3—41no—
414C22×C3C4×C22⋊D511yescharacteristic
415C22×C3#407611yescharacteristic
416C22×C3#409911yescharacteristic
417C3×C4—22no—
418C3×C4—42no—
419C3×C4—41no—
420C3×C4—41no—
421C3×C4C2×#68511yescharacteristic
422C3×C4C4×C22⋊D511yescharacteristic
423C3×C4#407511yescharacteristic
424C3×C4#407611yescharacteristic
425Dic3—202no—
426Dic3—202no—
427C2×C5—21no—
428C2×C5—21no—
429C2×C5—42no—
430C2×C5#521311yescharacteristic
431C2×C5#524211yescharacteristic
432C2×C5#524611yescharacteristic
433C23—21no—
434C23C2×#122011yescharacteristic
435Q8—21no—
436Q8—302no—
437Q8—302no—
438Q8—601no—
439Q8—601no—
440Q8#671412yes—
441D4—22no—
442D4—22no—
443D4—21no—
444D4#671412yes—
445C2×C4—21no—
446C2×C4—22no—
447C2×C4—22no—
448C2×C4—21no—
449C2×C4—22no—
450C2×C4—21no—
451C2×C4—42no—
452C2×C4—82no—
453C2×C4—302no—
454C2×C4—302no—
455C2×C4—601no—
456C2×C4—601no—
457C2×C4—601no—
458C2×C4—601no—
459C2×C4C2×C4×D1511yescharacteristic
460C2×C4C2×D6011yescharacteristic
461C2×C4#681711yescharacteristic
462C2×C4#681711yescharacteristic
463C2×C4C2×#122011yescharacteristic
464C2×C4C2×#122011yescharacteristic
465C8—302no—
466C8—601no—
467C2×C3—21no—
468C2×C3—21no—
469C2×C3—42no—
470C2×C3#6550211yescharacteristic
471C2×C3#6553111yescharacteristic
472C2×C3#6553511yescharacteristic
473C5#7166611yescharacteristic, complemented, Sylow
474C22—21no—
475C22—21no—
476C22—22no—
477C22—41no—
478C22—41no—
479C22C4×#122011yescharacteristic
480C22#9181911yescharacteristic
481C22#9184211yescharacteristic
482C4—22no—
483C4—42no—
484C4—41no—
485C4—41no—
486C4—602no—
487C4—602no—
488C4C2×#671411yescharacteristic
489C4C4×#122011yescharacteristic
490C4#9181811yescharacteristic
491C4#9181911yescharacteristic
492C3#1060245911yescharacteristic, complemented, Sylow
493C2—21no—
494C2—21no—
495C2—42no—
496C2#1175131411yescharacteristic
497C2#1175134311yescharacteristic
498C2#1175134711yescharacteristic
subgroupG/H
centerC22#91842
commutatorC2×C3×C4×C5C22×C4
FrattiniC22×C4C22×D15
FittingC3×C5×#520C2
socleC22×C3×C5#120
prime radicalC2×C3×C5×#41C22
cosocleC22×C3×C4×C5C23
hypercenter#520D15
hypocenterC3×C5#1882
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #520

Complements

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

NH ≅ G/Ntype
4. #11751105C2semidirect
5. #11751074C2semidirect
6. #11746353C2semidirect
7. #11746351C2semidirect
19. #91840C4semidirect
55. C2×Dic30C2×C4semidirect
72. #6710C2×C4semidirect
408. C3×C5#1882semidirect
473. C5#71666semidirect
492. C3#10602459semidirect

Maximal subgroups

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

subgroupindex
C3×C5×#5202
C2×#918402
C2×#911212
#117511052
#117510742
#117463532
#117463512
#106024593
#716665