Finite Group Databasebeta

← #14416892

Subgroups — #14416892

Order 1920.

968 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11756281C211yesmaximal, characteristic
2C2×#91331C211yesmaximal, characteristic, complemented
3C2×#91315C211yesmaximal, characteristic
4C5×#4995C211yesmaximal, characteristic, complemented
5C3×#65293C211yesmaximal, characteristic, complemented
6#11748394C211yesmaximal, characteristic
7#11748328C211yesmaximal, characteristic, complemented
8#11747274C212yesmaximal, complemented
9#11747273C212yesmaximal, complemented
10#11747272C212yesmaximal, complemented
11#11747271C212yesmaximal, complemented
12#10610931—31nomaximal, Hall
13C2×#6817C2211yescharacteristic
14C22×C3×Dic10C2211yescharacteristic
15C3×C5×#117C2211yescharacteristic
16C2×C4×Dic15C2211yescharacteristic
17#91782C2211yescharacteristic
18#91782C2211yescharacteristic
19#91776C2211yescharacteristic
20#91776C2211yescharacteristic
21#91770—21no—
22#91770—21no—
23#91761—21no—
24C42×D15—21no—
25C2×C5×C4⋊Dic3C2211yescharacteristic
26C2×C5×#208C2211yescharacteristic
27C2×C3×#681C2211yescharacteristic
28C2×#6668—21no—
29C2×Dic3×Dic5—21no—
30#91331—21no—
31#91331—21no—
32#91331C2212yes—
33#91331C2212yes—
34C2×#6662C2212yes—
35#91315C418yescomplemented
36C2×#6658C2212yes—
37C5×#808C2212yes—
38C5×#807C2212yes—
39C3×#3942C2212yes—
40C3×#3942C2212yes—
41#90975C2212yes—
42#90969C2212yes—
43#90954C2212yes—
44#90951C2212yes—
45#78946—51nomaximal, Hall
46C2×#4055—31no—
47#65856—31no—
48C2×#4033—31no—
49C5×#428—31no—
50M16×Dic5—31no—
51#65293D311yescharacteristic, complemented
52#65255—31no—
53#65178—32no—
54#65162—32no—
55#65142—32no—
56#65141—32no—
57C22×C3×C4×C5C2311yescharacteristic
58C2×#1220—21no—
59C22×Dic15D411yescharacteristic
60C22×Dic15D411yescharacteristic
61#6817—22no—
62#6817C2×C414yes—
63C2×D60C2311yescharacteristic
64C2×C4×D15—21no—
65C2×C4×D15—21no—
66C2×C4×D15—21no—
67C2×Dic30C2311yescharacteristic
68C22×C5×Dic3—21no—
69C22×C3×Dic5—21no—
70C2×C3×Dic10—21no—
71C2×C3×Dic10—22no—
72C2×C3×Dic10C2×C414yescomplemented
73C2×C3×Dic10C2312yes—
74C3×C5×C4⋊C4C2311yescharacteristic
75C3×C5×C4⋊C4C2311yescharacteristic
76C3×C5×C22⋊C4—21no—
77C3×C42×C5—21no—
78#6714—21no—
79#6714—21no—
80#6712—21no—
81#6712—21no—
82C4×Dic15—21no—
83C4×Dic15—21no—
84C4×Dic15D411yescharacteristic
85C4×Dic15D411yescharacteristic
86C4×Dic15C2311yescharacteristic
87C4×Dic15C2311yescharacteristic
88C5×C4⋊Dic3—21no—
89C5×C4⋊Dic3C2312yes—
90C5×#208C2×C414yescomplemented
91C2×C5×C3⋊C8C2312yes—
92C3×#681—22no—
93C2×C3×#168C2312yes—
94#6668—22no—
95#6668—22no—
96Dic3×Dic5—22no—
97Dic3×Dic5—22no—
98#6662C2×C418yescomplemented
99#6658C2×C418yescomplemented
100C2×Q8×Dic3—51no—
101C2×#942—51no—
102#5522—51no—
103C3×#419—51no—
104#5133—51no—
105Dic3×M16—51no—
106#4995D511yescharacteristic, complemented
107#4817—52no—
108#4790—52no—
109#4789—52no—
110#4788—52no—
111C2×#709—31no—
112C22×Dic10C2×D311yescharacteristic
113C2×C5×M16—31no—
114C5×#117C2×D311yescharacteristic
115C2×C5×C4⋊C4—31no—
116C2×#683—61no—
117C2×C4×Dic5—31no—
118C2×C4×Dic5—61no—
119C2×#681C2×D311yescharacteristic
120#4055—38no—
121C2×Dic20—32no—
122C2×#677—32no—
123#4039—31no—
124#4039—31no—
125#4033—32no—
126#4033—32no—
127#4033—31no—
128#4033—31no—
129#4033—61no—
130#4033—61no—
131#4027—61no—
132#4027—61no—
133#4018—61no—
134C42×D5—61no—
135C5×#107—32no—
136C5×#106—32no—
137#3947—32no—
138C8×Dic5—32no—
139#3942—32no—
140#3942C2×D312yes—
141#3942C2×D312yes—
142#3941—32no—
143C4×#137—151noSylow
144C23×C3×C5C2×D411yescharacteristic
145C22×D15—21no—
146C2×C3×C4×C5—21no—
147C2×C3×C4×C5—21no—
148C2×C3×C4×C5C22×C412yes—
149C2×C3×C4×C5C2×D411yescharacteristic
150C2×C3×C4×C5#4112yes—
151C2×C3×C4×C5C2411yescharacteristic
152#1220—24no—
153C2×Dic15—21no—
154C2×Dic15—22no—
155C2×Dic15—41no—
156C2×Dic15C2×D411yescharacteristic
157C2×Dic15C2×D411yescharacteristic
158C2×Dic15C2×D411yescharacteristic
159C2×Dic15C2×D411yescharacteristic
160D60—21no—
161D60C22×C412yes—
162C4×D15—24no—
163C4×D15—42no—
164C4×D15—42no—
165Dic30—21no—
166Dic30C22×C412yes—
167C2×C5×Dic3—22no—
168C2×C5×Dic3—42no—
169C2×C3×Dic5—24no—
170C2×C3×Dic5—22no—
171C3×Dic10—24no—
172C3×Dic10—22no—
173C3×Dic10C22×C414yes—
174C3×#168—22no—
175C5×C3⋊C8C22×C414yes—
176C22×C3×Q8—51no—
177C2×#235—51no—
178C2×C3×M16—51no—
179C3×#117C2×D511yescharacteristic
180Q8×Dic3—52no—
181Q8×Dic3—52no—
182Q8×Dic3—101no—
183Q8×Dic3—101no—
184C2×#216—52no—
185#942—58no—
186C2×#215—52no—
187C2×C4⋊Dic3—101no—
188C2×C4⋊Dic3C2×D511yescharacteristic
189C2×C4×Dic3—51no—
190C2×C4×Dic3—101no—
191C2×#208C2×D511yescharacteristic
192#893—51no—
193#893—51no—
194#887—51no—
195#887—51no—
196#67⋊C4—101no—
197#67⋊C4—101no—
198#872—101no—
199C42×D3—101no—
200C3×#103—52no—
201C3×#103—52no—
202#821—52no—
203#816—52no—
204#815—52no—
205C8×Dic3—52no—
206#808C2×D512yes—
207#807C2×D512yes—
208C22×C4×C5—61no—
209C22×C4×C5C22×D311yescharacteristic
210C2×C22⋊D5—61no—
211C22×Dic5—21no—
212C22×Dic5—31no—
213C22×Dic5—31no—
214#709—34no—
215#709—62no—
216C2×D20—31no—
217C2×C4×D5—61no—
218C2×C4×D5—61no—
219C2×C4×D5—61no—
220C2×Dic10—21no—
221C2×Dic10—22no—
222C2×Dic10—31no—
223C2×Dic10C4×D314yescomplemented
224C2×Dic10C22×D312yes—
225C5×M16—34no—
226C2×C5×C8—32no—
227C5×C4⋊C4—32no—
228C5×C4⋊C4—61no—
229C5×C4⋊C4C22×D311yescharacteristic
230C5×C4⋊C4C22×D311yescharacteristic
231C5×C22⋊C4—21no—
232C42×C5—21no—
233#685—61no—
234#685—61no—
235#683—62no—
236#683—62no—
237#683—61no—
238#683—61no—
239C4×Dic5—31no—
240C4×Dic5—31no—
241C4×Dic5—31no—
242C4×Dic5—31no—
243C4×Dic5—62no—
244C4×Dic5—62no—
245C4×Dic5—61no—
246C4×Dic5—61no—
247#681—22no—
248C2×#168C22×D312yes—
249Dic20—38no—
250#677—38no—
251C2×#137—151no—
252C4×#41—151no—
253C2×C4×Q8—151no—
254#441—152no—
255#440—152no—
256C4×Q16—152no—
257C4×SD16—152no—
258#428—31no—
259#419—51no—
260#417—151no—
261C4×M16—151no—
262C22×C3×C5—21no—
263C22×C3×C5—21no—
264C22×C3×C5C4×D412yes—
265C22×C3×C5C22×D411yescharacteristic
266C2×D15—22no—
267C2×D15—41no—
268C2×D15—41no—
269C3×C4×C5—41no—
270C3×C4×C5—41no—
271C3×C4×C5C4×D412yes—
272C3×C4×C5C23×C411yescharacteristic
273C3×C4×C5C2×#4111yescharacteristic
274Dic15—22no—
275Dic15—21no—
276Dic15—21no—
277Dic15—21no—
278Dic15#13712yescomplemented
279C3×Dic5—24no—
280C5×Dic3—42no—
281C2×C3×Q8—54no—
282C2×C3×Q8—52no—
283C2×C3×Q8—101no—
284C2×C3×Q8—102no—
285C22×C3×C4—101no—
286C22×C3×C4C22×D511yescharacteristic
287C2×#67—101no—
288C22×Dic3—21no—
289C22×Dic3—51no—
290C22×Dic3—51no—
291#235—54no—
292#235—102no—
293C2×D12—51no—
294C2×C4×D3—101no—
295C2×C4×D3—101no—
296C2×C4×D3—101no—
297C2×Dic6—51no—
298C3×M16—102no—
299C2×C3×C8—52no—
300C3×C4⋊C4C22×D511yescharacteristic
301C3×C4⋊C4C22×D511yescharacteristic
302C3×C22⋊C4—21no—
303C3×C42—21no—
304#216—58no—
305#215—58no—
306(C2×D3)⋊C4—101no—
307(C2×D3)⋊C4—101no—
308C4⋊Dic3—21no—
309C4⋊Dic3—102no—
310C4⋊Dic3—102no—
311C4⋊Dic3C22×D512yes—
312Dic3⋊C4—101no—
313Dic3⋊C4—101no—
314C4×Dic3—51no—
315C4×Dic3—51no—
316C4×Dic3—51no—
317C4×Dic3—51no—
318C4×Dic3—102no—
319C4×Dic3—102no—
320C4×Dic3—101no—
321C4×Dic3—101no—
322#208C4×D514yescomplemented
323C2×C3⋊C8C22×D512yes—
324C23×C5D3×D411yescharacteristic
325C22×D5—61no—
326C2×C4×C5—21no—
327C2×C4×C5—21no—
328C2×C4×C5—62no—
329C2×C4×C5—122no—
330C2×C4×C5C2×C4×D312yes—
331C2×C4×C5D3×D411yescharacteristic
332C2×C4×C5#23712yes—
333C2×C4×C5C23×D311yescharacteristic
334C22⋊D5—64no—
335C2×Dic5—24no—
336C2×Dic5—22no—
337C2×Dic5—31no—
338C2×Dic5—31no—
339C2×Dic5—31no—
340C2×Dic5—31no—
341C2×Dic5—62no—
342C2×Dic5—61no—
343C2×Dic5—121no—
344D20—32no—
345D20—61no—
346C4×D5—64no—
347C4×D5—122no—
348C4×D5—122no—
349Dic10—24no—
350Dic10—22no—
351Dic10—32no—
352Dic10—61no—
353Dic10C2×C4×D314yes—
354C5×C8—34no—
355#168—22no—
356C2×#41—151no—
357C22×Q8—51no—
358#137—158no—
359C2×Q16—152no—
360C2×SD16—152no—
361C2×M16—31no—
362C2×M16—51no—
363C4×Q8—152no—
364C4×Q8—152no—
365C4×Q8—151no—
366C4×Q8—151no—
367C4×Q8—301no—
368C4×Q8—301no—
369C4×D4—151no—
370C4×D4—151no—
371C4×D4—301no—
372C4×D4—301no—
373#117—301no—
374#117D3×D511yescharacteristic
375C2×C4⋊C4—31no—
376C2×C4⋊C4—301no—
377C2×C42—151no—
378C2×C42—301no—
379C2×C42—301no—
380#107—32no—
381#106—32no—
382#103—52no—
383#103—52no—
384#103—152no—
385#102—152no—
386#97—152no—
387C4×C8—152no—
388C2×C3×C5—22no—
389C2×C3×C5C2×C4×D411yescharacteristic
390C2×C3×C5C2×#13711yescharacteristic
391C2×C3×C5#57511yescharacteristic
392D15—42no—
393C23×C3D4×D511yescharacteristic
394C22×D3—101no—
395C3×Q8—54no—
396C3×Q8—102no—
397C3×Q8—104no—
398C2×C3×C4—21no—
399C2×C3×C4—21no—
400C2×C3×C4—102no—
401C2×C3×C4—104no—
402C2×C3×C4C2×C4×D512yes—
403C2×C3×C4D4×D511yescharacteristic
404C2×C3×C4#71312yes—
405C2×C3×C4C23×D511yescharacteristic
406#67—104no—
407C2×Dic3—22no—
408C2×Dic3—42no—
409C2×Dic3—51no—
410C2×Dic3—51no—
411C2×Dic3—51no—
412C2×Dic3—51no—
413C2×Dic3—102no—
414C2×Dic3—101no—
415C2×Dic3—201no—
416D12—52no—
417D12—101no—
418C4×D3—104no—
419C4×D3—202no—
420C4×D3—202no—
421Dic6—52no—
422Dic6—101no—
423C3×C8—102no—
424C3⋊C8C2×C4×D514yes—
425C22×C5—21no—
426C22×C5—21no—
427C22×C5#67⋊C412yes—
428C22×C5C2×D3×D411yescharacteristic
429C2×D5—62no—
430C2×D5—121no—
431C2×D5—121no—
432C4×C5—41no—
433C4×C5—41no—
434C4×C5—122no—
435C4×C5#67⋊C412yes—
436C4×C5C22×C4×D311yescharacteristic
437C4×C5C2×#23711yescharacteristic
438Dic5—24no—
439Dic5—32no—
440Dic5—62no—
441Dic5—61no—
442Dic5—61no—
443Dic5—61no—
444#41—154no—
445#41—302no—
446C2×Q8—54no—
447C2×Q8—52no—
448C2×Q8—102no—
449C2×Q8—101no—
450C2×Q8—151no—
451C2×D4—151no—
452C2×D4—301no—
453C22×C4—61no—
454C22×C4—101no—
455C22×C4—151no—
456C22×C4—151no—
457C22×C4—301no—
458C22×C4—301no—
459C22×C4—301no—
460C22×C4C2×D3×D511yescharacteristic
461Q16—158no—
462SD16—158no—
463M16—34no—
464M16—102no—
465C2×C8—32no—
466C2×C8—52no—
467C4⋊C4—32no—
468C4⋊C4—61no—
469C4⋊C4—301no—
470C4⋊C4—301no—
471C4⋊C4—302no—
472C4⋊C4—302no—
473C4⋊C4C2×D3×D511yescharacteristic
474C4⋊C4C2×D3×D511yescharacteristic
475C22⋊C4—21no—
476C22⋊C4—301no—
477C22⋊C4—301no—
478C42—21no—
479C42—151no—
480C42—151no—
481C42—151no—
482C42—151no—
483C42—301no—
484C42—301no—
485C42—302no—
486C42—302no—
487C3×C5C4×#13711yescharacteristic, complemented, Hall
488C22×C3—21no—
489C22×C3—21no—
490C22×C3#403912yes—
491C22×C3C2×D4×D511yescharacteristic
492C2×D3—102no—
493C2×D3—201no—
494C2×D3—201no—
495C3×C4—41no—
496C3×C4—41no—
497C3×C4—104no—
498C3×C4#403912yes—
499C3×C4C22×C4×D511yescharacteristic
500C3×C4C2×#71311yescharacteristic
501Dic3—42no—
502Dic3—52no—
503Dic3—102no—
504Dic3—101no—
505Dic3—101no—
506Dic3—101no—
507C2×C5—22no—
508C2×C5C2×#67⋊C411yescharacteristic
509C2×C5D3×#13711yescharacteristic
510C2×C5#577711yescharacteristic
511D5—122no—
512C23—301no—
513C23#679011yescharacteristic
514Q8—54no—
515Q8—104no—
516Q8—102no—
517Q8—152no—
518Q8—301no—
519D4—152no—
520D4—304no—
521D4—301no—
522C2×C4—21no—
523C2×C4—21no—
524C2×C4—62no—
525C2×C4—104no—
526C2×C4—102no—
527C2×C4—122no—
528C2×C4—151no—
529C2×C4—151no—
530C2×C4—151no—
531C2×C4—151no—
532C2×C4—302no—
533C2×C4—301no—
534C2×C4—304no—
535C2×C4—602no—
536C2×C4—602no—
537C2×C4—601no—
538C2×C4#676612yes—
539C2×C4C4×D3×D512yes—
540C2×C4#679011yescharacteristic
541C2×C4C22×D3×D511yescharacteristic
542C8—34no—
543C8—102no—
544C2×C3—22no—
545C2×C3C2×#403911yescharacteristic
546C2×C3#6610711yescharacteristic
547C2×C3D5×#13711yescharacteristic
548D3—202no—
549C5#7894611yescharacteristic, complemented, Sylow
550C22—21no—
551C22—21no—
552C22—302no—
553C22—601no—
554C22—601no—
555C22#9139612yes—
556C22C2×#679011yescharacteristic
557C4—41no—
558C4—41no—
559C4—104no—
560C4—122no—
561C4—152no—
562C4—302no—
563C4—301no—
564C4—301no—
565C4—301no—
566C4#9139612yes—
567C4C2×#676611yescharacteristic
568C4C2×C4×D3×D511yescharacteristic
569C3#1061093111yescharacteristic, complemented, Sylow
570C2—22no—
571C2—602no—
572C2C2×#9139611yescharacteristic
573C2#1175534211yescharacteristic
574C2#1175535211yescharacteristic
subgroupG/H
centerC22C2×#6790
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C22×D3×D5
FittingC3×C5×#117C22
socleC22×C3×C5C22×D4
prime radicalC2×#6817C22
cosocleC2×C3×C4×C5C24
hypercenter#117D3×D5
hypocenterC3×C5C4×#137
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
2. C2×#91331C2semidirect
4. C5×#4995C2semidirect
5. C3×#65293C2semidirect
7. #11748328C2semidirect
8. #11747274C2semidirect
9. #11747273C2semidirect
10. #11747272C2semidirect
11. #11747271C2semidirect
35. #91315C4semidirect
51. #65293D3semidirect
72. C2×C3×Dic10C2×C4semidirect
90. C5×#208C2×C4semidirect
98. #6662C2×C4semidirect
99. #6658C2×C4semidirect
106. #4995D5semidirect
223. C2×Dic10C4×D3semidirect
278. Dic15#137semidirect
322. #208C4×D5semidirect
487. C3×C5C4×#137semidirect
549. C5#78946semidirect
569. C3#10610931semidirect

Maximal subgroups

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

subgroupindex
#117562812
C2×#913312
C2×#913152
C5×#49952
C3×#652932
#117483942
#117483282
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