Finite Group Databasebeta

← #14297280

Subgroups — #14297280

Order 1920.

1096 conjugacy classes of subgroups.

The subgroup lattice has 544 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#66285C211yesmaximal, characteristic
2#11752454C212yesmaximal, complemented
3#11752453C212yesmaximal, complemented
4#11752435C211yesmaximal, characteristic, complemented
5#11752430C211yesmaximal, characteristic, complemented
6#10605016—31nomaximal, Hall
7C3×C5×H(5,𝔽2)C411yescharacteristic, complemented
8C3×#4150—22no—
9C3×#4150C412yescomplemented
10C3×D5×#41—22no—
11C3×D5×#41C2212yes—
12C3×#4144—22no—
13C3×#4144—22no—
14C3×#4144C411yescharacteristic, complemented
15C2×C3×#711—22no—
16C2×C3×#711—22no—
17C2×C3×#711C2211yescharacteristic
18#91989C2212yescomplemented
19C2×#6758C2211yescharacteristic, complemented
20C2×D5⋊(C3⋊C8)C2211yescharacteristic, complemented
21#91241—22no—
22#91240—22no—
23#91239—22no—
24#91238—22no—
25#91237—22no—
26#91236—22no—
27#72146—51nomaximal, Hall
28#66285D311yescharacteristic
29#65795—32no—
30#65794—32no—
31#65776—31no—
32#65771—31no—
33C3×C5×#41—22no—
34C3×C5×#41C2×C412yes—
35C2×C3×C5×D4—22no—
36C2×C3×C5×D4—22no—
37C2×C3×C5×D4C2×C411yescharacteristic
38C2×C3×C22⋊D5—24no—
39C2×C3×C22⋊D5—22no—
40C2×C3×C22⋊D5—22no—
41C2×C3×C22⋊D5D412yes—
42C22×C3×Dic5—22no—
43C22×C3×Dic5D412yes—
44C3×#713D412yes—
45C3×Q8×D5—22no—
46C3×Q8×D5D412yes—
47C3×#711—24no—
48C3×#711—24no—
49C3×#711—24no—
50C3×#711—22no—
51C3×#711—24no—
52C3×#711—22no—
53C3×#711—22no—
54C3×#711—22no—
55C3×#711—22no—
56C3×#711D412yes—
57C3×D4×D5—24no—
58C3×D4×D5—22no—
59C3×D4×D5—22no—
60C3×D4×D5D412yes—
61C3×#709—24no—
62C3×#709—22no—
63C3×#709—22no—
64C3×#709C2×C412yes—
65C2×C3×C4×D5—22no—
66C2×C3×C4×D5—22no—
67C2×C3×C4×D5C2311yescharacteristic
68C2×C3×Dic10—22no—
69C2×C3×Dic10—22no—
70C2×C3×Dic10C2×C411yescharacteristic
71#6763—22no—
72C2×C3⋊#170—21no—
73C2×C3⋊#170—21no—
74C4⋊ΓL(1,𝔽16)—22no—
75C4×ΓL(1,𝔽16)—22no—
76#6758—22no—
77D5⋊(C3⋊C8)—22no—
78C3×#585—51no—
79#5240—51no—
80#5236—52no—
81#5234—51no—
82#5233—52no—
83C5×H(5,𝔽2)Dic311yescharacteristic, complemented
84#4150—22no—
85#4150Dic312yescomplemented
86D5×#41—22no—
87D5×#41C2×D312yes—
88#4144—22no—
89#4144—22no—
90#4144Dic311yescharacteristic, complemented
91C2×#711—22no—
92C2×#711—22no—
93C2×#711C2×D311yescharacteristic
94#4130—32no—
95C2×#700—31no—
96C2×D5⋊C8—31no—
97C22⋊#170—62no—
98#4011—62no—
99#4010—62no—
100#4009—62no—
101#4008—62no—
102#4007—62no—
103#1794—151noSylow
104C23×C3×C5—22no—
105C23×C3×C5C22⋊C412yes—
106C22×C3×D5—22no—
107C22×C3×D5—22no—
108C22×C3×D5C2×D411yescharacteristic
109C2×ΓL(1,𝔽16)—42no—
110C3×C5×Q8C22⋊C412yes—
111C3×C5×D4—24no—
112C3×C5×D4—22no—
113C3×C5×D4—22no—
114C3×C5×D4C22⋊C412yes—
115C2×C3×C4×C5—22no—
116C2×C3×C4×C5—22no—
117C2×C3×C4×C5C22×C411yescharacteristic
118C3×C22⋊D5—24no—
119C3×C22⋊D5—24no—
120C3×C22⋊D5—22no—
121C3×C22⋊D5—24no—
122C3×C22⋊D5—22no—
123C3×C22⋊D5—22no—
124C2×C3×Dic5—24no—
125C2×C3×Dic5—22no—
126C2×C3×Dic5—22no—
127C2×C3×Dic5—22no—
128C2×C3×Dic5—22no—
129C2×C3×Dic5C22⋊C412yes—
130C2×C3×Dic5C2×D411yescharacteristic
131C3×D20—22no—
132C3×D20C22⋊C412yes—
133C3×C4×D5—24no—
134C3×C4×D5—22no—
135C3×C4×D5—22no—
136C3×C4×D5—22no—
137C3×C4×D5C2×D412yes—
138C3×C4×D5C2×D412yes—
139C3×Dic10—24no—
140C3×Dic10—22no—
141C3×Dic10—22no—
142C3×Dic10C22⋊C412yes—
143C3⋊#170—41no—
144C3⋊#170—41no—
145C3×2−1+4—52no—
146C3×2−1+4—102no—
147C3×H(5,𝔽2)—51no—
148C3×H(5,𝔽2)—102no—
149C3×H(5,𝔽2)—102no—
150C3×H(5,𝔽2)AGL(1,𝔽5)11yescharacteristic, complemented
151C2×C3×#41—51no—
152C2×C3×#41—52no—
153C2×C3×#41—102no—
154C2×C3×#41—102no—
155C2×C3×#41—102no—
156#926—52no—
157C2×#208—51no—
158C22×C3⋊C8—51no—
159#837—102no—
160#837—102no—
161#835—102no—
162#834—102no—
163#832—102no—
164#830—102no—
165C5×#41—22no—
166C5×#41C2×Dic312yes—
167C2×C5×D4—22no—
168C2×C5×D4—22no—
169C2×C5×D4C2×Dic311yescharacteristic
170C2×C22⋊D5—24no—
171C2×C22⋊D5—22no—
172C2×C22⋊D5—22no—
173C2×C22⋊D5#6712yes—
174C22×Dic5—22no—
175C22×Dic5#6712yes—
176#713#6712yes—
177Q8×D5—22no—
178Q8×D5#6712yes—
179#711—24no—
180#711—24no—
181#711—24no—
182#711—22no—
183#711—24no—
184#711—22no—
185#711—22no—
186#711—22no—
187#711—22no—
188#711#6712yes—
189D4×D5—24no—
190D4×D5—22no—
191D4×D5—22no—
192D4×D5#6712yes—
193#709—24no—
194#709—22no—
195#709—22no—
196#709C2×Dic312yes—
197C2×C4×D5—22no—
198C2×C4×D5—22no—
199C2×C4×D5C22×D311yescharacteristic
200C2×Dic10—22no—
201C2×Dic10—22no—
202C2×Dic10C2×Dic311yescharacteristic
203C22⋊AGL(1,𝔽5)—62no—
204C2×#170—61no—
205C2×#170—61no—
206C4⋊AGL(1,𝔽5)—62no—
207C4×AGL(1,𝔽5)—62no—
208#700—62no—
209D5⋊C8—62no—
210#585—51no—
211#421—152no—
212#416—152no—
213#410—151no—
214#408—151no—
215C22×C3×C5—22no—
216C22×C3×C5—22no—
217C22×C3×C5—42no—
218C22×C3×C5—82no—
219C22×C3×C5C2×C22⋊C411yescharacteristic
220C2×C3×D5—22no—
221C2×C3×D5—41no—
222C2×C3×D5—82no—
223C2×C3×D5—82no—
224C2×C3×D5#12012yes—
225ΓL(1,𝔽16)—82no—
226C3×C4×C5—22no—
227C3×C4×C5C2×C22⋊C412yes—
228C3×Dic5—22no—
229C3×Dic5—22no—
230C3×Dic5#12011yescharacteristic
231C3×Dic5#12011yescharacteristic
232C3×#41—22no—
233C3×#41—52no—
234C3×#41—52no—
235C3×#41—52no—
236C3×#41—102no—
237C3×#41—102no—
238C3×#41—104no—
239C3×#41—102no—
240C3×#41—102no—
241C3×#41—102no—
242C3×#41—104no—
243C3×#41—102no—
244C3×#41—104no—
245C3×#41—104no—
246C3×#41—104no—
247C3×#41—102no—
248C3×#41C2×AGL(1,𝔽5)12yes—
249C2×C3×Q8—51no—
250C2×C3×Q8—52no—
251C2×C3×Q8—102no—
252C2×C3×Q8—102no—
253C2×C3×Q8—102no—
254C2×C3×D4—22no—
255C2×C3×D4—22no—
256C2×C3×D4—52no—
257C2×C3×D4—52no—
258C2×C3×D4—102no—
259C2×C3×D4—102no—
260C2×C3×D4—104no—
261C2×C3×D4—102no—
262C2×C3×D4—104no—
263C2×C3×D4—102no—
264C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
265C22×C3×C4—51no—
266C22×C3×C4—52no—
267C22×C3×C4—102no—
268C22×C3×C4—102no—
269C22×C3×C4—102no—
270#217—102no—
271C4⋊Dic3—102no—
272C4×Dic3—102no—
273#208—102no—
274C2×C3⋊C8—101no—
275C2×C3⋊C8—101no—
276C2×C3⋊C8—102no—
277C23×C5—22no—
278C23×C5#21712yes—
279C22×D5—22no—
280C22×D5—22no—
281C22×D5C2×#6711yescharacteristic
282C2×AGL(1,𝔽5)—122no—
283C5×Q8#21712yes—
284C5×D4—24no—
285C5×D4—22no—
286C5×D4—22no—
287C5×D4#21712yes—
288C2×C4×C5—22no—
289C2×C4×C5—22no—
290C2×C4×C5C22×Dic311yescharacteristic
291C22⋊D5—24no—
292C22⋊D5—24no—
293C22⋊D5—22no—
294C22⋊D5—24no—
295C22⋊D5—22no—
296C22⋊D5—22no—
297C2×Dic5—24no—
298C2×Dic5—22no—
299C2×Dic5—22no—
300C2×Dic5—22no—
301C2×Dic5—22no—
302C2×Dic5#21712yes—
303C2×Dic5C2×#6711yescharacteristic
304D20—22no—
305D20#21712yes—
306C4×D5—24no—
307C4×D5—22no—
308C4×D5—22no—
309C4×D5—22no—
310C4×D5C2×#6712yes—
311C4×D5C2×#6712yes—
312Dic10—24no—
313Dic10—22no—
314Dic10—22no—
315Dic10#21712yes—
316#170—121no—
317#170—121no—
3182−1+4—52no—
3192−1+4—102no—
320H(5,𝔽2)—51no—
321H(5,𝔽2)—102no—
322H(5,𝔽2)—102no—
323H(5,𝔽2)ΓL(1,𝔽16)11yescharacteristic, complemented
324C2×#41—51no—
325C2×#41—52no—
326C2×#41—102no—
327C2×#41—102no—
328C2×#41—102no—
329C2×M16—151no—
330C22×C8—151no—
331#117—152no—
332#104—302no—
333#104—302no—
334#103—302no—
335#102—302no—
336#99—302no—
337C22⋊C8—302no—
338C2×C3×C5—21no—
339C2×C3×C5—42no—
340C2×C3×C5—42no—
341C2×C3×C5#37911yescharacteristic
342C3×D5—22no—
343C3×D5—42no—
344C23×C3—22no—
345C23×C3—51no—
346C23×C3—102no—
347C23×C3—102no—
348C23×C3C22⋊AGL(1,𝔽5)12yes—
349C3×Q8—52no—
350C3×Q8—102no—
351C3×Q8—104no—
352C3×Q8—102no—
353C3×Q8C22⋊AGL(1,𝔽5)12yes—
354C3×D4—24no—
355C3×D4—22no—
356C3×D4—22no—
357C3×D4—52no—
358C3×D4—102no—
359C3×D4—102no—
360C3×D4—104no—
361C3×D4—102no—
362C3×D4—104no—
363C3×D4—104no—
364C3×D4—102no—
365C3×D4C22⋊AGL(1,𝔽5)12yes—
366C2×C3×C4—22no—
367C2×C3×C4—22no—
368C2×C3×C4—51no—
369C2×C3×C4—52no—
370C2×C3×C4—52no—
371C2×C3×C4—52no—
372C2×C3×C4—102no—
373C2×C3×C4—102no—
374C2×C3×C4—104no—
375C2×C3×C4—102no—
376C2×C3×C4—102no—
377C2×C3×C4—102no—
378C2×C3×C4—104no—
379C2×C3×C4—102no—
380C2×C3×C4—102no—
381C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
382C2×Dic3—202no—
383C3⋊C8—201no—
384C3⋊C8—201no—
385C22×C5—22no—
386C22×C5—22no—
387C22×C5—42no—
388C22×C5—82no—
389C22×C5C2×#21711yescharacteristic
390C2×D5—22no—
391C2×D5—41no—
392C2×D5—82no—
393C2×D5—82no—
394C2×D5#95312yes—
395AGL(1,𝔽5)—242no—
396C4×C5—22no—
397C4×C5C2×#21712yes—
398Dic5—22no—
399Dic5—22no—
400Dic5#95311yescharacteristic
401Dic5#95311yescharacteristic
402#41—22no—
403#41—52no—
404#41—52no—
405#41—52no—
406#41—102no—
407#41—102no—
408#41—104no—
409#41—102no—
410#41—102no—
411#41—102no—
412#41—104no—
413#41—102no—
414#41—104no—
415#41—104no—
416#41—104no—
417#41—102no—
418#41C2×ΓL(1,𝔽16)12yes—
419C2×Q8—51no—
420C2×Q8—52no—
421C2×Q8—102no—
422C2×Q8—102no—
423C2×Q8—102no—
424C2×D4—22no—
425C2×D4—22no—
426C2×D4—52no—
427C2×D4—52no—
428C2×D4—102no—
429C2×D4—102no—
430C2×D4—104no—
431C2×D4—102no—
432C2×D4—104no—
433C2×D4—102no—
434C2×D4C2×ΓL(1,𝔽16)11yescharacteristic
435C22×C4—51no—
436C22×C4—52no—
437C22×C4—102no—
438C22×C4—102no—
439C22×C4—102no—
440M16—302no—
441C2×C8—301no—
442C2×C8—301no—
443C2×C8—302no—
444C4⋊C4—302no—
445C22⋊C4—302no—
446C42—302no—
447C3×C5#179411yescharacteristic, complemented, Hall
448C22×C3—22no—
449C22×C3—22no—
450C22×C3—42no—
451C22×C3—52no—
452C22×C3—82no—
453C22×C3—102no—
454C22×C3—201no—
455C22×C3—402no—
456C22×C3—402no—
457C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
458C3×C4—22no—
459C3×C4—51no—
460C3×C4—51no—
461C3×C4—102no—
462C3×C4—102no—
463C3×C4C2×C22⋊AGL(1,𝔽5)12yes—
464Dic3—402no—
465C2×C5—21no—
466C2×C5—42no—
467C2×C5—42no—
468C2×C5#524711yescharacteristic
469D5—22no—
470D5—42no—
471C23—22no—
472C23—51no—
473C23—102no—
474C23—102no—
475C23#676312yes—
476Q8—52no—
477Q8—102no—
478Q8—104no—
479Q8—102no—
480Q8#676312yes—
481D4—24no—
482D4—22no—
483D4—22no—
484D4—52no—
485D4—102no—
486D4—102no—
487D4—104no—
488D4—102no—
489D4—104no—
490D4—104no—
491D4—102no—
492D4#676312yes—
493C2×C4—22no—
494C2×C4—22no—
495C2×C4—51no—
496C2×C4—52no—
497C2×C4—52no—
498C2×C4—52no—
499C2×C4—102no—
500C2×C4—102no—
501C2×C4—104no—
502C2×C4—102no—
503C2×C4—102no—
504C2×C4—102no—
505C2×C4—104no—
506C2×C4—102no—
507C2×C4—102no—
508C2×C4—602no—
509C2×C4C22×ΓL(1,𝔽16)11yescharacteristic
510C8—601no—
511C8—601no—
512C2×C3—21no—
513C2×C3—42no—
514C2×C3—42no—
515C2×C3—102no—
516C2×C3—202no—
517C2×C3#6580011yescharacteristic
518C5#7214611yescharacteristic, complemented, Sylow
519C22—22no—
520C22—22no—
521C22—42no—
522C22—52no—
523C22—82no—
524C22—102no—
525C22—201no—
526C22—402no—
527C22—402no—
528C22C2×#676311yescharacteristic
529C4—22no—
530C4—51no—
531C4—51no—
532C4—102no—
533C4—102no—
534C4—1202no—
535C4C2×#676312yes—
536C3#1060501611yescharacteristic, complemented, Sylow
537C2—21no—
538C2—42no—
539C2—42no—
540C2—102no—
541C2—202no—
542C2#1175245911yescharacteristic
subgroupG/H
centerC2#11752459
commutatorC2×C3×C4×C5C22×C4
FrattiniC2×C4C22×ΓL(1,𝔽16)
FittingC3×C5×H(5,𝔽2)C4
socleC2×C3×C5#379
prime radicalC3×#66285C2
cosocleC2×C3×C4×D5C23
hypercenterH(5,𝔽2)ΓL(1,𝔽16)
hypocenterC3×C5#1794
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2 ◁ C2×C4 ◁ H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
2. #11752454C2semidirect
3. #11752453C2semidirect
4. #11752435C2semidirect
5. #11752430C2semidirect
7. C3×C5×H(5,𝔽2)C4semidirect
9. C3×#4150C4semidirect
14. C3×#4144C4semidirect
18. #91989C22semidirect
19. C2×#6758C22semidirect
20. C2×D5⋊(C3⋊C8)C22semidirect
83. C5×H(5,𝔽2)Dic3semidirect
85. #4150Dic3semidirect
90. #4144Dic3semidirect
150. C3×H(5,𝔽2)AGL(1,𝔽5)semidirect
323. H(5,𝔽2)ΓL(1,𝔽16)semidirect
447. C3×C5#1794semidirect
518. C5#72146semidirect
536. C3#10605016semidirect

Maximal subgroups

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

subgroupindex
C3×#662852
#117524542
#117524532
#117524352
#117524302
#106050163
#721465