Finite Group Databasebeta

← C5×#72114 #14302567

Subgroups — C5×#72114 #14302567

Order 1920.

664 conjugacy classes of subgroups.

The subgroup lattice has 520 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#575C211yesmaximal, characteristic
2C5×Dic3×#41C211yesmaximal, characteristic, complemented
3C5×#5240C211yesmaximal, characteristic, complemented
4C5×#5236C211yesmaximal, characteristic, complemented
5C2×C5×#837C211yesmaximal, characteristic, complemented
6C5×#4559C211yesmaximal, characteristic, complemented
7C5×#4555C211yesmaximal, characteristic, complemented
8C5×#1906—31nomaximal, Hall
9C2×C3×C5×#41C2211yescharacteristic
10C2×C3×C5×#41C2211yescharacteristic
11C3×C5×#137—21no—
12C3×C5×#137C412yescomplemented
13C3×C5×AGL(1,ℤ/8)—21no—
14C3×C5×AGL(1,ℤ/8)C412yescomplemented
15C3×C5×#135—22no—
16C2×C3×C5×M16C2211yescharacteristic
17C5×Q8×Dic3—21no—
18C5×D4×Dic3—21no—
19C5×D4×Dic3—22no—
20C5×#926—21no—
21C5×#926C2211yescharacteristic
22C5×#926C2211yescharacteristic
23C2×C4×C5×Dic3—21no—
24C2×C4×C5×Dic3C2211yescharacteristic
25C2×C5×#208C2211yescharacteristic
26C5×#837—22no—
27C5×#837—21no—
28C5×#837—21no—
29C5×#834—22no—
30#72114C511yesmaximal, characteristic, direct product factor, Hall
31C5×#575D311yescharacteristic
32C4×C5×#41—31no—
33C5×#421—31no—
34C2×C5×#104—31no—
35C5×#410—31no—
36C5×#344—31no—
37C5×#339—31no—
38C3×C5×#41—21no—
39C3×C5×#41—21no—
40C3×C5×#41—21no—
41C3×C5×#41—21no—
42C3×C5×#41—21no—
43C3×C5×#41—21no—
44C3×C5×#41C2×C412yes—
45C3×C5×#41D412yes—
46C2×C3×C5×Q8C2×C411yescharacteristic, complemented
47C2×C3×C5×Q8D411yescharacteristic
48C2×C3×C5×D4—21no—
49C2×C3×C5×D4—21no—
50C2×C3×C5×D4C2×C411yescharacteristic, complemented
51C2×C3×C5×D4D411yescharacteristic
52C22×C3×C4×C5—21no—
53C22×C3×C4×C5—21no—
54C22×C3×C4×C5C2311yescharacteristic
55C22×C5×Dic3—22no—
56C22×C5×Dic3—21no—
57C3×C5×Q16—22no—
58C3×C5×SD16—22no—
59C3×C5×SD16—22no—
60C3×C5×D8—22no—
61C3×C5×M16—21no—
62C3×C5×M16C2×C412yescomplemented
63C2×C3×C5×C8—21no—
64C5×#217—22no—
65C5×#217—21no—
66C5×#217D412yes—
67C5×C4⋊Dic3—22no—
68C5×C4⋊Dic3—21no—
69C5×C4⋊Dic3—21no—
70C4×C5×Dic3—22no—
71C4×C5×Dic3—21no—
72C4×C5×Dic3—21no—
73C4×C5×Dic3—21no—
74C4×C5×Dic3D411yescharacteristic
75C4×C5×Dic3D411yescharacteristic
76C5×#208—21no—
77C5×#208—21no—
78C2×C5×C3⋊C8—21no—
79C3×#575C2×C511yescharacteristic
80Dic3×#41C2×C511yescharacteristic, complemented
81#5240C2×C511yescharacteristic, complemented
82#5236C2×C511yescharacteristic, complemented
83C2×#837C2×C511yescharacteristic, complemented
84#4559C2×C511yescharacteristic, complemented
85#4555C2×C511yescharacteristic, complemented
86C2×C5×#41C2×D311yescharacteristic
87C2×C5×#41C2×D311yescharacteristic
88C5×#137—21no—
89C5×#137Dic312yescomplemented
90C5×AGL(1,ℤ/8)—21no—
91C5×AGL(1,ℤ/8)Dic312yescomplemented
92C5×#135—22no—
93C2×C5×M16—31no—
94C2×C5×M16C2×D311yescharacteristic
95C4×C5×Q8—61no—
96C4×C5×D4—62no—
97C4×C5×D4—61no—
98C5×#117—31no—
99C5×#117—31no—
100C5×#117—61no—
101C2×C42×C5—31no—
102C2×C42×C5—61no—
103C5×#104—61no—
104C5×#104—61no—
105C5×#104—62no—
106C5×#99—62no—
107#1906—31noSylow
108C23×C3×C5—21no—
109C23×C3×C5—21no—
110C23×C3×C5C2×D411yescharacteristic
111C3×C5×Q8—21no—
112C3×C5×Q8—21no—
113C3×C5×Q8—21no—
114C3×C5×Q8C22⋊C412yes—
115C3×C5×D4—22no—
116C3×C5×D4—22no—
117C3×C5×D4—21no—
118C3×C5×D4—21no—
119C3×C5×D4—21no—
120C3×C5×D4—41no—
121C3×C5×D4—41no—
122C3×C5×D4C22⋊C412yes—
123C2×C3×C4×C5—21no—
124C2×C3×C4×C5—22no—
125C2×C3×C4×C5—22no—
126C2×C3×C4×C5—21no—
127C2×C3×C4×C5—41no—
128C2×C3×C4×C5—41no—
129C2×C3×C4×C5C22×C411yescharacteristic
130C2×C3×C4×C5C2×D411yescharacteristic
131C2×C3×C4×C5C2×D411yescharacteristic
132C2×C3×C4×C5C2×D411yescharacteristic
133C2×C3×C4×C5#4111yescharacteristic
134C2×C3×C4×C5#4111yescharacteristic
135C2×C5×Dic3—21no—
136C2×C5×Dic3—22no—
137C2×C5×Dic3—21no—
138C2×C5×Dic3—42no—
139C2×C5×Dic3—44no—
140C2×C5×Dic3—41no—
141C3×C5×C8—22no—
142C5×C3⋊C8—41no—
143C2×C3×#41C22×C511yescharacteristic
144C2×C3×#41C22×C511yescharacteristic
145C3×#137—21no—
146C3×#137C4×C512yescomplemented
147C3×AGL(1,ℤ/8)—21no—
148C3×AGL(1,ℤ/8)C4×C512yescomplemented
149C3×#135—22no—
150C2×C3×M16C22×C511yescharacteristic
151Q8×Dic3—21no—
152D4×Dic3—21no—
153D4×Dic3—22no—
154#926—21no—
155#926C22×C511yescharacteristic
156#926C22×C511yescharacteristic
157C2×C4×Dic3—21no—
158C2×C4×Dic3C22×C511yescharacteristic
159C2×#208C22×C511yescharacteristic
160#837—22no—
161#837—21no—
162#837—21no—
163#834—22no—
164C5×#41—21no—
165C5×#41—21no—
166C5×#41—21no—
167C5×#41—21no—
168C5×#41—21no—
169C5×#41—21no—
170C5×#41C2×Dic312yes—
171C5×#41#6712yes—
172C2×C5×Q8C2×Dic311yescharacteristic, complemented
173C2×C5×Q8#6711yescharacteristic
174C2×C5×D4—21no—
175C2×C5×D4—21no—
176C2×C5×D4C2×Dic311yescharacteristic, complemented
177C2×C5×D4#6711yescharacteristic
178C22×C4×C5—21no—
179C22×C4×C5—21no—
180C22×C4×C5—62no—
181C22×C4×C5—61no—
182C22×C4×C5C22×D311yescharacteristic
183C5×Q16—22no—
184C5×SD16—22no—
185C5×SD16—22no—
186C5×D8—22no—
187C5×M16—21no—
188C5×M16—61no—
189C5×M16—61no—
190C5×M16C2×Dic312yescomplemented
191C2×C5×C8—21no—
192C2×C5×C8—61no—
193C5×C4⋊C4—62no—
194C5×C4⋊C4—61no—
195C5×C4⋊C4—61no—
196C5×C22⋊C4—32no—
197C5×C22⋊C4—62no—
198C5×C22⋊C4—61no—
199C42×C5—31no—
200C42×C5—31no—
201C42×C5—62no—
202C42×C5—61no—
203C42×C5—61no—
204C42×C5—61no—
205#575C5×D311yescharacteristic
206C4×#41—31no—
207#421—31no—
208C2×#104—31no—
209#410—31no—
210#344—31no—
211#339—31no—
212C22×C3×C5—22no—
213C22×C3×C5—41no—
214C22×C3×C5—42no—
215C22×C3×C5—41no—
216C22×C3×C5—81no—
217C22×C3×C5C4×D411yescharacteristic
218C22×C3×C5#12011yescharacteristic
219C22×C3×C5#12111yescharacteristic
220C3×C4×C5—22no—
221C3×C4×C5—41no—
222C3×C4×C5C2×C22⋊C411yescharacteristic
223C3×C4×C5C4×D411yescharacteristic
224C3×C4×C5#12111yescharacteristic
225C3×C4×C5#12311yescharacteristic
226C5×Dic3—22no—
227C5×Dic3—42no—
228C5×Dic3—41no—
229C5×Dic3—81no—
230C3×#41—21no—
231C3×#41—21no—
232C3×#41—21no—
233C3×#41—21no—
234C3×#41—21no—
235C3×#41—21no—
236C3×#41C2×C4×C512yes—
237C3×#41C5×D412yes—
238C2×C3×Q8C2×C4×C511yescharacteristic, complemented
239C2×C3×Q8C5×D411yescharacteristic
240C2×C3×D4—21no—
241C2×C3×D4—21no—
242C2×C3×D4C2×C4×C511yescharacteristic, complemented
243C2×C3×D4C5×D411yescharacteristic
244C22×C3×C4—21no—
245C22×C3×C4—21no—
246C22×C3×C4C23×C511yescharacteristic
247C22×Dic3—22no—
248C22×Dic3—21no—
249C3×Q16—22no—
250C3×SD16—22no—
251C3×SD16—22no—
252C3×D8—22no—
253C3×M16—21no—
254C3×M16C2×C4×C512yescomplemented
255C2×C3×C8—21no—
256#217—22no—
257#217—21no—
258#217C5×D412yes—
259C4⋊Dic3—22no—
260C4⋊Dic3—21no—
261C4⋊Dic3—21no—
262C4×Dic3—22no—
263C4×Dic3—21no—
264C4×Dic3—21no—
265C4×Dic3—21no—
266C4×Dic3C5×D411yescharacteristic
267C4×Dic3C5×D411yescharacteristic
268#208—21no—
269#208—21no—
270C2×C3⋊C8—21no—
271C23×C5—21no—
272C23×C5—21no—
273C23×C5D3×D411yescharacteristic
274C5×Q8—21no—
275C5×Q8—21no—
276C5×Q8—21no—
277C5×Q8#21712yes—
278C5×D4—22no—
279C5×D4—22no—
280C5×D4—21no—
281C5×D4—21no—
282C5×D4—21no—
283C5×D4—41no—
284C5×D4—41no—
285C5×D4#21712yes—
286C2×C4×C5—21no—
287C2×C4×C5—22no—
288C2×C4×C5—22no—
289C2×C4×C5—21no—
290C2×C4×C5—41no—
291C2×C4×C5—41no—
292C2×C4×C5—61no—
293C2×C4×C5—61no—
294C2×C4×C5—62no—
295C2×C4×C5—122no—
296C2×C4×C5—124no—
297C2×C4×C5—121no—
298C2×C4×C5D3×D411yescharacteristic
299C2×C4×C5#23711yescharacteristic
300C2×C4×C5#23711yescharacteristic
301C2×C4×C5C22×Dic311yescharacteristic
302C2×C4×C5C2×#6711yescharacteristic
303C2×C4×C5C2×#6711yescharacteristic
304C5×C8—22no—
305C5×C8—121no—
306C2×#41C2×C5×D311yescharacteristic
307C2×#41C2×C5×D311yescharacteristic
308#137—21no—
309#137C5×Dic312yescomplemented
310AGL(1,ℤ/8)—21no—
311AGL(1,ℤ/8)C5×Dic312yescomplemented
312#135—22no—
313C2×M16—31no—
314C2×M16C2×C5×D311yescharacteristic
315C4×Q8—61no—
316C4×D4—62no—
317C4×D4—61no—
318#117—31no—
319#117—31no—
320#117—61no—
321C2×C42—31no—
322C2×C42—61no—
323#104—61no—
324#104—61no—
325#104—62no—
326#99—62no—
327C2×C3×C5—21no—
328C2×C3×C5—21no—
329C2×C3×C5—21no—
330C2×C3×C5—42no—
331C2×C3×C5—81no—
332C2×C3×C5#38611yescharacteristic
333C23×C3—21no—
334C23×C3—21no—
335C23×C3C2×C5×D411yescharacteristic
336C3×Q8—21no—
337C3×Q8—21no—
338C3×Q8—21no—
339C3×Q8C5×C22⋊C412yes—
340C3×D4—22no—
341C3×D4—22no—
342C3×D4—21no—
343C3×D4—21no—
344C3×D4—21no—
345C3×D4—41no—
346C3×D4—41no—
347C3×D4C5×C22⋊C412yes—
348C2×C3×C4—21no—
349C2×C3×C4—22no—
350C2×C3×C4—22no—
351C2×C3×C4—21no—
352C2×C3×C4—41no—
353C2×C3×C4—41no—
354C2×C3×C4C22×C4×C511yescharacteristic
355C2×C3×C4C2×C5×D411yescharacteristic
356C2×C3×C4C2×C5×D411yescharacteristic
357C2×C3×C4C2×C5×D411yescharacteristic
358C2×C3×C4C5×#4111yescharacteristic
359C2×C3×C4C5×#4111yescharacteristic
360C2×Dic3—21no—
361C2×Dic3—22no—
362C2×Dic3—21no—
363C2×Dic3—42no—
364C2×Dic3—44no—
365C2×Dic3—41no—
366C3×C8—22no—
367C3⋊C8—41no—
368C22×C5—22no—
369C22×C5—41no—
370C22×C5—42no—
371C22×C5—41no—
372C22×C5—81no—
373C22×C5D4×Dic311yescharacteristic
374C22×C5#93711yescharacteristic
375C22×C5#93911yescharacteristic
376C4×C5—22no—
377C4×C5—41no—
378C4×C5—62no—
379C4×C5—121no—
380C4×C5—122no—
381C4×C5—241no—
382C4×C5D4×Dic311yescharacteristic
383C4×C5#93511yescharacteristic
384C4×C5#93911yescharacteristic
385C4×C5C2×#21711yescharacteristic
386#41—21no—
387#41—21no—
388#41—21no—
389#41—21no—
390#41—21no—
391#41—21no—
392#41C2×C5×Dic312yes—
393#41C5×#6712yes—
394C2×Q8C2×C5×Dic311yescharacteristic, complemented
395C2×Q8C5×#6711yescharacteristic
396C2×D4—21no—
397C2×D4—21no—
398C2×D4C2×C5×Dic311yescharacteristic, complemented
399C2×D4C5×#6711yescharacteristic
400C22×C4—21no—
401C22×C4—21no—
402C22×C4—62no—
403C22×C4—61no—
404C22×C4C22×C5×D311yescharacteristic
405Q16—22no—
406SD16—22no—
407SD16—22no—
408D8—22no—
409M16—21no—
410M16—61no—
411M16—61no—
412M16C2×C5×Dic312yescomplemented
413C2×C8—21no—
414C2×C8—61no—
415C4⋊C4—62no—
416C4⋊C4—61no—
417C4⋊C4—61no—
418C22⋊C4—32no—
419C22⋊C4—62no—
420C22⋊C4—61no—
421C42—31no—
422C42—31no—
423C42—62no—
424C42—61no—
425C42—61no—
426C42—61no—
427C3×C5#190611yescharacteristic, complemented, Hall
428C22×C3—22no—
429C22×C3—41no—
430C22×C3—42no—
431C22×C3—41no—
432C22×C3—81no—
433C22×C3C4×C5×D411yescharacteristic
434C22×C3C5×#12011yescharacteristic
435C22×C3C5×#12111yescharacteristic
436C3×C4—22no—
437C3×C4—41no—
438C3×C4C2×C5×C22⋊C411yescharacteristic
439C3×C4C4×C5×D411yescharacteristic
440C3×C4C5×#12111yescharacteristic
441C3×C4C5×#12311yescharacteristic
442Dic3—22no—
443Dic3—42no—
444Dic3—41no—
445Dic3—81no—
446C2×C5—21no—
447C2×C5—21no—
448C2×C5—21no—
449C2×C5—42no—
450C2×C5—81no—
451C2×C5#522011yescharacteristic
452C23—21no—
453C23—21no—
454C23C5×D3×D411yescharacteristic
455Q8—21no—
456Q8—21no—
457Q8—21no—
458Q8C5×#21712yes—
459D4—22no—
460D4—22no—
461D4—21no—
462D4—21no—
463D4—21no—
464D4—41no—
465D4—41no—
466D4C5×#21712yes—
467C2×C4—21no—
468C2×C4—22no—
469C2×C4—22no—
470C2×C4—21no—
471C2×C4—41no—
472C2×C4—41no—
473C2×C4—61no—
474C2×C4—61no—
475C2×C4—62no—
476C2×C4—122no—
477C2×C4—124no—
478C2×C4—121no—
479C2×C4C5×D3×D411yescharacteristic
480C2×C4C5×#23711yescharacteristic
481C2×C4C5×#23711yescharacteristic
482C2×C4C22×C5×Dic311yescharacteristic
483C2×C4C2×C5×#6711yescharacteristic
484C2×C4C2×C5×#6711yescharacteristic
485C8—22no—
486C8—121no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3—42no—
491C2×C3—81no—
492C2×C3C5×#38611yescharacteristic
493C5#7211411yescharacteristic, direct product factor, Sylow
494C22—22no—
495C22—41no—
496C22—42no—
497C22—41no—
498C22—81no—
499C22C5×D4×Dic311yescharacteristic
500C22C5×#93711yescharacteristic
501C22C5×#93911yescharacteristic
502C4—22no—
503C4—41no—
504C4—62no—
505C4—121no—
506C4—122no—
507C4—241no—
508C4C5×D4×Dic311yescharacteristic
509C4C5×#93511yescharacteristic
510C4C5×#93911yescharacteristic
511C4C2×C5×#21711yescharacteristic
512C3C5×#190611yescharacteristic, complemented, Sylow
513C2—21no—
514C2—21no—
515C2—21no—
516C2—42no—
517C2—81no—
518C2C5×#522011yescharacteristic
subgroupG/H
centerC4×C5#939
commutatorC2×C3×C4C22×C4×C5
FrattiniC22×C4C22×C5×D3
Fitting, prime radicalC3×C5×#575C2
socleC2×C3×C5#386
cosocleC22×C3×C4C23×C5
hypercenterC5×#575D3
hypocenterC3C5×#1906
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C5×#575

Complements

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

NH ≅ G/Ntype
2. C5×Dic3×#41C2semidirect
3. C5×#5240C2semidirect
4. C5×#5236C2semidirect
5. C2×C5×#837C2semidirect
6. C5×#4559C2semidirect
7. C5×#4555C2semidirect
12. C3×C5×#137C4semidirect
14. C3×C5×AGL(1,ℤ/8)C4semidirect
30. #72114C5direct (N × H)
46. C2×C3×C5×Q8C2×C4semidirect
50. C2×C3×C5×D4C2×C4semidirect
62. C3×C5×M16C2×C4semidirect
80. Dic3×#41C2×C5semidirect
81. #5240C2×C5semidirect
82. #5236C2×C5semidirect
83. C2×#837C2×C5semidirect
84. #4559C2×C5semidirect
85. #4555C2×C5semidirect
89. C5×#137Dic3semidirect
91. C5×AGL(1,ℤ/8)Dic3semidirect
146. C3×#137C4×C5semidirect
148. C3×AGL(1,ℤ/8)C4×C5semidirect
172. C2×C5×Q8C2×Dic3semidirect
176. C2×C5×D4C2×Dic3semidirect
190. C5×M16C2×Dic3semidirect
238. C2×C3×Q8C2×C4×C5semidirect
242. C2×C3×D4C2×C4×C5semidirect
254. C3×M16C2×C4×C5semidirect
309. #137C5×Dic3semidirect
311. AGL(1,ℤ/8)C5×Dic3semidirect
394. C2×Q8C2×C5×Dic3semidirect
398. C2×D4C2×C5×Dic3semidirect
412. M16C2×C5×Dic3semidirect
427. C3×C5#1906semidirect
493. C5#72114direct (N × H)
512. C3C5×#1906semidirect

Maximal subgroups

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

subgroupindex
C3×C5×#5752
C5×Dic3×#412
C5×#52402
C5×#52362
C2×C5×#8372
C5×#45592
C5×#45552
C5×#19063
#721145