Finite Group Databasebeta

← C5×#68778 #14302558

Subgroups — C5×#68778 #14302558

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
1C5×#5748C211yesmaximal, characteristic
2C4×C5×#235C211yesmaximal, characteristic, complemented
3C5×#5006C211yesmaximal, characteristic, complemented
4C5×#5005C211yesmaximal, characteristic, complemented
5C2×C5×#805C211yesmaximal, characteristic, complemented
6C3×C5×#339C211yesmaximal, characteristic, complemented
7C5×#4559C211yesmaximal, characteristic, complemented
8C5×#1906—31nomaximal, Hall
9C2×C5×#235C2211yescharacteristic
10C2×C5×#235C2211yescharacteristic
11C2×C3×C5×M16C2211yescharacteristic
12C3×C5×#117C2211yescharacteristic
13C2×C3×C42×C5C2211yescharacteristic
14C4×C5×#67—22no—
15C5×#926C2211yescharacteristic
16C2×C5×#208C2211yescharacteristic
17C5×#909—21no—
18C5×#909C412yescomplemented
19C5×#908—21no—
20C5×#908C412yescomplemented
21C5×#904—22no—
22C4×C5×D12—21no—
23C5×#872—21no—
24C42×C5×D3—21no—
25C4×C5×Dic6—21no—
26C5×#806—22no—
27C5×#805—22no—
28C5×#805—21no—
29C5×#805—21no—
30#68778C511yesmaximal, characteristic, direct product factor, Hall
31C5×#575—31no—
32C4×C5×#41—31no—
33C5×#421—31no—
34C2×C5×#104—31no—
35C5×#410—31no—
36C5×#344—31no—
37C5×#339D311yescharacteristic, complemented
38C22×C3×C4×C5—21no—
39C22×C3×C4×C5C2311yescharacteristic
40C2×C5×#67—21no—
41C2×C5×#67—21no—
42C5×#235—21no—
43C5×#235—21no—
44C5×#235—21no—
45C5×#235—21no—
46C5×#235—21no—
47C5×#235—21no—
48C5×#235C2×C412yes—
49C5×#235D412yes—
50C2×C5×D12C2×C411yescharacteristic, complemented
51C2×C5×D12D411yescharacteristic
52C2×C4×C5×D3—21no—
53C2×C4×C5×D3—22no—
54C2×C4×C5×D3—21no—
55C2×C5×Dic6C2×C411yescharacteristic, complemented
56C2×C5×Dic6D411yescharacteristic
57C3×C5×M16—21no—
58C3×C5×M16C2×C412yescomplemented
59C2×C3×C5×C8—21no—
60C3×C5×C4⋊C4—21no—
61C3×C5×C22⋊C4—21no—
62C3×C42×C5—21no—
63C3×C42×C5—21no—
64C3×C42×C5D411yescharacteristic
65C3×C42×C5D411yescharacteristic
66C5×#217D412yes—
67C5×(C2×D3)⋊C4—22no—
68C5×C4⋊Dic3—21no—
69C5×Dic3⋊C4—22no—
70C4×C5×Dic3—22no—
71C4×C5×Dic3—21no—
72C5×#208—21no—
73C5×#208—21no—
74C2×C5×C3⋊C8—21no—
75C5×Dic12—22no—
76C5×D24—22no—
77C5×#204—22no—
78C5×#204—22no—
79#5748C2×C511yescharacteristic
80C4×#235C2×C511yescharacteristic, complemented
81#5006C2×C511yescharacteristic, complemented
82#5005C2×C511yescharacteristic, complemented
83C2×#805C2×C511yescharacteristic, complemented
84C3×#339C2×C511yescharacteristic, complemented
85#4559C2×C511yescharacteristic, complemented
86C2×C5×#41—31no—
87C2×C5×#41—31no—
88C5×#137—32no—
89C5×#137—61no—
90C5×AGL(1,ℤ/8)—32no—
91C5×AGL(1,ℤ/8)—61no—
92C5×#135—62no—
93C2×C5×M16—31no—
94C2×C5×M16C2×D311yescharacteristic
95C4×C5×Q8—61no—
96C4×C5×D4—61no—
97C4×C5×D4—62no—
98C5×#117—31no—
99C5×#117—61no—
100C5×#117C2×D311yescharacteristic
101C2×C42×C5—61no—
102C2×C42×C5C2×D311yescharacteristic
103C5×#104—62no—
104C5×#104—61no—
105C5×#104—61no—
106C5×#99—62no—
107#1906—31noSylow
108C23×C3×C5C2×D411yescharacteristic
109C22×C5×D3—21no—
110C22×C5×D3—21no—
111C2×C3×C4×C5—21no—
112C2×C3×C4×C5—21no—
113C2×C3×C4×C5—42no—
114C2×C3×C4×C5—41no—
115C2×C3×C4×C5C22×C411yescharacteristic
116C2×C3×C4×C5C2×D411yescharacteristic
117C2×C3×C4×C5C2×D411yescharacteristic
118C2×C3×C4×C5C2×D411yescharacteristic
119C2×C3×C4×C5#4111yescharacteristic
120C2×C3×C4×C5#4111yescharacteristic
121C5×#67—22no—
122C5×#67—22no—
123C5×#67—41no—
124C5×#67—41no—
125C2×C5×Dic3—21no—
126C2×C5×Dic3—21no—
127C2×C5×Dic3—22no—
128C5×D12—21no—
129C5×D12—21no—
130C5×D12—21no—
131C5×D12C22⋊C412yes—
132C4×C5×D3—22no—
133C4×C5×D3—22no—
134C4×C5×D3—44no—
135C4×C5×D3—41no—
136C4×C5×D3—41no—
137C5×Dic6—21no—
138C5×Dic6—21no—
139C5×Dic6—21no—
140C5×Dic6C22⋊C412yes—
141C3×C5×C8—22no—
142C5×C3⋊C8—41no—
143C2×#235C22×C511yescharacteristic
144C2×#235C22×C511yescharacteristic
145C2×C3×M16C22×C511yescharacteristic
146C3×#117C22×C511yescharacteristic
147C2×C3×C42C22×C511yescharacteristic
148C4×#67—22no—
149#926C22×C511yescharacteristic
150C2×#208C22×C511yescharacteristic
151#909—21no—
152#909C4×C512yescomplemented
153#908—21no—
154#908C4×C512yescomplemented
155#904—22no—
156C4×D12—21no—
157#872—21no—
158C42×D3—21no—
159C4×Dic6—21no—
160#806—22no—
161#805—22no—
162#805—21no—
163#805—21no—
164C5×#41—32no—
165C5×#41—32no—
166C5×#41—61no—
167C5×#41—61no—
168C5×#41—61no—
169C5×#41—61no—
170C5×#41—61no—
171C5×#41—61no—
172C2×C5×Q8—31no—
173C2×C5×Q8—31no—
174C2×C5×D4—31no—
175C2×C5×D4—31no—
176C2×C5×D4—61no—
177C2×C5×D4—61no—
178C22×C4×C5—21no—
179C22×C4×C5—62no—
180C22×C4×C5—61no—
181C22×C4×C5—61no—
182C22×C4×C5C22×D311yescharacteristic
183C5×Q16—62no—
184C5×SD16—62no—
185C5×SD16—62no—
186C5×D8—62no—
187C5×M16—21no—
188C5×M16—61no—
189C5×M16—61no—
190C5×M16C4×D312yescomplemented
191C2×C5×C8—21no—
192C2×C5×C8—61no—
193C5×C4⋊C4—21no—
194C5×C4⋊C4—62no—
195C5×C4⋊C4—61no—
196C5×C22⋊C4—21no—
197C5×C22⋊C4—32no—
198C5×C22⋊C4—62no—
199C42×C5—21no—
200C42×C5—21no—
201C42×C5—62no—
202C42×C5—61no—
203C42×C5D1211yescharacteristic
204C42×C5D1211yescharacteristic
205#575—31no—
206C4×#41—31no—
207#421—31no—
208C2×#104—31no—
209#410—31no—
210#344—31no—
211#339C5×D311yescharacteristic, complemented
212C22×C3×C5—41no—
213C22×C3×C5C4×D411yescharacteristic
214C22×C3×C5#12011yescharacteristic
215C22×C3×C5#12111yescharacteristic
216C2×C5×D3—22no—
217C2×C5×D3—42no—
218C2×C5×D3—41no—
219C2×C5×D3—81no—
220C3×C4×C5—22no—
221C3×C4×C5—41no—
222C3×C4×C5—81no—
223C3×C4×C5C2×C22⋊C411yescharacteristic
224C3×C4×C5C4×D411yescharacteristic
225C3×C4×C5#12111yescharacteristic
226C3×C4×C5#12311yescharacteristic
227C5×Dic3—22no—
228C5×Dic3—42no—
229C5×Dic3—41no—
230C22×C3×C4—21no—
231C22×C3×C4C23×C511yescharacteristic
232C2×#67—21no—
233C2×#67—21no—
234#235—21no—
235#235—21no—
236#235—21no—
237#235—21no—
238#235—21no—
239#235—21no—
240#235C2×C4×C512yes—
241#235C5×D412yes—
242C2×D12C2×C4×C511yescharacteristic, complemented
243C2×D12C5×D411yescharacteristic
244C2×C4×D3—21no—
245C2×C4×D3—22no—
246C2×C4×D3—21no—
247C2×Dic6C2×C4×C511yescharacteristic, complemented
248C2×Dic6C5×D411yescharacteristic
249C3×M16—21no—
250C3×M16C2×C4×C512yescomplemented
251C2×C3×C8—21no—
252C3×C4⋊C4—21no—
253C3×C22⋊C4—21no—
254C3×C42—21no—
255C3×C42—21no—
256C3×C42C5×D411yescharacteristic
257C3×C42C5×D411yescharacteristic
258#217C5×D412yes—
259(C2×D3)⋊C4—22no—
260C4⋊Dic3—21no—
261Dic3⋊C4—22no—
262C4×Dic3—22no—
263C4×Dic3—21no—
264#208—21no—
265#208—21no—
266C2×C3⋊C8—21no—
267Dic12—22no—
268D24—22no—
269#204—22no—
270#204—22no—
271C23×C5—61no—
272C23×C5—61no—
273C23×C5D3×D411yescharacteristic
274C5×Q8—32no—
275C5×Q8—61no—
276C5×Q8—61no—
277C5×Q8—61no—
278C5×D4—32no—
279C5×D4—62no—
280C5×D4—62no—
281C5×D4—61no—
282C5×D4—61no—
283C5×D4—61no—
284C5×D4—121no—
285C5×D4—121no—
286C2×C4×C5—21no—
287C2×C4×C5—21no—
288C2×C4×C5—42no—
289C2×C4×C5—41no—
290C2×C4×C5—61no—
291C2×C4×C5—62no—
292C2×C4×C5—62no—
293C2×C4×C5—62no—
294C2×C4×C5—61no—
295C2×C4×C5—124no—
296C2×C4×C5—121no—
297C2×C4×C5—121no—
298C2×C4×C5C2×C4×D311yescharacteristic
299C2×C4×C5C2×D1211yescharacteristic
300C2×C4×C5#23511yescharacteristic
301C2×C4×C5D3×D411yescharacteristic
302C2×C4×C5D3×D411yescharacteristic
303C2×C4×C5#23911yescharacteristic
304C5×C8—22no—
305C5×C8—121no—
306C2×#41—31no—
307C2×#41—31no—
308#137—32no—
309#137—61no—
310AGL(1,ℤ/8)—32no—
311AGL(1,ℤ/8)—61no—
312#135—62no—
313C2×M16—31no—
314C2×M16C2×C5×D311yescharacteristic
315C4×Q8—61no—
316C4×D4—61no—
317C4×D4—62no—
318#117—31no—
319#117—61no—
320#117C2×C5×D311yescharacteristic
321C2×C42—61no—
322C2×C42C2×C5×D311yescharacteristic
323#104—62no—
324#104—61no—
325#104—61no—
326#99—62no—
327C2×C3×C5—21no—
328C2×C3×C5—21no—
329C2×C3×C5—21no—
330C2×C3×C5#38611yescharacteristic
331C5×D3—42no—
332C5×D3—81no—
333C23×C3C2×C5×D411yescharacteristic
334C22×D3—21no—
335C22×D3—21no—
336C2×C3×C4—21no—
337C2×C3×C4—21no—
338C2×C3×C4—42no—
339C2×C3×C4—41no—
340C2×C3×C4C22×C4×C511yescharacteristic
341C2×C3×C4C2×C5×D411yescharacteristic
342C2×C3×C4C2×C5×D411yescharacteristic
343C2×C3×C4C2×C5×D411yescharacteristic
344C2×C3×C4C5×#4111yescharacteristic
345C2×C3×C4C5×#4111yescharacteristic
346#67—22no—
347#67—22no—
348#67—41no—
349#67—41no—
350C2×Dic3—21no—
351C2×Dic3—21no—
352C2×Dic3—22no—
353D12—21no—
354D12—21no—
355D12—21no—
356D12C5×C22⋊C412yes—
357C4×D3—22no—
358C4×D3—22no—
359C4×D3—44no—
360C4×D3—41no—
361C4×D3—41no—
362Dic6—21no—
363Dic6—21no—
364Dic6—21no—
365Dic6C5×C22⋊C412yes—
366C3×C8—22no—
367C3⋊C8—41no—
368C22×C5—41no—
369C22×C5—62no—
370C22×C5—122no—
371C22×C5—121no—
372C22×C5—241no—
373C22×C5#88211yescharacteristic
374C22×C5#88411yescharacteristic
375C22×C5#89311yescharacteristic
376C4×C5—22no—
377C4×C5—41no—
378C4×C5—62no—
379C4×C5—81no—
380C4×C5—122no—
381C4×C5—121no—
382C4×C5C4×D1211yescharacteristic
383C4×C5D3×C22⋊C411yescharacteristic
384C4×C5#89411yescharacteristic
385C4×C5#89511yescharacteristic
386#41—32no—
387#41—32no—
388#41—61no—
389#41—61no—
390#41—61no—
391#41—61no—
392#41—61no—
393#41—61no—
394C2×Q8—31no—
395C2×Q8—31no—
396C2×D4—31no—
397C2×D4—31no—
398C2×D4—61no—
399C2×D4—61no—
400C22×C4—21no—
401C22×C4—62no—
402C22×C4—61no—
403C22×C4—61no—
404C22×C4C22×C5×D311yescharacteristic
405Q16—62no—
406SD16—62no—
407SD16—62no—
408D8—62no—
409M16—21no—
410M16—61no—
411M16—61no—
412M16C4×C5×D312yescomplemented
413C2×C8—21no—
414C2×C8—61no—
415C4⋊C4—21no—
416C4⋊C4—62no—
417C4⋊C4—61no—
418C22⋊C4—21no—
419C22⋊C4—32no—
420C22⋊C4—62no—
421C42—21no—
422C42—21no—
423C42—62no—
424C42—61no—
425C42C5×D1211yescharacteristic
426C42C5×D1211yescharacteristic
427C3×C5#190611yescharacteristic, complemented, Hall
428C22×C3—41no—
429C22×C3C4×C5×D411yescharacteristic
430C22×C3C5×#12011yescharacteristic
431C22×C3C5×#12111yescharacteristic
432C2×D3—22no—
433C2×D3—42no—
434C2×D3—41no—
435C2×D3—81no—
436C3×C4—22no—
437C3×C4—41no—
438C3×C4—81no—
439C3×C4C2×C5×C22⋊C411yescharacteristic
440C3×C4C4×C5×D411yescharacteristic
441C3×C4C5×#12111yescharacteristic
442C3×C4C5×#12311yescharacteristic
443Dic3—22no—
444Dic3—42no—
445Dic3—41no—
446C2×C5—21no—
447C2×C5—21no—
448C2×C5—21no—
449C2×C5—122no—
450C2×C5—241no—
451C2×C5#466511yescharacteristic
452C23—61no—
453C23—61no—
454C23C5×D3×D411yescharacteristic
455Q8—32no—
456Q8—61no—
457Q8—61no—
458Q8—61no—
459D4—32no—
460D4—62no—
461D4—62no—
462D4—61no—
463D4—61no—
464D4—61no—
465D4—121no—
466D4—121no—
467C2×C4—21no—
468C2×C4—21no—
469C2×C4—42no—
470C2×C4—41no—
471C2×C4—61no—
472C2×C4—62no—
473C2×C4—62no—
474C2×C4—62no—
475C2×C4—61no—
476C2×C4—124no—
477C2×C4—121no—
478C2×C4—121no—
479C2×C4C2×C4×C5×D311yescharacteristic
480C2×C4C2×C5×D1211yescharacteristic
481C2×C4C5×#23511yescharacteristic
482C2×C4C5×D3×D411yescharacteristic
483C2×C4C5×D3×D411yescharacteristic
484C2×C4C5×#23911yescharacteristic
485C8—22no—
486C8—121no—
487C2×C3—21no—
488C2×C3—21no—
489C2×C3—21no—
490C2×C3C5×#38611yescharacteristic
491D3—42no—
492D3—81no—
493C5#6877811yescharacteristic, direct product factor, Sylow
494C22—41no—
495C22—62no—
496C22—122no—
497C22—121no—
498C22—241no—
499C22C5×#88211yescharacteristic
500C22C5×#88411yescharacteristic
501C22C5×#89311yescharacteristic
502C4—22no—
503C4—41no—
504C4—62no—
505C4—81no—
506C4—122no—
507C4—121no—
508C4C4×C5×D1211yescharacteristic
509C4C5×D3×C22⋊C411yescharacteristic
510C4C5×#89411yescharacteristic
511C4C5×#89511yescharacteristic
512C3C5×#190611yescharacteristic, complemented, Sylow
513C2—21no—
514C2—21no—
515C2—21no—
516C2—122no—
517C2—241no—
518C2C5×#466511yescharacteristic
subgroupG/H
centerC4×C5#895
commutatorC2×C3×C4C22×C4×C5
FrattiniC22×C4C22×C5×D3
FittingC3×C5×#339C2
socleC2×C3×C5#386
prime radicalC5×#5748C2
cosocleC22×C3×C4C23×C5
hypercenterC5×#339D3
hypocenterC3C5×#1906
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C5×#339

Complements

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

NH ≅ G/Ntype
2. C4×C5×#235C2semidirect
3. C5×#5006C2semidirect
4. C5×#5005C2semidirect
5. C2×C5×#805C2semidirect
6. C3×C5×#339C2semidirect
7. C5×#4559C2semidirect
18. C5×#909C4semidirect
20. C5×#908C4semidirect
30. #68778C5direct (N × H)
37. C5×#339D3semidirect
50. C2×C5×D12C2×C4semidirect
55. C2×C5×Dic6C2×C4semidirect
58. C3×C5×M16C2×C4semidirect
80. C4×#235C2×C5semidirect
81. #5006C2×C5semidirect
82. #5005C2×C5semidirect
83. C2×#805C2×C5semidirect
84. C3×#339C2×C5semidirect
85. #4559C2×C5semidirect
152. #909C4×C5semidirect
154. #908C4×C5semidirect
190. C5×M16C4×D3semidirect
211. #339C5×D3semidirect
242. C2×D12C2×C4×C5semidirect
247. C2×Dic6C2×C4×C5semidirect
250. C3×M16C2×C4×C5semidirect
412. M16C4×C5×D3semidirect
427. C3×C5#1906semidirect
493. C5#68778direct (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
C5×#57482
C4×C5×#2352
C5×#50062
C5×#50052
C2×C5×#8052
C3×C5×#3392
C5×#45592
C5×#19063
#687785