Finite Group Databasebeta

← C5×#70012 #14301163

Subgroups — C5×#70012 #14301163

Order 1920.

620 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#5822C211yesmaximal, characteristic
2C5×#5745C211yesmaximal, characteristic, complemented
3C3×C5×#408C211yesmaximal, characteristic, complemented
4C5×#5236C211yesmaximal, characteristic, complemented
5C5×#5113C211yesmaximal, characteristic, complemented
6C5×#4556C211yesmaximal, characteristic, complemented
7C5×#4552C211yesmaximal, characteristic, complemented
8C5×#1881—31nomaximal, Hall
9C2×C3×C5×#41C2211yescharacteristic
10C2×C5×#235C2211yescharacteristic
11C2×C3×C5×M16C2211yescharacteristic
12C22×C3×C5×C8C2211yescharacteristic
13C5×#951—21no—
14C5×#950—22no—
15C5×#949—21no—
16C5×#942C412yescomplemented
17C5×#932C412yescomplemented
18C5×#926C2211yescharacteristic
19C2×C5×#208C2211yescharacteristic
20C2×C5×#208C2211yescharacteristic
21C5×#907—22no—
22C5×D3×M16—22no—
23C5×#901—22no—
24C3×C5×C22⋊C8—22no—
25C5×#837—22no—
26C5×#821—21no—
27C5×#816—21no—
28#70012C511yesmaximal, characteristic, direct product factor, Hall
29C5×#575—31no—
30C5×#568—31no—
31C5×#421—31no—
32C5×#416—31no—
33C5×#408D311yescharacteristic, complemented
34C5×#344—31no—
35C5×#341—31no—
36C3×C5×#41—21no—
37C3×C5×#41—21no—
38C3×C5×#41—21no—
39C3×C5×#41—21no—
40C2×C3×C5×Q8C2×C411yescharacteristic
41C2×C3×C5×D4—21no—
42C2×C3×C5×D4C2×C411yescharacteristic
43C22×C3×C4×C5—21no—
44C22×C3×C4×C5C2311yescharacteristic
45C2×C5×#67—21no—
46C5×#235—21no—
47C5×#235—21no—
48C5×#235C2×C412yes—
49C5×#235D412yes—
50C2×C5×D12D411yescharacteristic
51C2×C4×C5×D3—21no—
52C2×C5×Dic6D411yescharacteristic
53C3×C5×M16—21no—
54C3×C5×M16—21no—
55C2×C3×C5×C8—21no—
56C2×C3×C5×C8—21no—
57C2×C3×C5×C8—21no—
58C2×C3×C5×C8D412yes—
59C5×#217—21no—
60C5×#216—22no—
61C5×#215—22no—
62C5×#214—22no—
63C5×#213—22no—
64C5×C4⋊Dic3—21no—
65C4×C5×Dic3—21no—
66C5×#208—21no—
67C5×#208—21no—
68C5×#208C2×C412yes—
69C5×#208D412yes—
70C2×C5×C3⋊C8—21no—
71C2×C5×C3⋊C8—21no—
72C5×ΓL(1,𝔽25)—24no—
73C5×D3×C8—24no—
74#5822C2×C511yescharacteristic
75#5745C2×C511yescharacteristic, complemented
76C3×#408C2×C511yescharacteristic, complemented
77#5236C2×C511yescharacteristic, complemented
78#5113C2×C511yescharacteristic, complemented
79#4556C2×C511yescharacteristic, complemented
80#4552C2×C511yescharacteristic, complemented
81C2×C5×#41—31no—
82C2×C5×#41C2×D311yescharacteristic
83C5×#137—32no—
84C5×#137—61no—
85C5×AGL(1,ℤ/8)—32no—
86C5×AGL(1,ℤ/8)—61no—
87C5×#135—62no—
88C5×#131—62no—
89C5×#131—62no—
90C2×C5×M16—31no—
91C2×C5×M16—31no—
92C2×C5×M16—62no—
93C2×C5×M16C2×D311yescharacteristic
94C22×C5×C8C2×D311yescharacteristic
95C5×#117—31no—
96C5×#104—62no—
97C5×#103—61no—
98C5×#102—61no—
99C5×C22⋊C8—22no—
100#1881—31noSylow
101C23×C3×C5—21no—
102C23×C3×C5#4111yescharacteristic
103C22×C5×D3—21no—
104C3×C5×Q8—21no—
105C3×C5×Q8—21no—
106C3×C5×D4—21no—
107C3×C5×D4—21no—
108C3×C5×D4—41no—
109C3×C5×D4—41no—
110C2×C3×C4×C5—21no—
111C2×C3×C4×C5—41no—
112C2×C3×C4×C5—41no—
113C2×C3×C4×C5C22×C411yescharacteristic
114C2×C3×C4×C5C2×D411yescharacteristic
115C2×C3×C4×C5C2×D411yescharacteristic
116C2×C3×C4×C5C2×D411yescharacteristic
117C2×C3×C4×C5C2×D411yescharacteristic
118C2×C3×C4×C5#4111yescharacteristic
119C5×#67—22no—
120C5×#67—22no—
121C2×C5×Dic3—21no—
122C2×C5×Dic3—41no—
123C5×D12—21no—
124C5×D12C22⋊C412yes—
125C4×C5×D3—22no—
126C4×C5×D3—22no—
127C5×Dic6—21no—
128C5×Dic6C22⋊C412yes—
129C3×C5×C8—22no—
130C3×C5×C8—41no—
131C5×C3⋊C8—22no—
132C5×C3⋊C8—22no—
133C2×C3×#41C22×C511yescharacteristic
134C2×#235C22×C511yescharacteristic
135C2×C3×M16C22×C511yescharacteristic
136C22×C3×C8C22×C511yescharacteristic
137#951—21no—
138#950—22no—
139#949—21no—
140#942C4×C512yescomplemented
141#932C4×C512yescomplemented
142#926C22×C511yescharacteristic
143C2×#208C22×C511yescharacteristic
144C2×#208C22×C511yescharacteristic
145#907—22no—
146D3×M16—22no—
147#901—22no—
148C3×C22⋊C8—22no—
149#837—22no—
150#821—21no—
151#816—21no—
152C5×#41—21no—
153C5×#41—21no—
154C5×#41—21no—
155C5×#41—21no—
156C5×#41—32no—
157C5×#41—32no—
158C5×#41—61no—
159C5×#41—61no—
160C2×C5×Q8—31no—
161C2×C5×Q8C4×D311yescharacteristic
162C2×C5×D4—21no—
163C2×C5×D4—31no—
164C2×C5×D4—61no—
165C2×C5×D4C4×D311yescharacteristic
166C22×C4×C5—21no—
167C22×C4×C5—61no—
168C22×C4×C5C22×D311yescharacteristic
169C5×Q16—62no—
170C5×SD16—62no—
171C5×SD16—62no—
172C5×D8—62no—
173C5×M16—21no—
174C5×M16—21no—
175C5×M16—32no—
176C5×M16—32no—
177C5×M16—64no—
178C5×M16—61no—
179C5×M16—61no—
180C2×C5×C8—21no—
181C2×C5×C8—21no—
182C2×C5×C8—21no—
183C2×C5×C8—64no—
184C2×C5×C8—61no—
185C2×C5×C8—61no—
186C2×C5×C8#6712yes—
187C5×C4⋊C4—61no—
188C5×C22⋊C4—61no—
189C42×C5—61no—
190#575—31no—
191#568—31no—
192#421—31no—
193#416—31no—
194#408C5×D311yescharacteristic, complemented
195#344—31no—
196#341—31no—
197C22×C3×C5—41no—
198C22×C3×C5—42no—
199C22×C3×C5—41no—
200C22×C3×C5C4×D411yescharacteristic
201C22×C3×C5#12111yescharacteristic
202C22×C3×C5#12311yescharacteristic
203C2×C5×D3—22no—
204C2×C5×D3—81no—
205C3×C4×C5—41no—
206C3×C4×C5C2×C22⋊C411yescharacteristic
207C3×C4×C5C4×D411yescharacteristic
208C3×C4×C5#12011yescharacteristic
209C3×C4×C5#12111yescharacteristic
210C5×Dic3—22no—
211C5×Dic3—81no—
212C3×#41—21no—
213C3×#41—21no—
214C3×#41—21no—
215C3×#41—21no—
216C2×C3×Q8C2×C4×C511yescharacteristic
217C2×C3×D4—21no—
218C2×C3×D4C2×C4×C511yescharacteristic
219C22×C3×C4—21no—
220C22×C3×C4C23×C511yescharacteristic
221C2×#67—21no—
222#235—21no—
223#235—21no—
224#235C2×C4×C512yes—
225#235C5×D412yes—
226C2×D12C5×D411yescharacteristic
227C2×C4×D3—21no—
228C2×Dic6C5×D411yescharacteristic
229C3×M16—21no—
230C3×M16—21no—
231C2×C3×C8—21no—
232C2×C3×C8—21no—
233C2×C3×C8—21no—
234C2×C3×C8C5×D412yes—
235#217—21no—
236#216—22no—
237#215—22no—
238#214—22no—
239#213—22no—
240C4⋊Dic3—21no—
241C4×Dic3—21no—
242#208—21no—
243#208—21no—
244#208C2×C4×C512yes—
245#208C5×D412yes—
246C2×C3⋊C8—21no—
247C2×C3⋊C8—21no—
248ΓL(1,𝔽25)—24no—
249D3×C8—24no—
250C23×C5—21no—
251C23×C5—61no—
252C23×C5#23511yescharacteristic
253C5×Q8—21no—
254C5×Q8—21no—
255C5×Q8—32no—
256C5×Q8—61no—
257C5×D4—21no—
258C5×D4—21no—
259C5×D4—32no—
260C5×D4—41no—
261C5×D4—41no—
262C5×D4—62no—
263C5×D4—62no—
264C5×D4—61no—
265C2×C4×C5—21no—
266C2×C4×C5—41no—
267C2×C4×C5—41no—
268C2×C4×C5—61no—
269C2×C4×C5—62no—
270C2×C4×C5—62no—
271C2×C4×C5—121no—
272C2×C4×C5C2×C4×D311yescharacteristic
273C2×C4×C5D3×D411yescharacteristic
274C2×C4×C5D3×D411yescharacteristic
275C2×C4×C5D3×D411yescharacteristic
276C2×C4×C5#23711yescharacteristic
277C2×C4×C5C2×#6711yescharacteristic
278C5×C8—22no—
279C5×C8—41no—
280C5×C8—62no—
281C5×C8—62no—
282C2×#41—31no—
283C2×#41C2×C5×D311yescharacteristic
284#137—32no—
285#137—61no—
286AGL(1,ℤ/8)—32no—
287AGL(1,ℤ/8)—61no—
288#135—62no—
289#131—62no—
290#131—62no—
291C2×M16—31no—
292C2×M16—31no—
293C2×M16—62no—
294C2×M16C2×C5×D311yescharacteristic
295C22×C8C2×C5×D311yescharacteristic
296#117—31no—
297#104—62no—
298#103—61no—
299#102—61no—
300C22⋊C8—22no—
301C2×C3×C5—21no—
302C2×C3×C5—21no—
303C2×C3×C5—21no—
304C2×C3×C5—81no—
305C2×C3×C5#38611yescharacteristic
306C5×D3—42no—
307C23×C3—21no—
308C23×C3C5×#4111yescharacteristic
309C22×D3—21no—
310C3×Q8—21no—
311C3×Q8—21no—
312C3×D4—21no—
313C3×D4—21no—
314C3×D4—41no—
315C3×D4—41no—
316C2×C3×C4—21no—
317C2×C3×C4—41no—
318C2×C3×C4—41no—
319C2×C3×C4C22×C4×C511yescharacteristic
320C2×C3×C4C2×C5×D411yescharacteristic
321C2×C3×C4C2×C5×D411yescharacteristic
322C2×C3×C4C2×C5×D411yescharacteristic
323C2×C3×C4C2×C5×D411yescharacteristic
324C2×C3×C4C5×#4111yescharacteristic
325#67—22no—
326#67—22no—
327C2×Dic3—21no—
328C2×Dic3—41no—
329D12—21no—
330D12C5×C22⋊C412yes—
331C4×D3—22no—
332C4×D3—22no—
333Dic6—21no—
334Dic6C5×C22⋊C412yes—
335C3×C8—22no—
336C3×C8—41no—
337C3⋊C8—22no—
338C3⋊C8—22no—
339C22×C5—41no—
340C22×C5—42no—
341C22×C5—41no—
342C22×C5—62no—
343C22×C5—241no—
344C22×C5#88311yescharacteristic
345C22×C5#88411yescharacteristic
346C22×C5C4×#6711yescharacteristic
347C4×C5—41no—
348C4×C5—62no—
349C4×C5—241no—
350C4×C5D3×C22⋊C411yescharacteristic
351C4×C5#67⋊C411yescharacteristic
352C4×C5#93711yescharacteristic
353C4×C5#93811yescharacteristic
354#41—21no—
355#41—21no—
356#41—21no—
357#41—21no—
358#41—32no—
359#41—32no—
360#41—61no—
361#41—61no—
362C2×Q8—31no—
363C2×Q8C4×C5×D311yescharacteristic
364C2×D4—21no—
365C2×D4—31no—
366C2×D4—61no—
367C2×D4C4×C5×D311yescharacteristic
368C22×C4—21no—
369C22×C4—61no—
370C22×C4C22×C5×D311yescharacteristic
371Q16—62no—
372SD16—62no—
373SD16—62no—
374D8—62no—
375M16—21no—
376M16—21no—
377M16—32no—
378M16—32no—
379M16—64no—
380M16—61no—
381M16—61no—
382C2×C8—21no—
383C2×C8—21no—
384C2×C8—21no—
385C2×C8—64no—
386C2×C8—61no—
387C2×C8—61no—
388C2×C8C5×#6712yes—
389C4⋊C4—61no—
390C22⋊C4—61no—
391C42—61no—
392C3×C5#188111yescharacteristic, complemented, Hall
393C22×C3—41no—
394C22×C3—42no—
395C22×C3—41no—
396C22×C3C4×C5×D411yescharacteristic
397C22×C3C5×#12111yescharacteristic
398C22×C3C5×#12311yescharacteristic
399C2×D3—22no—
400C2×D3—81no—
401C3×C4—41no—
402C3×C4C2×C5×C22⋊C411yescharacteristic
403C3×C4C4×C5×D411yescharacteristic
404C3×C4C5×#12011yescharacteristic
405C3×C4C5×#12111yescharacteristic
406Dic3—22no—
407Dic3—81no—
408C2×C5—21no—
409C2×C5—21no—
410C2×C5—21no—
411C2×C5—81no—
412C2×C5—122no—
413C2×C5#495611yescharacteristic
414C23—21no—
415C23—61no—
416C23C5×#23511yescharacteristic
417Q8—21no—
418Q8—21no—
419Q8—32no—
420Q8—61no—
421D4—21no—
422D4—21no—
423D4—32no—
424D4—41no—
425D4—41no—
426D4—62no—
427D4—62no—
428D4—61no—
429C2×C4—21no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—61no—
433C2×C4—62no—
434C2×C4—62no—
435C2×C4—121no—
436C2×C4C2×C4×C5×D311yescharacteristic
437C2×C4C5×D3×D411yescharacteristic
438C2×C4C5×D3×D411yescharacteristic
439C2×C4C5×D3×D411yescharacteristic
440C2×C4C5×#23711yescharacteristic
441C2×C4C2×C5×#6711yescharacteristic
442C8—22no—
443C8—41no—
444C8—62no—
445C8—62no—
446C2×C3—21no—
447C2×C3—21no—
448C2×C3—21no—
449C2×C3—81no—
450C2×C3C5×#38611yescharacteristic
451D3—42no—
452C5#7001211yescharacteristic, direct product factor, Sylow
453C22—41no—
454C22—42no—
455C22—41no—
456C22—62no—
457C22—241no—
458C22C5×#88311yescharacteristic
459C22C5×#88411yescharacteristic
460C22C4×C5×#6711yescharacteristic
461C4—41no—
462C4—62no—
463C4—241no—
464C4C5×D3×C22⋊C411yescharacteristic
465C4C5×#67⋊C411yescharacteristic
466C4C5×#93711yescharacteristic
467C4C5×#93811yescharacteristic
468C3C5×#188111yescharacteristic, complemented, Sylow
469C2—21no—
470C2—21no—
471C2—21no—
472C2—81no—
473C2—122no—
474C2C5×#495611yescharacteristic
subgroupG/H
centerC4×C5#937
commutatorC2×C3×C4C22×C4×C5
FrattiniC22×C4C22×C5×D3
FittingC3×C5×#408C2
socleC2×C3×C5#386
prime radicalC5×#5822C2
cosocleC22×C3×C4C23×C5
hypercenterC5×#408D3
hypocenterC3C5×#1881
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C5×#408

Complements

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

NH ≅ G/Ntype
2. C5×#5745C2semidirect
3. C3×C5×#408C2semidirect
4. C5×#5236C2semidirect
5. C5×#5113C2semidirect
6. C5×#4556C2semidirect
7. C5×#4552C2semidirect
16. C5×#942C4semidirect
17. C5×#932C4semidirect
28. #70012C5direct (N × H)
33. C5×#408D3semidirect
75. #5745C2×C5semidirect
76. C3×#408C2×C5semidirect
77. #5236C2×C5semidirect
78. #5113C2×C5semidirect
79. #4556C2×C5semidirect
80. #4552C2×C5semidirect
140. #942C4×C5semidirect
141. #932C4×C5semidirect
194. #408C5×D3semidirect
392. C3×C5#1881semidirect
452. C5#70012direct (N × H)
468. C3C5×#1881semidirect

Maximal subgroups

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

subgroupindex
C5×#58222
C5×#57452
C3×C5×#4082
C5×#52362
C5×#51132
C5×#45562
C5×#45522
C5×#18813
#700125