Finite Group Databasebeta

← C2×C5×#3509 #11985524

Subgroups — C2×C5×#3509 #11985524

Order 1280.

7022 conjugacy classes of subgroups.

The subgroup lattice has 456 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3509C2132yesmaximal, direct product factor
2C2×C5×#557C214yesmaximal, complemented
3C2×C5×#556C214yesmaximal, complemented
4C2×C5×#552C218yesmaximal, complemented
5C2×C5×#541C214yesmaximal, complemented
6C2×C5×#536C212yesmaximal
7C2×C5×#533C211yesmaximal, characteristic, complemented
8C2×C5×#533C214yesmaximal, complemented
9C22×C5×C4⋊Q8C211yesmaximal, characteristic, complemented
10C2×C5×#519C212yesmaximal, complemented
11C2×C4×C5×#41C211yesmaximal, characteristic
12C22×C5×#41C2211yescharacteristic
13C23×C5×Q8C2212yes—
14C5×#557C22164yescomplemented
15C5×#556C22164yescomplemented
16C5×#552C221128yescomplemented
17C5×#541C22164yescomplemented
18C5×#536C22132yes—
19C5×#533C22116yescomplemented
20C5×#533C22164yescomplemented
21C2×C5×C4⋊Q8C2212yes—
22C2×C5×C4⋊Q8C2212yes—
23C2×C5×C4⋊Q8C2214yes—
24C2×C5×C4⋊Q8C2212yes—
25C2×C5×C4⋊Q8C2214yes—
26C2×C5×C4⋊Q8C22116yescomplemented
27C2×C5×#126C22116yes—
28C2×C5×#125C2212yes—
29C2×C5×#125C2218yes—
30C2×C5×#125C22116yes—
31C2×C5×#124C2214yes—
32C2×C5×#123C2218yes—
33C2×C5×C22⋊Q8C2218yes—
34C2×C5×C22⋊Q8C2214yes—
35C2×C5×C22⋊Q8C2218yes—
36C5×#519C22132yescomplemented
37C4×C5×#41C22116yes—
38C2×C4×C5×Q8C2212yes—
39C2×C4×C5×Q8C2218yes—
40C2×C4×C5×Q8C2218yes—
41C2×C4×C5×Q8C2218yes—
42C2×C4×C5×Q8C2214yes—
43C2×C4×C5×D4C2214yes—
44C2×C4×C5×D4C2212yes—
45C2×C5×#117C2211yescharacteristic
46C2×C5×#117C2212yes—
47C2×C5×#117C2212yes—
48C2×C5×#117C2218yes—
49C2×C5×#117C2218yes—
50C22×C5×C4⋊C4C2214yes—
51C22×C42×C5C2211yescharacteristic
52C22×C42×C5C2212yes—
53C2×#3509C511yesmaximal, characteristic, direct product factor, Sylow
54C2×C5×#41—28no—
55C2×C5×#41C2318yes—
56C22×C5×Q8—216no—
57C22×C5×Q8C2314yes—
58C22×C5×Q8C2314yes—
59C22×C5×Q8C2314yes—
60C22×C5×Q8C2311yescharacteristic
61C22×C5×Q8C23116yes—
62C22×C5×D4C2312yes—
63C22×C5×D4C2311yescharacteristic
64C23×C4×C5C2312yes—
65C23×C4×C5C2314yes—
66C23×C4×C5C2312yes—
67C23×C4×C5C2314yes—
68C23×C4×C5C2311yescharacteristic
69C5×C4⋊Q8C23116yes—
70C5×C4⋊Q8C23116yes—
71C5×C4⋊Q8C23132yes—
72C5×C4⋊Q8C23116yes—
73C5×C4⋊Q8C23132yes—
74C5×#126C231128yes—
75C5×#125C23116yes—
76C5×#125C23164yes—
77C5×#125C231128yes—
78C5×#124C23132yes—
79C5×#123C23164yes—
80C5×C22⋊Q8C23164yes—
81C5×C22⋊Q8C23132yes—
82C5×C22⋊Q8C23164yes—
83C4×C5×Q8C23116yes—
84C4×C5×Q8C23164yes—
85C4×C5×Q8C23164yes—
86C4×C5×Q8C23164yes—
87C4×C5×Q8C23132yes—
88C4×C5×D4C23132yes—
89C4×C5×D4C23116yes—
90C5×#117C2318yes—
91C5×#117C23116yes—
92C5×#117C23116yes—
93C5×#117C23164yes—
94C5×#117C23164yes—
95C2×C5×C4⋊C4C2318yes—
96C2×C5×C4⋊C4C2318yes—
97C2×C5×C4⋊C4C23132yes—
98C2×C5×C4⋊C4C2314yes—
99C2×C5×C4⋊C4C2312yes—
100C2×C5×C4⋊C4C2314yes—
101C2×C5×C4⋊C4C2314yes—
102C2×C5×C4⋊C4C23116yes—
103C2×C5×C4⋊C4C23116yes—
104C2×C5×C4⋊C4C2318yes—
105C2×C5×C22⋊C4C2312yes—
106C2×C5×C22⋊C4C2314yes—
107C2×C5×C22⋊C4C2314yes—
108C2×C5×C22⋊C4C23116yes—
109C2×C42×C5C2311yescharacteristic
110C2×C42×C5C2312yes—
111C2×C42×C5C2311yescharacteristic
112C2×C42×C5C2314yes—
113C2×C42×C5C2312yes—
114C2×C42×C5C2318yes—
115C2×C42×C5C23116yes—
116C2×C42×C5C23116yes—
117C2×C42×C5C2314yes—
118#3509C2×C5132yesdirect product factor
119C2×#557C2×C514yescomplemented
120C2×#556C2×C514yescomplemented
121C2×#552C2×C518yescomplemented
122C2×#541C2×C514yescomplemented
123C2×#536C2×C512yes—
124C2×#533C2×C511yescharacteristic, complemented
125C2×#533C2×C514yescomplemented
126C22×C4⋊Q8C2×C511yescharacteristic, complemented
127C2×#519C2×C512yescomplemented
128C2×C4×#41C2×C511yescharacteristic
129C24×C5C2412yes—
130C24×C5C2411yescharacteristic
131C5×#41—232no—
132C2×C5×Q8—216no—
133C2×C5×Q8—216no—
134C2×C5×Q8—264no—
135C2×C5×Q8—24no—
136C2×C5×Q8—216no—
137C2×C5×Q8C24116yes—
138C2×C5×Q8C24116yes—
139C2×C5×Q8C24116yes—
140C2×C5×Q8C2414yes—
141C2×C5×D4—28no—
142C2×C5×D4—24no—
143C2×C5×D4C2418yes—
144C2×C5×D4C2414yes—
145C22×C4×C5—216no—
146C22×C4×C5—28no—
147C22×C4×C5—28no—
148C22×C4×C5—216no—
149C22×C4×C5#4118yes—
150C22×C4×C5C2418yes—
151C22×C4×C5C24116yes—
152C22×C4×C5C2418yes—
153C22×C4×C5C24116yes—
154C22×C4×C5C2414yes—
155C22×C4×C5C2418yes—
156C22×C4×C5C2412yes—
157C22×C4×C5C2418yes—
158C22×C4×C5C2412yes—
159C22×C4×C5C2414yes—
160C22×C4×C5C2411yescharacteristic
161C22×C4×C5C2411yescharacteristic
162C22×C4×C5C2412yes—
163C5×C4⋊C4C24132yes—
164C5×C4⋊C4C24132yes—
165C5×C4⋊C4C24116yes—
166C5×C4⋊C4C2418yes—
167C5×C4⋊C4C24116yes—
168C5×C4⋊C4C24116yes—
169C5×C4⋊C4C24164yes—
170C5×C4⋊C4C24164yes—
171C5×C4⋊C4C24132yes—
172C5×C22⋊C4C2418yes—
173C5×C22⋊C4C24116yes—
174C5×C22⋊C4C24116yes—
175C5×C22⋊C4C24164yes—
176C42×C5C2414yes—
177C42×C5C2418yes—
178C42×C5C2414yes—
179C42×C5C24116yes—
180C42×C5C2418yes—
181C42×C5C24164yes—
182C42×C5C24116yes—
183C22×#41C22×C511yescharacteristic
184C23×Q8C22×C512yes—
185#557C22×C5164yescomplemented
186#556C22×C5164yescomplemented
187#552C22×C51128yescomplemented
188#541C22×C5164yescomplemented
189#536C22×C5132yes—
190#533C22×C5116yescomplemented
191#533C22×C5164yescomplemented
192C2×C4⋊Q8C22×C512yes—
193C2×C4⋊Q8C22×C512yes—
194C2×C4⋊Q8C22×C514yes—
195C2×C4⋊Q8C22×C512yes—
196C2×C4⋊Q8C22×C514yes—
197C2×C4⋊Q8C22×C5116yescomplemented
198C2×#126C22×C5116yes—
199C2×#125C22×C512yes—
200C2×#125C22×C518yes—
201C2×#125C22×C5116yes—
202C2×#124C22×C514yes—
203C2×#123C22×C518yes—
204C2×C22⋊Q8C22×C518yes—
205C2×C22⋊Q8C22×C514yes—
206C2×C22⋊Q8C22×C518yes—
207#519C22×C5132yescomplemented
208C4×#41C22×C5116yes—
209C2×C4×Q8C22×C512yes—
210C2×C4×Q8C22×C518yes—
211C2×C4×Q8C22×C518yes—
212C2×C4×Q8C22×C518yes—
213C2×C4×Q8C22×C514yes—
214C2×C4×D4C22×C514yes—
215C2×C4×D4C22×C512yes—
216C2×#117C22×C511yescharacteristic
217C2×#117C22×C512yes—
218C2×#117C22×C512yes—
219C2×#117C22×C518yes—
220C2×#117C22×C518yes—
221C22×C4⋊C4C22×C514yes—
222C22×C42C22×C511yescharacteristic
223C22×C42C22×C512yes—
224C23×C5—24no—
225C23×C5—24no—
226C23×C5—28no—
227C23×C5C2×#4114yes—
228C23×C5C2514yes—
229C23×C5C2512yes—
230C23×C5C2511yescharacteristic
231C5×Q8—232no—
232C5×Q8—232no—
233C5×Q8—28no—
234C5×Q8—232no—
235C5×D4—216no—
236C5×D4—28no—
237C2×C4×C5—24no—
238C2×C4×C5—232no—
239C2×C4×C5—216no—
240C2×C4×C5—28no—
241C2×C4×C5—24no—
242C2×C4×C5—216no—
243C2×C4×C5—216no—
244C2×C4×C5—216no—
245C2×C4×C5—232no—
246C2×C4×C5C2×#4114yes—
247C2×C4×C5C2×#4114yes—
248C2×C4×C5C2×#41116yes—
249C2×C4×C52−1+418yes—
250C2×C4×C5C2518yes—
251C2×C4×C5C25116yes—
252C2×C4×C5C2514yes—
253C2×C4×C5C25116yes—
254C2×C4×C5C2514yes—
255C2×C4×C5C2512yes—
256C2×C4×C5C2512yes—
257C2×C4×C5C2514yes—
258C2×#41—28no—
259C2×#41C23×C518yes—
260C22×Q8—216no—
261C22×Q8C23×C514yes—
262C22×Q8C23×C514yes—
263C22×Q8C23×C514yes—
264C22×Q8C23×C511yescharacteristic
265C22×Q8C23×C5116yes—
266C22×D4C23×C512yes—
267C22×D4C23×C511yescharacteristic
268C23×C4C23×C514yes—
269C23×C4C23×C512yes—
270C23×C4C23×C514yes—
271C23×C4C23×C512yes—
272C23×C4C23×C511yescharacteristic
273C4⋊Q8C23×C5116yes—
274C4⋊Q8C23×C5116yes—
275C4⋊Q8C23×C5132yes—
276C4⋊Q8C23×C5116yes—
277C4⋊Q8C23×C5132yes—
278#126C23×C51128yes—
279#125C23×C5116yes—
280#125C23×C5164yes—
281#125C23×C51128yes—
282#124C23×C5132yes—
283#123C23×C5164yes—
284C22⋊Q8C23×C5164yes—
285C22⋊Q8C23×C5132yes—
286C22⋊Q8C23×C5164yes—
287C4×Q8C23×C5116yes—
288C4×Q8C23×C5164yes—
289C4×Q8C23×C5164yes—
290C4×Q8C23×C5164yes—
291C4×Q8C23×C5132yes—
292C4×D4C23×C5132yes—
293C4×D4C23×C5116yes—
294#117C23×C518yes—
295#117C23×C5116yes—
296#117C23×C5116yes—
297#117C23×C5164yes—
298#117C23×C5164yes—
299C2×C4⋊C4C23×C518yes—
300C2×C4⋊C4C23×C518yes—
301C2×C4⋊C4C23×C5132yes—
302C2×C4⋊C4C23×C514yes—
303C2×C4⋊C4C23×C512yes—
304C2×C4⋊C4C23×C514yes—
305C2×C4⋊C4C23×C514yes—
306C2×C4⋊C4C23×C5116yes—
307C2×C4⋊C4C23×C5116yes—
308C2×C4⋊C4C23×C518yes—
309C2×C22⋊C4C23×C512yes—
310C2×C22⋊C4C23×C514yes—
311C2×C22⋊C4C23×C514yes—
312C2×C22⋊C4C23×C5116yes—
313C2×C42C23×C512yes—
314C2×C42C23×C5116yes—
315C2×C42C23×C511yescharacteristic
316C2×C42C23×C514yes—
317C2×C42C23×C514yes—
318C2×C42C23×C512yes—
319C2×C42C23×C518yes—
320C2×C42C23×C5116yes—
321C2×C42C23×C511yescharacteristic
322C22×C5—24no—
323C22×C5—28no—
324C22×C5—24no—
325C22×C5—28no—
326C22×C5—48no—
327C22×C5C22×#4114yes—
328C22×C5C22×#4112yes—
329C22×C5C2×2−1+414yes—
330C22×C5C2611yescharacteristic
331C4×C5—24no—
332C4×C5—216no—
333C4×C5—28no—
334C4×C5—24no—
335C4×C5—216no—
336C4×C5C22×#4114yes—
337C4×C5C22×#4114yes—
338C4×C5C2×2−1+418yes—
339C24C24×C512yes—
340C24C24×C511yescharacteristic
341#41—232no—
342C2×Q8—216no—
343C2×Q8—216no—
344C2×Q8—264no—
345C2×Q8—24no—
346C2×Q8—216no—
347C2×Q8C24×C5116yes—
348C2×Q8C24×C5116yes—
349C2×Q8C24×C5116yes—
350C2×Q8C24×C514yes—
351C2×D4—28no—
352C2×D4—24no—
353C2×D4C24×C518yes—
354C2×D4C24×C514yes—
355C22×C4—216no—
356C22×C4—28no—
357C22×C4—28no—
358C22×C4—216no—
359C22×C4C5×#4118yes—
360C22×C4C24×C514yes—
361C22×C4C24×C518yes—
362C22×C4C24×C512yes—
363C22×C4C24×C518yes—
364C22×C4C24×C512yes—
365C22×C4C24×C518yes—
366C22×C4C24×C5116yes—
367C22×C4C24×C518yes—
368C22×C4C24×C5116yes—
369C22×C4C24×C511yescharacteristic
370C22×C4C24×C511yescharacteristic
371C22×C4C24×C514yes—
372C22×C4C24×C512yes—
373C4⋊C4C24×C5132yes—
374C4⋊C4C24×C5132yes—
375C4⋊C4C24×C5116yes—
376C4⋊C4C24×C518yes—
377C4⋊C4C24×C5116yes—
378C4⋊C4C24×C5116yes—
379C4⋊C4C24×C5164yes—
380C4⋊C4C24×C5164yes—
381C4⋊C4C24×C5132yes—
382C22⋊C4C24×C518yes—
383C22⋊C4C24×C5116yes—
384C22⋊C4C24×C5116yes—
385C22⋊C4C24×C5164yes—
386C42C24×C514yes—
387C42C24×C518yes—
388C42C24×C514yes—
389C42C24×C5116yes—
390C42C24×C518yes—
391C42C24×C5164yes—
392C42C24×C5116yes—
393C2×C5—24no—
394C2×C5—44no—
395C2×C5#350914yesdirect product factor
396C2×C5C23×#4111yescharacteristic
397C2×C5C22×2−1+412yes—
398C23—24no—
399C23—24no—
400C23—28no—
401C23C2×C5×#4114yes—
402C23C25×C514yes—
403C23C25×C512yes—
404C23C25×C511yescharacteristic
405Q8—232no—
406Q8—232no—
407Q8—28no—
408Q8—232no—
409D4—216no—
410D4—28no—
411C2×C4—24no—
412C2×C4—232no—
413C2×C4—216no—
414C2×C4—28no—
415C2×C4—24no—
416C2×C4—216no—
417C2×C4—216no—
418C2×C4—216no—
419C2×C4—232no—
420C2×C4C2×C5×#4114yes—
421C2×C4C2×C5×#4114yes—
422C2×C4C2×C5×#41116yes—
423C2×C4C5×2−1+418yes—
424C2×C4C25×C518yes—
425C2×C4C25×C5116yes—
426C2×C4C25×C514yes—
427C2×C4C25×C5116yes—
428C2×C4C25×C514yes—
429C2×C4C25×C512yes—
430C2×C4C25×C512yes—
431C2×C4C25×C514yes—
432C5C2×#350911yescharacteristic, direct product factor, Sylow
433C22—24no—
434C22—28no—
435C22—24no—
436C22—28no—
437C22—48no—
438C22C22×C5×#4114yes—
439C22C22×C5×#4112yes—
440C22C2×C5×2−1+414yes—
441C22C26×C511yescharacteristic
442C4—24no—
443C4—216no—
444C4—28no—
445C4—24no—
446C4—216no—
447C4C22×C5×#4114yes—
448C4C22×C5×#4114yes—
449C4C2×C5×2−1+418yes—
450C2—24no—
451C2—44no—
452C2C5×#350914yesdirect product factor
453C2C23×C5×#4111yescharacteristic
454C2C22×C5×2−1+412yes—
subgroupG/H
center, socleC23×C5C25
commutator, Frattini, cosocleC22C26×C5
prime radicalC22×C5×#41C22
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C23×C5 ◁ G

Complements

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

NH ≅ G/Ntype
1. C5×#3509C2direct (N × H)
2. C2×C5×#557C2semidirect
3. C2×C5×#556C2semidirect
4. C2×C5×#552C2semidirect
5. C2×C5×#541C2semidirect
7. C2×C5×#533C2semidirect
8. C2×C5×#533C2semidirect
9. C22×C5×C4⋊Q8C2semidirect
10. C2×C5×#519C2semidirect
14. C5×#557C22semidirect
15. C5×#556C22semidirect
16. C5×#552C22semidirect
17. C5×#541C22semidirect
19. C5×#533C22semidirect
20. C5×#533C22semidirect
26. C2×C5×C4⋊Q8C22semidirect
36. C5×#519C22semidirect
53. C2×#3509C5direct (N × H)
118. #3509C2×C5direct (N × H)
119. C2×#557C2×C5semidirect
120. C2×#556C2×C5semidirect
121. C2×#552C2×C5semidirect
122. C2×#541C2×C5semidirect
124. C2×#533C2×C5semidirect
125. C2×#533C2×C5semidirect
126. C22×C4⋊Q8C2×C5semidirect
127. C2×#519C2×C5semidirect
185. #557C22×C5semidirect
186. #556C22×C5semidirect
187. #552C22×C5semidirect
188. #541C22×C5semidirect
190. #533C22×C5semidirect
191. #533C22×C5semidirect
197. C2×C4⋊Q8C22×C5semidirect
207. #519C22×C5semidirect
395. C2×C5#3509direct (N × H)
432. C5C2×#3509direct (N × H)
452. C2C5×#3509direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#35092
C2×C5×#5572
C2×C5×#5562
C2×C5×#5522
C2×C5×#5412
C2×C5×#5362
C2×C5×#5332
C22×C5×C4⋊Q82
C2×C5×#5192
C2×C4×C5×#412
C2×#35095