Finite Group Databasebeta

← C5×#36684 #11959404

Subgroups — C5×#36684 #11959404

Order 1280.

2138 conjugacy classes of subgroups.

The subgroup lattice has 442 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#529C212yesmaximal, complemented
2C5×#2823C214yesmaximal, complemented
3C5×#2822C211yesmaximal, characteristic, complemented
4C5×#2820C212yesmaximal, complemented
5C5×#2819C211yesmaximal, characteristic, complemented
6C5×#2814C214yesmaximal, complemented
7C5×#2813C214yesmaximal, complemented
8C5×#2396C214yesmaximal, complemented
9C5×#2392C214yesmaximal, complemented
10C5×#2392C214yesmaximal, complemented
11C5×#2285C211yesmaximal, characteristic, complemented
12C5×#529—216no—
13C2×C5×#126C2214yes—
14C2×C5×#125C2212yes—
15C2×C5×#124C2212yes—
16C2×C5×#123C2214yes—
17C2×C5×C22⋊Q8C2212yes—
18C2×C5×#121C2212yes—
19C2×C4×C5×Q8C2212yes—
20C2×C4×C5×D4C2214yes—
21C2×C4×C5×D4C2212yes—
22C2×C4×C5×D4C2214yes—
23C2×C5×#117C2214yes—
24C2×C5×#117C2212yes—
25C22×C42×C5C2211yescharacteristic
26C5×#400C2214yes—
27C5×#399C2214yes—
28C5×#398C2214yes—
29C5×#398C2218yes—
30C5×#398C2214yes—
31C5×#397C2214yes—
32C5×#397C2214yes—
33C5×#396C2218yes—
34C5×#396C2218yes—
35C5×#396C2218yes—
36C5×#395C2214yes—
37C5×#394C2218yes—
38C5×#394C2214yes—
39C5×#394C2218yes—
40C5×#394C2214yes—
41C5×#393C2214yes—
42C5×#392C2214yes—
43C5×#392C2214yes—
44C5×#384C2212yes—
45C5×#382C2211yescharacteristic
46C5×#382C2211yescharacteristic
47C5×#380C2214yes—
48C4×C5×C4⋊C4C2214yes—
49C4×C5×C4⋊C4C2214yes—
50C4×C5×C22⋊C4C2218yes—
51#36684C511yesmaximal, characteristic, direct product factor, Sylow
52C22×C5×Q8C2312yes—
53C22×C5×D4C2314yes—
54C22×C5×D4C2312yes—
55C22×C5×D4C2314yes—
56C23×C4×C5C2314yes—
57C23×C4×C5C2311yescharacteristic
58C23×C4×C5C2312yes—
59C5×#126—216no—
60C5×#125—28no—
61C5×#124—28no—
62C5×#123—216no—
63C5×C22⋊Q8—28no—
64C5×#121—28no—
65C4×C5×Q8—28no—
66C4×C5×D4—216no—
67C4×C5×D4—28no—
68C4×C5×D4—48no—
69C5×#117—216no—
70C5×#117—44no—
71C2×C5×C4⋊C4C2314yes—
72C2×C5×C4⋊C4C2314yes—
73C2×C5×C4⋊C4C2314yes—
74C2×C5×C4⋊C4C2314yes—
75C2×C5×C4⋊C4C2312yes—
76C2×C5×C4⋊C4C2312yes—
77C2×C5×C4⋊C4C2318yes—
78C2×C5×C4⋊C4C2318yes—
79C2×C5×C4⋊C4C2312yes—
80C2×C5×C4⋊C4C2312yes—
81C2×C5×C4⋊C4C2314yes—
82C2×C5×C22⋊C4C2314yes—
83C2×C5×C22⋊C4C2314yes—
84C2×C5×C22⋊C4C2314yes—
85C2×C5×C22⋊C4C2314yes—
86C2×C5×C22⋊C4C2314yes—
87C2×C5×C22⋊C4C2318yes—
88C2×C5×C22⋊C4C2318yes—
89C2×C5×C22⋊C4C2314yes—
90C2×C5×C22⋊C4C2318yes—
91C2×C42×C5D418yescomplemented
92C2×C42×C5C2314yes—
93C2×C42×C5C2314yes—
94C2×C42×C5C2311yescharacteristic
95C2×C42×C5C2311yescharacteristic
96C2×C42×C5C2312yes—
97C5×#95C2318yes—
98C5×#95C2318yes—
99C5×#95C2314yes—
100C5×#95C2314yes—
101C5×#95C2314yes—
102C5×#95C2314yes—
103C2×#529C2×C512yescomplemented
104#2823C2×C514yescomplemented
105#2822C2×C511yescharacteristic, complemented
106#2820C2×C512yescomplemented
107#2819C2×C511yescharacteristic, complemented
108#2814C2×C514yescomplemented
109#2813C2×C514yescomplemented
110#2396C2×C514yescomplemented
111#2392C2×C514yescomplemented
112#2392C2×C514yescomplemented
113#2285C2×C511yescharacteristic, complemented
114C24×C5C2414yes—
115C24×C5C2411yescharacteristic
116C2×C5×Q8—24no—
117C2×C5×Q8—44no—
118C2×C5×D4—28no—
119C2×C5×D4—24no—
120C2×C5×D4—44no—
121C2×C5×D4—44no—
122C2×C5×D4—48no—
123C2×C5×D4—48no—
124C22×C4×C5—22no—
125C22×C4×C5—22no—
126C22×C4×C5—28no—
127C22×C4×C5—28no—
128C22×C4×C5—28no—
129C22×C4×C5—44no—
130C22×C4×C5C2×D414yes—
131C22×C4×C5C2×D418yes—
132C22×C4×C5C2414yes—
133C22×C4×C5C2414yes—
134C22×C4×C5C2414yes—
135C22×C4×C5C2414yes—
136C22×C4×C5C2414yes—
137C22×C4×C5C2411yescharacteristic
138C22×C4×C5C2412yes—
139C22×C4×C5C2412yes—
140C22×C4×C5C2411yescharacteristic
141C5×C4⋊C4—28no—
142C5×C4⋊C4—28no—
143C5×C4⋊C4—24no—
144C5×C4⋊C4—28no—
145C5×C4⋊C4—24no—
146C5×C4⋊C4—28no—
147C5×C4⋊C4—48no—
148C5×C4⋊C4—48no—
149C5×C4⋊C4—42no—
150C5×C4⋊C4—42no—
151C5×C4⋊C4—44no—
152C5×C22⋊C4—28no—
153C5×C22⋊C4—28no—
154C5×C22⋊C4—28no—
155C5×C22⋊C4—28no—
156C5×C22⋊C4—28no—
157C5×C22⋊C4—48no—
158C5×C22⋊C4—48no—
159C5×C22⋊C4—44no—
160C5×C22⋊C4—48no—
161C42×C5—28no—
162C42×C5—28no—
163C42×C5C2×D418yes—
164C42×C5C2×D414yes—
165C42×C5C2×D414yes—
166#529—216no—
167C2×#126C22×C514yes—
168C2×#125C22×C512yes—
169C2×#124C22×C512yes—
170C2×#123C22×C514yes—
171C2×C22⋊Q8C22×C512yes—
172C2×#121C22×C512yes—
173C2×C4×Q8C22×C512yes—
174C2×C4×D4C22×C514yes—
175C2×C4×D4C22×C512yes—
176C2×C4×D4C22×C514yes—
177C2×#117C22×C514yes—
178C2×#117C22×C512yes—
179C22×C42C22×C511yescharacteristic
180#400C22×C514yes—
181#399C22×C514yes—
182#398C22×C514yes—
183#398C22×C514yes—
184#398C22×C518yes—
185#397C22×C514yes—
186#397C22×C514yes—
187#396C22×C518yes—
188#396C22×C518yes—
189#396C22×C518yes—
190#395C22×C514yes—
191#394C22×C514yes—
192#394C22×C514yes—
193#394C22×C518yes—
194#394C22×C518yes—
195#393C22×C514yes—
196#392C22×C514yes—
197#392C22×C514yes—
198#384C22×C512yes—
199#382C22×C511yescharacteristic
200#382C22×C511yescharacteristic
201#380C22×C514yes—
202C4×C4⋊C4C22×C514yes—
203C4×C4⋊C4C22×C514yes—
204C4×C22⋊C4C22×C518yes—
205C23×C5—21no—
206C23×C5—21no—
207C23×C5—24no—
208C23×C5—24no—
209C23×C5—24no—
210C23×C5—24no—
211C23×C5—28no—
212C23×C5—28no—
213C23×C5C22×D412yes—
214C23×C5C2511yescharacteristic
215C5×Q8—48no—
216C5×D4—48no—
217C5×D4—416no—
218C5×D4—416no—
219C2×C4×C5—21no—
220C2×C4×C5—21no—
221C2×C4×C5—24no—
222C2×C4×C5—24no—
223C2×C4×C5—28no—
224C2×C4×C5—24no—
225C2×C4×C5—24no—
226C2×C4×C5—24no—
227C2×C4×C5—24no—
228C2×C4×C5—24no—
229C2×C4×C5—24no—
230C2×C4×C5—24no—
231C2×C4×C5—28no—
232C2×C4×C5—28no—
233C2×C4×C5—28no—
234C2×C4×C5—28no—
235C2×C4×C5—48no—
236C2×C4×C5—416no—
237C2×C4×C5C22×D412yes—
238C2×C4×C5C22×D414yes—
239C2×C4×C5C22×D414yes—
240C2×C4×C5C22×D412yes—
241C2×C4×C5H(5,𝔽2)12yes—
242C2×C4×C52−1+412yes—
243C22×Q8C23×C512yes—
244C22×D4C23×C514yes—
245C22×D4C23×C512yes—
246C22×D4C23×C514yes—
247C23×C4C23×C514yes—
248C23×C4C23×C511yescharacteristic
249C23×C4C23×C512yes—
250#126—216no—
251#125—28no—
252#124—28no—
253#123—216no—
254C22⋊Q8—28no—
255#121—28no—
256C4×Q8—28no—
257C4×D4—216no—
258C4×D4—28no—
259C4×D4—48no—
260#117—216no—
261#117—44no—
262C2×C4⋊C4C23×C514yes—
263C2×C4⋊C4C23×C514yes—
264C2×C4⋊C4C23×C512yes—
265C2×C4⋊C4C23×C514yes—
266C2×C4⋊C4C23×C512yes—
267C2×C4⋊C4C23×C514yes—
268C2×C4⋊C4C23×C512yes—
269C2×C4⋊C4C23×C512yes—
270C2×C4⋊C4C23×C514yes—
271C2×C4⋊C4C23×C518yes—
272C2×C4⋊C4C23×C518yes—
273C2×C22⋊C4C23×C514yes—
274C2×C22⋊C4C23×C514yes—
275C2×C22⋊C4C23×C514yes—
276C2×C22⋊C4C23×C514yes—
277C2×C22⋊C4C23×C514yes—
278C2×C22⋊C4C23×C514yes—
279C2×C22⋊C4C23×C518yes—
280C2×C22⋊C4C23×C518yes—
281C2×C22⋊C4C23×C518yes—
282C2×C42C5×D418yescomplemented
283C2×C42C23×C514yes—
284C2×C42C23×C514yes—
285C2×C42C23×C512yes—
286C2×C42C23×C511yescharacteristic
287C2×C42C23×C511yescharacteristic
288#95C23×C518yes—
289#95C23×C518yes—
290#95C23×C514yes—
291#95C23×C514yes—
292#95C23×C514yes—
293#95C23×C514yes—
294C22×C5—22no—
295C22×C5—24no—
296C22×C5—42no—
297C22×C5—42no—
298C22×C5—44no—
299C22×C5—44no—
300C22×C5—44no—
301C22×C5—48no—
302C22×C5—48no—
303C22×C5C23×D411yescharacteristic
304C22×C5C2×H(5,𝔽2)11yescharacteristic
305C22×C5C2×2−1+411yescharacteristic
306C22×C5#58514yes—
307C4×C5—22no—
308C4×C5—24no—
309C4×C5—24no—
310C4×C5—44no—
311C4×C5—44no—
312C4×C5—44no—
313C4×C5—44no—
314C4×C5—44no—
315C4×C5#53714yes—
316C24C24×C514yes—
317C24C24×C511yescharacteristic
318C2×Q8—24no—
319C2×Q8—44no—
320C2×D4—28no—
321C2×D4—24no—
322C2×D4—44no—
323C2×D4—44no—
324C2×D4—48no—
325C2×D4—48no—
326C22×C4—22no—
327C22×C4—22no—
328C22×C4—28no—
329C22×C4—28no—
330C22×C4—28no—
331C22×C4—44no—
332C22×C4C2×C5×D414yes—
333C22×C4C2×C5×D418yes—
334C22×C4C24×C514yes—
335C22×C4C24×C514yes—
336C22×C4C24×C514yes—
337C22×C4C24×C514yes—
338C22×C4C24×C514yes—
339C22×C4C24×C512yes—
340C22×C4C24×C511yescharacteristic
341C22×C4C24×C512yes—
342C22×C4C24×C511yescharacteristic
343C4⋊C4—28no—
344C4⋊C4—28no—
345C4⋊C4—24no—
346C4⋊C4—28no—
347C4⋊C4—24no—
348C4⋊C4—28no—
349C4⋊C4—42no—
350C4⋊C4—42no—
351C4⋊C4—44no—
352C4⋊C4—48no—
353C4⋊C4—48no—
354C22⋊C4—28no—
355C22⋊C4—28no—
356C22⋊C4—28no—
357C22⋊C4—28no—
358C22⋊C4—28no—
359C22⋊C4—44no—
360C22⋊C4—48no—
361C22⋊C4—48no—
362C22⋊C4—48no—
363C42—28no—
364C42—28no—
365C42C2×C5×D418yes—
366C42C2×C5×D414yes—
367C42C2×C5×D414yes—
368C2×C5—42no—
369C2×C5—84no—
370C2×C5C2×#53711yescharacteristic
371C2×C5#344912yes—
372C2×C5#352912yes—
373C2×C5#353112yes—
374C23—21no—
375C23—21no—
376C23—24no—
377C23—24no—
378C23—24no—
379C23—24no—
380C23—28no—
381C23—28no—
382C23C22×C5×D412yes—
383C23C25×C511yescharacteristic
384Q8—48no—
385D4—48no—
386D4—416no—
387D4—416no—
388C2×C4—21no—
389C2×C4—21no—
390C2×C4—24no—
391C2×C4—24no—
392C2×C4—28no—
393C2×C4—24no—
394C2×C4—24no—
395C2×C4—24no—
396C2×C4—24no—
397C2×C4—24no—
398C2×C4—24no—
399C2×C4—24no—
400C2×C4—28no—
401C2×C4—28no—
402C2×C4—28no—
403C2×C4—28no—
404C2×C4—48no—
405C2×C4—416no—
406C2×C4C22×C5×D412yes—
407C2×C4C22×C5×D414yes—
408C2×C4C22×C5×D414yes—
409C2×C4C22×C5×D412yes—
410C2×C4C5×H(5,𝔽2)12yes—
411C2×C4C5×2−1+412yes—
412C5#3668411yescharacteristic, direct product factor, Sylow
413C22—22no—
414C22—24no—
415C22—42no—
416C22—42no—
417C22—44no—
418C22—44no—
419C22—44no—
420C22—48no—
421C22—48no—
422C22C23×C5×D411yescharacteristic
423C22C2×C5×H(5,𝔽2)11yescharacteristic
424C22C2×C5×2−1+411yescharacteristic
425C22C5×#58514yes—
426C4—22no—
427C4—24no—
428C4—24no—
429C4—44no—
430C4—44no—
431C4—44no—
432C4—44no—
433C4—44no—
434C4C5×#53714yes—
435C2—42no—
436C2—84no—
437C2C2×C5×#53711yescharacteristic
438C2C5×#344912yes—
439C2C5×#352912yes—
440C2C5×#353112yes—
subgroupG/H
centerC22×C4×C5C24
commutator, Frattini, cosocleC23C25×C5
socleC23×C5C25
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4×C5 ◁ G

Complements

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

NH ≅ G/Ntype
1. C2×C5×#529C2semidirect
2. C5×#2823C2semidirect
3. C5×#2822C2semidirect
4. C5×#2820C2semidirect
5. C5×#2819C2semidirect
6. C5×#2814C2semidirect
7. C5×#2813C2semidirect
8. C5×#2396C2semidirect
9. C5×#2392C2semidirect
10. C5×#2392C2semidirect
11. C5×#2285C2semidirect
51. #36684C5direct (N × H)
91. C2×C42×C5D4semidirect
103. C2×#529C2×C5semidirect
104. #2823C2×C5semidirect
105. #2822C2×C5semidirect
106. #2820C2×C5semidirect
107. #2819C2×C5semidirect
108. #2814C2×C5semidirect
109. #2813C2×C5semidirect
110. #2396C2×C5semidirect
111. #2392C2×C5semidirect
112. #2392C2×C5semidirect
113. #2285C2×C5semidirect
282. C2×C42C5×D4semidirect
412. C5#36684direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#5292
C5×#28232
C5×#28222
C5×#28202
C5×#28192
C5×#28142
C5×#28132
C5×#23962
C5×#23922
C5×#22852
#366845