Finite Group Databasebeta

← C5×#8520 #11931240

Subgroups — C5×#8520 #11931240

Order 1280.

976 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 976 conjugacy classes of subgroups — too many to draw. Only its 150 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#536C211yesmaximal, characteristic
2C5×#1785C211yesmaximal, characteristic
3C5×#1782C211yesmaximal, characteristic
4C5×#1511C214yesmaximal, complemented
5C22×C5×#41C2211yescharacteristic
6C23×C5×Q8C411yescharacteristic, complemented
7C5×#536—28no—
8C2×C5×C4⋊Q8—21no—
9C2×C5×#124—21no—
10C2×C5×#123—21no—
11C2×C5×#123—21no—
12C2×C5×C22⋊Q8C412yescomplemented
13C2×C5×C22⋊Q8C2212yes—
14C2×C5×#117C411yescharacteristic, complemented
15C2×C5×#101—21no—
16C2×C5×#100—21no—
17C2×C5×#99—21no—
18C2×C5×#99—21no—
19C5×#407C2212yes—
20C5×#381C2212yes—
21C5×#328—28no—
22#8520C511yesmaximal, characteristic, direct product factor, Sylow
23C2×C5×#41—24no—
24C2×C5×#41—42no—
25C2×C5×#41—42no—
26C22×C5×Q8—24no—
27C22×C5×Q8—21no—
28C22×C5×Q8—42no—
29C22×C5×Q8—42no—
30C22×C5×Q8C2×C412yes—
31C22×C5×Q8D411yescharacteristic
32C22×C5×D4D411yescharacteristic
33C22×C5×D4D411yescharacteristic
34C22×C5×D4D411yescharacteristic
35C23×C4×C5—21no—
36C23×C4×C5C2×C411yescharacteristic
37C23×C4×C5C2×C411yescharacteristic
38C23×C4×C5C2311yescharacteristic
39C2×C5×M16—22no—
40C5×C4⋊Q8—28no—
41C5×#124—28no—
42C5×#123—28no—
43C5×#123—28no—
44C5×C22⋊Q8—24no—
45C5×C22⋊Q8—24no—
46C5×C22⋊Q8—24no—
47C5×C22⋊Q8D418yes—
48C5×#117—24no—
49C2×C5×C4⋊C4—22no—
50C2×C5×C4⋊C4—22no—
51C2×C5×C4⋊C4C2×C412yes—
52C2×C5×C22⋊C4—21no—
53C2×C5×C22⋊C4—22no—
54C2×C5×C22⋊C4—22no—
55C2×C5×C22⋊C4—22no—
56C2×C42×C5—21no—
57C5×#101—42no—
58C5×#100—42no—
59C5×#99—42no—
60C5×#99—42no—
61C5×C22⋊C8—24no—
62C5×#95—24no—
63C2×#536C2×C511yescharacteristic
64#1785C2×C511yescharacteristic
65#1782C2×C511yescharacteristic
66#1511C2×C514yescomplemented
67C24×C5C22⋊C411yescharacteristic
68C24×C5C22⋊C411yescharacteristic
69C24×C5C2×D411yescharacteristic
70C5×#41—416no—
71C2×C5×Q8—24no—
72C2×C5×Q8—22no—
73C2×C5×Q8—22no—
74C2×C5×Q8—22no—
75C2×C5×Q8—41no—
76C2×C5×Q8—416no—
77C2×C5×Q8—42no—
78C2×C5×Q8—42no—
79C2×C5×Q8—42no—
80C2×C5×Q8—42no—
81C2×C5×Q8C22⋊C414yes—
82C2×C5×D4—22no—
83C2×C5×D4—22no—
84C2×C5×D4—22no—
85C2×C5×D4—22no—
86C2×C5×D4—22no—
87C2×C5×D4—22no—
88C2×C5×D4—41no—
89C2×C5×D4—41no—
90C2×C5×D4—41no—
91C22×C4×C5—22no—
92C22×C4×C5—22no—
93C22×C4×C5—22no—
94C22×C4×C5—24no—
95C22×C4×C5—22no—
96C22×C4×C5—22no—
97C22×C4×C5—22no—
98C22×C4×C5—41no—
99C22×C4×C5—41no—
100C22×C4×C5—41no—
101C22×C4×C5—41no—
102C22×C4×C5—42no—
103C22×C4×C5—42no—
104C22×C4×C5—42no—
105C22×C4×C5C22⋊C411yescharacteristic
106C22×C4×C5C22⋊C411yescharacteristic
107C22×C4×C5C22×C411yescharacteristic
108C22×C4×C5C2×D411yescharacteristic
109C22×C4×C5C2×D414yes—
110C5×M16—42no—
111C5×M16—42no—
112C2×C5×C8—42no—
113C5×C4⋊C4—28no—
114C5×C4⋊C4—22no—
115C5×C4⋊C4—28no—
116C5×C4⋊C4C22⋊C414yes—
117C5×C22⋊C4—24no—
118C5×C22⋊C4—28no—
119C5×C22⋊C4—28no—
120C5×C22⋊C4—42no—
121C5×C22⋊C4—42no—
122C42×C5—24no—
123C22×#41C22×C511yescharacteristic
124C23×Q8C4×C511yescharacteristic, complemented
125#536—28no—
126C2×C4⋊Q8—21no—
127C2×#124—21no—
128C2×#123—21no—
129C2×#123—21no—
130C2×C22⋊Q8C4×C512yescomplemented
131C2×C22⋊Q8C22×C512yes—
132C2×#117C4×C511yescharacteristic, complemented
133C2×#101—21no—
134C2×#100—21no—
135C2×#99—21no—
136C2×#99—21no—
137#407C22×C512yes—
138#381C22×C512yes—
139#328—28no—
140C23×C5—21no—
141C23×C5—21no—
142C23×C5—21no—
143C23×C5—21no—
144C23×C5—21no—
145C23×C5—21no—
146C23×C5—21no—
147C23×C5—42no—
148C23×C5—42no—
149C23×C5—42no—
150C23×C5#9912yes—
151C23×C5C2×C22⋊C411yescharacteristic
152C23×C5#12012yes—
153C5×Q8—44no—
154C5×Q8—44no—
155C5×Q8—44no—
156C5×Q8—48no—
157C5×D4—44no—
158C5×D4—44no—
159C5×D4—44no—
160C2×C4×C5—28no—
161C2×C4×C5—21no—
162C2×C4×C5—21no—
163C2×C4×C5—21no—
164C2×C4×C5—21no—
165C2×C4×C5—21no—
166C2×C4×C5—21no—
167C2×C4×C5—24no—
168C2×C4×C5—24no—
169C2×C4×C5—21no—
170C2×C4×C5—21no—
171C2×C4×C5—21no—
172C2×C4×C5—44no—
173C2×C4×C5—44no—
174C2×C4×C5—44no—
175C2×C4×C5—48no—
176C2×C4×C5—42no—
177C2×C4×C5—42no—
178C2×C4×C5—82no—
179C2×C4×C5—84no—
180C2×C4×C5#9912yes—
181C2×C4×C5C2×C22⋊C412yes—
182C2×C4×C5#12012yes—
183C5×C8—82no—
184C2×#41—24no—
185C2×#41—42no—
186C2×#41—42no—
187C22×Q8—24no—
188C22×Q8—21no—
189C22×Q8—42no—
190C22×Q8—42no—
191C22×Q8C2×C4×C512yes—
192C22×Q8C5×D411yescharacteristic
193C22×D4C5×D411yescharacteristic
194C22×D4C5×D411yescharacteristic
195C22×D4C5×D411yescharacteristic
196C23×C4—21no—
197C23×C4C2×C4×C511yescharacteristic
198C23×C4C2×C4×C511yescharacteristic
199C23×C4C23×C511yescharacteristic
200C2×M16—22no—
201C4⋊Q8—28no—
202#124—28no—
203#123—28no—
204#123—28no—
205C22⋊Q8—24no—
206C22⋊Q8—24no—
207C22⋊Q8—24no—
208C22⋊Q8C5×D418yes—
209#117—24no—
210C2×C4⋊C4—22no—
211C2×C4⋊C4—22no—
212C2×C4⋊C4C2×C4×C512yes—
213C2×C22⋊C4—21no—
214C2×C22⋊C4—22no—
215C2×C22⋊C4—22no—
216C2×C22⋊C4—22no—
217C2×C42—21no—
218#101—42no—
219#100—42no—
220#99—42no—
221#99—42no—
222C22⋊C8—24no—
223#95—24no—
224C22×C5—22no—
225C22×C5—24no—
226C22×C5—22no—
227C22×C5—22no—
228C22×C5—42no—
229C22×C5—42no—
230C22×C5—42no—
231C22×C5—42no—
232C22×C5—41no—
233C22×C5—81no—
234C22×C5—81no—
235C22×C5—81no—
236C22×C5—81no—
237C22×C5#37911yescharacteristic
238C22×C5C2×#9911yescharacteristic
239C22×C5C2×#9911yescharacteristic
240C4×C5—22no—
241C4×C5—22no—
242C4×C5—22no—
243C4×C5—44no—
244C4×C5—41no—
245C4×C5—44no—
246C4×C5—162no—
247C24C5×C22⋊C411yescharacteristic
248C24C5×C22⋊C411yescharacteristic
249C24C2×C5×D411yescharacteristic
250#41—416no—
251C2×Q8—24no—
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8—22no—
255C2×Q8—41no—
256C2×Q8—416no—
257C2×Q8—42no—
258C2×Q8—42no—
259C2×Q8—42no—
260C2×Q8—42no—
261C2×Q8C5×C22⋊C414yes—
262C2×D4—22no—
263C2×D4—22no—
264C2×D4—22no—
265C2×D4—22no—
266C2×D4—22no—
267C2×D4—22no—
268C2×D4—41no—
269C2×D4—41no—
270C2×D4—41no—
271C22×C4—22no—
272C22×C4—22no—
273C22×C4—22no—
274C22×C4—24no—
275C22×C4—22no—
276C22×C4—22no—
277C22×C4—22no—
278C22×C4—41no—
279C22×C4—41no—
280C22×C4—41no—
281C22×C4—41no—
282C22×C4—42no—
283C22×C4—42no—
284C22×C4—42no—
285C22×C4C5×C22⋊C411yescharacteristic
286C22×C4C5×C22⋊C411yescharacteristic
287C22×C4C22×C4×C511yescharacteristic
288C22×C4C2×C5×D411yescharacteristic
289C22×C4C2×C5×D414yes—
290M16—42no—
291M16—42no—
292C2×C8—42no—
293C4⋊C4—28no—
294C4⋊C4—22no—
295C4⋊C4—28no—
296C4⋊C4C5×C22⋊C414yes—
297C22⋊C4—24no—
298C22⋊C4—28no—
299C22⋊C4—28no—
300C22⋊C4—42no—
301C22⋊C4—42no—
302C42—24no—
303C2×C5—21no—
304C2×C5—21no—
305C2×C5—42no—
306C2×C5—42no—
307C2×C5—81no—
308C2×C5#178111yescharacteristic
309C2×C5#179612yes—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—42no—
318C23—42no—
319C23—42no—
320C23C5×#9912yes—
321C23C2×C5×C22⋊C411yescharacteristic
322C23C5×#12012yes—
323Q8—44no—
324Q8—44no—
325Q8—44no—
326Q8—48no—
327D4—44no—
328D4—44no—
329D4—44no—
330C2×C4—28no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—24no—
338C2×C4—24no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—44no—
343C2×C4—44no—
344C2×C4—48no—
345C2×C4—44no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—82no—
349C2×C4—84no—
350C2×C4C5×#9912yes—
351C2×C4C2×C5×C22⋊C412yes—
352C2×C4C5×#12012yes—
353C8—82no—
354C5#852011yescharacteristic, direct product factor, Sylow
355C22—22no—
356C22—22no—
357C22—22no—
358C22—24no—
359C22—42no—
360C22—42no—
361C22—42no—
362C22—41no—
363C22—42no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22—81no—
368C22C5×#37911yescharacteristic
369C22C2×C5×#9911yescharacteristic
370C22C2×C5×#9911yescharacteristic
371C4—22no—
372C4—22no—
373C4—22no—
374C4—41no—
375C4—44no—
376C4—44no—
377C4—162no—
378C2—21no—
379C2—21no—
380C2—42no—
381C2—42no—
382C2—81no—
383C2C5×#178111yescharacteristic
384C2C5×#179612yes—
subgroupG/H
center, socleC22×C5#379
commutatorC22×C4C22×C4×C5
Frattini, cosocleC23×C4C23×C5
prime radicalC22×C5×#41C22
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C5 ◁ C23×C4×C5 ◁ G

Complements

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

NH ≅ G/Ntype
4. C5×#1511C2semidirect
6. C23×C5×Q8C4semidirect
12. C2×C5×C22⋊Q8C4semidirect
14. C2×C5×#117C4semidirect
22. #8520C5direct (N × H)
66. #1511C2×C5semidirect
124. C23×Q8C4×C5semidirect
130. C2×C22⋊Q8C4×C5semidirect
132. C2×#117C4×C5semidirect
354. C5#8520direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#5362
C5×#17852
C5×#17822
C5×#15112
#85205