Finite Group Databasebeta

← C5×#8529 #11931249

Subgroups — C5×#8529 #11931249

Order 1280.

648 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 648 conjugacy classes of subgroups — too many to draw. Only its 142 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
1C5×#3553C211yesmaximal, characteristic
2C5×#1991C211yesmaximal, characteristic, complemented
3C5×#1534C211yesmaximal, characteristic, complemented
4C5×#1512C212yesmaximal, complemented
5C5×#1494C212yesmaximal, complemented
6C5×#564C411yescharacteristic, complemented
7C5×#563C412yescomplemented
8C5×#562C411yescharacteristic, complemented
9C5×#554—22no—
10C5×#553—22no—
11C5×#552—22no—
12C5×#541—22no—
13C5×#539—22no—
14C5×#539—22no—
15C5×#529C2212yes—
16C2×C5×#125C2211yescharacteristic
17C5×#432C2212yes—
18C2×C4×C5×C8C2211yescharacteristic
19C5×#382C2211yescharacteristic
20C5×#340—22no—
21C5×#330—22no—
22C5×#328—22no—
23C5×#327—22no—
24C5×Q8⋊C8—22no—
25C5×D4⋊C8—22no—
26#8529C511yesmaximal, characteristic, direct product factor, Sylow
27C22×C5×C8—21no—
28C5×C4⋊Q8—22no—
29C5×#126—22no—
30C5×#126—22no—
31C5×#126—22no—
32C5×#126—22no—
33C5×#126—22no—
34C5×#125—22no—
35C5×#125—22no—
36C5×#125—22no—
37C5×#125—22no—
38C5×#125—22no—
39C5×#125—22no—
40C5×#125C2×C411yescharacteristic
41C5×#125C2×C411yescharacteristic
42C5×#125C2×C412yes—
43C5×#125D412yes—
44C5×#124C2×C412yes—
45C5×#123—22no—
46C5×#123—22no—
47C5×#123—22no—
48C5×#123—22no—
49C5×#123—22no—
50C5×C22⋊Q8—22no—
51C5×C22⋊Q8—22no—
52C5×C22⋊Q8—22no—
53C5×C22⋊Q8—22no—
54C5×C22⋊Q8—22no—
55C5×C22⋊Q8—22no—
56C5×C22⋊Q8D412yes—
57C5×#121—22no—
58C5×#121—22no—
59C5×#121D412yes—
60C4×C5×Q8—22no—
61C4×C5×Q8D412yes—
62C4×C5×D4—22no—
63C4×C5×D4—22no—
64C4×C5×D4—22no—
65C4×C5×D4—22no—
66C4×C5×D4D412yes—
67C5×#117—22no—
68C5×#117—22no—
69C2×C5×C4⋊C4—22no—
70C2×C5×C4⋊C4—22no—
71C2×C5×C4⋊C4—21no—
72C2×C5×C4⋊C4D412yes—
73C2×C42×C5C2311yescharacteristic
74C5×C4⋊C8—22no—
75C5×C22⋊C8—22no—
76C5×#97—22no—
77C4×C5×C8—22no—
78C5×#95—22no—
79#3553C2×C511yescharacteristic
80#1991C2×C511yescharacteristic, complemented
81#1534C2×C511yescharacteristic, complemented
82#1512C2×C512yescomplemented
83#1494C2×C512yescomplemented
84C2×C5×Q8—22no—
85C2×C5×Q8C22⋊C412yes—
86C2×C5×D4—22no—
87C2×C5×D4—22no—
88C2×C5×D4C22⋊C412yes—
89C22×C4×C5—22no—
90C22×C4×C5—22no—
91C22×C4×C5—22no—
92C22×C4×C5—22no—
93C22×C4×C5—41no—
94C22×C4×C5C2×D411yescharacteristic
95C22×C4×C5C2×D412yes—
96C2×C5×C8—22no—
97C2×C5×C8—41no—
98C2×C5×C8—41no—
99C2×C5×C8—42no—
100C5×C4⋊C4—22no—
101C5×C4⋊C4—22no—
102C5×C4⋊C4—22no—
103C5×C4⋊C4—22no—
104C5×C4⋊C4—22no—
105C5×C4⋊C4—22no—
106C5×C4⋊C4—22no—
107C5×C4⋊C4—22no—
108C5×C4⋊C4—22no—
109C5×C4⋊C4—22no—
110C5×C4⋊C4—22no—
111C5×C4⋊C4—22no—
112C5×C4⋊C4—22no—
113C5×C4⋊C4—22no—
114C5×C4⋊C4—22no—
115C5×C4⋊C4—42no—
116C5×C4⋊C4C22⋊C412yes—
117C5×C4⋊C4C22⋊C412yes—
118C5×C4⋊C4C22⋊C412yes—
119C5×C4⋊C4C22⋊C412yes—
120C5×C22⋊C4—22no—
121C5×C22⋊C4—22no—
122C5×C22⋊C4—22no—
123C5×C22⋊C4—22no—
124C5×C22⋊C4—22no—
125C5×C22⋊C4—22no—
126C5×C22⋊C4—22no—
127C5×C22⋊C4—22no—
128C5×C22⋊C4—22no—
129C42×C5—22no—
130C42×C5—22no—
131C42×C5—22no—
132C42×C5C22×C411yescharacteristic
133C42×C5C2×D411yescharacteristic
134C42×C5C2×D412yes—
135#564C4×C511yescharacteristic, complemented
136#563C4×C512yescomplemented
137#562C4×C511yescharacteristic, complemented
138#554—22no—
139#553—22no—
140#552—22no—
141#541—22no—
142#539—22no—
143#539—22no—
144#529C22×C512yes—
145C2×#125C22×C511yescharacteristic
146#432C22×C512yes—
147C2×C4×C8C22×C511yescharacteristic
148#382C22×C511yescharacteristic
149#340—22no—
150#330—22no—
151#328—22no—
152#327—22no—
153Q8⋊C8—22no—
154D4⋊C8—22no—
155C23×C5—22no—
156C23×C5#12011yescharacteristic
157C5×Q8—42no—
158C5×Q8—82no—
159C5×D4—42no—
160C5×D4—42no—
161C5×D4—42no—
162C5×D4—82no—
163C2×C4×C5—21no—
164C2×C4×C5—21no—
165C2×C4×C5—22no—
166C2×C4×C5—22no—
167C2×C4×C5—22no—
168C2×C4×C5—22no—
169C2×C4×C5—22no—
170C2×C4×C5—42no—
171C2×C4×C5—82no—
172C2×C4×C5—82no—
173C2×C4×C5—82no—
174C2×C4×C5—82no—
175C2×C4×C5—82no—
176C2×C4×C5—81no—
177C2×C4×C5C2×C22⋊C411yescharacteristic
178C2×C4×C5C2×C22⋊C412yes—
179C2×C4×C5#12012yes—
180C2×C4×C5#12011yescharacteristic
181C5×C8—42no—
182C5×C8—82no—
183C22×C8—21no—
184C4⋊Q8—22no—
185#126—22no—
186#126—22no—
187#126—22no—
188#126—22no—
189#126—22no—
190#125—22no—
191#125—22no—
192#125—22no—
193#125—22no—
194#125—22no—
195#125—22no—
196#125C2×C4×C511yescharacteristic
197#125C2×C4×C511yescharacteristic
198#125C2×C4×C512yes—
199#125C5×D412yes—
200#124C2×C4×C512yes—
201#123—22no—
202#123—22no—
203#123—22no—
204#123—22no—
205#123—22no—
206C22⋊Q8—22no—
207C22⋊Q8—22no—
208C22⋊Q8—22no—
209C22⋊Q8—22no—
210C22⋊Q8—22no—
211C22⋊Q8—22no—
212C22⋊Q8C5×D412yes—
213#121—22no—
214#121—22no—
215#121C5×D412yes—
216C4×Q8—22no—
217C4×Q8C5×D412yes—
218C4×D4—22no—
219C4×D4—22no—
220C4×D4—22no—
221C4×D4—22no—
222C4×D4C5×D412yes—
223#117—22no—
224#117—22no—
225C2×C4⋊C4—22no—
226C2×C4⋊C4—22no—
227C2×C4⋊C4—21no—
228C2×C4⋊C4C5×D412yes—
229C2×C42C23×C511yescharacteristic
230C4⋊C8—22no—
231C22⋊C8—22no—
232#97—22no—
233C4×C8—22no—
234#95—22no—
235C22×C5—22no—
236C22×C5—42no—
237C22×C5—42no—
238C22×C5—42no—
239C22×C5#37911yescharacteristic
240C22×C5#45411yescharacteristic
241C22×C5#45611yescharacteristic
242C4×C5—21no—
243C4×C5—21no—
244C4×C5—22no—
245C4×C5—22no—
246C4×C5—42no—
247C4×C5—42no—
248C4×C5—42no—
249C4×C5—42no—
250C4×C5—42no—
251C4×C5—161no—
252C2×Q8—22no—
253C2×Q8C5×C22⋊C412yes—
254C2×D4—22no—
255C2×D4—22no—
256C2×D4C5×C22⋊C412yes—
257C22×C4—22no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—41no—
262C22×C4C2×C5×D411yescharacteristic
263C22×C4C2×C5×D412yes—
264C2×C8—22no—
265C2×C8—41no—
266C2×C8—41no—
267C2×C8—42no—
268C4⋊C4—22no—
269C4⋊C4—22no—
270C4⋊C4—22no—
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—22no—
274C4⋊C4—22no—
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—22no—
279C4⋊C4—22no—
280C4⋊C4—22no—
281C4⋊C4—22no—
282C4⋊C4—22no—
283C4⋊C4—42no—
284C4⋊C4C5×C22⋊C412yes—
285C4⋊C4C5×C22⋊C412yes—
286C4⋊C4C5×C22⋊C412yes—
287C4⋊C4C5×C22⋊C412yes—
288C22⋊C4—22no—
289C22⋊C4—22no—
290C22⋊C4—22no—
291C22⋊C4—22no—
292C22⋊C4—22no—
293C22⋊C4—22no—
294C22⋊C4—22no—
295C22⋊C4—22no—
296C22⋊C4—22no—
297C42—22no—
298C42—22no—
299C42—22no—
300C42C22×C4×C511yescharacteristic
301C42C2×C5×D412yes—
302C42C2×C5×D411yescharacteristic
303C2×C5—21no—
304C2×C5—21no—
305C2×C5—82no—
306C2×C5#179311yescharacteristic
307C2×C5#179412yes—
308C23—22no—
309C23C5×#12011yescharacteristic
310Q8—42no—
311Q8—82no—
312D4—42no—
313D4—42no—
314D4—42no—
315D4—82no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—42no—
324C2×C4—82no—
325C2×C4—82no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4—81no—
330C2×C4C2×C5×C22⋊C411yescharacteristic
331C2×C4C2×C5×C22⋊C412yes—
332C2×C4C5×#12012yes—
333C2×C4C5×#12011yescharacteristic
334C8—42no—
335C8—82no—
336C5#852911yescharacteristic, direct product factor, Sylow
337C22—22no—
338C22—42no—
339C22—42no—
340C22—42no—
341C22C5×#37911yescharacteristic
342C22C5×#45411yescharacteristic
343C22C5×#45611yescharacteristic
344C4—21no—
345C4—21no—
346C4—22no—
347C4—22no—
348C4—42no—
349C4—42no—
350C4—42no—
351C4—42no—
352C4—42no—
353C4—161no—
354C2—21no—
355C2—21no—
356C2—82no—
357C2C5×#179311yescharacteristic
358C2C5×#179412yes—
subgroupG/H
center, socleC22×C5#379
commutatorC42C22×C4×C5
Frattini, cosocleC2×C42C23×C5
prime radicalC5×#3553C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C5 ◁ C2×C42×C5 ◁ G

Complements

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

NH ≅ G/Ntype
2. C5×#1991C2semidirect
3. C5×#1534C2semidirect
4. C5×#1512C2semidirect
5. C5×#1494C2semidirect
6. C5×#564C4semidirect
7. C5×#563C4semidirect
8. C5×#562C4semidirect
26. #8529C5direct (N × H)
80. #1991C2×C5semidirect
81. #1534C2×C5semidirect
82. #1512C2×C5semidirect
83. #1494C2×C5semidirect
135. #564C4×C5semidirect
136. #563C4×C5semidirect
137. #562C4×C5semidirect
336. C5#8529direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#35532
C5×#19912
C5×#15342
C5×#15122
C5×#14942
#85295