Finite Group Databasebeta

← C5×#8505 #11931225

Subgroups — C5×#8505 #11931225

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×#3531C211yesmaximal, characteristic
2C5×#1995C211yesmaximal, characteristic, complemented
3C5×#1544C211yesmaximal, characteristic, complemented
4C5×#1500C212yesmaximal, complemented
5C5×#1495C212yesmaximal, complemented
6C5×#564C411yescharacteristic, complemented
7C5×#563—22no—
8C5×#563—22no—
9C5×#563C412yescomplemented
10C5×#562—22no—
11C5×#562C411yescharacteristic, complemented
12C5×#533—22no—
13C5×#533C2211yescharacteristic
14C5×#529—22no—
15C5×#529—22no—
16C5×#529C2212yes—
17C5×#431C2212yes—
18C5×#423C2211yescharacteristic
19C5×#376C2211yescharacteristic
20C5×#339—22no—
21C5×#332—22no—
22C5×#328—22no—
23C5×#327—22no—
24C5×Q8⋊C8—22no—
25C5×D4⋊C8—22no—
26#8505C511yesmaximal, characteristic, direct product factor, Sylow
27C2×C5×M16—21no—
28C5×C4⋊Q8—22no—
29C5×C4⋊Q8—22no—
30C5×C4⋊Q8D412yes—
31C5×#126—22no—
32C5×#126—22no—
33C5×#126—22no—
34C5×#126—22no—
35C5×#126—22no—
36C5×#125—22no—
37C5×#125—22no—
38C5×#125—22no—
39C5×#125—22no—
40C5×#125C2×C412yes—
41C5×#125C2×C411yescharacteristic
42C5×#125C2×C411yescharacteristic
43C5×#124—22no—
44C5×#124—22no—
45C5×#124C2×C412yes—
46C5×#123—22no—
47C5×#123—22no—
48C5×#123—22no—
49C5×#123—22no—
50C5×#123—22no—
51C5×C22⋊Q8—22no—
52C5×C22⋊Q8—22no—
53C5×C22⋊Q8—22no—
54C5×C22⋊Q8—22no—
55C5×C22⋊Q8—22no—
56C5×C22⋊Q8—22no—
57C5×C22⋊Q8—22no—
58C5×C22⋊Q8D412yes—
59C5×#121—22no—
60C5×#121—22no—
61C5×#121D412yes—
62C4×C5×Q8—22no—
63C4×C5×Q8—22no—
64C4×C5×Q8D412yes—
65C4×C5×D4—22no—
66C4×C5×D4—22no—
67C4×C5×D4D412yes—
68C5×#117—22no—
69C5×#117—22no—
70C5×#117D412yes—
71C2×C42×C5—22no—
72C2×C42×C5—21no—
73C2×C42×C5C2311yescharacteristic
74C5×C4⋊C8—22no—
75C5×C4⋊C8—22no—
76C5×C22⋊C8—22no—
77C4×C5×C8—22no—
78C5×#95—22no—
79#3531C2×C511yescharacteristic
80#1995C2×C511yescharacteristic, complemented
81#1544C2×C511yescharacteristic, complemented
82#1500C2×C512yescomplemented
83#1495C2×C512yescomplemented
84C2×C5×Q8—22no—
85C2×C5×Q8—22no—
86C2×C5×Q8C22⋊C412yes—
87C2×C5×D4—22no—
88C2×C5×D4—22no—
89C2×C5×D4C22⋊C412yes—
90C22×C4×C5—22no—
91C22×C4×C5—22no—
92C22×C4×C5—22no—
93C22×C4×C5—41no—
94C22×C4×C5C2×D411yescharacteristic
95C22×C4×C5C2×D412yes—
96C5×M16—42no—
97C2×C5×C8—42no—
98C2×C5×C8—41no—
99C5×C4⋊C4—22no—
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⋊C4C22⋊C412yes—
113C5×C4⋊C4C22⋊C412yes—
114C5×C4⋊C4C22⋊C412yes—
115C5×C4⋊C4C22⋊C412yes—
116C5×C22⋊C4—22no—
117C5×C22⋊C4—22no—
118C5×C22⋊C4—22no—
119C5×C22⋊C4—22no—
120C5×C22⋊C4—22no—
121C5×C22⋊C4—22no—
122C5×C22⋊C4—22no—
123C5×C22⋊C4—22no—
124C5×C22⋊C4—22no—
125C5×C22⋊C4—22no—
126C42×C5—22no—
127C42×C5—22no—
128C42×C5—22no—
129C42×C5—22no—
130C42×C5—22no—
131C42×C5—41no—
132C42×C5C22×C411yescharacteristic
133C42×C5C2×D411yescharacteristic
134C42×C5C2×D412yes—
135#564C4×C511yescharacteristic, complemented
136#563—22no—
137#563—22no—
138#563C4×C512yescomplemented
139#562—22no—
140#562C4×C511yescharacteristic, complemented
141#533—22no—
142#533C22×C511yescharacteristic
143#529—22no—
144#529—22no—
145#529C22×C512yes—
146#431C22×C512yes—
147#423C22×C511yescharacteristic
148#376C22×C511yescharacteristic
149#339—22no—
150#332—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×Q8—82no—
160C5×D4—42no—
161C5×D4—82no—
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—42no—
172C2×C4×C5—42no—
173C2×C4×C5—42no—
174C2×C4×C5—82no—
175C2×C4×C5—82no—
176C2×C4×C5—82no—
177C2×C4×C5C2×C22⋊C411yescharacteristic
178C2×C4×C5C2×C22⋊C412yes—
179C2×C4×C5#12012yes—
180C2×C4×C5#12011yescharacteristic
181C5×C8—82no—
182C5×C8—81no—
183C2×M16—21no—
184C4⋊Q8—22no—
185C4⋊Q8—22no—
186C4⋊Q8C5×D412yes—
187#126—22no—
188#126—22no—
189#126—22no—
190#126—22no—
191#126—22no—
192#125—22no—
193#125—22no—
194#125—22no—
195#125—22no—
196#125C2×C4×C512yes—
197#125C2×C4×C511yescharacteristic
198#125C2×C4×C511yescharacteristic
199#124—22no—
200#124—22no—
201#124C2×C4×C512yes—
202#123—22no—
203#123—22no—
204#123—22no—
205#123—22no—
206#123—22no—
207C22⋊Q8—22no—
208C22⋊Q8—22no—
209C22⋊Q8—22no—
210C22⋊Q8—22no—
211C22⋊Q8—22no—
212C22⋊Q8—22no—
213C22⋊Q8—22no—
214C22⋊Q8C5×D412yes—
215#121—22no—
216#121—22no—
217#121C5×D412yes—
218C4×Q8—22no—
219C4×Q8—22no—
220C4×Q8C5×D412yes—
221C4×D4—22no—
222C4×D4—22no—
223C4×D4C5×D412yes—
224#117—22no—
225#117—22no—
226#117C5×D412yes—
227C2×C42—22no—
228C2×C42—21no—
229C2×C42C23×C511yescharacteristic
230C4⋊C8—22no—
231C4⋊C8—22no—
232C22⋊C8—22no—
233C4×C8—22no—
234#95—22no—
235C22×C5—21no—
236C22×C5—22no—
237C22×C5—42no—
238C22×C5—42no—
239C22×C5—42no—
240C22×C5#37911yescharacteristic
241C4×C5—21no—
242C4×C5—22no—
243C4×C5—22no—
244C4×C5—42no—
245C4×C5—42no—
246C4×C5—42no—
247C4×C5—42no—
248C4×C5—42no—
249C4×C5—82no—
250C4×C5#45411yescharacteristic
251C4×C5#45611yescharacteristic
252C2×Q8—22no—
253C2×Q8—22no—
254C2×Q8C5×C22⋊C412yes—
255C2×D4—22no—
256C2×D4—22no—
257C2×D4C5×C22⋊C412yes—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—22no—
261C22×C4—41no—
262C22×C4C2×C5×D411yescharacteristic
263C22×C4C2×C5×D412yes—
264M16—42no—
265C2×C8—42no—
266C2×C8—41no—
267C4⋊C4—22no—
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⋊C4C5×C22⋊C412yes—
281C4⋊C4C5×C22⋊C412yes—
282C4⋊C4C5×C22⋊C412yes—
283C4⋊C4C5×C22⋊C412yes—
284C22⋊C4—22no—
285C22⋊C4—22no—
286C22⋊C4—22no—
287C22⋊C4—22no—
288C22⋊C4—22no—
289C22⋊C4—22no—
290C22⋊C4—22no—
291C22⋊C4—22no—
292C22⋊C4—22no—
293C22⋊C4—22no—
294C42—22no—
295C42—22no—
296C42—22no—
297C42—22no—
298C42—22no—
299C42—41no—
300C42C22×C4×C511yescharacteristic
301C42C2×C5×D412yes—
302C42C2×C5×D411yescharacteristic
303C2×C5—41no—
304C2×C5—82no—
305C2×C5#179311yescharacteristic
306C2×C5#179412yes—
307C23—22no—
308C23C5×#12011yescharacteristic
309Q8—42no—
310Q8—82no—
311Q8—82no—
312D4—42no—
313D4—82no—
314D4—82no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—22no—
318C2×C4—22no—
319C2×C4—22no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—42no—
323C2×C4—42no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—82no—
327C2×C4—82no—
328C2×C4—82no—
329C2×C4C2×C5×C22⋊C411yescharacteristic
330C2×C4C2×C5×C22⋊C412yes—
331C2×C4C5×#12012yes—
332C2×C4C5×#12011yescharacteristic
333C8—82no—
334C8—81no—
335C5#850511yescharacteristic, direct product factor, Sylow
336C22—21no—
337C22—22no—
338C22—42no—
339C22—42no—
340C22—42no—
341C22C5×#37911yescharacteristic
342C4—21no—
343C4—22no—
344C4—22no—
345C4—42no—
346C4—42no—
347C4—42no—
348C4—42no—
349C4—42no—
350C4—82no—
351C4C5×#45411yescharacteristic
352C4C5×#45611yescharacteristic
353C2—41no—
354C2—82no—
355C2C5×#179311yescharacteristic
356C2C5×#179412yes—
subgroupG/H
centerC2×C4×C5#120
commutatorC42C22×C4×C5
Frattini, cosocleC2×C42C23×C5
socleC22×C5#379
prime radicalC5×#3531C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C2×C4×C5 ◁ C2×C42×C5 ◁ G

Complements

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

NH ≅ G/Ntype
2. C5×#1995C2semidirect
3. C5×#1544C2semidirect
4. C5×#1500C2semidirect
5. C5×#1495C2semidirect
6. C5×#564C4semidirect
9. C5×#563C4semidirect
11. C5×#562C4semidirect
26. #8505C5direct (N × H)
80. #1995C2×C5semidirect
81. #1544C2×C5semidirect
82. #1500C2×C5semidirect
83. #1495C2×C5semidirect
135. #564C4×C5semidirect
138. #563C4×C5semidirect
140. #562C4×C5semidirect
335. C5#8505direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#35312
C5×#19952
C5×#15442
C5×#15002
C5×#14952
#85055