Finite Group Databasebeta

← C5×#21707 #11944427

Subgroups — C5×#21707 #11944427

Order 1280.

998 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3147C211yesmaximal, characteristic
2C2×C5×#444C212yesmaximal, complemented
3C2×C5×#443C212yesmaximal, complemented
4C5×#2932C212yesmaximal, complemented
5C5×#1975C212yesmaximal, complemented
6C5×#1974C212yesmaximal, complemented
7C5×#1963C212yesmaximal, complemented
8C5×#1848C211yesmaximal, characteristic, complemented
9C5×#1726C211yesmaximal, characteristic, complemented
10C2×C5×#135C2212yes—
11C2×C5×#135C2212yes—
12C2×C5×#131C2214yes—
13C5×#532C2212yes—
14C5×#495C412yescomplemented
15C5×#495C412yescomplemented
16C5×#494C411yescharacteristic, complemented
17C5×#493C411yescharacteristic, complemented
18C5×#492C412yescomplemented
19C5×#444—28no—
20C5×#443—28no—
21C5×#435—28no—
22C2×C5×#108C2212yes—
23C2×C5×#108C2212yes—
24C5×#423C2212yes—
25C2×C5×#104C2214yes—
26C2×C5×#104C2214yes—
27C5×#408C2214yes—
28C2×C5×#97C2211yescharacteristic
29C2×C5×#97C2211yescharacteristic
30C2×C5×#97C2211yescharacteristic
31C2×C4×C5×C8C2211yescharacteristic
32C2×C4×C5×C8C2211yescharacteristic
33C2×C4×C5×C8C2211yescharacteristic
34C5×#376C2211yescharacteristic
35#21707C511yesmaximal, characteristic, direct product factor, Sylow
36C2×C5×#41C2314yes—
37C5×#135—24no—
38C5×#135—24no—
39C5×#135—24no—
40C5×#135—24no—
41C2×C5×Q16C2×C412yes—
42C2×C5×Q16C2×C412yes—
43C2×C5×SD16C2×C414yes—
44C2×C5×SD16C2×C414yes—
45C2×C5×D8C2×C412yes—
46C2×C5×D8C2×C412yes—
47C5×#131—28no—
48C5×#131—28no—
49C2×C5×M16—24no—
50C2×C5×M16C2314yes—
51C22×C5×C8—24no—
52C22×C5×C8C2311yescharacteristic
53C22×C5×C8C2311yescharacteristic
54C22×C5×C8C2311yescharacteristic
55C22×C5×C8C2311yescharacteristic
56C5×C4⋊Q8C2×C412yes—
57C5×#127C2×C412yes—
58C5×#124C2×C414yes—
59C5×#121—24no—
60C4×C5×D4—24no—
61C2×C42×C5C2311yescharacteristic
62C2×C42×C5C2311yescharacteristic
63C2×C42×C5C2311yescharacteristic
64C5×#108—24no—
65C5×#108—24no—
66C5×C4⋊C8—24no—
67C5×#104—28no—
68C5×#104—28no—
69C5×C22⋊C8—28no—
70C5×#97—22no—
71C5×#97D414yes—
72C5×#97D414yescomplemented
73C4×C5×C8—22no—
74C4×C5×C8C2×C411yescharacteristic
75C4×C5×C8C2×C411yescharacteristic
76C4×C5×C8C2×C411yescharacteristic
77C4×C5×C8C2×C411yescharacteristic
78C4×C5×C8D414yescomplemented
79C5×#95—22no—
80#3147C2×C511yescharacteristic
81C2×#444C2×C512yescomplemented
82C2×#443C2×C512yescomplemented
83#2932C2×C512yescomplemented
84#1975C2×C512yescomplemented
85#1974C2×C512yescomplemented
86#1963C2×C512yescomplemented
87#1848C2×C511yescharacteristic, complemented
88#1726C2×C511yescharacteristic, complemented
89C5×#41—24no—
90C5×#41—24no—
91C5×#41—24no—
92C5×#41—24no—
93C2×C5×Q8C22×C414yes—
94C2×C5×D4—24no—
95C2×C5×D4—24no—
96C2×C5×D4C22×C414yes—
97C22×C4×C5—21no—
98C22×C4×C5—21no—
99C22×C4×C5—24no—
100C22×C4×C5#4112yes—
101C22×C4×C5C2411yescharacteristic
102C5×Q16—24no—
103C5×Q16—24no—
104C5×SD16—28no—
105C5×SD16—28no—
106C5×D8—24no—
107C5×D8—24no—
108C5×M16—24no—
109C5×M16—24no—
110C5×M16—48no—
111C2×C5×C8—21no—
112C2×C5×C8—21no—
113C2×C5×C8—24no—
114C2×C5×C8—48no—
115C2×C5×C8C22×C411yescharacteristic
116C2×C5×C8C22×C411yescharacteristic
117C2×C5×C8C22×C411yescharacteristic
118C2×C5×C8C22×C411yescharacteristic
119C2×C5×C8C2×D412yes—
120C2×C5×C8C2×D412yes—
121C2×C5×C8C2×D412yes—
122C2×C5×C8C2×D412yes—
123C2×C5×C8C2×D412yes—
124C2×C5×C8C2×D412yes—
125C2×C5×C8#4112yes—
126C2×C5×C8#4112yes—
127C5×C4⋊C4—24no—
128C5×C22⋊C4—28no—
129C42×C5—21no—
130C42×C5—21no—
131C42×C5C22×C411yescharacteristic
132C42×C5C22×C411yescharacteristic
133C42×C5C2×D412yes—
134C42×C5C2×D412yes—
135C42×C5C2×D412yes—
136C2×#135C22×C512yes—
137C2×#135C22×C512yes—
138C2×#131C22×C514yes—
139#532C22×C512yes—
140#495C4×C512yescomplemented
141#495C4×C512yescomplemented
142#494C4×C511yescharacteristic, complemented
143#493C4×C511yescharacteristic, complemented
144#492C4×C512yescomplemented
145#444—28no—
146#443—28no—
147#435—28no—
148C2×#108C22×C512yes—
149C2×#108C22×C512yes—
150#423C22×C512yes—
151C2×#104C22×C514yes—
152C2×#104C22×C514yes—
153#408C22×C514yes—
154C2×#97C22×C511yescharacteristic
155C2×#97C22×C511yescharacteristic
156C2×#97C22×C511yescharacteristic
157C2×C4×C8C22×C511yescharacteristic
158C2×C4×C8C22×C511yescharacteristic
159C2×C4×C8C22×C511yescharacteristic
160#376C22×C511yescharacteristic
161C23×C5—24no—
162C23×C5C2×#4111yescharacteristic
163C5×Q8—24no—
164C5×Q8—24no—
165C5×D4—24no—
166C5×D4—24no—
167C5×D4—44no—
168C5×D4—44no—
169C5×D4—84no—
170C2×C4×C5—24no—
171C2×C4×C5—22no—
172C2×C4×C5—22no—
173C2×C4×C5—42no—
174C2×C4×C5—42no—
175C2×C4×C5—42no—
176C2×C4×C5—44no—
177C2×C4×C5—44no—
178C2×C4×C5C4×D412yes—
179C2×C4×C5C4×D412yes—
180C2×C4×C5C23×C411yescharacteristic
181C2×C4×C5C22×D411yescharacteristic
182C2×C4×C5C22×D411yescharacteristic
183C2×C4×C5C22×D411yescharacteristic
184C2×C4×C5C2×#4111yescharacteristic
185C2×C4×C5C2×#4111yescharacteristic
186C5×C8—22no—
187C5×C8—44no—
188C5×C8C4×D412yes—
189C5×C8C4×D412yes—
190C5×C8C4×D412yes—
191C5×C8C4×D412yes—
192C5×C8#12714yes—
193C2×#41C23×C514yes—
194#135—24no—
195#135—24no—
196#135—24no—
197#135—24no—
198C2×Q16C2×C4×C512yes—
199C2×Q16C2×C4×C512yes—
200C2×SD16C2×C4×C514yes—
201C2×SD16C2×C4×C514yes—
202C2×D8C2×C4×C512yes—
203C2×D8C2×C4×C512yes—
204#131—28no—
205#131—28no—
206C2×M16—24no—
207C2×M16C23×C514yes—
208C22×C8—24no—
209C22×C8C23×C511yescharacteristic
210C22×C8C23×C511yescharacteristic
211C22×C8C23×C511yescharacteristic
212C22×C8C23×C511yescharacteristic
213C4⋊Q8C2×C4×C512yes—
214#127C2×C4×C512yes—
215#124C2×C4×C514yes—
216#121—24no—
217C4×D4—24no—
218C2×C42C23×C511yescharacteristic
219C2×C42C23×C511yescharacteristic
220C2×C42C23×C511yescharacteristic
221#108—24no—
222#108—24no—
223C4⋊C8—24no—
224#104—28no—
225#104—28no—
226C22⋊C8—28no—
227#97—22no—
228#97C5×D414yes—
229#97C5×D414yescomplemented
230C4×C8—22no—
231C4×C8C2×C4×C511yescharacteristic
232C4×C8C2×C4×C511yescharacteristic
233C4×C8C2×C4×C511yescharacteristic
234C4×C8C2×C4×C511yescharacteristic
235C4×C8C5×D414yescomplemented
236#95—22no—
237C22×C5—22no—
238C22×C5—44no—
239C22×C5—48no—
240C22×C5C2×C4×D411yescharacteristic
241C22×C5#53211yescharacteristic
242C22×C5#53211yescharacteristic
243C4×C5—22no—
244C4×C5—22no—
245C4×C5—42no—
246C4×C5—42no—
247C4×C5—44no—
248C4×C5C2×C4×D411yescharacteristic
249C4×C5C2×C4×D411yescharacteristic
250C4×C5C2×#12711yescharacteristic
251C4×C5#53211yescharacteristic
252#41—24no—
253#41—24no—
254#41—24no—
255#41—24no—
256C2×Q8C22×C4×C514yes—
257C2×D4—24no—
258C2×D4—24no—
259C2×D4C22×C4×C514yes—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—24no—
263C22×C4C5×#4112yes—
264C22×C4C24×C511yescharacteristic
265Q16—24no—
266Q16—24no—
267SD16—28no—
268SD16—28no—
269D8—24no—
270D8—24no—
271M16—24no—
272M16—24no—
273M16—48no—
274C2×C8—21no—
275C2×C8—21no—
276C2×C8—24no—
277C2×C8—48no—
278C2×C8C22×C4×C511yescharacteristic
279C2×C8C22×C4×C511yescharacteristic
280C2×C8C22×C4×C511yescharacteristic
281C2×C8C22×C4×C511yescharacteristic
282C2×C8C2×C5×D412yes—
283C2×C8C2×C5×D412yes—
284C2×C8C2×C5×D412yes—
285C2×C8C2×C5×D412yes—
286C2×C8C2×C5×D412yes—
287C2×C8C2×C5×D412yes—
288C2×C8C5×#4112yes—
289C2×C8C5×#4112yes—
290C4⋊C4—24no—
291C22⋊C4—28no—
292C42—21no—
293C42—21no—
294C42C22×C4×C511yescharacteristic
295C42C22×C4×C511yescharacteristic
296C42C2×C5×D412yes—
297C42C2×C5×D412yes—
298C42C2×C5×D412yes—
299C2×C5—21no—
300C2×C5—21no—
301C2×C5—84no—
302C2×C5C4×#12711yescharacteristic
303C2×C5#295712yes—
304C23—24no—
305C23C2×C5×#4111yescharacteristic
306Q8—24no—
307Q8—24no—
308D4—24no—
309D4—24no—
310D4—44no—
311D4—44no—
312D4—84no—
313C2×C4—24no—
314C2×C4—22no—
315C2×C4—22no—
316C2×C4—42no—
317C2×C4—42no—
318C2×C4—42no—
319C2×C4—44no—
320C2×C4—44no—
321C2×C4C4×C5×D412yes—
322C2×C4C4×C5×D412yes—
323C2×C4C23×C4×C511yescharacteristic
324C2×C4C22×C5×D411yescharacteristic
325C2×C4C22×C5×D411yescharacteristic
326C2×C4C22×C5×D411yescharacteristic
327C2×C4C2×C5×#4111yescharacteristic
328C2×C4C2×C5×#4111yescharacteristic
329C8—22no—
330C8—44no—
331C8C4×C5×D412yes—
332C8C4×C5×D412yes—
333C8C4×C5×D412yes—
334C8C4×C5×D412yes—
335C8C5×#12714yes—
336C5#2170711yescharacteristic, direct product factor, Sylow
337C22—22no—
338C22—44no—
339C22—48no—
340C22C2×C4×C5×D411yescharacteristic
341C22C5×#53211yescharacteristic
342C22C5×#53211yescharacteristic
343C4—22no—
344C4—22no—
345C4—42no—
346C4—42no—
347C4—44no—
348C4C2×C4×C5×D411yescharacteristic
349C4C2×C4×C5×D411yescharacteristic
350C4C2×C5×#12711yescharacteristic
351C4C5×#53211yescharacteristic
352C2—21no—
353C2—21no—
354C2—84no—
355C2C4×C5×#12711yescharacteristic
356C2C5×#295712yes—
subgroupG/H
centerC2×C4×C5C22×D4
commutatorC2×C4C23×C4×C5
Frattini, cosocleC22×C4C24×C5
socleC22×C5C2×C4×D4
prime radicalC5×#3147C2
LCSG ▷ C2×C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C4×C5 ◁ C2×C5×#97 ◁ G

Complements

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

NH ≅ G/Ntype
2. C2×C5×#444C2semidirect
3. C2×C5×#443C2semidirect
4. C5×#2932C2semidirect
5. C5×#1975C2semidirect
6. C5×#1974C2semidirect
7. C5×#1963C2semidirect
8. C5×#1848C2semidirect
9. C5×#1726C2semidirect
14. C5×#495C4semidirect
15. C5×#495C4semidirect
16. C5×#494C4semidirect
17. C5×#493C4semidirect
18. C5×#492C4semidirect
35. #21707C5direct (N × H)
72. C5×#97D4semidirect
78. C4×C5×C8D4semidirect
81. C2×#444C2×C5semidirect
82. C2×#443C2×C5semidirect
83. #2932C2×C5semidirect
84. #1975C2×C5semidirect
85. #1974C2×C5semidirect
86. #1963C2×C5semidirect
87. #1848C2×C5semidirect
88. #1726C2×C5semidirect
140. #495C4×C5semidirect
141. #495C4×C5semidirect
142. #494C4×C5semidirect
143. #493C4×C5semidirect
144. #492C4×C5semidirect
229. #97C5×D4semidirect
235. C4×C8C5×D4semidirect
336. C5#21707direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#31472
C2×C5×#4442
C2×C5×#4432
C5×#29322
C5×#19752
C5×#19742
C5×#19632
C5×#18482
C5×#17262
#217075