Finite Group Databasebeta

← C5×#9315 #11932035

Subgroups — C5×#9315 #11932035

Order 1280.

480 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3079C211yesmaximal, characteristic
2C5×#2988C211yesmaximal, characteristic
3C5×#1987C211yesmaximal, characteristic, complemented
4C5×#1806C211yesmaximal, characteristic
5C5×#1548C211yesmaximal, characteristic, complemented
6C5×#1303C211yesmaximal, characteristic, complemented
7C5×#1292C211yesmaximal, characteristic, complemented
8C2×C5×C4⋊Q8C2211yescharacteristic
9C4×C5×#41C2211yescharacteristic
10C5×#477—21no—
11C5×#476—21no—
12C5×#475C412yescomplemented
13C5×#474C412yescomplemented
14C5×C8×Q8—21no—
15C5×C8×D4—22no—
16C5×C8×D4—21no—
17C5×#431—21no—
18C5×#431C2211yescharacteristic
19C5×#428—21no—
20C5×#423C2211yescharacteristic
21C5×#423C2211yescharacteristic
22C2×C5×C4⋊C8—21no—
23C2×C5×C4⋊C8C2211yescharacteristic
24C2×C5×#104—21no—
25C5×#417—21no—
26C4×C5×C4⋊C4C2211yescharacteristic
27C5×#340—22no—
28C5×#339—21no—
29C5×#333—21no—
30C5×#333—21no—
31#9315C511yesmaximal, characteristic, direct product factor, Sylow
32C2×C5×#41D411yescharacteristic
33C22×C5×Q8—21no—
34C2×C5×M16—21no—
35C2×C5×M16—21no—
36C22×C5×C8—22no—
37C22×C5×C8D412yes—
38C5×C4⋊Q8—21no—
39C5×C4⋊Q8—21no—
40C5×C4⋊Q8—21no—
41C5×C4⋊Q8C2×C412yes—
42C4×C5×Q8C2×C411yescharacteristic
43C4×C5×Q8D411yescharacteristic
44C4×C5×D4—21no—
45C4×C5×D4—21no—
46C4×C5×D4C2×C411yescharacteristic
47C4×C5×D4D411yescharacteristic
48C5×#117—21no—
49C5×#117D411yescharacteristic
50C2×C5×C4⋊C4—21no—
51C2×C5×C4⋊C4—21no—
52C2×C5×C4⋊C4—21no—
53C2×C42×C5—21no—
54C2×C42×C5—21no—
55C2×C42×C5C2311yescharacteristic
56C5×#107—22no—
57C5×#106—22no—
58C5×C4⋊C8—22no—
59C5×C4⋊C8—21no—
60C5×C4⋊C8—21no—
61C5×C4⋊C8—21no—
62C5×C4⋊C8—21no—
63C5×C4⋊C8—21no—
64C5×C4⋊C8C2×C412yes—
65C5×C4⋊C8D411yescharacteristic
66C5×C4⋊C8D411yescharacteristic
67C5×#104—42no—
68C5×#103—22no—
69C5×#102—22no—
70C5×C22⋊C8—22no—
71C5×C22⋊C8—21no—
72C4×C5×C8—22no—
73C4×C5×C8—21no—
74C5×#95—21no—
75#3079C2×C511yescharacteristic
76#2988C2×C511yescharacteristic
77#1987C2×C511yescharacteristic, complemented
78#1806C2×C511yescharacteristic
79#1548C2×C511yescharacteristic, complemented
80#1303C2×C511yescharacteristic, complemented
81#1292C2×C511yescharacteristic, complemented
82C5×#41—21no—
83C5×#41—21no—
84C5×#41SD1612yes—
85C5×#41Q1612yes—
86C2×C5×Q8—22no—
87C2×C5×Q8—41no—
88C2×C5×Q8—82no—
89C2×C5×Q8C22⋊C411yescharacteristic
90C2×C5×D4—21no—
91C2×C5×D4C22⋊C411yescharacteristic
92C22×C4×C5—21no—
93C22×C4×C5—21no—
94C22×C4×C5—21no—
95C22×C4×C5—41no—
96C22×C4×C5—41no—
97C22×C4×C5C2×D411yescharacteristic
98C22×C4×C5C2×D411yescharacteristic
99C22×C4×C5#4111yescharacteristic
100C5×M16—41no—
101C5×M16—41no—
102C5×M16—41no—
103C5×M16—41no—
104C2×C5×C8—22no—
105C2×C5×C8—21no—
106C2×C5×C8—22no—
107C2×C5×C8—21no—
108C2×C5×C8—42no—
109C2×C5×C8—41no—
110C2×C5×C8—82no—
111C5×C4⋊C4—21no—
112C5×C4⋊C4—21no—
113C5×C4⋊C4—22no—
114C5×C4⋊C4—22no—
115C5×C4⋊C4—22no—
116C5×C4⋊C4—41no—
117C5×C4⋊C4—41no—
118C5×C4⋊C4—41no—
119C5×C4⋊C4C22⋊C411yescharacteristic
120C5×C4⋊C4C22⋊C411yescharacteristic
121C5×C22⋊C4—21no—
122C5×C22⋊C4—21no—
123C5×C22⋊C4—21no—
124C42×C5—21no—
125C42×C5—21no—
126C42×C5—21no—
127C42×C5—41no—
128C42×C5—41no—
129C42×C5C22×C411yescharacteristic
130C42×C5C2×D411yescharacteristic
131C42×C5C2×D411yescharacteristic
132C42×C5#4111yescharacteristic
133C2×C4⋊Q8C22×C511yescharacteristic
134C4×#41C22×C511yescharacteristic
135#477—21no—
136#476—21no—
137#475C4×C512yescomplemented
138#474C4×C512yescomplemented
139C8×Q8—21no—
140C8×D4—22no—
141C8×D4—21no—
142#431—21no—
143#431C22×C511yescharacteristic
144#428—21no—
145#423C22×C511yescharacteristic
146#423C22×C511yescharacteristic
147C2×C4⋊C8—21no—
148C2×C4⋊C8C22×C511yescharacteristic
149C2×#104—21no—
150#417—21no—
151C4×C4⋊C4C22×C511yescharacteristic
152#340—22no—
153#339—21no—
154#333—21no—
155#333—21no—
156C23×C5—21no—
157C23×C5#12111yescharacteristic
158C5×Q8—21no—
159C5×Q8—82no—
160C5×Q8—82no—
161C5×Q8#10312yes—
162C5×D4—22no—
163C5×D4—22no—
164C5×D4—21no—
165C5×D4#10312yes—
166C2×C4×C5—21no—
167C2×C4×C5—21no—
168C2×C4×C5—21no—
169C2×C4×C5—22no—
170C2×C4×C5—22no—
171C2×C4×C5—21no—
172C2×C4×C5—21no—
173C2×C4×C5—41no—
174C2×C4×C5—41no—
175C2×C4×C5—41no—
176C2×C4×C5—42no—
177C2×C4×C5—42no—
178C2×C4×C5—81no—
179C2×C4×C5—81no—
180C2×C4×C5—82no—
181C2×C4×C5—81no—
182C2×C4×C5C2×C22⋊C411yescharacteristic
183C2×C4×C5C4×D411yescharacteristic
184C2×C4×C5C4×D411yescharacteristic
185C2×C4×C5#12011yescharacteristic
186C2×C4×C5#12111yescharacteristic
187C2×C4×C5#12311yescharacteristic
188C2×C4×C5C2×SD1611yescharacteristic
189C2×C4×C5C2×Q1611yescharacteristic
190C2×C4×C5#13511yescharacteristic
191C2×C4×C5#13511yescharacteristic
192C5×C8—42no—
193C5×C8—42no—
194C5×C8—41no—
195C5×C8—41no—
196C5×C8—81no—
197C2×#41C5×D411yescharacteristic
198C22×Q8—21no—
199C2×M16—21no—
200C2×M16—21no—
201C22×C8—22no—
202C22×C8C5×D412yes—
203C4⋊Q8—21no—
204C4⋊Q8—21no—
205C4⋊Q8—21no—
206C4⋊Q8C2×C4×C512yes—
207C4×Q8C2×C4×C511yescharacteristic
208C4×Q8C5×D411yescharacteristic
209C4×D4—21no—
210C4×D4—21no—
211C4×D4C2×C4×C511yescharacteristic
212C4×D4C5×D411yescharacteristic
213#117—21no—
214#117C5×D411yescharacteristic
215C2×C4⋊C4—21no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4—21no—
218C2×C42—21no—
219C2×C42—21no—
220C2×C42C23×C511yescharacteristic
221#107—22no—
222#106—22no—
223C4⋊C8—22no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—21no—
227C4⋊C8—21no—
228C4⋊C8—21no—
229C4⋊C8C2×C4×C512yes—
230C4⋊C8C5×D411yescharacteristic
231C4⋊C8C5×D411yescharacteristic
232#104—42no—
233#103—22no—
234#102—22no—
235C22⋊C8—22no—
236C22⋊C8—21no—
237C4×C8—22no—
238C4×C8—21no—
239#95—21no—
240C22×C5—21no—
241C22×C5—21no—
242C22×C5—22no—
243C22×C5—41no—
244C22×C5—41no—
245C22×C5#38611yescharacteristic
246C22×C5#46511yescharacteristic
247C22×C5#46711yescharacteristic
248C4×C5—21no—
249C4×C5—22no—
250C4×C5—21no—
251C4×C5—21no—
252C4×C5—41no—
253C4×C5—41no—
254C4×C5—82no—
255C4×C5—82no—
256C4×C5C2×#10311yescharacteristic
257C4×C5#41611yescharacteristic
258C4×C5#45511yescharacteristic
259C4×C5#45611yescharacteristic
260C4×C5#46511yescharacteristic
261C4×C5#46711yescharacteristic
262#41—21no—
263#41—21no—
264#41C5×SD1612yes—
265#41C5×Q1612yes—
266C2×Q8—22no—
267C2×Q8—41no—
268C2×Q8—82no—
269C2×Q8C5×C22⋊C411yescharacteristic
270C2×D4—21no—
271C2×D4C5×C22⋊C411yescharacteristic
272C22×C4—21no—
273C22×C4—21no—
274C22×C4—21no—
275C22×C4—41no—
276C22×C4—41no—
277C22×C4C2×C5×D411yescharacteristic
278C22×C4C2×C5×D411yescharacteristic
279C22×C4C5×#4111yescharacteristic
280M16—41no—
281M16—41no—
282M16—41no—
283M16—41no—
284C2×C8—22no—
285C2×C8—21no—
286C2×C8—22no—
287C2×C8—21no—
288C2×C8—42no—
289C2×C8—41no—
290C2×C8—82no—
291C4⋊C4—21no—
292C4⋊C4—21no—
293C4⋊C4—22no—
294C4⋊C4—22no—
295C4⋊C4—22no—
296C4⋊C4—41no—
297C4⋊C4—41no—
298C4⋊C4—41no—
299C4⋊C4C5×C22⋊C411yescharacteristic
300C4⋊C4C5×C22⋊C411yescharacteristic
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—21no—
304C42—21no—
305C42—21no—
306C42—21no—
307C42—41no—
308C42—41no—
309C42C22×C4×C511yescharacteristic
310C42C2×C5×D411yescharacteristic
311C42C2×C5×D411yescharacteristic
312C42C5×#4111yescharacteristic
313C2×C5—21no—
314C2×C5—21no—
315C2×C5—42no—
316C2×C5#188211yescharacteristic
317C2×C5#189211yescharacteristic
318C2×C5#190311yescharacteristic
319C23—21no—
320C23C5×#12111yescharacteristic
321Q8—21no—
322Q8—82no—
323Q8—82no—
324Q8C5×#10312yes—
325D4—22no—
326D4—22no—
327D4—21no—
328D4C5×#10312yes—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4—22no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—41no—
339C2×C4—42no—
340C2×C4—42no—
341C2×C4—81no—
342C2×C4—82no—
343C2×C4—81no—
344C2×C4—81no—
345C2×C4C2×C5×C22⋊C411yescharacteristic
346C2×C4C4×C5×D411yescharacteristic
347C2×C4C4×C5×D411yescharacteristic
348C2×C4C5×#12011yescharacteristic
349C2×C4C5×#12111yescharacteristic
350C2×C4C5×#12311yescharacteristic
351C2×C4C2×C5×SD1611yescharacteristic
352C2×C4C2×C5×Q1611yescharacteristic
353C2×C4C5×#13511yescharacteristic
354C2×C4C5×#13511yescharacteristic
355C8—42no—
356C8—42no—
357C8—41no—
358C8—41no—
359C8—81no—
360C5#931511yescharacteristic, direct product factor, Sylow
361C22—21no—
362C22—21no—
363C22—22no—
364C22—41no—
365C22—41no—
366C22C5×#38611yescharacteristic
367C22C5×#46511yescharacteristic
368C22C5×#46711yescharacteristic
369C4—21no—
370C4—22no—
371C4—21no—
372C4—21no—
373C4—41no—
374C4—41no—
375C4—82no—
376C4—82no—
377C4C2×C5×#10311yescharacteristic
378C4C5×#41611yescharacteristic
379C4C5×#45511yescharacteristic
380C4C5×#45611yescharacteristic
381C4C5×#46511yescharacteristic
382C4C5×#46711yescharacteristic
383C2—21no—
384C2—21no—
385C2—42no—
386C2C5×#188211yescharacteristic
387C2C5×#189211yescharacteristic
388C2C5×#190311yescharacteristic
subgroupG/H
center, socleC22×C5#386
commutatorC42C22×C4×C5
Frattini, cosocleC2×C42C23×C5
prime radicalC2×C5×#41D4
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
3. C5×#1987C2semidirect
5. C5×#1548C2semidirect
6. C5×#1303C2semidirect
7. C5×#1292C2semidirect
12. C5×#475C4semidirect
13. C5×#474C4semidirect
31. #9315C5direct (N × H)
77. #1987C2×C5semidirect
79. #1548C2×C5semidirect
80. #1303C2×C5semidirect
81. #1292C2×C5semidirect
137. #475C4×C5semidirect
138. #474C4×C5semidirect
360. C5#9315direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#30792
C5×#29882
C5×#19872
C5×#18062
C5×#15482
C5×#13032
C5×#12922
#93155