Finite Group Databasebeta

← C5×#10539 #11933259

Subgroups — C5×#10539 #11933259

Order 1280.

490 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C5×#486C211yesmaximal, characteristic
2C5×#1987C211yesmaximal, characteristic, complemented
3C5×#1982C211yesmaximal, characteristic
4C5×#1939C211yesmaximal, characteristic, complemented
5C2×C5×#333C211yesmaximal, characteristic, complemented
6C2×C5×#331C211yesmaximal, characteristic
7C5×#1277C211yesmaximal, characteristic, complemented
8C2×C5×C4⋊Q8C2211yescharacteristic
9C2×C5×#127C2211yescharacteristic
10C5×#486—22no—
11C5×#486—22no—
12C2×C5×C4⋊C8—21no—
13C2×C5×C4⋊C8C2211yescharacteristic
14C2×C5×C4⋊C8C2211yescharacteristic
15C2×C5×C4⋊C8C2211yescharacteristic
16C2×C5×#102—21no—
17C2×C5×#102—21no—
18C2×C5×#102—21no—
19C2×C5×#102—21no—
20C2×C4×C5×C8C2211yescharacteristic
21C4×C5×C4⋊C4C2211yescharacteristic
22C5×#340—21no—
23C5×#340—21no—
24C5×#336—21no—
25C5×#333—22no—
26C5×#331—22no—
27#10539C511yesmaximal, characteristic, direct product factor, Sylow
28C22×C5×Q8—21no—
29C22×C5×D4—21no—
30C22×C5×D4—21no—
31C22×C5×D4—21no—
32C22×C5×C8—21no—
33C22×C5×C8—21no—
34C22×C5×C8—21no—
35C22×C5×C8D411yescharacteristic
36C22×C5×C8D411yescharacteristic
37C5×C4⋊Q8—21no—
38C5×C4⋊Q8—21no—
39C5×C4⋊Q8—21no—
40C5×C4⋊Q8D412yes—
41C5×#127—21no—
42C5×#127—21no—
43C5×#127—21no—
44C5×#127D412yes—
45C2×C5×C4⋊C4—21no—
46C2×C5×C4⋊C4—21no—
47C2×C5×C4⋊C4D411yescharacteristic
48C2×C5×C4⋊C4D411yescharacteristic
49C2×C42×C5—21no—
50C2×C42×C5C2311yescharacteristic
51C5×C4⋊C8—21no—
52C5×C4⋊C8—21no—
53C5×C4⋊C8—21no—
54C5×C4⋊C8—21no—
55C5×C4⋊C8—21no—
56C5×C4⋊C8—21no—
57C5×C4⋊C8—42no—
58C5×#102—24no—
59C5×#102—24no—
60C5×#102—42no—
61C5×#102—42no—
62C4×C5×C8—21no—
63C4×C5×C8—21no—
64C5×#95—21no—
65C2×#486C2×C511yescharacteristic
66#1987C2×C511yescharacteristic, complemented
67#1982C2×C511yescharacteristic
68#1939C2×C511yescharacteristic, complemented
69C2×#333C2×C511yescharacteristic, complemented
70C2×#331C2×C511yescharacteristic
71#1277C2×C511yescharacteristic, complemented
72C24×C5—41no—
73C2×C5×Q8—22no—
74C2×C5×Q8—41no—
75C2×C5×Q8—81no—
76C2×C5×Q8—81no—
77C2×C5×D4—22no—
78C2×C5×D4—22no—
79C2×C5×D4—22no—
80C2×C5×D4—41no—
81C2×C5×D4—41no—
82C2×C5×D4—41no—
83C2×C5×D4—41no—
84C2×C5×D4—41no—
85C2×C5×D4—41no—
86C2×C5×D4—41no—
87C2×C5×D4—41no—
88C2×C5×D4—41no—
89C2×C5×D4—41no—
90C2×C5×D4—41no—
91C2×C5×D4—81no—
92C2×C5×D4—81no—
93C22×C4×C5—21no—
94C22×C4×C5—21no—
95C22×C4×C5—41no—
96C22×C4×C5C2×D411yescharacteristic
97C22×C4×C5C2×D411yescharacteristic
98C22×C4×C5#4111yescharacteristic
99C2×C5×C8—21no—
100C2×C5×C8—21no—
101C2×C5×C8—21no—
102C2×C5×C8—21no—
103C2×C5×C8—21no—
104C2×C5×C8—21no—
105C2×C5×C8—41no—
106C2×C5×C8—41no—
107C2×C5×C8—41no—
108C2×C5×C8—41no—
109C2×C5×C8—41no—
110C2×C5×C8—41no—
111C2×C5×C8—41no—
112C2×C5×C8—41no—
113C2×C5×C8—41no—
114C5×C4⋊C4—21no—
115C5×C4⋊C4—22no—
116C5×C4⋊C4—22no—
117C5×C4⋊C4—21no—
118C5×C4⋊C4—21no—
119C5×C4⋊C4—41no—
120C5×C4⋊C4—41no—
121C5×C4⋊C4SD1612yes—
122C42×C5—41no—
123C42×C5—41no—
124C42×C5C2×D411yescharacteristic
125C42×C5C2×D411yescharacteristic
126C42×C5#4111yescharacteristic
127C42×C5#4111yescharacteristic
128C2×C4⋊Q8C22×C511yescharacteristic
129C2×#127C22×C511yescharacteristic
130#486—22no—
131#486—22no—
132C2×C4⋊C8—21no—
133C2×C4⋊C8C22×C511yescharacteristic
134C2×C4⋊C8C22×C511yescharacteristic
135C2×C4⋊C8C22×C511yescharacteristic
136C2×#102—21no—
137C2×#102—21no—
138C2×#102—21no—
139C2×#102—21no—
140C2×C4×C8C22×C511yescharacteristic
141C4×C4⋊C4C22×C511yescharacteristic
142#340—21no—
143#340—21no—
144#336—21no—
145#333—22no—
146#331—22no—
147C23×C5—42no—
148C23×C5—81no—
149C23×C5—81no—
150C23×C5—81no—
151C23×C5—81no—
152C23×C5—81no—
153C23×C5—81no—
154C23×C5#12111yescharacteristic
155C5×Q8—82no—
156C5×Q8—82no—
157C5×D4—42no—
158C5×D4—42no—
159C5×D4—42no—
160C5×D4—42no—
161C5×D4—82no—
162C5×D4—82no—
163C5×D4—81no—
164C5×D4—81no—
165C5×D4—81no—
166C5×D4—81no—
167C2×C4×C5—21no—
168C2×C4×C5—21no—
169C2×C4×C5—22no—
170C2×C4×C5—41no—
171C2×C4×C5—41no—
172C2×C4×C5—42no—
173C2×C4×C5—41no—
174C2×C4×C5—41no—
175C2×C4×C5—41no—
176C2×C4×C5—81no—
177C2×C4×C5—81no—
178C2×C4×C5#12011yescharacteristic
179C2×C4×C5#12111yescharacteristic
180C2×C4×C5#12111yescharacteristic
181C2×C4×C5#12311yescharacteristic
182C2×C4×C5#12311yescharacteristic
183C2×C4×C5#12411yescharacteristic
184C2×C4×C5C2×SD1611yescharacteristic
185C2×C4×C5#13511yescharacteristic
186C2×C4×C5#13511yescharacteristic
187C2×C4×C5#13511yescharacteristic
188C2×C4×C5AGL(1,ℤ/8)11yescharacteristic
189C2×C4×C5AGL(1,ℤ/8)11yescharacteristic
190C2×C4×C5#13711yescharacteristic
191C2×C4×C5#13711yescharacteristic
192C5×C8—41no—
193C5×C8—41no—
194C5×C8—81no—
195C5×C8—81no—
196C5×C8—81no—
197C22×Q8—21no—
198C22×D4—21no—
199C22×D4—21no—
200C22×D4—21no—
201C22×C8—21no—
202C22×C8—21no—
203C22×C8—21no—
204C22×C8C5×D411yescharacteristic
205C22×C8C5×D411yescharacteristic
206C4⋊Q8—21no—
207C4⋊Q8—21no—
208C4⋊Q8—21no—
209C4⋊Q8C5×D412yes—
210#127—21no—
211#127—21no—
212#127—21no—
213#127C5×D412yes—
214C2×C4⋊C4—21no—
215C2×C4⋊C4—21no—
216C2×C4⋊C4C5×D411yescharacteristic
217C2×C4⋊C4C5×D411yescharacteristic
218C2×C42—21no—
219C2×C42C23×C511yescharacteristic
220C4⋊C8—21no—
221C4⋊C8—21no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—21no—
225C4⋊C8—21no—
226C4⋊C8—42no—
227#102—24no—
228#102—24no—
229#102—42no—
230#102—42no—
231C4×C8—21no—
232C4×C8—21no—
233#95—21no—
234C22×C5—82no—
235C22×C5—82no—
236C22×C5—82no—
237C22×C5—161no—
238C22×C5—161no—
239C22×C5—161no—
240C22×C5—161no—
241C22×C5#39411yescharacteristic
242C22×C5#46311yescharacteristic
243C22×C5#46411yescharacteristic
244C22×C5#46511yescharacteristic
245C22×C5#46811yescharacteristic
246C22×C5#47211yescharacteristic
247C22×C5#47211yescharacteristic
248C4×C5—21no—
249C4×C5—21no—
250C4×C5—42no—
251C4×C5—81no—
252C4×C5—82no—
253C4×C5#44911yescharacteristic
254C4×C5#45211yescharacteristic
255C4×C5#46011yescharacteristic
256C4×C5#46111yescharacteristic
257C4×C5#46311yescharacteristic
258C4×C5#46411yescharacteristic
259C4×C5#48811yescharacteristic
260C4×C5#48911yescharacteristic
261C24—41no—
262C2×Q8—22no—
263C2×Q8—41no—
264C2×Q8—81no—
265C2×Q8—81no—
266C2×D4—22no—
267C2×D4—22no—
268C2×D4—22no—
269C2×D4—41no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—41no—
275C2×D4—41no—
276C2×D4—41no—
277C2×D4—41no—
278C2×D4—41no—
279C2×D4—41no—
280C2×D4—81no—
281C2×D4—81no—
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—41no—
285C22×C4C2×C5×D411yescharacteristic
286C22×C4C2×C5×D411yescharacteristic
287C22×C4C5×#4111yescharacteristic
288C2×C8—21no—
289C2×C8—21no—
290C2×C8—21no—
291C2×C8—21no—
292C2×C8—21no—
293C2×C8—21no—
294C2×C8—41no—
295C2×C8—41no—
296C2×C8—41no—
297C2×C8—41no—
298C2×C8—41no—
299C2×C8—41no—
300C2×C8—41no—
301C2×C8—41no—
302C2×C8—41no—
303C4⋊C4—21no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4—41no—
309C4⋊C4—41no—
310C4⋊C4C5×SD1612yes—
311C42—41no—
312C42—41no—
313C42C2×C5×D411yescharacteristic
314C42C2×C5×D411yescharacteristic
315C42C5×#4111yescharacteristic
316C42C5×#4111yescharacteristic
317C2×C5—162no—
318C2×C5#167711yescharacteristic
319C2×C5#167811yescharacteristic
320C2×C5#169711yescharacteristic
321C2×C5#169811yescharacteristic
322C2×C5#204211yescharacteristic
323C2×C5#204511yescharacteristic
324C2×C5#205511yescharacteristic
325C23—42no—
326C23—81no—
327C23—81no—
328C23—81no—
329C23—81no—
330C23—81no—
331C23—81no—
332C23C5×#12111yescharacteristic
333Q8—82no—
334Q8—82no—
335D4—42no—
336D4—42no—
337D4—42no—
338D4—42no—
339D4—82no—
340D4—82no—
341D4—81no—
342D4—81no—
343D4—81no—
344D4—81no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—22no—
348C2×C4—41no—
349C2×C4—41no—
350C2×C4—42no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—81no—
355C2×C4—81no—
356C2×C4C5×#12011yescharacteristic
357C2×C4C5×#12111yescharacteristic
358C2×C4C5×#12111yescharacteristic
359C2×C4C5×#12311yescharacteristic
360C2×C4C5×#12311yescharacteristic
361C2×C4C5×#12411yescharacteristic
362C2×C4C2×C5×SD1611yescharacteristic
363C2×C4C5×#13511yescharacteristic
364C2×C4C5×#13511yescharacteristic
365C2×C4C5×#13511yescharacteristic
366C2×C4C5×AGL(1,ℤ/8)11yescharacteristic
367C2×C4C5×AGL(1,ℤ/8)11yescharacteristic
368C2×C4C5×#13711yescharacteristic
369C2×C4C5×#13711yescharacteristic
370C8—41no—
371C8—41no—
372C8—81no—
373C8—81no—
374C8—81no—
375C5#1053911yescharacteristic, direct product factor, Sylow
376C22—82no—
377C22—82no—
378C22—82no—
379C22—161no—
380C22—161no—
381C22—161no—
382C22—161no—
383C22C5×#39411yescharacteristic
384C22C5×#46311yescharacteristic
385C22C5×#46411yescharacteristic
386C22C5×#46511yescharacteristic
387C22C5×#46811yescharacteristic
388C22C5×#47211yescharacteristic
389C22C5×#47211yescharacteristic
390C4—21no—
391C4—21no—
392C4—42no—
393C4—82no—
394C4—81no—
395C4C5×#44911yescharacteristic
396C4C5×#45211yescharacteristic
397C4C5×#46011yescharacteristic
398C4C5×#46111yescharacteristic
399C4C5×#46311yescharacteristic
400C4C5×#46411yescharacteristic
401C4C5×#48811yescharacteristic
402C4C5×#48911yescharacteristic
403C2—162no—
404C2C5×#167711yescharacteristic
405C2C5×#167811yescharacteristic
406C2C5×#169711yescharacteristic
407C2C5×#169811yescharacteristic
408C2C5×#204211yescharacteristic
409C2C5×#204511yescharacteristic
410C2C5×#205511yescharacteristic
subgroupG/H
center, socleC23×C5#121
commutator, Frattini, cosocleC2×C42C23×C5
prime radicalC2×C5×#127C22
LCSG ▷ C2×C42 ▷ C22 ▷ C1
UCSC1 ◁ C23×C5 ◁ C2×C42×C5 ◁ G

Complements

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

NH ≅ G/Ntype
2. C5×#1987C2semidirect
4. C5×#1939C2semidirect
5. C2×C5×#333C2semidirect
7. C5×#1277C2semidirect
27. #10539C5direct (N × H)
66. #1987C2×C5semidirect
68. #1939C2×C5semidirect
69. C2×#333C2×C5semidirect
71. #1277C2×C5semidirect
375. C5#10539direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C5×#4862
C5×#19872
C5×#19822
C5×#19392
C2×C5×#3332
C2×C5×#3312
C5×#12772
#105395