Finite Group Databasebeta

← C5×#10599 #11933319

Subgroups — C5×#10599 #11933319

Order 1280.

454 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#1982C211yesmaximal, characteristic
2C5×#1957C211yesmaximal, characteristic
3C5×#1948C211yesmaximal, characteristic, complemented
4C5×#1939C211yesmaximal, characteristic, complemented
5C2×C5×#335C211yesmaximal, characteristic, complemented
6C2×C5×#331C211yesmaximal, characteristic
7C5×#1296C211yesmaximal, characteristic, complemented
8C2×C5×#127C2211yescharacteristic
9C2×C5×#106—21no—
10C2×C5×C4⋊C8—21no—
11C2×C5×C4⋊C8C2211yescharacteristic
12C2×C5×C4⋊C8C2211yescharacteristic
13C2×C5×C4⋊C8C2211yescharacteristic
14C2×C5×#102—21no—
15C2×C5×#102—21no—
16C2×C5×#102—21no—
17C2×C5×#102—21no—
18C2×C4×C5×C8C2211yescharacteristic
19C5×#384C2211yescharacteristic
20C4×C5×C4⋊C4C2211yescharacteristic
21C5×#340—21no—
22C5×#336—21no—
23C5×#335—22no—
24C5×#331—22no—
25#10599C511yesmaximal, characteristic, direct product factor, Sylow
26C22×C5×D4—21no—
27C22×C5×D4—21no—
28C22×C5×D4—21no—
29C22×C5×C8—21no—
30C22×C5×C8—21no—
31C22×C5×C8—21no—
32C22×C5×C8D411yescharacteristic
33C22×C5×C8D411yescharacteristic
34C5×#127—21no—
35C5×#127—21no—
36C5×#127—21no—
37C5×#127Q812yes—
38C2×C5×C4⋊C4—21no—
39C2×C5×C4⋊C4—21no—
40C2×C5×C4⋊C4—21no—
41C2×C5×C4⋊C4D411yescharacteristic
42C2×C5×C4⋊C4D411yescharacteristic
43C2×C42×C5—21no—
44C2×C42×C5C2311yescharacteristic
45C5×#106—42no—
46C5×C4⋊C8—21no—
47C5×C4⋊C8—21no—
48C5×C4⋊C8—21no—
49C5×C4⋊C8—21no—
50C5×C4⋊C8—21no—
51C5×C4⋊C8—42no—
52C5×C4⋊C8D412yes—
53C5×#102—42no—
54C5×#102—42no—
55C5×#102—42no—
56C5×#102—42no—
57C4×C5×C8—21no—
58C4×C5×C8—21no—
59C5×#95—21no—
60#1982C2×C511yescharacteristic
61#1957C2×C511yescharacteristic
62#1948C2×C511yescharacteristic, complemented
63#1939C2×C511yescharacteristic, complemented
64C2×#335C2×C511yescharacteristic, complemented
65C2×#331C2×C511yescharacteristic
66#1296C2×C511yescharacteristic, complemented
67C24×C5—41no—
68C2×C5×D4—22no—
69C2×C5×D4—22no—
70C2×C5×D4—22no—
71C2×C5×D4—41no—
72C2×C5×D4—41no—
73C2×C5×D4—41no—
74C2×C5×D4—41no—
75C2×C5×D4—41no—
76C2×C5×D4—41no—
77C2×C5×D4—41no—
78C2×C5×D4—41no—
79C2×C5×D4—41no—
80C2×C5×D4—41no—
81C2×C5×D4—41no—
82C2×C5×D4—81no—
83C2×C5×D4—81no—
84C22×C4×C5—21no—
85C22×C4×C5—21no—
86C22×C4×C5—41no—
87C22×C4×C5C2×D411yescharacteristic
88C22×C4×C5C2×D411yescharacteristic
89C22×C4×C5#4111yescharacteristic
90C2×C5×C8—21no—
91C2×C5×C8—21no—
92C2×C5×C8—22no—
93C2×C5×C8—21no—
94C2×C5×C8—21no—
95C2×C5×C8—21no—
96C2×C5×C8—41no—
97C2×C5×C8—41no—
98C2×C5×C8—41no—
99C2×C5×C8—41no—
100C2×C5×C8—41no—
101C2×C5×C8—41no—
102C2×C5×C8—41no—
103C2×C5×C8—41no—
104C2×C5×C8SD1612yes—
105C5×C4⋊C4—21no—
106C5×C4⋊C4—21no—
107C5×C4⋊C4—21no—
108C5×C4⋊C4—41no—
109C5×C4⋊C4—41no—
110C5×C4⋊C4—41no—
111C5×C4⋊C4—41no—
112C5×C4⋊C4—81no—
113C5×C4⋊C4D812yes—
114C42×C5—41no—
115C42×C5—41no—
116C42×C5C2×D411yescharacteristic
117C42×C5C2×Q811yescharacteristic
118C42×C5#4111yescharacteristic
119C42×C5#4111yescharacteristic
120C2×#127C22×C511yescharacteristic
121C2×#106—21no—
122C2×C4⋊C8—21no—
123C2×C4⋊C8C22×C511yescharacteristic
124C2×C4⋊C8C22×C511yescharacteristic
125C2×C4⋊C8C22×C511yescharacteristic
126C2×#102—21no—
127C2×#102—21no—
128C2×#102—21no—
129C2×#102—21no—
130C2×C4×C8C22×C511yescharacteristic
131#384C22×C511yescharacteristic
132C4×C4⋊C4C22×C511yescharacteristic
133#340—21no—
134#336—21no—
135#335—22no—
136#331—22no—
137C23×C5—42no—
138C23×C5—81no—
139C23×C5—81no—
140C23×C5—81no—
141C23×C5—81no—
142C23×C5—81no—
143C23×C5—81no—
144C23×C5#12111yescharacteristic
145C5×D4—42no—
146C5×D4—42no—
147C5×D4—42no—
148C5×D4—42no—
149C5×D4—82no—
150C5×D4—82no—
151C5×D4—81no—
152C5×D4—81no—
153C5×D4—81no—
154C5×D4—81no—
155C2×C4×C5—21no—
156C2×C4×C5—21no—
157C2×C4×C5—22no—
158C2×C4×C5—41no—
159C2×C4×C5—41no—
160C2×C4×C5—41no—
161C2×C4×C5—41no—
162C2×C4×C5—41no—
163C2×C4×C5—81no—
164C2×C4×C5—81no—
165C2×C4×C5—81no—
166C2×C4×C5#12111yescharacteristic
167C2×C4×C5#12111yescharacteristic
168C2×C4×C5C22⋊Q811yescharacteristic
169C2×C4×C5C22⋊Q811yescharacteristic
170C2×C4×C5C22⋊Q811yescharacteristic
171C2×C4×C5#12611yescharacteristic
172C2×C4×C5C2×D811yescharacteristic
173C2×C4×C5C2×SD1611yescharacteristic
174C2×C4×C5#13511yescharacteristic
175C2×C4×C5#13511yescharacteristic
176C2×C4×C5AGL(1,ℤ/8)11yescharacteristic
177C2×C4×C5AGL(1,ℤ/8)11yescharacteristic
178C2×C4×C5AGL(1,ℤ/8)11yescharacteristic
179C2×C4×C5#13711yescharacteristic
180C5×C8—22no—
181C5×C8—42no—
182C5×C8—41no—
183C5×C8—81no—
184C5×C8—81no—
185C22×D4—21no—
186C22×D4—21no—
187C22×D4—21no—
188C22×C8—21no—
189C22×C8—21no—
190C22×C8—21no—
191C22×C8C5×D411yescharacteristic
192C22×C8C5×D411yescharacteristic
193#127—21no—
194#127—21no—
195#127—21no—
196#127C5×Q812yes—
197C2×C4⋊C4—21no—
198C2×C4⋊C4—21no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4C5×D411yescharacteristic
201C2×C4⋊C4C5×D411yescharacteristic
202C2×C42—21no—
203C2×C42C23×C511yescharacteristic
204#106—42no—
205C4⋊C8—21no—
206C4⋊C8—21no—
207C4⋊C8—21no—
208C4⋊C8—21no—
209C4⋊C8—21no—
210C4⋊C8—42no—
211C4⋊C8C5×D412yes—
212#102—42no—
213#102—42no—
214#102—42no—
215#102—42no—
216C4×C8—21no—
217C4×C8—21no—
218#95—21no—
219C22×C5—82no—
220C22×C5—82no—
221C22×C5—82no—
222C22×C5—161no—
223C22×C5—161no—
224C22×C5—161no—
225C22×C5—161no—
226C22×C5#39611yescharacteristic
227C22×C5#46011yescharacteristic
228C22×C5#46311yescharacteristic
229C22×C5#46611yescharacteristic
230C22×C5#46811yescharacteristic
231C22×C5#47111yescharacteristic
232C22×C5#47211yescharacteristic
233C4×C5—21no—
234C4×C5—21no—
235C4×C5—42no—
236C4×C5—81no—
237C4×C5—161no—
238C4×C5#45911yescharacteristic
239C4×C5#46411yescharacteristic
240C4×C5#46511yescharacteristic
241C4×C5#46911yescharacteristic
242C4×C5#47411yescharacteristic
243C4×C5#47611yescharacteristic
244C4×C5#47811yescharacteristic
245C4×C5#47811yescharacteristic
246C24—41no—
247C2×D4—22no—
248C2×D4—22no—
249C2×D4—22no—
250C2×D4—41no—
251C2×D4—41no—
252C2×D4—41no—
253C2×D4—41no—
254C2×D4—41no—
255C2×D4—41no—
256C2×D4—41no—
257C2×D4—41no—
258C2×D4—41no—
259C2×D4—41no—
260C2×D4—41no—
261C2×D4—81no—
262C2×D4—81no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—41no—
266C22×C4C2×C5×D411yescharacteristic
267C22×C4C2×C5×D411yescharacteristic
268C22×C4C5×#4111yescharacteristic
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—21no—
272C2×C8—22no—
273C2×C8—21no—
274C2×C8—21no—
275C2×C8—41no—
276C2×C8—41no—
277C2×C8—41no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—41no—
283C2×C8C5×SD1612yes—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—41no—
288C4⋊C4—41no—
289C4⋊C4—41no—
290C4⋊C4—41no—
291C4⋊C4—81no—
292C4⋊C4C5×D812yes—
293C42—41no—
294C42—41no—
295C42C2×C5×D411yescharacteristic
296C42C2×C5×Q811yescharacteristic
297C42C5×#4111yescharacteristic
298C42C5×#4111yescharacteristic
299C2×C5—162no—
300C2×C5#166811yescharacteristic
301C2×C5#168111yescharacteristic
302C2×C5#168611yescharacteristic
303C2×C5#169711yescharacteristic
304C2×C5#206311yescharacteristic
305C2×C5#206511yescharacteristic
306C2×C5#206711yescharacteristic
307C23—42no—
308C23—81no—
309C23—81no—
310C23—81no—
311C23—81no—
312C23—81no—
313C23—81no—
314C23C5×#12111yescharacteristic
315D4—42no—
316D4—42no—
317D4—42no—
318D4—42no—
319D4—82no—
320D4—82no—
321D4—81no—
322D4—81no—
323D4—81no—
324D4—81no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—22no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—81no—
334C2×C4—81no—
335C2×C4—81no—
336C2×C4C5×#12111yescharacteristic
337C2×C4C5×#12111yescharacteristic
338C2×C4C5×C22⋊Q811yescharacteristic
339C2×C4C5×C22⋊Q811yescharacteristic
340C2×C4C5×C22⋊Q811yescharacteristic
341C2×C4C5×#12611yescharacteristic
342C2×C4C2×C5×D811yescharacteristic
343C2×C4C2×C5×SD1611yescharacteristic
344C2×C4C5×#13511yescharacteristic
345C2×C4C5×#13511yescharacteristic
346C2×C4C5×AGL(1,ℤ/8)11yescharacteristic
347C2×C4C5×AGL(1,ℤ/8)11yescharacteristic
348C2×C4C5×AGL(1,ℤ/8)11yescharacteristic
349C2×C4C5×#13711yescharacteristic
350C8—22no—
351C8—42no—
352C8—41no—
353C8—81no—
354C8—81no—
355C5#1059911yescharacteristic, direct product factor, Sylow
356C22—82no—
357C22—82no—
358C22—82no—
359C22—161no—
360C22—161no—
361C22—161no—
362C22—161no—
363C22C5×#39611yescharacteristic
364C22C5×#46011yescharacteristic
365C22C5×#46311yescharacteristic
366C22C5×#46611yescharacteristic
367C22C5×#46811yescharacteristic
368C22C5×#47111yescharacteristic
369C22C5×#47211yescharacteristic
370C4—21no—
371C4—21no—
372C4—42no—
373C4—81no—
374C4—161no—
375C4C5×#45911yescharacteristic
376C4C5×#46411yescharacteristic
377C4C5×#46511yescharacteristic
378C4C5×#46911yescharacteristic
379C4C5×#47411yescharacteristic
380C4C5×#47611yescharacteristic
381C4C5×#47811yescharacteristic
382C4C5×#47811yescharacteristic
383C2—162no—
384C2C5×#166811yescharacteristic
385C2C5×#168111yescharacteristic
386C2C5×#168611yescharacteristic
387C2C5×#169711yescharacteristic
388C2C5×#206311yescharacteristic
389C2C5×#206511yescharacteristic
390C2C5×#206711yescharacteristic
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
3. C5×#1948C2semidirect
4. C5×#1939C2semidirect
5. C2×C5×#335C2semidirect
7. C5×#1296C2semidirect
25. #10599C5direct (N × H)
62. #1948C2×C5semidirect
63. #1939C2×C5semidirect
64. C2×#335C2×C5semidirect
66. #1296C2×C5semidirect
355. C5#10599direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#19822
C5×#19572
C5×#19482
C5×#19392
C2×C5×#3352
C2×C5×#3312
C5×#12962
#105995