Finite Group Databasebeta

← C5×#2746 #10616341

Subgroups — C5×#2746 #10616341

Order 640.

448 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#401C211yesmaximal, characteristic, complemented
2C5×#400C211yesmaximal, characteristic, complemented
3C5×#399C211yesmaximal, characteristic, complemented
4C5×#398C211yesmaximal, characteristic, complemented
5C5×#397C211yesmaximal, characteristic, complemented
6C5×#394C211yesmaximal, characteristic
7C5×#388C211yesmaximal, characteristic, complemented
8C5×#388C211yesmaximal, characteristic, complemented
9C5×#387C211yesmaximal, characteristic, complemented
10C5×#386C211yesmaximal, characteristic
11C5×#386C211yesmaximal, characteristic
12C5×#385C211yesmaximal, characteristic, complemented
13C5×#385C211yesmaximal, characteristic, complemented
14C5×#381C211yesmaximal, characteristic, complemented
15C5×#380C211yesmaximal, characteristic, complemented
16C22×C5×D4C2211yescharacteristic
17C23×C4×C5C2211yescharacteristic
18C23×C4×C5C2211yescharacteristic
19C2×C5×C4⋊C4C2211yescharacteristic
20C2×C5×C4⋊C4C2211yescharacteristic
21C2×C5×C4⋊C4C2211yescharacteristic
22C2×C5×C4⋊C4C2211yescharacteristic
23C2×C5×C4⋊C4C2211yescharacteristic
24C2×C5×C4⋊C4C2211yescharacteristic
25C2×C5×C4⋊C4C2211yescharacteristic
26C2×C5×C22⋊C4C2211yescharacteristic
27C2×C5×C22⋊C4C2211yescharacteristic
28C2×C5×C22⋊C4C2211yescharacteristic
29C2×C5×C22⋊C4C2211yescharacteristic
30C2×C5×C22⋊C4C2211yescharacteristic
31C2×C5×C22⋊C4C2211yescharacteristic
32C2×C5×C22⋊C4C2211yescharacteristic
33C2×C5×C22⋊C4C2211yescharacteristic
34C2×C5×C22⋊C4C2211yescharacteristic
35C2×C5×C22⋊C4C2211yescharacteristic
36C2×C42×C5C2211yescharacteristic
37C5×#95C2211yescharacteristic
38C5×#95C2211yescharacteristic
39C5×#95C2211yescharacteristic
40C5×#95C2211yescharacteristic
41C5×#95C2211yescharacteristic
42C5×#95C2211yescharacteristic
43C5×#95C2211yescharacteristic
44C5×#95C2211yescharacteristic
45C5×#95C2211yescharacteristic
46C5×#95C2211yescharacteristic
47C5×#95C2211yescharacteristic
48C5×#95C2211yescharacteristic
49C5×#95C2211yescharacteristic
50C5×#95C2211yescharacteristic
51#2746C511yesmaximal, characteristic, direct product factor, Sylow
52C24×C5C2311yescharacteristic
53C24×C5C2311yescharacteristic
54C2×C5×D4—22no—
55C2×C5×D4—22no—
56C2×C5×D4—41no—
57C22×C4×C5—22no—
58C22×C4×C5—22no—
59C22×C4×C5—41no—
60C22×C4×C5—41no—
61C22×C4×C5—41no—
62C22×C4×C5—41no—
63C22×C4×C5C2311yescharacteristic
64C22×C4×C5C2311yescharacteristic
65C22×C4×C5C2311yescharacteristic
66C22×C4×C5C2311yescharacteristic
67C22×C4×C5C2311yescharacteristic
68C22×C4×C5C2311yescharacteristic
69C22×C4×C5C2311yescharacteristic
70C22×C4×C5C2311yescharacteristic
71C22×C4×C5C2311yescharacteristic
72C22×C4×C5C2311yescharacteristic
73C22×C4×C5C2311yescharacteristic
74C22×C4×C5C2311yescharacteristic
75C22×C4×C5C2311yescharacteristic
76C5×C4⋊C4—22no—
77C5×C4⋊C4—41no—
78C5×C4⋊C4—41no—
79C5×C4⋊C4—41no—
80C5×C4⋊C4—41no—
81C5×C4⋊C4—41no—
82C5×C4⋊C4—41no—
83C5×C22⋊C4—22no—
84C5×C22⋊C4—22no—
85C5×C22⋊C4—22no—
86C5×C22⋊C4—22no—
87C5×C22⋊C4—41no—
88C5×C22⋊C4—41no—
89C5×C22⋊C4—41no—
90C5×C22⋊C4—41no—
91C5×C22⋊C4—41no—
92C5×C22⋊C4—41no—
93C42×C5—41no—
94#401C2×C511yescharacteristic, complemented
95#400C2×C511yescharacteristic, complemented
96#399C2×C511yescharacteristic, complemented
97#398C2×C511yescharacteristic, complemented
98#397C2×C511yescharacteristic, complemented
99#394C2×C511yescharacteristic
100#388C2×C511yescharacteristic, complemented
101#388C2×C511yescharacteristic, complemented
102#387C2×C511yescharacteristic, complemented
103#386C2×C511yescharacteristic
104#386C2×C511yescharacteristic
105#385C2×C511yescharacteristic, complemented
106#385C2×C511yescharacteristic, complemented
107#381C2×C511yescharacteristic, complemented
108#380C2×C511yescharacteristic, complemented
109C23×C5—21no—
110C23×C5—21no—
111C23×C5—21no—
112C23×C5—21no—
113C23×C5—21no—
114C23×C5—21no—
115C23×C5—21no—
116C23×C5—21no—
117C23×C5—21no—
118C23×C5—21no—
119C23×C5—21no—
120C23×C5—21no—
121C23×C5#4112yes—
122C23×C5#4112yes—
123C23×C5C2411yescharacteristic
124C5×D4—44no—
125C2×C4×C5—21no—
126C2×C4×C5—21no—
127C2×C4×C5—21no—
128C2×C4×C5—21no—
129C2×C4×C5—21no—
130C2×C4×C5—21no—
131C2×C4×C5—21no—
132C2×C4×C5—21no—
133C2×C4×C5—21no—
134C2×C4×C5—21no—
135C2×C4×C5—21no—
136C2×C4×C5—21no—
137C2×C4×C5—21no—
138C2×C4×C5—21no—
139C2×C4×C5—21no—
140C2×C4×C5—21no—
141C2×C4×C5—21no—
142C2×C4×C5—21no—
143C2×C4×C5—21no—
144C2×C4×C5—21no—
145C2×C4×C5—21no—
146C2×C4×C5—21no—
147C2×C4×C5—21no—
148C2×C4×C5—21no—
149C2×C4×C5—21no—
150C2×C4×C5—21no—
151C2×C4×C5—21no—
152C2×C4×C5—21no—
153C2×C4×C5—21no—
154C2×C4×C5—21no—
155C2×C4×C5—21no—
156C2×C4×C5—21no—
157C2×C4×C5—21no—
158C2×C4×C5—21no—
159C2×C4×C5—21no—
160C2×C4×C5—21no—
161C2×C4×C5—21no—
162C2×C4×C5—21no—
163C2×C4×C5—44no—
164C2×C4×C5—44no—
165C2×C4×C5#4112yes—
166C22×D4C22×C511yescharacteristic
167C23×C4C22×C511yescharacteristic
168C23×C4C22×C511yescharacteristic
169C2×C4⋊C4C22×C511yescharacteristic
170C2×C4⋊C4C22×C511yescharacteristic
171C2×C4⋊C4C22×C511yescharacteristic
172C2×C4⋊C4C22×C511yescharacteristic
173C2×C4⋊C4C22×C511yescharacteristic
174C2×C4⋊C4C22×C511yescharacteristic
175C2×C4⋊C4C22×C511yescharacteristic
176C2×C22⋊C4C22×C511yescharacteristic
177C2×C22⋊C4C22×C511yescharacteristic
178C2×C22⋊C4C22×C511yescharacteristic
179C2×C22⋊C4C22×C511yescharacteristic
180C2×C22⋊C4C22×C511yescharacteristic
181C2×C22⋊C4C22×C511yescharacteristic
182C2×C22⋊C4C22×C511yescharacteristic
183C2×C22⋊C4C22×C511yescharacteristic
184C2×C22⋊C4C22×C511yescharacteristic
185C2×C22⋊C4C22×C511yescharacteristic
186C2×C42C22×C511yescharacteristic
187#95C22×C511yescharacteristic
188#95C22×C511yescharacteristic
189#95C22×C511yescharacteristic
190#95C22×C511yescharacteristic
191#95C22×C511yescharacteristic
192#95C22×C511yescharacteristic
193#95C22×C511yescharacteristic
194#95C22×C511yescharacteristic
195#95C22×C511yescharacteristic
196#95C22×C511yescharacteristic
197#95C22×C511yescharacteristic
198#95C22×C511yescharacteristic
199#95C22×C511yescharacteristic
200#95C22×C511yescharacteristic
201C22×C5—22no—
202C22×C5—22no—
203C22×C5—22no—
204C22×C5—22no—
205C22×C5—22no—
206C22×C5—22no—
207C22×C5—41no—
208C22×C5—41no—
209C22×C5—41no—
210C22×C5—41no—
211C22×C5—41no—
212C22×C5—41no—
213C22×C5—41no—
214C22×C5—41no—
215C22×C5C2×#4111yescharacteristic
216C22×C5C2×#4111yescharacteristic
217C22×C5C2×#4111yescharacteristic
218C22×C5H(5,𝔽2)11yescharacteristic
219C22×C5H(5,𝔽2)11yescharacteristic
220C22×C5H(5,𝔽2)11yescharacteristic
221C22×C52−1+411yescharacteristic
222C4×C5—22no—
223C4×C5—41no—
224C4×C5—41no—
225C4×C5—41no—
226C4×C5—41no—
227C4×C5—41no—
228C4×C5—41no—
229C4×C5—41no—
230C4×C5—41no—
231C4×C5—41no—
232C4×C5—41no—
233C4×C5—41no—
234C4×C5—41no—
235C24C23×C511yescharacteristic
236C24C23×C511yescharacteristic
237C2×D4—22no—
238C2×D4—22no—
239C2×D4—41no—
240C22×C4—22no—
241C22×C4—22no—
242C22×C4—41no—
243C22×C4—41no—
244C22×C4—41no—
245C22×C4—41no—
246C22×C4C23×C511yescharacteristic
247C22×C4C23×C511yescharacteristic
248C22×C4C23×C511yescharacteristic
249C22×C4C23×C511yescharacteristic
250C22×C4C23×C511yescharacteristic
251C22×C4C23×C511yescharacteristic
252C22×C4C23×C511yescharacteristic
253C22×C4C23×C511yescharacteristic
254C22×C4C23×C511yescharacteristic
255C22×C4C23×C511yescharacteristic
256C22×C4C23×C511yescharacteristic
257C22×C4C23×C511yescharacteristic
258C22×C4C23×C511yescharacteristic
259C4⋊C4—22no—
260C4⋊C4—41no—
261C4⋊C4—41no—
262C4⋊C4—41no—
263C4⋊C4—41no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C22⋊C4—22no—
267C22⋊C4—22no—
268C22⋊C4—22no—
269C22⋊C4—22no—
270C22⋊C4—41no—
271C22⋊C4—41no—
272C22⋊C4—41no—
273C22⋊C4—41no—
274C22⋊C4—41no—
275C22⋊C4—41no—
276C42—41no—
277C2×C5—42no—
278C2×C5—42no—
279C2×C5#53911yescharacteristic
280C2×C5#53911yescharacteristic
281C2×C5#54211yescharacteristic
282C2×C5#55111yescharacteristic
283C2×C5#55111yescharacteristic
284C2×C5#55311yescharacteristic
285C2×C5#56011yescharacteristic
286C23—21no—
287C23—21no—
288C23—21no—
289C23—21no—
290C23—21no—
291C23—21no—
292C23—21no—
293C23—21no—
294C23—21no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23C5×#4112yes—
299C23C5×#4112yes—
300C23C24×C511yescharacteristic
301D4—44no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—44no—
341C2×C4—44no—
342C2×C4C5×#4112yes—
343C5#274611yescharacteristic, direct product factor, Sylow
344C22—22no—
345C22—22no—
346C22—22no—
347C22—22no—
348C22—22no—
349C22—22no—
350C22—41no—
351C22—41no—
352C22—41no—
353C22—41no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22C2×C5×#4111yescharacteristic
359C22C2×C5×#4111yescharacteristic
360C22C2×C5×#4111yescharacteristic
361C22C5×H(5,𝔽2)11yescharacteristic
362C22C5×H(5,𝔽2)11yescharacteristic
363C22C5×H(5,𝔽2)11yescharacteristic
364C22C5×2−1+411yescharacteristic
365C4—22no—
366C4—41no—
367C4—41no—
368C4—41no—
369C4—41no—
370C4—41no—
371C4—41no—
372C4—41no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—41no—
377C4—41no—
378C2—42no—
379C2—42no—
380C2C5×#53911yescharacteristic
381C2C5×#53911yescharacteristic
382C2C5×#54211yescharacteristic
383C2C5×#55111yescharacteristic
384C2C5×#55111yescharacteristic
385C2C5×#55311yescharacteristic
386C2C5×#56011yescharacteristic
subgroupG/H
center, socleC23×C5C24
commutator, Frattini, cosocleC23C24×C5
prime radicalC22×C5×D4C22
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C23×C5 ◁ G

Complements

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

NH ≅ G/Ntype
1. C5×#401C2semidirect
2. C5×#400C2semidirect
3. C5×#399C2semidirect
4. C5×#398C2semidirect
5. C5×#397C2semidirect
7. C5×#388C2semidirect
8. C5×#388C2semidirect
9. C5×#387C2semidirect
12. C5×#385C2semidirect
13. C5×#385C2semidirect
14. C5×#381C2semidirect
15. C5×#380C2semidirect
51. #2746C5direct (N × H)
94. #401C2×C5semidirect
95. #400C2×C5semidirect
96. #399C2×C5semidirect
97. #398C2×C5semidirect
98. #397C2×C5semidirect
100. #388C2×C5semidirect
101. #388C2×C5semidirect
102. #387C2×C5semidirect
105. #385C2×C5semidirect
106. #385C2×C5semidirect
107. #381C2×C5semidirect
108. #380C2×C5semidirect
343. C5#2746direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#4012
C5×#4002
C5×#3992
C5×#3982
C5×#3972
C5×#3942
C5×#3882
C5×#3872
C5×#3862
C5×#3852
C5×#3812
C5×#3802
#27465