Finite Group Databasebeta

← C5×#34973 #11957693

Subgroups — C5×#34973 #11957693

Order 1280.

1716 conjugacy classes of subgroups.

The subgroup lattice has 406 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#2524C214yesmaximal, complemented
2C5×#2519C214yesmaximal
3C5×#2469C214yesmaximal
4C5×#2462C214yesmaximal, complemented
5C5×#2402C211yesmaximal, characteristic, complemented
6C5×#2399C211yesmaximal, characteristic, complemented
7C5×#2396C212yesmaximal
8C4×C5×#125C214yesmaximal, complemented
9C4×C5×#126C214yesmaximal
10C5×#2298C211yesmaximal, characteristic
11C42×C5×Q8C212yesmaximal, complemented
12C2×C5×#126C2214yes—
13C2×C5×#125C2214yes—
14C2×C4×C5×Q8C2214yes—
15C2×C4×C5×Q8C2212yes—
16C2×C5×#117C2211yescharacteristic
17C5×#400C2214yes—
18C5×#400C2212yes—
19C5×#399C2212yes—
20C5×#398C2214yes—
21C5×#398C2214yes—
22C5×#398C2214yes—
23C5×#396C2214yes—
24C5×#395C2214yes—
25C5×#395C2212yes—
26C5×#393C2212yes—
27C5×#389C2214yes—
28C5×#389C2214yes—
29C5×#388C2214yes—
30C5×#388C2214yes—
31C5×#388C2214yes—
32C5×#388C2214yes—
33C5×#387C2214yes—
34C5×#387C2214yes—
35C5×#387C2214yes—
36C5×#387C2214yes—
37C5×#387C2214yes—
38C5×#387C2214yes—
39C5×#387C2214yes—
40C5×#387C2214yes—
41C5×#383C2214yes—
42C5×#382C2214yes—
43C4×C5×C4⋊C4C2214yes—
44C4×C5×C4⋊C4C2214yes—
45C4×C5×C4⋊C4C2214yes—
46C4×C5×C4⋊C4C2214yes—
47C4×C5×C4⋊C4C2212yes—
48C4×C5×C4⋊C4C2212yes—
49C4×C5×C4⋊C4C2214yes—
50C4×C5×C4⋊C4C2214yes—
51C4×C5×C4⋊C4C2214yes—
52C4×C5×C22⋊C4C2212yes—
53C4×C5×C22⋊C4C2214yes—
54C5×#376C2211yescharacteristic
55C43×C5C2211yescharacteristic
56C43×C5C2214yes—
57#34973C511yesmaximal, characteristic, direct product factor, Sylow
58C22×C5×Q8C2312yes—
59C23×C4×C5C2311yescharacteristic
60C5×#126—216no—
61C5×#125—216no—
62C4×C5×Q8—216no—
63C4×C5×Q8—28no—
64C5×#117—24no—
65C2×C5×C4⋊C4C2314yes—
66C2×C5×C4⋊C4C2314yes—
67C2×C5×C4⋊C4C2314yes—
68C2×C5×C4⋊C4C2312yes—
69C2×C5×C4⋊C4C2314yes—
70C2×C5×C4⋊C4C2314yes—
71C2×C5×C4⋊C4C2314yes—
72C2×C5×C4⋊C4C2314yes—
73C2×C5×C4⋊C4C2314yes—
74C2×C5×C4⋊C4C2314yes—
75C2×C5×C4⋊C4C2314yes—
76C2×C5×C4⋊C4C2312yes—
77C2×C5×C4⋊C4C2314yes—
78C2×C5×C22⋊C4C2314yes—
79C2×C5×C22⋊C4C2314yes—
80C2×C5×C22⋊C4C2314yes—
81C2×C5×C22⋊C4C2312yes—
82C2×C42×C5C2314yes—
83C2×C42×C5C2314yes—
84C2×C42×C5C2314yes—
85C2×C42×C5C2314yes—
86C2×C42×C5C2314yes—
87C2×C42×C5C2312yes—
88C2×C42×C5C2314yes—
89C2×C42×C5C2314yes—
90C2×C42×C5C2312yes—
91C2×C42×C5C2311yescharacteristic
92C2×C42×C5C2311yescharacteristic
93C5×#95C2314yes—
94C5×#95C2314yes—
95C5×#95C2314yes—
96C5×#95C2314yes—
97C5×#95C2314yes—
98C5×#95C2314yes—
99C5×#95C2312yes—
100C5×#95C2312yes—
101C5×#95C2314yes—
102C5×#95C2314yes—
103C5×#95C2314yes—
104C5×#95C2314yes—
105C5×#95C2314yes—
106C5×#95C2314yes—
107C5×#95C2314yes—
108#2524C2×C514yescomplemented
109#2519C2×C514yes—
110#2469C2×C514yes—
111#2462C2×C514yescomplemented
112#2402C2×C511yescharacteristic, complemented
113#2399C2×C511yescharacteristic, complemented
114#2396C2×C512yes—
115C4×#125C2×C514yescomplemented
116C4×#126C2×C514yes—
117#2298C2×C511yescharacteristic
118C42×Q8C2×C512yescomplemented
119C24×C5C2411yescharacteristic
120C2×C5×Q8—28no—
121C2×C5×Q8—24no—
122C22×C4×C5—22no—
123C22×C4×C5—24no—
124C22×C4×C5C2414yes—
125C22×C4×C5C2414yes—
126C22×C4×C5C2412yes—
127C22×C4×C5C2414yes—
128C22×C4×C5C2411yescharacteristic
129C22×C4×C5C2414yes—
130C22×C4×C5C2414yes—
131C22×C4×C5C2414yes—
132C22×C4×C5C2412yes—
133C22×C4×C5C2411yescharacteristic
134C5×C4⋊C4—24no—
135C5×C4⋊C4—28no—
136C5×C4⋊C4—28no—
137C5×C4⋊C4—28no—
138C5×C4⋊C4—24no—
139C5×C4⋊C4—28no—
140C5×C4⋊C4—28no—
141C5×C4⋊C4—28no—
142C5×C4⋊C4—28no—
143C5×C4⋊C4—28no—
144C5×C4⋊C4—28no—
145C5×C4⋊C4—28no—
146C5×C4⋊C4#41116yes—
147C5×C22⋊C4—24no—
148C5×C22⋊C4—28no—
149C5×C22⋊C4—28no—
150C5×C22⋊C4—28no—
151C42×C5—22no—
152C42×C5—28no—
153C42×C5—28no—
154C42×C5—28no—
155C42×C5—28no—
156C42×C5—28no—
157C42×C5—28no—
158C42×C5—24no—
159C42×C5—28no—
160C42×C5—24no—
161C42×C5#4114yes—
162C2×#126C22×C514yes—
163C2×#125C22×C514yes—
164C2×C4×Q8C22×C512yes—
165C2×C4×Q8C22×C514yes—
166C2×#117C22×C511yescharacteristic
167#400C22×C512yes—
168#400C22×C514yes—
169#399C22×C512yes—
170#398C22×C514yes—
171#398C22×C514yes—
172#398C22×C514yes—
173#396C22×C514yes—
174#395C22×C512yes—
175#395C22×C514yes—
176#393C22×C512yes—
177#389C22×C514yes—
178#389C22×C514yes—
179#388C22×C514yes—
180#388C22×C514yes—
181#388C22×C514yes—
182#388C22×C514yes—
183#387C22×C514yes—
184#387C22×C514yes—
185#387C22×C514yes—
186#387C22×C514yes—
187#387C22×C514yes—
188#387C22×C514yes—
189#387C22×C514yes—
190#387C22×C514yes—
191#383C22×C514yes—
192#382C22×C514yes—
193C4×C4⋊C4C22×C514yes—
194C4×C4⋊C4C22×C514yes—
195C4×C4⋊C4C22×C514yes—
196C4×C4⋊C4C22×C514yes—
197C4×C4⋊C4C22×C512yes—
198C4×C4⋊C4C22×C512yes—
199C4×C4⋊C4C22×C514yes—
200C4×C4⋊C4C22×C514yes—
201C4×C4⋊C4C22×C514yes—
202C4×C22⋊C4C22×C514yes—
203C4×C22⋊C4C22×C512yes—
204#376C22×C511yescharacteristic
205C43C22×C514yes—
206C43C22×C511yescharacteristic
207C23×C5—24no—
208C23×C5—21no—
209C23×C5—22no—
210C23×C5C2511yescharacteristic
211C5×Q8—216no—
212C2×C4×C5—24no—
213C2×C4×C5—24no—
214C2×C4×C5—24no—
215C2×C4×C5—24no—
216C2×C4×C5—24no—
217C2×C4×C5—24no—
218C2×C4×C5—24no—
219C2×C4×C5—22no—
220C2×C4×C5—24no—
221C2×C4×C5—24no—
222C2×C4×C5—24no—
223C2×C4×C5—24no—
224C2×C4×C5—24no—
225C2×C4×C5—24no—
226C2×C4×C5—24no—
227C2×C4×C5—24no—
228C2×C4×C5—21no—
229C2×C4×C5—24no—
230C2×C4×C5—24no—
231C2×C4×C5—22no—
232C2×C4×C5—44no—
233C2×C4×C5C2×#4118yes—
234C2×C4×C5C2×#4118yes—
235C2×C4×C5C2×#4118yes—
236C2×C4×C5C2×#4114yes—
237C2×C4×C5C2×#4112yes—
238C2×C4×C52−1+414yes—
239C22×Q8C23×C512yes—
240C23×C4C23×C511yescharacteristic
241#126—216no—
242#125—216no—
243C4×Q8—216no—
244C4×Q8—28no—
245#117—24no—
246C2×C4⋊C4C23×C512yes—
247C2×C4⋊C4C23×C514yes—
248C2×C4⋊C4C23×C514yes—
249C2×C4⋊C4C23×C514yes—
250C2×C4⋊C4C23×C512yes—
251C2×C4⋊C4C23×C514yes—
252C2×C4⋊C4C23×C514yes—
253C2×C4⋊C4C23×C514yes—
254C2×C4⋊C4C23×C514yes—
255C2×C4⋊C4C23×C514yes—
256C2×C4⋊C4C23×C514yes—
257C2×C4⋊C4C23×C514yes—
258C2×C4⋊C4C23×C514yes—
259C2×C22⋊C4C23×C512yes—
260C2×C22⋊C4C23×C514yes—
261C2×C22⋊C4C23×C514yes—
262C2×C22⋊C4C23×C514yes—
263C2×C42C23×C511yescharacteristic
264C2×C42C23×C514yes—
265C2×C42C23×C514yes—
266C2×C42C23×C514yes—
267C2×C42C23×C514yes—
268C2×C42C23×C514yes—
269C2×C42C23×C514yes—
270C2×C42C23×C512yes—
271C2×C42C23×C514yes—
272C2×C42C23×C512yes—
273C2×C42C23×C511yescharacteristic
274#95C23×C514yes—
275#95C23×C514yes—
276#95C23×C514yes—
277#95C23×C514yes—
278#95C23×C514yes—
279#95C23×C514yes—
280#95C23×C512yes—
281#95C23×C512yes—
282#95C23×C514yes—
283#95C23×C514yes—
284#95C23×C514yes—
285#95C23×C514yes—
286#95C23×C514yes—
287#95C23×C514yes—
288#95C23×C514yes—
289C22×C5—42no—
290C22×C5—41no—
291C22×C5—44no—
292C22×C5C22×#4111yescharacteristic
293C22×C5C22×#4114yes—
294C22×C5C2×2−1+412yes—
295C4×C5—28no—
296C4×C5—24no—
297C4×C5—28no—
298C4×C5—28no—
299C4×C5—44no—
300C4×C5—42no—
301C4×C5—41no—
302C4×C5—44no—
303C4×C5—44no—
304C4×C5#54114yes—
305C24C24×C511yescharacteristic
306C2×Q8—28no—
307C2×Q8—24no—
308C22×C4—22no—
309C22×C4—24no—
310C22×C4C24×C511yescharacteristic
311C22×C4C24×C514yes—
312C22×C4C24×C514yes—
313C22×C4C24×C514yes—
314C22×C4C24×C512yes—
315C22×C4C24×C514yes—
316C22×C4C24×C514yes—
317C22×C4C24×C514yes—
318C22×C4C24×C512yes—
319C22×C4C24×C511yescharacteristic
320C4⋊C4—24no—
321C4⋊C4—28no—
322C4⋊C4—28no—
323C4⋊C4—28no—
324C4⋊C4—24no—
325C4⋊C4—28no—
326C4⋊C4—28no—
327C4⋊C4—28no—
328C4⋊C4—28no—
329C4⋊C4—28no—
330C4⋊C4—28no—
331C4⋊C4—28no—
332C4⋊C4C5×#41116yes—
333C22⋊C4—24no—
334C22⋊C4—28no—
335C22⋊C4—28no—
336C22⋊C4—28no—
337C42—22no—
338C42—28no—
339C42—28no—
340C42—28no—
341C42—28no—
342C42—28no—
343C42—28no—
344C42—24no—
345C42—28no—
346C42—24no—
347C42C5×#4114yes—
348C2×C5—81no—
349C2×C5#344412yes—
350C2×C5C2×#54111yescharacteristic
351C2×C5#348214yes—
352C23—24no—
353C23—21no—
354C23—22no—
355C23C25×C511yescharacteristic
356Q8—216no—
357C2×C4—24no—
358C2×C4—24no—
359C2×C4—24no—
360C2×C4—24no—
361C2×C4—24no—
362C2×C4—24no—
363C2×C4—24no—
364C2×C4—22no—
365C2×C4—24no—
366C2×C4—24no—
367C2×C4—24no—
368C2×C4—24no—
369C2×C4—24no—
370C2×C4—24no—
371C2×C4—24no—
372C2×C4—24no—
373C2×C4—21no—
374C2×C4—24no—
375C2×C4—24no—
376C2×C4—22no—
377C2×C4—44no—
378C2×C4C2×C5×#4118yes—
379C2×C4C2×C5×#4118yes—
380C2×C4C2×C5×#4118yes—
381C2×C4C2×C5×#4114yes—
382C2×C4C2×C5×#4112yes—
383C2×C4C5×2−1+414yes—
384C5#3497311yescharacteristic, direct product factor, Sylow
385C22—42no—
386C22—41no—
387C22—44no—
388C22C22×C5×#4111yescharacteristic
389C22C22×C5×#4114yes—
390C22C2×C5×2−1+412yes—
391C4—28no—
392C4—24no—
393C4—28no—
394C4—28no—
395C4—44no—
396C4—42no—
397C4—41no—
398C4—44no—
399C4—44no—
400C4C5×#54114yes—
401C2—81no—
402C2C5×#344412yes—
403C2C2×C5×#54111yescharacteristic
404C2C5×#348214yes—
subgroupG/H
centerC22×C4×C5C24
commutator, Frattini, cosocleC23C25×C5
socleC23×C5C25
prime radicalC24×C5C24
LCSG ▷ C23 ▷ C1
UCSC1 ◁ C22×C4×C5 ◁ G

Complements

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

NH ≅ G/Ntype
1. C5×#2524C2semidirect
4. C5×#2462C2semidirect
5. C5×#2402C2semidirect
6. C5×#2399C2semidirect
8. C4×C5×#125C2semidirect
11. C42×C5×Q8C2semidirect
57. #34973C5direct (N × H)
108. #2524C2×C5semidirect
111. #2462C2×C5semidirect
112. #2402C2×C5semidirect
113. #2399C2×C5semidirect
115. C4×#125C2×C5semidirect
118. C42×Q8C2×C5semidirect
384. C5#34973direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#25242
C5×#25192
C5×#24692
C5×#24622
C5×#24022
C5×#23992
C5×#23962
C4×C5×#1252
C4×C5×#1262
C5×#22982
C42×C5×Q82
#349735