Finite Group Databasebeta

← C5×#19059 #11941779

Subgroups — C5×#19059 #11941779

Order 1280.

898 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#3553C211yesmaximal, characteristic, complemented
2C5×#3217C212yesmaximal, complemented
3C2×C5×#490C211yesmaximal, characteristic, complemented
4C5×#1658C212yesmaximal, complemented
5C5×#1657C214yesmaximal, complemented
6C5×#1656C212yesmaximal, complemented
7C5×#1534C211yesmaximal, characteristic, complemented
8C5×#1532C212yesmaximal, complemented
9C5×#564C2211yescharacteristic, complemented
10C5×#563C2212yescomplemented
11C5×#562C2211yescharacteristic, complemented
12C5×#554—22no—
13C5×#553—22no—
14C5×#552—22no—
15C5×#541—22no—
16C5×#539—22no—
17C5×#539—22no—
18C2×C5×#127C2211yescharacteristic
19C5×#529C2212yes—
20C2×C5×#125C2211yescharacteristic
21C5×#496C2214yes—
22C5×#490—22no—
23C5×#490C2212yes—
24C5×#490C2212yes—
25C5×#490C2214yes—
26C5×#460—22no—
27C5×#459—22no—
28C5×#450—22no—
29C5×#447—22no—
30C5×#432C2212yes—
31C2×C5×#102—22no—
32C2×C5×#97C2211yescharacteristic
33C5×#330C2214yescomplemented
34C5×#330C2212yescomplemented
35C5×#330C2212yescomplemented
36C5×#329C2214yescomplemented
37#19059C511yesmaximal, characteristic, direct product factor, Sylow
38C22×C5×D4—21no—
39C22×C5×D4—22no—
40C2×C5×SD16—24no—
41C2×C5×D8—24no—
42C22×C5×C8—21no—
43C5×C4⋊Q8—22no—
44C5×#127—22no—
45C5×#127—21no—
46C5×#127C2312yes—
47C5×#126—22no—
48C5×#126—22no—
49C5×#126—22no—
50C5×#126—22no—
51C5×#126—22no—
52C5×#125—22no—
53C5×#125—22no—
54C5×#125—22no—
55C5×#125—22no—
56C5×#125—22no—
57C5×#125—22no—
58C5×#125D412yes—
59C5×#125C2311yescharacteristic
60C5×#125C2311yescharacteristic
61C5×#125C2312yes—
62C5×#124C2312yes—
63C5×#123—22no—
64C5×#123—22no—
65C5×#123—22no—
66C5×#123—22no—
67C5×#123—22no—
68C5×C22⋊Q8—22no—
69C5×C22⋊Q8—22no—
70C5×C22⋊Q8—22no—
71C5×C22⋊Q8—22no—
72C5×C22⋊Q8—22no—
73C5×C22⋊Q8—22no—
74C5×C22⋊Q8D412yes—
75C5×#121—22no—
76C5×#121—22no—
77C5×#121D412yes—
78C4×C5×Q8—22no—
79C4×C5×Q8D412yes—
80C4×C5×D4—22no—
81C4×C5×D4—22no—
82C4×C5×D4—22no—
83C4×C5×D4—22no—
84C4×C5×D4D412yes—
85C5×#117—22no—
86C5×#117—22no—
87C2×C5×C4⋊C4—22no—
88C2×C5×C4⋊C4—22no—
89C2×C5×C4⋊C4D412yes—
90C2×C42×C5C2311yescharacteristic
91C5×C4⋊C8—22no—
92C5×#102—24no—
93C5×#102—24no—
94C5×#102—24no—
95C5×#102—24no—
96C5×C22⋊C8—22no—
97C5×#97—21no—
98C5×#97C2312yes—
99C5×#97C2314yes—
100#3553C2×C511yescharacteristic, complemented
101#3217C2×C512yescomplemented
102C2×#490C2×C511yescharacteristic, complemented
103#1658C2×C512yescomplemented
104#1657C2×C514yescomplemented
105#1656C2×C512yescomplemented
106#1534C2×C511yescharacteristic, complemented
107#1532C2×C512yescomplemented
108C24×C5—41no—
109C2×C5×Q8—22no—
110C2×C5×Q8C2×D412yes—
111C2×C5×D4—22no—
112C2×C5×D4—22no—
113C2×C5×D4—24no—
114C2×C5×D4—22no—
115C2×C5×D4—44no—
116C2×C5×D4—42no—
117C2×C5×D4—41no—
118C2×C5×D4—84no—
119C2×C5×D4C2×D412yes—
120C22×C4×C5—22no—
121C22×C4×C5—22no—
122C22×C4×C5—22no—
123C22×C4×C5—22no—
124C22×C4×C5C2×D411yescharacteristic
125C22×C4×C5C2×D412yes—
126C5×SD16—84no—
127C5×D8—84no—
128C2×C5×C8—22no—
129C2×C5×C8—24no—
130C2×C5×C8—42no—
131C5×C4⋊C4—22no—
132C5×C4⋊C4—22no—
133C5×C4⋊C4—22no—
134C5×C4⋊C4—22no—
135C5×C4⋊C4—22no—
136C5×C4⋊C4—22no—
137C5×C4⋊C4—22no—
138C5×C4⋊C4—22no—
139C5×C4⋊C4—22no—
140C5×C4⋊C4—22no—
141C5×C4⋊C4—22no—
142C5×C4⋊C4—22no—
143C5×C4⋊C4—22no—
144C5×C4⋊C4—22no—
145C5×C4⋊C4—22no—
146C5×C4⋊C4C2×D412yes—
147C5×C4⋊C4C2×D412yes—
148C5×C4⋊C4C2×D412yes—
149C5×C4⋊C4C2×D412yes—
150C5×C22⋊C4—22no—
151C5×C22⋊C4—22no—
152C5×C22⋊C4—22no—
153C5×C22⋊C4—22no—
154C5×C22⋊C4—22no—
155C5×C22⋊C4—22no—
156C5×C22⋊C4—22no—
157C5×C22⋊C4—22no—
158C5×C22⋊C4—22no—
159C42×C5—22no—
160C42×C5—22no—
161C42×C5—22no—
162C42×C5C2×D411yescharacteristic
163C42×C5C2×D412yes—
164C42×C5C2411yescharacteristic
165#564C22×C511yescharacteristic, complemented
166#563C22×C512yescomplemented
167#562C22×C511yescharacteristic, complemented
168#554—22no—
169#553—22no—
170#552—22no—
171#541—22no—
172#539—22no—
173#539—22no—
174C2×#127C22×C511yescharacteristic
175#529C22×C512yes—
176C2×#125C22×C511yescharacteristic
177#496C22×C514yes—
178#490—22no—
179#490C22×C512yes—
180#490C22×C512yes—
181#490C22×C514yes—
182#460—22no—
183#459—22no—
184#450—22no—
185#447—22no—
186#432C22×C512yes—
187C2×#102—22no—
188C2×#97C22×C511yescharacteristic
189#330C22×C512yescomplemented
190#330C22×C512yescomplemented
191#330C22×C514yescomplemented
192#329C22×C514yescomplemented
193C23×C5—22no—
194C23×C5—42no—
195C23×C5—84no—
196C23×C5—82no—
197C23×C5#12011yescharacteristic
198C5×Q8—42no—
199C5×Q8—82no—
200C5×D4—42no—
201C5×D4—44no—
202C5×D4—44no—
203C5×D4—44no—
204C5×D4—44no—
205C5×D4—82no—
206C5×D4—84no—
207C5×D4—82no—
208C2×C4×C5—22no—
209C2×C4×C5—22no—
210C2×C4×C5—22no—
211C2×C4×C5—22no—
212C2×C4×C5—22no—
213C2×C4×C5—22no—
214C2×C4×C5—42no—
215C2×C4×C5—82no—
216C2×C4×C5—82no—
217C2×C4×C5—82no—
218C2×C4×C5—82no—
219C2×C4×C5#12012yes—
220C2×C4×C5#12011yescharacteristic
221C2×C4×C5C22×D411yescharacteristic
222C2×C4×C5C22×D412yes—
223C5×C8—42no—
224C5×C8—44no—
225C22×D4—21no—
226C22×D4—22no—
227C2×SD16—24no—
228C2×D8—24no—
229C22×C8—21no—
230C4⋊Q8—22no—
231#127—22no—
232#127—21no—
233#127C23×C512yes—
234#126—22no—
235#126—22no—
236#126—22no—
237#126—22no—
238#126—22no—
239#125—22no—
240#125—22no—
241#125—22no—
242#125—22no—
243#125—22no—
244#125—22no—
245#125C5×D412yes—
246#125C23×C511yescharacteristic
247#125C23×C511yescharacteristic
248#125C23×C512yes—
249#124C23×C512yes—
250#123—22no—
251#123—22no—
252#123—22no—
253#123—22no—
254#123—22no—
255C22⋊Q8—22no—
256C22⋊Q8—22no—
257C22⋊Q8—22no—
258C22⋊Q8—22no—
259C22⋊Q8—22no—
260C22⋊Q8—22no—
261C22⋊Q8C5×D412yes—
262#121—22no—
263#121—22no—
264#121C5×D412yes—
265C4×Q8—22no—
266C4×Q8C5×D412yes—
267C4×D4—22no—
268C4×D4—22no—
269C4×D4—22no—
270C4×D4—22no—
271C4×D4C5×D412yes—
272#117—22no—
273#117—22no—
274C2×C4⋊C4—22no—
275C2×C4⋊C4—22no—
276C2×C4⋊C4C5×D412yes—
277C2×C42C23×C511yescharacteristic
278C4⋊C8—22no—
279#102—24no—
280#102—24no—
281#102—24no—
282#102—24no—
283C22⋊C8—22no—
284#97—21no—
285#97C23×C512yes—
286#97C23×C514yes—
287C22×C5—22no—
288C22×C5—42no—
289C22×C5—42no—
290C22×C5—42no—
291C22×C5—82no—
292C22×C5—84no—
293C22×C5—164no—
294C22×C5#45412yes—
295C22×C5C2×#12011yescharacteristic
296C4×C5—21no—
297C4×C5—21no—
298C4×C5—22no—
299C4×C5—22no—
300C4×C5—42no—
301C4×C5—42no—
302C4×C5—42no—
303C4×C5—42no—
304C4×C5—42no—
305C24—41no—
306C2×Q8—22no—
307C2×Q8C2×C5×D412yes—
308C2×D4—22no—
309C2×D4—22no—
310C2×D4—22no—
311C2×D4—24no—
312C2×D4—44no—
313C2×D4—42no—
314C2×D4—41no—
315C2×D4—84no—
316C2×D4C2×C5×D412yes—
317C22×C4—22no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—22no—
321C22×C4C2×C5×D411yescharacteristic
322C22×C4C2×C5×D412yes—
323SD16—84no—
324D8—84no—
325C2×C8—22no—
326C2×C8—24no—
327C2×C8—42no—
328C4⋊C4—22no—
329C4⋊C4—22no—
330C4⋊C4—22no—
331C4⋊C4—22no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—22no—
335C4⋊C4—22no—
336C4⋊C4—22no—
337C4⋊C4—22no—
338C4⋊C4—22no—
339C4⋊C4—22no—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4C2×C5×D412yes—
344C4⋊C4C2×C5×D412yes—
345C4⋊C4C2×C5×D412yes—
346C4⋊C4C2×C5×D412yes—
347C22⋊C4—22no—
348C22⋊C4—22no—
349C22⋊C4—22no—
350C22⋊C4—22no—
351C22⋊C4—22no—
352C22⋊C4—22no—
353C22⋊C4—22no—
354C22⋊C4—22no—
355C22⋊C4—22no—
356C42—22no—
357C42—22no—
358C42—22no—
359C42C2×C5×D411yescharacteristic
360C42C2×C5×D412yes—
361C42C24×C511yescharacteristic
362C2×C5—22no—
363C2×C5—82no—
364C2×C5—162no—
365C2×C5C2×#45411yescharacteristic
366C2×C5#302112yes—
367C23—22no—
368C23—42no—
369C23—84no—
370C23—82no—
371C23C5×#12011yescharacteristic
372Q8—42no—
373Q8—82no—
374D4—42no—
375D4—44no—
376D4—44no—
377D4—44no—
378D4—44no—
379D4—82no—
380D4—84no—
381D4—82no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—42no—
389C2×C4—82no—
390C2×C4—82no—
391C2×C4—82no—
392C2×C4—82no—
393C2×C4C5×#12012yes—
394C2×C4C5×#12011yescharacteristic
395C2×C4C22×C5×D411yescharacteristic
396C2×C4C22×C5×D412yes—
397C8—42no—
398C8—44no—
399C5#1905911yescharacteristic, direct product factor, Sylow
400C22—22no—
401C22—42no—
402C22—42no—
403C22—42no—
404C22—82no—
405C22—84no—
406C22—164no—
407C22C5×#45412yes—
408C22C2×C5×#12011yescharacteristic
409C4—21no—
410C4—21no—
411C4—22no—
412C4—22no—
413C4—42no—
414C4—42no—
415C4—42no—
416C4—42no—
417C4—42no—
418C2—22no—
419C2—82no—
420C2—162no—
421C2C2×C5×#45411yescharacteristic
422C2C5×#302112yes—
subgroupG/H
center, socleC22×C5C2×#120
commutator, Frattini, cosocleC42C24×C5
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
1. C5×#3553C2semidirect
2. C5×#3217C2semidirect
3. C2×C5×#490C2semidirect
4. C5×#1658C2semidirect
5. C5×#1657C2semidirect
6. C5×#1656C2semidirect
7. C5×#1534C2semidirect
8. C5×#1532C2semidirect
9. C5×#564C22semidirect
10. C5×#563C22semidirect
11. C5×#562C22semidirect
33. C5×#330C22semidirect
34. C5×#330C22semidirect
35. C5×#330C22semidirect
36. C5×#329C22semidirect
37. #19059C5direct (N × H)
100. #3553C2×C5semidirect
101. #3217C2×C5semidirect
102. C2×#490C2×C5semidirect
103. #1658C2×C5semidirect
104. #1657C2×C5semidirect
105. #1656C2×C5semidirect
106. #1534C2×C5semidirect
107. #1532C2×C5semidirect
165. #564C22×C5semidirect
166. #563C22×C5semidirect
167. #562C22×C5semidirect
189. #330C22×C5semidirect
190. #330C22×C5semidirect
191. #330C22×C5semidirect
192. #329C22×C5semidirect
399. C5#19059direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#35532
C5×#32172
C2×C5×#4902
C5×#16582
C5×#16572
C5×#16562
C5×#15342
C5×#15322
#190595