Finite Group Databasebeta

← C5×#10604615 #72221757

Subgroups — C5×#10604615 #72221757

Order 3200.

778 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 778 conjugacy classes of subgroups — too many to draw. Only its 132 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#66258C211yesmaximal, characteristic, complemented
2C5×#66049C211yesmaximal, characteristic
3C5×D5⋊#100C211yesmaximal, characteristic, complemented
4C5×#65769C212yesmaximal, complemented
5C5×#64863C212yesmaximal, complemented
6C2×C5×#711C2211yescharacteristic
7C2×C5×D4×D5C2211yescharacteristic
8C2×C5×C22⋊AGL(1,𝔽5)—22no—
9C5×D4×AGL(1,𝔽5)—24no—
10C5×#4130C2212yes—
11C5×D5⋊M16C2212yes—
12C52×#127C411yescharacteristic, complemented
13C5×#4086C411yescharacteristic, complemented
14C5×#4085—22no—
15C5×#4083—22no—
16C5×#4082C411yescharacteristic, complemented
17C5×#4021C411yescharacteristic, complemented
18C5×#4018C2211yescharacteristic
19C5×#4013—22no—
20C5×#4008—24no—
21#10604615C511yesmaximal, characteristic, direct product factor
22C5×#1911—51nomaximal
23C23×C5×D5D412yes—
24C22×C5×AGL(1,𝔽5)—24no—
25C2×C52×D4—22no—
26C2×C52×D4C2×C411yescharacteristic
27C2×C52×D4C2×C411yescharacteristic, complemented
28C2×C5×C22⋊D5—21no—
29C2×C5×C22⋊D5—22no—
30C2×C5×C22⋊D5D412yes—
31C22×C5×Dic5—21no—
32C5×#711—22no—
33C5×#711—22no—
34C5×D4×D5—22no—
35C5×D4×D5—22no—
36C2×C5×D20C2×C411yescharacteristic
37C2×C4×C5×D5C2311yescharacteristic
38C2×C5×Dic10C2×C411yescharacteristic, complemented
39C5×C22⋊AGL(1,𝔽5)—24no—
40C5×C22⋊AGL(1,𝔽5)—22no—
41C5×#704—22no—
42C5×C4⋊AGL(1,𝔽5)—22no—
43C4×C5×AGL(1,𝔽5)D414yescomplemented
44C5×#700—22no—
45C5×D5⋊C8—22no—
46C42×C52C2×C411yescharacteristic, complemented
47C5×C22⋊Dic5—22no—
48C5×#685—22no—
49C5×#685—21no—
50C5×#683—21no—
51C4×C5×Dic5C2×C411yescharacteristic, complemented
52#66258C2×C511yescharacteristic, complemented
53C5×#535—51no—
54C5×#518—51no—
55#66049C2×C511yescharacteristic
56D5⋊#100C2×C511yescharacteristic, complemented
57#65769C2×C512yescomplemented
58C5×#421—52no—
59C2×C5×#100—51no—
60#64863C2×C512yescomplemented
61C5×H(3,ℤ/4)—52no—
62C23×C52—21no—
63C23×C52C22⋊C412yes—
64C22×C5×D5—22no—
65C22×C5×D5—22no—
66C22×C5×D5—82no—
67C22×C5×D5C22⋊C412yes—
68C22×C5×D5C2×D411yescharacteristic
69C2×C5×AGL(1,𝔽5)—24no—
70C2×C5×AGL(1,𝔽5)—84no—
71C52×D4—22no—
72C52×D4—22no—
73C52×D4—82no—
74C2×C4×C52—21no—
75C2×C4×C52C22×C411yescharacteristic
76C5×C22⋊D5—24no—
77C5×C22⋊D5—41no—
78C5×C22⋊D5—41no—
79C5×C22⋊D5—82no—
80C2×C5×Dic5—21no—
81C2×C5×Dic5—21no—
82C2×C5×Dic5—42no—
83C2×C5×Dic5C2×D411yescharacteristic
84C5×D20—22no—
85C4×C5×D5C2×D412yes—
86C4×C5×D5#4112yes—
87C5×Dic10—22no—
88C5×#170—42no—
89C2×C5×#41—51no—
90C22×C5×D4—51no—
91C2×#711C22×C511yescharacteristic
92C2×D4×D5C22×C511yescharacteristic
93C2×C22⋊AGL(1,𝔽5)—22no—
94D4×AGL(1,𝔽5)—24no—
95#4130C22×C512yes—
96D5⋊M16C22×C512yes—
97C2×C5×M16—52no—
98C5×#127—41no—
99C5×#127—51no—
100C5×#127C4×C511yescharacteristic, complemented
101C5×#127AGL(1,𝔽5)11yescharacteristic, complemented
102C5×#124—51no—
103C5×#124—51no—
104C5×#121—102no—
105C5×#120—102no—
106C4×C5×D4—104no—
107C5×#117—52no—
108C5×#117—51no—
109C2×C5×C22⋊C4—102no—
110#4086C4×C511yescharacteristic, complemented
111#4085—22no—
112#4083—22no—
113#4082C4×C511yescharacteristic, complemented
114#4021C4×C511yescharacteristic, complemented
115#4018C22×C511yescharacteristic
116#4013—22no—
117#4008—24no—
118C5×#104—104no—
119C5×#100—102no—
120#1911—51noSylow
121C22×C52—22no—
122C22×C52—42no—
123C22×C52—42no—
124C22×C52—41no—
125C22×C52C2×C22⋊C411yescharacteristic
126C2×C5×D5—22no—
127C2×C5×D5—41no—
128C2×C5×D5—42no—
129C2×C5×D5—44no—
130C2×C5×D5—82no—
131C2×C5×D5#12011yescharacteristic
132C2×C5×D5#12311yescharacteristic
133C5×AGL(1,𝔽5)—44no—
134C4×C52—41no—
135C4×C52C4×D412yes—
136C5×Dic5—41no—
137C5×Dic5—41no—
138C5×Dic5#12112yes—
139C24×C5—52no—
140C23×D5C5×D412yes—
141C22×AGL(1,𝔽5)—24no—
142C5×#41—102no—
143C5×#41—102no—
144C2×C5×Q8—51no—
145C2×C5×D4—22no—
146C2×C5×D4—22no—
147C2×C5×D4—44no—
148C2×C5×D4—41no—
149C2×C5×D4—41no—
150C2×C5×D4—52no—
151C2×C5×D4—51no—
152C2×C5×D4—102no—
153C2×C5×D4—102no—
154C2×C5×D4—101no—
155C2×C5×D4—102no—
156C2×C5×D4C2×C4×C511yescharacteristic
157C2×C5×D4C2×C4×C511yescharacteristic, complemented
158C2×C5×D4C2×AGL(1,𝔽5)11yescharacteristic
159C2×C5×D4C2×AGL(1,𝔽5)11yescharacteristic, complemented
160C22×C4×C5—51no—
161C22×C4×C5—101no—
162C22×C4×C5—104no—
163C2×C22⋊D5—21no—
164C2×C22⋊D5—22no—
165C2×C22⋊D5C5×D412yes—
166C22×Dic5—21no—
167#711—22no—
168#711—22no—
169D4×D5—22no—
170D4×D5—22no—
171C2×D20C2×C4×C511yescharacteristic
172C2×C4×D5C23×C511yescharacteristic
173C2×Dic10C2×C4×C511yescharacteristic, complemented
174C22⋊AGL(1,𝔽5)—24no—
175C22⋊AGL(1,𝔽5)—22no—
176#704—22no—
177C4⋊AGL(1,𝔽5)—22no—
178C4×AGL(1,𝔽5)C5×D414yescomplemented
179#700—22no—
180D5⋊C8—22no—
181C5×M16—102no—
182C5×M16—102no—
183C2×C5×C8—102no—
184C5×C4⋊C4—101no—
185C5×C4⋊C4—102no—
186C5×C22⋊C4—101no—
187C5×C22⋊C4—102no—
188C5×C22⋊C4—102no—
189C5×C22⋊C4—104no—
190C5×C22⋊C4—102no—
191C42×C5—41no—
192C42×C5—54no—
193C42×C5—51no—
194C42×C5C2×C4×C511yescharacteristic, complemented
195C42×C5C2×AGL(1,𝔽5)11yescharacteristic, complemented
196C22⋊Dic5—22no—
197#685—22no—
198#685—21no—
199#683—21no—
200C4×Dic5C2×C4×C511yescharacteristic, complemented
201#535—51no—
202#518—51no—
203#421—52no—
204C2×#100—51no—
205H(3,ℤ/4)—52no—
206C2×C52—21no—
207C2×C52—42no—
208C2×C52—81no—
209C2×C52#38611yescharacteristic
210C5×D5—21no—
211C5×D5—21no—
212C5×D5—42no—
213C23×C5—21no—
214C23×C5—21no—
215C23×C5—42no—
216C23×C5—42no—
217C23×C5—51no—
218C23×C5—52no—
219C23×C5—102no—
220C23×C5—102no—
221C23×C5—402no—
222C23×C5C5×C22⋊C412yes—
223C23×C5C22⋊AGL(1,𝔽5)12yes—
224C22×D5—22no—
225C22×D5—22no—
226C22×D5—82no—
227C22×D5C5×C22⋊C412yes—
228C22×D5C2×C5×D411yescharacteristic
229C2×AGL(1,𝔽5)—24no—
230C2×AGL(1,𝔽5)—84no—
231C5×Q8—102no—
232C5×D4—22no—
233C5×D4—22no—
234C5×D4—22no—
235C5×D4—22no—
236C5×D4—82no—
237C5×D4—82no—
238C5×D4—82no—
239C5×D4—82no—
240C5×D4—88no—
241C5×D4—104no—
242C5×D4—102no—
243C5×D4—201no—
244C5×D4—201no—
245C5×D4—402no—
246C2×C4×C5—21no—
247C2×C4×C5—21no—
248C2×C4×C5—42no—
249C2×C4×C5—41no—
250C2×C4×C5—51no—
251C2×C4×C5—52no—
252C2×C4×C5—52no—
253C2×C4×C5—104no—
254C2×C4×C5—101no—
255C2×C4×C5—101no—
256C2×C4×C5—202no—
257C2×C4×C5—404no—
258C2×C4×C5C22×C4×C511yescharacteristic
259C2×C4×C5C22×AGL(1,𝔽5)11yescharacteristic
260C22⋊D5—24no—
261C22⋊D5—41no—
262C22⋊D5—41no—
263C22⋊D5—82no—
264C2×Dic5—21no—
265C2×Dic5—21no—
266C2×Dic5—42no—
267C2×Dic5C2×C5×D411yescharacteristic
268D20—22no—
269C4×D5C2×C5×D412yes—
270C4×D5C5×#4112yes—
271Dic10—22no—
272#170—42no—
273C5×C8—202no—
274C2×#41—51no—
275C22×D4—51no—
276C2×M16—52no—
277#127—51no—
278#127C5×AGL(1,𝔽5)11yescharacteristic, complemented
279#124—51no—
280#124—51no—
281#121—102no—
282#120—102no—
283C4×D4—104no—
284#117—52no—
285#117—51no—
286C2×C22⋊C4—102no—
287#104—104no—
288#100—102no—
289C52#191111yescharacteristic, complemented, Sylow
290C22×C5—22no—
291C22×C5—22no—
292C22×C5—42no—
293C22×C5—42no—
294C22×C5—42no—
295C22×C5—42no—
296C22×C5—41no—
297C22×C5—41no—
298C22×C5—41no—
299C22×C5—51no—
300C22×C5—51no—
301C22×C5—82no—
302C22×C5—84no—
303C22×C5—84no—
304C22×C5—82no—
305C22×C5—102no—
306C22×C5—201no—
307C22×C5—202no—
308C22×C5—204no—
309C22×C5—402no—
310C22×C5C2×C5×C22⋊C411yescharacteristic
311C22×C5C2×C22⋊AGL(1,𝔽5)11yescharacteristic
312C2×D5—22no—
313C2×D5—41no—
314C2×D5—44no—
315C2×D5—42no—
316C2×D5—82no—
317C2×D5C5×#12011yescharacteristic
318C2×D5C5×#12311yescharacteristic
319AGL(1,𝔽5)—44no—
320C4×C5—41no—
321C4×C5—41no—
322C4×C5—44no—
323C4×C5—42no—
324C4×C5—52no—
325C4×C5—201no—
326C4×C5—204no—
327C4×C5—201no—
328C4×C5C4×C5×D412yes—
329C4×C5D4×AGL(1,𝔽5)12yes—
330Dic5—41no—
331Dic5—41no—
332Dic5C5×#12112yes—
333C24—52no—
334#41—102no—
335#41—102no—
336C2×Q8—51no—
337C2×D4—22no—
338C2×D4—52no—
339C2×D4—51no—
340C2×D4—102no—
341C2×D4—102no—
342C2×D4—101no—
343C2×D4—102no—
344C2×D4C2×C5×AGL(1,𝔽5)11yescharacteristic
345C2×D4C2×C5×AGL(1,𝔽5)11yescharacteristic, complemented
346C22×C4—51no—
347C22×C4—101no—
348C22×C4—104no—
349M16—102no—
350M16—102no—
351C2×C8—102no—
352C4⋊C4—101no—
353C4⋊C4—102no—
354C22⋊C4—101no—
355C22⋊C4—102no—
356C22⋊C4—102no—
357C22⋊C4—104no—
358C22⋊C4—102no—
359C42—54no—
360C42—51no—
361C42C2×C5×AGL(1,𝔽5)11yescharacteristic, complemented
362C2×C5—21no—
363C2×C5—21no—
364C2×C5—42no—
365C2×C5—42no—
366C2×C5—42no—
367C2×C5—41no—
368C2×C5—81no—
369C2×C5—81no—
370C2×C5—101no—
371C2×C5—101no—
372C2×C5—162no—
373C2×C5—162no—
374C2×C5—202no—
375C2×C5C5×#38611yescharacteristic
376C2×C5#6577911yescharacteristic
377D5—21no—
378D5—21no—
379D5—42no—
380C23—21no—
381C23—51no—
382C23—52no—
383C23—102no—
384C23—102no—
385C23—402no—
386C23C5×C22⋊AGL(1,𝔽5)12yes—
387Q8—102no—
388D4—22no—
389D4—22no—
390D4—82no—
391D4—104no—
392D4—102no—
393D4—201no—
394D4—201no—
395D4—402no—
396C2×C4—21no—
397C2×C4—51no—
398C2×C4—52no—
399C2×C4—52no—
400C2×C4—104no—
401C2×C4—101no—
402C2×C4—101no—
403C2×C4—202no—
404C2×C4—404no—
405C2×C4C22×C5×AGL(1,𝔽5)11yescharacteristic
406C8—202no—
407C5—41no—
408C5C5×#191111yescharacteristic, complemented
409C5#1060461511yescharacteristic, direct product factor
410C22—22no—
411C22—42no—
412C22—42no—
413C22—41no—
414C22—51no—
415C22—51no—
416C22—102no—
417C22—201no—
418C22—202no—
419C22—204no—
420C22—402no—
421C22C2×C5×C22⋊AGL(1,𝔽5)11yescharacteristic
422C4—41no—
423C4—52no—
424C4—201no—
425C4—204no—
426C4—201no—
427C4C5×D4×AGL(1,𝔽5)12yes—
428C2—21no—
429C2—42no—
430C2—81no—
431C2—101no—
432C2—101no—
433C2—202no—
434C2C5×#6577911yescharacteristic
subgroupG/H
centerC2×C5#65779
commutatorC2×C4×C5C22×C4×C5
FrattiniC2×C4C22×C5×AGL(1,𝔽5)
FittingC52×#127C4
socleC2×C52#386
prime radicalC5×#66049C2
cosocleC2×C4×D5C23×C5
hypercenterC5×#127AGL(1,𝔽5)
hypocenterC5C5×#1911
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C5 ◁ C2×C4×C5 ◁ C5×#127

Complements

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

NH ≅ G/Ntype
1. C5×#66258C2semidirect
3. C5×D5⋊#100C2semidirect
4. C5×#65769C2semidirect
5. C5×#64863C2semidirect
12. C52×#127C4semidirect
13. C5×#4086C4semidirect
16. C5×#4082C4semidirect
17. C5×#4021C4semidirect
21. #10604615C5direct (N × H)
27. C2×C52×D4C2×C4semidirect
38. C2×C5×Dic10C2×C4semidirect
43. C4×C5×AGL(1,𝔽5)D4semidirect
46. C42×C52C2×C4semidirect
51. C4×C5×Dic5C2×C4semidirect
52. #66258C2×C5semidirect
56. D5⋊#100C2×C5semidirect
57. #65769C2×C5semidirect
60. #64863C2×C5semidirect
100. C5×#127C4×C5semidirect
101. C5×#127AGL(1,𝔽5)semidirect
110. #4086C4×C5semidirect
113. #4082C4×C5semidirect
114. #4021C4×C5semidirect
157. C2×C5×D4C2×C4×C5semidirect
159. C2×C5×D4C2×AGL(1,𝔽5)semidirect
173. C2×Dic10C2×C4×C5semidirect
178. C4×AGL(1,𝔽5)C5×D4semidirect
194. C42×C5C2×C4×C5semidirect
195. C42×C5C2×AGL(1,𝔽5)semidirect
200. C4×Dic5C2×C4×C5semidirect
278. #127C5×AGL(1,𝔽5)semidirect
289. C52#1911semidirect
345. C2×D4C2×C5×AGL(1,𝔽5)semidirect
361. C42C2×C5×AGL(1,𝔽5)semidirect
408. C5C5×#1911semidirect
409. C5#10604615direct (N × H)

Maximal subgroups

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

subgroupindex
C5×#662582
C5×#660492
C5×D5⋊#1002
C5×#657692
C5×#648632
#106046155
C5×#19115