Finite Group Databasebeta

← C5×#69711 #14288320

Subgroups — C5×#69711 #14288320

Order 1920.

432 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 432 conjugacy classes of subgroups — too many to draw. Only its 98 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×#5515C211yesmaximal, characteristic
2C3×C5×#398C211yesmaximal, characteristic, complemented
3C5×#4971C211yesmaximal, characteristic, complemented
4C5×#4967C211yesmaximal, characteristic
5C5×#4528C211yesmaximal, characteristic, complemented
6C5×#4452C211yesmaximal, characteristic, complemented
7C5×#4451C211yesmaximal, characteristic
8C5×#1609—31nomaximal, Hall
9C2×C3×C5×C4⋊C4—21no—
10C2×C3×C5×C4⋊C4C2211yescharacteristic
11C2×C3×C5×C4⋊C4C2211yescharacteristic
12C5×#930C2211yescharacteristic
13C5×#929—21no—
14C4×C5×#67—21no—
15C5×#924C2211yescharacteristic
16C5×#898—21no—
17C5×#896—21no—
18C5×#894—21no—
19C5×#889—21no—
20C3×C5×#95—21no—
21C3×C5×#95C2211yescharacteristic
22C5×#830C2211yescharacteristic
23C5×#830C2211yescharacteristic
24C5×#810—21no—
25C5×#809—21no—
26C5×#808—21no—
27C5×#807—21no—
28#69711C511yesmaximal, characteristic, direct product factor, Hall
29C5×#539—31no—
30C5×#483—31no—
31C5×#480—31no—
32C5×#398D311yescharacteristic, complemented
33C5×#328—31no—
34C5×#327—31no—
35C5×#324—31no—
36C22×C3×C4×C5—21no—
37C22×C3×C4×C5—21no—
38C22×C3×C4×C5D411yescharacteristic
39C22×C3×C4×C5D411yescharacteristic
40C22×C3×C4×C5C2311yescharacteristic
41C2×C5×#67—21no—
42C2×C5×D12D411yescharacteristic
43C2×C4×C5×D3—21no—
44C2×C5×Dic6D411yescharacteristic
45C3×C5×C4⋊C4—21no—
46C3×C5×C4⋊C4—21no—
47C3×C5×C4⋊C4—21no—
48C3×C5×C4⋊C4—21no—
49C3×C5×C4⋊C4—42no—
50C5×#217—21no—
51C5×(C2×D3)⋊C4—21no—
52C5×(C2×D3)⋊C4—21no—
53C5×(C2×D3)⋊C4—21no—
54C5×(C2×D3)⋊C4—21no—
55C5×C4⋊Dic3D411yescharacteristic
56C5×C4⋊Dic3D411yescharacteristic
57C5×Dic3⋊C4—21no—
58C5×Dic3⋊C4—21no—
59C5×Dic3⋊C4—21no—
60C5×Dic3⋊C4—21no—
61C4×C5×Dic3—21no—
62C2×C5×C3⋊C8—21no—
63C2×C5×C3⋊C8—21no—
64#5515C2×C511yescharacteristic
65C3×#398C2×C511yescharacteristic, complemented
66#4971C2×C511yescharacteristic, complemented
67#4967C2×C511yescharacteristic
68#4528C2×C511yescharacteristic, complemented
69#4452C2×C511yescharacteristic, complemented
70#4451C2×C511yescharacteristic
71C5×#126—61no—
72C5×#125—61no—
73C5×#123—61no—
74C5×#123—61no—
75C5×C22⋊Q8—31no—
76C5×C22⋊Q8—61no—
77C5×#121—31no—
78C4×C5×D4—61no—
79C2×C5×C4⋊C4—21no—
80C2×C5×C4⋊C4C2×D311yescharacteristic
81C2×C5×C4⋊C4C2×D311yescharacteristic
82C5×#107—61no—
83C5×#106—61no—
84C5×#103—61no—
85C5×#102—61no—
86C5×C22⋊C8—31no—
87C5×C22⋊C8—31no—
88C5×#95—21no—
89C5×#95C2×D311yescharacteristic
90#1609—31noSylow
91C23×C3×C5C2×D411yescharacteristic
92C22×C5×D3—21no—
93C2×C3×C4×C5—22no—
94C2×C3×C4×C5—21no—
95C2×C3×C4×C5—21no—
96C2×C3×C4×C5—21no—
97C2×C3×C4×C5—21no—
98C2×C3×C4×C5—41no—
99C2×C3×C4×C5—41no—
100C2×C3×C4×C5—41no—
101C2×C3×C4×C5—41no—
102C2×C3×C4×C5—41no—
103C2×C3×C4×C5—41no—
104C2×C3×C4×C5—41no—
105C2×C3×C4×C5C2×D411yescharacteristic
106C2×C3×C4×C5C2×D411yescharacteristic
107C5×#67—41no—
108C5×#67—41no—
109C2×C5×Dic3—21no—
110C2×C5×Dic3—21no—
111C2×C5×Dic3—21no—
112C5×D12—41no—
113C4×C5×D3—81no—
114C5×Dic6—41no—
115C5×C3⋊C8—41no—
116C5×C3⋊C8—41no—
117C2×C3×C4⋊C4—21no—
118C2×C3×C4⋊C4C22×C511yescharacteristic
119C2×C3×C4⋊C4C22×C511yescharacteristic
120#930C22×C511yescharacteristic
121#929—21no—
122C4×#67—21no—
123#924C22×C511yescharacteristic
124#898—21no—
125#896—21no—
126#894—21no—
127#889—21no—
128C3×#95—21no—
129C3×#95C22×C511yescharacteristic
130#830C22×C511yescharacteristic
131#830C22×C511yescharacteristic
132#810—21no—
133#809—21no—
134#808—21no—
135#807—21no—
136C2×C5×Q8—31no—
137C2×C5×D4—31no—
138C2×C5×D4—61no—
139C22×C4×C5—21no—
140C22×C4×C5—21no—
141C22×C4×C5—61no—
142C22×C4×C5#6711yescharacteristic
143C22×C4×C5#6711yescharacteristic
144C22×C4×C5C22×D311yescharacteristic
145C2×C5×C8—61no—
146C2×C5×C8—61no—
147C5×C4⋊C4—21no—
148C5×C4⋊C4—21no—
149C5×C4⋊C4—21no—
150C5×C4⋊C4—21no—
151C5×C4⋊C4—31no—
152C5×C4⋊C4—31no—
153C5×C4⋊C4—42no—
154C5×C4⋊C4—61no—
155C5×C4⋊C4—61no—
156C5×C4⋊C4—61no—
157C5×C4⋊C4—61no—
158C5×C22⋊C4—61no—
159C5×C22⋊C4—61no—
160C5×C22⋊C4—61no—
161C5×C22⋊C4—61no—
162C5×C22⋊C4—61no—
163C42×C5—61no—
164#539—31no—
165#483—31no—
166#480—31no—
167#398C5×D311yescharacteristic, complemented
168#328—31no—
169#327—31no—
170#324—31no—
171C22×C3×C5—21no—
172C22×C3×C5—21no—
173C22×C3×C5#12011yescharacteristic
174C22×C3×C5#13511yescharacteristic
175C22×C3×C5AGL(1,ℤ/8)11yescharacteristic
176C2×C5×D3—41no—
177C2×C5×D3—41no—
178C2×C5×D3—41no—
179C3×C4×C5—21no—
180C3×C4×C5—21no—
181C3×C4×C5—42no—
182C3×C4×C5—41no—
183C3×C4×C5—41no—
184C3×C4×C5—81no—
185C5×Dic3—41no—
186C5×Dic3—41no—
187C5×Dic3—41no—
188C22×C3×C4—21no—
189C22×C3×C4—21no—
190C22×C3×C4C5×D411yescharacteristic
191C22×C3×C4C5×D411yescharacteristic
192C22×C3×C4C23×C511yescharacteristic
193C2×#67—21no—
194C2×D12C5×D411yescharacteristic
195C2×C4×D3—21no—
196C2×Dic6C5×D411yescharacteristic
197C3×C4⋊C4—21no—
198C3×C4⋊C4—21no—
199C3×C4⋊C4—21no—
200C3×C4⋊C4—21no—
201C3×C4⋊C4—42no—
202#217—21no—
203(C2×D3)⋊C4—21no—
204(C2×D3)⋊C4—21no—
205(C2×D3)⋊C4—21no—
206(C2×D3)⋊C4—21no—
207C4⋊Dic3C5×D411yescharacteristic
208C4⋊Dic3C5×D411yescharacteristic
209Dic3⋊C4—21no—
210Dic3⋊C4—21no—
211Dic3⋊C4—21no—
212Dic3⋊C4—21no—
213C4×Dic3—21no—
214C2×C3⋊C8—21no—
215C2×C3⋊C8—21no—
216C23×C5—61no—
217C23×C5C2×#6711yescharacteristic
218C5×Q8—121no—
219C5×D4—121no—
220C5×D4—121no—
221C5×D4—121no—
222C2×C4×C5—22no—
223C2×C4×C5—21no—
224C2×C4×C5—21no—
225C2×C4×C5—21no—
226C2×C4×C5—21no—
227C2×C4×C5—41no—
228C2×C4×C5—41no—
229C2×C4×C5—41no—
230C2×C4×C5—41no—
231C2×C4×C5—41no—
232C2×C4×C5—41no—
233C2×C4×C5—41no—
234C2×C4×C5—61no—
235C2×C4×C5—61no—
236C2×C4×C5—61no—
237C2×C4×C5—241no—
238C2×C4×C5D3×D411yescharacteristic
239C2×C4×C5D3×D411yescharacteristic
240C5×C8—121no—
241C5×C8—121no—
242#126—61no—
243#125—61no—
244#123—61no—
245#123—61no—
246C22⋊Q8—31no—
247C22⋊Q8—61no—
248#121—31no—
249C4×D4—61no—
250C2×C4⋊C4—21no—
251C2×C4⋊C4C2×C5×D311yescharacteristic
252C2×C4⋊C4C2×C5×D311yescharacteristic
253#107—61no—
254#106—61no—
255#103—61no—
256#102—61no—
257C22⋊C8—31no—
258C22⋊C8—31no—
259#95—21no—
260#95C2×C5×D311yescharacteristic
261C2×C3×C5—21no—
262C2×C3×C5—21no—
263C2×C3×C5#44911yescharacteristic
264C2×C3×C5#45611yescharacteristic
265C2×C3×C5#45811yescharacteristic
266C5×D3—81no—
267C23×C3C2×C5×D411yescharacteristic
268C22×D3—21no—
269C2×C3×C4—22no—
270C2×C3×C4—21no—
271C2×C3×C4—21no—
272C2×C3×C4—21no—
273C2×C3×C4—21no—
274C2×C3×C4—41no—
275C2×C3×C4—41no—
276C2×C3×C4—41no—
277C2×C3×C4—41no—
278C2×C3×C4—41no—
279C2×C3×C4—41no—
280C2×C3×C4—41no—
281C2×C3×C4C2×C5×D411yescharacteristic
282C2×C3×C4C2×C5×D411yescharacteristic
283#67—41no—
284#67—41no—
285C2×Dic3—21no—
286C2×Dic3—21no—
287C2×Dic3—21no—
288D12—41no—
289C4×D3—81no—
290Dic6—41no—
291C3⋊C8—41no—
292C3⋊C8—41no—
293C22×C5—21no—
294C22×C5—21no—
295C22×C5—121no—
296C22×C5—121no—
297C22×C5—121no—
298C22×C5#93211yescharacteristic
299C22×C5#93711yescharacteristic
300C22×C5#95011yescharacteristic
301C4×C5—21no—
302C4×C5—21no—
303C4×C5—42no—
304C4×C5—41no—
305C4×C5—41no—
306C4×C5—81no—
307C4×C5—121no—
308C4×C5—121no—
309C4×C5—121no—
310C2×Q8—31no—
311C2×D4—31no—
312C2×D4—61no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—61no—
316C22×C4C5×#6711yescharacteristic
317C22×C4C5×#6711yescharacteristic
318C22×C4C22×C5×D311yescharacteristic
319C2×C8—61no—
320C2×C8—61no—
321C4⋊C4—21no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4—21no—
325C4⋊C4—31no—
326C4⋊C4—31no—
327C4⋊C4—42no—
328C4⋊C4—61no—
329C4⋊C4—61no—
330C4⋊C4—61no—
331C4⋊C4—61no—
332C22⋊C4—61no—
333C22⋊C4—61no—
334C22⋊C4—61no—
335C22⋊C4—61no—
336C22⋊C4—61no—
337C42—61no—
338C3×C5#160911yescharacteristic, complemented, Hall
339C22×C3—21no—
340C22×C3—21no—
341C22×C3C5×#12011yescharacteristic
342C22×C3C5×#13511yescharacteristic
343C22×C3C5×AGL(1,ℤ/8)11yescharacteristic
344C2×D3—41no—
345C2×D3—41no—
346C2×D3—41no—
347C3×C4—21no—
348C3×C4—21no—
349C3×C4—42no—
350C3×C4—41no—
351C3×C4—41no—
352C3×C4—81no—
353Dic3—41no—
354Dic3—41no—
355Dic3—41no—
356C2×C5—21no—
357C2×C5—21no—
358C2×C5—241no—
359C2×C5#503711yescharacteristic
360C2×C5#505311yescharacteristic
361C2×C5#509511yescharacteristic
362C23—61no—
363C23C2×C5×#6711yescharacteristic
364Q8—121no—
365D4—121no—
366D4—121no—
367D4—121no—
368C2×C4—22no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—41no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—61no—
381C2×C4—61no—
382C2×C4—61no—
383C2×C4—241no—
384C2×C4C5×D3×D411yescharacteristic
385C2×C4C5×D3×D411yescharacteristic
386C8—121no—
387C8—121no—
388C2×C3—21no—
389C2×C3—21no—
390C2×C3C5×#44911yescharacteristic
391C2×C3C5×#45611yescharacteristic
392C2×C3C5×#45811yescharacteristic
393D3—81no—
394C5#6971111yescharacteristic, direct product factor, Sylow
395C22—21no—
396C22—21no—
397C22—121no—
398C22—121no—
399C22—121no—
400C22C5×#93211yescharacteristic
401C22C5×#93711yescharacteristic
402C22C5×#95011yescharacteristic
403C4—21no—
404C4—21no—
405C4—42no—
406C4—41no—
407C4—41no—
408C4—81no—
409C4—121no—
410C4—121no—
411C4—121no—
412C3C5×#160911yescharacteristic, complemented, Sylow
413C2—21no—
414C2—21no—
415C2—241no—
416C2C5×#503711yescharacteristic
417C2C5×#505311yescharacteristic
418C2C5×#509511yescharacteristic
subgroupG/H
centerC22×C5#937
commutator, cosocleC22×C3×C4C23×C5
FrattiniC22×C4C22×C5×D3
FittingC3×C5×#398C2
socleC22×C3×C5#120
prime radicalC5×#930C22
hypercenterC5×#398D3
hypocenterC3C5×#1609
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C5×#398

Complements

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

NH ≅ G/Ntype
2. C3×C5×#398C2semidirect
3. C5×#4971C2semidirect
5. C5×#4528C2semidirect
6. C5×#4452C2semidirect
28. #69711C5direct (N × H)
32. C5×#398D3semidirect
65. C3×#398C2×C5semidirect
66. #4971C2×C5semidirect
68. #4528C2×C5semidirect
69. #4452C2×C5semidirect
167. #398C5×D3semidirect
338. C3×C5#1609semidirect
394. C5#69711direct (N × H)
412. C3C5×#1609semidirect

Maximal subgroups

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

subgroupindex
C5×#55152
C3×C5×#3982
C5×#49712
C5×#49672
C5×#45282
C5×#44522
C5×#44512
C5×#16093
#697115