Finite Group Databasebeta

← C5×#71844 #14288527

Subgroups — C5×#71844 #14288527

Order 1920.

576 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 114 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
1C3×C5×#534C211yesmaximal, characteristic
2C5×#5034C212yesmaximal, complemented
3C5×#4949C211yesmaximal, characteristic, complemented
4C5×#4537C212yesmaximal, complemented
5C5×#4525C211yesmaximal, characteristic, complemented
6C5×#1613—31nomaximal, Hall
7C22×C3×C5×D4—21no—
8C3×C5×#123—22no—
9C3×C5×#121—21no—
10C3×C5×#121C2212yes—
11C3×C5×#120—22no—
12C2×C3×C5×C22⋊C4C2211yescharacteristic
13C2×C5×#217—21no—
14C2×C5×C4⋊Dic3C2211yescharacteristic, complemented
15C2×C5×Dic3⋊C4—21no—
16C5×#832—22no—
17C5×C3⋊#95C2211yescharacteristic, complemented
18C5×#830C2212yescomplemented
19C5×#808—22no—
20#71844C511yesmaximal, characteristic, direct product factor, Hall
21C5×#534D311yescharacteristic
22C5×#481—32no—
23C5×#399—31no—
24C5×#327—32no—
25C5×#323—31no—
26C24×C3×C5—21no—
27C24×C3×C5D411yescharacteristic
28C2×C3×C5×D4—22no—
29C2×C3×C5×D4—22no—
30C2×C3×C5×D4—21no—
31C2×C3×C5×D4—42no—
32C2×C3×C5×D4—42no—
33C2×C3×C5×D4D412yes—
34C22×C3×C4×C5—21no—
35C22×C3×C4×C5D411yescharacteristic
36C22×C3×C4×C5C2311yescharacteristic
37C22×C5×Dic3—21no—
38C22×C5×Dic3—21no—
39C3×C5×C4⋊C4—21no—
40C3×C5×C4⋊C4D412yes—
41C3×C5×C22⋊C4—22no—
42C3×C5×C22⋊C4—22no—
43C3×C5×C22⋊C4—22no—
44C5×#217—42no—
45C5×C4⋊Dic3—22no—
46C5×Dic3⋊C4—42no—
47C2×C5×C3⋊C8—22no—
48C3×#534C2×C511yescharacteristic
49#5034C2×C512yescomplemented
50#4949C2×C511yescharacteristic, complemented
51#4537C2×C512yescomplemented
52#4525C2×C511yescharacteristic, complemented
53C22×C5×D4—21no—
54C5×#123—22no—
55C5×#121—21no—
56C5×#121C2×D312yes—
57C5×#120—22no—
58C2×C5×C4⋊C4—31no—
59C2×C5×C4⋊C4—61no—
60C2×C5×C22⋊C4—61no—
61C2×C5×C22⋊C4C2×D311yescharacteristic
62C5×#106—62no—
63C5×#102—62no—
64C5×C22⋊C8—32no—
65C5×#95—31no—
66#1613—31noSylow
67C23×C3×C5—21no—
68C23×C3×C5—21no—
69C23×C3×C5—22no—
70C23×C3×C5—41no—
71C23×C3×C5—41no—
72C23×C3×C5—42no—
73C23×C3×C5—81no—
74C23×C3×C5—81no—
75C23×C3×C5SD1612yes—
76C23×C3×C5C2×D411yescharacteristic
77C3×C5×D4—42no—
78C3×C5×D4—44no—
79C3×C5×D4—42no—
80C3×C5×D4—42no—
81C3×C5×D4—81no—
82C2×C3×C4×C5—21no—
83C2×C3×C4×C5—22no—
84C2×C3×C4×C5—41no—
85C2×C3×C4×C5—42no—
86C2×C3×C4×C5—41no—
87C2×C3×C4×C5C2×D412yes—
88C2×C5×Dic3—22no—
89C2×C5×Dic3—41no—
90C2×C5×Dic3—41no—
91C2×C5×Dic3—41no—
92C2×C5×Dic3—41no—
93C2×C5×Dic3—41no—
94C5×C3⋊C8—42no—
95C22×C3×D4—21no—
96C3×#123—22no—
97C3×#121—21no—
98C3×#121C22×C512yes—
99C3×#120—22no—
100C2×C3×C22⋊C4C22×C511yescharacteristic
101C2×#217—21no—
102C2×C4⋊Dic3C22×C511yescharacteristic, complemented
103C2×Dic3⋊C4—21no—
104#832—22no—
105C3⋊#95C22×C511yescharacteristic, complemented
106#830C22×C512yescomplemented
107#808—22no—
108C24×C5—21no—
109C24×C5#6711yescharacteristic
110C2×C5×D4—22no—
111C2×C5×D4—22no—
112C2×C5×D4—21no—
113C2×C5×D4—42no—
114C2×C5×D4—42no—
115C2×C5×D4#6712yes—
116C22×C4×C5—21no—
117C22×C4×C5—61no—
118C22×C4×C5—61no—
119C22×C4×C5#6711yescharacteristic
120C22×C4×C5C22×D311yescharacteristic
121C2×C5×C8—62no—
122C5×C4⋊C4—21no—
123C5×C4⋊C4—62no—
124C5×C4⋊C4—122no—
125C5×C4⋊C4#6712yes—
126C5×C22⋊C4—22no—
127C5×C22⋊C4—22no—
128C5×C22⋊C4—22no—
129C5×C22⋊C4—122no—
130#534C5×D311yescharacteristic
131#481—32no—
132#399—31no—
133#327—32no—
134#323—31no—
135C22×C3×C5—21no—
136C22×C3×C5—21no—
137C22×C3×C5—22no—
138C22×C3×C5—42no—
139C22×C3×C5—41no—
140C22×C3×C5—41no—
141C22×C3×C5—42no—
142C22×C3×C5—42no—
143C22×C3×C5—42no—
144C22×C3×C5—81no—
145C22×C3×C5—84no—
146C22×C3×C5#12011yescharacteristic
147C22×C3×C5C2×SD1611yescharacteristic
148C22×C3×C5AGL(1,ℤ/8)11yescharacteristic
149C3×C4×C5—22no—
150C3×C4×C5—42no—
151C3×C4×C5—41no—
152C5×Dic3—42no—
153C5×Dic3—81no—
154C24×C3—21no—
155C24×C3C5×D411yescharacteristic
156C2×C3×D4—22no—
157C2×C3×D4—22no—
158C2×C3×D4—21no—
159C2×C3×D4—42no—
160C2×C3×D4—42no—
161C2×C3×D4C5×D412yes—
162C22×C3×C4—21no—
163C22×C3×C4C5×D411yescharacteristic
164C22×C3×C4C23×C511yescharacteristic
165C22×Dic3—21no—
166C22×Dic3—21no—
167C3×C4⋊C4—21no—
168C3×C4⋊C4C5×D412yes—
169C3×C22⋊C4—22no—
170C3×C22⋊C4—22no—
171C3×C22⋊C4—22no—
172#217—42no—
173C4⋊Dic3—22no—
174Dic3⋊C4—42no—
175C2×C3⋊C8—22no—
176C23×C5—21no—
177C23×C5—21no—
178C23×C5—22no—
179C23×C5—41no—
180C23×C5—41no—
181C23×C5—42no—
182C23×C5—81no—
183C23×C5—81no—
184C23×C5#21412yes—
185C23×C5C2×#6711yescharacteristic
186C5×D4—42no—
187C5×D4—44no—
188C5×D4—42no—
189C5×D4—42no—
190C5×D4—81no—
191C2×C4×C5—21no—
192C2×C4×C5—22no—
193C2×C4×C5—41no—
194C2×C4×C5—42no—
195C2×C4×C5—41no—
196C2×C4×C5—62no—
197C2×C4×C5—121no—
198C2×C4×C5—121no—
199C2×C4×C5—121no—
200C2×C4×C5—121no—
201C2×C4×C5—121no—
202C2×C4×C5C2×#6712yes—
203C5×C8—122no—
204C22×D4—21no—
205#123—22no—
206#121—21no—
207#121C2×C5×D312yes—
208#120—22no—
209C2×C4⋊C4—31no—
210C2×C4⋊C4—61no—
211C2×C22⋊C4—61no—
212C2×C22⋊C4C2×C5×D311yescharacteristic
213#106—62no—
214#102—62no—
215C22⋊C8—32no—
216#95—31no—
217C2×C3×C5—21no—
218C2×C3×C5—21no—
219C2×C3×C5—42no—
220C2×C3×C5—82no—
221C2×C3×C5#45011yescharacteristic
222C2×C3×C5#45311yescharacteristic
223C2×C3×C5#45811yescharacteristic
224C23×C3—21no—
225C23×C3—21no—
226C23×C3—22no—
227C23×C3—41no—
228C23×C3—41no—
229C23×C3—42no—
230C23×C3—81no—
231C23×C3—81no—
232C23×C3C5×SD1612yes—
233C23×C3C2×C5×D411yescharacteristic
234C3×D4—42no—
235C3×D4—44no—
236C3×D4—42no—
237C3×D4—42no—
238C3×D4—81no—
239C2×C3×C4—21no—
240C2×C3×C4—22no—
241C2×C3×C4—41no—
242C2×C3×C4—42no—
243C2×C3×C4—41no—
244C2×C3×C4C2×C5×D412yes—
245C2×Dic3—22no—
246C2×Dic3—41no—
247C2×Dic3—41no—
248C2×Dic3—41no—
249C2×Dic3—41no—
250C2×Dic3—41no—
251C3⋊C8—42no—
252C22×C5—21no—
253C22×C5—21no—
254C22×C5—22no—
255C22×C5—42no—
256C22×C5—41no—
257C22×C5—41no—
258C22×C5—42no—
259C22×C5—42no—
260C22×C5—42no—
261C22×C5—81no—
262C22×C5—84no—
263C22×C5#93211yescharacteristic
264C22×C5C2×#21411yescharacteristic
265C22×C5#95311yescharacteristic
266C4×C5—22no—
267C4×C5—42no—
268C4×C5—41no—
269C4×C5—122no—
270C4×C5—241no—
271C24—21no—
272C24C5×#6711yescharacteristic
273C2×D4—22no—
274C2×D4—22no—
275C2×D4—21no—
276C2×D4—42no—
277C2×D4—42no—
278C2×D4C5×#6712yes—
279C22×C4—21no—
280C22×C4—61no—
281C22×C4—61no—
282C22×C4C5×#6711yescharacteristic
283C22×C4C22×C5×D311yescharacteristic
284C2×C8—62no—
285C4⋊C4—21no—
286C4⋊C4—62no—
287C4⋊C4—122no—
288C4⋊C4C5×#6712yes—
289C22⋊C4—22no—
290C22⋊C4—22no—
291C22⋊C4—22no—
292C22⋊C4—122no—
293C3×C5#161311yescharacteristic, complemented, Hall
294C22×C3—21no—
295C22×C3—21no—
296C22×C3—22no—
297C22×C3—42no—
298C22×C3—41no—
299C22×C3—41no—
300C22×C3—42no—
301C22×C3—42no—
302C22×C3—42no—
303C22×C3—81no—
304C22×C3—84no—
305C22×C3C5×#12011yescharacteristic
306C22×C3C2×C5×SD1611yescharacteristic
307C22×C3C5×AGL(1,ℤ/8)11yescharacteristic
308C3×C4—22no—
309C3×C4—42no—
310C3×C4—41no—
311Dic3—42no—
312Dic3—81no—
313C2×C5—21no—
314C2×C5—21no—
315C2×C5—42no—
316C2×C5—82no—
317C2×C5#521811yescharacteristic
318C2×C5#524111yescharacteristic
319C2×C5#524311yescharacteristic
320C23—21no—
321C23—21no—
322C23—22no—
323C23—41no—
324C23—41no—
325C23—42no—
326C23—81no—
327C23—81no—
328C23C5×#21412yes—
329C23C2×C5×#6711yescharacteristic
330D4—42no—
331D4—44no—
332D4—42no—
333D4—42no—
334D4—81no—
335C2×C4—21no—
336C2×C4—22no—
337C2×C4—41no—
338C2×C4—42no—
339C2×C4—41no—
340C2×C4—62no—
341C2×C4—121no—
342C2×C4—121no—
343C2×C4—121no—
344C2×C4—121no—
345C2×C4—121no—
346C2×C4C2×C5×#6712yes—
347C8—122no—
348C2×C3—21no—
349C2×C3—21no—
350C2×C3—42no—
351C2×C3—82no—
352C2×C3C5×#45011yescharacteristic
353C2×C3C5×#45311yescharacteristic
354C2×C3C5×#45811yescharacteristic
355C5#7184411yescharacteristic, direct product factor, Sylow
356C22—21no—
357C22—21no—
358C22—22no—
359C22—42no—
360C22—41no—
361C22—41no—
362C22—42no—
363C22—42no—
364C22—42no—
365C22—81no—
366C22—84no—
367C22C5×#93211yescharacteristic
368C22C2×C5×#21411yescharacteristic
369C22C5×#95311yescharacteristic
370C4—22no—
371C4—42no—
372C4—41no—
373C4—122no—
374C4—241no—
375C3C5×#161311yescharacteristic, complemented, Sylow
376C2—21no—
377C2—21no—
378C2—42no—
379C2—82no—
380C2C5×#521811yescharacteristic
381C2C5×#524111yescharacteristic
382C2C5×#524311yescharacteristic
subgroupG/H
centerC22×C5#953
commutator, cosocleC22×C3×C4C23×C5
FrattiniC22×C4C22×C5×D3
Fitting, prime radicalC3×C5×#534C2
socleC22×C3×C5#120
hypercenterC5×#534D3
hypocenterC3C5×#1613
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C5×#534

Complements

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

NH ≅ G/Ntype
2. C5×#5034C2semidirect
3. C5×#4949C2semidirect
4. C5×#4537C2semidirect
5. C5×#4525C2semidirect
14. C2×C5×C4⋊Dic3C22semidirect
17. C5×C3⋊#95C22semidirect
18. C5×#830C22semidirect
20. #71844C5direct (N × H)
49. #5034C2×C5semidirect
50. #4949C2×C5semidirect
51. #4537C2×C5semidirect
52. #4525C2×C5semidirect
102. C2×C4⋊Dic3C22×C5semidirect
105. C3⋊#95C22×C5semidirect
106. #830C22×C5semidirect
293. C3×C5#1613semidirect
355. C5#71844direct (N × H)
375. C3C5×#1613semidirect

Maximal subgroups

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

subgroupindex
C3×C5×#5342
C5×#50342
C5×#49492
C5×#45372
C5×#45252
C5×#16133
#718445