Finite Group Databasebeta

← C5×#68115 #14288008

Subgroups — C5×#68115 #14288008

Order 1920.

512 conjugacy classes of subgroups; only the 106 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 106 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×#5483C211yesmaximal, characteristic
2C5×#4947C211yesmaximal, characteristic
3C5×#4739C212yesmaximal, complemented
4C3×C5×#323C211yesmaximal, characteristic
5C5×#4467C212yesmaximal, complemented
6C5×#1602—31nomaximal, Hall
7C22×C5×Dic6C2211yescharacteristic
8C2×C3×C5×C22⋊C4C2211yescharacteristic
9C2×C5×#217—21no—
10C2×C5×#217—21no—
11C5×#924—21no—
12C5×#924C2212yes—
13C2×C5×Dic12—22no—
14C5×#878—22no—
15C5×#878—22no—
16C3×C5×C22⋊C8C2212yes—
17C5×C3⋊#95—21no—
18C5×C3⋊#95C2211yescharacteristic
19C5×#816—22no—
20#68115C511yesmaximal, characteristic, direct product factor, Hall
21C5×#543—31no—
22C5×#451—32no—
23C5×#393—31no—
24C5×#328—32no—
25C5×#323D311yescharacteristic
26C24×C3×C5D411yescharacteristic
27C22×C3×C4×C5D411yescharacteristic
28C22×C3×C4×C5C2311yescharacteristic
29C22×C5×Dic3—21no—
30C22×C5×Dic3—21no—
31C22×C5×Dic3—21no—
32C22×C5×Dic3—21no—
33C2×C5×Dic6—22no—
34C2×C5×Dic6—21no—
35C2×C5×Dic6—42no—
36C2×C5×Dic6D412yes—
37C2×C3×C5×C8—22no—
38C3×C5×C22⋊C4—22no—
39C5×#217—22no—
40C5×#217—22no—
41C5×#217—42no—
42C5×C4⋊Dic3—21no—
43C5×C4⋊Dic3D412yes—
44C5×Dic3⋊C4—22no—
45C5×Dic3⋊C4—22no—
46C5×Dic12—44no—
47#5483C2×C511yescharacteristic
48#4947C2×C511yescharacteristic
49#4739C2×C512yescomplemented
50C3×#323C2×C511yescharacteristic
51#4467C2×C512yescomplemented
52C22×C5×Q8—31no—
53C2×C5×Q16—62no—
54C5×C22⋊Q8—32no—
55C5×C22⋊Q8—62no—
56C5×C22⋊Q8—62no—
57C5×C22⋊Q8—61no—
58C2×C5×C22⋊C4—61no—
59C2×C5×C22⋊C4—61no—
60C2×C5×C22⋊C4C2×D311yescharacteristic
61C5×#103—62no—
62C5×C22⋊C8C2×D312yes—
63C5×#95—31no—
64C5×#95—61no—
65#1602—31noSylow
66C23×C3×C5—21no—
67C23×C3×C5—21no—
68C23×C3×C5—41no—
69C23×C3×C5—41no—
70C23×C3×C5Q1612yescomplemented
71C23×C3×C5C2×D411yescharacteristic
72C2×C3×C4×C5—21no—
73C2×C3×C4×C5—41no—
74C2×C3×C4×C5—41no—
75C2×C3×C4×C5C2×D412yes—
76C2×C5×Dic3—22no—
77C2×C5×Dic3—22no—
78C2×C5×Dic3—22no—
79C2×C5×Dic3—41no—
80C2×C5×Dic3—41no—
81C2×C5×Dic3—41no—
82C2×C5×Dic3—41no—
83C2×C5×Dic3—41no—
84C2×C5×Dic3—81no—
85C2×C5×Dic3—81no—
86C5×Dic6—24no—
87C5×Dic6—42no—
88C5×Dic6—42no—
89C5×Dic6—81no—
90C3×C5×C8—42no—
91C22×Dic6C22×C511yescharacteristic
92C2×C3×C22⋊C4C22×C511yescharacteristic
93C2×#217—21no—
94C2×#217—21no—
95#924—21no—
96#924C22×C512yes—
97C2×Dic12—22no—
98#878—22no—
99#878—22no—
100C3×C22⋊C8C22×C512yes—
101C3⋊#95—21no—
102C3⋊#95C22×C511yescharacteristic
103#816—22no—
104C24×C5D1211yescharacteristic
105C2×C5×Q8—32no—
106C2×C5×Q8—61no—
107C2×C5×Q8—62no—
108C2×C5×Q8—122no—
109C22×C4×C5—61no—
110C22×C4×C5—61no—
111C22×C4×C5—61no—
112C22×C4×C5—61no—
113C22×C4×C5D1211yescharacteristic
114C22×C4×C5C22×D311yescharacteristic
115C5×Q16—124no—
116C2×C5×C8—22no—
117C5×C4⋊C4—32no—
118C5×C4⋊C4—62no—
119C5×C4⋊C4—61no—
120C5×C4⋊C4—62no—
121C5×C22⋊C4—22no—
122C5×C22⋊C4—62no—
123C5×C22⋊C4—62no—
124C5×C22⋊C4—122no—
125#543—31no—
126#451—32no—
127#393—31no—
128#328—32no—
129#323C5×D311yescharacteristic
130C22×C3×C5—21no—
131C22×C3×C5—21no—
132C22×C3×C5—22no—
133C22×C3×C5—42no—
134C22×C3×C5—41no—
135C22×C3×C5—41no—
136C22×C3×C5—81no—
137C22×C3×C5#12011yescharacteristic
138C22×C3×C5C2×Q1611yescharacteristic
139C22×C3×C5#13711yescharacteristic
140C3×C4×C5—22no—
141C3×C4×C5—41no—
142C5×Dic3—42no—
143C5×Dic3—42no—
144C5×Dic3—42no—
145C5×Dic3—81no—
146C24×C3C5×D411yescharacteristic
147C22×C3×C4C5×D411yescharacteristic
148C22×C3×C4C23×C511yescharacteristic
149C22×Dic3—21no—
150C22×Dic3—21no—
151C22×Dic3—21no—
152C22×Dic3—21no—
153C2×Dic6—22no—
154C2×Dic6—21no—
155C2×Dic6—42no—
156C2×Dic6C5×D412yes—
157C2×C3×C8—22no—
158C3×C22⋊C4—22no—
159#217—22no—
160#217—22no—
161#217—42no—
162C4⋊Dic3—21no—
163C4⋊Dic3C5×D412yes—
164Dic3⋊C4—22no—
165Dic3⋊C4—22no—
166Dic12—44no—
167C23×C5—21no—
168C23×C5—21no—
169C23×C5—41no—
170C23×C5—41no—
171C23×C5Dic1212yescomplemented
172C23×C5C2×D1211yescharacteristic
173C5×Q8—64no—
174C5×Q8—122no—
175C5×Q8—122no—
176C5×Q8—241no—
177C2×C4×C5—21no—
178C2×C4×C5—41no—
179C2×C4×C5—41no—
180C2×C4×C5—62no—
181C2×C4×C5—62no—
182C2×C4×C5—62no—
183C2×C4×C5—121no—
184C2×C4×C5—121no—
185C2×C4×C5—121no—
186C2×C4×C5—121no—
187C2×C4×C5—121no—
188C2×C4×C5—241no—
189C2×C4×C5—241no—
190C2×C4×C5D3×D412yes—
191C5×C8—42no—
192C22×Q8—31no—
193C2×Q16—62no—
194C22⋊Q8—32no—
195C22⋊Q8—62no—
196C22⋊Q8—62no—
197C22⋊Q8—61no—
198C2×C22⋊C4—61no—
199C2×C22⋊C4—61no—
200C2×C22⋊C4C2×C5×D311yescharacteristic
201#103—62no—
202C22⋊C8C2×C5×D312yes—
203#95—31no—
204#95—61no—
205C2×C3×C5—21no—
206C2×C3×C5—21no—
207C2×C3×C5—42no—
208C2×C3×C5#45111yescharacteristic
209C2×C3×C5#45511yescharacteristic
210C2×C3×C5B(4,𝔽2)11yescharacteristic
211C23×C3—21no—
212C23×C3—21no—
213C23×C3—41no—
214C23×C3—41no—
215C23×C3C5×Q1612yescomplemented
216C23×C3C2×C5×D411yescharacteristic
217C2×C3×C4—21no—
218C2×C3×C4—41no—
219C2×C3×C4—41no—
220C2×C3×C4C2×C5×D412yes—
221C2×Dic3—22no—
222C2×Dic3—22no—
223C2×Dic3—22no—
224C2×Dic3—41no—
225C2×Dic3—41no—
226C2×Dic3—41no—
227C2×Dic3—41no—
228C2×Dic3—41no—
229C2×Dic3—81no—
230C2×Dic3—81no—
231Dic6—24no—
232Dic6—42no—
233Dic6—42no—
234Dic6—81no—
235C3×C8—42no—
236C22×C5—21no—
237C22×C5—21no—
238C22×C5—22no—
239C22×C5—42no—
240C22×C5—41no—
241C22×C5—41no—
242C22×C5—81no—
243C22×C5#88211yescharacteristic
244C22×C5C2×Dic1211yescharacteristic
245C22×C5#90911yescharacteristic
246C4×C5—22no—
247C4×C5—41no—
248C4×C5—122no—
249C4×C5—122no—
250C4×C5—122no—
251C4×C5—241no—
252C24C5×D1211yescharacteristic
253C2×Q8—32no—
254C2×Q8—61no—
255C2×Q8—62no—
256C2×Q8—122no—
257C22×C4—61no—
258C22×C4—61no—
259C22×C4—61no—
260C22×C4—61no—
261C22×C4C5×D1211yescharacteristic
262C22×C4C22×C5×D311yescharacteristic
263Q16—124no—
264C2×C8—22no—
265C4⋊C4—32no—
266C4⋊C4—62no—
267C4⋊C4—61no—
268C4⋊C4—62no—
269C22⋊C4—22no—
270C22⋊C4—62no—
271C22⋊C4—62no—
272C22⋊C4—122no—
273C3×C5#160211yescharacteristic, complemented, Hall
274C22×C3—21no—
275C22×C3—21no—
276C22×C3—22no—
277C22×C3—42no—
278C22×C3—41no—
279C22×C3—41no—
280C22×C3—81no—
281C22×C3C5×#12011yescharacteristic
282C22×C3C2×C5×Q1611yescharacteristic
283C22×C3C5×#13711yescharacteristic
284C3×C4—22no—
285C3×C4—41no—
286Dic3—42no—
287Dic3—42no—
288Dic3—42no—
289Dic3—81no—
290C2×C5—21no—
291C2×C5—21no—
292C2×C5—42no—
293C2×C5#473911yescharacteristic
294C2×C5#474111yescharacteristic
295C2×C5#474611yescharacteristic
296C23—21no—
297C23—21no—
298C23—41no—
299C23—41no—
300C23C5×Dic1212yescomplemented
301C23C2×C5×D1211yescharacteristic
302Q8—64no—
303Q8—122no—
304Q8—122no—
305Q8—241no—
306C2×C4—21no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4—62no—
310C2×C4—62no—
311C2×C4—62no—
312C2×C4—121no—
313C2×C4—121no—
314C2×C4—121no—
315C2×C4—121no—
316C2×C4—121no—
317C2×C4—241no—
318C2×C4—241no—
319C2×C4C5×D3×D412yes—
320C8—42no—
321C2×C3—21no—
322C2×C3—21no—
323C2×C3—42no—
324C2×C3C5×#45111yescharacteristic
325C2×C3C5×#45511yescharacteristic
326C2×C3C5×B(4,𝔽2)11yescharacteristic
327C5#6811511yescharacteristic, direct product factor, Sylow
328C22—21no—
329C22—21no—
330C22—22no—
331C22—42no—
332C22—41no—
333C22—41no—
334C22—81no—
335C22C5×#88211yescharacteristic
336C22C2×C5×Dic1211yescharacteristic
337C22C5×#90911yescharacteristic
338C4—22no—
339C4—41no—
340C4—122no—
341C4—122no—
342C4—122no—
343C4—241no—
344C3C5×#160211yescharacteristic, complemented, Sylow
345C2—21no—
346C2—21no—
347C2—42no—
348C2C5×#473911yescharacteristic
349C2C5×#474111yescharacteristic
350C2C5×#474611yescharacteristic
subgroupG/H
centerC22×C5#882
commutator, cosocleC22×C3×C4C23×C5
FrattiniC22×C4C22×C5×D3
FittingC3×C5×#323C2
socleC22×C3×C5#120
prime radicalC24×C3×C5D4
hypercenterC5×#323D3
hypocenterC3C5×#1602
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C5×#323

Complements

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

NH ≅ G/Ntype
3. C5×#4739C2semidirect
5. C5×#4467C2semidirect
20. #68115C5direct (N × H)
49. #4739C2×C5semidirect
51. #4467C2×C5semidirect
70. C23×C3×C5Q16semidirect
171. C23×C5Dic12semidirect
215. C23×C3C5×Q16semidirect
273. C3×C5#1602semidirect
300. C23C5×Dic12semidirect
327. C5#68115direct (N × H)
344. C3C5×#1602semidirect

Maximal subgroups

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

subgroupindex
C5×#54832
C5×#49472
C5×#47392
C3×C5×#3232
C5×#44672
C5×#16023
#681155