Finite Group Databasebeta

← C5×#69713 #14288422

Subgroups — C5×#69713 #14288422

Order 1920.

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

normal subgroups only

This group has 420 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×#5500C211yesmaximal, characteristic
2C3×C5×#398C211yesmaximal, characteristic
3C5×#4971C212yesmaximal
4C5×#4528C211yesmaximal, characteristic
5C5×#4452C212yesmaximal
6C5×#1611—31nomaximal, Hall
7C2×C3×C5×C4⋊C4—21no—
8C2×C3×C5×C4⋊C4C2211yescharacteristic
9C2×C3×C5×C4⋊C4C2211yescharacteristic
10C5×#926—21no—
11C5×#924—21no—
12C5×#924C2212yes—
13C5×#890—22no—
14C5×#888—22no—
15C3×C5×#95—21no—
16C3×C5×#95C2211yescharacteristic
17C5×#830C2212yes—
18C5×#810—22no—
19C5×#808—22no—
20#69713C511yesmaximal, characteristic, direct product factor, Hall
21C5×#544—31no—
22C5×#483—32no—
23C5×#398D311yescharacteristic
24C5×#328—32no—
25C5×#324—31no—
26C22×C3×C4×C5—21no—
27C22×C3×C4×C5—21no—
28C22×C3×C4×C5D411yescharacteristic
29C22×C3×C4×C5D411yescharacteristic
30C22×C3×C4×C5C2311yescharacteristic
31C2×C5×Dic6—21no—
32C2×C5×Dic6D412yes—
33C3×C5×C4⋊C4—22no—
34C3×C5×C4⋊C4—22no—
35C3×C5×C4⋊C4—42no—
36C5×#217—22no—
37C5×C4⋊Dic3—21no—
38C5×C4⋊Dic3—21no—
39C5×C4⋊Dic3—21no—
40C5×C4⋊Dic3D412yes—
41C5×Dic3⋊C4—22no—
42C5×Dic3⋊C4—22no—
43C4×C5×Dic3—22no—
44C2×C5×C3⋊C8—22no—
45#5500C2×C511yescharacteristic
46C3×#398C2×C511yescharacteristic
47#4971C2×C512yes—
48#4528C2×C511yescharacteristic
49#4452C2×C512yes—
50C5×C4⋊Q8—62no—
51C5×#125—62no—
52C5×C22⋊Q8—32no—
53C5×C22⋊Q8—61no—
54C5×#117—61no—
55C2×C5×C4⋊C4—21no—
56C2×C5×C4⋊C4C2×D311yescharacteristic
57C2×C5×C4⋊C4C2×D311yescharacteristic
58C5×#106—62no—
59C5×#103—62no—
60C5×C22⋊C8—32no—
61C5×#95—21no—
62C5×#95C2×D311yescharacteristic
63#1611—31noSylow
64C23×C3×C5C2×D411yescharacteristic
65C2×C3×C4×C5—21no—
66C2×C3×C4×C5—21no—
67C2×C3×C4×C5—22no—
68C2×C3×C4×C5—21no—
69C2×C3×C4×C5—41no—
70C2×C3×C4×C5—41no—
71C2×C3×C4×C5—41no—
72C2×C3×C4×C5—41no—
73C2×C3×C4×C5—41no—
74C2×C3×C4×C5—41no—
75C2×C3×C4×C5—41no—
76C2×C3×C4×C5SD1612yes—
77C2×C3×C4×C5C2×D412yes—
78C2×C5×Dic3—22no—
79C2×C5×Dic3—22no—
80C5×Dic6—42no—
81C5×Dic6—81no—
82C5×C3⋊C8—42no—
83C2×C3×C4⋊C4—21no—
84C2×C3×C4⋊C4C22×C511yescharacteristic
85C2×C3×C4⋊C4C22×C511yescharacteristic
86#926—21no—
87#924—21no—
88#924C22×C512yes—
89#890—22no—
90#888—22no—
91C3×#95—21no—
92C3×#95C22×C511yescharacteristic
93#830C22×C512yes—
94#810—22no—
95#808—22no—
96C2×C5×Q8—32no—
97C2×C5×Q8—61no—
98C22×C4×C5—21no—
99C22×C4×C5—21no—
100C22×C4×C5#6711yescharacteristic
101C22×C4×C5#6711yescharacteristic
102C22×C4×C5C22×D311yescharacteristic
103C2×C5×C8—62no—
104C5×C4⋊C4—22no—
105C5×C4⋊C4—22no—
106C5×C4⋊C4—32no—
107C5×C4⋊C4—42no—
108C5×C4⋊C4—61no—
109C5×C4⋊C4—61no—
110C5×C4⋊C4—61no—
111C5×C4⋊C4—62no—
112C5×C4⋊C4—62no—
113C5×C22⋊C4—62no—
114C42×C5—62no—
115#544—31no—
116#483—32no—
117#398C5×D311yescharacteristic
118#328—32no—
119#324—31no—
120C22×C3×C5—21no—
121C22×C3×C5—21no—
122C22×C3×C5#12011yescharacteristic
123C22×C3×C5C2×SD1611yescharacteristic
124C22×C3×C5#13711yescharacteristic
125C3×C4×C5—22no—
126C3×C4×C5—22no—
127C3×C4×C5—42no—
128C3×C4×C5—41no—
129C3×C4×C5—81no—
130C5×Dic3—42no—
131C5×Dic3—42no—
132C22×C3×C4—21no—
133C22×C3×C4—21no—
134C22×C3×C4C5×D411yescharacteristic
135C22×C3×C4C5×D411yescharacteristic
136C22×C3×C4C23×C511yescharacteristic
137C2×Dic6—21no—
138C2×Dic6C5×D412yes—
139C3×C4⋊C4—22no—
140C3×C4⋊C4—22no—
141C3×C4⋊C4—42no—
142#217—22no—
143C4⋊Dic3—21no—
144C4⋊Dic3—21no—
145C4⋊Dic3—21no—
146C4⋊Dic3C5×D412yes—
147Dic3⋊C4—22no—
148Dic3⋊C4—22no—
149C4×Dic3—22no—
150C2×C3⋊C8—22no—
151C23×C5C2×#6711yescharacteristic
152C5×Q8—122no—
153C5×Q8—241no—
154C2×C4×C5—21no—
155C2×C4×C5—21no—
156C2×C4×C5—22no—
157C2×C4×C5—21no—
158C2×C4×C5—41no—
159C2×C4×C5—41no—
160C2×C4×C5—41no—
161C2×C4×C5—41no—
162C2×C4×C5—41no—
163C2×C4×C5—41no—
164C2×C4×C5—41no—
165C2×C4×C5—62no—
166C2×C4×C5—62no—
167C2×C4×C5#21412yes—
168C2×C4×C5D3×D412yes—
169C5×C8—122no—
170C4⋊Q8—62no—
171#125—62no—
172C22⋊Q8—32no—
173C22⋊Q8—61no—
174#117—61no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4C2×C5×D311yescharacteristic
177C2×C4⋊C4C2×C5×D311yescharacteristic
178#106—62no—
179#103—62no—
180C22⋊C8—32no—
181#95—21no—
182#95C2×C5×D311yescharacteristic
183C2×C3×C5—21no—
184C2×C3×C5—21no—
185C2×C3×C5#44811yescharacteristic
186C2×C3×C5#45611yescharacteristic
187C2×C3×C5#45811yescharacteristic
188C23×C3C2×C5×D411yescharacteristic
189C2×C3×C4—21no—
190C2×C3×C4—21no—
191C2×C3×C4—22no—
192C2×C3×C4—21no—
193C2×C3×C4—41no—
194C2×C3×C4—41no—
195C2×C3×C4—41no—
196C2×C3×C4—41no—
197C2×C3×C4—41no—
198C2×C3×C4—41no—
199C2×C3×C4—41no—
200C2×C3×C4C5×SD1612yes—
201C2×C3×C4C2×C5×D412yes—
202C2×Dic3—22no—
203C2×Dic3—22no—
204Dic6—42no—
205Dic6—81no—
206C3⋊C8—42no—
207C22×C5—21no—
208C22×C5—21no—
209C22×C5C2×#21411yescharacteristic
210C22×C5#93711yescharacteristic
211C22×C5#95111yescharacteristic
212C4×C5—22no—
213C4×C5—22no—
214C4×C5—42no—
215C4×C5—41no—
216C4×C5—81no—
217C4×C5—122no—
218C4×C5—122no—
219C2×Q8—32no—
220C2×Q8—61no—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4C5×#6711yescharacteristic
224C22×C4C5×#6711yescharacteristic
225C22×C4C22×C5×D311yescharacteristic
226C2×C8—62no—
227C4⋊C4—22no—
228C4⋊C4—22no—
229C4⋊C4—32no—
230C4⋊C4—42no—
231C4⋊C4—61no—
232C4⋊C4—61no—
233C4⋊C4—61no—
234C4⋊C4—62no—
235C4⋊C4—62no—
236C22⋊C4—62no—
237C42—62no—
238C3×C5#161111yescharacteristic, complemented, Hall
239C22×C3—21no—
240C22×C3—21no—
241C22×C3C5×#12011yescharacteristic
242C22×C3C2×C5×SD1611yescharacteristic
243C22×C3C5×#13711yescharacteristic
244C3×C4—22no—
245C3×C4—22no—
246C3×C4—42no—
247C3×C4—41no—
248C3×C4—81no—
249Dic3—42no—
250Dic3—42no—
251C2×C5—21no—
252C2×C5—21no—
253C2×C5#504011yescharacteristic
254C2×C5#505311yescharacteristic
255C2×C5#509511yescharacteristic
256C23C2×C5×#6711yescharacteristic
257Q8—122no—
258Q8—241no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—41no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—62no—
271C2×C4—62no—
272C2×C4C5×#21412yes—
273C2×C4C5×D3×D412yes—
274C8—122no—
275C2×C3—21no—
276C2×C3—21no—
277C2×C3C5×#44811yescharacteristic
278C2×C3C5×#45611yescharacteristic
279C2×C3C5×#45811yescharacteristic
280C5#6971311yescharacteristic, direct product factor, Sylow
281C22—21no—
282C22—21no—
283C22C2×C5×#21411yescharacteristic
284C22C5×#93711yescharacteristic
285C22C5×#95111yescharacteristic
286C4—22no—
287C4—22no—
288C4—42no—
289C4—41no—
290C4—81no—
291C4—122no—
292C4—122no—
293C3C5×#161111yescharacteristic, complemented, Sylow
294C2—21no—
295C2—21no—
296C2C5×#504011yescharacteristic
297C2C5×#505311yescharacteristic
298C2C5×#509511yescharacteristic
subgroupG/H
centerC22×C5#937
commutator, cosocleC22×C3×C4C23×C5
FrattiniC22×C4C22×C5×D3
FittingC3×C5×#398C2
socleC22×C3×C5#120
prime radicalC23×C3×C5C2×D4
hypercenterC5×#398D3
hypocenterC3C5×#1611
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
20. #69713C5direct (N × H)
238. C3×C5#1611semidirect
280. C5#69713direct (N × H)
293. C3C5×#1611semidirect

Maximal subgroups

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

subgroupindex
C5×#55002
C3×C5×#3982
C5×#49712
C5×#45282
C5×#44522
C5×#16113
#697135