Finite Group Databasebeta

← C5×#68998 #14308950

Subgroups — C5×#68998 #14308950

Order 1920.

492 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 492 conjugacy classes of subgroups — too many to draw. Only its 134 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
2C5×#5120C212yesmaximal
3C5×#4973C211yesmaximal, characteristic
4C3×C5×#343C211yesmaximal, characteristic
5C5×#4555C212yesmaximal
6C5×#2028—31nomaximal, Hall
7C22×C5×Dic6C2211yescharacteristic
8C2×C3×C5×M16C2212yes—
9C2×C3×C5×C4⋊C4C2211yescharacteristic
10C5×#926C2212yes—
11C5×#924—22no—
12C2×C4×C5×Dic3C2211yescharacteristic
13C5×#890—22no—
14C5×#888—22no—
15C5×C3⋊#95—22no—
16C5×#816—22no—
17C5×#816—22no—
18#68998C511yesmaximal, characteristic, direct product factor, Hall
19C5×#544—31no—
20C5×#419—32no—
21C5×#391—31no—
22C5×#343D311yescharacteristic
23C5×#339—32no—
24C22×C3×C4×C5D412yes—
25C22×C3×C4×C5C2311yescharacteristic
26C22×C5×Dic3—21no—
27C22×C5×Dic3—21no—
28C2×C5×Dic6—22no—
29C2×C5×Dic6—22no—
30C2×C5×Dic6—21no—
31C2×C5×Dic6—22no—
32C2×C5×Dic6Q812yescomplemented
33C3×C5×M16—22no—
34C3×C5×M16—22no—
35C2×C3×C5×C8—22no—
36C3×C5×C4⋊C4—22no—
37C5×#217—22no—
38C5×C4⋊Dic3—22no—
39C5×C4⋊Dic3D412yes—
40C5×C4⋊Dic3D412yes—
41C5×Dic3⋊C4—24no—
42C4×C5×Dic3—22no—
43C4×C5×Dic3—22no—
44#5500C2×C511yescharacteristic
45#5120C2×C512yes—
46#4973C2×C511yescharacteristic
47C3×#343C2×C511yescharacteristic
48#4555C2×C512yes—
49C22×C5×Q8—31no—
50C2×C5×M16C2×D312yes—
51C5×C4⋊Q8—62no—
52C5×#125—62no—
53C5×C22⋊Q8—62no—
54C5×#117—32no—
55C2×C5×C4⋊C4C2×D311yescharacteristic
56C2×C42×C5—31no—
57C5×#103—62no—
58C5×#103—62no—
59C5×#95—62no—
60#2028—31noSylow
61C23×C3×C5C2×D411yescharacteristic
62C2×C3×C4×C5—22no—
63C2×C3×C4×C5—42no—
64C2×C3×C4×C5C2×D412yes—
65C2×C3×C4×C5C2×Q811yescharacteristic
66C2×C3×C4×C5#4112yes—
67C2×C3×C4×C5#4111yescharacteristic
68C2×C5×Dic3—22no—
69C2×C5×Dic3—22no—
70C2×C5×Dic3—22no—
71C2×C5×Dic3—22no—
72C2×C5×Dic3—41no—
73C2×C5×Dic3—41no—
74C2×C5×Dic3—41no—
75C2×C5×Dic3—41no—
76C5×Dic6—24no—
77C5×Dic6—42no—
78C5×Dic6—82no—
79C3×C5×C8—42no—
80C22×Dic6C22×C511yescharacteristic
81C2×C3×M16C22×C512yes—
82C2×C3×C4⋊C4C22×C511yescharacteristic
83#926C22×C512yes—
84#924—22no—
85C2×C4×Dic3C22×C511yescharacteristic
86#890—22no—
87#888—22no—
88C3⋊#95—22no—
89#816—22no—
90#816—22no—
91C2×C5×Q8—32no—
92C2×C5×Q8—62no—
93C2×C5×Q8—61no—
94C2×C5×Q8—62no—
95C2×C5×Q8—62no—
96C22×C4×C5—61no—
97C22×C4×C5—61no—
98C22×C4×C5D1212yes—
99C22×C4×C5C22×D311yescharacteristic
100C5×M16—22no—
101C5×M16—22no—
102C2×C5×C8—22no—
103C5×C4⋊C4—22no—
104C5×C4⋊C4—32no—
105C5×C4⋊C4—32no—
106C5×C4⋊C4—62no—
107C5×C4⋊C4—64no—
108C5×C22⋊C4—62no—
109C42×C5—62no—
110C42×C5—62no—
111#544—31no—
112#419—32no—
113#391—31no—
114#343C5×D311yescharacteristic
115#339—32no—
116C22×C3×C5—21no—
117C22×C3×C5—21no—
118C22×C3×C5#12011yescharacteristic
119C22×C3×C5C22⋊Q811yescharacteristic
120C22×C3×C5#12411yescharacteristic
121C3×C4×C5—42no—
122C3×C4×C5C22⋊Q812yes—
123C3×C4×C5#12412yes—
124C5×Dic3—42no—
125C5×Dic3—42no—
126C5×Dic3—42no—
127C5×Dic3—42no—
128C22×C3×C4C5×D412yes—
129C22×C3×C4C23×C511yescharacteristic
130C22×Dic3—21no—
131C22×Dic3—21no—
132C2×Dic6—22no—
133C2×Dic6—22no—
134C2×Dic6—21no—
135C2×Dic6—22no—
136C2×Dic6C5×Q812yescomplemented
137C3×M16—22no—
138C3×M16—22no—
139C2×C3×C8—22no—
140C3×C4⋊C4—22no—
141#217—22no—
142C4⋊Dic3—22no—
143C4⋊Dic3C5×D412yes—
144C4⋊Dic3C5×D412yes—
145Dic3⋊C4—24no—
146C4×Dic3—22no—
147C4×Dic3—22no—
148C23×C5C2×D1211yescharacteristic
149C5×Q8—64no—
150C5×Q8—122no—
151C5×Q8—242no—
152C2×C4×C5—22no—
153C2×C4×C5—42no—
154C2×C4×C5—62no—
155C2×C4×C5—62no—
156C2×C4×C5—62no—
157C2×C4×C5—62no—
158C2×C4×C5—121no—
159C2×C4×C5—121no—
160C2×C4×C5—121no—
161C2×C4×C5—121no—
162C2×C4×C5#23512yes—
163C2×C4×C5D3×D412yes—
164C2×C4×C5#23711yescharacteristic
165C2×C4×C5D3×Q811yescharacteristic
166C5×C8—42no—
167C22×Q8—31no—
168C2×M16C2×C5×D312yes—
169C4⋊Q8—62no—
170#125—62no—
171C22⋊Q8—62no—
172#117—32no—
173C2×C4⋊C4C2×C5×D311yescharacteristic
174C2×C42—31no—
175#103—62no—
176#103—62no—
177#95—62no—
178C2×C3×C5—21no—
179C2×C3×C5—21no—
180C2×C3×C5#39311yescharacteristic
181C2×C3×C5#45511yescharacteristic
182C2×C3×C5#45611yescharacteristic
183C23×C3C2×C5×D411yescharacteristic
184C2×C3×C4—22no—
185C2×C3×C4—42no—
186C2×C3×C4C2×C5×D412yes—
187C2×C3×C4C2×C5×Q811yescharacteristic
188C2×C3×C4C5×#4112yes—
189C2×C3×C4C5×#4111yescharacteristic
190C2×Dic3—22no—
191C2×Dic3—22no—
192C2×Dic3—22no—
193C2×Dic3—22no—
194C2×Dic3—41no—
195C2×Dic3—41no—
196C2×Dic3—41no—
197C2×Dic3—41no—
198Dic6—24no—
199Dic6—42no—
200Dic6—82no—
201C3×C8—42no—
202C22×C5—21no—
203C22×C5—21no—
204C22×C5#87511yescharacteristic
205C22×C5#88211yescharacteristic
206C22×C5#89711yescharacteristic
207C4×C5—42no—
208C4×C5—122no—
209C4×C5—122no—
210C4×C5—122no—
211C4×C5—122no—
212C4×C5#88512yes—
213C4×C5#89612yes—
214C2×Q8—32no—
215C2×Q8—62no—
216C2×Q8—61no—
217C2×Q8—62no—
218C2×Q8—62no—
219C22×C4—61no—
220C22×C4—61no—
221C22×C4C5×D1212yes—
222C22×C4C22×C5×D311yescharacteristic
223M16—22no—
224M16—22no—
225C2×C8—22no—
226C4⋊C4—22no—
227C4⋊C4—32no—
228C4⋊C4—32no—
229C4⋊C4—62no—
230C4⋊C4—64no—
231C22⋊C4—62no—
232C42—62no—
233C42—62no—
234C3×C5#202811yescharacteristic, complemented, Hall
235C22×C3—21no—
236C22×C3—21no—
237C22×C3C5×#12011yescharacteristic
238C22×C3C5×C22⋊Q811yescharacteristic
239C22×C3C5×#12411yescharacteristic
240C3×C4—42no—
241C3×C4C5×C22⋊Q812yes—
242C3×C4C5×#12412yes—
243Dic3—42no—
244Dic3—42no—
245Dic3—42no—
246Dic3—42no—
247C2×C5—21no—
248C2×C5—21no—
249C2×C5#467311yescharacteristic
250C2×C5#474611yescharacteristic
251C2×C5#475211yescharacteristic
252C23C2×C5×D1211yescharacteristic
253Q8—64no—
254Q8—122no—
255Q8—242no—
256C2×C4—22no—
257C2×C4—42no—
258C2×C4—62no—
259C2×C4—62no—
260C2×C4—62no—
261C2×C4—62no—
262C2×C4—121no—
263C2×C4—121no—
264C2×C4—121no—
265C2×C4—121no—
266C2×C4C5×#23512yes—
267C2×C4C5×D3×D412yes—
268C2×C4C5×#23711yescharacteristic
269C2×C4C5×D3×Q811yescharacteristic
270C8—42no—
271C2×C3—21no—
272C2×C3—21no—
273C2×C3C5×#39311yescharacteristic
274C2×C3C5×#45511yescharacteristic
275C2×C3C5×#45611yescharacteristic
276C5#6899811yescharacteristic, direct product factor, Sylow
277C22—21no—
278C22—21no—
279C22C5×#87511yescharacteristic
280C22C5×#88211yescharacteristic
281C22C5×#89711yescharacteristic
282C4—42no—
283C4—122no—
284C4—122no—
285C4—122no—
286C4—122no—
287C4C5×#88512yes—
288C4C5×#89612yes—
289C3C5×#202811yescharacteristic, complemented, Sylow
290C2—21no—
291C2—21no—
292C2C5×#467311yescharacteristic
293C2C5×#474611yescharacteristic
294C2C5×#475211yescharacteristic
subgroupG/H
centerC22×C5#882
commutator, cosocleC22×C3×C4C23×C5
FrattiniC22×C4C22×C5×D3
FittingC3×C5×#343C2
socleC22×C3×C5#120
prime radicalC23×C3×C5C2×D4
hypercenterC5×#343D3
hypocenterC3C5×#2028
LCSG ▷ C22×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C5×#343

Complements

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

NH ≅ G/Ntype
18. #68998C5direct (N × H)
32. C2×C5×Dic6Q8semidirect
136. C2×Dic6C5×Q8semidirect
234. C3×C5#2028semidirect
276. C5#68998direct (N × H)
289. C3C5×#2028semidirect

Maximal subgroups

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

subgroupindex
C5×#55002
C5×#51202
C5×#49732
C3×C5×#3432
C5×#45552
C5×#20283
#689985