Finite Group Databasebeta

← C5×#85322 #14511060

Subgroups — C5×#85322 #14511060

Order 1920.

424 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 84 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
1C2×C4×C5×SL(2,𝔽3)C211yesmaximal, characteristic
2C5×#5419C212yesmaximal
3C5×#5391C211yesmaximal, characteristic, complemented
4C5×C4⋊#226C211yesmaximal, characteristic, complemented
5C4×C5×#226C212yesmaximal, complemented
6C5×#3104—31nomaximal, Hall
7C22×C5×SL(2,𝔽3)C2211yescharacteristic
8C2×C5×#226C2212yes—
9C5×#926—41nomaximal
10C4×C5×SL(2,𝔽3)C2211yescharacteristic
11C4×C5×SL(2,𝔽3)C2211yescharacteristic
12C5×#859C2212yes—
13#85322C511yesmaximal, characteristic, direct product factor, Hall
14C5×#533—31no—
15C2×C4×C5×Q8D311yescharacteristic
16C5×#484—32no—
17C5×#477—31no—
18C5×#477—31no—
19C5×#462—31no—
20C5×#462—31no—
21C5×#451—32no—
22C4×C5×Q16—32no—
23C4×C5×Q16—32no—
24C5×#431—31no—
25C22×C3×C4×C5—41no—
26C2×C5×SL(2,𝔽3)—21no—
27C2×C5×SL(2,𝔽3)C2311yescharacteristic
28C5×#226—22no—
29C5×#217—42no—
30C5×C4⋊Dic3—41no—
31C5×C4⋊Dic3—41no—
32C4×C5×Dic3—42no—
33C2×C4×SL(2,𝔽3)C2×C511yescharacteristic
34#5419C2×C512yes—
35#5391C2×C511yescharacteristic, complemented
36C4⋊#226C2×C511yescharacteristic, complemented
37C4×#226C2×C512yescomplemented
38C22×C5×Q8C2×D311yescharacteristic
39C2×C5×Q16—32no—
40C2×C5×Q16—32no—
41C5×C4⋊Q8—31no—
42C5×C4⋊Q8—31no—
43C5×#125—61no—
44C5×C22⋊Q8—32no—
45C5×C22⋊Q8—61no—
46C4×C5×Q8—31no—
47C4×C5×Q8—31no—
48C4×C5×Q8—32no—
49C4×C5×Q8—61no—
50C4×C5×Q8C2×D311yescharacteristic
51C4×C5×Q8C2×D311yescharacteristic
52C5×#117—61no—
53C2×C5×C4⋊C4—31no—
54C2×C42×C5—31no—
55C5×#107—32no—
56C5×#107—32no—
57C5×C4⋊C8—31no—
58C5×C4⋊C8—31no—
59C5×#103—32no—
60C5×#103—32no—
61C5×#103—32no—
62C5×#103—32no—
63C5×C22⋊C8—32no—
64C4×C5×C8—32no—
65#3104—31noSylow
66C23×C3×C5—41no—
67C2×C3×C4×C5—44no—
68C2×C3×C4×C5—41no—
69C2×C3×C4×C5—41no—
70C2×C5×Dic3—42no—
71C2×C5×Dic3—42no—
72C5×SL(2,𝔽3)#4111yescharacteristic
73C22×SL(2,𝔽3)C22×C511yescharacteristic
74C2×#226C22×C512yes—
75#926—41no—
76C4×SL(2,𝔽3)C22×C511yescharacteristic
77C4×SL(2,𝔽3)C22×C511yescharacteristic
78#859C22×C512yes—
79C2×C5×Q8—21no—
80C2×C5×Q8—31no—
81C2×C5×Q8—32no—
82C2×C5×Q8—61no—
83C2×C5×Q8—61no—
84C2×C5×Q8C22×D311yescharacteristic
85C22×C4×C5—31no—
86C22×C4×C5—31no—
87C22×C4×C5S411yescharacteristic
88C5×Q16—62no—
89C5×Q16—64no—
90C5×Q16—62no—
91C2×C5×C8—32no—
92C2×C5×C8—32no—
93C5×C4⋊C4—31no—
94C5×C4⋊C4—31no—
95C5×C4⋊C4—32no—
96C5×C4⋊C4—61no—
97C5×C4⋊C4—62no—
98C5×C4⋊C4—62no—
99C5×C4⋊C4—61no—
100C5×C4⋊C4—61no—
101C5×C4⋊C4—61no—
102C5×C22⋊C4—62no—
103C42×C5—31no—
104C42×C5—31no—
105C42×C5—31no—
106C42×C5—31no—
107C42×C5—62no—
108#533—31no—
109C2×C4×Q8C5×D311yescharacteristic
110#484—32no—
111#477—31no—
112#477—31no—
113#462—31no—
114#462—31no—
115#451—32no—
116C4×Q16—32no—
117C4×Q16—32no—
118#431—31no—
119C22×C3×C5—42no—
120C22×C3×C5—41no—
121C22×C3×C5—81no—
122C22×C3×C5—81no—
123C3×C4×C5—42no—
124C3×C4×C5—42no—
125C5×Dic3—82no—
126C5×Dic3—82no—
127C22×C3×C4—41no—
128C2×SL(2,𝔽3)—21no—
129C2×SL(2,𝔽3)C23×C511yescharacteristic
130#226—22no—
131#217—42no—
132C4⋊Dic3—41no—
133C4⋊Dic3—41no—
134C4×Dic3—42no—
135C23×C5C2×S411yescharacteristic
136C5×Q8—31no—
137C5×Q8—32no—
138C5×Q8—61no—
139C5×Q8—62no—
140C5×Q8—62no—
141C5×Q8—122no—
142C5×Q8#23511yescharacteristic
143C2×C4×C5—31no—
144C2×C4×C5—31no—
145C2×C4×C5—31no—
146C2×C4×C5—31no—
147C2×C4×C5—61no—
148C2×C4×C5—61no—
149C2×C4×C5—62no—
150C2×C4×C5—62no—
151C2×C4×C5—121no—
152C2×C4×C5#22614yescomplemented
153C2×C4×C5C2×S411yescharacteristic
154C2×C4×C5C2×S411yescharacteristic
155C5×C8—62no—
156C5×C8—62no—
157C22×Q8C2×C5×D311yescharacteristic
158C2×Q16—32no—
159C2×Q16—32no—
160C4⋊Q8—31no—
161C4⋊Q8—31no—
162#125—61no—
163C22⋊Q8—32no—
164C22⋊Q8—61no—
165C4×Q8—31no—
166C4×Q8—31no—
167C4×Q8—32no—
168C4×Q8—61no—
169C4×Q8C2×C5×D311yescharacteristic
170C4×Q8C2×C5×D311yescharacteristic
171#117—61no—
172C2×C4⋊C4—31no—
173C2×C42—31no—
174#107—32no—
175#107—32no—
176C4⋊C8—31no—
177C4⋊C8—31no—
178#103—32no—
179#103—32no—
180#103—32no—
181#103—32no—
182C22⋊C8—32no—
183C4×C8—32no—
184C2×C3×C5—41no—
185C2×C3×C5—41no—
186C2×C3×C5—41no—
187C2×C3×C5—82no—
188C23×C3—41no—
189C2×C3×C4—44no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×Dic3—42no—
193C2×Dic3—42no—
194SL(2,𝔽3)C5×#4111yescharacteristic
195C22×C5—21no—
196C22×C5—21no—
197C22×C5C2×#22612yes—
198C22×C5C22×S411yescharacteristic
199C4×C5—31no—
200C4×C5—31no—
201C4×C5—61no—
202C4×C5—61no—
203C4×C5—61no—
204C4×C5—122no—
205C4×C5—122no—
206C4×C5C2×#22612yes—
207C4×C5C2×#22612yes—
208C2×Q8—21no—
209C2×Q8—31no—
210C2×Q8—32no—
211C2×Q8—61no—
212C2×Q8—61no—
213C2×Q8C22×C5×D311yescharacteristic
214C22×C4—31no—
215C22×C4—31no—
216C22×C4C5×S411yescharacteristic
217Q16—62no—
218Q16—64no—
219Q16—62no—
220C2×C8—32no—
221C2×C8—32no—
222C4⋊C4—31no—
223C4⋊C4—31no—
224C4⋊C4—32no—
225C4⋊C4—61no—
226C4⋊C4—62no—
227C4⋊C4—62no—
228C4⋊C4—61no—
229C4⋊C4—61no—
230C4⋊C4—61no—
231C22⋊C4—62no—
232C42—31no—
233C42—31no—
234C42—31no—
235C42—31no—
236C42—62no—
237C3×C5—41noHall
238C22×C3—42no—
239C22×C3—41no—
240C22×C3—81no—
241C22×C3—81no—
242C3×C4—42no—
243C3×C4—42no—
244Dic3—82no—
245Dic3—82no—
246C2×C5—22no—
247C2×C5#591211yescharacteristic
248C2×C5C22×#22611yescharacteristic
249C2×C5#592311yescharacteristic
250C23C2×C5×S411yescharacteristic
251Q8—31no—
252Q8—32no—
253Q8—61no—
254Q8—62no—
255Q8—62no—
256Q8—122no—
257Q8C5×#23511yescharacteristic
258C2×C4—31no—
259C2×C4—31no—
260C2×C4—31no—
261C2×C4—31no—
262C2×C4—61no—
263C2×C4—61no—
264C2×C4—62no—
265C2×C4—62no—
266C2×C4—121no—
267C2×C4C5×#22614yescomplemented
268C2×C4C2×C5×S411yescharacteristic
269C2×C4C2×C5×S411yescharacteristic
270C8—62no—
271C8—62no—
272C2×C3—41no—
273C2×C3—41no—
274C2×C3—41no—
275C2×C3—82no—
276C5#8532211yescharacteristic, direct product factor, Sylow
277C22—21no—
278C22—21no—
279C22C2×C5×#22612yes—
280C22C22×C5×S411yescharacteristic
281C4—31no—
282C4—31no—
283C4—61no—
284C4—61no—
285C4—61no—
286C4—122no—
287C4—122no—
288C4C2×C5×#22612yes—
289C4C2×C5×#22612yes—
290C3—41noSylow
291C2—22no—
292C2C5×#591211yescharacteristic
293C2C22×C5×#22611yescharacteristic
294C2C5×#592311yescharacteristic
subgroupG/H
centerC2×C4×C5C2×S4
commutator, cosocleC2×SL(2,𝔽3)C23×C5
FrattiniC22C22×C5×S4
FittingC2×C4×C5×Q8D3
socleC22×C5C22×S4
prime radicalC22×C5×SL(2,𝔽3)C22
hypercenterC22×C4×C5S4
hypocenterSL(2,𝔽3)C5×#41
LCSG ▷ C2×SL(2,𝔽3) ▷ SL(2,𝔽3)
UCSC1 ◁ C2×C4×C5 ◁ C22×C4×C5

Complements

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

NH ≅ G/Ntype
3. C5×#5391C2semidirect
4. C5×C4⋊#226C2semidirect
5. C4×C5×#226C2semidirect
13. #85322C5direct (N × H)
35. #5391C2×C5semidirect
36. C4⋊#226C2×C5semidirect
37. C4×#226C2×C5semidirect
152. C2×C4×C5#226semidirect
267. C2×C4C5×#226semidirect
276. C5#85322direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4×C5×SL(2,𝔽3)2
C5×#54192
C5×#53912
C5×C4⋊#2262
C4×C5×#2262
C5×#31043
C5×#9264
#853225