Finite Group Databasebeta

← D5×#5391 #14512808

Subgroups — D5×#5391 #14512808

Order 1920.

470 conjugacy classes of subgroups; only the 71 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 71 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
1C4×D5×SL(2,𝔽3)C211yesmaximal, characteristic
2#11752110C211yesmaximal, characteristic, complemented
3C5×#5391C211yesmaximal, characteristic, complemented
4D5×#859C212yesmaximal
5#11751598C212yesmaximal, complemented
6D5×#477—31nomaximal, Hall
7C2×D5×SL(2,𝔽3)C2211yescharacteristic
8C4×C5×SL(2,𝔽3)C2211yescharacteristic
9D5×C4⋊Dic3—41nomaximal
10Dic5×SL(2,𝔽3)C2211yescharacteristic
11#91184C2212yes—
12C5×#859C2212yes—
13C2×#5391—51nomaximal, Hall
14D5×C4⋊Q8—31no—
15C4×Q8×D5D311yescharacteristic
16C5×#477—31no—
17#65380—31no—
18#65310—31no—
19D5×#107—32no—
20#65164—32no—
21#65139—31no—
22D5×C4⋊C8—31no—
23D5×#103—32no—
24#65083—32no—
25C2×C3×C4×D5—41no—
26C2×C5×SL(2,𝔽3)C2311yescharacteristic
27C2×D5×Dic3—42no—
28D5×SL(2,𝔽3)Q812yescomplemented
29C4⋊Dic15—41no—
30C5×C4⋊Dic3—41no—
31#6668—42no—
32C2×C4×SL(2,𝔽3)—51no—
33C2×#859—52no—
34#5391—51no—
35#5391—52no—
36#5391D511yescharacteristic, direct product factor
37C2×Q8×D5—61no—
38C2×Q8×D5C2×D311yescharacteristic
39C5×C4⋊Q8—31no—
40C4×C5×Q8C2×D311yescharacteristic
41Q8×Dic5C2×D311yescharacteristic
42#4090—61no—
43C2×C8×D5—32no—
44D5×C4⋊C4—31no—
45D5×C4⋊C4—32no—
46D5×C4⋊C4—61no—
47#4034—32no—
48#4034—61no—
49#4033—31no—
50C42×D5—31no—
51#4015—31no—
52C4×Dic10—31no—
53C5×#107—32no—
54C5×C4⋊C8—31no—
55C5×#103—32no—
56#3967—32no—
57#3950—32no—
58#3948—32no—
59#3946—32no—
60#3942—32no—
61#3939—32no—
62#3939—32no—
63C4⋊#168—31no—
64C2×#477—151noSylow
65C22×C3×D5—41no—
66C2×C3×C4×C5—41no—
67C2×Dic15—42no—
68C2×C5×Dic3—42no—
69C2×C3×Dic5—41no—
70C3×C4×D5—44no—
71C5×SL(2,𝔽3)C2×Q811yescharacteristic, complemented
72D5×Dic3—84no—
73C22×SL(2,𝔽3)—51no—
74C2×C4⋊Dic3—201no—
75C4×SL(2,𝔽3)—51no—
76C4×SL(2,𝔽3)C2×D511yescharacteristic
77#859—52no—
78#859C2×D512yes—
79C2×C5×Q8—61no—
80C2×C5×Q8C22×D311yescharacteristic
81Q8×D5—32no—
82Q8×D5—124no—
83Q8×D5Dic612yes—
84C2×C4×D5—31no—
85C2×C4×D5—31no—
86C2×C4×D5—62no—
87C2×C4×D5S411yescharacteristic
88C2×Dic10—31no—
89C2×Dic10—62no—
90C2×Dic10—61no—
91C2×C5×C8—32no—
92C5×C4⋊C4—31no—
93C5×C4⋊C4—32no—
94C5×C4⋊C4—61no—
95C42×C5—31no—
96C4⋊Dic5—31no—
97C4⋊Dic5—32no—
98C4⋊Dic5—61no—
99#683—61no—
100#683—62no—
101#683—62no—
102C4×Dic5—31no—
103C4×Dic5—31no—
104C4×Dic5—31no—
105C2×#168—32no—
106C8×D5—64no—
107C2×C4⋊Q8—151no—
108C2×C4×Q8—51no—
109#477—31no—
110#477—152no—
111#477—151no—
112#477—152no—
113#477—151no—
114#477—151no—
115C2×#107—152no—
116C2×C4⋊C8—151no—
117C2×#103—152no—
118C22×C3×C5—41no—
119C2×C3×D5—42no—
120C2×C3×D5—42no—
121C2×C3×D5—42no—
122C3×C4×C5—42no—
123Dic15—82no—
124C3×Dic5—42no—
125C5×Dic3—82no—
126C22×C3×C4—201no—
127C22×Dic3—202no—
128C2×SL(2,𝔽3)—52no—
129C2×SL(2,𝔽3)C22×D511yescharacteristic
130C4⋊Dic3—41no—
131C4⋊Dic3—202no—
132C4⋊Dic3—201no—
133C22×D5C2×S411yescharacteristic
134C5×Q8—31no—
135C5×Q8—122no—
136C5×Q8C2×Dic611yescharacteristic
137C2×C4×C5—31no—
138C2×C4×C5—31no—
139C2×C4×C5—62no—
140C2×C4×C5C2×S411yescharacteristic
141C2×Dic5—31no—
142C2×Dic5—31no—
143C2×Dic5—62no—
144C2×Dic5C2×S411yescharacteristic
145C4×D5—32no—
146C4×D5—32no—
147C4×D5—62no—
148C4×D5—124no—
149C4×D5#22614yes—
150Dic10—32no—
151Dic10—61no—
152Dic10—124no—
153Dic10—122no—
154C5×C8—62no—
155#168—62no—
156C22×Q8—51no—
157C22×Q8—301no—
158C22×C8—152no—
159C4⋊Q8—31no—
160C4⋊Q8—151no—
161C4⋊Q8—152no—
162C4⋊Q8—301no—
163C4⋊Q8—301no—
164C4×Q8—51no—
165C4×Q8—151no—
166C4×Q8—151no—
167C4×Q8D3×D511yescharacteristic
168C2×C4⋊C4—151no—
169C2×C4⋊C4—152no—
170C2×C4⋊C4—301no—
171C2×C42—151no—
172#107—32no—
173#107—152no—
174#107—152no—
175#107—152no—
176C4⋊C8—31no—
177C4⋊C8—151no—
178C4⋊C8—152no—
179#103—32no—
180#103—152no—
181#103—152no—
182#103—152no—
183C2×C3×C5—41no—
184C2×C3×C5—41no—
185C2×C3×C5—41no—
186C3×D5—44no—
187C23×C3—201no—
188C2×C3×C4—41no—
189C2×C3×C4—204no—
190C2×C3×C4—201no—
191C2×Dic3—42no—
192C2×Dic3—202no—
193C2×Dic3—404no—
194SL(2,𝔽3)Q8×D511yescharacteristic, complemented
195C22×C5C22×S411yescharacteristic
196C2×D5A4⋊Q812yes—
197C2×D5C2×#22612yes—
198C2×D5#98612yes—
199C4×C5—31no—
200C4×C5—31no—
201C4×C5—61no—
202C4×C5—122no—
203C4×C5C2×#22612yes—
204Dic5—32no—
205Dic5—61no—
206Dic5—122no—
207Dic5C2×#22612yes—
208C2×Q8—52no—
209C2×Q8—61no—
210C2×Q8—151no—
211C2×Q8—152no—
212C2×Q8—301no—
213C2×Q8—302no—
214C2×Q8—604no—
215C2×Q8C2×D3×D511yescharacteristic
216C22×C4—51no—
217C22×C4—151no—
218C22×C4—151no—
219C22×C4—302no—
220C2×C8—32no—
221C2×C8—152no—
222C2×C8—304no—
223C4⋊C4—31no—
224C4⋊C4—32no—
225C4⋊C4—61no—
226C4⋊C4—151no—
227C4⋊C4—152no—
228C4⋊C4—301no—
229C4⋊C4—302no—
230C4⋊C4—302no—
231C4⋊C4—301no—
232C42—31no—
233C42—151no—
234C42—151no—
235C42—151no—
236C3×C5—41noHall
237C22×C3—41no—
238C22×C3—202no—
239C22×C3—202no—
240C22×C3—202no—
241C3×C4—42no—
242C3×C4—202no—
243Dic3—82no—
244Dic3—402no—
245C2×C5C2×A4⋊Q811yescharacteristic
246C2×C5C22×#22611yescharacteristic
247C2×C5C2×#98611yescharacteristic
248D5#539114yesdirect product factor
249C23—51no—
250Q8—31no—
251Q8—122no—
252Q8—152no—
253Q8—301no—
254Q8—602no—
255Q8—604no—
256Q8D5×Dic611yescharacteristic
257C2×C4—31no—
258C2×C4—31no—
259C2×C4—51no—
260C2×C4—54no—
261C2×C4—62no—
262C2×C4—152no—
263C2×C4—152no—
264C2×C4—151no—
265C2×C4—151no—
266C2×C4—302no—
267C2×C4—302no—
268C2×C4—604no—
269C2×C4D5×S411yescharacteristic
270C8—62no—
271C8—302no—
272C2×C3—41no—
273C2×C3—41no—
274C2×C3—41no—
275C2×C3—204no—
276C5C2×#539111yescharacteristic, complemented, Sylow
277C22—52no—
278C22—52no—
279C22—52no—
280C22C2×D5×S411yescharacteristic
281C4—31no—
282C4—31no—
283C4—52no—
284C4—61no—
285C4—122no—
286C4—152no—
287C4—301no—
288C4—602no—
289C4D5×#22612yes—
290C3—41noSylow
291C2—54no—
292C2D5×A4⋊Q811yescharacteristic
293C2C2×D5×#22611yescharacteristic
294C2D5×#98611yescharacteristic
subgroupG/H
center, FrattiniC22C2×D5×S4
commutator, cosocleC2×C5×SL(2,𝔽3)C23
FittingC4×C5×Q8C2×D3
socleC22×C5C22×S4
prime radicalC2×D5×SL(2,𝔽3)C22
hypercenterC2×C4D5×S4
hypocenterC5×SL(2,𝔽3)C2×Q8
LCSG ▷ C2×C5×SL(2,𝔽3) ▷ C5×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11752110C2semidirect
3. C5×#5391C2semidirect
5. #11751598C2semidirect
28. D5×SL(2,𝔽3)Q8semidirect
36. #5391D5direct (N × H)
71. C5×SL(2,𝔽3)C2×Q8semidirect
194. SL(2,𝔽3)Q8×D5semidirect
248. D5#5391direct (N × H)
276. C5C2×#5391semidirect

Maximal subgroups

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

subgroupindex
C4×D5×SL(2,𝔽3)2
#117521102
C5×#53912
D5×#8592
#117515982
D5×#4773
D5×C4⋊Dic34
C2×#53915