Finite Group Databasebeta

← C2×D5×#859 #14513334

Subgroups — C2×D5×#859 #14513334

Order 1920.

878 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 878 conjugacy classes of subgroups — too many to draw. Only its 139 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
1C22×D5×SL(2,𝔽3)C211yesmaximal, characteristic
2C2×#91184C211yesmaximal, characteristic, complemented
3C2×C5×#859C211yesmaximal, characteristic, complemented
4D5×#859C214yesmaximal, direct product factor
5C2×D5×#103—31nomaximal, Hall
6C22×C5×SL(2,𝔽3)C2211yescharacteristic
7C22×D5×Dic3—41nomaximal
8C2×D5×SL(2,𝔽3)C414yescomplemented
9C2×D5×SL(2,𝔽3)C2212yes—
10#91184C2212yescomplemented
11C5×#859C2212yescomplemented
12C22×#859—51nomaximal, Hall
13C22×Q8×D5D311yescharacteristic
14C22×C8×D5—31no—
15C2×D5×C4⋊C4—31no—
16C2×C5×#103—31no—
17C2×#3967—31no—
18C2×#3948—31no—
19C2×#3942—31no—
20D5×#103—34no—
21D5×#103—34no—
22C23×C3×D5—41no—
23C22×Dic15—41no—
24C22×C5×Dic3—41no—
25C2×C5×SL(2,𝔽3)C2×C412yescomplemented
26C2×C5×SL(2,𝔽3)C2311yescharacteristic
27C2×D5×Dic3—44no—
28C2×D5×Dic3—48no—
29D5×SL(2,𝔽3)C2×C414yescomplemented
30C23×SL(2,𝔽3)—51no—
31C2×#859—51no—
32C2×#859—54no—
33C2×#859D511yescharacteristic, direct product factor
34C22×C5×Q8C2×D311yescharacteristic
35C2×Q8×D5—32no—
36C2×Q8×D5—34no—
37C2×Q8×D5Dic314yescomplemented
38C2×Q8×D5C2×D312yes—
39C22×C4×D5—31no—
40C22×C4×D5—61no—
41C22×Dic10—31no—
42C22×C5×C8—31no—
43C2×C5×C4⋊C4—31no—
44C2×C4⋊Dic5—31no—
45C2×#683—61no—
46C22×#168—31no—
47C2×C8×D5—34no—
48C2×C8×D5—64no—
49D5×C4⋊C4—34no—
50D5×C4⋊C4—62no—
51C5×#103—32no—
52C5×#103—32no—
53#3967—32no—
54#3967—32no—
55#3948—34no—
56#3942—34no—
57C22×#103—151noSylow
58C23×C3×C5—41no—
59C22×C3×D5—44no—
60C22×C3×D5—44no—
61C22×C3×D5—44no—
62C22×C3×D5—42no—
63C2×Dic15—42no—
64C2×Dic15—44no—
65C2×C5×Dic3—42no—
66C2×C5×Dic3—44no—
67C5×SL(2,𝔽3)C22×C411yescharacteristic, complemented
68D5×Dic3—416no—
69C23×Dic3—201no—
70C22×SL(2,𝔽3)—54no—
71C22×SL(2,𝔽3)—52no—
72C22×SL(2,𝔽3)C2×D511yescharacteristic
73#859—52no—
74#859C2×D512yesdirect product factor
75C23×D5S411yescharacteristic
76C2×C5×Q8—32no—
77C2×C5×Q8—31no—
78C2×C5×Q8C2×Dic312yescomplemented
79C2×C5×Q8C22×D311yescharacteristic
80C22×C4×C5—31no—
81C22×C4×C5—61no—
82C22×Dic5—31no—
83C22×Dic5—61no—
84Q8×D5—38no—
85Q8×D5—34no—
86Q8×D5—64no—
87Q8×D5C2×Dic314yescomplemented
88C2×C4×D5—34no—
89C2×C4×D5—34no—
90C2×C4×D5—32no—
91C2×C4×D5—32no—
92C2×C4×D5—64no—
93C2×C4×D5—124no—
94C2×Dic10—34no—
95C2×Dic10—32no—
96C2×Dic10—62no—
97C2×Dic10—61no—
98C2×C5×C8—32no—
99C2×C5×C8—62no—
100C5×C4⋊C4—32no—
101C5×C4⋊C4—61no—
102C4⋊Dic5—32no—
103C4⋊Dic5—61no—
104#683—64no—
105C2×#168—32no—
106C2×#168—62no—
107C8×D5—68no—
108C23×Q8—51no—
109C23×C8—151no—
110C22×C4⋊C4—151no—
111C2×#103—31no—
112C2×#103—154no—
113C2×#103—154no—
114C2×#103—151no—
115C2×#103—151no—
116C2×#103—151no—
117C22×C3×C5—42no—
118C22×C3×C5—42no—
119C22×C3×C5—42no—
120C22×C3×C5—41no—
121C2×C3×D5—416no—
122C2×C3×D5—44no—
123C2×C3×D5—44no—
124C2×C3×D5—44no—
125Dic15—44no—
126C5×Dic3—44no—
127C24×C3—201no—
128C22×Dic3—41no—
129C22×Dic3—204no—
130C22×Dic3—201no—
131C22×Dic3—208no—
132C2×SL(2,𝔽3)—54no—
133C2×SL(2,𝔽3)C4×D512yescomplemented
134C2×SL(2,𝔽3)C22×D511yescharacteristic
135C23×C5C2×S411yescharacteristic
136C22×D5#22614yes—
137C22×D5GL(2,𝔽3)14yes—
138C22×D5SL(2,ℤ/4)14yes—
139C22×D5C2×S412yes—
140C5×Q8—32no—
141C5×Q8—31no—
142C5×Q8—61no—
143C5×Q8C22×Dic311yescharacteristic, complemented
144C2×C4×C5—32no—
145C2×C4×C5—32no—
146C2×C4×C5—31no—
147C2×C4×C5—31no—
148C2×C4×C5—62no—
149C2×C4×C5—122no—
150C2×Dic5—34no—
151C2×Dic5—32no—
152C2×Dic5—62no—
153C2×Dic5—122no—
154C4×D5—34no—
155C4×D5—34no—
156C4×D5—38no—
157C4×D5—128no—
158Dic10—34no—
159Dic10—62no—
160Dic10—64no—
161C5×C8—62no—
162#168—62no—
163C22×Q8—54no—
164C22×Q8—52no—
165C22×Q8—151no—
166C22×Q8—154no—
167C22×Q8—152no—
168C22×Q8D3×D511yescharacteristic
169C23×C4—151no—
170C23×C4—301no—
171C22×C8—31no—
172C22×C8—154no—
173C22×C8—151no—
174C22×C8—304no—
175C2×C4⋊C4—31no—
176C2×C4⋊C4—154no—
177C2×C4⋊C4—151no—
178C2×C4⋊C4—302no—
179C2×C4⋊C4—301no—
180#103—32no—
181#103—32no—
182#103—154no—
183#103—154no—
184#103—152no—
185#103—152no—
186C2×C3×C5—41no—
187C2×C3×C5—41no—
188C2×C3×C5—41no—
189C2×C3×C5—44no—
190C3×D5—48no—
191C23×C3—41no—
192C23×C3—202no—
193C23×C3—204no—
194C23×C3—204no—
195C23×C3—204no—
196C2×Dic3—42no—
197C2×Dic3—44no—
198C2×Dic3—202no—
199C2×Dic3—2016no—
200C2×Dic3—204no—
201SL(2,𝔽3)C2×C4×D511yescharacteristic, complemented
202C22×C5C2×#22612yes—
203C22×C5C2×GL(2,𝔽3)12yes—
204C22×C5C2×SL(2,ℤ/4)12yes—
205C22×C5C22×S411yescharacteristic
206C2×D5#859116yesdirect product factor
207C2×D5C2×#22614yes—
208C2×D5C2×GL(2,𝔽3)14yes—
209C2×D5C2×SL(2,ℤ/4)14yes—
210C4×C5—31no—
211C4×C5—31no—
212C4×C5—32no—
213C4×C5—122no—
214Dic5—34no—
215Dic5—122no—
216C24—51no—
217C2×Q8—31no—
218C2×Q8—32no—
219C2×Q8—54no—
220C2×Q8—152no—
221C2×Q8—154no—
222C2×Q8—154no—
223C2×Q8—158no—
224C2×Q8—301no—
225C2×Q8—302no—
226C2×Q8—304no—
227C2×Q8D5×Dic312yescomplemented
228C2×Q8C2×D3×D511yescharacteristic
229C22×C4—31no—
230C22×C4—61no—
231C22×C4—152no—
232C22×C4—152no—
233C22×C4—154no—
234C22×C4—154no—
235C22×C4—151no—
236C22×C4—304no—
237C22×C4—301no—
238C22×C4—604no—
239C2×C8—32no—
240C2×C8—62no—
241C2×C8—152no—
242C2×C8—308no—
243C2×C8—302no—
244C4⋊C4—32no—
245C4⋊C4—61no—
246C4⋊C4—152no—
247C4⋊C4—301no—
248C4⋊C4—304no—
249C3×C5—41noHall
250C22×C3—41no—
251C22×C3—42no—
252C22×C3—42no—
253C22×C3—42no—
254C22×C3—204no—
255C22×C3—2016no—
256C22×C3—204no—
257C22×C3—204no—
258Dic3—44no—
259Dic3—204no—
260C2×C5C2×#85914yescomplemented
261C2×C5C22×#22611yescharacteristic
262C2×C5C22×GL(2,𝔽3)11yescharacteristic
263C2×C5C22×SL(2,ℤ/4)11yescharacteristic
264D5C2×#85918yesdirect product factor
265C23—52no—
266C23—54no—
267C23—54no—
268C23—54no—
269C23D5×S411yescharacteristic
270Q8—32no—
271Q8—31no—
272Q8—61no—
273Q8—154no—
274Q8—302no—
275Q8—304no—
276Q8C2×D5×Dic311yescharacteristic, complemented
277C2×C4—31no—
278C2×C4—31no—
279C2×C4—32no—
280C2×C4—32no—
281C2×C4—62no—
282C2×C4—122no—
283C2×C4—154no—
284C2×C4—154no—
285C2×C4—158no—
286C2×C4—152no—
287C2×C4—154no—
288C2×C4—302no—
289C2×C4—608no—
290C2×C4—602no—
291C8—62no—
292C8—302no—
293C2×C3—44no—
294C2×C3—41no—
295C2×C3—41no—
296C2×C3—41no—
297C2×C3—208no—
298C5C22×#85911yescharacteristic, complemented, Sylow
299C22—516no—
300C22—54no—
301C22—54no—
302C22—54no—
303C22D5×#22612yes—
304C22D5×GL(2,𝔽3)12yes—
305C22D5×SL(2,ℤ/4)12yes—
306C22C2×D5×S411yescharacteristic
307C4—31no—
308C4—31no—
309C4—32no—
310C4—122no—
311C4—154no—
312C4—602no—
313C3—41noSylow
314C2—58no—
315C2D5×#85914yesdirect product factor
316C2C2×D5×#22611yescharacteristic
317C2C2×D5×GL(2,𝔽3)11yescharacteristic
318C2C2×D5×SL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, hypercenterC23D5×S4
commutator, hypocenterC5×SL(2,𝔽3)C22×C4
FrattiniC22C2×D5×S4
FittingC22×C5×Q8C2×D3
socleC23×C5C2×S4
prime radicalC22×D5×SL(2,𝔽3)C2
cosocleC2×C5×SL(2,𝔽3)C23
LCSG ▷ C5×SL(2,𝔽3)
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×#91184C2semidirect
3. C2×C5×#859C2semidirect
4. D5×#859C2direct (N × H)
8. C2×D5×SL(2,𝔽3)C4semidirect
10. #91184C22semidirect
11. C5×#859C22semidirect
25. C2×C5×SL(2,𝔽3)C2×C4semidirect
29. D5×SL(2,𝔽3)C2×C4semidirect
33. C2×#859D5direct (N × H)
37. C2×Q8×D5Dic3semidirect
67. C5×SL(2,𝔽3)C22×C4semidirect
74. #859C2×D5direct (N × H)
78. C2×C5×Q8C2×Dic3semidirect
87. Q8×D5C2×Dic3semidirect
133. C2×SL(2,𝔽3)C4×D5semidirect
143. C5×Q8C22×Dic3semidirect
201. SL(2,𝔽3)C2×C4×D5semidirect
206. C2×D5#859direct (N × H)
227. C2×Q8D5×Dic3semidirect
260. C2×C5C2×#859semidirect
264. D5C2×#859direct (N × H)
276. Q8C2×D5×Dic3semidirect
298. C5C22×#859semidirect
315. C2D5×#859direct (N × H)

Maximal subgroups

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

subgroupindex
C22×D5×SL(2,𝔽3)2
C2×#911842
C2×C5×#8592
D5×#8592
C2×D5×#1033
C22×D5×Dic34
C22×#8595