Finite Group Databasebeta

← D20×#226 #14512813

Subgroups — D20×#226 #14512813

Order 1920.

550 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 63 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×D5×#226C212yesmaximal, complemented
2SL(2,𝔽3)×D20C211yesmaximal, characteristic
3C4⋊(C5⋊#226)C211yesmaximal, characteristic, complemented
4C4×C5×#226C211yesmaximal, characteristic, complemented
5#11751607C212yesmaximal, complemented
6Q16×D20—31nomaximal, Hall
7C2×D5×SL(2,𝔽3)C2212yes—
8C2×C5⋊#226C2212yes—
9C2×C5×#226C2211yescharacteristic
10D5×#226—24no—
11C4×C5×SL(2,𝔽3)C2211yescharacteristic
12Dic3×D20—41nomaximal
13C5×#859C2211yescharacteristic
14D4×#226—51nomaximal, Hall
15C2×D5×Q16—32no—
16Q8×D20—31no—
17Q8×D20D311yescharacteristic
18C4×C5×Q16—31no—
19#65320—31no—
20#65320—31no—
21#65176—32no—
22#65097—32no—
23#65097—32no—
24#64988—31no—
25C8×D20—31no—
26C2×C3×D20—41no—
27C2×C5×SL(2,𝔽3)C2311yescharacteristic
28C2×D5×Dic3—42no—
29D5×SL(2,𝔽3)—22no—
30C5⋊#226—22no—
31C5×#226D412yescomplemented
32C4⋊Dic15—41no—
33C4×C5×Dic3—41no—
34#6665—42no—
35C22×#226—52no—
36D4×SL(2,𝔽3)—51no—
37#5419—52no—
38C4⋊#226—51no—
39C4×#226D511yescharacteristic, complemented
40C2×Q8×D5—32no—
41C2×Q8×D5C2×D312yes—
42C2×C5×Q16—31no—
43C4×C5×Q8—31no—
44C4×C5×Q8C2×D311yescharacteristic
45C2×#689—32no—
46C2×#689—32no—
47D5×Q16—64no—
48D5×Q16—64no—
49C2×Dic20—32no—
50C2×Dic20—32no—
51C2×C8×D5—32no—
52#4043—32no—
53#4043—32no—
54#4043—62no—
55C4×D20—31no—
56C4×D20—61no—
57#4015—31no—
58#4015—31no—
59#4015—61no—
60C5×#107—31no—
61C5×#103—31no—
62C5×#103—31no—
63C4×C5×C8—31no—
64#3952—32no—
65#3942—32no—
66#3942—32no—
67C4⋊#168—31no—
68D4×Q16—151noSylow
69C22×C3×D5—42no—
70C2×C3×C4×C5—41no—
71C2×Dic15—42no—
72C2×C5×Dic3—41no—
73C2×C5×Dic3—41no—
74C3×D20—44no—
75C5×SL(2,𝔽3)C2×D411yescharacteristic
76D5×Dic3—84no—
77C22×SL(2,𝔽3)—52no—
78C2×#226—52no—
79C2×#226—104no—
80C2×#226C2×D511yescharacteristic
81D4×Dic3—201no—
82C4×SL(2,𝔽3)C2×D511yescharacteristic
83#859C2×D511yescharacteristic
84C2×C5×Q8—31no—
85C2×C5×Q8C22×D311yescharacteristic
86Q8×D5—22no—
87Q8×D5—62no—
88Q8×D5—64no—
89Q8×D5—122no—
90C2×D20S411yescharacteristic
91C2×C4×D5—32no—
92C2×C4×D5—62no—
93C2×Dic10—32no—
94C2×Dic10—32no—
95C2×Dic10—62no—
96C5×Q16—32no—
97C5×Q16—32no—
98C2×C5×C8—31no—
99C2×C5×C8—31no—
100C5×C4⋊C4—31no—
101C5×C4⋊C4—31no—
102C5×C4⋊C4—61no—
103C42×C5—31no—
104C42×C5—61no—
105#689—64no—
106#689—62no—
107#689—62no—
108#685—32no—
109#685—62no—
110C4⋊Dic5—31no—
111C4⋊Dic5—61no—
112C4⋊Dic5—62no—
113C4⋊Dic5—61no—
114C4⋊Dic5—61no—
115C2×#168—32no—
116Dic20—64no—
117Dic20—122no—
118C8×D5—64no—
119C22×Q16—152no—
120D4×Q8—51no—
121D4×Q8—151no—
122#494—151no—
123#467—152no—
124#462—151no—
125#462—151no—
126#451—152no—
127#451—152no—
128C4×Q16—31no—
129C8×D4—151no—
130C22×C3×C5—41no—
131C2×C3×D5—44no—
132C2×C3×D5—82no—
133C2×C3×D5—82no—
134C3×C4×C5—42no—
135Dic15—82no—
136C5×Dic3—42no—
137C5×Dic3—81no—
138C2×C3×D4—201no—
139C22×Dic3—202no—
140C2×SL(2,𝔽3)—102no—
141C2×SL(2,𝔽3)C22×D511yescharacteristic
142#226—102no—
143#226D2012yesdirect product factor
144#217—202no—
145C4⋊Dic3—201no—
146C4×Dic3—41no—
147C22×D5C2×S412yes—
148C5×Q8—31no—
149C5×Q8—32no—
150C5×Q8—61no—
151C5×Q8D3×D411yescharacteristic
152C2×C4×C5—31no—
153C2×C4×C5—31no—
154C2×C4×C5—61no—
155C2×C4×C5—61no—
156C2×C4×C5C2×S411yescharacteristic
157C2×Dic5—32no—
158C2×Dic5—62no—
159D20#22614yesdirect product factor
160C4×D5—62no—
161C4×D5—62no—
162C4×D5—124no—
163Dic10—62no—
164Dic10—62no—
165Dic10—62no—
166Dic10—62no—
167Dic10—124no—
168C5×C8—32no—
169C5×C8—61no—
170#168—62no—
171C22×Q8—52no—
172C22×Q8—152no—
173C2×Q16—31no—
174C2×Q16—152no—
175C2×Q16—152no—
176C2×Q16—152no—
177C2×Q16—152no—
178C2×Q16—304no—
179C2×Q16—304no—
180C22×C8—152no—
181C4⋊Q8—151no—
182C4⋊Q8—151no—
183C4⋊Q8—301no—
184C22⋊Q8—152no—
185C22⋊Q8—152no—
186C22⋊Q8—302no—
187C4×Q8—31no—
188C4×Q8D3×D511yescharacteristic
189C4×D4—151no—
190C4×D4—301no—
191#107—31no—
192C4⋊C8—151no—
193#103—31no—
194#103—31no—
195#103—152no—
196#103—152no—
197C22⋊C8—152no—
198C4×C8—31no—
199C2×C3×C5—41no—
200C2×C3×C5—41no—
201C2×C3×C5—41no—
202C3×D5—84no—
203C23×C3—202no—
204C3×D4—204no—
205C2×C3×C4—41no—
206C2×Dic3—41no—
207C2×Dic3—41no—
208C2×Dic3—202no—
209C2×Dic3—404no—
210SL(2,𝔽3)C2×D2011yescharacteristic
211C22×C5C22×S411yescharacteristic
212C2×D5—22no—
213C2×D5—22no—
214C2×D5C2×#22614yescomplemented
215C4×C5—31no—
216C4×C5—31no—
217C4×C5—62no—
218C4×C5—61no—
219C4×C5—121no—
220C4×C5C2×#22612yescomplemented
221Dic5—62no—
222Dic5—122no—
223C2×Q8—31no—
224C2×Q8—102no—
225C2×Q8—152no—
226C2×Q8—152no—
227C2×Q8—302no—
228C2×Q8—302no—
229C2×Q8—304no—
230C2×Q8—602no—
231C2×Q8C2×D3×D511yescharacteristic
232C2×D4—51no—
233C22×C4—152no—
234C22×C4—302no—
235Q16—32no—
236Q16—32no—
237Q16—304no—
238Q16—302no—
239Q16—304no—
240Q16—302no—
241Q16—602no—
242C2×C8—31no—
243C2×C8—31no—
244C2×C8—152no—
245C2×C8—304no—
246C4⋊C4—31no—
247C4⋊C4—31no—
248C4⋊C4—61no—
249C4⋊C4—151no—
250C4⋊C4—301no—
251C4⋊C4—301no—
252C4⋊C4—302no—
253C4⋊C4—301no—
254C22⋊C4—152no—
255C22⋊C4—302no—
256C42—31no—
257C42—61no—
258C3×C5—41noHall
259C22×C3—41no—
260C22×C3—204no—
261C22×C3—402no—
262C22×C3—402no—
263C3×C4—42no—
264Dic3—42no—
265Dic3—81no—
266Dic3—402no—
267C2×C5D4×S411yescharacteristic
268C2×C5C22×#22611yescharacteristic
269C2×C5#592711yescharacteristic
270D5—24no—
271C23—52no—
272Q8—31no—
273Q8—32no—
274Q8—61no—
275Q8—302no—
276Q8—302no—
277Q8—302no—
278Q8—302no—
279Q8—604no—
280Q8D3×D2011yescharacteristic
281D4—54no—
282C2×C4—31no—
283C2×C4—31no—
284C2×C4—61no—
285C2×C4—61no—
286C2×C4—152no—
287C2×C4—302no—
288C2×C4—302no—
289C2×C4—302no—
290C2×C4—604no—
291C2×C4D5×S411yescharacteristic
292C8—32no—
293C8—61no—
294C8—302no—
295C2×C3—41no—
296C2×C3—41no—
297C2×C3—41no—
298C2×C3—404no—
299C5D4×#22611yescharacteristic, complemented, Sylow
300C22—54no—
301C22—102no—
302C22—102no—
303C22C2×D5×S411yescharacteristic
304C4—31no—
305C4—31no—
306C4—62no—
307C4—61no—
308C4—121no—
309C4—302no—
310C4—602no—
311C4D5×#22612yescomplemented
312C3—41noSylow
313C2—104no—
314C2S4×D2011yescharacteristic
315C2C2×D5×#22611yescharacteristic
316C2#1175677811yescharacteristic
subgroupG/H
center, FrattiniC22C2×D5×S4
commutator, cosocleC2×C5×SL(2,𝔽3)C23
FittingC4×C5×Q8C2×D3
socleC22×C5C22×S4
prime radicalSL(2,𝔽3)×D20C2
hypercenterC2×C4D5×S4
hypocenterC5×SL(2,𝔽3)C2×D4
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
1. C2×D5×#226C2semidirect
3. C4⋊(C5⋊#226)C2semidirect
4. C4×C5×#226C2semidirect
5. #11751607C2semidirect
31. C5×#226D4semidirect
39. C4×#226D5semidirect
143. #226D20direct (N × H)
159. D20#226direct (N × H)
214. C2×D5C2×#226semidirect
220. C4×C5C2×#226semidirect
299. C5D4×#226semidirect
311. C4D5×#226semidirect

Maximal subgroups

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

subgroupindex
C2×D5×#2262
SL(2,𝔽3)×D202
C4⋊(C5⋊#226)2
C4×C5×#2262
#117516072
Q16×D203
Dic3×D204
D4×#2265