Finite Group Databasebeta

← #14513721

Subgroups — #14513721

Order 1920.

418 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 61 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
2#11752138C212yesmaximal
3C4⋊(C5⋊#226)C211yesmaximal, characteristic, complemented
4#11752107C212yesmaximal, complemented
5#11752105C211yesmaximal, characteristic, complemented
6#10614179—31nomaximal, Hall
7C22×C5×SL(2,𝔽3)C2211yescharacteristic
8C2×C5⋊#226C2212yes—
9#91813—41nomaximal
10C4×C5×SL(2,𝔽3)C2211yescharacteristic
11C4×C5×SL(2,𝔽3)C2211yescharacteristic
12#91184C2212yes—
13#85319—51nomaximal, Hall
14C2×C4×C5×Q8D311yescharacteristic
15#65804—31no—
16#65517—32no—
17#65320—31no—
18#65320—31no—
19#65319—32no—
20#65319—32no—
21#65309—31no—
22#65309—31no—
23#65260—32no—
24#65215—31no—
25C22×C3×C4×C5—41no—
26C2×Dic30—41no—
27C2×C5×SL(2,𝔽3)—21no—
28C2×C5×SL(2,𝔽3)C2311yescharacteristic
29C5⋊#226—22no—
30#6719—42no—
31C4⋊Dic15—41no—
32#6712—42no—
33C2×C4×SL(2,𝔽3)D511yescharacteristic
34#5419—52no—
35C4⋊#226—51no—
36#5388—52no—
37#5386—51no—
38C22×C5×Q8C2×D311yescharacteristic
39C4×C5×Q8—31no—
40C4×C5×Q8—31no—
41C4×C5×Q8C2×D311yescharacteristic
42C4×C5×Q8C2×D311yescharacteristic
43C2×C5×C4⋊C4—31no—
44C2×C42×C5—31no—
45C2×#689—32no—
46C2×#689—32no—
47#4072—61no—
48#4070—32no—
49#4070—61no—
50#4016—61no—
51#4015—31no—
52#4015—31no—
53C4×Dic10—32no—
54C4×Dic10—61no—
55#3967—32no—
56#3967—32no—
57C22⋊#168—32no—
58#3942—32no—
59#3942—32no—
60#3940—32no—
61#3940—32no—
62C4⋊#168—31no—
63C4⋊#168—31no—
64#3935—32no—
65#3113—151noSylow
66C23×C3×C5—41no—
67C2×C3×C4×C5—41no—
68C2×C3×C4×C5—41no—
69C2×C3×C4×C5—82no—
70C2×Dic15—42no—
71C2×Dic15—42no—
72Dic30—82no—
73C5×SL(2,𝔽3)#4111yescharacteristic
74C22×SL(2,𝔽3)C2×D511yescharacteristic
75C2×#226—52no—
76#924—201no—
77C4×SL(2,𝔽3)C2×D511yescharacteristic
78C4×SL(2,𝔽3)C2×D511yescharacteristic
79#859—52no—
80C2×C5×Q8—21no—
81C2×C5×Q8—31no—
82C2×C5×Q8—61no—
83C2×C5×Q8C22×D311yescharacteristic
84C22×C4×C5—31no—
85C22×C4×C5—31no—
86C22×C4×C5S411yescharacteristic
87C2×Dic10—32no—
88C2×Dic10—61no—
89C5×C4⋊C4—31no—
90C5×C4⋊C4—31no—
91C5×C4⋊C4—61no—
92C42×C5—31no—
93C42×C5—31no—
94C42×C5—31no—
95C42×C5—31no—
96C22⋊Dic5—62no—
97#689—62no—
98#689—64no—
99#689—62no—
100C4⋊Dic5—32no—
101C4⋊Dic5—61no—
102C4⋊Dic5—61no—
103C4⋊Dic5—61no—
104#683—62no—
105#683—62no—
106C4×Dic5—62no—
107C2×#168—32no—
108C2×#168—32no—
109#533—151no—
110C2×C4×Q8D1511yescharacteristic
111#483—152no—
112#475—151no—
113#475—151no—
114#462—151no—
115#462—151no—
116#451—152no—
117#441—152no—
118#441—152no—
119#432—151no—
120C22×C3×C5—42no—
121C22×C3×C5—41no—
122C22×C3×C5—81no—
123C22×C3×C5—81no—
124C3×C4×C5—42no—
125C3×C4×C5—81no—
126Dic15—82no—
127Dic15—82no—
128C22×C3×C4—41no—
129C2×Dic6—201no—
130C2×SL(2,𝔽3)—21no—
131C2×SL(2,𝔽3)C22×D511yescharacteristic
132#226—102no—
133#217—202no—
134C4⋊Dic3—201no—
135Dic3⋊C4—202no—
136C23×C5C2×S411yescharacteristic
137C5×Q8—31no—
138C5×Q8—32no—
139C5×Q8—61no—
140C5×Q8#23511yescharacteristic
141C2×C4×C5—22no—
142C2×C4×C5—31no—
143C2×C4×C5—31no—
144C2×C4×C5—31no—
145C2×C4×C5—31no—
146C2×C4×C5—61no—
147C2×C4×C5—61no—
148C2×C4×C5—62no—
149C2×C4×C5C2×S411yescharacteristic
150C2×C4×C5C2×S411yescharacteristic
151C2×Dic5—62no—
152C2×Dic5—62no—
153Dic10—62no—
154Dic10—62no—
155Dic10—122no—
156#168—62no—
157#168—62no—
158C22×Q8C2×D1511yescharacteristic
159C2×Q16—152no—
160C2×Q16—152no—
161C4⋊Q8—151no—
162C4⋊Q8—151no—
163#125—301no—
164C22⋊Q8—152no—
165C22⋊Q8—301no—
166C4×Q8—31no—
167C4×Q8—31no—
168C4×Q8—152no—
169C4×Q8—301no—
170C4×Q8C2×D1511yescharacteristic
171C4×Q8C2×D1511yescharacteristic
172#117—301no—
173C2×C4⋊C4—31no—
174C2×C42—31no—
175#106—152no—
176#106—152no—
177C4⋊C8—151no—
178C4⋊C8—151no—
179#103—152no—
180#103—152no—
181#103—152no—
182#103—152no—
183C22⋊C8—152no—
184#97—152no—
185C2×C3×C5—41no—
186C2×C3×C5—41no—
187C2×C3×C5—41no—
188C2×C3×C5—82no—
189C23×C3—41no—
190C2×C3×C4—41no—
191C2×C3×C4—41no—
192C2×C3×C4—82no—
193C2×Dic3—202no—
194C2×Dic3—202no—
195Dic6—402no—
196SL(2,𝔽3)#70911yescharacteristic
197C22×C5—21no—
198C22×C5—21no—
199C22×C5#98412yes—
200C22×C5C22×S411yescharacteristic
201C4×C5—21no—
202C4×C5—31no—
203C4×C5—31no—
204C4×C5—61no—
205C4×C5—61no—
206C4×C5—61no—
207C4×C5#98312yes—
208Dic5—122no—
209Dic5—122no—
210C2×Q8—21no—
211C2×Q8—31no—
212C2×Q8—61no—
213C2×Q8—152no—
214C2×Q8—301no—
215C2×Q8C22×D1511yescharacteristic
216C22×C4—31no—
217C22×C4—31no—
218C22×C4C5⋊S411yescharacteristic
219Q16—302no—
220Q16—304no—
221Q16—302no—
222C2×C8—152no—
223C2×C8—152no—
224C4⋊C4—31no—
225C4⋊C4—31no—
226C4⋊C4—61no—
227C4⋊C4—152no—
228C4⋊C4—301no—
229C4⋊C4—301no—
230C4⋊C4—302no—
231C4⋊C4—302no—
232C4⋊C4—301no—
233C22⋊C4—302no—
234C42—31no—
235C42—31no—
236C42—31no—
237C42—31no—
238C42—302no—
239C3×C5—41noHall
240C22×C3—42no—
241C22×C3—41no—
242C22×C3—81no—
243C22×C3—81no—
244C3×C4—42no—
245C3×C4—81no—
246Dic3—402no—
247Dic3—402no—
248C2×C5—22no—
249C2×C5#591211yescharacteristic
250C2×C5C2×#98311yescharacteristic
251C2×C5C2×#98411yescharacteristic
252C23C2×C5⋊S411yescharacteristic
253Q8—31no—
254Q8—32no—
255Q8—61no—
256Q8—302no—
257Q8—302no—
258Q8—602no—
259Q8#681711yescharacteristic
260C2×C4—22no—
261C2×C4—31no—
262C2×C4—31no—
263C2×C4—31no—
264C2×C4—31no—
265C2×C4—61no—
266C2×C4—61no—
267C2×C4—62no—
268C2×C4—302no—
269C2×C4—302no—
270C2×C4C2×C5⋊S411yescharacteristic
271C2×C4C2×C5⋊S411yescharacteristic
272C8—302no—
273C8—302no—
274C2×C3—41no—
275C2×C3—41no—
276C2×C3—41no—
277C2×C3—82no—
278C5#8531911yescharacteristic, complemented, Sylow
279C22—21no—
280C22—21no—
281C22#9195412yes—
282C22C22×C5⋊S411yescharacteristic
283C4—21no—
284C4—31no—
285C4—31no—
286C4—61no—
287C4—61no—
288C4—61no—
289C4—602no—
290C4—602no—
291C4#9195312yes—
292C3—41noSylow
293C2—22no—
294C2#1175689611yescharacteristic
295C2C2×#9195311yescharacteristic
296C2C2×#9195411yescharacteristic
subgroupG/H
center, FrattiniC22C22×C5⋊S4
commutator, cosocleC2×C5×SL(2,𝔽3)C23
FittingC2×C4×C5×Q8D3
socleC22×C5C22×S4
prime radicalC22×C5×SL(2,𝔽3)C22
hypercenterC22×C4C5⋊S4
hypocenterC5×SL(2,𝔽3)#41
LCSG ▷ C2×C5×SL(2,𝔽3) ▷ C5×SL(2,𝔽3)
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
3. C4⋊(C5⋊#226)C2semidirect
4. #11752107C2semidirect
5. #11752105C2semidirect
278. C5#85319semidirect

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
#117521382
C4⋊(C5⋊#226)2
#117521072
#117521052
#106141793
#918134
#853195