Finite Group Databasebeta

← #14512680

Subgroups — #14512680

Order 1920.

420 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 51 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
1SL(2,𝔽3)×D20C211yesmaximal, characteristic
2#11752105C211yesmaximal, characteristic, complemented
3C4×C5×#226C211yesmaximal, characteristic, complemented
4#11751606C212yesmaximal, complemented
5#11751605C212yesmaximal, complemented
6#10611470—31nomaximal, Hall
7C2×D5×SL(2,𝔽3)C2212yes—
8C2×C5×#226C2211yescharacteristic
9C4×C5×SL(2,𝔽3)C2211yescharacteristic
10#91364—41nomaximal
11#91184C2212yes—
12C5×#859C2211yescharacteristic
13#85395—51nomaximal, Hall
14#65918—31no—
15Q8×D20D311yescharacteristic
16C4×C5×Q16—31no—
17#65475—32no—
18#65320—31no—
19#65309—31no—
20#65177—32no—
21#65105—32no—
22#65091—32no—
23#64977—31no—
24#64970—31no—
25C2×Dic30—41no—
26C2×C3×D20—41no—
27C2×C5×SL(2,𝔽3)C2311yescharacteristic
28D5×SL(2,𝔽3)—22no—
29C5×#226—21no—
30C4×C5×Dic3—41no—
31#6665—42no—
32#6665—42no—
33D4×SL(2,𝔽3)—51no—
34#5420—52no—
35#5419—52no—
36C4×#226D511yescharacteristic, complemented
37#5386—51no—
38C2×Q8×D5C2×D312yes—
39C2×C5×Q16—31no—
40C4×C5×Q8—31no—
41C4×C5×Q8C2×D311yescharacteristic
42#4092—32no—
43C2×#689—32no—
44C2×#676—32no—
45#4044—62no—
46#4043—32no—
47#4038—32no—
48#4021—61no—
49C4×D20—31no—
50#4015—31no—
51#4015—31no—
52C4×Dic10—61no—
53C5×#107—31no—
54C5×#103—31no—
55C5×#103—31no—
56C4×C5×C8—31no—
57#3967—32no—
58#3952—32no—
59#3948—32no—
60#3948—32no—
61#3942—32no—
62#3940—32no—
63C4⋊#168—31no—
64#3328—151noSylow
65C22×C3×D5—42no—
66C2×C3×C4×C5—41no—
67C2×Dic15—42no—
68Dic30—82no—
69C2×C5×Dic3—41no—
70C2×C5×Dic3—41no—
71C3×D20—82no—
72C5×SL(2,𝔽3)#4111yescharacteristic
73C22×SL(2,𝔽3)—52no—
74C2×#226C2×D511yescharacteristic
75#936—201no—
76C4×SL(2,𝔽3)C2×D511yescharacteristic
77#859—52no—
78#859C2×D511yescharacteristic
79C2×C5×Q8—31no—
80C2×C5×Q8C22×D311yescharacteristic
81Q8×D5—22no—
82Q8×D5—62no—
83C2×D20S411yescharacteristic
84C2×C4×D5—32no—
85C2×Dic10—32no—
86C2×Dic10—61no—
87C5×Q16—61no—
88C5×Q16—61no—
89C2×C5×C8—31no—
90C2×C5×C8—31no—
91C5×C4⋊C4—31no—
92C5×C4⋊C4—31no—
93C5×C4⋊C4—61no—
94C42×C5—31no—
95C42×C5—61no—
96#689—122no—
97#685—32no—
98#685—62no—
99#685—62no—
100C4⋊Dic5—31no—
101C4⋊Dic5—32no—
102C4⋊Dic5—61no—
103C4⋊Dic5—61no—
104#683—62no—
105C4×Dic5—62no—
106C2×#168—32no—
107#676—122no—
108#556—151no—
109D4×Q8—51no—
110#487—151no—
111#485—152no—
112#475—151no—
113#470—152no—
114#462—151no—
115#451—152no—
116C4×Q16—31no—
117#436—151no—
118#419—152no—
119C22×C3×C5—41no—
120C2×C3×D5—82no—
121C2×C3×D5—82no—
122C2×C3×D5—82no—
123C3×C4×C5—42no—
124Dic15—82no—
125C5×Dic3—81no—
126C5×Dic3—81no—
127C2×C3×D4—201no—
128C2×Dic6—201no—
129C2×SL(2,𝔽3)—102no—
130C2×SL(2,𝔽3)C22×D511yescharacteristic
131#226—21no—
132#217—202no—
133#217—202no—
134C4×Dic3—41no—
135C22×D5C2×S412yes—
136C5×Q8—31no—
137C5×Q8—61no—
138C5×Q8—61no—
139C5×Q8#23711yescharacteristic
140C2×C4×C5—31no—
141C2×C4×C5—31no—
142C2×C4×C5—61no—
143C2×C4×C5—61no—
144C2×C4×C5C2×S411yescharacteristic
145C2×Dic5—32no—
146C2×Dic5—62no—
147D20—22no—
148C4×D5—62no—
149C4×D5—62no—
150Dic10—62no—
151Dic10—62no—
152Dic10—122no—
153C5×C8—61no—
154C5×C8—61no—
155#168—62no—
156C22×Q8—52no—
157C2×Q16—31no—
158C2×Q16—152no—
159C2×M16—152no—
160C4⋊Q8—151no—
161C4⋊Q8—151no—
162#126—302no—
163#124—301no—
164C22⋊Q8—152no—
165C22⋊Q8—152no—
166C4×Q8—31no—
167C4×Q8—301no—
168C4×Q8D3×D511yescharacteristic
169C4×D4—151no—
170#117—152no—
171#107—31no—
172#106—152no—
173C4⋊C8—151no—
174#103—31no—
175#103—31no—
176#103—152no—
177#103—152no—
178#103—152no—
179#103—152no—
180C22⋊C8—152no—
181C4×C8—31no—
182C2×C3×C5—41no—
183C2×C3×C5—41no—
184C2×C3×C5—41no—
185C3×D5—162no—
186C23×C3—202no—
187C3×D4—402no—
188C2×C3×C4—41no—
189C2×Dic3—41no—
190C2×Dic3—41no—
191C2×Dic3—202no—
192Dic6—402no—
193SL(2,𝔽3)#70911yescharacteristic
194C22×C5C22×S411yescharacteristic
195C2×D5—22no—
196C2×D5—22no—
197C2×D5—22no—
198C4×C5—31no—
199C4×C5—31no—
200C4×C5—61no—
201C4×C5—121no—
202C4×C5—121no—
203C4×C5#98312yes—
204Dic5—62no—
205Dic5—122no—
206C2×Q8—31no—
207C2×Q8—102no—
208C2×Q8—152no—
209C2×Q8—301no—
210C2×Q8—302no—
211C2×Q8C2×D3×D511yescharacteristic
212C2×D4—51no—
213C22×C4—152no—
214Q16—61no—
215Q16—61no—
216Q16—602no—
217M16—602no—
218C2×C8—31no—
219C2×C8—31no—
220C2×C8—152no—
221C4⋊C4—31no—
222C4⋊C4—31no—
223C4⋊C4—61no—
224C4⋊C4—152no—
225C4⋊C4—151no—
226C4⋊C4—301no—
227C4⋊C4—301no—
228C4⋊C4—302no—
229C22⋊C4—152no—
230C22⋊C4—302no—
231C22⋊C4—302no—
232C42—31no—
233C42—61no—
234C42—302no—
235C3×C5—41noHall
236C22×C3—41no—
237C22×C3—402no—
238C22×C3—402no—
239C22×C3—402no—
240C3×C4—42no—
241Dic3—81no—
242Dic3—81no—
243Dic3—402no—
244C2×C5#591411yescharacteristic
245C2×C5C2×#98311yescharacteristic
246C2×C5#592711yescharacteristic
247D5—42no—
248C23—52no—
249Q8—31no—
250Q8—61no—
251Q8—61no—
252Q8—302no—
253Q8—302no—
254Q8—602no—
255Q8#676511yescharacteristic
256D4—102no—
257C2×C4—31no—
258C2×C4—31no—
259C2×C4—61no—
260C2×C4—61no—
261C2×C4—152no—
262C2×C4—302no—
263C2×C4—302no—
264C2×C4—302no—
265C2×C4D5×S411yescharacteristic
266C8—61no—
267C8—61no—
268C8—302no—
269C2×C3—41no—
270C2×C3—41no—
271C2×C3—41no—
272C2×C3—802no—
273C5#8539511yescharacteristic, complemented, Sylow
274C22—102no—
275C22—102no—
276C22—102no—
277C22C2×D5×S411yescharacteristic
278C4—31no—
279C4—31no—
280C4—61no—
281C4—121no—
282C4—121no—
283C4—302no—
284C4—602no—
285C4#9189712yes—
286C3—41noSylow
287C2—202no—
288C2#1175675011yescharacteristic
289C2C2×#9189711yescharacteristic
290C2#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)#41
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. #11752105C2semidirect
3. C4×C5×#226C2semidirect
4. #11751606C2semidirect
5. #11751605C2semidirect
36. C4×#226D5semidirect
273. C5#85395semidirect

Maximal subgroups

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

subgroupindex
SL(2,𝔽3)×D202
#117521052
C4×C5×#2262
#117516062
#117516052
#106114703
#913644
#853955