Finite Group Databasebeta

← #14512502

Subgroups — #14512502

Order 1920.

384 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 38 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×C5⋊#994C211yesmaximal, characteristic
2#11756720C212yesmaximal, complemented
3#10605012—31nomaximal, Hall
4C2×C5×#41⋊C3C411yescharacteristic, complemented
5C5⋊#994—22no—
6C2×C5⋊(#41⋊C3)C411yescharacteristic, complemented
7C2×C5⋊(#41⋊C3)C2211yescharacteristic
8#91885—22no—
9#91234—41nomaximal
10#73152—51nomaximal, Hall
11C2×#4149D311yescharacteristic
12#65784—32no—
13#65767—32no—
14#65761—31no—
15#65745—31no—
16C2×C3×Dic10—41no—
17C5×#41⋊C3—21no—
18C2×C5×SL(2,𝔽3)C2×C411yescharacteristic
19#6762—42no—
20C5⋊(#41⋊C3)—21no—
21C5⋊(#41⋊C3)D412yes—
22C2×#994—51no—
23#5423—52no—
24C2×C5×#41Dic311yescharacteristic
25#4149—22no—
26#4149—62no—
27C2×#713Dic311yescharacteristic
28C2×#713C2×D311yescharacteristic
29C2×D4×D5—31no—
30C2×D4×D5—31no—
31C2×#709—31no—
32C22×D20—31no—
33#4130—32no—
34D5⋊M16—32no—
35#4013—61no—
36#4011—61no—
37#4010—62no—
38#4010—62no—
39#4007—64no—
40#4005—61no—
41#3999—61no—
42#1795—151noSylow
43C2×C3×C4×C5—41no—
44C2×C3×Dic5—41no—
45C2×C3×Dic5—41no—
46C3×Dic10—82no—
47C5×SL(2,𝔽3)C22⋊C411yescharacteristic
48C3⋊#170—82no—
49#994—102no—
50C2×#41⋊C3—51no—
51C2×#41⋊C3—51no—
52C2×#41⋊C3AGL(1,𝔽5)11yescharacteristic, complemented
53GU(2,𝔽3)—102no—
54#836—201no—
55C23×D5—61no—
56C5×#41—21no—
57C5×#41—61no—
58C2×C5×Q8C2×Dic311yescharacteristic
59C2×C5×D4—31no—
60C22×C4×C5—31no—
61C2×C22⋊D5—31no—
62C2×C22⋊D5—31no—
63#713—21no—
64#713—32no—
65#713—61no—
66#713#6712yes—
67D4×D5—34no—
68D4×D5—62no—
69D4×D5—62no—
70D4×D5—62no—
71#709—62no—
72#709—62no—
73C2×D20—31no—
74C2×D20—31no—
75C2×D20—31no—
76C2×D20—62no—
77C2×D20—62no—
78C2×C4×D5—31no—
79C2×C4×D5—31no—
80C2×Dic10S411yescharacteristic
81C22⋊AGL(1,𝔽5)—62no—
82#704—62no—
83C4⋊AGL(1,𝔽5)—62no—
84C4×AGL(1,𝔽5)—62no—
85#700—62no—
86D5⋊C8—62no—
87C2×H(5,𝔽2)—51no—
88#421—152no—
89#418—152no—
90#413—151no—
91#410—151no—
92C22×C3×C5—41no—
93C3×C4×C5—81no—
94C3×Dic5—42no—
95C3×Dic5—81no—
96C2×C3×Q8—201no—
97#41⋊C3—21no—
98#41⋊C3—52no—
99#41⋊C3—101no—
100C2×SL(2,𝔽3)C2×AGL(1,𝔽5)11yescharacteristic
101#208—202no—
102C23×C5—31no—
103C22×D5—31no—
104C22×D5—31no—
105C22×D5—64no—
106C22×D5—242no—
107C2×AGL(1,𝔽5)—122no—
108C5×Q8—31no—
109C5×Q8#21711yescharacteristic
110C5×D4—32no—
111C5×D4—61no—
112C2×C4×C5—31no—
113C2×C4×C5—61no—
114C2×C4×C5—61no—
115C2×C4×C5SL(2,ℤ/4)11yescharacteristic
116C22⋊D5—62no—
117C22⋊D5—62no—
118C2×Dic5SL(2,ℤ/4)11yescharacteristic
119C2×Dic5C2×S411yescharacteristic
120D20—32no—
121D20—32no—
122D20—62no—
123D20—61no—
124D20—61no—
125C4×D5—32no—
126C4×D5—32no—
127C4×D5—61no—
128C4×D5—61no—
129Dic10—22no—
130#170—122no—
131H(5,𝔽2)—102no—
132H(5,𝔽2)—302no—
133C2×#41—51no—
134C2×#41—51no—
135C2×#41—151no—
136C2×#41ΓL(1,𝔽16)11yescharacteristic
137C22×D4—151no—
138C22×D4—151no—
139C22×D4—151no—
140C2×M16—152no—
141#117—152no—
142#104—302no—
143#104—302no—
144#102—304no—
145#101—301no—
146#100—301no—
147#99—301no—
148#99—301no—
149C2×C3×C5—41no—
150C2×C3×C5—81no—
151C3×Q8—402no—
152C2×C3×C4—41no—
153C2×C3×C4—201no—
154C2×C3×C4—201no—
155SL(2,𝔽3)C22⋊AGL(1,𝔽5)11yescharacteristic
156C3⋊C8—402no—
157C22×C5—61no—
158C22×C5—121no—
159C22×C5C2×SL(2,ℤ/4)11yescharacteristic
160C2×D5—32no—
161C2×D5—61no—
162C2×D5—121no—
163C2×D5—121no—
164C2×D5—244no—
165AGL(1,𝔽5)—242no—
166C4×C5—21no—
167C4×C5—31no—
168C4×C5—31no—
169Dic5—21no—
170Dic5GL(2,ℤ/4)12yes—
171C24—301no—
172#41—21no—
173#41—52no—
174#41—61no—
175#41—101no—
176#41—152no—
177#41—302no—
178#41—302no—
179#41—301no—
180C2×Q8—51no—
181C2×Q8C2×ΓL(1,𝔽16)11yescharacteristic
182C2×D4—31no—
183C2×D4—151no—
184C2×D4—151no—
185C2×D4—151no—
186C2×D4—151no—
187C2×D4—154no—
188C2×D4—151no—
189C2×D4—302no—
190C2×D4—302no—
191C2×D4—302no—
192C2×D4—302no—
193C2×D4—302no—
194C22×C4—31no—
195C22×C4—151no—
196C22×C4—151no—
197M16—302no—
198M16—302no—
199C2×C8—302no—
200C4⋊C4—302no—
201C22⋊C4—302no—
202C42—302no—
203C3×C5—41noHall
204C22×C3—41no—
205C3×C4—81no—
206C3×C4—202no—
207C3×C4—401no—
208C2×C5—21no—
209C2×C5—121no—
210C2×C5#543211yescharacteristic
211D5—62no—
212D5—121no—
213C23—31no—
214C23—151no—
215C23—151no—
216C23—304no—
217C23—1202no—
218Q8—31no—
219Q8—102no—
220Q8#676311yescharacteristic
221D4—32no—
222D4—61no—
223D4—152no—
224D4—152no—
225D4—302no—
226D4—301no—
227D4—301no—
228D4—302no—
229D4—302no—
230C2×C4—31no—
231C2×C4—51no—
232C2×C4—51no—
233C2×C4—61no—
234C2×C4—61no—
235C2×C4—152no—
236C2×C4—152no—
237C2×C4—301no—
238C2×C4—301no—
239C2×C4—602no—
240C2×C4A4⋊AGL(1,𝔽5)11yescharacteristic
241C8—602no—
242C2×C3—41no—
243C2×C3—81no—
244C5#7315211yescharacteristic, complemented, Sylow
245C22—61no—
246C22—121no—
247C22—152no—
248C22—301no—
249C22—601no—
250C22—601no—
251C22—1204no—
252C22C2×A4⋊AGL(1,𝔽5)11yescharacteristic
253C4—21no—
254C4—31no—
255C4—31no—
256C4—52no—
257C4—101no—
258C4—1202no—
259C3—41noSylow
260C2—21no—
261C2—121no—
262C2—302no—
263C2—601no—
264C2#1175672911yescharacteristic
subgroupG/H
centerC2#11756729
commutatorC2×C5×SL(2,𝔽3)C2×C4
FrattiniC22C2×A4⋊AGL(1,𝔽5)
FittingC2×C5×#41Dic3
socleC2×C5#5432
prime radicalC2×C5⋊#994C2
cosocleC2×C5⋊(#41⋊C3)C22
hypercenterC2×C4A4⋊AGL(1,𝔽5)
hypocenterC5×SL(2,𝔽3)C22⋊C4
LCSG ▷ C2×C5×SL(2,𝔽3) ▷ C5×SL(2,𝔽3)
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4

Complements

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

NH ≅ G/Ntype
2. #11756720C2semidirect
4. C2×C5×#41⋊C3C4semidirect
6. C2×C5⋊(#41⋊C3)C4semidirect
52. C2×#41⋊C3AGL(1,𝔽5)semidirect
244. C5#73152semidirect

Maximal subgroups

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

subgroupindex
C2×C5⋊#9942
#117567202
#106050123
#912344
#731525