Finite Group Databasebeta

← #14512503

Subgroups — #14512503

Order 1920.

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

normal subgroups only

This group has 386 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×#4150⋊C3C211yesmaximal, characteristic
2#11756722C212yesmaximal, complemented
3#10605013—31nomaximal, Hall
4C2×C5×#41⋊C3C411yescharacteristic, complemented
5#4150⋊C3—22no—
6C2×D5×SL(2,𝔽3)C411yescharacteristic, complemented
7C2×D5×SL(2,𝔽3)C2211yescharacteristic
8#91886—22no—
9#91228—41nomaximal
10#73156—51nomaximal, Hall
11C2×#4150D311yescharacteristic
12#65782—32no—
13#65769—32no—
14#65761—31no—
15#65745—31no—
16C2×C3×D20—41no—
17C5×#41⋊C3—21no—
18C2×C5×SL(2,𝔽3)C2×C411yescharacteristic
19#6763—42no—
20D5×SL(2,𝔽3)—21no—
21D5×SL(2,𝔽3)D412yes—
22C2×2−1+4⋊C3—51no—
23#5420—52no—
24C2×C5×#41Dic311yescharacteristic, complemented
25#4150—22no—
26#4150—62no—
27C2×Q8×D5Dic311yescharacteristic, complemented
28C2×Q8×D5C2×D311yescharacteristic
29C2×#711—31no—
30C2×#711—31no—
31C2×#709—31no—
32C22×Dic10—31no—
33#4130—32no—
34D5⋊M16—32no—
35#4013—61no—
36#4011—61no—
37#4009—62no—
38#4009—62no—
39#4008—64no—
40#4005—61no—
41#3999—61no—
42#1796—151noSylow
43C22×C3×D5—41no—
44C22×C3×D5—41no—
45C2×ΓL(1,𝔽16)—82no—
46C2×C3×C4×C5—41no—
47C3×D20—82no—
48C5×SL(2,𝔽3)C22⋊C411yescharacteristic
492−1+4⋊C3—102no—
50C2×#41⋊C3AGL(1,𝔽5)11yescharacteristic, complemented
51C22×SL(2,𝔽3)—51no—
52C22×SL(2,𝔽3)—51no—
53#859—102no—
54#834—201no—
55C5×#41—21no—
56C5×#41—61no—
57C2×C5×Q8C2×Dic311yescharacteristic
58C2×C5×D4—31no—
59C22×C4×C5—31no—
60C2×C22⋊D5—31no—
61C2×C22⋊D5—31no—
62C22×Dic5—61no—
63Q8×D5—21no—
64Q8×D5—32no—
65Q8×D5—61no—
66Q8×D5#6712yes—
67#711—34no—
68#711—62no—
69#711—62no—
70#711—62no—
71#709—62no—
72#709—62no—
73C2×D20S411yescharacteristic
74C2×C4×D5—31no—
75C2×C4×D5—31no—
76C2×Dic10—31no—
77C2×Dic10—31no—
78C2×Dic10—31no—
79C2×Dic10—62no—
80C2×Dic10—62no—
81C22⋊AGL(1,𝔽5)—62no—
82#704—62no—
83C4⋊AGL(1,𝔽5)—62no—
84C4×AGL(1,𝔽5)—62no—
85#700—62no—
86D5⋊C8—62no—
87C2×2−1+4—51no—
88#421—152no—
89#419—152no—
90#413—151no—
91#410—151no—
92C22×C3×C5—41no—
93C2×C3×D5—42no—
94C2×C3×D5—81no—
95C2×C3×D5—161no—
96C2×C3×D5—161no—
97ΓL(1,𝔽16)—162no—
98C3×C4×C5—81no—
99C2×C3×D4—201no—
100#41⋊C3—21no—
101C2×SL(2,𝔽3)—52no—
102C2×SL(2,𝔽3)—101no—
103C2×SL(2,𝔽3)C2×AGL(1,𝔽5)11yescharacteristic
104#217—202no—
105C23×C5—31no—
106C22×D5SL(2,ℤ/4)11yescharacteristic
107C22×D5C2×S411yescharacteristic
108C2×AGL(1,𝔽5)—122no—
109C5×Q8—31no—
110C5×Q8#21711yescharacteristic
111C5×D4—32no—
112C5×D4—61no—
113C2×C4×C5—31no—
114C2×C4×C5—61no—
115C2×C4×C5—61no—
116C2×C4×C5SL(2,ℤ/4)11yescharacteristic
117C22⋊D5—62no—
118C22⋊D5—62no—
119C2×Dic5—31no—
120C2×Dic5—31no—
121C2×Dic5—64no—
122D20—22no—
123C4×D5—32no—
124C4×D5—32no—
125C4×D5—61no—
126C4×D5—61no—
127Dic10—32no—
128Dic10—32no—
129Dic10—61no—
130Dic10—62no—
131Dic10—61no—
132#170—122no—
1332−1+4—102no—
1342−1+4—302no—
135C2×#41—151no—
136C2×#41—151no—
137C2×#41—151no—
138C2×#41ΓL(1,𝔽16)11yescharacteristic, complemented
139C22×Q8—51no—
140C22×Q8—51no—
141C22×Q8—151no—
142C2×M16—152no—
143#117—152no—
144#104—304no—
145#103—302no—
146#103—302no—
147#101—301no—
148#100—301no—
149#99—301no—
150#99—301no—
151C2×C3×C5—41no—
152C2×C3×C5—81no—
153C3×D5—82no—
154C3×D5—161no—
155C23×C3—201no—
156C23×C3—201no—
157C3×D4—402no—
158C2×C3×C4—41no—
159C2×Dic3—402no—
160SL(2,𝔽3)C22⋊AGL(1,𝔽5)11yescharacteristic
161C22×C5—61no—
162C22×C5—121no—
163C22×C5C2×SL(2,ℤ/4)11yescharacteristic
164C2×D5—21no—
165C2×D5—41no—
166C2×D5—41no—
167C2×D5GL(2,ℤ/4)12yes—
168AGL(1,𝔽5)—242no—
169C4×C5—21no—
170C4×C5—31no—
171C4×C5—31no—
172Dic5—32no—
173Dic5—61no—
174#41—21no—
175#41—61no—
176#41—154no—
177#41—302no—
178#41—302no—
179#41—302no—
180#41—302no—
181#41—302no—
182C2×Q8—52no—
183C2×Q8—101no—
184C2×Q8—151no—
185C2×Q8—151no—
186C2×Q8—151no—
187C2×Q8—152no—
188C2×Q8—302no—
189C2×Q8—302no—
190C2×Q8—301no—
191C2×Q8C2×ΓL(1,𝔽16)11yescharacteristic
192C2×D4—31no—
193C2×D4—51no—
194C2×D4—151no—
195C2×D4—151no—
196C22×C4—31no—
197C22×C4—151no—
198C22×C4—151no—
199C22×C4—301no—
200M16—302no—
201M16—302no—
202C2×C8—302no—
203C4⋊C4—302no—
204C22⋊C4—302no—
205C42—302no—
206C3×C5—41noHall
207C22×C3—41no—
208C22×C3—202no—
209C22×C3—401no—
210C22×C3—801no—
211C22×C3—801no—
212C3×C4—81no—
213Dic3—802no—
214C2×C5—21no—
215C2×C5—121no—
216C2×C5#543211yescharacteristic
217D5—22no—
218D5—41no—
219C23—31no—
220C23—51no—
221C23—51no—
222Q8—31no—
223Q8—152no—
224Q8—152no—
225Q8—302no—
226Q8—301no—
227Q8—301no—
228Q8#676311yescharacteristic
229D4—32no—
230D4—61no—
231D4—102no—
232D4—302no—
233D4—302no—
234C2×C4—31no—
235C2×C4—61no—
236C2×C4—61no—
237C2×C4—152no—
238C2×C4—152no—
239C2×C4—151no—
240C2×C4—151no—
241C2×C4—301no—
242C2×C4—301no—
243C2×C4—304no—
244C2×C4—602no—
245C2×C4A4⋊AGL(1,𝔽5)11yescharacteristic
246C8—602no—
247C2×C3—41no—
248C2×C3—81no—
249C2×C3—402no—
250C2×C3—801no—
251C5#7315611yescharacteristic, complemented, Sylow
252C22—52no—
253C22—61no—
254C22—101no—
255C22—121no—
256C22—201no—
257C22—201no—
258C22C2×A4⋊AGL(1,𝔽5)11yescharacteristic
259C4—21no—
260C4—31no—
261C4—31no—
262C4—152no—
263C4—301no—
264C4—1202no—
265C3—41noSylow
266C2—21no—
267C2—102no—
268C2—121no—
269C2—201no—
270C2#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×#4150⋊C3C2
cosocleC2×D5×SL(2,𝔽3)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. #11756722C2semidirect
4. C2×C5×#41⋊C3C4semidirect
6. C2×D5×SL(2,𝔽3)C4semidirect
24. C2×C5×#41Dic3semidirect
27. C2×Q8×D5Dic3semidirect
50. C2×#41⋊C3AGL(1,𝔽5)semidirect
138. C2×#41ΓL(1,𝔽16)semidirect
251. C5#73156semidirect

Maximal subgroups

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

subgroupindex
C2×#4150⋊C32
#117567222
#106050133
#912284
#731565