Finite Group Databasebeta

← #10617517⋊C3 #14512571

Subgroups — #10617517⋊C3 #14512571

Order 1920.

398 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 52 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
1C22×C5⋊(#41⋊C3)C211yesmaximal, characteristic
2C2×#4133⋊C3C212yesmaximal, complemented
3#10617517C311yesmaximal, characteristic, complemented, Hall
4C22×C5×SL(2,𝔽3)C411yescharacteristic
5C2×C5⋊(#41⋊C3)—22no—
6C2×C5⋊(#41⋊C3)C411yescharacteristic
7C2×C5⋊(#41⋊C3)C2211yescharacteristic
8#4133⋊C3—22no—
9C3×C22⋊#170—41nomaximal
10#2880⋊C3—51nomaximal, Hall
11C22×#713C2×C311yescharacteristic
12C2×#4133C2×C312yescomplemented
13#65787—32no—
14#65756—31no—
15D5⋊(C22⋊C8)—31no—
16C22×C3×Dic5—41no—
17C2×C5×SL(2,𝔽3)—21no—
18C2×C5×SL(2,𝔽3)C2×C411yescharacteristic
19C2×C3×#170—42no—
20C5⋊(#41⋊C3)—21no—
21C5⋊(#41⋊C3)D412yes—
22C22×#41⋊C3—51no—
23C2×#131⋊C3—52no—
24C22×C5×Q8C3×C411yescharacteristic
25C2×#713—22no—
26C2×#713—31no—
27C2×#713—31no—
28C2×#713—62no—
29C2×#713C3×C411yescharacteristic
30C2×#713C22×C311yescharacteristic
31C22×D20—31no—
32C22×C4×D5—31no—
33#4133—22no—
34#4133—62no—
35C2×#700—32no—
36C2×D5⋊C8—32no—
37C22⋊#170—31no—
38C22⋊#170A411yescharacteristic
39#4003—32no—
40#4003—32no—
41#2880—51noSylow
42C23×C3×C5—41no—
43C2×C3×Dic5—41no—
44C2×C3×Dic5—41no—
45C2×C3×Dic5—82no—
46C5×SL(2,𝔽3)C22⋊C411yescharacteristic
47C3×#170—82no—
48C2×#41⋊C3—51no—
49C2×#41⋊C3—51no—
50C2×#41⋊C3—102no—
51C22×SL(2,𝔽3)AGL(1,𝔽5)11yescharacteristic
52#131⋊C3—102no—
53C3×C22⋊C8—201no—
54C23×D5—31no—
55C2×C5×Q8—21no—
56C2×C5×Q8—31no—
57C2×C5×Q8—61no—
58C2×C5×Q8C2×C3×C411yescharacteristic
59C22×C4×C5—31no—
60C22×Dic5C2×A411yescharacteristic
61#713—21no—
62#713—32no—
63#713—62no—
64#713—62no—
65#713—62no—
66#713—62no—
67#713—61no—
68#713C3×D412yes—
69C2×D20—31no—
70C2×D20—31no—
71C2×D20—32no—
72C2×D20—62no—
73C2×D20—62no—
74C2×C4×D5—31no—
75C2×C4×D5—31no—
76C2×C4×D5—31no—
77C2×C4×D5—31no—
78C2×C4×D5—62no—
79C2×C4×D5—62no—
80C2×#170—32no—
81C2×#170C2×A412yes—
82#700—62no—
83#700—62no—
84D5⋊C8—62no—
85D5⋊C8—62no—
86C22×#41—51no—
87C2×#131—52no—
88#408—152no—
89#407—151no—
90C2×C22⋊C8—151no—
91C22×C3×C5—42no—
92C22×C3×C5—41no—
93C22×C3×C5—81no—
94C22×C3×C5—81no—
95C3×Dic5—42no—
96C3×Dic5—81no—
97C22×C3×C4—201no—
98#41⋊C3—52no—
99#41⋊C3—101no—
100C2×SL(2,𝔽3)—21no—
101C2×SL(2,𝔽3)C2×AGL(1,𝔽5)11yescharacteristic
102C2×C3×C8—202no—
103C23×C5C4×A411yescharacteristic
104C22×D5—31no—
105C22×D5—31no—
106C22×D5—62no—
107C22×D5—62no—
108C22×D5—121no—
109C5×Q8—31no—
110C5×Q8—61no—
111C5×Q8—61no—
112C5×Q8C3×C22⋊C411yescharacteristic
113C2×C4×C5—31no—
114C2×C4×C5—31no—
115C2×C4×C5—61no—
116C2×C4×C5—61no—
117C2×Dic5—22no—
118C2×Dic5C4×A411yescharacteristic
119C2×Dic5C22×A411yescharacteristic
120D20—32no—
121D20—32no—
122D20—61no—
123D20—61no—
124D20—64no—
125C4×D5—32no—
126C4×D5—32no—
127C4×D5—61no—
128C4×D5—61no—
129C4×D5—62no—
130C4×D5—62no—
131#170—22no—
132#170—62no—
133C2×#41—51no—
134C2×#41—51no—
135C2×#41—102no—
136C2×#41—151no—
137C2×#41—151no—
138C2×#41—302no—
139C22×Q8C3×AGL(1,𝔽5)11yescharacteristic
140C22×D4—151no—
141C23×C4—151no—
142#131—102no—
143#131—302no—
144C2×M16—152no—
145C22×C8—152no—
146C22⋊C8—51no—
147C22⋊C8—152no—
148C22⋊C8—151no—
149C22⋊C8—152no—
150C2×C3×C5—41no—
151C2×C3×C5—41no—
152C2×C3×C5—41no—
153C2×C3×C5—82no—
154C23×C3—41no—
155C2×C3×C4—201no—
156C2×C3×C4—201no—
157C2×C3×C4—402no—
158SL(2,𝔽3)C22⋊AGL(1,𝔽5)11yescharacteristic
159C3×C8—402no—
160C22×C5—21no—
161C22×C5—21no—
162C22×C5#131⋊C312yescomplemented
163C22×C5C2×C4×A411yescharacteristic
164C2×D5—32no—
165C2×D5—61no—
166C2×D5—61no—
167C2×D5—61no—
168C2×D5—61no—
169C2×D5—61no—
170C2×D5—124no—
171C4×C5—31no—
172C4×C5—31no—
173C4×C5—61no—
174Dic5—21no—
175Dic5D4×A412yes—
176C24—151no—
177#41—52no—
178#41—101no—
179#41—152no—
180#41—302no—
181#41—302no—
182#41—302no—
183#41—302no—
184#41—301no—
185C2×Q8—21no—
186C2×Q8—31no—
187C2×Q8—61no—
188C2×Q8C2×C3×AGL(1,𝔽5)11yescharacteristic
189C2×D4—151no—
190C2×D4—151no—
191C2×D4—152no—
192C2×D4—302no—
193C2×D4—302no—
194C22×C4—31no—
195C22×C4—51no—
196C22×C4—151no—
197C22×C4—151no—
198C22×C4—151no—
199C22×C4—151no—
200C22×C4—302no—
201C22×C4—302no—
202M16—302no—
203M16—302no—
204C2×C8—52no—
205C2×C8—152no—
206C2×C8—302no—
207C2×C8—302no—
208C3×C5—41noHall
209C22×C3—41no—
210C22×C3—42no—
211C22×C3—81no—
212C22×C3—81no—
213C3×C4—202no—
214C3×C4—401no—
215C2×C5—22no—
216C2×C5A4×C22⋊C411yescharacteristic
217C2×C5C2×#131⋊C311yescharacteristic
218C2×C5#568⋊C311yescharacteristic
219D5—62no—
220D5—121no—
221C23—151no—
222C23—151no—
223C23—302no—
224C23—302no—
225C23—601no—
226C23A4×AGL(1,𝔽5)11yescharacteristic
227Q8—31no—
228Q8—61no—
229Q8—61no—
230Q8C3×C22⋊AGL(1,𝔽5)11yescharacteristic
231D4—152no—
232D4—152no—
233D4—301no—
234D4—301no—
235D4—304no—
236C2×C4—31no—
237C2×C4—31no—
238C2×C4—51no—
239C2×C4—51no—
240C2×C4—61no—
241C2×C4—61no—
242C2×C4—102no—
243C2×C4—152no—
244C2×C4—152no—
245C2×C4—301no—
246C2×C4—301no—
247C2×C4—302no—
248C2×C4—302no—
249C8—102no—
250C8—302no—
251C2×C3—41no—
252C2×C3—41no—
253C2×C3—41no—
254C2×C3—82no—
255C5#2880⋊C311yescharacteristic, complemented, Sylow
256C22—21no—
257C22—21no—
258C22—152no—
259C22—301no—
260C22—301no—
261C22—301no—
262C22—301no—
263C22—301no—
264C22—604no—
265C22#4133⋊C312yescomplemented
266C22C2×A4×AGL(1,𝔽5)11yescharacteristic
267C4—31no—
268C4—31no—
269C4—52no—
270C4—61no—
271C4—101no—
272C3—41noSylow
273C2—22no—
274C2—302no—
275C2—601no—
276C2C2×#4133⋊C311yescharacteristic
277C2#66260⋊C311yescharacteristic
278C2A4×C22⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
center, FrattiniC22C2×A4×AGL(1,𝔽5)
commutatorC2×C5×Q8C2×C3×C4
FittingC22×C5×Q8C3×C4
socleC22×C5C2×C4×A4
prime radicalC22×C5⋊(#41⋊C3)C2
cosocleC2×#713C22×C3
hypercenterC23A4×AGL(1,𝔽5)
hypocenterC5×Q8C3×C22⋊C4
LCSG ▷ C2×C5×Q8 ▷ C5×Q8
UCSC1 ◁ C22 ◁ C23

Complements

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

NH ≅ G/Ntype
2. C2×#4133⋊C3C2semidirect
3. #10617517C3semidirect
12. C2×#4133C2×C3semidirect
162. C22×C5#131⋊C3semidirect
255. C5#2880⋊C3semidirect
265. C22#4133⋊C3semidirect

Maximal subgroups

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

subgroupindex
C22×C5⋊(#41⋊C3)2
C2×#4133⋊C32
#106175173
C3×C22⋊#1704
#2880⋊C35