Finite Group Databasebeta

← SL(2,𝔽3)×C22⋊AGL(1,𝔽5) #14512575

Subgroups — SL(2,𝔽3)×C22⋊AGL(1,𝔽5) #14512575

Order 1920.

550 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 550 conjugacy classes of subgroups — too many to draw. Only its 74 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×D5×SL(2,𝔽3)C211yesmaximal, characteristic
2C2×AGL(1,𝔽5)×SL(2,𝔽3)C212yesmaximal, complemented
3Q8×C22⋊AGL(1,𝔽5)C311yesmaximal, characteristic, complemented, Hall
4C22×C5×SL(2,𝔽3)C411yescharacteristic, complemented
5C2×C3×C22⋊AGL(1,𝔽5)—41nomaximal
6C2×D5×SL(2,𝔽3)—22no—
7C2×D5×SL(2,𝔽3)C411yescharacteristic, complemented
8C2×D5×SL(2,𝔽3)C2211yescharacteristic
9AGL(1,𝔽5)×SL(2,𝔽3)—22no—
10C22⋊C4×SL(2,𝔽3)—51nomaximal, Hall
11C22×Q8×D5C2×C311yescharacteristic
12C2×Q8×AGL(1,𝔽5)C2×C312yescomplemented
13(C2×AGL(1,𝔽5))⋊Q8—32no—
14#65765—31no—
15C4×C22⋊AGL(1,𝔽5)—31no—
16C23×C3×D5—41no—
17C22×C3×AGL(1,𝔽5)—42no—
18C2×C5×SL(2,𝔽3)—21no—
19C2×C5×SL(2,𝔽3)C2×C411yescharacteristic
20C3×C22⋊AGL(1,𝔽5)—44no—
21D5×SL(2,𝔽3)—21no—
22D5×SL(2,𝔽3)D412yes—
23C23×SL(2,𝔽3)—51no—
24C2×C4×SL(2,𝔽3)—52no—
25C22×C5×Q8C3×C411yescharacteristic, complemented
26C2×Q8×D5—22no—
27C2×Q8×D5—31no—
28C2×Q8×D5—31no—
29C2×Q8×D5—62no—
30C2×Q8×D5C3×C411yescharacteristic, complemented
31C2×Q8×D5C22×C311yescharacteristic
32C22×C4×D5—31no—
33C22×Dic10—31no—
34C2×C22⋊AGL(1,𝔽5)—31no—
35C2×C22⋊AGL(1,𝔽5)A411yescharacteristic
36Q8×AGL(1,𝔽5)—22no—
37Q8×AGL(1,𝔽5)—62no—
38C2×C4⋊AGL(1,𝔽5)—32no—
39C2×C4×AGL(1,𝔽5)—32no—
40(C2×AGL(1,𝔽5))⋊C4—32no—
41(C2×AGL(1,𝔽5))⋊C4—32no—
42Q8×C22⋊C4—51noSylow
43C23×C3×C5—41no—
44C22×C3×D5—41no—
45C22×C3×D5—41no—
46C22×C3×D5—42no—
47C22×C3×D5—42no—
48C22×C3×D5—82no—
49C22×C3×D5—82no—
50C2×C3×AGL(1,𝔽5)—44no—
51C2×C3×AGL(1,𝔽5)—82no—
52C2×C3×AGL(1,𝔽5)—82no—
53C5×SL(2,𝔽3)C22⋊C411yescharacteristic, complemented
54C22×SL(2,𝔽3)—51no—
55C22×SL(2,𝔽3)—51no—
56C22×SL(2,𝔽3)—102no—
57C22×SL(2,𝔽3)AGL(1,𝔽5)11yescharacteristic, complemented
58C2×C3×C22⋊C4—201no—
59C4×SL(2,𝔽3)—102no—
60C23×D5C2×A411yescharacteristic
61C22×AGL(1,𝔽5)—32no—
62C22×AGL(1,𝔽5)C2×A412yes—
63C2×C5×Q8—21no—
64C2×C5×Q8—31no—
65C2×C5×Q8—61no—
66C2×C5×Q8C2×C3×C411yescharacteristic
67C22×C4×C5—31no—
68C22×Dic5—31no—
69Q8×D5—21no—
70Q8×D5—32no—
71Q8×D5—62no—
72Q8×D5—62no—
73Q8×D5—62no—
74Q8×D5—62no—
75Q8×D5—61no—
76Q8×D5C3×D412yes—
77C2×C4×D5—31no—
78C2×C4×D5—31no—
79C2×C4×D5—31no—
80C2×C4×D5—31no—
81C2×C4×D5—62no—
82C2×C4×D5—62no—
83C2×Dic10—31no—
84C2×Dic10—32no—
85C2×Dic10—31no—
86C2×Dic10—62no—
87C2×Dic10—62no—
88C22⋊AGL(1,𝔽5)—62no—
89C22⋊AGL(1,𝔽5)SL(2,𝔽3)14yesdirect product factor
90C4⋊AGL(1,𝔽5)—62no—
91C4⋊AGL(1,𝔽5)—62no—
92C4×AGL(1,𝔽5)—62no—
93C4×AGL(1,𝔽5)—62no—
94C23×Q8—51no—
95C2×C4×Q8—52no—
96#391—152no—
97#380—151no—
98C4×C22⋊C4—151no—
99C22×C3×C5—42no—
100C22×C3×C5—41no—
101C22×C3×C5—81no—
102C22×C3×C5—81no—
103C2×C3×D5—42no—
104C2×C3×D5—42no—
105C2×C3×D5—41no—
106C2×C3×D5—41no—
107C2×C3×D5—42no—
108C2×C3×D5—84no—
109C2×C3×D5—84no—
110C2×C3×D5—81no—
111C2×C3×D5—81no—
112C3×AGL(1,𝔽5)—84no—
113C24×C3—201no—
114C22×C3×C4—202no—
115C2×SL(2,𝔽3)—21no—
116C2×SL(2,𝔽3)—52no—
117C2×SL(2,𝔽3)—101no—
118C2×SL(2,𝔽3)C2×AGL(1,𝔽5)11yescharacteristic
119C3×C22⋊C4—204no—
120C23×C5C4×A411yescharacteristic
121C22×D5—22no—
122C22×D5—22no—
123C22×D5C4×A411yescharacteristic
124C22×D5C2×SL(2,𝔽3)12yes—
125C22×D5#41⋊C312yes—
126C22×D5C22×A411yescharacteristic
127C2×AGL(1,𝔽5)—22no—
128C2×AGL(1,𝔽5)—22no—
129C2×AGL(1,𝔽5)—62no—
130C2×AGL(1,𝔽5)—62no—
131C2×AGL(1,𝔽5)—62no—
132C2×AGL(1,𝔽5)C2×SL(2,𝔽3)14yescomplemented
133C5×Q8—31no—
134C5×Q8—61no—
135C5×Q8—61no—
136C5×Q8C3×C22⋊C411yescharacteristic, complemented
137C2×C4×C5—31no—
138C2×C4×C5—31no—
139C2×C4×C5—61no—
140C2×C4×C5—61no—
141C2×Dic5—31no—
142C2×Dic5—31no—
143C2×Dic5—62no—
144C4×D5—32no—
145C4×D5—32no—
146C4×D5—62no—
147C4×D5—61no—
148C4×D5—61no—
149C4×D5—62no—
150Dic10—32no—
151Dic10—32no—
152Dic10—64no—
153Dic10—61no—
154Dic10—61no—
155C22×Q8—51no—
156C22×Q8—51no—
157C22×Q8—102no—
158C22×Q8—151no—
159C22×Q8—151no—
160C22×Q8—151no—
161C22×Q8—302no—
162C22×Q8C3×AGL(1,𝔽5)11yescharacteristic, complemented
163C23×C4—151no—
164C4×Q8—102no—
165C4×Q8—302no—
166C2×C4⋊C4—152no—
167C2×C22⋊C4—51no—
168C2×C22⋊C4—151no—
169C2×C42—152no—
170#95—152no—
171#95—152no—
172C2×C3×C5—41no—
173C2×C3×C5—41no—
174C2×C3×C5—41no—
175C2×C3×C5—82no—
176C3×D5—42no—
177C3×D5—42no—
178C3×D5—82no—
179C23×C3—41no—
180C23×C3—201no—
181C23×C3—202no—
182C23×C3—201no—
183C23×C3—202no—
184C23×C3—402no—
185C23×C3—402no—
186C2×C3×C4—204no—
187C2×C3×C4—402no—
188C2×C3×C4—402no—
189SL(2,𝔽3)C22⋊AGL(1,𝔽5)11yescharacteristic, direct product factor
190C22×C5—21no—
191C22×C5—21no—
192C22×C5C4×SL(2,𝔽3)12yescomplemented
193C22×C5C2×C4×A411yescharacteristic
194C2×D5—21no—
195C2×D5—21no—
196C2×D5—24no—
197C2×D5—24no—
198C2×D5C4×SL(2,𝔽3)12yescomplemented
199C2×D5D4×A412yes—
200C2×D5C22×SL(2,𝔽3)11yescharacteristic
201C2×D5C2×#41⋊C311yescharacteristic
202C2×D52−1+4⋊C312yes—
203AGL(1,𝔽5)—24no—
204AGL(1,𝔽5)—122no—
205C4×C5—31no—
206C4×C5—31no—
207C4×C5—61no—
208Dic5—32no—
209Dic5—61no—
210C24—51no—
211C2×Q8—21no—
212C2×Q8—31no—
213C2×Q8—52no—
214C2×Q8—61no—
215C2×Q8—101no—
216C2×Q8—152no—
217C2×Q8—151no—
218C2×Q8—151no—
219C2×Q8—152no—
220C2×Q8—302no—
221C2×Q8—302no—
222C2×Q8—302no—
223C2×Q8—302no—
224C2×Q8—302no—
225C2×Q8—301no—
226C2×Q8—302no—
227C2×Q8C2×C3×AGL(1,𝔽5)11yescharacteristic
228C22×C4—31no—
229C22×C4—52no—
230C22×C4—151no—
231C22×C4—151no—
232C22×C4—151no—
233C22×C4—151no—
234C22×C4—151no—
235C22×C4—152no—
236C22×C4—302no—
237C22×C4—302no—
238C4⋊C4—302no—
239C4⋊C4—302no—
240C22⋊C4—54no—
241C22⋊C4—302no—
242C42—302no—
243C42—302no—
244C3×C5—41noHall
245C22×C3—41no—
246C22×C3—42no—
247C22×C3—81no—
248C22×C3—81no—
249C22×C3—201no—
250C22×C3—202no—
251C22×C3—202no—
252C22×C3—201no—
253C22×C3—202no—
254C22×C3—401no—
255C22×C3—401no—
256C22×C3—404no—
257C22×C3—404no—
258C3×C4—404no—
259C2×C5—22no—
260C2×C5A4×C22⋊C411yescharacteristic
261C2×C5C2×C4×SL(2,𝔽3)11yescharacteristic
262C2×C5#519⋊C311yescharacteristic
263D5—22no—
264D5#548⋊C312yes—
265D5D4×SL(2,𝔽3)12yes—
266C23—51no—
267C23—51no—
268C23—52no—
269C23—52no—
270C23—102no—
271C23—102no—
272C23A4×AGL(1,𝔽5)11yescharacteristic
273Q8—31no—
274Q8—61no—
275Q8—61no—
276Q8—152no—
277Q8—152no—
278Q8—301no—
279Q8—301no—
280Q8—304no—
281Q8C3×C22⋊AGL(1,𝔽5)11yescharacteristic, complemented
282C2×C4—31no—
283C2×C4—31no—
284C2×C4—54no—
285C2×C4—61no—
286C2×C4—61no—
287C2×C4—102no—
288C2×C4—102no—
289C2×C4—152no—
290C2×C4—152no—
291C2×C4—151no—
292C2×C4—151no—
293C2×C4—301no—
294C2×C4—301no—
295C2×C4—302no—
296C2×C4—302no—
297C2×C4—302no—
298C2×C4—302no—
299C2×C4—302no—
300C2×C4—302no—
301C2×C3—41no—
302C2×C3—41no—
303C2×C3—41no—
304C2×C3—82no—
305C2×C3—202no—
306C2×C3—202no—
307C2×C3—402no—
308C5C22⋊C4×SL(2,𝔽3)11yescharacteristic, complemented, Sylow
309C22—21no—
310C22—21no—
311C22—51no—
312C22—51no—
313C22—52no—
314C22—52no—
315C22—52no—
316C22—104no—
317C22—104no—
318C22—101no—
319C22—101no—
320C22AGL(1,𝔽5)×SL(2,𝔽3)12yescomplemented
321C22C2×A4×AGL(1,𝔽5)11yescharacteristic
322C4—31no—
323C4—31no—
324C4—61no—
325C4—104no—
326C4—152no—
327C4—301no—
328C4—602no—
329C3—41noSylow
330C2—22no—
331C2—52no—
332C2—52no—
333C2—102no—
334C2C2×AGL(1,𝔽5)×SL(2,𝔽3)11yescharacteristic
335C2#66262⋊C311yescharacteristic
336C2A4×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×D5×SL(2,𝔽3)C2
cosocleC2×Q8×D5C22×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×AGL(1,𝔽5)×SL(2,𝔽3)C2semidirect
3. Q8×C22⋊AGL(1,𝔽5)C3semidirect
4. C22×C5×SL(2,𝔽3)C4semidirect
7. C2×D5×SL(2,𝔽3)C4semidirect
12. C2×Q8×AGL(1,𝔽5)C2×C3semidirect
25. C22×C5×Q8C3×C4semidirect
30. C2×Q8×D5C3×C4semidirect
53. C5×SL(2,𝔽3)C22⋊C4semidirect
57. C22×SL(2,𝔽3)AGL(1,𝔽5)semidirect
89. C22⋊AGL(1,𝔽5)SL(2,𝔽3)direct (N × H)
132. C2×AGL(1,𝔽5)C2×SL(2,𝔽3)semidirect
136. C5×Q8C3×C22⋊C4semidirect
162. C22×Q8C3×AGL(1,𝔽5)semidirect
189. SL(2,𝔽3)C22⋊AGL(1,𝔽5)direct (N × H)
192. C22×C5C4×SL(2,𝔽3)semidirect
198. C2×D5C4×SL(2,𝔽3)semidirect
281. Q8C3×C22⋊AGL(1,𝔽5)semidirect
308. C5C22⋊C4×SL(2,𝔽3)semidirect
320. C22AGL(1,𝔽5)×SL(2,𝔽3)semidirect

Maximal subgroups

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

subgroupindex
C22×D5×SL(2,𝔽3)2
C2×AGL(1,𝔽5)×SL(2,𝔽3)2
Q8×C22⋊AGL(1,𝔽5)3
C2×C3×C22⋊AGL(1,𝔽5)4
C22⋊C4×SL(2,𝔽3)5