Finite Group Databasebeta

← Q8×A4⋊AGL(1,𝔽5) #14514376

Subgroups — Q8×A4⋊AGL(1,𝔽5) #14514376

Order 1920.

702 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 702 conjugacy classes of subgroups — too many to draw. Only its 88 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
1Q8×D5×A4C211yesmaximal, characteristic
2C4⋊(A4⋊AGL(1,𝔽5))C213yesmaximal
3C4×A4⋊AGL(1,𝔽5)C213yesmaximal
4Q8×C22⋊AGL(1,𝔽5)—31nomaximal, Hall
5C2×A4⋊AGL(1,𝔽5)C2213yes—
6C2×A4⋊AGL(1,𝔽5)C2211yescharacteristic
7C5×Q8×A4C411yescharacteristic, complemented
8Q8×ΓL(1,𝔽16)—41nomaximal
9C4×D5×A4C2213yes—
10A4×Dic10C413yescomplemented
11Q8×SL(2,ℤ/4)—51nomaximal, Hall
12C22×Q8×D5D311yescharacteristic
13C2×Q8×AGL(1,𝔽5)—31no—
14C2×Q8×AGL(1,𝔽5)—31no—
15(C2×AGL(1,𝔽5))⋊Q8—33no—
16(C2×AGL(1,𝔽5))⋊Q8—33no—
17#65765—33no—
18C4×C22⋊AGL(1,𝔽5)—33no—
19C2×D5×A4C2311yescharacteristic
20A4⋊AGL(1,𝔽5)—23no—
21A4⋊AGL(1,𝔽5)Q812yesdirect product factor
22C3×Q8×D5—41no—
23C4×C5×A4C2×C413yes—
24C4⋊ΓL(1,𝔽16)—43no—
25C4×ΓL(1,𝔽16)—43no—
26A4×Dic5C2×C413yes—
27C2×Q8×A4—51no—
28C4⋊SL(2,ℤ/4)—53no—
29C4×SL(2,ℤ/4)—53no—
30C22×C5×Q8Dic311yescharacteristic, complemented
31C2×Q8×D5—31no—
32C2×Q8×D5—33no—
33C2×Q8×D5—31no—
34C2×Q8×D5—33no—
35C22×C4×D5C2×D313yes—
36C22×Dic10Dic313yescomplemented
37C2×C22⋊AGL(1,𝔽5)—31no—
38C2×C22⋊AGL(1,𝔽5)—33no—
39Q8×AGL(1,𝔽5)—61no—
40Q8×AGL(1,𝔽5)—63no—
41Q8×AGL(1,𝔽5)—63no—
42Q8×AGL(1,𝔽5)—61no—
43C2×C4⋊AGL(1,𝔽5)—33no—
44C2×C4⋊AGL(1,𝔽5)—33no—
45C2×C4×AGL(1,𝔽5)—33no—
46C2×C4×AGL(1,𝔽5)—33no—
47(C2×AGL(1,𝔽5))⋊C4—33no—
48(C2×AGL(1,𝔽5))⋊C4—33no—
49(C2×AGL(1,𝔽5))⋊C4—36no—
50Q8×C22⋊C4—151noSylow
51C2×C5×A4C22×C411yescharacteristic
52C2×ΓL(1,𝔽16)—41no—
53C2×ΓL(1,𝔽16)—43no—
54D5×A4C2×Q811yescharacteristic
55D5×A4#4111yescharacteristic
56C3×C5×Q8—41no—
57C3×C4×D5—43no—
58C3×Dic10—43no—
59Q8×A4—53no—
60Q8×A4AGL(1,𝔽5)11yescharacteristic, complemented
61C2×C4×A4—53no—
62C2×SL(2,ℤ/4)—51no—
63C2×SL(2,ℤ/4)—53no—
64Q8×Dic3—201no—
65C23×D5C22×D311yescharacteristic
66C22×AGL(1,𝔽5)—31no—
67C22×AGL(1,𝔽5)—31no—
68C22×AGL(1,𝔽5)—33no—
69C22×AGL(1,𝔽5)—33no—
70C2×C5×Q8—31no—
71C2×C5×Q8—33no—
72C22×C4×C5C2×Dic313yes—
73C22×Dic5C2×Dic313yes—
74Q8×D5—31no—
75Q8×D5—33no—
76Q8×D5—33no—
77Q8×D5—63no—
78Q8×D5—63no—
79Q8×D5—61no—
80Q8×D5S411yescharacteristic
81C2×C4×D5—33no—
82C2×C4×D5—33no—
83C2×C4×D5—33no—
84C2×C4×D5—33no—
85C2×Dic10—33no—
86C2×Dic10—33no—
87C2×Dic10—36no—
88C22⋊AGL(1,𝔽5)—32no—
89C22⋊AGL(1,𝔽5)—32no—
90C22⋊AGL(1,𝔽5)—63no—
91C22⋊AGL(1,𝔽5)—63no—
92C4⋊AGL(1,𝔽5)—63no—
93C4⋊AGL(1,𝔽5)—63no—
94C4⋊AGL(1,𝔽5)—63no—
95C4⋊AGL(1,𝔽5)—63no—
96C4×AGL(1,𝔽5)—63no—
97C4×AGL(1,𝔽5)—63no—
98C4×AGL(1,𝔽5)—63no—
99C4×AGL(1,𝔽5)—63no—
100C23×Q8—51no—
101C2×C4×Q8—151no—
102C2×C4×Q8—151no—
103#391—153no—
104#391—153no—
105#380—153no—
106C4×C22⋊C4—153no—
107C2×C3×D5—41no—
108C5×A4C4×Q811yescharacteristic, complemented
109ΓL(1,𝔽16)—42no—
110ΓL(1,𝔽16)—83no—
111C3×C4×C5—43no—
112C3×Dic5—43no—
113C22×A4—51no—
114C2×C3×Q8—201no—
115C4×A4—53no—
116C4×A4C2×AGL(1,𝔽5)13yes—
117SL(2,ℤ/4)—52no—
118SL(2,ℤ/4)—103no—
119C4⋊Dic3—203no—
120C4×Dic3—203no—
121C23×C5C22×Dic311yescharacteristic
122C22×D5—31no—
123C22×D5—31no—
124C22×D5—31no—
125C22×D5—31no—
126C22×D5D3×Q811yescharacteristic
127C22×D5#23911yescharacteristic
128C2×AGL(1,𝔽5)—32no—
129C2×AGL(1,𝔽5)—32no—
130C2×AGL(1,𝔽5)—61no—
131C2×AGL(1,𝔽5)—61no—
132C2×AGL(1,𝔽5)—61no—
133C2×AGL(1,𝔽5)—61no—
134C2×AGL(1,𝔽5)—63no—
135C2×AGL(1,𝔽5)—63no—
136C2×AGL(1,𝔽5)—63no—
137C2×AGL(1,𝔽5)—63no—
138C2×AGL(1,𝔽5)—63no—
139C2×AGL(1,𝔽5)—63no—
140C5×Q8—33no—
141C5×Q8—61no—
142C5×Q8SL(2,ℤ/4)11yescharacteristic, complemented
143C2×C4×C5—33no—
144C2×C4×C5—33no—
145C2×Dic5—33no—
146C2×Dic5—33no—
147C4×D5—33no—
148C4×D5—33no—
149C4×D5—33no—
150C4×D5—63no—
151C4×D5C2×S413yes—
152Dic10—33no—
153Dic10—36no—
154Dic10—63no—
155Dic10SL(2,ℤ/4)13yescomplemented
156C22×Q8—53no—
157C22×Q8—151no—
158C22×Q8—151no—
159C22×Q8—153no—
160C22×Q8—153no—
161C22×Q8ΓL(1,𝔽16)11yescharacteristic, complemented
162C23×C4—53no—
163C4×Q8—301no—
164C4×Q8—303no—
165C4×Q8—303no—
166C4×Q8—301no—
167C2×C4⋊C4—153no—
168C2×C4⋊C4—153no—
169C2×C22⋊C4—151no—
170C2×C22⋊C4—153no—
171C2×C42—153no—
172C2×C42—153no—
173#95—153no—
174#95—153no—
175#95—156no—
176C2×C3×C5—41no—
177C3×D5—41no—
178C3×D5—41no—
179C2×A4—51no—
180C2×A4—51no—
181C2×A4C22×AGL(1,𝔽5)11yescharacteristic
182C3×Q8—41no—
183C3×Q8—203no—
184C2×C3×C4—203no—
185C2×Dic3—201no—
186C2×Dic3—203no—
187C22×C5—31no—
188C22×C5—31no—
189C22×C5Q8×Dic311yescharacteristic, complemented
190C2×D5—31no—
191C2×D5—31no—
192C2×D5—31no—
193C2×D5—31no—
194C2×D5—31no—
195C2×D5—31no—
196C2×D5—31no—
197C2×D5—61no—
198C2×D5C22×S411yescharacteristic
199AGL(1,𝔽5)—62no—
200AGL(1,𝔽5)—62no—
201AGL(1,𝔽5)—123no—
202AGL(1,𝔽5)—123no—
203C4×C5—33no—
204C4×C5C2×SL(2,ℤ/4)13yes—
205Dic5—33no—
206Dic5C2×SL(2,ℤ/4)13yes—
207C24—51no—
208C2×Q8—31no—
209C2×Q8—33no—
210C2×Q8—51no—
211C2×Q8—153no—
212C2×Q8—153no—
213C2×Q8—156no—
214C2×Q8—153no—
215C2×Q8—153no—
216C2×Q8—151no—
217C2×Q8—301no—
218C2×Q8—303no—
219C2×Q8—303no—
220C22×C4—53no—
221C22×C4—153no—
222C22×C4—153no—
223C22×C4—153no—
224C22×C4—153no—
225C22×C4—151no—
226C22×C4—151no—
227C22×C4—153no—
228C22×C4—153no—
229C22×C4C2×ΓL(1,𝔽16)13yes—
230C4⋊C4—303no—
231C4⋊C4—303no—
232C4⋊C4—303no—
233C4⋊C4—303no—
234C22⋊C4—152no—
235C22⋊C4—152no—
236C22⋊C4—303no—
237C22⋊C4—303no—
238C42—303no—
239C42—303no—
240C42—303no—
241C42—303no—
242C3×C5—41noHall
243C22×C3—201no—
244A4Q8×AGL(1,𝔽5)11yescharacteristic, complemented
245C3×C4—43no—
246C3×C4—203no—
247Dic3—202no—
248Dic3—403no—
249C2×C5—31no—
250C2×C5—31no—
251C2×C5C22×SL(2,ℤ/4)11yescharacteristic
252D5—31no—
253D5—31no—
254D5Q8×S411yescharacteristic
255D5#591911yescharacteristic
256C23—51no—
257C23—51no—
258C23—151no—
259C23—151no—
260C23—151no—
261C23—151no—
262C23C22×ΓL(1,𝔽16)11yescharacteristic
263Q8—33no—
264Q8—53no—
265Q8—61no—
266Q8—156no—
267Q8—153no—
268Q8—303no—
269Q8A4⋊AGL(1,𝔽5)11yescharacteristic, direct product factor
270C2×C4—33no—
271C2×C4—33no—
272C2×C4—53no—
273C2×C4—153no—
274C2×C4—153no—
275C2×C4—153no—
276C2×C4—153no—
277C2×C4—153no—
278C2×C4—152no—
279C2×C4—152no—
280C2×C4—303no—
281C2×C4—301no—
282C2×C4—301no—
283C2×C4—301no—
284C2×C4—301no—
285C2×C4—303no—
286C2×C4—303no—
287C2×C4—303no—
288C2×C4—303no—
289C2×C4—303no—
290C2×C4—303no—
291C2×C3—41no—
292C2×C3—201no—
293C2×C3—201no—
294C5Q8×SL(2,ℤ/4)11yescharacteristic, complemented, Sylow
295C22—31no—
296C22—31no—
297C22—51no—
298C22—151no—
299C22—151no—
300C22—151no—
301C22—151no—
302C22—151no—
303C22—151no—
304C22—151no—
305C22—301no—
306C22Q8×ΓL(1,𝔽16)11yescharacteristic, complemented
307C4—33no—
308C4—53no—
309C4—153no—
310C4—302no—
311C4—302no—
312C4—603no—
313C4—603no—
314C4C2×A4⋊AGL(1,𝔽5)13yes—
315C3—41noSylow
316C2—31no—
317C2—31no—
318C2—51no—
319C2—51no—
320C2—151no—
321C2—151no—
322C2C22×A4⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
center, FrattiniC2C22×A4⋊AGL(1,𝔽5)
commutatorC2×C5×A4C22×C4
FittingC22×C5×Q8Dic3
socleC23×C5C22×Dic3
prime radical, cosocleC2×D5×A4C23
hypercenterQ8A4⋊AGL(1,𝔽5)
hypocenterC5×A4C4×Q8
LCSG ▷ C2×C5×A4 ▷ C5×A4
UCSC1 ◁ C2 ◁ Q8

Complements

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

NH ≅ G/Ntype
7. C5×Q8×A4C4semidirect
10. A4×Dic10C4semidirect
21. A4⋊AGL(1,𝔽5)Q8direct (N × H)
30. C22×C5×Q8Dic3semidirect
36. C22×Dic10Dic3semidirect
60. Q8×A4AGL(1,𝔽5)semidirect
108. C5×A4C4×Q8semidirect
142. C5×Q8SL(2,ℤ/4)semidirect
155. Dic10SL(2,ℤ/4)semidirect
161. C22×Q8ΓL(1,𝔽16)semidirect
189. C22×C5Q8×Dic3semidirect
244. A4Q8×AGL(1,𝔽5)semidirect
269. Q8A4⋊AGL(1,𝔽5)direct (N × H)
294. C5Q8×SL(2,ℤ/4)semidirect
306. C22Q8×ΓL(1,𝔽16)semidirect

Maximal subgroups

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

subgroupindex
Q8×D5×A42
C4⋊(A4⋊AGL(1,𝔽5))2
C4×A4⋊AGL(1,𝔽5)2
Q8×C22⋊AGL(1,𝔽5)3
Q8×ΓL(1,𝔽16)4
Q8×SL(2,ℤ/4)5