Finite Group Databasebeta

← A4×#4009 #14512568

Subgroups — A4×#4009 #14512568

Order 1920.

412 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 412 conjugacy classes of subgroups — too many to draw. Only its 54 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
2A4×C4⋊AGL(1,𝔽5)C211yesmaximal, characteristic
3A4×D5⋊C8C211yesmaximal, characteristic
4C22×#4009C311yesmaximal, characteristic, complemented, Hall
5C2×A4×AGL(1,𝔽5)—21no—
6C5×Q8×A4C411yescharacteristic, complemented
7C4×D5×A4—21no—
8C4×D5×A4C2211yescharacteristic
9A4×Dic10—21no—
10A4×Dic10C411yescharacteristic, complemented
11A4×#170—21no—
12C3×#4009—41nomaximal
13A4×#103—51nomaximal, Hall
14C22×Q8×D5C2×C311yescharacteristic
15C22×C4⋊AGL(1,𝔽5)C2×C311yescharacteristic
16C22×D5⋊C8C2×C311yescharacteristic
17C2×#4009—31no—
18C2×#4009—31no—
19C2×#4009—31no—
20C2×#4009—31no—
21C2×D5×A4D411yescharacteristic
22A4×AGL(1,𝔽5)—41no—
23C3×Q8×D5—41no—
24C4×C5×A4—21no—
25C4×C5×A4C2×C411yescharacteristic
26C3×C4⋊AGL(1,𝔽5)—41no—
27C3×D5⋊C8—41no—
28A4×Dic5—21no—
29A4×Dic5D411yescharacteristic
30C2×Q8×A4—51no—
31C2×C8×A4—51no—
32A4×C4⋊C4—51no—
33C23×AGL(1,𝔽5)—21no—
34C22×C5×Q8C3×C411yescharacteristic, complemented
35C2×Q8×D5—31no—
36C2×Q8×D5—31no—
37C2×Q8×D5—31no—
38C2×Q8×D5—31no—
39C2×Q8×D5—61no—
40C2×Q8×D5—61no—
41C22×C4×D5—21no—
42C22×C4×D5C22×C311yescharacteristic
43C22×Dic10—21no—
44C22×Dic10C3×C411yescharacteristic, complemented
45C22×#170—21no—
46C2×C4⋊AGL(1,𝔽5)—31no—
47C2×C4⋊AGL(1,𝔽5)—31no—
48C2×C4⋊AGL(1,𝔽5)—61no—
49C2×D5⋊C8—31no—
50C2×D5⋊C8—31no—
51C2×D5⋊C8—61no—
52#4009—32no—
53#4009—31no—
54#4009—31no—
55#4009—31no—
56#4009A411yescharacteristic, direct product factor
57C22×#103—51noSylow
58C2×C5×A4C22⋊C411yescharacteristic
59C2×C3×AGL(1,𝔽5)—81no—
60D5×A4SD1611yescharacteristic
61D5×A4Q1611yescharacteristic
62C3×C5×Q8—41no—
63C3×C4×D5—41no—
64C3×C4×D5—81no—
65C3×Dic10—41no—
66C3×Dic10—81no—
67C3×#170—81no—
68Q8×A4—51no—
69Q8×A4—101no—
70Q8×A4AGL(1,𝔽5)11yescharacteristic, complemented
71C2×C4×A4—51no—
72C2×C4×A4—101no—
73C2×C4×A4—101no—
74C8×A4—101no—
75C3×#103—201no—
76C23×D5C3×D411yescharacteristic
77C22×AGL(1,𝔽5)—41no—
78C22×AGL(1,𝔽5)—61no—
79C22×AGL(1,𝔽5)—61no—
80C22×AGL(1,𝔽5)—121no—
81C2×C5×Q8—31no—
82C2×C5×Q8—31no—
83C2×C5×Q8—61no—
84C22×C4×C5—21no—
85C22×C4×C5C2×C3×C411yescharacteristic
86C22×Dic5—21no—
87C22×Dic5C3×D411yescharacteristic
88Q8×D5—32no—
89Q8×D5—31no—
90Q8×D5—31no—
91Q8×D5—31no—
92Q8×D5—61no—
93Q8×D5—61no—
94Q8×D5—62no—
95Q8×D5—62no—
96Q8×D5—61no—
97Q8×D5—61no—
98Q8×D5C2×A411yescharacteristic
99C2×C4×D5—31no—
100C2×C4×D5—31no—
101C2×C4×D5—31no—
102C2×C4×D5—31no—
103C2×C4×D5—61no—
104C2×C4×D5—61no—
105C2×C4×D5—61no—
106C2×C4×D5—61no—
107C2×Dic10—31no—
108C2×Dic10—31no—
109C2×Dic10—61no—
110C2×Dic10—61no—
111C2×Dic10—61no—
112C2×Dic10—61no—
113C2×Dic10—61no—
114C2×#170—61no—
115C2×#170—61no—
116C4⋊AGL(1,𝔽5)—31no—
117C4⋊AGL(1,𝔽5)—61no—
118C4⋊AGL(1,𝔽5)—61no—
119C4⋊AGL(1,𝔽5)C2×A411yescharacteristic
120D5⋊C8—31no—
121D5⋊C8—61no—
122D5⋊C8—61no—
123D5⋊C8C2×A411yescharacteristic
124C23×Q8—51no—
125C23×C8—51no—
126C22×C4⋊C4—51no—
127C2×#103—151no—
128C2×#103—151no—
129C2×#103—151no—
130C2×#103—151no—
131C2×C3×D5—41no—
132C5×A4#10311yescharacteristic, complemented
133C3×AGL(1,𝔽5)—161no—
134C3×C4×C5—41no—
135C3×C4×C5—81no—
136C3×Dic5—41no—
137C3×Dic5—81no—
138C22×A4—51no—
139C2×C3×Q8—201no—
140C4×A4—21no—
141C4×A4—51no—
142C4×A4—101no—
143C4×A4—201no—
144C4×A4C2×AGL(1,𝔽5)11yescharacteristic
145C2×C3×C8—201no—
146C3×C4⋊C4—201no—
147C23×C5C3×C22⋊C411yescharacteristic
148C22×D5—31no—
149C22×D5—31no—
150C22×D5—31no—
151C22×D5—31no—
152C22×D5C3×SD1611yescharacteristic
153C22×D5C3×Q1611yescharacteristic
154C2×AGL(1,𝔽5)—21no—
155C2×AGL(1,𝔽5)—61no—
156C2×AGL(1,𝔽5)—121no—
157C2×AGL(1,𝔽5)—121no—
158C2×AGL(1,𝔽5)—121no—
159C2×AGL(1,𝔽5)—121no—
160C5×Q8—31no—
161C5×Q8—61no—
162C5×Q8—61no—
163C5×Q8C4×A411yescharacteristic, complemented
164C2×C4×C5—31no—
165C2×C4×C5—31no—
166C2×C4×C5—61no—
167C2×C4×C5—61no—
168C2×Dic5—31no—
169C2×Dic5—31no—
170C2×Dic5—61no—
171C2×Dic5—61no—
172C4×D5—21no—
173C4×D5—32no—
174C4×D5—31no—
175C4×D5—31no—
176C4×D5—31no—
177C4×D5—61no—
178C4×D5—61no—
179C4×D5—62no—
180C4×D5—61no—
181C4×D5C22×A411yescharacteristic
182Dic10—21no—
183Dic10—31no—
184Dic10—61no—
185Dic10—61no—
186Dic10—61no—
187Dic10—62no—
188Dic10—61no—
189Dic10—61no—
190Dic10C4×A411yescharacteristic, complemented
191#170—21no—
192#170—61no—
193C22×Q8—51no—
194C22×Q8—101no—
195C22×Q8—151no—
196C22×Q8—151no—
197C22×Q8—151no—
198C22×Q8—151no—
199C22×Q8—301no—
200C22×Q8—301no—
201C22×Q8C3×AGL(1,𝔽5)11yescharacteristic, complemented
202C23×C4—51no—
203C23×C4—101no—
204C23×C4—101no—
205C22×C8—101no—
206C22×C8—151no—
207C22×C8—151no—
208C22×C8—301no—
209C2×C4⋊C4—151no—
210C2×C4⋊C4—151no—
211C2×C4⋊C4—301no—
212#103—51no—
213#103—151no—
214#103—151no—
215#103—151no—
216#103—152no—
217C2×C3×C5—41no—
218C3×D5—41no—
219C3×D5—41no—
220C2×A4—51no—
221C2×A4—51no—
222C2×A4C22⋊AGL(1,𝔽5)11yescharacteristic
223C3×Q8—41no—
224C3×Q8—201no—
225C3×Q8—401no—
226C2×C3×C4—201no—
227C2×C3×C4—401no—
228C2×C3×C4—401no—
229C3×C8—401no—
230C22×C5—31no—
231C22×C5—31no—
232C22×C5C3×#10311yescharacteristic, complemented
233C2×D5—32no—
234C2×D5—31no—
235C2×D5—31no—
236C2×D5—31no—
237C2×D5—31no—
238C2×D5—31no—
239C2×D5—31no—
240C2×D5—31no—
241C2×D5D4×A411yescharacteristic
242AGL(1,𝔽5)—41no—
243AGL(1,𝔽5)—121no—
244C4×C5—21no—
245C4×C5—31no—
246C4×C5—61no—
247C4×C5C2×C4×A411yescharacteristic
248Dic5—21no—
249Dic5—31no—
250Dic5—61no—
251Dic5D4×A411yescharacteristic
252C24—51no—
253C2×Q8—31no—
254C2×Q8—31no—
255C2×Q8—51no—
256C2×Q8—61no—
257C2×Q8—151no—
258C2×Q8—151no—
259C2×Q8—151no—
260C2×Q8—151no—
261C2×Q8—152no—
262C2×Q8—151no—
263C2×Q8—301no—
264C2×Q8—301no—
265C2×Q8—301no—
266C2×Q8—301no—
267C2×Q8—301no—
268C2×Q8—301no—
269C2×Q8—301no—
270C2×Q8—301no—
271C2×Q8—302no—
272C2×Q8—302no—
273C2×Q8—301no—
274C22×C4—21no—
275C22×C4—51no—
276C22×C4—101no—
277C22×C4—151no—
278C22×C4—151no—
279C22×C4—151no—
280C22×C4—151no—
281C22×C4—201no—
282C22×C4—301no—
283C22×C4—301no—
284C22×C4—301no—
285C22×C4—301no—
286C22×C4—301no—
287C22×C4—301no—
288C22×C4—601no—
289C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
290C2×C8—51no—
291C2×C8—151no—
292C2×C8—301no—
293C2×C8—301no—
294C2×C8—301no—
295C2×C8—301no—
296C4⋊C4—51no—
297C4⋊C4—151no—
298C4⋊C4—301no—
299C4⋊C4—301no—
300C3×C5—41noHall
301C22×C3—201no—
302A4#400911yescharacteristic, direct product factor
303C3×C4—41no—
304C3×C4—81no—
305C3×C4—201no—
306C3×C4—401no—
307C3×C4—801no—
308C2×C5—31no—
309C2×C5—31no—
310C2×C5A4×C22⋊C411yescharacteristic
311D5—31no—
312D5—31no—
313D5A4×SD1611yescharacteristic
314D5A4×Q1611yescharacteristic
315C23—51no—
316C23—51no—
317C23—151no—
318C23—151no—
319C23—151no—
320C23—151no—
321C23C3×C22⋊AGL(1,𝔽5)11yescharacteristic
322Q8—31no—
323Q8—51no—
324Q8—61no—
325Q8—61no—
326Q8—101no—
327Q8—151no—
328Q8—301no—
329Q8—301no—
330Q8—301no—
331Q8—301no—
332Q8—301no—
333Q8—302no—
334Q8A4×AGL(1,𝔽5)11yescharacteristic, complemented
335C2×C4—31no—
336C2×C4—31no—
337C2×C4—51no—
338C2×C4—61no—
339C2×C4—61no—
340C2×C4—101no—
341C2×C4—101no—
342C2×C4—151no—
343C2×C4—151no—
344C2×C4—152no—
345C2×C4—151no—
346C2×C4—151no—
347C2×C4—151no—
348C2×C4—301no—
349C2×C4—301no—
350C2×C4—301no—
351C2×C4—302no—
352C2×C4—301no—
353C2×C4—301no—
354C2×C4—301no—
355C2×C4—601no—
356C2×C4—601no—
357C2×C4—601no—
358C2×C4—601no—
359C8—101no—
360C8—301no—
361C2×C3—41no—
362C2×C3—201no—
363C2×C3—201no—
364C5A4×#10311yescharacteristic, complemented, Sylow
365C22—31no—
366C22—31no—
367C22—51no—
368C22—151no—
369C22—151no—
370C22—151no—
371C22—151no—
372C22—151no—
373C22—151no—
374C22—152no—
375C22—151no—
376C22C3×#400911yescharacteristic, complemented
377C4—21no—
378C4—31no—
379C4—51no—
380C4—61no—
381C4—101no—
382C4—151no—
383C4—201no—
384C4—301no—
385C4—601no—
386C4C2×A4×AGL(1,𝔽5)11yescharacteristic
387C3—41noSylow
388C2—31no—
389C2—31no—
390C2—51no—
391C2—51no—
392C2—151no—
393C2—151no—
394C2A4×C22⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
centerC2A4×C22⋊AGL(1,𝔽5)
commutatorC22×C4×C5C2×C3×C4
FrattiniC4C2×A4×AGL(1,𝔽5)
FittingC22×C5×Q8C3×C4
socleC23×C5C3×C22⋊C4
prime radicalC2×D5×A4D4
cosocleC22×C4×D5C22×C3
hypercenterQ8A4×AGL(1,𝔽5)
hypocenterC22×C5C3×#103
LCSG ▷ C22×C4×C5 ▷ C23×C5 ▷ C22×C5
UCSC1 ◁ C2 ◁ C4 ◁ Q8

Complements

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

NH ≅ G/Ntype
4. C22×#4009C3semidirect
6. C5×Q8×A4C4semidirect
10. A4×Dic10C4semidirect
34. C22×C5×Q8C3×C4semidirect
44. C22×Dic10C3×C4semidirect
56. #4009A4direct (N × H)
70. Q8×A4AGL(1,𝔽5)semidirect
132. C5×A4#103semidirect
163. C5×Q8C4×A4semidirect
190. Dic10C4×A4semidirect
201. C22×Q8C3×AGL(1,𝔽5)semidirect
232. C22×C5C3×#103semidirect
302. A4#4009direct (N × H)
334. Q8A4×AGL(1,𝔽5)semidirect
364. C5A4×#103semidirect
376. C22C3×#4009semidirect

Maximal subgroups

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

subgroupindex
Q8×D5×A42
A4×C4⋊AGL(1,𝔽5)2
A4×D5⋊C82
C22×#40093
C3×#40094
A4×#1035