Finite Group Databasebeta

← A4×#4090 #14514195

Subgroups — A4×#4090 #14514195

Order 1920.

768 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 768 conjugacy classes of subgroups — too many to draw. Only its 111 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×C5×Q8×A4C211yesmaximal, characteristic
2C2×A4×Dic10C211yesmaximal, characteristic
3A4×#683C214yesmaximal
4C4×A4×Dic5C211yesmaximal, characteristic
5C22×#4090C311yesmaximal, characteristic, complemented, Hall
6C5×Q8×A4—22no—
7C2×C4×C5×A4C2212yes—
8C2×C4×C5×A4C2211yescharacteristic
9C2×A4×Dic5C2212yes—
10C2×A4×Dic5C2212yes—
11A4×Dic10—22no—
12C3×#4090—41nomaximal
13A4×C4⋊Q8—51nomaximal, Hall
14C23×C5×Q8C2×C311yescharacteristic
15C23×Dic10C2×C311yescharacteristic
16C2×#4090—31no—
17C2×#4090—31no—
18C2×#4090—34no—
19C2×#4090—31no—
20C2×#4090—31no—
21C22×#683C2×C314yes—
22C22×C4×Dic5C2×C311yescharacteristic
23C22×C5×A4C2311yescharacteristic
24C2×C3×C5×Q8—41no—
25C2×C3×Dic10—41no—
26C4×C5×A4—22no—
27C4×C5×A4D412yes—
28A4×Dic5—22no—
29A4×Dic5Q814yescomplemented
30C3×#683—44no—
31C3×C4×Dic5—41no—
32C2×Q8×A4—51no—
33C2×Q8×A4D511yescharacteristic
34A4×C4⋊C4—54no—
35C42×A4—51no—
36C22×C5×Q8—22no—
37C22×C5×Q8—31no—
38C22×C5×Q8—32no—
39C22×C5×Q8—31no—
40C22×C5×Q8—62no—
41C23×C4×C5C22×C312yes—
42C23×C4×C5C22×C311yescharacteristic
43C23×Dic5C22×C312yes—
44C23×Dic5C22×C312yes—
45C22×Dic10—22no—
46C22×Dic10—31no—
47C22×Dic10—31no—
48C22×Dic10—32no—
49C22×Dic10—62no—
50#4090—34no—
51#4090—34no—
52#4090—32no—
53#4090—34no—
54#4090—31no—
55#4090—31no—
56#4090—34no—
57#4090—31no—
58#4090A411yescharacteristic, direct product factor
59C2×#683—34no—
60C2×#683—34no—
61C2×#683—34no—
62C2×#683—34no—
63C2×C4×Dic5—31no—
64C2×C4×Dic5—31no—
65C2×C4×Dic5—32no—
66C22×C4⋊Q8—51noSylow
67C2×C5×A4C2×D411yescharacteristic
68C2×C5×A4C2×Q812yes—
69C3×C5×Q8—82no—
70C2×C3×C4×C5—41no—
71C2×C3×C4×C5—42no—
72C2×C3×Dic5—42no—
73C2×C3×Dic5—42no—
74C3×Dic10—82no—
75Q8×A4—22no—
76Q8×A4—102no—
77C2×C4×A4—52no—
78C2×C4×A4—52no—
79C2×C4×A4C2×D512yes—
80C2×C4×A4C2×D511yescharacteristic
81C3×C4⋊Q8—201no—
82C24×C5C23×C311yescharacteristic
83C2×C5×Q8—32no—
84C2×C5×Q8—32no—
85C2×C5×Q8—31no—
86C2×C5×Q8—62no—
87C2×C5×Q8—62no—
88C2×C5×Q8—62no—
89C2×C5×Q8—64no—
90C2×C5×Q8—62no—
91C2×C5×Q8—64no—
92C2×C5×Q8C2×A411yescharacteristic
93C22×C4×C5—22no—
94C22×C4×C5—31no—
95C22×C4×C5—31no—
96C22×C4×C5—32no—
97C22×C4×C5—32no—
98C22×C4×C5—32no—
99C22×C4×C5—62no—
100C22×C4×C5C3×D412yes—
101C22×Dic5—22no—
102C22×Dic5—32no—
103C22×Dic5—34no—
104C22×Dic5—32no—
105C22×Dic5—32no—
106C22×Dic5—32no—
107C22×Dic5—62no—
108C22×Dic5C3×Q814yescomplemented
109C2×Dic10—32no—
110C2×Dic10—31no—
111C2×Dic10—32no—
112C2×Dic10—62no—
113C2×Dic10—62no—
114C2×Dic10—64no—
115C2×Dic10—64no—
116C2×Dic10—62no—
117C2×Dic10—62no—
118C2×Dic10C2×A411yescharacteristic
119#683—38no—
120#683—34no—
121#683—34no—
122#683—34no—
123#683C2×A414yes—
124C4×Dic5—32no—
125C4×Dic5—31no—
126C4×Dic5—32no—
127C4×Dic5C2×A411yescharacteristic
128C23×Q8—51no—
129C23×Q8C3×D511yescharacteristic
130C2×C4⋊Q8—151no—
131C2×C4⋊Q8—151no—
132C2×C4⋊Q8—154no—
133C2×C4⋊Q8—151no—
134C2×C4⋊Q8—151no—
135C22×C4⋊C4—54no—
136C22×C42—51no—
137C22×C3×C5—41no—
138C5×A4C4⋊Q811yescharacteristic, complemented
139C3×C4×C5—42no—
140C3×C4×C5—82no—
141C3×Dic5—44no—
142C3×Dic5—82no—
143C22×A4C22×D511yescharacteristic
144C2×C3×Q8—41no—
145C2×C3×Q8—201no—
146C4×A4—22no—
147C4×A4—54no—
148C4×A4—102no—
149C4×A4C22⋊D512yes—
150C3×C4⋊C4—204no—
151C3×C42—201no—
152C23×C5—32no—
153C23×C5—31no—
154C23×C5—31no—
155C23×C5C2×C3×D411yescharacteristic
156C23×C5C2×C3×Q812yes—
157C5×Q8—22no—
158C5×Q8—64no—
159C5×Q8—64no—
160C5×Q8—62no—
161C2×C4×C5—32no—
162C2×C4×C5—31no—
163C2×C4×C5—32no—
164C2×C4×C5—32no—
165C2×C4×C5—32no—
166C2×C4×C5—32no—
167C2×C4×C5—62no—
168C2×C4×C5—62no—
169C2×C4×C5—62no—
170C2×C4×C5—62no—
171C2×C4×C5C22×A412yes—
172C2×C4×C5C22×A411yescharacteristic
173C2×Dic5—34no—
174C2×Dic5—32no—
175C2×Dic5—34no—
176C2×Dic5—34no—
177C2×Dic5—34no—
178C2×Dic5—32no—
179C2×Dic5—62no—
180C2×Dic5—62no—
181C2×Dic5—62no—
182C2×Dic5—62no—
183C2×Dic5C22×A412yes—
184C2×Dic5C22×A412yes—
185Dic10—22no—
186Dic10—64no—
187Dic10—64no—
188Dic10—62no—
189C22×Q8—22no—
190C22×Q8—31no—
191C22×Q8—32no—
192C22×Q8—31no—
193C22×Q8—62no—
194C22×Q8—102no—
195C22×Q8—151no—
196C22×Q8—151no—
197C22×Q8—152no—
198C22×Q8—302no—
199C23×C4—52no—
200C23×C4—52no—
201C23×C4C2×C3×D512yes—
202C23×C4C2×C3×D511yescharacteristic
203C4⋊Q8—51no—
204C4⋊Q8—151no—
205C4⋊Q8—154no—
206C4⋊Q8—151no—
207C4⋊Q8—154no—
208C4⋊Q8—154no—
209C4⋊Q8—154no—
210C4⋊Q8—152no—
211C4⋊Q8—151no—
212C2×C4⋊C4—154no—
213C2×C4⋊C4—154no—
214C2×C4⋊C4—154no—
215C2×C4⋊C4—154no—
216C2×C42—151no—
217C2×C42—152no—
218C2×C42—151no—
219C2×C3×C5—41no—
220C2×C3×C5—42no—
221C2×A4Q8×D512yes—
222C2×A4C2×C22⋊D511yescharacteristic
223C3×Q8—82no—
224C3×Q8—402no—
225C2×C3×C4—41no—
226C2×C3×C4—42no—
227C2×C3×C4—202no—
228C2×C3×C4—202no—
229C22×C5—31no—
230C22×C5—32no—
231C22×C5—32no—
232C22×C5—31no—
233C22×C5—31no—
234C22×C5—32no—
235C22×C5—32no—
236C22×C5C3×C4⋊Q811yescharacteristic, complemented
237C22×C5C23×A411yescharacteristic
238C4×C5—22no—
239C4×C5—32no—
240C4×C5—62no—
241C4×C5D4×A412yes—
242Dic5—22no—
243Dic5—34no—
244Dic5—62no—
245Dic5Q8×A414yescomplemented
246C24C22×C3×D511yescharacteristic
247C2×Q8—32no—
248C2×Q8—32no—
249C2×Q8—31no—
250C2×Q8—51no—
251C2×Q8—62no—
252C2×Q8—62no—
253C2×Q8—62no—
254C2×Q8—64no—
255C2×Q8—62no—
256C2×Q8—64no—
257C2×Q8—152no—
258C2×Q8—152no—
259C2×Q8—151no—
260C2×Q8—302no—
261C2×Q8—302no—
262C2×Q8—302no—
263C2×Q8—304no—
264C2×Q8—302no—
265C2×Q8—304no—
266C2×Q8D5×A411yescharacteristic
267C22×C4—22no—
268C22×C4—31no—
269C22×C4—31no—
270C22×C4—32no—
271C22×C4—32no—
272C22×C4—32no—
273C22×C4—54no—
274C22×C4—62no—
275C22×C4—102no—
276C22×C4—152no—
277C22×C4—152no—
278C22×C4—152no—
279C22×C4—154no—
280C22×C4—152no—
281C22×C4—302no—
282C22×C4C3×C22⋊D512yes—
283C4⋊C4—54no—
284C4⋊C4—154no—
285C4⋊C4—154no—
286C4⋊C4—158no—
287C4⋊C4—154no—
288C42—51no—
289C42—152no—
290C42—152no—
291C42—151no—
292C3×C5—41noHall
293C22×C3—41no—
294A4#409011yescharacteristic, direct product factor
295C3×C4—42no—
296C3×C4—82no—
297C3×C4—204no—
298C3×C4—402no—
299C2×C5—31no—
300C2×C5—32no—
301C2×C5—31no—
302C2×C5C2×D4×A411yescharacteristic
303C2×C5C2×Q8×A412yes—
304C23—31no—
305C23—32no—
306C23—31no—
307C23C3×Q8×D512yes—
308C23C2×C3×C22⋊D511yescharacteristic
309Q8—22no—
310Q8—64no—
311Q8—64no—
312Q8—62no—
313Q8—102no—
314Q8—304no—
315Q8—304no—
316Q8—302no—
317C2×C4—32no—
318C2×C4—31no—
319C2×C4—32no—
320C2×C4—32no—
321C2×C4—32no—
322C2×C4—32no—
323C2×C4—52no—
324C2×C4—52no—
325C2×C4—62no—
326C2×C4—62no—
327C2×C4—62no—
328C2×C4—62no—
329C2×C4—154no—
330C2×C4—154no—
331C2×C4—154no—
332C2×C4—152no—
333C2×C4—154no—
334C2×C4—152no—
335C2×C4—302no—
336C2×C4—302no—
337C2×C4—302no—
338C2×C4—302no—
339C2×C4C2×D5×A412yes—
340C2×C4C2×D5×A411yescharacteristic
341C2×C3—42no—
342C2×C3—41no—
343C5A4×C4⋊Q811yescharacteristic, complemented, Sylow
344C22—32no—
345C22—31no—
346C22—32no—
347C22—32no—
348C22—32no—
349C22—31no—
350C22—31no—
351C22C3×#409011yescharacteristic, complemented
352C22C22×D5×A411yescharacteristic
353C4—22no—
354C4—32no—
355C4—54no—
356C4—62no—
357C4—102no—
358C4—154no—
359C4—302no—
360C4A4×C22⋊D512yes—
361C3—41noSylow
362C2—31no—
363C2—32no—
364C2—31no—
365C2Q8×D5×A412yes—
366C2C2×A4×C22⋊D511yescharacteristic
subgroupG/H
center, FrattiniC22C22×D5×A4
commutator, socle, cosocleC24×C5C23×C3
FittingC23×C5×Q8C2×C3
prime radicalC22×C5×A4C23
hypercenterC2×Q8D5×A4
hypocenterC22×C5C3×C4⋊Q8
LCSG ▷ C24×C5 ▷ C22×C5
UCSC1 ◁ C22 ◁ C2×Q8

Complements

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

NH ≅ G/Ntype
5. C22×#4090C3semidirect
29. A4×Dic5Q8semidirect
58. #4090A4direct (N × H)
108. C22×Dic5C3×Q8semidirect
138. C5×A4C4⋊Q8semidirect
236. C22×C5C3×C4⋊Q8semidirect
245. Dic5Q8×A4semidirect
294. A4#4090direct (N × H)
343. C5A4×C4⋊Q8semidirect
351. C22C3×#4090semidirect

Maximal subgroups

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

subgroupindex
C2×C5×Q8×A42
C2×A4×Dic102
A4×#6832
C4×A4×Dic52
C22×#40903
C3×#40904
A4×C4⋊Q85