Finite Group Databasebeta

← A4×#4034 #14513994

Subgroups — A4×#4034 #14513994

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×A4×Dic10C212yesmaximal
2C5×A4×C4⋊C4C211yesmaximal, characteristic
3A4×C4⋊Dic5C211yesmaximal, characteristic
4A4×#683C212yesmaximal
5C4×A4×Dic5C211yesmaximal, characteristic
6C22×#4034C311yesmaximal, characteristic, complemented, Hall
7C2×C4×C5×A4C2212yes—
8C2×C4×C5×A4C2211yescharacteristic
9C2×A4×Dic5C2212yes—
10C2×A4×Dic5C2211yescharacteristic
11C2×A4×Dic5C2211yescharacteristic
12A4×Dic10—24no—
13C3×#4034—41nomaximal
14A4×C4⋊Q8—51nomaximal, Hall
15C23×Dic10C2×C312yes—
16C22×C5×C4⋊C4C2×C311yescharacteristic
17C22×C4⋊Dic5C2×C311yescharacteristic
18C22×#683C2×C312yes—
19C22×C4×Dic5C2×C311yescharacteristic
20C2×#4034—31no—
21C2×#4034—31no—
22C2×#4034—32no—
23C2×#4034—31no—
24C2×#4034—32no—
25C2×#4034—31no—
26C22×C5×A4C2311yescharacteristic
27C2×C3×Dic10—42no—
28C4×C5×A4—22no—
29C4×C5×A4Q812yescomplemented
30A4×Dic5—22no—
31A4×Dic5D412yes—
32A4×Dic5Q812yescomplemented
33C3×C5×C4⋊C4—41no—
34C3×C4⋊Dic5—41no—
35C3×#683—42no—
36C3×C4×Dic5—41no—
37C2×Q8×A4—52no—
38A4×C4⋊C4—52no—
39A4×C4⋊C4—51no—
40A4×C4⋊C4D511yescharacteristic
41C42×A4—51no—
42C23×C4×C5C22×C312yes—
43C23×C4×C5C22×C311yescharacteristic
44C23×Dic5C22×C312yes—
45C23×Dic5C22×C311yescharacteristic
46C23×Dic5C22×C311yescharacteristic
47C22×Dic10—24no—
48C22×Dic10—32no—
49C22×Dic10—32no—
50C22×Dic10—32no—
51C22×Dic10—32no—
52C22×Dic10—64no—
53C2×C5×C4⋊C4—31no—
54C2×C5×C4⋊C4—31no—
55C2×C5×C4⋊C4—32no—
56C2×C4⋊Dic5—31no—
57C2×C4⋊Dic5—31no—
58C2×C4⋊Dic5—32no—
59C2×#683—32no—
60C2×#683—32no—
61C2×#683—32no—
62C2×#683—32no—
63C2×C4×Dic5—31no—
64C2×C4×Dic5—31no—
65C2×C4×Dic5—31no—
66C2×C4×Dic5—31no—
67#4034—34no—
68#4034—34no—
69#4034—32no—
70#4034—32no—
71#4034—32no—
72#4034—32no—
73#4034—31no—
74#4034—31no—
75#4034—32no—
76#4034—31no—
77#4034A411yescharacteristic, direct product factor
78C22×C4⋊Q8—51noSylow
79C2×C5×A4C2×D411yescharacteristic
80C2×C5×A4C2×Q811yescharacteristic
81C2×C5×A4C2×Q811yescharacteristic
82C2×C3×C4×C5—41no—
83C2×C3×C4×C5—42no—
84C2×C3×Dic5—41no—
85C2×C3×Dic5—41no—
86C2×C3×Dic5—42no—
87C3×Dic10—84no—
88Q8×A4—104no—
89C2×C4×A4—51no—
90C2×C4×A4—51no—
91C2×C4×A4—52no—
92C2×C4×A4C2×D512yes—
93C2×C4×A4C2×D511yescharacteristic
94C3×C4⋊Q8—201no—
95C24×C5C23×C311yescharacteristic
96C22×C4×C5—22no—
97C22×C4×C5—31no—
98C22×C4×C5—31no—
99C22×C4×C5—32no—
100C22×C4×C5—32no—
101C22×C4×C5—32no—
102C22×C4×C5—62no—
103C22×C4×C5C3×Q812yescomplemented
104C22×Dic5—22no—
105C22×Dic5—31no—
106C22×Dic5—31no—
107C22×Dic5—31no—
108C22×Dic5—32no—
109C22×Dic5—32no—
110C22×Dic5—31no—
111C22×Dic5—32no—
112C22×Dic5—32no—
113C22×Dic5—62no—
114C22×Dic5C3×D412yes—
115C22×Dic5C3×Q812yescomplemented
116C2×Dic10—34no—
117C2×Dic10—32no—
118C2×Dic10—32no—
119C2×Dic10—32no—
120C2×Dic10—64no—
121C2×Dic10—64no—
122C2×Dic10—64no—
123C2×Dic10—64no—
124C2×Dic10—64no—
125C2×Dic10—64no—
126C2×Dic10—64no—
127C2×Dic10—64no—
128C2×Dic10C2×A412yes—
129C5×C4⋊C4—32no—
130C5×C4⋊C4—32no—
131C5×C4⋊C4—31no—
132C5×C4⋊C4C2×A411yescharacteristic
133C4⋊Dic5—32no—
134C4⋊Dic5—31no—
135C4⋊Dic5—32no—
136C4⋊Dic5C2×A411yescharacteristic
137#683—34no—
138#683—32no—
139#683—32no—
140#683—32no—
141#683C2×A412yes—
142C4×Dic5—32no—
143C4×Dic5—31no—
144C4×Dic5—31no—
145C4×Dic5—31no—
146C4×Dic5C2×A411yescharacteristic
147C23×Q8—52no—
148C2×C4⋊Q8—151no—
149C2×C4⋊Q8—151no—
150C2×C4⋊Q8—151no—
151C2×C4⋊Q8—152no—
152C2×C4⋊Q8—151no—
153C2×C4⋊Q8—152no—
154C22×C4⋊C4—52no—
155C22×C4⋊C4—51no—
156C22×C4⋊C4C3×D511yescharacteristic
157C22×C42—51no—
158C22×C3×C5—41no—
159C5×A4C4⋊Q811yescharacteristic, complemented
160C3×C4×C5—42no—
161C3×C4×C5—82no—
162C3×Dic5—42no—
163C3×Dic5—42no—
164C3×Dic5—82no—
165C22×A4C22×D511yescharacteristic
166C2×C3×Q8—202no—
167C4×A4—22no—
168C4×A4—52no—
169C4×A4—52no—
170C4×A4—102no—
171C4×A4Dic1012yescomplemented
172C3×C4⋊C4—41no—
173C3×C4⋊C4—202no—
174C3×C4⋊C4—201no—
175C3×C42—201no—
176C23×C5—31no—
177C23×C5—31no—
178C23×C5—31no—
179C23×C5—31no—
180C23×C5C2×C3×D411yescharacteristic
181C23×C5C2×C3×Q811yescharacteristic
182C23×C5C2×C3×Q811yescharacteristic
183C2×C4×C5—32no—
184C2×C4×C5—31no—
185C2×C4×C5—32no—
186C2×C4×C5—32no—
187C2×C4×C5—32no—
188C2×C4×C5—32no—
189C2×C4×C5—62no—
190C2×C4×C5—62no—
191C2×C4×C5—62no—
192C2×C4×C5—62no—
193C2×C4×C5C22×A412yes—
194C2×C4×C5C22×A411yescharacteristic
195C2×Dic5—32no—
196C2×Dic5—32no—
197C2×Dic5—31no—
198C2×Dic5—31no—
199C2×Dic5—32no—
200C2×Dic5—32no—
201C2×Dic5—32no—
202C2×Dic5—32no—
203C2×Dic5—32no—
204C2×Dic5—32no—
205C2×Dic5—32no—
206C2×Dic5—62no—
207C2×Dic5—62no—
208C2×Dic5—62no—
209C2×Dic5—62no—
210C2×Dic5C22×A412yes—
211C2×Dic5C22×A411yescharacteristic
212C2×Dic5C22×A411yescharacteristic
213Dic10—24no—
214Dic10—64no—
215Dic10—64no—
216Dic10—68no—
217Dic10—64no—
218C22×Q8—104no—
219C22×Q8—152no—
220C22×Q8—152no—
221C22×Q8—152no—
222C22×Q8—152no—
223C22×Q8—304no—
224C23×C4—51no—
225C23×C4—51no—
226C23×C4—52no—
227C23×C4C2×C3×D512yes—
228C23×C4C2×C3×D511yescharacteristic
229C4⋊Q8—51no—
230C4⋊Q8—151no—
231C4⋊Q8—152no—
232C4⋊Q8—151no—
233C4⋊Q8—152no—
234C4⋊Q8—152no—
235C4⋊Q8—152no—
236C4⋊Q8—154no—
237C4⋊Q8—154no—
238C4⋊Q8—152no—
239C4⋊Q8—151no—
240C2×C4⋊C4—31no—
241C2×C4⋊C4—31no—
242C2×C4⋊C4—32no—
243C2×C4⋊C4—152no—
244C2×C4⋊C4—151no—
245C2×C4⋊C4—152no—
246C2×C4⋊C4—152no—
247C2×C4⋊C4—152no—
248C2×C4⋊C4—152no—
249C2×C4⋊C4—151no—
250C2×C42—151no—
251C2×C42—151no—
252C2×C42—151no—
253C2×C42—151no—
254C2×C3×C5—41no—
255C2×C3×C5—41no—
256C2×C3×C5—41no—
257C2×A4C2×Dic1011yescharacteristic
258C2×A4D4×D511yescharacteristic
259C2×A4Q8×D511yescharacteristic
260C3×Q8—404no—
261C2×C3×C4—41no—
262C2×C3×C4—42no—
263C2×C3×C4—201no—
264C2×C3×C4—201no—
265C2×C3×C4—202no—
266C22×C5—31no—
267C22×C5—31no—
268C22×C5—31no—
269C22×C5—32no—
270C22×C5—31no—
271C22×C5—31no—
272C22×C5—31no—
273C22×C5—31no—
274C22×C5—31no—
275C22×C5—31no—
276C22×C5C3×C4⋊Q811yescharacteristic, complemented
277C22×C5C23×A411yescharacteristic
278C4×C5—22no—
279C4×C5—32no—
280C4×C5—62no—
281C4×C5Q8×A412yescomplemented
282Dic5—22no—
283Dic5—32no—
284Dic5—32no—
285Dic5—62no—
286Dic5D4×A412yes—
287Dic5Q8×A412yescomplemented
288C24C22×C3×D511yescharacteristic
289C2×Q8—52no—
290C2×Q8—152no—
291C2×Q8—152no—
292C2×Q8—154no—
293C2×Q8—152no—
294C2×Q8—304no—
295C2×Q8—304no—
296C2×Q8—304no—
297C2×Q8—304no—
298C2×Q8—304no—
299C2×Q8—304no—
300C2×Q8—304no—
301C2×Q8—304no—
302C22×C4—22no—
303C22×C4—31no—
304C22×C4—31no—
305C22×C4—32no—
306C22×C4—32no—
307C22×C4—32no—
308C22×C4—52no—
309C22×C4—52no—
310C22×C4—62no—
311C22×C4—102no—
312C22×C4—151no—
313C22×C4—151no—
314C22×C4—152no—
315C22×C4—151no—
316C22×C4—151no—
317C22×C4—152no—
318C22×C4—152no—
319C22×C4—152no—
320C22×C4—302no—
321C22×C4C3×Dic1012yescomplemented
322C4⋊C4—32no—
323C4⋊C4—32no—
324C4⋊C4—31no—
325C4⋊C4—52no—
326C4⋊C4—51no—
327C4⋊C4—152no—
328C4⋊C4—152no—
329C4⋊C4—152no—
330C4⋊C4—154no—
331C4⋊C4—152no—
332C4⋊C4—152no—
333C4⋊C4—151no—
334C4⋊C4D5×A411yescharacteristic
335C42—51no—
336C42—151no—
337C42—151no—
338C42—152no—
339C42—151no—
340C3×C5—41noHall
341C22×C3—41no—
342A4#403411yescharacteristic, direct product factor
343C3×C4—42no—
344C3×C4—82no—
345C3×C4—202no—
346C3×C4—202no—
347C3×C4—402no—
348C2×C5—31no—
349C2×C5—31no—
350C2×C5—31no—
351C2×C5—31no—
352C2×C5C2×D4×A411yescharacteristic
353C2×C5C2×Q8×A411yescharacteristic
354C2×C5C2×Q8×A411yescharacteristic
355C23—31no—
356C23—31no—
357C23—31no—
358C23—31no—
359C23C2×C3×Dic1011yescharacteristic
360C23C3×D4×D511yescharacteristic
361C23C3×Q8×D511yescharacteristic
362Q8—104no—
363Q8—304no—
364Q8—304no—
365Q8—304no—
366Q8—308no—
367C2×C4—32no—
368C2×C4—31no—
369C2×C4—32no—
370C2×C4—32no—
371C2×C4—32no—
372C2×C4—32no—
373C2×C4—51no—
374C2×C4—51no—
375C2×C4—52no—
376C2×C4—62no—
377C2×C4—62no—
378C2×C4—62no—
379C2×C4—62no—
380C2×C4—152no—
381C2×C4—152no—
382C2×C4—152no—
383C2×C4—152no—
384C2×C4—152no—
385C2×C4—152no—
386C2×C4—151no—
387C2×C4—151no—
388C2×C4—152no—
389C2×C4—152no—
390C2×C4—152no—
391C2×C4—302no—
392C2×C4—302no—
393C2×C4—302no—
394C2×C4—302no—
395C2×C4C2×D5×A412yes—
396C2×C4C2×D5×A411yescharacteristic
397C2×C3—41no—
398C2×C3—41no—
399C2×C3—41no—
400C5A4×C4⋊Q811yescharacteristic, complemented, Sylow
401C22—31no—
402C22—31no—
403C22—31no—
404C22—32no—
405C22—31no—
406C22—31no—
407C22—31no—
408C22—31no—
409C22—31no—
410C22—31no—
411C22C3×#403411yescharacteristic, complemented
412C22C22×D5×A411yescharacteristic
413C4—22no—
414C4—32no—
415C4—52no—
416C4—52no—
417C4—62no—
418C4—102no—
419C4—152no—
420C4—152no—
421C4—302no—
422C4A4×Dic1012yescomplemented
423C3—41noSylow
424C2—31no—
425C2—31no—
426C2—31no—
427C2—31no—
428C2C2×A4×Dic1011yescharacteristic
429C2D4×D5×A411yescharacteristic
430C2Q8×D5×A411yescharacteristic
subgroupG/H
center, FrattiniC22C22×D5×A4
commutator, socle, cosocleC24×C5C23×C3
FittingC22×C5×C4⋊C4C2×C3
prime radicalC22×C5×A4C23
hypercenterC4⋊C4D5×A4
hypocenterC22×C5C3×C4⋊Q8
LCSG ▷ C24×C5 ▷ C22×C5
UCSC1 ◁ C22 ◁ C4⋊C4

Complements

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

NH ≅ G/Ntype
6. C22×#4034C3semidirect
29. C4×C5×A4Q8semidirect
32. A4×Dic5Q8semidirect
77. #4034A4direct (N × H)
103. C22×C4×C5C3×Q8semidirect
115. C22×Dic5C3×Q8semidirect
159. C5×A4C4⋊Q8semidirect
171. C4×A4Dic10semidirect
276. C22×C5C3×C4⋊Q8semidirect
281. C4×C5Q8×A4semidirect
287. Dic5Q8×A4semidirect
321. C22×C4C3×Dic10semidirect
342. A4#4034direct (N × H)
400. C5A4×C4⋊Q8semidirect
411. C22C3×#4034semidirect
422. C4A4×Dic10semidirect

Maximal subgroups

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

subgroupindex
C2×A4×Dic102
C5×A4×C4⋊C42
A4×C4⋊Dic52
A4×#6832
C4×A4×Dic52
C22×#40343
C3×#40344
A4×C4⋊Q85