Finite Group Databasebeta

← C32×AGL(1,𝔽5)×S4 #72247708

Subgroups — C32×AGL(1,𝔽5)×S4 #72247708

Order 4320.

768 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 768 conjugacy classes of subgroups — too many to draw. Only its 108 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
1C32×A4⋊AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
2C32×D5×S4C211yesmaximal, characteristic
3C32×A4×AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
4C3×AGL(1,𝔽5)×S4C314yesmaximal, direct product factor
5C32×D4×AGL(1,𝔽5)—31nomaximal
6C32×D5×A4C2211yescharacteristic
7C32×C5⋊S4C411yescharacteristic, complemented
8C32×C5×S4C411yescharacteristic, complemented
9C32×D3×AGL(1,𝔽5)—41nomaximal
10C32×C4×S4—51nomaximal, Hall
11C22×C32×AGL(1,𝔽5)—31no—
12C22×C32×AGL(1,𝔽5)D311yescharacteristic, complemented
13C3×D5×S4C2×C314yes—
14C3×A4×AGL(1,𝔽5)—24no—
15C3×A4×AGL(1,𝔽5)C2×C314yescomplemented
16C3×A4⋊AGL(1,𝔽5)C2×C314yescomplemented
17C32×D4×D5—31no—
18C32×C22⋊AGL(1,𝔽5)—31no—
19C32×C22⋊AGL(1,𝔽5)—31no—
20C32×C4⋊AGL(1,𝔽5)—31no—
21C32×C4×AGL(1,𝔽5)—31no—
22C32×C5×A4C2×C411yescharacteristic, complemented
23C32×D3×D5—41no—
24C32×ΓL(1,𝔽16)—41no—
25C33×AGL(1,𝔽5)—41no—
26AGL(1,𝔽5)×S4C3211yescharacteristic, direct product factor
27C3×D4×AGL(1,𝔽5)—34no—
28C2×C32×S4—51no—
29C32×C4×A4—51no—
30C32×SL(2,ℤ/4)—51no—
31C22×C32×D5—31no—
32C22×C32×D5C2×D311yescharacteristic
33C2×C32×AGL(1,𝔽5)—31no—
34C2×C32×AGL(1,𝔽5)—31no—
35C2×C32×AGL(1,𝔽5)—31no—
36C2×C32×AGL(1,𝔽5)—31no—
37C2×C32×AGL(1,𝔽5)—61no—
38C2×C32×AGL(1,𝔽5)—61no—
39C3×D5×A4—24no—
40C3×D5×A4C22×C314yes—
41C3×C5⋊S4C3×C414yescomplemented
42C3×C5×S4C3×C414yescomplemented
43C3×D3×AGL(1,𝔽5)—44no—
44C32×C5×D4—31no—
45C32×C22⋊D5—31no—
46C32×C22⋊D5—31no—
47C32×D20—31no—
48C32×C4×D5—31no—
49C3×C4×S4—54no—
50C32×C4×D4—151no—
51C32×D15—41no—
52C32×C5×D3—41no—
53C33×D5—41no—
54C22×C3×AGL(1,𝔽5)—34no—
55C22×C3×AGL(1,𝔽5)C3×D314yescomplemented
56D5×S4C2×C3211yescharacteristic
57A4×AGL(1,𝔽5)—24no—
58A4×AGL(1,𝔽5)C2×C3211yescharacteristic, complemented
59A4⋊AGL(1,𝔽5)C2×C3211yescharacteristic, complemented
60C3×D4×D5—34no—
61C3×C22⋊AGL(1,𝔽5)—34no—
62C3×C22⋊AGL(1,𝔽5)—34no—
63C3×C4⋊AGL(1,𝔽5)—34no—
64C3×C4×AGL(1,𝔽5)—34no—
65C2×C32×A4—51no—
66C32×S4—51no—
67C32×S4AGL(1,𝔽5)11yescharacteristic, direct product factor
68C32×C4×D3—201no—
69C22×C32×C5—31no—
70C22×C32×C5C4×D311yescharacteristic, complemented
71C2×C32×D5—31no—
72C2×C32×D5—31no—
73C2×C32×D5—31no—
74C2×C32×D5—61no—
75C2×C32×D5—61no—
76C3×C5×A4—24no—
77C3×C5×A4C2×C3×C414yescomplemented
78C3×D3×D5—44no—
79C3×ΓL(1,𝔽16)—44no—
80C32×AGL(1,𝔽5)—31no—
81C32×AGL(1,𝔽5)—44no—
82C32×AGL(1,𝔽5)—61no—
83C32×AGL(1,𝔽5)—61no—
84C32×AGL(1,𝔽5)—84no—
85C32×AGL(1,𝔽5)S411yescharacteristic, direct product factor
86C32×C4×C5—31no—
87C32×Dic5—31no—
88D4×AGL(1,𝔽5)—31noHall
89C2×C3×S4—54no—
90C2×C32×D4—151no—
91C22×C32×C4—51no—
92C22×C32×C4—151no—
93C3×C4×A4—54no—
94C3×C4×A4—104no—
95C3×SL(2,ℤ/4)—54no—
96C32×C4⋊C4—151no—
97C32×C22⋊C4—151no—
98C32×C22⋊C4—151no—
99C32×C42—151no—
100C33×C5—41noHall
101C22×C3×D5—34no—
102C22×C3×D5C2×C3×D314yes—
103C2×C3×AGL(1,𝔽5)—34no—
104C2×C3×AGL(1,𝔽5)—34no—
105C2×C3×AGL(1,𝔽5)—34no—
106C2×C3×AGL(1,𝔽5)—34no—
107C2×C3×AGL(1,𝔽5)—64no—
108C2×C3×AGL(1,𝔽5)—64no—
109D5×A4—24no—
110D5×A4C22×C3211yescharacteristic
111C5⋊S4C32×C411yescharacteristic, complemented
112C5×S4C32×C411yescharacteristic, complemented
113D3×AGL(1,𝔽5)—41no—
114C3×C5×D4—34no—
115C3×C22⋊D5—34no—
116C3×C22⋊D5—34no—
117C3×D20—34no—
118C3×C4×D5—34no—
119C2×C32×D3—201no—
120C32×A4C2×AGL(1,𝔽5)11yescharacteristic, complemented
121C33×C4—201no—
122C32×Dic3—201no—
123C4×S4—51no—
124C3×C4×D4—154no—
125C2×C32×C5—31no—
126C2×C32×C5—61no—
127C3×D15—44no—
128C3×C5×D3—44no—
129C32×D5—31no—
130C32×D5—44no—
131C32×D5—61no—
132C32×D5—84no—
133C32×D5C2×S411yescharacteristic
134C22×AGL(1,𝔽5)—31no—
135C22×AGL(1,𝔽5)C32×D311yescharacteristic, complemented
136D4×D5—31no—
137C22⋊AGL(1,𝔽5)—31no—
138C22⋊AGL(1,𝔽5)—31no—
139C4⋊AGL(1,𝔽5)—31no—
140C4×AGL(1,𝔽5)—31no—
141C23×C32—51no—
142C23×C32—151no—
143C2×C3×A4—54no—
144C2×C3×A4—104no—
145C3×S4—54no—
146C3×S4C3×AGL(1,𝔽5)14yesdirect product factor
147C32×D4—31no—
148C32×D4—151no—
149C32×D4—151no—
150C32×D4—151no—
151C2×C32×C4—151no—
152C2×C32×C4—151no—
153C2×C32×C4—151no—
154C2×C32×C4—151no—
155C2×C32×C4—151no—
156C2×C32×C4—301no—
157C2×C32×C4—301no—
158C3×C4×D3—204no—
159C22×C3×C5—34no—
160C22×C3×C5C3×C4×D314yescomplemented
161C2×C3×D5—34no—
162C2×C3×D5—34no—
163C2×C3×D5—34no—
164C2×C3×D5—64no—
165C2×C3×D5—64no—
166C5×A4—24no—
167C5×A4C2×C32×C411yescharacteristic, complemented
168D3×D5—41no—
169ΓL(1,𝔽16)—41no—
170C3×AGL(1,𝔽5)—34no—
171C3×AGL(1,𝔽5)—41no—
172C3×AGL(1,𝔽5)—64no—
173C3×AGL(1,𝔽5)—64no—
174C3×AGL(1,𝔽5)—84no—
175C3×AGL(1,𝔽5)C3×S414yesdirect product factor
176C3×C4×C5—34no—
177C3×Dic5—34no—
178C2×C33—201no—
179C32×D3—41no—
180C32×D3—201no—
181C2×S4—51no—
182C2×C3×D4—154no—
183C22×C3×C4—54no—
184C22×C3×C4—154no—
185C4×A4—51no—
186C4×A4—104no—
187SL(2,ℤ/4)—51no—
188C3×C4⋊C4—154no—
189C3×C22⋊C4—154no—
190C3×C22⋊C4—154no—
191C3×C42—154no—
192C32×C5—44no—
193C32×C5—84no—
194C32×C5C4×S411yescharacteristic, complemented
195C22×D5—31no—
196C22×D5C2×C32×D311yescharacteristic
197C2×AGL(1,𝔽5)—31no—
198C2×AGL(1,𝔽5)—31no—
199C2×AGL(1,𝔽5)—31no—
200C2×AGL(1,𝔽5)—31no—
201C2×AGL(1,𝔽5)—61no—
202C2×AGL(1,𝔽5)—61no—
203C5×D4—31no—
204C22⋊D5—31no—
205C22⋊D5—31no—
206D20—31no—
207C4×D5—31no—
208C22×C32—31no—
209C22×C32—151no—
210C22×C32—151no—
211C22×C32—151no—
212C22×C32—301no—
213C22×C32—301no—
214C22×C32D3×AGL(1,𝔽5)11yescharacteristic, complemented
215C2×C3×D3—204no—
216C3×A4—24no—
217C3×A4C2×C3×AGL(1,𝔽5)14yescomplemented
218C32×C4—31no—
219C32×C4—51no—
220C32×C4—151no—
221C32×C4—151no—
222C32×C4—204no—
223C32×C4—301no—
224C32×C4—301no—
225C32×C4—404no—
226C3×Dic3—204no—
227C4×D4—151noSylow
228C2×C3×C5—34no—
229C2×C3×C5—64no—
230D15—41no—
231C3×D5—34no—
232C3×D5—41no—
233C3×D5—64no—
234C3×D5—84no—
235C3×D5C2×C3×S414yes—
236C5×D3—41no—
237C33—41noSylow
238C23×C3—54no—
239C23×C3—154no—
240C2×A4—51no—
241C2×A4—104no—
242S4—51no—
243S4C32×AGL(1,𝔽5)11yescharacteristic, direct product factor
244C3×D4—34no—
245C3×D4—154no—
246C3×D4—154no—
247C3×D4—154no—
248C2×C3×C4—154no—
249C2×C3×C4—154no—
250C2×C3×C4—154no—
251C2×C3×C4—154no—
252C2×C3×C4—154no—
253C2×C3×C4—304no—
254C2×C3×C4—304no—
255C4×D3—201no—
256C22×C5—31no—
257C22×C5C32×C4×D311yescharacteristic, complemented
258C2×D5—31no—
259C2×D5—31no—
260C2×D5—31no—
261C2×D5—61no—
262C2×D5—61no—
263AGL(1,𝔽5)—31no—
264AGL(1,𝔽5)—61no—
265AGL(1,𝔽5)—61no—
266AGL(1,𝔽5)C32×S411yescharacteristic, direct product factor
267C4×C5—31no—
268Dic5—31no—
269C2×C32—31no—
270C2×C32—51no—
271C2×C32—61no—
272C2×C32—151no—
273C2×C32—204no—
274C2×C32—301no—
275C2×C32—404no—
276C3×D3—44no—
277C3×D3—204no—
278C2×D4—151no—
279C22×C4—51no—
280C22×C4—151no—
281C4⋊C4—151no—
282C22⋊C4—151no—
283C22⋊C4—151no—
284C42—151no—
285C3×C5—41no—
286C3×C5—84no—
287C3×C5C3×C4×S414yescomplemented
288C22×C3—34no—
289C22×C3—154no—
290C22×C3—154no—
291C22×C3—154no—
292C22×C3—304no—
293C22×C3—304no—
294C22×C3C3×D3×AGL(1,𝔽5)14yescomplemented
295C2×D3—201no—
296A4—24no—
297A4C2×C32×AGL(1,𝔽5)11yescharacteristic, complemented
298C3×C4—34no—
299C3×C4—54no—
300C3×C4—154no—
301C3×C4—154no—
302C3×C4—201no—
303C3×C4—304no—
304C3×C4—304no—
305C3×C4—404no—
306Dic3—201no—
307C2×C5—31no—
308C2×C5—61no—
309D5—31no—
310D5—61no—
311D5C2×C32×S411yescharacteristic
312C32—44no—
313C32—84no—
314C32AGL(1,𝔽5)×S411yescharacteristic, direct product factor
315C23—51no—
316C23—151no—
317D4—31no—
318D4—151no—
319D4—151no—
320D4—151no—
321C2×C4—151no—
322C2×C4—151no—
323C2×C4—151no—
324C2×C4—151no—
325C2×C4—151no—
326C2×C4—301no—
327C2×C4—301no—
328C2×C3—34no—
329C2×C3—54no—
330C2×C3—64no—
331C2×C3—154no—
332C2×C3—201no—
333C2×C3—304no—
334C2×C3—404no—
335D3—41no—
336D3—201no—
337C5C32×C4×S411yescharacteristic, complemented, Sylow
338C22—31no—
339C22—151no—
340C22—151no—
341C22—151no—
342C22—301no—
343C22—301no—
344C22C32×D3×AGL(1,𝔽5)11yescharacteristic, complemented
345C4—31no—
346C4—51no—
347C4—151no—
348C4—151no—
349C4—301no—
350C4—301no—
351C3—41no—
352C3—84no—
353C3C3×AGL(1,𝔽5)×S414yesdirect product factor
354C2—31no—
355C2—51no—
356C2—61no—
357C2—151no—
358C2—301no—
subgroupG/H
center, hypercenterC32AGL(1,𝔽5)×S4
commutator, hypocenterC5×A4C2×C32×C4
FrattiniC1G
Fitting, socleC22×C32×C5C4×D3
prime radicalC32×D5×S4C2
cosocleD5×A4C22×C32
LCSG ▷ C5×A4
UCSC1 ◁ C32

Complements

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

NH ≅ G/Ntype
1. C32×A4⋊AGL(1,𝔽5)C2semidirect
3. C32×A4×AGL(1,𝔽5)C2semidirect
4. C3×AGL(1,𝔽5)×S4C3direct (N × H)
7. C32×C5⋊S4C4semidirect
8. C32×C5×S4C4semidirect
12. C22×C32×AGL(1,𝔽5)D3semidirect
15. C3×A4×AGL(1,𝔽5)C2×C3semidirect
16. C3×A4⋊AGL(1,𝔽5)C2×C3semidirect
22. C32×C5×A4C2×C4semidirect
26. AGL(1,𝔽5)×S4C32direct (N × H)
41. C3×C5⋊S4C3×C4semidirect
42. C3×C5×S4C3×C4semidirect
55. C22×C3×AGL(1,𝔽5)C3×D3semidirect
58. A4×AGL(1,𝔽5)C2×C32semidirect
59. A4⋊AGL(1,𝔽5)C2×C32semidirect
67. C32×S4AGL(1,𝔽5)direct (N × H)
70. C22×C32×C5C4×D3semidirect
77. C3×C5×A4C2×C3×C4semidirect
85. C32×AGL(1,𝔽5)S4direct (N × H)
111. C5⋊S4C32×C4semidirect
112. C5×S4C32×C4semidirect
120. C32×A4C2×AGL(1,𝔽5)semidirect
135. C22×AGL(1,𝔽5)C32×D3semidirect
146. C3×S4C3×AGL(1,𝔽5)direct (N × H)
160. C22×C3×C5C3×C4×D3semidirect
167. C5×A4C2×C32×C4semidirect
175. C3×AGL(1,𝔽5)C3×S4direct (N × H)
194. C32×C5C4×S4semidirect
214. C22×C32D3×AGL(1,𝔽5)semidirect
217. C3×A4C2×C3×AGL(1,𝔽5)semidirect
243. S4C32×AGL(1,𝔽5)direct (N × H)
257. C22×C5C32×C4×D3semidirect
266. AGL(1,𝔽5)C32×S4direct (N × H)
287. C3×C5C3×C4×S4semidirect
294. C22×C3C3×D3×AGL(1,𝔽5)semidirect
297. A4C2×C32×AGL(1,𝔽5)semidirect
314. C32AGL(1,𝔽5)×S4direct (N × H)
337. C5C32×C4×S4semidirect
344. C22C32×D3×AGL(1,𝔽5)semidirect
353. C3C3×AGL(1,𝔽5)×S4direct (N × H)

Maximal subgroups

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

subgroupindex
C32×A4⋊AGL(1,𝔽5)2
C32×D5×S42
C32×A4×AGL(1,𝔽5)2
C3×AGL(1,𝔽5)×S43
C32×D4×AGL(1,𝔽5)3
C32×D3×AGL(1,𝔽5)4
C32×C4×S45