Finite Group Databasebeta

← C22×C4×C5×S4 #14511406

Subgroups — C22×C4×C5×S4 #14511406

Order 1920.

2582 conjugacy classes of subgroups; only the 344 normal subgroups are shown.

normal subgroups only

This group has 2582 conjugacy classes of subgroups — too many to draw. Only its 344 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
1C23×C5×S4C211yesmaximal, characteristic
2C22×C4×C5×A4C211yesmaximal, characteristic, complemented
3C22×C5×SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
4C2×C4×C5×S4C2112yesmaximal, direct product factor
5C22×C4×C5×D4—31nomaximal, Hall
6C23×C5×A4C2211yescharacteristic
7C22×C5×S4C418yesdirect product factor
8C22×C5×S4C2216yes—
9C22×C4×C5×D3—41nomaximal
10C2×C4×C5×A4C2216yescomplemented
11C2×C5×SL(2,ℤ/4)C2216yescomplemented
12C4×C5×S4C22116yesdirect product factor
13C22×C4×S4C511yesmaximal, characteristic, direct product factor, Hall
14C23×C5×D4—31no—
15C24×C4×C5—31no—
16C24×C4×C5D311yescharacteristic, complemented
17C2×C4×C5×D4—312no—
18C2×C4×C5×D4—312no—
19C22×C5×C4⋊C4—31no—
20C22×C5×C22⋊C4—31no—
21C22×C5×C22⋊C4—31no—
22C22×C42×C5—31no—
23C23×C5×D3—41no—
24C22×C5×A4C2×C414yescomplemented
25C22×C5×A4C2313yes—
26C2×C5×S4C2×C4124yesdirect product factor
27C2×C5×S4C2314yes—
28C22×C3×C4×C5—41no—
29C22×C5×Dic3—41no—
30C2×C4×C5×D3—412no—
31C4×C5×A4C2314yescomplemented
32C5×SL(2,ℤ/4)C2314yescomplemented
33C23×S4C2×C511yescharacteristic
34C22×C4×A4C2×C511yescharacteristic, complemented
35C22×SL(2,ℤ/4)C2×C511yescharacteristic, complemented
36C2×C4×S4C2×C5112yesdirect product factor
37C25×C5—31no—
38C25×C5C2×D311yescharacteristic
39C22×C5×D4—36no—
40C22×C5×D4—38no—
41C22×C5×D4—38no—
42C22×C5×D4—36no—
43C23×C4×C5—31no—
44C23×C4×C5—31no—
45C23×C4×C5—31no—
46C23×C4×C5—36no—
47C23×C4×C5—31no—
48C23×C4×C5—31no—
49C23×C4×C5—312no—
50C23×C4×C5—61no—
51C23×C4×C5—66no—
52C23×C4×C5—61no—
53C23×C4×C5C2×D316yescomplemented
54C4×C5×D4—348no—
55C4×C5×D4—316no—
56C2×C5×C4⋊C4—312no—
57C2×C5×C22⋊C4—312no—
58C2×C5×C22⋊C4—36no—
59C2×C5×C22⋊C4—36no—
60C2×C42×C5—312no—
61C22×C4×D4—31noSylow
62C23×C3×C5—41no—
63C22×C5×D3—46no—
64C22×C5×D3—48no—
65C2×C5×A4C22×C416yescomplemented
66C2×C5×A4C2411yescharacteristic
67C5×S4C22×C418yesdirect product factor
68C2×C3×C4×C5—46no—
69C2×C5×Dic3—46no—
70C4×C5×D3—416no—
71C23×A4C22×C511yescharacteristic
72C22×S4C4×C518yesdirect product factor
73C22×S4C22×C516yes—
74C22×C4×D3—41no—
75C2×C4×A4C22×C516yescomplemented
76C2×SL(2,ℤ/4)C22×C516yescomplemented
77C4×S4C22×C5116yesdirect product factor
78C24×C5—31no—
79C24×C5—33no—
80C24×C5—34no—
81C24×C5—36no—
82C24×C5—38no—
83C24×C5—63no—
84C24×C5—64no—
85C24×C5—61no—
86C24×C5C4×D314yescomplemented
87C24×C5C22×D313yes—
88C2×C5×D4—312no—
89C2×C5×D4—348no—
90C2×C5×D4—324no—
91C2×C5×D4—324no—
92C2×C5×D4—34no—
93C22×C4×C5—36no—
94C22×C4×C5—38no—
95C22×C4×C5—31no—
96C22×C4×C5—36no—
97C22×C4×C5—36no—
98C22×C4×C5—36no—
99C22×C4×C5—312no—
100C22×C4×C5—36no—
101C22×C4×C5—36no—
102C22×C4×C5—316no—
103C22×C4×C5—64no—
104C22×C4×C5—63no—
105C22×C4×C5—612no—
106C22×C4×C5—624no—
107C22×C4×C5—64no—
108C22×C4×C5—63no—
109C22×C4×C5—61no—
110C22×C4×C5—612no—
111C22×C4×C5S411yescharacteristic, direct product factor
112C22×C4×C5C22×D314yescomplemented
113C5×C4⋊C4—316no—
114C5×C22⋊C4—316no—
115C5×C22⋊C4—312no—
116C5×C22⋊C4—34no—
117C42×C5—316no—
118C23×D4—31no—
119C24×C4—31no—
120C24×C4C5×D311yescharacteristic, complemented
121C2×C4×D4—312no—
122C2×C4×D4—312no—
123C22×C4⋊C4—31no—
124C22×C22⋊C4—31no—
125C22×C22⋊C4—31no—
126C22×C42—31no—
127C22×C3×C5—44no—
128C22×C3×C5—43no—
129C2×C5×D3—424no—
130C2×C5×D3—44no—
131C5×A4C23×C411yescharacteristic, complemented
132C3×C4×C5—44no—
133C5×Dic3—44no—
134C23×D3—41no—
135C22×A4C2×C4×C514yescomplemented
136C22×A4C23×C513yes—
137C2×S4C2×C4×C5124yesdirect product factor
138C2×S4C23×C514yes—
139C22×C3×C4—41no—
140C22×Dic3—41no—
141C2×C4×D3—412no—
142C4×A4C23×C514yescomplemented
143SL(2,ℤ/4)C23×C514yescomplemented
144C23×C5—34no—
145C23×C5—34no—
146C23×C5—33no—
147C23×C5—33no—
148C23×C5—33no—
149C23×C5—312no—
150C23×C5—36no—
151C23×C5—34no—
152C23×C5—324no—
153C23×C5—61no—
154C23×C5—66no—
155C23×C5—66no—
156C23×C5—612no—
157C23×C5—624no—
158C23×C5—68no—
159C23×C5—66no—
160C23×C5C2×C4×D316yescomplemented
161C23×C5C2×S411yescharacteristic
162C23×C5C23×D311yescharacteristic
163C5×D4—348no—
164C5×D4—38no—
165C5×D4—38no—
166C2×C4×C5—34no—
167C2×C4×C5—324no—
168C2×C4×C5—36no—
169C2×C4×C5—34no—
170C2×C4×C5—312no—
171C2×C4×C5—34no—
172C2×C4×C5—34no—
173C2×C4×C5—34no—
174C2×C4×C5—66no—
175C2×C4×C5—616no—
176C2×C4×C5—612no—
177C2×C4×C5—66no—
178C2×C4×C5—66no—
179C2×C4×C5—616no—
180C2×C4×C5C2×S416yesdirect product factor
181C25—31no—
182C25C2×C5×D311yescharacteristic
183C22×D4—36no—
184C22×D4—38no—
185C22×D4—38no—
186C22×D4—36no—
187C23×C4—31no—
188C23×C4—31no—
189C23×C4—31no—
190C23×C4—36no—
191C23×C4—31no—
192C23×C4—31no—
193C23×C4—312no—
194C23×C4—61no—
195C23×C4—66no—
196C23×C4—61no—
197C23×C4C2×C5×D316yescomplemented
198C4×D4—348no—
199C4×D4—316no—
200C2×C4⋊C4—312no—
201C2×C22⋊C4—312no—
202C2×C22⋊C4—36no—
203C2×C22⋊C4—36no—
204C2×C42—312no—
205C2×C3×C5—46no—
206C2×C3×C5—41no—
207C5×D3—48no—
208C23×C3—41no—
209C22×D3—46no—
210C22×D3—48no—
211C2×A4C22×C4×C516yescomplemented
212C2×A4C24×C511yescharacteristic
213S4C22×C4×C518yesdirect product factor
214C2×C3×C4—46no—
215C2×Dic3—46no—
216C4×D3—416no—
217C22×C5—36no—
218C22×C5—31no—
219C22×C5—36no—
220C22×C5—36no—
221C22×C5—312no—
222C22×C5—31no—
223C22×C5—33no—
224C22×C5—38no—
225C22×C5—63no—
226C22×C5—64no—
227C22×C5—64no—
228C22×C5—624no—
229C22×C5—624no—
230C22×C5—64no—
231C22×C5C4×S414yesdirect product factor
232C22×C5C22×C4×D311yescharacteristic, complemented
233C22×C5C22×S413yes—
234C4×C5—38no—
235C4×C5—34no—
236C4×C5—64no—
237C4×C5—64no—
238C4×C5C22×S414yesdirect product factor
239C24—31no—
240C24—33no—
241C24—34no—
242C24—36no—
243C24—38no—
244C24—63no—
245C24—64no—
246C24—61no—
247C24C4×C5×D314yescomplemented
248C24C22×C5×D313yes—
249C2×D4—312no—
250C2×D4—348no—
251C2×D4—324no—
252C2×D4—324no—
253C2×D4—34no—
254C22×C4—36no—
255C22×C4—38no—
256C22×C4—31no—
257C22×C4—36no—
258C22×C4—36no—
259C22×C4—36no—
260C22×C4—312no—
261C22×C4—36no—
262C22×C4—36no—
263C22×C4—316no—
264C22×C4—64no—
265C22×C4—63no—
266C22×C4—612no—
267C22×C4—624no—
268C22×C4—63no—
269C22×C4—64no—
270C22×C4—61no—
271C22×C4—612no—
272C22×C4C5×S411yescharacteristic, direct product factor
273C22×C4C22×C5×D314yescomplemented
274C4⋊C4—316no—
275C22⋊C4—316no—
276C22⋊C4—312no—
277C22⋊C4—34no—
278C42—316no—
279C3×C5—41noHall
280C22×C3—44no—
281C22×C3—43no—
282C2×D3—44no—
283C2×D3—424no—
284A4C23×C4×C511yescharacteristic, complemented
285C3×C4—44no—
286Dic3—44no—
287C2×C5—31no—
288C2×C5—31no—
289C2×C5—36no—
290C2×C5—68no—
291C2×C5C2×C4×S416yesdirect product factor
292C2×C5C23×S411yescharacteristic
293C23—34no—
294C23—34no—
295C23—33no—
296C23—33no—
297C23—33no—
298C23—312no—
299C23—36no—
300C23—34no—
301C23—324no—
302C23—61no—
303C23—66no—
304C23—66no—
305C23—612no—
306C23—624no—
307C23—68no—
308C23—66no—
309C23C2×C4×C5×D316yescomplemented
310C23C2×C5×S411yescharacteristic
311C23C23×C5×D311yescharacteristic
312D4—348no—
313D4—38no—
314D4—38no—
315C2×C4—34no—
316C2×C4—324no—
317C2×C4—36no—
318C2×C4—34no—
319C2×C4—312no—
320C2×C4—34no—
321C2×C4—34no—
322C2×C4—34no—
323C2×C4—66no—
324C2×C4—616no—
325C2×C4—66no—
326C2×C4—612no—
327C2×C4—66no—
328C2×C4—616no—
329C2×C4C2×C5×S416yesdirect product factor
330C2×C3—46no—
331C2×C3—41no—
332D3—48no—
333C5C22×C4×S411yescharacteristic, direct product factor, Sylow
334C22—36no—
335C22—31no—
336C22—36no—
337C22—36no—
338C22—312no—
339C22—31no—
340C22—33no—
341C22—38no—
342C22—63no—
343C22—64no—
344C22—64no—
345C22—624no—
346C22—624no—
347C22—64no—
348C22C4×C5×S414yesdirect product factor
349C22C22×C4×C5×D311yescharacteristic, complemented
350C22C22×C5×S413yes—
351C4—38no—
352C4—34no—
353C4—64no—
354C4—64no—
355C4C22×C5×S414yesdirect product factor
356C3—41noSylow
357C2—31no—
358C2—31no—
359C2—36no—
360C2—68no—
361C2C2×C4×C5×S416yesdirect product factor
362C2C23×C5×S411yescharacteristic
subgroupG/H
center, hypercenterC22×C4×C5S4
commutator, hypocenterA4C23×C4×C5
FrattiniC2C23×C5×S4
FittingC24×C4×C5D3
socleC25×C5C2×D3
prime radicalC23×C5×S4C2
cosocleC2×A4C24×C5
LCSG ▷ A4
UCSC1 ◁ C22×C4×C5

Complements

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

NH ≅ G/Ntype
2. C22×C4×C5×A4C2semidirect
3. C22×C5×SL(2,ℤ/4)C2semidirect
4. C2×C4×C5×S4C2direct (N × H)
7. C22×C5×S4C4direct (N × H)
10. C2×C4×C5×A4C22semidirect
11. C2×C5×SL(2,ℤ/4)C22semidirect
12. C4×C5×S4C22direct (N × H)
13. C22×C4×S4C5direct (N × H)
16. C24×C4×C5D3semidirect
24. C22×C5×A4C2×C4semidirect
26. C2×C5×S4C2×C4direct (N × H)
31. C4×C5×A4C23semidirect
32. C5×SL(2,ℤ/4)C23semidirect
34. C22×C4×A4C2×C5semidirect
35. C22×SL(2,ℤ/4)C2×C5semidirect
36. C2×C4×S4C2×C5direct (N × H)
53. C23×C4×C5C2×D3semidirect
65. C2×C5×A4C22×C4semidirect
67. C5×S4C22×C4direct (N × H)
72. C22×S4C4×C5direct (N × H)
75. C2×C4×A4C22×C5semidirect
76. C2×SL(2,ℤ/4)C22×C5semidirect
77. C4×S4C22×C5direct (N × H)
86. C24×C5C4×D3semidirect
111. C22×C4×C5S4direct (N × H)
112. C22×C4×C5C22×D3semidirect
120. C24×C4C5×D3semidirect
131. C5×A4C23×C4semidirect
135. C22×A4C2×C4×C5semidirect
137. C2×S4C2×C4×C5direct (N × H)
142. C4×A4C23×C5semidirect
143. SL(2,ℤ/4)C23×C5semidirect
160. C23×C5C2×C4×D3semidirect
180. C2×C4×C5C2×S4direct (N × H)
197. C23×C4C2×C5×D3semidirect
211. C2×A4C22×C4×C5semidirect
213. S4C22×C4×C5direct (N × H)
231. C22×C5C4×S4direct (N × H)
232. C22×C5C22×C4×D3semidirect
238. C4×C5C22×S4direct (N × H)
247. C24C4×C5×D3semidirect
272. C22×C4C5×S4direct (N × H)
273. C22×C4C22×C5×D3semidirect
284. A4C23×C4×C5semidirect
291. C2×C5C2×C4×S4direct (N × H)
309. C23C2×C4×C5×D3semidirect
329. C2×C4C2×C5×S4direct (N × H)
333. C5C22×C4×S4direct (N × H)
348. C22C4×C5×S4direct (N × H)
349. C22C22×C4×C5×D3semidirect
355. C4C22×C5×S4direct (N × H)
361. C2C2×C4×C5×S4direct (N × H)

Maximal subgroups

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

subgroupindex
C23×C5×S42
C22×C4×C5×A42
C22×C5×SL(2,ℤ/4)2
C2×C4×C5×S42
C22×C4×C5×D43
C22×C4×C5×D34
C22×C4×S45