Finite Group Databasebeta

← C2×C5×C4⋊SL(2,ℤ/4) #14511219

Subgroups — C2×C5×C4⋊SL(2,ℤ/4) #14511219

Order 1920.

998 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 998 conjugacy classes of subgroups — too many to draw. Only its 184 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
1C22×C4×C5×A4C211yesmaximal, characteristic
2C22×C5×SL(2,ℤ/4)C212yesmaximal
3C5×C4⋊SL(2,ℤ/4)C214yesmaximal, direct product factor
4C2×C5×#380—31nomaximal, Hall
5C23×C5×A4C2211yescharacteristic
6C2×C4×C5×A4C414yescomplemented
7C2×C4×C5×A4C2212yes—
8C2×C5×SL(2,ℤ/4)—24no—
9C2×C5×SL(2,ℤ/4)C2214yes—
10C2×C5×C4⋊Dic3—41nomaximal
11C2×C4⋊SL(2,ℤ/4)C511yesmaximal, characteristic, direct product factor, Hall
12C24×C4×C5D311yescharacteristic
13C22×C5×C4⋊C4—31no—
14C22×C5×C4⋊C4—31no—
15C22×C5×C22⋊C4—32no—
16C5×#380—34no—
17C5×#380—34no—
18C2×C5×#95—32no—
19C22×C5×A4C2×C412yes—
20C22×C5×A4D412yes—
21C22×C5×A4Q812yes—
22C22×C5×A4C2311yescharacteristic
23C22×C3×C4×C5—41no—
24C22×C5×Dic3—42no—
25C4×C5×A4C2×C414yescomplemented
26C5×SL(2,ℤ/4)—24no—
27C5×C4⋊Dic3—44no—
28C22×C4×A4C2×C511yescharacteristic
29C22×SL(2,ℤ/4)C2×C512yes—
30C4⋊SL(2,ℤ/4)C2×C514yesdirect product factor
31C25×C5C2×D311yescharacteristic
32C23×C4×C5—31no—
33C23×C4×C5—31no—
34C23×C4×C5—32no—
35C23×C4×C5—34no—
36C23×C4×C5—32no—
37C23×C4×C5—32no—
38C23×C4×C5Dic314yescomplemented
39C23×C4×C5C2×D312yes—
40C2×C5×C4⋊C4—34no—
41C2×C5×C4⋊C4—34no—
42C2×C5×C4⋊C4—61no—
43C2×C5×C4⋊C4—64no—
44C2×C5×C4⋊C4—62no—
45C2×C5×C4⋊C4—61no—
46C2×C5×C22⋊C4—34no—
47C2×C5×C22⋊C4—34no—
48C2×C5×C22⋊C4—64no—
49C2×C5×C22⋊C4—64no—
50C5×#95—38no—
51C2×#380—31noSylow
52C23×C3×C5—41no—
53C2×C5×A4C4⋊C414yescomplemented
54C2×C5×A4C22×C411yescharacteristic
55C2×C5×A4C2×D411yescharacteristic
56C2×C5×A4C2×Q811yescharacteristic
57C2×C3×C4×C5—44no—
58C2×C3×C4×C5—42no—
59C2×C5×Dic3—44no—
60C2×C5×Dic3—84no—
61C23×A4C22×C511yescharacteristic
62C2×C4×A4C4×C514yescomplemented
63C2×C4×A4C22×C512yes—
64C2×SL(2,ℤ/4)—24no—
65C2×SL(2,ℤ/4)C22×C514yes—
66C2×C4⋊Dic3—41no—
67C24×C5—31no—
68C24×C5—31no—
69C24×C5—32no—
70C24×C5—32no—
71C24×C5—32no—
72C24×C5Dic612yes—
73C24×C5D1212yes—
74C24×C5C2×Dic312yes—
75C24×C5C22×D311yescharacteristic
76C22×C4×C5—31no—
77C22×C4×C5—34no—
78C22×C4×C5—34no—
79C22×C4×C5—32no—
80C22×C4×C5—32no—
81C22×C4×C5—34no—
82C22×C4×C5—38no—
83C22×C4×C5—34no—
84C22×C4×C5—32no—
85C22×C4×C5—34no—
86C22×C4×C5—34no—
87C22×C4×C5—64no—
88C22×C4×C5—62no—
89C22×C4×C5—61no—
90C22×C4×C5—64no—
91C22×C4×C5—64no—
92C22×C4×C5—64no—
93C22×C4×C5—62no—
94C22×C4×C5—64no—
95C22×C4×C5—64no—
96C22×C4×C5—62no—
97C22×C4×C5C2×Dic314yescomplemented
98C22×C4×C5S411yescharacteristic
99C5×C4⋊C4—64no—
100C5×C4⋊C4—68no—
101C5×C4⋊C4—64no—
102C5×C22⋊C4—68no—
103C5×C22⋊C4—64no—
104C5×C22⋊C4—64no—
105C24×C4C5×D311yescharacteristic
106C22×C4⋊C4—31no—
107C22×C4⋊C4—31no—
108C22×C22⋊C4—32no—
109#380—34no—
110#380—34no—
111C2×#95—32no—
112C22×C3×C5—42no—
113C22×C3×C5—42no—
114C22×C3×C5—42no—
115C22×C3×C5—41no—
116C5×A4C2×C4⋊C411yescharacteristic, complemented
117C3×C4×C5—44no—
118C5×Dic3—84no—
119C22×A4C2×C4×C512yes—
120C22×A4C5×D412yes—
121C22×A4C5×Q812yes—
122C22×A4C23×C511yescharacteristic
123C22×C3×C4—41no—
124C22×Dic3—42no—
125C4×A4C2×C4×C514yescomplemented
126SL(2,ℤ/4)—24no—
127C4⋊Dic3—44no—
128C23×C5—32no—
129C23×C5—32no—
130C23×C5—32no—
131C23×C5—32no—
132C23×C5—32no—
133C23×C5—32no—
134C23×C5—31no—
135C23×C5—31no—
136C23×C5—32no—
137C23×C5—31no—
138C23×C5—32no—
139C23×C5—31no—
140C23×C5—32no—
141C23×C5—34no—
142C23×C5—34no—
143C23×C5—34no—
144C23×C5—31no—
145C23×C5—61no—
146C23×C5—61no—
147C23×C5—61no—
148C23×C5—64no—
149C23×C5C4⋊Dic314yescomplemented
150C23×C5C2×Dic611yescharacteristic
151C23×C5C2×D1211yescharacteristic
152C23×C5C22×Dic311yescharacteristic
153C23×C5C2×S411yescharacteristic
154C2×C4×C5—34no—
155C2×C4×C5—32no—
156C2×C4×C5—34no—
157C2×C4×C5—34no—
158C2×C4×C5—38no—
159C2×C4×C5—34no—
160C2×C4×C5—62no—
161C2×C4×C5—64no—
162C2×C4×C5—64no—
163C2×C4×C5—64no—
164C2×C4×C5—64no—
165C2×C4×C5—64no—
166C2×C4×C5—64no—
167C2×C4×C5—64no—
168C2×C4×C5—128no—
169C2×C4×C5—124no—
170C2×C4×C5—124no—
171C2×C4×C5SL(2,ℤ/4)14yescomplemented
172C2×C4×C5C2×S412yes—
173C25C2×C5×D311yescharacteristic
174C23×C4—31no—
175C23×C4—34no—
176C23×C4—31no—
177C23×C4—32no—
178C23×C4—32no—
179C23×C4—32no—
180C23×C4C5×Dic314yescomplemented
181C23×C4C2×C5×D312yes—
182C2×C4⋊C4—34no—
183C2×C4⋊C4—34no—
184C2×C4⋊C4—61no—
185C2×C4⋊C4—64no—
186C2×C4⋊C4—62no—
187C2×C4⋊C4—61no—
188C2×C22⋊C4—34no—
189C2×C22⋊C4—34no—
190C2×C22⋊C4—64no—
191C2×C22⋊C4—64no—
192#95—38no—
193C2×C3×C5—41no—
194C2×C3×C5—41no—
195C2×C3×C5—44no—
196C2×C3×C5—41no—
197C23×C3—41no—
198C2×A4C5×C4⋊C414yescomplemented
199C2×A4C22×C4×C511yescharacteristic
200C2×A4C2×C5×D411yescharacteristic
201C2×A4C2×C5×Q811yescharacteristic
202C2×C3×C4—44no—
203C2×C3×C4—42no—
204C2×Dic3—44no—
205C2×Dic3—84no—
206C22×C5—34no—
207C22×C5—31no—
208C22×C5—31no—
209C22×C5—31no—
210C22×C5—31no—
211C22×C5—31no—
212C22×C5—32no—
213C22×C5—31no—
214C22×C5—31no—
215C22×C5—32no—
216C22×C5—34no—
217C22×C5—34no—
218C22×C5—34no—
219C22×C5—34no—
220C22×C5—31no—
221C22×C5—31no—
222C22×C5—32no—
223C22×C5—62no—
224C22×C5—61no—
225C22×C5—62no—
226C22×C5—62no—
227C22×C5C2×C4⋊Dic311yescharacteristic, complemented
228C22×C5A4⋊Q812yes—
229C22×C5C4⋊S412yes—
230C22×C5C2×SL(2,ℤ/4)12yes—
231C22×C5C22×S411yescharacteristic
232C4×C5—34no—
233C4×C5—124no—
234C4×C5—124no—
235C4×C5C2×SL(2,ℤ/4)14yescomplemented
236C24—31no—
237C24—31no—
238C24—32no—
239C24—32no—
240C24—32no—
241C24C5×Dic612yes—
242C24C5×D1212yes—
243C24C2×C5×Dic312yes—
244C24C22×C5×D311yescharacteristic
245C22×C4—31no—
246C22×C4—34no—
247C22×C4—34no—
248C22×C4—32no—
249C22×C4—32no—
250C22×C4—32no—
251C22×C4—34no—
252C22×C4—38no—
253C22×C4—34no—
254C22×C4—34no—
255C22×C4—34no—
256C22×C4—64no—
257C22×C4—62no—
258C22×C4—61no—
259C22×C4—64no—
260C22×C4—64no—
261C22×C4—64no—
262C22×C4—62no—
263C22×C4—64no—
264C22×C4—64no—
265C22×C4—62no—
266C22×C4C2×C5×Dic314yescomplemented
267C22×C4C5×S411yescharacteristic
268C4⋊C4—64no—
269C4⋊C4—68no—
270C4⋊C4—64no—
271C22⋊C4—68no—
272C22⋊C4—64no—
273C22⋊C4—64no—
274C3×C5—41noHall
275C22×C3—42no—
276C22×C3—42no—
277C22×C3—42no—
278C22×C3—41no—
279A4C2×C5×C4⋊C411yescharacteristic, complemented
280C3×C4—44no—
281Dic3—84no—
282C2×C5—31no—
283C2×C5—31no—
284C2×C5—34no—
285C2×C5—31no—
286C2×C5—31no—
287C2×C5C4⋊SL(2,ℤ/4)14yesdirect product factor
288C2×C5C2×A4⋊Q811yescharacteristic
289C2×C5C2×C4⋊S411yescharacteristic
290C2×C5C22×SL(2,ℤ/4)11yescharacteristic
291C23—32no—
292C23—32no—
293C23—32no—
294C23—32no—
295C23—32no—
296C23—32no—
297C23—31no—
298C23—31no—
299C23—32no—
300C23—31no—
301C23—32no—
302C23—31no—
303C23—32no—
304C23—34no—
305C23—34no—
306C23—34no—
307C23—31no—
308C23—61no—
309C23—61no—
310C23—61no—
311C23—64no—
312C23C5×C4⋊Dic314yescomplemented
313C23C2×C5×Dic611yescharacteristic
314C23C2×C5×D1211yescharacteristic
315C23C22×C5×Dic311yescharacteristic
316C23C2×C5×S411yescharacteristic
317C2×C4—34no—
318C2×C4—32no—
319C2×C4—34no—
320C2×C4—34no—
321C2×C4—38no—
322C2×C4—34no—
323C2×C4—62no—
324C2×C4—64no—
325C2×C4—64no—
326C2×C4—64no—
327C2×C4—64no—
328C2×C4—64no—
329C2×C4—64no—
330C2×C4—64no—
331C2×C4—128no—
332C2×C4—124no—
333C2×C4—124no—
334C2×C4C5×SL(2,ℤ/4)14yescomplemented
335C2×C4C2×C5×S412yes—
336C2×C3—41no—
337C2×C3—41no—
338C2×C3—41no—
339C2×C3—44no—
340C5C2×C4⋊SL(2,ℤ/4)11yescharacteristic, direct product factor, Sylow
341C22—34no—
342C22—31no—
343C22—31no—
344C22—31no—
345C22—31no—
346C22—31no—
347C22—32no—
348C22—31no—
349C22—31no—
350C22—32no—
351C22—31no—
352C22—31no—
353C22—32no—
354C22—34no—
355C22—34no—
356C22—34no—
357C22—34no—
358C22—62no—
359C22—61no—
360C22—62no—
361C22—62no—
362C22C2×C5×C4⋊Dic311yescharacteristic, complemented
363C22C5×A4⋊Q812yes—
364C22C5×C4⋊S412yes—
365C22C2×C5×SL(2,ℤ/4)12yes—
366C22C22×C5×S411yescharacteristic
367C4—34no—
368C4—124no—
369C4—124no—
370C4C2×C5×SL(2,ℤ/4)14yescomplemented
371C3—41noSylow
372C2—31no—
373C2—31no—
374C2—34no—
375C2—31no—
376C2—31no—
377C2C5×C4⋊SL(2,ℤ/4)14yesdirect product factor
378C2C2×C5×A4⋊Q811yescharacteristic
379C2C2×C5×C4⋊S411yescharacteristic
380C2C22×C5×SL(2,ℤ/4)11yescharacteristic
subgroupG/H
centerC23×C5C2×S4
commutatorC2×A4C22×C4×C5
FrattiniC22C22×C5×S4
FittingC24×C4×C5D3
socleC25×C5C2×D3
prime radicalC23×C5×A4C22
cosocleC22×A4C23×C5
hypercenterC22×C4×C5S4
hypocenterA4C2×C5×C4⋊C4
LCSG ▷ C2×A4 ▷ A4
UCSC1 ◁ C23×C5 ◁ C22×C4×C5

Complements

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

NH ≅ G/Ntype
3. C5×C4⋊SL(2,ℤ/4)C2direct (N × H)
6. C2×C4×C5×A4C4semidirect
11. C2×C4⋊SL(2,ℤ/4)C5direct (N × H)
25. C4×C5×A4C2×C4semidirect
30. C4⋊SL(2,ℤ/4)C2×C5direct (N × H)
38. C23×C4×C5Dic3semidirect
53. C2×C5×A4C4⋊C4semidirect
62. C2×C4×A4C4×C5semidirect
97. C22×C4×C5C2×Dic3semidirect
116. C5×A4C2×C4⋊C4semidirect
125. C4×A4C2×C4×C5semidirect
149. C23×C5C4⋊Dic3semidirect
171. C2×C4×C5SL(2,ℤ/4)semidirect
180. C23×C4C5×Dic3semidirect
198. C2×A4C5×C4⋊C4semidirect
227. C22×C5C2×C4⋊Dic3semidirect
235. C4×C5C2×SL(2,ℤ/4)semidirect
266. C22×C4C2×C5×Dic3semidirect
279. A4C2×C5×C4⋊C4semidirect
287. C2×C5C4⋊SL(2,ℤ/4)direct (N × H)
312. C23C5×C4⋊Dic3semidirect
334. C2×C4C5×SL(2,ℤ/4)semidirect
340. C5C2×C4⋊SL(2,ℤ/4)direct (N × H)
362. C22C2×C5×C4⋊Dic3semidirect
370. C4C2×C5×SL(2,ℤ/4)semidirect
377. C2C5×C4⋊SL(2,ℤ/4)direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4×C5×A42
C22×C5×SL(2,ℤ/4)2
C5×C4⋊SL(2,ℤ/4)2
C2×C5×#3803
C2×C5×C4⋊Dic34
C2×C4⋊SL(2,ℤ/4)5