Finite Group Databasebeta

← C2×C4×C5×SL(2,ℤ/4) #14511216

Subgroups — C2×C4×C5×SL(2,ℤ/4) #14511216

Order 1920.

1158 conjugacy classes of subgroups; only the 216 normal subgroups are shown.

normal subgroups only

This group has 1158 conjugacy classes of subgroups — too many to draw. Only its 216 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
3C4×C5×SL(2,ℤ/4)C214yesmaximal, direct product factor
4C2×C4×C5×C22⋊C4—31nomaximal, Hall
5C23×C5×A4C2211yescharacteristic
6C2×C4×C5×A4C414yescomplemented
7C2×C4×C5×A4C2212yes—
8C2×C5×SL(2,ℤ/4)C418yesdirect product factor
9C2×C5×SL(2,ℤ/4)C2214yes—
10C2×C4×C5×Dic3—41nomaximal
11C2×C4×SL(2,ℤ/4)C511yesmaximal, characteristic, direct product factor, Hall
12C24×C4×C5D311yescharacteristic
13C22×C5×C22⋊C4—32no—
14C22×C42×C5—31no—
15C22×C42×C5—31no—
16C4×C5×C22⋊C4—34no—
17C4×C5×C22⋊C4—34no—
18C2×C5×#95—32no—
19C22×C5×A4C2×C414yes—
20C22×C5×A4C2×C412yes—
21C22×C5×A4C2311yescharacteristic
22C22×C3×C4×C5—41no—
23C22×C5×Dic3—42no—
24C4×C5×A4C2×C414yescomplemented
25C5×SL(2,ℤ/4)C2×C418yesdirect product factor
26C4×C5×Dic3—44no—
27C22×C4×A4C2×C511yescharacteristic
28C22×SL(2,ℤ/4)C2×C512yes—
29C4×SL(2,ℤ/4)C2×C514yesdirect product factor
30C25×C5C2×D311yescharacteristic
31C23×C4×C5—31no—
32C23×C4×C5—31no—
33C23×C4×C5—32no—
34C23×C4×C5—34no—
35C23×C4×C5—32no—
36C23×C4×C5—32no—
37C23×C4×C5Dic314yescomplemented
38C23×C4×C5C2×D312yes—
39C2×C5×C22⋊C4—34no—
40C2×C5×C22⋊C4—38no—
41C2×C5×C22⋊C4—38no—
42C2×C5×C22⋊C4—34no—
43C2×C42×C5—34no—
44C2×C42×C5—34no—
45C2×C42×C5—61no—
46C2×C42×C5—64no—
47C2×C42×C5—62no—
48C2×C42×C5—61no—
49C5×#95—38no—
50C2×C4×C22⋊C4—31noSylow
51C23×C3×C5—41no—
52C2×C5×A4C4214yescomplemented
53C2×C5×A4C22×C412yes—
54C2×C5×A4C22×C411yescharacteristic
55C2×C3×C4×C5—44no—
56C2×C3×C4×C5—42no—
57C2×C5×Dic3—44no—
58C2×C5×Dic3—48no—
59C23×A4C22×C511yescharacteristic
60C2×C4×A4C4×C514yescomplemented
61C2×C4×A4C22×C512yes—
62C2×SL(2,ℤ/4)C4×C518yesdirect product factor
63C2×SL(2,ℤ/4)C22×C514yes—
64C2×C4×Dic3—41no—
65C24×C5—31no—
66C24×C5—31no—
67C24×C5—34no—
68C24×C5—32no—
69C24×C5C4×D314yes—
70C24×C5C2×Dic312yes—
71C24×C5C22×D311yescharacteristic
72C22×C4×C5—31no—
73C22×C4×C5—34no—
74C22×C4×C5—34no—
75C22×C4×C5—32no—
76C22×C4×C5—32no—
77C22×C4×C5—34no—
78C22×C4×C5—38no—
79C22×C4×C5—34no—
80C22×C4×C5—32no—
81C22×C4×C5—34no—
82C22×C4×C5—34no—
83C22×C4×C5—38no—
84C22×C4×C5—38no—
85C22×C4×C5—64no—
86C22×C4×C5—62no—
87C22×C4×C5—61no—
88C22×C4×C5—64no—
89C22×C4×C5—62no—
90C22×C4×C5—64no—
91C22×C4×C5—64no—
92C22×C4×C5—62no—
93C22×C4×C5C2×Dic314yescomplemented
94C22×C4×C5S411yescharacteristic
95C5×C22⋊C4—316no—
96C5×C22⋊C4—38no—
97C5×C22⋊C4—38no—
98C42×C5—64no—
99C42×C5—68no—
100C42×C5—64no—
101C24×C4C5×D311yescharacteristic
102C22×C22⋊C4—32no—
103C22×C42—31no—
104C22×C42—31no—
105C4×C22⋊C4—34no—
106C4×C22⋊C4—34no—
107C2×#95—32no—
108C22×C3×C5—44no—
109C22×C3×C5—42no—
110C22×C3×C5—41no—
111C5×A4C2×C4211yescharacteristic, complemented
112C3×C4×C5—44no—
113C5×Dic3—48no—
114C22×A4C2×C4×C514yes—
115C22×A4C2×C4×C512yes—
116C22×A4C23×C511yescharacteristic
117C22×C3×C4—41no—
118C22×Dic3—42no—
119C4×A4C2×C4×C514yescomplemented
120SL(2,ℤ/4)C2×C4×C518yesdirect product factor
121C4×Dic3—44no—
122C23×C5—34no—
123C23×C5—34no—
124C23×C5—32no—
125C23×C5—32no—
126C23×C5—31no—
127C23×C5—31no—
128C23×C5—34no—
129C23×C5—32no—
130C23×C5—32no—
131C23×C5—31no—
132C23×C5—38no—
133C23×C5—34no—
134C23×C5—61no—
135C23×C5—62no—
136C23×C5—64no—
137C23×C5C4×Dic314yescomplemented
138C23×C5C2×C4×D312yes—
139C23×C5C22×Dic311yescharacteristic
140C23×C5C2×S411yescharacteristic
141C2×C4×C5—34no—
142C2×C4×C5—32no—
143C2×C4×C5—34no—
144C2×C4×C5—34no—
145C2×C4×C5—38no—
146C2×C4×C5—34no—
147C2×C4×C5—38no—
148C2×C4×C5—38no—
149C2×C4×C5—62no—
150C2×C4×C5—64no—
151C2×C4×C5—64no—
152C2×C4×C5—64no—
153C2×C4×C5—616no—
154C2×C4×C5—64no—
155C2×C4×C5—68no—
156C2×C4×C5—68no—
157C2×C4×C5—64no—
158C2×C4×C5SL(2,ℤ/4)14yesdirect product factor
159C2×C4×C5C2×S412yes—
160C25C2×C5×D311yescharacteristic
161C23×C4—31no—
162C23×C4—34no—
163C23×C4—31no—
164C23×C4—32no—
165C23×C4—32no—
166C23×C4—32no—
167C23×C4C5×Dic314yescomplemented
168C23×C4C2×C5×D312yes—
169C2×C22⋊C4—38no—
170C2×C22⋊C4—34no—
171C2×C22⋊C4—38no—
172C2×C22⋊C4—34no—
173C2×C42—34no—
174C2×C42—34no—
175C2×C42—61no—
176C2×C42—64no—
177C2×C42—62no—
178C2×C42—61no—
179#95—38no—
180C2×C3×C5—42no—
181C2×C3×C5—44no—
182C2×C3×C5—41no—
183C23×C3—41no—
184C2×A4C42×C514yescomplemented
185C2×A4C22×C4×C512yes—
186C2×A4C22×C4×C511yescharacteristic
187C2×C3×C4—44no—
188C2×C3×C4—42no—
189C2×Dic3—44no—
190C2×Dic3—48no—
191C22×C5—34no—
192C22×C5—32no—
193C22×C5—31no—
194C22×C5—32no—
195C22×C5—32no—
196C22×C5—34no—
197C22×C5—38no—
198C22×C5—34no—
199C22×C5—34no—
200C22×C5—31no—
201C22×C5—31no—
202C22×C5—32no—
203C22×C5—61no—
204C22×C5—64no—
205C22×C5—62no—
206C22×C5C2×C4×Dic311yescharacteristic, complemented
207C22×C5C4×S414yes—
208C22×C5C2×SL(2,ℤ/4)12yes—
209C22×C5C22×S411yescharacteristic
210C4×C5—34no—
211C4×C5—68no—
212C4×C5—68no—
213C4×C5C2×SL(2,ℤ/4)14yesdirect product factor
214C24—31no—
215C24—31no—
216C24—34no—
217C24—32no—
218C24C4×C5×D314yes—
219C24C2×C5×Dic312yes—
220C24C22×C5×D311yescharacteristic
221C22×C4—31no—
222C22×C4—34no—
223C22×C4—34no—
224C22×C4—32no—
225C22×C4—32no—
226C22×C4—32no—
227C22×C4—34no—
228C22×C4—38no—
229C22×C4—34no—
230C22×C4—34no—
231C22×C4—34no—
232C22×C4—38no—
233C22×C4—38no—
234C22×C4—64no—
235C22×C4—62no—
236C22×C4—61no—
237C22×C4—64no—
238C22×C4—62no—
239C22×C4—64no—
240C22×C4—64no—
241C22×C4—62no—
242C22×C4C2×C5×Dic314yescomplemented
243C22×C4C5×S411yescharacteristic
244C22⋊C4—316no—
245C22⋊C4—38no—
246C22⋊C4—38no—
247C42—64no—
248C42—68no—
249C42—64no—
250C3×C5—41noHall
251C22×C3—44no—
252C22×C3—42no—
253C22×C3—41no—
254A4C2×C42×C511yescharacteristic, complemented
255C3×C4—44no—
256Dic3—48no—
257C2×C5—32no—
258C2×C5—31no—
259C2×C5—34no—
260C2×C5—31no—
261C2×C5C4×SL(2,ℤ/4)14yesdirect product factor
262C2×C5C2×C4×S412yes—
263C2×C5C22×SL(2,ℤ/4)11yescharacteristic
264C23—34no—
265C23—34no—
266C23—32no—
267C23—32no—
268C23—31no—
269C23—31no—
270C23—34no—
271C23—32no—
272C23—32no—
273C23—31no—
274C23—38no—
275C23—34no—
276C23—61no—
277C23—62no—
278C23—64no—
279C23C4×C5×Dic314yescomplemented
280C23C2×C4×C5×D312yes—
281C23C22×C5×Dic311yescharacteristic
282C23C2×C5×S411yescharacteristic
283C2×C4—34no—
284C2×C4—32no—
285C2×C4—34no—
286C2×C4—34no—
287C2×C4—38no—
288C2×C4—34no—
289C2×C4—38no—
290C2×C4—38no—
291C2×C4—62no—
292C2×C4—64no—
293C2×C4—64no—
294C2×C4—64no—
295C2×C4—616no—
296C2×C4—64no—
297C2×C4—68no—
298C2×C4—68no—
299C2×C4—64no—
300C2×C4C5×SL(2,ℤ/4)14yesdirect product factor
301C2×C4C2×C5×S412yes—
302C2×C3—42no—
303C2×C3—41no—
304C2×C3—44no—
305C5C2×C4×SL(2,ℤ/4)11yescharacteristic, direct product factor, Sylow
306C22—34no—
307C22—32no—
308C22—31no—
309C22—32no—
310C22—32no—
311C22—34no—
312C22—31no—
313C22—31no—
314C22—32no—
315C22—38no—
316C22—34no—
317C22—34no—
318C22—61no—
319C22—64no—
320C22—62no—
321C22C2×C4×C5×Dic311yescharacteristic, complemented
322C22C4×C5×S414yes—
323C22C2×C5×SL(2,ℤ/4)12yes—
324C22C22×C5×S411yescharacteristic
325C4—34no—
326C4—68no—
327C4—68no—
328C4C2×C5×SL(2,ℤ/4)14yesdirect product factor
329C3—41noSylow
330C2—32no—
331C2—31no—
332C2—34no—
333C2—31no—
334C2C4×C5×SL(2,ℤ/4)14yesdirect product factor
335C2C2×C4×C5×S412yes—
336C2C22×C5×SL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, hypercenterC22×C4×C5S4
commutator, hypocenterA4C2×C42×C5
FrattiniC22C22×C5×S4
FittingC24×C4×C5D3
socleC25×C5C2×D3
prime radicalC23×C5×A4C22
cosocleC22×A4C23×C5
LCSG ▷ A4
UCSC1 ◁ C22×C4×C5

Complements

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

NH ≅ G/Ntype
3. C4×C5×SL(2,ℤ/4)C2direct (N × H)
6. C2×C4×C5×A4C4semidirect
8. C2×C5×SL(2,ℤ/4)C4direct (N × H)
11. C2×C4×SL(2,ℤ/4)C5direct (N × H)
24. C4×C5×A4C2×C4semidirect
25. C5×SL(2,ℤ/4)C2×C4direct (N × H)
29. C4×SL(2,ℤ/4)C2×C5direct (N × H)
37. C23×C4×C5Dic3semidirect
52. C2×C5×A4C42semidirect
60. C2×C4×A4C4×C5semidirect
62. C2×SL(2,ℤ/4)C4×C5direct (N × H)
93. C22×C4×C5C2×Dic3semidirect
111. C5×A4C2×C42semidirect
119. C4×A4C2×C4×C5semidirect
120. SL(2,ℤ/4)C2×C4×C5direct (N × H)
137. C23×C5C4×Dic3semidirect
158. C2×C4×C5SL(2,ℤ/4)direct (N × H)
167. C23×C4C5×Dic3semidirect
184. C2×A4C42×C5semidirect
206. C22×C5C2×C4×Dic3semidirect
213. C4×C5C2×SL(2,ℤ/4)direct (N × H)
242. C22×C4C2×C5×Dic3semidirect
254. A4C2×C42×C5semidirect
261. C2×C5C4×SL(2,ℤ/4)direct (N × H)
279. C23C4×C5×Dic3semidirect
300. C2×C4C5×SL(2,ℤ/4)direct (N × H)
305. C5C2×C4×SL(2,ℤ/4)direct (N × H)
321. C22C2×C4×C5×Dic3semidirect
328. C4C2×C5×SL(2,ℤ/4)direct (N × H)
334. C2C4×C5×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
C4×C5×SL(2,ℤ/4)2
C2×C4×C5×C22⋊C43
C2×C4×C5×Dic34
C2×C4×SL(2,ℤ/4)5