Finite Group Databasebeta

← C2×Dic5×SL(2,ℤ/4) #14513620

Subgroups — C2×Dic5×SL(2,ℤ/4) #14513620

Order 1920.

1158 conjugacy classes of subgroups; only the 167 normal subgroups are shown.

normal subgroups only

This group has 1158 conjugacy classes of subgroups — too many to draw. Only its 167 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×A4×Dic5C211yesmaximal, characteristic
2C22×A4⋊Dic5C211yesmaximal, characteristic
3C22×C5×SL(2,ℤ/4)C211yesmaximal, characteristic
4Dic5×SL(2,ℤ/4)C214yesmaximal, direct product factor
5C2×C22⋊C4×Dic5—31nomaximal, Hall
6C23×C5×A4C2211yescharacteristic
7C2×A4×Dic5C414yescomplemented
8C2×A4×Dic5C2212yes—
9C2×A4⋊Dic5C414yescomplemented
10C2×A4⋊Dic5C2212yes—
11C2×C5×SL(2,ℤ/4)C414yescomplemented
12C2×C5×SL(2,ℤ/4)C2212yes—
13C2×Dic3×Dic5—41nomaximal
14C2×C4×SL(2,ℤ/4)—51nomaximal, Hall
15C24×Dic5D311yescharacteristic
16C22×C5×C22⋊C4—31no—
17C22×C22⋊Dic5—31no—
18C22×C4×Dic5—31no—
19C22×C4×Dic5—31no—
20C2×#3963—31no—
21C2×#3963—31no—
22C22⋊C4×Dic5—34no—
23C22⋊C4×Dic5—34no—
24C22×C5×A4C2×C412yes—
25C22×C5×A4C2×C412yes—
26C22×C5×A4C2×C412yes—
27C22×C5×A4C2311yescharacteristic
28C22×Dic15—41no—
29C22×C5×Dic3—41no—
30C22×C3×Dic5—41no—
31A4×Dic5C2×C414yescomplemented
32A4⋊Dic5C2×C414yescomplemented
33C5×SL(2,ℤ/4)C2×C414yescomplemented
34Dic3×Dic5—44no—
35C22×C4×A4—51no—
36C22×SL(2,ℤ/4)—51no—
37C22×SL(2,ℤ/4)D511yescharacteristic
38C4×SL(2,ℤ/4)—54no—
39C25×C5C2×D311yescharacteristic
40C23×C4×C5—31no—
41C23×C4×C5—31no—
42C23×Dic5—31no—
43C23×Dic5—32no—
44C23×Dic5—31no—
45C23×Dic5—34no—
46C23×Dic5—31no—
47C23×Dic5—31no—
48C23×Dic5Dic314yescomplemented
49C23×Dic5C2×D312yes—
50C2×C5×C22⋊C4—34no—
51C2×C5×C22⋊C4—32no—
52C2×C5×C22⋊C4—34no—
53C2×C5×C22⋊C4—32no—
54C2×C22⋊Dic5—34no—
55C2×C22⋊Dic5—32no—
56C2×C22⋊Dic5—34no—
57C2×C22⋊Dic5—32no—
58C2×C4×Dic5—34no—
59C2×C4×Dic5—34no—
60C2×C4×Dic5—61no—
61C2×C4×Dic5—64no—
62C2×C4×Dic5—62no—
63C2×C4×Dic5—61no—
64#3963—34no—
65#3963—34no—
66C2×C4×C22⋊C4—151noSylow
67C23×C3×C5—41no—
68C2×C5×A4C4214yescomplemented
69C2×C5×A4C22×C411yescharacteristic
70C2×C5×A4C22×C411yescharacteristic
71C2×C5×A4C22×C411yescharacteristic
72C2×Dic15—42no—
73C2×Dic15—44no—
74C2×C5×Dic3—42no—
75C2×C5×Dic3—44no—
76C2×C3×Dic5—44no—
77C2×C3×Dic5—42no—
78C23×A4C2×D511yescharacteristic
79C2×C4×A4—54no—
80C2×C4×A4—52no—
81C2×SL(2,ℤ/4)—54no—
82C2×SL(2,ℤ/4)—52no—
83C2×SL(2,ℤ/4)Dic514yesdirect product factor
84C2×SL(2,ℤ/4)C2×D512yes—
85C2×C4×Dic3—201no—
86C24×C5—31no—
87C24×C5—31no—
88C24×C5—32no—
89C24×C5—32no—
90C24×C5—32no—
91C24×C5C4×D312yes—
92C24×C5C4×D312yes—
93C24×C5C2×Dic312yes—
94C24×C5C22×D311yescharacteristic
95C22×C4×C5—34no—
96C22×C4×C5—34no—
97C22×C4×C5—32no—
98C22×C4×C5—32no—
99C22×C4×C5—62no—
100C22×C4×C5—62no—
101C22×C4×C5—62no—
102C22×C4×C5—61no—
103C22×C4×C5—61no—
104C22×Dic5—31no—
105C22×Dic5—34no—
106C22×Dic5—34no—
107C22×Dic5—32no—
108C22×Dic5—32no—
109C22×Dic5—34no—
110C22×Dic5—38no—
111C22×Dic5—34no—
112C22×Dic5—32no—
113C22×Dic5—34no—
114C22×Dic5—34no—
115C22×Dic5—32no—
116C22×Dic5—32no—
117C22×Dic5—64no—
118C22×Dic5—62no—
119C22×Dic5—61no—
120C22×Dic5—62no—
121C22×Dic5—62no—
122C22×Dic5—62no—
123C22×Dic5—61no—
124C22×Dic5—61no—
125C22×Dic5C2×Dic314yescomplemented
126C22×Dic5S411yescharacteristic
127C5×C22⋊C4—38no—
128C5×C22⋊C4—34no—
129C5×C22⋊C4—34no—
130C22⋊Dic5—38no—
131C22⋊Dic5—34no—
132C22⋊Dic5—34no—
133C4×Dic5—64no—
134C4×Dic5—68no—
135C4×Dic5—64no—
136C24×C4—51no—
137C22×C22⋊C4—31no—
138C22×C22⋊C4—151no—
139C22×C42—151no—
140C22×C42—151no—
141C4×C22⋊C4—154no—
142C4×C22⋊C4—154no—
143C2×#95—151no—
144C2×#95—151no—
145C22×C3×C5—42no—
146C22×C3×C5—42no—
147C22×C3×C5—42no—
148C22×C3×C5—41no—
149C5×A4C2×C4211yescharacteristic, complemented
150Dic15—44no—
151C3×Dic5—44no—
152C5×Dic3—44no—
153C22×A4C4×D512yes—
154C22×A4C4×D512yes—
155C22×A4C2×Dic512yes—
156C22×A4C22×D511yescharacteristic
157C22×C3×C4—201no—
158C22×Dic3—41no—
159C22×Dic3—201no—
160C4×A4—54no—
161SL(2,ℤ/4)—54no—
162SL(2,ℤ/4)C2×Dic514yesdirect product factor
163C4×Dic3—204no—
164C23×C5—32no—
165C23×C5—32no—
166C23×C5—32no—
167C23×C5—32no—
168C23×C5—32no—
169C23×C5—32no—
170C23×C5—31no—
171C23×C5—31no—
172C23×C5—32no—
173C23×C5—31no—
174C23×C5—32no—
175C23×C5—31no—
176C23×C5—32no—
177C23×C5—31no—
178C23×C5—34no—
179C23×C5—34no—
180C23×C5—34no—
181C23×C5—61no—
182C23×C5—61no—
183C23×C5—61no—
184C23×C5—64no—
185C23×C5C4×Dic314yescomplemented
186C23×C5C2×C4×D311yescharacteristic
187C23×C5C2×C4×D311yescharacteristic
188C23×C5C22×Dic311yescharacteristic
189C23×C5C2×S411yescharacteristic
190C2×C4×C5—34no—
191C2×C4×C5—34no—
192C2×C4×C5—62no—
193C2×C4×C5—62no—
194C2×C4×C5—68no—
195C2×C4×C5—62no—
196C2×C4×C5—64no—
197C2×C4×C5—64no—
198C2×C4×C5—62no—
199C2×Dic5—34no—
200C2×Dic5—32no—
201C2×Dic5—34no—
202C2×Dic5—34no—
203C2×Dic5—38no—
204C2×Dic5—34no—
205C2×Dic5—34no—
206C2×Dic5—34no—
207C2×Dic5—62no—
208C2×Dic5—64no—
209C2×Dic5—62no—
210C2×Dic5—62no—
211C2×Dic5—68no—
212C2×Dic5—62no—
213C2×Dic5—64no—
214C2×Dic5—64no—
215C2×Dic5—62no—
216C2×Dic5SL(2,ℤ/4)14yesdirect product factor
217C2×Dic5C2×S412yes—
218C25D3×D511yescharacteristic
219C23×C4—31no—
220C23×C4—31no—
221C23×C4—54no—
222C23×C4—52no—
223C23×C4—151no—
224C23×C4—154no—
225C23×C4—151no—
226C23×C4—152no—
227C23×C4—151no—
228C23×C4—151no—
229C2×C22⋊C4—34no—
230C2×C22⋊C4—32no—
231C2×C22⋊C4—34no—
232C2×C22⋊C4—32no—
233C2×C22⋊C4—154no—
234C2×C22⋊C4—152no—
235C2×C22⋊C4—154no—
236C2×C22⋊C4—152no—
237C2×C42—154no—
238C2×C42—154no—
239C2×C42—301no—
240C2×C42—302no—
241C2×C42—304no—
242C2×C42—301no—
243#95—154no—
244#95—154no—
245C2×C3×C5—41no—
246C2×C3×C5—41no—
247C2×C3×C5—44no—
248C2×C3×C5—41no—
249C23×C3—41no—
250C2×A4C4×Dic514yescomplemented
251C2×A4C2×C4×D511yescharacteristic
252C2×A4C2×C4×D511yescharacteristic
253C2×A4C22×Dic511yescharacteristic
254C2×C3×C4—202no—
255C2×C3×C4—204no—
256C2×Dic3—42no—
257C2×Dic3—44no—
258C2×Dic3—202no—
259C2×Dic3—204no—
260C22×C5—34no—
261C22×C5—31no—
262C22×C5—31no—
263C22×C5—31no—
264C22×C5—31no—
265C22×C5—31no—
266C22×C5—31no—
267C22×C5—32no—
268C22×C5—31no—
269C22×C5—32no—
270C22×C5—34no—
271C22×C5—34no—
272C22×C5—34no—
273C22×C5—34no—
274C22×C5—31no—
275C22×C5—31no—
276C22×C5—32no—
277C22×C5—61no—
278C22×C5—62no—
279C22×C5—62no—
280C22×C5—62no—
281C22×C5C2×C4×Dic311yescharacteristic, complemented
282C22×C5C4×S412yes—
283C22×C5C4×S412yes—
284C22×C5C2×SL(2,ℤ/4)12yes—
285C22×C5C22×S411yescharacteristic
286C4×C5—64no—
287C4×C5—64no—
288Dic5—34no—
289Dic5—64no—
290Dic5—64no—
291Dic5C2×SL(2,ℤ/4)14yesdirect product factor
292C24—31no—
293C24—31no—
294C24—32no—
295C24—32no—
296C24—32no—
297C24D5×Dic312yes—
298C24D3×Dic512yes—
299C24#120012yes—
300C24C2×D3×D511yescharacteristic
301C22×C4—32no—
302C22×C4—32no—
303C22×C4—34no—
304C22×C4—34no—
305C22×C4—51no—
306C22×C4—54no—
307C22×C4—62no—
308C22×C4—61no—
309C22×C4—62no—
310C22×C4—62no—
311C22×C4—61no—
312C22×C4—152no—
313C22×C4—152no—
314C22×C4—154no—
315C22×C4—154no—
316C22×C4—151no—
317C22×C4—152no—
318C22×C4—154no—
319C22×C4—158no—
320C22×C4—154no—
321C22×C4—152no—
322C22×C4—152no—
323C22×C4—154no—
324C22×C4—154no—
325C22×C4—304no—
326C22×C4—301no—
327C22×C4—302no—
328C22×C4—302no—
329C22×C4—301no—
330C22×C4—302no—
331C22×C4—302no—
332C22×C4—301no—
333C22⋊C4—38no—
334C22⋊C4—34no—
335C22⋊C4—34no—
336C22⋊C4—158no—
337C22⋊C4—154no—
338C22⋊C4—154no—
339C42—304no—
340C42—308no—
341C42—304no—
342C3×C5—41noHall
343C22×C3—41no—
344C22×C3—42no—
345C22×C3—42no—
346C22×C3—42no—
347A4C2×C4×Dic511yescharacteristic, complemented
348C3×C4—204no—
349Dic3—44no—
350Dic3—204no—
351C2×C5—31no—
352C2×C5—31no—
353C2×C5—31no—
354C2×C5—34no—
355C2×C5—31no—
356C2×C5C4×SL(2,ℤ/4)14yescomplemented
357C2×C5C2×C4×S411yescharacteristic
358C2×C5C2×C4×S411yescharacteristic
359C2×C5C22×SL(2,ℤ/4)11yescharacteristic
360C23—31no—
361C23—31no—
362C23—32no—
363C23—32no—
364C23—32no—
365C23—32no—
366C23—32no—
367C23—32no—
368C23—32no—
369C23—31no—
370C23—32no—
371C23—31no—
372C23—32no—
373C23—31no—
374C23—34no—
375C23—34no—
376C23—34no—
377C23—61no—
378C23—64no—
379C23—61no—
380C23—61no—
381C23Dic3×Dic514yescomplemented
382C23C2×D5×Dic311yescharacteristic
383C23C2×D3×Dic511yescharacteristic
384C23C2×#120011yescharacteristic
385C23D5×S411yescharacteristic
386C2×C4—34no—
387C2×C4—34no—
388C2×C4—52no—
389C2×C4—54no—
390C2×C4—62no—
391C2×C4—68no—
392C2×C4—62no—
393C2×C4—62no—
394C2×C4—64no—
395C2×C4—62no—
396C2×C4—64no—
397C2×C4—154no—
398C2×C4—154no—
399C2×C4—158no—
400C2×C4—152no—
401C2×C4—154no—
402C2×C4—154no—
403C2×C4—154no—
404C2×C4—154no—
405C2×C4—302no—
406C2×C4—304no—
407C2×C4—302no—
408C2×C4—308no—
409C2×C4—302no—
410C2×C4—302no—
411C2×C4—304no—
412C2×C4—302no—
413C2×C4—304no—
414C2×C3—44no—
415C2×C3—41no—
416C2×C3—41no—
417C2×C3—41no—
418C5C2×C4×SL(2,ℤ/4)11yescharacteristic, complemented, Sylow
419C22—34no—
420C22—31no—
421C22—31no—
422C22—31no—
423C22—34no—
424C22—34no—
425C22—34no—
426C22—34no—
427C22—31no—
428C22—31no—
429C22—32no—
430C22—31no—
431C22—31no—
432C22—32no—
433C22—31no—
434C22—31no—
435C22—32no—
436C22—61no—
437C22—62no—
438C22—62no—
439C22—62no—
440C22C2×Dic3×Dic511yescharacteristic, complemented
441C22Dic5×S412yes—
442C22#9190112yes—
443C22D5×SL(2,ℤ/4)12yes—
444C22C2×D5×S411yescharacteristic
445C4—54no—
446C4—64no—
447C4—64no—
448C4—154no—
449C4—304no—
450C4—304no—
451C3—41noSylow
452C2—31no—
453C2—31no—
454C2—31no—
455C2—34no—
456C2—31no—
457C2Dic5×SL(2,ℤ/4)14yesdirect product factor
458C2C2×Dic5×S411yescharacteristic
459C2C2×#9190111yescharacteristic
460C2C2×D5×SL(2,ℤ/4)11yescharacteristic
subgroupG/H
center, hypercenterC23D5×S4
commutator, hypocenterC5×A4C2×C42
FrattiniC22C2×D5×S4
Fitting, socleC25×C5C2×D3
prime radicalC23×C5×A4C22
cosocleC22×C5×A4C23
LCSG ▷ C5×A4
UCSC1 ◁ C23

Complements

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

NH ≅ G/Ntype
4. Dic5×SL(2,ℤ/4)C2direct (N × H)
7. C2×A4×Dic5C4semidirect
9. C2×A4⋊Dic5C4semidirect
11. C2×C5×SL(2,ℤ/4)C4semidirect
31. A4×Dic5C2×C4semidirect
32. A4⋊Dic5C2×C4semidirect
33. C5×SL(2,ℤ/4)C2×C4semidirect
48. C23×Dic5Dic3semidirect
68. C2×C5×A4C42semidirect
83. C2×SL(2,ℤ/4)Dic5direct (N × H)
125. C22×Dic5C2×Dic3semidirect
149. C5×A4C2×C42semidirect
162. SL(2,ℤ/4)C2×Dic5direct (N × H)
185. C23×C5C4×Dic3semidirect
216. C2×Dic5SL(2,ℤ/4)direct (N × H)
250. C2×A4C4×Dic5semidirect
281. C22×C5C2×C4×Dic3semidirect
291. Dic5C2×SL(2,ℤ/4)direct (N × H)
347. A4C2×C4×Dic5semidirect
356. C2×C5C4×SL(2,ℤ/4)semidirect
381. C23Dic3×Dic5semidirect
418. C5C2×C4×SL(2,ℤ/4)semidirect
440. C22C2×Dic3×Dic5semidirect
457. C2Dic5×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×A4×Dic52
C22×A4⋊Dic52
C22×C5×SL(2,ℤ/4)2
Dic5×SL(2,ℤ/4)2
C2×C22⋊C4×Dic53
C2×Dic3×Dic54
C2×C4×SL(2,ℤ/4)5