Finite Group Databasebeta

← C5×#85312 #14511050

Subgroups — C5×#85312 #14511050

Order 1920.

674 conjugacy classes of subgroups; only the 92 normal subgroups are shown.

normal subgroups only

This group has 674 conjugacy classes of subgroups — too many to draw. Only its 92 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
1C2×C4×C5×S4C211yesmaximal, characteristic
2C2×C5×A4⋊Q8C211yesmaximal, characteristic, complemented
3C5×A4×C4⋊C4C211yesmaximal, characteristic, complemented
4C5×(C2×S4)⋊C4C212yesmaximal
5C5×C4⋊SL(2,ℤ/4)C212yesmaximal, complemented
6C5×#2417—31nomaximal, Hall
7C22×C5×S4C2211yescharacteristic
8C2×C4×C5×A4C2212yes—
9C2×C4×C5×A4C2211yescharacteristic
10C2×C5×SL(2,ℤ/4)C2212yes—
11C2×C5×SL(2,ℤ/4)C2211yescharacteristic
12C4×C5×S4—22no—
13C5×A4⋊Q8—22no—
14C5×#897—41nomaximal
15#85312C511yesmaximal, characteristic, direct product factor, Hall
16C2×C5×C22⋊Q8—31no—
17C2×C5×C22⋊Q8—31no—
18C2×C4×C5×D4—31no—
19C22×C5×C4⋊C4D311yescharacteristic, complemented
20C5×#398—32no—
21C5×#396—32no—
22C5×#391—31no—
23C5×#389—31no—
24C5×#386—32no—
25C5×#380—31no—
26C5×#380—32no—
27C22×C5×A4C2311yescharacteristic
28C2×C5×S4—21no—
29C2×C5×S4—21no—
30C2×C5×S4Q812yescomplemented
31C2×C4×C5×D3—41no—
32C2×C5×Dic6—41no—
33C4×C5×A4—22no—
34C4×C5×A4D412yes—
35C5×SL(2,ℤ/4)—22no—
36C5×SL(2,ℤ/4)—21no—
37C3×C5×C4⋊C4—41no—
38C5×(C2×D3)⋊C4—42no—
39C5×C4⋊Dic3—42no—
40C2×C4×S4C2×C511yescharacteristic
41C2×A4⋊Q8C2×C511yescharacteristic, complemented
42A4×C4⋊C4C2×C511yescharacteristic, complemented
43(C2×S4)⋊C4C2×C512yes—
44C4⋊SL(2,ℤ/4)C2×C512yescomplemented
45C22×C5×Q8—31no—
46C22×C5×D4—31no—
47C23×C4×C5—31no—
48C23×C4×C5C2×D312yes—
49C23×C4×C5C2×D311yescharacteristic
50C5×C22⋊Q8—61no—
51C5×C22⋊Q8—62no—
52C5×C22⋊Q8—62no—
53C5×C22⋊Q8—62no—
54C5×C22⋊Q8—61no—
55C4×C5×D4—62no—
56C4×C5×D4—62no—
57C2×C5×C4⋊C4—31no—
58C2×C5×C4⋊C4—31no—
59C2×C5×C4⋊C4—32no—
60C2×C5×C4⋊C4—31no—
61C2×C5×C4⋊C4—32no—
62C2×C5×C4⋊C4—32no—
63C2×C5×C4⋊C4—31no—
64C2×C5×C4⋊C4—32no—
65C2×C5×C22⋊C4—32no—
66C2×C5×C22⋊C4—31no—
67C2×C5×C22⋊C4—31no—
68C2×C5×C22⋊C4—32no—
69C2×C5×C22⋊C4—32no—
70C2×C42×C5—31no—
71C5×#95—32no—
72C5×#95—32no—
73C5×#95—32no—
74C5×#95—32no—
75#2417—31noSylow
76C22×C5×D3—41no—
77C2×C5×A4C2×D411yescharacteristic
78C2×C5×A4C2×Q811yescharacteristic
79C2×C5×A4#4111yescharacteristic
80C5×S4—22no—
81C2×C3×C4×C5—41no—
82C2×C3×C4×C5—42no—
83C2×C5×Dic3—41no—
84C2×C5×Dic3—42no—
85C4×C5×D3—82no—
86C5×Dic6—82no—
87C22×S4C22×C511yescharacteristic
88C2×C4×A4C22×C512yes—
89C2×C4×A4C22×C511yescharacteristic
90C2×SL(2,ℤ/4)C22×C512yes—
91C2×SL(2,ℤ/4)C22×C511yescharacteristic
92C4×S4—22no—
93A4⋊Q8—22no—
94#897—41no—
95C24×C5—31no—
96C24×C5C22×D311yescharacteristic
97C2×C5×Q8—61no—
98C2×C5×Q8—62no—
99C2×C5×Q8—62no—
100C2×C5×Q8—61no—
101C2×C5×D4—32no—
102C2×C5×D4—32no—
103C2×C5×D4—61no—
104C2×C5×D4—61no—
105C2×C5×D4—61no—
106C2×C5×D4—61no—
107C22×C4×C5—22no—
108C22×C4×C5—31no—
109C22×C4×C5—31no—
110C22×C4×C5—32no—
111C22×C4×C5—31no—
112C22×C4×C5—32no—
113C22×C4×C5—32no—
114C22×C4×C5—31no—
115C22×C4×C5—32no—
116C22×C4×C5—32no—
117C22×C4×C5—31no—
118C22×C4×C5—62no—
119C22×C4×C5—61no—
120C22×C4×C5—62no—
121C22×C4×C5—62no—
122C22×C4×C5—61no—
123C22×C4×C5D1212yes—
124C5×C4⋊C4—32no—
125C5×C4⋊C4—31no—
126C5×C4⋊C4—61no—
127C5×C4⋊C4—62no—
128C5×C4⋊C4—62no—
129C5×C4⋊C4—64no—
130C5×C4⋊C4—62no—
131C5×C4⋊C4—62no—
132C5×C4⋊C4—122no—
133C5×C4⋊C4S411yescharacteristic, complemented
134C5×C22⋊C4—62no—
135C5×C22⋊C4—62no—
136C5×C22⋊C4—61no—
137C5×C22⋊C4—62no—
138C5×C22⋊C4—61no—
139C5×C22⋊C4—62no—
140C5×C22⋊C4—62no—
141C5×C22⋊C4—122no—
142C42×C5—62no—
143C2×C22⋊Q8—31no—
144C2×C22⋊Q8—31no—
145C2×C4×D4—31no—
146C22×C4⋊C4C5×D311yescharacteristic, complemented
147#398—32no—
148#396—32no—
149#391—31no—
150#389—31no—
151#386—32no—
152#380—31no—
153#380—32no—
154C22×C3×C5—41no—
155C2×C5×D3—42no—
156C2×C5×D3—81no—
157C2×C5×D3—81no—
158C5×A4C22⋊Q811yescharacteristic, complemented
159C3×C4×C5—42no—
160C3×C4×C5—82no—
161C5×Dic3—82no—
162C5×Dic3—81no—
163C22×A4C23×C511yescharacteristic
164C2×S4—21no—
165C2×S4—21no—
166C2×S4C5×Q812yescomplemented
167C2×C4×D3—41no—
168C2×Dic6—41no—
169C4×A4—22no—
170C4×A4C5×D412yes—
171SL(2,ℤ/4)—22no—
172SL(2,ℤ/4)—21no—
173C3×C4⋊C4—41no—
174(C2×D3)⋊C4—42no—
175C4⋊Dic3—42no—
176C23×C5—31no—
177C23×C5—31no—
178C23×C5—31no—
179C23×C5—31no—
180C23×C5—32no—
181C23×C5—61no—
182C23×C5—61no—
183C23×C5—61no—
184C23×C5—61no—
185C23×C5—61no—
186C23×C5—61no—
187C23×C5C2×D1211yescharacteristic
188C23×C5#23711yescharacteristic
189C23×C5D3×Q811yescharacteristic
190C5×Q8—122no—
191C5×Q8—122no—
192C5×D4—62no—
193C5×D4—62no—
194C5×D4—62no—
195C5×D4—62no—
196C2×C4×C5—31no—
197C2×C4×C5—32no—
198C2×C4×C5—32no—
199C2×C4×C5—32no—
200C2×C4×C5—32no—
201C2×C4×C5—32no—
202C2×C4×C5—62no—
203C2×C4×C5—62no—
204C2×C4×C5—61no—
205C2×C4×C5—61no—
206C2×C4×C5—61no—
207C2×C4×C5—62no—
208C2×C4×C5—62no—
209C2×C4×C5—61no—
210C2×C4×C5—62no—
211C2×C4×C5—61no—
212C2×C4×C5—62no—
213C2×C4×C5—62no—
214C2×C4×C5—61no—
215C2×C4×C5—61no—
216C2×C4×C5—62no—
217C2×C4×C5—61no—
218C2×C4×C5—62no—
219C2×C4×C5—62no—
220C2×C4×C5—61no—
221C2×C4×C5—122no—
222C2×C4×C5—122no—
223C2×C4×C5C2×S412yes—
224C2×C4×C5C2×S411yescharacteristic
225C22×Q8—31no—
226C22×D4—31no—
227C23×C4—31no—
228C23×C4C2×C5×D312yes—
229C23×C4C2×C5×D311yescharacteristic
230C22⋊Q8—61no—
231C22⋊Q8—62no—
232C22⋊Q8—62no—
233C22⋊Q8—62no—
234C22⋊Q8—61no—
235C4×D4—62no—
236C4×D4—62no—
237C2×C4⋊C4—31no—
238C2×C4⋊C4—31no—
239C2×C4⋊C4—32no—
240C2×C4⋊C4—31no—
241C2×C4⋊C4—31no—
242C2×C4⋊C4—32no—
243C2×C4⋊C4—32no—
244C2×C4⋊C4—32no—
245C2×C22⋊C4—31no—
246C2×C22⋊C4—32no—
247C2×C22⋊C4—31no—
248C2×C22⋊C4—32no—
249C2×C22⋊C4—32no—
250C2×C42—31no—
251#95—32no—
252#95—32no—
253#95—32no—
254#95—32no—
255C2×C3×C5—41no—
256C2×C3×C5—41no—
257C2×C3×C5—41no—
258C5×D3—82no—
259C22×D3—41no—
260C2×A4C2×C5×D411yescharacteristic
261C2×A4C2×C5×Q811yescharacteristic
262C2×A4C5×#4111yescharacteristic
263S4—22no—
264C2×C3×C4—41no—
265C2×C3×C4—42no—
266C2×Dic3—41no—
267C2×Dic3—42no—
268C4×D3—82no—
269Dic6—82no—
270C22×C5—31no—
271C22×C5—31no—
272C22×C5—31no—
273C22×C5—31no—
274C22×C5—31no—
275C22×C5—31no—
276C22×C5—31no—
277C22×C5—31no—
278C22×C5—31no—
279C22×C5—61no—
280C22×C5—62no—
281C22×C5—62no—
282C22×C5—62no—
283C22×C5—121no—
284C22×C5—121no—
285C22×C5—121no—
286C22×C5—121no—
287C22×C5#89711yescharacteristic, complemented
288C22×C5C22×S411yescharacteristic
289C4×C5—22no—
290C4×C5—32no—
291C4×C5—62no—
292C4×C5—62no—
293C4×C5—121no—
294C4×C5—122no—
295C4×C5—121no—
296C4×C5—122no—
297C4×C5C4⋊S412yes—
298C24—31no—
299C24C22×C5×D311yescharacteristic
300C2×Q8—61no—
301C2×Q8—62no—
302C2×Q8—62no—
303C2×Q8—61no—
304C2×D4—32no—
305C2×D4—32no—
306C2×D4—61no—
307C2×D4—61no—
308C2×D4—61no—
309C2×D4—61no—
310C22×C4—22no—
311C22×C4—31no—
312C22×C4—31no—
313C22×C4—32no—
314C22×C4—31no—
315C22×C4—32no—
316C22×C4—32no—
317C22×C4—31no—
318C22×C4—32no—
319C22×C4—32no—
320C22×C4—31no—
321C22×C4—62no—
322C22×C4—61no—
323C22×C4—62no—
324C22×C4—62no—
325C22×C4—61no—
326C22×C4C5×D1212yes—
327C4⋊C4—32no—
328C4⋊C4—31no—
329C4⋊C4—61no—
330C4⋊C4—62no—
331C4⋊C4—62no—
332C4⋊C4—64no—
333C4⋊C4—62no—
334C4⋊C4—62no—
335C4⋊C4—122no—
336C4⋊C4C5×S411yescharacteristic, complemented
337C22⋊C4—62no—
338C22⋊C4—62no—
339C22⋊C4—61no—
340C22⋊C4—62no—
341C22⋊C4—61no—
342C22⋊C4—62no—
343C22⋊C4—62no—
344C22⋊C4—122no—
345C42—62no—
346C3×C5—41noHall
347C22×C3—41no—
348C2×D3—42no—
349C2×D3—81no—
350C2×D3—81no—
351A4C5×C22⋊Q811yescharacteristic, complemented
352C3×C4—42no—
353C3×C4—82no—
354Dic3—82no—
355Dic3—81no—
356C2×C5—31no—
357C2×C5—31no—
358C2×C5—31no—
359C2×C5—31no—
360C2×C5—122no—
361C2×C5C2×C4⋊S411yescharacteristic
362C2×C5#591411yescharacteristic
363C2×C5Q8×S411yescharacteristic
364C23—31no—
365C23—31no—
366C23—31no—
367C23—31no—
368C23—32no—
369C23—61no—
370C23—61no—
371C23—61no—
372C23—61no—
373C23—61no—
374C23—61no—
375C23C2×C5×D1211yescharacteristic
376C23C5×#23711yescharacteristic
377C23C5×D3×Q811yescharacteristic
378Q8—122no—
379Q8—122no—
380D4—62no—
381D4—62no—
382D4—62no—
383D4—62no—
384C2×C4—31no—
385C2×C4—32no—
386C2×C4—32no—
387C2×C4—32no—
388C2×C4—32no—
389C2×C4—32no—
390C2×C4—62no—
391C2×C4—62no—
392C2×C4—61no—
393C2×C4—61no—
394C2×C4—61no—
395C2×C4—62no—
396C2×C4—62no—
397C2×C4—61no—
398C2×C4—62no—
399C2×C4—61no—
400C2×C4—62no—
401C2×C4—62no—
402C2×C4—61no—
403C2×C4—61no—
404C2×C4—62no—
405C2×C4—61no—
406C2×C4—62no—
407C2×C4—62no—
408C2×C4—61no—
409C2×C4—122no—
410C2×C4—122no—
411C2×C4C2×C5×S412yes—
412C2×C4C2×C5×S411yescharacteristic
413C2×C3—41no—
414C2×C3—41no—
415C2×C3—41no—
416D3—82no—
417C5#8531211yescharacteristic, direct product factor, Sylow
418C22—31no—
419C22—31no—
420C22—31no—
421C22—31no—
422C22—31no—
423C22—31no—
424C22—31no—
425C22—31no—
426C22—31no—
427C22—61no—
428C22—62no—
429C22—62no—
430C22—62no—
431C22—121no—
432C22—121no—
433C22—121no—
434C22—121no—
435C22C5×#89711yescharacteristic, complemented
436C22C22×C5×S411yescharacteristic
437C4—22no—
438C4—32no—
439C4—62no—
440C4—62no—
441C4—121no—
442C4—122no—
443C4—121no—
444C4—122no—
445C4C5×C4⋊S412yes—
446C3—41noSylow
447C2—31no—
448C2—31no—
449C2—31no—
450C2—31no—
451C2—122no—
452C2C2×C5×C4⋊S411yescharacteristic
453C2C5×#591411yescharacteristic
454C2C5×Q8×S411yescharacteristic
subgroupG/H
centerC22×C5C22×S4
commutator, cosocleC22×A4C23×C5
FrattiniC22C22×C5×S4
FittingC22×C5×C4⋊C4D3
socleC24×C5C22×D3
prime radicalC22×C5×S4C22
hypercenterC5×C4⋊C4S4
hypocenterA4C5×C22⋊Q8
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C22×C5 ◁ C5×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C5×A4⋊Q8C2semidirect
3. C5×A4×C4⋊C4C2semidirect
5. C5×C4⋊SL(2,ℤ/4)C2semidirect
15. #85312C5direct (N × H)
19. C22×C5×C4⋊C4D3semidirect
30. C2×C5×S4Q8semidirect
41. C2×A4⋊Q8C2×C5semidirect
42. A4×C4⋊C4C2×C5semidirect
44. C4⋊SL(2,ℤ/4)C2×C5semidirect
133. C5×C4⋊C4S4semidirect
146. C22×C4⋊C4C5×D3semidirect
158. C5×A4C22⋊Q8semidirect
166. C2×S4C5×Q8semidirect
287. C22×C5#897semidirect
336. C4⋊C4C5×S4semidirect
351. A4C5×C22⋊Q8semidirect
417. C5#85312direct (N × H)
435. C22C5×#897semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C5×S42
C2×C5×A4⋊Q82
C5×A4×C4⋊C42
C5×(C2×S4)⋊C42
C5×C4⋊SL(2,ℤ/4)2
C5×#24173
C5×#8974
#853125