Finite Group Databasebeta

← C5×#85485 #14511223

Subgroups — C5×#85485 #14511223

Order 1920.

840 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 840 conjugacy classes of subgroups — too many to draw. Only its 88 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, complemented
2C2×C5×GL(2,ℤ/4)C211yesmaximal, characteristic
3C5×#5432C211yesmaximal, characteristic, complemented
4C5×(C2×S4)⋊C4C212yesmaximal, complemented
5C5×SL(2,ℤ/4)⋊C4C212yesmaximal, complemented
6C5×#2849—31nomaximal, Hall
7C23×C5×A4C2211yescharacteristic
8C22×C5×S4C2211yescharacteristic
9C2×C4×C5×A4—22no—
10C2×C4×C5×A4C2212yes—
11C5×GL(2,ℤ/4)—22no—
12C2×C5×SL(2,ℤ/4)C2211yescharacteristic
13C2×C5×SL(2,ℤ/4)C2212yes—
14C5×#929—41nomaximal
15#85485C511yesmaximal, characteristic, direct product factor, Hall
16C24×C4×C5D311yescharacteristic, complemented
17C2×C5×#123—31no—
18C2×C5×#123—31no—
19C2×C5×#120—31no—
20C5×#386—32no—
21C5×#386—32no—
22C5×#385—32no—
23C5×#385—32no—
24C5×#381—31no—
25C5×#381—31no—
26C5×#379—31no—
27C22×C5×A4—22no—
28C22×C5×A4D412yes—
29C22×C5×A4C2311yescharacteristic
30C2×C5×S4—21no—
31C2×C5×S4—22no—
32C22×C3×C4×C5—41no—
33C2×C5×#67—41no—
34C4×C5×A4—22no—
35C5×SL(2,ℤ/4)—21no—
36C5×SL(2,ℤ/4)—22no—
37C5×#217—41no—
38C5×(C2×D3)⋊C4—42no—
39C5×Dic3⋊C4—42no—
40C22×C4×A4C2×C511yescharacteristic, complemented
41C2×GL(2,ℤ/4)C2×C511yescharacteristic
42#5432C2×C511yescharacteristic, complemented
43(C2×S4)⋊C4C2×C512yescomplemented
44SL(2,ℤ/4)⋊C4C2×C512yescomplemented
45C25×C5C2×D311yescharacteristic
46C22×C5×D4—31no—
47C22×C5×D4—31no—
48C22×C5×D4—31no—
49C23×C4×C5—22no—
50C23×C4×C5—31no—
51C23×C4×C5—31no—
52C23×C4×C5—32no—
53C23×C4×C5—62no—
54C23×C4×C5C2×D312yes—
55C5×#123—64no—
56C5×#123—61no—
57C5×#123—62no—
58C5×#123—61no—
59C5×#120—62no—
60C5×#120—62no—
61C2×C5×C4⋊C4—32no—
62C2×C5×C4⋊C4—32no—
63C2×C5×C22⋊C4—31no—
64C2×C5×C22⋊C4—31no—
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×C5×C22⋊C4—32no—
71C2×C5×C22⋊C4—31no—
72C5×#95—32no—
73C5×#95—32no—
74C5×#95—32no—
75C5×#95—32no—
76#2849—31noSylow
77C23×C3×C5—41no—
78C22×C5×D3—41no—
79C2×C5×A4—22no—
80C2×C5×A4C2×D411yescharacteristic
81C2×C5×A4#4112yes—
82C5×S4—41no—
83C2×C3×C4×C5—42no—
84C2×C3×C4×C5—82no—
85C5×#67—82no—
86C2×C5×Dic3—41no—
87C2×C5×Dic3—42no—
88C23×A4C22×C511yescharacteristic
89C22×S4C22×C511yescharacteristic
90C2×C4×A4—22no—
91C2×C4×A4C22×C512yes—
92GL(2,ℤ/4)—22no—
93C2×SL(2,ℤ/4)C22×C511yescharacteristic
94C2×SL(2,ℤ/4)C22×C512yes—
95#929—41no—
96C24×C5—22no—
97C24×C5—31no—
98C24×C5—31no—
99C24×C5—32no—
100C24×C5—31no—
101C24×C5—62no—
102C24×C5#6712yes—
103C24×C5C22×D311yescharacteristic
104C2×C5×D4—62no—
105C2×C5×D4—61no—
106C2×C5×D4—62no—
107C2×C5×D4—62no—
108C2×C5×D4—61no—
109C2×C5×D4—62no—
110C2×C5×D4—64no—
111C2×C5×D4—62no—
112C2×C5×D4—61no—
113C2×C5×D4—61no—
114C22×C4×C5—22no—
115C22×C4×C5—32no—
116C22×C4×C5—34no—
117C22×C4×C5—31no—
118C22×C4×C5—32no—
119C22×C4×C5—32no—
120C22×C4×C5—32no—
121C22×C4×C5—32no—
122C22×C4×C5—31no—
123C22×C4×C5—31no—
124C22×C4×C5—31no—
125C22×C4×C5—62no—
126C22×C4×C5—62no—
127C22×C4×C5—61no—
128C22×C4×C5—62no—
129C22×C4×C5—64no—
130C22×C4×C5—64no—
131C22×C4×C5—62no—
132C22×C4×C5S411yescharacteristic, complemented
133C5×C4⋊C4—64no—
134C5×C4⋊C4—62no—
135C5×C4⋊C4—62no—
136C5×C22⋊C4—64no—
137C5×C22⋊C4—62no—
138C5×C22⋊C4—62no—
139C5×C22⋊C4—64no—
140C5×C22⋊C4—62no—
141C5×C22⋊C4—61no—
142C5×C22⋊C4—62no—
143C5×C22⋊C4—62no—
144C5×C22⋊C4—61no—
145C5×C22⋊C4—62no—
146C5×C22⋊C4—61no—
147C5×C22⋊C4—61no—
148C5×C22⋊C4—62no—
149C24×C4C5×D311yescharacteristic, complemented
150C2×#123—31no—
151C2×#123—31no—
152C2×#120—31no—
153#386—32no—
154#386—32no—
155#385—32no—
156#385—32no—
157#381—31no—
158#381—31no—
159#379—31no—
160C22×C3×C5—42no—
161C22×C3×C5—41no—
162C22×C3×C5—82no—
163C2×C5×D3—81no—
164C2×C5×D3—82no—
165C5×A4#12311yescharacteristic, complemented
166C3×C4×C5—82no—
167C5×Dic3—82no—
168C5×Dic3—81no—
169C22×A4—22no—
170C22×A4C5×D412yes—
171C22×A4C23×C511yescharacteristic
172C2×S4—21no—
173C2×S4—22no—
174C22×C3×C4—41no—
175C2×#67—41no—
176C4×A4—22no—
177SL(2,ℤ/4)—21no—
178SL(2,ℤ/4)—22no—
179#217—41no—
180(C2×D3)⋊C4—42no—
181Dic3⋊C4—42no—
182C23×C5—22no—
183C23×C5—31no—
184C23×C5—32no—
185C23×C5—31no—
186C23×C5—32no—
187C23×C5—34no—
188C23×C5—32no—
189C23×C5—32no—
190C23×C5—31no—
191C23×C5—62no—
192C23×C5—62no—
193C23×C5—62no—
194C23×C5—61no—
195C23×C5—64no—
196C23×C5—62no—
197C23×C5—64no—
198C23×C5—61no—
199C23×C5—61no—
200C23×C5—62no—
201C23×C5—62no—
202C23×C5—61no—
203C23×C5#23512yes—
204C23×C5C2×#6711yescharacteristic
205C23×C5C2×S411yescharacteristic
206C5×D4—122no—
207C5×D4—121no—
208C5×D4—124no—
209C5×D4—122no—
210C5×D4—121no—
211C5×D4—122no—
212C2×C4×C5—22no—
213C2×C4×C5—34no—
214C2×C4×C5—32no—
215C2×C4×C5—62no—
216C2×C4×C5—62no—
217C2×C4×C5—64no—
218C2×C4×C5—64no—
219C2×C4×C5—62no—
220C2×C4×C5—62no—
221C2×C4×C5—62no—
222C2×C4×C5—61no—
223C2×C4×C5—61no—
224C2×C4×C5—61no—
225C2×C4×C5—62no—
226C2×C4×C5—62no—
227C2×C4×C5—62no—
228C2×C4×C5—62no—
229C2×C4×C5—61no—
230C2×C4×C5—62no—
231C2×C4×C5—62no—
232C2×C4×C5—62no—
233C2×C4×C5—61no—
234C2×C4×C5C2×S412yes—
235C25C2×C5×D311yescharacteristic
236C22×D4—31no—
237C22×D4—31no—
238C22×D4—31no—
239C23×C4—22no—
240C23×C4—31no—
241C23×C4—31no—
242C23×C4—32no—
243C23×C4—62no—
244C23×C4C2×C5×D312yes—
245#123—64no—
246#123—61no—
247#123—62no—
248#123—61no—
249#120—62no—
250#120—62no—
251C2×C4⋊C4—32no—
252C2×C4⋊C4—32no—
253C2×C22⋊C4—31no—
254C2×C22⋊C4—31no—
255C2×C22⋊C4—31no—
256C2×C22⋊C4—31no—
257C2×C22⋊C4—32no—
258C2×C22⋊C4—32no—
259C2×C22⋊C4—32no—
260C2×C22⋊C4—31no—
261C2×C22⋊C4—32no—
262#95—32no—
263#95—32no—
264#95—32no—
265#95—32no—
266C2×C3×C5—42no—
267C2×C3×C5—41no—
268C2×C3×C5—82no—
269C5×D3—161no—
270C23×C3—41no—
271C22×D3—41no—
272C2×A4—22no—
273C2×A4C2×C5×D411yescharacteristic
274C2×A4C5×#4112yes—
275S4—41no—
276C2×C3×C4—42no—
277C2×C3×C4—82no—
278#67—82no—
279C2×Dic3—41no—
280C2×Dic3—42no—
281C22×C5—22no—
282C22×C5—32no—
283C22×C5—31no—
284C22×C5—32no—
285C22×C5—32no—
286C22×C5—31no—
287C22×C5—31no—
288C22×C5—34no—
289C22×C5—32no—
290C22×C5—62no—
291C22×C5—62no—
292C22×C5—64no—
293C22×C5—61no—
294C22×C5—62no—
295C22×C5—64no—
296C22×C5—62no—
297C22×C5—121no—
298C22×C5—121no—
299C22×C5—122no—
300C22×C5—122no—
301C22×C5—121no—
302C22×C5#92911yescharacteristic, complemented
303C22×C5GL(2,ℤ/4)12yes—
304C22×C5C22×S411yescharacteristic
305C4×C5—22no—
306C4×C5—62no—
307C4×C5—122no—
308C4×C5—121no—
309C4×C5—121no—
310C4×C5—122no—
311C4×C5—121no—
312C24—22no—
313C24—31no—
314C24—31no—
315C24—32no—
316C24—31no—
317C24—62no—
318C24C5×#6712yes—
319C24C22×C5×D311yescharacteristic
320C2×D4—62no—
321C2×D4—61no—
322C2×D4—62no—
323C2×D4—62no—
324C2×D4—61no—
325C2×D4—62no—
326C2×D4—64no—
327C2×D4—62no—
328C2×D4—61no—
329C2×D4—61no—
330C22×C4—22no—
331C22×C4—32no—
332C22×C4—34no—
333C22×C4—31no—
334C22×C4—32no—
335C22×C4—32no—
336C22×C4—31no—
337C22×C4—31no—
338C22×C4—32no—
339C22×C4—32no—
340C22×C4—31no—
341C22×C4—62no—
342C22×C4—62no—
343C22×C4—61no—
344C22×C4—62no—
345C22×C4—64no—
346C22×C4—64no—
347C22×C4—62no—
348C22×C4C5×S411yescharacteristic, complemented
349C4⋊C4—64no—
350C4⋊C4—62no—
351C4⋊C4—62no—
352C22⋊C4—64no—
353C22⋊C4—62no—
354C22⋊C4—62no—
355C22⋊C4—61no—
356C22⋊C4—62no—
357C22⋊C4—62no—
358C22⋊C4—61no—
359C22⋊C4—62no—
360C22⋊C4—64no—
361C22⋊C4—62no—
362C22⋊C4—61no—
363C22⋊C4—61no—
364C22⋊C4—62no—
365C3×C5—41noHall
366C22×C3—42no—
367C22×C3—41no—
368C22×C3—82no—
369C2×D3—81no—
370C2×D3—82no—
371A4C5×#12311yescharacteristic, complemented
372C3×C4—82no—
373Dic3—81no—
374Dic3—82no—
375C2×C5—22no—
376C2×C5—31no—
377C2×C5—31no—
378C2×C5—32no—
379C2×C5—62no—
380C2×C5—241no—
381C2×C5#591212yes—
382C2×C5C2×GL(2,ℤ/4)11yescharacteristic
383C23—22no—
384C23—31no—
385C23—32no—
386C23—31no—
387C23—32no—
388C23—34no—
389C23—32no—
390C23—32no—
391C23—31no—
392C23—62no—
393C23—62no—
394C23—62no—
395C23—61no—
396C23—64no—
397C23—62no—
398C23—64no—
399C23—61no—
400C23—61no—
401C23—62no—
402C23—62no—
403C23—61no—
404C23C5×#23512yes—
405C23C2×C5×#6711yescharacteristic
406C23C2×C5×S411yescharacteristic
407D4—122no—
408D4—121no—
409D4—124no—
410D4—122no—
411D4—122no—
412D4—121no—
413C2×C4—22no—
414C2×C4—34no—
415C2×C4—32no—
416C2×C4—62no—
417C2×C4—62no—
418C2×C4—64no—
419C2×C4—64no—
420C2×C4—62no—
421C2×C4—62no—
422C2×C4—62no—
423C2×C4—61no—
424C2×C4—61no—
425C2×C4—61no—
426C2×C4—62no—
427C2×C4—62no—
428C2×C4—62no—
429C2×C4—62no—
430C2×C4—61no—
431C2×C4—62no—
432C2×C4—62no—
433C2×C4—62no—
434C2×C4—61no—
435C2×C4C2×C5×S412yes—
436C2×C3—42no—
437C2×C3—41no—
438C2×C3—82no—
439D3—161no—
440C5#8548511yescharacteristic, direct product factor, Sylow
441C22—22no—
442C22—32no—
443C22—31no—
444C22—32no—
445C22—32no—
446C22—31no—
447C22—31no—
448C22—34no—
449C22—32no—
450C22—62no—
451C22—62no—
452C22—64no—
453C22—61no—
454C22—62no—
455C22—64no—
456C22—62no—
457C22—121no—
458C22—121no—
459C22—122no—
460C22—122no—
461C22—121no—
462C22C5×#92911yescharacteristic, complemented
463C22C5×GL(2,ℤ/4)12yes—
464C22C22×C5×S411yescharacteristic
465C4—22no—
466C4—62no—
467C4—121no—
468C4—121no—
469C4—122no—
470C4—122no—
471C4—121no—
472C3—41noSylow
473C2—22no—
474C2—31no—
475C2—31no—
476C2—32no—
477C2—62no—
478C2—241no—
479C2C5×#591212yes—
480C2C2×C5×GL(2,ℤ/4)11yescharacteristic
subgroupG/H
centerC22×C5C22×S4
commutator, cosocleC22×A4C23×C5
FrattiniC22C22×C5×S4
FittingC24×C4×C5D3
socleC24×C5C22×D3
prime radicalC2×C5×GL(2,ℤ/4)C2
hypercenterC22×C4×C5S4
hypocenterA4C5×#123
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C22×C5 ◁ C22×C4×C5

Complements

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

NH ≅ G/Ntype
1. C22×C4×C5×A4C2semidirect
3. C5×#5432C2semidirect
4. C5×(C2×S4)⋊C4C2semidirect
5. C5×SL(2,ℤ/4)⋊C4C2semidirect
15. #85485C5direct (N × H)
16. C24×C4×C5D3semidirect
40. C22×C4×A4C2×C5semidirect
42. #5432C2×C5semidirect
43. (C2×S4)⋊C4C2×C5semidirect
44. SL(2,ℤ/4)⋊C4C2×C5semidirect
132. C22×C4×C5S4semidirect
149. C24×C4C5×D3semidirect
165. C5×A4#123semidirect
302. C22×C5#929semidirect
348. C22×C4C5×S4semidirect
371. A4C5×#123semidirect
440. C5#85485direct (N × H)
462. C22C5×#929semidirect

Maximal subgroups

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

subgroupindex
C22×C4×C5×A42
C2×C5×GL(2,ℤ/4)2
C5×#54322
C5×(C2×S4)⋊C42
C5×SL(2,ℤ/4)⋊C42
C5×#28493
C5×#9294
#854855