Finite Group Databasebeta

← C5×#85301 #14511039

Subgroups — C5×#85301 #14511039

Order 1920.

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

normal subgroups only

This group has 768 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
1C2×C5×GL(2,ℤ/4)C211yesmaximal, characteristic
2C22×C5×SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
3C5×A4×C22⋊C4C211yesmaximal, characteristic, complemented
4C5×(C2×S4)⋊C4C212yesmaximal, complemented
5C5×C4⋊SL(2,ℤ/4)C212yesmaximal, complemented
6C5×#2412—31nomaximal, Hall
7C23×C5×A4C2211yescharacteristic
8C22×C5×S4C2211yescharacteristic
9C2×C4×C5×A4C2212yes—
10C5×GL(2,ℤ/4)—22no—
11C2×C5×SL(2,ℤ/4)—22no—
12C2×C5×SL(2,ℤ/4)C2212yes—
13C2×C5×SL(2,ℤ/4)C2211yescharacteristic
14C5×#886—41nomaximal
15#85301C511yesmaximal, characteristic, direct product factor, Hall
16C2×C5×#123—31no—
17C2×C5×#120—31no—
18C22×C5×C22⋊C4—31no—
19C22×C5×C22⋊C4D311yescharacteristic, complemented
20C5×#397—32no—
21C5×#394—32no—
22C5×#386—31no—
23C5×#386—32no—
24C5×#385—31no—
25C5×#381—31no—
26C5×#380—32no—
27C22×C5×A4—22no—
28C22×C5×A4D412yes—
29C22×C5×A4C2311yescharacteristic
30C2×C5×S4—21no—
31C2×C5×S4—22no—
32C2×C5×#67—41no—
33C22×C5×Dic3—41no—
34C4×C5×A4—22no—
35C5×SL(2,ℤ/4)—22no—
36C5×SL(2,ℤ/4)—21no—
37C3×C5×C22⋊C4—41no—
38C5×(C2×D3)⋊C4—42no—
39C5×C4⋊Dic3—42no—
40C2×GL(2,ℤ/4)C2×C511yescharacteristic
41C22×SL(2,ℤ/4)C2×C511yescharacteristic, complemented
42A4×C22⋊C4C2×C511yescharacteristic, complemented
43(C2×S4)⋊C4C2×C512yescomplemented
44C4⋊SL(2,ℤ/4)C2×C512yescomplemented
45C25×C5C2×D311yescharacteristic
46C22×C5×D4—31no—
47C22×C5×D4—31no—
48C22×C5×D4—31no—
49C23×C4×C5—31no—
50C23×C4×C5—31no—
51C23×C4×C5C2×D312yes—
52C5×#123—61no—
53C5×#123—62no—
54C5×#123—61no—
55C5×#120—62no—
56C5×#120—62no—
57C2×C5×C4⋊C4—32no—
58C2×C5×C4⋊C4—32no—
59C2×C5×C22⋊C4—31no—
60C2×C5×C22⋊C4—31no—
61C2×C5×C22⋊C4—32no—
62C2×C5×C22⋊C4—31no—
63C2×C5×C22⋊C4—32no—
64C2×C5×C22⋊C4—32no—
65C2×C5×C22⋊C4—31no—
66C2×C5×C22⋊C4—31no—
67C2×C5×C22⋊C4—32no—
68C2×C5×C22⋊C4—32no—
69C2×C5×C22⋊C4—62no—
70C2×C5×C22⋊C4—62no—
71C5×#95—32no—
72C5×#95—32no—
73C5×#95—32no—
74C5×#95—32no—
75#2412—31noSylow
76C23×C3×C5—41no—
77C22×C5×D3—41no—
78C2×C5×A4—22no—
79C2×C5×A4C2×D411yescharacteristic
80C2×C5×A4#4112yes—
81C5×S4—41no—
82C2×C3×C4×C5—42no—
83C5×#67—82no—
84C2×C5×Dic3—41no—
85C2×C5×Dic3—42no—
86C2×C5×Dic3—82no—
87C23×A4C22×C511yescharacteristic
88C22×S4C22×C511yescharacteristic
89C2×C4×A4C22×C512yes—
90GL(2,ℤ/4)—22no—
91C2×SL(2,ℤ/4)—22no—
92C2×SL(2,ℤ/4)C22×C511yescharacteristic
93C2×SL(2,ℤ/4)C22×C512yes—
94#886—41no—
95C24×C5—22no—
96C24×C5—31no—
97C24×C5—31no—
98C24×C5—32no—
99C24×C5—31no—
100C24×C5—62no—
101C24×C5D1212yes—
102C24×C5C22×D311yescharacteristic
103C2×C5×D4—61no—
104C2×C5×D4—62no—
105C2×C5×D4—62no—
106C2×C5×D4—61no—
107C2×C5×D4—62no—
108C2×C5×D4—62no—
109C2×C5×D4—61no—
110C2×C5×D4—62no—
111C2×C5×D4—61no—
112C2×C5×D4—122no—
113C22×C4×C5—22no—
114C22×C4×C5—32no—
115C22×C4×C5—32no—
116C22×C4×C5—32no—
117C22×C4×C5—32no—
118C22×C4×C5—31no—
119C22×C4×C5—31no—
120C22×C4×C5—31no—
121C22×C4×C5—62no—
122C22×C4×C5—62no—
123C22×C4×C5—62no—
124C22×C4×C5—61no—
125C22×C4×C5—62no—
126C22×C4×C5—61no—
127C22×C4×C5—122no—
128C5×C4⋊C4—62no—
129C5×C4⋊C4—62no—
130C5×C4⋊C4—122no—
131C5×C22⋊C4—31no—
132C5×C22⋊C4—32no—
133C5×C22⋊C4—61no—
134C5×C22⋊C4—62no—
135C5×C22⋊C4—62no—
136C5×C22⋊C4—64no—
137C5×C22⋊C4—61no—
138C5×C22⋊C4—62no—
139C5×C22⋊C4—62no—
140C5×C22⋊C4—62no—
141C5×C22⋊C4—61no—
142C5×C22⋊C4—62no—
143C5×C22⋊C4—122no—
144C5×C22⋊C4S411yescharacteristic, complemented
145C2×#123—31no—
146C2×#120—31no—
147C22×C22⋊C4—31no—
148C22×C22⋊C4C5×D311yescharacteristic, complemented
149#397—32no—
150#394—32no—
151#386—31no—
152#386—32no—
153#385—31no—
154#381—31no—
155#380—32no—
156C22×C3×C5—42no—
157C22×C3×C5—41no—
158C22×C3×C5—82no—
159C2×C5×D3—81no—
160C2×C5×D3—82no—
161C5×A4#12311yescharacteristic, complemented
162C3×C4×C5—82no—
163C5×Dic3—82no—
164C5×Dic3—81no—
165C22×A4—22no—
166C22×A4C5×D412yes—
167C22×A4C23×C511yescharacteristic
168C2×S4—21no—
169C2×S4—22no—
170C2×#67—41no—
171C22×Dic3—41no—
172C4×A4—22no—
173SL(2,ℤ/4)—22no—
174SL(2,ℤ/4)—21no—
175C3×C22⋊C4—41no—
176(C2×D3)⋊C4—42no—
177C4⋊Dic3—42no—
178C23×C5—22no—
179C23×C5—31no—
180C23×C5—31no—
181C23×C5—32no—
182C23×C5—32no—
183C23×C5—32no—
184C23×C5—32no—
185C23×C5—31no—
186C23×C5—62no—
187C23×C5—62no—
188C23×C5—61no—
189C23×C5—62no—
190C23×C5—62no—
191C23×C5—64no—
192C23×C5—62no—
193C23×C5—62no—
194C23×C5—61no—
195C23×C5—61no—
196C23×C5—62no—
197C23×C5—61no—
198C23×C5—122no—
199C23×C5C2×D1211yescharacteristic
200C23×C5#23712yes—
201C23×C5C2×S411yescharacteristic
202C5×D4—121no—
203C5×D4—122no—
204C5×D4—122no—
205C5×D4—124no—
206C5×D4—121no—
207C5×D4—122no—
208C2×C4×C5—32no—
209C2×C4×C5—62no—
210C2×C4×C5—62no—
211C2×C4×C5—62no—
212C2×C4×C5—62no—
213C2×C4×C5—62no—
214C2×C4×C5—62no—
215C2×C4×C5—61no—
216C2×C4×C5—61no—
217C2×C4×C5—61no—
218C2×C4×C5—62no—
219C2×C4×C5—62no—
220C2×C4×C5—62no—
221C2×C4×C5—62no—
222C2×C4×C5—62no—
223C2×C4×C5—61no—
224C2×C4×C5—62no—
225C2×C4×C5—61no—
226C2×C4×C5—122no—
227C2×C4×C5—124no—
228C2×C4×C5—122no—
229C2×C4×C5C2×S412yes—
230C25C2×C5×D311yescharacteristic
231C22×D4—31no—
232C22×D4—31no—
233C22×D4—31no—
234C23×C4—31no—
235C23×C4—31no—
236C23×C4C2×C5×D312yes—
237#123—61no—
238#123—62no—
239#123—61no—
240#120—62no—
241#120—62no—
242C2×C4⋊C4—32no—
243C2×C4⋊C4—32no—
244C2×C22⋊C4—31no—
245C2×C22⋊C4—31no—
246C2×C22⋊C4—32no—
247C2×C22⋊C4—32no—
248C2×C22⋊C4—31no—
249C2×C22⋊C4—31no—
250C2×C22⋊C4—32no—
251C2×C22⋊C4—32no—
252C2×C22⋊C4—31no—
253C2×C22⋊C4—32no—
254C2×C22⋊C4—62no—
255C2×C22⋊C4—62no—
256#95—32no—
257#95—32no—
258#95—32no—
259#95—32no—
260C2×C3×C5—41no—
261C2×C3×C5—42no—
262C2×C3×C5—82no—
263C5×D3—161no—
264C23×C3—41no—
265C22×D3—41no—
266C2×A4—22no—
267C2×A4C2×C5×D411yescharacteristic
268C2×A4C5×#4112yes—
269S4—41no—
270C2×C3×C4—42no—
271#67—82no—
272C2×Dic3—41no—
273C2×Dic3—42no—
274C2×Dic3—82no—
275C22×C5—22no—
276C22×C5—31no—
277C22×C5—32no—
278C22×C5—31no—
279C22×C5—31no—
280C22×C5—32no—
281C22×C5—32no—
282C22×C5—32no—
283C22×C5—62no—
284C22×C5—62no—
285C22×C5—64no—
286C22×C5—61no—
287C22×C5—62no—
288C22×C5—62no—
289C22×C5—62no—
290C22×C5—122no—
291C22×C5—121no—
292C22×C5—121no—
293C22×C5—122no—
294C22×C5—121no—
295C22×C5—122no—
296C22×C5#88611yescharacteristic, complemented
297C22×C5C4⋊S412yes—
298C22×C5C22×S411yescharacteristic
299C4×C5—22no—
300C4×C5—62no—
301C4×C5—122no—
302C4×C5—121no—
303C4×C5—121no—
304C4×C5—122no—
305C4×C5—121no—
306C24—22no—
307C24—31no—
308C24—31no—
309C24—32no—
310C24—31no—
311C24—62no—
312C24C5×D1212yes—
313C24C22×C5×D311yescharacteristic
314C2×D4—61no—
315C2×D4—62no—
316C2×D4—62no—
317C2×D4—61no—
318C2×D4—62no—
319C2×D4—62no—
320C2×D4—61no—
321C2×D4—62no—
322C2×D4—61no—
323C2×D4—122no—
324C22×C4—22no—
325C22×C4—32no—
326C22×C4—32no—
327C22×C4—31no—
328C22×C4—32no—
329C22×C4—32no—
330C22×C4—31no—
331C22×C4—31no—
332C22×C4—62no—
333C22×C4—62no—
334C22×C4—62no—
335C22×C4—61no—
336C22×C4—62no—
337C22×C4—61no—
338C22×C4—122no—
339C4⋊C4—62no—
340C4⋊C4—62no—
341C4⋊C4—122no—
342C22⋊C4—31no—
343C22⋊C4—32no—
344C22⋊C4—61no—
345C22⋊C4—62no—
346C22⋊C4—62no—
347C22⋊C4—64no—
348C22⋊C4—61no—
349C22⋊C4—62no—
350C22⋊C4—62no—
351C22⋊C4—62no—
352C22⋊C4—61no—
353C22⋊C4—62no—
354C22⋊C4—122no—
355C22⋊C4C5×S411yescharacteristic, complemented
356C3×C5—41noHall
357C22×C3—42no—
358C22×C3—41no—
359C22×C3—82no—
360C2×D3—81no—
361C2×D3—82no—
362A4C5×#12311yescharacteristic, complemented
363C3×C4—82no—
364Dic3—82no—
365Dic3—81no—
366C2×C5—22no—
367C2×C5—31no—
368C2×C5—31no—
369C2×C5—32no—
370C2×C5—62no—
371C2×C5—241no—
372C2×C5C2×C4⋊S411yescharacteristic
373C2×C5#591412yes—
374C23—22no—
375C23—31no—
376C23—31no—
377C23—32no—
378C23—32no—
379C23—32no—
380C23—32no—
381C23—31no—
382C23—62no—
383C23—62no—
384C23—61no—
385C23—62no—
386C23—62no—
387C23—64no—
388C23—62no—
389C23—62no—
390C23—61no—
391C23—61no—
392C23—62no—
393C23—61no—
394C23—122no—
395C23C2×C5×D1211yescharacteristic
396C23C5×#23712yes—
397C23C2×C5×S411yescharacteristic
398D4—121no—
399D4—122no—
400D4—122no—
401D4—124no—
402D4—122no—
403D4—121no—
404C2×C4—32no—
405C2×C4—62no—
406C2×C4—62no—
407C2×C4—62no—
408C2×C4—62no—
409C2×C4—62no—
410C2×C4—62no—
411C2×C4—61no—
412C2×C4—61no—
413C2×C4—61no—
414C2×C4—62no—
415C2×C4—62no—
416C2×C4—62no—
417C2×C4—62no—
418C2×C4—62no—
419C2×C4—61no—
420C2×C4—62no—
421C2×C4—61no—
422C2×C4—122no—
423C2×C4—124no—
424C2×C4—122no—
425C2×C4C2×C5×S412yes—
426C2×C3—41no—
427C2×C3—42no—
428C2×C3—82no—
429D3—161no—
430C5#8530111yescharacteristic, direct product factor, Sylow
431C22—22no—
432C22—31no—
433C22—32no—
434C22—31no—
435C22—31no—
436C22—32no—
437C22—32no—
438C22—32no—
439C22—62no—
440C22—62no—
441C22—64no—
442C22—61no—
443C22—62no—
444C22—62no—
445C22—62no—
446C22—122no—
447C22—121no—
448C22—121no—
449C22—122no—
450C22—121no—
451C22—122no—
452C22C5×#88611yescharacteristic, complemented
453C22C5×C4⋊S412yes—
454C22C22×C5×S411yescharacteristic
455C4—22no—
456C4—62no—
457C4—122no—
458C4—121no—
459C4—121no—
460C4—122no—
461C4—121no—
462C3—41noSylow
463C2—22no—
464C2—31no—
465C2—31no—
466C2—32no—
467C2—62no—
468C2—241no—
469C2C2×C5×C4⋊S411yescharacteristic
470C2C5×#591412yes—
subgroupG/H
centerC22×C5C22×S4
commutator, cosocleC22×A4C23×C5
FrattiniC22C22×C5×S4
FittingC22×C5×C22⋊C4D3
socleC24×C5C22×D3
prime radicalC2×C5×GL(2,ℤ/4)C2
hypercenterC5×C22⋊C4S4
hypocenterA4C5×#123
LCSG ▷ C22×A4 ▷ A4
UCSC1 ◁ C22×C5 ◁ C5×C22⋊C4

Complements

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

NH ≅ G/Ntype
2. C22×C5×SL(2,ℤ/4)C2semidirect
3. C5×A4×C22⋊C4C2semidirect
4. C5×(C2×S4)⋊C4C2semidirect
5. C5×C4⋊SL(2,ℤ/4)C2semidirect
15. #85301C5direct (N × H)
19. C22×C5×C22⋊C4D3semidirect
41. C22×SL(2,ℤ/4)C2×C5semidirect
42. A4×C22⋊C4C2×C5semidirect
43. (C2×S4)⋊C4C2×C5semidirect
44. C4⋊SL(2,ℤ/4)C2×C5semidirect
144. C5×C22⋊C4S4semidirect
148. C22×C22⋊C4C5×D3semidirect
161. C5×A4#123semidirect
296. C22×C5#886semidirect
355. C22⋊C4C5×S4semidirect
362. A4C5×#123semidirect
430. C5#85301direct (N × H)
452. C22C5×#886semidirect

Maximal subgroups

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

subgroupindex
C2×C5×GL(2,ℤ/4)2
C22×C5×SL(2,ℤ/4)2
C5×A4×C22⋊C42
C5×(C2×S4)⋊C42
C5×C4⋊SL(2,ℤ/4)2
C5×#24123
C5×#8864
#853015