Finite Group Databasebeta

← C5×#85307 #14511045

Subgroups — C5×#85307 #14511045

Order 1920.

690 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 690 conjugacy classes of subgroups — too many to draw. Only its 104 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
2C5×A4×C4⋊C4C211yesmaximal, characteristic, complemented
3C5×(C2×S4)⋊C4C212yesmaximal
4C5×C4⋊SL(2,ℤ/4)C211yesmaximal, characteristic, complemented
5C4×C5×SL(2,ℤ/4)C212yesmaximal, complemented
6C5×#2352—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×S4C414yescomplemented
13C5×#892—41nomaximal
14#85307C511yesmaximal, characteristic, direct product factor, Hall
15C2×C4×C5×D4—31no—
16C2×C5×#117—31no—
17C22×C5×C4⋊C4D311yescharacteristic, complemented
18C5×#388—32no—
19C5×#387—32no—
20C5×#386—32no—
21C5×#382—31no—
22C5×#380—31no—
23C5×#380—31no—
24C4×C5×C4⋊C4—31no—
25C4×C5×C22⋊C4—32no—
26C22×C5×A4C2311yescharacteristic
27C2×C5×S4—21no—
28C2×C5×S4—21no—
29C2×C5×S4C2×C412yes—
30C2×C4×C5×D3—41no—
31C4×C5×A4—22no—
32C4×C5×A4C2×C412yes—
33C5×SL(2,ℤ/4)—22no—
34C5×SL(2,ℤ/4)C2×C412yes—
35C3×C5×C4⋊C4—41no—
36C5×(C2×D3)⋊C4—42no—
37C5×C4⋊Dic3—41no—
38C4×C5×Dic3—42no—
39C2×C4×S4C2×C511yescharacteristic
40A4×C4⋊C4C2×C511yescharacteristic, complemented
41(C2×S4)⋊C4C2×C512yes—
42C4⋊SL(2,ℤ/4)C2×C511yescharacteristic, complemented
43C4×SL(2,ℤ/4)C2×C512yescomplemented
44C22×C5×D4—31no—
45C23×C4×C5—31no—
46C23×C4×C5C2×D312yes—
47C23×C4×C5C2×D311yescharacteristic
48C4×C5×D4—34no—
49C4×C5×D4—34no—
50C5×#117—61no—
51C5×#117—62no—
52C5×#117—61no—
53C2×C5×C4⋊C4—31no—
54C2×C5×C4⋊C4—31no—
55C2×C5×C4⋊C4—31no—
56C2×C5×C4⋊C4—32no—
57C2×C5×C4⋊C4—31no—
58C2×C5×C4⋊C4—31no—
59C2×C5×C22⋊C4—31no—
60C2×C5×C22⋊C4—31no—
61C2×C5×C22⋊C4—32no—
62C2×C5×C22⋊C4—32no—
63C2×C5×C22⋊C4—32no—
64C2×C42×C5—31no—
65C2×C42×C5—32no—
66C2×C42×C5—32no—
67C5×#95—32no—
68C5×#95—32no—
69C5×#95—32no—
70C5×#95—32no—
71C5×#95—32no—
72#2352—31noSylow
73C22×C5×D3—41no—
74C2×C5×A4C22×C411yescharacteristic
75C2×C5×A4#4111yescharacteristic
76C2×C5×A4#4111yescharacteristic
77C5×S4—22no—
78C2×C3×C4×C5—41no—
79C2×C3×C4×C5—42no—
80C2×C5×Dic3—41no—
81C2×C5×Dic3—42no—
82C4×C5×D3—44no—
83C22×S4C22×C511yescharacteristic
84C2×C4×A4C22×C512yes—
85C2×C4×A4C22×C511yescharacteristic
86C2×SL(2,ℤ/4)C22×C512yes—
87C2×SL(2,ℤ/4)C22×C511yescharacteristic
88C4×S4C4×C514yescomplemented
89#892—41no—
90C24×C5—31no—
91C24×C5C22×D311yescharacteristic
92C2×C5×D4—32no—
93C2×C5×D4—32no—
94C2×C5×D4—61no—
95C2×C5×D4—61no—
96C2×C5×D4—61no—
97C2×C5×D4—61no—
98C22×C4×C5—22no—
99C22×C4×C5—31no—
100C22×C4×C5—31no—
101C22×C4×C5—32no—
102C22×C4×C5—31no—
103C22×C4×C5—31no—
104C22×C4×C5—31no—
105C22×C4×C5—34no—
106C22×C4×C5—32no—
107C22×C4×C5—32no—
108C22×C4×C5—32no—
109C22×C4×C5—32no—
110C22×C4×C5—61no—
111C22×C4×C5—62no—
112C22×C4×C5—61no—
113C22×C4×C5—62no—
114C22×C4×C5C4×D312yes—
115C5×C4⋊C4—34no—
116C5×C4⋊C4—32no—
117C5×C4⋊C4—31no—
118C5×C4⋊C4—61no—
119C5×C4⋊C4—62no—
120C5×C4⋊C4—61no—
121C5×C4⋊C4—61no—
122C5×C4⋊C4S411yescharacteristic, complemented
123C5×C22⋊C4—34no—
124C5×C22⋊C4—32no—
125C5×C22⋊C4—32no—
126C5×C22⋊C4—62no—
127C5×C22⋊C4—62no—
128C5×C22⋊C4—62no—
129C5×C22⋊C4—62no—
130C5×C22⋊C4—122no—
131C42×C5—34no—
132C42×C5—62no—
133C42×C5—62no—
134C42×C5—122no—
135C2×C4×D4—31no—
136C2×#117—31no—
137C22×C4⋊C4C5×D311yescharacteristic, complemented
138#388—32no—
139#387—32no—
140#386—32no—
141#382—31no—
142#380—31no—
143#380—31no—
144C4×C4⋊C4—31no—
145C4×C22⋊C4—32no—
146C22×C3×C5—41no—
147C2×C5×D3—42no—
148C2×C5×D3—81no—
149C2×C5×D3—81no—
150C5×A4#11711yescharacteristic, complemented
151C3×C4×C5—42no—
152C3×C4×C5—82no—
153C5×Dic3—42no—
154C5×Dic3—82no—
155C22×A4C23×C511yescharacteristic
156C2×S4—21no—
157C2×S4—21no—
158C2×S4C2×C4×C512yes—
159C2×C4×D3—41no—
160C4×A4—22no—
161C4×A4C2×C4×C512yes—
162SL(2,ℤ/4)—22no—
163SL(2,ℤ/4)C2×C4×C512yes—
164C3×C4⋊C4—41no—
165(C2×D3)⋊C4—42no—
166C4⋊Dic3—41no—
167C4×Dic3—42no—
168C23×C5—31no—
169C23×C5—31no—
170C23×C5—31no—
171C23×C5—31no—
172C23×C5—32no—
173C23×C5—61no—
174C23×C5—61no—
175C23×C5—61no—
176C23×C5—61no—
177C23×C5—61no—
178C23×C5—61no—
179C23×C5C2×C4×D311yescharacteristic
180C23×C5#23711yescharacteristic
181C23×C5#23911yescharacteristic
182C5×D4—62no—
183C5×D4—62no—
184C5×D4—62no—
185C5×D4—62no—
186C2×C4×C5—31no—
187C2×C4×C5—32no—
188C2×C4×C5—32no—
189C2×C4×C5—32no—
190C2×C4×C5—32no—
191C2×C4×C5—32no—
192C2×C4×C5—32no—
193C2×C4×C5—32no—
194C2×C4×C5—62no—
195C2×C4×C5—62no—
196C2×C4×C5—61no—
197C2×C4×C5—64no—
198C2×C4×C5—61no—
199C2×C4×C5—61no—
200C2×C4×C5—61no—
201C2×C4×C5—61no—
202C2×C4×C5—61no—
203C2×C4×C5—61no—
204C2×C4×C5—64no—
205C2×C4×C5—62no—
206C2×C4×C5—62no—
207C2×C4×C5—62no—
208C2×C4×C5—62no—
209C2×C4×C5—62no—
210C2×C4×C5—62no—
211C2×C4×C5—62no—
212C2×C4×C5—62no—
213C2×C4×C5C2×S412yes—
214C2×C4×C5C2×S411yescharacteristic
215C22×D4—31no—
216C23×C4—31no—
217C23×C4C2×C5×D312yes—
218C23×C4C2×C5×D311yescharacteristic
219C4×D4—34no—
220C4×D4—34no—
221#117—61no—
222#117—62no—
223#117—61no—
224C2×C4⋊C4—31no—
225C2×C4⋊C4—31no—
226C2×C4⋊C4—31no—
227C2×C4⋊C4—32no—
228C2×C4⋊C4—31no—
229C2×C4⋊C4—31no—
230C2×C22⋊C4—31no—
231C2×C22⋊C4—31no—
232C2×C22⋊C4—32no—
233C2×C22⋊C4—32no—
234C2×C22⋊C4—32no—
235C2×C42—31no—
236C2×C42—32no—
237C2×C42—32no—
238#95—32no—
239#95—32no—
240#95—32no—
241#95—32no—
242#95—32no—
243C2×C3×C5—41no—
244C2×C3×C5—41no—
245C2×C3×C5—41no—
246C5×D3—82no—
247C22×D3—41no—
248C2×A4C22×C4×C511yescharacteristic
249C2×A4C5×#4111yescharacteristic
250C2×A4C5×#4111yescharacteristic
251S4—22no—
252C2×C3×C4—41no—
253C2×C3×C4—42no—
254C2×Dic3—41no—
255C2×Dic3—42no—
256C4×D3—44no—
257C22×C5—31no—
258C22×C5—31no—
259C22×C5—31no—
260C22×C5—31no—
261C22×C5—31no—
262C22×C5—31no—
263C22×C5—31no—
264C22×C5—31no—
265C22×C5—31no—
266C22×C5—61no—
267C22×C5—62no—
268C22×C5—62no—
269C22×C5—62no—
270C22×C5—121no—
271C22×C5—121no—
272C22×C5—121no—
273C22×C5—121no—
274C22×C5#89211yescharacteristic, complemented
275C22×C5C22×S411yescharacteristic
276C4×C5—22no—
277C4×C5—32no—
278C4×C5—62no—
279C4×C5—62no—
280C4×C5—62no—
281C4×C5—62no—
282C4×C5—122no—
283C4×C5—122no—
284C4×C5C4×S412yes—
285C24—31no—
286C24C22×C5×D311yescharacteristic
287C2×D4—32no—
288C2×D4—32no—
289C2×D4—61no—
290C2×D4—61no—
291C2×D4—61no—
292C2×D4—61no—
293C22×C4—22no—
294C22×C4—31no—
295C22×C4—31no—
296C22×C4—32no—
297C22×C4—31no—
298C22×C4—31no—
299C22×C4—31no—
300C22×C4—34no—
301C22×C4—32no—
302C22×C4—32no—
303C22×C4—32no—
304C22×C4—32no—
305C22×C4—61no—
306C22×C4—62no—
307C22×C4—61no—
308C22×C4—62no—
309C22×C4C4×C5×D312yes—
310C4⋊C4—34no—
311C4⋊C4—32no—
312C4⋊C4—31no—
313C4⋊C4—61no—
314C4⋊C4—61no—
315C4⋊C4—62no—
316C4⋊C4—61no—
317C4⋊C4C5×S411yescharacteristic, complemented
318C22⋊C4—34no—
319C22⋊C4—32no—
320C22⋊C4—32no—
321C22⋊C4—62no—
322C22⋊C4—62no—
323C22⋊C4—62no—
324C22⋊C4—62no—
325C22⋊C4—122no—
326C42—34no—
327C42—62no—
328C42—62no—
329C42—122no—
330C3×C5—41noHall
331C22×C3—41no—
332C2×D3—42no—
333C2×D3—81no—
334C2×D3—81no—
335A4C5×#11711yescharacteristic, complemented
336C3×C4—42no—
337C3×C4—82no—
338Dic3—42no—
339Dic3—82no—
340C2×C5—31no—
341C2×C5—31no—
342C2×C5—31no—
343C2×C5—31no—
344C2×C5—122no—
345C2×C5C2×C4×S411yescharacteristic
346C2×C5#591411yescharacteristic
347C2×C5#591911yescharacteristic
348C23—31no—
349C23—31no—
350C23—31no—
351C23—31no—
352C23—32no—
353C23—61no—
354C23—61no—
355C23—61no—
356C23—61no—
357C23—61no—
358C23—61no—
359C23C2×C4×C5×D311yescharacteristic
360C23C5×#23711yescharacteristic
361C23C5×#23911yescharacteristic
362D4—62no—
363D4—62no—
364D4—62no—
365D4—62no—
366C2×C4—31no—
367C2×C4—32no—
368C2×C4—32no—
369C2×C4—32no—
370C2×C4—32no—
371C2×C4—32no—
372C2×C4—32no—
373C2×C4—32no—
374C2×C4—62no—
375C2×C4—62no—
376C2×C4—61no—
377C2×C4—64no—
378C2×C4—61no—
379C2×C4—61no—
380C2×C4—61no—
381C2×C4—61no—
382C2×C4—61no—
383C2×C4—61no—
384C2×C4—64no—
385C2×C4—62no—
386C2×C4—62no—
387C2×C4—62no—
388C2×C4—62no—
389C2×C4—62no—
390C2×C4—62no—
391C2×C4—62no—
392C2×C4—62no—
393C2×C4C2×C5×S412yes—
394C2×C4C2×C5×S411yescharacteristic
395C2×C3—41no—
396C2×C3—41no—
397C2×C3—41no—
398D3—82no—
399C5#8530711yescharacteristic, direct product factor, Sylow
400C22—31no—
401C22—31no—
402C22—31no—
403C22—31no—
404C22—31no—
405C22—31no—
406C22—31no—
407C22—31no—
408C22—31no—
409C22—61no—
410C22—62no—
411C22—62no—
412C22—62no—
413C22—121no—
414C22—121no—
415C22—121no—
416C22—121no—
417C22C5×#89211yescharacteristic, complemented
418C22C22×C5×S411yescharacteristic
419C4—22no—
420C4—32no—
421C4—62no—
422C4—62no—
423C4—62no—
424C4—62no—
425C4—122no—
426C4—122no—
427C4C4×C5×S412yes—
428C3—41noSylow
429C2—31no—
430C2—31no—
431C2—31no—
432C2—31no—
433C2—122no—
434C2C2×C4×C5×S411yescharacteristic
435C2C5×#591411yescharacteristic
436C2C5×#591911yescharacteristic
subgroupG/H
centerC22×C5C22×S4
commutatorC2×A4C22×C4×C5
FrattiniC22C22×C5×S4
FittingC22×C5×C4⋊C4D3
socleC24×C5C22×D3
prime radicalC22×C5×S4C22
cosocleC22×A4C23×C5
hypercenterC5×C4⋊C4S4
hypocenterA4C5×#117
LCSG ▷ C2×A4 ▷ A4
UCSC1 ◁ C22×C5 ◁ C5×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C5×A4×C4⋊C4C2semidirect
4. C5×C4⋊SL(2,ℤ/4)C2semidirect
5. C4×C5×SL(2,ℤ/4)C2semidirect
12. C4×C5×S4C4semidirect
14. #85307C5direct (N × H)
17. C22×C5×C4⋊C4D3semidirect
40. A4×C4⋊C4C2×C5semidirect
42. C4⋊SL(2,ℤ/4)C2×C5semidirect
43. C4×SL(2,ℤ/4)C2×C5semidirect
88. C4×S4C4×C5semidirect
122. C5×C4⋊C4S4semidirect
137. C22×C4⋊C4C5×D3semidirect
150. C5×A4#117semidirect
274. C22×C5#892semidirect
317. C4⋊C4C5×S4semidirect
335. A4C5×#117semidirect
399. C5#85307direct (N × H)
417. C22C5×#892semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C5×S42
C5×A4×C4⋊C42
C5×(C2×S4)⋊C42
C5×C4⋊SL(2,ℤ/4)2
C4×C5×SL(2,ℤ/4)2
C5×#23523
C5×#8924
#853075