Finite Group Databasebeta

← C5×#70990 #14316673

Subgroups — C5×#70990 #14316673

Order 1920.

516 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 82 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
1C5×#5679C211yesmaximal, characteristic, complemented
2C3×C5×#455C211yesmaximal, characteristic, complemented
3C5×#5244C211yesmaximal, characteristic
4C5×#5243C211yesmaximal, characteristic, complemented
5C5×#4541C211yesmaximal, characteristic, complemented
6C5×#4476C211yesmaximal, characteristic, complemented
7C5×#4475C211yesmaximal, characteristic, complemented
8C5×#2197—31nomaximal, Hall
9C3×C5×H(5,𝔽2)C2211yescharacteristic
10C3×C5×#137—21no—
11C3×C5×#124C2211yescharacteristic
12C5×#946—21no—
13C5×#937—21no—
14C5×#937C2211yescharacteristic
15C5×#935—21no—
16C5×#935C2211yescharacteristic
17C5×#883—21no—
18C5×#67⋊C4—21no—
19C5×D3×C22⋊C4—21no—
20C5×#879—21no—
21C5×#872—21no—
22C3×C5×#104—21no—
23C3×C5×#100C2211yescharacteristic
24C5×#834—21no—
25C5×#834—21no—
26C5×#834C2211yescharacteristic
27C5×#834C2211yescharacteristic
28#70990C511yesmaximal, characteristic, direct product factor, Hall
29C5×#551—31no—
30C5×#458—31no—
31C5×B(4,𝔽2)—31no—
32C5×#455D311yescharacteristic, complemented
33C5×#354—31no—
34C5×#352—31no—
35C5×AΣL(1,𝔽16)—31no—
36C23×C5×D3D411yescharacteristic
37C3×C5×#41—21no—
38C3×C5×#41—41no—
39C2×C3×C5×Q8D411yescharacteristic, complemented
40C2×C3×C5×D4—21no—
41C2×C3×C5×D4—21no—
42C2×C3×C5×D4—41no—
43C2×C3×C5×D4C2311yescharacteristic
44C2×C5×#67—21no—
45C2×C5×#67—41no—
46C22×C5×Dic3D411yescharacteristic, complemented
47C2×C4×C5×D3—21no—
48C2×C4×C5×D3—21no—
49C3×C5×Q16—41no—
50C3×C5×SD16—41no—
51C3×C5×M16—21no—
52C3×C5×C22⋊C4—21no—
53C3×C5×C22⋊C4—21no—
54C3×C42×C5D411yescharacteristic, complemented
55C5×#217—21no—
56C5×#217—21no—
57C5×#217—21no—
58C5×#217—41no—
59C5×#217D411yescharacteristic
60C5×#217D411yescharacteristic, complemented
61C5×(C2×D3)⋊C4—21no—
62C5×(C2×D3)⋊C4—21no—
63C5×(C2×D3)⋊C4—21no—
64C5×(C2×D3)⋊C4—41no—
65C5×C4⋊Dic3—21no—
66C5×Dic3⋊C4—21no—
67C5×Dic3⋊C4—21no—
68C5×Dic3⋊C4—21no—
69C5×Dic3⋊C4—41no—
70C4×C5×Dic3—21no—
71#5679C2×C511yescharacteristic, complemented
72C3×#455C2×C511yescharacteristic, complemented
73#5244C2×C511yescharacteristic
74#5243C2×C511yescharacteristic, complemented
75#4541C2×C511yescharacteristic, complemented
76#4476C2×C511yescharacteristic, complemented
77#4475C2×C511yescharacteristic, complemented
78C5×H(5,𝔽2)C2×D311yescharacteristic
79C5×#137—21no—
80C5×#126—61no—
81C5×#124C2×D311yescharacteristic
82C5×#123—31no—
83C5×#123—61no—
84C5×#123—61no—
85C5×C22⋊Q8—61no—
86C5×#120—31no—
87C5×#120—61no—
88C4×C5×D4—61no—
89C5×#117—61no—
90C2×C5×C22⋊C4—61no—
91C5×#104—21no—
92C5×#100C2×D311yescharacteristic
93C5×#99—31no—
94C5×#99—31no—
95C5×#99—61no—
96C5×#99—61no—
97#2197—31noSylow
98C23×C3×C5—21no—
99C23×C3×C5—21no—
100C23×C3×C5C2×D411yescharacteristic
101C23×C3×C5C2×D411yescharacteristic
102C22×C5×D3—21no—
103C22×C5×D3—42no—
104C22×C5×D3—41no—
105C3×C5×Q8—21no—
106C3×C5×Q8—41no—
107C3×C5×D4—21no—
108C3×C5×D4—41no—
109C3×C5×D4—41no—
110C3×C5×D4—41no—
111C3×C5×D4—41no—
112C2×C3×C4×C5—21no—
113C2×C3×C4×C5—21no—
114C2×C3×C4×C5—21no—
115C2×C3×C4×C5—21no—
116C2×C3×C4×C5—41no—
117C2×C3×C4×C5C2×D411yescharacteristic
118C5×#67—42no—
119C5×#67—82no—
120C2×C5×Dic3—21no—
121C2×C5×Dic3—21no—
122C2×C5×Dic3—21no—
123C2×C5×Dic3—41no—
124C2×C5×Dic3—41no—
125C2×C5×Dic3—41no—
126C4×C5×D3—42no—
127C4×C5×D3—81no—
128C3×C5×C8—41no—
129C3×H(5,𝔽2)C22×C511yescharacteristic
130C3×#137—21no—
131C3×#124C22×C511yescharacteristic
132#946—21no—
133#937—21no—
134#937C22×C511yescharacteristic
135#935—21no—
136#935C22×C511yescharacteristic
137#883—21no—
138#67⋊C4—21no—
139D3×C22⋊C4—21no—
140#879—21no—
141#872—21no—
142C3×#104—21no—
143C3×#100C22×C511yescharacteristic
144#834—21no—
145#834—21no—
146#834C22×C511yescharacteristic
147#834C22×C511yescharacteristic
148C24×C5—31no—
149C5×#41—21no—
150C5×#41—41no—
151C2×C5×Q8#6711yescharacteristic, complemented
152C2×C5×D4—21no—
153C2×C5×D4—21no—
154C2×C5×D4—41no—
155C2×C5×D4—61no—
156C2×C5×D4—121no—
157C2×C5×D4C22×D311yescharacteristic
158C22×C4×C5—31no—
159C22×C4×C5—61no—
160C22×C4×C5—61no—
161C5×Q16—41no—
162C5×SD16—41no—
163C5×M16—21no—
164C5×C4⋊C4—61no—
165C5×C4⋊C4—61no—
166C5×C4⋊C4—61no—
167C5×C4⋊C4—61no—
168C5×C4⋊C4—121no—
169C5×C22⋊C4—21no—
170C5×C22⋊C4—21no—
171C5×C22⋊C4—31no—
172C5×C22⋊C4—31no—
173C5×C22⋊C4—61no—
174C5×C22⋊C4—61no—
175C5×C22⋊C4—61no—
176C5×C22⋊C4—61no—
177C5×C22⋊C4—61no—
178C5×C22⋊C4—61no—
179C5×C22⋊C4—121no—
180C5×C22⋊C4—121no—
181C42×C5—61no—
182C42×C5#6711yescharacteristic, complemented
183#551—31no—
184#458—31no—
185B(4,𝔽2)—31no—
186#455C5×D311yescharacteristic, complemented
187#354—31no—
188#352—31no—
189AΣL(1,𝔽16)—31no—
190C22×C3×C5—21no—
191C22×C3×C5—21no—
192C22×C3×C5—41no—
193C22×C3×C5—41no—
194C22×C3×C5—41no—
195C22×C3×C5—81no—
196C22×C3×C5—81no—
197C22×C3×C5#12011yescharacteristic
198C2×C5×D3—22no—
199C2×C5×D3—41no—
200C2×C5×D3—41no—
201C2×C5×D3—42no—
202C2×C5×D3—81no—
203C3×C4×C5—21no—
204C3×C4×C5—41no—
205C3×C4×C5—41no—
206C3×C4×C5—41no—
207C5×Dic3—41no—
208C5×Dic3—41no—
209C5×Dic3—41no—
210C5×Dic3—81no—
211C5×Dic3—81no—
212C23×D3C5×D411yescharacteristic
213C3×#41—21no—
214C3×#41—41no—
215C2×C3×Q8C5×D411yescharacteristic, complemented
216C2×C3×D4—21no—
217C2×C3×D4—21no—
218C2×C3×D4—41no—
219C2×C3×D4C23×C511yescharacteristic
220C2×#67—21no—
221C2×#67—41no—
222C22×Dic3C5×D411yescharacteristic, complemented
223C2×C4×D3—21no—
224C2×C4×D3—21no—
225C3×Q16—41no—
226C3×SD16—41no—
227C3×M16—21no—
228C3×C22⋊C4—21no—
229C3×C22⋊C4—21no—
230C3×C42C5×D411yescharacteristic, complemented
231#217—21no—
232#217—21no—
233#217—21no—
234#217—41no—
235#217C5×D411yescharacteristic
236#217C5×D411yescharacteristic, complemented
237(C2×D3)⋊C4—21no—
238(C2×D3)⋊C4—21no—
239(C2×D3)⋊C4—21no—
240(C2×D3)⋊C4—41no—
241C4⋊Dic3—21no—
242Dic3⋊C4—21no—
243Dic3⋊C4—21no—
244Dic3⋊C4—21no—
245Dic3⋊C4—41no—
246C4×Dic3—21no—
247C23×C5—21no—
248C23×C5—21no—
249C23×C5—61no—
250C23×C5—121no—
251C23×C5—122no—
252C23×C5D3×D411yescharacteristic
253C23×C5D3×D411yescharacteristic
254C5×Q8—21no—
255C5×Q8—41no—
256C5×D4—21no—
257C5×D4—41no—
258C5×D4—41no—
259C5×D4—41no—
260C5×D4—41no—
261C5×D4—122no—
262C5×D4—242no—
263C2×C4×C5—21no—
264C2×C4×C5—21no—
265C2×C4×C5—21no—
266C2×C4×C5—21no—
267C2×C4×C5—41no—
268C2×C4×C5—61no—
269C2×C4×C5—61no—
270C2×C4×C5—61no—
271C2×C4×C5—122no—
272C2×C4×C5—121no—
273C2×C4×C5—121no—
274C2×C4×C5—121no—
275C2×C4×C5—241no—
276C2×C4×C5C2×#6711yescharacteristic
277C5×C8—41no—
278H(5,𝔽2)C2×C5×D311yescharacteristic
279#137—21no—
280#126—61no—
281#124C2×C5×D311yescharacteristic
282#123—31no—
283#123—61no—
284#123—61no—
285C22⋊Q8—61no—
286#120—31no—
287#120—61no—
288C4×D4—61no—
289#117—61no—
290C2×C22⋊C4—61no—
291#104—21no—
292#100C2×C5×D311yescharacteristic
293#99—31no—
294#99—31no—
295#99—61no—
296#99—61no—
297C2×C3×C5—21no—
298C2×C3×C5—41no—
299C2×C3×C5—41no—
300C2×C3×C5—81no—
301C2×C3×C5B(4,𝔽2)11yescharacteristic
302C5×D3—42no—
303C23×C3—21no—
304C23×C3—21no—
305C23×C3C2×C5×D411yescharacteristic
306C23×C3C2×C5×D411yescharacteristic
307C22×D3—21no—
308C22×D3—42no—
309C22×D3—41no—
310C3×Q8—21no—
311C3×Q8—41no—
312C3×D4—21no—
313C3×D4—41no—
314C3×D4—41no—
315C3×D4—41no—
316C3×D4—41no—
317C2×C3×C4—21no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—41no—
322C2×C3×C4C2×C5×D411yescharacteristic
323#67—42no—
324#67—82no—
325C2×Dic3—21no—
326C2×Dic3—21no—
327C2×Dic3—21no—
328C2×Dic3—41no—
329C2×Dic3—41no—
330C2×Dic3—41no—
331C4×D3—42no—
332C4×D3—81no—
333C3×C8—41no—
334C22×C5—21no—
335C22×C5—21no—
336C22×C5—41no—
337C22×C5—41no—
338C22×C5—41no—
339C22×C5—62no—
340C22×C5—81no—
341C22×C5—81no—
342C22×C5—121no—
343C22×C5—122no—
344C22×C5—121no—
345C22×C5—241no—
346C22×C5#93711yescharacteristic
347C4×C5—21no—
348C4×C5—41no—
349C4×C5—41no—
350C4×C5—41no—
351C4×C5—121no—
352C4×C5—121no—
353C4×C5—121no—
354C4×C5—241no—
355C4×C5—241no—
356C24—31no—
357#41—21no—
358#41—41no—
359C2×Q8C5×#6711yescharacteristic, complemented
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—41no—
363C2×D4—61no—
364C2×D4—121no—
365C2×D4C22×C5×D311yescharacteristic
366C22×C4—31no—
367C22×C4—61no—
368C22×C4—61no—
369Q16—41no—
370SD16—41no—
371M16—21no—
372C4⋊C4—61no—
373C4⋊C4—61no—
374C4⋊C4—61no—
375C4⋊C4—61no—
376C4⋊C4—121no—
377C22⋊C4—21no—
378C22⋊C4—21no—
379C22⋊C4—31no—
380C22⋊C4—31no—
381C22⋊C4—61no—
382C22⋊C4—61no—
383C22⋊C4—61no—
384C22⋊C4—61no—
385C22⋊C4—61no—
386C22⋊C4—61no—
387C22⋊C4—121no—
388C22⋊C4—121no—
389C42—61no—
390C42C5×#6711yescharacteristic, complemented
391C3×C5#219711yescharacteristic, complemented, Hall
392C22×C3—21no—
393C22×C3—21no—
394C22×C3—41no—
395C22×C3—41no—
396C22×C3—41no—
397C22×C3—81no—
398C22×C3—81no—
399C22×C3C5×#12011yescharacteristic
400C2×D3—22no—
401C2×D3—41no—
402C2×D3—41no—
403C2×D3—42no—
404C2×D3—81no—
405C3×C4—21no—
406C3×C4—41no—
407C3×C4—41no—
408C3×C4—41no—
409Dic3—41no—
410Dic3—41no—
411Dic3—41no—
412Dic3—81no—
413Dic3—81no—
414C2×C5—21no—
415C2×C5—41no—
416C2×C5—41no—
417C2×C5—81no—
418C2×C5—122no—
419C2×C5#503211yescharacteristic
420C23—21no—
421C23—21no—
422C23—61no—
423C23—121no—
424C23—122no—
425C23C5×D3×D411yescharacteristic
426C23C5×D3×D411yescharacteristic
427Q8—21no—
428Q8—41no—
429D4—21no—
430D4—41no—
431D4—41no—
432D4—41no—
433D4—41no—
434D4—122no—
435D4—242no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—41no—
441C2×C4—61no—
442C2×C4—61no—
443C2×C4—61no—
444C2×C4—122no—
445C2×C4—121no—
446C2×C4—121no—
447C2×C4—121no—
448C2×C4—241no—
449C2×C4C2×C5×#6711yescharacteristic
450C8—41no—
451C2×C3—21no—
452C2×C3—41no—
453C2×C3—41no—
454C2×C3—81no—
455C2×C3C5×B(4,𝔽2)11yescharacteristic
456D3—42no—
457C5#7099011yescharacteristic, direct product factor, Sylow
458C22—21no—
459C22—21no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—62no—
464C22—81no—
465C22—81no—
466C22—121no—
467C22—122no—
468C22—121no—
469C22—241no—
470C22C5×#93711yescharacteristic
471C4—21no—
472C4—41no—
473C4—41no—
474C4—41no—
475C4—121no—
476C4—121no—
477C4—121no—
478C4—241no—
479C4—241no—
480C3C5×#219711yescharacteristic, complemented, Sylow
481C2—21no—
482C2—41no—
483C2—41no—
484C2—81no—
485C2—122no—
486C2C5×#503211yescharacteristic
subgroupG/H
centerC2×C5#5032
commutator, cosocleC2×C3×D4C23×C5
FrattiniC2×D4C22×C5×D3
FittingC3×C5×#455C2
socleC2×C3×C5B(4,𝔽2)
prime radicalC5×#5244C2
hypercenterC5×#455D3
hypocenterC3C5×#2197
LCSG ▷ C2×C3×D4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C5 ◁ C22×C5 ◁ C2×C5×D4 ◁ C5×#455

Complements

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

NH ≅ G/Ntype
1. C5×#5679C2semidirect
2. C3×C5×#455C2semidirect
4. C5×#5243C2semidirect
5. C5×#4541C2semidirect
6. C5×#4476C2semidirect
7. C5×#4475C2semidirect
28. #70990C5direct (N × H)
32. C5×#455D3semidirect
39. C2×C3×C5×Q8D4semidirect
46. C22×C5×Dic3D4semidirect
54. C3×C42×C5D4semidirect
60. C5×#217D4semidirect
71. #5679C2×C5semidirect
72. C3×#455C2×C5semidirect
74. #5243C2×C5semidirect
75. #4541C2×C5semidirect
76. #4476C2×C5semidirect
77. #4475C2×C5semidirect
151. C2×C5×Q8#67semidirect
182. C42×C5#67semidirect
186. #455C5×D3semidirect
215. C2×C3×Q8C5×D4semidirect
222. C22×Dic3C5×D4semidirect
230. C3×C42C5×D4semidirect
236. #217C5×D4semidirect
359. C2×Q8C5×#67semidirect
390. C42C5×#67semidirect
391. C3×C5#2197semidirect
457. C5#70990direct (N × H)
480. C3C5×#2197semidirect

Maximal subgroups

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

subgroupindex
C5×#56792
C3×C5×#4552
C5×#52442
C5×#52432
C5×#45412
C5×#44762
C5×#44752
C5×#21973
#709905