Finite Group Databasebeta

← C5×#69697 #14287897

Subgroups — C5×#69697 #14287897

Order 1920.

640 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 98 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×#5494C211yesmaximal, characteristic, complemented
2C3×C5×#392C211yesmaximal, characteristic, complemented
3C5×#5218C211yesmaximal, characteristic, complemented
4C5×#5217C211yesmaximal, characteristic, complemented
5C5×#4525C211yesmaximal, characteristic, complemented
6C5×#4452C211yesmaximal, characteristic, complemented
7C5×#4451C211yesmaximal, characteristic, complemented
8C5×#1600—31nomaximal, Hall
9C22×C3×C5×D4—21no—
10C22×C3×C5×D4C2211yescharacteristic
11C2×C3×C5×C22⋊C4—21no—
12C2×C3×C5×C22⋊C4C2211yescharacteristic
13C5×#953—21no—
14C2×C5×#214—21no—
15C2×C5×#213—21no—
16C5×#930C2211yescharacteristic
17C4×C5×#67—21no—
18C5×#924C2211yescharacteristic
19C5×#885—21no—
20C5×#884—21no—
21C5×#883—21no—
22C5×#879—21no—
23C3×C5×#95C2211yescharacteristic
24C5×#832—21no—
25C5×#832—21no—
26C5×#830C2211yescharacteristic
27C5×#830C2211yescharacteristic
28#69697C511yesmaximal, characteristic, direct product factor, Hall
29C5×#538—31no—
30C5×#450—31no—
31C5×#447—31no—
32C5×#392D311yescharacteristic, complemented
33C5×#328—31no—
34C5×#327—31no—
35C5×#323—31no—
36C24×C3×C5—21no—
37C24×C3×C5D411yescharacteristic
38C2×C3×C5×D4—21no—
39C2×C3×C5×D4—21no—
40C2×C3×C5×D4—42no—
41C2×C3×C5×D4—42no—
42C2×C3×C5×D4—42no—
43C2×C3×C5×D4—42no—
44C22×C3×C4×C5—21no—
45C22×C3×C4×C5D411yescharacteristic
46C22×C3×C4×C5C2311yescharacteristic
47C2×C5×#67—21no—
48C2×C5×#67—21no—
49C2×C5×#67—21no—
50C2×C5×D12D411yescharacteristic, complemented
51C2×C4×C5×D3—21no—
52C2×C5×Dic6D411yescharacteristic, complemented
53C3×C5×C22⋊C4—21no—
54C3×C5×C22⋊C4—21no—
55C3×C5×C22⋊C4—42no—
56C5×#217—21no—
57C5×#217—21no—
58C5×#217—21no—
59C5×#214—42no—
60C5×#213—42no—
61C5×(C2×D3)⋊C4—21no—
62C5×(C2×D3)⋊C4—21no—
63C5×C4⋊Dic3D411yescharacteristic, complemented
64C5×C4⋊Dic3D411yescharacteristic, complemented
65C5×Dic3⋊C4—21no—
66C5×Dic3⋊C4—21no—
67C4×C5×Dic3—21no—
68C2×C5×C3⋊C8—21no—
69C2×C5×C3⋊C8—21no—
70#5494C2×C511yescharacteristic, complemented
71C3×#392C2×C511yescharacteristic, complemented
72#5218C2×C511yescharacteristic, complemented
73#5217C2×C511yescharacteristic, complemented
74#4525C2×C511yescharacteristic, complemented
75#4452C2×C511yescharacteristic, complemented
76#4451C2×C511yescharacteristic, complemented
77C22×C5×D4—21no—
78C22×C5×D4C2×D311yescharacteristic
79C2×C5×SD16—61no—
80C2×C5×D8—61no—
81C5×#126—61no—
82C5×#124—61no—
83C5×#123—61no—
84C5×C22⋊Q8—31no—
85C5×#121—31no—
86C5×#121—61no—
87C5×#120—61no—
88C4×C5×D4—61no—
89C2×C5×C22⋊C4—21no—
90C2×C5×C22⋊C4C2×D311yescharacteristic
91C5×#102—61no—
92C5×#102—61no—
93C5×C22⋊C8—31no—
94C5×C22⋊C8—31no—
95C5×#95C2×D311yescharacteristic
96#1600—31noSylow
97C23×C3×C5—22no—
98C23×C3×C5—21no—
99C23×C3×C5—21no—
100C23×C3×C5—21no—
101C23×C3×C5—41no—
102C23×C3×C5—41no—
103C23×C3×C5—41no—
104C23×C3×C5—41no—
105C23×C3×C5—41no—
106C23×C3×C5—41no—
107C23×C3×C5—41no—
108C23×C3×C5—41no—
109C23×C3×C5C2×D411yescharacteristic
110C22×C5×D3—21no—
111C3×C5×D4—22no—
112C3×C5×D4—22no—
113C3×C5×D4—41no—
114C3×C5×D4—41no—
115C3×C5×D4—48no—
116C2×C3×C4×C5—22no—
117C2×C3×C4×C5—21no—
118C2×C3×C4×C5—41no—
119C2×C3×C4×C5—41no—
120C2×C3×C4×C5—41no—
121C2×C3×C4×C5—41no—
122C2×C3×C4×C5C2×D411yescharacteristic
123C2×C3×C4×C5C2×D411yescharacteristic
124C5×#67—41no—
125C5×#67—41no—
126C5×#67—81no—
127C5×#67—81no—
128C2×C5×Dic3—21no—
129C2×C5×Dic3—21no—
130C2×C5×Dic3—21no—
131C5×D12—41no—
132C4×C5×D3—81no—
133C5×Dic6—41no—
134C5×C3⋊C8—41no—
135C5×C3⋊C8—41no—
136C22×C3×D4—21no—
137C22×C3×D4C22×C511yescharacteristic
138C2×C3×C22⋊C4—21no—
139C2×C3×C22⋊C4C22×C511yescharacteristic
140#953—21no—
141C2×#214—21no—
142C2×#213—21no—
143#930C22×C511yescharacteristic
144C4×#67—21no—
145#924C22×C511yescharacteristic
146#885—21no—
147#884—21no—
148#883—21no—
149#879—21no—
150C3×#95C22×C511yescharacteristic
151#832—21no—
152#832—21no—
153#830C22×C511yescharacteristic
154#830C22×C511yescharacteristic
155C24×C5—21no—
156C24×C5#6711yescharacteristic
157C2×C5×Q8—31no—
158C2×C5×D4—21no—
159C2×C5×D4—21no—
160C2×C5×D4—31no—
161C2×C5×D4—42no—
162C2×C5×D4—42no—
163C2×C5×D4—42no—
164C2×C5×D4—42no—
165C2×C5×D4—61no—
166C2×C5×D4—61no—
167C2×C5×D4—61no—
168C22×C4×C5—21no—
169C22×C4×C5—61no—
170C22×C4×C5#6711yescharacteristic
171C22×C4×C5C22×D311yescharacteristic
172C5×SD16—122no—
173C5×D8—122no—
174C2×C5×C8—61no—
175C2×C5×C8—61no—
176C5×C4⋊C4—31no—
177C5×C4⋊C4—31no—
178C5×C4⋊C4—61no—
179C5×C4⋊C4—61no—
180C5×C22⋊C4—21no—
181C5×C22⋊C4—21no—
182C5×C22⋊C4—42no—
183C5×C22⋊C4—61no—
184C5×C22⋊C4—61no—
185C5×C22⋊C4—61no—
186C5×C22⋊C4—61no—
187C5×C22⋊C4—61no—
188C42×C5—61no—
189#538—31no—
190#450—31no—
191#447—31no—
192#392C5×D311yescharacteristic, complemented
193#328—31no—
194#327—31no—
195#323—31no—
196C22×C3×C5—21no—
197C22×C3×C5—21no—
198C22×C3×C5—42no—
199C22×C3×C5—42no—
200C22×C3×C5—42no—
201C22×C3×C5—41no—
202C22×C3×C5—41no—
203C22×C3×C5—41no—
204C22×C3×C5—42no—
205C22×C3×C5—81no—
206C22×C3×C5—81no—
207C22×C3×C5—81no—
208C22×C3×C5—81no—
209C22×C3×C5—81no—
210C22×C3×C5#12011yescharacteristic
211C22×C3×C5#13511yescharacteristic
212C22×C3×C5AGL(1,ℤ/8)11yescharacteristic
213C2×C5×D3—41no—
214C2×C5×D3—41no—
215C2×C5×D3—41no—
216C3×C4×C5—21no—
217C3×C4×C5—21no—
218C3×C4×C5—42no—
219C3×C4×C5—41no—
220C5×Dic3—41no—
221C5×Dic3—41no—
222C5×Dic3—41no—
223C24×C3—21no—
224C24×C3C5×D411yescharacteristic
225C2×C3×D4—21no—
226C2×C3×D4—21no—
227C2×C3×D4—42no—
228C2×C3×D4—42no—
229C2×C3×D4—42no—
230C2×C3×D4—42no—
231C22×C3×C4—21no—
232C22×C3×C4C5×D411yescharacteristic
233C22×C3×C4C23×C511yescharacteristic
234C2×#67—21no—
235C2×#67—21no—
236C2×#67—21no—
237C2×D12C5×D411yescharacteristic, complemented
238C2×C4×D3—21no—
239C2×Dic6C5×D411yescharacteristic, complemented
240C3×C22⋊C4—21no—
241C3×C22⋊C4—21no—
242C3×C22⋊C4—42no—
243#217—21no—
244#217—21no—
245#217—21no—
246#214—42no—
247#213—42no—
248(C2×D3)⋊C4—21no—
249(C2×D3)⋊C4—21no—
250C4⋊Dic3C5×D411yescharacteristic, complemented
251C4⋊Dic3C5×D411yescharacteristic, complemented
252Dic3⋊C4—21no—
253Dic3⋊C4—21no—
254C4×Dic3—21no—
255C2×C3⋊C8—21no—
256C2×C3⋊C8—21no—
257C23×C5—22no—
258C23×C5—21no—
259C23×C5—21no—
260C23×C5—21no—
261C23×C5—41no—
262C23×C5—41no—
263C23×C5—41no—
264C23×C5—41no—
265C23×C5—41no—
266C23×C5—41no—
267C23×C5—41no—
268C23×C5—41no—
269C23×C5—61no—
270C23×C5C2×#6711yescharacteristic
271C5×Q8—121no—
272C5×D4—22no—
273C5×D4—22no—
274C5×D4—41no—
275C5×D4—41no—
276C5×D4—48no—
277C5×D4—121no—
278C5×D4—121no—
279C5×D4—121no—
280C5×D4—241no—
281C5×D4—241no—
282C2×C4×C5—22no—
283C2×C4×C5—21no—
284C2×C4×C5—41no—
285C2×C4×C5—41no—
286C2×C4×C5—41no—
287C2×C4×C5—41no—
288C2×C4×C5—61no—
289C2×C4×C5—61no—
290C2×C4×C5—61no—
291C2×C4×C5—241no—
292C2×C4×C5D3×D411yescharacteristic
293C2×C4×C5D3×D411yescharacteristic
294C5×C8—121no—
295C5×C8—121no—
296C22×D4—21no—
297C22×D4C2×C5×D311yescharacteristic
298C2×SD16—61no—
299C2×D8—61no—
300#126—61no—
301#124—61no—
302#123—61no—
303C22⋊Q8—31no—
304#121—31no—
305#121—61no—
306#120—61no—
307C4×D4—61no—
308C2×C22⋊C4—21no—
309C2×C22⋊C4C2×C5×D311yescharacteristic
310#102—61no—
311#102—61no—
312C22⋊C8—31no—
313C22⋊C8—31no—
314#95C2×C5×D311yescharacteristic
315C2×C3×C5—21no—
316C2×C3×C5—21no—
317C2×C3×C5—81no—
318C2×C3×C5—82no—
319C2×C3×C5#44911yescharacteristic
320C2×C3×C5#45311yescharacteristic
321C2×C3×C5B(4,𝔽2)11yescharacteristic
322C5×D3—81no—
323C23×C3—22no—
324C23×C3—21no—
325C23×C3—21no—
326C23×C3—21no—
327C23×C3—41no—
328C23×C3—41no—
329C23×C3—41no—
330C23×C3—41no—
331C23×C3—41no—
332C23×C3—41no—
333C23×C3—41no—
334C23×C3—41no—
335C23×C3C2×C5×D411yescharacteristic
336C22×D3—21no—
337C3×D4—22no—
338C3×D4—22no—
339C3×D4—41no—
340C3×D4—41no—
341C3×D4—48no—
342C2×C3×C4—22no—
343C2×C3×C4—21no—
344C2×C3×C4—41no—
345C2×C3×C4—41no—
346C2×C3×C4—41no—
347C2×C3×C4—41no—
348C2×C3×C4C2×C5×D411yescharacteristic
349C2×C3×C4C2×C5×D411yescharacteristic
350#67—41no—
351#67—41no—
352#67—81no—
353#67—81no—
354C2×Dic3—21no—
355C2×Dic3—21no—
356C2×Dic3—21no—
357D12—41no—
358C4×D3—81no—
359Dic6—41no—
360C3⋊C8—41no—
361C3⋊C8—41no—
362C22×C5—21no—
363C22×C5—21no—
364C22×C5—42no—
365C22×C5—42no—
366C22×C5—42no—
367C22×C5—42no—
368C22×C5—41no—
369C22×C5—41no—
370C22×C5—41no—
371C22×C5—81no—
372C22×C5—81no—
373C22×C5—81no—
374C22×C5—81no—
375C22×C5—81no—
376C22×C5—121no—
377C22×C5—121no—
378C22×C5—121no—
379C22×C5#93211yescharacteristic
380C22×C5#93711yescharacteristic
381C22×C5#95011yescharacteristic
382C4×C5—21no—
383C4×C5—21no—
384C4×C5—42no—
385C4×C5—41no—
386C4×C5—121no—
387C4×C5—121no—
388C4×C5—121no—
389C24—21no—
390C24C5×#6711yescharacteristic
391C2×Q8—31no—
392C2×D4—21no—
393C2×D4—21no—
394C2×D4—31no—
395C2×D4—42no—
396C2×D4—42no—
397C2×D4—42no—
398C2×D4—42no—
399C2×D4—61no—
400C2×D4—61no—
401C2×D4—61no—
402C22×C4—21no—
403C22×C4—61no—
404C22×C4C5×#6711yescharacteristic
405C22×C4C22×C5×D311yescharacteristic
406SD16—122no—
407D8—122no—
408C2×C8—61no—
409C2×C8—61no—
410C4⋊C4—31no—
411C4⋊C4—31no—
412C4⋊C4—61no—
413C4⋊C4—61no—
414C22⋊C4—21no—
415C22⋊C4—21no—
416C22⋊C4—42no—
417C22⋊C4—61no—
418C22⋊C4—61no—
419C22⋊C4—61no—
420C22⋊C4—61no—
421C22⋊C4—61no—
422C42—61no—
423C3×C5#160011yescharacteristic, complemented, Hall
424C22×C3—21no—
425C22×C3—21no—
426C22×C3—42no—
427C22×C3—42no—
428C22×C3—42no—
429C22×C3—41no—
430C22×C3—41no—
431C22×C3—41no—
432C22×C3—42no—
433C22×C3—81no—
434C22×C3—81no—
435C22×C3—81no—
436C22×C3—81no—
437C22×C3—81no—
438C22×C3C5×#12011yescharacteristic
439C22×C3C5×#13511yescharacteristic
440C22×C3C5×AGL(1,ℤ/8)11yescharacteristic
441C2×D3—41no—
442C2×D3—41no—
443C2×D3—41no—
444C3×C4—21no—
445C3×C4—21no—
446C3×C4—42no—
447C3×C4—41no—
448Dic3—41no—
449Dic3—41no—
450Dic3—41no—
451C2×C5—21no—
452C2×C5—21no—
453C2×C5—81no—
454C2×C5—82no—
455C2×C5—241no—
456C2×C5#503211yescharacteristic
457C2×C5#503711yescharacteristic
458C2×C5#507711yescharacteristic
459C23—22no—
460C23—21no—
461C23—21no—
462C23—21no—
463C23—41no—
464C23—41no—
465C23—41no—
466C23—41no—
467C23—41no—
468C23—41no—
469C23—41no—
470C23—41no—
471C23—61no—
472C23C2×C5×#6711yescharacteristic
473Q8—121no—
474D4—22no—
475D4—22no—
476D4—41no—
477D4—41no—
478D4—48no—
479D4—121no—
480D4—121no—
481D4—121no—
482D4—241no—
483D4—241no—
484C2×C4—22no—
485C2×C4—21no—
486C2×C4—41no—
487C2×C4—41no—
488C2×C4—41no—
489C2×C4—41no—
490C2×C4—61no—
491C2×C4—61no—
492C2×C4—61no—
493C2×C4—241no—
494C2×C4C5×D3×D411yescharacteristic
495C2×C4C5×D3×D411yescharacteristic
496C8—121no—
497C8—121no—
498C2×C3—21no—
499C2×C3—21no—
500C2×C3—81no—
501C2×C3—82no—
502C2×C3C5×#44911yescharacteristic
503C2×C3C5×#45311yescharacteristic
504C2×C3C5×B(4,𝔽2)11yescharacteristic
505D3—81no—
506C5#6969711yescharacteristic, direct product factor, Sylow
507C22—21no—
508C22—21no—
509C22—42no—
510C22—42no—
511C22—42no—
512C22—42no—
513C22—41no—
514C22—41no—
515C22—41no—
516C22—81no—
517C22—81no—
518C22—81no—
519C22—81no—
520C22—81no—
521C22—121no—
522C22—121no—
523C22—121no—
524C22C5×#93211yescharacteristic
525C22C5×#93711yescharacteristic
526C22C5×#95011yescharacteristic
527C4—21no—
528C4—21no—
529C4—42no—
530C4—41no—
531C4—121no—
532C4—121no—
533C4—121no—
534C3C5×#160011yescharacteristic, complemented, Sylow
535C2—21no—
536C2—21no—
537C2—81no—
538C2—82no—
539C2—241no—
540C2C5×#503211yescharacteristic
541C2C5×#503711yescharacteristic
542C2C5×#507711yescharacteristic
subgroupG/H
centerC22×C5#937
commutator, cosocleC22×C3×C4C23×C5
FrattiniC22×C4C22×C5×D3
FittingC3×C5×#392C2
socleC22×C3×C5#120
hypercenterC5×#392D3
hypocenterC3C5×#1600
LCSG ▷ C22×C3×C4 ▷ C22×C3 ▷ C3
UCSC1 ◁ C22×C5 ◁ C22×C4×C5 ◁ C5×#392

Complements

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

NH ≅ G/Ntype
1. C5×#5494C2semidirect
2. C3×C5×#392C2semidirect
3. C5×#5218C2semidirect
4. C5×#5217C2semidirect
5. C5×#4525C2semidirect
6. C5×#4452C2semidirect
7. C5×#4451C2semidirect
28. #69697C5direct (N × H)
32. C5×#392D3semidirect
50. C2×C5×D12D4semidirect
52. C2×C5×Dic6D4semidirect
63. C5×C4⋊Dic3D4semidirect
64. C5×C4⋊Dic3D4semidirect
70. #5494C2×C5semidirect
71. C3×#392C2×C5semidirect
72. #5218C2×C5semidirect
73. #5217C2×C5semidirect
74. #4525C2×C5semidirect
75. #4452C2×C5semidirect
76. #4451C2×C5semidirect
192. #392C5×D3semidirect
237. C2×D12C5×D4semidirect
239. C2×Dic6C5×D4semidirect
250. C4⋊Dic3C5×D4semidirect
251. C4⋊Dic3C5×D4semidirect
423. C3×C5#1600semidirect
506. C5#69697direct (N × H)
534. C3C5×#1600semidirect

Maximal subgroups

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

subgroupindex
C5×#54942
C3×C5×#3922
C5×#52182
C5×#52172
C5×#45252
C5×#44522
C5×#44512
C5×#16003
#696975