Finite Group Databasebeta

← C5×#71001 #14316808

Subgroups — C5×#71001 #14316808

Order 1920.

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

normal subgroups only

This group has 520 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×#5666C211yesmaximal, characteristic, complemented
2C3×C5×#458C211yesmaximal, characteristic, complemented
3C5×#5244C211yesmaximal, characteristic, complemented
4C5×#5241C211yesmaximal, characteristic, complemented
5C5×#4539C211yesmaximal, characteristic, complemented
6C5×#4474C211yesmaximal, characteristic, complemented
7C5×#4471C211yesmaximal, characteristic, complemented
8C5×#2200—31nomaximal, Hall
9C3×C5×H(5,𝔽2)C2211yescharacteristic
10C3×C5×#123—21no—
11C3×C5×#123C2211yescharacteristic
12C5×#949—21no—
13C5×#940C2211yescharacteristic
14C5×#937—21no—
15C5×#937C2211yescharacteristic
16C5×#930—21no—
17C5×#898—21no—
18C5×#895—21no—
19C5×#884—21no—
20C5×#883—21no—
21C5×#882—21no—
22C3×C5×#99—21no—
23C3×C5×#99C2211yescharacteristic
24C5×#837—21no—
25C5×#834—21no—
26C5×#834C2211yescharacteristic
27C5×#833C2211yescharacteristic
28#71001C511yesmaximal, characteristic, direct product factor, Hall
29C5×#560—31no—
30C5×#458D311yescharacteristic, complemented
31C5×B(4,𝔽2)—31no—
32C5×#453—31no—
33C5×#353—31no—
34C5×#352—31no—
35C5×AΣL(1,𝔽16)—31no—
36C23×C5×D3D411yescharacteristic
37C3×C5×#41—21no—
38C3×C5×#41—41no—
39C2×C3×C5×D4—21no—
40C2×C3×C5×D4—21no—
41C2×C3×C5×D4—41no—
42C2×C3×C5×D4C2311yescharacteristic
43C22×C3×C4×C5D411yescharacteristic, complemented
44C2×C5×#67—21no—
45C2×C5×#67—21no—
46C2×C5×#67—21no—
47C2×C5×#67—41no—
48C2×C5×D12—21no—
49C2×C5×D12—21no—
50C2×C5×D12D411yescharacteristic, complemented
51C2×C4×C5×D3—21no—
52C3×C5×C4⋊C4—21no—
53C3×C5×C4⋊C4—41no—
54C3×C5×C22⋊C4—21no—
55C3×C5×C22⋊C4—21no—
56C3×C5×C22⋊C4—41no—
57C3×C5×C22⋊C4D411yescharacteristic, complemented
58C5×#217—21no—
59C5×#217D411yescharacteristic
60C5×#215—41no—
61C5×#213—41no—
62C5×(C2×D3)⋊C4—21no—
63C5×(C2×D3)⋊C4—21no—
64C5×(C2×D3)⋊C4—21no—
65C5×(C2×D3)⋊C4—21no—
66C5×(C2×D3)⋊C4—41no—
67C5×C4⋊Dic3—21no—
68C5×Dic3⋊C4—21no—
69C4×C5×Dic3D411yescharacteristic, complemented
70C5×#208—21no—
71#5666C2×C511yescharacteristic, complemented
72C3×#458C2×C511yescharacteristic, complemented
73#5244C2×C511yescharacteristic, complemented
74#5241C2×C511yescharacteristic, complemented
75#4539C2×C511yescharacteristic, complemented
76#4474C2×C511yescharacteristic, complemented
77#4471C2×C511yescharacteristic, complemented
78C5×H(5,𝔽2)C2×D311yescharacteristic
79C5×AGL(1,ℤ/8)—61no—
80C5×#127—31no—
81C5×#126—61no—
82C5×#123—21no—
83C5×#123—61no—
84C5×#123C2×D311yescharacteristic
85C5×#121—61no—
86C5×#121—61no—
87C5×#121—61no—
88C5×#120—31no—
89C5×#120—61no—
90C5×#120—61no—
91C5×#104—61no—
92C5×#100—31no—
93C5×#99—21no—
94C5×#99—31no—
95C5×#99—61no—
96C5×#99C2×D311yescharacteristic
97#2200—31noSylow
98C23×C3×C5—21no—
99C23×C3×C5—21no—
100C23×C3×C5C2×D411yescharacteristic
101C23×C3×C5C2×D411yescharacteristic
102C22×C5×D3—21no—
103C22×C5×D3—21no—
104C22×C5×D3—41no—
105C22×C5×D3—81no—
106C3×C5×Q8—21no—
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×C5—41no—
118C2×C3×C4×C5—41no—
119C2×C3×C4×C5C2×D411yescharacteristic
120C5×#67—81no—
121C5×#67—81no—
122C5×#67—81no—
123C5×#67—82no—
124C2×C5×Dic3—21no—
125C2×C5×Dic3—21no—
126C2×C5×Dic3—41no—
127C5×D12—41no—
128C5×D12—81no—
129C5×D12—81no—
130C4×C5×D3—81no—
131C5×C3⋊C8—41no—
132C3×H(5,𝔽2)C22×C511yescharacteristic
133C3×#123—21no—
134C3×#123C22×C511yescharacteristic
135#949—21no—
136#940C22×C511yescharacteristic
137#937—21no—
138#937C22×C511yescharacteristic
139#930—21no—
140#898—21no—
141#895—21no—
142#884—21no—
143#883—21no—
144#882—21no—
145C3×#99—21no—
146C3×#99C22×C511yescharacteristic
147#837—21no—
148#834—21no—
149#834C22×C511yescharacteristic
150#833C22×C511yescharacteristic
151C24×C5—31no—
152C5×#41—21no—
153C5×#41—41no—
154C2×C5×D4—21no—
155C2×C5×D4—21no—
156C2×C5×D4—31no—
157C2×C5×D4—41no—
158C2×C5×D4—61no—
159C2×C5×D4—61no—
160C2×C5×D4—61no—
161C2×C5×D4—61no—
162C2×C5×D4—61no—
163C2×C5×D4—121no—
164C2×C5×D4C22×D311yescharacteristic
165C22×C4×C5—61no—
166C22×C4×C5#6711yescharacteristic, complemented
167C5×SD16—121no—
168C5×D8—121no—
169C5×M16—61no—
170C5×C4⋊C4—21no—
171C5×C4⋊C4—41no—
172C5×C4⋊C4—61no—
173C5×C4⋊C4—61no—
174C5×C22⋊C4—21no—
175C5×C22⋊C4—21no—
176C5×C22⋊C4—31no—
177C5×C22⋊C4—41no—
178C5×C22⋊C4—61no—
179C5×C22⋊C4—61no—
180C5×C22⋊C4—61no—
181C5×C22⋊C4—61no—
182C5×C22⋊C4—61no—
183C5×C22⋊C4—121no—
184C5×C22⋊C4#6711yescharacteristic, complemented
185C42×C5—31no—
186#560—31no—
187#458C5×D311yescharacteristic, complemented
188B(4,𝔽2)—31no—
189#453—31no—
190#353—31no—
191#352—31no—
192AΣL(1,𝔽16)—31no—
193C22×C3×C5—21no—
194C22×C3×C5—21no—
195C22×C3×C5—41no—
196C22×C3×C5—41no—
197C22×C3×C5—41no—
198C22×C3×C5—81no—
199C22×C3×C5—81no—
200C22×C3×C5#12011yescharacteristic
201C2×C5×D3—41no—
202C2×C5×D3—41no—
203C2×C5×D3—41no—
204C2×C5×D3—41no—
205C2×C5×D3—42no—
206C2×C5×D3—41no—
207C2×C5×D3—41no—
208C2×C5×D3—81no—
209C3×C4×C5—21no—
210C3×C4×C5—41no—
211C3×C4×C5—41no—
212C3×C4×C5—41no—
213C3×C4×C5—81no—
214C5×Dic3—41no—
215C5×Dic3—41no—
216C5×Dic3—81no—
217C23×D3C5×D411yescharacteristic
218C3×#41—21no—
219C3×#41—41no—
220C2×C3×D4—21no—
221C2×C3×D4—21no—
222C2×C3×D4—41no—
223C2×C3×D4C23×C511yescharacteristic
224C22×C3×C4C5×D411yescharacteristic, complemented
225C2×#67—21no—
226C2×#67—21no—
227C2×#67—21no—
228C2×#67—41no—
229C2×D12—21no—
230C2×D12—21no—
231C2×D12C5×D411yescharacteristic, complemented
232C2×C4×D3—21no—
233C3×C4⋊C4—21no—
234C3×C4⋊C4—41no—
235C3×C22⋊C4—21no—
236C3×C22⋊C4—21no—
237C3×C22⋊C4—41no—
238C3×C22⋊C4C5×D411yescharacteristic, complemented
239#217—21no—
240#217C5×D411yescharacteristic
241#215—41no—
242#213—41no—
243(C2×D3)⋊C4—21no—
244(C2×D3)⋊C4—21no—
245(C2×D3)⋊C4—21no—
246(C2×D3)⋊C4—21no—
247(C2×D3)⋊C4—41no—
248C4⋊Dic3—21no—
249Dic3⋊C4—21no—
250C4×Dic3C5×D411yescharacteristic, complemented
251#208—21no—
252C23×C5—21no—
253C23×C5—21no—
254C23×C5—61no—
255C23×C5—61no—
256C23×C5—121no—
257C23×C5—241no—
258C23×C5D3×D411yescharacteristic
259C23×C5C2×#6711yescharacteristic
260C5×Q8—21no—
261C5×D4—21no—
262C5×D4—41no—
263C5×D4—41no—
264C5×D4—41no—
265C5×D4—41no—
266C5×D4—121no—
267C5×D4—241no—
268C5×D4—241no—
269C5×D4—241no—
270C5×D4—241no—
271C5×D4—241no—
272C5×D4—242no—
273C2×C4×C5—21no—
274C2×C4×C5—21no—
275C2×C4×C5—21no—
276C2×C4×C5—21no—
277C2×C4×C5—41no—
278C2×C4×C5—41no—
279C2×C4×C5—41no—
280C2×C4×C5—61no—
281C2×C4×C5—61no—
282C2×C4×C5—121no—
283C2×C4×C5—241no—
284C2×C4×C5D3×D411yescharacteristic
285C5×C8—121no—
286H(5,𝔽2)C2×C5×D311yescharacteristic
287AGL(1,ℤ/8)—61no—
288#127—31no—
289#126—61no—
290#123—21no—
291#123—61no—
292#123C2×C5×D311yescharacteristic
293#121—61no—
294#121—61no—
295#121—61no—
296#120—31no—
297#120—61no—
298#120—61no—
299#104—61no—
300#100—31no—
301#99—21no—
302#99—31no—
303#99—61no—
304#99C2×C5×D311yescharacteristic
305C2×C3×C5—21no—
306C2×C3×C5—41no—
307C2×C3×C5—41no—
308C2×C3×C5—81no—
309C2×C3×C5B(4,𝔽2)11yescharacteristic
310C5×D3—81no—
311C5×D3—81no—
312C23×C3—21no—
313C23×C3—21no—
314C23×C3C2×C5×D411yescharacteristic
315C23×C3C2×C5×D411yescharacteristic
316C22×D3—21no—
317C22×D3—21no—
318C22×D3—41no—
319C22×D3—81no—
320C3×Q8—21no—
321C3×D4—21no—
322C3×D4—41no—
323C3×D4—41no—
324C3×D4—41no—
325C3×D4—41no—
326C2×C3×C4—21no—
327C2×C3×C4—21no—
328C2×C3×C4—21no—
329C2×C3×C4—21no—
330C2×C3×C4—41no—
331C2×C3×C4—41no—
332C2×C3×C4—41no—
333C2×C3×C4C2×C5×D411yescharacteristic
334#67—81no—
335#67—81no—
336#67—81no—
337#67—82no—
338C2×Dic3—21no—
339C2×Dic3—21no—
340C2×Dic3—41no—
341D12—41no—
342D12—81no—
343D12—81no—
344C4×D3—81no—
345C3⋊C8—41no—
346C22×C5—21no—
347C22×C5—21no—
348C22×C5—41no—
349C22×C5—41no—
350C22×C5—41no—
351C22×C5—81no—
352C22×C5—81no—
353C22×C5—121no—
354C22×C5—121no—
355C22×C5—121no—
356C22×C5—121no—
357C22×C5—122no—
358C22×C5—121no—
359C22×C5—121no—
360C22×C5—241no—
361C22×C5#93711yescharacteristic
362C4×C5—21no—
363C4×C5—41no—
364C4×C5—41no—
365C4×C5—41no—
366C4×C5—81no—
367C4×C5—121no—
368C4×C5—121no—
369C4×C5—241no—
370C24—31no—
371#41—21no—
372#41—41no—
373C2×D4—21no—
374C2×D4—21no—
375C2×D4—31no—
376C2×D4—41no—
377C2×D4—61no—
378C2×D4—61no—
379C2×D4—61no—
380C2×D4—61no—
381C2×D4—61no—
382C2×D4—121no—
383C2×D4C22×C5×D311yescharacteristic
384C22×C4—61no—
385C22×C4C5×#6711yescharacteristic, complemented
386SD16—121no—
387D8—121no—
388M16—61no—
389C4⋊C4—21no—
390C4⋊C4—41no—
391C4⋊C4—61no—
392C4⋊C4—61no—
393C22⋊C4—21no—
394C22⋊C4—21no—
395C22⋊C4—31no—
396C22⋊C4—41no—
397C22⋊C4—61no—
398C22⋊C4—61no—
399C22⋊C4—61no—
400C22⋊C4—61no—
401C22⋊C4—61no—
402C22⋊C4—121no—
403C22⋊C4C5×#6711yescharacteristic, complemented
404C42—31no—
405C3×C5#220011yescharacteristic, complemented, Hall
406C22×C3—21no—
407C22×C3—21no—
408C22×C3—41no—
409C22×C3—41no—
410C22×C3—41no—
411C22×C3—81no—
412C22×C3—81no—
413C22×C3C5×#12011yescharacteristic
414C2×D3—41no—
415C2×D3—41no—
416C2×D3—41no—
417C2×D3—41no—
418C2×D3—42no—
419C2×D3—41no—
420C2×D3—41no—
421C2×D3—81no—
422C3×C4—21no—
423C3×C4—41no—
424C3×C4—41no—
425C3×C4—41no—
426C3×C4—81no—
427Dic3—41no—
428Dic3—41no—
429Dic3—81no—
430C2×C5—21no—
431C2×C5—41no—
432C2×C5—41no—
433C2×C5—81no—
434C2×C5—241no—
435C2×C5—241no—
436C2×C5#503211yescharacteristic
437C23—21no—
438C23—21no—
439C23—61no—
440C23—61no—
441C23—121no—
442C23—241no—
443C23C5×D3×D411yescharacteristic
444C23C2×C5×#6711yescharacteristic
445Q8—21no—
446D4—21no—
447D4—41no—
448D4—41no—
449D4—41no—
450D4—41no—
451D4—121no—
452D4—241no—
453D4—241no—
454D4—241no—
455D4—241no—
456D4—241no—
457D4—242no—
458C2×C4—21no—
459C2×C4—21no—
460C2×C4—21no—
461C2×C4—21no—
462C2×C4—41no—
463C2×C4—41no—
464C2×C4—41no—
465C2×C4—61no—
466C2×C4—61no—
467C2×C4—121no—
468C2×C4—241no—
469C2×C4C5×D3×D411yescharacteristic
470C8—121no—
471C2×C3—21no—
472C2×C3—41no—
473C2×C3—41no—
474C2×C3—81no—
475C2×C3C5×B(4,𝔽2)11yescharacteristic
476D3—81no—
477D3—81no—
478C5#7100111yescharacteristic, direct product factor, Sylow
479C22—21no—
480C22—21no—
481C22—41no—
482C22—41no—
483C22—41no—
484C22—81no—
485C22—81no—
486C22—121no—
487C22—121no—
488C22—121no—
489C22—121no—
490C22—122no—
491C22—121no—
492C22—121no—
493C22—241no—
494C22C5×#93711yescharacteristic
495C4—21no—
496C4—41no—
497C4—41no—
498C4—41no—
499C4—81no—
500C4—121no—
501C4—121no—
502C4—241no—
503C3C5×#220011yescharacteristic, complemented, Sylow
504C2—21no—
505C2—41no—
506C2—41no—
507C2—81no—
508C2—241no—
509C2—241no—
510C2C5×#503211yescharacteristic
subgroupG/H
centerC2×C5#5032
commutator, cosocleC2×C3×D4C23×C5
FrattiniC2×D4C22×C5×D3
FittingC3×C5×#458C2
socleC2×C3×C5B(4,𝔽2)
hypercenterC5×#458D3
hypocenterC3C5×#2200
LCSG ▷ C2×C3×D4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C5 ◁ C22×C5 ◁ C2×C5×D4 ◁ C5×#458

Complements

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

NH ≅ G/Ntype
1. C5×#5666C2semidirect
2. C3×C5×#458C2semidirect
3. C5×#5244C2semidirect
4. C5×#5241C2semidirect
5. C5×#4539C2semidirect
6. C5×#4474C2semidirect
7. C5×#4471C2semidirect
28. #71001C5direct (N × H)
30. C5×#458D3semidirect
43. C22×C3×C4×C5D4semidirect
50. C2×C5×D12D4semidirect
57. C3×C5×C22⋊C4D4semidirect
69. C4×C5×Dic3D4semidirect
71. #5666C2×C5semidirect
72. C3×#458C2×C5semidirect
73. #5244C2×C5semidirect
74. #5241C2×C5semidirect
75. #4539C2×C5semidirect
76. #4474C2×C5semidirect
77. #4471C2×C5semidirect
166. C22×C4×C5#67semidirect
184. C5×C22⋊C4#67semidirect
187. #458C5×D3semidirect
224. C22×C3×C4C5×D4semidirect
231. C2×D12C5×D4semidirect
238. C3×C22⋊C4C5×D4semidirect
250. C4×Dic3C5×D4semidirect
385. C22×C4C5×#67semidirect
403. C22⋊C4C5×#67semidirect
405. C3×C5#2200semidirect
478. C5#71001direct (N × H)
503. C3C5×#2200semidirect

Maximal subgroups

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

subgroupindex
C5×#56662
C3×C5×#4582
C5×#52442
C5×#52412
C5×#45392
C5×#44742
C5×#44712
C5×#22003
#710015