Finite Group Databasebeta

← C5×#71002 #14316730

Subgroups — C5×#71002 #14316730

Order 1920.

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

normal subgroups only

This group has 492 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×#5664C211yesmaximal, characteristic, complemented
2C3×C5×#458C211yesmaximal, characteristic, complemented
3C5×#5243C211yesmaximal, characteristic, complemented
4C5×#5241C211yesmaximal, characteristic
5C5×#4539C211yesmaximal, characteristic, complemented
6C5×#4474C211yesmaximal, characteristic, complemented
7C5×#4472C211yesmaximal, characteristic, complemented
8C5×#2198—31nomaximal, Hall
9C3×C5×H(5,𝔽2)C2211yescharacteristic
10C3×C5×#123—21no—
11C3×C5×#123C2211yescharacteristic
12C5×#949—21no—
13C5×#940C2211yescharacteristic
14C5×#935—21no—
15C5×#935C2211yescharacteristic
16C4×C5×#67—21no—
17C5×#894—21no—
18C5×#887—21no—
19C5×#885—21no—
20C5×#885—21no—
21C5×#67⋊C4—21no—
22C3×C5×#99—21no—
23C3×C5×#99C2211yescharacteristic
24C5×#837—21no—
25C5×#834—21no—
26C5×#834C2211yescharacteristic
27C5×#833C2211yescharacteristic
28#71002C511yesmaximal, characteristic, direct product factor, Hall
29C5×#559—31no—
30C5×#458—31no—
31C5×#458D311yescharacteristic, complemented
32C5×#453—31no—
33C5×#353—31no—
34C5×#352—31no—
35C5×#352—31no—
36C3×C5×#41—21no—
37C3×C5×#41—41no—
38C2×C3×C5×D4—21no—
39C2×C3×C5×D4—21no—
40C2×C3×C5×D4—41no—
41C2×C3×C5×D4C2311yescharacteristic
42C22×C3×C4×C5D411yescharacteristic
43C2×C5×#67—21no—
44C2×C5×#67—21no—
45C22×C5×Dic3D411yescharacteristic
46C2×C5×D12D411yescharacteristic
47C2×C4×C5×D3—21no—
48C2×C5×Dic6—21no—
49C3×C5×C4⋊C4—21no—
50C3×C5×C4⋊C4—41no—
51C3×C5×C22⋊C4—21no—
52C3×C5×C22⋊C4—21no—
53C3×C5×C22⋊C4—41no—
54C3×C5×C22⋊C4D411yescharacteristic
55C5×#217—21no—
56C5×#217—21no—
57C5×#217—41no—
58C5×#217D411yescharacteristic
59C5×#215—41no—
60C5×#213—41no—
61C5×(C2×D3)⋊C4—21no—
62C5×(C2×D3)⋊C4—21no—
63C5×(C2×D3)⋊C4—21no—
64C5×Dic3⋊C4—21no—
65C5×Dic3⋊C4—21no—
66C5×Dic3⋊C4—41no—
67C4×C5×Dic3—21no—
68C4×C5×Dic3—21no—
69C4×C5×Dic3D411yescharacteristic
70C5×#208—21no—
71#5664C2×C511yescharacteristic, complemented
72C3×#458C2×C511yescharacteristic, complemented
73#5243C2×C511yescharacteristic, complemented
74#5241C2×C511yescharacteristic
75#4539C2×C511yescharacteristic, complemented
76#4474C2×C511yescharacteristic, complemented
77#4472C2×C511yescharacteristic, complemented
78C5×H(5,𝔽2)C2×D311yescharacteristic
79C5×AGL(1,ℤ/8)—61no—
80C5×#127—31no—
81C5×#124—61no—
82C5×#124—61no—
83C5×#123—21no—
84C5×#123—31no—
85C5×#123—61no—
86C5×#123—61no—
87C5×#123C2×D311yescharacteristic
88C4×C5×Q8—61no—
89C4×C5×D4—61no—
90C4×C5×D4—61no—
91C5×#104—61no—
92C5×#100—31no—
93C5×#99—21no—
94C5×#99—31no—
95C5×#99—61no—
96C5×#99C2×D311yescharacteristic
97#2198—31noSylow
98C23×C3×C5—21no—
99C23×C3×C5—21no—
100C23×C3×C5C2×D411yescharacteristic
101C23×C3×C5C2×D411yescharacteristic
102C22×C5×D3—21no—
103C3×C5×Q8—21no—
104C3×C5×D4—21no—
105C3×C5×D4—41no—
106C3×C5×D4—41no—
107C3×C5×D4—41no—
108C3×C5×D4—41no—
109C2×C3×C4×C5—21no—
110C2×C3×C4×C5—21no—
111C2×C3×C4×C5—21no—
112C2×C3×C4×C5—21no—
113C2×C3×C4×C5—41no—
114C2×C3×C4×C5—41no—
115C2×C3×C4×C5—41no—
116C2×C3×C4×C5C2×D411yescharacteristic
117C5×#67—42no—
118C5×#67—42no—
119C2×C5×Dic3—21no—
120C2×C5×Dic3—21no—
121C2×C5×Dic3—21no—
122C2×C5×Dic3—41no—
123C2×C5×Dic3—41no—
124C5×D12—41no—
125C4×C5×D3—81no—
126C5×Dic6—42no—
127C5×C3⋊C8—41no—
128C3×H(5,𝔽2)C22×C511yescharacteristic
129C3×#123—21no—
130C3×#123C22×C511yescharacteristic
131#949—21no—
132#940C22×C511yescharacteristic
133#935—21no—
134#935C22×C511yescharacteristic
135C4×#67—21no—
136#894—21no—
137#887—21no—
138#885—21no—
139#885—21no—
140#67⋊C4—21no—
141C3×#99—21no—
142C3×#99C22×C511yescharacteristic
143#837—21no—
144#834—21no—
145#834C22×C511yescharacteristic
146#833C22×C511yescharacteristic
147C5×#41—21no—
148C5×#41—41no—
149C2×C5×Q8—61no—
150C2×C5×D4—21no—
151C2×C5×D4—21no—
152C2×C5×D4—31no—
153C2×C5×D4—41no—
154C2×C5×D4—61no—
155C2×C5×D4—61no—
156C2×C5×D4C22×D311yescharacteristic
157C22×C4×C5—31no—
158C22×C4×C5—61no—
159C22×C4×C5#6711yescharacteristic
160C5×SD16—121no—
161C5×D8—121no—
162C5×M16—61no—
163C5×C4⋊C4—21no—
164C5×C4⋊C4—41no—
165C5×C4⋊C4—61no—
166C5×C4⋊C4—61no—
167C5×C4⋊C4—121no—
168C5×C22⋊C4—21no—
169C5×C22⋊C4—21no—
170C5×C22⋊C4—31no—
171C5×C22⋊C4—41no—
172C5×C22⋊C4—61no—
173C5×C22⋊C4—61no—
174C5×C22⋊C4—61no—
175C5×C22⋊C4—61no—
176C5×C22⋊C4—61no—
177C5×C22⋊C4—121no—
178C5×C22⋊C4#6711yescharacteristic
179C42×C5—31no—
180C42×C5—61no—
181C42×C5—61no—
182#559—31no—
183#458—31no—
184#458C5×D311yescharacteristic, complemented
185#453—31no—
186#353—31no—
187#352—31no—
188#352—31no—
189C22×C3×C5—21no—
190C22×C3×C5—21no—
191C22×C3×C5—41no—
192C22×C3×C5—41no—
193C22×C3×C5—41no—
194C22×C3×C5—81no—
195C22×C3×C5—81no—
196C22×C3×C5#12011yescharacteristic
197C2×C5×D3—41no—
198C2×C5×D3—41no—
199C2×C5×D3—41no—
200C3×C4×C5—21no—
201C3×C4×C5—41no—
202C3×C4×C5—41no—
203C3×C4×C5—41no—
204C3×C4×C5—81no—
205C5×Dic3—22no—
206C5×Dic3—41no—
207C5×Dic3—41no—
208C5×Dic3—81no—
209C3×#41—21no—
210C3×#41—41no—
211C2×C3×D4—21no—
212C2×C3×D4—21no—
213C2×C3×D4—41no—
214C2×C3×D4C23×C511yescharacteristic
215C22×C3×C4C5×D411yescharacteristic
216C2×#67—21no—
217C2×#67—21no—
218C22×Dic3C5×D411yescharacteristic
219C2×D12C5×D411yescharacteristic
220C2×C4×D3—21no—
221C2×Dic6—21no—
222C3×C4⋊C4—21no—
223C3×C4⋊C4—41no—
224C3×C22⋊C4—21no—
225C3×C22⋊C4—21no—
226C3×C22⋊C4—41no—
227C3×C22⋊C4C5×D411yescharacteristic
228#217—21no—
229#217—21no—
230#217—41no—
231#217C5×D411yescharacteristic
232#215—41no—
233#213—41no—
234(C2×D3)⋊C4—21no—
235(C2×D3)⋊C4—21no—
236(C2×D3)⋊C4—21no—
237Dic3⋊C4—21no—
238Dic3⋊C4—21no—
239Dic3⋊C4—41no—
240C4×Dic3—21no—
241C4×Dic3—21no—
242C4×Dic3C5×D411yescharacteristic
243#208—21no—
244C23×C5—21no—
245C23×C5—21no—
246C23×C5—61no—
247C23×C5D3×D411yescharacteristic
248C23×C5C2×#6711yescharacteristic
249C5×Q8—21no—
250C5×Q8—122no—
251C5×D4—21no—
252C5×D4—41no—
253C5×D4—41no—
254C5×D4—41no—
255C5×D4—41no—
256C5×D4—121no—
257C5×D4—122no—
258C5×D4—122no—
259C2×C4×C5—21no—
260C2×C4×C5—21no—
261C2×C4×C5—21no—
262C2×C4×C5—21no—
263C2×C4×C5—41no—
264C2×C4×C5—41no—
265C2×C4×C5—41no—
266C2×C4×C5—61no—
267C2×C4×C5—61no—
268C2×C4×C5—61no—
269C2×C4×C5—121no—
270C2×C4×C5—121no—
271C2×C4×C5—241no—
272C2×C4×C5D3×D411yescharacteristic
273C5×C8—121no—
274H(5,𝔽2)C2×C5×D311yescharacteristic
275AGL(1,ℤ/8)—61no—
276#127—31no—
277#124—61no—
278#124—61no—
279#123—21no—
280#123—31no—
281#123—61no—
282#123—61no—
283#123C2×C5×D311yescharacteristic
284C4×Q8—61no—
285C4×D4—61no—
286C4×D4—61no—
287#104—61no—
288#100—31no—
289#99—21no—
290#99—31no—
291#99—61no—
292#99C2×C5×D311yescharacteristic
293C2×C3×C5—21no—
294C2×C3×C5—41no—
295C2×C3×C5—41no—
296C2×C3×C5—81no—
297C2×C3×C5B(4,𝔽2)11yescharacteristic
298C5×D3—81no—
299C23×C3—21no—
300C23×C3—21no—
301C23×C3C2×C5×D411yescharacteristic
302C23×C3C2×C5×D411yescharacteristic
303C22×D3—21no—
304C3×Q8—21no—
305C3×D4—21no—
306C3×D4—41no—
307C3×D4—41no—
308C3×D4—41no—
309C3×D4—41no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—41no—
315C2×C3×C4—41no—
316C2×C3×C4—41no—
317C2×C3×C4C2×C5×D411yescharacteristic
318#67—42no—
319#67—42no—
320C2×Dic3—21no—
321C2×Dic3—21no—
322C2×Dic3—21no—
323C2×Dic3—41no—
324C2×Dic3—41no—
325D12—41no—
326C4×D3—81no—
327Dic6—42no—
328C3⋊C8—41no—
329C22×C5—21no—
330C22×C5—21no—
331C22×C5—41no—
332C22×C5—41no—
333C22×C5—41no—
334C22×C5—81no—
335C22×C5—81no—
336C22×C5—121no—
337C22×C5—121no—
338C22×C5—121no—
339C22×C5#93711yescharacteristic
340C4×C5—21no—
341C4×C5—41no—
342C4×C5—41no—
343C4×C5—41no—
344C4×C5—62no—
345C4×C5—81no—
346C4×C5—121no—
347C4×C5—121no—
348C4×C5—241no—
349#41—21no—
350#41—41no—
351C2×Q8—61no—
352C2×D4—21no—
353C2×D4—21no—
354C2×D4—31no—
355C2×D4—41no—
356C2×D4—61no—
357C2×D4—61no—
358C2×D4C22×C5×D311yescharacteristic
359C22×C4—31no—
360C22×C4—61no—
361C22×C4C5×#6711yescharacteristic
362SD16—121no—
363D8—121no—
364M16—61no—
365C4⋊C4—21no—
366C4⋊C4—41no—
367C4⋊C4—61no—
368C4⋊C4—61no—
369C4⋊C4—121no—
370C22⋊C4—21no—
371C22⋊C4—21no—
372C22⋊C4—31no—
373C22⋊C4—41no—
374C22⋊C4—61no—
375C22⋊C4—61no—
376C22⋊C4—61no—
377C22⋊C4—61no—
378C22⋊C4—61no—
379C22⋊C4—121no—
380C22⋊C4C5×#6711yescharacteristic
381C42—31no—
382C42—61no—
383C42—61no—
384C3×C5#219811yescharacteristic, complemented, Hall
385C22×C3—21no—
386C22×C3—21no—
387C22×C3—41no—
388C22×C3—41no—
389C22×C3—41no—
390C22×C3—81no—
391C22×C3—81no—
392C22×C3C5×#12011yescharacteristic
393C2×D3—41no—
394C2×D3—41no—
395C2×D3—41no—
396C3×C4—21no—
397C3×C4—41no—
398C3×C4—41no—
399C3×C4—41no—
400C3×C4—81no—
401Dic3—22no—
402Dic3—41no—
403Dic3—41no—
404Dic3—81no—
405C2×C5—21no—
406C2×C5—41no—
407C2×C5—41no—
408C2×C5—81no—
409C2×C5—241no—
410C2×C5#503211yescharacteristic
411C23—21no—
412C23—21no—
413C23—61no—
414C23C5×D3×D411yescharacteristic
415C23C2×C5×#6711yescharacteristic
416Q8—21no—
417Q8—122no—
418D4—21no—
419D4—41no—
420D4—41no—
421D4—41no—
422D4—41no—
423D4—121no—
424D4—122no—
425D4—122no—
426C2×C4—21no—
427C2×C4—21no—
428C2×C4—21no—
429C2×C4—21no—
430C2×C4—41no—
431C2×C4—41no—
432C2×C4—41no—
433C2×C4—61no—
434C2×C4—61no—
435C2×C4—61no—
436C2×C4—121no—
437C2×C4—121no—
438C2×C4—241no—
439C2×C4C5×D3×D411yescharacteristic
440C8—121no—
441C2×C3—21no—
442C2×C3—41no—
443C2×C3—41no—
444C2×C3—81no—
445C2×C3C5×B(4,𝔽2)11yescharacteristic
446D3—81no—
447C5#7100211yescharacteristic, direct product factor, Sylow
448C22—21no—
449C22—21no—
450C22—41no—
451C22—41no—
452C22—41no—
453C22—81no—
454C22—81no—
455C22—121no—
456C22—121no—
457C22—121no—
458C22C5×#93711yescharacteristic
459C4—21no—
460C4—41no—
461C4—41no—
462C4—41no—
463C4—62no—
464C4—81no—
465C4—121no—
466C4—121no—
467C4—241no—
468C3C5×#219811yescharacteristic, complemented, Sylow
469C2—21no—
470C2—41no—
471C2—41no—
472C2—81no—
473C2—241no—
474C2C5×#503211yescharacteristic
subgroupG/H
centerC2×C5#5032
commutator, cosocleC2×C3×D4C23×C5
FrattiniC2×D4C22×C5×D3
FittingC3×C5×#458C2
socleC2×C3×C5B(4,𝔽2)
prime radicalC5×#5241C2
hypercenterC5×#458D3
hypocenterC3C5×#2198
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×#5664C2semidirect
2. C3×C5×#458C2semidirect
3. C5×#5243C2semidirect
5. C5×#4539C2semidirect
6. C5×#4474C2semidirect
7. C5×#4472C2semidirect
28. #71002C5direct (N × H)
31. C5×#458D3semidirect
71. #5664C2×C5semidirect
72. C3×#458C2×C5semidirect
73. #5243C2×C5semidirect
75. #4539C2×C5semidirect
76. #4474C2×C5semidirect
77. #4472C2×C5semidirect
184. #458C5×D3semidirect
384. C3×C5#2198semidirect
447. C5#71002direct (N × H)
468. C3C5×#2198semidirect

Maximal subgroups

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

subgroupindex
C5×#56642
C3×C5×#4582
C5×#52432
C5×#52412
C5×#45392
C5×#44742
C5×#44722
C5×#21983
#710025