Finite Group Databasebeta

← C5×#70998 #14316670

Subgroups — C5×#70998 #14316670

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×#5587C211yesmaximal, characteristic, complemented
2C3×C5×B(4,𝔽2)C211yesmaximal, characteristic
3C5×#5243C211yesmaximal, characteristic, complemented
4C5×#5242C211yesmaximal, characteristic, complemented
5C5×#4536C211yesmaximal, characteristic, complemented
6C5×#4473C211yesmaximal, characteristic, complemented
7C5×#4472C211yesmaximal, characteristic, complemented
8C5×#2197—31nomaximal, Hall
9C3×C5×H(5,𝔽2)C2211yescharacteristic
10C3×C5×#120—21no—
11C3×C5×#120C2211yescharacteristic
12C2×C5×#217—21no—
13C5×#951—21no—
14C5×#936C2211yescharacteristic
15C5×#935—21no—
16C5×#935—21no—
17C5×#935C2211yescharacteristic
18C5×D4×Dic3—21no—
19C5×#879—21no—
20C5×#878—21no—
21C5×#877—21no—
22C3×C5×#99—21no—
23C3×C5×#99C2211yescharacteristic
24C5×#837—21no—
25C5×#834—21no—
26C5×#834C2211yescharacteristic
27C5×#833C2211yescharacteristic
28#70998C511yesmaximal, characteristic, direct product factor, Hall
29C5×#551—31no—
30C5×#458—31no—
31C5×B(4,𝔽2)D311yescharacteristic
32C5×#455—31no—
33C5×#354—31no—
34C5×#352—31no—
35C5×AΣL(1,𝔽16)—31no—
36C24×C3×C5D411yescharacteristic
37C3×C5×#41—21no—
38C3×C5×#41—41no—
39C2×C3×C5×D4—21no—
40C2×C3×C5×D4—21no—
41C2×C3×C5×D4—21no—
42C2×C3×C5×D4—41no—
43C2×C3×C5×D4—41no—
44C2×C3×C5×D4C2311yescharacteristic
45C22×C5×Dic3—21no—
46C22×C5×Dic3—21no—
47C22×C5×Dic3D411yescharacteristic, complemented
48C2×C5×Dic6D411yescharacteristic, complemented
49C3×C5×C22⋊C4—21no—
50C3×C5×C22⋊C4—21no—
51C3×C5×C22⋊C4—41no—
52C3×C5×C22⋊C4D411yescharacteristic
53C5×#217—21no—
54C5×#217—21no—
55C5×#217—21no—
56C5×#217—21no—
57C5×#217—21no—
58C5×#217—21no—
59C5×#217—41no—
60C5×#217D411yescharacteristic, complemented
61C5×#216—41no—
62C5×#214—41no—
63C5×C4⋊Dic3—21no—
64C5×Dic3⋊C4—21no—
65C5×Dic3⋊C4—21no—
66C5×Dic3⋊C4—21no—
67C5×Dic3⋊C4—41no—
68C4×C5×Dic3—21no—
69C4×C5×Dic3D411yescharacteristic, complemented
70C5×#208—21no—
71#5587C2×C511yescharacteristic, complemented
72C3×B(4,𝔽2)C2×C511yescharacteristic
73#5243C2×C511yescharacteristic, complemented
74#5242C2×C511yescharacteristic, complemented
75#4536C2×C511yescharacteristic, complemented
76#4473C2×C511yescharacteristic, complemented
77#4472C2×C511yescharacteristic, complemented
78C5×H(5,𝔽2)C2×D311yescharacteristic
79C5×#137—61no—
80C5×#126—61no—
81C5×#124—31no—
82C5×#123—31no—
83C5×#123—61no—
84C5×#123—61no—
85C5×C22⋊Q8—61no—
86C5×#120—21no—
87C5×#120C2×D311yescharacteristic
88C4×C5×D4—61no—
89C5×#117—61no—
90C2×C5×C22⋊C4—61no—
91C5×#104—61no—
92C5×#100—31no—
93C5×#99—21no—
94C5×#99—31no—
95C5×#99—61no—
96C5×#99C2×D311yescharacteristic
97#2197—31noSylow
98C23×C3×C5—21no—
99C23×C3×C5—21no—
100C23×C3×C5—21no—
101C23×C3×C5—41no—
102C23×C3×C5—42no—
103C23×C3×C5C2×D411yescharacteristic
104C23×C3×C5C2×D411yescharacteristic
105C3×C5×Q8—21no—
106C3×C5×D4—21no—
107C3×C5×D4—41no—
108C3×C5×D4—42no—
109C3×C5×D4—41no—
110C3×C5×D4—41no—
111C3×C5×D4—41no—
112C3×C5×D4—82no—
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×C5C2×D411yescharacteristic
119C2×C5×Dic3—21no—
120C2×C5×Dic3—21no—
121C2×C5×Dic3—21no—
122C2×C5×Dic3—21no—
123C2×C5×Dic3—42no—
124C2×C5×Dic3—41no—
125C2×C5×Dic3—41no—
126C2×C5×Dic3—81no—
127C5×Dic6—41no—
128C5×C3⋊C8—41no—
129C3×H(5,𝔽2)C22×C511yescharacteristic
130C3×#120—21no—
131C3×#120C22×C511yescharacteristic
132C2×#217—21no—
133#951—21no—
134#936C22×C511yescharacteristic
135#935—21no—
136#935—21no—
137#935C22×C511yescharacteristic
138D4×Dic3—21no—
139#879—21no—
140#878—21no—
141#877—21no—
142C3×#99—21no—
143C3×#99C22×C511yescharacteristic
144#837—21no—
145#834—21no—
146#834C22×C511yescharacteristic
147#833C22×C511yescharacteristic
148C24×C5#6711yescharacteristic
149C5×#41—21no—
150C5×#41—41no—
151C2×C5×Q8—31no—
152C2×C5×D4—21no—
153C2×C5×D4—21no—
154C2×C5×D4—21no—
155C2×C5×D4—41no—
156C2×C5×D4—41no—
157C2×C5×D4C22×D311yescharacteristic
158C22×C4×C5—31no—
159C22×C4×C5—61no—
160C22×C4×C5—61no—
161C5×Q16—121no—
162C5×SD16—121no—
163C5×M16—61no—
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—41no—
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#6711yescharacteristic
181C42×C5—31no—
182C42×C5—61no—
183#551—31no—
184#458—31no—
185B(4,𝔽2)C5×D311yescharacteristic
186#455—31no—
187#354—31no—
188#352—31no—
189AΣL(1,𝔽16)—31no—
190C22×C3×C5—22no—
191C22×C3×C5—21no—
192C22×C3×C5—21no—
193C22×C3×C5—41no—
194C22×C3×C5—41no—
195C22×C3×C5—41no—
196C22×C3×C5—42no—
197C22×C3×C5—41no—
198C22×C3×C5—41no—
199C22×C3×C5—81no—
200C22×C3×C5—81no—
201C22×C3×C5—81no—
202C22×C3×C5#12011yescharacteristic
203C3×C4×C5—21no—
204C3×C4×C5—41no—
205C3×C4×C5—41no—
206C3×C4×C5—81no—
207C5×Dic3—41no—
208C5×Dic3—41no—
209C5×Dic3—41no—
210C5×Dic3—41no—
211C5×Dic3—81no—
212C24×C3C5×D411yescharacteristic
213C3×#41—21no—
214C3×#41—41no—
215C2×C3×D4—21no—
216C2×C3×D4—21no—
217C2×C3×D4—21no—
218C2×C3×D4—41no—
219C2×C3×D4—41no—
220C2×C3×D4C23×C511yescharacteristic
221C22×Dic3—21no—
222C22×Dic3—21no—
223C22×Dic3C5×D411yescharacteristic, complemented
224C2×Dic6C5×D411yescharacteristic, complemented
225C3×C22⋊C4—21no—
226C3×C22⋊C4—21no—
227C3×C22⋊C4—41no—
228C3×C22⋊C4C5×D411yescharacteristic
229#217—21no—
230#217—21no—
231#217—21no—
232#217—21no—
233#217—21no—
234#217—21no—
235#217—41no—
236#217C5×D411yescharacteristic, complemented
237#216—41no—
238#214—41no—
239C4⋊Dic3—21no—
240Dic3⋊C4—21no—
241Dic3⋊C4—21no—
242Dic3⋊C4—21no—
243Dic3⋊C4—41no—
244C4×Dic3—21no—
245C4×Dic3C5×D411yescharacteristic, complemented
246#208—21no—
247C23×C5—21no—
248C23×C5—21no—
249C23×C5—21no—
250C23×C5—41no—
251C23×C5—42no—
252C23×C5D3×D411yescharacteristic
253C23×C5C2×#6711yescharacteristic
254C5×Q8—21no—
255C5×Q8—121no—
256C5×D4—21no—
257C5×D4—41no—
258C5×D4—42no—
259C5×D4—41no—
260C5×D4—41no—
261C5×D4—41no—
262C5×D4—82no—
263C2×C4×C5—21no—
264C2×C4×C5—21no—
265C2×C4×C5—21no—
266C2×C4×C5—41no—
267C2×C4×C5—41no—
268C2×C4×C5—61no—
269C2×C4×C5—61no—
270C2×C4×C5—61no—
271C2×C4×C5—61no—
272C2×C4×C5—122no—
273C2×C4×C5—121no—
274C2×C4×C5—121no—
275C2×C4×C5—241no—
276C2×C4×C5D3×D411yescharacteristic
277C5×C8—121no—
278H(5,𝔽2)C2×C5×D311yescharacteristic
279#137—61no—
280#126—61no—
281#124—31no—
282#123—31no—
283#123—61no—
284#123—61no—
285C22⋊Q8—61no—
286#120—21no—
287#120C2×C5×D311yescharacteristic
288C4×D4—61no—
289#117—61no—
290C2×C22⋊C4—61no—
291#104—61no—
292#100—31no—
293#99—21no—
294#99—31no—
295#99—61no—
296#99C2×C5×D311yescharacteristic
297C2×C3×C5—21no—
298C2×C3×C5—41no—
299C2×C3×C5—41no—
300C2×C3×C5—42no—
301C2×C3×C5—81no—
302C2×C3×C5B(4,𝔽2)11yescharacteristic
303C23×C3—21no—
304C23×C3—21no—
305C23×C3—21no—
306C23×C3—41no—
307C23×C3—42no—
308C23×C3C2×C5×D411yescharacteristic
309C23×C3C2×C5×D411yescharacteristic
310C3×Q8—21no—
311C3×D4—21no—
312C3×D4—41no—
313C3×D4—42no—
314C3×D4—41no—
315C3×D4—41no—
316C3×D4—41no—
317C3×D4—82no—
318C2×C3×C4—21no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—41no—
322C2×C3×C4—41no—
323C2×C3×C4C2×C5×D411yescharacteristic
324C2×Dic3—21no—
325C2×Dic3—21no—
326C2×Dic3—21no—
327C2×Dic3—21no—
328C2×Dic3—42no—
329C2×Dic3—41no—
330C2×Dic3—41no—
331C2×Dic3—81no—
332Dic6—41no—
333C3⋊C8—41no—
334C22×C5—22no—
335C22×C5—21no—
336C22×C5—21no—
337C22×C5—41no—
338C22×C5—41no—
339C22×C5—41no—
340C22×C5—42no—
341C22×C5—41no—
342C22×C5—41no—
343C22×C5—81no—
344C22×C5—81no—
345C22×C5—81no—
346C22×C5#93711yescharacteristic
347C4×C5—21no—
348C4×C5—41no—
349C4×C5—41no—
350C4×C5—81no—
351C4×C5—121no—
352C4×C5—121no—
353C4×C5—121no—
354C4×C5—121no—
355C4×C5—241no—
356C24C5×#6711yescharacteristic
357#41—21no—
358#41—41no—
359C2×Q8—31no—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—21no—
363C2×D4—41no—
364C2×D4—41no—
365C2×D4C22×C5×D311yescharacteristic
366C22×C4—31no—
367C22×C4—61no—
368C22×C4—61no—
369Q16—121no—
370SD16—121no—
371M16—61no—
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—41no—
381C22⋊C4—61no—
382C22⋊C4—61no—
383C22⋊C4—61no—
384C22⋊C4—61no—
385C22⋊C4—61no—
386C22⋊C4—61no—
387C22⋊C4—121no—
388C22⋊C4C5×#6711yescharacteristic
389C42—31no—
390C42—61no—
391C3×C5#219711yescharacteristic, complemented, Hall
392C22×C3—22no—
393C22×C3—21no—
394C22×C3—21no—
395C22×C3—41no—
396C22×C3—41no—
397C22×C3—41no—
398C22×C3—42no—
399C22×C3—41no—
400C22×C3—41no—
401C22×C3—81no—
402C22×C3—81no—
403C22×C3—81no—
404C22×C3C5×#12011yescharacteristic
405C3×C4—21no—
406C3×C4—41no—
407C3×C4—41no—
408C3×C4—81no—
409Dic3—41no—
410Dic3—41no—
411Dic3—41no—
412Dic3—41no—
413Dic3—81no—
414C2×C5—21no—
415C2×C5—41no—
416C2×C5—41no—
417C2×C5—42no—
418C2×C5—81no—
419C2×C5#503211yescharacteristic
420C23—21no—
421C23—21no—
422C23—21no—
423C23—41no—
424C23—42no—
425C23C5×D3×D411yescharacteristic
426C23C2×C5×#6711yescharacteristic
427Q8—21no—
428Q8—121no—
429D4—21no—
430D4—41no—
431D4—41no—
432D4—41no—
433D4—42no—
434D4—41no—
435D4—82no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—41no—
440C2×C4—41no—
441C2×C4—61no—
442C2×C4—61no—
443C2×C4—61no—
444C2×C4—61no—
445C2×C4—122no—
446C2×C4—121no—
447C2×C4—121no—
448C2×C4—241no—
449C2×C4C5×D3×D411yescharacteristic
450C8—121no—
451C2×C3—21no—
452C2×C3—41no—
453C2×C3—41no—
454C2×C3—42no—
455C2×C3—81no—
456C2×C3C5×B(4,𝔽2)11yescharacteristic
457C5#7099811yescharacteristic, direct product factor, Sylow
458C22—22no—
459C22—21no—
460C22—21no—
461C22—41no—
462C22—41no—
463C22—41no—
464C22—42no—
465C22—41no—
466C22—41no—
467C22—81no—
468C22—81no—
469C22—81no—
470C22C5×#93711yescharacteristic
471C4—21no—
472C4—41no—
473C4—41no—
474C4—81no—
475C4—121no—
476C4—121no—
477C4—121no—
478C4—121no—
479C4—241no—
480C3C5×#219711yescharacteristic, complemented, Sylow
481C2—21no—
482C2—41no—
483C2—41no—
484C2—42no—
485C2—81no—
486C2C5×#503211yescharacteristic
subgroupG/H
centerC2×C5#5032
commutator, cosocleC2×C3×D4C23×C5
FrattiniC2×D4C22×C5×D3
Fitting, prime radicalC3×C5×B(4,𝔽2)C2
socleC2×C3×C5B(4,𝔽2)
hypercenterC5×B(4,𝔽2)D3
hypocenterC3C5×#2197
LCSG ▷ C2×C3×D4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2×C5 ◁ C22×C5 ◁ C2×C5×D4 ◁ C5×B(4,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C5×#5587C2semidirect
3. C5×#5243C2semidirect
4. C5×#5242C2semidirect
5. C5×#4536C2semidirect
6. C5×#4473C2semidirect
7. C5×#4472C2semidirect
28. #70998C5direct (N × H)
47. C22×C5×Dic3D4semidirect
48. C2×C5×Dic6D4semidirect
60. C5×#217D4semidirect
69. C4×C5×Dic3D4semidirect
71. #5587C2×C5semidirect
73. #5243C2×C5semidirect
74. #5242C2×C5semidirect
75. #4536C2×C5semidirect
76. #4473C2×C5semidirect
77. #4472C2×C5semidirect
223. C22×Dic3C5×D4semidirect
224. C2×Dic6C5×D4semidirect
236. #217C5×D4semidirect
245. C4×Dic3C5×D4semidirect
391. C3×C5#2197semidirect
457. C5#70998direct (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×#55872
C3×C5×B(4,𝔽2)2
C5×#52432
C5×#52422
C5×#45362
C5×#44732
C5×#44722
C5×#21973
#709985