Finite Group Databasebeta

← C5×#70997 #14316769

Subgroups — C5×#70997 #14316769

Order 1920.

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

normal subgroups only

This group has 540 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×#5594C211yesmaximal, characteristic, complemented
2C3×C5×B(4,𝔽2)C211yesmaximal, characteristic, complemented
3C5×#5244C211yesmaximal, characteristic, complemented
4C5×#5242C211yesmaximal, characteristic, complemented
5C5×#4536C211yesmaximal, characteristic, complemented
6C5×#4473C211yesmaximal, characteristic, complemented
7C5×#4471C211yesmaximal, characteristic, complemented
8C5×#2199—31nomaximal, Hall
9C3×C5×H(5,𝔽2)C2211yescharacteristic
10C3×C5×#120—21no—
11C3×C5×#120C2211yescharacteristic
12C5×#953—21no—
13C5×#951—21no—
14C5×#937—21no—
15C5×#937—21no—
16C5×#937C2211yescharacteristic
17C5×#936—21no—
18C5×#936C2211yescharacteristic
19C5×#885—21no—
20C5×#885—21no—
21C5×#885—21no—
22C3×C5×#99—21no—
23C3×C5×#99C2211yescharacteristic
24C5×#837—21no—
25C5×#834—21no—
26C5×#834C2211yescharacteristic
27C5×#833C2211yescharacteristic
28#70997C511yesmaximal, characteristic, direct product factor, Hall
29C5×H(3,𝔽4)—31no—
30C5×B(4,𝔽2)—31no—
31C5×B(4,𝔽2)D311yescharacteristic, complemented
32C5×#455—31no—
33C5×#354—31no—
34C5×AΣL(1,𝔽16)—31no—
35C5×AΣL(1,𝔽16)—31no—
36C24×C3×C5D411yescharacteristic, complemented
37C23×C5×D3D411yescharacteristic, complemented
38C3×C5×#41—21no—
39C3×C5×#41—41no—
40C2×C3×C5×D4—21no—
41C2×C3×C5×D4—21no—
42C2×C3×C5×D4—21no—
43C2×C3×C5×D4—41no—
44C2×C3×C5×D4—41no—
45C2×C3×C5×D4C2311yescharacteristic
46C2×C5×#67—21no—
47C2×C5×#67—21no—
48C2×C5×#67—21no—
49C2×C5×#67—41no—
50C2×C5×Dic6—21no—
51C2×C5×Dic6D411yescharacteristic, complemented
52C3×C5×C22⋊C4—21no—
53C3×C5×C22⋊C4—21no—
54C3×C5×C22⋊C4—41no—
55C3×C5×C22⋊C4D411yescharacteristic, complemented
56C5×#217—21no—
57C5×#217—21no—
58C5×#217—21no—
59C5×#217—21no—
60C5×#217—21no—
61C5×#217D411yescharacteristic, complemented
62C5×#216—41no—
63C5×#214—41no—
64C5×(C2×D3)⋊C4—21no—
65C5×(C2×D3)⋊C4—21no—
66C5×(C2×D3)⋊C4—21no—
67C5×(C2×D3)⋊C4—41no—
68C4×C5×Dic3—21no—
69C4×C5×Dic3D411yescharacteristic, complemented
70C5×#208—21no—
71#5594C2×C511yescharacteristic, complemented
72C3×B(4,𝔽2)C2×C511yescharacteristic, complemented
73#5244C2×C511yescharacteristic, complemented
74#5242C2×C511yescharacteristic, complemented
75#4536C2×C511yescharacteristic, complemented
76#4473C2×C511yescharacteristic, complemented
77#4471C2×C511yescharacteristic, complemented
78C5×H(5,𝔽2)C2×D311yescharacteristic
79C5×#137—61no—
80C5×#124—31no—
81C5×#124—61no—
82C5×#124—61no—
83C5×#124—61no—
84C5×#124—61no—
85C5×#120—21no—
86C5×#120—31no—
87C5×#120—61no—
88C5×#120—61no—
89C5×#120—61no—
90C5×#120C2×D311yescharacteristic
91C5×#104—61no—
92C5×#100—31no—
93C5×#99—21no—
94C5×#99—31no—
95C5×#99—61no—
96C5×#99C2×D311yescharacteristic
97#2199—31noSylow
98C23×C3×C5—21no—
99C23×C3×C5—21no—
100C23×C3×C5—21no—
101C23×C3×C5—41no—
102C23×C3×C5—81no—
103C23×C3×C5C2×D411yescharacteristic
104C23×C3×C5C2×D411yescharacteristic
105C22×C5×D3—21no—
106C22×C5×D3—41no—
107C22×C5×D3—81no—
108C3×C5×Q8—21no—
109C3×C5×D4—21no—
110C3×C5×D4—41no—
111C3×C5×D4—41no—
112C3×C5×D4—41no—
113C3×C5×D4—41no—
114C3×C5×D4—81no—
115C3×C5×D4—82no—
116C2×C3×C4×C5—21no—
117C2×C3×C4×C5—21no—
118C2×C3×C4×C5—21no—
119C2×C3×C4×C5—41no—
120C2×C3×C4×C5—41no—
121C2×C3×C4×C5C2×D411yescharacteristic
122C5×#67—81no—
123C5×#67—81no—
124C5×#67—81no—
125C5×#67—82no—
126C2×C5×Dic3—21no—
127C2×C5×Dic3—21no—
128C2×C5×Dic3—21no—
129C2×C5×Dic3—41no—
130C5×Dic6—41no—
131C5×Dic6—81no—
132C5×C3⋊C8—41no—
133C3×H(5,𝔽2)C22×C511yescharacteristic
134C3×#120—21no—
135C3×#120C22×C511yescharacteristic
136#953—21no—
137#951—21no—
138#937—21no—
139#937—21no—
140#937C22×C511yescharacteristic
141#936—21no—
142#936C22×C511yescharacteristic
143#885—21no—
144#885—21no—
145#885—21no—
146C3×#99—21no—
147C3×#99C22×C511yescharacteristic
148#837—21no—
149#834—21no—
150#834C22×C511yescharacteristic
151#833C22×C511yescharacteristic
152C24×C5—31no—
153C24×C5#6711yescharacteristic, complemented
154C5×#41—21no—
155C5×#41—41no—
156C2×C5×Q8—31no—
157C2×C5×Q8—61no—
158C2×C5×D4—21no—
159C2×C5×D4—21no—
160C2×C5×D4—21no—
161C2×C5×D4—41no—
162C2×C5×D4—41no—
163C2×C5×D4—61no—
164C2×C5×D4—61no—
165C2×C5×D4—61no—
166C2×C5×D4—121no—
167C2×C5×D4C22×D311yescharacteristic
168C5×Q16—121no—
169C5×SD16—121no—
170C5×M16—61no—
171C5×C22⋊C4—21no—
172C5×C22⋊C4—21no—
173C5×C22⋊C4—31no—
174C5×C22⋊C4—41no—
175C5×C22⋊C4—61no—
176C5×C22⋊C4—61no—
177C5×C22⋊C4—61no—
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—
186C42×C5—61no—
187H(3,𝔽4)—31no—
188B(4,𝔽2)—31no—
189B(4,𝔽2)C5×D311yescharacteristic, complemented
190#455—31no—
191#354—31no—
192AΣL(1,𝔽16)—31no—
193AΣL(1,𝔽16)—31no—
194C22×C3×C5—21no—
195C22×C3×C5—21no—
196C22×C3×C5—41no—
197C22×C3×C5—41no—
198C22×C3×C5—41no—
199C22×C3×C5—41no—
200C22×C3×C5—42no—
201C22×C3×C5—41no—
202C22×C3×C5—41no—
203C22×C3×C5—81no—
204C22×C3×C5—81no—
205C22×C3×C5—81no—
206C22×C3×C5#12011yescharacteristic
207C2×C5×D3—41no—
208C2×C5×D3—41no—
209C2×C5×D3—42no—
210C2×C5×D3—41no—
211C2×C5×D3—81no—
212C3×C4×C5—21no—
213C3×C4×C5—41no—
214C3×C4×C5—41no—
215C3×C4×C5—81no—
216C5×Dic3—41no—
217C5×Dic3—41no—
218C5×Dic3—41no—
219C5×Dic3—81no—
220C24×C3C5×D411yescharacteristic, complemented
221C23×D3C5×D411yescharacteristic, complemented
222C3×#41—21no—
223C3×#41—41no—
224C2×C3×D4—21no—
225C2×C3×D4—21no—
226C2×C3×D4—21no—
227C2×C3×D4—41no—
228C2×C3×D4—41no—
229C2×C3×D4C23×C511yescharacteristic
230C2×#67—21no—
231C2×#67—21no—
232C2×#67—21no—
233C2×#67—41no—
234C2×Dic6—21no—
235C2×Dic6C5×D411yescharacteristic, complemented
236C3×C22⋊C4—21no—
237C3×C22⋊C4—21no—
238C3×C22⋊C4—41no—
239C3×C22⋊C4C5×D411yescharacteristic, complemented
240#217—21no—
241#217—21no—
242#217—21no—
243#217—21no—
244#217—21no—
245#217C5×D411yescharacteristic, complemented
246#216—41no—
247#214—41no—
248(C2×D3)⋊C4—21no—
249(C2×D3)⋊C4—21no—
250(C2×D3)⋊C4—21no—
251(C2×D3)⋊C4—41no—
252C4×Dic3—21no—
253C4×Dic3C5×D411yescharacteristic, complemented
254#208—21no—
255C23×C5—21no—
256C23×C5—21no—
257C23×C5—21no—
258C23×C5—41no—
259C23×C5—61no—
260C23×C5—81no—
261C23×C5—121no—
262C23×C5—241no—
263C23×C5D3×D411yescharacteristic
264C23×C5C2×#6711yescharacteristic
265C5×Q8—21no—
266C5×Q8—121no—
267C5×Q8—241no—
268C5×D4—21no—
269C5×D4—41no—
270C5×D4—41no—
271C5×D4—41no—
272C5×D4—41no—
273C5×D4—81no—
274C5×D4—82no—
275C5×D4—241no—
276C5×D4—241no—
277C5×D4—241no—
278C5×D4—242no—
279C2×C4×C5—21no—
280C2×C4×C5—21no—
281C2×C4×C5—21no—
282C2×C4×C5—41no—
283C2×C4×C5—41no—
284C2×C4×C5—61no—
285C2×C4×C5—61no—
286C2×C4×C5—61no—
287C2×C4×C5—121no—
288C2×C4×C5D3×D411yescharacteristic
289C5×C8—121no—
290H(5,𝔽2)C2×C5×D311yescharacteristic
291#137—61no—
292#124—31no—
293#124—61no—
294#124—61no—
295#124—61no—
296#124—61no—
297#120—21no—
298#120—31no—
299#120—61no—
300#120—61no—
301#120—61no—
302#120C2×C5×D311yescharacteristic
303#104—61no—
304#100—31no—
305#99—21no—
306#99—31no—
307#99—61no—
308#99C2×C5×D311yescharacteristic
309C2×C3×C5—21no—
310C2×C3×C5—41no—
311C2×C3×C5—41no—
312C2×C3×C5—81no—
313C2×C3×C5—81no—
314C2×C3×C5B(4,𝔽2)11yescharacteristic
315C5×D3—81no—
316C23×C3—21no—
317C23×C3—21no—
318C23×C3—21no—
319C23×C3—41no—
320C23×C3—81no—
321C23×C3C2×C5×D411yescharacteristic
322C23×C3C2×C5×D411yescharacteristic
323C22×D3—21no—
324C22×D3—41no—
325C22×D3—81no—
326C3×Q8—21no—
327C3×D4—21no—
328C3×D4—41no—
329C3×D4—41no—
330C3×D4—41no—
331C3×D4—41no—
332C3×D4—81no—
333C3×D4—82no—
334C2×C3×C4—21no—
335C2×C3×C4—21no—
336C2×C3×C4—21no—
337C2×C3×C4—41no—
338C2×C3×C4—41no—
339C2×C3×C4C2×C5×D411yescharacteristic
340#67—81no—
341#67—81no—
342#67—81no—
343#67—82no—
344C2×Dic3—21no—
345C2×Dic3—21no—
346C2×Dic3—21no—
347C2×Dic3—41no—
348Dic6—41no—
349Dic6—81no—
350C3⋊C8—41no—
351C22×C5—21no—
352C22×C5—21no—
353C22×C5—41no—
354C22×C5—41no—
355C22×C5—41no—
356C22×C5—41no—
357C22×C5—42no—
358C22×C5—41no—
359C22×C5—41no—
360C22×C5—81no—
361C22×C5—81no—
362C22×C5—81no—
363C22×C5—121no—
364C22×C5—121no—
365C22×C5—122no—
366C22×C5—121no—
367C22×C5—241no—
368C22×C5#93711yescharacteristic
369C4×C5—21no—
370C4×C5—41no—
371C4×C5—41no—
372C4×C5—81no—
373C4×C5—121no—
374C4×C5—121no—
375C4×C5—121no—
376C4×C5—241no—
377C24—31no—
378C24C5×#6711yescharacteristic, complemented
379#41—21no—
380#41—41no—
381C2×Q8—31no—
382C2×Q8—61no—
383C2×D4—21no—
384C2×D4—21no—
385C2×D4—21no—
386C2×D4—41no—
387C2×D4—41no—
388C2×D4—61no—
389C2×D4—61no—
390C2×D4—61no—
391C2×D4—121no—
392C2×D4C22×C5×D311yescharacteristic
393Q16—121no—
394SD16—121no—
395M16—61no—
396C22⋊C4—21no—
397C22⋊C4—21no—
398C22⋊C4—31no—
399C22⋊C4—41no—
400C22⋊C4—61no—
401C22⋊C4—61no—
402C22⋊C4—61no—
403C22⋊C4—61no—
404C22⋊C4—61no—
405C22⋊C4—61no—
406C22⋊C4—61no—
407C22⋊C4—61no—
408C22⋊C4—121no—
409C22⋊C4C5×#6711yescharacteristic, complemented
410C42—31no—
411C42—61no—
412C3×C5#219911yescharacteristic, complemented, Hall
413C22×C3—21no—
414C22×C3—21no—
415C22×C3—41no—
416C22×C3—41no—
417C22×C3—41no—
418C22×C3—41no—
419C22×C3—42no—
420C22×C3—41no—
421C22×C3—41no—
422C22×C3—81no—
423C22×C3—81no—
424C22×C3—81no—
425C22×C3C5×#12011yescharacteristic
426C2×D3—41no—
427C2×D3—41no—
428C2×D3—42no—
429C2×D3—41no—
430C2×D3—81no—
431C3×C4—21no—
432C3×C4—41no—
433C3×C4—41no—
434C3×C4—81no—
435Dic3—41no—
436Dic3—41no—
437Dic3—41no—
438Dic3—81no—
439C2×C5—21no—
440C2×C5—41no—
441C2×C5—41no—
442C2×C5—81no—
443C2×C5—81no—
444C2×C5—241no—
445C2×C5#503211yescharacteristic
446C23—21no—
447C23—21no—
448C23—21no—
449C23—41no—
450C23—61no—
451C23—81no—
452C23—121no—
453C23—241no—
454C23C5×D3×D411yescharacteristic
455C23C2×C5×#6711yescharacteristic
456Q8—21no—
457Q8—121no—
458Q8—241no—
459D4—21no—
460D4—41no—
461D4—41no—
462D4—41no—
463D4—41no—
464D4—81no—
465D4—82no—
466D4—241no—
467D4—241no—
468D4—241no—
469D4—242no—
470C2×C4—21no—
471C2×C4—21no—
472C2×C4—21no—
473C2×C4—41no—
474C2×C4—41no—
475C2×C4—61no—
476C2×C4—61no—
477C2×C4—61no—
478C2×C4—121no—
479C2×C4C5×D3×D411yescharacteristic
480C8—121no—
481C2×C3—21no—
482C2×C3—41no—
483C2×C3—41no—
484C2×C3—81no—
485C2×C3—81no—
486C2×C3C5×B(4,𝔽2)11yescharacteristic
487D3—81no—
488C5#7099711yescharacteristic, direct product factor, Sylow
489C22—21no—
490C22—21no—
491C22—41no—
492C22—41no—
493C22—41no—
494C22—41no—
495C22—42no—
496C22—41no—
497C22—41no—
498C22—81no—
499C22—81no—
500C22—81no—
501C22—121no—
502C22—121no—
503C22—122no—
504C22—121no—
505C22—241no—
506C22C5×#93711yescharacteristic
507C4—21no—
508C4—41no—
509C4—41no—
510C4—81no—
511C4—121no—
512C4—121no—
513C4—121no—
514C4—241no—
515C3C5×#219911yescharacteristic, complemented, Sylow
516C2—21no—
517C2—41no—
518C2—41no—
519C2—81no—
520C2—81no—
521C2—241no—
522C2C5×#503211yescharacteristic
subgroupG/H
centerC2×C5#5032
commutator, cosocleC2×C3×D4C23×C5
FrattiniC2×D4C22×C5×D3
FittingC3×C5×B(4,𝔽2)C2
socleC2×C3×C5B(4,𝔽2)
hypercenterC5×B(4,𝔽2)D3
hypocenterC3C5×#2199
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×#5594C2semidirect
2. C3×C5×B(4,𝔽2)C2semidirect
3. C5×#5244C2semidirect
4. C5×#5242C2semidirect
5. C5×#4536C2semidirect
6. C5×#4473C2semidirect
7. C5×#4471C2semidirect
28. #70997C5direct (N × H)
31. C5×B(4,𝔽2)D3semidirect
36. C24×C3×C5D4semidirect
37. C23×C5×D3D4semidirect
51. C2×C5×Dic6D4semidirect
55. C3×C5×C22⋊C4D4semidirect
61. C5×#217D4semidirect
69. C4×C5×Dic3D4semidirect
71. #5594C2×C5semidirect
72. C3×B(4,𝔽2)C2×C5semidirect
73. #5244C2×C5semidirect
74. #5242C2×C5semidirect
75. #4536C2×C5semidirect
76. #4473C2×C5semidirect
77. #4471C2×C5semidirect
153. C24×C5#67semidirect
184. C5×C22⋊C4#67semidirect
189. B(4,𝔽2)C5×D3semidirect
220. C24×C3C5×D4semidirect
221. C23×D3C5×D4semidirect
235. C2×Dic6C5×D4semidirect
239. C3×C22⋊C4C5×D4semidirect
245. #217C5×D4semidirect
253. C4×Dic3C5×D4semidirect
378. C24C5×#67semidirect
409. C22⋊C4C5×#67semidirect
412. C3×C5#2199semidirect
488. C5#70997direct (N × H)
515. C3C5×#2199semidirect

Maximal subgroups

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

subgroupindex
C5×#55942
C3×C5×B(4,𝔽2)2
C5×#52442
C5×#52422
C5×#45362
C5×#44732
C5×#44712
C5×#21993
#709975