Finite Group Databasebeta

← C5×#72116 #14301160

Subgroups — C5×#72116 #14301160

Order 1920.

620 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#575C211yesmaximal, characteristic
2C5×#5819C211yesmaximal, characteristic, complemented
3C5×#5236C211yesmaximal, characteristic, complemented
4C5×#5234C211yesmaximal, characteristic, complemented
5C5×#5233C211yesmaximal, characteristic, complemented
6C5×#4559C211yesmaximal, characteristic, complemented
7C5×#4558C211yesmaximal, characteristic, complemented
8C5×#1881—31nomaximal, Hall
9C2×C3×C5×#41C2211yescharacteristic
10C2×C3×C5×#41C2211yescharacteristic
11C3×C5×#137—21no—
12C3×C5×#137C412yescomplemented
13C3×C5×AGL(1,ℤ/8)—21no—
14C3×C5×AGL(1,ℤ/8)C412yescomplemented
15C3×C5×#135—22no—
16C2×C3×C5×M16C2211yescharacteristic
17C5×#948—22no—
18C5×#948—22no—
19C5×#926C2211yescharacteristic
20C2×C5×#208—22no—
21C2×C5×#208C2211yescharacteristic
22C2×C5×#208C2211yescharacteristic
23C22×C5×C3⋊C8C2211yescharacteristic
24C5×#837—22no—
25C5×#835—21no—
26C5×#832—21no—
27C5×#830—22no—
28#72116C511yesmaximal, characteristic, direct product factor, Hall
29C5×#575D311yescharacteristic
30C5×#568—31no—
31C5×#421—31no—
32C5×#416—31no—
33C5×#408—31no—
34C5×#344—31no—
35C5×#341—31no—
36C3×C5×#41—21no—
37C3×C5×#41—21no—
38C3×C5×#41—21no—
39C3×C5×#41—21no—
40C3×C5×#41—21no—
41C3×C5×#41—21no—
42C3×C5×#41C2×C412yes—
43C3×C5×#41D412yes—
44C2×C3×C5×Q8C2×C411yescharacteristic
45C2×C3×C5×Q8D411yescharacteristic
46C2×C3×C5×D4—21no—
47C2×C3×C5×D4—21no—
48C2×C3×C5×D4C2×C411yescharacteristic
49C2×C3×C5×D4D411yescharacteristic
50C22×C3×C4×C5—21no—
51C22×C3×C4×C5—21no—
52C22×C3×C4×C5C2311yescharacteristic
53C3×C5×Q16—22no—
54C3×C5×SD16—22no—
55C3×C5×SD16—22no—
56C3×C5×D8—22no—
57C3×C5×M16—21no—
58C3×C5×M16C2×C412yes—
59C2×C3×C5×C8—21no—
60C5×#217—21no—
61C5×C4⋊Dic3—21no—
62C4×C5×Dic3—21no—
63C5×#208—24no—
64C5×#208—21no—
65C5×#208—21no—
66C5×#208—21no—
67C5×#208D412yes—
68C2×C5×C3⋊C8—24no—
69C2×C5×C3⋊C8—21no—
70C2×C5×C3⋊C8—21no—
71C2×C5×C3⋊C8—21no—
72C2×C5×C3⋊C8—21no—
73C2×C5×C3⋊C8D412yes—
74C3×#575C2×C511yescharacteristic
75#5819C2×C511yescharacteristic, complemented
76#5236C2×C511yescharacteristic, complemented
77#5234C2×C511yescharacteristic, complemented
78#5233C2×C511yescharacteristic, complemented
79#4559C2×C511yescharacteristic, complemented
80#4558C2×C511yescharacteristic, complemented
81C2×C5×#41C2×D311yescharacteristic
82C2×C5×#41C2×D311yescharacteristic
83C5×#137—21no—
84C5×#137Dic312yescomplemented
85C5×AGL(1,ℤ/8)—21no—
86C5×AGL(1,ℤ/8)Dic312yescomplemented
87C5×#135—22no—
88C5×#131—62no—
89C5×#131—62no—
90C2×C5×M16—31no—
91C2×C5×M16—31no—
92C2×C5×M16—62no—
93C2×C5×M16C2×D311yescharacteristic
94C22×C5×C8—31no—
95C5×#117—31no—
96C5×#104—62no—
97C5×#103—61no—
98C5×#102—61no—
99C5×C22⋊C8—62no—
100#1881—31noSylow
101C23×C3×C5—21no—
102C23×C3×C5—21no—
103C23×C3×C5#4111yescharacteristic
104C3×C5×Q8—21no—
105C3×C5×Q8—21no—
106C3×C5×Q8—21no—
107C3×C5×Q8C22⋊C412yes—
108C3×C5×D4—22no—
109C3×C5×D4—22no—
110C3×C5×D4—21no—
111C3×C5×D4—21no—
112C3×C5×D4—21no—
113C3×C5×D4—41no—
114C3×C5×D4—41no—
115C3×C5×D4C22⋊C412yes—
116C2×C3×C4×C5—21no—
117C2×C3×C4×C5—22no—
118C2×C3×C4×C5—22no—
119C2×C3×C4×C5—21no—
120C2×C3×C4×C5—41no—
121C2×C3×C4×C5—41no—
122C2×C3×C4×C5C22×C411yescharacteristic
123C2×C3×C4×C5C2×D411yescharacteristic
124C2×C3×C4×C5C2×D411yescharacteristic
125C2×C3×C4×C5C2×D411yescharacteristic
126C2×C3×C4×C5C2×D411yescharacteristic
127C2×C3×C4×C5#4111yescharacteristic
128C2×C5×Dic3—41no—
129C3×C5×C8—22no—
130C5×C3⋊C8—22no—
131C5×C3⋊C8—22no—
132C5×C3⋊C8—41no—
133C2×C3×#41C22×C511yescharacteristic
134C2×C3×#41C22×C511yescharacteristic
135C3×#137—21no—
136C3×#137C4×C512yescomplemented
137C3×AGL(1,ℤ/8)—21no—
138C3×AGL(1,ℤ/8)C4×C512yescomplemented
139C3×#135—22no—
140C2×C3×M16C22×C511yescharacteristic
141#948—22no—
142#948—22no—
143#926C22×C511yescharacteristic
144C2×#208—22no—
145C2×#208C22×C511yescharacteristic
146C2×#208C22×C511yescharacteristic
147C22×C3⋊C8C22×C511yescharacteristic
148#837—22no—
149#835—21no—
150#832—21no—
151#830—22no—
152C5×#41—21no—
153C5×#41—21no—
154C5×#41—21no—
155C5×#41—21no—
156C5×#41—21no—
157C5×#41—21no—
158C5×#41C2×Dic312yes—
159C5×#41#6712yes—
160C2×C5×Q8C2×Dic311yescharacteristic
161C2×C5×Q8#6711yescharacteristic
162C2×C5×D4—21no—
163C2×C5×D4—21no—
164C2×C5×D4C2×Dic311yescharacteristic
165C2×C5×D4#6711yescharacteristic
166C22×C4×C5—21no—
167C22×C4×C5—21no—
168C22×C4×C5C22×D311yescharacteristic
169C5×Q16—22no—
170C5×SD16—22no—
171C5×SD16—22no—
172C5×D8—22no—
173C5×M16—21no—
174C5×M16—32no—
175C5×M16—64no—
176C5×M16—61no—
177C5×M16—61no—
178C5×M16—61no—
179C5×M16C2×Dic312yes—
180C2×C5×C8—21no—
181C2×C5×C8—32no—
182C2×C5×C8—64no—
183C2×C5×C8—61no—
184C2×C5×C8—61no—
185C2×C5×C8—61no—
186C2×C5×C8—61no—
187C5×C4⋊C4—61no—
188C5×C22⋊C4—61no—
189C42×C5—61no—
190#575C5×D311yescharacteristic
191#568—31no—
192#421—31no—
193#416—31no—
194#408—31no—
195#344—31no—
196#341—31no—
197C22×C3×C5—22no—
198C22×C3×C5—41no—
199C22×C3×C5—42no—
200C22×C3×C5—41no—
201C22×C3×C5—81no—
202C22×C3×C5C4×D411yescharacteristic
203C22×C3×C5#12111yescharacteristic
204C22×C3×C5#12311yescharacteristic
205C3×C4×C5—22no—
206C3×C4×C5—41no—
207C3×C4×C5C2×C22⋊C411yescharacteristic
208C3×C4×C5C4×D411yescharacteristic
209C3×C4×C5#12011yescharacteristic
210C3×C4×C5#12111yescharacteristic
211C5×Dic3—81no—
212C3×#41—21no—
213C3×#41—21no—
214C3×#41—21no—
215C3×#41—21no—
216C3×#41—21no—
217C3×#41—21no—
218C3×#41C2×C4×C512yes—
219C3×#41C5×D412yes—
220C2×C3×Q8C2×C4×C511yescharacteristic
221C2×C3×Q8C5×D411yescharacteristic
222C2×C3×D4—21no—
223C2×C3×D4—21no—
224C2×C3×D4C2×C4×C511yescharacteristic
225C2×C3×D4C5×D411yescharacteristic
226C22×C3×C4—21no—
227C22×C3×C4—21no—
228C22×C3×C4C23×C511yescharacteristic
229C3×Q16—22no—
230C3×SD16—22no—
231C3×SD16—22no—
232C3×D8—22no—
233C3×M16—21no—
234C3×M16C2×C4×C512yes—
235C2×C3×C8—21no—
236#217—21no—
237C4⋊Dic3—21no—
238C4×Dic3—21no—
239#208—24no—
240#208—21no—
241#208—21no—
242#208—21no—
243#208C5×D412yes—
244C2×C3⋊C8—24no—
245C2×C3⋊C8—21no—
246C2×C3⋊C8—21no—
247C2×C3⋊C8—21no—
248C2×C3⋊C8—21no—
249C2×C3⋊C8C5×D412yes—
250C23×C5—21no—
251C23×C5—21no—
252C23×C5#23711yescharacteristic
253C5×Q8—21no—
254C5×Q8—21no—
255C5×Q8—21no—
256C5×Q8#21712yes—
257C5×D4—22no—
258C5×D4—22no—
259C5×D4—21no—
260C5×D4—21no—
261C5×D4—21no—
262C5×D4—41no—
263C5×D4—41no—
264C5×D4#21712yes—
265C2×C4×C5—21no—
266C2×C4×C5—22no—
267C2×C4×C5—22no—
268C2×C4×C5—21no—
269C2×C4×C5—41no—
270C2×C4×C5—41no—
271C2×C4×C5—121no—
272C2×C4×C5D3×D411yescharacteristic
273C2×C4×C5D3×D411yescharacteristic
274C2×C4×C5#23711yescharacteristic
275C2×C4×C5C22×Dic311yescharacteristic
276C2×C4×C5C2×#6711yescharacteristic
277C2×C4×C5C2×#6711yescharacteristic
278C5×C8—22no—
279C5×C8—62no—
280C5×C8—62no—
281C5×C8—121no—
282C2×#41C2×C5×D311yescharacteristic
283C2×#41C2×C5×D311yescharacteristic
284#137—21no—
285#137C5×Dic312yescomplemented
286AGL(1,ℤ/8)—21no—
287AGL(1,ℤ/8)C5×Dic312yescomplemented
288#135—22no—
289#131—62no—
290#131—62no—
291C2×M16—31no—
292C2×M16—31no—
293C2×M16—62no—
294C2×M16C2×C5×D311yescharacteristic
295C22×C8—31no—
296#117—31no—
297#104—62no—
298#103—61no—
299#102—61no—
300C22⋊C8—62no—
301C2×C3×C5—21no—
302C2×C3×C5—21no—
303C2×C3×C5—21no—
304C2×C3×C5—42no—
305C2×C3×C5—81no—
306C2×C3×C5#38611yescharacteristic
307C23×C3—21no—
308C23×C3—21no—
309C23×C3C5×#4111yescharacteristic
310C3×Q8—21no—
311C3×Q8—21no—
312C3×Q8—21no—
313C3×Q8C5×C22⋊C412yes—
314C3×D4—22no—
315C3×D4—22no—
316C3×D4—21no—
317C3×D4—21no—
318C3×D4—21no—
319C3×D4—41no—
320C3×D4—41no—
321C3×D4C5×C22⋊C412yes—
322C2×C3×C4—21no—
323C2×C3×C4—22no—
324C2×C3×C4—22no—
325C2×C3×C4—21no—
326C2×C3×C4—41no—
327C2×C3×C4—41no—
328C2×C3×C4C22×C4×C511yescharacteristic
329C2×C3×C4C2×C5×D411yescharacteristic
330C2×C3×C4C2×C5×D411yescharacteristic
331C2×C3×C4C2×C5×D411yescharacteristic
332C2×C3×C4C2×C5×D411yescharacteristic
333C2×C3×C4C5×#4111yescharacteristic
334C2×Dic3—41no—
335C3×C8—22no—
336C3⋊C8—22no—
337C3⋊C8—22no—
338C3⋊C8—41no—
339C22×C5—22no—
340C22×C5—41no—
341C22×C5—42no—
342C22×C5—41no—
343C22×C5—81no—
344C22×C5D4×Dic311yescharacteristic
345C22×C5#93511yescharacteristic
346C22×C5#93911yescharacteristic
347C4×C5—22no—
348C4×C5—41no—
349C4×C5—241no—
350C4×C5D4×Dic311yescharacteristic
351C4×C5#93711yescharacteristic
352C4×C5#93911yescharacteristic
353C4×C5C2×#21711yescharacteristic
354#41—21no—
355#41—21no—
356#41—21no—
357#41—21no—
358#41—21no—
359#41—21no—
360#41C2×C5×Dic312yes—
361#41C5×#6712yes—
362C2×Q8C2×C5×Dic311yescharacteristic
363C2×Q8C5×#6711yescharacteristic
364C2×D4—21no—
365C2×D4—21no—
366C2×D4C2×C5×Dic311yescharacteristic
367C2×D4C5×#6711yescharacteristic
368C22×C4—21no—
369C22×C4—21no—
370C22×C4C22×C5×D311yescharacteristic
371Q16—22no—
372SD16—22no—
373SD16—22no—
374D8—22no—
375M16—21no—
376M16—32no—
377M16—64no—
378M16—61no—
379M16—61no—
380M16—61no—
381M16C2×C5×Dic312yes—
382C2×C8—21no—
383C2×C8—32no—
384C2×C8—64no—
385C2×C8—61no—
386C2×C8—61no—
387C2×C8—61no—
388C2×C8—61no—
389C4⋊C4—61no—
390C22⋊C4—61no—
391C42—61no—
392C3×C5#188111yescharacteristic, complemented, Hall
393C22×C3—22no—
394C22×C3—41no—
395C22×C3—42no—
396C22×C3—41no—
397C22×C3—81no—
398C22×C3C4×C5×D411yescharacteristic
399C22×C3C5×#12111yescharacteristic
400C22×C3C5×#12311yescharacteristic
401C3×C4—22no—
402C3×C4—41no—
403C3×C4C2×C5×C22⋊C411yescharacteristic
404C3×C4C4×C5×D411yescharacteristic
405C3×C4C5×#12011yescharacteristic
406C3×C4C5×#12111yescharacteristic
407Dic3—81no—
408C2×C5—21no—
409C2×C5—21no—
410C2×C5—21no—
411C2×C5—42no—
412C2×C5—81no—
413C2×C5#522011yescharacteristic
414C23—21no—
415C23—21no—
416C23C5×#23711yescharacteristic
417Q8—21no—
418Q8—21no—
419Q8—21no—
420Q8C5×#21712yes—
421D4—22no—
422D4—22no—
423D4—21no—
424D4—21no—
425D4—21no—
426D4—41no—
427D4—41no—
428D4C5×#21712yes—
429C2×C4—21no—
430C2×C4—22no—
431C2×C4—22no—
432C2×C4—21no—
433C2×C4—41no—
434C2×C4—41no—
435C2×C4—121no—
436C2×C4C5×D3×D411yescharacteristic
437C2×C4C5×D3×D411yescharacteristic
438C2×C4C5×#23711yescharacteristic
439C2×C4C22×C5×Dic311yescharacteristic
440C2×C4C2×C5×#6711yescharacteristic
441C2×C4C2×C5×#6711yescharacteristic
442C8—22no—
443C8—62no—
444C8—62no—
445C8—121no—
446C2×C3—21no—
447C2×C3—21no—
448C2×C3—21no—
449C2×C3—42no—
450C2×C3—81no—
451C2×C3C5×#38611yescharacteristic
452C5#7211611yescharacteristic, direct product factor, Sylow
453C22—22no—
454C22—41no—
455C22—42no—
456C22—41no—
457C22—81no—
458C22C5×D4×Dic311yescharacteristic
459C22C5×#93511yescharacteristic
460C22C5×#93911yescharacteristic
461C4—22no—
462C4—41no—
463C4—241no—
464C4C5×D4×Dic311yescharacteristic
465C4C5×#93711yescharacteristic
466C4C5×#93911yescharacteristic
467C4C2×C5×#21711yescharacteristic
468C3C5×#188111yescharacteristic, complemented, Sylow
469C2—21no—
470C2—21no—
471C2—21no—
472C2—42no—
473C2—81no—
474C2C5×#522011yescharacteristic
subgroupG/H
centerC4×C5#937
commutatorC2×C3×C4C22×C4×C5
FrattiniC22×C4C22×C5×D3
Fitting, prime radicalC3×C5×#575C2
socleC2×C3×C5#386
cosocleC22×C3×C4C23×C5
hypercenterC5×#575D3
hypocenterC3C5×#1881
LCSG ▷ C2×C3×C4 ▷ C2×C3 ▷ C3
UCSC1 ◁ C4×C5 ◁ C22×C4×C5 ◁ C5×#575

Complements

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

NH ≅ G/Ntype
2. C5×#5819C2semidirect
3. C5×#5236C2semidirect
4. C5×#5234C2semidirect
5. C5×#5233C2semidirect
6. C5×#4559C2semidirect
7. C5×#4558C2semidirect
12. C3×C5×#137C4semidirect
14. C3×C5×AGL(1,ℤ/8)C4semidirect
28. #72116C5direct (N × H)
75. #5819C2×C5semidirect
76. #5236C2×C5semidirect
77. #5234C2×C5semidirect
78. #5233C2×C5semidirect
79. #4559C2×C5semidirect
80. #4558C2×C5semidirect
84. C5×#137Dic3semidirect
86. C5×AGL(1,ℤ/8)Dic3semidirect
136. C3×#137C4×C5semidirect
138. C3×AGL(1,ℤ/8)C4×C5semidirect
285. #137C5×Dic3semidirect
287. AGL(1,ℤ/8)C5×Dic3semidirect
392. C3×C5#1881semidirect
452. C5#72116direct (N × H)
468. C3C5×#1881semidirect

Maximal subgroups

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

subgroupindex
C3×C5×#5752
C5×#58192
C5×#52362
C5×#52342
C5×#52332
C5×#45592
C5×#45582
C5×#18813
#721165