Finite Group Databasebeta

← C3×#10604459 #14313210

Subgroups — C3×#10604459 #14313210

Order 1920.

560 conjugacy classes of subgroups; only the 104 normal subgroups are shown.

normal subgroups only

This group has 560 conjugacy classes of subgroups — too many to draw. Only its 104 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
1C3×#65923C211yesmaximal, characteristic
2C3×D5⋊#100C211yesmaximal, characteristic, complemented
3C3×D5⋊#99C211yesmaximal, characteristic, complemented
4C3×#64934C212yesmaximal, complemented
5C3×#64865C212yesmaximal, complemented
6#10604459C311yesmaximal, characteristic, direct product factor, Hall
7C22×C3×C22⋊D5C411yescharacteristic, complemented
8C2×C3×D4×D5C2211yescharacteristic
9C2×C3×C22⋊AGL(1,𝔽5)—21no—
10C3×D5⋊M16—21no—
11C3×C5×#120C411yescharacteristic, complemented
12C3×#4099C2211yescharacteristic
13C3×#4084—21no—
14C3×#4083—21no—
15C3×#4081C411yescharacteristic, complemented
16C3×#4032C2211yescharacteristic
17C3×#4030—21no—
18C3×#4029—21no—
19C3×D5×C22⋊C4C411yescharacteristic, complemented
20C3×#4013—21no—
21C3×#4013C2212yes—
22C3×#4011C2212yes—
23C3×#3999—21no—
24C3×#2121—51nomaximal, Hall
25#65923C2×C311yescharacteristic
26D5⋊#100C2×C311yescharacteristic, complemented
27D5⋊#99C2×C311yescharacteristic, complemented
28#64934C2×C312yescomplemented
29#64865C2×C312yescomplemented
30C24×C3×C5C2×C411yescharacteristic, complemented
31C23×C3×D5C2×C411yescharacteristic
32C23×C3×D5D411yescharacteristic
33C22×C3×AGL(1,𝔽5)—41no—
34C2×C3×C5×D4—21no—
35C2×C3×C5×D4C2×C411yescharacteristic
36C2×C3×C22⋊D5—21no—
37C2×C3×C22⋊D5—21no—
38C2×C3×C22⋊D5—21no—
39C2×C3×C22⋊D5—41no—
40C2×C3×C22⋊D5—41no—
41C2×C3×C22⋊D5D411yescharacteristic
42C2×C3×C22⋊D5C2311yescharacteristic
43C22×C3×Dic5C2×C411yescharacteristic, complemented
44C3×D4×D5—41no—
45C3×D4×D5—41no—
46C2×C3×D20D411yescharacteristic
47C2×C3×C4×D5—21no—
48C2×C3×C4×D5D411yescharacteristic
49C3×C22⋊AGL(1,𝔽5)—22no—
50C3×C22⋊AGL(1,𝔽5)—41no—
51C3×#704—22no—
52C3×#700—41no—
53C3×D5⋊C8—41no—
54C3×C5×C22⋊C4—21no—
55C3×C5×C22⋊C4C2×C411yescharacteristic, complemented
56C3×C22⋊Dic5—21no—
57C3×C22⋊Dic5—21no—
58C3×C22⋊Dic5C2×C411yescharacteristic, complemented
59C3×#685—21no—
60C3×#685—21no—
61C3×C4⋊Dic5—21no—
62C3×#683—21no—
63C3×#534—51no—
64C2×C3×#100—51no—
65C2×C3×#99—51no—
66C3×#352—52no—
67C3×AΣL(1,𝔽16)—52no—
68C22×C22⋊D5C3×C411yescharacteristic, complemented
69C2×D4×D5C22×C311yescharacteristic
70C2×C22⋊AGL(1,𝔽5)—21no—
71D5⋊M16—21no—
72C5×#120C3×C411yescharacteristic, complemented
73#4099C22×C311yescharacteristic
74#4084—21no—
75#4083—21no—
76#4081C3×C411yescharacteristic, complemented
77#4032C22×C311yescharacteristic
78#4030—21no—
79#4029—21no—
80D5×C22⋊C4C3×C411yescharacteristic, complemented
81#4013—21no—
82#4013C22×C312yes—
83#4011C22×C312yes—
84#3999—21no—
85#2121—51noSylow
86C23×C3×C5—21no—
87C23×C3×C5—41no—
88C23×C3×C5—81no—
89C23×C3×C5C22⋊C411yescharacteristic
90C23×C3×C5C22×C411yescharacteristic
91C22×C3×D5—21no—
92C22×C3×D5—21no—
93C22×C3×D5—21no—
94C22×C3×D5—21no—
95C22×C3×D5—81no—
96C22×C3×D5—81no—
97C22×C3×D5C22⋊C411yescharacteristic
98C22×C3×D5C22⋊C411yescharacteristic
99C22×C3×D5C2×D411yescharacteristic
100C2×C3×AGL(1,𝔽5)—42no—
101C2×C3×AGL(1,𝔽5)—82no—
102C3×C5×D4—41no—
103C3×C5×D4—81no—
104C2×C3×C4×C5—21no—
105C2×C3×C4×C5C22⋊C411yescharacteristic
106C3×C22⋊D5—41no—
107C3×C22⋊D5—41no—
108C3×C22⋊D5—42no—
109C3×C22⋊D5—41no—
110C3×C22⋊D5—41no—
111C3×C22⋊D5—81no—
112C2×C3×Dic5—21no—
113C2×C3×Dic5—21no—
114C2×C3×Dic5—41no—
115C2×C3×Dic5C2×D411yescharacteristic
116C3×D20—41no—
117C3×C4×D5—21no—
118C3×C4×D5—21no—
119C3×C4×D5—81no—
120C3×#170—42no—
121C22×C3×D4—51no—
122C22×C3×D4—51no—
123C2×C3×M16—101no—
124C3×#123—51no—
125C3×#123—51no—
126C3×#123—101no—
127C3×#121—101no—
128C3×#121—101no—
129C3×#120—51no—
130C3×#120—101no—
131C3×#120AGL(1,𝔽5)11yescharacteristic, complemented
132C2×C3×C22⋊C4—51no—
133C2×C3×C22⋊C4—101no—
134C3×#100—52no—
135C3×#100—101no—
136C3×#99—52no—
137C3×#99—101no—
138C24×C5C2×C3×C411yescharacteristic, complemented
139C23×D5C2×C3×C411yescharacteristic
140C23×D5C3×D411yescharacteristic
141C22×AGL(1,𝔽5)—41no—
142C2×C5×D4—21no—
143C2×C5×D4C2×C3×C411yescharacteristic
144C2×C22⋊D5—21no—
145C2×C22⋊D5—21no—
146C2×C22⋊D5—21no—
147C2×C22⋊D5—41no—
148C2×C22⋊D5—41no—
149C2×C22⋊D5C3×D411yescharacteristic
150C2×C22⋊D5C23×C311yescharacteristic
151C22×Dic5C2×C3×C411yescharacteristic, complemented
152D4×D5—41no—
153D4×D5—41no—
154C2×D20C3×D411yescharacteristic
155C2×C4×D5—21no—
156C2×C4×D5C3×D411yescharacteristic
157C22⋊AGL(1,𝔽5)—22no—
158C22⋊AGL(1,𝔽5)—41no—
159#704—22no—
160#700—41no—
161D5⋊C8—41no—
162C5×C22⋊C4—21no—
163C5×C22⋊C4C2×C3×C411yescharacteristic, complemented
164C22⋊Dic5—21no—
165C22⋊Dic5—21no—
166C22⋊Dic5C2×C3×C411yescharacteristic, complemented
167#685—21no—
168#685—21no—
169C4⋊Dic5—21no—
170#683—21no—
171#534—51no—
172C2×#100—51no—
173C2×#99—51no—
174#352—52no—
175AΣL(1,𝔽16)—52no—
176C22×C3×C5—21no—
177C22×C3×C5—21no—
178C22×C3×C5—41no—
179C22×C3×C5—41no—
180C22×C3×C5—41no—
181C22×C3×C5—41no—
182C22×C3×C5—41no—
183C22×C3×C5—82no—
184C22×C3×C5C2×C22⋊C411yescharacteristic
185C2×C3×D5—21no—
186C2×C3×D5—21no—
187C2×C3×D5—41no—
188C2×C3×D5—42no—
189C2×C3×D5—41no—
190C2×C3×D5—41no—
191C2×C3×D5—81no—
192C2×C3×D5—81no—
193C2×C3×D5—81no—
194C2×C3×D5#9911yescharacteristic
195C2×C3×D5#9911yescharacteristic
196C3×AGL(1,𝔽5)—82no—
197C3×C4×C5—21no—
198C3×C4×C5—41no—
199C3×Dic5—21no—
200C3×Dic5—41no—
201C3×Dic5—41no—
202C24×C3—51no—
203C24×C3—51no—
204C24×C3C2×AGL(1,𝔽5)11yescharacteristic, complemented
205C2×C3×D4—21no—
206C2×C3×D4—51no—
207C2×C3×D4—51no—
208C2×C3×D4—51no—
209C2×C3×D4—101no—
210C2×C3×D4—101no—
211C2×C3×D4—101no—
212C2×C3×D4—201no—
213C2×C3×D4—201no—
214C2×C3×D4—201no—
215C2×C3×D4—201no—
216C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
217C22×C3×C4—51no—
218C22×C3×C4—51no—
219C22×C3×C4—101no—
220C22×C3×C4—201no—
221C3×M16—102no—
222C3×M16—201no—
223C2×C3×C8—201no—
224C3×C4⋊C4—101no—
225C3×C4⋊C4—101no—
226C3×C22⋊C4—21no—
227C3×C22⋊C4—51no—
228C3×C22⋊C4—101no—
229C3×C22⋊C4—101no—
230C3×C22⋊C4—101no—
231C3×C22⋊C4—101no—
232C3×C22⋊C4—102no—
233C3×C22⋊C4—201no—
234C3×C22⋊C4C2×AGL(1,𝔽5)11yescharacteristic, complemented
235C23×C5—21no—
236C23×C5—41no—
237C23×C5—81no—
238C23×C5C3×C22⋊C411yescharacteristic
239C23×C5C22×C3×C411yescharacteristic
240C22×D5—21no—
241C22×D5—21no—
242C22×D5—21no—
243C22×D5—21no—
244C22×D5—81no—
245C22×D5—81no—
246C22×D5C3×C22⋊C411yescharacteristic
247C22×D5C3×C22⋊C411yescharacteristic
248C22×D5C2×C3×D411yescharacteristic
249C2×AGL(1,𝔽5)—42no—
250C2×AGL(1,𝔽5)—82no—
251C5×D4—41no—
252C5×D4—81no—
253C2×C4×C5—21no—
254C2×C4×C5C3×C22⋊C411yescharacteristic
255C22⋊D5—41no—
256C22⋊D5—41no—
257C22⋊D5—42no—
258C22⋊D5—41no—
259C22⋊D5—41no—
260C22⋊D5—81no—
261C2×Dic5—21no—
262C2×Dic5—21no—
263C2×Dic5—41no—
264C2×Dic5C2×C3×D411yescharacteristic
265D20—41no—
266C4×D5—21no—
267C4×D5—21no—
268C4×D5—81no—
269#170—42no—
270C22×D4—51no—
271C22×D4—51no—
272C2×M16—101no—
273#123—51no—
274#123—51no—
275#123—101no—
276#121—101no—
277#121—101no—
278#120—51no—
279#120—101no—
280#120C3×AGL(1,𝔽5)11yescharacteristic, complemented
281C2×C22⋊C4—51no—
282C2×C22⋊C4—101no—
283#100—52no—
284#100—101no—
285#99—52no—
286#99—101no—
287C2×C3×C5—21no—
288C2×C3×C5—41no—
289C2×C3×C5—41no—
290C2×C3×C5—81no—
291C2×C3×C5C2×#9911yescharacteristic
292C3×D5—21no—
293C3×D5—21no—
294C3×D5—41no—
295C3×D5—41no—
296C23×C3—21no—
297C23×C3—41no—
298C23×C3—51no—
299C23×C3—51no—
300C23×C3—51no—
301C23×C3—81no—
302C23×C3—101no—
303C23×C3—101no—
304C23×C3—101no—
305C23×C3—101no—
306C23×C3—401no—
307C23×C3—401no—
308C23×C3C22⋊AGL(1,𝔽5)11yescharacteristic
309C23×C3C22×AGL(1,𝔽5)11yescharacteristic
310C3×D4—41no—
311C3×D4—81no—
312C3×D4—201no—
313C3×D4—201no—
314C3×D4—201no—
315C3×D4—201no—
316C3×D4—201no—
317C3×D4—202no—
318C3×D4—401no—
319C2×C3×C4—21no—
320C2×C3×C4—51no—
321C2×C3×C4—101no—
322C2×C3×C4—101no—
323C2×C3×C4—101no—
324C2×C3×C4—101no—
325C2×C3×C4—201no—
326C2×C3×C4—202no—
327C2×C3×C4—401no—
328C2×C3×C4—402no—
329C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
330C3×C8—202no—
331C22×C5—21no—
332C22×C5—21no—
333C22×C5—41no—
334C22×C5—41no—
335C22×C5—41no—
336C22×C5—41no—
337C22×C5—41no—
338C22×C5—82no—
339C22×C5C2×C3×C22⋊C411yescharacteristic
340C2×D5—21no—
341C2×D5—21no—
342C2×D5—41no—
343C2×D5—42no—
344C2×D5—41no—
345C2×D5—41no—
346C2×D5—81no—
347C2×D5—81no—
348C2×D5—81no—
349C2×D5C3×#9911yescharacteristic
350C2×D5C3×#9911yescharacteristic
351AGL(1,𝔽5)—82no—
352C4×C5—21no—
353C4×C5—41no—
354Dic5—21no—
355Dic5—41no—
356Dic5—41no—
357C24—51no—
358C24—51no—
359C24C2×C3×AGL(1,𝔽5)11yescharacteristic, complemented
360C2×D4—21no—
361C2×D4—51no—
362C2×D4—51no—
363C2×D4—51no—
364C2×D4—101no—
365C2×D4—101no—
366C2×D4—101no—
367C2×D4—201no—
368C2×D4—201no—
369C2×D4—201no—
370C2×D4—201no—
371C2×D4C2×C3×AGL(1,𝔽5)11yescharacteristic
372C22×C4—51no—
373C22×C4—51no—
374C22×C4—101no—
375C22×C4—201no—
376M16—102no—
377M16—201no—
378C2×C8—201no—
379C4⋊C4—101no—
380C4⋊C4—101no—
381C22⋊C4—21no—
382C22⋊C4—51no—
383C22⋊C4—101no—
384C22⋊C4—101no—
385C22⋊C4—101no—
386C22⋊C4—101no—
387C22⋊C4—102no—
388C22⋊C4—201no—
389C22⋊C4C2×C3×AGL(1,𝔽5)11yescharacteristic, complemented
390C3×C5#212111yescharacteristic, complemented, Hall
391C22×C3—21no—
392C22×C3—21no—
393C22×C3—41no—
394C22×C3—41no—
395C22×C3—41no—
396C22×C3—41no—
397C22×C3—41no—
398C22×C3—51no—
399C22×C3—51no—
400C22×C3—82no—
401C22×C3—101no—
402C22×C3—101no—
403C22×C3—201no—
404C22×C3—201no—
405C22×C3—201no—
406C22×C3—202no—
407C22×C3—401no—
408C22×C3—401no—
409C22×C3—401no—
410C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
411C3×C4—21no—
412C3×C4—41no—
413C3×C4—101no—
414C3×C4—201no—
415C3×C4—201no—
416C3×C4—402no—
417C2×C5—21no—
418C2×C5—41no—
419C2×C5—41no—
420C2×C5—81no—
421C2×C5C2×C3×#9911yescharacteristic
422D5—21no—
423D5—21no—
424D5—41no—
425D5—41no—
426C23—21no—
427C23—41no—
428C23—51no—
429C23—51no—
430C23—51no—
431C23—81no—
432C23—101no—
433C23—101no—
434C23—101no—
435C23—101no—
436C23—401no—
437C23—401no—
438C23C3×C22⋊AGL(1,𝔽5)11yescharacteristic
439C23C22×C3×AGL(1,𝔽5)11yescharacteristic
440D4—41no—
441D4—81no—
442D4—201no—
443D4—201no—
444D4—201no—
445D4—201no—
446D4—201no—
447D4—202no—
448D4—401no—
449C2×C4—21no—
450C2×C4—51no—
451C2×C4—101no—
452C2×C4—101no—
453C2×C4—101no—
454C2×C4—101no—
455C2×C4—201no—
456C2×C4—202no—
457C2×C4—401no—
458C2×C4—402no—
459C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
460C8—202no—
461C2×C3—21no—
462C2×C3—41no—
463C2×C3—41no—
464C2×C3—81no—
465C2×C3—101no—
466C2×C3—101no—
467C2×C3—201no—
468C2×C3—201no—
469C2×C3D5⋊#9911yescharacteristic
470C5C3×#212111yescharacteristic, complemented, Sylow
471C22—21no—
472C22—21no—
473C22—41no—
474C22—41no—
475C22—41no—
476C22—41no—
477C22—41no—
478C22—51no—
479C22—51no—
480C22—82no—
481C22—101no—
482C22—101no—
483C22—201no—
484C22—201no—
485C22—201no—
486C22—202no—
487C22—401no—
488C22—401no—
489C22—401no—
490C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
491C4—21no—
492C4—41no—
493C4—101no—
494C4—201no—
495C4—201no—
496C4—402no—
497C3#1060445911yescharacteristic, direct product factor, Sylow
498C2—21no—
499C2—41no—
500C2—41no—
501C2—81no—
502C2—101no—
503C2—101no—
504C2—201no—
505C2—201no—
506C2C3×D5⋊#9911yescharacteristic
subgroupG/H
centerC2×C3D5⋊#99
commutatorC23×C5C22×C3×C4
FrattiniC23C22×C3×AGL(1,𝔽5)
FittingC3×C5×#120C4
socleC2×C3×C5C2×#99
prime radicalC3×#65923C2
cosocleC2×C22⋊D5C23×C3
hypercenterC3×#120AGL(1,𝔽5)
hypocenterC5C3×#2121
LCSG ▷ C23×C5 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×D4 ◁ C3×#120

Complements

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

NH ≅ G/Ntype
2. C3×D5⋊#100C2semidirect
3. C3×D5⋊#99C2semidirect
4. C3×#64934C2semidirect
5. C3×#64865C2semidirect
6. #10604459C3direct (N × H)
7. C22×C3×C22⋊D5C4semidirect
11. C3×C5×#120C4semidirect
15. C3×#4081C4semidirect
19. C3×D5×C22⋊C4C4semidirect
26. D5⋊#100C2×C3semidirect
27. D5⋊#99C2×C3semidirect
28. #64934C2×C3semidirect
29. #64865C2×C3semidirect
30. C24×C3×C5C2×C4semidirect
43. C22×C3×Dic5C2×C4semidirect
55. C3×C5×C22⋊C4C2×C4semidirect
58. C3×C22⋊Dic5C2×C4semidirect
68. C22×C22⋊D5C3×C4semidirect
72. C5×#120C3×C4semidirect
76. #4081C3×C4semidirect
80. D5×C22⋊C4C3×C4semidirect
131. C3×#120AGL(1,𝔽5)semidirect
138. C24×C5C2×C3×C4semidirect
151. C22×Dic5C2×C3×C4semidirect
163. C5×C22⋊C4C2×C3×C4semidirect
166. C22⋊Dic5C2×C3×C4semidirect
204. C24×C3C2×AGL(1,𝔽5)semidirect
234. C3×C22⋊C4C2×AGL(1,𝔽5)semidirect
280. #120C3×AGL(1,𝔽5)semidirect
359. C24C2×C3×AGL(1,𝔽5)semidirect
389. C22⋊C4C2×C3×AGL(1,𝔽5)semidirect
390. C3×C5#2121semidirect
470. C5C3×#2121semidirect
497. C3#10604459direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#659232
C3×D5⋊#1002
C3×D5⋊#992
C3×#649342
C3×#648652
#106044593
C3×#21215