Finite Group Databasebeta

← C3×#10601793 #14316668

Subgroups — C3×#10601793 #14316668

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
1C3×#65993C211yesmaximal, characteristic, complemented
2C3×C5×#458C211yesmaximal, characteristic, complemented
3C3×#65533C211yesmaximal, characteristic
4C3×#65531C211yesmaximal, characteristic, complemented
5C3×#64759C211yesmaximal, characteristic, complemented
6C3×#64693C211yesmaximal, characteristic, complemented
7C3×#64691C211yesmaximal, characteristic, complemented
8#10601793C311yesmaximal, characteristic, direct product factor, Hall
9C3×C5×H(5,𝔽2)C2211yescharacteristic
10C3×C5×#123—21no—
11C3×C5×#123C2211yescharacteristic
12C3×#4097—21no—
13C3×#4083—21no—
14C3×#4083C2211yescharacteristic
15C3×#4082C2211yescharacteristic
16C3×C4×C22⋊D5—21no—
17C3×#4042—21no—
18C3×#4038—21no—
19C3×#4029—21no—
20C3×D5×C22⋊C4—21no—
21C3×#4025—21no—
22C3×C5×#99—21no—
23C3×C5×#99C2211yescharacteristic
24C3×#3969—21no—
25C3×#3966—21no—
26C3×#3966C2211yescharacteristic
27C3×#3965C2211yescharacteristic
28C3×#2197—51nomaximal, Hall
29#65993C2×C311yescharacteristic, complemented
30C5×#458C2×C311yescharacteristic, complemented
31#65533C2×C311yescharacteristic
32#65531C2×C311yescharacteristic, complemented
33#64759C2×C311yescharacteristic, complemented
34#64693C2×C311yescharacteristic, complemented
35#64691C2×C311yescharacteristic, complemented
36C23×C3×D5D411yescharacteristic
37C3×C5×#41—21no—
38C3×C5×#41—41no—
39C2×C3×C5×D4—21no—
40C2×C3×C5×D4—21no—
41C2×C3×C5×D4—41no—
42C2×C3×C5×D4C2311yescharacteristic
43C22×C3×C4×C5D411yescharacteristic, complemented
44C2×C3×C22⋊D5—21no—
45C2×C3×C22⋊D5—41no—
46C2×C3×C4×D5—21no—
47C2×C3×C4×D5—21no—
48C2×C3×Dic10D411yescharacteristic, complemented
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, complemented
55C3×C22⋊Dic5—21no—
56C3×C22⋊Dic5—21no—
57C3×C22⋊Dic5—21no—
58C3×C22⋊Dic5D411yescharacteristic
59C3×#689—41no—
60C3×#687—41no—
61C3×#685—21no—
62C3×#685—21no—
63C3×#685—21no—
64C3×#685—41no—
65C3×C4⋊Dic5—21no—
66C3×#683—21no—
67C3×#683—21no—
68C3×C4×Dic5—21no—
69C3×C4×Dic5D411yescharacteristic, complemented
70C3×#681—21no—
71C3×#551—51no—
72C3×#458D511yescharacteristic, complemented
73C3×B(4,𝔽2)—51no—
74C3×#455—51no—
75C3×#354—51no—
76C3×#352—51no—
77C3×AΣL(1,𝔽16)—51no—
78C5×H(5,𝔽2)C22×C311yescharacteristic
79C5×#123—21no—
80C5×#123C22×C311yescharacteristic
81#4097—21no—
82#4083—21no—
83#4083C22×C311yescharacteristic
84#4082C22×C311yescharacteristic
85C4×C22⋊D5—21no—
86#4042—21no—
87#4038—21no—
88#4029—21no—
89D5×C22⋊C4—21no—
90#4025—21no—
91C5×#99—21no—
92C5×#99C22×C311yescharacteristic
93#3969—21no—
94#3966—21no—
95#3966C22×C311yescharacteristic
96#3965C22×C311yescharacteristic
97#2197—51noSylow
98C23×C3×C5—21no—
99C23×C3×C5—21no—
100C23×C3×C5C2×D411yescharacteristic
101C23×C3×C5C2×D411yescharacteristic
102C22×C3×D5—21no—
103C22×C3×D5—42no—
104C22×C3×D5—41no—
105C3×C5×Q8—21no—
106C3×C5×D4—21no—
107C3×C5×D4—41no—
108C3×C5×D4—41no—
109C3×C5×D4—41no—
110C3×C5×D4—41no—
111C2×C3×C4×C5—21no—
112C2×C3×C4×C5—21no—
113C2×C3×C4×C5—21no—
114C2×C3×C4×C5—21no—
115C2×C3×C4×C5—41no—
116C2×C3×C4×C5—41no—
117C2×C3×C4×C5—41no—
118C2×C3×C4×C5C2×D411yescharacteristic
119C3×C22⋊D5—42no—
120C3×C22⋊D5—82no—
121C2×C3×Dic5—21no—
122C2×C3×Dic5—21no—
123C2×C3×Dic5—21no—
124C2×C3×Dic5—41no—
125C3×C4×D5—42no—
126C3×C4×D5—81no—
127C3×Dic10—41no—
128C3×#168—41no—
129C3×H(5,𝔽2)C2×D511yescharacteristic
130C3×#137—101no—
131C3×#126—101no—
132C3×#124—51no—
133C3×#123—21no—
134C3×#123—101no—
135C3×#123C2×D511yescharacteristic
136C3×C22⋊Q8—101no—
137C3×#120—51no—
138C3×#120—101no—
139C3×C4×D4—101no—
140C3×#117—101no—
141C2×C3×C22⋊C4—101no—
142C3×#104—101no—
143C3×#100—51no—
144C3×#99—21no—
145C3×#99—51no—
146C3×#99—101no—
147C3×#99C2×D511yescharacteristic
148C23×D5C3×D411yescharacteristic
149C5×#41—21no—
150C5×#41—41no—
151C2×C5×D4—21no—
152C2×C5×D4—21no—
153C2×C5×D4—41no—
154C2×C5×D4C23×C311yescharacteristic
155C22×C4×C5C3×D411yescharacteristic, complemented
156C2×C22⋊D5—21no—
157C2×C22⋊D5—41no—
158C2×C4×D5—21no—
159C2×C4×D5—21no—
160C2×Dic10C3×D411yescharacteristic, complemented
161C5×C4⋊C4—21no—
162C5×C4⋊C4—41no—
163C5×C22⋊C4—21no—
164C5×C22⋊C4—21no—
165C5×C22⋊C4—41no—
166C5×C22⋊C4C3×D411yescharacteristic, complemented
167C22⋊Dic5—21no—
168C22⋊Dic5—21no—
169C22⋊Dic5—21no—
170C22⋊Dic5C3×D411yescharacteristic
171#689—41no—
172#687—41no—
173#685—21no—
174#685—21no—
175#685—21no—
176#685—41no—
177C4⋊Dic5—21no—
178#683—21no—
179#683—21no—
180C4×Dic5—21no—
181C4×Dic5C3×D411yescharacteristic, complemented
182#681—21no—
183#551—51no—
184#458C3×D511yescharacteristic, complemented
185B(4,𝔽2)—51no—
186#455—51no—
187#354—51no—
188#352—51no—
189AΣL(1,𝔽16)—51no—
190C22×C3×C5—21no—
191C22×C3×C5—21no—
192C22×C3×C5—41no—
193C22×C3×C5—41no—
194C22×C3×C5—41no—
195C22×C3×C5—81no—
196C22×C3×C5—81no—
197C22×C3×C5#12011yescharacteristic
198C2×C3×D5—22no—
199C2×C3×D5—41no—
200C2×C3×D5—41no—
201C2×C3×D5—42no—
202C2×C3×D5—81no—
203C3×C4×C5—21no—
204C3×C4×C5—41no—
205C3×C4×C5—41no—
206C3×C4×C5—41no—
207C3×C4×C5—81no—
208C3×Dic5—41no—
209C3×Dic5—41no—
210C3×Dic5—41no—
211C3×Dic5—81no—
212C24×C3—51no—
213C3×#41—21no—
214C3×#41—41no—
215C2×C3×Q8—51no—
216C2×C3×D4—21no—
217C2×C3×D4—21no—
218C2×C3×D4—41no—
219C2×C3×D4—101no—
220C2×C3×D4—201no—
221C2×C3×D4C22×D511yescharacteristic
222C22×C3×C4—101no—
223C22×C3×C4—101no—
224C22×C3×C4C22⋊D511yescharacteristic, complemented
225C3×Q16—201no—
226C3×SD16—201no—
227C3×M16—101no—
228C3×C4⋊C4—21no—
229C3×C4⋊C4—41no—
230C3×C4⋊C4—101no—
231C3×C4⋊C4—101no—
232C3×C4⋊C4—101no—
233C3×C22⋊C4—21no—
234C3×C22⋊C4—21no—
235C3×C22⋊C4—41no—
236C3×C22⋊C4—51no—
237C3×C22⋊C4—101no—
238C3×C22⋊C4—101no—
239C3×C22⋊C4—101no—
240C3×C22⋊C4—101no—
241C3×C22⋊C4—101no—
242C3×C22⋊C4—101no—
243C3×C22⋊C4—201no—
244C3×C22⋊C4C22⋊D511yescharacteristic, complemented
245C3×C42—51no—
246C3×C42—101no—
247C23×C5—21no—
248C23×C5—21no—
249C23×C5C2×C3×D411yescharacteristic
250C23×C5C2×C3×D411yescharacteristic
251C22×D5—21no—
252C22×D5—42no—
253C22×D5—41no—
254C5×Q8—21no—
255C5×D4—21no—
256C5×D4—41no—
257C5×D4—41no—
258C5×D4—41no—
259C5×D4—41no—
260C2×C4×C5—21no—
261C2×C4×C5—21no—
262C2×C4×C5—21no—
263C2×C4×C5—21no—
264C2×C4×C5—41no—
265C2×C4×C5—41no—
266C2×C4×C5—41no—
267C2×C4×C5C2×C3×D411yescharacteristic
268C22⋊D5—42no—
269C22⋊D5—82no—
270C2×Dic5—21no—
271C2×Dic5—21no—
272C2×Dic5—21no—
273C2×Dic5—41no—
274C4×D5—42no—
275C4×D5—81no—
276Dic10—41no—
277#168—41no—
278H(5,𝔽2)C2×C3×D511yescharacteristic
279#137—101no—
280#126—101no—
281#124—51no—
282#123—21no—
283#123—101no—
284#123C2×C3×D511yescharacteristic
285C22⋊Q8—101no—
286#120—51no—
287#120—101no—
288C4×D4—101no—
289#117—101no—
290C2×C22⋊C4—101no—
291#104—101no—
292#100—51no—
293#99—21no—
294#99—51no—
295#99—101no—
296#99C2×C3×D511yescharacteristic
297C2×C3×C5—21no—
298C2×C3×C5—41no—
299C2×C3×C5—41no—
300C2×C3×C5—81no—
301C2×C3×C5B(4,𝔽2)11yescharacteristic
302C3×D5—42no—
303C23×C3—21no—
304C23×C3—21no—
305C23×C3—101no—
306C23×C3—201no—
307C23×C3—202no—
308C23×C3D4×D511yescharacteristic
309C23×C3C2×C22⋊D511yescharacteristic
310C3×Q8—21no—
311C3×Q8—201no—
312C3×D4—21no—
313C3×D4—41no—
314C3×D4—41no—
315C3×D4—41no—
316C3×D4—41no—
317C3×D4—202no—
318C3×D4—402no—
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—21no—
322C2×C3×C4—21no—
323C2×C3×C4—41no—
324C2×C3×C4—41no—
325C2×C3×C4—41no—
326C2×C3×C4—101no—
327C2×C3×C4—101no—
328C2×C3×C4—101no—
329C2×C3×C4—202no—
330C2×C3×C4—201no—
331C2×C3×C4—401no—
332C2×C3×C4D4×D511yescharacteristic
333C3×C8—201no—
334C22×C5—21no—
335C22×C5—21no—
336C22×C5—41no—
337C22×C5—41no—
338C22×C5—41no—
339C22×C5—81no—
340C22×C5—81no—
341C22×C5C3×#12011yescharacteristic
342C2×D5—22no—
343C2×D5—41no—
344C2×D5—41no—
345C2×D5—42no—
346C2×D5—81no—
347C4×C5—21no—
348C4×C5—41no—
349C4×C5—41no—
350C4×C5—41no—
351C4×C5—81no—
352Dic5—41no—
353Dic5—41no—
354Dic5—41no—
355Dic5—81no—
356C24—51no—
357#41—21no—
358#41—41no—
359C2×Q8—51no—
360C2×D4—21no—
361C2×D4—21no—
362C2×D4—41no—
363C2×D4—101no—
364C2×D4—201no—
365C2×D4C22×C3×D511yescharacteristic
366C22×C4—101no—
367C22×C4—101no—
368C22×C4C3×C22⋊D511yescharacteristic, complemented
369Q16—201no—
370SD16—201no—
371M16—101no—
372C4⋊C4—21no—
373C4⋊C4—41no—
374C4⋊C4—101no—
375C4⋊C4—101no—
376C4⋊C4—101no—
377C22⋊C4—21no—
378C22⋊C4—21no—
379C22⋊C4—41no—
380C22⋊C4—51no—
381C22⋊C4—101no—
382C22⋊C4—101no—
383C22⋊C4—101no—
384C22⋊C4—101no—
385C22⋊C4—101no—
386C22⋊C4—101no—
387C22⋊C4—201no—
388C22⋊C4C3×C22⋊D511yescharacteristic, complemented
389C42—51no—
390C42—101no—
391C3×C5#219711yescharacteristic, complemented, Hall
392C22×C3—21no—
393C22×C3—21no—
394C22×C3—41no—
395C22×C3—41no—
396C22×C3—41no—
397C22×C3—81no—
398C22×C3—81no—
399C22×C3—102no—
400C22×C3—201no—
401C22×C3—202no—
402C22×C3—201no—
403C22×C3—401no—
404C22×C3#408311yescharacteristic
405C3×C4—21no—
406C3×C4—41no—
407C3×C4—41no—
408C3×C4—41no—
409C3×C4—81no—
410C3×C4—201no—
411C3×C4—201no—
412C3×C4—201no—
413C3×C4—401no—
414C2×C5—21no—
415C2×C5—41no—
416C2×C5—41no—
417C2×C5—81no—
418C2×C5C3×B(4,𝔽2)11yescharacteristic
419D5—42no—
420C23—21no—
421C23—21no—
422C23—101no—
423C23—201no—
424C23—202no—
425C23C3×D4×D511yescharacteristic
426C23C2×C3×C22⋊D511yescharacteristic
427Q8—21no—
428Q8—201no—
429D4—21no—
430D4—41no—
431D4—41no—
432D4—41no—
433D4—41no—
434D4—202no—
435D4—402no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—21no—
439C2×C4—21no—
440C2×C4—41no—
441C2×C4—41no—
442C2×C4—41no—
443C2×C4—101no—
444C2×C4—101no—
445C2×C4—101no—
446C2×C4—202no—
447C2×C4—201no—
448C2×C4—401no—
449C2×C4C3×D4×D511yescharacteristic
450C8—201no—
451C2×C3—21no—
452C2×C3—41no—
453C2×C3—41no—
454C2×C3—81no—
455C2×C3—202no—
456C2×C3#6532111yescharacteristic
457C5C3×#219711yescharacteristic, complemented, Sylow
458C22—21no—
459C22—21no—
460C22—41no—
461C22—41no—
462C22—41no—
463C22—81no—
464C22—81no—
465C22—102no—
466C22—201no—
467C22—202no—
468C22—201no—
469C22—401no—
470C22C3×#408311yescharacteristic
471C4—21no—
472C4—41no—
473C4—41no—
474C4—41no—
475C4—81no—
476C4—201no—
477C4—201no—
478C4—201no—
479C4—401no—
480C3#1060179311yescharacteristic, direct product factor, Sylow
481C2—21no—
482C2—41no—
483C2—41no—
484C2—81no—
485C2—202no—
486C2C3×#6532111yescharacteristic
subgroupG/H
centerC2×C3#65321
commutator, cosocleC2×C5×D4C23×C3
FrattiniC2×D4C22×C3×D5
FittingC3×C5×#458C2
socleC2×C3×C5B(4,𝔽2)
prime radicalC3×#65533C2
hypercenterC3×#458D5
hypocenterC5C3×#2197
LCSG ▷ C2×C5×D4 ▷ C22×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C22×C3 ◁ C2×C3×D4 ◁ C3×#458

Complements

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

NH ≅ G/Ntype
1. C3×#65993C2semidirect
2. C3×C5×#458C2semidirect
4. C3×#65531C2semidirect
5. C3×#64759C2semidirect
6. C3×#64693C2semidirect
7. C3×#64691C2semidirect
8. #10601793C3direct (N × H)
29. #65993C2×C3semidirect
30. C5×#458C2×C3semidirect
32. #65531C2×C3semidirect
33. #64759C2×C3semidirect
34. #64693C2×C3semidirect
35. #64691C2×C3semidirect
43. C22×C3×C4×C5D4semidirect
48. C2×C3×Dic10D4semidirect
54. C3×C5×C22⋊C4D4semidirect
69. C3×C4×Dic5D4semidirect
72. C3×#458D5semidirect
155. C22×C4×C5C3×D4semidirect
160. C2×Dic10C3×D4semidirect
166. C5×C22⋊C4C3×D4semidirect
181. C4×Dic5C3×D4semidirect
184. #458C3×D5semidirect
224. C22×C3×C4C22⋊D5semidirect
244. C3×C22⋊C4C22⋊D5semidirect
368. C22×C4C3×C22⋊D5semidirect
388. C22⋊C4C3×C22⋊D5semidirect
391. C3×C5#2197semidirect
457. C5C3×#2197semidirect
480. C3#10601793direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#659932
C3×C5×#4582
C3×#655332
C3×#655312
C3×#647592
C3×#646932
C3×#646912
#106017933
C3×#21975