Finite Group Databasebeta

← C3×#10602682 #14288045

Subgroups — C3×#10602682 #14288045

Order 1920.

488 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#539C211yesmaximal, characteristic
2C3×#65339C211yesmaximal, characteristic, complemented
3C3×#65329C211yesmaximal, characteristic
4C3×#65268C211yesmaximal, characteristic, complemented
5C3×#64758C211yesmaximal, characteristic, complemented
6C3×#64757C211yesmaximal, characteristic
7C3×#64748C211yesmaximal, characteristic, complemented
8#10602682C311yesmaximal, characteristic, direct product factor, Hall
9C22×C3×Dic10C2211yescharacteristic
10C3×C5×#126—21no—
11C3×C5×#125—21no—
12C3×C5×#123—21no—
13C3×C5×#123—21no—
14C3×C5×C22⋊Q8—21no—
15C3×C5×C22⋊Q8C2211yescharacteristic
16C3×C5×#121C2211yescharacteristic
17C3×C4×C5×D4—21no—
18C2×C3×C5×C4⋊C4C2211yescharacteristic
19C2×C3×#689—21no—
20C2×C3×#687—21no—
21C2×C3×#683—21no—
22C3×#3963—21no—
23C3×#3963C2211yescharacteristic
24C3×C22⋊#168C2211yescharacteristic
25C3×C22⋊#168C2211yescharacteristic
26C3×#3942—21no—
27C3×#3942—21no—
28C3×#1603—51nomaximal, Hall
29C5×#539C2×C311yescharacteristic
30#65339C2×C311yescharacteristic, complemented
31#65329C2×C311yescharacteristic
32#65268C2×C311yescharacteristic, complemented
33#64758C2×C311yescharacteristic, complemented
34#64757C2×C311yescharacteristic
35#64748C2×C311yescharacteristic, complemented
36C2×C3×C5×Q8D411yescharacteristic
37C2×C3×C5×D4—21no—
38C2×C3×C5×D4D411yescharacteristic
39C22×C3×C4×C5—21no—
40C22×C3×C4×C5D411yescharacteristic
41C22×C3×C4×C5D411yescharacteristic
42C22×C3×C4×C5C2311yescharacteristic
43C22×C3×Dic5—21no—
44C22×C3×Dic5—21no—
45C2×C3×Dic10—21no—
46C2×C3×Dic10—21no—
47C2×C3×Dic10—42no—
48C3×C5×C4⋊C4—21no—
49C3×C5×C4⋊C4—21no—
50C3×C5×C4⋊C4—21no—
51C3×C5×C4⋊C4—21no—
52C3×C5×C4⋊C4—21no—
53C3×C5×C4⋊C4—21no—
54C3×C5×C4⋊C4D411yescharacteristic
55C3×C5×C4⋊C4D411yescharacteristic
56C3×C5×C22⋊C4—21no—
57C3×C5×C22⋊C4—21no—
58C3×C5×C22⋊C4—21no—
59C3×C5×C22⋊C4—21no—
60C3×C5×C22⋊C4—21no—
61C3×C42×C5—21no—
62C3×#689—42no—
63C3×#687—42no—
64C3×#683—42no—
65C2×C3×#168—21no—
66C2×C3×#168—21no—
67C3×#539D511yescharacteristic
68C3×#451—51no—
69C3×#448—51no—
70C3×#395—51no—
71C3×#328—51no—
72C3×#327—51no—
73C3×#324—51no—
74C22×Dic10C22×C311yescharacteristic
75C5×#126—21no—
76C5×#125—21no—
77C5×#123—21no—
78C5×#123—21no—
79C5×C22⋊Q8—21no—
80C5×C22⋊Q8C22×C311yescharacteristic
81C5×#121C22×C311yescharacteristic
82C4×C5×D4—21no—
83C2×C5×C4⋊C4C22×C311yescharacteristic
84C2×#689—21no—
85C2×#687—21no—
86C2×#683—21no—
87#3963—21no—
88#3963C22×C311yescharacteristic
89C22⋊#168C22×C311yescharacteristic
90C22⋊#168C22×C311yescharacteristic
91#3942—21no—
92#3942—21no—
93#1603—51noSylow
94C23×C3×C5—21no—
95C23×C3×C5C2×D411yescharacteristic
96C3×C5×Q8—41no—
97C3×C5×D4—41no—
98C3×C5×D4—41no—
99C3×C5×D4—41no—
100C2×C3×C4×C5—21no—
101C2×C3×C4×C5—21no—
102C2×C3×C4×C5—21no—
103C2×C3×C4×C5—21no—
104C2×C3×C4×C5—21no—
105C2×C3×C4×C5—21no—
106C2×C3×C4×C5—21no—
107C2×C3×C4×C5—41no—
108C2×C3×C4×C5—41no—
109C2×C3×C4×C5—81no—
110C2×C3×C4×C5C2×D411yescharacteristic
111C2×C3×C4×C5C2×D411yescharacteristic
112C2×C3×Dic5—22no—
113C2×C3×Dic5—22no—
114C2×C3×Dic5—41no—
115C2×C3×Dic5—41no—
116C2×C3×Dic5—41no—
117C2×C3×Dic5—41no—
118C3×Dic10—22no—
119C3×Dic10—22no—
120C3×Dic10—41no—
121C3×Dic10—41no—
122C3×#168—41no—
123C3×#168—41no—
124C22×C3×Q8—51no—
125C2×C3×Q16—101no—
126C2×C3×SD16—101no—
127C3×#126—21no—
128C3×#125—21no—
129C3×#123—21no—
130C3×#123—21no—
131C3×C22⋊Q8—21no—
132C3×C22⋊Q8C2×D511yescharacteristic
133C3×#121C2×D511yescharacteristic
134C3×C4×D4—21no—
135C2×C3×C4⋊C4—101no—
136C2×C3×C4⋊C4C2×D511yescharacteristic
137C3×#103—101no—
138C3×#103—101no—
139C3×C22⋊C8—51no—
140C3×C22⋊C8—51no—
141C3×#95—51no—
142C3×#95—101no—
143C2×C5×Q8C3×D411yescharacteristic
144C2×C5×D4—21no—
145C2×C5×D4C3×D411yescharacteristic
146C22×C4×C5—21no—
147C22×C4×C5C3×D411yescharacteristic
148C22×C4×C5C3×D411yescharacteristic
149C22×C4×C5C23×C311yescharacteristic
150C22×Dic5—21no—
151C22×Dic5—21no—
152C2×Dic10—21no—
153C2×Dic10—21no—
154C2×Dic10—42no—
155C5×C4⋊C4—21no—
156C5×C4⋊C4—21no—
157C5×C4⋊C4—21no—
158C5×C4⋊C4—21no—
159C5×C4⋊C4—21no—
160C5×C4⋊C4—21no—
161C5×C4⋊C4C3×D411yescharacteristic
162C5×C4⋊C4C3×D411yescharacteristic
163C5×C22⋊C4—21no—
164C5×C22⋊C4—21no—
165C5×C22⋊C4—21no—
166C5×C22⋊C4—21no—
167C5×C22⋊C4—21no—
168C42×C5—21no—
169#689—42no—
170#687—42no—
171#683—42no—
172C2×#168—21no—
173C2×#168—21no—
174#539C3×D511yescharacteristic
175#451—51no—
176#448—51no—
177#395—51no—
178#328—51no—
179#327—51no—
180#324—51no—
181C22×C3×C5—21no—
182C22×C3×C5—21no—
183C22×C3×C5—41no—
184C22×C3×C5—41no—
185C22×C3×C5—41no—
186C22×C3×C5#12011yescharacteristic
187C22×C3×C5#13511yescharacteristic
188C22×C3×C5#13711yescharacteristic
189C3×C4×C5—21no—
190C3×C4×C5—21no—
191C3×C4×C5—41no—
192C3×C4×C5—41no—
193C3×C4×C5—41no—
194C3×C4×C5—41no—
195C3×C4×C5—41no—
196C3×Dic5—42no—
197C3×Dic5—42no—
198C2×C3×Q8—101no—
199C2×C3×Q8—101no—
200C2×C3×Q8—202no—
201C2×C3×Q8C22⋊D511yescharacteristic
202C2×C3×D4—21no—
203C2×C3×D4C22⋊D511yescharacteristic
204C22×C3×C4—21no—
205C22×C3×C4—101no—
206C22×C3×C4—101no—
207C22×C3×C4C22⋊D511yescharacteristic
208C22×C3×C4C22⋊D511yescharacteristic
209C22×C3×C4C22×D511yescharacteristic
210C3×Q16—202no—
211C3×SD16—202no—
212C2×C3×C8—101no—
213C2×C3×C8—101no—
214C3×C4⋊C4—21no—
215C3×C4⋊C4—21no—
216C3×C4⋊C4—21no—
217C3×C4⋊C4—21no—
218C3×C4⋊C4—21no—
219C3×C4⋊C4—21no—
220C3×C4⋊C4—202no—
221C3×C4⋊C4C22⋊D511yescharacteristic
222C3×C4⋊C4C22⋊D511yescharacteristic
223C3×C22⋊C4—21no—
224C3×C22⋊C4—21no—
225C3×C22⋊C4—21no—
226C3×C22⋊C4—21no—
227C3×C22⋊C4—21no—
228C3×C42—21no—
229C23×C5—21no—
230C23×C5C2×C3×D411yescharacteristic
231C5×Q8—41no—
232C5×D4—41no—
233C5×D4—41no—
234C5×D4—41no—
235C2×C4×C5—21no—
236C2×C4×C5—21no—
237C2×C4×C5—21no—
238C2×C4×C5—21no—
239C2×C4×C5—21no—
240C2×C4×C5—21no—
241C2×C4×C5—21no—
242C2×C4×C5—41no—
243C2×C4×C5—41no—
244C2×C4×C5—81no—
245C2×C4×C5C2×C3×D411yescharacteristic
246C2×C4×C5C2×C3×D411yescharacteristic
247C2×Dic5—22no—
248C2×Dic5—22no—
249C2×Dic5—41no—
250C2×Dic5—41no—
251C2×Dic5—41no—
252C2×Dic5—41no—
253Dic10—22no—
254Dic10—22no—
255Dic10—41no—
256Dic10—41no—
257#168—41no—
258#168—41no—
259C22×Q8—51no—
260C2×Q16—101no—
261C2×SD16—101no—
262#126—21no—
263#125—21no—
264#123—21no—
265#123—21no—
266C22⋊Q8—21no—
267C22⋊Q8C2×C3×D511yescharacteristic
268#121C2×C3×D511yescharacteristic
269C4×D4—21no—
270C2×C4⋊C4—101no—
271C2×C4⋊C4C2×C3×D511yescharacteristic
272#103—101no—
273#103—101no—
274C22⋊C8—51no—
275C22⋊C8—51no—
276#95—51no—
277#95—101no—
278C2×C3×C5—21no—
279C2×C3×C5—21no—
280C2×C3×C5—81no—
281C2×C3×C5#45211yescharacteristic
282C2×C3×C5#45611yescharacteristic
283C2×C3×C5B(4,𝔽2)11yescharacteristic
284C23×C3—21no—
285C23×C3C2×C22⋊D511yescharacteristic
286C3×Q8—41no—
287C3×Q8—102no—
288C3×Q8—102no—
289C3×Q8—201no—
290C3×Q8—201no—
291C3×D4—41no—
292C3×D4—41no—
293C3×D4—41no—
294C2×C3×C4—21no—
295C2×C3×C4—21no—
296C2×C3×C4—21no—
297C2×C3×C4—21no—
298C2×C3×C4—21no—
299C2×C3×C4—21no—
300C2×C3×C4—21no—
301C2×C3×C4—41no—
302C2×C3×C4—41no—
303C2×C3×C4—81no—
304C2×C3×C4—102no—
305C2×C3×C4—102no—
306C2×C3×C4—201no—
307C2×C3×C4—201no—
308C2×C3×C4—201no—
309C2×C3×C4—201no—
310C2×C3×C4C2×C22⋊D511yescharacteristic
311C2×C3×C4C2×C22⋊D511yescharacteristic
312C3×C8—201no—
313C3×C8—201no—
314C22×C5—21no—
315C22×C5—21no—
316C22×C5—41no—
317C22×C5—41no—
318C22×C5—41no—
319C22×C5C3×#12011yescharacteristic
320C22×C5C3×#13511yescharacteristic
321C22×C5C3×#13711yescharacteristic
322C4×C5—21no—
323C4×C5—21no—
324C4×C5—41no—
325C4×C5—41no—
326C4×C5—41no—
327C4×C5—41no—
328C4×C5—41no—
329Dic5—42no—
330Dic5—42no—
331C2×Q8—101no—
332C2×Q8—101no—
333C2×Q8—202no—
334C2×Q8C3×C22⋊D511yescharacteristic
335C2×D4—21no—
336C2×D4C3×C22⋊D511yescharacteristic
337C22×C4—21no—
338C22×C4—101no—
339C22×C4—101no—
340C22×C4C3×C22⋊D511yescharacteristic
341C22×C4C3×C22⋊D511yescharacteristic
342C22×C4C22×C3×D511yescharacteristic
343Q16—202no—
344SD16—202no—
345C2×C8—101no—
346C2×C8—101no—
347C4⋊C4—21no—
348C4⋊C4—21no—
349C4⋊C4—21no—
350C4⋊C4—21no—
351C4⋊C4—21no—
352C4⋊C4—21no—
353C4⋊C4—202no—
354C4⋊C4C3×C22⋊D511yescharacteristic
355C4⋊C4C3×C22⋊D511yescharacteristic
356C22⋊C4—21no—
357C22⋊C4—21no—
358C22⋊C4—21no—
359C22⋊C4—21no—
360C22⋊C4—21no—
361C42—21no—
362C3×C5#160311yescharacteristic, complemented, Hall
363C22×C3—21no—
364C22×C3—21no—
365C22×C3—41no—
366C22×C3—41no—
367C22×C3—41no—
368C22×C3#409611yescharacteristic
369C22×C3#409711yescharacteristic
370C22×C3#409911yescharacteristic
371C3×C4—21no—
372C3×C4—21no—
373C3×C4—41no—
374C3×C4—41no—
375C3×C4—41no—
376C3×C4—41no—
377C3×C4—41no—
378C3×C4—202no—
379C3×C4—202no—
380C2×C5—21no—
381C2×C5—21no—
382C2×C5—81no—
383C2×C5C3×#45211yescharacteristic
384C2×C5C3×#45611yescharacteristic
385C2×C5C3×B(4,𝔽2)11yescharacteristic
386C23—21no—
387C23C2×C3×C22⋊D511yescharacteristic
388Q8—41no—
389Q8—102no—
390Q8—102no—
391Q8—201no—
392Q8—201no—
393D4—41no—
394D4—41no—
395D4—41no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—81no—
406C2×C4—102no—
407C2×C4—102no—
408C2×C4—201no—
409C2×C4—201no—
410C2×C4—201no—
411C2×C4—201no—
412C2×C4C2×C3×C22⋊D511yescharacteristic
413C2×C4C2×C3×C22⋊D511yescharacteristic
414C8—201no—
415C8—201no—
416C2×C3—21no—
417C2×C3—21no—
418C2×C3—81no—
419C2×C3#6552811yescharacteristic
420C2×C3#6553311yescharacteristic
421C2×C3#6553511yescharacteristic
422C5C3×#160311yescharacteristic, complemented, Sylow
423C22—21no—
424C22—21no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22C3×#409611yescharacteristic
429C22C3×#409711yescharacteristic
430C22C3×#409911yescharacteristic
431C4—21no—
432C4—21no—
433C4—41no—
434C4—41no—
435C4—41no—
436C4—41no—
437C4—41no—
438C4—202no—
439C4—202no—
440C3#1060268211yescharacteristic, direct product factor, Sylow
441C2—21no—
442C2—21no—
443C2—81no—
444C2C3×#6552811yescharacteristic
445C2C3×#6553311yescharacteristic
446C2C3×#6553511yescharacteristic
subgroupG/H
centerC22×C3#4099
commutator, cosocleC22×C4×C5C23×C3
FrattiniC22×C4C22×C3×D5
FittingC3×C5×#539C2
socleC22×C3×C5#120
prime radicalC3×C5×#121C22
hypercenterC3×#539D5
hypocenterC5C3×#1603
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#539

Complements

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

NH ≅ G/Ntype
2. C3×#65339C2semidirect
4. C3×#65268C2semidirect
5. C3×#64758C2semidirect
7. C3×#64748C2semidirect
8. #10602682C3direct (N × H)
30. #65339C2×C3semidirect
32. #65268C2×C3semidirect
33. #64758C2×C3semidirect
35. #64748C2×C3semidirect
362. C3×C5#1603semidirect
422. C5C3×#1603semidirect
440. C3#10602682direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#5392
C3×#653392
C3×#653292
C3×#652682
C3×#647582
C3×#647572
C3×#647482
#106026823
C3×#16035