Finite Group Databasebeta

← C3×#10599087 #14288048

Subgroups — C3×#10599087 #14288048

Order 1920.

488 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 98 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×#65936C211yesmaximal, characteristic
2C3×#65329C211yesmaximal, characteristic
3C3×#65028C211yesmaximal, characteristic, complemented
4C3×#64945C211yesmaximal, characteristic, complemented
5C3×C5×#327C211yesmaximal, characteristic
6C3×#64689C211yesmaximal, characteristic, complemented
7C3×#64672C211yesmaximal, characteristic, complemented
8#10599087C311yesmaximal, characteristic, direct product factor, Hall
9C22×C3×Dic10C2211yescharacteristic
10C3×C5×#121C2211yescharacteristic
11C3×#4081—21no—
12C3×#4081—21no—
13C3×D4×Dic5—21no—
14C2×C3×#687—21no—
15C2×C3×C4⋊Dic5—21no—
16C3×#4070C2211yescharacteristic
17C2×C3×#683C2211yescharacteristic
18C2×C3×Dic20—21no—
19C3×#4035—21no—
20C3×#4025—21no—
21C3×#4024—21no—
22C3×C5×C22⋊C8C2211yescharacteristic
23C3×C5×#95C2211yescharacteristic
24C3×#3963—21no—
25C3×C22⋊#168C2211yescharacteristic
26C3×#3948—21no—
27C3×#3942—21no—
28C3×#1603—51nomaximal, Hall
29#65936C2×C311yescharacteristic
30#65329C2×C311yescharacteristic
31#65028C2×C311yescharacteristic, complemented
32#64945C2×C311yescharacteristic, complemented
33C5×#327C2×C311yescharacteristic
34#64689C2×C311yescharacteristic, complemented
35#64672C2×C311yescharacteristic, complemented
36C2×C3×C5×D4—21no—
37C2×C3×C5×D4D411yescharacteristic
38C22×C3×C4×C5—21no—
39C22×C3×C4×C5C2311yescharacteristic
40C22×C3×Dic5—21no—
41C22×C3×Dic5—21no—
42C22×C3×Dic5D411yescharacteristic
43C22×C3×Dic5D411yescharacteristic
44C2×C3×Dic10—21no—
45C2×C3×Dic10—21no—
46C2×C3×Dic10—42no—
47C2×C3×Dic10D411yescharacteristic
48C2×C3×C5×C8—21no—
49C3×C5×C4⋊C4D411yescharacteristic
50C3×C5×C22⋊C4—21no—
51C3×C22⋊Dic5—21no—
52C3×C22⋊Dic5—21no—
53C3×C22⋊Dic5—21no—
54C3×C22⋊Dic5—21no—
55C3×#687—42no—
56C3×C4⋊Dic5—21no—
57C3×C4⋊Dic5—42no—
58C3×C4⋊Dic5D411yescharacteristic
59C3×#683—21no—
60C3×#683—21no—
61C3×#683—21no—
62C3×#683—21no—
63C3×#683—21no—
64C3×C4×Dic5—21no—
65C2×C3×#168—21no—
66C3×Dic20—42no—
67C3×#539—51no—
68C3×#451—51no—
69C3×#448—51no—
70C3×#395—51no—
71C3×#328—51no—
72C3×#327D511yescharacteristic
73C3×#324—51no—
74C22×Dic10C22×C311yescharacteristic
75C5×#121C22×C311yescharacteristic
76#4081—21no—
77#4081—21no—
78D4×Dic5—21no—
79C2×#687—21no—
80C2×C4⋊Dic5—21no—
81#4070C22×C311yescharacteristic
82C2×#683C22×C311yescharacteristic
83C2×Dic20—21no—
84#4035—21no—
85#4025—21no—
86#4024—21no—
87C5×C22⋊C8C22×C311yescharacteristic
88C5×#95C22×C311yescharacteristic
89#3963—21no—
90C22⋊#168C22×C311yescharacteristic
91#3948—21no—
92#3942—21no—
93#1603—51noSylow
94C23×C3×C5—21no—
95C23×C3×C5C2×D411yescharacteristic
96C3×C5×D4—41no—
97C3×C5×D4—41no—
98C3×C5×D4—41no—
99C2×C3×C4×C5—22no—
100C2×C3×C4×C5—21no—
101C2×C3×C4×C5—41no—
102C2×C3×C4×C5—41no—
103C2×C3×C4×C5—41no—
104C2×C3×C4×C5C2×D411yescharacteristic
105C2×C3×C4×C5C2×D411yescharacteristic
106C2×C3×Dic5—22no—
107C2×C3×Dic5—21no—
108C2×C3×Dic5—21no—
109C2×C3×Dic5—21no—
110C2×C3×Dic5—21no—
111C2×C3×Dic5—21no—
112C2×C3×Dic5—21no—
113C2×C3×Dic5—41no—
114C2×C3×Dic5—41no—
115C2×C3×Dic5—41no—
116C2×C3×Dic5—81no—
117C3×Dic10—22no—
118C3×Dic10—22no—
119C3×Dic10—41no—
120C3×Dic10—41no—
121C3×Dic10—41no—
122C3×C5×C8—41no—
123C3×#168—41no—
124C22×C3×Q8—51no—
125C2×C3×Q16—101no—
126C2×C3×SD16—101no—
127C3×#126—101no—
128C3×#125—101no—
129C3×#123—101no—
130C3×#123—101no—
131C3×C22⋊Q8—51no—
132C3×C22⋊Q8—101no—
133C3×#121C2×D511yescharacteristic
134C3×C4×D4—101no—
135C2×C3×C4⋊C4—51no—
136C2×C3×C4⋊C4—101no—
137C3×#103—101no—
138C3×#103—101no—
139C3×C22⋊C8—51no—
140C3×C22⋊C8C2×D511yescharacteristic
141C3×#95—101no—
142C3×#95C2×D511yescharacteristic
143C2×C5×D4—21no—
144C2×C5×D4C3×D411yescharacteristic
145C22×C4×C5—21no—
146C22×C4×C5C23×C311yescharacteristic
147C22×Dic5—21no—
148C22×Dic5—21no—
149C22×Dic5C3×D411yescharacteristic
150C22×Dic5C3×D411yescharacteristic
151C2×Dic10—21no—
152C2×Dic10—21no—
153C2×Dic10—42no—
154C2×Dic10C3×D411yescharacteristic
155C2×C5×C8—21no—
156C5×C4⋊C4C3×D411yescharacteristic
157C5×C22⋊C4—21no—
158C22⋊Dic5—21no—
159C22⋊Dic5—21no—
160C22⋊Dic5—21no—
161C22⋊Dic5—21no—
162#687—42no—
163C4⋊Dic5—21no—
164C4⋊Dic5—42no—
165C4⋊Dic5C3×D411yescharacteristic
166#683—21no—
167#683—21no—
168#683—21no—
169#683—21no—
170#683—21no—
171C4×Dic5—21no—
172C2×#168—21no—
173Dic20—42no—
174#539—51no—
175#451—51no—
176#448—51no—
177#395—51no—
178#328—51no—
179#327C3×D511yescharacteristic
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—42no—
192C3×C4×C5—41no—
193C3×Dic5—41no—
194C3×Dic5—41no—
195C3×Dic5—42no—
196C3×Dic5—41no—
197C3×Dic5—41no—
198C2×C3×Q8—51no—
199C2×C3×Q8—101no—
200C2×C3×Q8—101no—
201C2×C3×Q8—202no—
202C2×C3×D4—21no—
203C2×C3×D4D2011yescharacteristic
204C22×C3×C4—21no—
205C22×C3×C4—51no—
206C22×C3×C4—51no—
207C22×C3×C4—101no—
208C22×C3×C4—101no—
209C22×C3×C4C22×D511yescharacteristic
210C3×Q16—202no—
211C3×SD16—202no—
212C2×C3×C8—21no—
213C2×C3×C8—101no—
214C3×C4⋊C4—51no—
215C3×C4⋊C4—101no—
216C3×C4⋊C4—101no—
217C3×C4⋊C4—101no—
218C3×C4⋊C4—101no—
219C3×C4⋊C4—101no—
220C3×C4⋊C4—101no—
221C3×C4⋊C4—202no—
222C3×C4⋊C4D2011yescharacteristic
223C3×C22⋊C4—21no—
224C3×C22⋊C4—101no—
225C3×C22⋊C4—101no—
226C3×C22⋊C4—101no—
227C3×C22⋊C4—101no—
228C3×C42—101no—
229C23×C5—21no—
230C23×C5C2×C3×D411yescharacteristic
231C5×D4—41no—
232C5×D4—41no—
233C5×D4—41no—
234C2×C4×C5—22no—
235C2×C4×C5—21no—
236C2×C4×C5—41no—
237C2×C4×C5—41no—
238C2×C4×C5—41no—
239C2×C4×C5C2×C3×D411yescharacteristic
240C2×C4×C5C2×C3×D411yescharacteristic
241C2×Dic5—22no—
242C2×Dic5—21no—
243C2×Dic5—21no—
244C2×Dic5—21no—
245C2×Dic5—21no—
246C2×Dic5—21no—
247C2×Dic5—21no—
248C2×Dic5—41no—
249C2×Dic5—41no—
250C2×Dic5—41no—
251C2×Dic5—81no—
252Dic10—22no—
253Dic10—22no—
254Dic10—41no—
255Dic10—41no—
256Dic10—41no—
257C5×C8—41no—
258#168—41no—
259C22×Q8—51no—
260C2×Q16—101no—
261C2×SD16—101no—
262#126—101no—
263#125—101no—
264#123—101no—
265#123—101no—
266C22⋊Q8—51no—
267C22⋊Q8—101no—
268#121C2×C3×D511yescharacteristic
269C4×D4—101no—
270C2×C4⋊C4—51no—
271C2×C4⋊C4—101no—
272#103—101no—
273#103—101no—
274C22⋊C8—51no—
275C22⋊C8C2×C3×D511yescharacteristic
276#95—101no—
277#95C2×C3×D511yescharacteristic
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×C3D4×D511yescharacteristic
286C3×Q8—102no—
287C3×Q8—102no—
288C3×Q8—201no—
289C3×Q8—201no—
290C3×Q8—201no—
291C3×D4—41no—
292C3×D4—41no—
293C3×D4—41no—
294C2×C3×C4—22no—
295C2×C3×C4—21no—
296C2×C3×C4—41no—
297C2×C3×C4—41no—
298C2×C3×C4—41no—
299C2×C3×C4—102no—
300C2×C3×C4—101no—
301C2×C3×C4—101no—
302C2×C3×C4—101no—
303C2×C3×C4—101no—
304C2×C3×C4—101no—
305C2×C3×C4—101no—
306C2×C3×C4—201no—
307C2×C3×C4—201no—
308C2×C3×C4—201no—
309C2×C3×C4—401no—
310C2×C3×C4C2×D2011yescharacteristic
311C2×C3×C4D4×D511yescharacteristic
312C3×C8—41no—
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—42no—
325C4×C5—41no—
326Dic5—41no—
327Dic5—41no—
328Dic5—42no—
329Dic5—41no—
330Dic5—41no—
331C2×Q8—51no—
332C2×Q8—101no—
333C2×Q8—101no—
334C2×Q8—202no—
335C2×D4—21no—
336C2×D4C3×D2011yescharacteristic
337C22×C4—21no—
338C22×C4—51no—
339C22×C4—51no—
340C22×C4—101no—
341C22×C4—101no—
342C22×C4C22×C3×D511yescharacteristic
343Q16—202no—
344SD16—202no—
345C2×C8—21no—
346C2×C8—101no—
347C4⋊C4—51no—
348C4⋊C4—101no—
349C4⋊C4—101no—
350C4⋊C4—101no—
351C4⋊C4—101no—
352C4⋊C4—101no—
353C4⋊C4—101no—
354C4⋊C4—202no—
355C4⋊C4C3×D2011yescharacteristic
356C22⋊C4—21no—
357C22⋊C4—101no—
358C22⋊C4—101no—
359C22⋊C4—101no—
360C22⋊C4—101no—
361C42—101no—
362C3×C5#160311yescharacteristic, complemented, Hall
363C22×C3—21no—
364C22×C3—21no—
365C22×C3—41no—
366C22×C3—41no—
367C22×C3—41no—
368C22×C3#402811yescharacteristic
369C22×C3#405811yescharacteristic
370C22×C3#406111yescharacteristic
371C3×C4—21no—
372C3×C4—21no—
373C3×C4—42no—
374C3×C4—41no—
375C3×C4—202no—
376C3×C4—201no—
377C3×C4—201no—
378C3×C4—201no—
379C3×C4—201no—
380C2×C5—21no—
381C2×C5—21no—
382C2×C5—81no—
383C2×C5C3×#45211yescharacteristic
384C2×C5C3×#45611yescharacteristic
385C2×C5C3×B(4,𝔽2)11yescharacteristic
386C23—21no—
387C23C3×D4×D511yescharacteristic
388Q8—102no—
389Q8—102no—
390Q8—201no—
391Q8—201no—
392Q8—201no—
393D4—41no—
394D4—41no—
395D4—41no—
396C2×C4—22no—
397C2×C4—21no—
398C2×C4—41no—
399C2×C4—41no—
400C2×C4—41no—
401C2×C4—102no—
402C2×C4—101no—
403C2×C4—101no—
404C2×C4—101no—
405C2×C4—101no—
406C2×C4—101no—
407C2×C4—101no—
408C2×C4—201no—
409C2×C4—201no—
410C2×C4—201no—
411C2×C4—401no—
412C2×C4C2×C3×D2011yescharacteristic
413C2×C4C3×D4×D511yescharacteristic
414C8—41no—
415C8—201no—
416C2×C3—21no—
417C2×C3—21no—
418C2×C3—81no—
419C2×C3#6503011yescharacteristic
420C2×C3#6507211yescharacteristic
421C2×C3#6511511yescharacteristic
422C5C3×#160311yescharacteristic, complemented, Sylow
423C22—21no—
424C22—21no—
425C22—41no—
426C22—41no—
427C22—41no—
428C22C3×#402811yescharacteristic
429C22C3×#405811yescharacteristic
430C22C3×#406111yescharacteristic
431C4—21no—
432C4—21no—
433C4—42no—
434C4—41no—
435C4—202no—
436C4—201no—
437C4—201no—
438C4—201no—
439C4—201no—
440C3#1059908711yescharacteristic, direct product factor, Sylow
441C2—21no—
442C2—21no—
443C2—81no—
444C2C3×#6503011yescharacteristic
445C2C3×#6507211yescharacteristic
446C2C3×#6511511yescharacteristic
subgroupG/H
centerC22×C3#4028
commutator, cosocleC22×C4×C5C23×C3
FrattiniC22×C4C22×C3×D5
FittingC3×C5×#327C2
socleC22×C3×C5#120
prime radicalC3×C5×#121C22
hypercenterC3×#327D5
hypocenterC5C3×#1603
LCSG ▷ C22×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C22×C3×C4 ◁ C3×#327

Complements

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

NH ≅ G/Ntype
3. C3×#65028C2semidirect
4. C3×#64945C2semidirect
6. C3×#64689C2semidirect
7. C3×#64672C2semidirect
8. #10599087C3direct (N × H)
31. #65028C2×C3semidirect
32. #64945C2×C3semidirect
34. #64689C2×C3semidirect
35. #64672C2×C3semidirect
362. C3×C5#1603semidirect
422. C5C3×#1603semidirect
440. C3#10599087direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#659362
C3×#653292
C3×#650282
C3×#649452
C3×C5×#3272
C3×#646892
C3×#646722
#105990873
C3×#16035