Finite Group Databasebeta

← C3×#10601454 #14303608

Subgroups — C3×#10601454 #14303608

Order 1920.

372 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#427C211yesmaximal, characteristic
2C3×#65288C211yesmaximal, characteristic
3C3×#65286C211yesmaximal, characteristic
4C3×#64772C211yesmaximal, characteristic
5C3×#64770C211yesmaximal, characteristic
6C3×#64752C211yesmaximal, characteristic
7C3×#64752C211yesmaximal, characteristic
8#10601454C311yesmaximal, characteristic, direct product factor, Hall
9C22×C3×C5×C8C2211yescharacteristic
10C3×C5×#117C2211yescharacteristic
11C3×C5×#117C2211yescharacteristic
12C2×C3×C4×Dic5C2211yescharacteristic
13C2×C3×#681C2211yescharacteristic
14C2×C3×#681C2211yescharacteristic
15C22×C3×#168C2211yescharacteristic
16C3×C5×#107C411yescharacteristic, complemented
17C3×C5×#107C411yescharacteristic, complemented
18C3×C5×#106C411yescharacteristic, complemented
19C3×C5×#106C411yescharacteristic, complemented
20C3×C4⋊#168—21no—
21C3×C4⋊#168—21no—
22C3×#3935—21no—
23C3×C4×#168—21no—
24C3×#1922—51nomaximal, Hall
25C5×#427C2×C311yescharacteristic
26#65288C2×C311yescharacteristic
27#65286C2×C311yescharacteristic
28#64772C2×C311yescharacteristic
29#64770C2×C311yescharacteristic
30#64752C2×C311yescharacteristic
31#64752C2×C311yescharacteristic
32C22×C3×C4×C5C2311yescharacteristic
33C22×C3×Dic5—21no—
34C2×C3×C5×C8—21no—
35C2×C3×C5×C8—21no—
36C2×C3×C5×C8C2×C411yescharacteristic
37C2×C3×C5×C8C2×C411yescharacteristic
38C3×C5×C4⋊C4C2×C411yescharacteristic
39C3×C5×C4⋊C4C2×C411yescharacteristic
40C3×C5×C4⋊C4C2×C411yescharacteristic
41C3×C5×C4⋊C4C2×C411yescharacteristic
42C3×C5×C22⋊C4—21no—
43C3×C5×C22⋊C4—21no—
44C3×C42×C5—21no—
45C3×C42×C5—21no—
46C3×C4×Dic5—21no—
47C3×C4×Dic5—21no—
48C3×#681—21no—
49C3×#681—21no—
50C3×#681—21no—
51C3×#681Q812yes—
52C2×C3×#168—21no—
53C2×C3×#168—21no—
54C2×C3×#168—21no—
55C2×C3×#168—21no—
56C2×C3×#168D411yescharacteristic
57C2×C3×#168D411yescharacteristic
58C3×#427D511yescharacteristic
59C3×#424—51no—
60C3×#405—51no—
61C3×#341—51no—
62C3×#339—51no—
63C3×#339—51no—
64C3×#336—51no—
65C22×C5×C8C22×C311yescharacteristic
66C5×#117C22×C311yescharacteristic
67C5×#117C22×C311yescharacteristic
68C2×C4×Dic5C22×C311yescharacteristic
69C2×#681C22×C311yescharacteristic
70C2×#681C22×C311yescharacteristic
71C22×#168C22×C311yescharacteristic
72C5×#107C3×C411yescharacteristic, complemented
73C5×#107C3×C411yescharacteristic, complemented
74C5×#106C3×C411yescharacteristic, complemented
75C5×#106C3×C411yescharacteristic, complemented
76C4⋊#168—21no—
77C4⋊#168—21no—
78#3935—21no—
79C4×#168—21no—
80#1922—51noSylow
81C23×C3×C5#4111yescharacteristic
82C2×C3×C4×C5—21no—
83C2×C3×C4×C5—21no—
84C2×C3×C4×C5—21no—
85C2×C3×C4×C5—21no—
86C2×C3×C4×C5C22×C411yescharacteristic
87C2×C3×C4×C5C2×D411yescharacteristic
88C2×C3×C4×C5C2×Q811yescharacteristic
89C2×C3×C4×C5#4111yescharacteristic
90C2×C3×C4×C5#4111yescharacteristic
91C2×C3×C4×C5#4111yescharacteristic
92C2×C3×Dic5—22no—
93C2×C3×Dic5—41no—
94C2×C3×Dic5—41no—
95C3×C5×C8—21no—
96C3×C5×C8—21no—
97C3×#168—22no—
98C3×#168—21no—
99C3×#168—21no—
100C3×#168—41no—
101C2×C3×M16—51no—
102C2×C3×M16—51no—
103C22×C3×C8—51no—
104C22×C3×C8C2×D511yescharacteristic
105C3×#117C2×D511yescharacteristic
106C3×#117C2×D511yescharacteristic
107C2×C3×C42—51no—
108C3×#107Dic511yescharacteristic, complemented
109C3×#107Dic511yescharacteristic, complemented
110C3×#106Dic511yescharacteristic, complemented
111C3×#106Dic511yescharacteristic, complemented
112C3×C4⋊C8—101no—
113C3×C4⋊C8—101no—
114C3×#97—101no—
115C3×C4×C8—101no—
116C22×C4×C5C23×C311yescharacteristic
117C22×Dic5—21no—
118C2×C5×C8—21no—
119C2×C5×C8—21no—
120C2×C5×C8C2×C3×C411yescharacteristic
121C2×C5×C8C2×C3×C411yescharacteristic
122C5×C4⋊C4C2×C3×C411yescharacteristic
123C5×C4⋊C4C2×C3×C411yescharacteristic
124C5×C4⋊C4C2×C3×C411yescharacteristic
125C5×C4⋊C4C2×C3×C411yescharacteristic
126C5×C22⋊C4—21no—
127C5×C22⋊C4—21no—
128C42×C5—21no—
129C42×C5—21no—
130C4×Dic5—21no—
131C4×Dic5—21no—
132#681—21no—
133#681—21no—
134#681—21no—
135#681C3×Q812yes—
136C2×#168—21no—
137C2×#168—21no—
138C2×#168—21no—
139C2×#168—21no—
140C2×#168C3×D411yescharacteristic
141C2×#168C3×D411yescharacteristic
142#427C3×D511yescharacteristic
143#424—51no—
144#405—51no—
145#341—51no—
146#339—51no—
147#339—51no—
148#336—51no—
149C22×C3×C5—21no—
150C22×C3×C5—21no—
151C22×C3×C5C4×D411yescharacteristic
152C22×C3×C5#12511yescharacteristic
153C22×C3×C5#12611yescharacteristic
154C3×C4×C5—41no—
155C3×C4×C5—41no—
156C3×C4×C5—41no—
157C3×C4×C5—41no—
158C3×C4×C5#11711yescharacteristic
159C3×C4×C5C4×Q811yescharacteristic
160C3×C4×C5C22⋊Q811yescharacteristic
161C3×C4×C5#12311yescharacteristic
162C3×Dic5—42no—
163C22×C3×C4—101no—
164C22×C3×C4C22×D511yescharacteristic
165C3×M16—52no—
166C3×M16—101no—
167C3×M16—101no—
168C3×M16—101no—
169C2×C3×C8—21no—
170C2×C3×C8—21no—
171C2×C3×C8—51no—
172C2×C3×C8—51no—
173C2×C3×C8—101no—
174C2×C3×C8—101no—
175C2×C3×C8—101no—
176C2×C3×C8—101no—
177C2×C3×C8C2×Dic511yescharacteristic
178C2×C3×C8C2×Dic511yescharacteristic
179C3×C4⋊C4C2×Dic511yescharacteristic
180C3×C4⋊C4C2×Dic511yescharacteristic
181C3×C4⋊C4C2×Dic511yescharacteristic
182C3×C4⋊C4C2×Dic511yescharacteristic
183C3×C22⋊C4—21no—
184C3×C22⋊C4—21no—
185C3×C42—21no—
186C3×C42—21no—
187C3×C42—101no—
188C3×C42—101no—
189C23×C5C3×#4111yescharacteristic
190C2×C4×C5—21no—
191C2×C4×C5—21no—
192C2×C4×C5—21no—
193C2×C4×C5—21no—
194C2×C4×C5C22×C3×C411yescharacteristic
195C2×C4×C5C2×C3×D411yescharacteristic
196C2×C4×C5C2×C3×Q811yescharacteristic
197C2×C4×C5C3×#4111yescharacteristic
198C2×C4×C5C3×#4111yescharacteristic
199C2×C4×C5C3×#4111yescharacteristic
200C2×Dic5—22no—
201C2×Dic5—41no—
202C2×Dic5—41no—
203C5×C8—21no—
204C5×C8—21no—
205#168—22no—
206#168—21no—
207#168—21no—
208#168—41no—
209C2×M16—51no—
210C2×M16—51no—
211C22×C8—51no—
212C22×C8C2×C3×D511yescharacteristic
213#117C2×C3×D511yescharacteristic
214#117C2×C3×D511yescharacteristic
215C2×C42—51no—
216#107C3×Dic511yescharacteristic, complemented
217#107C3×Dic511yescharacteristic, complemented
218#106C3×Dic511yescharacteristic, complemented
219#106C3×Dic511yescharacteristic, complemented
220C4⋊C8—101no—
221C4⋊C8—101no—
222#97—101no—
223C4×C8—101no—
224C2×C3×C5—21no—
225C2×C3×C5—21no—
226C2×C3×C5#38711yescharacteristic
227C2×C3×C5#44311yescharacteristic
228C2×C3×C5#44411yescharacteristic
229C23×C3#71111yescharacteristic
230C2×C3×C4—21no—
231C2×C3×C4—21no—
232C2×C3×C4—21no—
233C2×C3×C4—21no—
234C2×C3×C4—102no—
235C2×C3×C4—201no—
236C2×C3×C4—201no—
237C2×C3×C4#70911yescharacteristic
238C2×C3×C4#70911yescharacteristic
239C2×C3×C4D4×D511yescharacteristic
240C2×C3×C4Q8×D511yescharacteristic
241C2×C3×C4#71311yescharacteristic
242C2×C3×C4C22×Dic511yescharacteristic
243C3×C8—21no—
244C3×C8—21no—
245C3×C8—101no—
246C3×C8—101no—
247C3×C8—102no—
248C3×C8—201no—
249C22×C5—21no—
250C22×C5—21no—
251C22×C5C3×C4×D411yescharacteristic
252C22×C5C3×#12511yescharacteristic
253C22×C5C3×#12611yescharacteristic
254C4×C5—41no—
255C4×C5—41no—
256C4×C5—41no—
257C4×C5—41no—
258C4×C5C3×#11711yescharacteristic
259C4×C5C3×C4×Q811yescharacteristic
260C4×C5C3×C22⋊Q811yescharacteristic
261C4×C5C3×#12311yescharacteristic
262Dic5—42no—
263C22×C4—101no—
264C22×C4C22×C3×D511yescharacteristic
265M16—52no—
266M16—101no—
267M16—101no—
268M16—101no—
269C2×C8—21no—
270C2×C8—21no—
271C2×C8—51no—
272C2×C8—51no—
273C2×C8—101no—
274C2×C8—101no—
275C2×C8—101no—
276C2×C8—101no—
277C2×C8C2×C3×Dic511yescharacteristic
278C2×C8C2×C3×Dic511yescharacteristic
279C4⋊C4C2×C3×Dic511yescharacteristic
280C4⋊C4C2×C3×Dic511yescharacteristic
281C4⋊C4C2×C3×Dic511yescharacteristic
282C4⋊C4C2×C3×Dic511yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—21no—
285C42—21no—
286C42—21no—
287C42—101no—
288C42—101no—
289C3×C5#192211yescharacteristic, complemented, Hall
290C22×C3—21no—
291C22×C3—21no—
292C22×C3#403511yescharacteristic
293C22×C3#404411yescharacteristic
294C22×C3D4×Dic511yescharacteristic
295C3×C4—41no—
296C3×C4—41no—
297C3×C4—41no—
298C3×C4—41no—
299C3×C4—202no—
300C3×C4#404011yescharacteristic
301C3×C4#404211yescharacteristic
302C3×C4#407211yescharacteristic
303C3×C4Q8×Dic511yescharacteristic
304C2×C5—21no—
305C2×C5—21no—
306C2×C5C3×#38711yescharacteristic
307C2×C5C3×#44311yescharacteristic
308C2×C5C3×#44411yescharacteristic
309C23C3×#71111yescharacteristic
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—102no—
315C2×C4—201no—
316C2×C4—201no—
317C2×C4C3×#70911yescharacteristic
318C2×C4C3×#70911yescharacteristic
319C2×C4C3×D4×D511yescharacteristic
320C2×C4C3×Q8×D511yescharacteristic
321C2×C4C3×#71311yescharacteristic
322C2×C4C22×C3×Dic511yescharacteristic
323C8—21no—
324C8—21no—
325C8—101no—
326C8—101no—
327C8—102no—
328C8—201no—
329C2×C3—21no—
330C2×C3—21no—
331C2×C3#6527011yescharacteristic
332C2×C3#6548511yescharacteristic
333C2×C3#6548611yescharacteristic
334C5C3×#192211yescharacteristic, complemented, Sylow
335C22—21no—
336C22—21no—
337C22C3×#403511yescharacteristic
338C22C3×#404411yescharacteristic
339C22C3×D4×Dic511yescharacteristic
340C4—41no—
341C4—41no—
342C4—41no—
343C4—41no—
344C4—202no—
345C4C3×#404011yescharacteristic
346C4C3×#404211yescharacteristic
347C4C3×#407211yescharacteristic
348C4C3×Q8×Dic511yescharacteristic
349C3#1060145411yescharacteristic, direct product factor, Sylow
350C2—21no—
351C2—21no—
352C2C3×#6527011yescharacteristic
353C2C3×#6548511yescharacteristic
354C2C3×#6548611yescharacteristic
subgroupG/H
centerC2×C3×C4D4×D5
commutatorC2×C4×C5C22×C3×C4
FrattiniC22×C4C22×C3×D5
FittingC3×C5×#427C2
socleC22×C3×C5C4×D4
prime radicalC23×C3×C5#41
cosocleC22×C4×C5C23×C3
hypercenterC3×#427D5
hypocenterC5C3×#1922
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3×C4 ◁ C22×C3×C4 ◁ C3×#427

Complements

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

NH ≅ G/Ntype
8. #10601454C3direct (N × H)
16. C3×C5×#107C4semidirect
17. C3×C5×#107C4semidirect
18. C3×C5×#106C4semidirect
19. C3×C5×#106C4semidirect
72. C5×#107C3×C4semidirect
73. C5×#107C3×C4semidirect
74. C5×#106C3×C4semidirect
75. C5×#106C3×C4semidirect
108. C3×#107Dic5semidirect
109. C3×#107Dic5semidirect
110. C3×#106Dic5semidirect
111. C3×#106Dic5semidirect
216. #107C3×Dic5semidirect
217. #107C3×Dic5semidirect
218. #106C3×Dic5semidirect
219. #106C3×Dic5semidirect
289. C3×C5#1922semidirect
334. C5C3×#1922semidirect
349. C3#10601454direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#4272
C3×#652882
C3×#652862
C3×#647722
C3×#647702
C3×#647522
#106014543
C3×#19225