Finite Group Databasebeta

← C3×#10602547 #14282756

Subgroups — C3×#10602547 #14282756

Order 1920.

476 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#529C211yesmaximal, characteristic
2C3×#65221C211yesmaximal, characteristic, complemented
3C3×#65214C211yesmaximal, characteristic, complemented
4C3×#64758C212yesmaximal, complemented
5C3×#64757C212yesmaximal
6#10602547C311yesmaximal, characteristic, direct product factor, Hall
7C3×C5×#126—21no—
8C3×C5×#125C411yescharacteristic, complemented
9C3×C5×#124C411yescharacteristic, complemented
10C3×C5×#123—21no—
11C3×C5×C22⋊Q8C2211yescharacteristic
12C3×C5×#121C2211yescharacteristic
13C3×C4×C5×Q8C411yescharacteristic, complemented
14C3×C4×C5×D4—21no—
15C3×C4×C5×D4C411yescharacteristic, complemented
16C3×C5×#117—21no—
17C2×C3×C42×C5C2211yescharacteristic
18C2×C3×C4×Dic5—21no—
19C3×#3963—21no—
20C3×#3963C2212yes—
21C3×C22⋊#168C2212yes—
22C3×C4⋊#168—21no—
23C3×C4×#168—21no—
24C3×#1500—51nomaximal, Hall
25C5×#529C2×C311yescharacteristic
26#65221C2×C311yescharacteristic, complemented
27#65214C2×C311yescharacteristic, complemented
28#64758C2×C312yescomplemented
29#64757C2×C312yes—
30C2×C3×C5×Q8C2×C411yescharacteristic
31C2×C3×C5×D4—21no—
32C2×C3×C5×D4C2×C411yescharacteristic
33C22×C3×C4×C5—21no—
34C22×C3×C4×C5D411yescharacteristic
35C22×C3×C4×C5D411yescharacteristic
36C22×C3×C4×C5C2311yescharacteristic
37C22×C3×Dic5—22no—
38C3×C5×C4⋊C4—21no—
39C3×C5×C4⋊C4—21no—
40C3×C5×C4⋊C4—21no—
41C3×C5×C4⋊C4—21no—
42C3×C5×C4⋊C4C2×C411yescharacteristic
43C3×C5×C4⋊C4C2×C411yescharacteristic
44C3×C5×C22⋊C4—21no—
45C3×C5×C22⋊C4—21no—
46C3×C5×C22⋊C4—21no—
47C3×C5×C22⋊C4—21no—
48C3×C5×C22⋊C4—21no—
49C3×C42×C5—21no—
50C3×C42×C5C2×C411yescharacteristic
51C3×C42×C5C2×C411yescharacteristic
52C3×C42×C5D411yescharacteristic
53C3×C42×C5D411yescharacteristic
54C3×C4×Dic5—24no—
55C2×C3×#168—22no—
56C3×#529D511yescharacteristic
57C3×#431—51no—
58C3×#376—51no—
59C3×#328—52no—
60C3×#327—52no—
61C5×#126—21no—
62C5×#125C3×C411yescharacteristic, complemented
63C5×#124C3×C411yescharacteristic, complemented
64C5×#123—21no—
65C5×C22⋊Q8C22×C311yescharacteristic
66C5×#121C22×C311yescharacteristic
67C4×C5×Q8C3×C411yescharacteristic, complemented
68C4×C5×D4—21no—
69C4×C5×D4C3×C411yescharacteristic, complemented
70C5×#117—21no—
71C2×C42×C5C22×C311yescharacteristic
72C2×C4×Dic5—21no—
73#3963—21no—
74#3963C22×C312yes—
75C22⋊#168C22×C312yes—
76C4⋊#168—21no—
77C4×#168—21no—
78#1500—51noSylow
79C23×C3×C5—21no—
80C23×C3×C5C2×D411yescharacteristic
81C3×C5×Q8—41no—
82C3×C5×D4—41no—
83C3×C5×D4—42no—
84C2×C3×C4×C5—22no—
85C2×C3×C4×C5—22no—
86C2×C3×C4×C5—21no—
87C2×C3×C4×C5—21no—
88C2×C3×C4×C5—21no—
89C2×C3×C4×C5—41no—
90C2×C3×C4×C5—42no—
91C2×C3×C4×C5C22⋊C411yescharacteristic
92C2×C3×C4×C5C22⋊C411yescharacteristic
93C2×C3×C4×C5C22⋊C411yescharacteristic
94C2×C3×C4×C5C22⋊C411yescharacteristic
95C2×C3×C4×C5C22×C411yescharacteristic
96C2×C3×C4×C5C2×D411yescharacteristic
97C2×C3×Dic5—24no—
98C2×C3×Dic5—42no—
99C2×C3×Dic5—42no—
100C3×#168—42no—
101C3×#126—21no—
102C3×#125Dic511yescharacteristic, complemented
103C3×#124Dic511yescharacteristic, complemented
104C3×#123—21no—
105C3×C22⋊Q8C2×D511yescharacteristic
106C3×#121C2×D511yescharacteristic
107C3×C4×Q8Dic511yescharacteristic, complemented
108C3×C4×D4—21no—
109C3×C4×D4Dic511yescharacteristic, complemented
110C3×#117—21no—
111C2×C3×C42—101no—
112C2×C3×C42C2×D511yescharacteristic
113C3×C4⋊C8—101no—
114C3×C22⋊C8—52no—
115C3×C4×C8—101no—
116C3×#95—52no—
117C3×#95—101no—
118C2×C5×Q8C2×C3×C411yescharacteristic
119C2×C5×D4—21no—
120C2×C5×D4C2×C3×C411yescharacteristic
121C22×C4×C5—21no—
122C22×C4×C5C3×D411yescharacteristic
123C22×C4×C5C3×D411yescharacteristic
124C22×C4×C5C23×C311yescharacteristic
125C22×Dic5—22no—
126C5×C4⋊C4—21no—
127C5×C4⋊C4—21no—
128C5×C4⋊C4—21no—
129C5×C4⋊C4—21no—
130C5×C4⋊C4C2×C3×C411yescharacteristic
131C5×C4⋊C4C2×C3×C411yescharacteristic
132C5×C22⋊C4—21no—
133C5×C22⋊C4—21no—
134C5×C22⋊C4—21no—
135C5×C22⋊C4—21no—
136C5×C22⋊C4—21no—
137C42×C5—21no—
138C42×C5C2×C3×C411yescharacteristic
139C42×C5C2×C3×C411yescharacteristic
140C42×C5C3×D411yescharacteristic
141C42×C5C3×D411yescharacteristic
142C4×Dic5—24no—
143C2×#168—22no—
144#529C3×D511yescharacteristic
145#431—51no—
146#376—51no—
147#328—52no—
148#327—52no—
149C22×C3×C5—21no—
150C22×C3×C5—21no—
151C22×C3×C5—41no—
152C22×C3×C5—41no—
153C22×C3×C5—41no—
154C22×C3×C5C2×C22⋊C411yescharacteristic
155C22×C3×C5#13512yes—
156C3×C4×C5—21no—
157C3×C4×C5—21no—
158C3×C4×C5—21no—
159C3×C4×C5—21no—
160C3×C4×C5—21no—
161C3×C4×C5—41no—
162C3×C4×C5—41no—
163C3×C4×C5—41no—
164C3×C4×C5#10412yes—
165C3×Dic5—44no—
166C2×C3×Q8C2×Dic511yescharacteristic
167C2×C3×D4—21no—
168C2×C3×D4C2×Dic511yescharacteristic
169C22×C3×C4—21no—
170C22×C3×C4—102no—
171C22×C3×C4C22⋊D511yescharacteristic
172C22×C3×C4C22⋊D511yescharacteristic
173C22×C3×C4C22×D511yescharacteristic
174C2×C3×C8—102no—
175C3×C4⋊C4—21no—
176C3×C4⋊C4—21no—
177C3×C4⋊C4—21no—
178C3×C4⋊C4—21no—
179C3×C4⋊C4C2×Dic511yescharacteristic
180C3×C4⋊C4C2×Dic511yescharacteristic
181C3×C22⋊C4—21no—
182C3×C22⋊C4—21no—
183C3×C22⋊C4—21no—
184C3×C22⋊C4—21no—
185C3×C22⋊C4—21no—
186C3×C42—21no—
187C3×C42—104no—
188C3×C42C2×Dic511yescharacteristic
189C3×C42C2×Dic511yescharacteristic
190C3×C42C22⋊D511yescharacteristic
191C3×C42C22⋊D511yescharacteristic
192C23×C5—21no—
193C23×C5C2×C3×D411yescharacteristic
194C5×Q8—41no—
195C5×D4—41no—
196C5×D4—42no—
197C2×C4×C5—22no—
198C2×C4×C5—22no—
199C2×C4×C5—21no—
200C2×C4×C5—21no—
201C2×C4×C5—21no—
202C2×C4×C5—41no—
203C2×C4×C5—42no—
204C2×C4×C5C3×C22⋊C411yescharacteristic
205C2×C4×C5C3×C22⋊C411yescharacteristic
206C2×C4×C5C3×C22⋊C411yescharacteristic
207C2×C4×C5C3×C22⋊C411yescharacteristic
208C2×C4×C5C22×C3×C411yescharacteristic
209C2×C4×C5C2×C3×D411yescharacteristic
210C2×Dic5—24no—
211C2×Dic5—42no—
212C2×Dic5—42no—
213#168—42no—
214#126—21no—
215#125C3×Dic511yescharacteristic, complemented
216#124C3×Dic511yescharacteristic, complemented
217#123—21no—
218C22⋊Q8C2×C3×D511yescharacteristic
219#121C2×C3×D511yescharacteristic
220C4×Q8C3×Dic511yescharacteristic, complemented
221C4×D4—21no—
222C4×D4C3×Dic511yescharacteristic, complemented
223#117—21no—
224C2×C42—101no—
225C2×C42C2×C3×D511yescharacteristic
226C4⋊C8—101no—
227C22⋊C8—52no—
228C4×C8—101no—
229#95—52no—
230#95—101no—
231C2×C3×C5—22no—
232C2×C3×C5—81no—
233C2×C3×C5#41011yescharacteristic
234C2×C3×C5#41611yescharacteristic
235C2×C3×C5C2×#10411yescharacteristic
236C23×C3—21no—
237C23×C3C2×C22⋊D511yescharacteristic
238C3×Q8—41no—
239C3×D4—41no—
240C3×D4—42no—
241C2×C3×C4—22no—
242C2×C3×C4—22no—
243C2×C3×C4—21no—
244C2×C3×C4—21no—
245C2×C3×C4—21no—
246C2×C3×C4—41no—
247C2×C3×C4—42no—
248C2×C3×C4—104no—
249C2×C3×C4—202no—
250C2×C3×C4—202no—
251C2×C3×C4C22⋊Dic511yescharacteristic
252C2×C3×C4C22⋊Dic511yescharacteristic
253C2×C3×C4C22⋊Dic511yescharacteristic
254C2×C3×C4C22⋊Dic511yescharacteristic
255C2×C3×C4C22×Dic511yescharacteristic
256C2×C3×C4C2×C22⋊D511yescharacteristic
257C3×C8—202no—
258C22×C5—21no—
259C22×C5—21no—
260C22×C5—41no—
261C22×C5—41no—
262C22×C5—41no—
263C22×C5C2×C3×C22⋊C411yescharacteristic
264C22×C5C3×#13512yes—
265C4×C5—21no—
266C4×C5—21no—
267C4×C5—21no—
268C4×C5—21no—
269C4×C5—21no—
270C4×C5—41no—
271C4×C5—41no—
272C4×C5—41no—
273C4×C5C3×#10412yes—
274Dic5—44no—
275C2×Q8C2×C3×Dic511yescharacteristic
276C2×D4—21no—
277C2×D4C2×C3×Dic511yescharacteristic
278C22×C4—21no—
279C22×C4—102no—
280C22×C4C3×C22⋊D511yescharacteristic
281C22×C4C3×C22⋊D511yescharacteristic
282C22×C4C22×C3×D511yescharacteristic
283C2×C8—102no—
284C4⋊C4—21no—
285C4⋊C4—21no—
286C4⋊C4—21no—
287C4⋊C4—21no—
288C4⋊C4C2×C3×Dic511yescharacteristic
289C4⋊C4C2×C3×Dic511yescharacteristic
290C22⋊C4—21no—
291C22⋊C4—21no—
292C22⋊C4—21no—
293C22⋊C4—21no—
294C22⋊C4—21no—
295C42—21no—
296C42—104no—
297C42C2×C3×Dic511yescharacteristic
298C42C2×C3×Dic511yescharacteristic
299C42C3×C22⋊D511yescharacteristic
300C42C3×C22⋊D511yescharacteristic
301C3×C5#150011yescharacteristic, complemented, Hall
302C22×C3—21no—
303C22×C3—21no—
304C22×C3—41no—
305C22×C3—41no—
306C22×C3—41no—
307C22×C3#409612yes—
308C22×C3C2×C22⋊Dic511yescharacteristic
309C3×C4—21no—
310C3×C4—21no—
311C3×C4—21no—
312C3×C4—21no—
313C3×C4—21no—
314C3×C4—41no—
315C3×C4—41no—
316C3×C4—41no—
317C3×C4—204no—
318C3×C4#396912yes—
319C2×C5—22no—
320C2×C5—81no—
321C2×C5C3×#41011yescharacteristic
322C2×C5C3×#41611yescharacteristic
323C2×C5C2×C3×#10411yescharacteristic
324C23—21no—
325C23C2×C3×C22⋊D511yescharacteristic
326Q8—41no—
327D4—41no—
328D4—42no—
329C2×C4—22no—
330C2×C4—22no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—41no—
335C2×C4—42no—
336C2×C4—104no—
337C2×C4—202no—
338C2×C4—202no—
339C2×C4C3×C22⋊Dic511yescharacteristic
340C2×C4C3×C22⋊Dic511yescharacteristic
341C2×C4C3×C22⋊Dic511yescharacteristic
342C2×C4C3×C22⋊Dic511yescharacteristic
343C2×C4C22×C3×Dic511yescharacteristic
344C2×C4C2×C3×C22⋊D511yescharacteristic
345C8—202no—
346C2×C3—22no—
347C2×C3—81no—
348C2×C3#6552211yescharacteristic
349C2×C3C2×#396911yescharacteristic
350C2×C3#6552911yescharacteristic
351C5C3×#150011yescharacteristic, complemented, Sylow
352C22—21no—
353C22—21no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22C3×#409612yes—
358C22C2×C3×C22⋊Dic511yescharacteristic
359C4—21no—
360C4—21no—
361C4—21no—
362C4—21no—
363C4—21no—
364C4—41no—
365C4—41no—
366C4—41no—
367C4—204no—
368C4C3×#396912yes—
369C3#1060254711yescharacteristic, direct product factor, Sylow
370C2—22no—
371C2—81no—
372C2C3×#6552211yescharacteristic
373C2C2×C3×#396911yescharacteristic
374C2C3×#6552911yescharacteristic
subgroupG/H
centerC2×C3×C4C22⋊Dic5
commutatorC2×C4×C5C22×C3×C4
FrattiniC22×C4C22×C3×D5
FittingC3×C5×#529C2
socleC22×C3×C5C2×C22⋊C4
prime radicalC3×C5×#121C22
cosocleC22×C4×C5C23×C3
hypercenterC3×#529D5
hypocenterC5C3×#1500
LCSG ▷ C2×C4×C5 ▷ C22×C5 ▷ C5
UCSC1 ◁ C2×C3×C4 ◁ C2×C3×C42 ◁ C3×#529

Complements

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

NH ≅ G/Ntype
2. C3×#65221C2semidirect
3. C3×#65214C2semidirect
4. C3×#64758C2semidirect
6. #10602547C3direct (N × H)
8. C3×C5×#125C4semidirect
9. C3×C5×#124C4semidirect
13. C3×C4×C5×Q8C4semidirect
15. C3×C4×C5×D4C4semidirect
26. #65221C2×C3semidirect
27. #65214C2×C3semidirect
28. #64758C2×C3semidirect
62. C5×#125C3×C4semidirect
63. C5×#124C3×C4semidirect
67. C4×C5×Q8C3×C4semidirect
69. C4×C5×D4C3×C4semidirect
102. C3×#125Dic5semidirect
103. C3×#124Dic5semidirect
107. C3×C4×Q8Dic5semidirect
109. C3×C4×D4Dic5semidirect
215. #125C3×Dic5semidirect
216. #124C3×Dic5semidirect
220. C4×Q8C3×Dic5semidirect
222. C4×D4C3×Dic5semidirect
301. C3×C5#1500semidirect
351. C5C3×#1500semidirect
369. C3#10602547direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#5292
C3×#652212
C3×#652142
C3×#647582
C3×#647572
#106025473
C3×#15005