Finite Group Databasebeta

← C3×C5×#3550 #14276584

Subgroups — C3×C5×#3550 #14276584

Order 1920.

2004 conjugacy classes of subgroups.

The subgroup lattice has 420 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#563C214yesmaximal, complemented
2C3×C5×#559C212yesmaximal, complemented
3C3×C5×Q82C212yesmaximal, complemented
4C3×C5×#556C214yesmaximal, complemented
5C3×C5×D4×Q8C214yesmaximal, complemented
6C3×C5×#542C218yesmaximal, complemented
7C3×C5×#536C214yesmaximal, complemented
8C3×C5×#533C212yesmaximal, complemented
9C3×C5×#532C211yesmaximal, characteristic
10C5×#3550C311yesmaximal, characteristic, direct product factor, Hall
11C2×C3×C5×#41C2212yes—
12C22×C3×C5×Q8C2214yes—
13C3×C5×C4⋊Q8C2211yescharacteristic
14C3×C5×C4⋊Q8C2212yes—
15C3×C5×C4⋊Q8C2212yes—
16C3×C5×C4⋊Q8C2218yes—
17C3×C5×C4⋊Q8C2218yes—
18C3×C5×#127C2211yescharacteristic
19C3×C5×#126C22116yes—
20C3×C5×#125C2214yes—
21C3×C5×#124C2212yes—
22C3×C5×#124C2218yes—
23C3×C5×#124C2218yes—
24C3×C5×#123C2218yes—
25C3×C5×#123C2218yes—
26C3×C5×C22⋊Q8C2214yes—
27C3×C5×C22⋊Q8C2214yes—
28C3×C5×C22⋊Q8C2218yes—
29C3×C5×C22⋊Q8C22116yes—
30C3×C5×#121C2214yes—
31C3×C4×C5×Q8C2214yes—
32C3×C4×C5×Q8C2214yes—
33C3×C4×C5×Q8C2218yes—
34C3×C4×C5×D4C2214yes—
35C3×C4×C5×D4C2218yes—
36C3×C5×#117C2214yes—
37C3×C5×#117C2214yes—
38C2×C3×C42×C5C2211yescharacteristic
39C3×#3550C511yesmaximal, characteristic, direct product factor, Hall
40C5×#563C2×C314yescomplemented
41C5×#559C2×C312yescomplemented
42C5×Q82C2×C312yescomplemented
43C5×#556C2×C314yescomplemented
44C5×D4×Q8C2×C314yescomplemented
45C5×#542C2×C318yescomplemented
46C5×#536C2×C314yescomplemented
47C5×#533C2×C312yescomplemented
48C5×#532C2×C311yescharacteristic
49C3×C5×#41—44no—
50C2×C3×C5×Q8—48no—
51C2×C3×C5×Q8C2314yes—
52C2×C3×C5×Q8C2314yes—
53C2×C3×C5×Q8C2318yes—
54C2×C3×C5×Q8C2312yes—
55C2×C3×C5×D4C2314yes—
56C2×C3×C5×D4C2314yes—
57C2×C3×C5×D4C2312yes—
58C22×C3×C4×C5C2312yes—
59C22×C3×C4×C5C2311yescharacteristic
60C22×C3×C4×C5C2314yes—
61C22×C3×C4×C5C2318yes—
62C3×C5×C4⋊C4C2314yes—
63C3×C5×C4⋊C4C2314yes—
64C3×C5×C4⋊C4C2314yes—
65C3×C5×C4⋊C4C2314yes—
66C3×C5×C4⋊C4C2314yes—
67C3×C5×C4⋊C4C2318yes—
68C3×C5×C4⋊C4C23116yes—
69C3×C5×C4⋊C4C2318yes—
70C3×C5×C22⋊C4C2318yes—
71C3×C5×C22⋊C4C2318yes—
72C3×C5×C22⋊C4C2318yes—
73C3×C5×C22⋊C4C23116yes—
74C3×C42×C5C2312yes—
75C3×C42×C5C2311yescharacteristic
76C3×C42×C5C2311yescharacteristic
77C3×C42×C5C2318yes—
78C3×C42×C5C2318yes—
79C3×#563C2×C514yescomplemented
80C3×#559C2×C512yescomplemented
81C3×Q82C2×C512yescomplemented
82C3×#556C2×C514yescomplemented
83C3×D4×Q8C2×C514yescomplemented
84C3×#542C2×C518yescomplemented
85C3×#536C2×C514yescomplemented
86C3×#533C2×C512yescomplemented
87C3×#532C2×C511yescharacteristic
88C2×C5×#41C22×C312yes—
89C22×C5×Q8C22×C314yes—
90C5×C4⋊Q8C22×C311yescharacteristic
91C5×C4⋊Q8C22×C312yes—
92C5×C4⋊Q8C22×C312yes—
93C5×C4⋊Q8C22×C318yes—
94C5×C4⋊Q8C22×C318yes—
95C5×#127C22×C311yescharacteristic
96C5×#126C22×C3116yes—
97C5×#125C22×C314yes—
98C5×#124C22×C312yes—
99C5×#124C22×C318yes—
100C5×#124C22×C318yes—
101C5×#123C22×C318yes—
102C5×#123C22×C318yes—
103C5×C22⋊Q8C22×C314yes—
104C5×C22⋊Q8C22×C314yes—
105C5×C22⋊Q8C22×C318yes—
106C5×C22⋊Q8C22×C3116yes—
107C5×#121C22×C314yes—
108C4×C5×Q8C22×C314yes—
109C4×C5×Q8C22×C314yes—
110C4×C5×Q8C22×C318yes—
111C4×C5×D4C22×C314yes—
112C4×C5×D4C22×C318yes—
113C5×#117C22×C314yes—
114C5×#117C22×C314yes—
115C2×C42×C5C22×C311yescharacteristic
116#3550C3×C511yescharacteristic, direct product factor, Sylow
117C23×C3×C5C2414yes—
118C23×C3×C5C2411yescharacteristic
119C3×C5×Q8—28no—
120C3×C5×Q8—28no—
121C3×C5×Q8—48no—
122C3×C5×Q8—42no—
123C3×C5×D4—28no—
124C3×C5×D4—42no—
125C3×C5×D4—44no—
126C2×C3×C4×C5—28no—
127C2×C3×C4×C5—22no—
128C2×C3×C4×C5—24no—
129C2×C3×C4×C5—48no—
130C2×C3×C4×C5C2418yes—
131C2×C3×C4×C5C2412yes—
132C2×C3×C4×C5C2414yes—
133C2×C3×C4×C5C2411yescharacteristic
134C2×C3×C4×C5C2418yes—
135C2×C3×C4×C5C2411yescharacteristic
136C2×C3×C4×C5C2412yes—
137C2×C3×#41C22×C512yes—
138C22×C3×Q8C22×C514yes—
139C3×C4⋊Q8C22×C511yescharacteristic
140C3×C4⋊Q8C22×C518yes—
141C3×C4⋊Q8C22×C518yes—
142C3×C4⋊Q8C22×C512yes—
143C3×C4⋊Q8C22×C512yes—
144C3×#127C22×C511yescharacteristic
145C3×#126C22×C5116yes—
146C3×#125C22×C514yes—
147C3×#124C22×C518yes—
148C3×#124C22×C512yes—
149C3×#124C22×C518yes—
150C3×#123C22×C518yes—
151C3×#123C22×C518yes—
152C3×C22⋊Q8C22×C5116yes—
153C3×C22⋊Q8C22×C514yes—
154C3×C22⋊Q8C22×C514yes—
155C3×C22⋊Q8C22×C518yes—
156C3×#121C22×C514yes—
157C3×C4×Q8C22×C514yes—
158C3×C4×Q8C22×C518yes—
159C3×C4×Q8C22×C514yes—
160C3×C4×D4C22×C518yes—
161C3×C4×D4C22×C514yes—
162C3×#117C22×C514yes—
163C3×#117C22×C514yes—
164C2×C3×C42C22×C511yescharacteristic
165C5×#41—44no—
166C2×C5×Q8—48no—
167C2×C5×Q8C23×C314yes—
168C2×C5×Q8C23×C314yes—
169C2×C5×Q8C23×C318yes—
170C2×C5×Q8C23×C312yes—
171C2×C5×D4C23×C314yes—
172C2×C5×D4C23×C314yes—
173C2×C5×D4C23×C312yes—
174C22×C4×C5C23×C312yes—
175C22×C4×C5C23×C311yescharacteristic
176C22×C4×C5C23×C314yes—
177C22×C4×C5C23×C318yes—
178C5×C4⋊C4C23×C314yes—
179C5×C4⋊C4C23×C314yes—
180C5×C4⋊C4C23×C314yes—
181C5×C4⋊C4C23×C314yes—
182C5×C4⋊C4C23×C314yes—
183C5×C4⋊C4C23×C318yes—
184C5×C4⋊C4C23×C3116yes—
185C5×C4⋊C4C23×C318yes—
186C5×C22⋊C4C23×C318yes—
187C5×C22⋊C4C23×C318yes—
188C5×C22⋊C4C23×C318yes—
189C5×C22⋊C4C23×C3116yes—
190C42×C5C23×C312yes—
191C42×C5C23×C311yescharacteristic
192C42×C5C23×C311yescharacteristic
193C42×C5C23×C318yes—
194C42×C5C23×C318yes—
195#563C2×C3×C514yescomplemented
196#559C2×C3×C512yescomplemented
197Q82C2×C3×C512yescomplemented
198#556C2×C3×C514yescomplemented
199D4×Q8C2×C3×C514yescomplemented
200#542C2×C3×C518yescomplemented
201#536C2×C3×C514yescomplemented
202#533C2×C3×C512yescomplemented
203#532C2×C3×C511yescharacteristic
204C22×C3×C5—24no—
205C22×C3×C5—21no—
206C22×C3×C5—22no—
207C22×C3×C5—28no—
208C22×C3×C5C2511yescharacteristic
209C3×C4×C5—21no—
210C3×C4×C5—24no—
211C3×C4×C5—28no—
212C3×C4×C5—22no—
213C3×C4×C5—28no—
214C3×C4×C5H(5,𝔽2)12yes—
215C3×C4×C52−1+414yes—
216C3×#41—44no—
217C2×C3×Q8—48no—
218C2×C3×Q8C23×C514yes—
219C2×C3×Q8C23×C514yes—
220C2×C3×Q8C23×C518yes—
221C2×C3×Q8C23×C512yes—
222C2×C3×D4C23×C514yes—
223C2×C3×D4C23×C514yes—
224C2×C3×D4C23×C512yes—
225C22×C3×C4C23×C512yes—
226C22×C3×C4C23×C511yescharacteristic
227C22×C3×C4C23×C514yes—
228C22×C3×C4C23×C518yes—
229C3×C4⋊C4C23×C514yes—
230C3×C4⋊C4C23×C514yes—
231C3×C4⋊C4C23×C514yes—
232C3×C4⋊C4C23×C514yes—
233C3×C4⋊C4C23×C514yes—
234C3×C4⋊C4C23×C518yes—
235C3×C4⋊C4C23×C5116yes—
236C3×C4⋊C4C23×C518yes—
237C3×C22⋊C4C23×C518yes—
238C3×C22⋊C4C23×C518yes—
239C3×C22⋊C4C23×C518yes—
240C3×C22⋊C4C23×C5116yes—
241C3×C42C23×C512yes—
242C3×C42C23×C511yescharacteristic
243C3×C42C23×C511yescharacteristic
244C3×C42C23×C518yes—
245C3×C42C23×C518yes—
246C23×C5C24×C314yes—
247C23×C5C24×C311yescharacteristic
248C5×Q8—28no—
249C5×Q8—28no—
250C5×Q8—48no—
251C5×Q8—42no—
252C5×D4—28no—
253C5×D4—42no—
254C5×D4—44no—
255C2×C4×C5—28no—
256C2×C4×C5—22no—
257C2×C4×C5—24no—
258C2×C4×C5—48no—
259C2×C4×C5C24×C318yes—
260C2×C4×C5C24×C312yes—
261C2×C4×C5C24×C314yes—
262C2×C4×C5C24×C311yescharacteristic
263C2×C4×C5C24×C318yes—
264C2×C4×C5C24×C311yescharacteristic
265C2×C4×C5C24×C312yes—
266C2×#41C22×C3×C512yes—
267C22×Q8C22×C3×C514yes—
268C4⋊Q8C22×C3×C511yescharacteristic
269C4⋊Q8C22×C3×C512yes—
270C4⋊Q8C22×C3×C512yes—
271C4⋊Q8C22×C3×C518yes—
272C4⋊Q8C22×C3×C518yes—
273#127C22×C3×C511yescharacteristic
274#126C22×C3×C5116yes—
275#125C22×C3×C514yes—
276#124C22×C3×C512yes—
277#124C22×C3×C518yes—
278#124C22×C3×C518yes—
279#123C22×C3×C518yes—
280#123C22×C3×C518yes—
281C22⋊Q8C22×C3×C514yes—
282C22⋊Q8C22×C3×C514yes—
283C22⋊Q8C22×C3×C518yes—
284C22⋊Q8C22×C3×C5116yes—
285#121C22×C3×C514yes—
286C4×Q8C22×C3×C514yes—
287C4×Q8C22×C3×C514yes—
288C4×Q8C22×C3×C518yes—
289C4×D4C22×C3×C514yes—
290C4×D4C22×C3×C518yes—
291#117C22×C3×C514yes—
292#117C22×C3×C514yes—
293C2×C42C22×C3×C511yescharacteristic
294C2×C3×C5—44no—
295C2×C3×C5—41no—
296C2×C3×C5C2×H(5,𝔽2)11yescharacteristic
297C2×C3×C5C2×2−1+412yes—
298C23×C3C24×C514yes—
299C23×C3C24×C511yescharacteristic
300C3×Q8—28no—
301C3×Q8—28no—
302C3×Q8—48no—
303C3×Q8—42no—
304C3×D4—28no—
305C3×D4—42no—
306C3×D4—44no—
307C2×C3×C4—28no—
308C2×C3×C4—22no—
309C2×C3×C4—24no—
310C2×C3×C4—48no—
311C2×C3×C4C24×C518yes—
312C2×C3×C4C24×C512yes—
313C2×C3×C4C24×C514yes—
314C2×C3×C4C24×C511yescharacteristic
315C2×C3×C4C24×C518yes—
316C2×C3×C4C24×C511yescharacteristic
317C2×C3×C4C24×C512yes—
318C22×C5—24no—
319C22×C5—21no—
320C22×C5—22no—
321C22×C5—28no—
322C22×C5C25×C311yescharacteristic
323C4×C5—21no—
324C4×C5—24no—
325C4×C5—28no—
326C4×C5—22no—
327C4×C5—28no—
328C4×C5C3×H(5,𝔽2)12yes—
329C4×C5C3×2−1+414yes—
330#41—44no—
331C2×Q8—48no—
332C2×Q8C23×C3×C514yes—
333C2×Q8C23×C3×C514yes—
334C2×Q8C23×C3×C518yes—
335C2×Q8C23×C3×C512yes—
336C2×D4C23×C3×C514yes—
337C2×D4C23×C3×C514yes—
338C2×D4C23×C3×C512yes—
339C22×C4C23×C3×C512yes—
340C22×C4C23×C3×C511yescharacteristic
341C22×C4C23×C3×C514yes—
342C22×C4C23×C3×C518yes—
343C4⋊C4C23×C3×C514yes—
344C4⋊C4C23×C3×C514yes—
345C4⋊C4C23×C3×C514yes—
346C4⋊C4C23×C3×C514yes—
347C4⋊C4C23×C3×C514yes—
348C4⋊C4C23×C3×C518yes—
349C4⋊C4C23×C3×C5116yes—
350C4⋊C4C23×C3×C518yes—
351C22⋊C4C23×C3×C518yes—
352C22⋊C4C23×C3×C518yes—
353C22⋊C4C23×C3×C518yes—
354C22⋊C4C23×C3×C5116yes—
355C42C23×C3×C512yes—
356C42C23×C3×C511yescharacteristic
357C42C23×C3×C511yescharacteristic
358C42C23×C3×C518yes—
359C42C23×C3×C518yes—
360C3×C5#355011yescharacteristic, direct product factor, Hall
361C22×C3—24no—
362C22×C3—21no—
363C22×C3—22no—
364C22×C3—28no—
365C22×C3C25×C511yescharacteristic
366C3×C4—21no—
367C3×C4—24no—
368C3×C4—28no—
369C3×C4—22no—
370C3×C4—28no—
371C3×C4C5×H(5,𝔽2)12yes—
372C3×C4C5×2−1+414yes—
373C2×C5—44no—
374C2×C5—41no—
375C2×C5C2×C3×H(5,𝔽2)11yescharacteristic
376C2×C5C2×C3×2−1+412yes—
377C23C24×C3×C514yes—
378C23C24×C3×C511yescharacteristic
379Q8—28no—
380Q8—28no—
381Q8—48no—
382Q8—42no—
383D4—28no—
384D4—42no—
385D4—44no—
386C2×C4—28no—
387C2×C4—22no—
388C2×C4—24no—
389C2×C4—48no—
390C2×C4C24×C3×C518yes—
391C2×C4C24×C3×C512yes—
392C2×C4C24×C3×C514yes—
393C2×C4C24×C3×C511yescharacteristic
394C2×C4C24×C3×C518yes—
395C2×C4C24×C3×C511yescharacteristic
396C2×C4C24×C3×C512yes—
397C2×C3—44no—
398C2×C3—41no—
399C2×C3C2×C5×H(5,𝔽2)11yescharacteristic
400C2×C3C2×C5×2−1+412yes—
401C5C3×#355011yescharacteristic, direct product factor, Sylow
402C22—24no—
403C22—21no—
404C22—22no—
405C22—28no—
406C22C25×C3×C511yescharacteristic
407C4—21no—
408C4—24no—
409C4—28no—
410C4—22no—
411C4—28no—
412C4C3×C5×H(5,𝔽2)12yes—
413C4C3×C5×2−1+414yes—
414C3C5×#355011yescharacteristic, direct product factor, Sylow
415C2—44no—
416C2—41no—
417C2C2×C3×C5×H(5,𝔽2)11yescharacteristic
418C2C2×C3×C5×2−1+412yes—
subgroupG/H
center, socleC22×C3×C5C25
commutator, Frattini, cosocleC22C25×C3×C5
prime radicalC3×C5×#532C2
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C3×C5 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×C5×#563C2semidirect
2. C3×C5×#559C2semidirect
3. C3×C5×Q82C2semidirect
4. C3×C5×#556C2semidirect
5. C3×C5×D4×Q8C2semidirect
6. C3×C5×#542C2semidirect
7. C3×C5×#536C2semidirect
8. C3×C5×#533C2semidirect
10. C5×#3550C3direct (N × H)
39. C3×#3550C5direct (N × H)
40. C5×#563C2×C3semidirect
41. C5×#559C2×C3semidirect
42. C5×Q82C2×C3semidirect
43. C5×#556C2×C3semidirect
44. C5×D4×Q8C2×C3semidirect
45. C5×#542C2×C3semidirect
46. C5×#536C2×C3semidirect
47. C5×#533C2×C3semidirect
79. C3×#563C2×C5semidirect
80. C3×#559C2×C5semidirect
81. C3×Q82C2×C5semidirect
82. C3×#556C2×C5semidirect
83. C3×D4×Q8C2×C5semidirect
84. C3×#542C2×C5semidirect
85. C3×#536C2×C5semidirect
86. C3×#533C2×C5semidirect
116. #3550C3×C5direct (N × H)
195. #563C2×C3×C5semidirect
196. #559C2×C3×C5semidirect
197. Q82C2×C3×C5semidirect
198. #556C2×C3×C5semidirect
199. D4×Q8C2×C3×C5semidirect
200. #542C2×C3×C5semidirect
201. #536C2×C3×C5semidirect
202. #533C2×C3×C5semidirect
360. C3×C5#3550direct (N × H)
401. C5C3×#3550direct (N × H)
414. C3C5×#3550direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#5632
C3×C5×#5592
C3×C5×Q822
C3×C5×#5562
C3×C5×D4×Q82
C3×C5×#5422
C3×C5×#5362
C3×C5×#5332
C3×C5×#5322
C5×#35503
C3×#35505