Finite Group Databasebeta

← C3×C5×#3567 #14276601

Subgroups — C3×C5×#3567 #14276601

Order 1920.

1900 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#564C211yesmaximal, characteristic, complemented
2C3×C5×#562C213yesmaximal, complemented
3C3×C5×#553C216yesmaximal, complemented
4C3×C5×#552C216yesmaximal, complemented
5C3×C5×#540C216yesmaximal, complemented
6C3×C5×#539C216yesmaximal, complemented
7C3×C5×#529C213yesmaximal, complemented
8C5×#3567C311yesmaximal, characteristic, direct product factor, Hall
9C3×C5×#127C2212yes—
10C3×C5×#126C2216yes—
11C3×C5×#126C2212yes—
12C3×C5×#126C2216yes—
13C3×C5×#125C2213yes—
14C3×C5×#125C2216yes—
15C3×C5×#125C2216yes—
16C3×C5×#125C2216yes—
17C3×C5×#124C2213yes—
18C3×C5×#123C2216yes—
19C3×C5×#123C2216yes—
20C3×C5×#123C2216yes—
21C3×C5×#123C2216yes—
22C3×C5×#123C2216yes—
23C3×C5×C22⋊Q8C2213yes—
24C3×C5×C22⋊Q8C2216yes—
25C3×C5×C22⋊Q8C2216yes—
26C3×C5×#121C2213yes—
27C3×C5×#121C2216yes—
28C3×C5×#121C2216yes—
29C3×C5×#121C2216yes—
30C3×C5×#121C2216yes—
31C3×C4×C5×Q8C2213yes—
32C3×C4×C5×D4C2216yes—
33C3×C4×C5×D4C2213yes—
34C3×C4×C5×D4C2216yes—
35C3×C4×C5×D4C2216yes—
36C3×C5×#117C2216yes—
37C3×C5×#117C2216yes—
38C2×C3×C5×C4⋊C4C2216yes—
39C2×C3×C42×C5C2211yescharacteristic
40C3×#3567C511yesmaximal, characteristic, direct product factor, Hall
41C5×#564C2×C311yescharacteristic, complemented
42C5×#562C2×C313yescomplemented
43C5×#553C2×C316yescomplemented
44C5×#552C2×C316yescomplemented
45C5×#540C2×C316yescomplemented
46C5×#539C2×C316yescomplemented
47C5×#529C2×C313yescomplemented
48C2×C3×C5×Q8C2313yes—
49C2×C3×C5×D4C2316yes—
50C2×C3×C5×D4C2313yes—
51C2×C3×C5×D4C2316yes—
52C2×C3×C5×D4C2316yes—
53C22×C3×C4×C5C2313yes—
54C22×C3×C4×C5C2316yes—
55C22×C3×C4×C5C2316yes—
56C22×C3×C4×C5C2316yes—
57C3×C5×C4⋊C4C2316yes—
58C3×C5×C4⋊C4C2316yes—
59C3×C5×C4⋊C4C2316yes—
60C3×C5×C4⋊C4C2316yes—
61C3×C5×C4⋊C4C2316yes—
62C3×C5×C4⋊C4C2316yes—
63C3×C5×C4⋊C4C2316yes—
64C3×C5×C4⋊C4C2316yes—
65C3×C5×C4⋊C4C2313yes—
66C3×C5×C4⋊C4C2313yes—
67C3×C5×C22⋊C4C2316yes—
68C3×C5×C22⋊C4C2316yes—
69C3×C5×C22⋊C4C2316yes—
70C3×C5×C22⋊C4C2316yes—
71C3×C5×C22⋊C4C2316yes—
72C3×C5×C22⋊C4C2316yes—
73C3×C5×C22⋊C4C2316yes—
74C3×C42×C5C2313yes—
75C3×C42×C5C2311yescharacteristic
76C3×C42×C5C2312yes—
77C3×C42×C5C2316yes—
78C3×C42×C5C2312yes—
79C3×#564C2×C511yescharacteristic, complemented
80C3×#562C2×C513yescomplemented
81C3×#553C2×C516yescomplemented
82C3×#552C2×C516yescomplemented
83C3×#540C2×C516yescomplemented
84C3×#539C2×C516yescomplemented
85C3×#529C2×C513yescomplemented
86C5×#127C22×C312yes—
87C5×#126C22×C316yes—
88C5×#126C22×C312yes—
89C5×#126C22×C316yes—
90C5×#125C22×C313yes—
91C5×#125C22×C316yes—
92C5×#125C22×C316yes—
93C5×#125C22×C316yes—
94C5×#124C22×C313yes—
95C5×#123C22×C316yes—
96C5×#123C22×C316yes—
97C5×#123C22×C316yes—
98C5×#123C22×C316yes—
99C5×#123C22×C316yes—
100C5×C22⋊Q8C22×C313yes—
101C5×C22⋊Q8C22×C316yes—
102C5×C22⋊Q8C22×C316yes—
103C5×#121C22×C313yes—
104C5×#121C22×C316yes—
105C5×#121C22×C316yes—
106C5×#121C22×C316yes—
107C5×#121C22×C316yes—
108C4×C5×Q8C22×C313yes—
109C4×C5×D4C22×C316yes—
110C4×C5×D4C22×C313yes—
111C4×C5×D4C22×C316yes—
112C4×C5×D4C22×C316yes—
113C5×#117C22×C316yes—
114C5×#117C22×C316yes—
115C2×C5×C4⋊C4C22×C316yes—
116C2×C42×C5C22×C311yescharacteristic
117#3567C3×C511yescharacteristic, direct product factor, Sylow
118C23×C3×C5C2416yes—
119C23×C3×C5C2411yescharacteristic
120C3×C5×Q8—43no—
121C3×C5×D4—46no—
122C3×C5×D4—46no—
123C3×C5×D4—46no—
124C3×C5×D4—43no—
125C2×C3×C4×C5—46no—
126C2×C3×C4×C5—43no—
127C2×C3×C4×C5—46no—
128C2×C3×C4×C5—46no—
129C2×C3×C4×C5C2413yes—
130C2×C3×C4×C5C2413yes—
131C2×C3×C4×C5C2416yes—
132C2×C3×C4×C5C2416yes—
133C2×C3×C4×C5C2416yes—
134C3×#127C22×C512yes—
135C3×#126C22×C516yes—
136C3×#126C22×C512yes—
137C3×#126C22×C516yes—
138C3×#125C22×C513yes—
139C3×#125C22×C516yes—
140C3×#125C22×C516yes—
141C3×#125C22×C516yes—
142C3×#124C22×C513yes—
143C3×#123C22×C516yes—
144C3×#123C22×C516yes—
145C3×#123C22×C516yes—
146C3×#123C22×C516yes—
147C3×#123C22×C516yes—
148C3×C22⋊Q8C22×C516yes—
149C3×C22⋊Q8C22×C516yes—
150C3×C22⋊Q8C22×C513yes—
151C3×#121C22×C516yes—
152C3×#121C22×C516yes—
153C3×#121C22×C513yes—
154C3×#121C22×C516yes—
155C3×#121C22×C516yes—
156C3×C4×Q8C22×C513yes—
157C3×C4×D4C22×C516yes—
158C3×C4×D4C22×C513yes—
159C3×C4×D4C22×C516yes—
160C3×C4×D4C22×C516yes—
161C3×#117C22×C516yes—
162C3×#117C22×C516yes—
163C2×C3×C4⋊C4C22×C516yes—
164C2×C3×C42C22×C511yescharacteristic
165C2×C5×Q8C23×C313yes—
166C2×C5×D4C23×C316yes—
167C2×C5×D4C23×C313yes—
168C2×C5×D4C23×C316yes—
169C2×C5×D4C23×C316yes—
170C22×C4×C5C23×C313yes—
171C22×C4×C5C23×C316yes—
172C22×C4×C5C23×C316yes—
173C22×C4×C5C23×C316yes—
174C5×C4⋊C4C23×C316yes—
175C5×C4⋊C4C23×C316yes—
176C5×C4⋊C4C23×C316yes—
177C5×C4⋊C4C23×C316yes—
178C5×C4⋊C4C23×C316yes—
179C5×C4⋊C4C23×C316yes—
180C5×C4⋊C4C23×C316yes—
181C5×C4⋊C4C23×C316yes—
182C5×C4⋊C4C23×C313yes—
183C5×C4⋊C4C23×C313yes—
184C5×C22⋊C4C23×C316yes—
185C5×C22⋊C4C23×C316yes—
186C5×C22⋊C4C23×C316yes—
187C5×C22⋊C4C23×C316yes—
188C5×C22⋊C4C23×C316yes—
189C5×C22⋊C4C23×C316yes—
190C5×C22⋊C4C23×C316yes—
191C42×C5C23×C313yes—
192C42×C5C23×C311yescharacteristic
193C42×C5C23×C312yes—
194C42×C5C23×C316yes—
195C42×C5C23×C312yes—
196#564C2×C3×C511yescharacteristic, complemented
197#562C2×C3×C513yescomplemented
198#553C2×C3×C516yescomplemented
199#552C2×C3×C516yescomplemented
200#540C2×C3×C516yescomplemented
201#539C2×C3×C516yescomplemented
202#529C2×C3×C513yescomplemented
203C22×C3×C5—26no—
204C22×C3×C5—26no—
205C22×C3×C5—23no—
206C22×C3×C5—26no—
207C22×C3×C5C2511yescharacteristic
208C3×C4×C5—23no—
209C3×C4×C5—26no—
210C3×C4×C5—26no—
211C3×C4×C5—26no—
212C3×C4×C5—23no—
213C2×C3×Q8C23×C513yes—
214C2×C3×D4C23×C516yes—
215C2×C3×D4C23×C513yes—
216C2×C3×D4C23×C516yes—
217C2×C3×D4C23×C516yes—
218C22×C3×C4C23×C513yes—
219C22×C3×C4C23×C516yes—
220C22×C3×C4C23×C516yes—
221C22×C3×C4C23×C516yes—
222C3×C4⋊C4C23×C516yes—
223C3×C4⋊C4C23×C516yes—
224C3×C4⋊C4C23×C516yes—
225C3×C4⋊C4C23×C516yes—
226C3×C4⋊C4C23×C516yes—
227C3×C4⋊C4C23×C516yes—
228C3×C4⋊C4C23×C516yes—
229C3×C4⋊C4C23×C516yes—
230C3×C4⋊C4C23×C513yes—
231C3×C4⋊C4C23×C513yes—
232C3×C22⋊C4C23×C516yes—
233C3×C22⋊C4C23×C516yes—
234C3×C22⋊C4C23×C516yes—
235C3×C22⋊C4C23×C516yes—
236C3×C22⋊C4C23×C516yes—
237C3×C22⋊C4C23×C516yes—
238C3×C22⋊C4C23×C516yes—
239C3×C42C23×C513yes—
240C3×C42C23×C511yescharacteristic
241C3×C42C23×C512yes—
242C3×C42C23×C516yes—
243C3×C42C23×C512yes—
244C23×C5C24×C316yes—
245C23×C5C24×C311yescharacteristic
246C5×Q8—43no—
247C5×D4—46no—
248C5×D4—46no—
249C5×D4—46no—
250C5×D4—43no—
251C2×C4×C5—46no—
252C2×C4×C5—43no—
253C2×C4×C5—46no—
254C2×C4×C5—46no—
255C2×C4×C5C24×C313yes—
256C2×C4×C5C24×C313yes—
257C2×C4×C5C24×C316yes—
258C2×C4×C5C24×C316yes—
259C2×C4×C5C24×C316yes—
260#127C22×C3×C512yes—
261#126C22×C3×C516yes—
262#126C22×C3×C512yes—
263#126C22×C3×C516yes—
264#125C22×C3×C513yes—
265#125C22×C3×C516yes—
266#125C22×C3×C516yes—
267#125C22×C3×C516yes—
268#124C22×C3×C513yes—
269#123C22×C3×C516yes—
270#123C22×C3×C516yes—
271#123C22×C3×C516yes—
272#123C22×C3×C516yes—
273#123C22×C3×C516yes—
274C22⋊Q8C22×C3×C513yes—
275C22⋊Q8C22×C3×C516yes—
276C22⋊Q8C22×C3×C516yes—
277#121C22×C3×C513yes—
278#121C22×C3×C516yes—
279#121C22×C3×C516yes—
280#121C22×C3×C516yes—
281#121C22×C3×C516yes—
282C4×Q8C22×C3×C513yes—
283C4×D4C22×C3×C516yes—
284C4×D4C22×C3×C513yes—
285C4×D4C22×C3×C516yes—
286C4×D4C22×C3×C516yes—
287#117C22×C3×C516yes—
288#117C22×C3×C516yes—
289C2×C4⋊C4C22×C3×C516yes—
290C2×C42C22×C3×C511yescharacteristic
291C2×C3×C5—46no—
292C2×C3×C5—41no—
293C2×C3×C5#58513yes—
294C23×C3C24×C516yes—
295C23×C3C24×C511yescharacteristic
296C3×Q8—43no—
297C3×D4—46no—
298C3×D4—46no—
299C3×D4—46no—
300C3×D4—43no—
301C2×C3×C4—46no—
302C2×C3×C4—43no—
303C2×C3×C4—46no—
304C2×C3×C4—46no—
305C2×C3×C4C24×C513yes—
306C2×C3×C4C24×C513yes—
307C2×C3×C4C24×C516yes—
308C2×C3×C4C24×C516yes—
309C2×C3×C4C24×C516yes—
310C22×C5—26no—
311C22×C5—26no—
312C22×C5—23no—
313C22×C5—26no—
314C22×C5C25×C311yescharacteristic
315C4×C5—23no—
316C4×C5—26no—
317C4×C5—26no—
318C4×C5—26no—
319C4×C5—23no—
320C2×Q8C23×C3×C513yes—
321C2×D4C23×C3×C516yes—
322C2×D4C23×C3×C513yes—
323C2×D4C23×C3×C516yes—
324C2×D4C23×C3×C516yes—
325C22×C4C23×C3×C513yes—
326C22×C4C23×C3×C516yes—
327C22×C4C23×C3×C516yes—
328C22×C4C23×C3×C516yes—
329C4⋊C4C23×C3×C516yes—
330C4⋊C4C23×C3×C516yes—
331C4⋊C4C23×C3×C516yes—
332C4⋊C4C23×C3×C516yes—
333C4⋊C4C23×C3×C516yes—
334C4⋊C4C23×C3×C516yes—
335C4⋊C4C23×C3×C516yes—
336C4⋊C4C23×C3×C516yes—
337C4⋊C4C23×C3×C513yes—
338C4⋊C4C23×C3×C513yes—
339C22⋊C4C23×C3×C516yes—
340C22⋊C4C23×C3×C516yes—
341C22⋊C4C23×C3×C516yes—
342C22⋊C4C23×C3×C516yes—
343C22⋊C4C23×C3×C516yes—
344C22⋊C4C23×C3×C516yes—
345C22⋊C4C23×C3×C516yes—
346C42C23×C3×C513yes—
347C42C23×C3×C511yescharacteristic
348C42C23×C3×C512yes—
349C42C23×C3×C516yes—
350C42C23×C3×C512yes—
351C3×C5#356711yescharacteristic, direct product factor, Hall
352C22×C3—26no—
353C22×C3—26no—
354C22×C3—23no—
355C22×C3—26no—
356C22×C3C25×C511yescharacteristic
357C3×C4—23no—
358C3×C4—26no—
359C3×C4—26no—
360C3×C4—26no—
361C3×C4—23no—
362C2×C5—46no—
363C2×C5—41no—
364C2×C5C3×#58513yes—
365C23C24×C3×C516yes—
366C23C24×C3×C511yescharacteristic
367Q8—43no—
368D4—46no—
369D4—46no—
370D4—46no—
371D4—43no—
372C2×C4—46no—
373C2×C4—43no—
374C2×C4—46no—
375C2×C4—46no—
376C2×C4C24×C3×C513yes—
377C2×C4C24×C3×C513yes—
378C2×C4C24×C3×C516yes—
379C2×C4C24×C3×C516yes—
380C2×C4C24×C3×C516yes—
381C2×C3—46no—
382C2×C3—41no—
383C2×C3C5×#58513yes—
384C5C3×#356711yescharacteristic, direct product factor, Sylow
385C22—26no—
386C22—26no—
387C22—23no—
388C22—26no—
389C22C25×C3×C511yescharacteristic
390C4—23no—
391C4—26no—
392C4—26no—
393C4—26no—
394C4—23no—
395C3C5×#356711yescharacteristic, direct product factor, Sylow
396C2—46no—
397C2—41no—
398C2C3×C5×#58513yes—
subgroupG/H
center, socleC22×C3×C5C25
commutator, Frattini, cosocleC22C25×C3×C5
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×#564C2semidirect
2. C3×C5×#562C2semidirect
3. C3×C5×#553C2semidirect
4. C3×C5×#552C2semidirect
5. C3×C5×#540C2semidirect
6. C3×C5×#539C2semidirect
7. C3×C5×#529C2semidirect
8. C5×#3567C3direct (N × H)
40. C3×#3567C5direct (N × H)
41. C5×#564C2×C3semidirect
42. C5×#562C2×C3semidirect
43. C5×#553C2×C3semidirect
44. C5×#552C2×C3semidirect
45. C5×#540C2×C3semidirect
46. C5×#539C2×C3semidirect
47. C5×#529C2×C3semidirect
79. C3×#564C2×C5semidirect
80. C3×#562C2×C5semidirect
81. C3×#553C2×C5semidirect
82. C3×#552C2×C5semidirect
83. C3×#540C2×C5semidirect
84. C3×#539C2×C5semidirect
85. C3×#529C2×C5semidirect
117. #3567C3×C5direct (N × H)
196. #564C2×C3×C5semidirect
197. #562C2×C3×C5semidirect
198. #553C2×C3×C5semidirect
199. #552C2×C3×C5semidirect
200. #540C2×C3×C5semidirect
201. #539C2×C3×C5semidirect
202. #529C2×C3×C5semidirect
351. C3×C5#3567direct (N × H)
384. C5C3×#3567direct (N × H)
395. C3C5×#3567direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#5642
C3×C5×#5622
C3×C5×#5532
C3×C5×#5522
C3×C5×#5402
C3×C5×#5392
C3×C5×#5292
C5×#35673
C3×#35675