Finite Group Databasebeta

← C3×C5×#3529 #14276563

Subgroups — C3×C5×#3529 #14276563

Order 1920.

2116 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#563C213yesmaximal, complemented
2C3×C5×#562C216yesmaximal, complemented
3C3×C5×H(3,𝔽4)C211yesmaximal, characteristic, complemented
4C3×C5×#560C216yesmaximal, complemented
5C3×C5×#532C213yesmaximal, complemented
6C3×C5×#529C216yesmaximal, complemented
7C3×C5×#525C216yesmaximal, complemented
8C5×#3529C311yesmaximal, characteristic, direct product factor, Hall
9C2×C3×C5×#41C2213yes—
10C2×C3×C5×#41C2211yescharacteristic, complemented
11C23×C3×C4×C5C2212yescomplemented
12C3×C5×C4⋊Q8C2213yescomplemented
13C3×C5×#127C2213yes—
14C3×C5×#126C22112yescomplemented
15C3×C5×#125C2216yescomplemented
16C3×C5×#124C2216yes—
17C3×C5×#124C2213yes—
18C3×C5×#124C2213yescomplemented
19C3×C5×#123C2216yes—
20C3×C5×#123C2216yescomplemented
21C3×C5×#123C22112yescomplemented
22C3×C5×C22⋊Q8C22112yescomplemented
23C3×C5×C22⋊Q8C2216yescomplemented
24C3×C5×#121C22112yes—
25C3×C5×#121C2216yes—
26C3×C5×#121C22112yescomplemented
27C3×C5×#120C2216yes—
28C3×C5×#120C2216yescomplemented
29C3×C4×C5×Q8C2212yescomplemented
30C3×C4×C5×D4C2216yes—
31C3×C4×C5×D4C22112yescomplemented
32C3×C5×#117C2216yescomplemented
33C2×C3×C42×C5C2213yescomplemented
34C3×#3529C511yesmaximal, characteristic, direct product factor, Hall
35C5×#563C2×C313yescomplemented
36C5×#562C2×C316yescomplemented
37C5×H(3,𝔽4)C2×C311yescharacteristic, complemented
38C5×#560C2×C316yescomplemented
39C5×#532C2×C313yescomplemented
40C5×#529C2×C316yescomplemented
41C5×#525C2×C316yescomplemented
42C24×C3×C5C2312yes—
43C3×C5×#41—46no—
44C3×C5×#41—42no—
45C2×C3×C5×Q8C2312yes—
46C2×C3×C5×Q8C2313yes—
47C2×C3×C5×Q8C2311yescharacteristic
48C2×C3×C5×D4C2316yes—
49C2×C3×C5×D4C23112yes—
50C2×C3×C5×D4C2313yes—
51C2×C3×C5×D4C2316yes—
52C2×C3×C5×D4C2313yes—
53C22×C3×C4×C5—44no—
54C22×C3×C4×C5C2313yes—
55C22×C3×C4×C5C2316yes—
56C22×C3×C4×C5C2316yes—
57C22×C3×C4×C5C2316yes—
58C3×C5×C4⋊C4C23112yes—
59C3×C5×C4⋊C4C2316yes—
60C3×C5×C4⋊C4C23112yes—
61C3×C5×C4⋊C4C2316yes—
62C3×C5×C22⋊C4C2316yes—
63C3×C5×C22⋊C4C2316yes—
64C3×C5×C22⋊C4C23112yes—
65C3×C5×C22⋊C4C23112yes—
66C3×C5×C22⋊C4C23112yes—
67C3×C42×C5C2313yes—
68C3×C42×C5C2313yes—
69C3×C42×C5C2316yes—
70C3×#563C2×C513yescomplemented
71C3×#562C2×C516yescomplemented
72C3×H(3,𝔽4)C2×C511yescharacteristic, complemented
73C3×#560C2×C516yescomplemented
74C3×#532C2×C513yescomplemented
75C3×#529C2×C516yescomplemented
76C3×#525C2×C516yescomplemented
77C2×C5×#41C22×C313yes—
78C2×C5×#41C22×C311yescharacteristic, complemented
79C23×C4×C5C22×C312yescomplemented
80C5×C4⋊Q8C22×C313yescomplemented
81C5×#127C22×C313yes—
82C5×#126C22×C3112yescomplemented
83C5×#125C22×C316yescomplemented
84C5×#124C22×C316yes—
85C5×#124C22×C313yes—
86C5×#124C22×C313yescomplemented
87C5×#123C22×C316yes—
88C5×#123C22×C316yescomplemented
89C5×#123C22×C3112yescomplemented
90C5×C22⋊Q8C22×C3112yescomplemented
91C5×C22⋊Q8C22×C316yescomplemented
92C5×#121C22×C3112yes—
93C5×#121C22×C316yes—
94C5×#121C22×C3112yescomplemented
95C5×#120C22×C316yes—
96C5×#120C22×C316yescomplemented
97C4×C5×Q8C22×C312yescomplemented
98C4×C5×D4C22×C316yes—
99C4×C5×D4C22×C3112yescomplemented
100C5×#117C22×C316yescomplemented
101C2×C42×C5C22×C313yescomplemented
102#3529C3×C511yescharacteristic, direct product factor, Sylow
103C23×C3×C5—42no—
104C23×C3×C5—44no—
105C23×C3×C5C2416yes—
106C23×C3×C5C2413yes—
107C3×C5×Q8—42no—
108C3×C5×Q8—43no—
109C3×C5×Q8—41no—
110C3×C5×D4—412no—
111C3×C5×D4—46no—
112C3×C5×D4—46no—
113C3×C5×D4—43no—
114C3×C5×D4—43no—
115C2×C3×C4×C5—212no—
116C2×C3×C4×C5—26no—
117C2×C3×C4×C5—46no—
118C2×C3×C4×C5—46no—
119C2×C3×C4×C5C2416yes—
120C2×C3×C4×C5C2416yes—
121C2×C3×C4×C5C2413yes—
122C2×C3×C4×C5C2416yes—
123C2×C3×C4×C5C2411yescharacteristic
124C2×C3×#41C22×C513yes—
125C2×C3×#41C22×C511yescharacteristic, complemented
126C23×C3×C4C22×C512yescomplemented
127C3×C4⋊Q8C22×C513yescomplemented
128C3×#127C22×C513yes—
129C3×#126C22×C5112yescomplemented
130C3×#125C22×C516yescomplemented
131C3×#124C22×C516yes—
132C3×#124C22×C513yes—
133C3×#124C22×C513yescomplemented
134C3×#123C22×C516yes—
135C3×#123C22×C5112yescomplemented
136C3×#123C22×C516yescomplemented
137C3×C22⋊Q8C22×C516yescomplemented
138C3×C22⋊Q8C22×C5112yescomplemented
139C3×#121C22×C516yes—
140C3×#121C22×C5112yes—
141C3×#121C22×C5112yescomplemented
142C3×#120C22×C516yes—
143C3×#120C22×C516yescomplemented
144C3×C4×Q8C22×C512yescomplemented
145C3×C4×D4C22×C516yes—
146C3×C4×D4C22×C5112yescomplemented
147C3×#117C22×C516yescomplemented
148C2×C3×C42C22×C513yescomplemented
149C24×C5C23×C312yes—
150C5×#41—46no—
151C5×#41—42no—
152C2×C5×Q8C23×C312yes—
153C2×C5×Q8C23×C313yes—
154C2×C5×Q8C23×C311yescharacteristic
155C2×C5×D4C23×C316yes—
156C2×C5×D4C23×C3112yes—
157C2×C5×D4C23×C313yes—
158C2×C5×D4C23×C316yes—
159C2×C5×D4C23×C313yes—
160C22×C4×C5—44no—
161C22×C4×C5C23×C313yes—
162C22×C4×C5C23×C316yes—
163C22×C4×C5C23×C316yes—
164C22×C4×C5C23×C316yes—
165C5×C4⋊C4C23×C3112yes—
166C5×C4⋊C4C23×C316yes—
167C5×C4⋊C4C23×C3112yes—
168C5×C4⋊C4C23×C316yes—
169C5×C22⋊C4C23×C316yes—
170C5×C22⋊C4C23×C316yes—
171C5×C22⋊C4C23×C3112yes—
172C5×C22⋊C4C23×C3112yes—
173C5×C22⋊C4C23×C3112yes—
174C42×C5C23×C316yes—
175C42×C5C23×C313yes—
176C42×C5C23×C313yes—
177#563C2×C3×C513yescomplemented
178#562C2×C3×C516yescomplemented
179H(3,𝔽4)C2×C3×C511yescharacteristic, complemented
180#560C2×C3×C516yescomplemented
181#532C2×C3×C513yescomplemented
182#529C2×C3×C516yescomplemented
183#525C2×C3×C516yescomplemented
184C22×C3×C5—26no—
185C22×C3×C5—23no—
186C22×C3×C5—26no—
187C22×C3×C5—212no—
188C22×C3×C5—48no—
189C22×C3×C5C2511yescharacteristic
190C3×C4×C5—26no—
191C3×C4×C5—23no—
192C3×C4×C5—26no—
193C3×C4×C5—26no—
194C3×C4×C5H(5,𝔽2)12yes—
195C24×C3C23×C512yes—
196C3×#41—46no—
197C3×#41—42no—
198C2×C3×Q8C23×C512yes—
199C2×C3×Q8C23×C513yes—
200C2×C3×Q8C23×C511yescharacteristic
201C2×C3×D4C23×C516yes—
202C2×C3×D4C23×C5112yes—
203C2×C3×D4C23×C513yes—
204C2×C3×D4C23×C516yes—
205C2×C3×D4C23×C513yes—
206C22×C3×C4—44no—
207C22×C3×C4C23×C513yes—
208C22×C3×C4C23×C516yes—
209C22×C3×C4C23×C516yes—
210C22×C3×C4C23×C516yes—
211C3×C4⋊C4C23×C5112yes—
212C3×C4⋊C4C23×C516yes—
213C3×C4⋊C4C23×C5112yes—
214C3×C4⋊C4C23×C516yes—
215C3×C22⋊C4C23×C516yes—
216C3×C22⋊C4C23×C516yes—
217C3×C22⋊C4C23×C5112yes—
218C3×C22⋊C4C23×C5112yes—
219C3×C22⋊C4C23×C5112yes—
220C3×C42C23×C516yes—
221C3×C42C23×C513yes—
222C3×C42C23×C513yes—
223C23×C5—42no—
224C23×C5—44no—
225C23×C5C24×C316yes—
226C23×C5C24×C313yes—
227C5×Q8—42no—
228C5×Q8—43no—
229C5×Q8—41no—
230C5×D4—412no—
231C5×D4—46no—
232C5×D4—46no—
233C5×D4—43no—
234C5×D4—43no—
235C2×C4×C5—212no—
236C2×C4×C5—26no—
237C2×C4×C5—46no—
238C2×C4×C5—46no—
239C2×C4×C5C24×C316yes—
240C2×C4×C5C24×C316yes—
241C2×C4×C5C24×C313yes—
242C2×C4×C5C24×C316yes—
243C2×C4×C5C24×C311yescharacteristic
244C2×#41C22×C3×C513yes—
245C2×#41C22×C3×C511yescharacteristic, complemented
246C23×C4C22×C3×C512yescomplemented
247C4⋊Q8C22×C3×C513yescomplemented
248#127C22×C3×C513yes—
249#126C22×C3×C5112yescomplemented
250#125C22×C3×C516yescomplemented
251#124C22×C3×C516yes—
252#124C22×C3×C513yes—
253#124C22×C3×C513yescomplemented
254#123C22×C3×C516yes—
255#123C22×C3×C516yescomplemented
256#123C22×C3×C5112yescomplemented
257C22⋊Q8C22×C3×C5112yescomplemented
258C22⋊Q8C22×C3×C516yescomplemented
259#121C22×C3×C5112yes—
260#121C22×C3×C516yes—
261#121C22×C3×C5112yescomplemented
262#120C22×C3×C516yes—
263#120C22×C3×C516yescomplemented
264C4×Q8C22×C3×C512yescomplemented
265C4×D4C22×C3×C516yes—
266C4×D4C22×C3×C5112yescomplemented
267#117C22×C3×C516yescomplemented
268C2×C42C22×C3×C513yescomplemented
269C2×C3×C5—43no—
270C2×C3×C5—46no—
271C2×C3×C5C2×H(5,𝔽2)11yescharacteristic
272C2×C3×C5#58512yes—
273C23×C3—42no—
274C23×C3—44no—
275C23×C3C24×C516yes—
276C23×C3C24×C513yes—
277C3×Q8—42no—
278C3×Q8—43no—
279C3×Q8—41no—
280C3×D4—412no—
281C3×D4—46no—
282C3×D4—46no—
283C3×D4—43no—
284C3×D4—43no—
285C2×C3×C4—212no—
286C2×C3×C4—26no—
287C2×C3×C4—46no—
288C2×C3×C4—46no—
289C2×C3×C4C24×C516yes—
290C2×C3×C4C24×C516yes—
291C2×C3×C4C24×C513yes—
292C2×C3×C4C24×C516yes—
293C2×C3×C4C24×C511yescharacteristic
294C22×C5—26no—
295C22×C5—23no—
296C22×C5—26no—
297C22×C5—212no—
298C22×C5—48no—
299C22×C5C25×C311yescharacteristic
300C4×C5—26no—
301C4×C5—23no—
302C4×C5—26no—
303C4×C5—26no—
304C4×C5C3×H(5,𝔽2)12yes—
305C24C23×C3×C512yes—
306#41—46no—
307#41—42no—
308C2×Q8C23×C3×C512yes—
309C2×Q8C23×C3×C513yes—
310C2×Q8C23×C3×C511yescharacteristic
311C2×D4C23×C3×C516yes—
312C2×D4C23×C3×C5112yes—
313C2×D4C23×C3×C513yes—
314C2×D4C23×C3×C516yes—
315C2×D4C23×C3×C513yes—
316C22×C4—44no—
317C22×C4C23×C3×C513yes—
318C22×C4C23×C3×C516yes—
319C22×C4C23×C3×C516yes—
320C22×C4C23×C3×C516yes—
321C4⋊C4C23×C3×C5112yes—
322C4⋊C4C23×C3×C516yes—
323C4⋊C4C23×C3×C5112yes—
324C4⋊C4C23×C3×C516yes—
325C22⋊C4C23×C3×C516yes—
326C22⋊C4C23×C3×C516yes—
327C22⋊C4C23×C3×C5112yes—
328C22⋊C4C23×C3×C5112yes—
329C22⋊C4C23×C3×C5112yes—
330C42C23×C3×C516yes—
331C42C23×C3×C513yes—
332C42C23×C3×C513yes—
333C3×C5#352911yescharacteristic, direct product factor, Hall
334C22×C3—26no—
335C22×C3—23no—
336C22×C3—26no—
337C22×C3—212no—
338C22×C3—48no—
339C22×C3C25×C511yescharacteristic
340C3×C4—26no—
341C3×C4—23no—
342C3×C4—26no—
343C3×C4—26no—
344C3×C4C5×H(5,𝔽2)12yes—
345C2×C5—46no—
346C2×C5—43no—
347C2×C5C2×C3×H(5,𝔽2)11yescharacteristic
348C2×C5C3×#58512yes—
349C23—42no—
350C23—44no—
351C23C24×C3×C516yes—
352C23C24×C3×C513yes—
353Q8—43no—
354Q8—41no—
355Q8—42no—
356D4—46no—
357D4—43no—
358D4—43no—
359D4—412no—
360D4—46no—
361C2×C4—212no—
362C2×C4—26no—
363C2×C4—46no—
364C2×C4—46no—
365C2×C4C24×C3×C516yes—
366C2×C4C24×C3×C516yes—
367C2×C4C24×C3×C513yes—
368C2×C4C24×C3×C516yes—
369C2×C4C24×C3×C511yescharacteristic
370C2×C3—46no—
371C2×C3—43no—
372C2×C3C2×C5×H(5,𝔽2)11yescharacteristic
373C2×C3C5×#58512yes—
374C5C3×#352911yescharacteristic, direct product factor, Sylow
375C22—26no—
376C22—23no—
377C22—26no—
378C22—212no—
379C22—48no—
380C22C25×C3×C511yescharacteristic
381C4—26no—
382C4—23no—
383C4—26no—
384C4—26no—
385C4C3×C5×H(5,𝔽2)12yes—
386C3C5×#352911yescharacteristic, direct product factor, Sylow
387C2—46no—
388C2—43no—
389C2C2×C3×C5×H(5,𝔽2)11yescharacteristic
390C2C3×C5×#58512yes—
subgroupG/H
centerC2×C3×C4×C5C24
commutator, Frattini, cosocleC22C25×C3×C5
socleC22×C3×C5C25
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C2×C3×C4×C5 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×C5×#563C2semidirect
2. C3×C5×#562C2semidirect
3. C3×C5×H(3,𝔽4)C2semidirect
4. C3×C5×#560C2semidirect
5. C3×C5×#532C2semidirect
6. C3×C5×#529C2semidirect
7. C3×C5×#525C2semidirect
8. C5×#3529C3direct (N × H)
10. C2×C3×C5×#41C22semidirect
11. C23×C3×C4×C5C22semidirect
12. C3×C5×C4⋊Q8C22semidirect
14. C3×C5×#126C22semidirect
15. C3×C5×#125C22semidirect
18. C3×C5×#124C22semidirect
20. C3×C5×#123C22semidirect
21. C3×C5×#123C22semidirect
22. C3×C5×C22⋊Q8C22semidirect
23. C3×C5×C22⋊Q8C22semidirect
26. C3×C5×#121C22semidirect
28. C3×C5×#120C22semidirect
29. C3×C4×C5×Q8C22semidirect
31. C3×C4×C5×D4C22semidirect
32. C3×C5×#117C22semidirect
33. C2×C3×C42×C5C22semidirect
34. C3×#3529C5direct (N × H)
35. C5×#563C2×C3semidirect
36. C5×#562C2×C3semidirect
37. C5×H(3,𝔽4)C2×C3semidirect
38. C5×#560C2×C3semidirect
39. C5×#532C2×C3semidirect
40. C5×#529C2×C3semidirect
41. C5×#525C2×C3semidirect
70. C3×#563C2×C5semidirect
71. C3×#562C2×C5semidirect
72. C3×H(3,𝔽4)C2×C5semidirect
73. C3×#560C2×C5semidirect
74. C3×#532C2×C5semidirect
75. C3×#529C2×C5semidirect
76. C3×#525C2×C5semidirect
78. C2×C5×#41C22×C3semidirect
79. C23×C4×C5C22×C3semidirect
80. C5×C4⋊Q8C22×C3semidirect
82. C5×#126C22×C3semidirect
83. C5×#125C22×C3semidirect
86. C5×#124C22×C3semidirect
88. C5×#123C22×C3semidirect
89. C5×#123C22×C3semidirect
90. C5×C22⋊Q8C22×C3semidirect
91. C5×C22⋊Q8C22×C3semidirect
94. C5×#121C22×C3semidirect
96. C5×#120C22×C3semidirect
97. C4×C5×Q8C22×C3semidirect
99. C4×C5×D4C22×C3semidirect
100. C5×#117C22×C3semidirect
101. C2×C42×C5C22×C3semidirect
102. #3529C3×C5direct (N × H)
125. C2×C3×#41C22×C5semidirect
126. C23×C3×C4C22×C5semidirect
127. C3×C4⋊Q8C22×C5semidirect
129. C3×#126C22×C5semidirect
130. C3×#125C22×C5semidirect
133. C3×#124C22×C5semidirect
135. C3×#123C22×C5semidirect
136. C3×#123C22×C5semidirect
137. C3×C22⋊Q8C22×C5semidirect
138. C3×C22⋊Q8C22×C5semidirect
141. C3×#121C22×C5semidirect
143. C3×#120C22×C5semidirect
144. C3×C4×Q8C22×C5semidirect
146. C3×C4×D4C22×C5semidirect
147. C3×#117C22×C5semidirect
148. C2×C3×C42C22×C5semidirect
177. #563C2×C3×C5semidirect
178. #562C2×C3×C5semidirect
179. H(3,𝔽4)C2×C3×C5semidirect
180. #560C2×C3×C5semidirect
181. #532C2×C3×C5semidirect
182. #529C2×C3×C5semidirect
183. #525C2×C3×C5semidirect
245. C2×#41C22×C3×C5semidirect
246. C23×C4C22×C3×C5semidirect
247. C4⋊Q8C22×C3×C5semidirect
249. #126C22×C3×C5semidirect
250. #125C22×C3×C5semidirect
253. #124C22×C3×C5semidirect
255. #123C22×C3×C5semidirect
256. #123C22×C3×C5semidirect
257. C22⋊Q8C22×C3×C5semidirect
258. C22⋊Q8C22×C3×C5semidirect
261. #121C22×C3×C5semidirect
263. #120C22×C3×C5semidirect
264. C4×Q8C22×C3×C5semidirect
266. C4×D4C22×C3×C5semidirect
267. #117C22×C3×C5semidirect
268. C2×C42C22×C3×C5semidirect
333. C3×C5#3529direct (N × H)
374. C5C3×#3529direct (N × H)
386. C3C5×#3529direct (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×#5622
C3×C5×H(3,𝔽4)2
C3×C5×#5602
C3×C5×#5322
C3×C5×#5292
C3×C5×#5252
C5×#35293
C3×#35295