Finite Group Databasebeta

← C3×C5×#1598 #14274632

Subgroups — C3×C5×#1598 #14274632

Order 1920.

620 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#537C211yesmaximal, characteristic, complemented
2C3×C5×#447C212yesmaximal, complemented
3C3×C5×#394C211yesmaximal, characteristic, complemented
4C3×C5×#327C212yesmaximal, complemented
5C3×C5×#324C211yesmaximal, characteristic, complemented
6C5×#1598C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×C5×#41—21no—
8C22×C3×C5×D4C2211yescharacteristic
9C2×C3×C5×D8—22no—
10C3×C5×C22⋊Q8—22no—
11C3×C5×#121—22no—
12C3×C5×#121—21no—
13C3×C5×#121C2212yes—
14C2×C3×C5×C4⋊C4C2211yescharacteristic
15C2×C3×C5×C22⋊C4—21no—
16C2×C3×C5×C22⋊C4—21no—
17C3×C5×#102—22no—
18C3×C5×C22⋊C8C2212yes—
19C3×C5×#95C2211yescharacteristic
20C3×#1598C511yesmaximal, characteristic, direct product factor, Hall
21C5×#537C2×C311yescharacteristic, complemented
22C5×#447C2×C312yescomplemented
23C5×#394C2×C311yescharacteristic, complemented
24C5×#327C2×C312yescomplemented
25C5×#324C2×C311yescharacteristic, complemented
26C24×C3×C5—21no—
27C3×C5×#41—42no—
28C3×C5×#41—42no—
29C2×C3×C5×Q8—21no—
30C2×C3×C5×D4—21no—
31C2×C3×C5×D4—22no—
32C2×C3×C5×D4—21no—
33C2×C3×C5×D4—22no—
34C2×C3×C5×D4—42no—
35C2×C3×C5×D4D412yes—
36C22×C3×C4×C5—22no—
37C22×C3×C4×C5—21no—
38C22×C3×C4×C5D411yescharacteristic
39C22×C3×C4×C5D411yescharacteristic
40C22×C3×C4×C5C2311yescharacteristic
41C3×C5×D8—44no—
42C2×C3×C5×C8—22no—
43C3×C5×C4⋊C4—21no—
44C3×C5×C4⋊C4—22no—
45C3×C5×C4⋊C4D412yes—
46C3×C5×C22⋊C4—22no—
47C3×C5×C22⋊C4—22no—
48C3×C5×C22⋊C4—42no—
49C3×C5×C22⋊C4—42no—
50C3×#537C2×C511yescharacteristic, complemented
51C3×#447C2×C512yescomplemented
52C3×#394C2×C511yescharacteristic, complemented
53C3×#327C2×C512yescomplemented
54C3×#324C2×C511yescharacteristic, complemented
55C2×C5×#41—21no—
56C22×C5×D4C22×C311yescharacteristic
57C2×C5×D8—22no—
58C5×C22⋊Q8—22no—
59C5×#121—22no—
60C5×#121—21no—
61C5×#121C22×C312yes—
62C2×C5×C4⋊C4C22×C311yescharacteristic
63C2×C5×C22⋊C4—21no—
64C2×C5×C22⋊C4—21no—
65C5×#102—22no—
66C5×C22⋊C8C22×C312yes—
67C5×#95C22×C311yescharacteristic
68#1598C3×C511yescharacteristic, direct product factor, Sylow
69C23×C3×C5—22no—
70C23×C3×C5—22no—
71C23×C3×C5—41no—
72C23×C3×C5—41no—
73C23×C3×C5—41no—
74C23×C3×C5—41no—
75C23×C3×C5—41no—
76C23×C3×C5—41no—
77C23×C3×C5C2×D411yescharacteristic
78C3×C5×Q8—42no—
79C3×C5×D4—24no—
80C3×C5×D4—42no—
81C3×C5×D4—42no—
82C3×C5×D4—42no—
83C3×C5×D4—42no—
84C3×C5×D4—42no—
85C3×C5×D4—81no—
86C2×C3×C4×C5—21no—
87C2×C3×C4×C5—21no—
88C2×C3×C4×C5—21no—
89C2×C3×C4×C5—22no—
90C2×C3×C4×C5—41no—
91C2×C3×C4×C5—44no—
92C2×C3×C4×C5—41no—
93C2×C3×C4×C5—41no—
94C2×C3×C4×C5—41no—
95C2×C3×C4×C5—41no—
96C2×C3×C4×C5D812yescomplemented
97C2×C3×C4×C5C2×D412yes—
98C3×C5×C8—42no—
99C2×C3×#41—21no—
100C22×C3×D4C22×C511yescharacteristic
101C2×C3×D8—22no—
102C3×C22⋊Q8—22no—
103C3×#121—22no—
104C3×#121—21no—
105C3×#121C22×C512yes—
106C2×C3×C4⋊C4C22×C511yescharacteristic
107C2×C3×C22⋊C4—21no—
108C2×C3×C22⋊C4—21no—
109C3×#102—22no—
110C3×C22⋊C8C22×C512yes—
111C3×#95C22×C511yescharacteristic
112C24×C5—21no—
113C5×#41—42no—
114C5×#41—42no—
115C2×C5×Q8—21no—
116C2×C5×D4—21no—
117C2×C5×D4—22no—
118C2×C5×D4—21no—
119C2×C5×D4—22no—
120C2×C5×D4—42no—
121C2×C5×D4C3×D412yes—
122C22×C4×C5—22no—
123C22×C4×C5—21no—
124C22×C4×C5C3×D411yescharacteristic
125C22×C4×C5C3×D411yescharacteristic
126C22×C4×C5C23×C311yescharacteristic
127C5×D8—44no—
128C2×C5×C8—22no—
129C5×C4⋊C4—21no—
130C5×C4⋊C4—22no—
131C5×C4⋊C4C3×D412yes—
132C5×C22⋊C4—22no—
133C5×C22⋊C4—22no—
134C5×C22⋊C4—42no—
135C5×C22⋊C4—42no—
136#537C2×C3×C511yescharacteristic, complemented
137#447C2×C3×C512yescomplemented
138#394C2×C3×C511yescharacteristic, complemented
139#327C2×C3×C512yescomplemented
140#324C2×C3×C511yescharacteristic, complemented
141C22×C3×C5—21no—
142C22×C3×C5—21no—
143C22×C3×C5—42no—
144C22×C3×C5—42no—
145C22×C3×C5—42no—
146C22×C3×C5—42no—
147C22×C3×C5—42no—
148C22×C3×C5—42no—
149C22×C3×C5—81no—
150C22×C3×C5—81no—
151C22×C3×C5—81no—
152C22×C3×C5—81no—
153C22×C3×C5#12011yescharacteristic
154C22×C3×C5C2×D811yescharacteristic
155C22×C3×C5AGL(1,ℤ/8)11yescharacteristic
156C3×C4×C5—22no—
157C3×C4×C5—22no—
158C3×C4×C5—42no—
159C3×C4×C5—41no—
160C3×C4×C5—81no—
161C24×C3—21no—
162C3×#41—42no—
163C3×#41—42no—
164C2×C3×Q8—21no—
165C2×C3×D4—21no—
166C2×C3×D4—22no—
167C2×C3×D4—21no—
168C2×C3×D4—22no—
169C2×C3×D4—42no—
170C2×C3×D4C5×D412yes—
171C22×C3×C4—22no—
172C22×C3×C4—21no—
173C22×C3×C4C5×D411yescharacteristic
174C22×C3×C4C5×D411yescharacteristic
175C22×C3×C4C23×C511yescharacteristic
176C3×D8—44no—
177C2×C3×C8—22no—
178C3×C4⋊C4—21no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4C5×D412yes—
181C3×C22⋊C4—22no—
182C3×C22⋊C4—22no—
183C3×C22⋊C4—42no—
184C3×C22⋊C4—42no—
185C23×C5—22no—
186C23×C5—22no—
187C23×C5—41no—
188C23×C5—41no—
189C23×C5—41no—
190C23×C5—41no—
191C23×C5—41no—
192C23×C5—41no—
193C23×C5C2×C3×D411yescharacteristic
194C5×Q8—42no—
195C5×D4—24no—
196C5×D4—42no—
197C5×D4—42no—
198C5×D4—42no—
199C5×D4—42no—
200C5×D4—42no—
201C5×D4—81no—
202C2×C4×C5—21no—
203C2×C4×C5—21no—
204C2×C4×C5—21no—
205C2×C4×C5—22no—
206C2×C4×C5—41no—
207C2×C4×C5—44no—
208C2×C4×C5—41no—
209C2×C4×C5—41no—
210C2×C4×C5—41no—
211C2×C4×C5—41no—
212C2×C4×C5C3×D812yescomplemented
213C2×C4×C5C2×C3×D412yes—
214C5×C8—42no—
215C2×#41—21no—
216C22×D4C22×C3×C511yescharacteristic
217C2×D8—22no—
218C22⋊Q8—22no—
219#121—22no—
220#121—21no—
221#121C22×C3×C512yes—
222C2×C4⋊C4C22×C3×C511yescharacteristic
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225#102—22no—
226C22⋊C8C22×C3×C512yes—
227#95C22×C3×C511yescharacteristic
228C2×C3×C5—21no—
229C2×C3×C5—21no—
230C2×C3×C5—82no—
231C2×C3×C5—82no—
232C2×C3×C5#44711yescharacteristic
233C2×C3×C5#45411yescharacteristic
234C2×C3×C5B(4,𝔽2)11yescharacteristic
235C23×C3—22no—
236C23×C3—22no—
237C23×C3—41no—
238C23×C3—41no—
239C23×C3—41no—
240C23×C3—41no—
241C23×C3—41no—
242C23×C3—41no—
243C23×C3C2×C5×D411yescharacteristic
244C3×Q8—42no—
245C3×D4—24no—
246C3×D4—42no—
247C3×D4—42no—
248C3×D4—42no—
249C3×D4—42no—
250C3×D4—42no—
251C3×D4—81no—
252C2×C3×C4—21no—
253C2×C3×C4—21no—
254C2×C3×C4—21no—
255C2×C3×C4—22no—
256C2×C3×C4—41no—
257C2×C3×C4—44no—
258C2×C3×C4—41no—
259C2×C3×C4—41no—
260C2×C3×C4—41no—
261C2×C3×C4—41no—
262C2×C3×C4C5×D812yescomplemented
263C2×C3×C4C2×C5×D412yes—
264C3×C8—42no—
265C22×C5—21no—
266C22×C5—21no—
267C22×C5—42no—
268C22×C5—42no—
269C22×C5—42no—
270C22×C5—42no—
271C22×C5—42no—
272C22×C5—42no—
273C22×C5—81no—
274C22×C5—81no—
275C22×C5—81no—
276C22×C5—81no—
277C22×C5C3×#12011yescharacteristic
278C22×C5C2×C3×D811yescharacteristic
279C22×C5C3×AGL(1,ℤ/8)11yescharacteristic
280C4×C5—22no—
281C4×C5—22no—
282C4×C5—42no—
283C4×C5—41no—
284C4×C5—81no—
285C24—21no—
286#41—42no—
287#41—42no—
288C2×Q8—21no—
289C2×D4—21no—
290C2×D4—22no—
291C2×D4—21no—
292C2×D4—22no—
293C2×D4—42no—
294C2×D4C3×C5×D412yes—
295C22×C4—22no—
296C22×C4—21no—
297C22×C4C3×C5×D411yescharacteristic
298C22×C4C3×C5×D411yescharacteristic
299C22×C4C23×C3×C511yescharacteristic
300D8—44no—
301C2×C8—22no—
302C4⋊C4—21no—
303C4⋊C4—22no—
304C4⋊C4C3×C5×D412yes—
305C22⋊C4—22no—
306C22⋊C4—22no—
307C22⋊C4—42no—
308C22⋊C4—42no—
309C3×C5#159811yescharacteristic, direct product factor, Hall
310C22×C3—21no—
311C22×C3—21no—
312C22×C3—42no—
313C22×C3—42no—
314C22×C3—42no—
315C22×C3—42no—
316C22×C3—42no—
317C22×C3—42no—
318C22×C3—81no—
319C22×C3—81no—
320C22×C3—81no—
321C22×C3—81no—
322C22×C3C5×#12011yescharacteristic
323C22×C3C2×C5×D811yescharacteristic
324C22×C3C5×AGL(1,ℤ/8)11yescharacteristic
325C3×C4—22no—
326C3×C4—22no—
327C3×C4—42no—
328C3×C4—41no—
329C3×C4—81no—
330C2×C5—21no—
331C2×C5—21no—
332C2×C5—82no—
333C2×C5—82no—
334C2×C5C3×#44711yescharacteristic
335C2×C5C3×#45411yescharacteristic
336C2×C5C3×B(4,𝔽2)11yescharacteristic
337C23—22no—
338C23—22no—
339C23—41no—
340C23—41no—
341C23—41no—
342C23—41no—
343C23—41no—
344C23—41no—
345C23C2×C3×C5×D411yescharacteristic
346Q8—42no—
347D4—24no—
348D4—42no—
349D4—42no—
350D4—42no—
351D4—42no—
352D4—42no—
353D4—81no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—22no—
358C2×C4—41no—
359C2×C4—44no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4C3×C5×D812yescomplemented
365C2×C4C2×C3×C5×D412yes—
366C8—42no—
367C2×C3—21no—
368C2×C3—21no—
369C2×C3—82no—
370C2×C3—82no—
371C2×C3C5×#44711yescharacteristic
372C2×C3C5×#45411yescharacteristic
373C2×C3C5×B(4,𝔽2)11yescharacteristic
374C5C3×#159811yescharacteristic, direct product factor, Sylow
375C22—21no—
376C22—21no—
377C22—42no—
378C22—42no—
379C22—42no—
380C22—42no—
381C22—42no—
382C22—42no—
383C22—81no—
384C22—81no—
385C22—81no—
386C22—81no—
387C22C3×C5×#12011yescharacteristic
388C22C2×C3×C5×D811yescharacteristic
389C22C3×C5×AGL(1,ℤ/8)11yescharacteristic
390C4—22no—
391C4—22no—
392C4—41no—
393C4—42no—
394C4—81no—
395C3C5×#159811yescharacteristic, direct product factor, Sylow
396C2—21no—
397C2—21no—
398C2—82no—
399C2—82no—
400C2C3×C5×#44711yescharacteristic
401C2C3×C5×#45411yescharacteristic
402C2C3×C5×B(4,𝔽2)11yescharacteristic
subgroupG/H
center, socleC22×C3×C5#120
commutator, Frattini, cosocleC22×C4C23×C3×C5
LCSG ▷ C22×C4 ▷ C22 ▷ C1
UCSC1 ◁ C22×C3×C5 ◁ C22×C3×C4×C5 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×C5×#537C2semidirect
2. C3×C5×#447C2semidirect
3. C3×C5×#394C2semidirect
4. C3×C5×#327C2semidirect
5. C3×C5×#324C2semidirect
6. C5×#1598C3direct (N × H)
20. C3×#1598C5direct (N × H)
21. C5×#537C2×C3semidirect
22. C5×#447C2×C3semidirect
23. C5×#394C2×C3semidirect
24. C5×#327C2×C3semidirect
25. C5×#324C2×C3semidirect
50. C3×#537C2×C5semidirect
51. C3×#447C2×C5semidirect
52. C3×#394C2×C5semidirect
53. C3×#327C2×C5semidirect
54. C3×#324C2×C5semidirect
68. #1598C3×C5direct (N × H)
96. C2×C3×C4×C5D8semidirect
136. #537C2×C3×C5semidirect
137. #447C2×C3×C5semidirect
138. #394C2×C3×C5semidirect
139. #327C2×C3×C5semidirect
140. #324C2×C3×C5semidirect
212. C2×C4×C5C3×D8semidirect
262. C2×C3×C4C5×D8semidirect
309. C3×C5#1598direct (N × H)
364. C2×C4C3×C5×D8semidirect
374. C5C3×#1598direct (N × H)
395. C3C5×#1598direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#5372
C3×C5×#4472
C3×C5×#3942
C3×C5×#3272
C3×C5×#3242
C5×#15983
C3×#15985