Finite Group Databasebeta

← C3×C5×#1628 #14274662

Subgroups — C3×C5×#1628 #14274662

Order 1920.

432 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#556C211yesmaximal, characteristic, complemented
2C3×C5×#486C211yesmaximal, characteristic, complemented
3C3×C5×#460C211yesmaximal, characteristic, complemented
4C3×C5×#459C211yesmaximal, characteristic
5C3×C5×#332C211yesmaximal, characteristic, complemented
6C3×C5×Q8⋊C8C211yesmaximal, characteristic, complemented
7C3×C5×D4⋊C8C211yesmaximal, characteristic, complemented
8C5×#1628C311yesmaximal, characteristic, direct product factor, Hall
9C2×C3×C5×SD16—21no—
10C2×C3×C5×D8—21no—
11C3×C5×C4⋊Q8C2211yescharacteristic
12C3×C5×#127C2211yescharacteristic
13C3×C5×#126—22no—
14C3×C5×#124—21no—
15C3×C5×C22⋊Q8—21no—
16C3×C4×C5×Q8—21no—
17C3×C4×C5×Q8C2211yescharacteristic
18C3×C4×C5×D4C2211yescharacteristic
19C3×C5×#117—21no—
20C3×C5×C4⋊C8C2211yescharacteristic
21C3×C5×C4⋊C8C2211yescharacteristic
22C3×C5×#102—21no—
23C3×C5×#102—21no—
24C3×C5×#102—22no—
25C3×C4×C5×C8C2211yescharacteristic
26C3×#1628C511yesmaximal, characteristic, direct product factor, Hall
27C5×#556C2×C311yescharacteristic, complemented
28C5×#486C2×C311yescharacteristic, complemented
29C5×#460C2×C311yescharacteristic, complemented
30C5×#459C2×C311yescharacteristic
31C5×#332C2×C311yescharacteristic, complemented
32C5×Q8⋊C8C2×C311yescharacteristic, complemented
33C5×D4⋊C8C2×C311yescharacteristic, complemented
34C2×C3×C5×Q8—21no—
35C2×C3×C5×Q8D411yescharacteristic
36C2×C3×C5×D4—21no—
37C2×C3×C5×D4—21no—
38C2×C3×C5×D4—21no—
39C2×C3×C5×D4D411yescharacteristic
40C22×C3×C4×C5—21no—
41C3×C5×SD16—42no—
42C3×C5×D8—42no—
43C2×C3×C5×C8—21no—
44C2×C3×C5×C8—21no—
45C2×C3×C5×C8—21no—
46C3×C5×C4⋊C4—22no—
47C3×C5×C4⋊C4—21no—
48C3×C5×C4⋊C4—21no—
49C3×C5×C4⋊C4D411yescharacteristic
50C3×C5×C4⋊C4D411yescharacteristic
51C3×C5×C4⋊C4D412yes—
52C3×C5×C22⋊C4—22no—
53C3×C5×C22⋊C4—22no—
54C3×C5×C22⋊C4—21no—
55C3×C42×C5—22no—
56C3×C42×C5—21no—
57C3×C42×C5C2311yescharacteristic
58C3×#556C2×C511yescharacteristic, complemented
59C3×#486C2×C511yescharacteristic, complemented
60C3×#460C2×C511yescharacteristic, complemented
61C3×#459C2×C511yescharacteristic
62C3×#332C2×C511yescharacteristic, complemented
63C3×Q8⋊C8C2×C511yescharacteristic, complemented
64C3×D4⋊C8C2×C511yescharacteristic, complemented
65C2×C5×SD16—21no—
66C2×C5×D8—21no—
67C5×C4⋊Q8C22×C311yescharacteristic
68C5×#127C22×C311yescharacteristic
69C5×#126—22no—
70C5×#124—21no—
71C5×C22⋊Q8—21no—
72C4×C5×Q8—21no—
73C4×C5×Q8C22×C311yescharacteristic
74C4×C5×D4C22×C311yescharacteristic
75C5×#117—21no—
76C5×C4⋊C8C22×C311yescharacteristic
77C5×C4⋊C8C22×C311yescharacteristic
78C5×#102—21no—
79C5×#102—21no—
80C5×#102—22no—
81C4×C5×C8C22×C311yescharacteristic
82#1628C3×C511yescharacteristic, direct product factor, Sylow
83C23×C3×C5—21no—
84C23×C3×C5—41no—
85C3×C5×Q8—21no—
86C3×C5×Q8—21no—
87C3×C5×Q8—41no—
88C3×C5×Q8—41no—
89C3×C5×D4—21no—
90C3×C5×D4—21no—
91C3×C5×D4—42no—
92C3×C5×D4—41no—
93C3×C5×D4—41no—
94C3×C5×D4—41no—
95C3×C5×D4—41no—
96C2×C3×C4×C5—21no—
97C2×C3×C4×C5—21no—
98C2×C3×C4×C5—21no—
99C2×C3×C4×C5—22no—
100C2×C3×C4×C5—41no—
101C2×C3×C4×C5—41no—
102C2×C3×C4×C5C2×D411yescharacteristic
103C2×C3×C4×C5C2×D411yescharacteristic
104C2×C3×C4×C5C2×D411yescharacteristic
105C3×C5×C8—41no—
106C3×C5×C8—41no—
107C3×C5×C8—41no—
108C2×C3×SD16—21no—
109C2×C3×D8—21no—
110C3×C4⋊Q8C22×C511yescharacteristic
111C3×#127C22×C511yescharacteristic
112C3×#126—22no—
113C3×#124—21no—
114C3×C22⋊Q8—21no—
115C3×C4×Q8—21no—
116C3×C4×Q8C22×C511yescharacteristic
117C3×C4×D4C22×C511yescharacteristic
118C3×#117—21no—
119C3×C4⋊C8C22×C511yescharacteristic
120C3×C4⋊C8C22×C511yescharacteristic
121C3×#102—21no—
122C3×#102—21no—
123C3×#102—22no—
124C3×C4×C8C22×C511yescharacteristic
125C2×C5×Q8—21no—
126C2×C5×Q8C3×D411yescharacteristic
127C2×C5×D4—21no—
128C2×C5×D4—21no—
129C2×C5×D4—21no—
130C2×C5×D4C3×D411yescharacteristic
131C22×C4×C5—21no—
132C5×SD16—42no—
133C5×D8—42no—
134C2×C5×C8—21no—
135C2×C5×C8—21no—
136C2×C5×C8—21no—
137C5×C4⋊C4—22no—
138C5×C4⋊C4—21no—
139C5×C4⋊C4—21no—
140C5×C4⋊C4C3×D411yescharacteristic
141C5×C4⋊C4C3×D411yescharacteristic
142C5×C4⋊C4C3×D412yes—
143C5×C22⋊C4—22no—
144C5×C22⋊C4—22no—
145C5×C22⋊C4—21no—
146C42×C5—22no—
147C42×C5—21no—
148C42×C5C23×C311yescharacteristic
149#556C2×C3×C511yescharacteristic, complemented
150#486C2×C3×C511yescharacteristic, complemented
151#460C2×C3×C511yescharacteristic, complemented
152#459C2×C3×C511yescharacteristic
153#332C2×C3×C511yescharacteristic, complemented
154Q8⋊C8C2×C3×C511yescharacteristic, complemented
155D4⋊C8C2×C3×C511yescharacteristic, complemented
156C22×C3×C5—41no—
157C22×C3×C5—41no—
158C22×C3×C5—41no—
159C22×C3×C5—81no—
160C22×C3×C5—81no—
161C22×C3×C5—81no—
162C22×C3×C5#12011yescharacteristic
163C3×C4×C5—21no—
164C3×C4×C5—41no—
165C3×C4×C5—41no—
166C3×C4×C5—42no—
167C3×C4×C5—41no—
168C3×C4×C5#13511yescharacteristic
169C3×C4×C5#13511yescharacteristic
170C3×C4×C5AGL(1,ℤ/8)11yescharacteristic
171C3×C4×C5AGL(1,ℤ/8)11yescharacteristic
172C2×C3×Q8—21no—
173C2×C3×Q8C5×D411yescharacteristic
174C2×C3×D4—21no—
175C2×C3×D4—21no—
176C2×C3×D4—21no—
177C2×C3×D4C5×D411yescharacteristic
178C22×C3×C4—21no—
179C3×SD16—42no—
180C3×D8—42no—
181C2×C3×C8—21no—
182C2×C3×C8—21no—
183C2×C3×C8—21no—
184C3×C4⋊C4—22no—
185C3×C4⋊C4—21no—
186C3×C4⋊C4—21no—
187C3×C4⋊C4C5×D411yescharacteristic
188C3×C4⋊C4C5×D411yescharacteristic
189C3×C4⋊C4C5×D412yes—
190C3×C22⋊C4—22no—
191C3×C22⋊C4—22no—
192C3×C22⋊C4—21no—
193C3×C42—22no—
194C3×C42—21no—
195C3×C42C23×C511yescharacteristic
196C23×C5—21no—
197C23×C5—41no—
198C5×Q8—21no—
199C5×Q8—21no—
200C5×Q8—41no—
201C5×Q8—41no—
202C5×D4—21no—
203C5×D4—21no—
204C5×D4—42no—
205C5×D4—41no—
206C5×D4—41no—
207C5×D4—41no—
208C5×D4—41no—
209C2×C4×C5—21no—
210C2×C4×C5—21no—
211C2×C4×C5—21no—
212C2×C4×C5—22no—
213C2×C4×C5—41no—
214C2×C4×C5—41no—
215C2×C4×C5C2×C3×D411yescharacteristic
216C2×C4×C5C2×C3×D411yescharacteristic
217C2×C4×C5C2×C3×D411yescharacteristic
218C5×C8—41no—
219C5×C8—41no—
220C5×C8—41no—
221C2×SD16—21no—
222C2×D8—21no—
223C4⋊Q8C22×C3×C511yescharacteristic
224#127C22×C3×C511yescharacteristic
225#126—22no—
226#124—21no—
227C22⋊Q8—21no—
228C4×Q8—21no—
229C4×Q8C22×C3×C511yescharacteristic
230C4×D4C22×C3×C511yescharacteristic
231#117—21no—
232C4⋊C8C22×C3×C511yescharacteristic
233C4⋊C8C22×C3×C511yescharacteristic
234#102—21no—
235#102—21no—
236#102—22no—
237C4×C8C22×C3×C511yescharacteristic
238C2×C3×C5—81no—
239C2×C3×C5—161no—
240C2×C3×C5#44911yescharacteristic
241C2×C3×C5#44911yescharacteristic
242C2×C3×C5#45411yescharacteristic
243C23×C3—21no—
244C23×C3—41no—
245C3×Q8—21no—
246C3×Q8—21no—
247C3×Q8—41no—
248C3×Q8—41no—
249C3×D4—21no—
250C3×D4—21no—
251C3×D4—42no—
252C3×D4—41no—
253C3×D4—41no—
254C3×D4—41no—
255C3×D4—41no—
256C2×C3×C4—21no—
257C2×C3×C4—21no—
258C2×C3×C4—21no—
259C2×C3×C4—22no—
260C2×C3×C4—41no—
261C2×C3×C4—41no—
262C2×C3×C4C2×C5×D411yescharacteristic
263C2×C3×C4C2×C5×D411yescharacteristic
264C2×C3×C4C2×C5×D411yescharacteristic
265C3×C8—41no—
266C3×C8—41no—
267C3×C8—41no—
268C22×C5—41no—
269C22×C5—41no—
270C22×C5—41no—
271C22×C5—81no—
272C22×C5—81no—
273C22×C5—81no—
274C22×C5C3×#12011yescharacteristic
275C4×C5—21no—
276C4×C5—41no—
277C4×C5—41no—
278C4×C5—42no—
279C4×C5—41no—
280C4×C5C3×#13511yescharacteristic
281C4×C5C3×#13511yescharacteristic
282C4×C5C3×AGL(1,ℤ/8)11yescharacteristic
283C4×C5C3×AGL(1,ℤ/8)11yescharacteristic
284C2×Q8—21no—
285C2×Q8C3×C5×D411yescharacteristic
286C2×D4—21no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4C3×C5×D411yescharacteristic
290C22×C4—21no—
291SD16—42no—
292D8—42no—
293C2×C8—21no—
294C2×C8—21no—
295C2×C8—21no—
296C4⋊C4—22no—
297C4⋊C4—21no—
298C4⋊C4—21no—
299C4⋊C4C3×C5×D411yescharacteristic
300C4⋊C4C3×C5×D411yescharacteristic
301C4⋊C4C3×C5×D412yes—
302C22⋊C4—22no—
303C22⋊C4—22no—
304C22⋊C4—21no—
305C42—22no—
306C42—21no—
307C42C23×C3×C511yescharacteristic
308C3×C5#162811yescharacteristic, direct product factor, Hall
309C22×C3—41no—
310C22×C3—41no—
311C22×C3—41no—
312C22×C3—81no—
313C22×C3—81no—
314C22×C3—81no—
315C22×C3C5×#12011yescharacteristic
316C3×C4—21no—
317C3×C4—41no—
318C3×C4—41no—
319C3×C4—42no—
320C3×C4—41no—
321C3×C4C5×#13511yescharacteristic
322C3×C4C5×#13511yescharacteristic
323C3×C4C5×AGL(1,ℤ/8)11yescharacteristic
324C3×C4C5×AGL(1,ℤ/8)11yescharacteristic
325C2×C5—81no—
326C2×C5—161no—
327C2×C5C3×#44911yescharacteristic
328C2×C5C3×#44911yescharacteristic
329C2×C5C3×#45411yescharacteristic
330C23—21no—
331C23—41no—
332Q8—21no—
333Q8—21no—
334Q8—41no—
335Q8—41no—
336D4—21no—
337D4—21no—
338D4—42no—
339D4—41no—
340D4—41no—
341D4—41no—
342D4—41no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—41no—
348C2×C4—41no—
349C2×C4C2×C3×C5×D411yescharacteristic
350C2×C4C2×C3×C5×D411yescharacteristic
351C2×C4C2×C3×C5×D411yescharacteristic
352C8—41no—
353C8—41no—
354C8—41no—
355C2×C3—81no—
356C2×C3—161no—
357C2×C3C5×#44911yescharacteristic
358C2×C3C5×#44911yescharacteristic
359C2×C3C5×#45411yescharacteristic
360C5C3×#162811yescharacteristic, direct product factor, Sylow
361C22—41no—
362C22—41no—
363C22—41no—
364C22—81no—
365C22—81no—
366C22—81no—
367C22C3×C5×#12011yescharacteristic
368C4—21no—
369C4—41no—
370C4—41no—
371C4—42no—
372C4—41no—
373C4C3×C5×#13511yescharacteristic
374C4C3×C5×#13511yescharacteristic
375C4C3×C5×AGL(1,ℤ/8)11yescharacteristic
376C4C3×C5×AGL(1,ℤ/8)11yescharacteristic
377C3C5×#162811yescharacteristic, direct product factor, Sylow
378C2—81no—
379C2—161no—
380C2C3×C5×#44911yescharacteristic
381C2C3×C5×#44911yescharacteristic
382C2C3×C5×#45411yescharacteristic
subgroupG/H
center, socleC22×C3×C5#120
commutator, Frattini, cosocleC42C23×C3×C5
prime radicalC3×C5×#459C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C3×C5 ◁ C3×C42×C5 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×C5×#556C2semidirect
2. C3×C5×#486C2semidirect
3. C3×C5×#460C2semidirect
5. C3×C5×#332C2semidirect
6. C3×C5×Q8⋊C8C2semidirect
7. C3×C5×D4⋊C8C2semidirect
8. C5×#1628C3direct (N × H)
26. C3×#1628C5direct (N × H)
27. C5×#556C2×C3semidirect
28. C5×#486C2×C3semidirect
29. C5×#460C2×C3semidirect
31. C5×#332C2×C3semidirect
32. C5×Q8⋊C8C2×C3semidirect
33. C5×D4⋊C8C2×C3semidirect
58. C3×#556C2×C5semidirect
59. C3×#486C2×C5semidirect
60. C3×#460C2×C5semidirect
62. C3×#332C2×C5semidirect
63. C3×Q8⋊C8C2×C5semidirect
64. C3×D4⋊C8C2×C5semidirect
82. #1628C3×C5direct (N × H)
149. #556C2×C3×C5semidirect
150. #486C2×C3×C5semidirect
151. #460C2×C3×C5semidirect
153. #332C2×C3×C5semidirect
154. Q8⋊C8C2×C3×C5semidirect
155. D4⋊C8C2×C3×C5semidirect
308. C3×C5#1628direct (N × H)
360. C5C3×#1628direct (N × H)
377. C3C5×#1628direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#5562
C3×C5×#4862
C3×C5×#4602
C3×C5×#4592
C3×C5×#3322
C3×C5×Q8⋊C82
C3×C5×D4⋊C82
C5×#16283
C3×#16285