Finite Group Databasebeta

← C3×C5×#1622 #14274656

Subgroups — C3×C5×#1622 #14274656

Order 1920.

472 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#554C211yesmaximal, characteristic, complemented
2C3×C5×#492C211yesmaximal, characteristic, complemented
3C3×C5×#460C211yesmaximal, characteristic, complemented
4C3×C5×#459C211yesmaximal, characteristic
5C3×C5×#333C211yesmaximal, characteristic, complemented
6C3×C5×Q8⋊C8C211yesmaximal, characteristic, complemented
7C3×C5×D4⋊C8C211yesmaximal, characteristic, complemented
8C5×#1622C311yesmaximal, characteristic, direct product factor, Hall
9C2×C3×C5×SD16—21no—
10C2×C3×C5×SD16—22no—
11C2×C3×C5×D8—21no—
12C3×C5×C4⋊Q8C2211yescharacteristic
13C3×C5×#127C2211yescharacteristic
14C3×C5×#125—21no—
15C3×C5×C22⋊Q8—21no—
16C3×C5×C22⋊Q8—22no—
17C3×C4×C5×Q8C2211yescharacteristic
18C3×C4×C5×D4—21no—
19C3×C4×C5×D4C2211yescharacteristic
20C2×C3×C5×C4⋊C4—21no—
21C3×C5×C4⋊C8C2211yescharacteristic
22C3×C5×C4⋊C8C2211yescharacteristic
23C3×C5×#102—21no—
24C3×C5×#102—21no—
25C3×C4×C5×C8C2211yescharacteristic
26C3×#1622C511yesmaximal, characteristic, direct product factor, Hall
27C5×#554C2×C311yescharacteristic, complemented
28C5×#492C2×C311yescharacteristic, complemented
29C5×#460C2×C311yescharacteristic, complemented
30C5×#459C2×C311yescharacteristic
31C5×#333C2×C311yescharacteristic, complemented
32C5×Q8⋊C8C2×C311yescharacteristic, complemented
33C5×D4⋊C8C2×C311yescharacteristic, complemented
34C2×C3×C5×Q8D411yescharacteristic
35C2×C3×C5×Q8D412yes—
36C2×C3×C5×D4—21no—
37C2×C3×C5×D4—21no—
38C2×C3×C5×D4—21no—
39C2×C3×C5×D4D411yescharacteristic
40C22×C3×C4×C5—22no—
41C22×C3×C4×C5—21no—
42C3×C5×SD16—44no—
43C3×C5×SD16—42no—
44C3×C5×D8—42no—
45C2×C3×C5×C8—21no—
46C2×C3×C5×C8—21no—
47C2×C3×C5×C8—21no—
48C3×C5×C4⋊C4—22no—
49C3×C5×C4⋊C4—22no—
50C3×C5×C4⋊C4—21no—
51C3×C5×C4⋊C4—21no—
52C3×C5×C4⋊C4—21no—
53C3×C5×C4⋊C4D411yescharacteristic
54C3×C5×C4⋊C4D411yescharacteristic
55C3×C5×C22⋊C4—22no—
56C3×C5×C22⋊C4—21no—
57C3×C42×C5—21no—
58C3×C42×C5C2311yescharacteristic
59C3×#554C2×C511yescharacteristic, complemented
60C3×#492C2×C511yescharacteristic, complemented
61C3×#460C2×C511yescharacteristic, complemented
62C3×#459C2×C511yescharacteristic
63C3×#333C2×C511yescharacteristic, complemented
64C3×Q8⋊C8C2×C511yescharacteristic, complemented
65C3×D4⋊C8C2×C511yescharacteristic, complemented
66C2×C5×SD16—21no—
67C2×C5×SD16—22no—
68C2×C5×D8—21no—
69C5×C4⋊Q8C22×C311yescharacteristic
70C5×#127C22×C311yescharacteristic
71C5×#125—21no—
72C5×C22⋊Q8—21no—
73C5×C22⋊Q8—22no—
74C4×C5×Q8C22×C311yescharacteristic
75C4×C5×D4—21no—
76C4×C5×D4C22×C311yescharacteristic
77C2×C5×C4⋊C4—21no—
78C5×C4⋊C8C22×C311yescharacteristic
79C5×C4⋊C8C22×C311yescharacteristic
80C5×#102—21no—
81C5×#102—21no—
82C4×C5×C8C22×C311yescharacteristic
83#1622C3×C511yescharacteristic, direct product factor, Sylow
84C23×C3×C5—21no—
85C23×C3×C5—41no—
86C3×C5×Q8—21no—
87C3×C5×Q8—21no—
88C3×C5×Q8—42no—
89C3×C5×D4—21no—
90C3×C5×D4—42no—
91C3×C5×D4—41no—
92C3×C5×D4—41no—
93C3×C5×D4—41no—
94C3×C5×D4—41no—
95C3×C5×D4SD1612yescomplemented
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×C5—82no—
103C2×C3×C4×C5C2×D411yescharacteristic
104C2×C3×C4×C5C2×D411yescharacteristic
105C2×C3×C4×C5C2×D411yescharacteristic
106C3×C5×C8—22no—
107C3×C5×C8—41no—
108C3×C5×C8—41no—
109C2×C3×SD16—21no—
110C2×C3×SD16—22no—
111C2×C3×D8—21no—
112C3×C4⋊Q8C22×C511yescharacteristic
113C3×#127C22×C511yescharacteristic
114C3×#125—21no—
115C3×C22⋊Q8—21no—
116C3×C22⋊Q8—22no—
117C3×C4×Q8C22×C511yescharacteristic
118C3×C4×D4—21no—
119C3×C4×D4C22×C511yescharacteristic
120C2×C3×C4⋊C4—21no—
121C3×C4⋊C8C22×C511yescharacteristic
122C3×C4⋊C8C22×C511yescharacteristic
123C3×#102—21no—
124C3×#102—21no—
125C3×C4×C8C22×C511yescharacteristic
126C2×C5×Q8C3×D411yescharacteristic
127C2×C5×Q8C3×D412yes—
128C2×C5×D4—21no—
129C2×C5×D4—21no—
130C2×C5×D4—21no—
131C2×C5×D4C3×D411yescharacteristic
132C22×C4×C5—22no—
133C22×C4×C5—21no—
134C5×SD16—44no—
135C5×SD16—42no—
136C5×D8—42no—
137C2×C5×C8—21no—
138C2×C5×C8—21no—
139C2×C5×C8—21no—
140C5×C4⋊C4—22no—
141C5×C4⋊C4—22no—
142C5×C4⋊C4—21no—
143C5×C4⋊C4—21no—
144C5×C4⋊C4—21no—
145C5×C4⋊C4C3×D411yescharacteristic
146C5×C4⋊C4C3×D411yescharacteristic
147C5×C22⋊C4—22no—
148C5×C22⋊C4—21no—
149C42×C5—21no—
150C42×C5C23×C311yescharacteristic
151#554C2×C3×C511yescharacteristic, complemented
152#492C2×C3×C511yescharacteristic, complemented
153#460C2×C3×C511yescharacteristic, complemented
154#459C2×C3×C511yescharacteristic
155#333C2×C3×C511yescharacteristic, complemented
156Q8⋊C8C2×C3×C511yescharacteristic, complemented
157D4⋊C8C2×C3×C511yescharacteristic, complemented
158C22×C3×C5—22no—
159C22×C3×C5—41no—
160C22×C3×C5—41no—
161C22×C3×C5—81no—
162C22×C3×C5—81no—
163C22×C3×C5—81no—
164C22×C3×C5#12011yescharacteristic
165C3×C4×C5—21no—
166C3×C4×C5—41no—
167C3×C4×C5—41no—
168C3×C4×C5—41no—
169C3×C4×C5—42no—
170C3×C4×C5C2×SD1611yescharacteristic
171C3×C4×C5#13511yescharacteristic
172C3×C4×C5AGL(1,ℤ/8)11yescharacteristic
173C3×C4×C5#13711yescharacteristic
174C2×C3×Q8C5×D411yescharacteristic
175C2×C3×Q8C5×D412yes—
176C2×C3×D4—21no—
177C2×C3×D4—21no—
178C2×C3×D4—21no—
179C2×C3×D4C5×D411yescharacteristic
180C22×C3×C4—22no—
181C22×C3×C4—21no—
182C3×SD16—44no—
183C3×SD16—42no—
184C3×D8—42no—
185C2×C3×C8—21no—
186C2×C3×C8—21no—
187C2×C3×C8—21no—
188C3×C4⋊C4—22no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—21no—
191C3×C4⋊C4—21no—
192C3×C4⋊C4—21no—
193C3×C4⋊C4C5×D411yescharacteristic
194C3×C4⋊C4C5×D411yescharacteristic
195C3×C22⋊C4—22no—
196C3×C22⋊C4—21no—
197C3×C42—21no—
198C3×C42C23×C511yescharacteristic
199C23×C5—21no—
200C23×C5—41no—
201C5×Q8—21no—
202C5×Q8—21no—
203C5×Q8—42no—
204C5×D4—21no—
205C5×D4—42no—
206C5×D4—41no—
207C5×D4—41no—
208C5×D4—41no—
209C5×D4—41no—
210C5×D4C3×SD1612yescomplemented
211C2×C4×C5—21no—
212C2×C4×C5—21no—
213C2×C4×C5—21no—
214C2×C4×C5—22no—
215C2×C4×C5—41no—
216C2×C4×C5—41no—
217C2×C4×C5—82no—
218C2×C4×C5C2×C3×D411yescharacteristic
219C2×C4×C5C2×C3×D411yescharacteristic
220C2×C4×C5C2×C3×D411yescharacteristic
221C5×C8—22no—
222C5×C8—41no—
223C5×C8—41no—
224C2×SD16—21no—
225C2×SD16—22no—
226C2×D8—21no—
227C4⋊Q8C22×C3×C511yescharacteristic
228#127C22×C3×C511yescharacteristic
229#125—21no—
230C22⋊Q8—21no—
231C22⋊Q8—22no—
232C4×Q8C22×C3×C511yescharacteristic
233C4×D4—21no—
234C4×D4C22×C3×C511yescharacteristic
235C2×C4⋊C4—21no—
236C4⋊C8C22×C3×C511yescharacteristic
237C4⋊C8C22×C3×C511yescharacteristic
238#102—21no—
239#102—21no—
240C4×C8C22×C3×C511yescharacteristic
241C2×C3×C5—42no—
242C2×C3×C5—161no—
243C2×C3×C5#44811yescharacteristic
244C2×C3×C5#44911yescharacteristic
245C2×C3×C5#45311yescharacteristic
246C23×C3—21no—
247C23×C3—41no—
248C3×Q8—21no—
249C3×Q8—21no—
250C3×Q8—42no—
251C3×D4—21no—
252C3×D4—42no—
253C3×D4—41no—
254C3×D4—41no—
255C3×D4—41no—
256C3×D4—41no—
257C3×D4C5×SD1612yescomplemented
258C2×C3×C4—21no—
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—22no—
262C2×C3×C4—41no—
263C2×C3×C4—41no—
264C2×C3×C4—82no—
265C2×C3×C4C2×C5×D411yescharacteristic
266C2×C3×C4C2×C5×D411yescharacteristic
267C2×C3×C4C2×C5×D411yescharacteristic
268C3×C8—22no—
269C3×C8—41no—
270C3×C8—41no—
271C22×C5—22no—
272C22×C5—41no—
273C22×C5—41no—
274C22×C5—81no—
275C22×C5—81no—
276C22×C5—81no—
277C22×C5C3×#12011yescharacteristic
278C4×C5—21no—
279C4×C5—41no—
280C4×C5—41no—
281C4×C5—41no—
282C4×C5—42no—
283C4×C5C2×C3×SD1611yescharacteristic
284C4×C5C3×#13511yescharacteristic
285C4×C5C3×AGL(1,ℤ/8)11yescharacteristic
286C4×C5C3×#13711yescharacteristic
287C2×Q8C3×C5×D411yescharacteristic
288C2×Q8C3×C5×D412yes—
289C2×D4—21no—
290C2×D4—21no—
291C2×D4—21no—
292C2×D4C3×C5×D411yescharacteristic
293C22×C4—22no—
294C22×C4—21no—
295SD16—44no—
296SD16—42no—
297D8—42no—
298C2×C8—21no—
299C2×C8—21no—
300C2×C8—21no—
301C4⋊C4—22no—
302C4⋊C4—22no—
303C4⋊C4—21no—
304C4⋊C4—21no—
305C4⋊C4—21no—
306C4⋊C4C3×C5×D411yescharacteristic
307C4⋊C4C3×C5×D411yescharacteristic
308C22⋊C4—22no—
309C22⋊C4—21no—
310C42—21no—
311C42C23×C3×C511yescharacteristic
312C3×C5#162211yescharacteristic, direct product factor, Hall
313C22×C3—22no—
314C22×C3—41no—
315C22×C3—41no—
316C22×C3—81no—
317C22×C3—81no—
318C22×C3—81no—
319C22×C3C5×#12011yescharacteristic
320C3×C4—21no—
321C3×C4—41no—
322C3×C4—41no—
323C3×C4—41no—
324C3×C4—42no—
325C3×C4C2×C5×SD1611yescharacteristic
326C3×C4C5×#13511yescharacteristic
327C3×C4C5×AGL(1,ℤ/8)11yescharacteristic
328C3×C4C5×#13711yescharacteristic
329C2×C5—42no—
330C2×C5—161no—
331C2×C5C3×#44811yescharacteristic
332C2×C5C3×#44911yescharacteristic
333C2×C5C3×#45311yescharacteristic
334C23—21no—
335C23—41no—
336Q8—21no—
337Q8—21no—
338Q8—42no—
339D4—21no—
340D4—42no—
341D4—41no—
342D4—41no—
343D4—41no—
344D4—41no—
345D4C3×C5×SD1612yescomplemented
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—22no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—82no—
353C2×C4C2×C3×C5×D411yescharacteristic
354C2×C4C2×C3×C5×D411yescharacteristic
355C2×C4C2×C3×C5×D411yescharacteristic
356C8—22no—
357C8—41no—
358C8—41no—
359C2×C3—42no—
360C2×C3—161no—
361C2×C3C5×#44811yescharacteristic
362C2×C3C5×#44911yescharacteristic
363C2×C3C5×#45311yescharacteristic
364C5C3×#162211yescharacteristic, direct product factor, Sylow
365C22—22no—
366C22—41no—
367C22—41no—
368C22—81no—
369C22—81no—
370C22—81no—
371C22C3×C5×#12011yescharacteristic
372C4—21no—
373C4—41no—
374C4—41no—
375C4—41no—
376C4—42no—
377C4C2×C3×C5×SD1611yescharacteristic
378C4C3×C5×#13511yescharacteristic
379C4C3×C5×AGL(1,ℤ/8)11yescharacteristic
380C4C3×C5×#13711yescharacteristic
381C3C5×#162211yescharacteristic, direct product factor, Sylow
382C2—42no—
383C2—161no—
384C2C3×C5×#44811yescharacteristic
385C2C3×C5×#44911yescharacteristic
386C2C3×C5×#45311yescharacteristic
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×#554C2semidirect
2. C3×C5×#492C2semidirect
3. C3×C5×#460C2semidirect
5. C3×C5×#333C2semidirect
6. C3×C5×Q8⋊C8C2semidirect
7. C3×C5×D4⋊C8C2semidirect
8. C5×#1622C3direct (N × H)
26. C3×#1622C5direct (N × H)
27. C5×#554C2×C3semidirect
28. C5×#492C2×C3semidirect
29. C5×#460C2×C3semidirect
31. C5×#333C2×C3semidirect
32. C5×Q8⋊C8C2×C3semidirect
33. C5×D4⋊C8C2×C3semidirect
59. C3×#554C2×C5semidirect
60. C3×#492C2×C5semidirect
61. C3×#460C2×C5semidirect
63. C3×#333C2×C5semidirect
64. C3×Q8⋊C8C2×C5semidirect
65. C3×D4⋊C8C2×C5semidirect
83. #1622C3×C5direct (N × H)
95. C3×C5×D4SD16semidirect
151. #554C2×C3×C5semidirect
152. #492C2×C3×C5semidirect
153. #460C2×C3×C5semidirect
155. #333C2×C3×C5semidirect
156. Q8⋊C8C2×C3×C5semidirect
157. D4⋊C8C2×C3×C5semidirect
210. C5×D4C3×SD16semidirect
257. C3×D4C5×SD16semidirect
312. C3×C5#1622direct (N × H)
345. D4C3×C5×SD16semidirect
364. C5C3×#1622direct (N × H)
381. C3C5×#1622direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#5542
C3×C5×#4922
C3×C5×#4602
C3×C5×#4592
C3×C5×#3332
C3×C5×Q8⋊C82
C3×C5×D4⋊C82
C5×#16223
C3×#16225