Finite Group Databasebeta

← C3×C5×#1626 #14274660

Subgroups — C3×C5×#1626 #14274660

Order 1920.

420 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#557C211yesmaximal, characteristic, complemented
2C3×C5×#486C211yesmaximal, characteristic
3C3×C5×#460C211yesmaximal, characteristic
4C3×C5×#460C211yesmaximal, characteristic
5C3×C5×#332C211yesmaximal, characteristic, complemented
6C3×C5×Q8⋊C8C211yesmaximal, characteristic, complemented
7C3×C5×Q8⋊C8C211yesmaximal, characteristic, complemented
8C5×#1626C311yesmaximal, characteristic, direct product factor, Hall
9C2×C3×C5×SD16—21no—
10C2×C3×C5×SD16—21no—
11C3×C5×C4⋊Q8—21no—
12C3×C5×C4⋊Q8C2211yescharacteristic
13C3×C5×#127C2211yescharacteristic
14C3×C5×#125—22no—
15C3×C5×#125—21no—
16C3×C4×C5×Q8—21no—
17C3×C4×C5×Q8—21no—
18C3×C4×C5×Q8C2211yescharacteristic
19C3×C4×C5×Q8C2211yescharacteristic
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×#1626C511yesmaximal, characteristic, direct product factor, Hall
27C5×#557C2×C311yescharacteristic, complemented
28C5×#486C2×C311yescharacteristic
29C5×#460C2×C311yescharacteristic
30C5×#460C2×C311yescharacteristic
31C5×#332C2×C311yescharacteristic, complemented
32C5×Q8⋊C8C2×C311yescharacteristic, complemented
33C5×Q8⋊C8C2×C311yescharacteristic, complemented
34C2×C3×C5×Q8—21no—
35C2×C3×C5×Q8D411yescharacteristic
36C2×C3×C5×Q8D411yescharacteristic
37C2×C3×C5×D4—21no—
38C2×C3×C5×D4—21no—
39C2×C3×C5×D4—21no—
40C3×C5×SD16—42no—
41C3×C5×SD16—42no—
42C2×C3×C5×C8—21no—
43C2×C3×C5×C8—21no—
44C2×C3×C5×C8—21no—
45C3×C5×C4⋊C4—22no—
46C3×C5×C4⋊C4—22no—
47C3×C5×C4⋊C4—22no—
48C3×C5×C4⋊C4—21no—
49C3×C5×C4⋊C4—21no—
50C3×C5×C4⋊C4—21no—
51C3×C5×C4⋊C4D411yescharacteristic
52C3×C5×C4⋊C4D411yescharacteristic
53C3×C5×C4⋊C4D412yes—
54C3×C42×C5—22no—
55C3×C42×C5—21no—
56C3×C42×C5—21no—
57C3×C42×C5C2311yescharacteristic
58C3×#557C2×C511yescharacteristic, complemented
59C3×#486C2×C511yescharacteristic
60C3×#460C2×C511yescharacteristic
61C3×#460C2×C511yescharacteristic
62C3×#332C2×C511yescharacteristic, complemented
63C3×Q8⋊C8C2×C511yescharacteristic, complemented
64C3×Q8⋊C8C2×C511yescharacteristic, complemented
65C2×C5×SD16—21no—
66C2×C5×SD16—21no—
67C5×C4⋊Q8—21no—
68C5×C4⋊Q8C22×C311yescharacteristic
69C5×#127C22×C311yescharacteristic
70C5×#125—22no—
71C5×#125—21no—
72C4×C5×Q8—21no—
73C4×C5×Q8—21no—
74C4×C5×Q8C22×C311yescharacteristic
75C4×C5×Q8C22×C311yescharacteristic
76C5×C4⋊C8C22×C311yescharacteristic
77C5×C4⋊C8C22×C311yescharacteristic
78C5×#102—21no—
79C5×#102—21no—
80C5×#102—22no—
81C4×C5×C8C22×C311yescharacteristic
82#1626C3×C511yescharacteristic, direct product factor, Sylow
83C23×C3×C5—41no—
84C3×C5×Q8—21no—
85C3×C5×Q8—21no—
86C3×C5×Q8—21no—
87C3×C5×Q8—41no—
88C3×C5×Q8—41no—
89C3×C5×Q8SD1612yescomplemented
90C3×C5×D4—42no—
91C3×C5×D4—41no—
92C3×C5×D4—41no—
93C3×C5×D4—41no—
94C3×C5×D4—41no—
95C2×C3×C4×C5—21no—
96C2×C3×C4×C5—21no—
97C2×C3×C4×C5—21no—
98C2×C3×C4×C5—21no—
99C2×C3×C4×C5—22no—
100C2×C3×C4×C5C2×D411yescharacteristic
101C2×C3×C4×C5C2×D411yescharacteristic
102C2×C3×C4×C5C2×D411yescharacteristic
103C3×C5×C8—41no—
104C3×C5×C8—41no—
105C3×C5×C8—41no—
106C2×C3×SD16—21no—
107C2×C3×SD16—21no—
108C3×C4⋊Q8—21no—
109C3×C4⋊Q8C22×C511yescharacteristic
110C3×#127C22×C511yescharacteristic
111C3×#125—22no—
112C3×#125—21no—
113C3×C4×Q8—21no—
114C3×C4×Q8—21no—
115C3×C4×Q8C22×C511yescharacteristic
116C3×C4×Q8C22×C511yescharacteristic
117C3×C4⋊C8C22×C511yescharacteristic
118C3×C4⋊C8C22×C511yescharacteristic
119C3×#102—21no—
120C3×#102—21no—
121C3×#102—22no—
122C3×C4×C8C22×C511yescharacteristic
123C2×C5×Q8—21no—
124C2×C5×Q8C3×D411yescharacteristic
125C2×C5×Q8C3×D411yescharacteristic
126C2×C5×D4—21no—
127C2×C5×D4—21no—
128C2×C5×D4—21no—
129C5×SD16—42no—
130C5×SD16—42no—
131C2×C5×C8—21no—
132C2×C5×C8—21no—
133C2×C5×C8—21no—
134C5×C4⋊C4—22no—
135C5×C4⋊C4—22no—
136C5×C4⋊C4—22no—
137C5×C4⋊C4—21no—
138C5×C4⋊C4—21no—
139C5×C4⋊C4—21no—
140C5×C4⋊C4C3×D411yescharacteristic
141C5×C4⋊C4C3×D411yescharacteristic
142C5×C4⋊C4C3×D412yes—
143C42×C5—22no—
144C42×C5—21no—
145C42×C5—21no—
146C42×C5C23×C311yescharacteristic
147#557C2×C3×C511yescharacteristic, complemented
148#486C2×C3×C511yescharacteristic
149#460C2×C3×C511yescharacteristic
150#460C2×C3×C511yescharacteristic
151#332C2×C3×C511yescharacteristic, complemented
152Q8⋊C8C2×C3×C511yescharacteristic, complemented
153Q8⋊C8C2×C3×C511yescharacteristic, complemented
154C22×C3×C5—81no—
155C22×C3×C5—81no—
156C22×C3×C5—81no—
157C22×C3×C5#12011yescharacteristic
158C3×C4×C5—22no—
159C3×C4×C5—21no—
160C3×C4×C5—41no—
161C3×C4×C5—42no—
162C3×C4×C5—41no—
163C3×C4×C5—41no—
164C3×C4×C5C2×SD1611yescharacteristic
165C3×C4×C5#13511yescharacteristic
166C3×C4×C5AGL(1,ℤ/8)11yescharacteristic
167C3×C4×C5#13711yescharacteristic
168C2×C3×Q8—21no—
169C2×C3×Q8C5×D411yescharacteristic
170C2×C3×Q8C5×D411yescharacteristic
171C2×C3×D4—21no—
172C2×C3×D4—21no—
173C2×C3×D4—21no—
174C3×SD16—42no—
175C3×SD16—42no—
176C2×C3×C8—21no—
177C2×C3×C8—21no—
178C2×C3×C8—21no—
179C3×C4⋊C4—22no—
180C3×C4⋊C4—22no—
181C3×C4⋊C4—22no—
182C3×C4⋊C4—21no—
183C3×C4⋊C4—21no—
184C3×C4⋊C4—21no—
185C3×C4⋊C4C5×D411yescharacteristic
186C3×C4⋊C4C5×D411yescharacteristic
187C3×C4⋊C4C5×D412yes—
188C3×C42—22no—
189C3×C42—21no—
190C3×C42—21no—
191C3×C42C23×C511yescharacteristic
192C23×C5—41no—
193C5×Q8—21no—
194C5×Q8—21no—
195C5×Q8—21no—
196C5×Q8—41no—
197C5×Q8—41no—
198C5×Q8C3×SD1612yescomplemented
199C5×D4—42no—
200C5×D4—41no—
201C5×D4—41no—
202C5×D4—41no—
203C5×D4—41no—
204C2×C4×C5—21no—
205C2×C4×C5—21no—
206C2×C4×C5—21no—
207C2×C4×C5—21no—
208C2×C4×C5—22no—
209C2×C4×C5C2×C3×D411yescharacteristic
210C2×C4×C5C2×C3×D411yescharacteristic
211C2×C4×C5C2×C3×D411yescharacteristic
212C5×C8—41no—
213C5×C8—41no—
214C5×C8—41no—
215C2×SD16—21no—
216C2×SD16—21no—
217C4⋊Q8—21no—
218C4⋊Q8C22×C3×C511yescharacteristic
219#127C22×C3×C511yescharacteristic
220#125—22no—
221#125—21no—
222C4×Q8—21no—
223C4×Q8—21no—
224C4×Q8C22×C3×C511yescharacteristic
225C4×Q8C22×C3×C511yescharacteristic
226C4⋊C8C22×C3×C511yescharacteristic
227C4⋊C8C22×C3×C511yescharacteristic
228#102—21no—
229#102—21no—
230#102—22no—
231C4×C8C22×C3×C511yescharacteristic
232C2×C3×C5—161no—
233C2×C3×C5#44811yescharacteristic
234C2×C3×C5#44911yescharacteristic
235C2×C3×C5#45411yescharacteristic
236C23×C3—41no—
237C3×Q8—21no—
238C3×Q8—21no—
239C3×Q8—21no—
240C3×Q8—41no—
241C3×Q8—41no—
242C3×Q8C5×SD1612yescomplemented
243C3×D4—42no—
244C3×D4—41no—
245C3×D4—41no—
246C3×D4—41no—
247C3×D4—41no—
248C2×C3×C4—21no—
249C2×C3×C4—21no—
250C2×C3×C4—21no—
251C2×C3×C4—21no—
252C2×C3×C4—22no—
253C2×C3×C4C2×C5×D411yescharacteristic
254C2×C3×C4C2×C5×D411yescharacteristic
255C2×C3×C4C2×C5×D411yescharacteristic
256C3×C8—41no—
257C3×C8—41no—
258C3×C8—41no—
259C22×C5—81no—
260C22×C5—81no—
261C22×C5—81no—
262C22×C5C3×#12011yescharacteristic
263C4×C5—22no—
264C4×C5—21no—
265C4×C5—41no—
266C4×C5—42no—
267C4×C5—41no—
268C4×C5—41no—
269C4×C5C2×C3×SD1611yescharacteristic
270C4×C5C3×#13511yescharacteristic
271C4×C5C3×AGL(1,ℤ/8)11yescharacteristic
272C4×C5C3×#13711yescharacteristic
273C2×Q8—21no—
274C2×Q8C3×C5×D411yescharacteristic
275C2×Q8C3×C5×D411yescharacteristic
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—21no—
279SD16—42no—
280SD16—42no—
281C2×C8—21no—
282C2×C8—21no—
283C2×C8—21no—
284C4⋊C4—22no—
285C4⋊C4—22no—
286C4⋊C4—22no—
287C4⋊C4—21no—
288C4⋊C4—21no—
289C4⋊C4—21no—
290C4⋊C4C3×C5×D411yescharacteristic
291C4⋊C4C3×C5×D411yescharacteristic
292C4⋊C4C3×C5×D412yes—
293C42—22no—
294C42—21no—
295C42—21no—
296C42C23×C3×C511yescharacteristic
297C3×C5#162611yescharacteristic, direct product factor, Hall
298C22×C3—81no—
299C22×C3—81no—
300C22×C3—81no—
301C22×C3C5×#12011yescharacteristic
302C3×C4—22no—
303C3×C4—21no—
304C3×C4—41no—
305C3×C4—42no—
306C3×C4—41no—
307C3×C4—41no—
308C3×C4C2×C5×SD1611yescharacteristic
309C3×C4C5×#13511yescharacteristic
310C3×C4C5×AGL(1,ℤ/8)11yescharacteristic
311C3×C4C5×#13711yescharacteristic
312C2×C5—161no—
313C2×C5C3×#44811yescharacteristic
314C2×C5C3×#44911yescharacteristic
315C2×C5C3×#45411yescharacteristic
316C23—41no—
317Q8—21no—
318Q8—21no—
319Q8—21no—
320Q8—41no—
321Q8—41no—
322Q8C3×C5×SD1612yescomplemented
323D4—42no—
324D4—41no—
325D4—41no—
326D4—41no—
327D4—41no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—22no—
333C2×C4C2×C3×C5×D411yescharacteristic
334C2×C4C2×C3×C5×D411yescharacteristic
335C2×C4C2×C3×C5×D411yescharacteristic
336C8—41no—
337C8—41no—
338C8—41no—
339C2×C3—161no—
340C2×C3C5×#44811yescharacteristic
341C2×C3C5×#44911yescharacteristic
342C2×C3C5×#45411yescharacteristic
343C5C3×#162611yescharacteristic, direct product factor, Sylow
344C22—81no—
345C22—81no—
346C22—81no—
347C22C3×C5×#12011yescharacteristic
348C4—21no—
349C4—22no—
350C4—41no—
351C4—42no—
352C4—41no—
353C4—41no—
354C4C2×C3×C5×SD1611yescharacteristic
355C4C3×C5×#13511yescharacteristic
356C4C3×C5×AGL(1,ℤ/8)11yescharacteristic
357C4C3×C5×#13711yescharacteristic
358C3C5×#162611yescharacteristic, direct product factor, Sylow
359C2—161no—
360C2C3×C5×#44811yescharacteristic
361C2C3×C5×#44911yescharacteristic
362C2C3×C5×#45411yescharacteristic
subgroupG/H
center, socleC22×C3×C5#120
commutator, Frattini, cosocleC42C23×C3×C5
prime radicalC3×C5×#127C22
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×#557C2semidirect
5. C3×C5×#332C2semidirect
6. C3×C5×Q8⋊C8C2semidirect
7. C3×C5×Q8⋊C8C2semidirect
8. C5×#1626C3direct (N × H)
26. C3×#1626C5direct (N × H)
27. C5×#557C2×C3semidirect
31. C5×#332C2×C3semidirect
32. C5×Q8⋊C8C2×C3semidirect
33. C5×Q8⋊C8C2×C3semidirect
58. C3×#557C2×C5semidirect
62. C3×#332C2×C5semidirect
63. C3×Q8⋊C8C2×C5semidirect
64. C3×Q8⋊C8C2×C5semidirect
82. #1626C3×C5direct (N × H)
89. C3×C5×Q8SD16semidirect
147. #557C2×C3×C5semidirect
151. #332C2×C3×C5semidirect
152. Q8⋊C8C2×C3×C5semidirect
153. Q8⋊C8C2×C3×C5semidirect
198. C5×Q8C3×SD16semidirect
242. C3×Q8C5×SD16semidirect
297. C3×C5#1626direct (N × H)
322. Q8C3×C5×SD16semidirect
343. C5C3×#1626direct (N × H)
358. C3C5×#1626direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#5572
C3×C5×#4862
C3×C5×#4602
C3×C5×#3322
C3×C5×Q8⋊C82
C5×#16263
C3×#16265