Finite Group Databasebeta

← C3×C5×#1610 #14274644

Subgroups — C3×C5×#1610 #14274644

Order 1920.

508 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#537C211yesmaximal, characteristic
2C3×C5×#480C212yesmaximal, complemented
3C3×C5×#398C211yesmaximal, characteristic, complemented
4C3×C5×#327C212yesmaximal, complemented
5C3×C5×#324C211yesmaximal, characteristic, complemented
6C5×#1610C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×C5×#41—21no—
8C3×C5×C22⋊Q8—22no—
9C3×C5×#121—22no—
10C3×C5×#121—21no—
11C3×C5×#121C2212yes—
12C2×C3×C5×C4⋊C4—21no—
13C2×C3×C5×C4⋊C4C2211yescharacteristic
14C2×C3×C5×C4⋊C4C2211yescharacteristic
15C3×C5×#107—22no—
16C3×C5×#102—22no—
17C3×C5×C22⋊C8C2212yes—
18C3×C5×#95—21no—
19C3×C5×#95C2211yescharacteristic
20C3×#1610C511yesmaximal, characteristic, direct product factor, Hall
21C5×#537C2×C311yescharacteristic
22C5×#480C2×C312yescomplemented
23C5×#398C2×C311yescharacteristic, complemented
24C5×#327C2×C312yescomplemented
25C5×#324C2×C311yescharacteristic, complemented
26C3×C5×#41—42no—
27C3×C5×#41—42no—
28C2×C3×C5×Q8—21no—
29C2×C3×C5×D4—21no—
30C2×C3×C5×D4—22no—
31C2×C3×C5×D4—21no—
32C2×C3×C5×D4D412yes—
33C22×C3×C4×C5—22no—
34C22×C3×C4×C5—21no—
35C22×C3×C4×C5—21no—
36C22×C3×C4×C5D411yescharacteristic
37C22×C3×C4×C5D411yescharacteristic
38C22×C3×C4×C5C2311yescharacteristic
39C2×C3×C5×C8—22no—
40C3×C5×C4⋊C4—21no—
41C3×C5×C4⋊C4—22no—
42C3×C5×C4⋊C4—22no—
43C3×C5×C4⋊C4—42no—
44C3×C5×C4⋊C4D412yes—
45C3×C5×C22⋊C4—22no—
46C3×C5×C22⋊C4—22no—
47C3×#537C2×C511yescharacteristic
48C3×#480C2×C512yescomplemented
49C3×#398C2×C511yescharacteristic, complemented
50C3×#327C2×C512yescomplemented
51C3×#324C2×C511yescharacteristic, complemented
52C2×C5×#41—21no—
53C5×C22⋊Q8—22no—
54C5×#121—22no—
55C5×#121—21no—
56C5×#121C22×C312yes—
57C2×C5×C4⋊C4—21no—
58C2×C5×C4⋊C4C22×C311yescharacteristic
59C2×C5×C4⋊C4C22×C311yescharacteristic
60C5×#107—22no—
61C5×#102—22no—
62C5×C22⋊C8C22×C312yes—
63C5×#95—21no—
64C5×#95C22×C311yescharacteristic
65#1610C3×C511yescharacteristic, direct product factor, Sylow
66C23×C3×C5—22no—
67C23×C3×C5C2×D411yescharacteristic
68C3×C5×Q8—42no—
69C3×C5×D4—42no—
70C3×C5×D4—42no—
71C3×C5×D4—42no—
72C3×C5×D4—42no—
73C3×C5×D4—81no—
74C2×C3×C4×C5—21no—
75C2×C3×C4×C5—21no—
76C2×C3×C4×C5—22no—
77C2×C3×C4×C5—21no—
78C2×C3×C4×C5—22no—
79C2×C3×C4×C5—41no—
80C2×C3×C4×C5—44no—
81C2×C3×C4×C5—41no—
82C2×C3×C4×C5—41no—
83C2×C3×C4×C5—41no—
84C2×C3×C4×C5—41no—
85C2×C3×C4×C5—41no—
86C2×C3×C4×C5—41no—
87C2×C3×C4×C5D812yes—
88C2×C3×C4×C5C2×D412yes—
89C3×C5×C8—42no—
90C2×C3×#41—21no—
91C3×C22⋊Q8—22no—
92C3×#121—22no—
93C3×#121—21no—
94C3×#121C22×C512yes—
95C2×C3×C4⋊C4—21no—
96C2×C3×C4⋊C4C22×C511yescharacteristic
97C2×C3×C4⋊C4C22×C511yescharacteristic
98C3×#107—22no—
99C3×#102—22no—
100C3×C22⋊C8C22×C512yes—
101C3×#95—21no—
102C3×#95C22×C511yescharacteristic
103C5×#41—42no—
104C5×#41—42no—
105C2×C5×Q8—21no—
106C2×C5×D4—21no—
107C2×C5×D4—22no—
108C2×C5×D4—21no—
109C2×C5×D4C3×D412yes—
110C22×C4×C5—22no—
111C22×C4×C5—21no—
112C22×C4×C5—21no—
113C22×C4×C5C3×D411yescharacteristic
114C22×C4×C5C3×D411yescharacteristic
115C22×C4×C5C23×C311yescharacteristic
116C2×C5×C8—22no—
117C5×C4⋊C4—21no—
118C5×C4⋊C4—22no—
119C5×C4⋊C4—22no—
120C5×C4⋊C4—42no—
121C5×C4⋊C4C3×D412yes—
122C5×C22⋊C4—22no—
123C5×C22⋊C4—22no—
124#537C2×C3×C511yescharacteristic
125#480C2×C3×C512yescomplemented
126#398C2×C3×C511yescharacteristic, complemented
127#327C2×C3×C512yescomplemented
128#324C2×C3×C511yescharacteristic, complemented
129C22×C3×C5—21no—
130C22×C3×C5—21no—
131C22×C3×C5—42no—
132C22×C3×C5—42no—
133C22×C3×C5—42no—
134C22×C3×C5#12011yescharacteristic
135C22×C3×C5C2×D811yescharacteristic
136C22×C3×C5AGL(1,ℤ/8)11yescharacteristic
137C3×C4×C5—22no—
138C3×C4×C5—22no—
139C3×C4×C5—42no—
140C3×C4×C5—42no—
141C3×C4×C5—41no—
142C3×C4×C5—81no—
143C3×#41—42no—
144C3×#41—42no—
145C2×C3×Q8—21no—
146C2×C3×D4—21no—
147C2×C3×D4—22no—
148C2×C3×D4—21no—
149C2×C3×D4C5×D412yes—
150C22×C3×C4—22no—
151C22×C3×C4—21no—
152C22×C3×C4—21no—
153C22×C3×C4C5×D411yescharacteristic
154C22×C3×C4C5×D411yescharacteristic
155C22×C3×C4C23×C511yescharacteristic
156C2×C3×C8—22no—
157C3×C4⋊C4—21no—
158C3×C4⋊C4—22no—
159C3×C4⋊C4—22no—
160C3×C4⋊C4—42no—
161C3×C4⋊C4C5×D412yes—
162C3×C22⋊C4—22no—
163C3×C22⋊C4—22no—
164C23×C5—22no—
165C23×C5C2×C3×D411yescharacteristic
166C5×Q8—42no—
167C5×D4—42no—
168C5×D4—42no—
169C5×D4—42no—
170C5×D4—42no—
171C5×D4—81no—
172C2×C4×C5—21no—
173C2×C4×C5—21no—
174C2×C4×C5—22no—
175C2×C4×C5—21no—
176C2×C4×C5—22no—
177C2×C4×C5—41no—
178C2×C4×C5—44no—
179C2×C4×C5—41no—
180C2×C4×C5—41no—
181C2×C4×C5—41no—
182C2×C4×C5—41no—
183C2×C4×C5—41no—
184C2×C4×C5—41no—
185C2×C4×C5C3×D812yes—
186C2×C4×C5C2×C3×D412yes—
187C5×C8—42no—
188C2×#41—21no—
189C22⋊Q8—22no—
190#121—22no—
191#121—21no—
192#121C22×C3×C512yes—
193C2×C4⋊C4—21no—
194C2×C4⋊C4C22×C3×C511yescharacteristic
195C2×C4⋊C4C22×C3×C511yescharacteristic
196#107—22no—
197#102—22no—
198C22⋊C8C22×C3×C512yes—
199#95—21no—
200#95C22×C3×C511yescharacteristic
201C2×C3×C5—21no—
202C2×C3×C5—21no—
203C2×C3×C5—82no—
204C2×C3×C5#44711yescharacteristic
205C2×C3×C5#45611yescharacteristic
206C2×C3×C5#45811yescharacteristic
207C23×C3—22no—
208C23×C3C2×C5×D411yescharacteristic
209C3×Q8—42no—
210C3×D4—42no—
211C3×D4—42no—
212C3×D4—42no—
213C3×D4—42no—
214C3×D4—81no—
215C2×C3×C4—21no—
216C2×C3×C4—21no—
217C2×C3×C4—22no—
218C2×C3×C4—21no—
219C2×C3×C4—22no—
220C2×C3×C4—41no—
221C2×C3×C4—44no—
222C2×C3×C4—41no—
223C2×C3×C4—41no—
224C2×C3×C4—41no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4C5×D812yes—
229C2×C3×C4C2×C5×D412yes—
230C3×C8—42no—
231C22×C5—21no—
232C22×C5—21no—
233C22×C5—42no—
234C22×C5—42no—
235C22×C5—42no—
236C22×C5C3×#12011yescharacteristic
237C22×C5C2×C3×D811yescharacteristic
238C22×C5C3×AGL(1,ℤ/8)11yescharacteristic
239C4×C5—22no—
240C4×C5—22no—
241C4×C5—42no—
242C4×C5—42no—
243C4×C5—41no—
244C4×C5—81no—
245#41—42no—
246#41—42no—
247C2×Q8—21no—
248C2×D4—21no—
249C2×D4—22no—
250C2×D4—21no—
251C2×D4C3×C5×D412yes—
252C22×C4—22no—
253C22×C4—21no—
254C22×C4—21no—
255C22×C4C3×C5×D411yescharacteristic
256C22×C4C3×C5×D411yescharacteristic
257C22×C4C23×C3×C511yescharacteristic
258C2×C8—22no—
259C4⋊C4—21no—
260C4⋊C4—22no—
261C4⋊C4—22no—
262C4⋊C4—42no—
263C4⋊C4C3×C5×D412yes—
264C22⋊C4—22no—
265C22⋊C4—22no—
266C3×C5#161011yescharacteristic, direct product factor, Hall
267C22×C3—21no—
268C22×C3—21no—
269C22×C3—42no—
270C22×C3—42no—
271C22×C3—42no—
272C22×C3C5×#12011yescharacteristic
273C22×C3C2×C5×D811yescharacteristic
274C22×C3C5×AGL(1,ℤ/8)11yescharacteristic
275C3×C4—22no—
276C3×C4—22no—
277C3×C4—42no—
278C3×C4—42no—
279C3×C4—41no—
280C3×C4—81no—
281C2×C5—21no—
282C2×C5—21no—
283C2×C5—82no—
284C2×C5C3×#44711yescharacteristic
285C2×C5C3×#45611yescharacteristic
286C2×C5C3×#45811yescharacteristic
287C23—22no—
288C23C2×C3×C5×D411yescharacteristic
289Q8—42no—
290D4—42no—
291D4—42no—
292D4—42no—
293D4—42no—
294D4—81no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—22no—
298C2×C4—21no—
299C2×C4—22no—
300C2×C4—41no—
301C2×C4—44no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4C3×C5×D812yes—
309C2×C4C2×C3×C5×D412yes—
310C8—42no—
311C2×C3—21no—
312C2×C3—21no—
313C2×C3—82no—
314C2×C3C5×#44711yescharacteristic
315C2×C3C5×#45611yescharacteristic
316C2×C3C5×#45811yescharacteristic
317C5C3×#161011yescharacteristic, direct product factor, Sylow
318C22—21no—
319C22—21no—
320C22—42no—
321C22—42no—
322C22—42no—
323C22C3×C5×#12011yescharacteristic
324C22C2×C3×C5×D811yescharacteristic
325C22C3×C5×AGL(1,ℤ/8)11yescharacteristic
326C4—22no—
327C4—22no—
328C4—42no—
329C4—41no—
330C4—42no—
331C4—81no—
332C3C5×#161011yescharacteristic, direct product factor, Sylow
333C2—21no—
334C2—21no—
335C2—82no—
336C2C3×C5×#44711yescharacteristic
337C2C3×C5×#45611yescharacteristic
338C2C3×C5×#45811yescharacteristic
subgroupG/H
center, socleC22×C3×C5#120
commutator, Frattini, cosocleC22×C4C23×C3×C5
prime radicalC3×C5×#537C2
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
2. C3×C5×#480C2semidirect
3. C3×C5×#398C2semidirect
4. C3×C5×#327C2semidirect
5. C3×C5×#324C2semidirect
6. C5×#1610C3direct (N × H)
20. C3×#1610C5direct (N × H)
22. C5×#480C2×C3semidirect
23. C5×#398C2×C3semidirect
24. C5×#327C2×C3semidirect
25. C5×#324C2×C3semidirect
48. C3×#480C2×C5semidirect
49. C3×#398C2×C5semidirect
50. C3×#327C2×C5semidirect
51. C3×#324C2×C5semidirect
65. #1610C3×C5direct (N × H)
125. #480C2×C3×C5semidirect
126. #398C2×C3×C5semidirect
127. #327C2×C3×C5semidirect
128. #324C2×C3×C5semidirect
266. C3×C5#1610direct (N × H)
317. C5C3×#1610direct (N × H)
332. C3C5×#1610direct (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×#4802
C3×C5×#3982
C3×C5×#3272
C3×C5×#3242
C5×#16103
C3×#16105