Finite Group Databasebeta

← C3×C5×#2190 #14275224

Subgroups — C3×C5×#2190 #14275224

Order 1920.

452 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#575C211yesmaximal, characteristic, complemented
2C3×C5×#509C212yesmaximal, complemented
3C3×C5×#469C211yesmaximal, characteristic, complemented
4C3×C5×#360C212yesmaximal, complemented
5C3×C5×#349C211yesmaximal, characteristic, complemented
6C5×#2190C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×C5×#41—21no—
8C3×C5×#137—22no—
9C3×C5×AGL(1,ℤ/8)—22no—
10C3×C5×#135—21no—
11C3×C5×#135C2212yes—
12C2×C3×C5×D8C2211yescharacteristic, complemented
13C2×C3×C5×M16C2211yescharacteristic
14C3×C5×#121—21no—
15C3×C5×SD32—22no—
16C3×C5×D16—22no—
17C3×C5×M32C2212yescomplemented
18C3×C5×#106C2211yescharacteristic, complemented
19C3×C5×#102—21no—
20C3×#2190C511yesmaximal, characteristic, direct product factor, Hall
21C5×#575C2×C311yescharacteristic, complemented
22C5×#509C2×C312yescomplemented
23C5×#469C2×C311yescharacteristic, complemented
24C5×#360C2×C312yescomplemented
25C5×#349C2×C311yescharacteristic, complemented
26C3×C5×#41—22no—
27C3×C5×#41—22no—
28C3×C5×#41—41no—
29C3×C5×#41—41no—
30C2×C3×C5×Q8—21no—
31C2×C3×C5×D4—21no—
32C2×C3×C5×D4—21no—
33C2×C3×C5×D4—41no—
34C2×C3×C5×D4—41no—
35C22×C3×C4×C5—41no—
36C22×C3×C4×C5D411yescharacteristic
37C3×C5×Q16—21no—
38C3×C5×Q16D412yescomplemented
39C3×C5×SD16—22no—
40C3×C5×SD16—22no—
41C3×C5×D8—21no—
42C3×C5×D8—22no—
43C3×C5×D8D412yescomplemented
44C3×C5×M16—22no—
45C2×C3×C5×C8D411yescharacteristic
46C2×C3×C5×C8C2311yescharacteristic
47C3×C5×C4⋊C4—21no—
48C3×C5×C22⋊C4—41no—
49C3×C5×C16—22no—
50C3×#575C2×C511yescharacteristic, complemented
51C3×#509C2×C512yescomplemented
52C3×#469C2×C511yescharacteristic, complemented
53C3×#360C2×C512yescomplemented
54C3×#349C2×C511yescharacteristic, complemented
55C2×C5×#41—21no—
56C5×#137—22no—
57C5×AGL(1,ℤ/8)—22no—
58C5×#135—21no—
59C5×#135C22×C312yes—
60C2×C5×D8C22×C311yescharacteristic, complemented
61C2×C5×M16C22×C311yescharacteristic
62C5×#121—21no—
63C5×SD32—22no—
64C5×D16—22no—
65C5×M32C22×C312yescomplemented
66C5×#106C22×C311yescharacteristic, complemented
67C5×#102—21no—
68#2190C3×C511yescharacteristic, direct product factor, Sylow
69C23×C3×C5—41no—
70C23×C3×C5—41no—
71C23×C3×C5D811yescharacteristic
72C3×C5×Q8—22no—
73C3×C5×Q8—41no—
74C3×C5×D4—22no—
75C3×C5×D4—41no—
76C3×C5×D4—42no—
77C3×C5×D4—42no—
78C3×C5×D4—41no—
79C3×C5×D4—41no—
80C3×C5×D4—82no—
81C2×C3×C4×C5—21no—
82C2×C3×C4×C5—21no—
83C2×C3×C4×C5—42no—
84C2×C3×C4×C5—41no—
85C2×C3×C4×C5—42no—
86C2×C3×C4×C5—41no—
87C2×C3×C4×C5D811yescharacteristic
88C2×C3×C4×C5C2×D411yescharacteristic
89C3×C5×C8—21no—
90C3×C5×C8C2×D412yes—
91C2×C3×#41—21no—
92C3×#137—22no—
93C3×AGL(1,ℤ/8)—22no—
94C3×#135—21no—
95C3×#135C22×C512yes—
96C2×C3×D8C22×C511yescharacteristic, complemented
97C2×C3×M16C22×C511yescharacteristic
98C3×#121—21no—
99C3×SD32—22no—
100C3×D16—22no—
101C3×M32C22×C512yescomplemented
102C3×#106C22×C511yescharacteristic, complemented
103C3×#102—21no—
104C5×#41—22no—
105C5×#41—22no—
106C5×#41—41no—
107C5×#41—41no—
108C2×C5×Q8—21no—
109C2×C5×D4—21no—
110C2×C5×D4—21no—
111C2×C5×D4—41no—
112C2×C5×D4—41no—
113C22×C4×C5—41no—
114C22×C4×C5C3×D411yescharacteristic
115C5×Q16—21no—
116C5×Q16C3×D412yescomplemented
117C5×SD16—22no—
118C5×SD16—22no—
119C5×D8—21no—
120C5×D8—22no—
121C5×D8C3×D412yescomplemented
122C5×M16—22no—
123C2×C5×C8C3×D411yescharacteristic
124C2×C5×C8C23×C311yescharacteristic
125C5×C4⋊C4—21no—
126C5×C22⋊C4—41no—
127C5×C16—22no—
128#575C2×C3×C511yescharacteristic, complemented
129#509C2×C3×C512yescomplemented
130#469C2×C3×C511yescharacteristic, complemented
131#360C2×C3×C512yescomplemented
132#349C2×C3×C511yescharacteristic, complemented
133C22×C3×C5—21no—
134C22×C3×C5—41no—
135C22×C3×C5—42no—
136C22×C3×C5—81no—
137C22×C3×C5—81no—
138C22×C3×C5—82no—
139C22×C3×C5—81no—
140C22×C3×C5C2×D811yescharacteristic
141C3×C4×C5—21no—
142C3×C4×C5—42no—
143C3×C4×C5—81no—
144C3×C4×C5#12011yescharacteristic
145C3×C4×C5AGL(1,ℤ/8)11yescharacteristic
146C3×#41—22no—
147C3×#41—22no—
148C3×#41—41no—
149C3×#41—41no—
150C2×C3×Q8—21no—
151C2×C3×D4—21no—
152C2×C3×D4—21no—
153C2×C3×D4—41no—
154C2×C3×D4—41no—
155C22×C3×C4—41no—
156C22×C3×C4C5×D411yescharacteristic
157C3×Q16—21no—
158C3×Q16C5×D412yescomplemented
159C3×SD16—22no—
160C3×SD16—22no—
161C3×D8—21no—
162C3×D8—22no—
163C3×D8C5×D412yescomplemented
164C3×M16—22no—
165C2×C3×C8C5×D411yescharacteristic
166C2×C3×C8C23×C511yescharacteristic
167C3×C4⋊C4—21no—
168C3×C22⋊C4—41no—
169C3×C16—22no—
170C23×C5—41no—
171C23×C5—41no—
172C23×C5C3×D811yescharacteristic
173C5×Q8—22no—
174C5×Q8—41no—
175C5×D4—22no—
176C5×D4—41no—
177C5×D4—42no—
178C5×D4—42no—
179C5×D4—41no—
180C5×D4—41no—
181C5×D4—82no—
182C2×C4×C5—21no—
183C2×C4×C5—21no—
184C2×C4×C5—42no—
185C2×C4×C5—41no—
186C2×C4×C5—42no—
187C2×C4×C5—41no—
188C2×C4×C5C3×D811yescharacteristic
189C2×C4×C5C2×C3×D411yescharacteristic
190C5×C8—21no—
191C5×C8C2×C3×D412yes—
192C2×#41—21no—
193#137—22no—
194AGL(1,ℤ/8)—22no—
195#135—21no—
196#135C22×C3×C512yes—
197C2×D8C22×C3×C511yescharacteristic, complemented
198C2×M16C22×C3×C511yescharacteristic
199#121—21no—
200SD32—22no—
201D16—22no—
202M32C22×C3×C512yescomplemented
203#106C22×C3×C511yescharacteristic, complemented
204#102—21no—
205C2×C3×C5—21no—
206C2×C3×C5—41no—
207C2×C3×C5—82no—
208C2×C3×C5—161no—
209C2×C3×C5#44711yescharacteristic
210C23×C3—41no—
211C23×C3—41no—
212C23×C3C5×D811yescharacteristic
213C3×Q8—22no—
214C3×Q8—41no—
215C3×D4—22no—
216C3×D4—41no—
217C3×D4—42no—
218C3×D4—42no—
219C3×D4—41no—
220C3×D4—41no—
221C3×D4—82no—
222C2×C3×C4—21no—
223C2×C3×C4—21no—
224C2×C3×C4—42no—
225C2×C3×C4—41no—
226C2×C3×C4—42no—
227C2×C3×C4—41no—
228C2×C3×C4C5×D811yescharacteristic
229C2×C3×C4C2×C5×D411yescharacteristic
230C3×C8—21no—
231C3×C8C2×C5×D412yes—
232C22×C5—21no—
233C22×C5—41no—
234C22×C5—42no—
235C22×C5—81no—
236C22×C5—81no—
237C22×C5—82no—
238C22×C5—81no—
239C22×C5C2×C3×D811yescharacteristic
240C4×C5—21no—
241C4×C5—42no—
242C4×C5—81no—
243C4×C5C3×#12011yescharacteristic
244C4×C5C3×AGL(1,ℤ/8)11yescharacteristic
245#41—22no—
246#41—22no—
247#41—41no—
248#41—41no—
249C2×Q8—21no—
250C2×D4—21no—
251C2×D4—21no—
252C2×D4—41no—
253C2×D4—41no—
254C22×C4—41no—
255C22×C4C3×C5×D411yescharacteristic
256Q16—21no—
257Q16C3×C5×D412yescomplemented
258SD16—22no—
259SD16—22no—
260D8—21no—
261D8—22no—
262D8C3×C5×D412yescomplemented
263M16—22no—
264C2×C8C3×C5×D411yescharacteristic
265C2×C8C23×C3×C511yescharacteristic
266C4⋊C4—21no—
267C22⋊C4—41no—
268C16—22no—
269C3×C5#219011yescharacteristic, direct product factor, Hall
270C22×C3—21no—
271C22×C3—41no—
272C22×C3—42no—
273C22×C3—81no—
274C22×C3—81no—
275C22×C3—82no—
276C22×C3—81no—
277C22×C3C2×C5×D811yescharacteristic
278C3×C4—21no—
279C3×C4—42no—
280C3×C4—81no—
281C3×C4C5×#12011yescharacteristic
282C3×C4C5×AGL(1,ℤ/8)11yescharacteristic
283C2×C5—21no—
284C2×C5—41no—
285C2×C5—82no—
286C2×C5—161no—
287C2×C5C3×#44711yescharacteristic
288C23—41no—
289C23—41no—
290C23C3×C5×D811yescharacteristic
291Q8—22no—
292Q8—41no—
293D4—22no—
294D4—41no—
295D4—42no—
296D4—42no—
297D4—41no—
298D4—41no—
299D4—82no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—42no—
303C2×C4—41no—
304C2×C4—42no—
305C2×C4—41no—
306C2×C4C3×C5×D811yescharacteristic
307C2×C4C2×C3×C5×D411yescharacteristic
308C8—21no—
309C8C2×C3×C5×D412yes—
310C2×C3—21no—
311C2×C3—41no—
312C2×C3—82no—
313C2×C3—161no—
314C2×C3C5×#44711yescharacteristic
315C5C3×#219011yescharacteristic, direct product factor, Sylow
316C22—21no—
317C22—41no—
318C22—42no—
319C22—81no—
320C22—81no—
321C22—82no—
322C22—81no—
323C22C2×C3×C5×D811yescharacteristic
324C4—21no—
325C4—42no—
326C4—81no—
327C4C3×C5×#12011yescharacteristic
328C4C3×C5×AGL(1,ℤ/8)11yescharacteristic
329C3C5×#219011yescharacteristic, direct product factor, Sylow
330C2—21no—
331C2—41no—
332C2—82no—
333C2—161no—
334C2C3×C5×#44711yescharacteristic
subgroupG/H
center, socleC2×C3×C5#447
commutator, Frattini, cosocleC2×C8C23×C3×C5
LCSG ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1
UCSC1 ◁ C2×C3×C5 ◁ C2×C3×C4×C5 ◁ C2×C3×C5×M16 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×C5×#575C2semidirect
2. C3×C5×#509C2semidirect
3. C3×C5×#469C2semidirect
4. C3×C5×#360C2semidirect
5. C3×C5×#349C2semidirect
6. C5×#2190C3direct (N × H)
12. C2×C3×C5×D8C22semidirect
17. C3×C5×M32C22semidirect
18. C3×C5×#106C22semidirect
20. C3×#2190C5direct (N × H)
21. C5×#575C2×C3semidirect
22. C5×#509C2×C3semidirect
23. C5×#469C2×C3semidirect
24. C5×#360C2×C3semidirect
25. C5×#349C2×C3semidirect
38. C3×C5×Q16D4semidirect
43. C3×C5×D8D4semidirect
50. C3×#575C2×C5semidirect
51. C3×#509C2×C5semidirect
52. C3×#469C2×C5semidirect
53. C3×#360C2×C5semidirect
54. C3×#349C2×C5semidirect
60. C2×C5×D8C22×C3semidirect
65. C5×M32C22×C3semidirect
66. C5×#106C22×C3semidirect
68. #2190C3×C5direct (N × H)
96. C2×C3×D8C22×C5semidirect
101. C3×M32C22×C5semidirect
102. C3×#106C22×C5semidirect
116. C5×Q16C3×D4semidirect
121. C5×D8C3×D4semidirect
128. #575C2×C3×C5semidirect
129. #509C2×C3×C5semidirect
130. #469C2×C3×C5semidirect
131. #360C2×C3×C5semidirect
132. #349C2×C3×C5semidirect
158. C3×Q16C5×D4semidirect
163. C3×D8C5×D4semidirect
197. C2×D8C22×C3×C5semidirect
202. M32C22×C3×C5semidirect
203. #106C22×C3×C5semidirect
257. Q16C3×C5×D4semidirect
262. D8C3×C5×D4semidirect
269. C3×C5#2190direct (N × H)
315. C5C3×#2190direct (N × H)
329. C3C5×#2190direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#5752
C3×C5×#5092
C3×C5×#4692
C3×C5×#3602
C3×C5×#3492
C5×#21903
C3×#21905