Finite Group Databasebeta

← C3×C5×#2126 #14275160

Subgroups — C3×C5×#2126 #14275160

Order 1920.

516 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#532C211yesmaximal, characteristic
2C3×C5×#410C212yesmaximal, complemented
3C3×C5×#354C212yesmaximal, complemented
4C3×C5×#353C212yesmaximal, complemented
5C5×#2126C311yesmaximal, characteristic, direct product factor, Hall
6C2×C3×C5×#41C411yescharacteristic, complemented
7C2×C3×C5×#41C2211yescharacteristic
8C3×C5×C4⋊Q8C411yescharacteristic, complemented
9C3×C5×#127C2211yescharacteristic
10C3×C5×#124C411yescharacteristic, complemented
11C3×C5×#124C2211yescharacteristic
12C3×C5×#121—22no—
13C3×C4×C5×D4—22no—
14C3×C5×#117—22no—
15C2×C3×C42×C5C411yescharacteristic, complemented
16C3×C5×#99—22no—
17C3×C5×#99C2214yes—
18C3×#2126C511yesmaximal, characteristic, direct product factor, Hall
19C5×#532C2×C311yescharacteristic
20C5×#410C2×C312yescomplemented
21C5×#354C2×C312yescomplemented
22C5×#353C2×C312yescomplemented
23C3×C5×#41—41no—
24C3×C5×#41—41no—
25C3×C5×#41—41no—
26C3×C5×#41—41no—
27C2×C3×C5×Q8C2×C411yescharacteristic
28C2×C3×C5×Q8C2×C411yescharacteristic
29C2×C3×C5×D4—21no—
30C2×C3×C5×D4—22no—
31C2×C3×C5×D4C2×C411yescharacteristic
32C2×C3×C5×D4D412yes—
33C2×C3×C5×D4C2311yescharacteristic
34C22×C3×C4×C5—21no—
35C22×C3×C4×C5—21no—
36C22×C3×C4×C5C2×C411yescharacteristic
37C22×C3×C4×C5D412yes—
38C3×C5×C4⋊C4—22no—
39C3×C5×C4⋊C4—42no—
40C3×C5×C22⋊C4—22no—
41C3×C5×C22⋊C4—22no—
42C3×C5×C22⋊C4—24no—
43C3×C42×C5—21no—
44C3×C42×C5—42no—
45C3×C42×C5C2×C411yescharacteristic
46C3×C42×C5C2×C411yescharacteristic
47C3×#532C2×C511yescharacteristic
48C3×#410C2×C512yescomplemented
49C3×#354C2×C512yescomplemented
50C3×#353C2×C512yescomplemented
51C2×C5×#41C3×C411yescharacteristic, complemented
52C2×C5×#41C22×C311yescharacteristic
53C5×C4⋊Q8C3×C411yescharacteristic, complemented
54C5×#127C22×C311yescharacteristic
55C5×#124C3×C411yescharacteristic, complemented
56C5×#124C22×C311yescharacteristic
57C5×#121—22no—
58C4×C5×D4—22no—
59C5×#117—22no—
60C2×C42×C5C3×C411yescharacteristic, complemented
61C5×#99—22no—
62C5×#99C22×C314yes—
63#2126C3×C511yescharacteristic, direct product factor, Sylow
64C23×C3×C5—21no—
65C23×C3×C5C22⋊C411yescharacteristic
66C23×C3×C5C2×D412yes—
67C3×C5×Q8—41no—
68C3×C5×Q8—41no—
69C3×C5×D4—42no—
70C3×C5×D4—41no—
71C3×C5×D4—44no—
72C3×C5×D4—41no—
73C3×C5×D4—42no—
74C2×C3×C4×C5—21no—
75C2×C3×C4×C5—22no—
76C2×C3×C4×C5—21no—
77C2×C3×C4×C5—22no—
78C2×C3×C4×C5—21no—
79C2×C3×C4×C5—21no—
80C2×C3×C4×C5—21no—
81C2×C3×C4×C5—42no—
82C2×C3×C4×C5—41no—
83C2×C3×C4×C5—41no—
84C2×C3×C4×C5—44no—
85C2×C3×C4×C5C22⋊C412yes—
86C2×C3×C4×C5C22⋊C411yescharacteristic
87C2×C3×C4×C5C22×C411yescharacteristic
88C2×C3×#41C4×C511yescharacteristic, complemented
89C2×C3×#41C22×C511yescharacteristic
90C3×C4⋊Q8C4×C511yescharacteristic, complemented
91C3×#127C22×C511yescharacteristic
92C3×#124C4×C511yescharacteristic, complemented
93C3×#124C22×C511yescharacteristic
94C3×#121—22no—
95C3×C4×D4—22no—
96C3×#117—22no—
97C2×C3×C42C4×C511yescharacteristic, complemented
98C3×#99—22no—
99C3×#99C22×C514yes—
100C5×#41—41no—
101C5×#41—41no—
102C5×#41—41no—
103C5×#41—41no—
104C2×C5×Q8C2×C3×C411yescharacteristic
105C2×C5×Q8C2×C3×C411yescharacteristic
106C2×C5×D4—21no—
107C2×C5×D4—22no—
108C2×C5×D4C2×C3×C411yescharacteristic
109C2×C5×D4C3×D412yes—
110C2×C5×D4C23×C311yescharacteristic
111C22×C4×C5—21no—
112C22×C4×C5—21no—
113C22×C4×C5C2×C3×C411yescharacteristic
114C22×C4×C5C3×D412yes—
115C5×C4⋊C4—22no—
116C5×C4⋊C4—42no—
117C5×C22⋊C4—22no—
118C5×C22⋊C4—22no—
119C5×C22⋊C4—24no—
120C42×C5—21no—
121C42×C5—42no—
122C42×C5C2×C3×C411yescharacteristic
123C42×C5C2×C3×C411yescharacteristic
124#532C2×C3×C511yescharacteristic
125#410C2×C3×C512yescomplemented
126#354C2×C3×C512yescomplemented
127#353C2×C3×C512yescomplemented
128C22×C3×C5—22no—
129C22×C3×C5—21no—
130C22×C3×C5—41no—
131C22×C3×C5—42no—
132C22×C3×C5—42no—
133C22×C3×C5—41no—
134C22×C3×C5C2×C22⋊C411yescharacteristic
135C3×C4×C5—22no—
136C3×C4×C5—22no—
137C3×C4×C5—21no—
138C3×C4×C5—41no—
139C3×C4×C5—41no—
140C3×C4×C5—84no—
141C3×C4×C5#9911yescharacteristic
142C3×C4×C5#9911yescharacteristic
143C3×#41—41no—
144C3×#41—41no—
145C3×#41—41no—
146C3×#41—41no—
147C2×C3×Q8C2×C4×C511yescharacteristic
148C2×C3×Q8C2×C4×C511yescharacteristic
149C2×C3×D4—21no—
150C2×C3×D4—22no—
151C2×C3×D4C2×C4×C511yescharacteristic
152C2×C3×D4C5×D412yes—
153C2×C3×D4C23×C511yescharacteristic
154C22×C3×C4—21no—
155C22×C3×C4—21no—
156C22×C3×C4C2×C4×C511yescharacteristic
157C22×C3×C4C5×D412yes—
158C3×C4⋊C4—22no—
159C3×C4⋊C4—42no—
160C3×C22⋊C4—22no—
161C3×C22⋊C4—22no—
162C3×C22⋊C4—24no—
163C3×C42—21no—
164C3×C42—42no—
165C3×C42C2×C4×C511yescharacteristic
166C3×C42C2×C4×C511yescharacteristic
167C23×C5—21no—
168C23×C5C3×C22⋊C411yescharacteristic
169C23×C5C2×C3×D412yes—
170C5×Q8—41no—
171C5×Q8—41no—
172C5×D4—42no—
173C5×D4—41no—
174C5×D4—41no—
175C5×D4—44no—
176C5×D4—42no—
177C2×C4×C5—21no—
178C2×C4×C5—22no—
179C2×C4×C5—22no—
180C2×C4×C5—21no—
181C2×C4×C5—21no—
182C2×C4×C5—21no—
183C2×C4×C5—21no—
184C2×C4×C5—42no—
185C2×C4×C5—41no—
186C2×C4×C5—41no—
187C2×C4×C5—44no—
188C2×C4×C5C3×C22⋊C412yes—
189C2×C4×C5C3×C22⋊C411yescharacteristic
190C2×C4×C5C22×C3×C411yescharacteristic
191C2×#41C3×C4×C511yescharacteristic, complemented
192C2×#41C22×C3×C511yescharacteristic
193C4⋊Q8C3×C4×C511yescharacteristic, complemented
194#127C22×C3×C511yescharacteristic
195#124C3×C4×C511yescharacteristic, complemented
196#124C22×C3×C511yescharacteristic
197#121—22no—
198C4×D4—22no—
199#117—22no—
200C2×C42C3×C4×C511yescharacteristic, complemented
201#99—22no—
202#99C22×C3×C514yes—
203C2×C3×C5—21no—
204C2×C3×C5—42no—
205C2×C3×C5—41no—
206C2×C3×C5—81no—
207C2×C3×C5C2×#9911yescharacteristic
208C23×C3—21no—
209C23×C3C5×C22⋊C411yescharacteristic
210C23×C3C2×C5×D412yes—
211C3×Q8—41no—
212C3×Q8—41no—
213C3×D4—42no—
214C3×D4—41no—
215C3×D4—41no—
216C3×D4—44no—
217C3×D4—42no—
218C2×C3×C4—21no—
219C2×C3×C4—22no—
220C2×C3×C4—22no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—21no—
224C2×C3×C4—21no—
225C2×C3×C4—42no—
226C2×C3×C4—41no—
227C2×C3×C4—41no—
228C2×C3×C4—44no—
229C2×C3×C4C5×C22⋊C412yes—
230C2×C3×C4C5×C22⋊C411yescharacteristic
231C2×C3×C4C22×C4×C511yescharacteristic
232C22×C5—22no—
233C22×C5—21no—
234C22×C5—41no—
235C22×C5—42no—
236C22×C5—42no—
237C22×C5—41no—
238C22×C5C2×C3×C22⋊C411yescharacteristic
239C4×C5—22no—
240C4×C5—22no—
241C4×C5—21no—
242C4×C5—41no—
243C4×C5—41no—
244C4×C5—84no—
245C4×C5C3×#9911yescharacteristic
246C4×C5C3×#9911yescharacteristic
247#41—41no—
248#41—41no—
249#41—41no—
250#41—41no—
251C2×Q8C2×C3×C4×C511yescharacteristic
252C2×Q8C2×C3×C4×C511yescharacteristic
253C2×D4—21no—
254C2×D4—22no—
255C2×D4C2×C3×C4×C511yescharacteristic
256C2×D4C3×C5×D412yes—
257C2×D4C23×C3×C511yescharacteristic
258C22×C4—21no—
259C22×C4—21no—
260C22×C4C2×C3×C4×C511yescharacteristic
261C22×C4C3×C5×D412yes—
262C4⋊C4—22no—
263C4⋊C4—42no—
264C22⋊C4—22no—
265C22⋊C4—22no—
266C22⋊C4—24no—
267C42—21no—
268C42—42no—
269C42C2×C3×C4×C511yescharacteristic
270C42C2×C3×C4×C511yescharacteristic
271C3×C5#212611yescharacteristic, direct product factor, Hall
272C22×C3—22no—
273C22×C3—21no—
274C22×C3—41no—
275C22×C3—42no—
276C22×C3—42no—
277C22×C3—41no—
278C22×C3C2×C5×C22⋊C411yescharacteristic
279C3×C4—22no—
280C3×C4—22no—
281C3×C4—21no—
282C3×C4—41no—
283C3×C4—41no—
284C3×C4—84no—
285C3×C4C5×#9911yescharacteristic
286C3×C4C5×#9911yescharacteristic
287C2×C5—21no—
288C2×C5—42no—
289C2×C5—41no—
290C2×C5—81no—
291C2×C5C2×C3×#9911yescharacteristic
292C23—21no—
293C23C3×C5×C22⋊C411yescharacteristic
294C23C2×C3×C5×D412yes—
295Q8—41no—
296Q8—41no—
297D4—42no—
298D4—41no—
299D4—41no—
300D4—44no—
301D4—42no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—22no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—42no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—44no—
313C2×C4C3×C5×C22⋊C412yes—
314C2×C4C3×C5×C22⋊C411yescharacteristic
315C2×C4C22×C3×C4×C511yescharacteristic
316C2×C3—21no—
317C2×C3—42no—
318C2×C3—41no—
319C2×C3—81no—
320C2×C3C2×C5×#9911yescharacteristic
321C5C3×#212611yescharacteristic, direct product factor, Sylow
322C22—22no—
323C22—21no—
324C22—41no—
325C22—42no—
326C22—42no—
327C22—41no—
328C22C2×C3×C5×C22⋊C411yescharacteristic
329C4—22no—
330C4—22no—
331C4—21no—
332C4—41no—
333C4—41no—
334C4—84no—
335C4C3×C5×#9911yescharacteristic
336C4C3×C5×#9911yescharacteristic
337C3C5×#212611yescharacteristic, direct product factor, Sylow
338C2—21no—
339C2—42no—
340C2—41no—
341C2—81no—
342C2C2×C3×C5×#9911yescharacteristic
subgroupG/H
centerC3×C4×C5#99
commutatorC2×C4C22×C3×C4×C5
Frattini, cosocleC2×D4C23×C3×C5
socleC2×C3×C5C2×#99
prime radicalC3×C5×#532C2
LCSG ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1
UCSC1 ◁ C3×C4×C5 ◁ C2×C3×C4×C5 ◁ C2×C3×C5×#41 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×C5×#410C2semidirect
3. C3×C5×#354C2semidirect
4. C3×C5×#353C2semidirect
5. C5×#2126C3direct (N × H)
6. C2×C3×C5×#41C4semidirect
8. C3×C5×C4⋊Q8C4semidirect
10. C3×C5×#124C4semidirect
15. C2×C3×C42×C5C4semidirect
18. C3×#2126C5direct (N × H)
20. C5×#410C2×C3semidirect
21. C5×#354C2×C3semidirect
22. C5×#353C2×C3semidirect
48. C3×#410C2×C5semidirect
49. C3×#354C2×C5semidirect
50. C3×#353C2×C5semidirect
51. C2×C5×#41C3×C4semidirect
53. C5×C4⋊Q8C3×C4semidirect
55. C5×#124C3×C4semidirect
60. C2×C42×C5C3×C4semidirect
63. #2126C3×C5direct (N × H)
88. C2×C3×#41C4×C5semidirect
90. C3×C4⋊Q8C4×C5semidirect
92. C3×#124C4×C5semidirect
97. C2×C3×C42C4×C5semidirect
125. #410C2×C3×C5semidirect
126. #354C2×C3×C5semidirect
127. #353C2×C3×C5semidirect
191. C2×#41C3×C4×C5semidirect
193. C4⋊Q8C3×C4×C5semidirect
195. #124C3×C4×C5semidirect
200. C2×C42C3×C4×C5semidirect
271. C3×C5#2126direct (N × H)
321. C5C3×#2126direct (N × H)
337. C3C5×#2126direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#5322
C3×C5×#4102
C3×C5×#3542
C3×C5×#3532
C5×#21263
C3×#21265