Finite Group Databasebeta

← C3×C5×#2132 #14275166

Subgroups — C3×C5×#2132 #14275166

Order 1920.

500 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×#532C211yesmaximal, characteristic
2C3×C5×#413C212yesmaximal, complemented
3C3×C5×#356C212yesmaximal, complemented
4C3×C5×#355C212yesmaximal, complemented
5C5×#2132C311yesmaximal, characteristic, direct product factor, Hall
6C2×C3×C5×#41C411yescharacteristic
7C2×C3×C5×#41C2211yescharacteristic
8C2×C3×C5×M16—22no—
9C3×C5×C4⋊Q8C2211yescharacteristic
10C3×C5×#127C411yescharacteristic
11C3×C5×#124C411yescharacteristic
12C3×C5×#124C2211yescharacteristic
13C3×C5×#121—22no—
14C3×C4×C5×D4—22no—
15C2×C3×C42×C5C411yescharacteristic
16C3×C5×#101C2214yes—
17C3×C5×#100—22no—
18C3×#2132C511yesmaximal, characteristic, direct product factor, Hall
19C5×#532C2×C311yescharacteristic
20C5×#413C2×C312yescomplemented
21C5×#356C2×C312yescomplemented
22C5×#355C2×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×Q8C2311yescharacteristic
29C2×C3×C5×D4—22no—
30C2×C3×C5×D4—21no—
31C2×C3×C5×D4C2×C411yescharacteristic
32C2×C3×C5×D4C2×C411yescharacteristic
33C2×C3×C5×D4D412yes—
34C22×C3×C4×C5—21no—
35C22×C3×C4×C5—21no—
36C22×C3×C4×C5C2×C411yescharacteristic
37C22×C3×C4×C5D412yes—
38C3×C5×M16—24no—
39C3×C5×M16—42no—
40C2×C3×C5×C8—42no—
41C3×C5×C4⋊C4—22no—
42C3×C5×C22⋊C4—22no—
43C3×C5×C22⋊C4—22no—
44C3×C42×C5—21no—
45C3×C42×C5C2×C411yescharacteristic
46C3×C42×C5C2×C411yescharacteristic
47C3×#532C2×C511yescharacteristic
48C3×#413C2×C512yescomplemented
49C3×#356C2×C512yescomplemented
50C3×#355C2×C512yescomplemented
51C2×C5×#41C3×C411yescharacteristic
52C2×C5×#41C22×C311yescharacteristic
53C2×C5×M16—22no—
54C5×C4⋊Q8C22×C311yescharacteristic
55C5×#127C3×C411yescharacteristic
56C5×#124C3×C411yescharacteristic
57C5×#124C22×C311yescharacteristic
58C5×#121—22no—
59C4×C5×D4—22no—
60C2×C42×C5C3×C411yescharacteristic
61C5×#101C22×C314yes—
62C5×#100—22no—
63#2132C3×C511yescharacteristic, direct product factor, Sylow
64C23×C3×C5—21no—
65C23×C3×C5C22⋊C412yes—
66C23×C3×C5C22⋊C411yescharacteristic
67C3×C5×Q8—41no—
68C3×C5×Q8—41no—
69C3×C5×D4—41no—
70C3×C5×D4—42no—
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×C5C22⋊C411yescharacteristic
85C2×C3×C4×C5C22×C411yescharacteristic
86C2×C3×C4×C5C2×D412yes—
87C3×C5×C8—44no—
88C2×C3×#41C4×C511yescharacteristic
89C2×C3×#41C22×C511yescharacteristic
90C2×C3×M16—22no—
91C3×C4⋊Q8C22×C511yescharacteristic
92C3×#127C4×C511yescharacteristic
93C3×#124C4×C511yescharacteristic
94C3×#124C22×C511yescharacteristic
95C3×#121—22no—
96C3×C4×D4—22no—
97C2×C3×C42C4×C511yescharacteristic
98C3×#101C22×C514yes—
99C3×#100—22no—
100C5×#41—41no—
101C5×#41—41no—
102C5×#41—41no—
103C5×#41—41no—
104C2×C5×Q8C2×C3×C411yescharacteristic
105C2×C5×Q8C23×C311yescharacteristic
106C2×C5×D4—22no—
107C2×C5×D4—21no—
108C2×C5×D4C2×C3×C411yescharacteristic
109C2×C5×D4C2×C3×C411yescharacteristic
110C2×C5×D4C3×D412yes—
111C22×C4×C5—21no—
112C22×C4×C5—21no—
113C22×C4×C5C2×C3×C411yescharacteristic
114C22×C4×C5C3×D412yes—
115C5×M16—24no—
116C5×M16—42no—
117C2×C5×C8—42no—
118C5×C4⋊C4—22no—
119C5×C22⋊C4—22no—
120C5×C22⋊C4—22no—
121C42×C5—21no—
122C42×C5C2×C3×C411yescharacteristic
123C42×C5C2×C3×C411yescharacteristic
124#532C2×C3×C511yescharacteristic
125#413C2×C3×C512yescomplemented
126#356C2×C3×C512yescomplemented
127#355C2×C3×C512yescomplemented
128C22×C3×C5—21no—
129C22×C3×C5—22no—
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#9911yescharacteristic
141C3×C4×C5#9911yescharacteristic
142C3×#41—41no—
143C3×#41—41no—
144C3×#41—41no—
145C3×#41—41no—
146C2×C3×Q8C2×C4×C511yescharacteristic
147C2×C3×Q8C23×C511yescharacteristic
148C2×C3×D4—22no—
149C2×C3×D4—21no—
150C2×C3×D4C2×C4×C511yescharacteristic
151C2×C3×D4C2×C4×C511yescharacteristic
152C2×C3×D4C5×D412yes—
153C22×C3×C4—21no—
154C22×C3×C4—21no—
155C22×C3×C4C2×C4×C511yescharacteristic
156C22×C3×C4C5×D412yes—
157C3×M16—24no—
158C3×M16—42no—
159C2×C3×C8—42no—
160C3×C4⋊C4—22no—
161C3×C22⋊C4—22no—
162C3×C22⋊C4—22no—
163C3×C42—21no—
164C3×C42C2×C4×C511yescharacteristic
165C3×C42C2×C4×C511yescharacteristic
166C23×C5—21no—
167C23×C5C3×C22⋊C412yes—
168C23×C5C3×C22⋊C411yescharacteristic
169C5×Q8—41no—
170C5×Q8—41no—
171C5×D4—41no—
172C5×D4—42no—
173C5×D4—41no—
174C5×D4—44no—
175C5×D4—42no—
176C2×C4×C5—21no—
177C2×C4×C5—22no—
178C2×C4×C5—22no—
179C2×C4×C5—21no—
180C2×C4×C5—21no—
181C2×C4×C5—21no—
182C2×C4×C5—21no—
183C2×C4×C5—42no—
184C2×C4×C5—41no—
185C2×C4×C5—41no—
186C2×C4×C5C3×C22⋊C411yescharacteristic
187C2×C4×C5C22×C3×C411yescharacteristic
188C2×C4×C5C2×C3×D412yes—
189C5×C8—44no—
190C2×#41C3×C4×C511yescharacteristic
191C2×#41C22×C3×C511yescharacteristic
192C2×M16—22no—
193C4⋊Q8C22×C3×C511yescharacteristic
194#127C3×C4×C511yescharacteristic
195#124C3×C4×C511yescharacteristic
196#124C22×C3×C511yescharacteristic
197#121—22no—
198C4×D4—22no—
199C2×C42C3×C4×C511yescharacteristic
200#101C22×C3×C514yes—
201#100—22no—
202C2×C3×C5—21no—
203C2×C3×C5—41no—
204C2×C3×C5—42no—
205C2×C3×C5—81no—
206C2×C3×C5C2×#9911yescharacteristic
207C23×C3—21no—
208C23×C3C5×C22⋊C412yes—
209C23×C3C5×C22⋊C411yescharacteristic
210C3×Q8—41no—
211C3×Q8—41no—
212C3×D4—41no—
213C3×D4—42no—
214C3×D4—41no—
215C3×D4—44no—
216C3×D4—42no—
217C2×C3×C4—21no—
218C2×C3×C4—22no—
219C2×C3×C4—22no—
220C2×C3×C4—21no—
221C2×C3×C4—21no—
222C2×C3×C4—21no—
223C2×C3×C4—21no—
224C2×C3×C4—42no—
225C2×C3×C4—41no—
226C2×C3×C4—41no—
227C2×C3×C4C5×C22⋊C411yescharacteristic
228C2×C3×C4C22×C4×C511yescharacteristic
229C2×C3×C4C2×C5×D412yes—
230C3×C8—44no—
231C22×C5—21no—
232C22×C5—22no—
233C22×C5—41no—
234C22×C5—42no—
235C22×C5—42no—
236C22×C5—41no—
237C22×C5C2×C3×C22⋊C411yescharacteristic
238C4×C5—22no—
239C4×C5—22no—
240C4×C5—21no—
241C4×C5—41no—
242C4×C5—41no—
243C4×C5C3×#9911yescharacteristic
244C4×C5C3×#9911yescharacteristic
245#41—41no—
246#41—41no—
247#41—41no—
248#41—41no—
249C2×Q8C2×C3×C4×C511yescharacteristic
250C2×Q8C23×C3×C511yescharacteristic
251C2×D4—22no—
252C2×D4—21no—
253C2×D4C2×C3×C4×C511yescharacteristic
254C2×D4C2×C3×C4×C511yescharacteristic
255C2×D4C3×C5×D412yes—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4C2×C3×C4×C511yescharacteristic
259C22×C4C3×C5×D412yes—
260M16—24no—
261M16—42no—
262C2×C8—42no—
263C4⋊C4—22no—
264C22⋊C4—22no—
265C22⋊C4—22no—
266C42—21no—
267C42C2×C3×C4×C511yescharacteristic
268C42C2×C3×C4×C511yescharacteristic
269C3×C5#213211yescharacteristic, direct product factor, Hall
270C22×C3—21no—
271C22×C3—22no—
272C22×C3—41no—
273C22×C3—42no—
274C22×C3—42no—
275C22×C3—41no—
276C22×C3C2×C5×C22⋊C411yescharacteristic
277C3×C4—22no—
278C3×C4—22no—
279C3×C4—21no—
280C3×C4—41no—
281C3×C4—41no—
282C3×C4C5×#9911yescharacteristic
283C3×C4C5×#9911yescharacteristic
284C2×C5—21no—
285C2×C5—41no—
286C2×C5—42no—
287C2×C5—81no—
288C2×C5C2×C3×#9911yescharacteristic
289C23—21no—
290C23C3×C5×C22⋊C412yes—
291C23C3×C5×C22⋊C411yescharacteristic
292Q8—41no—
293Q8—41no—
294D4—41no—
295D4—42no—
296D4—41no—
297D4—44no—
298D4—42no—
299C2×C4—21no—
300C2×C4—22no—
301C2×C4—22no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—42no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4C3×C5×C22⋊C411yescharacteristic
310C2×C4C22×C3×C4×C511yescharacteristic
311C2×C4C2×C3×C5×D412yes—
312C8—44no—
313C2×C3—21no—
314C2×C3—41no—
315C2×C3—42no—
316C2×C3—81no—
317C2×C3C2×C5×#9911yescharacteristic
318C5C3×#213211yescharacteristic, direct product factor, Sylow
319C22—21no—
320C22—22no—
321C22—41no—
322C22—42no—
323C22—42no—
324C22—41no—
325C22C2×C3×C5×C22⋊C411yescharacteristic
326C4—22no—
327C4—22no—
328C4—21no—
329C4—41no—
330C4—41no—
331C4C3×C5×#9911yescharacteristic
332C4C3×C5×#9911yescharacteristic
333C3C5×#213211yescharacteristic, direct product factor, Sylow
334C2—21no—
335C2—41no—
336C2—42no—
337C2—81no—
338C2C2×C3×C5×#9911yescharacteristic
subgroupG/H
centerC3×C4×C5#99
commutatorC2×C4C22×C3×C4×C5
Frattini, cosocleC2×Q8C23×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×#413C2semidirect
3. C3×C5×#356C2semidirect
4. C3×C5×#355C2semidirect
5. C5×#2132C3direct (N × H)
18. C3×#2132C5direct (N × H)
20. C5×#413C2×C3semidirect
21. C5×#356C2×C3semidirect
22. C5×#355C2×C3semidirect
48. C3×#413C2×C5semidirect
49. C3×#356C2×C5semidirect
50. C3×#355C2×C5semidirect
63. #2132C3×C5direct (N × H)
125. #413C2×C3×C5semidirect
126. #356C2×C3×C5semidirect
127. #355C2×C3×C5semidirect
269. C3×C5#2132direct (N × H)
318. C5C3×#2132direct (N × H)
333. C3C5×#2132direct (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×#4132
C3×C5×#3562
C3×C5×#3552
C5×#21323
C3×#21325