Finite Group Databasebeta

← C3×C5×#1616 #14274650

Subgroups — C3×C5×#1616 #14274650

Order 1920.

492 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#537C211yesmaximal, characteristic
2C3×C5×#481C212yesmaximal, complemented
3C3×C5×#400C211yesmaximal, characteristic, complemented
4C3×C5×#327C212yesmaximal, complemented
5C3×C5×#324C211yesmaximal, characteristic, complemented
6C5×#1616C311yesmaximal, 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⋊C4C2211yescharacteristic
13C2×C3×C5×C4⋊C4C2211yescharacteristic
14C3×C5×#106—22no—
15C3×C5×#102—22no—
16C3×C5×C22⋊C8C2212yes—
17C3×C5×#95—21no—
18C3×C5×#95—21no—
19C3×C5×#95C2211yescharacteristic
20C3×#1616C511yesmaximal, characteristic, direct product factor, Hall
21C5×#537C2×C311yescharacteristic
22C5×#481C2×C312yescomplemented
23C5×#400C2×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⋊C4D412yes—
44C3×C5×C22⋊C4—22no—
45C3×C5×C22⋊C4—22no—
46C3×#537C2×C511yescharacteristic
47C3×#481C2×C512yescomplemented
48C3×#400C2×C511yescharacteristic, complemented
49C3×#327C2×C512yescomplemented
50C3×#324C2×C511yescharacteristic, complemented
51C2×C5×#41—21no—
52C5×C22⋊Q8—22no—
53C5×#121—22no—
54C5×#121—21no—
55C5×#121C22×C312yes—
56C2×C5×C4⋊C4C22×C311yescharacteristic
57C2×C5×C4⋊C4C22×C311yescharacteristic
58C5×#106—22no—
59C5×#102—22no—
60C5×C22⋊C8C22×C312yes—
61C5×#95—21no—
62C5×#95—21no—
63C5×#95C22×C311yescharacteristic
64#1616C3×C511yescharacteristic, direct product factor, Sylow
65C23×C3×C5—22no—
66C23×C3×C5C2×D411yescharacteristic
67C3×C5×Q8—42no—
68C3×C5×D4—42no—
69C3×C5×D4—42no—
70C3×C5×D4—42no—
71C3×C5×D4—42no—
72C3×C5×D4—81no—
73C2×C3×C4×C5—21no—
74C2×C3×C4×C5—21no—
75C2×C3×C4×C5—21no—
76C2×C3×C4×C5—22no—
77C2×C3×C4×C5—41no—
78C2×C3×C4×C5—44no—
79C2×C3×C4×C5—41no—
80C2×C3×C4×C5—41no—
81C2×C3×C4×C5—41no—
82C2×C3×C4×C5—42no—
83C2×C3×C4×C5—41no—
84C2×C3×C4×C5—41no—
85C2×C3×C4×C5SD1612yes—
86C2×C3×C4×C5C2×D412yes—
87C3×C5×C8—42no—
88C2×C3×#41—21no—
89C3×C22⋊Q8—22no—
90C3×#121—22no—
91C3×#121—21no—
92C3×#121C22×C512yes—
93C2×C3×C4⋊C4C22×C511yescharacteristic
94C2×C3×C4⋊C4C22×C511yescharacteristic
95C3×#106—22no—
96C3×#102—22no—
97C3×C22⋊C8C22×C512yes—
98C3×#95—21no—
99C3×#95—21no—
100C3×#95C22×C511yescharacteristic
101C5×#41—42no—
102C5×#41—42no—
103C2×C5×Q8—21no—
104C2×C5×D4—21no—
105C2×C5×D4—22no—
106C2×C5×D4—21no—
107C2×C5×D4C3×D412yes—
108C22×C4×C5—22no—
109C22×C4×C5—21no—
110C22×C4×C5—21no—
111C22×C4×C5C3×D411yescharacteristic
112C22×C4×C5C3×D411yescharacteristic
113C22×C4×C5C23×C311yescharacteristic
114C2×C5×C8—22no—
115C5×C4⋊C4—21no—
116C5×C4⋊C4—22no—
117C5×C4⋊C4—22no—
118C5×C4⋊C4C3×D412yes—
119C5×C22⋊C4—22no—
120C5×C22⋊C4—22no—
121#537C2×C3×C511yescharacteristic
122#481C2×C3×C512yescomplemented
123#400C2×C3×C511yescharacteristic, complemented
124#327C2×C3×C512yescomplemented
125#324C2×C3×C511yescharacteristic, complemented
126C22×C3×C5—21no—
127C22×C3×C5—21no—
128C22×C3×C5—42no—
129C22×C3×C5—42no—
130C22×C3×C5—42no—
131C22×C3×C5#12011yescharacteristic
132C22×C3×C5C2×SD1611yescharacteristic
133C22×C3×C5AGL(1,ℤ/8)11yescharacteristic
134C3×C4×C5—22no—
135C3×C4×C5—22no—
136C3×C4×C5—42no—
137C3×C4×C5—41no—
138C3×C4×C5—81no—
139C3×C4×C5—81no—
140C3×#41—42no—
141C3×#41—42no—
142C2×C3×Q8—21no—
143C2×C3×D4—21no—
144C2×C3×D4—22no—
145C2×C3×D4—21no—
146C2×C3×D4C5×D412yes—
147C22×C3×C4—22no—
148C22×C3×C4—21no—
149C22×C3×C4—21no—
150C22×C3×C4C5×D411yescharacteristic
151C22×C3×C4C5×D411yescharacteristic
152C22×C3×C4C23×C511yescharacteristic
153C2×C3×C8—22no—
154C3×C4⋊C4—21no—
155C3×C4⋊C4—22no—
156C3×C4⋊C4—22no—
157C3×C4⋊C4C5×D412yes—
158C3×C22⋊C4—22no—
159C3×C22⋊C4—22no—
160C23×C5—22no—
161C23×C5C2×C3×D411yescharacteristic
162C5×Q8—42no—
163C5×D4—42no—
164C5×D4—42no—
165C5×D4—42no—
166C5×D4—42no—
167C5×D4—81no—
168C2×C4×C5—21no—
169C2×C4×C5—21no—
170C2×C4×C5—21no—
171C2×C4×C5—22no—
172C2×C4×C5—41no—
173C2×C4×C5—44no—
174C2×C4×C5—41no—
175C2×C4×C5—41no—
176C2×C4×C5—41no—
177C2×C4×C5—42no—
178C2×C4×C5—41no—
179C2×C4×C5—41no—
180C2×C4×C5C3×SD1612yes—
181C2×C4×C5C2×C3×D412yes—
182C5×C8—42no—
183C2×#41—21no—
184C22⋊Q8—22no—
185#121—22no—
186#121—21no—
187#121C22×C3×C512yes—
188C2×C4⋊C4C22×C3×C511yescharacteristic
189C2×C4⋊C4C22×C3×C511yescharacteristic
190#106—22no—
191#102—22no—
192C22⋊C8C22×C3×C512yes—
193#95—21no—
194#95—21no—
195#95C22×C3×C511yescharacteristic
196C2×C3×C5—21no—
197C2×C3×C5—21no—
198C2×C3×C5—82no—
199C2×C3×C5#45011yescharacteristic
200C2×C3×C5#45411yescharacteristic
201C2×C3×C5#45811yescharacteristic
202C23×C3—22no—
203C23×C3C2×C5×D411yescharacteristic
204C3×Q8—42no—
205C3×D4—42no—
206C3×D4—42no—
207C3×D4—42no—
208C3×D4—42no—
209C3×D4—81no—
210C2×C3×C4—21no—
211C2×C3×C4—21no—
212C2×C3×C4—21no—
213C2×C3×C4—22no—
214C2×C3×C4—41no—
215C2×C3×C4—44no—
216C2×C3×C4—41no—
217C2×C3×C4—41no—
218C2×C3×C4—41no—
219C2×C3×C4—42no—
220C2×C3×C4—41no—
221C2×C3×C4—41no—
222C2×C3×C4C5×SD1612yes—
223C2×C3×C4C2×C5×D412yes—
224C3×C8—42no—
225C22×C5—21no—
226C22×C5—21no—
227C22×C5—42no—
228C22×C5—42no—
229C22×C5—42no—
230C22×C5C3×#12011yescharacteristic
231C22×C5C2×C3×SD1611yescharacteristic
232C22×C5C3×AGL(1,ℤ/8)11yescharacteristic
233C4×C5—22no—
234C4×C5—22no—
235C4×C5—42no—
236C4×C5—41no—
237C4×C5—81no—
238C4×C5—81no—
239#41—42no—
240#41—42no—
241C2×Q8—21no—
242C2×D4—21no—
243C2×D4—22no—
244C2×D4—21no—
245C2×D4C3×C5×D412yes—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4C3×C5×D411yescharacteristic
250C22×C4C3×C5×D411yescharacteristic
251C22×C4C23×C3×C511yescharacteristic
252C2×C8—22no—
253C4⋊C4—21no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4C3×C5×D412yes—
257C22⋊C4—22no—
258C22⋊C4—22no—
259C3×C5#161611yescharacteristic, direct product factor, Hall
260C22×C3—21no—
261C22×C3—21no—
262C22×C3—42no—
263C22×C3—42no—
264C22×C3—42no—
265C22×C3C5×#12011yescharacteristic
266C22×C3C2×C5×SD1611yescharacteristic
267C22×C3C5×AGL(1,ℤ/8)11yescharacteristic
268C3×C4—22no—
269C3×C4—22no—
270C3×C4—42no—
271C3×C4—41no—
272C3×C4—81no—
273C3×C4—81no—
274C2×C5—21no—
275C2×C5—21no—
276C2×C5—82no—
277C2×C5C3×#45011yescharacteristic
278C2×C5C3×#45411yescharacteristic
279C2×C5C3×#45811yescharacteristic
280C23—22no—
281C23C2×C3×C5×D411yescharacteristic
282Q8—42no—
283D4—42no—
284D4—42no—
285D4—42no—
286D4—42no—
287D4—81no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—22no—
292C2×C4—41no—
293C2×C4—44no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—42no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4C3×C5×SD1612yes—
301C2×C4C2×C3×C5×D412yes—
302C8—42no—
303C2×C3—21no—
304C2×C3—21no—
305C2×C3—82no—
306C2×C3C5×#45011yescharacteristic
307C2×C3C5×#45411yescharacteristic
308C2×C3C5×#45811yescharacteristic
309C5C3×#161611yescharacteristic, direct product factor, Sylow
310C22—21no—
311C22—21no—
312C22—42no—
313C22—42no—
314C22—42no—
315C22C3×C5×#12011yescharacteristic
316C22C2×C3×C5×SD1611yescharacteristic
317C22C3×C5×AGL(1,ℤ/8)11yescharacteristic
318C4—22no—
319C4—22no—
320C4—42no—
321C4—41no—
322C4—81no—
323C4—81no—
324C3C5×#161611yescharacteristic, direct product factor, Sylow
325C2—21no—
326C2—21no—
327C2—82no—
328C2C3×C5×#45011yescharacteristic
329C2C3×C5×#45411yescharacteristic
330C2C3×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×#481C2semidirect
3. C3×C5×#400C2semidirect
4. C3×C5×#327C2semidirect
5. C3×C5×#324C2semidirect
6. C5×#1616C3direct (N × H)
20. C3×#1616C5direct (N × H)
22. C5×#481C2×C3semidirect
23. C5×#400C2×C3semidirect
24. C5×#327C2×C3semidirect
25. C5×#324C2×C3semidirect
47. C3×#481C2×C5semidirect
48. C3×#400C2×C5semidirect
49. C3×#327C2×C5semidirect
50. C3×#324C2×C5semidirect
64. #1616C3×C5direct (N × H)
122. #481C2×C3×C5semidirect
123. #400C2×C3×C5semidirect
124. #327C2×C3×C5semidirect
125. #324C2×C3×C5semidirect
259. C3×C5#1616direct (N × H)
309. C5C3×#1616direct (N × H)
324. C3C5×#1616direct (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×#4812
C3×C5×#4002
C3×C5×#3272
C3×C5×#3242
C5×#16163
C3×#16165