Finite Group Databasebeta

← C3×C5×#1642 #14274676

Subgroups — C3×C5×#1642 #14274676

Order 1920.

500 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#550C211yesmaximal, characteristic
2C3×C5×#486C211yesmaximal, characteristic, complemented
3C3×C5×#475C211yesmaximal, characteristic, complemented
4C3×C5×#461C211yesmaximal, characteristic, complemented
5C3×C5×#331C211yesmaximal, characteristic, complemented
6C3×C5×Q8⋊C8C211yesmaximal, characteristic, complemented
7C3×C5×D4⋊C8C211yesmaximal, characteristic, complemented
8C5×#1642C311yesmaximal, characteristic, direct product factor, Hall
9C2×C3×C5×#41—21no—
10C2×C3×C5×SD16—21no—
11C3×C5×C4⋊Q8C2211yescharacteristic
12C3×C5×#127—21no—
13C3×C5×#127C2211yescharacteristic
14C3×C5×#121—22no—
15C3×C5×#121—21no—
16C3×C4×C5×Q8C2211yescharacteristic
17C3×C4×C5×D4—21no—
18C3×C4×C5×D4C2211yescharacteristic
19C3×C5×#106—21no—
20C3×C5×C4⋊C8C2211yescharacteristic
21C3×C5×C4⋊C8C2211yescharacteristic
22C3×C5×#103—21no—
23C3×C5×#103—21no—
24C3×C5×#102—22no—
25C3×C4×C5×C8C2211yescharacteristic
26C3×#1642C511yesmaximal, characteristic, direct product factor, Hall
27C5×#550C2×C311yescharacteristic
28C5×#486C2×C311yescharacteristic, complemented
29C5×#475C2×C311yescharacteristic, complemented
30C5×#461C2×C311yescharacteristic, complemented
31C5×#331C2×C311yescharacteristic, complemented
32C5×Q8⋊C8C2×C311yescharacteristic, complemented
33C5×D4⋊C8C2×C311yescharacteristic, complemented
34C3×C5×#41—42no—
35C3×C5×#41—42no—
36C2×C3×C5×Q8—21no—
37C2×C3×C5×Q8D411yescharacteristic
38C2×C3×C5×D4—22no—
39C2×C3×C5×D4—22no—
40C2×C3×C5×D4—21no—
41C2×C3×C5×D4—21no—
42C2×C3×C5×D4D411yescharacteristic
43C2×C3×C5×D4D412yes—
44C22×C3×C4×C5—22no—
45C22×C3×C4×C5—21no—
46C3×C5×SD16—42no—
47C2×C3×C5×C8—21no—
48C2×C3×C5×C8—21no—
49C2×C3×C5×C8—21no—
50C3×C5×C4⋊C4—21no—
51C3×C5×C4⋊C4—21no—
52C3×C5×C4⋊C4—21no—
53C3×C5×C4⋊C4D411yescharacteristic
54C3×C5×C4⋊C4D411yescharacteristic
55C3×C5×C22⋊C4—22no—
56C3×C5×C22⋊C4—21no—
57C3×C42×C5—21no—
58C3×C42×C5C2311yescharacteristic
59C3×#550C2×C511yescharacteristic
60C3×#486C2×C511yescharacteristic, complemented
61C3×#475C2×C511yescharacteristic, complemented
62C3×#461C2×C511yescharacteristic, complemented
63C3×#331C2×C511yescharacteristic, complemented
64C3×Q8⋊C8C2×C511yescharacteristic, complemented
65C3×D4⋊C8C2×C511yescharacteristic, complemented
66C2×C5×#41—21no—
67C2×C5×SD16—21no—
68C5×C4⋊Q8C22×C311yescharacteristic
69C5×#127—21no—
70C5×#127C22×C311yescharacteristic
71C5×#121—22no—
72C5×#121—21no—
73C4×C5×Q8C22×C311yescharacteristic
74C4×C5×D4—21no—
75C4×C5×D4C22×C311yescharacteristic
76C5×#106—21no—
77C5×C4⋊C8C22×C311yescharacteristic
78C5×C4⋊C8C22×C311yescharacteristic
79C5×#103—21no—
80C5×#103—21no—
81C5×#102—22no—
82C4×C5×C8C22×C311yescharacteristic
83#1642C3×C511yescharacteristic, direct product factor, Sylow
84C23×C3×C5—21no—
85C23×C3×C5—22no—
86C3×C5×Q8—21no—
87C3×C5×Q8—41no—
88C3×C5×Q8—41no—
89C3×C5×Q8SD1612yescomplemented
90C3×C5×D4—21no—
91C3×C5×D4—21no—
92C3×C5×D4—44no—
93C3×C5×D4—42no—
94C3×C5×D4—41no—
95C3×C5×D4—41no—
96C3×C5×D4—41no—
97C3×C5×D4—41no—
98C3×C5×D4—82no—
99C2×C3×C4×C5—21no—
100C2×C3×C4×C5—21no—
101C2×C3×C4×C5—21no—
102C2×C3×C4×C5—44no—
103C2×C3×C4×C5—41no—
104C2×C3×C4×C5—41no—
105C2×C3×C4×C5—41no—
106C2×C3×C4×C5C2×D411yescharacteristic
107C2×C3×C4×C5C2×D411yescharacteristic
108C2×C3×C4×C5C2×D411yescharacteristic
109C3×C5×C8—41no—
110C3×C5×C8—41no—
111C3×C5×C8—41no—
112C2×C3×#41—21no—
113C2×C3×SD16—21no—
114C3×C4⋊Q8C22×C511yescharacteristic
115C3×#127—21no—
116C3×#127C22×C511yescharacteristic
117C3×#121—22no—
118C3×#121—21no—
119C3×C4×Q8C22×C511yescharacteristic
120C3×C4×D4—21no—
121C3×C4×D4C22×C511yescharacteristic
122C3×#106—21no—
123C3×C4⋊C8C22×C511yescharacteristic
124C3×C4⋊C8C22×C511yescharacteristic
125C3×#103—21no—
126C3×#103—21no—
127C3×#102—22no—
128C3×C4×C8C22×C511yescharacteristic
129C5×#41—42no—
130C5×#41—42no—
131C2×C5×Q8—21no—
132C2×C5×Q8C3×D411yescharacteristic
133C2×C5×D4—22no—
134C2×C5×D4—22no—
135C2×C5×D4—21no—
136C2×C5×D4—21no—
137C2×C5×D4C3×D411yescharacteristic
138C2×C5×D4C3×D412yes—
139C22×C4×C5—22no—
140C22×C4×C5—21no—
141C5×SD16—42no—
142C2×C5×C8—21no—
143C2×C5×C8—21no—
144C2×C5×C8—21no—
145C5×C4⋊C4—21no—
146C5×C4⋊C4—21no—
147C5×C4⋊C4—21no—
148C5×C4⋊C4C3×D411yescharacteristic
149C5×C4⋊C4C3×D411yescharacteristic
150C5×C22⋊C4—22no—
151C5×C22⋊C4—21no—
152C42×C5—21no—
153C42×C5C23×C311yescharacteristic
154#550C2×C3×C511yescharacteristic
155#486C2×C3×C511yescharacteristic, complemented
156#475C2×C3×C511yescharacteristic, complemented
157#461C2×C3×C511yescharacteristic, complemented
158#331C2×C3×C511yescharacteristic, complemented
159Q8⋊C8C2×C3×C511yescharacteristic, complemented
160D4⋊C8C2×C3×C511yescharacteristic, complemented
161C22×C3×C5—41no—
162C22×C3×C5—42no—
163C22×C3×C5—42no—
164C22×C3×C5—41no—
165C22×C3×C5—42no—
166C22×C3×C5—41no—
167C22×C3×C5#12011yescharacteristic
168C3×C4×C5—22no—
169C3×C4×C5—21no—
170C3×C4×C5—41no—
171C3×C4×C5—41no—
172C3×C4×C5—81no—
173C3×C4×C5C2×SD1611yescharacteristic
174C3×C4×C5#13511yescharacteristic
175C3×C4×C5AGL(1,ℤ/8)11yescharacteristic
176C3×C4×C5AGL(1,ℤ/8)11yescharacteristic
177C3×#41—42no—
178C3×#41—42no—
179C2×C3×Q8—21no—
180C2×C3×Q8C5×D411yescharacteristic
181C2×C3×D4—22no—
182C2×C3×D4—22no—
183C2×C3×D4—21no—
184C2×C3×D4—21no—
185C2×C3×D4C5×D411yescharacteristic
186C2×C3×D4C5×D412yes—
187C22×C3×C4—22no—
188C22×C3×C4—21no—
189C3×SD16—42no—
190C2×C3×C8—21no—
191C2×C3×C8—21no—
192C2×C3×C8—21no—
193C3×C4⋊C4—21no—
194C3×C4⋊C4—21no—
195C3×C4⋊C4—21no—
196C3×C4⋊C4C5×D411yescharacteristic
197C3×C4⋊C4C5×D411yescharacteristic
198C3×C22⋊C4—22no—
199C3×C22⋊C4—21no—
200C3×C42—21no—
201C3×C42C23×C511yescharacteristic
202C23×C5—21no—
203C23×C5—22no—
204C5×Q8—21no—
205C5×Q8—41no—
206C5×Q8—41no—
207C5×Q8C3×SD1612yescomplemented
208C5×D4—21no—
209C5×D4—21no—
210C5×D4—42no—
211C5×D4—44no—
212C5×D4—41no—
213C5×D4—41no—
214C5×D4—41no—
215C5×D4—41no—
216C5×D4—82no—
217C2×C4×C5—21no—
218C2×C4×C5—21no—
219C2×C4×C5—21no—
220C2×C4×C5—44no—
221C2×C4×C5—41no—
222C2×C4×C5—41no—
223C2×C4×C5—41no—
224C2×C4×C5C2×C3×D411yescharacteristic
225C2×C4×C5C2×C3×D411yescharacteristic
226C2×C4×C5C2×C3×D411yescharacteristic
227C5×C8—41no—
228C5×C8—41no—
229C5×C8—41no—
230C2×#41—21no—
231C2×SD16—21no—
232C4⋊Q8C22×C3×C511yescharacteristic
233#127—21no—
234#127C22×C3×C511yescharacteristic
235#121—22no—
236#121—21no—
237C4×Q8C22×C3×C511yescharacteristic
238C4×D4—21no—
239C4×D4C22×C3×C511yescharacteristic
240#106—21no—
241C4⋊C8C22×C3×C511yescharacteristic
242C4⋊C8C22×C3×C511yescharacteristic
243#103—21no—
244#103—21no—
245#102—22no—
246C4×C8C22×C3×C511yescharacteristic
247C2×C3×C5—81no—
248C2×C3×C5—82no—
249C2×C3×C5#44911yescharacteristic
250C2×C3×C5#45011yescharacteristic
251C2×C3×C5#45511yescharacteristic
252C23×C3—21no—
253C23×C3—22no—
254C3×Q8—21no—
255C3×Q8—41no—
256C3×Q8—41no—
257C3×Q8C5×SD1612yescomplemented
258C3×D4—21no—
259C3×D4—21no—
260C3×D4—42no—
261C3×D4—44no—
262C3×D4—41no—
263C3×D4—41no—
264C3×D4—41no—
265C3×D4—41no—
266C3×D4—82no—
267C2×C3×C4—21no—
268C2×C3×C4—21no—
269C2×C3×C4—21no—
270C2×C3×C4—44no—
271C2×C3×C4—41no—
272C2×C3×C4—41no—
273C2×C3×C4—41no—
274C2×C3×C4C2×C5×D411yescharacteristic
275C2×C3×C4C2×C5×D411yescharacteristic
276C2×C3×C4C2×C5×D411yescharacteristic
277C3×C8—41no—
278C3×C8—41no—
279C3×C8—41no—
280C22×C5—41no—
281C22×C5—42no—
282C22×C5—42no—
283C22×C5—41no—
284C22×C5—42no—
285C22×C5—41no—
286C22×C5C3×#12011yescharacteristic
287C4×C5—22no—
288C4×C5—21no—
289C4×C5—41no—
290C4×C5—41no—
291C4×C5—81no—
292C4×C5C2×C3×SD1611yescharacteristic
293C4×C5C3×#13511yescharacteristic
294C4×C5C3×AGL(1,ℤ/8)11yescharacteristic
295C4×C5C3×AGL(1,ℤ/8)11yescharacteristic
296#41—42no—
297#41—42no—
298C2×Q8—21no—
299C2×Q8C3×C5×D411yescharacteristic
300C2×D4—22no—
301C2×D4—22no—
302C2×D4—21no—
303C2×D4—21no—
304C2×D4C3×C5×D411yescharacteristic
305C2×D4C3×C5×D412yes—
306C22×C4—22no—
307C22×C4—21no—
308SD16—42no—
309C2×C8—21no—
310C2×C8—21no—
311C2×C8—21no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4C3×C5×D411yescharacteristic
316C4⋊C4C3×C5×D411yescharacteristic
317C22⋊C4—22no—
318C22⋊C4—21no—
319C42—21no—
320C42C23×C3×C511yescharacteristic
321C3×C5#164211yescharacteristic, direct product factor, Hall
322C22×C3—41no—
323C22×C3—42no—
324C22×C3—42no—
325C22×C3—41no—
326C22×C3—42no—
327C22×C3—41no—
328C22×C3C5×#12011yescharacteristic
329C3×C4—22no—
330C3×C4—21no—
331C3×C4—41no—
332C3×C4—41no—
333C3×C4—81no—
334C3×C4C2×C5×SD1611yescharacteristic
335C3×C4C5×#13511yescharacteristic
336C3×C4C5×AGL(1,ℤ/8)11yescharacteristic
337C3×C4C5×AGL(1,ℤ/8)11yescharacteristic
338C2×C5—81no—
339C2×C5—82no—
340C2×C5C3×#44911yescharacteristic
341C2×C5C3×#45011yescharacteristic
342C2×C5C3×#45511yescharacteristic
343C23—21no—
344C23—22no—
345Q8—21no—
346Q8—41no—
347Q8—41no—
348Q8C3×C5×SD1612yescomplemented
349D4—21no—
350D4—21no—
351D4—42no—
352D4—44no—
353D4—41no—
354D4—41no—
355D4—41no—
356D4—41no—
357D4—82no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—44no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4C2×C3×C5×D411yescharacteristic
366C2×C4C2×C3×C5×D411yescharacteristic
367C2×C4C2×C3×C5×D411yescharacteristic
368C8—41no—
369C8—41no—
370C8—41no—
371C2×C3—81no—
372C2×C3—82no—
373C2×C3C5×#44911yescharacteristic
374C2×C3C5×#45011yescharacteristic
375C2×C3C5×#45511yescharacteristic
376C5C3×#164211yescharacteristic, direct product factor, Sylow
377C22—41no—
378C22—42no—
379C22—42no—
380C22—41no—
381C22—42no—
382C22—41no—
383C22C3×C5×#12011yescharacteristic
384C4—22no—
385C4—21no—
386C4—41no—
387C4—41no—
388C4—81no—
389C4C2×C3×C5×SD1611yescharacteristic
390C4C3×C5×#13511yescharacteristic
391C4C3×C5×AGL(1,ℤ/8)11yescharacteristic
392C4C3×C5×AGL(1,ℤ/8)11yescharacteristic
393C3C5×#164211yescharacteristic, direct product factor, Sylow
394C2—81no—
395C2—82no—
396C2C3×C5×#44911yescharacteristic
397C2C3×C5×#45011yescharacteristic
398C2C3×C5×#45511yescharacteristic
subgroupG/H
center, socleC22×C3×C5#120
commutator, Frattini, cosocleC42C23×C3×C5
prime radicalC3×C5×#550C2
LCSG ▷ C42 ▷ C22 ▷ C1
UCSC1 ◁ C22×C3×C5 ◁ C3×C42×C5 ◁ G

Complements

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

NH ≅ G/Ntype
2. C3×C5×#486C2semidirect
3. C3×C5×#475C2semidirect
4. C3×C5×#461C2semidirect
5. C3×C5×#331C2semidirect
6. C3×C5×Q8⋊C8C2semidirect
7. C3×C5×D4⋊C8C2semidirect
8. C5×#1642C3direct (N × H)
26. C3×#1642C5direct (N × H)
28. C5×#486C2×C3semidirect
29. C5×#475C2×C3semidirect
30. C5×#461C2×C3semidirect
31. C5×#331C2×C3semidirect
32. C5×Q8⋊C8C2×C3semidirect
33. C5×D4⋊C8C2×C3semidirect
60. C3×#486C2×C5semidirect
61. C3×#475C2×C5semidirect
62. C3×#461C2×C5semidirect
63. C3×#331C2×C5semidirect
64. C3×Q8⋊C8C2×C5semidirect
65. C3×D4⋊C8C2×C5semidirect
83. #1642C3×C5direct (N × H)
89. C3×C5×Q8SD16semidirect
155. #486C2×C3×C5semidirect
156. #475C2×C3×C5semidirect
157. #461C2×C3×C5semidirect
158. #331C2×C3×C5semidirect
159. Q8⋊C8C2×C3×C5semidirect
160. D4⋊C8C2×C3×C5semidirect
207. C5×Q8C3×SD16semidirect
257. C3×Q8C5×SD16semidirect
321. C3×C5#1642direct (N × H)
348. Q8C3×C5×SD16semidirect
376. C5C3×#1642direct (N × H)
393. C3C5×#1642direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#5502
C3×C5×#4862
C3×C5×#4752
C3×C5×#4612
C3×C5×#3312
C3×C5×Q8⋊C82
C3×C5×D4⋊C82
C5×#16423
C3×#16425