Finite Group Databasebeta

← C3×C5×#1654 #14274688

Subgroups — C3×C5×#1654 #14274688

Order 1920.

480 conjugacy classes of subgroups.

The subgroup lattice has 392 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#547C211yesmaximal, characteristic
2C3×C5×#498C211yesmaximal, characteristic, complemented
3C3×C5×#474C211yesmaximal, characteristic, complemented
4C3×C5×#461C211yesmaximal, characteristic, complemented
5C3×C5×#332C211yesmaximal, characteristic, complemented
6C3×C5×D4⋊C8C211yesmaximal, characteristic, complemented
7C3×C5×D4⋊C8C211yesmaximal, characteristic, complemented
8C5×#1654C311yesmaximal, characteristic, direct product factor, Hall
9C2×C3×C5×#41—21no—
10C2×C3×C5×SD16—21no—
11C3×C5×C4⋊Q8C2211yescharacteristic
12C3×C5×C4⋊Q8C2211yescharacteristic
13C3×C5×#123—22no—
14C3×C5×C22⋊Q8—21no—
15C3×C5×#121—21no—
16C3×C4×C5×D4C2211yescharacteristic
17C3×C4×C5×D4C2211yescharacteristic
18C2×C3×C5×C4⋊C4—21no—
19C3×C5×#107—21no—
20C3×C5×#106—22no—
21C3×C5×C4⋊C8C2211yescharacteristic
22C3×C5×C4⋊C8C2211yescharacteristic
23C3×C5×#103—21no—
24C3×C5×#102—21no—
25C3×C4×C5×C8C2211yescharacteristic
26C3×#1654C511yesmaximal, characteristic, direct product factor, Hall
27C5×#547C2×C311yescharacteristic
28C5×#498C2×C311yescharacteristic, complemented
29C5×#474C2×C311yescharacteristic, complemented
30C5×#461C2×C311yescharacteristic, complemented
31C5×#332C2×C311yescharacteristic, complemented
32C5×D4⋊C8C2×C311yescharacteristic, complemented
33C5×D4⋊C8C2×C311yescharacteristic, complemented
34C3×C5×#41—42no—
35C3×C5×#41—42no—
36C2×C3×C5×Q8—21no—
37C2×C3×C5×Q8—21no—
38C2×C3×C5×D4—22no—
39C2×C3×C5×D4D411yescharacteristic
40C2×C3×C5×D4D411yescharacteristic
41C22×C3×C4×C5—22no—
42C22×C3×C4×C5—21no—
43C22×C3×C4×C5—21no—
44C3×C5×SD16—42no—
45C2×C3×C5×C8—21no—
46C2×C3×C5×C8—21no—
47C2×C3×C5×C8—21no—
48C3×C5×C4⋊C4—22no—
49C3×C5×C4⋊C4—21no—
50C3×C5×C4⋊C4—21no—
51C3×C5×C4⋊C4—21no—
52C3×C5×C4⋊C4D411yescharacteristic
53C3×C5×C4⋊C4D411yescharacteristic
54C3×C5×C4⋊C4D412yes—
55C3×C5×C22⋊C4—22no—
56C3×C5×C22⋊C4—21no—
57C3×C5×C22⋊C4—21no—
58C3×C42×C5C2311yescharacteristic
59C3×#547C2×C511yescharacteristic
60C3×#498C2×C511yescharacteristic, complemented
61C3×#474C2×C511yescharacteristic, complemented
62C3×#461C2×C511yescharacteristic, complemented
63C3×#332C2×C511yescharacteristic, complemented
64C3×D4⋊C8C2×C511yescharacteristic, complemented
65C3×D4⋊C8C2×C511yescharacteristic, complemented
66C2×C5×#41—21no—
67C2×C5×SD16—21no—
68C5×C4⋊Q8C22×C311yescharacteristic
69C5×C4⋊Q8C22×C311yescharacteristic
70C5×#123—22no—
71C5×C22⋊Q8—21no—
72C5×#121—21no—
73C4×C5×D4C22×C311yescharacteristic
74C4×C5×D4C22×C311yescharacteristic
75C2×C5×C4⋊C4—21no—
76C5×#107—21no—
77C5×#106—22no—
78C5×C4⋊C8C22×C311yescharacteristic
79C5×C4⋊C8C22×C311yescharacteristic
80C5×#103—21no—
81C5×#102—21no—
82C4×C5×C8C22×C311yescharacteristic
83#1654C3×C511yescharacteristic, direct product factor, Sylow
84C23×C3×C5—21no—
85C23×C3×C5—21no—
86C3×C5×Q8—41no—
87C3×C5×Q8—41no—
88C3×C5×Q8—41no—
89C3×C5×Q8—41no—
90C3×C5×D4—21no—
91C3×C5×D4—21no—
92C3×C5×D4—21no—
93C3×C5×D4—44no—
94C3×C5×D4SD1612yescomplemented
95C2×C3×C4×C5—21no—
96C2×C3×C4×C5—21no—
97C2×C3×C4×C5—22no—
98C2×C3×C4×C5—44no—
99C2×C3×C4×C5—41no—
100C2×C3×C4×C5—41no—
101C2×C3×C4×C5—41no—
102C2×C3×C4×C5—41no—
103C2×C3×C4×C5—41no—
104C2×C3×C4×C5C2×D411yescharacteristic
105C2×C3×C4×C5C2×D411yescharacteristic
106C2×C3×C4×C5C2×D411yescharacteristic
107C3×C5×C8—22no—
108C3×C5×C8—41no—
109C3×C5×C8—41no—
110C2×C3×#41—21no—
111C2×C3×SD16—21no—
112C3×C4⋊Q8C22×C511yescharacteristic
113C3×C4⋊Q8C22×C511yescharacteristic
114C3×#123—22no—
115C3×C22⋊Q8—21no—
116C3×#121—21no—
117C3×C4×D4C22×C511yescharacteristic
118C3×C4×D4C22×C511yescharacteristic
119C2×C3×C4⋊C4—21no—
120C3×#107—21no—
121C3×#106—22no—
122C3×C4⋊C8C22×C511yescharacteristic
123C3×C4⋊C8C22×C511yescharacteristic
124C3×#103—21no—
125C3×#102—21no—
126C3×C4×C8C22×C511yescharacteristic
127C5×#41—42no—
128C5×#41—42no—
129C2×C5×Q8—21no—
130C2×C5×Q8—21no—
131C2×C5×D4—22no—
132C2×C5×D4C3×D411yescharacteristic
133C2×C5×D4C3×D411yescharacteristic
134C22×C4×C5—22no—
135C22×C4×C5—21no—
136C22×C4×C5—21no—
137C5×SD16—42no—
138C2×C5×C8—21no—
139C2×C5×C8—21no—
140C2×C5×C8—21no—
141C5×C4⋊C4—22no—
142C5×C4⋊C4—21no—
143C5×C4⋊C4—21no—
144C5×C4⋊C4—21no—
145C5×C4⋊C4C3×D411yescharacteristic
146C5×C4⋊C4C3×D411yescharacteristic
147C5×C4⋊C4C3×D412yes—
148C5×C22⋊C4—22no—
149C5×C22⋊C4—21no—
150C5×C22⋊C4—21no—
151C42×C5C23×C311yescharacteristic
152#547C2×C3×C511yescharacteristic
153#498C2×C3×C511yescharacteristic, complemented
154#474C2×C3×C511yescharacteristic, complemented
155#461C2×C3×C511yescharacteristic, complemented
156#332C2×C3×C511yescharacteristic, complemented
157D4⋊C8C2×C3×C511yescharacteristic, complemented
158D4⋊C8C2×C3×C511yescharacteristic, complemented
159C22×C3×C5—22no—
160C22×C3×C5—41no—
161C22×C3×C5—41no—
162C22×C3×C5—41no—
163C22×C3×C5—41no—
164C22×C3×C5—41no—
165C22×C3×C5#12011yescharacteristic
166C3×C4×C5—21no—
167C3×C4×C5—41no—
168C3×C4×C5—41no—
169C3×C4×C5—42no—
170C3×C4×C5—81no—
171C3×C4×C5C2×SD1611yescharacteristic
172C3×C4×C5#13511yescharacteristic
173C3×C4×C5AGL(1,ℤ/8)11yescharacteristic
174C3×C4×C5AGL(1,ℤ/8)11yescharacteristic
175C3×#41—42no—
176C3×#41—42no—
177C2×C3×Q8—21no—
178C2×C3×Q8—21no—
179C2×C3×D4—22no—
180C2×C3×D4C5×D411yescharacteristic
181C2×C3×D4C5×D411yescharacteristic
182C22×C3×C4—22no—
183C22×C3×C4—21no—
184C22×C3×C4—21no—
185C3×SD16—42no—
186C2×C3×C8—21no—
187C2×C3×C8—21no—
188C2×C3×C8—21no—
189C3×C4⋊C4—22no—
190C3×C4⋊C4—21no—
191C3×C4⋊C4—21no—
192C3×C4⋊C4—21no—
193C3×C4⋊C4C5×D411yescharacteristic
194C3×C4⋊C4C5×D412yes—
195C3×C4⋊C4C5×D411yescharacteristic
196C3×C22⋊C4—22no—
197C3×C22⋊C4—21no—
198C3×C22⋊C4—21no—
199C3×C42C23×C511yescharacteristic
200C23×C5—21no—
201C23×C5—21no—
202C5×Q8—41no—
203C5×Q8—41no—
204C5×Q8—41no—
205C5×Q8—41no—
206C5×D4—21no—
207C5×D4—21no—
208C5×D4—21no—
209C5×D4—44no—
210C5×D4C3×SD1612yescomplemented
211C2×C4×C5—21no—
212C2×C4×C5—21no—
213C2×C4×C5—22no—
214C2×C4×C5—44no—
215C2×C4×C5—41no—
216C2×C4×C5—41no—
217C2×C4×C5—41no—
218C2×C4×C5—41no—
219C2×C4×C5—41no—
220C2×C4×C5C2×C3×D411yescharacteristic
221C2×C4×C5C2×C3×D411yescharacteristic
222C2×C4×C5C2×C3×D411yescharacteristic
223C5×C8—22no—
224C5×C8—41no—
225C5×C8—41no—
226C2×#41—21no—
227C2×SD16—21no—
228C4⋊Q8C22×C3×C511yescharacteristic
229C4⋊Q8C22×C3×C511yescharacteristic
230#123—22no—
231C22⋊Q8—21no—
232#121—21no—
233C4×D4C22×C3×C511yescharacteristic
234C4×D4C22×C3×C511yescharacteristic
235C2×C4⋊C4—21no—
236#107—21no—
237#106—22no—
238C4⋊C8C22×C3×C511yescharacteristic
239C4⋊C8C22×C3×C511yescharacteristic
240#103—21no—
241#102—21no—
242C4×C8C22×C3×C511yescharacteristic
243C2×C3×C5—42no—
244C2×C3×C5—81no—
245C2×C3×C5#44911yescharacteristic
246C2×C3×C5#45011yescharacteristic
247C2×C3×C5#45611yescharacteristic
248C23×C3—21no—
249C23×C3—21no—
250C3×Q8—41no—
251C3×Q8—41no—
252C3×Q8—41no—
253C3×Q8—41no—
254C3×D4—21no—
255C3×D4—21no—
256C3×D4—21no—
257C3×D4—44no—
258C3×D4C5×SD1612yescomplemented
259C2×C3×C4—21no—
260C2×C3×C4—21no—
261C2×C3×C4—22no—
262C2×C3×C4—44no—
263C2×C3×C4—41no—
264C2×C3×C4—41no—
265C2×C3×C4—41no—
266C2×C3×C4—41no—
267C2×C3×C4—41no—
268C2×C3×C4C2×C5×D411yescharacteristic
269C2×C3×C4C2×C5×D411yescharacteristic
270C2×C3×C4C2×C5×D411yescharacteristic
271C3×C8—22no—
272C3×C8—41no—
273C3×C8—41no—
274C22×C5—22no—
275C22×C5—41no—
276C22×C5—41no—
277C22×C5—41no—
278C22×C5—41no—
279C22×C5—41no—
280C22×C5C3×#12011yescharacteristic
281C4×C5—21no—
282C4×C5—41no—
283C4×C5—41no—
284C4×C5—42no—
285C4×C5—81no—
286C4×C5C2×C3×SD1611yescharacteristic
287C4×C5C3×#13511yescharacteristic
288C4×C5C3×AGL(1,ℤ/8)11yescharacteristic
289C4×C5C3×AGL(1,ℤ/8)11yescharacteristic
290#41—42no—
291#41—42no—
292C2×Q8—21no—
293C2×Q8—21no—
294C2×D4—22no—
295C2×D4C3×C5×D411yescharacteristic
296C2×D4C3×C5×D411yescharacteristic
297C22×C4—22no—
298C22×C4—21no—
299C22×C4—21no—
300SD16—42no—
301C2×C8—21no—
302C2×C8—21no—
303C2×C8—21no—
304C4⋊C4—22no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—21no—
308C4⋊C4C3×C5×D411yescharacteristic
309C4⋊C4C3×C5×D412yes—
310C4⋊C4C3×C5×D411yescharacteristic
311C22⋊C4—22no—
312C22⋊C4—21no—
313C22⋊C4—21no—
314C42C23×C3×C511yescharacteristic
315C3×C5#165411yescharacteristic, direct product factor, Hall
316C22×C3—22no—
317C22×C3—41no—
318C22×C3—41no—
319C22×C3—41no—
320C22×C3—41no—
321C22×C3—41no—
322C22×C3C5×#12011yescharacteristic
323C3×C4—21no—
324C3×C4—41no—
325C3×C4—41no—
326C3×C4—42no—
327C3×C4—81no—
328C3×C4C2×C5×SD1611yescharacteristic
329C3×C4C5×#13511yescharacteristic
330C3×C4C5×AGL(1,ℤ/8)11yescharacteristic
331C3×C4C5×AGL(1,ℤ/8)11yescharacteristic
332C2×C5—42no—
333C2×C5—81no—
334C2×C5C3×#44911yescharacteristic
335C2×C5C3×#45011yescharacteristic
336C2×C5C3×#45611yescharacteristic
337C23—21no—
338C23—21no—
339Q8—41no—
340Q8—41no—
341Q8—41no—
342Q8—41no—
343D4—21no—
344D4—21no—
345D4—21no—
346D4—44no—
347D4C3×C5×SD1612yescomplemented
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—22no—
351C2×C4—44no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—41no—
357C2×C4C2×C3×C5×D411yescharacteristic
358C2×C4C2×C3×C5×D411yescharacteristic
359C2×C4C2×C3×C5×D411yescharacteristic
360C8—22no—
361C8—41no—
362C8—41no—
363C2×C3—42no—
364C2×C3—81no—
365C2×C3C5×#44911yescharacteristic
366C2×C3C5×#45011yescharacteristic
367C2×C3C5×#45611yescharacteristic
368C5C3×#165411yescharacteristic, direct product factor, Sylow
369C22—22no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22C3×C5×#12011yescharacteristic
376C4—21no—
377C4—41no—
378C4—41no—
379C4—42no—
380C4—81no—
381C4C2×C3×C5×SD1611yescharacteristic
382C4C3×C5×#13511yescharacteristic
383C4C3×C5×AGL(1,ℤ/8)11yescharacteristic
384C4C3×C5×AGL(1,ℤ/8)11yescharacteristic
385C3C5×#165411yescharacteristic, direct product factor, Sylow
386C2—42no—
387C2—81no—
388C2C3×C5×#44911yescharacteristic
389C2C3×C5×#45011yescharacteristic
390C2C3×C5×#45611yescharacteristic
subgroupG/H
center, socleC22×C3×C5#120
commutator, Frattini, cosocleC42C23×C3×C5
prime radicalC3×C5×#547C2
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×#498C2semidirect
3. C3×C5×#474C2semidirect
4. C3×C5×#461C2semidirect
5. C3×C5×#332C2semidirect
6. C3×C5×D4⋊C8C2semidirect
7. C3×C5×D4⋊C8C2semidirect
8. C5×#1654C3direct (N × H)
26. C3×#1654C5direct (N × H)
28. C5×#498C2×C3semidirect
29. C5×#474C2×C3semidirect
30. C5×#461C2×C3semidirect
31. C5×#332C2×C3semidirect
32. C5×D4⋊C8C2×C3semidirect
33. C5×D4⋊C8C2×C3semidirect
60. C3×#498C2×C5semidirect
61. C3×#474C2×C5semidirect
62. C3×#461C2×C5semidirect
63. C3×#332C2×C5semidirect
64. C3×D4⋊C8C2×C5semidirect
65. C3×D4⋊C8C2×C5semidirect
83. #1654C3×C5direct (N × H)
94. C3×C5×D4SD16semidirect
153. #498C2×C3×C5semidirect
154. #474C2×C3×C5semidirect
155. #461C2×C3×C5semidirect
156. #332C2×C3×C5semidirect
157. D4⋊C8C2×C3×C5semidirect
158. D4⋊C8C2×C3×C5semidirect
210. C5×D4C3×SD16semidirect
258. C3×D4C5×SD16semidirect
315. C3×C5#1654direct (N × H)
347. D4C3×C5×SD16semidirect
368. C5C3×#1654direct (N × H)
385. C3C5×#1654direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#5472
C3×C5×#4982
C3×C5×#4742
C3×C5×#4612
C3×C5×#3322
C3×C5×D4⋊C82
C5×#16543
C3×#16545