Finite Group Databasebeta

← C3×C5×#3548 #14276582

Subgroups — C3×C5×#3548 #14276582

Order 1920.

2228 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C5×#562C214yesmaximal, complemented
2C3×C5×#554C214yesmaximal, complemented
3C3×C5×#550C214yesmaximal, complemented
4C3×C5×#547C214yesmaximal, complemented
5C3×C5×#540C214yesmaximal, complemented
6C3×C5×#537C218yesmaximal, complemented
7C3×C5×#533C211yesmaximal, characteristic, complemented
8C3×C5×#532C212yesmaximal, complemented
9C5×#3548C311yesmaximal, characteristic, direct product factor, Hall
10C2×C3×C5×#41C2214yes—
11C2×C3×C5×#41C2218yes—
12C3×C5×C4⋊Q8C2212yes—
13C3×C5×C4⋊Q8C2212yes—
14C3×C5×#127C2212yes—
15C3×C5×#127C2214yes—
16C3×C5×#125C2212yes—
17C3×C5×#125C2214yes—
18C3×C5×#124C2214yes—
19C3×C5×#123C22116yes—
20C3×C5×C22⋊Q8C2214yes—
21C3×C5×C22⋊Q8C2218yes—
22C3×C5×C22⋊Q8C22116yes—
23C3×C5×#121C2214yes—
24C3×C5×#121C2214yes—
25C3×C5×#121C22116yes—
26C3×C5×#121C2218yes—
27C3×C5×#121C22116yes—
28C3×C4×C5×Q8C2214yes—
29C3×C4×C5×D4C2214yes—
30C3×C4×C5×D4C2214yes—
31C3×C4×C5×D4C2218yes—
32C3×C5×#117C2212yes—
33C2×C3×C5×C4⋊C4C2218yes—
34C2×C3×C42×C5C2211yescharacteristic
35C3×#3548C511yesmaximal, characteristic, direct product factor, Hall
36C5×#562C2×C314yescomplemented
37C5×#554C2×C314yescomplemented
38C5×#550C2×C314yescomplemented
39C5×#547C2×C314yescomplemented
40C5×#540C2×C314yescomplemented
41C5×#537C2×C318yescomplemented
42C5×#533C2×C311yescharacteristic, complemented
43C5×#532C2×C312yescomplemented
44C3×C5×#41—48no—
45C3×C5×#41—416no—
46C2×C3×C5×Q8C2314yes—
47C2×C3×C5×Q8C2314yes—
48C2×C3×C5×D4C2314yes—
49C2×C3×C5×D4C2314yes—
50C2×C3×C5×D4C2318yes—
51C2×C3×C5×D4C23116yes—
52C2×C3×C5×D4C2314yes—
53C22×C3×C4×C5C2311yescharacteristic
54C22×C3×C4×C5C2312yes—
55C22×C3×C4×C5C2318yes—
56C22×C3×C4×C5C23116yes—
57C3×C5×C4⋊C4C2314yes—
58C3×C5×C4⋊C4C2314yes—
59C3×C5×C4⋊C4C23116yes—
60C3×C5×C4⋊C4C2314yes—
61C3×C5×C4⋊C4C2314yes—
62C3×C5×C4⋊C4C2318yes—
63C3×C5×C22⋊C4C2314yes—
64C3×C5×C22⋊C4C23116yes—
65C3×C5×C22⋊C4C2318yes—
66C3×C5×C22⋊C4C2318yes—
67C3×C42×C5C2311yescharacteristic
68C3×C42×C5C2312yes—
69C3×C42×C5C2311yescharacteristic
70C3×C42×C5C2314yes—
71C3×#562C2×C514yescomplemented
72C3×#554C2×C514yescomplemented
73C3×#550C2×C514yescomplemented
74C3×#547C2×C514yescomplemented
75C3×#540C2×C514yescomplemented
76C3×#537C2×C518yescomplemented
77C3×#533C2×C511yescharacteristic, complemented
78C3×#532C2×C512yescomplemented
79C2×C5×#41C22×C314yes—
80C2×C5×#41C22×C318yes—
81C5×C4⋊Q8C22×C312yes—
82C5×C4⋊Q8C22×C312yes—
83C5×#127C22×C312yes—
84C5×#127C22×C314yes—
85C5×#125C22×C312yes—
86C5×#125C22×C314yes—
87C5×#124C22×C314yes—
88C5×#123C22×C3116yes—
89C5×C22⋊Q8C22×C314yes—
90C5×C22⋊Q8C22×C318yes—
91C5×C22⋊Q8C22×C3116yes—
92C5×#121C22×C314yes—
93C5×#121C22×C314yes—
94C5×#121C22×C3116yes—
95C5×#121C22×C318yes—
96C5×#121C22×C3116yes—
97C4×C5×Q8C22×C314yes—
98C4×C5×D4C22×C314yes—
99C4×C5×D4C22×C314yes—
100C4×C5×D4C22×C318yes—
101C5×#117C22×C312yes—
102C2×C5×C4⋊C4C22×C318yes—
103C2×C42×C5C22×C311yescharacteristic
104#3548C3×C511yescharacteristic, direct product factor, Sylow
105C23×C3×C5C2411yescharacteristic
106C23×C3×C5C2418yes—
107C3×C5×Q8—28no—
108C3×C5×Q8—44no—
109C3×C5×D4—28no—
110C3×C5×D4—28no—
111C3×C5×D4—48no—
112C3×C5×D4—44no—
113C3×C5×D4—416no—
114C2×C3×C4×C5—216no—
115C2×C3×C4×C5—24no—
116C2×C3×C4×C5—22no—
117C2×C3×C4×C5—416no—
118C2×C3×C4×C5C2411yescharacteristic
119C2×C3×C4×C5C2412yes—
120C2×C3×C4×C5C2418yes—
121C2×C3×C4×C5C2418yes—
122C2×C3×C4×C5C2412yes—
123C2×C3×C4×C5C2411yescharacteristic
124C2×C3×#41C22×C514yes—
125C2×C3×#41C22×C518yes—
126C3×C4⋊Q8C22×C512yes—
127C3×C4⋊Q8C22×C512yes—
128C3×#127C22×C512yes—
129C3×#127C22×C514yes—
130C3×#125C22×C512yes—
131C3×#125C22×C514yes—
132C3×#124C22×C514yes—
133C3×#123C22×C5116yes—
134C3×C22⋊Q8C22×C5116yes—
135C3×C22⋊Q8C22×C514yes—
136C3×C22⋊Q8C22×C518yes—
137C3×#121C22×C5116yes—
138C3×#121C22×C514yes—
139C3×#121C22×C514yes—
140C3×#121C22×C5116yes—
141C3×#121C22×C518yes—
142C3×C4×Q8C22×C514yes—
143C3×C4×D4C22×C514yes—
144C3×C4×D4C22×C514yes—
145C3×C4×D4C22×C518yes—
146C3×#117C22×C512yes—
147C2×C3×C4⋊C4C22×C518yes—
148C2×C3×C42C22×C511yescharacteristic
149C5×#41—48no—
150C5×#41—416no—
151C2×C5×Q8C23×C314yes—
152C2×C5×Q8C23×C314yes—
153C2×C5×D4C23×C314yes—
154C2×C5×D4C23×C314yes—
155C2×C5×D4C23×C318yes—
156C2×C5×D4C23×C3116yes—
157C2×C5×D4C23×C314yes—
158C22×C4×C5C23×C311yescharacteristic
159C22×C4×C5C23×C312yes—
160C22×C4×C5C23×C318yes—
161C22×C4×C5C23×C3116yes—
162C5×C4⋊C4C23×C314yes—
163C5×C4⋊C4C23×C314yes—
164C5×C4⋊C4C23×C3116yes—
165C5×C4⋊C4C23×C314yes—
166C5×C4⋊C4C23×C314yes—
167C5×C4⋊C4C23×C318yes—
168C5×C22⋊C4C23×C314yes—
169C5×C22⋊C4C23×C3116yes—
170C5×C22⋊C4C23×C318yes—
171C5×C22⋊C4C23×C318yes—
172C42×C5C23×C311yescharacteristic
173C42×C5C23×C312yes—
174C42×C5C23×C311yescharacteristic
175C42×C5C23×C314yes—
176#562C2×C3×C514yescomplemented
177#554C2×C3×C514yescomplemented
178#550C2×C3×C514yescomplemented
179#547C2×C3×C514yescomplemented
180#540C2×C3×C514yescomplemented
181#537C2×C3×C518yescomplemented
182#533C2×C3×C511yescharacteristic, complemented
183#532C2×C3×C512yescomplemented
184C22×C3×C5—22no—
185C22×C3×C5—28no—
186C22×C3×C5—28no—
187C22×C3×C5—21no—
188C22×C3×C5—28no—
189C22×C3×C5C2511yescharacteristic
190C3×C4×C5—28no—
191C3×C4×C5—22no—
192C3×C4×C5—28no—
193C3×C4×C5—21no—
194C3×C4×C5H(5,𝔽2)14yes—
195C3×C4×C52−1+412yes—
196C3×#41—48no—
197C3×#41—416no—
198C2×C3×Q8C23×C514yes—
199C2×C3×Q8C23×C514yes—
200C2×C3×D4C23×C514yes—
201C2×C3×D4C23×C514yes—
202C2×C3×D4C23×C518yes—
203C2×C3×D4C23×C5116yes—
204C2×C3×D4C23×C514yes—
205C22×C3×C4C23×C511yescharacteristic
206C22×C3×C4C23×C512yes—
207C22×C3×C4C23×C518yes—
208C22×C3×C4C23×C5116yes—
209C3×C4⋊C4C23×C514yes—
210C3×C4⋊C4C23×C514yes—
211C3×C4⋊C4C23×C5116yes—
212C3×C4⋊C4C23×C514yes—
213C3×C4⋊C4C23×C514yes—
214C3×C4⋊C4C23×C518yes—
215C3×C22⋊C4C23×C514yes—
216C3×C22⋊C4C23×C5116yes—
217C3×C22⋊C4C23×C518yes—
218C3×C22⋊C4C23×C518yes—
219C3×C42C23×C511yescharacteristic
220C3×C42C23×C512yes—
221C3×C42C23×C511yescharacteristic
222C3×C42C23×C514yes—
223C23×C5C24×C311yescharacteristic
224C23×C5C24×C318yes—
225C5×Q8—28no—
226C5×Q8—44no—
227C5×D4—28no—
228C5×D4—28no—
229C5×D4—48no—
230C5×D4—44no—
231C5×D4—416no—
232C2×C4×C5—216no—
233C2×C4×C5—24no—
234C2×C4×C5—22no—
235C2×C4×C5—416no—
236C2×C4×C5C24×C311yescharacteristic
237C2×C4×C5C24×C312yes—
238C2×C4×C5C24×C318yes—
239C2×C4×C5C24×C318yes—
240C2×C4×C5C24×C312yes—
241C2×C4×C5C24×C311yescharacteristic
242C2×#41C22×C3×C514yes—
243C2×#41C22×C3×C518yes—
244C4⋊Q8C22×C3×C512yes—
245C4⋊Q8C22×C3×C512yes—
246#127C22×C3×C512yes—
247#127C22×C3×C514yes—
248#125C22×C3×C512yes—
249#125C22×C3×C514yes—
250#124C22×C3×C514yes—
251#123C22×C3×C5116yes—
252C22⋊Q8C22×C3×C514yes—
253C22⋊Q8C22×C3×C518yes—
254C22⋊Q8C22×C3×C5116yes—
255#121C22×C3×C514yes—
256#121C22×C3×C514yes—
257#121C22×C3×C5116yes—
258#121C22×C3×C518yes—
259#121C22×C3×C5116yes—
260C4×Q8C22×C3×C514yes—
261C4×D4C22×C3×C514yes—
262C4×D4C22×C3×C514yes—
263C4×D4C22×C3×C518yes—
264#117C22×C3×C512yes—
265C2×C4⋊C4C22×C3×C518yes—
266C2×C42C22×C3×C511yescharacteristic
267C2×C3×C5—41no—
268C2×C3×C5—48no—
269C2×C3×C5C2×H(5,𝔽2)12yes—
270C2×C3×C5C2×2−1+411yescharacteristic
271C23×C3C24×C511yescharacteristic
272C23×C3C24×C518yes—
273C3×Q8—28no—
274C3×Q8—44no—
275C3×D4—28no—
276C3×D4—28no—
277C3×D4—48no—
278C3×D4—44no—
279C3×D4—416no—
280C2×C3×C4—216no—
281C2×C3×C4—24no—
282C2×C3×C4—22no—
283C2×C3×C4—416no—
284C2×C3×C4C24×C511yescharacteristic
285C2×C3×C4C24×C512yes—
286C2×C3×C4C24×C518yes—
287C2×C3×C4C24×C518yes—
288C2×C3×C4C24×C512yes—
289C2×C3×C4C24×C511yescharacteristic
290C22×C5—22no—
291C22×C5—28no—
292C22×C5—28no—
293C22×C5—21no—
294C22×C5—28no—
295C22×C5C25×C311yescharacteristic
296C4×C5—28no—
297C4×C5—22no—
298C4×C5—28no—
299C4×C5—21no—
300C4×C5C3×H(5,𝔽2)14yes—
301C4×C5C3×2−1+412yes—
302#41—48no—
303#41—416no—
304C2×Q8C23×C3×C514yes—
305C2×Q8C23×C3×C514yes—
306C2×D4C23×C3×C514yes—
307C2×D4C23×C3×C514yes—
308C2×D4C23×C3×C518yes—
309C2×D4C23×C3×C5116yes—
310C2×D4C23×C3×C514yes—
311C22×C4C23×C3×C511yescharacteristic
312C22×C4C23×C3×C512yes—
313C22×C4C23×C3×C518yes—
314C22×C4C23×C3×C5116yes—
315C4⋊C4C23×C3×C514yes—
316C4⋊C4C23×C3×C514yes—
317C4⋊C4C23×C3×C5116yes—
318C4⋊C4C23×C3×C514yes—
319C4⋊C4C23×C3×C514yes—
320C4⋊C4C23×C3×C518yes—
321C22⋊C4C23×C3×C514yes—
322C22⋊C4C23×C3×C5116yes—
323C22⋊C4C23×C3×C518yes—
324C22⋊C4C23×C3×C518yes—
325C42C23×C3×C511yescharacteristic
326C42C23×C3×C512yes—
327C42C23×C3×C511yescharacteristic
328C42C23×C3×C514yes—
329C3×C5#354811yescharacteristic, direct product factor, Hall
330C22×C3—22no—
331C22×C3—28no—
332C22×C3—28no—
333C22×C3—21no—
334C22×C3—28no—
335C22×C3C25×C511yescharacteristic
336C3×C4—28no—
337C3×C4—22no—
338C3×C4—28no—
339C3×C4—21no—
340C3×C4C5×H(5,𝔽2)14yes—
341C3×C4C5×2−1+412yes—
342C2×C5—41no—
343C2×C5—48no—
344C2×C5C2×C3×H(5,𝔽2)12yes—
345C2×C5C2×C3×2−1+411yescharacteristic
346C23C24×C3×C511yescharacteristic
347C23C24×C3×C518yes—
348Q8—28no—
349Q8—44no—
350D4—28no—
351D4—28no—
352D4—48no—
353D4—44no—
354D4—416no—
355C2×C4—216no—
356C2×C4—24no—
357C2×C4—22no—
358C2×C4—416no—
359C2×C4C24×C3×C511yescharacteristic
360C2×C4C24×C3×C512yes—
361C2×C4C24×C3×C518yes—
362C2×C4C24×C3×C518yes—
363C2×C4C24×C3×C512yes—
364C2×C4C24×C3×C511yescharacteristic
365C2×C3—41no—
366C2×C3—48no—
367C2×C3C2×C5×H(5,𝔽2)12yes—
368C2×C3C2×C5×2−1+411yescharacteristic
369C5C3×#354811yescharacteristic, direct product factor, Sylow
370C22—22no—
371C22—28no—
372C22—28no—
373C22—21no—
374C22—28no—
375C22C25×C3×C511yescharacteristic
376C4—28no—
377C4—22no—
378C4—28no—
379C4—21no—
380C4C3×C5×H(5,𝔽2)14yes—
381C4C3×C5×2−1+412yes—
382C3C5×#354811yescharacteristic, direct product factor, Sylow
383C2—41no—
384C2—48no—
385C2C2×C3×C5×H(5,𝔽2)12yes—
386C2C2×C3×C5×2−1+411yescharacteristic
subgroupG/H
center, socleC22×C3×C5C25
commutator, Frattini, cosocleC22C25×C3×C5
LCSG ▷ C22 ▷ C1
UCSC1 ◁ C22×C3×C5 ◁ G

Complements

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

NH ≅ G/Ntype
1. C3×C5×#562C2semidirect
2. C3×C5×#554C2semidirect
3. C3×C5×#550C2semidirect
4. C3×C5×#547C2semidirect
5. C3×C5×#540C2semidirect
6. C3×C5×#537C2semidirect
7. C3×C5×#533C2semidirect
8. C3×C5×#532C2semidirect
9. C5×#3548C3direct (N × H)
35. C3×#3548C5direct (N × H)
36. C5×#562C2×C3semidirect
37. C5×#554C2×C3semidirect
38. C5×#550C2×C3semidirect
39. C5×#547C2×C3semidirect
40. C5×#540C2×C3semidirect
41. C5×#537C2×C3semidirect
42. C5×#533C2×C3semidirect
43. C5×#532C2×C3semidirect
71. C3×#562C2×C5semidirect
72. C3×#554C2×C5semidirect
73. C3×#550C2×C5semidirect
74. C3×#547C2×C5semidirect
75. C3×#540C2×C5semidirect
76. C3×#537C2×C5semidirect
77. C3×#533C2×C5semidirect
78. C3×#532C2×C5semidirect
104. #3548C3×C5direct (N × H)
176. #562C2×C3×C5semidirect
177. #554C2×C3×C5semidirect
178. #550C2×C3×C5semidirect
179. #547C2×C3×C5semidirect
180. #540C2×C3×C5semidirect
181. #537C2×C3×C5semidirect
182. #533C2×C3×C5semidirect
183. #532C2×C3×C5semidirect
329. C3×C5#3548direct (N × H)
369. C5C3×#3548direct (N × H)
382. C3C5×#3548direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C5×#5622
C3×C5×#5542
C3×C5×#5502
C3×C5×#5472
C3×C5×#5402
C3×C5×#5372
C3×C5×#5332
C3×C5×#5322
C5×#35483
C3×#35485