Finite Group Databasebeta

← D15×#324 #14280767

Subgroups — D15×#324 #14280767

Order 1920.

656 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 125 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4⋊C4×D15C211yesmaximal, characteristic
2D15×C22⋊C8C212yesmaximal
3C3×C5×#324C211yesmaximal, characteristic, complemented
4#11746349C211yesmaximal, characteristic, complemented
5#11746290C212yesmaximal, complemented
6D5×#324—31nomaximal, Hall
7C22×C4×D15C411yescharacteristic
8C22×C4×D15C411yescharacteristic
9C22×C4×D15C2211yescharacteristic
10C2×C3×C5×C4⋊C4C2211yescharacteristic
11C2×C4⋊Dic15C411yescharacteristic
12C2×#6712C411yescharacteristic
13C2×#6712C2211yescharacteristic
14C2×C8×D15—22no—
15C4⋊C4×D15—22no—
16C4⋊C4×D15—22no—
17C3×C5×C22⋊C8C2212yes—
18#91114C2212yes—
19#91104—22no—
20D3×#324—51nomaximal, Hall
21C2×D5×C4⋊C4—31no—
22D5×C22⋊C8—32no—
23C5×#324D311yescharacteristic, complemented
24#64748—31no—
25#64689—32no—
26C23×D15C2×C411yescharacteristic
27C22×C3×C4×C5C2×C411yescharacteristic
28C22×C3×C4×C5C2×C411yescharacteristic
29C22×C3×C4×C5C2311yescharacteristic
30C22×Dic15C2×C411yescharacteristic
31C22×Dic15C2×C411yescharacteristic
32C22×Dic15C2×C411yescharacteristic
33C2×C4×D15—21no—
34C2×C4×D15—21no—
35C2×C4×D15—22no—
36C2×C4×D15—21no—
37C2×C4×D15—21no—
38C2×C4×D15—42no—
39C2×C4×D15—42no—
40C2×C4×D15C814yescomplemented
41C2×C4×D15D412yes—
42C2×C4×D15D412yes—
43C2×C3×C5×C8—22no—
44C3×C5×C4⋊C4—22no—
45C4⋊Dic15—22no—
46#6712—22no—
47#6712—22no—
48C2×C3⋊#168—22no—
49C8×D15—44no—
50C2×D3×C4⋊C4—51no—
51D3×C22⋊C8—52no—
52C3×#324D511yescharacteristic, complemented
53#4528—51no—
54#4469—52no—
55C22×C4×D5—31no—
56C22×C4×D5—31no—
57C22×C4×D5—31no—
58C2×C5×C4⋊C4C2×D311yescharacteristic
59C2×C4⋊Dic5—31no—
60C2×#683—31no—
61C2×#683—31no—
62C2×C8×D5—62no—
63D5×C4⋊C4—62no—
64D5×C4⋊C4—62no—
65C5×C22⋊C8C2×D312yes—
66C22⋊#168—32no—
67#3952—62no—
68C2×#324—151noSylow
69C23×C3×C5C22×C411yescharacteristic
70C22×D15—22no—
71C22×D15—22no—
72C22×D15C22⋊C411yescharacteristic
73C22×D15C22⋊C411yescharacteristic
74C22×D15C2×C812yes—
75C22×D15M1612yes—
76C2×C3×C4×C5—21no—
77C2×C3×C4×C5—21no—
78C2×C3×C4×C5—21no—
79C2×C3×C4×C5—41no—
80C2×C3×C4×C5—41no—
81C2×C3×C4×C5C2×C812yescomplemented
82C2×C3×C4×C5C2×D412yes—
83C2×Dic15—21no—
84C2×Dic15—21no—
85C2×Dic15—21no—
86C2×Dic15—41no—
87C2×Dic15—41no—
88C2×Dic15C22⋊C412yes—
89C2×Dic15C2×C812yescomplemented
90C4×D15—24no—
91C4×D15—24no—
92C4×D15—24no—
93C4×D15—42no—
94C4×D15—42no—
95C4×D15—42no—
96C3×C5×C8—42no—
97C3⋊#168—42no—
98C22×C4×D3—51no—
99C22×C4×D3—51no—
100C22×C4×D3—51no—
101C2×C3×C4⋊C4C2×D511yescharacteristic
102C2×C4⋊Dic3—51no—
103C2×Dic3⋊C4—51no—
104C2×Dic3⋊C4—51no—
105C2×D3×C8—102no—
106D3×C4⋊C4—102no—
107D3×C4⋊C4—102no—
108C3×C22⋊C8C2×D512yes—
109#830—52no—
110(C2×D3)⋊C8—102no—
111C23×D5—31no—
112C22×C4×C5C4×D311yescharacteristic
113C22×C4×C5C4×D311yescharacteristic
114C22×C4×C5C22×D311yescharacteristic
115C22×Dic5—31no—
116C22×Dic5—31no—
117C22×Dic5—31no—
118C2×C4×D5—34no—
119C2×C4×D5—32no—
120C2×C4×D5—32no—
121C2×C4×D5—62no—
122C2×C4×D5—61no—
123C2×C4×D5—61no—
124C2×C4×D5—61no—
125C2×C4×D5—61no—
126C2×C4×D5—122no—
127C2×C4×D5—122no—
128C2×C5×C8—22no—
129C5×C4⋊C4—22no—
130C4⋊Dic5—62no—
131#683—62no—
132#683—62no—
133C2×#168—62no—
134C8×D5—124no—
135C22×C4⋊C4—151no—
136C2×C22⋊C8—152no—
137#324—151no—
138#324—152no—
139#324D1511yescharacteristic, direct product factor
140C22×C3×C5—21no—
141C22×C3×C5—21no—
142C22×C3×C5C2×C22⋊C411yescharacteristic
143C22×C3×C5C22×C811yescharacteristic
144C22×C3×C5C2×M1611yescharacteristic
145C2×D15—21no—
146C2×D15—21no—
147C2×D15—24no—
148C2×D15—24no—
149C2×D15C22⋊C812yes—
150C2×D15C22⋊C812yes—
151C2×D15#9912yes—
152C2×D15#10112yes—
153C3×C4×C5—22no—
154C3×C4×C5—22no—
155C3×C4×C5—41no—
156Dic15—22no—
157Dic15—22no—
158Dic15—41no—
159C23×D3—51no—
160C22×C3×C4C4×D511yescharacteristic
161C22×C3×C4C4×D511yescharacteristic
162C22×C3×C4C22×D511yescharacteristic
163C22×Dic3—51no—
164C22×Dic3—51no—
165C22×Dic3—51no—
166C2×C4×D3—54no—
167C2×C4×D3—52no—
168C2×C4×D3—52no—
169C2×C4×D3—101no—
170C2×C4×D3—101no—
171C2×C4×D3—102no—
172C2×C4×D3—101no—
173C2×C4×D3—101no—
174C2×C4×D3—202no—
175C2×C4×D3—202no—
176C2×C3×C8—22no—
177C3×C4⋊C4—22no—
178C4⋊Dic3—102no—
179Dic3⋊C4—102no—
180Dic3⋊C4—102no—
181C2×C3⋊C8—102no—
182D3×C8—204no—
183C23×C5C2×C4×D311yescharacteristic
184C22×D5—31no—
185C22×D5—31no—
186C22×D5—32no—
187C22×D5—32no—
188C22×D5—62no—
189C22×D5—62no—
190C2×C4×C5—21no—
191C2×C4×C5—21no—
192C2×C4×C5—21no—
193C2×C4×C5—41no—
194C2×C4×C5—41no—
195C2×C4×C5D3×C812yescomplemented
196C2×C4×C5D3×D412yes—
197C2×Dic5—32no—
198C2×Dic5—32no—
199C2×Dic5—61no—
200C2×Dic5—61no—
201C2×Dic5—61no—
202C2×Dic5—121no—
203C2×Dic5—121no—
204C4×D5—64no—
205C4×D5—64no—
206C4×D5—64no—
207C4×D5—122no—
208C4×D5—122no—
209C4×D5—122no—
210C5×C8—42no—
211#168—122no—
212C23×C4—151no—
213C23×C4—151no—
214C23×C4—151no—
215C22×C8—302no—
216C2×C4⋊C4—151no—
217C2×C4⋊C4—151no—
218C2×C4⋊C4—151no—
219C2×C4⋊C4—302no—
220C2×C4⋊C4—302no—
221C2×C4⋊C4C2×D1511yescharacteristic
222C22⋊C8—152no—
223C22⋊C8—302no—
224C22⋊C8C2×D1512yes—
225C2×C3×C5—21no—
226C2×C3×C5—21no—
227C2×C3×C5C2×C22⋊C811yescharacteristic
228C2×C3×C5C2×#9911yescharacteristic
229C2×C3×C5C2×#10111yescharacteristic
230D15—22no—
231D15#32414yesdirect product factor
232C23×C3C2×C4×D511yescharacteristic
233C22×D3—51no—
234C22×D3—51no—
235C22×D3—52no—
236C22×D3—52no—
237C22×D3—102no—
238C22×D3—102no—
239C2×C3×C4—21no—
240C2×C3×C4—21no—
241C2×C3×C4—21no—
242C2×C3×C4—41no—
243C2×C3×C4—41no—
244C2×C3×C4C8×D512yescomplemented
245C2×C3×C4D4×D512yes—
246C2×Dic3—52no—
247C2×Dic3—52no—
248C2×Dic3—101no—
249C2×Dic3—101no—
250C2×Dic3—101no—
251C2×Dic3—201no—
252C2×Dic3—201no—
253C4×D3—104no—
254C4×D3—104no—
255C4×D3—104no—
256C4×D3—202no—
257C4×D3—202no—
258C4×D3—202no—
259C3×C8—42no—
260C3⋊C8—202no—
261C22×C5—21no—
262C22×C5—21no—
263C22×C5D3×C22⋊C411yescharacteristic
264C22×C5C2×D3×C811yescharacteristic
265C22×C5D3×M1611yescharacteristic
266C2×D5—32no—
267C2×D5—32no—
268C2×D5—32no—
269C2×D5—32no—
270C2×D5—64no—
271C2×D5—64no—
272C2×D5—61no—
273C2×D5—61no—
274C4×C5—22no—
275C4×C5—22no—
276C4×C5—41no—
277Dic5—62no—
278Dic5—62no—
279Dic5—121no—
280C24—151no—
281C22×C4—151no—
282C22×C4—151no—
283C22×C4—154no—
284C22×C4—152no—
285C22×C4—152no—
286C22×C4—151no—
287C22×C4—301no—
288C22×C4—301no—
289C22×C4—302no—
290C22×C4—301no—
291C22×C4—301no—
292C22×C4—602no—
293C22×C4—602no—
294C22×C4C4×D1511yescharacteristic
295C22×C4C4×D1511yescharacteristic
296C22×C4C22×D1511yescharacteristic
297C2×C8—22no—
298C2×C8—302no—
299C2×C8—604no—
300C4⋊C4—22no—
301C4⋊C4—302no—
302C4⋊C4—302no—
303C4⋊C4—302no—
304C3×C5C2×#32411yescharacteristic, complemented, Hall
305C22×C3—21no—
306C22×C3—21no—
307C22×C3D5×C22⋊C411yescharacteristic
308C22×C3C2×C8×D511yescharacteristic
309C22×C3D5×M1611yescharacteristic
310C2×D3—52no—
311C2×D3—52no—
312C2×D3—52no—
313C2×D3—52no—
314C2×D3—104no—
315C2×D3—104no—
316C2×D3—101no—
317C2×D3—101no—
318C3×C4—22no—
319C3×C4—22no—
320C3×C4—41no—
321Dic3—102no—
322Dic3—102no—
323Dic3—201no—
324C2×C5—21no—
325C2×C5—21no—
326C2×C5D3×C22⋊C811yescharacteristic
327C2×C5D3×#9911yescharacteristic
328C2×C5D3×#10111yescharacteristic
329D5—34no—
330D5—62no—
331C23—151no—
332C23—151no—
333C23—152no—
334C23—152no—
335C23—302no—
336C23—302no—
337C23C2×C4×D1511yescharacteristic
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—41no—
342C2×C4—41no—
343C2×C4—152no—
344C2×C4—152no—
345C2×C4—304no—
346C2×C4—304no—
347C2×C4—304no—
348C2×C4—301no—
349C2×C4—301no—
350C2×C4—301no—
351C2×C4—602no—
352C2×C4—602no—
353C2×C4—602no—
354C2×C4—601no—
355C2×C4—601no—
356C2×C4C8×D1512yescomplemented
357C2×C4D4×D1512yes—
358C8—42no—
359C8—602no—
360C2×C3—21no—
361C2×C3—21no—
362C2×C3D5×C22⋊C811yescharacteristic
363C2×C3D5×#9911yescharacteristic
364C2×C3D5×#10111yescharacteristic
365D3—54no—
366D3—102no—
367C5D3×#32411yescharacteristic, complemented, Sylow
368C22—21no—
369C22—21no—
370C22—152no—
371C22—152no—
372C22—152no—
373C22—152no—
374C22—304no—
375C22—304no—
376C22—301no—
377C22—301no—
378C22C22⋊C4×D1511yescharacteristic
379C22C2×C8×D1511yescharacteristic
380C22M16×D1511yescharacteristic
381C4—22no—
382C4—22no—
383C4—41no—
384C4—302no—
385C4—302no—
386C4—601no—
387C3D5×#32411yescharacteristic, complemented, Sylow
388C2—21no—
389C2—21no—
390C2—154no—
391C2—302no—
392C2D15×C22⋊C811yescharacteristic
393C2D15×#9911yescharacteristic
394C2D15×#10111yescharacteristic
subgroupG/H
centerC22C22⋊C4×D15
commutatorC22×C3×C5C22×C8
FrattiniC22×C4C22×D15
FittingC3×C5×#324C2
socleC22×C3×C5C2×C22⋊C4
prime radicalC23×D15C2×C4
cosocleC22×C3×C4×C5C23
hypercenter#324D15
hypocenterC3×C5C2×#324
LCSG ▷ C22×C3×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ #324

Complements

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

NH ≅ G/Ntype
3. C3×C5×#324C2semidirect
4. #11746349C2semidirect
5. #11746290C2semidirect
23. C5×#324D3semidirect
40. C2×C4×D15C8semidirect
52. C3×#324D5semidirect
81. C2×C3×C4×C5C2×C8semidirect
89. C2×Dic15C2×C8semidirect
139. #324D15direct (N × H)
195. C2×C4×C5D3×C8semidirect
231. D15#324direct (N × H)
244. C2×C3×C4C8×D5semidirect
304. C3×C5C2×#324semidirect
356. C2×C4C8×D15semidirect
367. C5D3×#324semidirect
387. C3D5×#324semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×D152
D15×C22⋊C82
C3×C5×#3242
#117463492
#117462902
D5×#3243
D3×#3245