Finite Group Databasebeta

← D3×D5×ΓL(1,𝔽16) #72231117

Subgroups — D3×D5×ΓL(1,𝔽16) #72231117

Order 3600.

486 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 486 conjugacy classes of subgroups — too many to draw. Only its 94 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
1C3×D3×D52C211yesmaximal, characteristic
2#14268443C211yesmaximal, characteristic, complemented
3D15×ΓL(1,𝔽16)C211yesmaximal, characteristic, complemented
4D3×C3⋊#1042C211yesmaximal, characteristic, complemented
5C5×D3×ΓL(1,𝔽16)C211yesmaximal, characteristic, complemented
6D5×C3⋊ΓL(1,𝔽16)C211yesmaximal, characteristic, complemented
7C3×D5×ΓL(1,𝔽16)C211yesmaximal, characteristic, complemented
8C2×D5×ΓL(1,𝔽16)—31nomaximal
9D3×D5×AGL(1,𝔽5)—31nomaximal
10C3×D15⋊D5C411yescharacteristic, complemented
11C3×D5×D15C411yescharacteristic, complemented
12C3×D5×D15C2211yescharacteristic
13C3×D3×C5⋊D5C411yescharacteristic, complemented
14C3×C5×D3×D5C411yescharacteristic, complemented
15C3×C5×D3×D5C2211yescharacteristic
16C32×D52C2211yescharacteristic
17C3⋊(C3⋊#1042)C2211yescharacteristic, complemented
18C5×C3⋊ΓL(1,𝔽16)C2211yescharacteristic, complemented
19C3×C3⋊#1042C2211yescharacteristic, complemented
20C3×C5×ΓL(1,𝔽16)C2211yescharacteristic, complemented
21C2×D3×ΓL(1,𝔽16)—51nomaximal
22D3×D5×Dic3—51nomaximal
23C2×C3×D52—31no—
24C2×C3⋊#1042—31no—
25C2×C5×ΓL(1,𝔽16)—31no—
26D3×D52D311yescharacteristic
27D3×#1042—31no—
28ΓL(1,𝔽16)⋊D5—31no—
29D5×ΓL(1,𝔽16)—31no—
30D5×ΓL(1,𝔽16)—31no—
31D5×ΓL(1,𝔽16)—31no—
32D5×ΓL(1,𝔽16)—31no—
33D5×ΓL(1,𝔽16)—61no—
34D5×ΓL(1,𝔽16)D311yescharacteristic, direct product factor
35AGL(1,𝔽5)×D15—31no—
36C5×D3×AGL(1,𝔽5)—31no—
37C3×D5×AGL(1,𝔽5)—31no—
38C3×C3⋊(C5⋊D5)C2×C411yescharacteristic, complemented
39C3×C5×D15C2×C411yescharacteristic, complemented
40C3×C5×D15C2×C411yescharacteristic, complemented
41C3×C52×D3C2×C411yescharacteristic, complemented
42C32×C5⋊D5C2×C411yescharacteristic, complemented
43C32×C5×D5C2×C411yescharacteristic, complemented
44C32×C5×D5C2311yescharacteristic
45C2×D5×AGL(1,𝔽5)—91noHall
46C2×C3×D3×D5—51no—
47C2×C3×D3×D5—51no—
48C2×C3⋊ΓL(1,𝔽16)—51no—
49C2×C3×ΓL(1,𝔽16)—51no—
50D3×ΓL(1,𝔽16)—51no—
51D3×ΓL(1,𝔽16)—51no—
52D3×ΓL(1,𝔽16)—51no—
53D3×ΓL(1,𝔽16)D511yescharacteristic, direct product factor
54#67154—51no—
55D3×Dic15—51no—
56Dic3×D15—51no—
57C5×D3×Dic3—51no—
58D5×C3⋊Dic3—51no—
59C3×D5×Dic3—51no—
60C2×C3×C5⋊D5—31no—
61C2×C3×C5×D5—31no—
62C2×C3×C5×D5—31no—
63D15⋊D5Dic311yescharacteristic, complemented
64D5×D15Dic311yescharacteristic, complemented
65D5×D15C2×D311yescharacteristic
66D3×C5⋊D5Dic311yescharacteristic, complemented
67C5×D3×D5Dic311yescharacteristic, complemented
68C5×D3×D5C2×D311yescharacteristic
69C3×D52—21no—
70C3×D52—31no—
71C3×D52—31no—
72C3×D52—31no—
73C3×D52C2×D311yescharacteristic
74C3×D52C2×D311yescharacteristic
75C3⋊#1042—31no—
76C3⋊#1042—31no—
77C3⋊#1042—61no—
78C3⋊#1042C2×D311yescharacteristic, complemented
79C5×ΓL(1,𝔽16)—31no—
80C5×ΓL(1,𝔽16)—31no—
81C5×ΓL(1,𝔽16)—61no—
82C5×ΓL(1,𝔽16)C2×D311yescharacteristic, complemented
83C3×#1042—31no—
84C3×C5×AGL(1,𝔽5)—31no—
85C22×ΓL(1,𝔽16)—151no—
86C2×D3×AGL(1,𝔽5)—151no—
87C2×D5×Dic3—151no—
88C4×D3×D5—151no—
89C32×C52C22×C411yescharacteristic, complemented, Hall
90C2×D52—31no—
91C2×#1042—91no—
92C2×C5×AGL(1,𝔽5)—91no—
93D5×AGL(1,𝔽5)—31no—
94D5×AGL(1,𝔽5)—91no—
95D5×AGL(1,𝔽5)—91no—
96D5×AGL(1,𝔽5)—91no—
97C2×C3×D15—51no—
98C2×C3×D15—51no—
99C2×C3×C5×D3—51no—
100C2×C3×C5×D3—51no—
101C2×C32×D5—51no—
102C2×C32×D5—51no—
103C3×D3×D5—51no—
104C3×D3×D5—51no—
105C3×D3×D5—51no—
106C3×D3×D5—51no—
107C3×D3×D5—51no—
108C3×D3×D5—51no—
109C3×D3×D5—201no—
110C3×D3×D5AGL(1,𝔽5)11yescharacteristic, complemented
111C3×D3×D5C2×D511yescharacteristic
112C3⋊ΓL(1,𝔽16)—51no—
113C3⋊ΓL(1,𝔽16)C2×D511yescharacteristic, complemented
114C3×ΓL(1,𝔽16)—51no—
115C3×ΓL(1,𝔽16)C2×D511yescharacteristic, complemented
116C3⋊Dic15—51no—
117C5×C3⋊Dic3—51no—
118C3×Dic15—51no—
119C3×C5×Dic3—51no—
120C2×C3×C52—31no—
121C3⋊(C5⋊D5)C2×Dic311yescharacteristic, complemented
122C5×D15C2×Dic311yescharacteristic, complemented
123C5×D15C2×Dic311yescharacteristic, complemented
124C52×D3C2×Dic311yescharacteristic, complemented
125C3×C5⋊D5—21no—
126C3×C5⋊D5—31no—
127C3×C5⋊D5C4×D311yescharacteristic, complemented
128C3×C5⋊D5C2×Dic311yescharacteristic, complemented
129C3×C5×D5—21no—
130C3×C5×D5—21no—
131C3×C5×D5—31no—
132C3×C5×D5—31no—
133C3×C5×D5C4×D311yescharacteristic, complemented
134C3×C5×D5C2×Dic311yescharacteristic, complemented
135C3×C5×D5C22×D311yescharacteristic
136C3×C5×D5C22×D311yescharacteristic
137C2×D3×Dic3—251noHall
138C22×C3×D5—151no—
139C22×C3×D5—151no—
140C2×D3×D5—51no—
141C2×D3×D5—51no—
142C2×ΓL(1,𝔽16)—31no—
143C2×ΓL(1,𝔽16)—51no—
144C2×ΓL(1,𝔽16)—151no—
145C2×ΓL(1,𝔽16)—151no—
146C2×ΓL(1,𝔽16)—151no—
147C2×ΓL(1,𝔽16)—151no—
148C2×ΓL(1,𝔽16)—151no—
149C2×ΓL(1,𝔽16)—301no—
150C2×C3×AGL(1,𝔽5)—151no—
151D3×AGL(1,𝔽5)—31no—
152D3×AGL(1,𝔽5)—151no—
153D3×AGL(1,𝔽5)—151no—
154D3×AGL(1,𝔽5)—151no—
155C2×Dic15—151no—
156C4×D15—151no—
157C2×C5×Dic3—151no—
158C4×C5×D3—151no—
159C3×C4×D5—151no—
160#1200—151no—
161D3×Dic5—151no—
162D5×Dic3—51no—
163D5×Dic3—151no—
164D5×Dic3—151no—
165D5×Dic3—151no—
166D5×Dic3—151no—
167D5×Dic3—301no—
168C2×C5⋊D5—31no—
169C2×C5×D5—31no—
170C2×C5×D5—31no—
171D52—31no—
172D52—31no—
173D52—31no—
174D52D3211yescharacteristic
175#1042—31no—
176#1042—91no—
177C5×AGL(1,𝔽5)—31no—
178C5×AGL(1,𝔽5)—91no—
179C2×C32×C5—51no—
180C2×C32×C5—51no—
181C3×D15—51no—
182C3×D15—51no—
183C3×D15—201no—
184C3×D15C4×D511yescharacteristic, complemented
185C3×D15C2×AGL(1,𝔽5)11yescharacteristic, complemented
186C3×C5×D3—41no—
187C3×C5×D3—51no—
188C3×C5×D3—51no—
189C3×C5×D3C4×D511yescharacteristic, complemented
190C3×C5×D3C2×AGL(1,𝔽5)11yescharacteristic, complemented
191C32×D5—51no—
192C32×D5—51no—
193C32×D5—201no—
194C32×D5C2×AGL(1,𝔽5)11yescharacteristic, complemented
195C32×D5C22×D511yescharacteristic
196C22×AGL(1,𝔽5)—451no—
197C2×C4×D5—451no—
198C3×C52—21no—
199C3×C52C2×C4×D311yescharacteristic, complemented
200C3×C52C22×Dic311yescharacteristic, complemented
201C22×C3×D3—251no—
202C2×C3⋊Dic3—251no—
203C2×C3×Dic3—251no—
204D3×Dic3—51no—
205D3×Dic3—251no—
206D3×Dic3—251no—
207D3×Dic3—251no—
208C22×C3×C5—151no—
209C22×C3×C5—151no—
210C2×D15—51no—
211C2×D15—51no—
212C2×C5×D3—51no—
213C2×C5×D3—51no—
214C2×C3×D5—31no—
215C2×C3×D5—31no—
216C2×C3×D5—51no—
217C2×C3×D5—51no—
218C2×C3×D5—51no—
219C2×C3×D5—51no—
220C2×C3×D5—101no—
221C2×C3×D5—101no—
222C2×C3×D5—151no—
223C2×C3×D5—151no—
224C2×C3×D5—151no—
225C2×C3×D5—151no—
226C2×C3×D5—151no—
227C2×C3×D5—151no—
228C2×C3×D5—151no—
229C2×C3×D5—151no—
230C2×C3×D5—601no—
231D3×D5—51no—
232D3×D5—51no—
233D3×D5—51no—
234D3×D5—51no—
235D3×D5—51no—
236D3×D5—51no—
237D3×D5—201no—
238D3×D5ΓL(1,𝔽16)11yescharacteristic, direct product factor
239D3×D5D3×D511yescharacteristic
240ΓL(1,𝔽16)—31no—
241ΓL(1,𝔽16)—31no—
242ΓL(1,𝔽16)—51no—
243ΓL(1,𝔽16)—61no—
244ΓL(1,𝔽16)—151no—
245ΓL(1,𝔽16)—151no—
246ΓL(1,𝔽16)—301no—
247ΓL(1,𝔽16)D3×D511yescharacteristic, direct product factor
248C3×AGL(1,𝔽5)—31no—
249C3×AGL(1,𝔽5)—151no—
250C3×C4×C5—151no—
251Dic15—51no—
252Dic15—151no—
253Dic15—151no—
254Dic15—301no—
255C3×Dic5—151no—
256C5×Dic3—51no—
257C5×Dic3—151no—
258C5×Dic3—151no—
259C5×Dic3—301no—
260C2×C52—31no—
261C5⋊D5—31no—
262C5⋊D5D3×Dic311yescharacteristic, complemented
263C5×D5—31no—
264C5×D5—31no—
265C5×D5D3×Dic311yescharacteristic, complemented
266C5×D5C2×D3211yescharacteristic
267C22×Dic3—751no—
268C2×C4×D3—751no—
269C32×C5—41no—
270C32×C5C2×C4×D511yescharacteristic, complemented
271C32×C5C22×AGL(1,𝔽5)11yescharacteristic, complemented
272C22×D5—151no—
273C22×D5—151no—
274C2×AGL(1,𝔽5)—91no—
275C2×AGL(1,𝔽5)—151no—
276C2×AGL(1,𝔽5)—451no—
277C2×AGL(1,𝔽5)—451no—
278C2×AGL(1,𝔽5)—451no—
279C2×AGL(1,𝔽5)—451no—
280C2×C4×C5—451no—
281C2×Dic5—451no—
282C4×D5—151no—
283C4×D5—451no—
284C4×D5—451no—
285C4×D5—451no—
286C22×C32—251no—
287C2×C3×D3—51no—
288C2×C3×D3—51no—
289C2×C3×D3—251no—
290C2×C3×D3—251no—
291C2×C3×D3—251no—
292C2×C3×D3—251no—
293C3⋊Dic3—51no—
294C3⋊Dic3—251no—
295C3×Dic3—51no—
296C3×Dic3—251no—
297C2×C3×C5—31no—
298C2×C3×C5—31no—
299C2×C3×C5—51no—
300C2×C3×C5—51no—
301C2×C3×C5—51no—
302C2×C3×C5—51no—
303C2×C3×C5—101no—
304C2×C3×C5—101no—
305C2×C3×C5—121no—
306C2×C3×C5—151no—
307C2×C3×C5—151no—
308D15—51no—
309D15—51no—
310D15—201no—
311D15D5×Dic311yescharacteristic, complemented
312D15C2×ΓL(1,𝔽16)11yescharacteristic, complemented
313C3×D5—21no—
314C3×D5—21no—
315C3×D5—31no—
316C3×D5—31no—
317C3×D5—51no—
318C3×D5—51no—
319C3×D5—51no—
320C3×D5—51no—
321C3×D5—101no—
322C3×D5—101no—
323C3×D5—151no—
324C3×D5—151no—
325C3×D5—201no—
326C3×D5—201no—
327C3×D5—401no—
328C3×D5—601no—
329C3×D5D3×AGL(1,𝔽5)11yescharacteristic, complemented
330C3×D5C2×ΓL(1,𝔽16)11yescharacteristic, complemented
331C3×D5C2×D3×D511yescharacteristic
332C3×D5C2×D3×D511yescharacteristic
333C5×D3—41no—
334C5×D3—51no—
335C5×D3—51no—
336C5×D3D5×Dic311yescharacteristic, complemented
337C5×D3C2×ΓL(1,𝔽16)11yescharacteristic, complemented
338C52C2×D3×Dic311yescharacteristic, complemented, Sylow
339C23×C3—751no—
340C22×D3—251no—
341C2×C3×C4—751no—
342C2×Dic3—151no—
343C2×Dic3—251no—
344C2×Dic3—751no—
345C2×Dic3—751no—
346C2×Dic3—751no—
347C2×Dic3—751no—
348C2×Dic3—751no—
349C2×Dic3—1501no—
350C4×D3—151no—
351C4×D3—751no—
352C4×D3—751no—
353C4×D3—751no—
354C22×C5—151no—
355C22×C5—151no—
356C2×D5—31no—
357C2×D5—31no—
358C2×D5—51no—
359C2×D5—51no—
360C2×D5—151no—
361C2×D5—151no—
362C2×D5—151no—
363C2×D5—151no—
364C2×D5—151no—
365C2×D5—151no—
366C2×D5—151no—
367C2×D5—151no—
368C2×D5—601no—
369AGL(1,𝔽5)—31no—
370AGL(1,𝔽5)—91no—
371AGL(1,𝔽5)—151no—
372AGL(1,𝔽5)—451no—
373C4×C5—151no—
374C4×C5—451no—
375Dic5—151no—
376Dic5—451no—
377C2×C32—51no—
378C2×C32—51no—
379C2×C32—251no—
380C3×D3—51no—
381C3×D3—51no—
382C3×D3—251no—
383C3×D3D5×AGL(1,𝔽5)11yescharacteristic, complemented
384C22×C4—2251noSylow
385C3×C5—21no—
386C3×C5—21no—
387C3×C5—41no—
388C3×C5—41no—
389C3×C5—81no—
390C3×C5C4×D3×D511yescharacteristic, complemented
391C3×C5C2×D5×Dic311yescharacteristic, complemented
392C3×C5C2×D3×AGL(1,𝔽5)11yescharacteristic, complemented
393C3×C5C22×ΓL(1,𝔽16)11yescharacteristic, complemented
394C22×C3—151no—
395C22×C3—151no—
396C22×C3—251no—
397C22×C3—251no—
398C22×C3—501no—
399C22×C3—751no—
400C22×C3—751no—
401C22×C3—751no—
402C22×C3—751no—
403C2×D3—51no—
404C2×D3—51no—
405C2×D3—251no—
406C2×D3—251no—
407C2×D3—251no—
408C2×D3—251no—
409C3×C4—151no—
410C3×C4—751no—
411Dic3—51no—
412Dic3—151no—
413Dic3—151no—
414Dic3—251no—
415Dic3—301no—
416Dic3—751no—
417Dic3—751no—
418Dic3—1501no—
419C2×C5—31no—
420C2×C5—31no—
421C2×C5—51no—
422C2×C5—51no—
423C2×C5—121no—
424C2×C5—151no—
425C2×C5—151no—
426D5—31no—
427D5—31no—
428D5—51no—
429D5—51no—
430D5—151no—
431D5—151no—
432D5—201no—
433D5—601no—
434D5D3×ΓL(1,𝔽16)11yescharacteristic, direct product factor
435D5D32×D511yescharacteristic
436C32C2×D5×AGL(1,𝔽5)11yescharacteristic, complemented, Sylow
437C23—751no—
438C2×C4—451no—
439C2×C4—751no—
440C2×C4—2251no—
441C2×C4—2251no—
442C2×C4—2251no—
443C2×C4—2251no—
444C2×C3—31no—
445C2×C3—51no—
446C2×C3—51no—
447C2×C3—51no—
448C2×C3—51no—
449C2×C3—101no—
450C2×C3—101no—
451C2×C3—151no—
452C2×C3—151no—
453C2×C3—251no—
454C2×C3—251no—
455C2×C3—501no—
456C2×C3—751no—
457D3—51no—
458D3—51no—
459D3—251no—
460D3D5×ΓL(1,𝔽16)11yescharacteristic, direct product factor
461C5—41no—
462C5D3×D5×Dic311yescharacteristic, complemented
463C5C2×D3×ΓL(1,𝔽16)11yescharacteristic, complemented
464C22—151no—
465C22—151no—
466C22—251no—
467C22—751no—
468C22—751no—
469C22—751no—
470C22—751no—
471C4—151no—
472C4—451no—
473C4—751no—
474C4—2251no—
475C3—21no—
476C3D3×D5×AGL(1,𝔽5)11yescharacteristic, complemented
477C3C2×D5×ΓL(1,𝔽16)11yescharacteristic, complemented
478C2—31no—
479C2—51no—
480C2—51no—
481C2—151no—
482C2—151no—
483C2—251no—
484C2—751no—
subgroupG/H
center, Frattini, hypercenterC1G
commutator, Fitting, socle, hypocenterC32×C52C22×C4
prime radicalC3×D3×D52C2
cosocleC32×C5×D5C23
LCSG ▷ C32×C52
UCSC1

Complements

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

NH ≅ G/Ntype
2. #14268443C2semidirect
3. D15×ΓL(1,𝔽16)C2semidirect
4. D3×C3⋊#1042C2semidirect
5. C5×D3×ΓL(1,𝔽16)C2semidirect
6. D5×C3⋊ΓL(1,𝔽16)C2semidirect
7. C3×D5×ΓL(1,𝔽16)C2semidirect
10. C3×D15⋊D5C4semidirect
11. C3×D5×D15C4semidirect
13. C3×D3×C5⋊D5C4semidirect
14. C3×C5×D3×D5C4semidirect
17. C3⋊(C3⋊#1042)C22semidirect
18. C5×C3⋊ΓL(1,𝔽16)C22semidirect
19. C3×C3⋊#1042C22semidirect
20. C3×C5×ΓL(1,𝔽16)C22semidirect
34. D5×ΓL(1,𝔽16)D3direct (N × H)
38. C3×C3⋊(C5⋊D5)C2×C4semidirect
39. C3×C5×D15C2×C4semidirect
40. C3×C5×D15C2×C4semidirect
41. C3×C52×D3C2×C4semidirect
42. C32×C5⋊D5C2×C4semidirect
43. C32×C5×D5C2×C4semidirect
53. D3×ΓL(1,𝔽16)D5direct (N × H)
63. D15⋊D5Dic3semidirect
64. D5×D15Dic3semidirect
66. D3×C5⋊D5Dic3semidirect
67. C5×D3×D5Dic3semidirect
78. C3⋊#1042C2×D3semidirect
82. C5×ΓL(1,𝔽16)C2×D3semidirect
89. C32×C52C22×C4semidirect
110. C3×D3×D5AGL(1,𝔽5)semidirect
113. C3⋊ΓL(1,𝔽16)C2×D5semidirect
115. C3×ΓL(1,𝔽16)C2×D5semidirect
121. C3⋊(C5⋊D5)C2×Dic3semidirect
122. C5×D15C2×Dic3semidirect
123. C5×D15C2×Dic3semidirect
124. C52×D3C2×Dic3semidirect
127. C3×C5⋊D5C4×D3semidirect
128. C3×C5⋊D5C2×Dic3semidirect
133. C3×C5×D5C4×D3semidirect
134. C3×C5×D5C2×Dic3semidirect
184. C3×D15C4×D5semidirect
185. C3×D15C2×AGL(1,𝔽5)semidirect
189. C3×C5×D3C4×D5semidirect
190. C3×C5×D3C2×AGL(1,𝔽5)semidirect
194. C32×D5C2×AGL(1,𝔽5)semidirect
199. C3×C52C2×C4×D3semidirect
200. C3×C52C22×Dic3semidirect
238. D3×D5ΓL(1,𝔽16)direct (N × H)
247. ΓL(1,𝔽16)D3×D5direct (N × H)
262. C5⋊D5D3×Dic3semidirect
265. C5×D5D3×Dic3semidirect
270. C32×C5C2×C4×D5semidirect
271. C32×C5C22×AGL(1,𝔽5)semidirect
311. D15D5×Dic3semidirect
312. D15C2×ΓL(1,𝔽16)semidirect
329. C3×D5D3×AGL(1,𝔽5)semidirect
330. C3×D5C2×ΓL(1,𝔽16)semidirect
336. C5×D3D5×Dic3semidirect
337. C5×D3C2×ΓL(1,𝔽16)semidirect
338. C52C2×D3×Dic3semidirect
383. C3×D3D5×AGL(1,𝔽5)semidirect
390. C3×C5C4×D3×D5semidirect
391. C3×C5C2×D5×Dic3semidirect
392. C3×C5C2×D3×AGL(1,𝔽5)semidirect
393. C3×C5C22×ΓL(1,𝔽16)semidirect
434. D5D3×ΓL(1,𝔽16)direct (N × H)
436. C32C2×D5×AGL(1,𝔽5)semidirect
460. D3D5×ΓL(1,𝔽16)direct (N × H)
462. C5D3×D5×Dic3semidirect
463. C5C2×D3×ΓL(1,𝔽16)semidirect
476. C3D3×D5×AGL(1,𝔽5)semidirect
477. C3C2×D5×ΓL(1,𝔽16)semidirect

Maximal subgroups

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

subgroupindex
C3×D3×D522
#142684432
D15×ΓL(1,𝔽16)2
D3×C3⋊#10422
C5×D3×ΓL(1,𝔽16)2
D5×C3⋊ΓL(1,𝔽16)2
C3×D5×ΓL(1,𝔽16)2
C2×D5×ΓL(1,𝔽16)3
D3×D5×AGL(1,𝔽5)3
C2×D3×ΓL(1,𝔽16)5
D3×D5×Dic35