Finite Group Databasebeta

← C3×#11751727 #72230436

Subgroups — C3×#11751727 #72230436

Order 2880.

564 conjugacy classes of subgroups.

The subgroup lattice has 470 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#91989C211yesmaximal, characteristic
2C2×C32×C4⋊AGL(1,𝔽5)C211yesmaximal, characteristic
3C3×D5×#208C211yesmaximal, characteristic
4C3×#91160C212yesmaximal, complemented
5C3×#91159C212yesmaximal, complemented
6C3×#65727—31nomaximal
7#11751727C311yesmaximal, characteristic, direct product factor
8C22×C32×AGL(1,𝔽5)—21no—
9C2×C32×C4×D5C2211yescharacteristic
10C3×#6763—21no—
11C3×C4⋊ΓL(1,𝔽16)C2211yescharacteristic
12C3×C4⋊ΓL(1,𝔽16)C2211yescharacteristic
13C3×C4×ΓL(1,𝔽16)—21no—
14C32×C4⋊AGL(1,𝔽5)—21no—
15C32×C4⋊AGL(1,𝔽5)C2212yes—
16C3×#6710C411yescharacteristic, complemented
17C3×C5×#208C411yescharacteristic, complemented
18C3×#6649C412yescomplemented
19C3×D5×C3⋊C8C2212yes—
20C3×#4964—51nomaximal, Hall
21#91989C2×C311yescharacteristic
22C3×#4130—31no—
23C2×C3×C4⋊AGL(1,𝔽5)—21no—
24C2×C3×C4⋊AGL(1,𝔽5)D311yescharacteristic
25C2×C3×C4⋊AGL(1,𝔽5)C2×C311yescharacteristic
26C3×D5×M16—31no—
27D5×#208C2×C311yescharacteristic
28C3×#3994—32no—
29C3×#3993—32no—
30#91160C2×C312yescomplemented
31#91159C2×C312yescomplemented
32C22×C32×D5D411yescharacteristic
33C2×C3×ΓL(1,𝔽16)—21no—
34C2×C3×ΓL(1,𝔽16)—21no—
35C2×C32×AGL(1,𝔽5)—22no—
36C2×C32×AGL(1,𝔽5)—42no—
37C2×C32×C4×C5C2×C411yescharacteristic
38C2×C32×Dic5Q811yescharacteristic
39C32×C4×D5C2×C411yescharacteristic
40C32×C4×D5D411yescharacteristic
41C32×C4×D5Q811yescharacteristic
42C32×C4×D5C2311yescharacteristic
43C3×C3⋊#168C2×C412yescomplemented
44C3×C5×C3⋊C8C2×C412yescomplemented
45#65727—31noHall
46C2×C32×C4⋊C4—51no—
47C3×#926—51no—
48C2×C3×#208—51no—
49C3×#808—52no—
50C3×#807—52no—
51C22×C3×AGL(1,𝔽5)—21no—
52C22×C3×AGL(1,𝔽5)—21no—
53C22×C3×AGL(1,𝔽5)—21no—
54C22×C3×AGL(1,𝔽5)—21no—
55C2×C3×C4×D5—21no—
56C2×C3×C4×D5C2×D311yescharacteristic
57C2×C3×C4×D5C22×C311yescharacteristic
58#6763—21no—
59C4⋊ΓL(1,𝔽16)C22×C311yescharacteristic
60C4⋊ΓL(1,𝔽16)C22×C311yescharacteristic
61C4×ΓL(1,𝔽16)—21no—
62C3×C22⋊AGL(1,𝔽5)—61no—
63C3×C4⋊AGL(1,𝔽5)—21no—
64C3×C4⋊AGL(1,𝔽5)—21no—
65C3×C4⋊AGL(1,𝔽5)—22no—
66C3×C4⋊AGL(1,𝔽5)—21no—
67C3×C4⋊AGL(1,𝔽5)—21no—
68C3×C4⋊AGL(1,𝔽5)—31no—
69C3×C4⋊AGL(1,𝔽5)—31no—
70C3×C4⋊AGL(1,𝔽5)C2×D312yes—
71C3×C4⋊AGL(1,𝔽5)C22×C312yes—
72C3×C4×AGL(1,𝔽5)—61no—
73C3×C5×M16—31no—
74#6710C3×C411yescharacteristic, complemented
75C5×#208C3×C411yescharacteristic, complemented
76C3×#681—31no—
77C3×#676—32no—
78C3×C8×D5—32no—
79#6649C3×C412yescomplemented
80D5×C3⋊C8C22×C312yes—
81C3×#428—151no—
82#4964—51no—
83C22×C32×C5C4⋊C411yescharacteristic
84C2×C32×D5—21no—
85C2×C32×D5—21no—
86C2×C32×D5C4⋊C411yescharacteristic
87C2×C32×D5C2×D411yescharacteristic
88C3×ΓL(1,𝔽16)—41no—
89C3×ΓL(1,𝔽16)—41no—
90C32×AGL(1,𝔽5)—42no—
91C32×C4×C5C4⋊C411yescharacteristic
92C32×C4×C5C22×C411yescharacteristic
93C32×Dic5C4⋊C411yescharacteristic
94C32×Dic5C2×Q811yescharacteristic
95#4130—31no—
96C2×C4⋊AGL(1,𝔽5)C3×D311yescharacteristic
97D5×M16—31no—
98#3994—32no—
99#3993—32no—
100C22×C32×C4—51no—
101C22×C32×C4—101no—
102C32×C4⋊C4—52no—
103C32×C4⋊C4—101no—
104C3×#217—101no—
105C3×C4⋊Dic3—51no—
106C3×C4⋊Dic3—51no—
107C3×C4×Dic3—101no—
108C3×#208—51no—
109C3×#208—52no—
110C3×#208AGL(1,𝔽5)11yescharacteristic, complemented
111C2×C3×C3⋊C8—52no—
112C22×C3×D5—21no—
113C22×C3×D5#6711yescharacteristic
114C22×C3×D5C3×D411yescharacteristic
115C2×ΓL(1,𝔽16)—21no—
116C2×ΓL(1,𝔽16)—21no—
117C2×C3×AGL(1,𝔽5)—22no—
118C2×C3×AGL(1,𝔽5)—22no—
119C2×C3×AGL(1,𝔽5)—22no—
120C2×C3×AGL(1,𝔽5)—22no—
121C2×C3×AGL(1,𝔽5)—42no—
122C2×C3×AGL(1,𝔽5)—42no—
123C2×C3×AGL(1,𝔽5)—42no—
124C2×C3×AGL(1,𝔽5)—42no—
125C2×C3×AGL(1,𝔽5)—61no—
126C2×C3×AGL(1,𝔽5)—61no—
127C2×C3×C4×C5—21no—
128C2×C3×C4×C5C4×D311yescharacteristic
129C2×C3×C4×C5C2×C3×C411yescharacteristic
130C2×C3×Dic5—21no—
131C2×C3×Dic5Dic611yescharacteristic
132C2×C3×Dic5C3×Q811yescharacteristic
133C3×C4×D5—21no—
134C3×C4×D5—21no—
135C3×C4×D5—21no—
136C3×C4×D5—21no—
137C3×C4×D5Dic611yescharacteristic
138C3×C4×D5C4×D311yescharacteristic
139C3×C4×D5#6711yescharacteristic
140C3×C4×D5C2×C3×C411yescharacteristic
141C3×C4×D5C3×D411yescharacteristic
142C3×C4×D5C3×Q811yescharacteristic
143C3×C4×D5C22×D311yescharacteristic
144C3×C4×D5C23×C311yescharacteristic
145C3×C5×C8—32no—
146C3⋊#168C2×C3×C412yescomplemented
147C3×#168—32no—
148C5×C3⋊C8C2×C3×C412yescomplemented
149C2×C3×M16—151no—
150C3×#117—151no—
151C2×C3×C4⋊C4—51no—
152C2×C3×C4⋊C4—51no—
153C2×C3×C4⋊C4—101no—
154#926—51no—
155C2×#208—51no—
156C3×#107—152no—
157C3×#106—152no—
158#808—52no—
159#807—52no—
160C2×C32×C5—21no—
161C2×C32×C5C2×C4⋊C411yescharacteristic
162C32×D5—21no—
163C32×D5AGL(1,ℤ/8)11yescharacteristic
164C32×D5#13711yescharacteristic
165C22×AGL(1,𝔽5)—21no—
166C2×C4×D5C2×C3×D311yescharacteristic
167C22⋊AGL(1,𝔽5)—61no—
168C4⋊AGL(1,𝔽5)—21no—
169C4⋊AGL(1,𝔽5)—31no—
170C4⋊AGL(1,𝔽5)—31no—
171C4⋊AGL(1,𝔽5)C2×C3×D312yes—
172C4×AGL(1,𝔽5)—61no—
173C5×M16—31no—
174#681—31no—
175#676—32no—
176C8×D5—32no—
177C23×C32—51no—
178C2×C32×C4—51no—
179C2×C32×C4—51no—
180C2×C32×C4—51no—
181C2×C32×C4—51no—
182C2×C32×C4—51no—
183C2×C32×C4—102no—
184C2×C32×C4—202no—
185C2×C32×C4C2×AGL(1,𝔽5)11yescharacteristic
186C2×C3×Dic3—101no—
187C2×C3×Dic3—101no—
188C3×C3⋊C8—52no—
189C3×C3⋊C8C2×AGL(1,𝔽5)12yescomplemented
190#428—151noSylow
191C22×C3×C5—21no—
192C22×C3×C5Dic3⋊C411yescharacteristic
193C22×C3×C5C3×C4⋊C411yescharacteristic
194C2×C3×D5—21no—
195C2×C3×D5—21no—
196C2×C3×D5—21no—
197C2×C3×D5—21no—
198C2×C3×D5—21no—
199C2×C3×D5—21no—
200C2×C3×D5—21no—
201C2×C3×D5—21no—
202C2×C3×D5—21no—
203C2×C3×D5—21no—
204C2×C3×D5Dic3⋊C411yescharacteristic
205C2×C3×D5C3×C4⋊C411yescharacteristic
206C2×C3×D5C2×#6711yescharacteristic
207C2×C3×D5C2×C3×D411yescharacteristic
208ΓL(1,𝔽16)—41no—
209ΓL(1,𝔽16)—41no—
210C3×AGL(1,𝔽5)—42no—
211C3×AGL(1,𝔽5)—42no—
212C3×AGL(1,𝔽5)—42no—
213C3×AGL(1,𝔽5)—42no—
214C3×AGL(1,𝔽5)—121no—
215C3×AGL(1,𝔽5)—121no—
216C3×C4×C5—21no—
217C3×C4×C5—21no—
218C3×C4×C5Dic3⋊C411yescharacteristic
219C3×C4×C5C3×C4⋊C411yescharacteristic
220C3×C4×C5C2×C4×D311yescharacteristic
221C3×C4×C5C22×C3×C411yescharacteristic
222C3×Dic5—21no—
223C3×Dic5—21no—
224C3×Dic5Dic3⋊C411yescharacteristic
225C3×Dic5C3×C4⋊C411yescharacteristic
226C3×Dic5C2×Dic611yescharacteristic
227C3×Dic5C2×C3×Q811yescharacteristic
228C22×C3×C4—51no—
229C22×C3×C4—51no—
230C22×C3×C4—101no—
231C22×C3×C4—101no—
232C22×C3×C4—101no—
233C22×C3×C4—101no—
234C22×C3×C4—101no—
235C3×M16—31no—
236C3×M16—152no—
237C3×M16—151no—
238C2×C3×C8—152no—
239C3×C4⋊C4—52no—
240C3×C4⋊C4—52no—
241C3×C4⋊C4—102no—
242C3×C4⋊C4—101no—
243C3×C4⋊C4—101no—
244C3×C4⋊C4—101no—
245C3×C4⋊C4—101no—
246C3×C4⋊C4—151no—
247C3×C4⋊C4—151no—
248C3×C22⋊C4—301no—
249C3×C42—301no—
250#217—101no—
251C4⋊Dic3—51no—
252C4⋊Dic3—51no—
253C4×Dic3—101no—
254#208—51no—
255#208—52no—
256#208C3×AGL(1,𝔽5)11yescharacteristic, complemented
257C2×C3⋊C8—52no—
258C32×C5#42811yescharacteristic, complemented, Hall
259C22×D5C3×#6711yescharacteristic
260C2×AGL(1,𝔽5)—22no—
261C2×AGL(1,𝔽5)—42no—
262C2×AGL(1,𝔽5)—61no—
263C2×AGL(1,𝔽5)—61no—
264C2×C4×C5C3×C4×D311yescharacteristic
265C2×Dic5C3×Dic611yescharacteristic
266C4×D5C3×Dic611yescharacteristic
267C4×D5C3×C4×D311yescharacteristic
268C4×D5C3×#6711yescharacteristic
269C4×D5C22×C3×D311yescharacteristic
270C5×C8—32no—
271#168—32no—
272C22×C32—51no—
273C22×C32—51no—
274C22×C32—101no—
275C22×C32—101no—
276C22×C32C4⋊AGL(1,𝔽5)11yescharacteristic
277C32×C4—51no—
278C32×C4—51no—
279C32×C4—202no—
280C32×C4C4⋊AGL(1,𝔽5)11yescharacteristic
281C32×C4C22×AGL(1,𝔽5)11yescharacteristic
282C3×Dic3—201no—
283C3×Dic3—201no—
284C2×M16—151no—
285#117—151no—
286C2×C4⋊C4—51no—
287#107—152no—
288#106—152no—
289C2×C3×C5—21no—
290C2×C3×C5—21no—
291C2×C3×C5—21no—
292C2×C3×C5—21no—
293C2×C3×C5—21no—
294C2×C3×C5C2×Dic3⋊C411yescharacteristic
295C2×C3×C5C2×C3×C4⋊C411yescharacteristic
296C3×D5—21no—
297C3×D5—21no—
298C3×D5—21no—
299C3×D5—21no—
300C3×D5—21no—
301C3×D5—21no—
302C3×D5#93211yescharacteristic
303C3×D5#94211yescharacteristic
304C3×D5C3×AGL(1,ℤ/8)11yescharacteristic
305C3×D5C3×#13711yescharacteristic
306C23×C3—51no—
307C23×C3—51no—
308C23×C3—101no—
309C2×C3×C4—21no—
310C2×C3×C4—51no—
311C2×C3×C4—51no—
312C2×C3×C4—51no—
313C2×C3×C4—51no—
314C2×C3×C4—51no—
315C2×C3×C4—51no—
316C2×C3×C4—51no—
317C2×C3×C4—51no—
318C2×C3×C4—51no—
319C2×C3×C4—51no—
320C2×C3×C4—101no—
321C2×C3×C4—101no—
322C2×C3×C4—101no—
323C2×C3×C4—101no—
324C2×C3×C4—101no—
325C2×C3×C4—102no—
326C2×C3×C4—102no—
327C2×C3×C4—102no—
328C2×C3×C4—102no—
329C2×C3×C4—202no—
330C2×C3×C4—202no—
331C2×C3×C4—202no—
332C2×C3×C4—202no—
333C2×C3×C4—301no—
334C2×C3×C4—301no—
335C2×C3×C4D3×AGL(1,𝔽5)11yescharacteristic
336C2×C3×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
337C2×Dic3—101no—
338C2×Dic3—101no—
339C3×C8—32no—
340C3×C8—152no—
341C3⋊C8—52no—
342C3⋊C8C2×C3×AGL(1,𝔽5)12yescomplemented
343C22×C5C3×Dic3⋊C411yescharacteristic
344C2×D5—21no—
345C2×D5—21no—
346C2×D5C3×Dic3⋊C411yescharacteristic
347C2×D5C2×C3×#6711yescharacteristic
348AGL(1,𝔽5)—42no—
349AGL(1,𝔽5)—121no—
350AGL(1,𝔽5)—121no—
351C4×C5C3×Dic3⋊C411yescharacteristic
352C4×C5C2×C3×C4×D311yescharacteristic
353Dic5C3×Dic3⋊C411yescharacteristic
354Dic5C2×C3×Dic611yescharacteristic
355C2×C32—21no—
356C2×C32—51no—
357C2×C32—51no—
358C2×C32—101no—
359C2×C32C2×C4⋊AGL(1,𝔽5)11yescharacteristic
360C22×C4—51no—
361C22×C4—101no—
362M16—31no—
363M16—152no—
364M16—151no—
365C2×C8—152no—
366C4⋊C4—52no—
367C4⋊C4—101no—
368C4⋊C4—151no—
369C4⋊C4—151no—
370C22⋊C4—301no—
371C42—301no—
372C3×C5—21no—
373C3×C5#496411yescharacteristic, complemented
374C3×C5C3×#42811yescharacteristic, complemented
375C22×C3—21no—
376C22×C3—51no—
377C22×C3—51no—
378C22×C3—51no—
379C22×C3—51no—
380C22×C3—101no—
381C22×C3—101no—
382C22×C3—101no—
383C22×C3—101no—
384C22×C3—101no—
385C22×C3—101no—
386C22×C3—101no—
387C22×C3—101no—
388C22×C3—101no—
389C22×C3—101no—
390C22×C3Dic3⋊AGL(1,𝔽5)11yescharacteristic
391C22×C3C3×C4⋊AGL(1,𝔽5)11yescharacteristic
392C3×C4—21no—
393C3×C4—21no—
394C3×C4—51no—
395C3×C4—51no—
396C3×C4—51no—
397C3×C4—51no—
398C3×C4—101no—
399C3×C4—101no—
400C3×C4—202no—
401C3×C4—202no—
402C3×C4—202no—
403C3×C4—202no—
404C3×C4—601no—
405C3×C4—601no—
406C3×C4Dic3⋊AGL(1,𝔽5)11yescharacteristic
407C3×C4C3×C4⋊AGL(1,𝔽5)11yescharacteristic
408C3×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
409C3×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
410Dic3—201no—
411Dic3—201no—
412C2×C5—21no—
413C2×C5C2×C3×Dic3⋊C411yescharacteristic
414D5—21no—
415D5C3×#93211yescharacteristic
416D5C3×#94211yescharacteristic
417C32#6572711yescharacteristic, complemented, Sylow
418C23—51no—
419C2×C4—51no—
420C2×C4—51no—
421C2×C4—51no—
422C2×C4—51no—
423C2×C4—51no—
424C2×C4—102no—
425C2×C4—202no—
426C2×C4—301no—
427C2×C4—301no—
428C2×C4C3×D3×AGL(1,𝔽5)11yescharacteristic
429C8—32no—
430C8—152no—
431C2×C3—21no—
432C2×C3—21no—
433C2×C3—21no—
434C2×C3—21no—
435C2×C3—21no—
436C2×C3—51no—
437C2×C3—51no—
438C2×C3—51no—
439C2×C3—51no—
440C2×C3—101no—
441C2×C3—101no—
442C2×C3—101no—
443C2×C3—101no—
444C2×C3—101no—
445C2×C3—101no—
446C2×C3C2×Dic3⋊AGL(1,𝔽5)11yescharacteristic
447C2×C3C2×C3×C4⋊AGL(1,𝔽5)11yescharacteristic
448C5C3×#496411yescharacteristic, complemented, Sylow
449C22—51no—
450C22—51no—
451C22—101no—
452C22—101no—
453C22C3×Dic3⋊AGL(1,𝔽5)11yescharacteristic
454C4—51no—
455C4—51no—
456C4—202no—
457C4—601no—
458C4—601no—
459C4C3×Dic3⋊AGL(1,𝔽5)11yescharacteristic
460C4C2×C3×D3×AGL(1,𝔽5)11yescharacteristic
461C3—21no—
462C3#1175172711yescharacteristic, direct product factor
463C3C3×#6572711yescharacteristic, complemented
464C2—21no—
465C2—51no—
466C2—51no—
467C2—101no—
468C2C2×C3×Dic3⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
centerC2×C3C2×Dic3⋊AGL(1,𝔽5)
commutatorC3×C4×C5C22×C3×C4
FrattiniC4C2×C3×D3×AGL(1,𝔽5)
FittingC2×C32×C4×C5C2×C4
socleC2×C32×C5C2×C4⋊C4
prime radicalC22×C32×D5D4
cosocleC3×C4×D5C23×C3
hypercenterC2×C3×C4D3×AGL(1,𝔽5)
hypocenterC3×C5C3×#428
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C2×C3 ◁ C2×C3×C4

Complements

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

NH ≅ G/Ntype
4. C3×#91160C2semidirect
5. C3×#91159C2semidirect
7. #11751727C3direct (N × H)
16. C3×#6710C4semidirect
17. C3×C5×#208C4semidirect
18. C3×#6649C4semidirect
30. #91160C2×C3semidirect
31. #91159C2×C3semidirect
43. C3×C3⋊#168C2×C4semidirect
44. C3×C5×C3⋊C8C2×C4semidirect
74. #6710C3×C4semidirect
75. C5×#208C3×C4semidirect
79. #6649C3×C4semidirect
110. C3×#208AGL(1,𝔽5)semidirect
146. C3⋊#168C2×C3×C4semidirect
148. C5×C3⋊C8C2×C3×C4semidirect
189. C3×C3⋊C8C2×AGL(1,𝔽5)semidirect
256. #208C3×AGL(1,𝔽5)semidirect
258. C32×C5#428semidirect
342. C3⋊C8C2×C3×AGL(1,𝔽5)semidirect
373. C3×C5#4964semidirect
374. C3×C5C3×#428semidirect
417. C32#65727semidirect
448. C5C3×#4964semidirect
462. C3#11751727direct (N × H)
463. C3C3×#65727semidirect

Maximal subgroups

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

subgroupindex
C3×#919892
C2×C32×C4⋊AGL(1,𝔽5)2
C3×D5×#2082
C3×#911602
C3×#911592
C3×#657273
#117517273
C3×#49645