Finite Group Databasebeta

← C3×#10604418 #14297729

Subgroups — C3×#10604418 #14297729

Order 1920.

712 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×C4×D4×D5C211yesmaximal, characteristic
2C2×C3×#4007C212yesmaximal
3C3×#65689C211yesmaximal, characteristic, complemented
4C3×D5⋊(C4⋊C8)C211yesmaximal, characteristic, complemented
5C3×#64874C212yesmaximal, complemented
6#10604418C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×D4×D5C2211yescharacteristic
8C22×C3×C4×D5—21no—
9C2×C3×C4⋊AGL(1,𝔽5)C2212yes—
10C2×C3×D5⋊C8C2212yes—
11C3×C4×C5×D4C411yescharacteristic, complemented
12C3×D4×Dic5C411yescharacteristic, complemented
13C3×C4×C22⋊D5—21no—
14C3×#4039C411yescharacteristic, complemented
15C3×D5×C4⋊C4C2211yescharacteristic
16C3×#4027—21no—
17C3×D5×C22⋊C4—21no—
18C3×C4×D20C411yescharacteristic, complemented
19C3×C42×D5C2211yescharacteristic
20C3×#4007—24no—
21C3×(C2×AGL(1,𝔽5))⋊C4—21no—
22C3×(C2×AGL(1,𝔽5))⋊C4—21no—
23C3×#4004—21no—
24C3×C4⋊#170—21no—
25C3×#1804—51nomaximal, Hall
26C4×D4×D5C2×C311yescharacteristic
27C2×#4007C2×C312yes—
28#65689C2×C311yescharacteristic, complemented
29D5⋊(C4⋊C8)C2×C311yescharacteristic, complemented
30#64874C2×C312yescomplemented
31C23×C3×D5—21no—
32C22×C3×AGL(1,𝔽5)—22no—
33C2×C3×C5×D4C2×C411yescharacteristic
34C22×C3×C4×C5—21no—
35C2×C3×C22⋊D5—21no—
36C22×C3×Dic5—21no—
37C3×D4×D5—21no—
38C3×D4×D5—21no—
39C3×D4×D5D412yes—
40C3×D4×D5Q812yes—
41C2×C3×D20C2×C411yescharacteristic
42C2×C3×C4×D5—21no—
43C2×C3×C4×D5—21no—
44C2×C3×C4×D5—21no—
45C2×C3×C4×D5—44no—
46C2×C3×C4×D5D411yescharacteristic
47C2×C3×C4×D5D411yescharacteristic
48C2×C3×C4×D5C2311yescharacteristic
49C2×C3×#170—22no—
50C3×C4⋊AGL(1,𝔽5)—22no—
51C3×C4⋊AGL(1,𝔽5)—22no—
52C3×D5⋊C8—22no—
53C3×D5⋊C8—22no—
54C3×C5×C4⋊C4C2×C411yescharacteristic
55C3×C5×C22⋊C4—21no—
56C3×C42×C5C2×C411yescharacteristic
57C3×C22⋊Dic5—21no—
58C3×#685—21no—
59C3×#685—21no—
60C3×C4⋊Dic5C2×C411yescharacteristic
61C3×#683—21no—
62C3×C4×Dic5C2×C411yescharacteristic
63C3×C4×Dic5D411yescharacteristic
64C3×C4×Dic5D411yescharacteristic
65C2×C3×C4×D4—51no—
66C2×C3×C4⋊C8—51no—
67C2×C3×#102—52no—
68C3×#382—51no—
69C3×#340—52no—
70C2×D4×D5C22×C311yescharacteristic
71C22×C4×D5—21no—
72C2×C4⋊AGL(1,𝔽5)C22×C312yes—
73C2×D5⋊C8C22×C312yes—
74C4×C5×D4C3×C411yescharacteristic, complemented
75D4×Dic5C3×C411yescharacteristic, complemented
76C4×C22⋊D5—21no—
77#4039C3×C411yescharacteristic, complemented
78D5×C4⋊C4C22×C311yescharacteristic
79#4027—21no—
80D5×C22⋊C4—21no—
81C4×D20C3×C411yescharacteristic, complemented
82C42×D5C22×C311yescharacteristic
83#4007—24no—
84(C2×AGL(1,𝔽5))⋊C4—21no—
85(C2×AGL(1,𝔽5))⋊C4—21no—
86#4004—21no—
87C4⋊#170—21no—
88#1804—51noSylow
89C23×C3×C5—21no—
90C22×C3×D5—21no—
91C22×C3×D5—22no—
92C22×C3×D5—22no—
93C22×C3×D5—42no—
94C22×C3×D5—42no—
95C22×C3×D5C2×D411yescharacteristic
96C2×C3×AGL(1,𝔽5)—42no—
97C2×C3×AGL(1,𝔽5)—42no—
98C2×C3×AGL(1,𝔽5)—42no—
99C3×C5×D4—21no—
100C3×C5×D4C4⋊C412yes—
101C2×C3×C4×C5—21no—
102C2×C3×C4×C5—42no—
103C2×C3×C4×C5C22⋊C411yescharacteristic
104C2×C3×C4×C5C22⋊C411yescharacteristic
105C2×C3×C4×C5C22×C411yescharacteristic
106C3×C22⋊D5—22no—
107C3×C22⋊D5—22no—
108C2×C3×Dic5—21no—
109C2×C3×Dic5—42no—
110C2×C3×Dic5C22⋊C411yescharacteristic
111C2×C3×Dic5C22⋊C411yescharacteristic
112C2×C3×Dic5C2×D411yescharacteristic
113C3×D20—21no—
114C3×D20C4⋊C412yes—
115C3×C4×D5—22no—
116C3×C4×D5—22no—
117C3×C4×D5—42no—
118C3×C4×D5—42no—
119C3×C4×D5—42no—
120C3×C4×D5C2×D411yescharacteristic
121C3×C4×D5C2×Q811yescharacteristic
122C3×C4×D5#4111yescharacteristic
123C3×C4×D5#4111yescharacteristic
124C3×#170—42no—
125C22×C3×D4—51no—
126C23×C3×C4—101no—
127C22×C3×C8—52no—
128C3×C4×D4—51no—
129C3×C4×D4—51no—
130C3×C4×D4—51no—
131C3×C4×D4—101no—
132C3×C4×D4—101no—
133C3×C4×D4AGL(1,𝔽5)11yescharacteristic, complemented
134C2×C3×C4⋊C4—51no—
135C2×C3×C4⋊C4—52no—
136C2×C3×C22⋊C4—101no—
137C2×C3×C42—51no—
138C3×C4⋊C8—101no—
139C3×C4⋊C8—101no—
140C3×#102—104no—
141C3×#95—101no—
142C3×#95—101no—
143C23×D5—21no—
144C22×AGL(1,𝔽5)—22no—
145C2×C5×D4C2×C3×C411yescharacteristic
146C22×C4×C5—21no—
147C2×C22⋊D5—21no—
148C22×Dic5—21no—
149D4×D5—21no—
150D4×D5—21no—
151D4×D5C3×D412yes—
152D4×D5C3×Q812yes—
153C2×D20C2×C3×C411yescharacteristic
154C2×C4×D5—21no—
155C2×C4×D5—21no—
156C2×C4×D5—21no—
157C2×C4×D5—44no—
158C2×C4×D5C3×D411yescharacteristic
159C2×C4×D5C3×D411yescharacteristic
160C2×C4×D5C23×C311yescharacteristic
161C2×#170—22no—
162C4⋊AGL(1,𝔽5)—22no—
163C4⋊AGL(1,𝔽5)—22no—
164D5⋊C8—22no—
165D5⋊C8—22no—
166C5×C4⋊C4C2×C3×C411yescharacteristic
167C5×C22⋊C4—21no—
168C42×C5C2×C3×C411yescharacteristic
169C22⋊Dic5—21no—
170#685—21no—
171#685—21no—
172C4⋊Dic5C2×C3×C411yescharacteristic
173#683—21no—
174C4×Dic5C2×C3×C411yescharacteristic
175C4×Dic5C3×D411yescharacteristic
176C4×Dic5C3×D411yescharacteristic
177C2×C4×D4—51no—
178C2×C4⋊C8—51no—
179C2×#102—52no—
180#382—51no—
181#340—52no—
182C22×C3×C5—22no—
183C22×C3×C5—41no—
184C22×C3×C5—41no—
185C22×C3×C5C2×C22⋊C411yescharacteristic
186C2×C3×D5—22no—
187C2×C3×D5—41no—
188C2×C3×D5—41no—
189C2×C3×D5—44no—
190C2×C3×D5—44no—
191C2×C3×D5#12111yescharacteristic
192C2×C3×D5C22⋊Q811yescharacteristic
193C2×C3×D5#13512yes—
194C2×C3×D5AGL(1,ℤ/8)12yes—
195C3×AGL(1,𝔽5)—82no—
196C3×C4×C5—21no—
197C3×C4×C5—21no—
198C3×C4×C5—41no—
199C3×C4×C5C2×C4⋊C411yescharacteristic
200C3×C4×C5#11711yescharacteristic
201C3×Dic5—21no—
202C3×Dic5—21no—
203C3×Dic5—41no—
204C3×Dic5#12111yescharacteristic
205C3×Dic5C22⋊Q811yescharacteristic
206C24×C3—101no—
207C2×C3×D4—52no—
208C2×C3×D4—52no—
209C2×C3×D4—51no—
210C2×C3×D4—101no—
211C2×C3×D4—101no—
212C2×C3×D4—101no—
213C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
214C22×C3×C4—21no—
215C22×C3×C4—51no—
216C22×C3×C4—51no—
217C22×C3×C4—51no—
218C22×C3×C4—101no—
219C22×C3×C4—101no—
220C22×C3×C4—101no—
221C22×C3×C4—101no—
222C22×C3×C4—102no—
223C22×C3×C4—204no—
224C2×C3×C8—102no—
225C2×C3×C8—102no—
226C2×C3×C8—102no—
227C3×C4⋊C4—51no—
228C3×C4⋊C4—101no—
229C3×C4⋊C4—102no—
230C3×C4⋊C4—102no—
231C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
232C3×C22⋊C4—21no—
233C3×C22⋊C4—101no—
234C3×C22⋊C4—101no—
235C3×C22⋊C4—101no—
236C3×C42—51no—
237C3×C42—51no—
238C3×C42—51no—
239C3×C42C2×AGL(1,𝔽5)11yescharacteristic
240C23×C5—21no—
241C22×D5—21no—
242C22×D5—22no—
243C22×D5—22no—
244C22×D5—42no—
245C22×D5—42no—
246C22×D5C2×C3×D411yescharacteristic
247C2×AGL(1,𝔽5)—42no—
248C2×AGL(1,𝔽5)—42no—
249C2×AGL(1,𝔽5)—42no—
250C5×D4—21no—
251C5×D4C3×C4⋊C412yes—
252C2×C4×C5—21no—
253C2×C4×C5—42no—
254C2×C4×C5C3×C22⋊C411yescharacteristic
255C2×C4×C5C3×C22⋊C411yescharacteristic
256C2×C4×C5C22×C3×C411yescharacteristic
257C22⋊D5—22no—
258C22⋊D5—22no—
259C2×Dic5—21no—
260C2×Dic5—42no—
261C2×Dic5C3×C22⋊C411yescharacteristic
262C2×Dic5C3×C22⋊C411yescharacteristic
263C2×Dic5C2×C3×D411yescharacteristic
264D20—21no—
265D20C3×C4⋊C412yes—
266C4×D5—22no—
267C4×D5—22no—
268C4×D5—42no—
269C4×D5—42no—
270C4×D5—42no—
271C4×D5C2×C3×D411yescharacteristic
272C4×D5C2×C3×Q811yescharacteristic
273C4×D5C3×#4111yescharacteristic
274C4×D5C3×#4111yescharacteristic
275#170—42no—
276C22×D4—51no—
277C23×C4—101no—
278C22×C8—52no—
279C4×D4—51no—
280C4×D4—51no—
281C4×D4—51no—
282C4×D4—101no—
283C4×D4—101no—
284C4×D4C3×AGL(1,𝔽5)11yescharacteristic, complemented
285C2×C4⋊C4—51no—
286C2×C4⋊C4—52no—
287C2×C22⋊C4—101no—
288C2×C42—51no—
289C4⋊C8—101no—
290C4⋊C8—101no—
291#102—104no—
292#95—101no—
293#95—101no—
294C2×C3×C5—42no—
295C2×C3×C5#38011yescharacteristic
296C2×C3×C5#41611yescharacteristic
297C2×C3×C5#41811yescharacteristic
298C3×D5—42no—
299C3×D5#46312yes—
300C3×D5#47812yes—
301C23×C3—21no—
302C23×C3—51no—
303C23×C3—102no—
304C23×C3—102no—
305C23×C3—101no—
306C23×C3—202no—
307C23×C3—202no—
308C3×D4—21no—
309C3×D4—52no—
310C3×D4—102no—
311C3×D4—102no—
312C3×D4—101no—
313C3×D4C4⋊AGL(1,𝔽5)12yes—
314C2×C3×C4—21no—
315C2×C3×C4—42no—
316C2×C3×C4—51no—
317C2×C3×C4—51no—
318C2×C3×C4—51no—
319C2×C3×C4—51no—
320C2×C3×C4—51no—
321C2×C3×C4—51no—
322C2×C3×C4—51no—
323C2×C3×C4—102no—
324C2×C3×C4—102no—
325C2×C3×C4—101no—
326C2×C3×C4—202no—
327C2×C3×C4—202no—
328C2×C3×C4—202no—
329C2×C3×C4—202no—
330C2×C3×C4—202no—
331C2×C3×C4—202no—
332C2×C3×C4—202no—
333C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
334C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
335C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
336C3×C8—202no—
337C22×C5—22no—
338C22×C5—41no—
339C22×C5—41no—
340C22×C5C2×C3×C22⋊C411yescharacteristic
341C2×D5—22no—
342C2×D5—41no—
343C2×D5—41no—
344C2×D5—44no—
345C2×D5—44no—
346C2×D5C3×#12111yescharacteristic
347C2×D5C3×C22⋊Q811yescharacteristic
348C2×D5C3×#13512yes—
349C2×D5C3×AGL(1,ℤ/8)12yes—
350AGL(1,𝔽5)—82no—
351C4×C5—21no—
352C4×C5—21no—
353C4×C5—41no—
354C4×C5C2×C3×C4⋊C411yescharacteristic
355C4×C5C3×#11711yescharacteristic
356Dic5—21no—
357Dic5—21no—
358Dic5—41no—
359Dic5C3×#12111yescharacteristic
360Dic5C3×C22⋊Q811yescharacteristic
361C24—101no—
362C2×D4—52no—
363C2×D4—52no—
364C2×D4—51no—
365C2×D4—101no—
366C2×D4—101no—
367C2×D4—101no—
368C2×D4C2×C3×AGL(1,𝔽5)11yescharacteristic
369C22×C4—21no—
370C22×C4—51no—
371C22×C4—51no—
372C22×C4—51no—
373C22×C4—101no—
374C22×C4—101no—
375C22×C4—101no—
376C22×C4—101no—
377C22×C4—102no—
378C22×C4—204no—
379C2×C8—102no—
380C2×C8—102no—
381C2×C8—102no—
382C4⋊C4—51no—
383C4⋊C4—101no—
384C4⋊C4—102no—
385C4⋊C4—102no—
386C4⋊C4C2×C3×AGL(1,𝔽5)11yescharacteristic
387C22⋊C4—21no—
388C22⋊C4—101no—
389C22⋊C4—101no—
390C22⋊C4—101no—
391C42—51no—
392C42—51no—
393C42—51no—
394C42C2×C3×AGL(1,𝔽5)11yescharacteristic
395C3×C5#180411yescharacteristic, complemented, Hall
396C22×C3—22no—
397C22×C3—41no—
398C22×C3—41no—
399C22×C3—52no—
400C22×C3—51no—
401C22×C3—51no—
402C22×C3—52no—
403C22×C3—102no—
404C22×C3—204no—
405C22×C3—204no—
406C22×C3—201no—
407C22×C3—201no—
408C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
409C3×C4—21no—
410C3×C4—21no—
411C3×C4—41no—
412C3×C4—51no—
413C3×C4—51no—
414C3×C4—101no—
415C3×C4—101no—
416C3×C4—201no—
417C3×C4—402no—
418C3×C4C2×C4⋊AGL(1,𝔽5)11yescharacteristic
419C3×C4#413011yescharacteristic
420C2×C5—42no—
421C2×C5C3×#38011yescharacteristic
422C2×C5C3×#41611yescharacteristic
423C2×C5C3×#41811yescharacteristic
424D5—42no—
425D5C3×#46312yes—
426D5C3×#47812yes—
427C23—21no—
428C23—51no—
429C23—102no—
430C23—102no—
431C23—101no—
432C23—202no—
433C23—202no—
434D4—21no—
435D4—52no—
436D4—102no—
437D4—102no—
438D4—101no—
439D4C3×C4⋊AGL(1,𝔽5)12yes—
440C2×C4—21no—
441C2×C4—42no—
442C2×C4—51no—
443C2×C4—51no—
444C2×C4—51no—
445C2×C4—51no—
446C2×C4—51no—
447C2×C4—51no—
448C2×C4—51no—
449C2×C4—102no—
450C2×C4—102no—
451C2×C4—101no—
452C2×C4—202no—
453C2×C4—202no—
454C2×C4—202no—
455C2×C4—202no—
456C2×C4—202no—
457C2×C4—202no—
458C2×C4—202no—
459C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
460C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
461C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
462C8—202no—
463C2×C3—42no—
464C2×C3—52no—
465C2×C3—52no—
466C2×C3—202no—
467C2×C3#6576511yescharacteristic
468C2×C3#6576711yescharacteristic
469C2×C3#6579411yescharacteristic
470C5C3×#180411yescharacteristic, complemented, Sylow
471C22—22no—
472C22—41no—
473C22—41no—
474C22—51no—
475C22—51no—
476C22—52no—
477C22—52no—
478C22—102no—
479C22—204no—
480C22—204no—
481C22—201no—
482C22—201no—
483C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
484C4—21no—
485C4—21no—
486C4—41no—
487C4—51no—
488C4—51no—
489C4—101no—
490C4—101no—
491C4—201no—
492C4—402no—
493C4C2×C3×C4⋊AGL(1,𝔽5)11yescharacteristic
494C4C3×#413011yescharacteristic
495C3#1060441811yescharacteristic, direct product factor, Sylow
496C2—42no—
497C2—52no—
498C2—52no—
499C2—202no—
500C2C3×#6576511yescharacteristic
501C2C3×#6576711yescharacteristic
502C2C3×#6579411yescharacteristic
subgroupG/H
centerC22×C3C2×C22⋊AGL(1,𝔽5)
commutatorC2×C4×C5C22×C3×C4
FrattiniC2×C4C22×C3×AGL(1,𝔽5)
FittingC3×C4×C5×D4C4
socleC22×C3×C5C2×C22⋊C4
prime radicalC2×C3×D4×D5C22
cosocleC2×C4×D5C23×C3
hypercenterC3×C4×D4AGL(1,𝔽5)
hypocenterC5C3×#1804
LCSG ▷ C2×C4×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C22×C3 ◁ C3×C42 ◁ C3×C4×D4

Complements

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

NH ≅ G/Ntype
3. C3×#65689C2semidirect
4. C3×D5⋊(C4⋊C8)C2semidirect
5. C3×#64874C2semidirect
6. #10604418C3direct (N × H)
11. C3×C4×C5×D4C4semidirect
12. C3×D4×Dic5C4semidirect
14. C3×#4039C4semidirect
18. C3×C4×D20C4semidirect
28. #65689C2×C3semidirect
29. D5⋊(C4⋊C8)C2×C3semidirect
30. #64874C2×C3semidirect
74. C4×C5×D4C3×C4semidirect
75. D4×Dic5C3×C4semidirect
77. #4039C3×C4semidirect
81. C4×D20C3×C4semidirect
133. C3×C4×D4AGL(1,𝔽5)semidirect
284. C4×D4C3×AGL(1,𝔽5)semidirect
395. C3×C5#1804semidirect
470. C5C3×#1804semidirect
495. C3#10604418direct (N × H)

Maximal subgroups

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

subgroupindex
C3×C4×D4×D52
C2×C3×#40072
C3×#656892
C3×D5⋊(C4⋊C8)2
C3×#648742
#106044183
C3×#18045