Finite Group Databasebeta

← C3×#10603919 #14300322

Subgroups — C3×#10603919 #14300322

Order 1920.

748 conjugacy classes of subgroups.

The subgroup lattice has 512 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C3×#66258C211yesmaximal, characteristic
2C3×#65779C211yesmaximal, characteristic
3C3×#65699C212yesmaximal, complemented
4C3×D5×#99C211yesmaximal, characteristic
5C3×#64866C212yesmaximal, complemented
6#10603919C311yesmaximal, characteristic, direct product factor, Hall
7C23×C3×AGL(1,𝔽5)C2211yescharacteristic
8C2×C3×D4×D5C2211yescharacteristic
9C2×C3×C22⋊AGL(1,𝔽5)—21no—
10C2×C3×C22⋊AGL(1,𝔽5)—21no—
11C2×C3×C22⋊AGL(1,𝔽5)C2211yescharacteristic
12C2×C3×C22⋊AGL(1,𝔽5)C2212yes—
13C3×D4×AGL(1,𝔽5)—22no—
14C3×D4×AGL(1,𝔽5)—22no—
15C3×#4130—21no—
16C2×C3×C4⋊AGL(1,𝔽5)—22no—
17C3×D5×C22⋊C4C2212yes—
18C3×(C2×AGL(1,𝔽5))⋊C4—22no—
19C3×(C2×AGL(1,𝔽5))⋊C4—21no—
20C3×C5×#99C411yescharacteristic, complemented
21C3×#3966C411yescharacteristic, complemented
22C3×#3938C412yescomplemented
23C3×#1867—51nomaximal, Hall
24#66258C2×C311yescharacteristic
25#65779C2×C311yescharacteristic
26#65699C2×C312yescomplemented
27D5×#99C2×C311yescharacteristic
28#64866C2×C312yescomplemented
29C23×C3×D5D411yescharacteristic
30C23×C3×D5C2311yescharacteristic
31C22×C3×AGL(1,𝔽5)—22no—
32C22×C3×AGL(1,𝔽5)—21no—
33C22×C3×AGL(1,𝔽5)—21no—
34C22×C3×AGL(1,𝔽5)—21no—
35C22×C3×AGL(1,𝔽5)—21no—
36C22×C3×AGL(1,𝔽5)—42no—
37C22×C3×AGL(1,𝔽5)D412yes—
38C22×C3×AGL(1,𝔽5)D412yes—
39C2×C3×C5×D4C2×C411yescharacteristic
40C2×C3×C22⋊D5C2×C411yescharacteristic
41C2×C3×C22⋊D5Q811yescharacteristic
42C3×D4×D5—22no—
43C3×D4×D5—22no—
44C2×C3×D20Q811yescharacteristic
45C2×C3×C4×D5—22no—
46C2×C3×C4×D5D411yescharacteristic
47C3×C22⋊AGL(1,𝔽5)—22no—
48C3×C22⋊AGL(1,𝔽5)—22no—
49C3×C22⋊AGL(1,𝔽5)—22no—
50C3×C22⋊AGL(1,𝔽5)—22no—
51C3×C22⋊AGL(1,𝔽5)—42no—
52C3×C4⋊AGL(1,𝔽5)—21no—
53C3×C4⋊AGL(1,𝔽5)—21no—
54C3×C4⋊AGL(1,𝔽5)—44no—
55C3×C4×AGL(1,𝔽5)—22no—
56C3×C5×C22⋊C4C2×C412yescomplemented
57C3×C22⋊Dic5C2×C412yescomplemented
58C3×#685—22no—
59C3×#518—51no—
60C2×C3×#99—51no—
61C3×#386—51no—
62C3×#385—52no—
63C3×#342—52no—
64C23×AGL(1,𝔽5)C22×C311yescharacteristic
65C2×D4×D5C22×C311yescharacteristic
66C2×C22⋊AGL(1,𝔽5)—21no—
67C2×C22⋊AGL(1,𝔽5)—21no—
68C2×C22⋊AGL(1,𝔽5)C22×C311yescharacteristic
69C2×C22⋊AGL(1,𝔽5)C22×C312yes—
70D4×AGL(1,𝔽5)—22no—
71D4×AGL(1,𝔽5)—22no—
72#4130—21no—
73C2×C4⋊AGL(1,𝔽5)—22no—
74D5×C22⋊C4C22×C312yes—
75(C2×AGL(1,𝔽5))⋊C4—22no—
76(C2×AGL(1,𝔽5))⋊C4—21no—
77C5×#99C3×C411yescharacteristic, complemented
78#3966C3×C411yescharacteristic, complemented
79#3938C3×C412yescomplemented
80#1867—51noSylow
81C23×C3×C5C4⋊C411yescharacteristic, complemented
82C23×C3×C5C22×C411yescharacteristic
83C22×C3×D5—21no—
84C22×C3×D5—21no—
85C22×C3×D5—41no—
86C22×C3×D5—41no—
87C22×C3×D5—41no—
88C22×C3×D5—41no—
89C22×C3×D5C4⋊C411yescharacteristic, complemented
90C22×C3×D5C2×D412yes—
91C22×C3×D5C2×D411yescharacteristic
92C22×C3×D5C2×Q811yescharacteristic
93C22×C3×D5#4112yes—
94C2×C3×AGL(1,𝔽5)—24no—
95C2×C3×AGL(1,𝔽5)—22no—
96C2×C3×AGL(1,𝔽5)—22no—
97C2×C3×AGL(1,𝔽5)—22no—
98C2×C3×AGL(1,𝔽5)—42no—
99C2×C3×AGL(1,𝔽5)—41no—
100C2×C3×AGL(1,𝔽5)—41no—
101C2×C3×AGL(1,𝔽5)—42no—
102C2×C3×AGL(1,𝔽5)—42no—
103C2×C3×AGL(1,𝔽5)—41no—
104C2×C3×AGL(1,𝔽5)—41no—
105C2×C3×AGL(1,𝔽5)—41no—
106C2×C3×AGL(1,𝔽5)—81no—
107C2×C3×AGL(1,𝔽5)—81no—
108C3×C5×D4—22no—
109C2×C3×C4×C5—22no—
110C2×C3×C4×C5C4⋊C411yescharacteristic, complemented
111C3×C22⋊D5—22no—
112C3×C22⋊D5—22no—
113C2×C3×Dic5—22no—
114C2×C3×Dic5C4⋊C411yescharacteristic, complemented
115C3×D20—22no—
116C3×C4×D5—21no—
117C3×C4×D5—21no—
118C3×C4×D5—42no—
119C3×C4×D5—42no—
120C22×C3×D4—51no—
121C23×C3×C4—51no—
122C3×C4×D4—102no—
123C3×C4×D4—102no—
124C3×#117—101no—
125C2×C3×C4⋊C4—102no—
126C2×C3×C22⋊C4—52no—
127C2×C3×C22⋊C4—52no—
128C2×C3×C22⋊C4—51no—
129C2×C3×C22⋊C4—101no—
130C2×C3×C22⋊C4—101no—
131C3×#99—51no—
132C3×#99—52no—
133C3×#99AGL(1,𝔽5)11yescharacteristic, complemented
134C3×#95—102no—
135C3×#95—101no—
136C23×D5C3×D411yescharacteristic
137C23×D5C23×C311yescharacteristic
138C22×AGL(1,𝔽5)—22no—
139C22×AGL(1,𝔽5)—21no—
140C22×AGL(1,𝔽5)—21no—
141C22×AGL(1,𝔽5)—21no—
142C22×AGL(1,𝔽5)—21no—
143C22×AGL(1,𝔽5)—42no—
144C22×AGL(1,𝔽5)C3×D412yes—
145C22×AGL(1,𝔽5)C3×D412yes—
146C2×C5×D4C2×C3×C411yescharacteristic
147C2×C22⋊D5C2×C3×C411yescharacteristic
148C2×C22⋊D5C3×Q811yescharacteristic
149D4×D5—22no—
150D4×D5—22no—
151C2×D20C3×Q811yescharacteristic
152C2×C4×D5—22no—
153C2×C4×D5C3×D411yescharacteristic
154C22⋊AGL(1,𝔽5)—22no—
155C22⋊AGL(1,𝔽5)—22no—
156C22⋊AGL(1,𝔽5)—22no—
157C22⋊AGL(1,𝔽5)—22no—
158C22⋊AGL(1,𝔽5)—42no—
159C4⋊AGL(1,𝔽5)—21no—
160C4⋊AGL(1,𝔽5)—21no—
161C4⋊AGL(1,𝔽5)—44no—
162C4×AGL(1,𝔽5)—22no—
163C5×C22⋊C4C2×C3×C412yescomplemented
164C22⋊Dic5C2×C3×C412yescomplemented
165#685—22no—
166#518—51no—
167C2×#99—51no—
168#386—51no—
169#385—52no—
170#342—52no—
171C22×C3×C5—21no—
172C22×C3×C5—41no—
173C22×C3×C5—41no—
174C22×C3×C5C2×C4⋊C411yescharacteristic
175C22×C3×C5C4×D412yes—
176C2×C3×D5—22no—
177C2×C3×D5—21no—
178C2×C3×D5—22no—
179C2×C3×D5—21no—
180C2×C3×D5—21no—
181C2×C3×D5—41no—
182C2×C3×D5—42no—
183C2×C3×D5—41no—
184C2×C3×D5—41no—
185C2×C3×D5—41no—
186C2×C3×D5—81no—
187C2×C3×D5#12011yescharacteristic
188C2×C3×D5C22⋊Q812yes—
189C2×C3×D5#12311yescharacteristic
190C3×AGL(1,𝔽5)—42no—
191C3×AGL(1,𝔽5)—42no—
192C3×AGL(1,𝔽5)—42no—
193C3×AGL(1,𝔽5)—81no—
194C3×C4×C5—21no—
195C3×C4×C5—42no—
196C3×Dic5—21no—
197C3×Dic5—42no—
198C24×C3—51no—
199C24×C3—51no—
200C2×C3×D4—51no—
201C2×C3×D4—51no—
202C2×C3×D4—51no—
203C2×C3×D4—102no—
204C2×C3×D4—102no—
205C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
206C22×C3×C4—51no—
207C22×C3×C4—52no—
208C22×C3×C4—52no—
209C22×C3×C4—102no—
210C22×C3×C4—101no—
211C22×C3×C4—102no—
212C22×C3×C4—101no—
213C22×C3×C4—101no—
214C22×C3×C4—101no—
215C22×C3×C4—202no—
216C3×C4⋊C4—101no—
217C3×C4⋊C4—101no—
218C3×C4⋊C4—204no—
219C3×C22⋊C4—52no—
220C3×C22⋊C4—102no—
221C3×C22⋊C4—102no—
222C3×C22⋊C4—102no—
223C3×C22⋊C4—102no—
224C3×C22⋊C4—102no—
225C3×C22⋊C4—202no—
226C3×C22⋊C4C2×AGL(1,𝔽5)12yescomplemented
227C3×C42—102no—
228C23×C5C3×C4⋊C411yescharacteristic, complemented
229C23×C5C22×C3×C411yescharacteristic
230C22×D5—21no—
231C22×D5—21no—
232C22×D5—41no—
233C22×D5—41no—
234C22×D5—41no—
235C22×D5—41no—
236C22×D5C3×C4⋊C411yescharacteristic, complemented
237C22×D5C2×C3×D412yes—
238C22×D5C2×C3×D411yescharacteristic
239C22×D5C2×C3×Q811yescharacteristic
240C22×D5C3×#4112yes—
241C2×AGL(1,𝔽5)—24no—
242C2×AGL(1,𝔽5)—22no—
243C2×AGL(1,𝔽5)—22no—
244C2×AGL(1,𝔽5)—22no—
245C2×AGL(1,𝔽5)—42no—
246C2×AGL(1,𝔽5)—41no—
247C2×AGL(1,𝔽5)—41no—
248C2×AGL(1,𝔽5)—42no—
249C2×AGL(1,𝔽5)—42no—
250C2×AGL(1,𝔽5)—41no—
251C2×AGL(1,𝔽5)—41no—
252C2×AGL(1,𝔽5)—41no—
253C2×AGL(1,𝔽5)—81no—
254C2×AGL(1,𝔽5)—81no—
255C5×D4—22no—
256C2×C4×C5—22no—
257C2×C4×C5C3×C4⋊C411yescharacteristic, complemented
258C22⋊D5—22no—
259C22⋊D5—22no—
260C2×Dic5—22no—
261C2×Dic5C3×C4⋊C411yescharacteristic, complemented
262D20—22no—
263C4×D5—21no—
264C4×D5—21no—
265C4×D5—42no—
266C4×D5—42no—
267C22×D4—51no—
268C23×C4—51no—
269C4×D4—102no—
270C4×D4—102no—
271#117—101no—
272C2×C4⋊C4—102no—
273C2×C22⋊C4—52no—
274C2×C22⋊C4—52no—
275C2×C22⋊C4—51no—
276C2×C22⋊C4—101no—
277C2×C22⋊C4—101no—
278#99—51no—
279#99—52no—
280#99C3×AGL(1,𝔽5)11yescharacteristic, complemented
281#95—102no—
282#95—101no—
283C2×C3×C5—22no—
284C2×C3×C5—21no—
285C2×C3×C5—41no—
286C2×C3×C5#38511yescharacteristic
287C3×D5—21no—
288C3×D5—22no—
289C3×D5—41no—
290C3×D5B(4,𝔽2)11yescharacteristic
291C3×D5#45811yescharacteristic
292C23×C3—51no—
293C23×C3—52no—
294C23×C3—51no—
295C23×C3—52no—
296C23×C3—51no—
297C23×C3—101no—
298C23×C3—101no—
299C23×C3—201no—
300C23×C3—201no—
301C23×C3—201no—
302C23×C3—201no—
303C23×C3C4⋊AGL(1,𝔽5)11yescharacteristic, complemented
304C23×C3C22×AGL(1,𝔽5)11yescharacteristic
305C3×D4—22no—
306C3×D4—102no—
307C3×D4—102no—
308C3×D4—102no—
309C2×C3×C4—22no—
310C2×C3×C4—51no—
311C2×C3×C4—101no—
312C2×C3×C4—101no—
313C2×C3×C4—102no—
314C2×C3×C4—102no—
315C2×C3×C4—102no—
316C2×C3×C4—104no—
317C2×C3×C4—102no—
318C2×C3×C4—202no—
319C2×C3×C4—202no—
320C2×C3×C4—201no—
321C2×C3×C4—202no—
322C2×C3×C4—201no—
323C2×C3×C4—202no—
324C2×C3×C4—202no—
325C2×C3×C4—201no—
326C2×C3×C4—201no—
327C2×C3×C4—201no—
328C2×C3×C4—401no—
329C2×C3×C4—401no—
330C2×C3×C4C4⋊AGL(1,𝔽5)11yescharacteristic, complemented
331C22×C5—21no—
332C22×C5—41no—
333C22×C5—41no—
334C22×C5C2×C3×C4⋊C411yescharacteristic
335C22×C5C3×C4×D412yes—
336C2×D5—22no—
337C2×D5—21no—
338C2×D5—22no—
339C2×D5—21no—
340C2×D5—21no—
341C2×D5—41no—
342C2×D5—42no—
343C2×D5—41no—
344C2×D5—41no—
345C2×D5—41no—
346C2×D5—81no—
347C2×D5C3×#12011yescharacteristic
348C2×D5C3×C22⋊Q812yes—
349C2×D5C3×#12311yescharacteristic
350AGL(1,𝔽5)—42no—
351AGL(1,𝔽5)—42no—
352AGL(1,𝔽5)—42no—
353AGL(1,𝔽5)—81no—
354C4×C5—21no—
355C4×C5—42no—
356Dic5—21no—
357Dic5—42no—
358C24—51no—
359C24—51no—
360C2×D4—51no—
361C2×D4—51no—
362C2×D4—51no—
363C2×D4—102no—
364C2×D4—102no—
365C2×D4C2×C3×AGL(1,𝔽5)11yescharacteristic
366C22×C4—51no—
367C22×C4—52no—
368C22×C4—52no—
369C22×C4—102no—
370C22×C4—101no—
371C22×C4—102no—
372C22×C4—101no—
373C22×C4—101no—
374C22×C4—101no—
375C22×C4—202no—
376C4⋊C4—101no—
377C4⋊C4—101no—
378C4⋊C4—204no—
379C22⋊C4—52no—
380C22⋊C4—102no—
381C22⋊C4—102no—
382C22⋊C4—102no—
383C22⋊C4—102no—
384C22⋊C4—102no—
385C22⋊C4—202no—
386C22⋊C4C2×C3×AGL(1,𝔽5)12yescomplemented
387C42—102no—
388C3×C5#186711yescharacteristic, complemented, Hall
389C22×C3—21no—
390C22×C3—41no—
391C22×C3—41no—
392C22×C3—51no—
393C22×C3—52no—
394C22×C3—51no—
395C22×C3—102no—
396C22×C3—101no—
397C22×C3—102no—
398C22×C3—101no—
399C22×C3—101no—
400C22×C3—202no—
401C22×C3—201no—
402C22×C3—201no—
403C22×C3—201no—
404C22×C3—201no—
405C22×C3—401no—
406C22×C3C2×C4⋊AGL(1,𝔽5)11yescharacteristic
407C22×C3D4×AGL(1,𝔽5)12yes—
408C3×C4—21no—
409C3×C4—42no—
410C3×C4—101no—
411C3×C4—202no—
412C3×C4—202no—
413C3×C4—202no—
414C3×C4—202no—
415C3×C4—401no—
416C2×C5—22no—
417C2×C5—21no—
418C2×C5—41no—
419C2×C5C3×#38511yescharacteristic
420D5—21no—
421D5—22no—
422D5—41no—
423D5C3×B(4,𝔽2)11yescharacteristic
424D5C3×#45811yescharacteristic
425C23—51no—
426C23—52no—
427C23—51no—
428C23—52no—
429C23—51no—
430C23—101no—
431C23—101no—
432C23—201no—
433C23—201no—
434C23—201no—
435C23—201no—
436C23C3×C4⋊AGL(1,𝔽5)11yescharacteristic, complemented
437C23C22×C3×AGL(1,𝔽5)11yescharacteristic
438D4—22no—
439D4—102no—
440D4—102no—
441D4—102no—
442C2×C4—22no—
443C2×C4—51no—
444C2×C4—101no—
445C2×C4—101no—
446C2×C4—102no—
447C2×C4—102no—
448C2×C4—102no—
449C2×C4—104no—
450C2×C4—102no—
451C2×C4—202no—
452C2×C4—202no—
453C2×C4—201no—
454C2×C4—202no—
455C2×C4—201no—
456C2×C4—202no—
457C2×C4—202no—
458C2×C4—201no—
459C2×C4—201no—
460C2×C4—201no—
461C2×C4—401no—
462C2×C4—401no—
463C2×C4C3×C4⋊AGL(1,𝔽5)11yescharacteristic, complemented
464C2×C3—22no—
465C2×C3—21no—
466C2×C3—41no—
467C2×C3—51no—
468C2×C3—51no—
469C2×C3—101no—
470C2×C3—102no—
471C2×C3—201no—
472C2×C3#6569911yescharacteristic
473C5C3×#186711yescharacteristic, complemented, Sylow
474C22—21no—
475C22—41no—
476C22—41no—
477C22—51no—
478C22—51no—
479C22—52no—
480C22—102no—
481C22—101no—
482C22—102no—
483C22—101no—
484C22—101no—
485C22—202no—
486C22—201no—
487C22—201no—
488C22—201no—
489C22—201no—
490C22—401no—
491C22C2×C3×C4⋊AGL(1,𝔽5)11yescharacteristic
492C22C3×D4×AGL(1,𝔽5)12yes—
493C4—21no—
494C4—42no—
495C4—101no—
496C4—202no—
497C4—202no—
498C4—202no—
499C4—202no—
500C4—401no—
501C3#1060391911yescharacteristic, direct product factor, Sylow
502C2—22no—
503C2—21no—
504C2—41no—
505C2—51no—
506C2—51no—
507C2—101no—
508C2—102no—
509C2—201no—
510C2C3×#6569911yescharacteristic
subgroupG/H
centerC2×C3#65699
commutatorC23×C5C22×C3×C4
FrattiniC23C22×C3×AGL(1,𝔽5)
FittingC3×C5×#99C4
socleC2×C3×C5#385
prime radicalC2×C3×D4×D5C22
cosocleC23×D5C23×C3
hypercenterC3×#99AGL(1,𝔽5)
hypocenterC5C3×#1867
LCSG ▷ C23×C5 ▷ C2×C5 ▷ C5
UCSC1 ◁ C2×C3 ◁ C23×C3 ◁ C3×#99

Complements

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

NH ≅ G/Ntype
3. C3×#65699C2semidirect
5. C3×#64866C2semidirect
6. #10603919C3direct (N × H)
20. C3×C5×#99C4semidirect
21. C3×#3966C4semidirect
22. C3×#3938C4semidirect
26. #65699C2×C3semidirect
28. #64866C2×C3semidirect
56. C3×C5×C22⋊C4C2×C4semidirect
57. C3×C22⋊Dic5C2×C4semidirect
77. C5×#99C3×C4semidirect
78. #3966C3×C4semidirect
79. #3938C3×C4semidirect
81. C23×C3×C5C4⋊C4semidirect
89. C22×C3×D5C4⋊C4semidirect
110. C2×C3×C4×C5C4⋊C4semidirect
114. C2×C3×Dic5C4⋊C4semidirect
133. C3×#99AGL(1,𝔽5)semidirect
163. C5×C22⋊C4C2×C3×C4semidirect
164. C22⋊Dic5C2×C3×C4semidirect
226. C3×C22⋊C4C2×AGL(1,𝔽5)semidirect
228. C23×C5C3×C4⋊C4semidirect
236. C22×D5C3×C4⋊C4semidirect
257. C2×C4×C5C3×C4⋊C4semidirect
261. C2×Dic5C3×C4⋊C4semidirect
280. #99C3×AGL(1,𝔽5)semidirect
303. C23×C3C4⋊AGL(1,𝔽5)semidirect
330. C2×C3×C4C4⋊AGL(1,𝔽5)semidirect
386. C22⋊C4C2×C3×AGL(1,𝔽5)semidirect
388. C3×C5#1867semidirect
436. C23C3×C4⋊AGL(1,𝔽5)semidirect
463. C2×C4C3×C4⋊AGL(1,𝔽5)semidirect
473. C5C3×#1867semidirect
501. C3#10603919direct (N × H)

Maximal subgroups

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

subgroupindex
C3×#662582
C3×#657792
C3×#656992
C3×D5×#992
C3×#648662
#106039193
C3×#18675