Finite Group Databasebeta

← C3×#10606109 #14321076

Subgroups — C3×#10606109 #14321076

Order 1920.

1312 conjugacy classes of subgroups.

The subgroup lattice has 536 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×C4×Dic5C211yesmaximal, characteristic
2C2×C3×#4023C211yesmaximal, characteristic
3C2×C3×#4023C211yesmaximal, characteristic
4C3×C5×#385C211yesmaximal, characteristic
5C3×#65499C211yesmaximal, characteristic
6C3×C4⋊C4×Dic5C212yesmaximal, complemented
7C3×#65261C212yesmaximal, complemented
8C3×#65239C211yesmaximal, characteristic
9C3×#65235C211yesmaximal, characteristic
10C3×#64938C212yesmaximal, complemented
11C3×#64936C212yesmaximal, complemented
12#10606109C311yesmaximal, characteristic, direct product factor, Hall
13C23×C3×C4×C5C2211yescharacteristic
14C23×C3×Dic5C2211yescharacteristic
15C23×C3×Dic5C2211yescharacteristic
16C2×C3×C5×C4⋊C4C2212yes—
17C2×C3×C5×C22⋊C4C2211yescharacteristic
18C2×C3×C5×C22⋊C4C2211yescharacteristic
19C2×C3×C22⋊Dic5C2211yescharacteristic
20C2×C3×C22⋊Dic5C2211yescharacteristic
21C2×C3×C4⋊Dic5C2212yes—
22C2×C3×#683C2211yescharacteristic
23C2×C3×#683C2212yes—
24C2×C3×#683C2212yes—
25C2×C3×#683C2211yescharacteristic
26C2×C3×#683C2212yes—
27C2×C3×C4×Dic5—24no—
28C2×C3×C4×Dic5C2212yes—
29C2×C3×C4×Dic5C2212yes—
30C2×C3×C4×Dic5C2211yescharacteristic
31C2×C3×C4×Dic5C2211yescharacteristic
32C2×C3×C4×Dic5C2212yes—
33C3×#4023—24no—
34C3×#4023C418yescomplemented
35C3×C5×#95C2212yes—
36C3×#3963C2212yes—
37C3×#3963C2212yes—
38C3×#3963C2212yes—
39C3×#2296—51nomaximal, Hall
40C22×C4×Dic5C2×C311yescharacteristic
41C2×#4023C2×C311yescharacteristic
42C2×#4023C2×C311yescharacteristic
43C5×#385C2×C311yescharacteristic
44#65499C2×C311yescharacteristic
45C4⋊C4×Dic5C2×C312yescomplemented
46#65261C2×C312yescomplemented
47#65239C2×C311yescharacteristic
48#65235C2×C311yescharacteristic
49#64938C2×C312yescomplemented
50#64936C2×C312yescomplemented
51C24×C3×C5C2311yescharacteristic
52C22×C3×C4×C5—22no—
53C22×C3×C4×C5—24no—
54C22×C3×C4×C5C2311yescharacteristic
55C22×C3×C4×C5C2311yescharacteristic
56C22×C3×C4×C5C2312yes—
57C22×C3×C4×C5C2312yes—
58C22×C3×Dic5—22no—
59C22×C3×Dic5—24no—
60C22×C3×Dic5C2×C414yes—
61C22×C3×Dic5D414yes—
62C22×C3×Dic5Q814yes—
63C22×C3×Dic5C2311yescharacteristic
64C22×C3×Dic5C2311yescharacteristic
65C22×C3×Dic5C2312yes—
66C22×C3×Dic5C2312yes—
67C22×C3×Dic5C2311yescharacteristic
68C22×C3×Dic5C2311yescharacteristic
69C3×C5×C4⋊C4—24no—
70C3×C5×C22⋊C4—22no—
71C3×C5×C22⋊C4C2×C414yes—
72C3×C22⋊Dic5—22no—
73C3×C22⋊Dic5C2×C414yes—
74C3×C4⋊Dic5—24no—
75C3×#683—24no—
76C3×#683—22no—
77C3×#683—22no—
78C3×#683—24no—
79C3×#683C2×C418yescomplemented
80C3×C4×Dic5—22no—
81C3×C4×Dic5—22no—
82C3×C4×Dic5—24no—
83C3×C4×Dic5—24no—
84C3×C4×Dic5C2×C418yescomplemented
85C2×C3×#117—51no—
86C2×C3×#117—51no—
87C22×C3×C42—51no—
88C3×#389—52no—
89C3×#389—52no—
90C3×#385—51no—
91C3×#385—51no—
92C3×#385—51no—
93C3×#385D511yescharacteristic
94C3×C4×C4⋊C4—52no—
95C3×C4×C4⋊C4—52no—
96C23×C4×C5C22×C311yescharacteristic
97C23×Dic5C22×C311yescharacteristic
98C23×Dic5C22×C311yescharacteristic
99C2×C5×C4⋊C4C22×C312yes—
100C2×C5×C22⋊C4C22×C311yescharacteristic
101C2×C5×C22⋊C4C22×C311yescharacteristic
102C2×C22⋊Dic5C22×C311yescharacteristic
103C2×C22⋊Dic5C22×C311yescharacteristic
104C2×C4⋊Dic5C22×C312yes—
105C2×#683C22×C311yescharacteristic
106C2×#683C22×C311yescharacteristic
107C2×#683C22×C312yes—
108C2×#683C22×C312yes—
109C2×#683C22×C312yes—
110C2×C4×Dic5—24no—
111C2×C4×Dic5C22×C311yescharacteristic
112C2×C4×Dic5C22×C311yescharacteristic
113C2×C4×Dic5C22×C312yes—
114C2×C4×Dic5C22×C312yes—
115C2×C4×Dic5C22×C312yes—
116#4023—24no—
117#4023C3×C418yescomplemented
118C5×#95C22×C312yes—
119#3963C22×C312yes—
120#3963C22×C312yes—
121#3963C22×C312yes—
122#2296—51noSylow
123C23×C3×C5—22no—
124C23×C3×C5—22no—
125C23×C3×C5C22×C412yes—
126C23×C3×C5C2×D412yes—
127C23×C3×C5C2×Q812yes—
128C23×C3×C5C2411yescharacteristic
129C2×C3×C4×C5—28no—
130C2×C3×C4×C5—22no—
131C2×C3×C4×C5—22no—
132C2×C3×C4×C5—22no—
133C2×C3×C4×C5—22no—
134C2×C3×C4×C5—21no—
135C2×C3×C4×C5—21no—
136C2×C3×C4×C5—22no—
137C2×C3×C4×C5—22no—
138C2×C3×C4×C5C22×C414yes—
139C2×C3×C4×C5#4114yes—
140C2×C3×Dic5—28no—
141C2×C3×Dic5—22no—
142C2×C3×Dic5—22no—
143C2×C3×Dic5—22no—
144C2×C3×Dic5—22no—
145C2×C3×Dic5—21no—
146C2×C3×Dic5—21no—
147C2×C3×Dic5—22no—
148C2×C3×Dic5—22no—
149C2×C3×Dic5C4⋊C4116yescomplemented
150C2×C3×Dic5C22×C414yes—
151C2×C3×Dic5C22×C412yes—
152C2×C3×Dic5C22×C412yes—
153C2×C3×Dic5C2×D412yes—
154C2×C3×Dic5C2×D412yes—
155C2×C3×Dic5C2×Q812yes—
156C2×C3×Dic5C2×Q812yes—
157C2×C3×Dic5#4114yes—
158C23×C3×C4—51no—
159C23×C3×C4—51no—
160C23×C3×C4C2×D511yescharacteristic
161C3×#117—58no—
162C3×#117—104no—
163C2×C3×C4⋊C4—52no—
164C2×C3×C4⋊C4—52no—
165C2×C3×C4⋊C4—52no—
166C2×C3×C4⋊C4—52no—
167C2×C3×C4⋊C4—51no—
168C2×C3×C4⋊C4—51no—
169C2×C3×C4⋊C4C2×D512yes—
170C2×C3×C22⋊C4—51no—
171C2×C3×C22⋊C4—51no—
172C2×C3×C22⋊C4C2×D511yescharacteristic
173C2×C3×C22⋊C4C2×D511yescharacteristic
174C2×C3×C42—52no—
175C2×C3×C42—52no—
176C2×C3×C42—51no—
177C2×C3×C42—51no—
178C2×C3×C42—52no—
179C2×C3×C42—104no—
180C3×#95—52no—
181C3×#95—52no—
182C3×#95—52no—
183C3×#95C2×D512yes—
184C24×C5C23×C311yescharacteristic
185C22×C4×C5—22no—
186C22×C4×C5—24no—
187C22×C4×C5C23×C311yescharacteristic
188C22×C4×C5C23×C311yescharacteristic
189C22×C4×C5C23×C312yes—
190C22×C4×C5C23×C312yes—
191C22×Dic5—22no—
192C22×Dic5—24no—
193C22×Dic5C2×C3×C414yes—
194C22×Dic5C3×D414yes—
195C22×Dic5C3×Q814yes—
196C22×Dic5C23×C311yescharacteristic
197C22×Dic5C23×C311yescharacteristic
198C22×Dic5C23×C312yes—
199C22×Dic5C23×C312yes—
200C22×Dic5C23×C311yescharacteristic
201C22×Dic5C23×C311yescharacteristic
202C5×C4⋊C4—24no—
203C5×C22⋊C4—22no—
204C5×C22⋊C4C2×C3×C414yes—
205C22⋊Dic5—22no—
206C22⋊Dic5C2×C3×C414yes—
207C4⋊Dic5—24no—
208#683—24no—
209#683—22no—
210#683—22no—
211#683—24no—
212#683C2×C3×C418yescomplemented
213C4×Dic5—22no—
214C4×Dic5—22no—
215C4×Dic5—24no—
216C4×Dic5—24no—
217C4×Dic5C2×C3×C418yescomplemented
218C2×#117—51no—
219C2×#117—51no—
220C22×C42—51no—
221#389—52no—
222#389—52no—
223#385—51no—
224#385—51no—
225#385—51no—
226#385C3×D511yescharacteristic
227C4×C4⋊C4—52no—
228C4×C4⋊C4—52no—
229C22×C3×C5—24no—
230C22×C3×C5—22no—
231C22×C3×C5—22no—
232C22×C3×C5—24no—
233C22×C3×C5C2×C4⋊C414yes—
234C22×C3×C5C23×C411yescharacteristic
235C22×C3×C5C22×D411yescharacteristic
236C22×C3×C5C22×Q811yescharacteristic
237C22×C3×C5C2×#4112yes—
238C22×C3×C5C2×#4112yes—
239C3×C4×C5—24no—
240C3×C4×C5—24no—
241C3×C4×C5—41no—
242C3×C4×C5—41no—
243C3×Dic5—24no—
244C3×Dic5—24no—
245C3×Dic5—41no—
246C3×Dic5—41no—
247C3×Dic5C2×C4⋊C414yes—
248C3×Dic5C2×C4⋊C414yes—
249C24×C3C22×D511yescharacteristic
250C22×C3×C4—22no—
251C22×C3×C4—24no—
252C22×C3×C4—54no—
253C22×C3×C4—54no—
254C22×C3×C4—54no—
255C22×C3×C4—51no—
256C22×C3×C4—51no—
257C22×C3×C4—52no—
258C22×C3×C4—52no—
259C22×C3×C4—51no—
260C22×C3×C4—51no—
261C22×C3×C4—104no—
262C22×C3×C4—102no—
263C22×C3×C4C22×D511yescharacteristic
264C22×C3×C4C22×D511yescharacteristic
265C22×C3×C4C22×D512yes—
266C22×C3×C4C22×D512yes—
267C3×C4⋊C4—24no—
268C3×C4⋊C4—58no—
269C3×C4⋊C4—104no—
270C3×C4⋊C4—102no—
271C3×C4⋊C4—102no—
272C3×C4⋊C4—104no—
273C3×C4⋊C4—104no—
274C3×C22⋊C4—22no—
275C3×C22⋊C4—54no—
276C3×C22⋊C4—102no—
277C3×C22⋊C4C4×D514yes—
278C3×C42—58no—
279C3×C42—104no—
280C3×C42—104no—
281C3×C42—102no—
282C3×C42—102no—
283C23×C5—22no—
284C23×C5—22no—
285C23×C5C22×C3×C412yes—
286C23×C5C2×C3×D412yes—
287C23×C5C2×C3×Q812yes—
288C23×C5C24×C311yescharacteristic
289C2×C4×C5—28no—
290C2×C4×C5—22no—
291C2×C4×C5—22no—
292C2×C4×C5—22no—
293C2×C4×C5—22no—
294C2×C4×C5—21no—
295C2×C4×C5—21no—
296C2×C4×C5—22no—
297C2×C4×C5—22no—
298C2×C4×C5C22×C3×C414yes—
299C2×C4×C5C3×#4114yes—
300C2×Dic5—28no—
301C2×Dic5—22no—
302C2×Dic5—22no—
303C2×Dic5—22no—
304C2×Dic5—22no—
305C2×Dic5—21no—
306C2×Dic5—21no—
307C2×Dic5—22no—
308C2×Dic5—22no—
309C2×Dic5C3×C4⋊C4116yescomplemented
310C2×Dic5C22×C3×C414yes—
311C2×Dic5C22×C3×C412yes—
312C2×Dic5C22×C3×C412yes—
313C2×Dic5C2×C3×D412yes—
314C2×Dic5C2×C3×D412yes—
315C2×Dic5C2×C3×Q812yes—
316C2×Dic5C2×C3×Q812yes—
317C2×Dic5C3×#4114yes—
318C23×C4—51no—
319C23×C4—51no—
320C23×C4C2×C3×D511yescharacteristic
321#117—58no—
322#117—104no—
323C2×C4⋊C4—52no—
324C2×C4⋊C4—51no—
325C2×C4⋊C4—51no—
326C2×C4⋊C4—52no—
327C2×C4⋊C4—52no—
328C2×C4⋊C4—52no—
329C2×C4⋊C4C2×C3×D512yes—
330C2×C22⋊C4—51no—
331C2×C22⋊C4—51no—
332C2×C22⋊C4C2×C3×D511yescharacteristic
333C2×C22⋊C4C2×C3×D511yescharacteristic
334C2×C42—52no—
335C2×C42—51no—
336C2×C42—51no—
337C2×C42—52no—
338C2×C42—52no—
339C2×C42—104no—
340#95—52no—
341#95—52no—
342#95—52no—
343#95C2×C3×D512yes—
344C2×C3×C5—24no—
345C2×C3×C5C22×C4⋊C411yescharacteristic
346C2×C3×C5C4×#4112yes—
347C2×C3×C5#52511yescharacteristic
348C2×C3×C5#52511yescharacteristic
349C2×C3×C5#53312yes—
350C23×C3—22no—
351C23×C3—22no—
352C23×C3C2×C4×D512yes—
353C23×C3D4×D512yes—
354C23×C3Q8×D512yes—
355C23×C3C23×D511yescharacteristic
356C2×C3×C4—28no—
357C2×C3×C4—22no—
358C2×C3×C4—22no—
359C2×C3×C4—22no—
360C2×C3×C4—22no—
361C2×C3×C4—21no—
362C2×C3×C4—21no—
363C2×C3×C4—22no—
364C2×C3×C4—22no—
365C2×C3×C4—52no—
366C2×C3×C4—52no—
367C2×C3×C4—52no—
368C2×C3×C4—52no—
369C2×C3×C4—52no—
370C2×C3×C4—52no—
371C2×C3×C4—516no—
372C2×C3×C4—54no—
373C2×C3×C4—54no—
374C2×C3×C4—108no—
375C2×C3×C4—102no—
376C2×C3×C4—102no—
377C2×C3×C4—102no—
378C2×C3×C4—102no—
379C2×C3×C4—101no—
380C2×C3×C4—101no—
381C2×C3×C4—102no—
382C2×C3×C4—102no—
383C2×C3×C4C2×C4×D514yes—
384C2×C3×C4#71114yes—
385C22×C5—24no—
386C22×C5—22no—
387C22×C5—22no—
388C22×C5—24no—
389C22×C5C2×C3×C4⋊C414yes—
390C22×C5C23×C3×C411yescharacteristic
391C22×C5C22×C3×D411yescharacteristic
392C22×C5C22×C3×Q811yescharacteristic
393C22×C5C2×C3×#4112yes—
394C22×C5C2×C3×#4112yes—
395C4×C5—24no—
396C4×C5—24no—
397C4×C5—41no—
398C4×C5—41no—
399Dic5—24no—
400Dic5—24no—
401Dic5—41no—
402Dic5—41no—
403Dic5C2×C3×C4⋊C414yes—
404Dic5C2×C3×C4⋊C414yes—
405C24C22×C3×D511yescharacteristic
406C22×C4—22no—
407C22×C4—24no—
408C22×C4—54no—
409C22×C4—54no—
410C22×C4—54no—
411C22×C4—51no—
412C22×C4—51no—
413C22×C4—52no—
414C22×C4—52no—
415C22×C4—51no—
416C22×C4—51no—
417C22×C4—104no—
418C22×C4—102no—
419C22×C4C22×C3×D511yescharacteristic
420C22×C4C22×C3×D511yescharacteristic
421C22×C4C22×C3×D512yes—
422C22×C4C22×C3×D512yes—
423C4⋊C4—24no—
424C4⋊C4—58no—
425C4⋊C4—102no—
426C4⋊C4—102no—
427C4⋊C4—104no—
428C4⋊C4—104no—
429C4⋊C4—104no—
430C22⋊C4—22no—
431C22⋊C4—54no—
432C22⋊C4—102no—
433C22⋊C4C3×C4×D514yes—
434C42—58no—
435C42—104no—
436C42—104no—
437C42—102no—
438C42—102no—
439C3×C5#229611yescharacteristic, complemented, Hall
440C22×C3—24no—
441C22×C3—22no—
442C22×C3—24no—
443C22×C3—22no—
444C22×C3D5×C4⋊C414yes—
445C22×C3C22×C4×D511yescharacteristic
446C22×C3C2×D4×D511yescharacteristic
447C22×C3C2×#71112yes—
448C22×C3C2×Q8×D511yescharacteristic
449C22×C3D5×#4112yes—
450C3×C4—24no—
451C3×C4—24no—
452C3×C4—41no—
453C3×C4—41no—
454C3×C4—54no—
455C3×C4—54no—
456C3×C4—104no—
457C3×C4—104no—
458C3×C4—201no—
459C3×C4—201no—
460C2×C5—24no—
461C2×C5C22×C3×C4⋊C411yescharacteristic
462C2×C5C3×C4×#4112yes—
463C2×C5C3×#52511yescharacteristic
464C2×C5C3×#52511yescharacteristic
465C2×C5C3×#53312yes—
466C23—22no—
467C23—22no—
468C23C2×C3×C4×D512yes—
469C23C3×D4×D512yes—
470C23C3×Q8×D512yes—
471C23C23×C3×D511yescharacteristic
472C2×C4—28no—
473C2×C4—22no—
474C2×C4—22no—
475C2×C4—22no—
476C2×C4—22no—
477C2×C4—21no—
478C2×C4—21no—
479C2×C4—22no—
480C2×C4—22no—
481C2×C4—52no—
482C2×C4—52no—
483C2×C4—52no—
484C2×C4—52no—
485C2×C4—52no—
486C2×C4—52no—
487C2×C4—516no—
488C2×C4—54no—
489C2×C4—54no—
490C2×C4—108no—
491C2×C4—102no—
492C2×C4—102no—
493C2×C4—102no—
494C2×C4—102no—
495C2×C4—101no—
496C2×C4—101no—
497C2×C4—102no—
498C2×C4—102no—
499C2×C4C2×C3×C4×D514yes—
500C2×C4C3×#71114yes—
501C2×C3—24no—
502C2×C3C2×D5×C4⋊C411yescharacteristic
503C2×C3C4×#71112yes—
504C2×C3#6592011yescharacteristic
505C2×C3#6595612yes—
506C2×C3#6599011yescharacteristic
507C5C3×#229611yescharacteristic, complemented, Sylow
508C22—24no—
509C22—22no—
510C22—24no—
511C22—22no—
512C22C3×D5×C4⋊C414yes—
513C22C22×C3×C4×D511yescharacteristic
514C22C2×C3×D4×D511yescharacteristic
515C22C2×C3×#71112yes—
516C22C2×C3×Q8×D511yescharacteristic
517C22C3×D5×#4112yes—
518C4—24no—
519C4—24no—
520C4—41no—
521C4—41no—
522C4—54no—
523C4—54no—
524C4—104no—
525C4—104no—
526C4—201no—
527C4—201no—
528C3#1060610911yescharacteristic, direct product factor, Sylow
529C2—24no—
530C2C2×C3×D5×C4⋊C411yescharacteristic
531C2C3×C4×#71112yes—
532C2C3×#6592011yescharacteristic
533C2C3×#6595612yes—
534C2C3×#6599011yescharacteristic
subgroupG/H
centerC23×C3C23×D5
commutatorC22×C5C23×C3×C4
FrattiniC23C23×C3×D5
FittingC3×C5×#385C2
socleC23×C3×C5C24
prime radicalC24×C3×C5C23
cosocleC23×C5C24×C3
hypercenterC3×#385D5
hypocenterC5C3×#2296
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C3×#385

Complements

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

NH ≅ G/Ntype
6. C3×C4⋊C4×Dic5C2semidirect
7. C3×#65261C2semidirect
10. C3×#64938C2semidirect
11. C3×#64936C2semidirect
12. #10606109C3direct (N × H)
34. C3×#4023C4semidirect
45. C4⋊C4×Dic5C2×C3semidirect
46. #65261C2×C3semidirect
49. #64938C2×C3semidirect
50. #64936C2×C3semidirect
79. C3×#683C2×C4semidirect
84. C3×C4×Dic5C2×C4semidirect
117. #4023C3×C4semidirect
149. C2×C3×Dic5C4⋊C4semidirect
212. #683C2×C3×C4semidirect
217. C4×Dic5C2×C3×C4semidirect
309. C2×Dic5C3×C4⋊C4semidirect
439. C3×C5#2296semidirect
507. C5C3×#2296semidirect
528. C3#10606109direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×C4×Dic52
C2×C3×#40232
C3×C5×#3852
C3×#654992
C3×C4⋊C4×Dic52
C3×#652612
C3×#652392
C3×#652352
C3×#649382
C3×#649362
#106061093
C3×#22965