Finite Group Databasebeta

← C3×#10614219 #14323691

Subgroups — C3×#10614219 #14323691

Order 1920.

1112 conjugacy classes of subgroups.

The subgroup lattice has 576 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C4×C5×Q8C211yesmaximal, characteristic, complemented
2C2×C3×#4092C211yesmaximal, characteristic
3C2×C3×#4018C211yesmaximal, characteristic
4C3×#65519C211yesmaximal, characteristic, complemented
5C3×#65279C212yesmaximal
6C3×#65276C211yesmaximal, characteristic
7C3×#65262C211yesmaximal, characteristic, complemented
8C3×#65261C212yesmaximal, complemented
9C3×#65229C212yesmaximal
10C3×C4×C4⋊Dic5C211yesmaximal, characteristic, complemented
11C3×#65222C212yesmaximal, complemented
12#10614219C311yesmaximal, characteristic, direct product factor, Hall
13C22×C3×C5×Q8C2211yescharacteristic
14C22×C3×C4×D5C2211yescharacteristic
15C3×C4×C5×Q8—24no—
16C2×C3×C5×C4⋊C4C2212yes—
17C2×C3×C5×C4⋊C4C2211yescharacteristic
18C2×C3×C42×C5C2212yes—
19C2×C3×C42×C5C2211yescharacteristic
20C3×#4092C418yescomplemented
21C2×C3×#685C2211yescharacteristic
22C2×C3×#685C2211yescharacteristic
23C2×C3×#685C2212yes—
24C2×C3×#685C2212yes—
25C2×C3×C4⋊Dic5C2211yescharacteristic
26C2×C3×C4⋊Dic5C2211yescharacteristic
27C2×C3×#683C2211yescharacteristic
28C2×C3×#683C2211yescharacteristic
29C2×C3×#683C2212yes—
30C2×C3×#683C2212yes—
31C2×C3×C4×Dic5C2211yescharacteristic
32C2×C3×C4×Dic5C2212yes—
33C3×#4018—24no—
34C3×#3963C2212yes—
35C3×#3963C2212yes—
36C3×#3963C2212yes—
37C3×#3963C2212yes—
38C3×#3963C2212yes—
39C3×#2323—51nomaximal, Hall
40C2×C4×C5×Q8C2×C311yescharacteristic, complemented
41C2×#4092C2×C311yescharacteristic
42C2×#4018C2×C311yescharacteristic
43#65519C2×C311yescharacteristic, complemented
44#65279C2×C312yes—
45#65276C2×C311yescharacteristic
46#65262C2×C311yescharacteristic, complemented
47#65261C2×C312yescomplemented
48#65229C2×C312yes—
49C4×C4⋊Dic5C2×C311yescharacteristic, complemented
50#65222C2×C312yescomplemented
51C23×C3×D5C2311yescharacteristic
52C2×C3×C5×Q8—22no—
53C2×C3×C5×Q8—22no—
54C2×C3×C5×Q8C2×C414yes—
55C22×C3×C4×C5C2312yes—
56C22×C3×C4×C5C2312yes—
57C22×C3×C4×C5C2311yescharacteristic
58C22×C3×C4×C5C2311yescharacteristic
59C22×C3×C4×C5C2311yescharacteristic
60C22×C3×Dic5C2311yescharacteristic
61C22×C3×Dic5C2312yes—
62C22×C3×Dic5C2311yescharacteristic
63C22×C3×Dic5C2311yescharacteristic
64C22×C3×Dic5C2312yes—
65C2×C3×C4×D5—22no—
66C2×C3×C4×D5—22no—
67C2×C3×C4×D5C2×C414yes—
68C3×C5×C4⋊C4—24no—
69C3×C5×C4⋊C4—22no—
70C3×C42×C5—24no—
71C3×C42×C5D414yes—
72C3×#685—22no—
73C3×#685—22no—
74C3×#685—42no—
75C3×#685C2×C418yes—
76C3×C4⋊Dic5—22no—
77C3×C4⋊Dic5C2×C414yes—
78C3×#683—22no—
79C3×#683—22no—
80C3×#683—42no—
81C3×#683C2×C418yes—
82C3×C4×Dic5—22no—
83C3×C4×Dic5—42no—
84C2×C3×C22⋊Q8—51no—
85C2×C3×C4×Q8D511yescharacteristic, complemented
86C2×C3×#117—51no—
87C3×#391—51no—
88C3×#389—52no—
89C3×#389—51no—
90C3×#388—52no—
91C3×#388—52no—
92C3×#387—52no—
93C3×#380—51no—
94C3×C4×C4⋊C4—51no—
95C22×C5×Q8C22×C311yescharacteristic
96C22×C4×D5C22×C311yescharacteristic
97C4×C5×Q8—24no—
98C2×C5×C4⋊C4C22×C312yes—
99C2×C5×C4⋊C4C22×C311yescharacteristic
100C2×C42×C5C22×C312yes—
101C2×C42×C5C22×C311yescharacteristic
102#4092C3×C418yescomplemented
103C2×#685C22×C311yescharacteristic
104C2×#685C22×C311yescharacteristic
105C2×#685C22×C312yes—
106C2×#685C22×C312yes—
107C2×C4⋊Dic5C22×C311yescharacteristic
108C2×C4⋊Dic5C22×C311yescharacteristic
109C2×#683C22×C311yescharacteristic
110C2×#683C22×C311yescharacteristic
111C2×#683C22×C312yes—
112C2×#683C22×C312yes—
113C2×C4×Dic5C22×C311yescharacteristic
114C2×C4×Dic5C22×C312yes—
115#4018—24no—
116#3963C22×C312yes—
117#3963C22×C312yes—
118#3963C22×C312yes—
119#3963C22×C312yes—
120#3963C22×C312yes—
121#2323—51noSylow
122C23×C3×C5C2411yescharacteristic
123C22×C3×D5—21no—
124C22×C3×D5—21no—
125C22×C3×D5—21no—
126C22×C3×D5—21no—
127C22×C3×D5—21no—
128C22×C3×D5—21no—
129C22×C3×D5C22×C412yes—
130C3×C5×Q8—28no—
131C2×C3×C4×C5—22no—
132C2×C3×C4×C5—22no—
133C2×C3×C4×C5—21no—
134C2×C3×C4×C5—21no—
135C2×C3×C4×C5—21no—
136C2×C3×C4×C5—21no—
137C2×C3×C4×C5—22no—
138C2×C3×C4×C5—22no—
139C2×C3×C4×C5—22no—
140C2×C3×C4×C5C22×C414yes—
141C2×C3×C4×C5C22×C412yes—
142C2×C3×C4×C5C2×D412yes—
143C2×C3×C4×C5C2×D412yes—
144C2×C3×C4×C5C2×D412yes—
145C2×C3×C4×C5#4112yes—
146C2×C3×Dic5—21no—
147C2×C3×Dic5—21no—
148C2×C3×Dic5—22no—
149C2×C3×Dic5—22no—
150C2×C3×Dic5—21no—
151C2×C3×Dic5—21no—
152C2×C3×Dic5—22no—
153C2×C3×Dic5—22no—
154C2×C3×Dic5—21no—
155C2×C3×Dic5—21no—
156C2×C3×Dic5—21no—
157C2×C3×Dic5—21no—
158C2×C3×Dic5—22no—
159C2×C3×Dic5C22×C412yes—
160C2×C3×Dic5C22×C414yes—
161C3×C4×D5—28no—
162C22×C3×Q8C2×D511yescharacteristic
163C23×C3×C4—51no—
164C3×C22⋊Q8—58no—
165C3×C4×Q8—24no—
166C3×#117—104no—
167C2×C3×C4⋊C4—51no—
168C2×C3×C4⋊C4—52no—
169C2×C3×C4⋊C4—51no—
170C2×C3×C4⋊C4—51no—
171C2×C3×C4⋊C4—51no—
172C2×C3×C4⋊C4—52no—
173C2×C3×C4⋊C4C2×D511yescharacteristic
174C2×C3×C4⋊C4C2×D512yes—
175C2×C3×C22⋊C4—52no—
176C2×C3×C22⋊C4—51no—
177C2×C3×C22⋊C4—51no—
178C2×C3×C22⋊C4—52no—
179C2×C3×C42—51no—
180C2×C3×C42—52no—
181C2×C3×C42C2×D512yes—
182C2×C3×C42C2×D511yescharacteristic
183C3×#95—52no—
184C3×#95—52no—
185C3×#95—52no—
186C3×#95—52no—
187C3×#95—52no—
188C23×D5C23×C311yescharacteristic
189C2×C5×Q8—22no—
190C2×C5×Q8—22no—
191C2×C5×Q8C2×C3×C414yes—
192C22×C4×C5C23×C312yes—
193C22×C4×C5C23×C312yes—
194C22×C4×C5C23×C311yescharacteristic
195C22×C4×C5C23×C311yescharacteristic
196C22×C4×C5C23×C311yescharacteristic
197C22×Dic5C23×C311yescharacteristic
198C22×Dic5C23×C312yes—
199C22×Dic5C23×C311yescharacteristic
200C22×Dic5C23×C311yescharacteristic
201C22×Dic5C23×C312yes—
202C2×C4×D5—22no—
203C2×C4×D5—22no—
204C2×C4×D5C2×C3×C414yes—
205C5×C4⋊C4—24no—
206C5×C4⋊C4—22no—
207C42×C5—24no—
208C42×C5C3×D414yes—
209#685—22no—
210#685—22no—
211#685—42no—
212#685C2×C3×C418yes—
213C4⋊Dic5—22no—
214C4⋊Dic5C2×C3×C414yes—
215#683—22no—
216#683—22no—
217#683—42no—
218#683C2×C3×C418yes—
219C4×Dic5—22no—
220C4×Dic5—42no—
221C2×C22⋊Q8—51no—
222C2×C4×Q8C3×D511yescharacteristic, complemented
223C2×#117—51no—
224#391—51no—
225#389—52no—
226#389—51no—
227#388—52no—
228#388—52no—
229#387—52no—
230#380—51no—
231C4×C4⋊C4—51no—
232C22×C3×C5C23×C411yescharacteristic
233C22×C3×C5C22×D411yescharacteristic
234C22×C3×C5C2×#4111yescharacteristic
235C22×C3×C5H(5,𝔽2)11yescharacteristic
236C22×C3×C52−1+411yescharacteristic
237C22×C3×C52−1+411yescharacteristic
238C22×C3×C52−1+411yescharacteristic
239C2×C3×D5—22no—
240C2×C3×D5—22no—
241C2×C3×D5—22no—
242C2×C3×D5—41no—
243C2×C3×D5—41no—
244C2×C3×D5—41no—
245C2×C3×D5—41no—
246C3×C4×C5—24no—
247C3×C4×C5—22no—
248C3×C4×C5—22no—
249C3×C4×C5—42no—
250C3×C4×C5C4×D414yes—
251C3×Dic5—22no—
252C3×Dic5—24no—
253C3×Dic5—41no—
254C3×Dic5—41no—
255C3×Dic5—42no—
256C24×C3—51no—
257C2×C3×Q8—22no—
258C2×C3×Q8—22no—
259C2×C3×Q8C4×D514yes—
260C22×C3×C4—52no—
261C22×C3×C4—51no—
262C22×C3×C4—54no—
263C22×C3×C4—52no—
264C22×C3×C4—51no—
265C22×C3×C4—51no—
266C22×C3×C4—102no—
267C22×C3×C4—102no—
268C22×C3×C4C22×D512yes—
269C22×C3×C4C22×D512yes—
270C22×C3×C4C22×D511yescharacteristic
271C22×C3×C4C22×D511yescharacteristic
272C22×C3×C4C22×D511yescharacteristic
273C3×C4⋊C4—24no—
274C3×C4⋊C4—22no—
275C3×C4⋊C4—58no—
276C3×C4⋊C4—54no—
277C3×C4⋊C4—102no—
278C3×C4⋊C4—102no—
279C3×C4⋊C4—102no—
280C3×C4⋊C4—202no—
281C3×C22⋊C4—58no—
282C3×C22⋊C4—102no—
283C3×C22⋊C4—102no—
284C3×C22⋊C4—202no—
285C3×C42—24no—
286C3×C42—102no—
287C3×C42—202no—
288C3×C42C22⋊D514yes—
289C23×C5C24×C311yescharacteristic
290C22×D5—21no—
291C22×D5—21no—
292C22×D5—21no—
293C22×D5—21no—
294C22×D5—21no—
295C22×D5—21no—
296C22×D5C22×C3×C412yes—
297C5×Q8—28no—
298C2×C4×C5—22no—
299C2×C4×C5—22no—
300C2×C4×C5—21no—
301C2×C4×C5—21no—
302C2×C4×C5—21no—
303C2×C4×C5—21no—
304C2×C4×C5—22no—
305C2×C4×C5—22no—
306C2×C4×C5—22no—
307C2×C4×C5C22×C3×C414yes—
308C2×C4×C5C22×C3×C412yes—
309C2×C4×C5C2×C3×D412yes—
310C2×C4×C5C2×C3×D412yes—
311C2×C4×C5C2×C3×D412yes—
312C2×C4×C5C3×#4112yes—
313C2×Dic5—21no—
314C2×Dic5—21no—
315C2×Dic5—22no—
316C2×Dic5—22no—
317C2×Dic5—21no—
318C2×Dic5—21no—
319C2×Dic5—22no—
320C2×Dic5—22no—
321C2×Dic5—21no—
322C2×Dic5—21no—
323C2×Dic5—21no—
324C2×Dic5—22no—
325C2×Dic5—21no—
326C2×Dic5C22×C3×C412yes—
327C2×Dic5C22×C3×C414yes—
328C4×D5—28no—
329C22×Q8C2×C3×D511yescharacteristic
330C23×C4—51no—
331C22⋊Q8—58no—
332C4×Q8—24no—
333#117—104no—
334C2×C4⋊C4—51no—
335C2×C4⋊C4—52no—
336C2×C4⋊C4—51no—
337C2×C4⋊C4—51no—
338C2×C4⋊C4—51no—
339C2×C4⋊C4—52no—
340C2×C4⋊C4C2×C3×D511yescharacteristic
341C2×C4⋊C4C2×C3×D512yes—
342C2×C22⋊C4—52no—
343C2×C22⋊C4—51no—
344C2×C22⋊C4—51no—
345C2×C22⋊C4—52no—
346C2×C42—51no—
347C2×C42—52no—
348C2×C42C2×C3×D512yes—
349C2×C42C2×C3×D511yescharacteristic
350#95—52no—
351#95—52no—
352#95—52no—
353#95—52no—
354#95—52no—
355C2×C3×C5C2×C4×D411yescharacteristic
356C2×C3×C5#51911yescharacteristic
357C2×C3×C5#52011yescharacteristic
358C2×C3×C5#53611yescharacteristic
359C2×C3×C5#53711yescharacteristic
360C2×C3×C5#54111yescharacteristic
361C2×C3×C5#54211yescharacteristic
362C3×D5—42no—
363C23×C3—52no—
364C23×C3—101no—
365C23×C3—101no—
366C23×C3—101no—
367C23×C3—101no—
368C23×C3—101no—
369C23×C3—101no—
370C23×C3C23×D511yescharacteristic
371C3×Q8—28no—
372C2×C3×C4—22no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—22no—
378C2×C3×C4—22no—
379C2×C3×C4—22no—
380C2×C3×C4—22no—
381C2×C3×C4—54no—
382C2×C3×C4—52no—
383C2×C3×C4—101no—
384C2×C3×C4—102no—
385C2×C3×C4—102no—
386C2×C3×C4—101no—
387C2×C3×C4—108no—
388C2×C3×C4—102no—
389C2×C3×C4—101no—
390C2×C3×C4—101no—
391C2×C3×C4—102no—
392C2×C3×C4—102no—
393C2×C3×C4—101no—
394C2×C3×C4—101no—
395C2×C3×C4—101no—
396C2×C3×C4—101no—
397C2×C3×C4C2×C4×D514yes—
398C2×C3×C4C2×C4×D512yes—
399C2×C3×C4#70912yes—
400C2×C3×C4C2×C22⋊D512yes—
401C2×C3×C4C2×C22⋊D512yes—
402C2×C3×C4C2×C22⋊D512yes—
403C22×C5C23×C3×C411yescharacteristic
404C22×C5C22×C3×D411yescharacteristic
405C22×C5C2×C3×#4111yescharacteristic
406C22×C5C3×H(5,𝔽2)11yescharacteristic
407C22×C5C3×2−1+411yescharacteristic
408C22×C5C3×2−1+411yescharacteristic
409C22×C5C3×2−1+411yescharacteristic
410C2×D5—22no—
411C2×D5—22no—
412C2×D5—22no—
413C2×D5—41no—
414C2×D5—41no—
415C2×D5—41no—
416C2×D5—41no—
417C4×C5—24no—
418C4×C5—22no—
419C4×C5—22no—
420C4×C5—42no—
421C4×C5C3×C4×D414yes—
422Dic5—22no—
423Dic5—24no—
424Dic5—41no—
425Dic5—41no—
426Dic5—42no—
427C24—51no—
428C2×Q8—22no—
429C2×Q8—22no—
430C2×Q8C3×C4×D514yes—
431C22×C4—52no—
432C22×C4—51no—
433C22×C4—54no—
434C22×C4—52no—
435C22×C4—51no—
436C22×C4—51no—
437C22×C4—102no—
438C22×C4—102no—
439C22×C4C22×C3×D512yes—
440C22×C4C22×C3×D512yes—
441C22×C4C22×C3×D511yescharacteristic
442C22×C4C22×C3×D511yescharacteristic
443C22×C4C22×C3×D511yescharacteristic
444C4⋊C4—24no—
445C4⋊C4—22no—
446C4⋊C4—58no—
447C4⋊C4—54no—
448C4⋊C4—102no—
449C4⋊C4—102no—
450C4⋊C4—102no—
451C4⋊C4—202no—
452C22⋊C4—58no—
453C22⋊C4—102no—
454C22⋊C4—102no—
455C22⋊C4—202no—
456C42—24no—
457C42—102no—
458C42—202no—
459C42C3×C22⋊D514yes—
460C3×C5#232311yescharacteristic, complemented, Hall
461C22×C3—102no—
462C22×C3—102no—
463C22×C3—102no—
464C22×C3—201no—
465C22×C3—201no—
466C22×C3—201no—
467C22×C3—201no—
468C22×C3C22×C4×D511yescharacteristic
469C22×C3C2×#70911yescharacteristic
470C22×C3#414711yescharacteristic
471C22×C3#414711yescharacteristic
472C22×C3#414911yescharacteristic
473C22×C3#415011yescharacteristic
474C22×C3C22×C22⋊D511yescharacteristic
475C3×C4—24no—
476C3×C4—22no—
477C3×C4—22no—
478C3×C4—42no—
479C3×C4—102no—
480C3×C4—104no—
481C3×C4—201no—
482C3×C4—201no—
483C3×C4—202no—
484C3×C4C4×C22⋊D514yes—
485C2×C5C2×C3×C4×D411yescharacteristic
486C2×C5C3×#51911yescharacteristic
487C2×C5C3×#52011yescharacteristic
488C2×C5C3×#53611yescharacteristic
489C2×C5C3×#53711yescharacteristic
490C2×C5C3×#54111yescharacteristic
491C2×C5C3×#54211yescharacteristic
492D5—42no—
493C23—52no—
494C23—101no—
495C23—101no—
496C23—101no—
497C23—101no—
498C23—101no—
499C23—101no—
500C23C23×C3×D511yescharacteristic
501Q8—28no—
502C2×C4—22no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—22no—
508C2×C4—22no—
509C2×C4—22no—
510C2×C4—22no—
511C2×C4—54no—
512C2×C4—52no—
513C2×C4—101no—
514C2×C4—102no—
515C2×C4—102no—
516C2×C4—101no—
517C2×C4—108no—
518C2×C4—102no—
519C2×C4—101no—
520C2×C4—101no—
521C2×C4—102no—
522C2×C4—102no—
523C2×C4—101no—
524C2×C4—101no—
525C2×C4—101no—
526C2×C4—101no—
527C2×C4C2×C3×C4×D514yes—
528C2×C4C2×C3×C4×D512yes—
529C2×C4C3×#70912yes—
530C2×C4C2×C3×C22⋊D512yes—
531C2×C4C2×C3×C22⋊D512yes—
532C2×C4C2×C3×C22⋊D512yes—
533C2×C3—202no—
534C2×C3#6590611yescharacteristic
535C2×C3#6590811yescharacteristic
536C2×C3#6591611yescharacteristic
537C2×C3#6591711yescharacteristic
538C2×C3C2×C4×C22⋊D511yescharacteristic
539C2×C3#6615011yescharacteristic
540C2×C3#6616711yescharacteristic
541C5C3×#232311yescharacteristic, complemented, Sylow
542C22—102no—
543C22—102no—
544C22—102no—
545C22—201no—
546C22—201no—
547C22—201no—
548C22—201no—
549C22C22×C3×C4×D511yescharacteristic
550C22C2×C3×#70911yescharacteristic
551C22C3×#414711yescharacteristic
552C22C3×#414711yescharacteristic
553C22C3×#414911yescharacteristic
554C22C3×#415011yescharacteristic
555C22C22×C3×C22⋊D511yescharacteristic
556C4—24no—
557C4—22no—
558C4—22no—
559C4—42no—
560C4—102no—
561C4—104no—
562C4—201no—
563C4—201no—
564C4—202no—
565C4C3×C4×C22⋊D514yes—
566C3#1061421911yescharacteristic, direct product factor, Sylow
567C2—202no—
568C2C3×#6590611yescharacteristic
569C2C3×#6590811yescharacteristic
570C2C3×#6591611yescharacteristic
571C2C3×#6591711yescharacteristic
572C2C2×C3×C4×C22⋊D511yescharacteristic
573C2C3×#6615011yescharacteristic
574C2C3×#6616711yescharacteristic
subgroupG/H
centerC23×C3C23×D5
commutatorC22×C5C23×C3×C4
FrattiniC23C23×C3×D5
FittingC2×C3×C4×C5×Q8C2
socleC23×C3×C5C24
prime radicalC23×C3×D5C23
cosocleC23×C5C24×C3
hypercenterC2×C3×C4×Q8D5
hypocenterC5C3×#2323
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C2×C3×C4×Q8

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×C5×Q8C2semidirect
4. C3×#65519C2semidirect
7. C3×#65262C2semidirect
8. C3×#65261C2semidirect
10. C3×C4×C4⋊Dic5C2semidirect
11. C3×#65222C2semidirect
12. #10614219C3direct (N × H)
20. C3×#4092C4semidirect
40. C2×C4×C5×Q8C2×C3semidirect
43. #65519C2×C3semidirect
46. #65262C2×C3semidirect
47. #65261C2×C3semidirect
49. C4×C4⋊Dic5C2×C3semidirect
50. #65222C2×C3semidirect
85. C2×C3×C4×Q8D5semidirect
102. #4092C3×C4semidirect
222. C2×C4×Q8C3×D5semidirect
460. C3×C5#2323semidirect
541. C5C3×#2323semidirect
566. C3#10614219direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4×C5×Q82
C2×C3×#40922
C2×C3×#40182
C3×#655192
C3×#652792
C3×#652762
C3×#652622
C3×#652612
C3×#652292
C3×C4×C4⋊Dic52
C3×#652222
#106142193
C3×#23235