Finite Group Databasebeta

← C3×#10613966 #14323709

Subgroups — C3×#10613966 #14323709

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×C5×#117C211yesmaximal, characteristic
2C2×C3×#4070C211yesmaximal, characteristic
3C2×C3×C4×Dic10C211yesmaximal, characteristic, complemented
4C3×#65501C211yesmaximal, characteristic
5C3×#65267C212yesmaximal, complemented
6C3×#65261C212yesmaximal, complemented
7C3×#65234C212yesmaximal
8C3×#65233C212yesmaximal
9C3×C4×C4⋊Dic5C211yesmaximal, characteristic, complemented
10C3×#65218C211yesmaximal, characteristic, complemented
11C3×#65217C211yesmaximal, characteristic, complemented
12#10613966C311yesmaximal, characteristic, direct product factor, Hall
13C23×C3×C4×C5C2211yescharacteristic
14C22×C3×Dic10C2211yescharacteristic
15C3×C5×#117—24no—
16C2×C3×C5×C4⋊C4C2211yescharacteristic
17C2×C3×C5×C4⋊C4C2211yescharacteristic
18C2×C3×C5×C22⋊C4C2211yescharacteristic
19C2×C3×C5×C22⋊C4C2211yescharacteristic
20C2×C3×C42×C5C2211yescharacteristic
21C2×C3×C42×C5C2211yescharacteristic
22C2×C3×C22⋊Dic5C2212yes—
23C2×C3×C22⋊Dic5C2212yes—
24C2×C3×C4⋊Dic5C2211yescharacteristic
25C2×C3×C4⋊Dic5C2211yescharacteristic
26C2×C3×C4⋊Dic5C2211yescharacteristic
27C3×#4070C418yescomplemented
28C2×C3×#683C2212yes—
29C2×C3×#683C2212yes—
30C2×C3×#683C2212yes—
31C2×C3×C4×Dic5C2212yes—
32C2×C3×C4×Dic5C2212yes—
33C3×C4×Dic10—24no—
34C3×#3963C2212yes—
35C3×#3963C2212yes—
36C3×#3963C2212yes—
37C3×#3963C2212yes—
38C3×#3963C2212yes—
39C3×#2323—51nomaximal, Hall
40C2×C5×#117C2×C311yescharacteristic
41C2×#4070C2×C311yescharacteristic
42C2×C4×Dic10C2×C311yescharacteristic, complemented
43#65501C2×C311yescharacteristic
44#65267C2×C312yescomplemented
45#65261C2×C312yescomplemented
46#65234C2×C312yes—
47#65233C2×C312yes—
48C4×C4⋊Dic5C2×C311yescharacteristic, complemented
49#65218C2×C311yescharacteristic, complemented
50#65217C2×C311yescharacteristic, complemented
51C24×C3×C5C2311yescharacteristic
52C22×C3×C4×C5—22no—
53C22×C3×C4×C5—22no—
54C22×C3×C4×C5C2×C414yes—
55C22×C3×C4×C5C2311yescharacteristic
56C22×C3×C4×C5C2311yescharacteristic
57C22×C3×C4×C5C2311yescharacteristic
58C22×C3×C4×C5C2311yescharacteristic
59C22×C3×C4×C5C2311yescharacteristic
60C22×C3×C4×C5C2311yescharacteristic
61C22×C3×Dic5C2312yes—
62C22×C3×Dic5C2312yes—
63C22×C3×Dic5C2312yes—
64C22×C3×Dic5C2312yes—
65C2×C3×Dic10—22no—
66C2×C3×Dic10—22no—
67C2×C3×Dic10C2×C414yes—
68C3×C5×C4⋊C4—22no—
69C3×C5×C4⋊C4—22no—
70C3×C5×C22⋊C4—22no—
71C3×C5×C22⋊C4—22no—
72C3×C42×C5—22no—
73C3×C42×C5D414yes—
74C3×C22⋊Dic5—42no—
75C3×C22⋊Dic5C2×C418yes—
76C3×C4⋊Dic5—22no—
77C3×C4⋊Dic5—22no—
78C3×C4⋊Dic5C2×C414yes—
79C3×#683—24no—
80C3×#683—42no—
81C3×#683C2×C418yes—
82C3×C4×Dic5—24no—
83C3×C4×Dic5—42no—
84C2×C3×C22⋊Q8—51no—
85C2×C3×C4×Q8—51no—
86C2×C3×#117D511yescharacteristic
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—
95C23×C4×C5C22×C311yescharacteristic
96C22×Dic10C22×C311yescharacteristic
97C5×#117—24no—
98C2×C5×C4⋊C4C22×C311yescharacteristic
99C2×C5×C4⋊C4C22×C311yescharacteristic
100C2×C5×C22⋊C4C22×C311yescharacteristic
101C2×C5×C22⋊C4C22×C311yescharacteristic
102C2×C42×C5C22×C311yescharacteristic
103C2×C42×C5C22×C311yescharacteristic
104C2×C22⋊Dic5C22×C312yes—
105C2×C22⋊Dic5C22×C312yes—
106C2×C4⋊Dic5C22×C311yescharacteristic
107C2×C4⋊Dic5C22×C311yescharacteristic
108C2×C4⋊Dic5C22×C311yescharacteristic
109#4070C3×C418yescomplemented
110C2×#683C22×C312yes—
111C2×#683C22×C312yes—
112C2×#683C22×C312yes—
113C2×C4×Dic5C22×C312yes—
114C2×C4×Dic5C22×C312yes—
115C4×Dic10—24no—
116#3963C22×C312yes—
117#3963C22×C312yes—
118#3963C22×C312yes—
119#3963C22×C312yes—
120#3963C22×C312yes—
121#2323—51noSylow
122C23×C3×C5—21no—
123C23×C3×C5—21no—
124C23×C3×C5—21no—
125C23×C3×C5—21no—
126C23×C3×C5—21no—
127C23×C3×C5—21no—
128C23×C3×C5C22×C412yes—
129C23×C3×C5C2411yescharacteristic
130C2×C3×C4×C5—21no—
131C2×C3×C4×C5—21no—
132C2×C3×C4×C5—21no—
133C2×C3×C4×C5—21no—
134C2×C3×C4×C5—21no—
135C2×C3×C4×C5—21no—
136C2×C3×C4×C5—28no—
137C2×C3×C4×C5—21no—
138C2×C3×C4×C5—21no—
139C2×C3×C4×C5—21no—
140C2×C3×C4×C5—21no—
141C2×C3×C4×C5—21no—
142C2×C3×C4×C5—21no—
143C2×C3×C4×C5C22×C412yes—
144C2×C3×C4×C5C22×C412yes—
145C2×C3×C4×C5C2×D412yes—
146C2×C3×C4×C5C2×D412yes—
147C2×C3×C4×C5C2×D412yes—
148C2×C3×C4×C5#4112yes—
149C2×C3×Dic5—22no—
150C2×C3×Dic5—22no—
151C2×C3×Dic5—22no—
152C2×C3×Dic5—22no—
153C2×C3×Dic5—22no—
154C2×C3×Dic5—22no—
155C2×C3×Dic5—22no—
156C2×C3×Dic5—22no—
157C2×C3×Dic5—22no—
158C2×C3×Dic5—22no—
159C2×C3×Dic5C22×C414yes—
160C2×C3×Dic5C22×C414yes—
161C3×Dic10—28no—
162C22×C3×Q8—51no—
163C23×C3×C4C2×D511yescharacteristic
164C3×C22⋊Q8—58no—
165C3×C4×Q8—104no—
166C3×#117—24no—
167C2×C3×C4⋊C4—51no—
168C2×C3×C4⋊C4—52no—
169C2×C3×C4⋊C4—51no—
170C2×C3×C4⋊C4—51no—
171C2×C3×C4⋊C4—52no—
172C2×C3×C4⋊C4—52no—
173C2×C3×C4⋊C4C2×D511yescharacteristic
174C2×C3×C4⋊C4C2×D511yescharacteristic
175C2×C3×C22⋊C4—52no—
176C2×C3×C22⋊C4—52no—
177C2×C3×C22⋊C4C2×D511yescharacteristic
178C2×C3×C22⋊C4C2×D511yescharacteristic
179C2×C3×C42—52no—
180C2×C3×C42—52no—
181C2×C3×C42C2×D511yescharacteristic
182C2×C3×C42C2×D511yescharacteristic
183C3×#95—52no—
184C3×#95—52no—
185C3×#95—52no—
186C3×#95—52no—
187C3×#95—52no—
188C24×C5C23×C311yescharacteristic
189C22×C4×C5—22no—
190C22×C4×C5—22no—
191C22×C4×C5C2×C3×C414yes—
192C22×C4×C5C23×C311yescharacteristic
193C22×C4×C5C23×C311yescharacteristic
194C22×C4×C5C23×C311yescharacteristic
195C22×C4×C5C23×C311yescharacteristic
196C22×C4×C5C23×C311yescharacteristic
197C22×C4×C5C23×C311yescharacteristic
198C22×Dic5C23×C312yes—
199C22×Dic5C23×C312yes—
200C22×Dic5C23×C312yes—
201C22×Dic5C23×C312yes—
202C2×Dic10—22no—
203C2×Dic10—22no—
204C2×Dic10C2×C3×C414yes—
205C5×C4⋊C4—22no—
206C5×C4⋊C4—22no—
207C5×C22⋊C4—22no—
208C5×C22⋊C4—22no—
209C42×C5—22no—
210C42×C5C3×D414yes—
211C22⋊Dic5—42no—
212C22⋊Dic5C2×C3×C418yes—
213C4⋊Dic5—22no—
214C4⋊Dic5—22no—
215C4⋊Dic5C2×C3×C414yes—
216#683—24no—
217#683—42no—
218#683C2×C3×C418yes—
219C4×Dic5—24no—
220C4×Dic5—42no—
221C2×C22⋊Q8—51no—
222C2×C4×Q8—51no—
223C2×#117C3×D511yescharacteristic
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×C5—22no—
233C22×C3×C5—22no—
234C22×C3×C5—22no—
235C22×C3×C5—41no—
236C22×C3×C5—41no—
237C22×C3×C5—41no—
238C22×C3×C5—41no—
239C22×C3×C5C23×C411yescharacteristic
240C22×C3×C5C22×D411yescharacteristic
241C22×C3×C5C2×#4111yescharacteristic
242C22×C3×C5H(5,𝔽2)11yescharacteristic
243C22×C3×C52−1+411yescharacteristic
244C22×C3×C52−1+411yescharacteristic
245C22×C3×C52−1+411yescharacteristic
246C3×C4×C5—22no—
247C3×C4×C5—22no—
248C3×C4×C5—22no—
249C3×C4×C5—41no—
250C3×C4×C5—41no—
251C3×C4×C5C4×D414yes—
252C3×Dic5—24no—
253C3×Dic5—24no—
254C3×Dic5—42no—
255C3×Dic5—42no—
256C24×C3C22×D511yescharacteristic
257C2×C3×Q8—54no—
258C2×C3×Q8—102no—
259C2×C3×Q8—102no—
260C22×C3×C4—22no—
261C22×C3×C4—22no—
262C22×C3×C4—52no—
263C22×C3×C4—52no—
264C22×C3×C4—52no—
265C22×C3×C4—52no—
266C22×C3×C4C4×D514yes—
267C22×C3×C4C22×D511yescharacteristic
268C22×C3×C4C22×D511yescharacteristic
269C22×C3×C4C22×D511yescharacteristic
270C22×C3×C4C22×D511yescharacteristic
271C22×C3×C4C22×D511yescharacteristic
272C22×C3×C4C22×D511yescharacteristic
273C3×C4⋊C4—22no—
274C3×C4⋊C4—22no—
275C3×C4⋊C4—58no—
276C3×C4⋊C4—54no—
277C3×C4⋊C4—102no—
278C3×C4⋊C4—104no—
279C3×C4⋊C4—102no—
280C3×C4⋊C4—202no—
281C3×C22⋊C4—22no—
282C3×C22⋊C4—22no—
283C3×C22⋊C4—58no—
284C3×C22⋊C4—202no—
285C3×C42—22no—
286C3×C42—104no—
287C3×C42—202no—
288C3×C42C22⋊D514yes—
289C23×C5—21no—
290C23×C5—21no—
291C23×C5—21no—
292C23×C5—21no—
293C23×C5—21no—
294C23×C5—21no—
295C23×C5C22×C3×C412yes—
296C23×C5C24×C311yescharacteristic
297C2×C4×C5—21no—
298C2×C4×C5—21no—
299C2×C4×C5—21no—
300C2×C4×C5—21no—
301C2×C4×C5—21no—
302C2×C4×C5—21no—
303C2×C4×C5—28no—
304C2×C4×C5—21no—
305C2×C4×C5—21no—
306C2×C4×C5—21no—
307C2×C4×C5—21no—
308C2×C4×C5—21no—
309C2×C4×C5—21no—
310C2×C4×C5C22×C3×C412yes—
311C2×C4×C5C22×C3×C412yes—
312C2×C4×C5C2×C3×D412yes—
313C2×C4×C5C2×C3×D412yes—
314C2×C4×C5C2×C3×D412yes—
315C2×C4×C5C3×#4112yes—
316C2×Dic5—22no—
317C2×Dic5—22no—
318C2×Dic5—22no—
319C2×Dic5—22no—
320C2×Dic5—22no—
321C2×Dic5—22no—
322C2×Dic5—22no—
323C2×Dic5—22no—
324C2×Dic5—22no—
325C2×Dic5—22no—
326C2×Dic5C22×C3×C414yes—
327C2×Dic5C22×C3×C414yes—
328Dic10—28no—
329C22×Q8—51no—
330C23×C4C2×C3×D511yescharacteristic
331C22⋊Q8—58no—
332C4×Q8—104no—
333#117—24no—
334C2×C4⋊C4—51no—
335C2×C4⋊C4—52no—
336C2×C4⋊C4—51no—
337C2×C4⋊C4—52no—
338C2×C4⋊C4—51no—
339C2×C4⋊C4—52no—
340C2×C4⋊C4C2×C3×D511yescharacteristic
341C2×C4⋊C4C2×C3×D511yescharacteristic
342C2×C22⋊C4—52no—
343C2×C22⋊C4—52no—
344C2×C22⋊C4C2×C3×D511yescharacteristic
345C2×C22⋊C4C2×C3×D511yescharacteristic
346C2×C42—52no—
347C2×C42—52no—
348C2×C42C2×C3×D511yescharacteristic
349C2×C42C2×C3×D511yescharacteristic
350#95—52no—
351#95—52no—
352#95—52no—
353#95—52no—
354#95—52no—
355C2×C3×C5—42no—
356C2×C3×C5C2×C4×D411yescharacteristic
357C2×C3×C5#51911yescharacteristic
358C2×C3×C5#52011yescharacteristic
359C2×C3×C5#53611yescharacteristic
360C2×C3×C5#53711yescharacteristic
361C2×C3×C5#54111yescharacteristic
362C2×C3×C5#54211yescharacteristic
363C23×C3—21no—
364C23×C3—21no—
365C23×C3—21no—
366C23×C3—21no—
367C23×C3—21no—
368C23×C3—21no—
369C23×C3C2×C4×D512yes—
370C23×C3C23×D511yescharacteristic
371C3×Q8—108no—
372C2×C3×C4—21no—
373C2×C3×C4—21no—
374C2×C3×C4—21no—
375C2×C3×C4—21no—
376C2×C3×C4—21no—
377C2×C3×C4—21no—
378C2×C3×C4—28no—
379C2×C3×C4—21no—
380C2×C3×C4—21no—
381C2×C3×C4—21no—
382C2×C3×C4—21no—
383C2×C3×C4—21no—
384C2×C3×C4—21no—
385C2×C3×C4—54no—
386C2×C3×C4—54no—
387C2×C3×C4—102no—
388C2×C3×C4—102no—
389C2×C3×C4—102no—
390C2×C3×C4—102no—
391C2×C3×C4—102no—
392C2×C3×C4—102no—
393C2×C3×C4—102no—
394C2×C3×C4—102no—
395C2×C3×C4—102no—
396C2×C3×C4—102no—
397C2×C3×C4C2×C4×D512yes—
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×C5—22no—
404C22×C5—22no—
405C22×C5—22no—
406C22×C5—41no—
407C22×C5—41no—
408C22×C5—41no—
409C22×C5—41no—
410C22×C5C23×C3×C411yescharacteristic
411C22×C5C22×C3×D411yescharacteristic
412C22×C5C2×C3×#4111yescharacteristic
413C22×C5C3×H(5,𝔽2)11yescharacteristic
414C22×C5C3×2−1+411yescharacteristic
415C22×C5C3×2−1+411yescharacteristic
416C22×C5C3×2−1+411yescharacteristic
417C4×C5—22no—
418C4×C5—22no—
419C4×C5—22no—
420C4×C5—41no—
421C4×C5—41no—
422C4×C5C3×C4×D414yes—
423Dic5—24no—
424Dic5—24no—
425Dic5—42no—
426Dic5—42no—
427C24C22×C3×D511yescharacteristic
428C2×Q8—54no—
429C2×Q8—102no—
430C2×Q8—102no—
431C22×C4—22no—
432C22×C4—22no—
433C22×C4—52no—
434C22×C4—52no—
435C22×C4—52no—
436C22×C4—52no—
437C22×C4C3×C4×D514yes—
438C22×C4C22×C3×D511yescharacteristic
439C22×C4C22×C3×D511yescharacteristic
440C22×C4C22×C3×D511yescharacteristic
441C22×C4C22×C3×D511yescharacteristic
442C22×C4C22×C3×D511yescharacteristic
443C22×C4C22×C3×D511yescharacteristic
444C4⋊C4—22no—
445C4⋊C4—22no—
446C4⋊C4—58no—
447C4⋊C4—54no—
448C4⋊C4—102no—
449C4⋊C4—104no—
450C4⋊C4—102no—
451C4⋊C4—202no—
452C22⋊C4—22no—
453C22⋊C4—22no—
454C22⋊C4—58no—
455C22⋊C4—202no—
456C42—22no—
457C42—104no—
458C42—202no—
459C42C3×C22⋊D514yes—
460C3×C5#232311yescharacteristic, complemented, Hall
461C22×C3—22no—
462C22×C3—22no—
463C22×C3—22no—
464C22×C3—41no—
465C22×C3—41no—
466C22×C3—41no—
467C22×C3—41no—
468C22×C3C22×C4×D511yescharacteristic
469C22×C3C2×#70911yescharacteristic
470C22×C3#414911yescharacteristic
471C22×C3#415011yescharacteristic
472C22×C3#415011yescharacteristic
473C22×C3#415011yescharacteristic
474C22×C3C22×C22⋊D511yescharacteristic
475C3×C4—22no—
476C3×C4—22no—
477C3×C4—22no—
478C3×C4—41no—
479C3×C4—41no—
480C3×C4—104no—
481C3×C4—104no—
482C3×C4—202no—
483C3×C4—202no—
484C3×C4C4×C22⋊D514yes—
485C2×C5—42no—
486C2×C5C2×C3×C4×D411yescharacteristic
487C2×C5C3×#51911yescharacteristic
488C2×C5C3×#52011yescharacteristic
489C2×C5C3×#53611yescharacteristic
490C2×C5C3×#53711yescharacteristic
491C2×C5C3×#54111yescharacteristic
492C2×C5C3×#54211yescharacteristic
493C23—21no—
494C23—21no—
495C23—21no—
496C23—21no—
497C23—21no—
498C23—21no—
499C23C2×C3×C4×D512yes—
500C23C23×C3×D511yescharacteristic
501Q8—108no—
502C2×C4—21no—
503C2×C4—21no—
504C2×C4—21no—
505C2×C4—21no—
506C2×C4—21no—
507C2×C4—21no—
508C2×C4—28no—
509C2×C4—21no—
510C2×C4—21no—
511C2×C4—21no—
512C2×C4—21no—
513C2×C4—21no—
514C2×C4—21no—
515C2×C4—54no—
516C2×C4—54no—
517C2×C4—102no—
518C2×C4—102no—
519C2×C4—102no—
520C2×C4—102no—
521C2×C4—102no—
522C2×C4—102no—
523C2×C4—102no—
524C2×C4—102no—
525C2×C4—102no—
526C2×C4—102no—
527C2×C4C2×C3×C4×D512yes—
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—42no—
534C2×C3#6585011yescharacteristic
535C2×C3#6585211yescharacteristic
536C2×C3#6585711yescharacteristic
537C2×C3#6586811yescharacteristic
538C2×C3C2×C4×C22⋊D511yescharacteristic
539C2×C3#6615911yescharacteristic
540C2×C3#6616711yescharacteristic
541C5C3×#232311yescharacteristic, complemented, Sylow
542C22—22no—
543C22—22no—
544C22—22no—
545C22—41no—
546C22—41no—
547C22—41no—
548C22—41no—
549C22C22×C3×C4×D511yescharacteristic
550C22C2×C3×#70911yescharacteristic
551C22C3×#414911yescharacteristic
552C22C3×#415011yescharacteristic
553C22C3×#415011yescharacteristic
554C22C3×#415011yescharacteristic
555C22C22×C3×C22⋊D511yescharacteristic
556C4—22no—
557C4—22no—
558C4—22no—
559C4—41no—
560C4—41no—
561C4—104no—
562C4—104no—
563C4—202no—
564C4—202no—
565C4C3×C4×C22⋊D514yes—
566C3#1061396611yescharacteristic, direct product factor, Sylow
567C2—42no—
568C2C3×#6585011yescharacteristic
569C2C3×#6585211yescharacteristic
570C2C3×#6585711yescharacteristic
571C2C3×#6586811yescharacteristic
572C2C2×C3×C4×C22⋊D511yescharacteristic
573C2C3×#6615911yescharacteristic
574C2C3×#6616711yescharacteristic
subgroupG/H
centerC23×C3C23×D5
commutatorC22×C5C23×C3×C4
FrattiniC23C23×C3×D5
FittingC2×C3×C5×#117C2
socleC23×C3×C5C24
prime radicalC24×C3×C5C23
cosocleC23×C5C24×C3
hypercenterC2×C3×#117D5
hypocenterC5C3×#2323
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C2×C3×#117

Complements

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

NH ≅ G/Ntype
3. C2×C3×C4×Dic10C2semidirect
5. C3×#65267C2semidirect
6. C3×#65261C2semidirect
9. C3×C4×C4⋊Dic5C2semidirect
10. C3×#65218C2semidirect
11. C3×#65217C2semidirect
12. #10613966C3direct (N × H)
27. C3×#4070C4semidirect
42. C2×C4×Dic10C2×C3semidirect
44. #65267C2×C3semidirect
45. #65261C2×C3semidirect
48. C4×C4⋊Dic5C2×C3semidirect
49. #65218C2×C3semidirect
50. #65217C2×C3semidirect
109. #4070C3×C4semidirect
460. C3×C5#2323semidirect
541. C5C3×#2323semidirect
566. C3#10613966direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C5×#1172
C2×C3×#40702
C2×C3×C4×Dic102
C3×#655012
C3×#652672
C3×#652612
C3×#652342
C3×#652332
C3×C4×C4⋊Dic52
C3×#652182
C3×#652172
#106139663
C3×#23235