Finite Group Databasebeta

← C3×#10604417 #14297639

Subgroups — C3×#10604417 #14297639

Order 1920.

744 conjugacy classes of subgroups.

The subgroup lattice has 560 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×#65688C211yesmaximal, characteristic, complemented
4C3×D5⋊(C4⋊C8)C211yesmaximal, characteristic, complemented
5C3×#64874C212yesmaximal, complemented
6#10604417C311yesmaximal, characteristic, direct product factor, Hall
7C2×C3×D4×D5C2211yescharacteristic
8C22×C3×C4×D5—21no—
9C2×C3×C4⋊AGL(1,𝔽5)—21no—
10C2×C3×C4⋊AGL(1,𝔽5)—21no—
11C2×C3×C4⋊AGL(1,𝔽5)C2212yes—
12C2×C3×D5⋊C8C2212yes—
13C3×C4×C5×D4C411yescharacteristic, complemented
14C3×D4×Dic5C411yescharacteristic, complemented
15C3×C4×C22⋊D5—21no—
16C3×#4039C411yescharacteristic, complemented
17C3×D5×C4⋊C4C2211yescharacteristic
18C3×#4027—21no—
19C3×D5×C22⋊C4—21no—
20C3×C4×D20C411yescharacteristic, complemented
21C3×C42×D5C2211yescharacteristic
22C3×#4007—24no—
23C3×#4004—21no—
24C3×C4⋊#170—21no—
25C3×#1802—51nomaximal, Hall
26C4×D4×D5C2×C311yescharacteristic
27C2×#4007C2×C312yes—
28#65688C2×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—42no—
46C2×C3×C4×D5—42no—
47C2×C3×C4×D5D411yescharacteristic
48C2×C3×C4×D5D411yescharacteristic
49C2×C3×C4×D5C2311yescharacteristic
50C2×C3×#170—22no—
51C3×C4⋊AGL(1,𝔽5)—22no—
52C3×C4⋊AGL(1,𝔽5)—22no—
53C3×C4⋊AGL(1,𝔽5)—42no—
54C3×C4⋊AGL(1,𝔽5)—42no—
55C3×D5⋊C8—22no—
56C3×D5⋊C8—22no—
57C3×C5×C4⋊C4C2×C411yescharacteristic
58C3×C5×C22⋊C4—21no—
59C3×C42×C5C2×C411yescharacteristic
60C3×C22⋊Dic5—21no—
61C3×#685—21no—
62C3×#685—21no—
63C3×C4⋊Dic5C2×C411yescharacteristic
64C3×#683—21no—
65C3×C4×Dic5C2×C411yescharacteristic
66C3×C4×Dic5D411yescharacteristic
67C3×C4×Dic5D411yescharacteristic
68C2×C3×C4×D4—51no—
69C2×C3×C4⋊C8—51no—
70C2×C3×#102—52no—
71C3×#384—51no—
72C3×#340—52no—
73C2×D4×D5C22×C311yescharacteristic
74C22×C4×D5—21no—
75C2×C4⋊AGL(1,𝔽5)—21no—
76C2×C4⋊AGL(1,𝔽5)—21no—
77C2×C4⋊AGL(1,𝔽5)C22×C312yes—
78C2×D5⋊C8C22×C312yes—
79C4×C5×D4C3×C411yescharacteristic, complemented
80D4×Dic5C3×C411yescharacteristic, complemented
81C4×C22⋊D5—21no—
82#4039C3×C411yescharacteristic, complemented
83D5×C4⋊C4C22×C311yescharacteristic
84#4027—21no—
85D5×C22⋊C4—21no—
86C4×D20C3×C411yescharacteristic, complemented
87C42×D5C22×C311yescharacteristic
88#4007—24no—
89#4004—21no—
90C4⋊#170—21no—
91#1802—51noSylow
92C23×C3×C5—21no—
93C22×C3×D5—21no—
94C22×C3×D5—22no—
95C22×C3×D5—22no—
96C22×C3×D5—41no—
97C22×C3×D5—41no—
98C22×C3×D5—41no—
99C22×C3×D5—41no—
100C22×C3×D5C2×D411yescharacteristic
101C2×C3×AGL(1,𝔽5)—42no—
102C2×C3×AGL(1,𝔽5)—42no—
103C2×C3×AGL(1,𝔽5)—42no—
104C3×C5×D4—21no—
105C3×C5×D4C4⋊C412yescomplemented
106C2×C3×C4×C5—21no—
107C2×C3×C4×C5—42no—
108C2×C3×C4×C5C22⋊C411yescharacteristic
109C2×C3×C4×C5C22⋊C411yescharacteristic
110C2×C3×C4×C5C22×C411yescharacteristic
111C3×C22⋊D5—22no—
112C3×C22⋊D5—22no—
113C2×C3×Dic5—21no—
114C2×C3×Dic5—42no—
115C2×C3×Dic5C22⋊C411yescharacteristic
116C2×C3×Dic5C22⋊C411yescharacteristic
117C2×C3×Dic5C2×D411yescharacteristic
118C3×D20—21no—
119C3×D20C4⋊C412yescomplemented
120C3×C4×D5—21no—
121C3×C4×D5—21no—
122C3×C4×D5—41no—
123C3×C4×D5—41no—
124C3×C4×D5—42no—
125C3×C4×D5—42no—
126C3×C4×D5D812yes—
127C3×C4×D5SD1612yes—
128C3×C4×D5C2×D411yescharacteristic
129C3×C4×D5C2×Q811yescharacteristic
130C3×C4×D5#4111yescharacteristic
131C3×C4×D5#4111yescharacteristic
132C3×#170—42no—
133C22×C3×D4—51no—
134C23×C3×C4—101no—
135C22×C3×C8—52no—
136C3×C4×D4—51no—
137C3×C4×D4—51no—
138C3×C4×D4—51no—
139C3×C4×D4—101no—
140C3×C4×D4—101no—
141C3×C4×D4AGL(1,𝔽5)11yescharacteristic, complemented
142C2×C3×C4⋊C4—51no—
143C2×C3×C4⋊C4—52no—
144C2×C3×C4⋊C4—101no—
145C2×C3×C4⋊C4—101no—
146C2×C3×C22⋊C4—101no—
147C2×C3×C42—51no—
148C3×C4⋊C8—101no—
149C3×C4⋊C8—101no—
150C3×#102—104no—
151C23×D5—21no—
152C22×AGL(1,𝔽5)—22no—
153C2×C5×D4C2×C3×C411yescharacteristic
154C22×C4×C5—21no—
155C2×C22⋊D5—21no—
156C22×Dic5—21no—
157D4×D5—21no—
158D4×D5—21no—
159D4×D5C3×D412yes—
160D4×D5C3×Q812yes—
161C2×D20C2×C3×C411yescharacteristic
162C2×C4×D5—21no—
163C2×C4×D5—21no—
164C2×C4×D5—21no—
165C2×C4×D5—42no—
166C2×C4×D5—42no—
167C2×C4×D5C3×D411yescharacteristic
168C2×C4×D5C3×D411yescharacteristic
169C2×C4×D5C23×C311yescharacteristic
170C2×#170—22no—
171C4⋊AGL(1,𝔽5)—22no—
172C4⋊AGL(1,𝔽5)—22no—
173C4⋊AGL(1,𝔽5)—42no—
174C4⋊AGL(1,𝔽5)—42no—
175D5⋊C8—22no—
176D5⋊C8—22no—
177C5×C4⋊C4C2×C3×C411yescharacteristic
178C5×C22⋊C4—21no—
179C42×C5C2×C3×C411yescharacteristic
180C22⋊Dic5—21no—
181#685—21no—
182#685—21no—
183C4⋊Dic5C2×C3×C411yescharacteristic
184#683—21no—
185C4×Dic5C2×C3×C411yescharacteristic
186C4×Dic5C3×D411yescharacteristic
187C4×Dic5C3×D411yescharacteristic
188C2×C4×D4—51no—
189C2×C4⋊C8—51no—
190C2×#102—52no—
191#384—51no—
192#340—52no—
193C22×C3×C5—22no—
194C22×C3×C5—41no—
195C22×C3×C5—41no—
196C22×C3×C5C2×C22⋊C411yescharacteristic
197C2×C3×D5—22no—
198C2×C3×D5—41no—
199C2×C3×D5—41no—
200C2×C3×D5—42no—
201C2×C3×D5—42no—
202C2×C3×D5—42no—
203C2×C3×D5—42no—
204C2×C3×D5#12111yescharacteristic
205C2×C3×D5C22⋊Q811yescharacteristic
206C2×C3×D5C2×D811yescharacteristic
207C2×C3×D5C2×SD1611yescharacteristic
208C2×C3×D5AGL(1,ℤ/8)11yescharacteristic
209C2×C3×D5#13711yescharacteristic
210C3×AGL(1,𝔽5)—82no—
211C3×C4×C5—21no—
212C3×C4×C5—41no—
213C3×C4×C5#10212yescomplemented
214C3×C4×C5C2×C4⋊C411yescharacteristic
215C3×C4×C5#11711yescharacteristic
216C3×Dic5—21no—
217C3×Dic5—41no—
218C3×Dic5#10212yescomplemented
219C3×Dic5#12111yescharacteristic
220C3×Dic5C22⋊Q811yescharacteristic
221C24×C3—101no—
222C2×C3×D4—52no—
223C2×C3×D4—52no—
224C2×C3×D4—51no—
225C2×C3×D4—101no—
226C2×C3×D4—101no—
227C2×C3×D4—101no—
228C2×C3×D4C2×AGL(1,𝔽5)11yescharacteristic
229C22×C3×C4—21no—
230C22×C3×C4—51no—
231C22×C3×C4—51no—
232C22×C3×C4—51no—
233C22×C3×C4—101no—
234C22×C3×C4—101no—
235C22×C3×C4—101no—
236C22×C3×C4—101no—
237C22×C3×C4—102no—
238C22×C3×C4—202no—
239C22×C3×C4—202no—
240C2×C3×C8—102no—
241C2×C3×C8—102no—
242C2×C3×C8—102no—
243C3×C4⋊C4—51no—
244C3×C4⋊C4—101no—
245C3×C4⋊C4—102no—
246C3×C4⋊C4—102no—
247C3×C4⋊C4—202no—
248C3×C4⋊C4—202no—
249C3×C4⋊C4C2×AGL(1,𝔽5)11yescharacteristic
250C3×C22⋊C4—21no—
251C3×C22⋊C4—101no—
252C3×C22⋊C4—101no—
253C3×C22⋊C4—101no—
254C3×C42—51no—
255C3×C42—51no—
256C3×C42—51no—
257C3×C42C2×AGL(1,𝔽5)11yescharacteristic
258C23×C5—21no—
259C22×D5—21no—
260C22×D5—22no—
261C22×D5—22no—
262C22×D5—41no—
263C22×D5—41no—
264C22×D5—41no—
265C22×D5—41no—
266C22×D5C2×C3×D411yescharacteristic
267C2×AGL(1,𝔽5)—42no—
268C2×AGL(1,𝔽5)—42no—
269C2×AGL(1,𝔽5)—42no—
270C5×D4—21no—
271C5×D4C3×C4⋊C412yescomplemented
272C2×C4×C5—21no—
273C2×C4×C5—42no—
274C2×C4×C5C3×C22⋊C411yescharacteristic
275C2×C4×C5C3×C22⋊C411yescharacteristic
276C2×C4×C5C22×C3×C411yescharacteristic
277C22⋊D5—22no—
278C22⋊D5—22no—
279C2×Dic5—21no—
280C2×Dic5—42no—
281C2×Dic5C3×C22⋊C411yescharacteristic
282C2×Dic5C3×C22⋊C411yescharacteristic
283C2×Dic5C2×C3×D411yescharacteristic
284D20—21no—
285D20C3×C4⋊C412yescomplemented
286C4×D5—21no—
287C4×D5—21no—
288C4×D5—41no—
289C4×D5—41no—
290C4×D5—42no—
291C4×D5—42no—
292C4×D5C3×D812yes—
293C4×D5C3×SD1612yes—
294C4×D5C2×C3×D411yescharacteristic
295C4×D5C2×C3×Q811yescharacteristic
296C4×D5C3×#4111yescharacteristic
297C4×D5C3×#4111yescharacteristic
298#170—42no—
299C22×D4—51no—
300C23×C4—101no—
301C22×C8—52no—
302C4×D4—51no—
303C4×D4—51no—
304C4×D4—51no—
305C4×D4—101no—
306C4×D4—101no—
307C4×D4C3×AGL(1,𝔽5)11yescharacteristic, complemented
308C2×C4⋊C4—51no—
309C2×C4⋊C4—52no—
310C2×C4⋊C4—101no—
311C2×C4⋊C4—101no—
312C2×C22⋊C4—101no—
313C2×C42—51no—
314C4⋊C8—101no—
315C4⋊C8—101no—
316#102—104no—
317C2×C3×C5—42no—
318C2×C3×C5#38011yescharacteristic
319C2×C3×C5C2×#10211yescharacteristic
320C2×C3×C5#41711yescharacteristic
321C3×D5—42no—
322C3×D5#45911yescharacteristic
323C3×D5#46111yescharacteristic
324C3×D5#47411yescharacteristic
325C3×D5#47611yescharacteristic
326C23×C3—21no—
327C23×C3—51no—
328C23×C3—102no—
329C23×C3—102no—
330C23×C3—101no—
331C23×C3—201no—
332C23×C3—201no—
333C23×C3—201no—
334C23×C3—201no—
335C3×D4—21no—
336C3×D4—52no—
337C3×D4—102no—
338C3×D4—102no—
339C3×D4—101no—
340C3×D4C4⋊AGL(1,𝔽5)12yescomplemented
341C2×C3×C4—21no—
342C2×C3×C4—42no—
343C2×C3×C4—51no—
344C2×C3×C4—51no—
345C2×C3×C4—51no—
346C2×C3×C4—51no—
347C2×C3×C4—51no—
348C2×C3×C4—51no—
349C2×C3×C4—52no—
350C2×C3×C4—52no—
351C2×C3×C4—51no—
352C2×C3×C4—101no—
353C2×C3×C4—101no—
354C2×C3×C4—101no—
355C2×C3×C4—202no—
356C2×C3×C4—201no—
357C2×C3×C4—201no—
358C2×C3×C4—202no—
359C2×C3×C4—202no—
360C2×C3×C4—202no—
361C2×C3×C4—202no—
362C2×C3×C4—202no—
363C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
364C2×C3×C4C22⋊AGL(1,𝔽5)11yescharacteristic
365C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
366C3×C8—202no—
367C22×C5—22no—
368C22×C5—41no—
369C22×C5—41no—
370C22×C5C2×C3×C22⋊C411yescharacteristic
371C2×D5—22no—
372C2×D5—41no—
373C2×D5—41no—
374C2×D5—42no—
375C2×D5—42no—
376C2×D5—42no—
377C2×D5—42no—
378C2×D5C3×#12111yescharacteristic
379C2×D5C3×C22⋊Q811yescharacteristic
380C2×D5C2×C3×D811yescharacteristic
381C2×D5C2×C3×SD1611yescharacteristic
382C2×D5C3×AGL(1,ℤ/8)11yescharacteristic
383C2×D5C3×#13711yescharacteristic
384AGL(1,𝔽5)—82no—
385C4×C5—21no—
386C4×C5—41no—
387C4×C5C3×#10212yescomplemented
388C4×C5C2×C3×C4⋊C411yescharacteristic
389C4×C5C3×#11711yescharacteristic
390Dic5—21no—
391Dic5—41no—
392Dic5C3×#10212yescomplemented
393Dic5C3×#12111yescharacteristic
394Dic5C3×C22⋊Q811yescharacteristic
395C24—101no—
396C2×D4—52no—
397C2×D4—52no—
398C2×D4—51no—
399C2×D4—101no—
400C2×D4—101no—
401C2×D4—101no—
402C2×D4C2×C3×AGL(1,𝔽5)11yescharacteristic
403C22×C4—21no—
404C22×C4—51no—
405C22×C4—51no—
406C22×C4—51no—
407C22×C4—101no—
408C22×C4—101no—
409C22×C4—101no—
410C22×C4—101no—
411C22×C4—102no—
412C22×C4—202no—
413C22×C4—202no—
414C2×C8—102no—
415C2×C8—102no—
416C2×C8—102no—
417C4⋊C4—51no—
418C4⋊C4—101no—
419C4⋊C4—102no—
420C4⋊C4—102no—
421C4⋊C4—202no—
422C4⋊C4—202no—
423C4⋊C4C2×C3×AGL(1,𝔽5)11yescharacteristic
424C22⋊C4—21no—
425C22⋊C4—101no—
426C22⋊C4—101no—
427C22⋊C4—101no—
428C42—51no—
429C42—51no—
430C42—51no—
431C42C2×C3×AGL(1,𝔽5)11yescharacteristic
432C3×C5#180211yescharacteristic, complemented, Hall
433C22×C3—22no—
434C22×C3—41no—
435C22×C3—41no—
436C22×C3—51no—
437C22×C3—51no—
438C22×C3—51no—
439C22×C3—51no—
440C22×C3—51no—
441C22×C3—51no—
442C22×C3—102no—
443C22×C3—202no—
444C22×C3—202no—
445C22×C3—202no—
446C22×C3—202no—
447C22×C3—201no—
448C22×C3—201no—
449C22×C3C2×C22⋊AGL(1,𝔽5)11yescharacteristic
450C3×C4—21no—
451C3×C4—41no—
452C3×C4—51no—
453C3×C4—51no—
454C3×C4—52no—
455C3×C4—101no—
456C3×C4—201no—
457C3×C4—402no—
458C3×C4#400712yescomplemented
459C3×C4C2×C4⋊AGL(1,𝔽5)11yescharacteristic
460C3×C4#413011yescharacteristic
461C2×C5—42no—
462C2×C5C3×#38011yescharacteristic
463C2×C5C2×C3×#10211yescharacteristic
464C2×C5C3×#41711yescharacteristic
465D5—42no—
466D5C3×#45911yescharacteristic
467D5C3×#46111yescharacteristic
468D5C3×#47411yescharacteristic
469D5C3×#47611yescharacteristic
470C23—21no—
471C23—51no—
472C23—102no—
473C23—102no—
474C23—101no—
475C23—201no—
476C23—201no—
477C23—201no—
478C23—201no—
479D4—21no—
480D4—52no—
481D4—102no—
482D4—102no—
483D4—101no—
484D4C3×C4⋊AGL(1,𝔽5)12yescomplemented
485C2×C4—21no—
486C2×C4—42no—
487C2×C4—51no—
488C2×C4—51no—
489C2×C4—51no—
490C2×C4—51no—
491C2×C4—51no—
492C2×C4—51no—
493C2×C4—52no—
494C2×C4—52no—
495C2×C4—51no—
496C2×C4—101no—
497C2×C4—101no—
498C2×C4—101no—
499C2×C4—202no—
500C2×C4—201no—
501C2×C4—201no—
502C2×C4—202no—
503C2×C4—202no—
504C2×C4—202no—
505C2×C4—202no—
506C2×C4—202no—
507C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
508C2×C4C3×C22⋊AGL(1,𝔽5)11yescharacteristic
509C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
510C8—202no—
511C2×C3—42no—
512C2×C3—51no—
513C2×C3—51no—
514C2×C3—51no—
515C2×C3—51no—
516C2×C3—202no—
517C2×C3#6576511yescharacteristic
518C2×C3C2×#400711yescharacteristic
519C2×C3#6579311yescharacteristic
520C5C3×#180211yescharacteristic, complemented, Sylow
521C22—22no—
522C22—41no—
523C22—41no—
524C22—51no—
525C22—51no—
526C22—51no—
527C22—51no—
528C22—51no—
529C22—51no—
530C22—102no—
531C22—202no—
532C22—202no—
533C22—202no—
534C22—202no—
535C22—201no—
536C22—201no—
537C22C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
538C4—21no—
539C4—41no—
540C4—51no—
541C4—51no—
542C4—52no—
543C4—101no—
544C4—201no—
545C4—402no—
546C4C3×#400712yescomplemented
547C4C2×C3×C4⋊AGL(1,𝔽5)11yescharacteristic
548C4C3×#413011yescharacteristic
549C3#1060441711yescharacteristic, direct product factor, Sylow
550C2—42no—
551C2—51no—
552C2—51no—
553C2—51no—
554C2—51no—
555C2—202no—
556C2C3×#6576511yescharacteristic
557C2C2×C3×#400711yescharacteristic
558C2C3×#6579311yescharacteristic
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×#1802
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×#65688C2semidirect
4. C3×D5⋊(C4⋊C8)C2semidirect
5. C3×#64874C2semidirect
6. #10604417C3direct (N × H)
13. C3×C4×C5×D4C4semidirect
14. C3×D4×Dic5C4semidirect
16. C3×#4039C4semidirect
20. C3×C4×D20C4semidirect
28. #65688C2×C3semidirect
29. D5⋊(C4⋊C8)C2×C3semidirect
30. #64874C2×C3semidirect
79. C4×C5×D4C3×C4semidirect
80. D4×Dic5C3×C4semidirect
82. #4039C3×C4semidirect
86. C4×D20C3×C4semidirect
105. C3×C5×D4C4⋊C4semidirect
119. C3×D20C4⋊C4semidirect
141. C3×C4×D4AGL(1,𝔽5)semidirect
213. C3×C4×C5#102semidirect
218. C3×Dic5#102semidirect
271. C5×D4C3×C4⋊C4semidirect
285. D20C3×C4⋊C4semidirect
307. C4×D4C3×AGL(1,𝔽5)semidirect
340. C3×D4C4⋊AGL(1,𝔽5)semidirect
387. C4×C5C3×#102semidirect
392. Dic5C3×#102semidirect
432. C3×C5#1802semidirect
458. C3×C4#4007semidirect
484. D4C3×C4⋊AGL(1,𝔽5)semidirect
520. C5C3×#1802semidirect
546. C4C3×#4007semidirect
549. C3#10604417direct (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×#656882
C3×D5⋊(C4⋊C8)2
C3×#648742
#106044173
C3×#18025