Finite Group Databasebeta

← C3×#10604757 #14295286

Subgroups — C3×#10604757 #14295286

Order 1920.

696 conjugacy classes of subgroups.

The subgroup lattice has 516 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×D5×M16C211yesmaximal, characteristic
2C3×C4×C22⋊AGL(1,𝔽5)C211yesmaximal, characteristic
3C3×D5⋊(C22⋊C8)C211yesmaximal, characteristic
4C2×C3×#3992C211yesmaximal, characteristic, complemented
5C2×C3×C8×AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
6C3×(C2×AGL(1,𝔽5))⋊C8C211yesmaximal, characteristic, complemented
7C3×(C2×AGL(1,𝔽5))⋊C8C211yesmaximal, characteristic, complemented
8#10604757C311yesmaximal, characteristic, direct product factor, Hall
9C22×C3×C4×D5C2211yescharacteristic
10C2×C3×C22⋊AGL(1,𝔽5)C412yes—
11C2×C3×C4×AGL(1,𝔽5)C2211yescharacteristic
12C2×C3×C4×AGL(1,𝔽5)C2211yescharacteristic
13C2×C3×D5⋊C8C2211yescharacteristic
14C2×C3×D5⋊C8C2211yescharacteristic
15C2×C3×C5×M16C411yescharacteristic, complemented
16C2×C3×#681C411yescharacteristic, complemented
17C3×D5×M16—22no—
18C3×D5×M16—22no—
19C2×C3×#676C411yescharacteristic, complemented
20C2×C3×#676C411yescharacteristic, complemented
21C2×C3×C8×D5C2211yescharacteristic
22C2×C3×C8×D5C2211yescharacteristic
23C3×C22⋊#170C412yescomplemented
24C3×(C2×AGL(1,𝔽5))⋊C4C412yes—
25C3×#4003C412yescomplemented
26C3×#3992—22no—
27C3×C8×AGL(1,𝔽5)—22no—
28C3×#1753—51nomaximal, Hall
29C2×D5×M16C2×C311yescharacteristic
30C4×C22⋊AGL(1,𝔽5)C2×C311yescharacteristic
31D5⋊(C22⋊C8)C2×C311yescharacteristic
32C2×#3992C2×C311yescharacteristic, complemented
33C2×C8×AGL(1,𝔽5)C2×C311yescharacteristic, complemented
34(C2×AGL(1,𝔽5))⋊C8C2×C311yescharacteristic, complemented
35(C2×AGL(1,𝔽5))⋊C8C2×C311yescharacteristic, complemented
36C23×C3×D5C2×C411yescharacteristic
37C22×C3×AGL(1,𝔽5)C2×C412yes—
38C22×C3×AGL(1,𝔽5)C2×C412yes—
39C22×C3×C4×C5C2×C411yescharacteristic
40C22×C3×Dic5C2×C411yescharacteristic
41C2×C3×C4×D5—21no—
42C2×C3×C4×D5—21no—
43C2×C3×C4×D5—21no—
44C2×C3×C4×D5—21no—
45C2×C3×C4×D5C2×C411yescharacteristic
46C2×C3×C4×D5C2×C411yescharacteristic
47C2×C3×C4×D5C2×C411yescharacteristic
48C2×C3×C4×D5C2311yescharacteristic
49C3×C22⋊AGL(1,𝔽5)—24no—
50C2×C3×#170C2×C412yes—
51C2×C3×#170C2×C412yes—
52C3×C4×AGL(1,𝔽5)—21no—
53C3×C4×AGL(1,𝔽5)—21no—
54C3×C4×AGL(1,𝔽5)—21no—
55C3×C4×AGL(1,𝔽5)—21no—
56C3×D5⋊C8—21no—
57C3×D5⋊C8—21no—
58C3×D5⋊C8—21no—
59C3×D5⋊C8—21no—
60C3×C5×M16—22no—
61C2×C3×C5×C8C2×C411yescharacteristic
62C2×C3×C5×C8C2×C411yescharacteristic
63C3×#681—22no—
64C2×C3×#168C2×C411yescharacteristic
65C2×C3×#168C2×C411yescharacteristic
66C3×#676—22no—
67C3×#676—22no—
68C3×C8×D5—21no—
69C3×C8×D5—21no—
70C3×C8×D5D412yes—
71C3×C8×D5D412yes—
72C22×C3×M16—51no—
73C2×C3×C22⋊C8—51no—
74C2×C3×#97—51no—
75C2×C3×C4×C8—51no—
76C3×C4×C22⋊C4—51no—
77C3×#336—51no—
78C3×#336—51no—
79C22×C4×D5C22×C311yescharacteristic
80C2×C22⋊AGL(1,𝔽5)C3×C412yes—
81C2×C4×AGL(1,𝔽5)C22×C311yescharacteristic
82C2×C4×AGL(1,𝔽5)C22×C311yescharacteristic
83C2×D5⋊C8C22×C311yescharacteristic
84C2×D5⋊C8C22×C311yescharacteristic
85C2×C5×M16C3×C411yescharacteristic, complemented
86C2×#681C3×C411yescharacteristic, complemented
87D5×M16—22no—
88D5×M16—22no—
89C2×#676C3×C411yescharacteristic, complemented
90C2×#676C3×C411yescharacteristic, complemented
91C2×C8×D5C22×C311yescharacteristic
92C2×C8×D5C22×C311yescharacteristic
93C22⋊#170C3×C412yescomplemented
94(C2×AGL(1,𝔽5))⋊C4C3×C412yes—
95#4003C3×C412yescomplemented
96#3992—22no—
97C8×AGL(1,𝔽5)—22no—
98#1753—51noSylow
99C23×C3×C5C4211yescharacteristic
100C22×C3×D5—21no—
101C22×C3×D5—21no—
102C22×C3×D5—22no—
103C22×C3×D5—22no—
104C22×C3×D5C4211yescharacteristic
105C22×C3×D5C22×C411yescharacteristic
106C2×C3×AGL(1,𝔽5)—22no—
107C2×C3×AGL(1,𝔽5)—22no—
108C2×C3×AGL(1,𝔽5)—22no—
109C2×C3×AGL(1,𝔽5)—22no—
110C2×C3×AGL(1,𝔽5)—22no—
111C2×C3×AGL(1,𝔽5)M1614yescomplemented
112C2×C3×C4×C5—21no—
113C2×C3×C4×C5—21no—
114C2×C3×C4×C5C4211yescharacteristic
115C2×C3×C4×C5C22×C411yescharacteristic
116C2×C3×Dic5—21no—
117C2×C3×Dic5—21no—
118C2×C3×Dic5C4211yescharacteristic
119C2×C3×Dic5C22×C411yescharacteristic
120C3×C4×D5—21no—
121C3×C4×D5—21no—
122C3×C4×D5—21no—
123C3×C4×D5—21no—
124C3×C4×D5—21no—
125C3×C4×D5—21no—
126C3×C4×D5C2×D411yescharacteristic
127C3×C4×D5C2×D411yescharacteristic
128C3×C4×D5#4111yescharacteristic
129C3×C4×D5#4111yescharacteristic
130C3×#170—22no—
131C3×#170—22no—
132C3×C5×C8—21no—
133C3×C5×C8C22⋊C412yescomplemented
134C3×#168—21no—
135C3×#168C22⋊C412yescomplemented
136C23×C3×C4—51no—
137C2×C3×M16—51no—
138C2×C3×M16—51no—
139C2×C3×M16—51no—
140C2×C3×M16—102no—
141C2×C3×M16—102no—
142C2×C3×M16AGL(1,𝔽5)11yescharacteristic, complemented
143C22×C3×C8—51no—
144C22×C3×C8—51no—
145C22×C3×C8—51no—
146C22×C3×C8—51no—
147C2×C3×C22⋊C4—52no—
148C2×C3×C42—51no—
149C2×C3×C42—51no—
150C3×C22⋊C8—52no—
151C3×C22⋊C8—52no—
152C3×#97—102no—
153C3×C4×C8—102no—
154C3×#95—52no—
155C23×D5C2×C3×C411yescharacteristic
156C22×AGL(1,𝔽5)C2×C3×C412yes—
157C22×AGL(1,𝔽5)C2×C3×C412yes—
158C22×C4×C5C2×C3×C411yescharacteristic
159C22×Dic5C2×C3×C411yescharacteristic
160C2×C4×D5—21no—
161C2×C4×D5—21no—
162C2×C4×D5—21no—
163C2×C4×D5—21no—
164C2×C4×D5C2×C3×C411yescharacteristic
165C2×C4×D5C2×C3×C411yescharacteristic
166C2×C4×D5C2×C3×C411yescharacteristic
167C2×C4×D5C23×C311yescharacteristic
168C22⋊AGL(1,𝔽5)—24no—
169C2×#170C2×C3×C412yes—
170C2×#170C2×C3×C412yes—
171C4×AGL(1,𝔽5)—21no—
172C4×AGL(1,𝔽5)—21no—
173C4×AGL(1,𝔽5)—21no—
174C4×AGL(1,𝔽5)—21no—
175D5⋊C8—21no—
176D5⋊C8—21no—
177D5⋊C8—21no—
178D5⋊C8—21no—
179C5×M16—22no—
180C2×C5×C8C2×C3×C411yescharacteristic
181C2×C5×C8C2×C3×C411yescharacteristic
182#681—22no—
183C2×#168C2×C3×C411yescharacteristic
184C2×#168C2×C3×C411yescharacteristic
185#676—22no—
186#676—22no—
187C8×D5—21no—
188C8×D5—21no—
189C8×D5C3×D412yes—
190C8×D5C3×D412yes—
191C22×M16—51no—
192C2×C22⋊C8—51no—
193C2×#97—51no—
194C2×C4×C8—51no—
195C4×C22⋊C4—51no—
196#336—51no—
197#336—51no—
198C22×C3×C5—21no—
199C22×C3×C5—21no—
200C22×C3×C5—21no—
201C22×C3×C5C2×C4211yescharacteristic
202C2×C3×D5—21no—
203C2×C3×D5—21no—
204C2×C3×D5—21no—
205C2×C3×D5—42no—
206C2×C3×D5—42no—
207C2×C3×D5C4×D411yescharacteristic
208C2×C3×D5C4×D411yescharacteristic
209C2×C3×D5C2×M1612yes—
210C2×C3×D5#13112yes—
211C3×AGL(1,𝔽5)—24no—
212C3×AGL(1,𝔽5)—42no—
213C3×C4×C5—21no—
214C3×C4×C5C2×C22⋊C411yescharacteristic
215C3×C4×C5#11711yescharacteristic
216C3×Dic5—21no—
217C3×Dic5C4×D411yescharacteristic
218C3×Dic5C4×D411yescharacteristic
219C24×C3—51no—
220C22×C3×C4—51no—
221C22×C3×C4—52no—
222C22×C3×C4—51no—
223C22×C3×C4—51no—
224C22×C3×C4—51no—
225C22×C3×C4—51no—
226C22×C3×C4—52no—
227C22×C3×C4—101no—
228C22×C3×C4—101no—
229C22×C3×C4—101no—
230C22×C3×C4—101no—
231C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
232C3×M16—22no—
233C3×M16—102no—
234C3×M16—102no—
235C3×M16—102no—
236C2×C3×C8—51no—
237C2×C3×C8—52no—
238C2×C3×C8—52no—
239C2×C3×C8—52no—
240C2×C3×C8—52no—
241C2×C3×C8—51no—
242C2×C3×C8—101no—
243C2×C3×C8—101no—
244C2×C3×C8—101no—
245C2×C3×C8—101no—
246C2×C3×C8—101no—
247C2×C3×C8—101no—
248C2×C3×C8C2×AGL(1,𝔽5)11yescharacteristic
249C2×C3×C8C2×AGL(1,𝔽5)11yescharacteristic
250C3×C22⋊C4—104no—
251C3×C42—101no—
252C3×C42—101no—
253C3×C42—101no—
254C3×C42—101no—
255C23×C5C3×C4211yescharacteristic
256C22×D5—21no—
257C22×D5—21no—
258C22×D5—22no—
259C22×D5—22no—
260C22×D5C3×C4211yescharacteristic
261C22×D5C22×C3×C411yescharacteristic
262C2×AGL(1,𝔽5)—22no—
263C2×AGL(1,𝔽5)—22no—
264C2×AGL(1,𝔽5)—22no—
265C2×AGL(1,𝔽5)—22no—
266C2×AGL(1,𝔽5)—22no—
267C2×AGL(1,𝔽5)C3×M1614yescomplemented
268C2×C4×C5—21no—
269C2×C4×C5—21no—
270C2×C4×C5C3×C4211yescharacteristic
271C2×C4×C5C22×C3×C411yescharacteristic
272C2×Dic5—21no—
273C2×Dic5—21no—
274C2×Dic5C3×C4211yescharacteristic
275C2×Dic5C22×C3×C411yescharacteristic
276C4×D5—21no—
277C4×D5—21no—
278C4×D5—21no—
279C4×D5—21no—
280C4×D5—21no—
281C4×D5—21no—
282C4×D5C2×C3×D411yescharacteristic
283C4×D5C2×C3×D411yescharacteristic
284C4×D5C3×#4111yescharacteristic
285C4×D5C3×#4111yescharacteristic
286#170—22no—
287#170—22no—
288C5×C8—21no—
289C5×C8C3×C22⋊C412yescomplemented
290#168—21no—
291#168C3×C22⋊C412yescomplemented
292C23×C4—51no—
293C2×M16—51no—
294C2×M16—51no—
295C2×M16—51no—
296C2×M16—102no—
297C2×M16—102no—
298C2×M16C3×AGL(1,𝔽5)11yescharacteristic, complemented
299C22×C8—51no—
300C22×C8—51no—
301C22×C8—51no—
302C22×C8—51no—
303C2×C22⋊C4—52no—
304C2×C42—51no—
305C2×C42—51no—
306C22⋊C8—52no—
307C22⋊C8—52no—
308#97—102no—
309C4×C8—102no—
310#95—52no—
311C2×C3×C5—41no—
312C2×C3×C5C4×C22⋊C411yescharacteristic
313C2×C3×C5C4×M1611yescharacteristic
314C2×C3×C5#40511yescharacteristic
315C3×D5—41no—
316C3×D5#43512yes—
317C3×D5#43612yes—
318C23×C3—51no—
319C23×C3—51no—
320C23×C3—102no—
321C23×C3—101no—
322C23×C3—101no—
323C23×C3—102no—
324C23×C3C4×AGL(1,𝔽5)11yescharacteristic
325C2×C3×C4—21no—
326C2×C3×C4—21no—
327C2×C3×C4—51no—
328C2×C3×C4—51no—
329C2×C3×C4—51no—
330C2×C3×C4—51no—
331C2×C3×C4—51no—
332C2×C3×C4—54no—
333C2×C3×C4—51no—
334C2×C3×C4—101no—
335C2×C3×C4—101no—
336C2×C3×C4—101no—
337C2×C3×C4—101no—
338C2×C3×C4—101no—
339C2×C3×C4—101no—
340C2×C3×C4—102no—
341C2×C3×C4—102no—
342C2×C3×C4—101no—
343C2×C3×C4—101no—
344C2×C3×C4—102no—
345C2×C3×C4—102no—
346C2×C3×C4—102no—
347C2×C3×C4C4×AGL(1,𝔽5)11yescharacteristic
348C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
349C3×C8—21no—
350C3×C8—52no—
351C3×C8—102no—
352C3×C8—102no—
353C3×C8—101no—
354C3×C8C22⋊AGL(1,𝔽5)12yescomplemented
355C22×C5—21no—
356C22×C5—21no—
357C22×C5—21no—
358C22×C5C2×C3×C4211yescharacteristic
359C2×D5—21no—
360C2×D5—21no—
361C2×D5—21no—
362C2×D5—42no—
363C2×D5—42no—
364C2×D5C3×C4×D411yescharacteristic
365C2×D5C3×C4×D411yescharacteristic
366C2×D5C2×C3×M1612yes—
367C2×D5C3×#13112yes—
368AGL(1,𝔽5)—24no—
369AGL(1,𝔽5)—42no—
370C4×C5—21no—
371C4×C5C2×C3×C22⋊C411yescharacteristic
372C4×C5C3×#11711yescharacteristic
373Dic5—21no—
374Dic5C3×C4×D411yescharacteristic
375Dic5C3×C4×D411yescharacteristic
376C24—51no—
377C22×C4—51no—
378C22×C4—52no—
379C22×C4—51no—
380C22×C4—51no—
381C22×C4—51no—
382C22×C4—51no—
383C22×C4—52no—
384C22×C4—101no—
385C22×C4—101no—
386C22×C4—101no—
387C22×C4—101no—
388C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
389M16—22no—
390M16—102no—
391M16—102no—
392M16—102no—
393C2×C8—51no—
394C2×C8—52no—
395C2×C8—52no—
396C2×C8—52no—
397C2×C8—52no—
398C2×C8—51no—
399C2×C8—101no—
400C2×C8—101no—
401C2×C8—101no—
402C2×C8—101no—
403C2×C8—101no—
404C2×C8—101no—
405C2×C8C2×C3×AGL(1,𝔽5)11yescharacteristic
406C2×C8C2×C3×AGL(1,𝔽5)11yescharacteristic
407C22⋊C4—104no—
408C42—101no—
409C42—101no—
410C42—101no—
411C42—101no—
412C3×C5#175311yescharacteristic, complemented, Hall
413C22×C3—21no—
414C22×C3—21no—
415C22×C3—21no—
416C22×C3—52no—
417C22×C3—52no—
418C22×C3—51no—
419C22×C3—51no—
420C22×C3—101no—
421C22×C3—101no—
422C22×C3—101no—
423C22×C3—202no—
424C22×C3—202no—
425C22×C3C2×C4×AGL(1,𝔽5)11yescharacteristic
426C3×C4—21no—
427C3×C4—51no—
428C3×C4—51no—
429C3×C4—104no—
430C3×C4—101no—
431C3×C4—202no—
432C3×C4#413011yescharacteristic
433C3×C4C2×C22⋊AGL(1,𝔽5)11yescharacteristic
434C2×C5—41no—
435C2×C5C3×C4×C22⋊C411yescharacteristic
436C2×C5C3×C4×M1611yescharacteristic
437C2×C5C3×#40511yescharacteristic
438D5—41no—
439D5C3×#43512yes—
440D5C3×#43612yes—
441C23—51no—
442C23—51no—
443C23—102no—
444C23—101no—
445C23—101no—
446C23—102no—
447C23C3×C4×AGL(1,𝔽5)11yescharacteristic
448C2×C4—21no—
449C2×C4—21no—
450C2×C4—51no—
451C2×C4—51no—
452C2×C4—51no—
453C2×C4—51no—
454C2×C4—51no—
455C2×C4—54no—
456C2×C4—51no—
457C2×C4—101no—
458C2×C4—101no—
459C2×C4—101no—
460C2×C4—101no—
461C2×C4—101no—
462C2×C4—101no—
463C2×C4—102no—
464C2×C4—102no—
465C2×C4—101no—
466C2×C4—101no—
467C2×C4—102no—
468C2×C4—102no—
469C2×C4—102no—
470C2×C4C3×C4×AGL(1,𝔽5)11yescharacteristic
471C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
472C8—21no—
473C8—52no—
474C8—102no—
475C8—102no—
476C8—101no—
477C8C3×C22⋊AGL(1,𝔽5)12yescomplemented
478C2×C3—41no—
479C2×C3—52no—
480C2×C3—52no—
481C2×C3—201no—
482C2×C3M16×AGL(1,𝔽5)11yescharacteristic
483C2×C3#6572811yescharacteristic
484C2×C3C4×C22⋊AGL(1,𝔽5)11yescharacteristic
485C5C3×#175311yescharacteristic, complemented, Sylow
486C22—21no—
487C22—21no—
488C22—21no—
489C22—52no—
490C22—52no—
491C22—51no—
492C22—51no—
493C22—101no—
494C22—101no—
495C22—101no—
496C22—202no—
497C22—202no—
498C22C2×C3×C4×AGL(1,𝔽5)11yescharacteristic
499C4—21no—
500C4—51no—
501C4—51no—
502C4—104no—
503C4—101no—
504C4—202no—
505C4C3×#413011yescharacteristic
506C4C2×C3×C22⋊AGL(1,𝔽5)11yescharacteristic
507C3#1060475711yescharacteristic, direct product factor, Sylow
508C2—41no—
509C2—52no—
510C2—52no—
511C2—201no—
512C2C3×M16×AGL(1,𝔽5)11yescharacteristic
513C2C3×#6572811yescharacteristic
514C2C3×C4×C22⋊AGL(1,𝔽5)11yescharacteristic
subgroupG/H
centerC2×C3×C4C22×AGL(1,𝔽5)
commutatorC22×C5C2×C3×C42
FrattiniC2×C4C22×C3×AGL(1,𝔽5)
FittingC2×C3×C5×M16C4
socleC22×C3×C5C2×C42
prime radicalC23×C3×D5C2×C4
cosocleC2×C4×D5C23×C3
hypercenterC2×C3×M16AGL(1,𝔽5)
hypocenterC5C3×#1753
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C2×C3×C4 ◁ C2×C3×M16

Complements

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

NH ≅ G/Ntype
4. C2×C3×#3992C2semidirect
5. C2×C3×C8×AGL(1,𝔽5)C2semidirect
6. C3×(C2×AGL(1,𝔽5))⋊C8C2semidirect
7. C3×(C2×AGL(1,𝔽5))⋊C8C2semidirect
8. #10604757C3direct (N × H)
15. C2×C3×C5×M16C4semidirect
16. C2×C3×#681C4semidirect
19. C2×C3×#676C4semidirect
20. C2×C3×#676C4semidirect
23. C3×C22⋊#170C4semidirect
25. C3×#4003C4semidirect
32. C2×#3992C2×C3semidirect
33. C2×C8×AGL(1,𝔽5)C2×C3semidirect
34. (C2×AGL(1,𝔽5))⋊C8C2×C3semidirect
35. (C2×AGL(1,𝔽5))⋊C8C2×C3semidirect
85. C2×C5×M16C3×C4semidirect
86. C2×#681C3×C4semidirect
89. C2×#676C3×C4semidirect
90. C2×#676C3×C4semidirect
93. C22⋊#170C3×C4semidirect
95. #4003C3×C4semidirect
111. C2×C3×AGL(1,𝔽5)M16semidirect
133. C3×C5×C8C22⋊C4semidirect
135. C3×#168C22⋊C4semidirect
142. C2×C3×M16AGL(1,𝔽5)semidirect
267. C2×AGL(1,𝔽5)C3×M16semidirect
289. C5×C8C3×C22⋊C4semidirect
291. #168C3×C22⋊C4semidirect
298. C2×M16C3×AGL(1,𝔽5)semidirect
354. C3×C8C22⋊AGL(1,𝔽5)semidirect
412. C3×C5#1753semidirect
477. C8C3×C22⋊AGL(1,𝔽5)semidirect
485. C5C3×#1753semidirect
507. C3#10604757direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×D5×M162
C3×C4×C22⋊AGL(1,𝔽5)2
C3×D5⋊(C22⋊C8)2
C2×C3×#39922
C2×C3×C8×AGL(1,𝔽5)2
C3×(C2×AGL(1,𝔽5))⋊C82
#106047573
C3×#17535