Finite Group Databasebeta

← C3×#10603915 #14295292

Subgroups — C3×#10603915 #14295292

Order 1920.

696 conjugacy classes of subgroups.

The subgroup lattice has 508 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
3C2×C3×#3992C211yesmaximal, characteristic, complemented
4C2×C3×C8×AGL(1,𝔽5)C211yesmaximal, characteristic, complemented
5C3×D5×C22⋊C8C211yesmaximal, characteristic
6C3×(C2×AGL(1,𝔽5))⋊C8C211yesmaximal, characteristic, complemented
7C3×(C2×AGL(1,𝔽5))⋊C8C211yesmaximal, characteristic, complemented
8#10603915C311yesmaximal, characteristic, direct product factor, Hall
9C22×C3×C4×D5C2211yescharacteristic
10C2×C3×C22⋊AGL(1,𝔽5)C412yes—
11C2×C3×#704C412yescomplemented
12C2×C3×C4×AGL(1,𝔽5)C2211yescharacteristic
13C2×C3×C4×AGL(1,𝔽5)C2211yescharacteristic
14C3×D5⋊M16—22no—
15C3×D5⋊M16—22no—
16C2×C3×#700C412yescomplemented
17C2×C3×D5⋊C8C2211yescharacteristic
18C2×C3×D5⋊C8C2211yescharacteristic
19C2×C3×C8×D5C2211yescharacteristic
20C2×C3×C8×D5C2211yescharacteristic
21C3×(C2×AGL(1,𝔽5))⋊C4C412yes—
22C3×#3992—22no—
23C3×C8×AGL(1,𝔽5)—22no—
24C3×C5×C22⋊C8C411yescharacteristic, complemented
25C3×C22⋊#168C411yescharacteristic, complemented
26C3×#3952C411yescharacteristic, complemented
27C3×#3952C411yescharacteristic, complemented
28C3×#1753—51nomaximal, Hall
29C2×D5⋊M16C2×C311yescharacteristic
30C4×C22⋊AGL(1,𝔽5)C2×C311yescharacteristic
31C2×#3992C2×C311yescharacteristic, complemented
32C2×C8×AGL(1,𝔽5)C2×C311yescharacteristic, complemented
33D5×C22⋊C8C2×C311yescharacteristic
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—
50C3×#704—24no—
51C2×C3×#170C2×C412yes—
52C2×C3×#170C2×C412yes—
53C3×C4×AGL(1,𝔽5)—21no—
54C3×C4×AGL(1,𝔽5)—21no—
55C3×C4×AGL(1,𝔽5)—21no—
56C3×C4×AGL(1,𝔽5)—21no—
57C3×#700—24no—
58C3×D5⋊C8—21no—
59C3×D5⋊C8—21no—
60C3×D5⋊C8D412yes—
61C3×D5⋊C8D412yes—
62C2×C3×C5×C8C2×C411yescharacteristic
63C2×C3×C5×C8C2×C411yescharacteristic
64C2×C3×#168C2×C411yescharacteristic
65C2×C3×#168C2×C411yescharacteristic
66C3×C8×D5—21no—
67C3×C8×D5—21no—
68C3×C8×D5—21no—
69C3×C8×D5—21no—
70C22×C3×M16—51no—
71C2×C3×C22⋊C8—51no—
72C2×C3×#97—51no—
73C2×C3×C4×C8—51no—
74C3×C4×C22⋊C4—51no—
75C3×#336—51no—
76C3×#336—51no—
77C22×C4×D5C22×C311yescharacteristic
78C2×C22⋊AGL(1,𝔽5)C3×C412yes—
79C2×#704C3×C412yescomplemented
80C2×C4×AGL(1,𝔽5)C22×C311yescharacteristic
81C2×C4×AGL(1,𝔽5)C22×C311yescharacteristic
82D5⋊M16—22no—
83D5⋊M16—22no—
84C2×#700C3×C412yescomplemented
85C2×D5⋊C8C22×C311yescharacteristic
86C2×D5⋊C8C22×C311yescharacteristic
87C2×C8×D5C22×C311yescharacteristic
88C2×C8×D5C22×C311yescharacteristic
89(C2×AGL(1,𝔽5))⋊C4C3×C412yes—
90#3992—22no—
91C8×AGL(1,𝔽5)—22no—
92C5×C22⋊C8C3×C411yescharacteristic, complemented
93C22⋊#168C3×C411yescharacteristic, complemented
94#3952C3×C411yescharacteristic, complemented
95#3952C3×C411yescharacteristic, complemented
96#1753—51noSylow
97C23×C3×C5C4211yescharacteristic
98C22×C3×D5—21no—
99C22×C3×D5—21no—
100C22×C3×D5—22no—
101C22×C3×D5—22no—
102C22×C3×D5C4211yescharacteristic
103C22×C3×D5C22×C411yescharacteristic
104C2×C3×AGL(1,𝔽5)—22no—
105C2×C3×AGL(1,𝔽5)—22no—
106C2×C3×AGL(1,𝔽5)—22no—
107C2×C3×AGL(1,𝔽5)—22no—
108C2×C3×AGL(1,𝔽5)—22no—
109C2×C3×AGL(1,𝔽5)M1614yescomplemented
110C2×C3×C4×C5—21no—
111C2×C3×C4×C5—21no—
112C2×C3×C4×C5C4211yescharacteristic
113C2×C3×C4×C5C22×C411yescharacteristic
114C2×C3×Dic5—21no—
115C2×C3×Dic5—21no—
116C2×C3×Dic5C4211yescharacteristic
117C2×C3×Dic5C22×C411yescharacteristic
118C3×C4×D5—21no—
119C3×C4×D5—21no—
120C3×C4×D5—21no—
121C3×C4×D5—21no—
122C3×C4×D5—21no—
123C3×C4×D5—21no—
124C3×C4×D5C2×D411yescharacteristic
125C3×C4×D5C2×D411yescharacteristic
126C3×C4×D5#4111yescharacteristic
127C3×C4×D5#4111yescharacteristic
128C3×#170—22no—
129C3×#170C22⋊C414yescomplemented
130C3×C5×C8—21no—
131C3×C5×C8—21no—
132C3×#168—21no—
133C3×#168—21no—
134C23×C3×C4—51no—
135C2×C3×M16—52no—
136C2×C3×M16—52no—
137C2×C3×M16—102no—
138C2×C3×M16—102no—
139C22×C3×C8—51no—
140C22×C3×C8—51no—
141C22×C3×C8—51no—
142C22×C3×C8—51no—
143C2×C3×C22⋊C4—52no—
144C2×C3×C42—51no—
145C2×C3×C42—51no—
146C3×C22⋊C8—51no—
147C3×C22⋊C8—51no—
148C3×C22⋊C8—51no—
149C3×C22⋊C8AGL(1,𝔽5)11yescharacteristic, complemented
150C3×#97—102no—
151C3×C4×C8—102no—
152C3×#95—52no—
153C23×D5C2×C3×C411yescharacteristic
154C22×AGL(1,𝔽5)C2×C3×C412yes—
155C22×AGL(1,𝔽5)C2×C3×C412yes—
156C22×C4×C5C2×C3×C411yescharacteristic
157C22×Dic5C2×C3×C411yescharacteristic
158C2×C4×D5—21no—
159C2×C4×D5—21no—
160C2×C4×D5—21no—
161C2×C4×D5—21no—
162C2×C4×D5C2×C3×C411yescharacteristic
163C2×C4×D5C2×C3×C411yescharacteristic
164C2×C4×D5C2×C3×C411yescharacteristic
165C2×C4×D5C23×C311yescharacteristic
166C22⋊AGL(1,𝔽5)—24no—
167#704—24no—
168C2×#170C2×C3×C412yes—
169C2×#170C2×C3×C412yes—
170C4×AGL(1,𝔽5)—21no—
171C4×AGL(1,𝔽5)—21no—
172C4×AGL(1,𝔽5)—21no—
173C4×AGL(1,𝔽5)—21no—
174#700—24no—
175D5⋊C8—21no—
176D5⋊C8—21no—
177D5⋊C8C3×D412yes—
178D5⋊C8C3×D412yes—
179C2×C5×C8C2×C3×C411yescharacteristic
180C2×C5×C8C2×C3×C411yescharacteristic
181C2×#168C2×C3×C411yescharacteristic
182C2×#168C2×C3×C411yescharacteristic
183C8×D5—21no—
184C8×D5—21no—
185C8×D5—21no—
186C8×D5—21no—
187C22×M16—51no—
188C2×C22⋊C8—51no—
189C2×#97—51no—
190C2×C4×C8—51no—
191C4×C22⋊C4—51no—
192#336—51no—
193#336—51no—
194C22×C3×C5—21no—
195C22×C3×C5—21no—
196C22×C3×C5—21no—
197C22×C3×C5C2×C4211yescharacteristic
198C2×C3×D5—21no—
199C2×C3×D5—21no—
200C2×C3×D5—21no—
201C2×C3×D5—42no—
202C2×C3×D5—42no—
203C2×C3×D5C4×D411yescharacteristic
204C2×C3×D5C4×D411yescharacteristic
205C2×C3×D5C2×M1612yes—
206C2×C3×D5#13112yes—
207C3×AGL(1,𝔽5)—24no—
208C3×AGL(1,𝔽5)—42no—
209C3×C4×C5—21no—
210C3×C4×C5C4×D411yescharacteristic
211C3×C4×C5C4×D411yescharacteristic
212C3×Dic5—21no—
213C3×Dic5C2×C22⋊C411yescharacteristic
214C3×Dic5#11711yescharacteristic
215C24×C3—51no—
216C22×C3×C4—51no—
217C22×C3×C4—52no—
218C22×C3×C4—51no—
219C22×C3×C4—51no—
220C22×C3×C4—51no—
221C22×C3×C4—51no—
222C22×C3×C4—52no—
223C22×C3×C4—101no—
224C22×C3×C4—101no—
225C22×C3×C4—101no—
226C22×C3×C4—101no—
227C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
228C3×M16—104no—
229C3×M16—104no—
230C2×C3×C8—52no—
231C2×C3×C8—52no—
232C2×C3×C8—52no—
233C2×C3×C8—52no—
234C2×C3×C8—51no—
235C2×C3×C8—51no—
236C2×C3×C8—101no—
237C2×C3×C8—101no—
238C2×C3×C8—101no—
239C2×C3×C8—101no—
240C2×C3×C8—101no—
241C2×C3×C8—101no—
242C2×C3×C8C2×AGL(1,𝔽5)11yescharacteristic
243C2×C3×C8C2×AGL(1,𝔽5)11yescharacteristic
244C3×C22⋊C4—104no—
245C3×C42—101no—
246C3×C42—101no—
247C3×C42—101no—
248C3×C42—101no—
249C23×C5C3×C4211yescharacteristic
250C22×D5—21no—
251C22×D5—21no—
252C22×D5—22no—
253C22×D5—22no—
254C22×D5C3×C4211yescharacteristic
255C22×D5C22×C3×C411yescharacteristic
256C2×AGL(1,𝔽5)—22no—
257C2×AGL(1,𝔽5)—22no—
258C2×AGL(1,𝔽5)—22no—
259C2×AGL(1,𝔽5)—22no—
260C2×AGL(1,𝔽5)—22no—
261C2×AGL(1,𝔽5)C3×M1614yescomplemented
262C2×C4×C5—21no—
263C2×C4×C5—21no—
264C2×C4×C5C3×C4211yescharacteristic
265C2×C4×C5C22×C3×C411yescharacteristic
266C2×Dic5—21no—
267C2×Dic5—21no—
268C2×Dic5C3×C4211yescharacteristic
269C2×Dic5C22×C3×C411yescharacteristic
270C4×D5—21no—
271C4×D5—21no—
272C4×D5—21no—
273C4×D5—21no—
274C4×D5—21no—
275C4×D5—21no—
276C4×D5C2×C3×D411yescharacteristic
277C4×D5C2×C3×D411yescharacteristic
278C4×D5C3×#4111yescharacteristic
279C4×D5C3×#4111yescharacteristic
280#170—22no—
281#170C3×C22⋊C414yescomplemented
282C5×C8—21no—
283C5×C8—21no—
284#168—21no—
285#168—21no—
286C23×C4—51no—
287C2×M16—52no—
288C2×M16—52no—
289C2×M16—102no—
290C2×M16—102no—
291C22×C8—51no—
292C22×C8—51no—
293C22×C8—51no—
294C22×C8—51no—
295C2×C22⋊C4—52no—
296C2×C42—51no—
297C2×C42—51no—
298C22⋊C8—51no—
299C22⋊C8—51no—
300C22⋊C8—51no—
301C22⋊C8C3×AGL(1,𝔽5)11yescharacteristic, complemented
302#97—102no—
303C4×C8—102no—
304#95—52no—
305C2×C3×C5—41no—
306C2×C3×C5C4×C22⋊C411yescharacteristic
307C2×C3×C5C4×M1611yescharacteristic
308C2×C3×C5#40511yescharacteristic
309C3×D5—41no—
310C3×D5#43512yes—
311C3×D5#43612yes—
312C23×C3—51no—
313C23×C3—51no—
314C23×C3—102no—
315C23×C3—101no—
316C23×C3—101no—
317C23×C3—102no—
318C23×C3C4×AGL(1,𝔽5)11yescharacteristic
319C2×C3×C4—21no—
320C2×C3×C4—21no—
321C2×C3×C4—51no—
322C2×C3×C4—51no—
323C2×C3×C4—51no—
324C2×C3×C4—51no—
325C2×C3×C4—51no—
326C2×C3×C4—54no—
327C2×C3×C4—51no—
328C2×C3×C4—101no—
329C2×C3×C4—101no—
330C2×C3×C4—101no—
331C2×C3×C4—101no—
332C2×C3×C4—101no—
333C2×C3×C4—101no—
334C2×C3×C4—102no—
335C2×C3×C4—102no—
336C2×C3×C4—101no—
337C2×C3×C4—101no—
338C2×C3×C4—102no—
339C2×C3×C4—102no—
340C2×C3×C4—102no—
341C2×C3×C4C4×AGL(1,𝔽5)11yescharacteristic
342C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
343C3×C8—21no—
344C3×C8—21no—
345C3×C8—54no—
346C3×C8—102no—
347C3×C8—101no—
348C3×C8—101no—
349C22×C5—21no—
350C22×C5—21no—
351C22×C5—21no—
352C22×C5C2×C3×C4211yescharacteristic
353C2×D5—21no—
354C2×D5—21no—
355C2×D5—21no—
356C2×D5—42no—
357C2×D5—42no—
358C2×D5C3×C4×D411yescharacteristic
359C2×D5C3×C4×D411yescharacteristic
360C2×D5C2×C3×M1612yes—
361C2×D5C3×#13112yes—
362AGL(1,𝔽5)—24no—
363AGL(1,𝔽5)—42no—
364C4×C5—21no—
365C4×C5C3×C4×D411yescharacteristic
366C4×C5C3×C4×D411yescharacteristic
367Dic5—21no—
368Dic5C2×C3×C22⋊C411yescharacteristic
369Dic5C3×#11711yescharacteristic
370C24—51no—
371C22×C4—51no—
372C22×C4—52no—
373C22×C4—51no—
374C22×C4—51no—
375C22×C4—51no—
376C22×C4—51no—
377C22×C4—52no—
378C22×C4—101no—
379C22×C4—101no—
380C22×C4—101no—
381C22×C4—101no—
382C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
383M16—104no—
384M16—104no—
385C2×C8—52no—
386C2×C8—52no—
387C2×C8—52no—
388C2×C8—52no—
389C2×C8—51no—
390C2×C8—51no—
391C2×C8—101no—
392C2×C8—101no—
393C2×C8—101no—
394C2×C8—101no—
395C2×C8—101no—
396C2×C8—101no—
397C2×C8C2×C3×AGL(1,𝔽5)11yescharacteristic
398C2×C8C2×C3×AGL(1,𝔽5)11yescharacteristic
399C22⋊C4—104no—
400C42—101no—
401C42—101no—
402C42—101no—
403C42—101no—
404C3×C5#175311yescharacteristic, complemented, Hall
405C22×C3—21no—
406C22×C3—21no—
407C22×C3—21no—
408C22×C3—51no—
409C22×C3—51no—
410C22×C3—52no—
411C22×C3—52no—
412C22×C3—101no—
413C22×C3—101no—
414C22×C3—101no—
415C22×C3—202no—
416C22×C3—202no—
417C22×C3C2×C4×AGL(1,𝔽5)11yescharacteristic
418C3×C4—21no—
419C3×C4—51no—
420C3×C4—51no—
421C3×C4—104no—
422C3×C4—101no—
423C3×C4—202no—
424C3×C4D4×AGL(1,𝔽5)11yescharacteristic
425C3×C4D4×AGL(1,𝔽5)11yescharacteristic
426C2×C5—41no—
427C2×C5C3×C4×C22⋊C411yescharacteristic
428C2×C5C3×C4×M1611yescharacteristic
429C2×C5C3×#40511yescharacteristic
430D5—41no—
431D5C3×#43512yes—
432D5C3×#43612yes—
433C23—51no—
434C23—51no—
435C23—102no—
436C23—101no—
437C23—101no—
438C23—102no—
439C23C3×C4×AGL(1,𝔽5)11yescharacteristic
440C2×C4—21no—
441C2×C4—21no—
442C2×C4—51no—
443C2×C4—51no—
444C2×C4—51no—
445C2×C4—51no—
446C2×C4—51no—
447C2×C4—54no—
448C2×C4—51no—
449C2×C4—101no—
450C2×C4—101no—
451C2×C4—101no—
452C2×C4—101no—
453C2×C4—101no—
454C2×C4—101no—
455C2×C4—102no—
456C2×C4—102no—
457C2×C4—101no—
458C2×C4—101no—
459C2×C4—102no—
460C2×C4—102no—
461C2×C4—102no—
462C2×C4C3×C4×AGL(1,𝔽5)11yescharacteristic
463C2×C4C22×C3×AGL(1,𝔽5)11yescharacteristic
464C8—21no—
465C8—21no—
466C8—54no—
467C8—102no—
468C8—101no—
469C8—101no—
470C2×C3—41no—
471C2×C3—52no—
472C2×C3—52no—
473C2×C3—201no—
474C2×C3C22⋊C4×AGL(1,𝔽5)11yescharacteristic
475C2×C3#6571811yescharacteristic
476C2×C3M16×AGL(1,𝔽5)11yescharacteristic
477C5C3×#175311yescharacteristic, complemented, Sylow
478C22—21no—
479C22—21no—
480C22—21no—
481C22—51no—
482C22—51no—
483C22—52no—
484C22—52no—
485C22—101no—
486C22—101no—
487C22—101no—
488C22—202no—
489C22—202no—
490C22C2×C3×C4×AGL(1,𝔽5)11yescharacteristic
491C4—21no—
492C4—51no—
493C4—51no—
494C4—104no—
495C4—101no—
496C4—202no—
497C4C3×D4×AGL(1,𝔽5)11yescharacteristic
498C4C3×D4×AGL(1,𝔽5)11yescharacteristic
499C3#1060391511yescharacteristic, direct product factor, Sylow
500C2—41no—
501C2—52no—
502C2—52no—
503C2—201no—
504C2C3×C22⋊C4×AGL(1,𝔽5)11yescharacteristic
505C2C3×#6571811yescharacteristic
506C2C3×M16×AGL(1,𝔽5)11yescharacteristic
subgroupG/H
centerC2×C3×C4C22×AGL(1,𝔽5)
commutatorC22×C5C2×C3×C42
FrattiniC2×C4C22×C3×AGL(1,𝔽5)
FittingC3×C5×C22⋊C8C4
socleC22×C3×C5C2×C42
prime radicalC23×C3×D5C2×C4
cosocleC2×C4×D5C23×C3
hypercenterC3×C22⋊C8AGL(1,𝔽5)
hypocenterC5C3×#1753
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C2×C3×C4 ◁ C3×C22⋊C8

Complements

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

NH ≅ G/Ntype
3. C2×C3×#3992C2semidirect
4. C2×C3×C8×AGL(1,𝔽5)C2semidirect
6. C3×(C2×AGL(1,𝔽5))⋊C8C2semidirect
7. C3×(C2×AGL(1,𝔽5))⋊C8C2semidirect
8. #10603915C3direct (N × H)
11. C2×C3×#704C4semidirect
16. C2×C3×#700C4semidirect
24. C3×C5×C22⋊C8C4semidirect
25. C3×C22⋊#168C4semidirect
26. C3×#3952C4semidirect
27. C3×#3952C4semidirect
31. C2×#3992C2×C3semidirect
32. C2×C8×AGL(1,𝔽5)C2×C3semidirect
34. (C2×AGL(1,𝔽5))⋊C8C2×C3semidirect
35. (C2×AGL(1,𝔽5))⋊C8C2×C3semidirect
79. C2×#704C3×C4semidirect
84. C2×#700C3×C4semidirect
92. C5×C22⋊C8C3×C4semidirect
93. C22⋊#168C3×C4semidirect
94. #3952C3×C4semidirect
95. #3952C3×C4semidirect
109. C2×C3×AGL(1,𝔽5)M16semidirect
129. C3×#170C22⋊C4semidirect
149. C3×C22⋊C8AGL(1,𝔽5)semidirect
261. C2×AGL(1,𝔽5)C3×M16semidirect
281. #170C3×C22⋊C4semidirect
301. C22⋊C8C3×AGL(1,𝔽5)semidirect
404. C3×C5#1753semidirect
477. C5C3×#1753semidirect
499. C3#10603915direct (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
C2×C3×#39922
C2×C3×C8×AGL(1,𝔽5)2
C3×D5×C22⋊C82
C3×(C2×AGL(1,𝔽5))⋊C82
#106039153
C3×#17535