Finite Group Databasebeta

← C3×#10604227 #14295288

Subgroups — C3×#10604227 #14295288

Order 1920.

696 conjugacy classes of subgroups.

The subgroup lattice has 504 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#704C211yesmaximal, characteristic
2C2×C3×C22⋊#170C211yesmaximal, characteristic
3C2×C3×#4002C211yesmaximal, characteristic, complemented
4C2×C3×C4×#170C211yesmaximal, characteristic, complemented
5C3×C22⋊C4×Dic5C211yesmaximal, characteristic
6C3×#64918C211yesmaximal, characteristic, complemented
7C3×#64918C211yesmaximal, characteristic, complemented
8#10604227C311yesmaximal, characteristic, direct product factor, Hall
9C23×C3×Dic5C2211yescharacteristic
10C2×C3×#704—22no—
11C2×C3×#704—22no—
12C2×C3×#704C414yescomplemented
13C22×C3×#170C2211yescharacteristic
14C22×C3×#170C2211yescharacteristic
15C22×C3×#170C2211yescharacteristic
16C22×C3×#170C2211yescharacteristic
17C2×C3×C5×C22⋊C4C411yescharacteristic
18C2×C3×C22⋊Dic5C411yescharacteristic
19C2×C3×C4×Dic5C2211yescharacteristic
20C2×C3×C4×Dic5C2211yescharacteristic
21C3×C22⋊#170C414yescomplemented
22C3×#4002—22no—
23C3×C4×#170—22no—
24C3×#3963C411yescharacteristic
25C3×#3963C411yescharacteristic
26C3×#1753—51nomaximal, Hall
27C22×#704C2×C311yescharacteristic
28C2×C22⋊#170C2×C311yescharacteristic
29C2×#4002C2×C311yescharacteristic, complemented
30C2×C4×#170C2×C311yescharacteristic, complemented
31C22⋊C4×Dic5C2×C311yescharacteristic
32#64918C2×C311yescharacteristic, complemented
33#64918C2×C311yescharacteristic, complemented
34C24×C3×C5C2×C411yescharacteristic
35C22×C3×C4×C5C2×C411yescharacteristic
36C22×C3×C4×C5C2×C411yescharacteristic
37C22×C3×Dic5—21no—
38C22×C3×Dic5—21no—
39C22×C3×Dic5—21no—
40C22×C3×Dic5—21no—
41C22×C3×Dic5C2×C411yescharacteristic
42C22×C3×Dic5C2×C411yescharacteristic
43C22×C3×Dic5C2×C412yes—
44C22×C3×Dic5C2×C412yes—
45C22×C3×Dic5C2×C411yescharacteristic
46C22×C3×Dic5C2311yescharacteristic
47C3×#704—28no—
48C2×C3×#170—21no—
49C2×C3×#170—21no—
50C2×C3×#170—21no—
51C2×C3×#170—21no—
52C2×C3×#170—21no—
53C2×C3×#170—21no—
54C2×C3×#170C2×C412yes—
55C2×C3×#170C2×C412yes—
56C2×C3×#170C2×C412yes—
57C2×C3×#170C2×C412yes—
58C2×C3×#170D412yes—
59C2×C3×#170D412yes—
60C3×C5×C22⋊C4—22no—
61C3×C22⋊Dic5—22no—
62C3×C4×Dic5—21no—
63C3×C4×Dic5—21no—
64C3×C4×Dic5—21no—
65C3×C4×Dic5—21no—
66C22×C3×M16—51no—
67C2×C3×C22⋊C8—51no—
68C2×C3×#97—51no—
69C2×C3×C4×C8—51no—
70C3×C4×C22⋊C4—51no—
71C3×#336—51no—
72C3×#336—51no—
73C23×Dic5C22×C311yescharacteristic
74C2×#704—22no—
75C2×#704—22no—
76C2×#704C3×C414yescomplemented
77C22×#170C22×C311yescharacteristic
78C22×#170C22×C311yescharacteristic
79C22×#170C22×C311yescharacteristic
80C22×#170C22×C311yescharacteristic
81C2×C5×C22⋊C4C3×C411yescharacteristic
82C2×C22⋊Dic5C3×C411yescharacteristic
83C2×C4×Dic5C22×C311yescharacteristic
84C2×C4×Dic5C22×C311yescharacteristic
85C22⋊#170C3×C414yescomplemented
86#4002—22no—
87C4×#170—22no—
88#3963C3×C411yescharacteristic
89#3963C3×C411yescharacteristic
90#1753—51noSylow
91C23×C3×C5—21no—
92C23×C3×C5—21no—
93C23×C3×C5—21no—
94C23×C3×C5—21no—
95C23×C3×C5—21no—
96C23×C3×C5—21no—
97C23×C3×C5C4212yes—
98C23×C3×C5C22×C411yescharacteristic
99C2×C3×C4×C5—21no—
100C2×C3×C4×C5—21no—
101C2×C3×C4×C5—21no—
102C2×C3×C4×C5—21no—
103C2×C3×C4×C5—21no—
104C2×C3×C4×C5M1612yescomplemented
105C2×C3×Dic5—21no—
106C2×C3×Dic5—21no—
107C2×C3×Dic5—21no—
108C2×C3×Dic5—21no—
109C2×C3×Dic5—22no—
110C2×C3×Dic5—22no—
111C2×C3×Dic5—22no—
112C2×C3×Dic5—22no—
113C2×C3×Dic5—21no—
114C2×C3×Dic5—21no—
115C2×C3×Dic5—21no—
116C2×C3×Dic5C4212yes—
117C2×C3×Dic5M1612yescomplemented
118C2×C3×Dic5C22×C411yescharacteristic
119C2×C3×Dic5C22×C411yescharacteristic
120C2×C3×Dic5C2×D411yescharacteristic
121C2×C3×Dic5C2×D411yescharacteristic
122C2×C3×Dic5#4111yescharacteristic
123C2×C3×Dic5#4111yescharacteristic
124C3×#170—22no—
125C3×#170—22no—
126C3×#170—22no—
127C3×#170C22⋊C414yescomplemented
128C23×C3×C4—51no—
129C2×C3×M16—54no—
130C2×C3×M16—102no—
131C2×C3×M16—102no—
132C22×C3×C8—51no—
133C22×C3×C8—51no—
134C22×C3×C8—51no—
135C22×C3×C8—51no—
136C2×C3×C22⋊C4—51no—
137C2×C3×C22⋊C4AGL(1,𝔽5)11yescharacteristic
138C2×C3×C42—51no—
139C2×C3×C42—51no—
140C3×C22⋊C8—54no—
141C3×#97—102no—
142C3×C4×C8—102no—
143C3×#95—51no—
144C3×#95—51no—
145C24×C5C2×C3×C411yescharacteristic
146C22×C4×C5C2×C3×C411yescharacteristic
147C22×C4×C5C2×C3×C411yescharacteristic
148C22×Dic5—21no—
149C22×Dic5—21no—
150C22×Dic5—21no—
151C22×Dic5—21no—
152C22×Dic5C2×C3×C411yescharacteristic
153C22×Dic5C2×C3×C411yescharacteristic
154C22×Dic5C2×C3×C412yes—
155C22×Dic5C2×C3×C412yes—
156C22×Dic5C2×C3×C411yescharacteristic
157C22×Dic5C23×C311yescharacteristic
158#704—28no—
159C2×#170—21no—
160C2×#170—21no—
161C2×#170—21no—
162C2×#170—21no—
163C2×#170—21no—
164C2×#170—21no—
165C2×#170C2×C3×C412yes—
166C2×#170C2×C3×C412yes—
167C2×#170C2×C3×C412yes—
168C2×#170C2×C3×C412yes—
169C2×#170C3×D412yes—
170C2×#170C3×D412yes—
171C5×C22⋊C4—22no—
172C22⋊Dic5—22no—
173C4×Dic5—21no—
174C4×Dic5—21no—
175C4×Dic5—21no—
176C4×Dic5—21no—
177C22×M16—51no—
178C2×C22⋊C8—51no—
179C2×#97—51no—
180C2×C4×C8—51no—
181C4×C22⋊C4—51no—
182#336—51no—
183#336—51no—
184C22×C3×C5—22no—
185C22×C3×C5—22no—
186C22×C3×C5—22no—
187C22×C3×C5—41no—
188C22×C3×C5—41no—
189C22×C3×C5—41no—
190C22×C3×C5—41no—
191C22×C3×C5C2×C4211yescharacteristic
192C22×C3×C5C4×D411yescharacteristic
193C22×C3×C5C4×D411yescharacteristic
194C22×C3×C5C2×M1611yescharacteristic
195C22×C3×C5C2×M1611yescharacteristic
196C22×C3×C5#13111yescharacteristic
197C22×C3×C5#13111yescharacteristic
198C3×C4×C5—22no—
199C3×C4×C5—41no—
200C3×Dic5—22no—
201C3×Dic5—22no—
202C3×Dic5—41no—
203C3×Dic5C2×C22⋊C411yescharacteristic
204C3×Dic5#11711yescharacteristic
205C3×Dic5C4×D411yescharacteristic
206C3×Dic5C4×D411yescharacteristic
207C24×C3C2×AGL(1,𝔽5)11yescharacteristic
208C22×C3×C4—51no—
209C22×C3×C4—52no—
210C22×C3×C4—52no—
211C22×C3×C4—51no—
212C22×C3×C4—51no—
213C22×C3×C4—51no—
214C22×C3×C4—101no—
215C22×C3×C4—101no—
216C22×C3×C4—101no—
217C22×C3×C4—101no—
218C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
219C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
220C3×M16—108no—
221C2×C3×C8—52no—
222C2×C3×C8—52no—
223C2×C3×C8—52no—
224C2×C3×C8—52no—
225C2×C3×C8—52no—
226C2×C3×C8—52no—
227C2×C3×C8—101no—
228C2×C3×C8—101no—
229C2×C3×C8—101no—
230C2×C3×C8—101no—
231C2×C3×C8—101no—
232C2×C3×C8—101no—
233C3×C22⋊C4—22no—
234C3×C22⋊C4—102no—
235C3×C42—101no—
236C3×C42—101no—
237C3×C42—101no—
238C3×C42—101no—
239C23×C5—21no—
240C23×C5—21no—
241C23×C5—21no—
242C23×C5—21no—
243C23×C5—21no—
244C23×C5—21no—
245C23×C5C3×C4212yes—
246C23×C5C22×C3×C411yescharacteristic
247C2×C4×C5—21no—
248C2×C4×C5—21no—
249C2×C4×C5—21no—
250C2×C4×C5—21no—
251C2×C4×C5—21no—
252C2×C4×C5C3×M1612yescomplemented
253C2×Dic5—21no—
254C2×Dic5—21no—
255C2×Dic5—21no—
256C2×Dic5—21no—
257C2×Dic5—22no—
258C2×Dic5—22no—
259C2×Dic5—22no—
260C2×Dic5—22no—
261C2×Dic5—21no—
262C2×Dic5—21no—
263C2×Dic5—21no—
264C2×Dic5C3×C4212yes—
265C2×Dic5C3×M1612yescomplemented
266C2×Dic5C22×C3×C411yescharacteristic
267C2×Dic5C22×C3×C411yescharacteristic
268C2×Dic5C2×C3×D411yescharacteristic
269C2×Dic5C2×C3×D411yescharacteristic
270C2×Dic5C3×#4111yescharacteristic
271C2×Dic5C3×#4111yescharacteristic
272#170—22no—
273#170—22no—
274#170—22no—
275#170C3×C22⋊C414yescomplemented
276C23×C4—51no—
277C2×M16—54no—
278C2×M16—102no—
279C2×M16—102no—
280C22×C8—51no—
281C22×C8—51no—
282C22×C8—51no—
283C22×C8—51no—
284C2×C22⋊C4—51no—
285C2×C22⋊C4C3×AGL(1,𝔽5)11yescharacteristic
286C2×C42—51no—
287C2×C42—51no—
288C22⋊C8—54no—
289#97—102no—
290C4×C8—102no—
291#95—51no—
292#95—51no—
293C2×C3×C5—42no—
294C2×C3×C5C4×C22⋊C411yescharacteristic
295C2×C3×C5C4×M1611yescharacteristic
296C2×C3×C5#40511yescharacteristic
297C2×C3×C5#43511yescharacteristic
298C2×C3×C5#43511yescharacteristic
299C2×C3×C5#43611yescharacteristic
300C2×C3×C5#43611yescharacteristic
301C23×C3—21no—
302C23×C3—21no—
303C23×C3—21no—
304C23×C3—21no—
305C23×C3—21no—
306C23×C3—21no—
307C23×C3C4×AGL(1,𝔽5)12yes—
308C23×C3C22×AGL(1,𝔽5)11yescharacteristic
309C2×C3×C4—21no—
310C2×C3×C4—21no—
311C2×C3×C4—21no—
312C2×C3×C4—21no—
313C2×C3×C4—21no—
314C2×C3×C4—51no—
315C2×C3×C4—51no—
316C2×C3×C4—51no—
317C2×C3×C4—51no—
318C2×C3×C4—51no—
319C2×C3×C4—51no—
320C2×C3×C4—52no—
321C2×C3×C4—52no—
322C2×C3×C4—102no—
323C2×C3×C4—102no—
324C2×C3×C4—102no—
325C2×C3×C4—102no—
326C2×C3×C4—101no—
327C2×C3×C4—101no—
328C2×C3×C4—101no—
329C2×C3×C4—101no—
330C2×C3×C4—101no—
331C2×C3×C4—101no—
332C2×C3×C4—101no—
333C2×C3×C4#70412yescomplemented
334C3×C8—54no—
335C3×C8—102no—
336C3×C8—102no—
337C3×C8—102no—
338C22×C5—22no—
339C22×C5—22no—
340C22×C5—22no—
341C22×C5—41no—
342C22×C5—41no—
343C22×C5—41no—
344C22×C5—41no—
345C22×C5C2×C3×C4211yescharacteristic
346C22×C5C3×C4×D411yescharacteristic
347C22×C5C3×C4×D411yescharacteristic
348C22×C5C2×C3×M1611yescharacteristic
349C22×C5C2×C3×M1611yescharacteristic
350C22×C5C3×#13111yescharacteristic
351C22×C5C3×#13111yescharacteristic
352C4×C5—22no—
353C4×C5—41no—
354Dic5—22no—
355Dic5—22no—
356Dic5—41no—
357Dic5C2×C3×C22⋊C411yescharacteristic
358Dic5C3×#11711yescharacteristic
359Dic5C3×C4×D411yescharacteristic
360Dic5C3×C4×D411yescharacteristic
361C24C2×C3×AGL(1,𝔽5)11yescharacteristic
362C22×C4—51no—
363C22×C4—52no—
364C22×C4—52no—
365C22×C4—51no—
366C22×C4—51no—
367C22×C4—51no—
368C22×C4—101no—
369C22×C4—101no—
370C22×C4—101no—
371C22×C4—101no—
372C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
373C22×C4C2×C3×AGL(1,𝔽5)11yescharacteristic
374M16—108no—
375C2×C8—52no—
376C2×C8—52no—
377C2×C8—52no—
378C2×C8—52no—
379C2×C8—52no—
380C2×C8—52no—
381C2×C8—101no—
382C2×C8—101no—
383C2×C8—101no—
384C2×C8—101no—
385C2×C8—101no—
386C2×C8—101no—
387C22⋊C4—22no—
388C22⋊C4—102no—
389C42—101no—
390C42—101no—
391C42—101no—
392C42—101no—
393C3×C5#175311yescharacteristic, complemented, Hall
394C22×C3—22no—
395C22×C3—22no—
396C22×C3—22no—
397C22×C3—41no—
398C22×C3—41no—
399C22×C3—41no—
400C22×C3—41no—
401C22×C3D5⋊M1611yescharacteristic
402C22×C3C2×C4×AGL(1,𝔽5)11yescharacteristic
403C22×C3#413111yescharacteristic
404C22×C3#413111yescharacteristic
405C22×C3D4×AGL(1,𝔽5)11yescharacteristic
406C22×C3D4×AGL(1,𝔽5)11yescharacteristic
407C22×C3C2×#70411yescharacteristic
408C3×C4—22no—
409C3×C4—41no—
410C3×C4—51no—
411C3×C4—51no—
412C3×C4—51no—
413C3×C4—51no—
414C3×C4—102no—
415C3×C4—102no—
416C3×C4—201no—
417C2×C5—42no—
418C2×C5C3×C4×C22⋊C411yescharacteristic
419C2×C5C3×C4×M1611yescharacteristic
420C2×C5C3×#40511yescharacteristic
421C2×C5C3×#43511yescharacteristic
422C2×C5C3×#43511yescharacteristic
423C2×C5C3×#43611yescharacteristic
424C2×C5C3×#43611yescharacteristic
425C23—21no—
426C23—21no—
427C23—21no—
428C23—21no—
429C23—21no—
430C23—21no—
431C23C3×C4×AGL(1,𝔽5)12yes—
432C23C22×C3×AGL(1,𝔽5)11yescharacteristic
433C2×C4—21no—
434C2×C4—21no—
435C2×C4—21no—
436C2×C4—21no—
437C2×C4—21no—
438C2×C4—51no—
439C2×C4—51no—
440C2×C4—51no—
441C2×C4—51no—
442C2×C4—51no—
443C2×C4—51no—
444C2×C4—52no—
445C2×C4—52no—
446C2×C4—102no—
447C2×C4—102no—
448C2×C4—102no—
449C2×C4—102no—
450C2×C4—101no—
451C2×C4—101no—
452C2×C4—101no—
453C2×C4—101no—
454C2×C4—101no—
455C2×C4—101no—
456C2×C4—101no—
457C2×C4C3×#70412yescomplemented
458C8—54no—
459C8—102no—
460C8—102no—
461C8—102no—
462C2×C3—42no—
463C2×C3#6569111yescharacteristic
464C2×C3#6569411yescharacteristic
465C2×C3#6569511yescharacteristic
466C2×C3C22⋊C4×AGL(1,𝔽5)11yescharacteristic
467C2×C3C4×#70411yescharacteristic
468C2×C3#6577311yescharacteristic
469C2×C3#6577411yescharacteristic
470C5C3×#175311yescharacteristic, complemented, Sylow
471C22—22no—
472C22—22no—
473C22—22no—
474C22—41no—
475C22—41no—
476C22—41no—
477C22—41no—
478C22C3×D5⋊M1611yescharacteristic
479C22C2×C3×C4×AGL(1,𝔽5)11yescharacteristic
480C22C3×#413111yescharacteristic
481C22C3×#413111yescharacteristic
482C22C3×D4×AGL(1,𝔽5)11yescharacteristic
483C22C3×D4×AGL(1,𝔽5)11yescharacteristic
484C22C2×C3×#70411yescharacteristic
485C4—22no—
486C4—41no—
487C4—51no—
488C4—51no—
489C4—51no—
490C4—51no—
491C4—102no—
492C4—102no—
493C4—201no—
494C3#1060422711yescharacteristic, direct product factor, Sylow
495C2—42no—
496C2C3×#6569111yescharacteristic
497C2C3×#6569411yescharacteristic
498C2C3×#6569511yescharacteristic
499C2C3×C22⋊C4×AGL(1,𝔽5)11yescharacteristic
500C2C3×C4×#70411yescharacteristic
501C2C3×#6577311yescharacteristic
502C2C3×#6577411yescharacteristic
subgroupG/H
centerC23×C3C22×AGL(1,𝔽5)
commutatorC22×C5C2×C3×C42
FrattiniC23C22×C3×AGL(1,𝔽5)
FittingC2×C3×C5×C22⋊C4C4
socleC23×C3×C5C22×C4
prime radicalC24×C3×C5C2×C4
cosocleC22×Dic5C23×C3
hypercenterC2×C3×C22⋊C4AGL(1,𝔽5)
hypocenterC5C3×#1753
LCSG ▷ C22×C5 ▷ C5
UCSC1 ◁ C23×C3 ◁ C2×C3×C22⋊C4

Complements

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

NH ≅ G/Ntype
3. C2×C3×#4002C2semidirect
4. C2×C3×C4×#170C2semidirect
6. C3×#64918C2semidirect
7. C3×#64918C2semidirect
8. #10604227C3direct (N × H)
12. C2×C3×#704C4semidirect
21. C3×C22⋊#170C4semidirect
29. C2×#4002C2×C3semidirect
30. C2×C4×#170C2×C3semidirect
32. #64918C2×C3semidirect
33. #64918C2×C3semidirect
76. C2×#704C3×C4semidirect
85. C22⋊#170C3×C4semidirect
104. C2×C3×C4×C5M16semidirect
117. C2×C3×Dic5M16semidirect
127. C3×#170C22⋊C4semidirect
252. C2×C4×C5C3×M16semidirect
265. C2×Dic5C3×M16semidirect
275. #170C3×C22⋊C4semidirect
333. C2×C3×C4#704semidirect
393. C3×C5#1753semidirect
457. C2×C4C3×#704semidirect
470. C5C3×#1753semidirect
494. C3#10604227direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#7042
C2×C3×C22⋊#1702
C2×C3×#40022
C2×C3×C4×#1702
C3×C22⋊C4×Dic52
C3×#649182
#106042273
C3×#17535