Finite Group Databasebeta

← #14411432

Subgroups — #14411432

Order 1920.

1128 conjugacy classes of subgroups.

The subgroup lattice has 548 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6739C211yesmaximal, characteristic
2C2×D3×#704C211yesmaximal, characteristic
3D3×#4023C211yesmaximal, characteristic
4#11752350C211yesmaximal, characteristic, complemented
5C3×#65691C211yesmaximal, characteristic, complemented
6#11751995C211yesmaximal, characteristic, complemented
7#11751988C211yesmaximal, characteristic, complemented
8#11751907C212yesmaximal, complemented
9C4×#6739C212yesmaximal, complemented
10#11751870C212yesmaximal, complemented
11#11751868C212yesmaximal, complemented
12#10617050—31nomaximal, Hall
13C22×D3×Dic5C2211yescharacteristic
14C2×#6762C2211yescharacteristic
15C22×C3⋊#170C2211yescharacteristic
16C2×C3×#704C2211yescharacteristic
17C22×C3×#170C2211yescharacteristic
18C2×#6739C418yescomplemented
19C2×#6739C2212yes—
20C2×#6739C2211yescharacteristic
21C2×#6739C2211yescharacteristic
22C2×#6739C2211yescharacteristic
23C2×#6739C2211yescharacteristic
24D3×#704C418yescomplemented
25C2×D3×#170C2212yes—
26#91766C411yescharacteristic
27C5×D3×C22⋊C4C411yescharacteristic
28C3×#4023C2211yescharacteristic
29D3×C22⋊Dic5C411yescharacteristic
30#91532C411yescharacteristic
31#91528C2211yescharacteristic
32#91400C412yes—
33D3×#683C412yes—
34#91398C2212yes—
35C4×D3×Dic5C2212yes—
36#91231C2212yescomplemented
37C4×C3⋊#170C2212yescomplemented
38C3×#4002C2212yescomplemented
39C3×C4×#170C2212yescomplemented
40#91169C2212yescomplemented
41#91169C2212yescomplemented
42Dic3×#170C2212yescomplemented
43Dic3×#170C2212yescomplemented
44#78548—51nomaximal, Hall
45C22×#704—31no—
46C22×#704—31no—
47C2×#4023—31no—
48C4×#704—32no—
49C4×#704—32no—
50#65691—31no—
51#65691—32no—
52#65691—32no—
53#65691—31no—
54#65691—31no—
55#65691D311yescharacteristic, complemented
56C23×C5×D3C2×C411yescharacteristic
57C22×Dic15C2×C411yescharacteristic
58C2×C4×C5×D3C2×C412yes—
59C22×C3×Dic5C2311yescharacteristic
60C2×D3×Dic5C2×C412yes—
61C2×D3×Dic5C2×C411yescharacteristic
62C2×D3×Dic5C2×C411yescharacteristic
63C2×D3×Dic5C2×C414yes—
64C2×D3×Dic5C2×C414yes—
65C2×D3×Dic5C2311yescharacteristic
66C2×D3×Dic5C2311yescharacteristic
67#6762C2×C414yescomplemented
68C2×C3⋊#170C2×C414yescomplemented
69C2×C3⋊#170C2312yes—
70C2×C3⋊#170C2311yescharacteristic
71C2×C3⋊#170C2311yescharacteristic
72C3×#704C2×C414yescomplemented
73C2×C3×#170C2×C414yescomplemented
74C2×C3×#170C2312yes—
75C2×C3×#170C2311yescharacteristic
76C2×C3×#170C2311yescharacteristic
77#6739C2×C418yes—
78#6739C2×C414yes—
79#6739C2×C414yes—
80#6739C2×C414yescomplemented
81#6739C2×C414yescomplemented
82D3×#170C2×C418yescomplemented
83C3×C5×C22⋊C4C2×C411yescharacteristic
84#6719C2×C411yescharacteristic
85#6712C2×C412yes—
86C4×Dic15C2×C412yes—
87C5×#217C2×C411yescharacteristic
88C5×(C2×D3)⋊C4C2×C412yes—
89C3×C22⋊Dic5C2×C411yescharacteristic
90C3×#683C2×C412yes—
91C3×C4×Dic5C2312yes—
92#6669C2×C412yes—
93#6668C2×C412yes—
94(C2×D3)⋊Dic5C2×C412yes—
95Dic3×Dic5C2×C412yes—
96Dic3×Dic5C2312yes—
97C2×D3×M16—51no—
98C22×ΓL(1,𝔽25)—51no—
99D3×#117—51no—
100C3×#405—51no—
101#5122—51no—
102#5101—51no—
103#4997—51no—
104#4708—52no—
105#4707—52no—
106#4689—52no—
107C4×ΓL(1,𝔽25)—52no—
108C23×Dic5—31no—
109C2×#704—31no—
110C2×#704—38no—
111C2×#704—38no—
112C2×#704—31no—
113C2×#704—31no—
114C2×#704—32no—
115C2×#704—31no—
116C2×#704—31no—
117C2×#704C2×D311yescharacteristic
118C22×#170—31no—
119C22×#170—32no—
120C22×#170C2×D311yescharacteristic
121C2×C5×C22⋊C4—31no—
122C2×C22⋊Dic5—31no—
123C2×#683—32no—
124C2×C4×Dic5—32no—
125#4023—31no—
126#4023—31no—
127#4023—31no—
128#4023—32no—
129#4023—32no—
130#4023C2×D311yescharacteristic
131#4002—32no—
132#4002—32no—
133#4002—32no—
134#4002C2×D312yescomplemented
135C4×#170—32no—
136C4×#170—32no—
137C4×#170—32no—
138C4×#170C2×D312yescomplemented
139#2873—151noSylow
140C23×C3×C5C22×C411yescharacteristic
141C22×C5×D3—42no—
142C22×C5×D3C4214yes—
143C22×C5×D3C22×C411yescharacteristic
144C22×C5×D3C22×C411yescharacteristic
145C2×C3×C4×C5C22×C412yes—
146C2×Dic15C4214yes—
147C2×Dic15C22×C412yes—
148C2×Dic15C22×C411yescharacteristic
149C2×Dic15C22×C411yescharacteristic
150C2×C5×Dic3C22×C412yes—
151C4×C5×D3—42no—
152C2×C3×Dic5C22×C412yes—
153C2×C3×Dic5C22×C411yescharacteristic
154C2×C3×Dic5C22×C412yes—
155C2×C3×Dic5C22×C412yes—
156C2×C3×Dic5C2411yescharacteristic
157D3×Dic5—42no—
158D3×Dic5C4214yes—
159D3×Dic5C4214yes—
160D3×Dic5C22×C412yes—
161D3×Dic5C22×C412yes—
162D3×Dic5C22×C412yes—
163D3×Dic5C22×C412yes—
164C3⋊#170C22×C414yes—
165C3⋊#170C22×C412yes—
166C3⋊#170C22×C412yes—
167C3×#170C22×C414yes—
168C3×#170C22×C412yes—
169C3×#170C22×C412yes—
170C22×C4×D3—51no—
171C2×C3×M16—51no—
172C22×C3×C8—51no—
173C3×#117—51no—
174#926—51no—
175C2×#208—51no—
176C22×C3⋊C8—51no—
177D3×M16—58no—
178C2×ΓL(1,𝔽25)—58no—
179C2×ΓL(1,𝔽25)—51no—
180C2×ΓL(1,𝔽25)—52no—
181C2×ΓL(1,𝔽25)—51no—
182C2×ΓL(1,𝔽25)—51no—
183C2×ΓL(1,𝔽25)—51no—
184C2×D3×C8—52no—
185#892—52no—
186D3×C4⋊C4—52no—
187D3×C22⋊C4—51no—
188D3×C22⋊C4AGL(1,𝔽5)11yescharacteristic
189#877—51no—
190#877—51no—
191#872—52no—
192C42×D3—52no—
193C3×#97—52no—
194C3×C4×C8—52no—
195#815—52no—
196#815—52no—
197C8×Dic3—52no—
198C8×Dic3—52no—
199#803—52no—
200C4×C3⋊C8—52no—
201C24×C5—31no—
202C22×C4×C5—32no—
203C22×Dic5—31no—
204C22×Dic5—34no—
205C22×Dic5—34no—
206C22×Dic5—31no—
207C22×Dic5—31no—
208C22×Dic5—31no—
209C22×Dic5—31no—
210C22×Dic5—32no—
211C22×Dic5C22×D311yescharacteristic
212#704—34no—
213#704—38no—
214#704—34no—
215#704—34no—
216#704—34no—
217#704—34no—
218#704C4×D314yescomplemented
219C2×#170—34no—
220C2×#170—31no—
221C2×#170—31no—
222C2×#170—32no—
223C2×#170—38no—
224C2×#170C4×D314yescomplemented
225C2×#170C22×D312yes—
226C2×#170C22×D311yescharacteristic
227C2×#170C22×D311yescharacteristic
228C5×C22⋊C4—31no—
229C5×C22⋊C4—32no—
230C5×C22⋊C4C4×D311yescharacteristic
231C22⋊Dic5—31no—
232C22⋊Dic5—32no—
233C22⋊Dic5C4×D311yescharacteristic
234#683—32no—
235#683—32no—
236#683—32no—
237#683C4×D312yes—
238C4×Dic5—32no—
239C4×Dic5—32no—
240C4×Dic5—32no—
241C4×Dic5C22×D312yes—
242C22×M16—151no—
243C22×M16—151no—
244C2×#117—151no—
245#405—51no—
246#405—152no—
247#405—152no—
248#405—151no—
249#405—151no—
250#405—151no—
251C4×M16—152no—
252C4×M16—152no—
253C22×C3×C5—22no—
254C22×C3×C5C2×C4212yes—
255C22×C3×C5C23×C411yescharacteristic
256C2×C5×D3—22no—
257C2×C5×D3—22no—
258C2×C5×D3—44no—
259C2×C5×D3C2×C4212yes—
260C2×C5×D3C2×C4212yes—
261C3×C4×C5—22no—
262Dic15—22no—
263Dic15C2×C4212yes—
264Dic15C2×C4212yes—
265C3×Dic5—22no—
266C3×Dic5C2×C4212yes—
267C3×Dic5C23×C411yescharacteristic
268C3×Dic5C23×C411yescharacteristic
269C5×Dic3—22no—
270C23×D3C2×AGL(1,𝔽5)11yescharacteristic
271C22×C3×C4—51no—
272C22×Dic3—51no—
273C2×C4×D3—54no—
274C2×C4×D3—54no—
275C2×C4×D3—51no—
276C2×C4×D3—51no—
277C2×C4×D3—51no—
278C2×C4×D3—51no—
279C2×C4×D3—52no—
280C2×C4×D3C2×AGL(1,𝔽5)12yes—
281C3×M16—54no—
282C2×C3×C8—51no—
283C2×C3×C8—52no—
284C2×C3×C8—51no—
285C2×C3×C8—54no—
286C3×C4⋊C4—52no—
287C3×C22⋊C4—51no—
288C3×C22⋊C4C2×AGL(1,𝔽5)11yescharacteristic
289C3×C42—52no—
290#217—51no—
291#217C2×AGL(1,𝔽5)11yescharacteristic
292(C2×D3)⋊C4—52no—
293(C2×D3)⋊C4C2×AGL(1,𝔽5)12yes—
294C4⋊Dic3—52no—
295Dic3⋊C4—52no—
296Dic3⋊C4—52no—
297C4×Dic3—52no—
298C4×Dic3—52no—
299C4×Dic3—52no—
300#208—54no—
301C2×C3⋊C8—51no—
302C2×C3⋊C8—52no—
303C2×C3⋊C8—51no—
304C2×C3⋊C8—54no—
305ΓL(1,𝔽25)—54no—
306ΓL(1,𝔽25)—54no—
307ΓL(1,𝔽25)—58no—
308ΓL(1,𝔽25)—54no—
309ΓL(1,𝔽25)—54no—
310D3×C8—58no—
311C23×C5—31no—
312C23×C5—31no—
313C23×C5—34no—
314C23×C5—122no—
315C23×C5C2×C4×D311yescharacteristic
316C2×C4×C5—32no—
317C2×C4×C5—122no—
318C2×C4×C5C2×C4×D312yes—
319C2×Dic5—34no—
320C2×Dic5—31no—
321C2×Dic5—31no—
322C2×Dic5—34no—
323C2×Dic5—34no—
324C2×Dic5—32no—
325C2×Dic5—32no—
326C2×Dic5—32no—
327C2×Dic5—32no—
328C2×Dic5—32no—
329C2×Dic5—122no—
330C2×Dic5C2×C4×D312yes—
331C2×Dic5C2×C4×D311yescharacteristic
332C2×Dic5C2×C4×D312yes—
333C2×Dic5C2×C4×D312yes—
334C2×Dic5C23×D311yescharacteristic
335#170—34no—
336#170—32no—
337#170—32no—
338#170C2×C4×D314yes—
339#170C2×C4×D312yes—
340#170C2×C4×D312yes—
341C23×C4—151no—
342C2×M16—51no—
343C2×M16—158no—
344C2×M16—151no—
345C2×M16—152no—
346C2×M16—151no—
347C2×M16—151no—
348C2×M16—151no—
349C2×M16—158no—
350C2×M16—151no—
351C22×C8—51no—
352C22×C8—152no—
353C22×C8—151no—
354#117—51no—
355#117—152no—
356#117—151no—
357#117—152no—
358#117—151no—
359#117—151no—
360C2×C4⋊C4—152no—
361C2×C22⋊C4—31no—
362C2×C22⋊C4—151no—
363C2×C42—152no—
364#97—52no—
365#97—152no—
366#97—152no—
367#97—152no—
368C4×C8—52no—
369C4×C8—152no—
370C4×C8—152no—
371C4×C8—152no—
372C2×C3×C5—22no—
373C2×C3×C5C22×C4211yescharacteristic
374C2×C3×C5#56812yes—
375C5×D3—22no—
376C5×D3—22no—
377C23×C3C22×AGL(1,𝔽5)11yescharacteristic
378C22×D3—42no—
379C22×D3C4×AGL(1,𝔽5)14yes—
380C22×D3C22×AGL(1,𝔽5)11yescharacteristic
381C22×D3C22×AGL(1,𝔽5)11yescharacteristic
382C2×C3×C4—52no—
383C2×C3×C4—52no—
384C2×C3×C4—51no—
385C2×C3×C4—51no—
386C2×C3×C4—52no—
387C2×C3×C4C22×AGL(1,𝔽5)12yes—
388C2×Dic3—54no—
389C2×Dic3—51no—
390C2×Dic3—51no—
391C2×Dic3—52no—
392C2×Dic3C22×AGL(1,𝔽5)12yes—
393C4×D3—42no—
394C4×D3—54no—
395C4×D3—54no—
396C4×D3—52no—
397C4×D3—52no—
398C4×D3—52no—
399C4×D3—52no—
400C4×D3—202no—
401C3×C8—54no—
402C3×C8—52no—
403C3×C8—52no—
404C3⋊C8—54no—
405C3⋊C8—52no—
406C3⋊C8—52no—
407C22×C5—22no—
408C22×C5—32no—
409C22×C5—32no—
410C22×C5—62no—
411C22×C5—62no—
412C22×C5—124no—
413C22×C5C42×D312yes—
414C22×C5C22×C4×D311yescharacteristic
415C4×C5—22no—
416C4×C5—62no—
417Dic5—22no—
418Dic5—32no—
419Dic5—32no—
420Dic5—62no—
421Dic5C42×D312yes—
422Dic5C22×C4×D311yescharacteristic
423Dic5C22×C4×D311yescharacteristic
424C24—31no—
425C22×C4—32no—
426C22×C4—51no—
427C22×C4—151no—
428C22×C4—151no—
429C22×C4—151no—
430C22×C4—151no—
431C22×C4—154no—
432C22×C4—154no—
433C22×C4—152no—
434C22×C4—151no—
435M16—54no—
436M16—154no—
437M16—154no—
438M16—154no—
439M16—154no—
440M16—158no—
441M16—154no—
442C2×C8—51no—
443C2×C8—52no—
444C2×C8—51no—
445C2×C8—54no—
446C2×C8—158no—
447C2×C8—151no—
448C2×C8—152no—
449C2×C8—151no—
450C2×C8—154no—
451C4⋊C4—52no—
452C4⋊C4—152no—
453C4⋊C4—152no—
454C4⋊C4—152no—
455C22⋊C4—31no—
456C22⋊C4—32no—
457C22⋊C4—51no—
458C22⋊C4—152no—
459C22⋊C4—151no—
460C22⋊C4D3×AGL(1,𝔽5)11yescharacteristic
461C42—52no—
462C42—152no—
463C42—152no—
464C42—152no—
465C3×C5#287311yescharacteristic, complemented, Hall
466C22×C3—22no—
467C22×C3C2×C4×AGL(1,𝔽5)12yes—
468C22×C3C23×AGL(1,𝔽5)11yescharacteristic
469C2×D3—22no—
470C2×D3—22no—
471C2×D3—44no—
472C2×D3C2×C4×AGL(1,𝔽5)12yes—
473C2×D3C2×C4×AGL(1,𝔽5)12yes—
474C3×C4—22no—
475C3×C4—52no—
476C3×C4—51no—
477C3×C4—51no—
478C3×C4—102no—
479Dic3—22no—
480Dic3—52no—
481Dic3—52no—
482Dic3—102no—
483C2×C5—22no—
484C2×C5—62no—
485C2×C5—62no—
486C2×C5C2×C42×D311yescharacteristic
487C2×C5#575012yes—
488C23—31no—
489C23—31no—
490C23—34no—
491C23—122no—
492C23C2×D3×AGL(1,𝔽5)11yescharacteristic
493C2×C4—32no—
494C2×C4—52no—
495C2×C4—52no—
496C2×C4—51no—
497C2×C4—51no—
498C2×C4—52no—
499C2×C4—122no—
500C2×C4—152no—
501C2×C4—152no—
502C2×C4—152no—
503C2×C4—152no—
504C2×C4—154no—
505C2×C4—154no—
506C2×C4—154no—
507C2×C4—151no—
508C2×C4—151no—
509C2×C4—152no—
510C2×C4—602no—
511C2×C4C2×D3×AGL(1,𝔽5)12yes—
512C8—52no—
513C8—52no—
514C8—54no—
515C8—154no—
516C8—152no—
517C8—152no—
518C2×C3—22no—
519C2×C3C22×C4×AGL(1,𝔽5)11yescharacteristic
520C2×C3#6625612yes—
521D3—22no—
522D3—22no—
523C5#7854811yescharacteristic, complemented, Sylow
524C22—22no—
525C22—32no—
526C22—32no—
527C22—62no—
528C22—62no—
529C22—124no—
530C22C4×D3×AGL(1,𝔽5)12yes—
531C22C22×D3×AGL(1,𝔽5)11yescharacteristic
532C4—22no—
533C4—51no—
534C4—51no—
535C4—52no—
536C4—62no—
537C4—102no—
538C4—152no—
539C4—152no—
540C4—302no—
541C3#1061705011yescharacteristic, complemented, Sylow
542C2—22no—
543C2—62no—
544C2—62no—
545C2C2×C4×D3×AGL(1,𝔽5)11yescharacteristic
546C2#1175684112yes—
subgroupG/H
center, FrattiniC22C22×D3×AGL(1,𝔽5)
commutatorC2×C3×C5C22×C42
FittingC3×C5×C22⋊C4C2×C4
socleC22×C3×C5C23×C4
prime radicalC23×C5×D3C2×C4
cosocleC2×C3×Dic5C24
hypercenterC22⋊C4D3×AGL(1,𝔽5)
hypocenterC3×C5#2873
LCSG ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22⋊C4

Complements

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

NH ≅ G/Ntype
4. #11752350C2semidirect
5. C3×#65691C2semidirect
6. #11751995C2semidirect
7. #11751988C2semidirect
8. #11751907C2semidirect
9. C4×#6739C2semidirect
10. #11751870C2semidirect
11. #11751868C2semidirect
18. C2×#6739C4semidirect
24. D3×#704C4semidirect
36. #91231C22semidirect
37. C4×C3⋊#170C22semidirect
38. C3×#4002C22semidirect
39. C3×C4×#170C22semidirect
40. #91169C22semidirect
41. #91169C22semidirect
42. Dic3×#170C22semidirect
43. Dic3×#170C22semidirect
55. #65691D3semidirect
67. #6762C2×C4semidirect
68. C2×C3⋊#170C2×C4semidirect
72. C3×#704C2×C4semidirect
73. C2×C3×#170C2×C4semidirect
80. #6739C2×C4semidirect
81. #6739C2×C4semidirect
82. D3×#170C2×C4semidirect
134. #4002C2×D3semidirect
138. C4×#170C2×D3semidirect
218. #704C4×D3semidirect
224. C2×#170C4×D3semidirect
465. C3×C5#2873semidirect
523. C5#78548semidirect
541. C3#10617050semidirect

Maximal subgroups

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

subgroupindex
C22×#67392
C2×D3×#7042
D3×#40232
#117523502
C3×#656912
#117519952
#117519882
#117519072
C4×#67392
#117518702
#117518682
#106170503
#785485