Finite Group Databasebeta

← #14417089

Subgroups — #14417089

Order 1920.

1012 conjugacy classes of subgroups.

The subgroup lattice has 556 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C5×#5523C211yesmaximal, characteristic, complemented
2C3×#65855C211yesmaximal, characteristic, complemented
3C2×#91298C211yesmaximal, characteristic
4#11751098C211yesmaximal, characteristic, complemented
5#11748472C211yesmaximal, characteristic, complemented
6#11748382C211yesmaximal, characteristic, complemented
7C2×#90964C211yesmaximal, characteristic, complemented
8#11747048C212yesmaximal, complemented
9C4×#6655C212yesmaximal, complemented
10#11747021C212yesmaximal, complemented
11C4×#6652C212yesmaximal, complemented
12#10614295—31nomaximal, Hall
13C2×C5×#235C2211yescharacteristic
14C22×C3×D20C2211yescharacteristic
15C3×C5×#117C2211yescharacteristic
16C2×#6710C2211yescharacteristic
17C22×C3⋊#168C2211yescharacteristic
18C2×C5×C4⋊Dic3C2211yescharacteristic
19C5×#892—21no—
20C5×D3×C4⋊C4—21no—
21C5×#67⋊C4—22no—
22C4×C5×D12C2212yescomplemented
23C4×C5×Dic6C2212yescomplemented
24C3×#4072C2211yescharacteristic
25C3×#4039—21no—
26C3×#4039—21no—
27C3×D5×C22⋊C4—22no—
28C3×C4×D20C2212yes—
29C3×C4×D20C2212yes—
30C2×#6655C2212yes—
31#91298C418yescomplemented
32C2×#6652C2212yes—
33#91087C2212yescomplemented
34C4×C3⋊#168C2212yescomplemented
35#90989C2212yescomplemented
36#90986C2212yescomplemented
37#90972C2211yescharacteristic, complemented
38#90972C2211yescharacteristic, complemented
39#90966C2211yescharacteristic, complemented
40#90966C2211yescharacteristic, complemented
41#90964C2212yes—
42#90964C2212yes—
43#82537—51nomaximal, Hall
44C5×#520—31no—
45C2×#4095—31no—
46#65855D311yescharacteristic, complemented
47#65522—31no—
48#65315—32no—
49C4×#688—32no—
50#65303—32no—
51C4×#686—32no—
52#65286—31no—
53C2×#3941—31no—
54#65253—31no—
55C23×C3×D5—21no—
56C22×C3×C4×C5C2311yescharacteristic
57C2×C5×#67—21no—
58C22×C5×Dic3—21no—
59C5×#235—22no—
60C5×#235C2×C414yescomplemented
61C2×C5×D12C2311yescharacteristic
62C2×C4×C5×D3—21no—
63C2×C4×C5×D3—22no—
64C2×C5×Dic6C2311yescharacteristic
65C2×C3×D20—21no—
66C2×C3×D20—22no—
67C2×C3×D20C2×C414yescomplemented
68C2×C3×D20C2312yes—
69C2×C3×C4×D5—22no—
70C2×C3×C4×D5—22no—
71C3×C5×C4⋊C4D411yescharacteristic
72C3×C5×C4⋊C4D411yescharacteristic
73C3×C5×C22⋊C4D412yes—
74C3×C42×C5C2312yes—
75#6710C2×C414yescomplemented
76C2×C3⋊#168—22no—
77C2×C3⋊#168C2311yescharacteristic
78C2×C3⋊#168C2311yescharacteristic
79C2×C3⋊#168C2312yes—
80C5×(C2×D3)⋊C4—22no—
81C5×C4⋊Dic3—21no—
82C5×C4⋊Dic3C2312yes—
83C5×Dic3⋊C4—22no—
84C4×C5×Dic3—22no—
85C3×C22⋊Dic5—21no—
86C3×#685—22no—
87C3×#685—22no—
88C3×C4⋊Dic5C2311yescharacteristic
89C3×C4⋊Dic5C2311yescharacteristic
90C3×C4×Dic5—21no—
91#6655C2×C418yescomplemented
92#6652C2×C418yescomplemented
93C3×#518—51no—
94C2×#932—51no—
95#5523D511yescharacteristic, complemented
96C2×#832—51no—
97#5018—52no—
98C4×#214—52no—
99#5014—52no—
100C4×#213—52no—
101#5002—51no—
102#4997—51no—
103#4995—51no—
104C2×C5×#41—31no—
105C22×D20C2×D311yescharacteristic
106C4×C5×Q8—32no—
107C4×C5×D4—32no—
108C4×C5×D4—62no—
109C5×#117—61no—
110C5×#117C2×D311yescharacteristic
111C2×C5×C4⋊C4—31no—
112C2×C5×C4⋊C4—61no—
113#4095—38no—
114C2×#688—32no—
115C2×#686—32no—
116#4072C2×D311yescharacteristic
117C2×#681—31no—
118C22×#168—31no—
119#4039—21no—
120#4039—21no—
121D5×C22⋊C4—22no—
122C4×D20C2×D312yes—
123C4×D20C2×D312yes—
124#3967—31no—
125#3967—31no—
126#3964—31no—
127#3964—31no—
128#3941—32no—
129#3941—32no—
130#3940—32no—
131#3939—32no—
132#3935—32no—
133C4×#168—32no—
134#2946—151noSylow
135C23×C3×C5C2×D411yescharacteristic
136C22×C5×D3—21no—
137C22×C3×D5—22no—
138C22×C3×D5—24no—
139C22×C3×D5—81no—
140C22×C3×D5—81no—
141C2×C3×C4×C5C22×C412yes—
142C2×C3×C4×C5C2×D412yes—
143C2×C3×C4×C5C2×D412yes—
144C2×C3×C4×C5C2×D411yescharacteristic
145C2×C3×C4×C5#4112yes—
146C2×C3×C4×C5C2411yescharacteristic
147C5×#67—24no—
148C2×C5×Dic3—21no—
149C2×C5×Dic3—22no—
150C2×C5×Dic3—81no—
151C5×D12—21no—
152C5×D12C22×C412yes—
153C4×C5×D3—24no—
154C4×C5×D3—82no—
155C5×Dic6—21no—
156C5×Dic6C22×C412yes—
157C2×C3×Dic5—21no—
158C2×C3×Dic5—21no—
159C3×D20—24no—
160C3×D20—22no—
161C3×D20C22×C414yes—
162C3×C4×D5—82no—
163C3×C4×D5—82no—
164C3⋊#168—21no—
165C3⋊#168—21no—
166C3⋊#168C22×C414yes—
167C22×C3×D4—51no—
168C2×#235C2×D511yescharacteristic
169C3×C4×D4—52no—
170C3×C4×D4—52no—
171C3×C4×D4—101no—
172C3×C4×D4—101no—
173C3×#117—51no—
174C3×#117C2×D511yescharacteristic
175C2×C3×C22⋊C4—102no—
176C2×#214—52no—
177#932—58no—
178C2×#213—52no—
179C2×C4⋊Dic3C2×D511yescharacteristic
180C2×#208—51no—
181C22×C3⋊C8—51no—
182#892—21no—
183D3×C4⋊C4—21no—
184#67⋊C4—22no—
185C4×D12C2×D512yescomplemented
186C4×Dic6C2×D512yescomplemented
187#832—52no—
188#832—52no—
189#810—51no—
190#810—51no—
191#809—51no—
192#809—51no—
193#808—52no—
194#807—52no—
195#803—52no—
196C4×C3⋊C8—52no—
197C23×D5—21no—
198C5×#41—34no—
199C5×#41—62no—
200C2×C5×Q8—31no—
201C2×C5×D4—31no—
202C2×C5×D4—61no—
203C22×C4×C5—61no—
204C22×C4×C5—61no—
205C22×C4×C5—62no—
206C22×C4×C5C22×D311yescharacteristic
207C2×D20—22no—
208C2×D20—21no—
209C2×D20C4×D314yescomplemented
210C2×D20C22×D312yes—
211C2×C4×D5—22no—
212C2×C4×D5—22no—
213C5×C4⋊C4—32no—
214C5×C4⋊C4—61no—
215C5×C4⋊C4—62no—
216C5×C4⋊C4#6711yescharacteristic
217C5×C4⋊C4#6711yescharacteristic
218C5×C22⋊C4—62no—
219C5×C22⋊C4#6712yes—
220C42×C5—62no—
221C42×C5C22×D312yes—
222C22⋊Dic5—21no—
223#688—38no—
224#686—38no—
225#685—22no—
226#685—22no—
227C4⋊Dic5C22×D311yescharacteristic
228C4⋊Dic5C22×D311yescharacteristic
229C4×Dic5—21no—
230#681—34no—
231C2×#168—32no—
232C2×#168—31no—
233C2×#168—31no—
234C2×#168—62no—
235C2×AGL(1,ℤ/8)—151no—
236#520—31no—
237#518—51no—
238#442—152no—
239#440—152no—
240C4×SD16—152no—
241C4×D8—152no—
242#428—151no—
243#416—151no—
244C2×#102—151no—
245#405—151no—
246C22×C3×C5—21no—
247C22×C3×C5—21no—
248C22×C3×C5C4×D412yes—
249C22×C3×C5C22×D411yescharacteristic
250C2×C5×D3—22no—
251C2×C5×D3—41no—
252C2×C5×D3—41no—
253C2×C3×D5—24no—
254C2×C3×D5—42no—
255C2×C3×D5—42no—
256C2×C3×D5—84no—
257C3×C4×C5—22no—
258C3×C4×C5—22no—
259C3×C4×C5C4×D412yes—
260C3×C4×C5C23×C411yescharacteristic
261C3×C4×C5C2×#4111yescharacteristic
262C3×Dic5—41no—
263C3×Dic5—41no—
264C5×Dic3—22no—
265C5×Dic3—42no—
266C24×C3—101no—
267C2×C3×D4—54no—
268C2×C3×D4—52no—
269C2×C3×D4—102no—
270C2×C3×D4—101no—
271C22×C3×C4—102no—
272C22×C3×C4—102no—
273C22×C3×C4C22×D511yescharacteristic
274C2×#67—21no—
275C22×Dic3—21no—
276#235—22no—
277#235C4×D514yescomplemented
278C2×D12C22×D511yescharacteristic
279C2×C4×D3—21no—
280C2×C4×D3—22no—
281C2×Dic6C22×D511yescharacteristic
282C3×C4⋊C4—51no—
283C3×C4⋊C4—51no—
284C3×C4⋊C4C22⋊D511yescharacteristic
285C3×C4⋊C4C22⋊D511yescharacteristic
286C3×C22⋊C4—101no—
287C3×C22⋊C4—102no—
288C3×C22⋊C4—102no—
289C3×C22⋊C4C22⋊D512yes—
290C3×C42—101no—
291C3×C42C22×D512yes—
292#214—58no—
293#213—58no—
294(C2×D3)⋊C4—22no—
295C4⋊Dic3—21no—
296C4⋊Dic3C22×D512yes—
297Dic3⋊C4—22no—
298C4×Dic3—22no—
299#208—54no—
300C2×C3⋊C8—52no—
301C2×C3⋊C8—51no—
302C2×C3⋊C8—51no—
303C2×C3⋊C8—102no—
304C23×C5—61no—
305C23×C5C2×#6711yescharacteristic
306C22×D5—22no—
307C22×D5—24no—
308C22×D5—81no—
309C22×D5—81no—
310C5×Q8—32no—
311C5×Q8—61no—
312C5×D4—32no—
313C5×D4—64no—
314C5×D4—61no—
315C2×C4×C5—61no—
316C2×C4×C5—64no—
317C2×C4×C5—62no—
318C2×C4×C5—242no—
319C2×C4×C5—241no—
320C2×C4×C5C2×C4×D312yes—
321C2×C4×C5#23512yes—
322C2×C4×C5C2×#6712yes—
323C2×C4×C5C2×#6712yes—
324C2×C4×C5C2×#6711yescharacteristic
325C2×C4×C5C23×D311yescharacteristic
326C2×Dic5—21no—
327C2×Dic5—21no—
328D20—24no—
329D20—22no—
330D20C2×C4×D314yes—
331C4×D5—82no—
332C4×D5—82no—
333#168—34no—
334#168—61no—
335#168—61no—
336C2×#41—31no—
337C22×D4—51no—
338AGL(1,ℤ/8)—158no—
339C2×SD16—152no—
340C2×D8—152no—
341C2×M16—151no—
342C22×C8—151no—
343C4×Q8—32no—
344C4×D4—32no—
345C4×D4—52no—
346C4×D4—52no—
347C4×D4—62no—
348C4×D4—101no—
349C4×D4—101no—
350#117—51no—
351#117—61no—
352#117D3×D511yescharacteristic
353C2×C4⋊C4—31no—
354C2×C4⋊C4—61no—
355C2×C22⋊C4—102no—
356#107—152no—
357#106—152no—
358#103—151no—
359#103—151no—
360#102—152no—
361#102—152no—
362#102—151no—
363#102—151no—
364#97—152no—
365C4×C8—152no—
366C2×C3×C5—22no—
367C2×C3×C5C2×C4×D411yescharacteristic
368C2×C3×C5#57611yescharacteristic
369C2×C3×C5#57711yescharacteristic
370C3×D5—44no—
371C5×D3—42no—
372C23×C3—102no—
373C23×C3—104no—
374C23×C3—401no—
375C23×C3—401no—
376C23×C3C2×C22⋊D511yescharacteristic
377C22×D3—21no—
378C3×D4—54no—
379C3×D4—102no—
380C3×D4—104no—
381C2×C3×C4—101no—
382C2×C3×C4—101no—
383C2×C3×C4—402no—
384C2×C3×C4—402no—
385C2×C3×C4C2×C4×D512yes—
386C2×C3×C4#70912yes—
387C2×C3×C4C2×C22⋊D512yes—
388C2×C3×C4C2×C22⋊D512yes—
389C2×C3×C4C2×C22⋊D511yescharacteristic
390C2×C3×C4C23×D511yescharacteristic
391#67—24no—
392C2×Dic3—21no—
393C2×Dic3—22no—
394C2×Dic3—81no—
395D12—21no—
396D12C2×C4×D512yes—
397C4×D3—24no—
398C4×D3—82no—
399Dic6—21no—
400Dic6C2×C4×D512yes—
401C3⋊C8—54no—
402C3⋊C8—101no—
403C3⋊C8—101no—
404C22×C5—21no—
405C22×C5—21no—
406C22×C5—62no—
407C22×C5—121no—
408C22×C5—121no—
409C22×C5C4×#6712yes—
410C22×C5C22×#6711yescharacteristic
411C2×D5—24no—
412C2×D5—42no—
413C2×D5—42no—
414C2×D5—84no—
415C4×C5—22no—
416C4×C5—22no—
417C4×C5—62no—
418C4×C5—122no—
419C4×C5C4×#6712yes—
420C4×C5C22×C4×D311yescharacteristic
421C4×C5C2×#23511yescharacteristic
422Dic5—41no—
423Dic5—41no—
424C24—101no—
425#41—34no—
426#41—62no—
427C2×Q8—31no—
428C2×D4—31no—
429C2×D4—54no—
430C2×D4—52no—
431C2×D4—61no—
432C2×D4—102no—
433C2×D4—101no—
434C22×C4—61no—
435C22×C4—61no—
436C22×C4—62no—
437C22×C4—102no—
438C22×C4—102no—
439C22×C4C2×D3×D511yescharacteristic
440SD16—158no—
441D8—158no—
442M16—154no—
443C2×C8—152no—
444C2×C8—151no—
445C2×C8—151no—
446C2×C8—302no—
447C4⋊C4—32no—
448C4⋊C4—51no—
449C4⋊C4—51no—
450C4⋊C4—61no—
451C4⋊C4—62no—
452C4⋊C4#120111yescharacteristic
453C4⋊C4#120111yescharacteristic
454C22⋊C4—62no—
455C22⋊C4—101no—
456C22⋊C4—102no—
457C22⋊C4—102no—
458C22⋊C4#120112yes—
459C42—62no—
460C42—101no—
461C42C2×D3×D512yes—
462C3×C5#294611yescharacteristic, complemented, Hall
463C22×C3—21no—
464C22×C3—21no—
465C22×C3—104no—
466C22×C3—202no—
467C22×C3—202no—
468C22×C3—404no—
469C22×C3C4×C22⋊D512yes—
470C22×C3C22×C22⋊D511yescharacteristic
471C2×D3—22no—
472C2×D3—41no—
473C2×D3—41no—
474C3×C4—22no—
475C3×C4—22no—
476C3×C4—201no—
477C3×C4—201no—
478C3×C4C4×C22⋊D512yes—
479C3×C4C22×C4×D511yescharacteristic
480C3×C4C2×#70911yescharacteristic
481Dic3—22no—
482Dic3—42no—
483C2×C5—22no—
484C2×C5—122no—
485C2×C5C2×C4×#6711yescharacteristic
486C2×C5#583511yescharacteristic
487C2×C5#583611yescharacteristic
488D5—44no—
489C23—61no—
490C23—102no—
491C23—104no—
492C23—401no—
493C23—401no—
494C23C2×#120111yescharacteristic
495Q8—32no—
496Q8—61no—
497D4—32no—
498D4—54no—
499D4—64no—
500D4—61no—
501D4—104no—
502D4—102no—
503C2×C4—61no—
504C2×C4—64no—
505C2×C4—62no—
506C2×C4—101no—
507C2×C4—101no—
508C2×C4—242no—
509C2×C4—241no—
510C2×C4—402no—
511C2×C4—402no—
512C2×C4#677112yes—
513C2×C4C4×D3×D512yes—
514C2×C4C2×#120112yes—
515C2×C4C2×#120112yes—
516C2×C4C2×#120111yescharacteristic
517C2×C4C22×D3×D511yescharacteristic
518C8—154no—
519C8—301no—
520C8—301no—
521C2×C3—22no—
522C2×C3—204no—
523C2×C3C2×C4×C22⋊D511yescharacteristic
524C2×C3#6616911yescharacteristic
525C2×C3#6617111yescharacteristic
526D3—42no—
527C5#8253711yescharacteristic, complemented, Sylow
528C22—21no—
529C22—21no—
530C22—62no—
531C22—104no—
532C22—121no—
533C22—121no—
534C22—202no—
535C22—202no—
536C22—404no—
537C22C4×#120112yes—
538C22C22×#120111yescharacteristic
539C4—22no—
540C4—22no—
541C4—62no—
542C4—122no—
543C4—201no—
544C4—201no—
545C4C4×#120112yes—
546C4C2×#677111yescharacteristic
547C4C2×C4×D3×D511yescharacteristic
548C3#1061429511yescharacteristic, complemented, Sylow
549C2—22no—
550C2—122no—
551C2—204no—
552C2C2×C4×#120111yescharacteristic
553C2#1175530111yescharacteristic
554C2#1175530311yescharacteristic
subgroupG/H
centerC22C22×#1201
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C22×D3×D5
FittingC3×C5×#117C22
socleC22×C3×C5C22×D4
prime radicalC2×#91298C2
cosocleC2×C3×C4×C5C24
hypercenter#117D3×D5
hypocenterC3×C5#2946
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ #117

Complements

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

NH ≅ G/Ntype
1. C5×#5523C2semidirect
2. C3×#65855C2semidirect
4. #11751098C2semidirect
5. #11748472C2semidirect
6. #11748382C2semidirect
7. C2×#90964C2semidirect
8. #11747048C2semidirect
9. C4×#6655C2semidirect
10. #11747021C2semidirect
11. C4×#6652C2semidirect
22. C4×C5×D12C22semidirect
23. C4×C5×Dic6C22semidirect
31. #91298C4semidirect
33. #91087C22semidirect
34. C4×C3⋊#168C22semidirect
35. #90989C22semidirect
36. #90986C22semidirect
37. #90972C22semidirect
38. #90972C22semidirect
39. #90966C22semidirect
40. #90966C22semidirect
46. #65855D3semidirect
60. C5×#235C2×C4semidirect
67. C2×C3×D20C2×C4semidirect
75. #6710C2×C4semidirect
91. #6655C2×C4semidirect
92. #6652C2×C4semidirect
95. #5523D5semidirect
185. C4×D12C2×D5semidirect
186. C4×Dic6C2×D5semidirect
209. C2×D20C4×D3semidirect
277. #235C4×D5semidirect
462. C3×C5#2946semidirect
527. C5#82537semidirect
548. C3#10614295semidirect

Maximal subgroups

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

subgroupindex
C5×#55232
C3×#658552
C2×#912982
#117510982
#117484722
#117483822
C2×#909642
#117470482
C4×#66552
#117470212
C4×#66522
#106142953
#825375