Finite Group Databasebeta

← #14417792

Subgroups — #14417792

Order 1920.

968 conjugacy classes of subgroups.

The subgroup lattice has 560 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11756282C211yesmaximal, characteristic, complemented
2C3×#65857C211yesmaximal, characteristic, complemented
3C2×#91311C211yesmaximal, characteristic
4C5×#4997C211yesmaximal, characteristic, complemented
5C2×#90987C211yesmaximal, characteristic, complemented
6#11748413C211yesmaximal, characteristic, complemented
7#11748369C211yesmaximal, characteristic, complemented
8#11747091C212yesmaximal, complemented
9C4×#6658C212yesmaximal, complemented
10C4×#6657C212yesmaximal, complemented
11#11747023C212yesmaximal, complemented
12#10610305—31nomaximal, Hall
13C2×#6817C2211yescharacteristic
14C2×C3×#709C2211yescharacteristic
15C3×C5×#117C2211yescharacteristic
16C2×C4⋊Dic15C2211yescharacteristic
17#91781—21no—
18C4⋊C4×D15—21no—
19#91770—22no—
20C4×D60C2212yes—
21C4×Dic30C2212yes—
22C2×C5×#208C2211yescharacteristic
23C22×C5×C3⋊C8C2211yescharacteristic
24C2×C3×C4⋊Dic5C2211yescharacteristic
25C3×#4038—21no—
26C3×D5×C4⋊C4—21no—
27C3×#4027—22no—
28C3×C4×D20C2212yes—
29C3×C4×Dic10C2212yes—
30C2×#6662C2211yescharacteristic
31C2×#6658C2211yescharacteristic
32#91311C418yescomplemented
33C2×#6657C2211yescharacteristic
34C2×#6653C2211yescharacteristic
35C5×#803C2212yes—
36C4×C5×C3⋊C8C2212yes—
37#90987C2212yes—
38#90987C2212yes—
39#90974C2212yes—
40#90971C2212yes—
41#90968C2212yes—
42#90965C2212yes—
43#82558—51nomaximal, Hall
44C2×#4050—31no—
45#65857—31no—
46#65857D311yescharacteristic, complemented
47C5×#405—31no—
48#65422—31no—
49#65422—31no—
50C2×#3949—31no—
51#65005—32no—
52#65000—32no—
53C4×#677—32no—
54C4×#677—32no—
55C22×C3×C4×C5C2311yescharacteristic
56C2×#1220—21no—
57C22×Dic15—21no—
58#6817—22no—
59#6817C2×C414yescomplemented
60C2×D60C2311yescharacteristic
61C2×C4×D15—21no—
62C2×C4×D15—22no—
63C2×Dic30C2311yescharacteristic
64C2×C3×C22⋊D5—21no—
65C22×C3×Dic5—21no—
66C3×#709—22no—
67C3×#709C2×C414yescomplemented
68C2×C3×D20C2311yescharacteristic
69C2×C3×C4×D5—21no—
70C2×C3×C4×D5—22no—
71C2×C3×Dic10C2311yescharacteristic
72C3×C5×C4⋊C4D411yescharacteristic
73C3×C5×C4⋊C4D411yescharacteristic
74C3×C5×C22⋊C4D412yes—
75C3×C42×C5C2312yes—
76#6714—22no—
77C4⋊Dic15—21no—
78C4⋊Dic15C2312yes—
79#6712—22no—
80C4×Dic15—22no—
81C5×#208—22no—
82C2×C5×C3⋊C8C2×C414yescomplemented
83C2×C5×C3⋊C8C2312yes—
84C2×C5×C3⋊C8C2311yescharacteristic
85C2×C5×C3⋊C8C2311yescharacteristic
86C3×#685—22no—
87C3×C4⋊Dic5—21no—
88C3×C4⋊Dic5C2312yes—
89C3×#683—22no—
90C3×C4×Dic5—22no—
91#6662C2×C414yescomplemented
92#6658C2×C414yescomplemented
93#6657C2×C414yescomplemented
94#6653C2×C414yescomplemented
95C3×#520—51no—
96C2×#950—51no—
97#5523—51no—
98#5232—51no—
99#5030—52no—
100C4×#215—52no—
101C4×#214—52no—
102#5014—52no—
103#4997D511yescharacteristic, complemented
104#4966—51no—
105C2×#808—51no—
106C2×#709—31no—
107C2×#709C2×D311yescharacteristic
108C2×C5×M16—31no—
109C22×C5×C8—31no—
110C5×#117C2×D311yescharacteristic
111C2×C4⋊Dic5—31no—
112C2×C4⋊Dic5C2×D311yescharacteristic
113C2×Dic20—31no—
114#4050—38no—
115C2×D40—31no—
116C2×#677—31no—
117C2×#677—31no—
118#4038—21no—
119#4038—61no—
120D5×C4⋊C4—21no—
121D5×C4⋊C4—61no—
122#4027—22no—
123#4027—62no—
124C4×D20—32no—
125C4×D20C2×D312yes—
126C4×Dic10—32no—
127C4×Dic10C2×D312yes—
128C5×#97—32no—
129C4×C5×C8—32no—
130#3953—32no—
131#3953—32no—
132#3949—32no—
133#3949—32no—
134#3948—32no—
135#3948—32no—
136#2952—151noSylow
137C23×C3×C5C2×D411yescharacteristic
138C22×D15—21no—
139C22×C3×D5—21no—
140C2×C3×C4×C5C22×C412yes—
141C2×C3×C4×C5C2×D412yes—
142C2×C3×C4×C5C2×D412yes—
143C2×C3×C4×C5C2×D411yescharacteristic
144C2×C3×C4×C5#4112yes—
145C2×C3×C4×C5C2411yescharacteristic
146#1220—24no—
147C2×Dic15—21no—
148C2×Dic15—22no—
149C2×Dic15—81no—
150D60—21no—
151D60C22×C412yes—
152C4×D15—24no—
153C4×D15—82no—
154Dic30—21no—
155Dic30C22×C412yes—
156C3×C22⋊D5—24no—
157C2×C3×Dic5—21no—
158C2×C3×Dic5—22no—
159C2×C3×Dic5—81no—
160C3×D20—21no—
161C3×D20C22×C412yes—
162C3×C4×D5—24no—
163C3×C4×D5—82no—
164C3×Dic10—21no—
165C3×Dic10C22×C412yes—
166C5×C3⋊C8—22no—
167C5×C3⋊C8C22×C412yes—
168C5×C3⋊C8C22×C412yes—
169C2×C3×#41—51no—
170C2×#235—51no—
171C3×C4×Q8—52no—
172C3×C4×D4—52no—
173C3×C4×D4—102no—
174C3×#117—101no—
175C3×#117C2×D511yescharacteristic
176C2×C3×C4⋊C4—51no—
177C2×C3×C4⋊C4—101no—
178#950—58no—
179C2×#216—51no—
180C2×#215—51no—
181C2×#214—51no—
182C2×#213—51no—
183C2×C4⋊Dic3—51no—
184C2×#208C2×D511yescharacteristic
185C22×C3⋊C8C2×D511yescharacteristic
186#892—101no—
187D3×C4⋊C4—101no—
188#67⋊C4—102no—
189C4×D12—52no—
190C4×Dic6—52no—
191#835—52no—
192#832—52no—
193#810—52no—
194#809—52no—
195#808—52no—
196#808—52no—
197#803C2×D512yes—
198C4×C3⋊C8C2×D512yes—
199C22×C4×C5C22×D311yescharacteristic
200C2×C22⋊D5—21no—
201C2×C22⋊D5—61no—
202C22×Dic5—21no—
203C22×Dic5—61no—
204#709—22no—
205#709—34no—
206#709—62no—
207#709C4×D314yescomplemented
208C2×D20—31no—
209C2×D20C22×D311yescharacteristic
210C2×C4×D5—21no—
211C2×C4×D5—22no—
212C2×C4×D5—61no—
213C2×C4×D5—62no—
214C2×Dic10—31no—
215C2×Dic10C22×D311yescharacteristic
216C5×M16—62no—
217C2×C5×C8—34no—
218C2×C5×C8—31no—
219C2×C5×C8—31no—
220C2×C5×C8—32no—
221C5×C4⋊C4#6711yescharacteristic
222C5×C4⋊C4#6711yescharacteristic
223C5×C22⋊C4#6712yes—
224C42×C5C22×D312yes—
225#685—22no—
226#685—62no—
227C4⋊Dic5—21no—
228C4⋊Dic5—32no—
229C4⋊Dic5—61no—
230C4⋊Dic5C22×D312yes—
231#683—22no—
232#683—62no—
233C4×Dic5—22no—
234C4×Dic5—62no—
235Dic20—34no—
236D40—34no—
237#677—34no—
238#677—34no—
239C2×#135—151no—
240#520—51no—
241#520—151no—
242#442—152no—
243#441—152no—
244C4×SD16—152no—
245C4×SD16—152no—
246C2×#106—151no—
247#417—151no—
248#417—151no—
249#405—31no—
250C22×C3×C5—21no—
251C22×C3×C5—21no—
252C22×C3×C5C4×D412yes—
253C22×C3×C5C22×D411yescharacteristic
254C2×D15—22no—
255C2×D15—41no—
256C2×D15—41no—
257C2×C3×D5—22no—
258C2×C3×D5—41no—
259C2×C3×D5—41no—
260C3×C4×C5—22no—
261C3×C4×C5—22no—
262C3×C4×C5C4×D412yes—
263C3×C4×C5C23×C411yescharacteristic
264C3×C4×C5C2×#4111yescharacteristic
265Dic15—22no—
266Dic15—42no—
267C3×Dic5—22no—
268C3×Dic5—42no—
269C3×#41—54no—
270C3×#41—102no—
271C2×C3×Q8—51no—
272C2×C3×D4—51no—
273C2×C3×D4—101no—
274C22×C3×C4—101no—
275C22×C3×C4—101no—
276C22×C3×C4—102no—
277C22×C3×C4C22×D511yescharacteristic
278C2×#67—101no—
279C22×Dic3—101no—
280#235—54no—
281#235—102no—
282C2×D12—51no—
283C2×C4×D3—101no—
284C2×C4×D3—102no—
285C2×Dic6—51no—
286C3×C4⋊C4—52no—
287C3×C4⋊C4—102no—
288C3×C4⋊C4—101no—
289C3×C4⋊C4D2011yescharacteristic
290C3×C4⋊C4D2011yescharacteristic
291C3×C22⋊C4—102no—
292C3×C22⋊C4D2012yes—
293C3×C42—102no—
294C3×C42C22×D512yes—
295#216—54no—
296#215—54no—
297#214—54no—
298#213—54no—
299(C2×D3)⋊C4—102no—
300C4⋊Dic3—52no—
301C4⋊Dic3—101no—
302Dic3⋊C4—102no—
303C4×Dic3—102no—
304#208—22no—
305C2×C3⋊C8C4×D514yescomplemented
306C2×C3⋊C8C22×D512yes—
307C2×C3⋊C8C22×D511yescharacteristic
308C2×C3⋊C8C22×D511yescharacteristic
309C23×C5C2×#6711yescharacteristic
310C22×D5—21no—
311C22×D5—61no—
312C2×C4×C5C2×C4×D312yes—
313C2×C4×C5#23512yes—
314C2×C4×C5C2×#6712yes—
315C2×C4×C5C2×#6712yes—
316C2×C4×C5C2×#6711yescharacteristic
317C2×C4×C5C23×D311yescharacteristic
318C22⋊D5—24no—
319C22⋊D5—64no—
320C2×Dic5—21no—
321C2×Dic5—22no—
322C2×Dic5—61no—
323C2×Dic5—62no—
324C2×Dic5—81no—
325C2×Dic5—241no—
326D20—21no—
327D20—32no—
328D20—61no—
329D20C2×C4×D312yes—
330C4×D5—24no—
331C4×D5—64no—
332C4×D5—82no—
333C4×D5—242no—
334Dic10—21no—
335Dic10—32no—
336Dic10—61no—
337Dic10C2×C4×D312yes—
338C5×C8—32no—
339C5×C8—32no—
340C5×C8—62no—
341C2×#41—51no—
342C2×#41—151no—
343#135—158no—
344C2×Q16—151no—
345C2×SD16—151no—
346C2×SD16—151no—
347C2×D8—151no—
348C2×M16—31no—
349C22×C8—31no—
350C4×Q8—52no—
351C4×Q8—152no—
352C4×D4—52no—
353C4×D4—102no—
354C4×D4—152no—
355C4×D4—302no—
356#117—101no—
357#117—301no—
358#117D3×D511yescharacteristic
359C2×C4⋊C4—51no—
360C2×C4⋊C4—101no—
361C2×C4⋊C4—151no—
362C2×C4⋊C4—301no—
363#106—152no—
364#106—152no—
365#103—152no—
366#103—152no—
367#102—152no—
368#102—152no—
369#97—32no—
370C4×C8—32no—
371C2×C3×C5—22no—
372C2×C3×C5C2×C4×D411yescharacteristic
373C2×C3×C5#57711yescharacteristic
374C2×C3×C5#57711yescharacteristic
375D15—42no—
376C3×D5—42no—
377C23×C3—101no—
378C23×C3C2×D2011yescharacteristic
379C22×D3—101no—
380C3×Q8—52no—
381C3×Q8—101no—
382C3×D4—52no—
383C3×D4—104no—
384C3×D4—101no—
385C2×C3×C4—101no—
386C2×C3×C4—104no—
387C2×C3×C4—102no—
388C2×C3×C4—402no—
389C2×C3×C4—401no—
390C2×C3×C4C2×C4×D512yes—
391C2×C3×C4C2×D2012yes—
392C2×C3×C4C2×D2012yes—
393C2×C3×C4C2×D2011yescharacteristic
394C2×C3×C4#70912yes—
395C2×C3×C4C23×D511yescharacteristic
396#67—104no—
397C2×Dic3—101no—
398C2×Dic3—102no—
399C2×Dic3—401no—
400D12—52no—
401D12—101no—
402C4×D3—104no—
403C4×D3—402no—
404Dic6—52no—
405Dic6—101no—
406C3⋊C8—22no—
407C3⋊C8C2×C4×D512yes—
408C3⋊C8C2×C4×D512yes—
409C22×C5—21no—
410C22×C5—21no—
411C22×C5C4×#6712yes—
412C22×C5C22×#6711yescharacteristic
413C2×D5—22no—
414C2×D5—41no—
415C2×D5—41no—
416C2×D5—62no—
417C2×D5—121no—
418C2×D5—121no—
419C4×C5—22no—
420C4×C5—22no—
421C4×C5C4×#6712yes—
422C4×C5C22×C4×D311yescharacteristic
423C4×C5C2×#23511yescharacteristic
424Dic5—22no—
425Dic5—42no—
426Dic5—62no—
427Dic5—122no—
428#41—54no—
429#41—102no—
430#41—154no—
431#41—302no—
432C2×Q8—51no—
433C2×Q8—151no—
434C2×D4—51no—
435C2×D4—101no—
436C2×D4—151no—
437C2×D4—301no—
438C22×C4—101no—
439C22×C4—101no—
440C22×C4—102no—
441C22×C4—301no—
442C22×C4—301no—
443C22×C4—302no—
444C22×C4C2×D3×D511yescharacteristic
445Q16—154no—
446SD16—154no—
447SD16—154no—
448D8—154no—
449M16—62no—
450C2×C8—34no—
451C2×C8—31no—
452C2×C8—31no—
453C2×C8—32no—
454C4⋊C4—52no—
455C4⋊C4—101no—
456C4⋊C4—102no—
457C4⋊C4—152no—
458C4⋊C4—301no—
459C4⋊C4—302no—
460C4⋊C4#120211yescharacteristic
461C4⋊C4#120211yescharacteristic
462C22⋊C4—102no—
463C22⋊C4—302no—
464C22⋊C4#120212yes—
465C42—102no—
466C42—302no—
467C42C2×D3×D512yes—
468C3×C5#295211yescharacteristic, complemented, Hall
469C22×C3—21no—
470C22×C3—21no—
471C22×C3—102no—
472C22×C3—201no—
473C22×C3—201no—
474C22×C3C4×D2012yes—
475C22×C3C22×D2011yescharacteristic
476C2×D3—102no—
477C2×D3—201no—
478C2×D3—201no—
479C3×C4—22no—
480C3×C4—22no—
481C3×C4—102no—
482C3×C4—202no—
483C3×C4C4×D2012yes—
484C3×C4C22×C4×D511yescharacteristic
485C3×C4C2×#70911yescharacteristic
486Dic3—102no—
487Dic3—202no—
488C2×C5—22no—
489C2×C5C2×C4×#6711yescharacteristic
490C2×C5#583611yescharacteristic
491C2×C5#583711yescharacteristic
492D5—42no—
493D5—122no—
494C23—101no—
495C23—301no—
496C23C2×#120211yescharacteristic
497Q8—52no—
498Q8—101no—
499Q8—152no—
500Q8—301no—
501D4—52no—
502D4—104no—
503D4—101no—
504D4—152no—
505D4—304no—
506D4—301no—
507C2×C4—101no—
508C2×C4—104no—
509C2×C4—102no—
510C2×C4—301no—
511C2×C4—304no—
512C2×C4—302no—
513C2×C4—402no—
514C2×C4—401no—
515C2×C4—1202no—
516C2×C4—1201no—
517C2×C4#677112yes—
518C2×C4C4×D3×D512yes—
519C2×C4C2×#120212yes—
520C2×C4C2×#120212yes—
521C2×C4C2×#120211yescharacteristic
522C2×C4C22×D3×D511yescharacteristic
523C8—32no—
524C8—32no—
525C8—62no—
526C2×C3—22no—
527C2×C3—202no—
528C2×C3C2×C4×D2011yescharacteristic
529C2×C3#6608511yescharacteristic
530C2×C3#6608511yescharacteristic
531D3—202no—
532C5#8255811yescharacteristic, complemented, Sylow
533C22—21no—
534C22—21no—
535C22—102no—
536C22—201no—
537C22—201no—
538C22—302no—
539C22—601no—
540C22—601no—
541C22C4×#120212yes—
542C22C22×#120211yescharacteristic
543C4—22no—
544C4—22no—
545C4—102no—
546C4—202no—
547C4—302no—
548C4—602no—
549C4C4×#120212yes—
550C4C2×#677111yescharacteristic
551C4C2×C4×D3×D511yescharacteristic
552C3#1061030511yescharacteristic, complemented, Sylow
553C2—22no—
554C2—202no—
555C2—602no—
556C2C2×C4×#120211yescharacteristic
557C2#1175530911yescharacteristic
558C2#1175531011yescharacteristic
subgroupG/H
centerC22C22×#1202
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C22×D3×D5
FittingC3×C5×#117C22
socleC22×C3×C5C22×D4
prime radicalC2×#91311C2
cosocleC2×C3×C4×C5C24
hypercenter#117D3×D5
hypocenterC3×C5#2952
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. #11756282C2semidirect
2. C3×#65857C2semidirect
4. C5×#4997C2semidirect
5. C2×#90987C2semidirect
6. #11748413C2semidirect
7. #11748369C2semidirect
8. #11747091C2semidirect
9. C4×#6658C2semidirect
10. C4×#6657C2semidirect
11. #11747023C2semidirect
32. #91311C4semidirect
46. #65857D3semidirect
59. #6817C2×C4semidirect
67. C3×#709C2×C4semidirect
82. C2×C5×C3⋊C8C2×C4semidirect
91. #6662C2×C4semidirect
92. #6658C2×C4semidirect
93. #6657C2×C4semidirect
94. #6653C2×C4semidirect
103. #4997D5semidirect
207. #709C4×D3semidirect
305. C2×C3⋊C8C4×D5semidirect
468. C3×C5#2952semidirect
532. C5#82558semidirect
552. C3#10610305semidirect

Maximal subgroups

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

subgroupindex
#117562822
C3×#658572
C2×#913112
C5×#49972
C2×#909872
#117484132
#117483692
#117470912
C4×#66582
C4×#66572
#117470232
#106103053
#825585