Finite Group Databasebeta

← #14417790

Subgroups — #14417790

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
1C5×#5523C211yesmaximal, characteristic, complemented
2C3×#65857C211yesmaximal, characteristic, complemented
3C2×#91297C211yesmaximal, characteristic
4#11751098C211yesmaximal, characteristic, complemented
5C2×#90986C211yesmaximal, characteristic, complemented
6#11748384C211yesmaximal, characteristic, complemented
7#11748368C211yesmaximal, characteristic, complemented
8#11747089C212yesmaximal, complemented
9C4×#6656C212yesmaximal, complemented
10C4×#6655C212yesmaximal, complemented
11#11747021C212yesmaximal, complemented
12#10614289—31nomaximal, Hall
13C2×C5×#235C2211yescharacteristic
14C2×C3×#709C2211yescharacteristic
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×D12C2212yes—
23C4×C5×Dic6C2212yes—
24C2×C3×C4⋊Dic5C2211yescharacteristic
25C3×#4038—21no—
26C3×D5×C4⋊C4—21no—
27C3×#4027—22no—
28C3×C4×D20C2212yes—
29C3×C4×Dic10C2212yes—
30C2×#6661C2211yescharacteristic
31C2×#6656C2211yescharacteristic
32C2×#6655C2211yescharacteristic
33#91297C418yescomplemented
34C2×#6652C2211yescharacteristic
35#91087C2212yes—
36C4×C3⋊#168C2212yes—
37#90986C2212yes—
38#90986C2212yes—
39#90972C2212yes—
40#90970C2212yes—
41#90966C2212yes—
42#90964C2212yes—
43#82558—51nomaximal, Hall
44C5×#520—31no—
45C2×#4096—31no—
46#65857D311yescharacteristic, complemented
47#65521—31no—
48#65319—32no—
49C4×#688—32no—
50C4×#687—32no—
51#65303—32no—
52#65286—31no—
53#65255—31no—
54C2×#3940—31no—
55C22×C3×C4×C5C2311yescharacteristic
56C2×C5×#67—21no—
57C22×C5×Dic3—21no—
58C5×#235—22no—
59C5×#235C2×C414yescomplemented
60C2×C5×D12C2311yescharacteristic
61C2×C4×C5×D3—21no—
62C2×C4×C5×D3—22no—
63C2×C5×Dic6C2311yescharacteristic
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#6710—22no—
77C2×C3⋊#168C2×C414yescomplemented
78C2×C3⋊#168C2312yes—
79C2×C3⋊#168C2311yescharacteristic
80C2×C3⋊#168C2311yescharacteristic
81C5×(C2×D3)⋊C4—22no—
82C5×C4⋊Dic3—21no—
83C5×C4⋊Dic3C2312yes—
84C5×Dic3⋊C4—22no—
85C4×C5×Dic3—22no—
86C3×#685—22no—
87C3×C4⋊Dic5—21no—
88C3×C4⋊Dic5C2312yes—
89C3×#683—22no—
90C3×C4×Dic5—22no—
91#6661C2×C414yescomplemented
92#6656C2×C414yescomplemented
93#6655C2×C414yescomplemented
94#6652C2×C414yescomplemented
95C3×#520—51no—
96C2×#950—51no—
97#5523D511yescharacteristic, complemented
98#5232—51no—
99#5030—52no—
100C4×#215—52no—
101C4×#214—52no—
102#5014—52no—
103#4997—51no—
104#4966—51no—
105C2×#808—51no—
106C2×C5×#41—31no—
107C2×#709C2×D311yescharacteristic
108C4×C5×Q8—32no—
109C4×C5×D4—32no—
110C4×C5×D4—62no—
111C5×#117—61no—
112C5×#117C2×D311yescharacteristic
113C2×C5×C4⋊C4—31no—
114C2×C5×C4⋊C4—61no—
115#4096—38no—
116C2×#689—31no—
117C2×#688—31no—
118C2×#687—31no—
119C2×#686—31no—
120C2×C4⋊Dic5C2×D311yescharacteristic
121C2×#681—31no—
122C22×#168—31no—
123#4038—21no—
124D5×C4⋊C4—21no—
125#4027—22no—
126C4×D20C2×D312yes—
127C4×Dic10C2×D312yes—
128#3967—32no—
129#3964—32no—
130#3942—32no—
131#3941—32no—
132#3940—32no—
133#3940—32no—
134#3935—32no—
135C4×#168—32no—
136#2952—151noSylow
137C23×C3×C5C2×D411yescharacteristic
138C22×C5×D3—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
146C5×#67—24no—
147C2×C5×Dic3—21no—
148C2×C5×Dic3—22no—
149C2×C5×Dic3—81no—
150C5×D12—21no—
151C5×D12C22×C412yes—
152C4×C5×D3—24no—
153C4×C5×D3—82no—
154C5×Dic6—21no—
155C5×Dic6C22×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—
166C3⋊#168—22no—
167C3⋊#168C22×C412yes—
168C3⋊#168C22×C412yes—
169C2×C3×#41—51no—
170C2×#235C2×D511yescharacteristic
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⋊Dic3C2×D511yescharacteristic
184C2×#208—51no—
185C22×C3⋊C8—51no—
186#892—21no—
187D3×C4⋊C4—21no—
188#67⋊C4—22no—
189C4×D12C2×D512yes—
190C4×Dic6C2×D512yes—
191#835—52no—
192#832—52no—
193#810—52no—
194#809—52no—
195#808—52no—
196#808—52no—
197#803—52no—
198C4×C3⋊C8—52no—
199C5×#41—34no—
200C5×#41—62no—
201C2×C5×Q8—31no—
202C2×C5×D4—31no—
203C2×C5×D4—61no—
204C22×C4×C5—61no—
205C22×C4×C5—61no—
206C22×C4×C5—62no—
207C22×C4×C5C22×D311yescharacteristic
208C2×C22⋊D5—21no—
209C22×Dic5—21no—
210#709—22no—
211#709C4×D314yescomplemented
212C2×D20C22×D311yescharacteristic
213C2×C4×D5—21no—
214C2×C4×D5—22no—
215C2×Dic10C22×D311yescharacteristic
216C5×C4⋊C4—32no—
217C5×C4⋊C4—61no—
218C5×C4⋊C4—62no—
219C5×C4⋊C4#6711yescharacteristic
220C5×C4⋊C4#6711yescharacteristic
221C5×C22⋊C4—62no—
222C5×C22⋊C4#6712yes—
223C42×C5—62no—
224C42×C5C22×D312yes—
225#689—34no—
226#688—34no—
227#687—34no—
228#686—34no—
229#685—22no—
230C4⋊Dic5—21no—
231C4⋊Dic5C22×D312yes—
232#683—22no—
233C4×Dic5—22no—
234#681—62no—
235C2×#168—34no—
236C2×#168—31no—
237C2×#168—31no—
238C2×#168—32no—
239C2×#135—151no—
240#520—31no—
241#520—51no—
242#442—152no—
243#441—152no—
244C4×SD16—152no—
245C4×SD16—152no—
246C2×#106—151no—
247#417—151no—
248#417—151no—
249#405—151no—
250C22×C3×C5—21no—
251C22×C3×C5—21no—
252C22×C3×C5C4×D412yes—
253C22×C3×C5C22×D411yescharacteristic
254C2×C5×D3—22no—
255C2×C5×D3—41no—
256C2×C5×D3—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
265C3×Dic5—22no—
266C3×Dic5—42no—
267C5×Dic3—22no—
268C5×Dic3—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—21no—
279C22×Dic3—21no—
280#235—22no—
281#235C4×D514yescomplemented
282C2×D12C22×D511yescharacteristic
283C2×C4×D3—21no—
284C2×C4×D3—22no—
285C2×Dic6C22×D511yescharacteristic
286C3×C4⋊C4—52no—
287C3×C4⋊C4—102no—
288C3×C4⋊C4—101no—
289C3×C4⋊C4C22⋊D511yescharacteristic
290C3×C4⋊C4C22⋊D511yescharacteristic
291C3×C22⋊C4—102no—
292C3×C22⋊C4C22⋊D512yes—
293C3×C42—102no—
294C3×C42C22×D512yes—
295#216—54no—
296#215—54no—
297#214—54no—
298#213—54no—
299(C2×D3)⋊C4—22no—
300C4⋊Dic3—21no—
301C4⋊Dic3C22×D512yes—
302Dic3⋊C4—22no—
303C4×Dic3—22no—
304#208—102no—
305C2×C3⋊C8—54no—
306C2×C3⋊C8—51no—
307C2×C3⋊C8—51no—
308C2×C3⋊C8—52no—
309C23×C5—61no—
310C23×C5C2×#6711yescharacteristic
311C22×D5—21no—
312C5×Q8—32no—
313C5×Q8—61no—
314C5×D4—32no—
315C5×D4—64no—
316C5×D4—61no—
317C2×C4×C5—61no—
318C2×C4×C5—64no—
319C2×C4×C5—62no—
320C2×C4×C5—242no—
321C2×C4×C5—241no—
322C2×C4×C5C2×C4×D312yes—
323C2×C4×C5#23512yes—
324C2×C4×C5C2×#6712yes—
325C2×C4×C5C2×#6712yes—
326C2×C4×C5C2×#6711yescharacteristic
327C2×C4×C5C23×D311yescharacteristic
328C22⋊D5—24no—
329C2×Dic5—21no—
330C2×Dic5—22no—
331C2×Dic5—81no—
332D20—21no—
333D20C2×C4×D312yes—
334C4×D5—24no—
335C4×D5—82no—
336Dic10—21no—
337Dic10C2×C4×D312yes—
338#168—32no—
339#168—32no—
340#168—62no—
341C2×#41—31no—
342C2×#41—51no—
343#135—158no—
344C2×Q16—151no—
345C2×SD16—151no—
346C2×SD16—151no—
347C2×D8—151no—
348C2×M16—151no—
349C22×C8—151no—
350C4×Q8—32no—
351C4×Q8—52no—
352C4×D4—32no—
353C4×D4—52no—
354C4×D4—62no—
355C4×D4—102no—
356#117—61no—
357#117—101no—
358#117D3×D511yescharacteristic
359C2×C4⋊C4—31no—
360C2×C4⋊C4—51no—
361C2×C4⋊C4—61no—
362C2×C4⋊C4—101no—
363#106—152no—
364#106—152no—
365#103—152no—
366#103—152no—
367#102—152no—
368#102—152no—
369#97—152no—
370C4×C8—152no—
371C2×C3×C5—22no—
372C2×C3×C5C2×C4×D411yescharacteristic
373C2×C3×C5#57711yescharacteristic
374C2×C3×C5#57711yescharacteristic
375C3×D5—42no—
376C5×D3—42no—
377C23×C3—101no—
378C23×C3C2×C22⋊D511yescharacteristic
379C22×D3—21no—
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×C4#70912yes—
392C2×C3×C4C2×C22⋊D512yes—
393C2×C3×C4C2×C22⋊D512yes—
394C2×C3×C4C2×C22⋊D511yescharacteristic
395C2×C3×C4C23×D511yescharacteristic
396#67—24no—
397C2×Dic3—21no—
398C2×Dic3—22no—
399C2×Dic3—81no—
400D12—21no—
401D12C2×C4×D512yes—
402C4×D3—24no—
403C4×D3—82no—
404Dic6—21no—
405Dic6C2×C4×D512yes—
406C3⋊C8—52no—
407C3⋊C8—52no—
408C3⋊C8—102no—
409C22×C5—21no—
410C22×C5—21no—
411C22×C5—62no—
412C22×C5—121no—
413C22×C5—121no—
414C22×C5C4×#6712yes—
415C22×C5C22×#6711yescharacteristic
416C2×D5—22no—
417C2×D5—41no—
418C2×D5—41no—
419C4×C5—22no—
420C4×C5—22no—
421C4×C5—62no—
422C4×C5—122no—
423C4×C5C4×#6712yes—
424C4×C5C22×C4×D311yescharacteristic
425C4×C5C2×#23511yescharacteristic
426Dic5—22no—
427Dic5—42no—
428#41—34no—
429#41—54no—
430#41—62no—
431#41—102no—
432C2×Q8—31no—
433C2×Q8—51no—
434C2×D4—31no—
435C2×D4—51no—
436C2×D4—61no—
437C2×D4—101no—
438C22×C4—61no—
439C22×C4—61no—
440C22×C4—62no—
441C22×C4—101no—
442C22×C4—101no—
443C22×C4—102no—
444C22×C4C2×D3×D511yescharacteristic
445Q16—154no—
446SD16—154no—
447SD16—154no—
448D8—154no—
449M16—302no—
450C2×C8—154no—
451C2×C8—151no—
452C2×C8—151no—
453C2×C8—152no—
454C4⋊C4—32no—
455C4⋊C4—52no—
456C4⋊C4—61no—
457C4⋊C4—62no—
458C4⋊C4—101no—
459C4⋊C4—102no—
460C4⋊C4#120111yescharacteristic
461C4⋊C4#120111yescharacteristic
462C22⋊C4—62no—
463C22⋊C4—102no—
464C22⋊C4#120112yes—
465C42—62no—
466C42—102no—
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×C22⋊D512yes—
475C22×C3C22×C22⋊D511yescharacteristic
476C2×D3—22no—
477C2×D3—41no—
478C2×D3—41no—
479C3×C4—22no—
480C3×C4—22no—
481C3×C4—102no—
482C3×C4—202no—
483C3×C4C4×C22⋊D512yes—
484C3×C4C22×C4×D511yescharacteristic
485C3×C4C2×#70911yescharacteristic
486Dic3—22no—
487Dic3—42no—
488C2×C5—22no—
489C2×C5—122no—
490C2×C5C2×C4×#6711yescharacteristic
491C2×C5#583611yescharacteristic
492C2×C5#583711yescharacteristic
493D5—42no—
494C23—61no—
495C23—101no—
496C23C2×#120111yescharacteristic
497Q8—32no—
498Q8—52no—
499Q8—61no—
500Q8—101no—
501D4—32no—
502D4—52no—
503D4—64no—
504D4—61no—
505D4—104no—
506D4—101no—
507C2×C4—61no—
508C2×C4—64no—
509C2×C4—62no—
510C2×C4—101no—
511C2×C4—104no—
512C2×C4—102no—
513C2×C4—242no—
514C2×C4—241no—
515C2×C4—402no—
516C2×C4—401no—
517C2×C4#677112yes—
518C2×C4C4×D3×D512yes—
519C2×C4C2×#120112yes—
520C2×C4C2×#120112yes—
521C2×C4C2×#120111yescharacteristic
522C2×C4C22×D3×D511yescharacteristic
523C8—152no—
524C8—152no—
525C8—302no—
526C2×C3—22no—
527C2×C3—202no—
528C2×C3C2×C4×C22⋊D511yescharacteristic
529C2×C3#6617011yescharacteristic
530C2×C3#6617111yescharacteristic
531D3—42no—
532C5#8255811yescharacteristic, complemented, Sylow
533C22—21no—
534C22—21no—
535C22—62no—
536C22—102no—
537C22—121no—
538C22—121no—
539C22—201no—
540C22—201no—
541C22C4×#120112yes—
542C22C22×#120111yescharacteristic
543C4—22no—
544C4—22no—
545C4—62no—
546C4—102no—
547C4—122no—
548C4—202no—
549C4C4×#120112yes—
550C4C2×#677111yescharacteristic
551C4C2×C4×D3×D511yescharacteristic
552C3#1061428911yescharacteristic, complemented, Sylow
553C2—22no—
554C2—122no—
555C2—202no—
556C2C2×C4×#120111yescharacteristic
557C2#1175530311yescharacteristic
558C2#1175530411yescharacteristic
subgroupG/H
centerC22C22×#1201
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C22×D3×D5
FittingC3×C5×#117C22
socleC22×C3×C5C22×D4
prime radicalC2×#91297C2
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. C5×#5523C2semidirect
2. C3×#65857C2semidirect
4. #11751098C2semidirect
5. C2×#90986C2semidirect
6. #11748384C2semidirect
7. #11748368C2semidirect
8. #11747089C2semidirect
9. C4×#6656C2semidirect
10. C4×#6655C2semidirect
11. #11747021C2semidirect
33. #91297C4semidirect
46. #65857D3semidirect
59. C5×#235C2×C4semidirect
67. C3×#709C2×C4semidirect
77. C2×C3⋊#168C2×C4semidirect
91. #6661C2×C4semidirect
92. #6656C2×C4semidirect
93. #6655C2×C4semidirect
94. #6652C2×C4semidirect
97. #5523D5semidirect
211. #709C4×D3semidirect
281. #235C4×D5semidirect
468. C3×C5#2952semidirect
532. C5#82558semidirect
552. C3#10614289semidirect

Maximal subgroups

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

subgroupindex
C5×#55232
C3×#658572
C2×#912972
#117510982
C2×#909862
#117483842
#117483682
#117470892
C4×#66562
C4×#66552
#117470212
#106142893
#825585