Finite Group Databasebeta

← #14415310

Subgroups — #14415310

Order 1920.

1000 conjugacy classes of subgroups.

The subgroup lattice has 552 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91927C212yesmaximal, complemented
2#11754330C211yesmaximal, characteristic
3#11752417C211yesmaximal, characteristic, complemented
4C2×C3×C4×#170C211yesmaximal, characteristic, complemented
5#11751999C212yesmaximal, complemented
6#11751914C211yesmaximal, characteristic, complemented
7#11751913C211yesmaximal, characteristic, complemented
8#11751912C212yesmaximal, complemented
9#11751893C211yesmaximal, characteristic, complemented
10#11751876C212yesmaximal, complemented
11D12×#170C211yesmaximal, characteristic, complemented
12#10617227—31nomaximal, Hall
13C2×#6782C2212yes—
14C2×#6762C2212yes—
15C22×C3×#170C2211yescharacteristic
16C22×C3×#170C2211yescharacteristic
17C2×#6740C2212yes—
18C2×#6739C2212yes—
19C2×#6738C2212yes—
20#91927—28no—
21C2×D3×#170C2212yes—
22#91819C411yescharacteristic
23C5×#930C411yescharacteristic
24C2×C3×C4×Dic5C2211yescharacteristic
25#91566C412yes—
26#91468C411yescharacteristic
27#91451C411yescharacteristic
28#91418C2211yescharacteristic
29Dic5×D12C2211yescharacteristic
30#91405C2212yes—
31#91350C412yes—
32#91233C2212yes—
33C4⋊(C3⋊#170)C2211yescharacteristic
34C4⋊(C3⋊#170)C2211yescharacteristic
35C3×C4×#170C2211yescharacteristic
36C3×C4×#170C2211yescharacteristic
37C3×C4×#170C2211yescharacteristic
38C3×C4×#170C2211yescharacteristic
39#91171C2212yes—
40#91171C2212yes—
41(C2×D3)⋊#170C2212yes—
42(C2×D3)⋊#170C2212yes—
43#78458—51nomaximal, Hall
44C2×#4131—32no—
45#65930—31no—
46#65775—32no—
47#65774—31no—
48D4×#170—31no—
49#65758—31no—
50C2×C4×#170D311yescharacteristic, complemented
51#65705—31no—
52#65702—31no—
53#65695—32no—
54#65692—32no—
55C22×C3×C4×C5C2×C411yescharacteristic
56C2×#1220C2×C412yes—
57C2×D60C2×C411yescharacteristic
58C2×C5×#67C2×C412yes—
59C2×C5×D12C2×C411yescharacteristic
60C22×C3×Dic5C2×C411yescharacteristic
61C22×C3×Dic5C2311yescharacteristic
62C2×#1204C2×C412yes—
63C2×#1203C2×C412yes—
64C2×#1203C2×C412yes—
65C2×#1200C2312yes—
66#6782—24no—
67#6782—24no—
68C2×D3×Dic5C2312yes—
69#6762—24no—
70C2×C3⋊#170C2312yes—
71C2×C3⋊#170C2312yes—
72C2×C3×#170—22no—
73C2×C3×#170D414yescomplemented
74C2×C3×#170C2311yescharacteristic
75C2×C3×#170C2311yescharacteristic
76C2×C3×#170C2311yescharacteristic
77C2×C3×#170C2311yescharacteristic
78#6740—24no—
79#6739—24no—
80#6738—24no—
81D3×#170—24no—
82#6714C2×C412yes—
83C4⋊Dic15C2×C411yescharacteristic
84C5×(C2×D3)⋊C4C2×C412yes—
85C5×C4⋊Dic3C2×C411yescharacteristic
86C3×C4×Dic5C2×C411yescharacteristic
87C3×C4×Dic5C2×C411yescharacteristic
88C3×C4×Dic5C2311yescharacteristic
89C3×C4×Dic5C2311yescharacteristic
90#6668C2×C412yes—
91#6667C2×C412yes—
92(C2×D3)⋊Dic5C2×C412yes—
93C2×#901—52no—
94#5477—51no—
95C2×C3×C4×C8—51no—
96#5110—52no—
97#4924—51no—
98#4688—52no—
99#4688—51no—
100#4688—51no—
101C8×D12—51no—
102C8×D12—52no—
103C8×D12—51no—
104C2×#711—32no—
105C2×#704—32no—
106C2×#704—32no—
107C22×#170—32no—
108C22×#170C2×D311yescharacteristic
109C22×#170C2×D311yescharacteristic
110#4131—68no—
111C2×#700—32no—
112C2×D5⋊C8—32no—
113C5×#121—31no—
114#4086—31no—
115#4085—32no—
116D4×Dic5—31no—
117#4076—31no—
118C2×C4×Dic5C2×D311yescharacteristic
119#4039—31no—
120#4034—31no—
121#4031—32no—
122#4027—32no—
123C22⋊#170—32no—
124C22⋊#170—32no—
125#4004—32no—
126#4003—32no—
127#4003—32no—
128C4⋊#170—31no—
129C4⋊#170—31no—
130C4×#170C2×D311yescharacteristic
131C4×#170C2×D311yescharacteristic
132C4×#170C2×D311yescharacteristic
133C4×#170C2×D311yescharacteristic
134#2931—151noSylow
135C23×C3×C5C22×C411yescharacteristic
136C22×D15C22×C412yes—
137C22×C5×D3C22×C412yes—
138C2×C3×C4×C5—22no—
139C2×C3×C4×C5C22×C411yescharacteristic
140C2×C3×C4×C5C22×C411yescharacteristic
141#1220—24no—
142C2×Dic15C22×C412yes—
143D60—22no—
144C5×#67—24no—
145C2×C5×Dic3C22×C412yes—
146C5×D12—22no—
147C2×C3×Dic5—22no—
148C2×C3×Dic5C22×C411yescharacteristic
149C2×C3×Dic5C22×C411yescharacteristic
150C2×C3×Dic5C22×C411yescharacteristic
151C2×C3×Dic5C2×D412yes—
152C2×C3×Dic5#4112yes—
153C2×C3×Dic5C2411yescharacteristic
154#1204—24no—
155#1203—24no—
156#1203—24no—
157#1200—22no—
158#1200—22no—
159D3×Dic5—22no—
160D3×Dic5—22no—
161C3⋊#170—22no—
162C3⋊#170—22no—
163C3×#170—21no—
164C3×#170—21no—
165C3×#170C2×D412yes—
166C3×#170C2×D412yes—
167C2×#235—52no—
168C22×C3×C8—51no—
169C22×C3×C8—51no—
170C2×C3×C42—51no—
171#930—51no—
172#930—52no—
173#930AGL(1,𝔽5)11yescharacteristic
174C2×#208—52no—
175#901—108no—
176C2×ΓL(1,𝔽25)—52no—
177C2×ΓL(1,𝔽25)—52no—
178C2×D3×C8—52no—
179C2×D3×C8—52no—
180#875—52no—
181#874—51no—
182C4×D12—51no—
183C4×D12—52no—
184C4×D12—51no—
185#869—51no—
186C3×C4×C8—51no—
187C3×C4×C8—51no—
188C3×C4×C8—51no—
189C3×C4×C8—51no—
190(C2×D3)⋊C8—52no—
191(C2×D3)⋊C8—52no—
192(C2×D3)⋊C8—52no—
193(C2×D3)⋊C8—52no—
194C4⋊(C3⋊C8)—51no—
195C4⋊(C3⋊C8)—51no—
196C4⋊(C3⋊C8)—52no—
197C2×C5×D4—32no—
198C2×C5×D4—31no—
199C22×C4×C5C4×D311yescharacteristic
200C2×C22⋊D5—32no—
201C2×C22⋊D5—32no—
202C2×C22⋊D5—32no—
203C22×Dic5—32no—
204C22×Dic5C4×D311yescharacteristic
205C22×Dic5C22×D311yescharacteristic
206#711—64no—
207#711—64no—
208C2×D20—31no—
209C2×C4×D5—32no—
210C2×Dic10—32no—
211#704—64no—
212#704—64no—
213C2×#170—22no—
214C2×#170—32no—
215C2×#170—32no—
216C2×#170—64no—
217C2×#170D1214yescomplemented
218C2×#170C22×D311yescharacteristic
219C2×#170C22×D311yescharacteristic
220C2×#170C22×D311yescharacteristic
221C2×#170C22×D311yescharacteristic
222#700—64no—
223D5⋊C8—64no—
224C5×C4⋊C4—31no—
225C5×C22⋊C4—32no—
226C22⋊Dic5—32no—
227#685—32no—
228#685—32no—
229C4⋊Dic5—31no—
230#683—32no—
231C4×Dic5C4×D311yescharacteristic
232C4×Dic5C4×D311yescharacteristic
233C4×Dic5C22×D311yescharacteristic
234C4×Dic5C22×D311yescharacteristic
235C2×#131—152no—
236#532—151no—
237#436—152no—
238#436—151no—
239#436—151no—
240C8×D4—152no—
241C8×D4—151no—
242C8×D4—151no—
243#423—151no—
244#408—152no—
245C2×C4×C8—51no—
246C22×C3×C5—21no—
247C22×C3×C5—21no—
248C22×C3×C5C4×D412yes—
249C22×C3×C5C23×C411yescharacteristic
250C2×D15—22no—
251C2×D15—22no—
252C2×D15—22no—
253C2×C5×D3—22no—
254C2×C5×D3—22no—
255C2×C5×D3—22no—
256C3×C4×C5—21no—
257C3×C4×C5#13112yescomplemented
258Dic15—22no—
259C3×Dic5—21no—
260C3×Dic5C4×D412yes—
261C3×Dic5#13112yescomplemented
262C3×Dic5C22×D411yescharacteristic
263C3×Dic5C2×#4111yescharacteristic
264C5×Dic3—22no—
265C22×C3×C4—51no—
266C22×C3×C4—51no—
267C22×C3×C4C2×AGL(1,𝔽5)11yescharacteristic
268C2×#67—52no—
269C2×#67C2×AGL(1,𝔽5)12yes—
270#235—104no—
271#235—104no—
272C2×D12—52no—
273C2×D12—52no—
274C2×D12—51no—
275C2×D12C2×AGL(1,𝔽5)11yescharacteristic
276C2×C4×D3—52no—
277C2×C4×D3—52no—
278C2×Dic6—52no—
279C2×C3×C8—51no—
280C2×C3×C8—51no—
281C2×C3×C8—54no—
282C2×C3×C8—51no—
283C2×C3×C8—51no—
284C2×C3×C8—102no—
285C3×C42—51no—
286C3×C42—51no—
287C3×C42—51no—
288C3×C42—51no—
289(C2×D3)⋊C4—52no—
290(C2×D3)⋊C4—52no—
291(C2×D3)⋊C4—52no—
292(C2×D3)⋊C4C2×AGL(1,𝔽5)12yes—
293C4⋊Dic3—51no—
294C4⋊Dic3—52no—
295C4⋊Dic3C2×AGL(1,𝔽5)11yescharacteristic
296#208—104no—
297C2×C3⋊C8—52no—
298C2×C3⋊C8—52no—
299ΓL(1,𝔽25)—104no—
300ΓL(1,𝔽25)—104no—
301D3×C8—104no—
302D3×C8—104no—
303C23×C5—32no—
304C23×C5C2×C4×D311yescharacteristic
305C22×D5—32no—
306C5×D4—64no—
307C5×D4—62no—
308C2×C4×C5—22no—
309C2×C4×C5—32no—
310C2×C4×C5C2×C4×D311yescharacteristic
311C2×C4×C5C2×C4×D311yescharacteristic
312C22⋊D5—64no—
313C22⋊D5—64no—
314C22⋊D5—64no—
315C2×Dic5—22no—
316C2×Dic5—32no—
317C2×Dic5—62no—
318C2×Dic5—62no—
319C2×Dic5C2×C4×D311yescharacteristic
320C2×Dic5C2×C4×D311yescharacteristic
321C2×Dic5C2×C4×D311yescharacteristic
322C2×Dic5C2×D1212yes—
323C2×Dic5#23512yes—
324C2×Dic5C23×D311yescharacteristic
325D20—62no—
326C4×D5—62no—
327C4×D5—62no—
328Dic10—64no—
329#170—21no—
330#170—21no—
331#170—62no—
332#170—62no—
333#170C2×D1212yes—
334#170C2×D1212yes—
335C2×#41—152no—
336#131—308no—
337C2×M16—152no—
338C2×M16—152no—
339C2×M16—152no—
340C22×C8—51no—
341C22×C8—51no—
342C22×C8—152no—
343C22×C8—152no—
344C4⋊Q8—151no—
345#127—151no—
346#124—152no—
347#121—31no—
348#121—152no—
349#121—151no—
350C4×D4—152no—
351C4×D4—151no—
352C4×D4—151no—
353C2×C42—51no—
354C4⋊C8—151no—
355C4⋊C8—151no—
356C4⋊C8—152no—
357C22⋊C8—152no—
358C22⋊C8—152no—
359C22⋊C8—152no—
360C22⋊C8—152no—
361C4×C8—51no—
362C4×C8—51no—
363C4×C8—51no—
364C4×C8—51no—
365C2×C3×C5—22no—
366C2×C3×C5C2×C4×D411yescharacteristic
367C2×C3×C5C2×#13111yescharacteristic
368C2×C3×C5C2×#13111yescharacteristic
369D15—42no—
370C5×D3—42no—
371C23×C3C22×AGL(1,𝔽5)11yescharacteristic
372C22×D3—52no—
373C22×D3C22×AGL(1,𝔽5)12yes—
374C2×C3×C4—22no—
375C2×C3×C4—52no—
376C2×C3×C4—52no—
377C2×C3×C4—51no—
378C2×C3×C4—51no—
379C2×C3×C4—51no—
380C2×C3×C4—51no—
381C2×C3×C4—102no—
382C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
383C2×C3×C4C22×AGL(1,𝔽5)11yescharacteristic
384#67—24no—
385#67—104no—
386C2×Dic3—52no—
387C2×Dic3C22×AGL(1,𝔽5)12yes—
388D12—22no—
389D12—104no—
390D12—104no—
391D12—102no—
392C4×D3—102no—
393C4×D3—102no—
394C4×D3—102no—
395C4×D3—102no—
396Dic6—104no—
397C3×C8—52no—
398C3×C8—52no—
399C3×C8—101no—
400C3×C8—101no—
401C3⋊C8—102no—
402C3⋊C8—102no—
403C22×C5—21no—
404C22×C5—21no—
405C22×C5—62no—
406C22×C5—62no—
407C22×C5—62no—
408C22×C5C4×D1212yes—
409C22×C5C22×C4×D311yescharacteristic
410C2×D5—62no—
411C2×D5—62no—
412C2×D5—62no—
413C4×C5—21no—
414C4×C5—62no—
415C4×C5#90112yescomplemented
416Dic5—21no—
417Dic5—62no—
418Dic5C4×D1212yes—
419Dic5#90112yescomplemented
420Dic5C22×D1211yescharacteristic
421Dic5C2×#23511yescharacteristic
422#41—304no—
423#41—304no—
424C2×Q8—152no—
425C2×D4—32no—
426C2×D4—31no—
427C2×D4—152no—
428C2×D4—152no—
429C2×D4—152no—
430C2×D4—151no—
431C22×C4—51no—
432C22×C4—51no—
433C22×C4—152no—
434C22×C4—152no—
435C22×C4D3×AGL(1,𝔽5)11yescharacteristic
436M16—304no—
437M16—304no—
438M16—304no—
439C2×C8—51no—
440C2×C8—51no—
441C2×C8—54no—
442C2×C8—51no—
443C2×C8—51no—
444C2×C8—102no—
445C2×C8—152no—
446C2×C8—152no—
447C2×C8—304no—
448C2×C8—304no—
449C4⋊C4—31no—
450C4⋊C4—152no—
451C4⋊C4—151no—
452C22⋊C4—32no—
453C22⋊C4—152no—
454C22⋊C4—152no—
455C22⋊C4—152no—
456C42—51no—
457C42—51no—
458C42—51no—
459C42—51no—
460C3×C5#293111yescharacteristic, complemented, Hall
461C22×C3—21no—
462C22×C3—21no—
463C22×C3D4×AGL(1,𝔽5)12yes—
464C22×C3C23×AGL(1,𝔽5)11yescharacteristic
465C2×D3—22no—
466C2×D3—22no—
467C2×D3—22no—
468C2×D3—102no—
469C2×D3—102no—
470C2×D3—102no—
471C3×C4—21no—
472C3×C4—52no—
473C3×C4—51no—
474C3×C4—51no—
475C3×C4—52no—
476C3×C4—101no—
477C3×C4#413112yescomplemented
478Dic3—22no—
479Dic3—102no—
480C2×C5—22no—
481C2×C5—122no—
482C2×C5C2×C4×D1211yescharacteristic
483C2×C5C2×#90111yescharacteristic
484C2×C5C2×#90111yescharacteristic
485D5—122no—
486C23—32no—
487C23—152no—
488C23C2×D3×AGL(1,𝔽5)11yescharacteristic
489Q8—304no—
490D4—64no—
491D4—62no—
492D4—304no—
493D4—304no—
494D4—304no—
495D4—302no—
496C2×C4—22no—
497C2×C4—32no—
498C2×C4—52no—
499C2×C4—52no—
500C2×C4—51no—
501C2×C4—51no—
502C2×C4—51no—
503C2×C4—51no—
504C2×C4—102no—
505C2×C4—152no—
506C2×C4—302no—
507C2×C4—302no—
508C2×C4—302no—
509C2×C4—302no—
510C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
511C2×C4C2×D3×AGL(1,𝔽5)11yescharacteristic
512C8—52no—
513C8—52no—
514C8—101no—
515C8—101no—
516C8—302no—
517C8—302no—
518C2×C3—22no—
519C2×C3C2×#413111yescharacteristic
520C2×C3C2×D4×AGL(1,𝔽5)11yescharacteristic
521C2×C3D5⋊#13111yescharacteristic
522D3—42no—
523D3—202no—
524C5#7845811yescharacteristic, complemented, Sylow
525C22—21no—
526C22—21no—
527C22—62no—
528C22—62no—
529C22—62no—
530C22—302no—
531C22—302no—
532C22—302no—
533C22AGL(1,𝔽5)×D1212yes—
534C22C22×D3×AGL(1,𝔽5)11yescharacteristic
535C4—21no—
536C4—52no—
537C4—51no—
538C4—51no—
539C4—52no—
540C4—62no—
541C4—101no—
542C4—302no—
543C4#9192712yescomplemented
544C3#1061722711yescharacteristic, complemented, Sylow
545C2—22no—
546C2—122no—
547C2—602no—
548C2D5⋊#90111yescharacteristic
549C2C2×AGL(1,𝔽5)×D1211yescharacteristic
550C2C2×#9192711yescharacteristic
subgroupG/H
center, FrattiniC22C22×D3×AGL(1,𝔽5)
commutator, socleC22×C3×C5C23×C4
FittingC22×C3×C4×C5C2×C4
prime radical#11754330C2
cosocleC2×C3×Dic5C24
hypercenterC22×C4D3×AGL(1,𝔽5)
hypocenterC3×C5#2931
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4

Complements

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

NH ≅ G/Ntype
1. C2×#91927C2semidirect
3. #11752417C2semidirect
4. C2×C3×C4×#170C2semidirect
5. #11751999C2semidirect
6. #11751914C2semidirect
7. #11751913C2semidirect
8. #11751912C2semidirect
9. #11751893C2semidirect
10. #11751876C2semidirect
11. D12×#170C2semidirect
50. C2×C4×#170D3semidirect
73. C2×C3×#170D4semidirect
217. C2×#170D12semidirect
257. C3×C4×C5#131semidirect
261. C3×Dic5#131semidirect
415. C4×C5#901semidirect
419. Dic5#901semidirect
460. C3×C5#2931semidirect
477. C3×C4#4131semidirect
524. C5#78458semidirect
543. C4#91927semidirect
544. C3#10617227semidirect

Maximal subgroups

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

subgroupindex
C2×#919272
#117543302
#117524172
C2×C3×C4×#1702
#117519992
#117519142
#117519132
#117519122
#117518932
#117518762
D12×#1702
#106172273
#784585