Finite Group Databasebeta

← C2×#11756595 #14510035

Subgroups — C2×#11756595 #14510035

Order 1920.

2920 conjugacy classes of subgroups.

The subgroup lattice has 484 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C3×C5×H(5,𝔽2)C211yesmaximal, characteristic, complemented
2C2×#92101C211yesmaximal, characteristic, complemented
3#11756595C2116yesmaximal, direct product factor
4C2×#91840C213yesmaximal, complemented
5C2×#91839C213yesmaximal, complemented
6C2×#91837C211yesmaximal, characteristic, complemented
7C22×#6716C213yesmaximal, complemented
8C2×#91821C213yesmaximal, complemented
9C2×#66170—31nomaximal, Hall
10C3×C5×H(5,𝔽2)—24no—
11C3×C5×H(5,𝔽2)C2218yescomplemented
12C2×C3×C5×#41—21no—
13C2×C3×C5×#41C2213yes—
14C2×C3×C5×#41C2211yescharacteristic
15C22×C3×C5×D4—23no—
16C22×C3×C5×D4C2213yes—
17#92101—24no—
18#92101C2218yescomplemented
19C2×Q8×D15—21no—
20C2×#6819—23no—
21C2×#6817C2213yescomplemented
22C22×Dic30C2213yescomplemented
23#91840C22124yescomplemented
24#91839C22124yescomplemented
25#91837C2218yescomplemented
26C2×#6718C2213yescomplemented
27C2×#6717C2211yescharacteristic, complemented
28C2×#6716C2216yescomplemented
29C2×#6716C2213yescomplemented
30C2×#6716C22124yescomplemented
31#91821C22124yescomplemented
32C2×#6715C2213yescomplemented
33C2×#6710C2213yescomplemented
34C22×C3⋊#168C2213yescomplemented
35C2×#5836—51nomaximal, Hall
36C2×C5×H(5,𝔽2)D311yescharacteristic, complemented
37C2×#4150—31no—
38#66170—316no—
39C2×#4097—33no—
40C2×#4096—33no—
41C2×#4094—31no—
42C22×#687—33no—
43C2×#4078—33no—
44C24×C3×C5—23no—
45C3×C5×#41—28no—
46C3×C5×#41—26no—
47C3×C5×#41D414yes—
48C3×C5×#41C23112yes—
49C3×C5×#41C2314yes—
50C2×C3×C5×Q8D411yescharacteristic
51C2×C3×C5×Q8C2311yescharacteristic
52C2×C3×C5×D4—23no—
53C2×C3×C5×D4—224no—
54C2×C3×C5×D4—23no—
55C2×C3×C5×D4—26no—
56C2×C3×C5×D4D413yes—
57C2×C3×C5×D4C2313yes—
58C2×C3×C5×D4C23112yes—
59C22×C3×C4×C5—23no—
60C22×C3×C4×C5C2313yes—
61C2×#1220—23no—
62C22×Dic15—23no—
63Q8×D15—28no—
64#6819—224no—
65#6817—26no—
66#6817C23112yescomplemented
67C2×D60C2311yescharacteristic
68C2×C4×D15—23no—
69C2×Dic30—23no—
70C2×Dic30—26no—
71C2×Dic30C2313yes—
72C2×Dic30C23112yescomplemented
73#6718C23112yescomplemented
74#6717C2314yescomplemented
75#6716C23124yescomplemented
76#6716C23112yescomplemented
77#6715C23112yescomplemented
78#6710C23112yescomplemented
79C2×C3⋊#168C2311yescharacteristic
80C2×C3⋊#168C2313yes—
81C2×C3⋊#168C23112yescomplemented
82C2×C3×H(5,𝔽2)D511yescharacteristic, complemented
83C2×#1010—51no—
84#5836—516no—
85C2×#951—53no—
86C2×#950—53no—
87C2×#948—51no—
88C22×#214—53no—
89C2×#932—53no—
90C5×H(5,𝔽2)—24no—
91C5×H(5,𝔽2)C2×D318yescomplemented
92C2×C5×#41—21no—
93C2×C5×#41C2×D313yes—
94C2×C5×#41C2×D311yescharacteristic
95C22×C5×D4—23no—
96C22×C5×D4C2×D313yes—
97#4150—38no—
98#4150—64no—
99C2×Q8×D5—61no—
100C2×#711—63no—
101C2×#709—33no—
102C22×Dic10—33no—
103#4097—324no—
104#4096—324no—
105#4094—38no—
106C2×#689—33no—
107C2×#688—31no—
108C2×#687—36no—
109C2×#687—33no—
110C2×#687—324no—
111#4078—324no—
112C2×#686—33no—
113C2×#681—33no—
114C22×#168—33no—
115C2×#577—151noSylow
116C23×C3×C5—23no—
117C23×C3×C5—212no—
118C23×C3×C5—86no—
119C23×C3×C5C2×D413yes—
120C22×D15—21no—
121C3×C5×Q8—21no—
122C3×C5×Q8C2×D414yes—
123C3×C5×Q8C2412yes—
124C3×C5×D4—212no—
125C3×C5×D4—23no—
126C3×C5×D4—212no—
127C3×C5×D4C2×D4112yes—
128C3×C5×D4C2416yes—
129C2×C3×C4×C5—21no—
130C2×C3×C4×C5—212no—
131C2×C3×C4×C5C2×D413yes—
132C2×C3×C4×C5C2×D416yes—
133C2×C3×C4×C5C2411yescharacteristic
134C2×C3×C4×C5C2416yes—
135#1220—212no—
136C2×Dic15—23no—
137C2×Dic15—212no—
138D60—21no—
139D60C2412yes—
140C4×D15—212no—
141Dic30—23no—
142Dic30—212no—
143Dic30C2416yes—
144C3⋊#168C2412yes—
145C3⋊#168C2416yes—
146C3×H(5,𝔽2)—24no—
147C3×H(5,𝔽2)C2×D518yescomplemented
148C2×C3×#41—21no—
149C2×C3×#41C2×D513yes—
150C2×C3×#41C2×D511yescharacteristic
151C22×C3×D4—23no—
152C22×C3×D4C2×D513yes—
153#1010—58no—
154#1010—104no—
155C2×D3×Q8—101no—
156C2×#237—103no—
157C2×#235—53no—
158C22×Dic6—53no—
159#951—524no—
160#950—524no—
161#948—58no—
162C2×#216—53no—
163C2×#215—51no—
164C2×#214—56no—
165C2×#214—53no—
166C2×#214—524no—
167#932—524no—
168C2×#213—53no—
169C2×#208—53no—
170C22×C3⋊C8—53no—
171C24×C5—23no—
172C5×#41—28no—
173C5×#41—26no—
174C5×#41#6714yes—
175C5×#41C22×D3112yes—
176C5×#41C22×D314yes—
177C2×C5×Q8#6711yescharacteristic
178C2×C5×Q8C22×D311yescharacteristic
179C2×C5×D4—23no—
180C2×C5×D4—224no—
181C2×C5×D4—23no—
182C2×C5×D4—26no—
183C2×C5×D4#6713yes—
184C2×C5×D4C22×D313yes—
185C2×C5×D4C22×D3112yes—
186C22×C4×C5—23no—
187C22×C4×C5C22×D313yes—
188C2×C22⋊D5—63no—
189C22×Dic5—63no—
190Q8×D5—68no—
191#711—624no—
192#709—312no—
193#709—66no—
194C2×D20—31no—
195C2×C4×D5—63no—
196C2×Dic10—312no—
197C2×Dic10—33no—
198C2×Dic10—63no—
199C2×Dic10—66no—
200#689—312no—
201#688—34no—
202#687—312no—
203#687—324no—
204#686—312no—
205#681—312no—
206C2×#168—312no—
207C2×#168—31no—
208C2×#168—33no—
209C2×2−1+4—151no—
210C2×H(5,𝔽2)D1511yescharacteristic, complemented
211#577—1516no—
212C2×#137—153no—
213C2×AGL(1,ℤ/8)—153no—
214C2×#135—153no—
215C22×SD16—153no—
216C2×#131—151no—
217C22×C3×C5—26no—
218C22×C3×C5—26no—
219C22×C3×C5—46no—
220C22×C3×C5—812no—
221C22×C3×C5C22×D416yes—
222C22×C3×C5C22×D411yescharacteristic
223C2×D15—22no—
224C2×D15—42no—
225C3×C4×C5—22no—
226C3×C4×C5C22×D411yescharacteristic
227C3×C4×C5C22×D416yes—
228C3×C4×C5C2511yescharacteristic
229Dic15—26no—
230C24×C3—23no—
231C3×#41—28no—
232C3×#41—26no—
233C3×#41C22⋊D514yes—
234C3×#41C22×D5112yes—
235C3×#41C22×D514yes—
236C2×C3×Q8C22⋊D511yescharacteristic
237C2×C3×Q8C22×D511yescharacteristic
238C2×C3×D4—23no—
239C2×C3×D4—224no—
240C2×C3×D4—23no—
241C2×C3×D4—26no—
242C2×C3×D4C22⋊D513yes—
243C2×C3×D4C22×D513yes—
244C2×C3×D4C22×D5112yes—
245C22×C3×C4—23no—
246C22×C3×C4C22×D513yes—
247C2×#67—103no—
248C22×Dic3—103no—
249D3×Q8—108no—
250#237—1024no—
251#235—512no—
252#235—106no—
253C2×D12—51no—
254C2×C4×D3—103no—
255C2×Dic6—53no—
256C2×Dic6—512no—
257C2×Dic6—103no—
258C2×Dic6—106no—
259#216—512no—
260#215—54no—
261#214—524no—
262#214—512no—
263#213—512no—
264#208—512no—
265C2×C3⋊C8—51no—
266C2×C3⋊C8—53no—
267C2×C3⋊C8—512no—
268C23×C5—23no—
269C23×C5—212no—
270C23×C5—86no—
271C23×C5C2×#6713yes—
272C22×D5—61no—
273C5×Q8—21no—
274C5×Q8C2×#6714yes—
275C5×Q8C23×D312yes—
276C5×D4—212no—
277C5×D4—23no—
278C5×D4—212no—
279C5×D4C2×#67112yes—
280C5×D4C23×D316yes—
281C2×C4×C5—21no—
282C2×C4×C5—212no—
283C2×C4×C5C2×#6713yes—
284C2×C4×C5C2×#6716yes—
285C2×C4×C5C23×D311yescharacteristic
286C2×C4×C5C23×D316yes—
287C22⋊D5—612no—
288C2×Dic5—63no—
289C2×Dic5—612no—
290D20—32no—
291D20—61no—
292C4×D5—612no—
293Dic10—36no—
294Dic10—63no—
295Dic10—612no—
296#168—32no—
297#168—36no—
2982−1+4—158no—
2992−1+4—304no—
300H(5,𝔽2)—24no—
301H(5,𝔽2)C2×D1518yescomplemented
302C2×#41—21no—
303C2×#41—153no—
304C2×#41—303no—
305C2×#41C2×D1513yes—
306C2×#41C2×D1511yescharacteristic
307C22×Q8—153no—
308C22×Q8—301no—
309C22×D4—23no—
310C22×D4C2×D1513yes—
311#137—1524no—
312AGL(1,ℤ/8)—1524no—
313#135—1524no—
314C2×Q16—153no—
315C2×SD16—151no—
316C2×SD16—153no—
317C2×SD16—156no—
318C2×SD16—1524no—
319C2×D8—153no—
320#131—158no—
321C2×M16—153no—
322C22×C8—153no—
323C2×C3×C5—26no—
324C2×C3×C5—46no—
325C2×C3×C5#57712yescomplemented
326C2×C3×C5C23×D411yescharacteristic
327D15—42no—
328C23×C3—23no—
329C23×C3—212no—
330C23×C3—86no—
331C23×C3C2×C22⋊D513yes—
332C22×D3—101no—
333C3×Q8—21no—
334C3×Q8C2×C22⋊D514yes—
335C3×Q8C23×D512yes—
336C3×D4—212no—
337C3×D4—23no—
338C3×D4—212no—
339C3×D4C2×C22⋊D5112yes—
340C3×D4C23×D516yes—
341C2×C3×C4—21no—
342C2×C3×C4—212no—
343C2×C3×C4C2×C22⋊D513yes—
344C2×C3×C4C2×C22⋊D516yes—
345C2×C3×C4C23×D511yescharacteristic
346C2×C3×C4C23×D516yes—
347#67—1012no—
348C2×Dic3—103no—
349C2×Dic3—1012no—
350D12—52no—
351D12—101no—
352C4×D3—1012no—
353Dic6—56no—
354Dic6—103no—
355Dic6—1012no—
356C3⋊C8—52no—
357C3⋊C8—56no—
358C22×C5—26no—
359C22×C5—26no—
360C22×C5—46no—
361C22×C5—812no—
362C22×C5C22×#6716yes—
363C22×C5C22×#6711yescharacteristic
364C2×D5—62no—
365C2×D5—122no—
366C4×C5—22no—
367C4×C5C22×#6711yescharacteristic
368C4×C5C22×#6716yes—
369C4×C5C24×D311yescharacteristic
370Dic5—66no—
371C24—23no—
372#41—28no—
373#41—26no—
374#41—1512no—
375#41—3024no—
376#41—306no—
377#41#122014yes—
378#41C22×D15112yes—
379#41C22×D1514yes—
380C2×Q8—153no—
381C2×Q8—1512no—
382C2×Q8—308no—
383C2×Q8—303no—
384C2×Q8—306no—
385C2×Q8#122011yescharacteristic
386C2×Q8C22×D1511yescharacteristic
387C2×D4—23no—
388C2×D4—224no—
389C2×D4—23no—
390C2×D4—26no—
391C2×D4—151no—
392C2×D4—303no—
393C2×D4#122013yes—
394C2×D4C22×D1513yes—
395C2×D4C22×D15112yes—
396C22×C4—23no—
397C22×C4—303no—
398C22×C4—303no—
399C22×C4C22×D1513yes—
400Q16—1512no—
401SD16—154no—
402SD16—1512no—
403SD16—1524no—
404D8—1512no—
405M16—1512no—
406C2×C8—151no—
407C2×C8—153no—
408C2×C8—1512no—
409C3×C5C2×#57711yescharacteristic, complemented, Hall
410C22×C3—26no—
411C22×C3—26no—
412C22×C3—46no—
413C22×C3—812no—
414C22×C3C22×C22⋊D516yes—
415C22×C3C22×C22⋊D511yescharacteristic
416C2×D3—102no—
417C2×D3—202no—
418C3×C4—22no—
419C3×C4C22×C22⋊D511yescharacteristic
420C3×C4C22×C22⋊D516yes—
421C3×C4C24×D511yescharacteristic
422Dic3—106no—
423C2×C5—26no—
424C2×C5—46no—
425C2×C5#583612yescomplemented
426C2×C5C23×#6711yescharacteristic
427D5—122no—
428C23—23no—
429C23—212no—
430C23—86no—
431C23—301no—
432C23C2×#122013yes—
433Q8—21no—
434Q8—156no—
435Q8—3012no—
436Q8—303no—
437Q8C2×#122014yes—
438Q8C23×D1512yes—
439D4—212no—
440D4—212no—
441D4—23no—
442D4—152no—
443D4—3012no—
444D4—301no—
445D4C2×#1220112yes—
446D4C23×D1516yes—
447C2×C4—21no—
448C2×C4—212no—
449C2×C4—3012no—
450C2×C4—303no—
451C2×C4—3012no—
452C2×C4C2×#122016yes—
453C2×C4C2×#122013yes—
454C2×C4C23×D1516yes—
455C2×C4C23×D1511yescharacteristic
456C8—152no—
457C8—156no—
458C2×C3—26no—
459C2×C3—46no—
460C2×C3#6617012yescomplemented
461C2×C3C23×C22⋊D511yescharacteristic
462D3—202no—
463C5C2×#583611yescharacteristic, complemented, Sylow
464C22—26no—
465C22—26no—
466C22—46no—
467C22—812no—
468C22—302no—
469C22—602no—
470C22C22×#122016yes—
471C22C22×#122011yescharacteristic
472C4—22no—
473C4—306no—
474C4C22×#122016yes—
475C4C22×#122011yescharacteristic
476C4C24×D1511yescharacteristic
477C3C2×#6617011yescharacteristic, complemented, Sylow
478C2—26no—
479C2—46no—
480C2—602no—
481C2#1175659512yesdirect product factor
482C2C23×#122011yescharacteristic
subgroupG/H
centerC22C22×#1220
commutator, cosocleC3×C4×C5C25
FrattiniC4C24×D15
FittingC2×C3×C5×H(5,𝔽2)C2
socleC22×C3×C5C22×D4
hypercenterC2×H(5,𝔽2)D15
hypocenterC3×C5C2×#577
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×H(5,𝔽2)

Complements

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

NH ≅ G/Ntype
1. C2×C3×C5×H(5,𝔽2)C2semidirect
2. C2×#92101C2semidirect
3. #11756595C2direct (N × H)
4. C2×#91840C2semidirect
5. C2×#91839C2semidirect
6. C2×#91837C2semidirect
7. C22×#6716C2semidirect
8. C2×#91821C2semidirect
11. C3×C5×H(5,𝔽2)C22semidirect
18. #92101C22semidirect
21. C2×#6817C22semidirect
22. C22×Dic30C22semidirect
23. #91840C22semidirect
24. #91839C22semidirect
25. #91837C22semidirect
26. C2×#6718C22semidirect
27. C2×#6717C22semidirect
28. C2×#6716C22semidirect
29. C2×#6716C22semidirect
30. C2×#6716C22semidirect
31. #91821C22semidirect
32. C2×#6715C22semidirect
33. C2×#6710C22semidirect
34. C22×C3⋊#168C22semidirect
36. C2×C5×H(5,𝔽2)D3semidirect
66. #6817C23semidirect
72. C2×Dic30C23semidirect
73. #6718C23semidirect
74. #6717C23semidirect
75. #6716C23semidirect
76. #6716C23semidirect
77. #6715C23semidirect
78. #6710C23semidirect
81. C2×C3⋊#168C23semidirect
82. C2×C3×H(5,𝔽2)D5semidirect
91. C5×H(5,𝔽2)C2×D3semidirect
147. C3×H(5,𝔽2)C2×D5semidirect
210. C2×H(5,𝔽2)D15semidirect
301. H(5,𝔽2)C2×D15semidirect
325. C2×C3×C5#577semidirect
409. C3×C5C2×#577semidirect
425. C2×C5#5836semidirect
460. C2×C3#66170semidirect
463. C5C2×#5836semidirect
477. C3C2×#66170semidirect
481. C2#11756595direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C5×H(5,𝔽2)2
C2×#921012
#117565952
C2×#918402
C2×#918392
C2×#918372
C22×#67162
C2×#918212
C2×#661703
C2×#58365