Finite Group Databasebeta

← C2×#11748089 #14429326

Subgroups — C2×#11748089 #14429326

Order 1920.

936 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
1C2×#91790C211yesmaximal, characteristic, complemented
2C22×C5×#204C211yesmaximal, characteristic, complemented
3C2×#91321C211yesmaximal, characteristic, complemented
4C2×#91304C211yesmaximal, characteristic
5C2×C3×#3951C211yesmaximal, characteristic, complemented
6C2×#90962C211yesmaximal, characteristic, complemented
7C2×#90959C211yesmaximal, characteristic, complemented
8#11748089C218yesmaximal, direct product factor
9C2×#65466—31nomaximal, Hall
10C2×#6817C2211yescharacteristic
11C22×C5×D12C2211yescharacteristic
12C22×C5×Dic6C2211yescharacteristic
13C22×C3×C5×C8C2211yescharacteristic
14C2×#6710C2211yescharacteristic
15#91790—22no—
16#91790C2214yescomplemented
17C2×#6705—21no—
18C2×C8×D15—21no—
19C2×C5×#204—21no—
20C2×C5×#204—21no—
21C2×C5×#204—22no—
22C2×C5×#204C2214yescomplemented
23C2×C3×#681C2211yescharacteristic
24C2×C3×#681C2211yescharacteristic
25#91321—22no—
26#91321C2214yescomplemented
27C2×#6663—21no—
28C2×#6660—21no—
29C2×#6659—21no—
30#91304—22no—
31#91304C2214yes—
32C2×#6654—21no—
33C3×#3951C2214yescomplemented
34#90962C2214yescomplemented
35#90959C2214yescomplemented
36C2×#4898—51nomaximal, Hall
37C22×C5×SD16—31no—
38C2×#4088—31no—
39C2×#4078—31no—
40C2×#4047—31no—
41C2×#3968—31no—
42C2×#3965—31no—
43#65466—38no—
44C2×#3951D311yescharacteristic, complemented
45C23×C5×D3—21no—
46C22×C3×C4×C5C2311yescharacteristic
47C2×#1220—21no—
48#6817—21no—
49#6817—21no—
50#6817D412yes—
51#6817C2312yes—
52C2×D60D411yescharacteristic
53C2×C4×D15—21no—
54C2×Dic30D411yescharacteristic
55C22×C5×Dic3—21no—
56C2×C5×D12—21no—
57C2×C5×D12—21no—
58C2×C5×D12—21no—
59C2×C5×D12—21no—
60C2×C5×D12—21no—
61C2×C5×D12C2312yes—
62C2×C5×Dic6—21no—
63C2×C5×Dic6—21no—
64C2×C5×Dic6—21no—
65C2×C5×Dic6—21no—
66C2×C5×Dic6—21no—
67C2×C5×Dic6C2312yes—
68C2×C3×C5×C8D411yescharacteristic
69C2×C3×C5×C8D411yescharacteristic
70C2×C3×C5×C8D412yes—
71C2×C3×C5×C8C2312yes—
72#6710—21no—
73#6710C2312yes—
74C2×C3⋊#168—21no—
75#6705—24no—
76C8×D15—24no—
77C5×#204—24no—
78C5×#204—24no—
79C3×#681—21no—
80C3×#681—21no—
81C3×#681C2312yes—
82C3×#681C2312yes—
83C2×C3×#168—21no—
84C2×C3×#168—21no—
85#6663—24no—
86#6660—24no—
87#6659—24no—
88#6654—24no—
89C2×#909—51no—
90C2×#908—51no—
91C22×#204D511yescharacteristic, complemented
92C2×#901—51no—
93C2×C3×#108—51no—
94C2×#824—51no—
95C2×#823—51no—
96#4898—58no—
97C22×C5×Q8—31no—
98C22×C5×D4—31no—
99C2×#709—31no—
100C2×C5×SD16—34no—
101C2×C5×SD16—61no—
102C2×C5×SD16—61no—
103C2×C5×SD16—62no—
104C22×C5×C8C2×D311yescharacteristic
105C2×#689—61no—
106#4088—34no—
107#4088—62no—
108C2×#688—61no—
109C2×#687—61no—
110#4078—34no—
111#4078—62no—
112C2×#686—61no—
113C2×#681—31no—
114C2×#681C2×D311yescharacteristic
115C2×#681C2×D311yescharacteristic
116#4047—34no—
117#4047—62no—
118C2×#676—61no—
119C2×C8×D5—61no—
120#3968—34no—
121#3965—34no—
122#3951C2×D314yescomplemented
123C2×#471—151noSylow
124C23×C3×C5#4111yescharacteristic
125C22×D15—21no—
126C22×C5×D3—22no—
127C22×C5×D3—41no—
128C22×C5×D3—41no—
129C22×C5×D3—82no—
130C2×C3×C4×C5C2×D411yescharacteristic
131C2×C3×C4×C5C2×D411yescharacteristic
132C2×C3×C4×C5C2×D411yescharacteristic
133C2×C3×C4×C5C2×D411yescharacteristic
134C2×C3×C4×C5#4111yescharacteristic
135C2×C3×C4×C5C2411yescharacteristic
136#1220—22no—
137#1220—22no—
138C2×Dic15—21no—
139D60—21no—
140D60C2×D412yes—
141C4×D15—22no—
142C4×D15—22no—
143Dic30—21no—
144Dic30C2×D412yes—
145C2×C5×Dic3—22no—
146C2×C5×Dic3—41no—
147C2×C5×Dic3—41no—
148C5×D12—22no—
149C5×D12—22no—
150C5×D12—41no—
151C5×D12—41no—
152C5×Dic6—22no—
153C5×Dic6—22no—
154C5×Dic6—41no—
155C5×Dic6—41no—
156C3×C5×C8C2×D412yes—
157C3×C5×C8C2×D412yes—
158C3⋊#168—22no—
159C3×#168—22no—
160C3×#168—22no—
161C2×#235—51no—
162C22×D12C2×D511yescharacteristic
163C22×Dic6C2×D511yescharacteristic
164C2×C3×M16—51no—
165C2×C3×M16—51no—
166C22×C3×C8C2×D511yescharacteristic
167C2×#208—51no—
168#909—54no—
169#909—102no—
170#908—54no—
171#908—102no—
172C2×Dic12—101no—
173C2×D24—101no—
174C2×#204—21no—
175C2×#204—21no—
176C2×#204—22no—
177C2×#204—101no—
178C2×#204—101no—
179C2×#204C2×D514yescomplemented
180#901—54no—
181#901—102no—
182C2×ΓL(1,𝔽25)—101no—
183C2×D3×C8—101no—
184C3×#108—54no—
185#824—54no—
186#823—54no—
187C24×C5—61no—
188C2×C5×Q8—32no—
189C2×C5×Q8—61no—
190C2×C5×Q8—61no—
191C2×C5×Q8—61no—
192C2×C5×Q8—61no—
193C2×C5×Q8—61no—
194C2×C5×D4—32no—
195C2×C5×D4—61no—
196C2×C5×D4—61no—
197C2×C5×D4—61no—
198C2×C5×D4—61no—
199C2×C5×D4—61no—
200C22×C4×C5—61no—
201C22×C4×C5C22×D311yescharacteristic
202C2×C22⋊D5—61no—
203#709—32no—
204#709—32no—
205#709—61no—
206#709—61no—
207C2×D20—31no—
208C2×C4×D5—61no—
209C2×Dic10—31no—
210C5×SD16—64no—
211C5×SD16—64no—
212C2×C5×C8D1211yescharacteristic
213C2×C5×C8D1211yescharacteristic
214C2×C5×C8D1212yes—
215C2×C5×C8C22×D312yes—
216#689—64no—
217#688—64no—
218#687—64no—
219#686—64no—
220#681—21no—
221#681—21no—
222#681—32no—
223#681—61no—
224#681C22×D312yes—
225#681C22×D312yes—
226C2×#168—21no—
227C2×#168—21no—
228C2×#168—61no—
229#676—64no—
230C8×D5—64no—
231C2×#137—151no—
232C2×AGL(1,ℤ/8)—151no—
233C22×SD16—31no—
234C2×#131—151no—
235#471—158no—
236C2×#108—51no—
237C2×#101—151no—
238C2×#100—151no—
239C22×C3×C5—22no—
240C22×C3×C5#12111yescharacteristic
241C22×C3×C5#12111yescharacteristic
242C22×C3×C5C2×#4111yescharacteristic
243C2×D15—22no—
244C2×D15—42no—
245C2×C5×D3—42no—
246C2×C5×D3—82no—
247C2×C5×D3—82no—
248C2×C5×D3—82no—
249C3×C4×C5#12111yescharacteristic
250C3×C4×C5#12111yescharacteristic
251C3×C4×C5C22×D411yescharacteristic
252C3×C4×C5C22×D411yescharacteristic
253Dic15—22no—
254C5×Dic3—42no—
255C23×D3—21no—
256C22×C3×C4C22×D511yescharacteristic
257C2×#67—101no—
258C22×Dic3—21no—
259#235—52no—
260#235—52no—
261#235—101no—
262#235—101no—
263C2×D12—21no—
264C2×D12—21no—
265C2×D12—21no—
266C2×D12—21no—
267C2×D12—21no—
268C2×D12—51no—
269C2×D12C22×D512yes—
270C2×C4×D3—101no—
271C2×Dic6—21no—
272C2×Dic6—21no—
273C2×Dic6—21no—
274C2×Dic6—21no—
275C2×Dic6—21no—
276C2×Dic6—51no—
277C2×Dic6C22×D512yes—
278C3×M16—52no—
279C3×M16—52no—
280C3×M16—101no—
281C3×M16—101no—
282C2×C3×C8—101no—
283C2×C3×C8—101no—
284C2×C3×C8C22⋊D511yescharacteristic
285C2×C3×C8C22⋊D511yescharacteristic
286C2×C3×C8C22⋊D512yes—
287C2×C3×C8C22×D512yes—
288#208—52no—
289#208—101no—
290C2×C3⋊C8—101no—
291Dic12—104no—
292D24—104no—
293#204—24no—
294#204—24no—
295#204—104no—
296#204—104no—
297ΓL(1,𝔽25)—104no—
298D3×C8—104no—
299C23×C5—62no—
300C23×C5—121no—
301C23×C5—121no—
302C23×C5—242no—
303C23×C5#23911yescharacteristic
304C22×D5—61no—
305C5×Q8—62no—
306C5×Q8—62no—
307C5×Q8—121no—
308C5×Q8—121no—
309C5×D4—62no—
310C5×D4—62no—
311C5×D4—121no—
312C5×D4—121no—
313C2×C4×C5—62no—
314C2×C4×C5—121no—
315C2×C4×C5—121no—
316C2×C4×C5C2×D1211yescharacteristic
317C2×C4×C5C2×D1211yescharacteristic
318C2×C4×C5D3×D411yescharacteristic
319C2×C4×C5D3×D411yescharacteristic
320C2×C4×C5#23911yescharacteristic
321C2×C4×C5C23×D311yescharacteristic
322C22⋊D5—62no—
323C22⋊D5—62no—
324C2×Dic5—61no—
325D20—32no—
326D20—61no—
327C4×D5—62no—
328C4×D5—62no—
329Dic10—32no—
330Dic10—61no—
331C5×C8C2×D1212yes—
332C5×C8C2×D1212yes—
333#168—22no—
334#168—22no—
335#168—62no—
336C2×#41—151no—
337C22×Q8—31no—
338C22×D4—31no—
339#137—154no—
340#137—302no—
341AGL(1,ℤ/8)—154no—
342AGL(1,ℤ/8)—302no—
343C2×Q16—301no—
344C2×SD16—34no—
345C2×SD16—61no—
346C2×SD16—61no—
347C2×SD16—62no—
348C2×SD16—301no—
349C2×SD16—301no—
350C2×D8—301no—
351#131—154no—
352#131—302no—
353C2×M16—51no—
354C2×M16—51no—
355C2×M16—151no—
356C2×M16—301no—
357C22×C8—301no—
358C22×C8D3×D511yescharacteristic
359#108—54no—
360#101—154no—
361#100—154no—
362C2×C3×C5—21no—
363C2×C3×C5—21no—
364C2×C3×C5#47112yescomplemented
365C2×C3×C5C2×#12111yescharacteristic
366D15—42no—
367C5×D3—82no—
368C23×C3#71111yescharacteristic
369C22×D3—22no—
370C22×D3—41no—
371C22×D3—41no—
372C22×D3—82no—
373C22×D3—101no—
374C2×C3×C4D4×D511yescharacteristic
375C2×C3×C4D4×D511yescharacteristic
376C2×C3×C4#71111yescharacteristic
377C2×C3×C4C2×C22⋊D511yescharacteristic
378C2×C3×C4C2×C22⋊D511yescharacteristic
379C2×C3×C4C23×D511yescharacteristic
380#67—102no—
381#67—102no—
382C2×Dic3—22no—
383C2×Dic3—41no—
384C2×Dic3—41no—
385C2×Dic3—101no—
386D12—22no—
387D12—22no—
388D12—41no—
389D12—41no—
390D12—52no—
391D12—101no—
392C4×D3—102no—
393C4×D3—102no—
394Dic6—22no—
395Dic6—22no—
396Dic6—41no—
397Dic6—41no—
398Dic6—52no—
399Dic6—101no—
400C3×C8—102no—
401C3×C8—102no—
402C3×C8C2×C22⋊D512yes—
403C3×C8C2×C22⋊D512yes—
404C3⋊C8—102no—
405C22×C5—22no—
406C22×C5—122no—
407C22×C5—242no—
408C22×C5—242no—
409C22×C5—242no—
410C22×C5#89511yescharacteristic
411C22×C5#89511yescharacteristic
412C22×C5C2×#23911yescharacteristic
413C2×D5—62no—
414C2×D5—122no—
415C4×C5—122no—
416C4×C5#89511yescharacteristic
417C4×C5#89511yescharacteristic
418C4×C5C22×D1211yescharacteristic
419C4×C5C2×D3×D411yescharacteristic
420Dic5—62no—
421C24—61no—
422#41—152no—
423#41—152no—
424#41—301no—
425#41—301no—
426C2×Q8—32no—
427C2×Q8—61no—
428C2×Q8—61no—
429C2×Q8—61no—
430C2×Q8—61no—
431C2×Q8—61no—
432C2×Q8—151no—
433C2×D4—32no—
434C2×D4—61no—
435C2×D4—61no—
436C2×D4—61no—
437C2×D4—61no—
438C2×D4—61no—
439C2×D4—151no—
440C2×D4—301no—
441C22×C4—61no—
442C22×C4—301no—
443C22×C4C2×D3×D511yescharacteristic
444Q16—304no—
445SD16—64no—
446SD16—64no—
447SD16—304no—
448SD16—304no—
449D8—304no—
450M16—52no—
451M16—52no—
452M16—101no—
453M16—101no—
454M16—152no—
455M16—301no—
456M16—304no—
457C2×C8—101no—
458C2×C8—101no—
459C2×C8—301no—
460C2×C8—304no—
461C2×C8#120311yescharacteristic
462C2×C8#120311yescharacteristic
463C2×C8#120312yes—
464C2×C8C2×D3×D512yes—
465C3×C5C2×#47111yescharacteristic, complemented, Hall
466C22×C3—22no—
467C22×C3#408411yescharacteristic
468C22×C3#408411yescharacteristic
469C22×C3C2×#71111yescharacteristic
470C2×D3—42no—
471C2×D3—82no—
472C2×D3—82no—
473C2×D3—82no—
474C2×D3—102no—
475C2×D3—202no—
476C3×C4#408411yescharacteristic
477C3×C4#408411yescharacteristic
478C3×C4C2×D4×D511yescharacteristic
479C3×C4C22×C22⋊D511yescharacteristic
480Dic3—42no—
481Dic3—102no—
482C2×C5—21no—
483C2×C5—21no—
484C2×C5—242no—
485C2×C5#489812yescomplemented
486C2×C5C2×#89511yescharacteristic
487D5—122no—
488C23—62no—
489C23—121no—
490C23—121no—
491C23—242no—
492C23—301no—
493C23#676811yescharacteristic
494Q8—62no—
495Q8—62no—
496Q8—121no—
497Q8—121no—
498Q8—152no—
499Q8—301no—
500D4—62no—
501D4—62no—
502D4—121no—
503D4—121no—
504D4—152no—
505D4—302no—
506D4—302no—
507D4—301no—
508C2×C4—62no—
509C2×C4—121no—
510C2×C4—121no—
511C2×C4—301no—
512C2×C4—302no—
513C2×C4—302no—
514C2×C4#676811yescharacteristic
515C2×C4#677711yescharacteristic
516C2×C4#677711yescharacteristic
517C2×C4C2×#120311yescharacteristic
518C2×C4C2×#120311yescharacteristic
519C2×C4C22×D3×D511yescharacteristic
520C8—102no—
521C8—102no—
522C8—302no—
523C8C2×#120312yes—
524C8C2×#120312yes—
525C2×C3—21no—
526C2×C3—21no—
527C2×C3#6546612yescomplemented
528C2×C3C2×#408411yescharacteristic
529D3—82no—
530D3—202no—
531C5C2×#489811yescharacteristic, complemented, Sylow
532C22—22no—
533C22—122no—
534C22—242no—
535C22—242no—
536C22—242no—
537C22—302no—
538C22—602no—
539C22#9146611yescharacteristic
540C22#9146611yescharacteristic
541C22C2×#676811yescharacteristic
542C4—122no—
543C4—302no—
544C4#9146611yescharacteristic
545C4#9146611yescharacteristic
546C4C2×#677711yescharacteristic
547C4C22×#120311yescharacteristic
548C3C2×#6546611yescharacteristic, complemented, Sylow
549C2—21no—
550C2—21no—
551C2—242no—
552C2—602no—
553C2#1174808912yesdirect product factor
554C2C2×#9146611yescharacteristic
subgroupG/H
centerC22#91466
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC22×C3×C5×C8C22
socleC22×C3×C5#121
prime radicalC2×#91304C2
hypercenterC22×C8D3×D5
hypocenterC3×C5C2×#471
LCSG ▷ C2×C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8

Complements

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

NH ≅ G/Ntype
1. C2×#91790C2semidirect
2. C22×C5×#204C2semidirect
3. C2×#91321C2semidirect
5. C2×C3×#3951C2semidirect
6. C2×#90962C2semidirect
7. C2×#90959C2semidirect
8. #11748089C2direct (N × H)
16. #91790C22semidirect
22. C2×C5×#204C22semidirect
26. #91321C22semidirect
33. C3×#3951C22semidirect
34. #90962C22semidirect
35. #90959C22semidirect
44. C2×#3951D3semidirect
91. C22×#204D5semidirect
122. #3951C2×D3semidirect
179. C2×#204C2×D5semidirect
364. C2×C3×C5#471semidirect
465. C3×C5C2×#471semidirect
485. C2×C5#4898semidirect
527. C2×C3#65466semidirect
531. C5C2×#4898semidirect
548. C3C2×#65466semidirect
553. C2#11748089direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#917902
C22×C5×#2042
C2×#913212
C2×#913042
C2×C3×#39512
C2×#909622
C2×#909592
#117480892
C2×#654663
C2×#48985