Finite Group Databasebeta

← C2×#11749198 #14412520

Subgroups — C2×#11749198 #14412520

Order 1920.

1632 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C3×#713C211yesmaximal, characteristic
2C2×D5×C4⋊Dic3C211yesmaximal, characteristic, complemented
3C22×#6649C211yesmaximal, characteristic, complemented
4C2×#91119C211yesmaximal, characteristic, complemented
5C2×C5×#835C211yesmaximal, characteristic, complemented
6#11749198C218yesmaximal, direct product factor
7C2×#90965C211yesmaximal, characteristic, complemented
8C2×#90964C211yesmaximal, characteristic, complemented
9C2×#65092—31nomaximal, Hall
10C22×C3×C5×Q8C2211yescharacteristic
11C2×C3×#713—24no—
12C2×C3×#713—22no—
13C2×C3×#713C418yescomplemented
14C2×C3×#713C2214yes—
15C22×C3×D20—21no—
16C22×C3×D20C2211yescharacteristic
17C22×C3×C4×D5—21no—
18C22×C3×C4×D5C2211yescharacteristic
19C22×D5×Dic3—21no—
20C2×C4⋊Dic15C2211yescharacteristic
21C22×C3⋊#168C2211yescharacteristic
22C2×C5×C4⋊Dic3C2211yescharacteristic
23C22×C5×C3⋊C8C2211yescharacteristic
24C2×#6668—21no—
25D5×C4⋊Dic3—22no—
26D5×C4⋊Dic3C2214yescomplemented
27C2×#6649—24no—
28C2×#6649C2214yescomplemented
29#91119C2214yescomplemented
30C5×#835C2214yescomplemented
31#90965C2214yescomplemented
32#90964C2214yescomplemented
33C2×#5232—51nomaximal, Hall
34C22×#713D311yescharacteristic
35C22×#676—31no—
36C2×D5×C4⋊C4—31no—
37C2×C5×#103—31no—
38C2×#3967—31no—
39C2×#3953—31no—
40C2×#3941—31no—
41#65092—38no—
42C23×C3×D5—21no—
43C23×C3×D5D411yescharacteristic
44C2×C3×C5×Q8—22no—
45C2×C3×C5×Q8—21no—
46C2×C3×C5×Q8C2×C414yescomplemented
47C2×C3×C5×Q8C2312yes—
48C22×C3×C4×C5—21no—
49C22×C3×C4×C5C2311yescharacteristic
50C22×Dic15—21no—
51C22×C5×Dic3—21no—
52C22×C3×Dic5D411yescharacteristic
53C3×#713—216no—
54C3×#713—28no—
55C3×#713C2×C4116yescomplemented
56C2×C3×D20—28no—
57C2×C3×D20—24no—
58C2×C3×D20—21no—
59C2×C3×D20—22no—
60C2×C3×D20C2×C414yescomplemented
61C2×C3×D20C2312yes—
62C2×C3×C4×D5—28no—
63C2×C3×C4×D5—24no—
64C2×C3×C4×D5C2×C414yes—
65C2×C3×C4×D5D412yes—
66C2×C3×C4×D5D414yes—
67C2×C3×C4×D5C2312yes—
68C2×D5×Dic3—24no—
69C2×D5×Dic3—44no—
70C4⋊Dic15—21no—
71C4⋊Dic15C2312yes—
72C2×C3⋊#168—22no—
73C2×C3⋊#168C2312yes—
74C5×C4⋊Dic3—21no—
75C5×C4⋊Dic3C2312yes—
76C2×C5×C3⋊C8—22no—
77C2×C5×C3⋊C8C2312yes—
78#6668—24no—
79#6649—28no—
80C22×C3×#41—51no—
81C22×C4⋊Dic3—51no—
82C22×#208—51no—
83#5232—58no—
84C2×#835—51no—
85C2×#835D511yescharacteristic, complemented
86C2×#832—51no—
87C2×#832—51no—
88C22×C5×Q8C2×D311yescharacteristic
89C2×#713—24no—
90C2×#713—22no—
91C2×#713Dic318yescomplemented
92C2×#713C2×D314yes—
93C22×D20—21no—
94C22×D20C2×D311yescharacteristic
95C22×C4×D5—21no—
96C22×C4×D5—61no—
97C22×C4×D5C2×D311yescharacteristic
98C22×C5×C8—31no—
99C2×C5×C4⋊C4—31no—
100C2×C4⋊Dic5—31no—
101C2×#683—61no—
102C22×#168—31no—
103C2×#676—34no—
104C2×#676—64no—
105D5×C4⋊C4—34no—
106D5×C4⋊C4—62no—
107C5×#103—34no—
108#3967—34no—
109#3953—34no—
110#3941—34no—
111C2×#417—151noSylow
112C23×C3×C5C2×D411yescharacteristic
113C22×C3×D5—22no—
114C22×C3×D5—22no—
115C22×C3×D5—24no—
116C22×C3×D5—22no—
117C22×C3×D5—42no—
118C22×C3×D5—42no—
119C22×C3×D5C22⋊C414yes—
120C22×C3×D5C2×D412yes—
121C3×C5×Q8—24no—
122C3×C5×Q8—22no—
123C3×C5×Q8C22×C414yescomplemented
124C2×C3×C4×C5—24no—
125C2×C3×C4×C5—22no—
126C2×C3×C4×C5C22×C412yes—
127C2×C3×C4×C5C2×D411yescharacteristic
128C2×C3×C4×C5C2×D412yes—
129C2×C3×C4×C5C2411yescharacteristic
130C2×Dic15—22no—
131C2×Dic15—42no—
132C2×C5×Dic3—22no—
133C2×C5×Dic3—42no—
134C2×C3×Dic5C22⋊C414yes—
135C2×C3×Dic5C2×D412yes—
136C3×D20—216no—
137C3×D20—24no—
138C3×D20—22no—
139C3×D20C22×C414yescomplemented
140C3×C4×D5—216no—
141C3×C4×D5C22⋊C418yes—
142C3×C4×D5C22×C414yes—
143C3×C4×D5C2×D414yes—
144D5×Dic3—48no—
145C3⋊#168—22no—
146C5×C3⋊C8—22no—
147C2×C3×#41—58no—
148C2×C3×#41—54no—
149C2×C3×#41—104no—
150C2×C3×#41—102no—
151C22×C3×Q8C2×D511yescharacteristic
152C22×C3×D4—51no—
153C22×C3×D4—101no—
154C23×C3×C4—51no—
155C23×C3×C4—101no—
156C23×Dic3—101no—
157C2×C4⋊Dic3—51no—
158C2×C4⋊Dic3—54no—
159C2×C4⋊Dic3—102no—
160C2×C4⋊Dic3—101no—
161C2×C4⋊Dic3C2×D511yescharacteristic
162C2×#208—54no—
163C2×#208—104no—
164C22×C3⋊C8—51no—
165C22×C3⋊C8C2×D511yescharacteristic
166#835—54no—
167#835C2×D514yescomplemented
168#832—54no—
169#832—54no—
170C23×D5—21no—
171C23×D5#6711yescharacteristic
172C2×C5×Q8—22no—
173C2×C5×Q8—21no—
174C2×C5×Q8C2×Dic314yescomplemented
175C2×C5×Q8C22×D312yes—
176C22×C4×C5—21no—
177C22×C4×C5—61no—
178C22×C4×C5C22×D311yescharacteristic
179C22×Dic5—61no—
180C22×Dic5#6711yescharacteristic
181#713—216no—
182#713—28no—
183#713C2×Dic3116yescomplemented
184C2×D20—28no—
185C2×D20—24no—
186C2×D20—21no—
187C2×D20—22no—
188C2×D20C2×Dic314yescomplemented
189C2×D20C22×D312yes—
190C2×C4×D5—28no—
191C2×C4×D5—24no—
192C2×C4×D5—64no—
193C2×C4×D5—124no—
194C2×C4×D5C2×Dic314yes—
195C2×C4×D5#6712yes—
196C2×C4×D5#6714yes—
197C2×C4×D5C22×D312yes—
198C2×C5×C8—32no—
199C2×C5×C8—62no—
200C5×C4⋊C4—32no—
201C5×C4⋊C4—61no—
202C4⋊Dic5—32no—
203C4⋊Dic5—61no—
204#683—64no—
205C2×#168—32no—
206C2×#168—62no—
207#676—68no—
208C22×#41—51no—
209C22×M16—151no—
210C22×C4⋊C4—151no—
211#417—158no—
212C2×#103—31no—
213C2×#103—151no—
214C2×#102—151no—
215C2×#102—151no—
216C22×C3×C5C2×C22⋊C412yes—
217C22×C3×C5AGL(1,ℤ/8)12yes—
218C22×C3×C5#13712yes—
219C22×C3×C5C22×D411yescharacteristic
220C2×C3×D5—22no—
221C2×C3×D5—22no—
222C2×C3×D5—28no—
223C2×C3×D5—24no—
224C2×C3×D5—42no—
225C2×C3×D5—42no—
226C2×C3×D5—48no—
227C2×C3×D5C2×C22⋊C414yes—
228C3×C4×C5—24no—
229C3×C4×C5C2×C22⋊C412yes—
230C3×C4×C5C23×C411yescharacteristic
231C3×C4×C5C22×D411yescharacteristic
232Dic15—42no—
233C3×Dic5C2×C22⋊C414yes—
234C5×Dic3—42no—
235C24×C3—51no—
236C24×C3—101no—
237C3×#41—516no—
238C3×#41—108no—
239C3×#41—1016no—
240C2×C3×Q8—21no—
241C2×C3×Q8—22no—
242C2×C3×Q8C4×D514yescomplemented
243C2×C3×Q8C22×D512yes—
244C2×C3×D4—54no—
245C2×C3×D4—52no—
246C2×C3×D4—104no—
247C2×C3×D4—108no—
248C2×C3×D4—102no—
249C2×C3×D4—101no—
250C22×C3×C4—21no—
251C22×C3×C4—51no—
252C22×C3×C4—54no—
253C22×C3×C4—54no—
254C22×C3×C4—52no—
255C22×C3×C4—52no—
256C22×C3×C4—104no—
257C22×C3×C4—108no—
258C22×C3×C4C22×D511yescharacteristic
259C22×Dic3—21no—
260C22×Dic3—104no—
261C22×Dic3—101no—
262C22×Dic3—204no—
263C4⋊Dic3—21no—
264C4⋊Dic3—52no—
265C4⋊Dic3—104no—
266C4⋊Dic3—101no—
267C4⋊Dic3C22×D512yes—
268#208—108no—
269C2×C3⋊C8—22no—
270C2×C3⋊C8—52no—
271C2×C3⋊C8—102no—
272C2×C3⋊C8C22×D512yes—
273C23×C5C2×#6711yescharacteristic
274C22×D5—22no—
275C22×D5—22no—
276C22×D5—24no—
277C22×D5—22no—
278C22×D5—42no—
279C22×D5—42no—
280C22×D5#21714yes—
281C22×D5C2×#6712yes—
282C5×Q8—24no—
283C5×Q8—22no—
284C5×Q8C22×Dic314yescomplemented
285C2×C4×C5—24no—
286C2×C4×C5—22no—
287C2×C4×C5—62no—
288C2×C4×C5—122no—
289C2×C4×C5C22×Dic312yes—
290C2×C4×C5C2×#6711yescharacteristic
291C2×C4×C5C2×#6712yes—
292C2×C4×C5C23×D311yescharacteristic
293C2×Dic5—62no—
294C2×Dic5—122no—
295C2×Dic5#21714yes—
296C2×Dic5C2×#6712yes—
297D20—216no—
298D20—24no—
299D20—22no—
300D20C22×Dic314yescomplemented
301C4×D5—216no—
302C4×D5—128no—
303C4×D5#21718yes—
304C4×D5C22×Dic314yes—
305C4×D5C2×#6714yes—
306C5×C8—62no—
307#168—62no—
308C2×#41—58no—
309C2×#41—54no—
310C2×#41—102no—
311C2×#41—104no—
312C22×Q8D3×D511yescharacteristic
313C22×D4—51no—
314C22×D4—101no—
315C23×C4—51no—
316C23×C4—101no—
317C23×C4—301no—
318C2×M16—154no—
319C2×M16—304no—
320C22×C8—31no—
321C22×C8—151no—
322C2×C4⋊C4—31no—
323C2×C4⋊C4—154no—
324C2×C4⋊C4—151no—
325C2×C4⋊C4—302no—
326C2×C4⋊C4—301no—
327#103—34no—
328#103—154no—
329#102—154no—
330#102—154no—
331C2×C3×C5#41714yescomplemented
332C2×C3×C5C22×C22⋊C411yescharacteristic
333C2×C3×C5C2×AGL(1,ℤ/8)11yescharacteristic
334C2×C3×C5C2×#13711yescharacteristic
335C3×D5—24no—
336C3×D5—44no—
337C23×C3—52no—
338C23×C3—54no—
339C23×C3—102no—
340C23×C3—102no—
341C23×C3—102no—
342C23×C3—104no—
343C23×C3—202no—
344C23×C3—202no—
345C23×C3D4×D511yescharacteristic
346C3×Q8—22no—
347C3×Q8—24no—
348C3×Q8C2×C4×D514yescomplemented
349C3×D4—54no—
350C3×D4—1016no—
351C3×D4—102no—
352C3×D4—104no—
353C2×C3×C4—22no—
354C2×C3×C4—24no—
355C2×C3×C4—52no—
356C2×C3×C4—54no—
357C2×C3×C4—54no—
358C2×C3×C4—54no—
359C2×C3×C4—58no—
360C2×C3×C4—1016no—
361C2×C3×C4C2×C4×D512yes—
362C2×C3×C4D4×D512yes—
363C2×C3×C4D4×D511yescharacteristic
364C2×C3×C4C23×D511yescharacteristic
365C2×Dic3—22no—
366C2×Dic3—42no—
367C2×Dic3—102no—
368C2×Dic3—208no—
369C2×Dic3—202no—
370C3⋊C8—22no—
371C3⋊C8—102no—
372C22×C5#94912yes—
373C22×C5#95112yes—
374C22×C5C2×#21712yes—
375C22×C5C22×#6711yescharacteristic
376C2×D5—28no—
377C2×D5—22no—
378C2×D5—22no—
379C2×D5—24no—
380C2×D5—48no—
381C2×D5—42no—
382C2×D5—42no—
383C2×D5C2×#21714yes—
384C4×C5—24no—
385C4×C5—122no—
386C4×C5C2×#21712yes—
387C4×C5C23×Dic311yescharacteristic
388C4×C5C22×#6711yescharacteristic
389Dic5—122no—
390Dic5C2×#21714yes—
391C24—51no—
392C24—101no—
393#41—516no—
394#41—108no—
395#41—1016no—
396C2×Q8—21no—
397C2×Q8—22no—
398C2×Q8D5×Dic314yescomplemented
399C2×Q8C2×D3×D512yes—
400C2×D4—54no—
401C2×D4—52no—
402C2×D4—104no—
403C2×D4—108no—
404C2×D4—102no—
405C2×D4—101no—
406C22×C4—21no—
407C22×C4—51no—
408C22×C4—54no—
409C22×C4—52no—
410C22×C4—54no—
411C22×C4—52no—
412C22×C4—61no—
413C22×C4—104no—
414C22×C4—108no—
415C22×C4—304no—
416C22×C4—301no—
417C22×C4—604no—
418C22×C4C2×D3×D511yescharacteristic
419M16—308no—
420C2×C8—32no—
421C2×C8—62no—
422C2×C8—152no—
423C2×C8—302no—
424C4⋊C4—32no—
425C4⋊C4—61no—
426C4⋊C4—152no—
427C4⋊C4—304no—
428C4⋊C4—301no—
429C3×C5C2×#41711yescharacteristic, complemented, Hall
430C22×C3—54no—
431C22×C3—102no—
432C22×C3—102no—
433C22×C3—108no—
434C22×C3—104no—
435C22×C3—202no—
436C22×C3—202no—
437C22×C3—208no—
438C22×C3D5×C22⋊C412yes—
439C22×C3#406012yes—
440C22×C3#406412yes—
441C22×C3C2×D4×D511yescharacteristic
442C3×C4—24no—
443C3×C4—54no—
444C3×C4D5×C22⋊C412yes—
445C3×C4C22×C4×D511yescharacteristic
446C3×C4C2×D4×D511yescharacteristic
447Dic3—42no—
448Dic3—202no—
449C2×C5#523214yescomplemented
450C2×C5C2×#94911yescharacteristic
451C2×C5C2×#95111yescharacteristic
452C2×C5C22×#21711yescharacteristic
453D5—24no—
454D5—44no—
455C23—52no—
456C23—54no—
457C23—102no—
458C23—102no—
459C23—102no—
460C23—104no—
461C23—202no—
462C23—202no—
463C23D5×#6711yescharacteristic
464Q8—22no—
465Q8—24no—
466Q8C2×D5×Dic314yescomplemented
467D4—54no—
468D4—1016no—
469D4—102no—
470D4—104no—
471C2×C4—22no—
472C2×C4—24no—
473C2×C4—52no—
474C2×C4—54no—
475C2×C4—54no—
476C2×C4—54no—
477C2×C4—58no—
478C2×C4—62no—
479C2×C4—1016no—
480C2×C4—122no—
481C2×C4—302no—
482C2×C4—608no—
483C2×C4—602no—
484C2×C4C2×D5×Dic312yes—
485C2×C4D5×#6712yes—
486C2×C4D5×#6711yescharacteristic
487C2×C4C22×D3×D511yescharacteristic
488C8—62no—
489C8—302no—
490C2×C3—104no—
491C2×C3—204no—
492C2×C3#6509214yescomplemented
493C2×C3C2×D5×C22⋊C411yescharacteristic
494C2×C3C2×#406011yescharacteristic
495C2×C3C2×#406411yescharacteristic
496C5C2×#523211yescharacteristic, complemented, Sylow
497C22—54no—
498C22—102no—
499C22—108no—
500C22—102no—
501C22—104no—
502C22—202no—
503C22—208no—
504C22—202no—
505C22#9150212yes—
506C22#9150812yes—
507C22D5×#21712yes—
508C22C2×D5×#6711yescharacteristic
509C4—24no—
510C4—54no—
511C4—122no—
512C4—602no—
513C4D5×#21712yes—
514C4C22×D5×Dic311yescharacteristic
515C4C2×D5×#6711yescharacteristic
516C3C2×#6509211yescharacteristic, complemented, Sylow
517C2—104no—
518C2—204no—
519C2#1174919814yesdirect product factor
520C2C2×#9150211yescharacteristic
521C2C2×#9150811yescharacteristic
522C2C2×D5×#21711yescharacteristic
subgroupG/H
centerC23D5×#67
commutatorC3×C4×C5C23×C4
FrattiniC2×C4C22×D3×D5
FittingC22×C3×C5×Q8C22
socleC23×C3×C5C2×D4
prime radicalC22×C3×#713C2
cosocleC2×C3×C4×C5C24
hypercenterC22×Q8D3×D5
hypocenterC3×C5C2×#417
LCSG ▷ C3×C4×C5 ▷ C2×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C22×C4 ◁ C22×Q8

Complements

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

NH ≅ G/Ntype
2. C2×D5×C4⋊Dic3C2semidirect
3. C22×#6649C2semidirect
4. C2×#91119C2semidirect
5. C2×C5×#835C2semidirect
6. #11749198C2direct (N × H)
7. C2×#90965C2semidirect
8. C2×#90964C2semidirect
13. C2×C3×#713C4semidirect
26. D5×C4⋊Dic3C22semidirect
28. C2×#6649C22semidirect
29. #91119C22semidirect
30. C5×#835C22semidirect
31. #90965C22semidirect
32. #90964C22semidirect
46. C2×C3×C5×Q8C2×C4semidirect
55. C3×#713C2×C4semidirect
60. C2×C3×D20C2×C4semidirect
85. C2×#835D5semidirect
91. C2×#713Dic3semidirect
123. C3×C5×Q8C22×C4semidirect
139. C3×D20C22×C4semidirect
167. #835C2×D5semidirect
174. C2×C5×Q8C2×Dic3semidirect
183. #713C2×Dic3semidirect
188. C2×D20C2×Dic3semidirect
242. C2×C3×Q8C4×D5semidirect
284. C5×Q8C22×Dic3semidirect
300. D20C22×Dic3semidirect
331. C2×C3×C5#417semidirect
348. C3×Q8C2×C4×D5semidirect
398. C2×Q8D5×Dic3semidirect
429. C3×C5C2×#417semidirect
449. C2×C5#5232semidirect
466. Q8C2×D5×Dic3semidirect
492. C2×C3#65092semidirect
496. C5C2×#5232semidirect
516. C3C2×#65092semidirect
519. C2#11749198direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C3×#7132
C2×D5×C4⋊Dic32
C22×#66492
C2×#911192
C2×C5×#8352
#117491982
C2×#909652
C2×#909642
C2×#650923
C2×#52325