Finite Group Databasebeta

← C2×#11749273 #14427947

Subgroups — C2×#11749273 #14427947

Order 1920.

1088 conjugacy classes of subgroups.

The subgroup lattice has 496 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6717C211yesmaximal, characteristic, complemented
2C2×C3×#4092C211yesmaximal, characteristic, complemented
3C2×#91450C211yesmaximal, characteristic
4C22×#6649C211yesmaximal, characteristic, complemented
5C2×C5×#835C211yesmaximal, characteristic, complemented
6#11749273C218yesmaximal, direct product factor
7C2×#90989C211yesmaximal, characteristic, complemented
8C2×#90968C211yesmaximal, characteristic, complemented
9C2×#65101—31nomaximal, Hall
10C22×C3×C5×Q8C2211yescharacteristic
11C22×D60C2211yescharacteristic
12C22×C3×C4×D5C2211yescharacteristic
13C22×#1202—21no—
14C2×#6717—24no—
15C2×#6717C2214yescomplemented
16C22×C3⋊#168C2211yescharacteristic
17C2×C5×C4⋊Dic3C2211yescharacteristic
18C22×C5×C3⋊C8C2211yescharacteristic
19C3×#4092—22no—
20C3×#4092C2214yescomplemented
21C2×C3×#685—21no—
22C2×C3×C4⋊Dic5C2211yescharacteristic
23C2×C3×#683—21no—
24C2×#6667—21no—
25#91450—22no—
26#91450C2214yes—
27C2×#6649—24no—
28C2×#6649C2214yescomplemented
29C5×#835C2214yescomplemented
30#90989C2214yescomplemented
31#90968C2214yescomplemented
32C2×#5049—51nomaximal, Hall
33C2×#4092D311yescharacteristic, complemented
34C22×#688—31no—
35C22×#676—31no—
36C2×#4041—31no—
37C2×C5×#103—31no—
38C2×#3953—31no—
39C2×#3939—31no—
40#65101—38no—
41C23×D15—21no—
42C23×C3×D5D411yescharacteristic
43C2×C3×C5×Q8—21no—
44C2×C3×C5×Q8—22no—
45C2×C3×C5×Q8—22no—
46C2×C3×C5×Q8C2312yes—
47C22×C3×C4×C5—21no—
48C22×C3×C4×C5C2311yescharacteristic
49C2×D60—22no—
50C2×D60—22no—
51C2×D60—21no—
52C2×D60C2312yes—
53C22×C5×Dic3—21no—
54C22×C3×Dic5—21no—
55C22×C3×Dic5D411yescharacteristic
56C2×C3×C4×D5—22no—
57C2×C3×C4×D5—22no—
58C2×C3×C4×D5D412yes—
59C2×C3×C4×D5C2312yes—
60C2×#1202—24no—
61C2×#1202—44no—
62#6717—28no—
63C2×C3⋊#168D414yescomplemented
64C2×C3⋊#168C2312yes—
65C5×C4⋊Dic3—21no—
66C5×C4⋊Dic3C2312yes—
67C2×C5×C3⋊C8—22no—
68C2×C5×C3⋊C8C2312yes—
69C3×#685—24no—
70C3×C4⋊Dic5—21no—
71C3×C4⋊Dic5C2312yes—
72C3×#683—24no—
73#6667—24no—
74#6649—28no—
75C2×C3×C22⋊Q8—51no—
76C22×#215—51no—
77C2×#930—51no—
78C22×#208—51no—
79C2×#835D511yescharacteristic, complemented
80#5049—58no—
81C2×#809—51no—
82C2×#807—51no—
83C22×C5×Q8C2×D311yescharacteristic
84C22×D20—31no—
85C22×D20—61no—
86C22×C4×D5C2×D311yescharacteristic
87C22×C5×C8—31no—
88C2×C5×C4⋊C4—31no—
89#4092—22no—
90#4092C2×D314yescomplemented
91C2×#688—34no—
92C2×#688—64no—
93C2×#685—21no—
94C2×#685—61no—
95C2×C4⋊Dic5C2×D311yescharacteristic
96C2×#683—21no—
97C22×#168—31no—
98C2×#676—34no—
99C2×#676—64no—
100#4041—34no—
101#4041—62no—
102C5×#103—34no—
103#3953—34no—
104#3939—34no—
105C2×#468—151noSylow
106C23×C3×C5C2×D411yescharacteristic
107C22×D15—22no—
108C22×D15—42no—
109C22×D15—42no—
110C22×D15—42no—
111C22×C3×D5—22no—
112C22×C3×D5—22no—
113C22×C3×D5—22no—
114C22×C3×D5C2×D412yes—
115C3×C5×Q8—22no—
116C3×C5×Q8—22no—
117C3×C5×Q8—42no—
118C2×C3×C4×C5—22no—
119C2×C3×C4×C5—42no—
120C2×C3×C4×C5C2×D411yescharacteristic
121C2×C3×C4×C5C2×D412yes—
122C2×C3×C4×C5#4112yes—
123C2×C3×C4×C5C2411yescharacteristic
124D60—22no—
125D60—22no—
126D60—42no—
127C2×C5×Dic3—22no—
128C2×C5×Dic3—42no—
129C2×C3×Dic5—22no—
130C2×C3×Dic5—22no—
131C2×C3×Dic5—42no—
132C2×C3×Dic5C2×D412yes—
133C3×C4×D5—24no—
134C3×C4×D5—22no—
135C3×C4×D5—22no—
136#1202—48no—
137C3⋊#168C2×D414yescomplemented
138C5×C3⋊C8—22no—
139C22×C3×Q8C2×D511yescharacteristic
140C23×C3×C4—51no—
141C22×#67—101no—
142C22×D12—51no—
143C3×C22⋊Q8—54no—
144C3×C22⋊Q8—102no—
145C2×C3×C4⋊C4—51no—
146C2×C3×C4⋊C4—101no—
147C2×C3×C22⋊C4—101no—
148C2×#215—54no—
149C2×#215—104no—
150#930—54no—
151#930—102no—
152C2×(C2×D3)⋊C4—101no—
153C2×C4⋊Dic3C2×D511yescharacteristic
154C2×#208—54no—
155C2×#208—104no—
156C22×C3⋊C8—51no—
157C22×C3⋊C8C2×D511yescharacteristic
158#835C2×D514yescomplemented
159#809—54no—
160#807—54no—
161C23×D5—61no—
162C23×D5#6711yescharacteristic
163C2×C5×Q8—21no—
164C2×C5×Q8—22no—
165C2×C5×Q8—22no—
166C2×C5×Q8C22×D312yes—
167C22×C4×C5—21no—
168C22×C4×C5—61no—
169C22×C4×C5C22×D311yescharacteristic
170C22×Dic5—21no—
171C22×Dic5#6711yescharacteristic
172C2×D20—32no—
173C2×D20—62no—
174C2×D20—62no—
175C2×D20—64no—
176C2×D20—61no—
177C2×D20—124no—
178C2×C4×D5—22no—
179C2×C4×D5—22no—
180C2×C4×D5#6712yes—
181C2×C4×D5C22×D312yes—
182C2×C5×C8—32no—
183C2×C5×C8—62no—
184C5×C4⋊C4—32no—
185C5×C4⋊C4—61no—
186#688—68no—
187#685—24no—
188#685—64no—
189C4⋊Dic5—21no—
190C4⋊Dic5C22×D312yes—
191#683—24no—
192C2×#168—34no—
193C2×#168—32no—
194#676—68no—
195C22×SD16—151no—
196C22×M16—151no—
197C2×C22⋊Q8—51no—
198C2×#121—151no—
199#468—158no—
200C2×#107—151no—
201C2×#103—31no—
202C2×#102—151no—
203C22×C3×C5#12112yes—
204C22×C3×C5AGL(1,ℤ/8)12yes—
205C22×C3×C5#13712yes—
206C22×C3×C5C22×D411yescharacteristic
207C2×D15—42no—
208C2×D15—42no—
209C2×D15—42no—
210C2×D15—84no—
211C2×C3×D5—22no—
212C2×C3×D5—22no—
213C2×C3×D5—22no—
214C2×C3×D5—44no—
215C3×C4×C5—42no—
216C3×C4×C5#12112yes—
217C3×C4×C5C22×D411yescharacteristic
218C3×C4×C5C2×#4111yescharacteristic
219C3×Dic5—22no—
220C3×Dic5—42no—
221C5×Dic3—42no—
222C24×C3—51no—
223C23×D3—101no—
224C2×C3×Q8—21no—
225C2×C3×Q8—22no—
226C2×C3×Q8—22no—
227C2×C3×Q8C22×D512yes—
228C22×C3×C4—21no—
229C22×C3×C4—51no—
230C22×C3×C4—52no—
231C22×C3×C4—52no—
232C22×C3×C4—102no—
233C22×C3×C4—102no—
234C22×C3×C4—101no—
235C22×C3×C4C22×D511yescharacteristic
236C2×#67—104no—
237C2×#67—204no—
238C22×Dic3—21no—
239C2×D12—52no—
240C2×D12—102no—
241C2×D12—102no—
242C2×D12—101no—
243C3×C4⋊C4—52no—
244C3×C4⋊C4—104no—
245C3×C4⋊C4—101no—
246C3×C22⋊C4—104no—
247#215—108no—
248(C2×D3)⋊C4—104no—
249C4⋊Dic3—21no—
250C4⋊Dic3C22×D512yes—
251#208—108no—
252C2×C3⋊C8—22no—
253C2×C3⋊C8—54no—
254C2×C3⋊C8—52no—
255C2×C3⋊C8C22×D512yes—
256C23×C5C2×#6711yescharacteristic
257C22×D5—22no—
258C22×D5—22no—
259C22×D5—22no—
260C22×D5—62no—
261C22×D5—122no—
262C22×D5—122no—
263C22×D5—122no—
264C22×D5C2×#6712yes—
265C5×Q8—22no—
266C5×Q8—22no—
267C5×Q8—42no—
268C2×C4×C5—22no—
269C2×C4×C5—42no—
270C2×C4×C5—62no—
271C2×C4×C5—122no—
272C2×C4×C5D3×D412yes—
273C2×C4×C5#23712yes—
274C2×C4×C5C2×#6711yescharacteristic
275C2×C4×C5C23×D311yescharacteristic
276C2×Dic5—22no—
277C2×Dic5—22no—
278C2×Dic5—42no—
279C2×Dic5C2×#6712yes—
280D20—62no—
281D20—62no—
282D20—128no—
283D20—122no—
284C4×D5—24no—
285C4×D5—22no—
286C4×D5—22no—
287C5×C8—62no—
288#168—34no—
289C22×Q8D3×D511yescharacteristic
290C22×D4—151no—
291C22×D4—301no—
292C23×C4—51no—
293C2×SD16—154no—
294C2×SD16—304no—
295C2×M16—154no—
296C2×M16—304no—
297C22×C8—31no—
298C22×C8—151no—
299C22⋊Q8—54no—
300C22⋊Q8—102no—
301#121—154no—
302#121—302no—
303C2×C4⋊C4—31no—
304C2×C4⋊C4—51no—
305C2×C4⋊C4—101no—
306C2×C22⋊C4—101no—
307C2×C22⋊C4—301no—
308#107—154no—
309#103—34no—
310#102—154no—
311C2×C3×C5#46814yescomplemented
312C2×C3×C5C2×#12111yescharacteristic
313C2×C3×C5C2×AGL(1,ℤ/8)11yescharacteristic
314C2×C3×C5C2×#13711yescharacteristic
315D15—82no—
316C3×D5—42no—
317C23×C3—52no—
318C23×C3—102no—
319C23×C3—102no—
320C23×C3—102no—
321C23×C3D4×D511yescharacteristic
322C22×D3—102no—
323C22×D3—202no—
324C22×D3—202no—
325C22×D3—202no—
326C3×Q8—22no—
327C3×Q8—22no—
328C3×Q8—42no—
329C2×C3×C4—22no—
330C2×C3×C4—42no—
331C2×C3×C4—52no—
332C2×C3×C4—102no—
333C2×C3×C4—102no—
334C2×C3×C4—102no—
335C2×C3×C4—104no—
336C2×C3×C4—102no—
337C2×C3×C4—202no—
338C2×C3×C4#70912yes—
339C2×C3×C4D4×D511yescharacteristic
340C2×C3×C4D4×D512yes—
341C2×C3×C4C23×D511yescharacteristic
342#67—208no—
343C2×Dic3—22no—
344C2×Dic3—42no—
345D12—102no—
346D12—102no—
347D12—202no—
348C3⋊C8—22no—
349C3⋊C8—54no—
350C22×C5#93912yes—
351C22×C5#94212yes—
352C22×C5#94912yes—
353C22×C5C22×#6711yescharacteristic
354C2×D5—22no—
355C2×D5—22no—
356C2×D5—22no—
357C2×D5—44no—
358C2×D5—122no—
359C2×D5—122no—
360C2×D5—122no—
361C2×D5—244no—
362C4×C5—42no—
363C4×C5—122no—
364C4×C5#93912yes—
365C4×C5C2×D3×D411yescharacteristic
366C4×C5C2×#23711yescharacteristic
367Dic5—22no—
368Dic5—42no—
369C24—51no—
370C24—301no—
371C2×Q8—21no—
372C2×Q8—22no—
373C2×Q8—22no—
374C2×Q8C2×D3×D512yes—
375C2×D4—152no—
376C2×D4—304no—
377C2×D4—302no—
378C2×D4—301no—
379C2×D4—302no—
380C2×D4—604no—
381C22×C4—21no—
382C22×C4—51no—
383C22×C4—52no—
384C22×C4—52no—
385C22×C4—61no—
386C22×C4—102no—
387C22×C4—102no—
388C22×C4—101no—
389C22×C4C2×D3×D511yescharacteristic
390SD16—308no—
391M16—308no—
392C2×C8—32no—
393C2×C8—62no—
394C2×C8—154no—
395C2×C8—152no—
396C4⋊C4—32no—
397C4⋊C4—52no—
398C4⋊C4—61no—
399C4⋊C4—104no—
400C4⋊C4—101no—
401C22⋊C4—104no—
402C22⋊C4—304no—
403C3×C5C2×#46811yescharacteristic, complemented, Hall
404C22×C3—102no—
405C22×C3—102no—
406C22×C3—102no—
407C22×C3—204no—
408C22×C3#403012yes—
409C22×C3#406012yes—
410C22×C3#406412yes—
411C22×C3C2×D4×D511yescharacteristic
412C2×D3—202no—
413C2×D3—202no—
414C2×D3—202no—
415C2×D3—404no—
416C3×C4—42no—
417C3×C4—102no—
418C3×C4—202no—
419C3×C4#403012yes—
420C3×C4C2×#70911yescharacteristic
421C3×C4C2×D4×D511yescharacteristic
422Dic3—42no—
423C2×C5#504914yescomplemented
424C2×C5C2×#93911yescharacteristic
425C2×C5C2×#94211yescharacteristic
426C2×C5C2×#94911yescharacteristic
427D5—42no—
428D5—242no—
429C23—52no—
430C23—102no—
431C23—102no—
432C23—102no—
433C23—302no—
434C23—602no—
435C23—602no—
436C23—602no—
437C23D5×#6711yescharacteristic
438Q8—22no—
439Q8—22no—
440Q8—42no—
441D4—302no—
442D4—302no—
443D4—608no—
444D4—602no—
445C2×C4—22no—
446C2×C4—42no—
447C2×C4—52no—
448C2×C4—62no—
449C2×C4—102no—
450C2×C4—102no—
451C2×C4—102no—
452C2×C4—104no—
453C2×C4—102no—
454C2×C4—122no—
455C2×C4—202no—
456C2×C4#677912yes—
457C2×C4D5×#6711yescharacteristic
458C2×C4#679012yes—
459C2×C4C22×D3×D511yescharacteristic
460C8—62no—
461C8—154no—
462C2×C3—202no—
463C2×C3#6510114yescomplemented
464C2×C3C2×#403011yescharacteristic
465C2×C3C2×#406011yescharacteristic
466C2×C3C2×#406411yescharacteristic
467D3—402no—
468C5C2×#504911yescharacteristic, complemented, Sylow
469C22—102no—
470C22—102no—
471C22—102no—
472C22—204no—
473C22—602no—
474C22—602no—
475C22—602no—
476C22—1204no—
477C22#9150212yes—
478C22#9151512yes—
479C22#9155012yes—
480C22C2×D5×#6711yescharacteristic
481C4—42no—
482C4—102no—
483C4—122no—
484C4—202no—
485C4#9155012yes—
486C4C2×#677911yescharacteristic
487C4C2×#679011yescharacteristic
488C3C2×#6510111yescharacteristic, complemented, Sylow
489C2—202no—
490C2—1202no—
491C2#1174927314yesdirect product factor
492C2C2×#9150211yescharacteristic
493C2C2×#9151511yescharacteristic
494C2C2×#9155011yescharacteristic
subgroupG/H
centerC23D5×#67
commutator, cosocleC2×C3×C4×C5C24
FrattiniC2×C4C22×D3×D5
FittingC22×C3×C5×Q8C22
socleC23×C3×C5C2×D4
prime radicalC2×#91450C2
hypercenterC22×Q8D3×D5
hypocenterC3×C5C2×#468
LCSG ▷ C2×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
1. C22×#6717C2semidirect
2. C2×C3×#4092C2semidirect
4. C22×#6649C2semidirect
5. C2×C5×#835C2semidirect
6. #11749273C2direct (N × H)
7. C2×#90989C2semidirect
8. C2×#90968C2semidirect
15. C2×#6717C22semidirect
20. C3×#4092C22semidirect
28. C2×#6649C22semidirect
29. C5×#835C22semidirect
30. #90989C22semidirect
31. #90968C22semidirect
33. C2×#4092D3semidirect
63. C2×C3⋊#168D4semidirect
79. C2×#835D5semidirect
90. #4092C2×D3semidirect
137. C3⋊#168C2×D4semidirect
158. #835C2×D5semidirect
311. C2×C3×C5#468semidirect
403. C3×C5C2×#468semidirect
423. C2×C5#5049semidirect
463. C2×C3#65101semidirect
468. C5C2×#5049semidirect
488. C3C2×#65101semidirect
491. C2#11749273direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#67172
C2×C3×#40922
C2×#914502
C22×#66492
C2×C5×#8352
#117492732
C2×#909892
C2×#909682
C2×#651013
C2×#50495