Finite Group Databasebeta

← C2×#11753332 #14481775

Subgroups — C2×#11753332 #14481775

Order 1920.

2160 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×C4⋊C4×D15C211yesmaximal, characteristic
2C2×C5×#888C211yesmaximal, characteristic, complemented
3C2×C3×#4036C211yesmaximal, characteristic, complemented
4C2×#91468C211yesmaximal, characteristic, complemented
5C2×#91386C211yesmaximal, characteristic
6C2×#91383C212yesmaximal
7C2×#91381C211yesmaximal, characteristic, complemented
8C2×#91377C212yesmaximal
9C2×#91347C211yesmaximal, characteristic
10C2×#91339C212yesmaximal, complemented
11C2×#91329C212yesmaximal, complemented
12#11753332C2116yesmaximal, direct product factor
13C2×#66063—31nomaximal, Hall
14C22×C4×D15C2212yes—
15C22×C4×D15C2211yescharacteristic
16C22×C5×Dic6C2212yes—
17C22×#1204C2212yes—
18C2×C3×C5×C4⋊C4C2211yescharacteristic
19C2×C4⋊Dic15C2211yescharacteristic
20C2×#6712C2212yes—
21C4⋊C4×D15C2218yes—
22C2×C5×C4⋊Dic3C2211yescharacteristic
23C2×C5×Dic3⋊C4C2212yes—
24C2×C4×C5×Dic3C2211yescharacteristic
25C5×#888C2218yescomplemented
26C2×C3×C4⋊Dic5C2211yescharacteristic
27C2×C3×C4⋊Dic5C2212yes—
28C2×C3×#683C2212yes—
29C2×C3×C4×Dic5C2211yescharacteristic
30C3×#4036C2218yescomplemented
31C2×#6669C2212yes—
32C2×#6669C2211yescharacteristic
33C2×#6669C2211yescharacteristic
34C2×#6668C2212yes—
35C2×#6668C2212yes—
36C2×#6667C2211yescharacteristic
37C2×#6667C2212yes—
38C2×#6667C2211yescharacteristic
39C2×Dic3×Dic5C2212yes—
40#91468C2218yescomplemented
41#91386C2218yes—
42#91383C22116yes—
43#91381C2218yescomplemented
44#91377C22116yes—
45#91347C2218yes—
46#91339C22116yescomplemented
47#91329C22116yescomplemented
48C2×#5689—51nomaximal, Hall
49C2×C5×C4⋊Q8—31no—
50C2×#4092—32no—
51C2×#4090—32no—
52#66063—316no—
53C2×#4042—32no—
54C2×#4038—31no—
55C2×D5×C4⋊C4—31no—
56C2×#4036D311yescharacteristic, complemented
57C2×#4035—32no—
58C2×#4034—31no—
59C2×#4018—31no—
60C2×#4016—31no—
61C23×D15C2311yescharacteristic
62C22×C3×C4×C5C2312yes—
63C22×C3×C4×C5C2311yescharacteristic
64C22×Dic15C2312yes—
65C22×Dic15C2311yescharacteristic
66C2×C4×D15—44no—
67C2×C4×D15Q818yescomplemented
68C2×C4×D15C2318yes—
69C2×C4×D15C2314yes—
70C22×C5×Dic3C2311yescharacteristic
71C22×C5×Dic3C2311yescharacteristic
72C22×C5×Dic3C2312yes—
73C2×C5×Dic6—44no—
74C2×C5×Dic6C2318yes—
75C22×C3×Dic5C2312yes—
76C22×C3×Dic5C2311yescharacteristic
77C22×C3×Dic5C2311yescharacteristic
78C2×#1204—44no—
79C2×#1204C2318yes—
80C3×C5×C4⋊C4C2314yes—
81C4⋊Dic15C2314yes—
82#6712C2318yes—
83C5×C4⋊Dic3C2314yes—
84C5×Dic3⋊C4C2318yes—
85C4×C5×Dic3C2314yes—
86C3×C4⋊Dic5C2318yes—
87C3×C4⋊Dic5C2314yes—
88C3×#683C2318yes—
89C3×C4×Dic5C2314yes—
90#6669C2314yes—
91#6669C2314yes—
92#6669C2318yes—
93#6668C2318yes—
94#6668C2318yes—
95#6667C2318yes—
96#6667C2314yes—
97#6667C2314yes—
98Dic3×Dic5C2318yes—
99C2×C3×#125—51no—
100#5689—516no—
101C2×#897—52no—
102C2×#896—52no—
103C2×#892—51no—
104C2×D3×C4⋊C4—51no—
105C2×#890—51no—
106C2×#889—52no—
107C2×#888—52no—
108C2×#888D511yescharacteristic, complemented
109C2×#872—51no—
110C2×#869—51no—
111C22×C5×Q8—32no—
112C22×C4×D5—31no—
113C22×C4×D5—32no—
114C22×Dic10—32no—
115C5×C4⋊Q8—38no—
116C2×C5×C4⋊C4—31no—
117C2×C5×C4⋊C4—32no—
118C2×C5×C4⋊C4C2×D311yescharacteristic
119C2×C42×C5—31no—
120#4092—316no—
121#4090—316no—
122C2×#685—31no—
123C2×#685—31no—
124C2×#685—32no—
125C2×C4⋊Dic5—31no—
126C2×C4⋊Dic5C2×D311yescharacteristic
127C2×C4⋊Dic5C2×D312yes—
128C2×#683—32no—
129C2×#683—32no—
130C2×#683—32no—
131C2×#683—31no—
132C2×#683—31no—
133C2×#683—32no—
134C2×#683C2×D312yes—
135C2×C4×Dic5—32no—
136C2×C4×Dic5C2×D311yescharacteristic
137#4042—316no—
138#4038—38no—
139D5×C4⋊C4—38no—
140#4036C2×D318yescomplemented
141#4035—316no—
142#4034—38no—
143#4018—38no—
144#4016—38no—
145C2×#544—151noSylow
146C23×C3×C5C2411yescharacteristic
147C22×D15—22no—
148C22×D15—22no—
149C22×D15C2×Q814yes—
150C22×D15C2412yes—
151C2×C3×C4×C5—24no—
152C2×C3×C4×C5C2×Q814yes—
153C2×C3×C4×C5C2414yes—
154C2×C3×C4×C5C2412yes—
155C2×Dic15—24no—
156C2×Dic15C2×Q814yes—
157C2×Dic15C2414yes—
158C2×Dic15C2412yes—
159C4×D15—48no—
160C4×D15C2×Q8116yescomplemented
161C2×C5×Dic3—24no—
162C2×C5×Dic3—22no—
163C2×C5×Dic3—22no—
164C2×C5×Dic3C2414yes—
165C2×C5×Dic3C2412yes—
166C2×C5×Dic3C2412yes—
167C5×Dic6—48no—
168C2×C3×Dic5—22no—
169C2×C3×Dic5—24no—
170C2×C3×Dic5—22no—
171C2×C3×Dic5C2414yes—
172C2×C3×Dic5C2412yes—
173C2×C3×Dic5C2412yes—
174#1204—48no—
175C22×C4×D3—51no—
176C22×C4×D3—52no—
177C22×Dic6—52no—
178C22×Dic6C2×D512yes—
179C3×#125—58no—
180C2×C3×C4⋊C4—52no—
181C2×C3×C4⋊C4—52no—
182C2×C3×C4⋊C4—51no—
183C2×C3×C4⋊C4C2×D511yescharacteristic
184C2×C3×C42—51no—
185C2×(C2×D3)⋊C4—51no—
186C2×(C2×D3)⋊C4—51no—
187C2×(C2×D3)⋊C4—52no—
188C2×C4⋊Dic3—51no—
189C2×C4⋊Dic3—52no—
190C2×C4⋊Dic3—52no—
191C2×C4⋊Dic3C2×D511yescharacteristic
192C2×Dic3⋊C4—52no—
193C2×Dic3⋊C4—51no—
194C2×Dic3⋊C4—51no—
195C2×Dic3⋊C4—52no—
196C2×Dic3⋊C4C2×D512yes—
197C2×C4×Dic3—52no—
198C2×C4×Dic3C2×D511yescharacteristic
199#897—516no—
200#896—516no—
201#892—58no—
202D3×C4⋊C4—58no—
203#890—58no—
204#889—516no—
205#888—516no—
206#888C2×D518yescomplemented
207#872—58no—
208#869—58no—
209C23×D5—31no—
210C2×C5×Q8—38no—
211C2×C5×Q8—124no—
212C22×C4×C5—31no—
213C22×C4×C5—31no—
214C22×C4×C5—32no—
215C22×C4×C5C22×D312yes—
216C22×C4×C5C22×D311yescharacteristic
217C22×Dic5—31no—
218C22×Dic5—32no—
219C22×Dic5C22×D311yescharacteristic
220C22×Dic5C22×D312yes—
221C22×Dic5C22×D311yescharacteristic
222C2×C4×D5—38no—
223C2×C4×D5—34no—
224C2×C4×D5—38no—
225C2×C4×D5—124no—
226C2×Dic10—38no—
227C2×Dic10—124no—
228C5×C4⋊C4—38no—
229C5×C4⋊C4—34no—
230C5×C4⋊C4C22×D314yes—
231C42×C5—34no—
232#685—34no—
233#685—34no—
234#685—38no—
235C4⋊Dic5—34no—
236C4⋊Dic5C22×D314yes—
237C4⋊Dic5C22×D318yes—
238#683—38no—
239#683—38no—
240#683—38no—
241#683—38no—
242#683—34no—
243#683—34no—
244#683C22×D318yes—
245C4×Dic5—38no—
246C4×Dic5C22×D314yes—
247#544—1516no—
248C2×C4⋊Q8—31no—
249C2×C4⋊Q8—151no—
250C2×C4⋊Q8—152no—
251C2×#125—51no—
252C2×#125—151no—
253C2×#125—152no—
254C2×C22⋊Q8—152no—
255C2×C22⋊Q8—152no—
256C2×#117—151no—
257C2×#117—151no—
258C22×C4⋊C4—151no—
259C22×C3×C5C22×Q812yes—
260C22×C3×C5H(5,𝔽2)12yes—
261C22×C3×C52−1+412yes—
262C22×C3×C5C2511yescharacteristic
263C2×D15—28no—
264C2×D15—22no—
265C2×D15—22no—
266C2×D15C22×Q814yes—
267C3×C4×C5—24no—
268C3×C4×C5C22×Q814yes—
269Dic15—24no—
270Dic15C22×Q814yes—
271C3×Dic5—22no—
272C3×Dic5—24no—
273C3×Dic5—22no—
274C5×Dic3—24no—
275C5×Dic3—22no—
276C5×Dic3—22no—
277C23×D3—51no—
278C22×C3×C4—52no—
279C22×C3×C4—51no—
280C22×C3×C4—51no—
281C22×C3×C4C22×D512yes—
282C22×C3×C4C22×D511yescharacteristic
283C22×Dic3—51no—
284C22×Dic3—52no—
285C22×Dic3C22×D512yes—
286C22×Dic3C22×D511yescharacteristic
287C22×Dic3C22×D511yescharacteristic
288C2×C4×D3—58no—
289C2×C4×D3—54no—
290C2×C4×D3—58no—
291C2×C4×D3—204no—
292C2×Dic6—44no—
293C2×Dic6—58no—
294C2×Dic6—204no—
295C2×Dic6C22×D518yes—
296C3×C4⋊C4—58no—
297C3×C4⋊C4—58no—
298C3×C4⋊C4—54no—
299C3×C4⋊C4C22×D514yes—
300C3×C42—54no—
301(C2×D3)⋊C4—54no—
302(C2×D3)⋊C4—54no—
303(C2×D3)⋊C4—58no—
304C4⋊Dic3—54no—
305C4⋊Dic3—58no—
306C4⋊Dic3—58no—
307C4⋊Dic3C22×D514yes—
308Dic3⋊C4—58no—
309Dic3⋊C4—54no—
310Dic3⋊C4—54no—
311Dic3⋊C4—58no—
312Dic3⋊C4C22×D518yes—
313C4×Dic3—58no—
314C4×Dic3C22×D514yes—
315C23×C5C23×D311yescharacteristic
316C22×D5—34no—
317C22×D5—32no—
318C22×D5—62no—
319C22×D5—62no—
320C5×Q8—128no—
321C2×C4×C5—24no—
322C2×C4×C5—34no—
323C2×C4×C5—32no—
324C2×C4×C5—32no—
325C2×C4×C5—64no—
326C2×C4×C5—62no—
327C2×C4×C5—62no—
328C2×C4×C5D3×Q814yes—
329C2×C4×C5C23×D314yes—
330C2×C4×C5C23×D312yes—
331C2×Dic5—22no—
332C2×Dic5—24no—
333C2×Dic5—22no—
334C2×Dic5—34no—
335C2×Dic5—32no—
336C2×Dic5—34no—
337C2×Dic5—64no—
338C2×Dic5C23×D314yes—
339C2×Dic5C23×D312yes—
340C2×Dic5C23×D312yes—
341C4×D5—316no—
342C4×D5—128no—
343Dic10—128no—
344C22×Q8—32no—
345C22×Q8—152no—
346C23×C4—151no—
347C23×C4—152no—
348C4⋊Q8—38no—
349C4⋊Q8—158no—
350C4⋊Q8—1516no—
351#125—58no—
352#125—158no—
353#125—1516no—
354C22⋊Q8—1516no—
355C22⋊Q8—1516no—
356#117—158no—
357#117—158no—
358C2×C4⋊C4—31no—
359C2×C4⋊C4—32no—
360C2×C4⋊C4—51no—
361C2×C4⋊C4—52no—
362C2×C4⋊C4—52no—
363C2×C4⋊C4—151no—
364C2×C4⋊C4—151no—
365C2×C4⋊C4—152no—
366C2×C4⋊C4—152no—
367C2×C4⋊C4—152no—
368C2×C4⋊C4—152no—
369C2×C4⋊C4—151no—
370C2×C4⋊C4—158no—
371C2×C4⋊C4D3×D511yescharacteristic
372C2×C22⋊C4—151no—
373C2×C22⋊C4—151no—
374C2×C22⋊C4—152no—
375C2×C42—31no—
376C2×C42—51no—
377C2×C42—152no—
378C2×C3×C5#54414yescomplemented
379C2×C3×C5C23×Q811yescharacteristic
380C2×C3×C5C2×H(5,𝔽2)11yescharacteristic
381C2×C3×C5C2×2−1+411yescharacteristic
382D15—24no—
383C23×C3C23×D511yescharacteristic
384C22×D3—54no—
385C22×D3—52no—
386C22×D3—102no—
387C22×D3—102no—
388C2×C3×C4—24no—
389C2×C3×C4—54no—
390C2×C3×C4—52no—
391C2×C3×C4—52no—
392C2×C3×C4—102no—
393C2×C3×C4—104no—
394C2×C3×C4—102no—
395C2×C3×C4Q8×D514yes—
396C2×C3×C4C23×D514yes—
397C2×C3×C4C23×D512yes—
398C2×Dic3—22no—
399C2×Dic3—24no—
400C2×Dic3—22no—
401C2×Dic3—54no—
402C2×Dic3—52no—
403C2×Dic3—54no—
404C2×Dic3—104no—
405C2×Dic3C23×D514yes—
406C2×Dic3C23×D512yes—
407C2×Dic3C23×D512yes—
408C4×D3—516no—
409C4×D3—208no—
410Dic6—48no—
411Dic6—208no—
412C22×C5C2×D3×Q812yes—
413C22×C5#100912yes—
414C22×C5#101012yes—
415C22×C5C24×D311yescharacteristic
416C2×D5—34no—
417C2×D5—68no—
418C2×D5—62no—
419C2×D5—62no—
420C4×C5—24no—
421C4×C5—64no—
422C4×C5—62no—
423C4×C5—62no—
424C4×C5C2×D3×Q814yes—
425Dic5—22no—
426Dic5—24no—
427Dic5—22no—
428Dic5—34no—
429Dic5—64no—
430C24—151no—
431C2×Q8—38no—
432C2×Q8—124no—
433C2×Q8—158no—
434C2×Q8—604no—
435C22×C4—31no—
436C22×C4—31no—
437C22×C4—32no—
438C22×C4—51no—
439C22×C4—51no—
440C22×C4—52no—
441C22×C4—151no—
442C22×C4—158no—
443C22×C4—154no—
444C22×C4—152no—
445C22×C4—158no—
446C22×C4—604no—
447C22×C4C2×D3×D512yes—
448C22×C4C2×D3×D511yescharacteristic
449C4⋊C4—34no—
450C4⋊C4—38no—
451C4⋊C4—54no—
452C4⋊C4—58no—
453C4⋊C4—58no—
454C4⋊C4—154no—
455C4⋊C4—154no—
456C4⋊C4—158no—
457C4⋊C4—158no—
458C4⋊C4—158no—
459C4⋊C4—158no—
460C4⋊C4—154no—
461C4⋊C4C2×D3×D514yes—
462C22⋊C4—154no—
463C22⋊C4—154no—
464C22⋊C4—158no—
465C42—34no—
466C42—54no—
467C42—158no—
468C3×C5C2×#54411yescharacteristic, complemented, Hall
469C22×C3#414412yes—
470C22×C3C2×Q8×D512yes—
471C22×C3#414712yes—
472C22×C3C24×D511yescharacteristic
473C2×D3—54no—
474C2×D3—108no—
475C2×D3—102no—
476C2×D3—102no—
477C3×C4—24no—
478C3×C4—102no—
479C3×C4—104no—
480C3×C4—102no—
481C3×C4C2×Q8×D514yes—
482Dic3—22no—
483Dic3—24no—
484Dic3—22no—
485Dic3—54no—
486Dic3—104no—
487C2×C5#568914yescomplemented
488C2×C5C22×D3×Q811yescharacteristic
489C2×C5C2×#100911yescharacteristic
490C2×C5C2×#101011yescharacteristic
491D5—64no—
492C23—154no—
493C23—152no—
494C23—302no—
495C23—302no—
496C23C22×D3×D511yescharacteristic
497Q8—128no—
498Q8—608no—
499C2×C4—24no—
500C2×C4—32no—
501C2×C4—32no—
502C2×C4—34no—
503C2×C4—52no—
504C2×C4—52no—
505C2×C4—54no—
506C2×C4—62no—
507C2×C4—64no—
508C2×C4—62no—
509C2×C4—102no—
510C2×C4—102no—
511C2×C4—104no—
512C2×C4—154no—
513C2×C4—152no—
514C2×C4—1516no—
515C2×C4—154no—
516C2×C4—304no—
517C2×C4—608no—
518C2×C4D15⋊Q814yes—
519C2×C4C22×D3×D514yes—
520C2×C4C22×D3×D512yes—
521C2×C3#6606314yescomplemented
522C2×C3C2×#414411yescharacteristic
523C2×C3C22×Q8×D511yescharacteristic
524C2×C3C2×#414711yescharacteristic
525D3—104no—
526C5C2×#568911yescharacteristic, complemented, Sylow
527C22—154no—
528C22—308no—
529C22—302no—
530C22—302no—
531C22C2×D15⋊Q812yes—
532C22#9202712yes—
533C22#9202812yes—
534C22C23×D3×D511yescharacteristic
535C4—24no—
536C4—62no—
537C4—64no—
538C4—62no—
539C4—102no—
540C4—102no—
541C4—104no—
542C4—154no—
543C4—304no—
544C4C2×D15⋊Q814yes—
545C3C2×#6606311yescharacteristic, complemented, Sylow
546C2—304no—
547C2#1175333214yesdirect product factor
548C2C22×D15⋊Q811yescharacteristic
549C2C2×#9202711yescharacteristic
550C2C2×#9202811yescharacteristic
subgroupG/H
centerC23C22×D3×D5
commutator, cosocleC22×C3×C5C25
FrattiniC22C23×D3×D5
FittingC2×C3×C5×C4⋊C4C22
socleC23×C3×C5C24
prime radicalC23×D15C23
hypercenterC2×C4⋊C4D3×D5
hypocenterC3×C5C2×#544
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. C2×C5×#888C2semidirect
3. C2×C3×#4036C2semidirect
4. C2×#91468C2semidirect
7. C2×#91381C2semidirect
10. C2×#91339C2semidirect
11. C2×#91329C2semidirect
12. #11753332C2direct (N × H)
25. C5×#888C22semidirect
30. C3×#4036C22semidirect
40. #91468C22semidirect
43. #91381C22semidirect
46. #91339C22semidirect
47. #91329C22semidirect
56. C2×#4036D3semidirect
67. C2×C4×D15Q8semidirect
108. C2×#888D5semidirect
140. #4036C2×D3semidirect
160. C4×D15C2×Q8semidirect
206. #888C2×D5semidirect
378. C2×C3×C5#544semidirect
468. C3×C5C2×#544semidirect
487. C2×C5#5689semidirect
521. C2×C3#66063semidirect
526. C5C2×#5689semidirect
545. C3C2×#66063semidirect
547. C2#11753332direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×D152
C2×C5×#8882
C2×C3×#40362
C2×#914682
C2×#913862
C2×#913832
C2×#913812
C2×#913772
C2×#913472
C2×#913392
C2×#913292
#117533322
C2×#660633
C2×#56895