Finite Group Databasebeta

← C2×#11752474 #14495408

Subgroups — C2×#11752474 #14495408

Order 1920.

2048 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91764C211yesmaximal, characteristic
2C2×C5×#870C211yesmaximal, characteristic, complemented
3C2×C3×#4016C211yesmaximal, characteristic, complemented
4C2×#91433C212yesmaximal
5C2×#91429C212yesmaximal, complemented
6C2×#91419C212yesmaximal, complemented
7C2×#91345C212yesmaximal, complemented
8C2×#91337C212yesmaximal, complemented
9C2×#91328C212yesmaximal, complemented
10#11752474C2116yesmaximal, direct product factor
11C2×#66033—31nomaximal, Hall
12C22×D60C2211yescharacteristic
13C22×Dic30C2211yescharacteristic
14C22×#1204C2212yes—
15C22×#1200C2212yes—
16C2×C3×C42×C5C2211yescharacteristic
17C2×#6714C2212yes—
18C2×#6714C2212yes—
19#91764C2218yes—
20C2×C5×C4⋊Dic3C2211yescharacteristic
21C2×C5×C4⋊Dic3C2211yescharacteristic
22C2×C5×Dic3⋊C4C2212yes—
23C2×C5×Dic3⋊C4C2212yes—
24C5×#870C2218yescomplemented
25C2×C3×C4⋊Dic5C2211yescharacteristic
26C2×C3×C4⋊Dic5C2211yescharacteristic
27C2×C3×#683C2212yes—
28C2×C3×#683C2212yes—
29C3×#4016C2218yescomplemented
30C2×Dic3⋊Dic5C2212yes—
31C2×#6668C2212yes—
32C2×#6667C2212yes—
33C2×#6667C2212yes—
34C2×#6667C2212yes—
35C2×Dic3×Dic5C2212yes—
36#91433C22116yes—
37#91429C22116yescomplemented
38#91419C22116yescomplemented
39#91345C22116yescomplemented
40#91337C22116yescomplemented
41#91328C22116yescomplemented
42C2×#5699—51nomaximal, Hall
43C2×C5×#125—31no—
44#66033—316no—
45C2×#4044—32no—
46C2×#4044—32no—
47C2×#4043—32no—
48C2×#4042—32no—
49C2×#4040—32no—
50C2×#4034—32no—
51C2×#4021—31no—
52C2×#4016D311yescharacteristic, complemented
53C23×D15C2312yes—
54C22×C3×C4×C5C2311yescharacteristic
55C22×C3×C4×C5C2311yescharacteristic
56C22×C3×C4×C5C2311yescharacteristic
57C22×Dic15C2312yes—
58C2×D60—42no—
59C2×D60C2314yes—
60C2×Dic30—42no—
61C2×Dic30C2314yes—
62C22×C5×Dic3C2312yes—
63C22×C5×Dic3C2312yes—
64C22×C3×Dic5C2312yes—
65C22×C3×Dic5C2312yes—
66C2×#1204—44no—
67C2×#1204C2318yes—
68C2×#1200—44no—
69C2×#1200C2318yes—
70C3×C42×C5C2314yes—
71#6714C2318yes—
72#6714C2318yes—
73C5×C4⋊Dic3C2314yes—
74C5×C4⋊Dic3C2314yes—
75C5×Dic3⋊C4C2318yes—
76C5×Dic3⋊C4C2318yes—
77C3×C4⋊Dic5C2314yes—
78C3×C4⋊Dic5C2314yes—
79C3×#683C2318yes—
80C3×#683C2318yes—
81Dic3⋊Dic5C2318yes—
82#6668C2318yes—
83#6667C2318yes—
84#6667C2318yes—
85#6667C2318yes—
86Dic3×Dic5C2318yes—
87C2×C3×#125—51no—
88#5699—516no—
89C2×#898—52no—
90C2×#898—52no—
91C2×#897—52no—
92C2×#896—52no—
93C2×#894—52no—
94C2×#888—52no—
95C2×#875—51no—
96C2×#870D511yescharacteristic, complemented
97C22×D20—31no—
98C22×C4×D5—32no—
99C22×Dic10—31no—
100C22×Dic10—32no—
101C5×#125—38no—
102C2×C5×C4⋊C4—32no—
103C2×C5×C4⋊C4—31no—
104C2×C5×C4⋊C4—32no—
105C2×C5×C4⋊C4—31no—
106C2×C42×C5C2×D311yescharacteristic
107C2×#685—32no—
108C2×#685—32no—
109C2×#685—32no—
110C2×#685—32no—
111C2×#685—32no—
112C2×C4⋊Dic5—32no—
113C2×C4⋊Dic5C2×D311yescharacteristic
114C2×C4⋊Dic5C2×D311yescharacteristic
115C2×#683—32no—
116C2×#683C2×D312yes—
117C2×#683C2×D312yes—
118C2×C4×Dic5—32no—
119#4044—316no—
120#4044—316no—
121#4043—316no—
122#4042—316no—
123#4040—316no—
124#4034—316no—
125#4021—38no—
126#4016C2×D318yescomplemented
127C2×#563—151noSylow
128C23×C3×C5C2411yescharacteristic
129C22×D15—24no—
130C22×D15—24no—
131C22×D15—24no—
132C22×D15C2414yes—
133C2×C3×C4×C5—22no—
134C2×C3×C4×C5—22no—
135C2×C3×C4×C5—22no—
136C2×C3×C4×C5C2412yes—
137C2×C3×C4×C5C2412yes—
138C2×C3×C4×C5C2412yes—
139C2×Dic15—24no—
140C2×Dic15C2414yes—
141D60—44no—
142Dic30—44no—
143C2×C5×Dic3—24no—
144C2×C5×Dic3—24no—
145C2×C5×Dic3C2414yes—
146C2×C5×Dic3C2414yes—
147C2×C3×Dic5—24no—
148C2×C3×Dic5—24no—
149C2×C3×Dic5C2414yes—
150C2×C3×Dic5C2414yes—
151#1204—48no—
152#1200—48no—
153C22×D12—51no—
154C22×C4×D3—52no—
155C22×Dic6—51no—
156C22×Dic6—52no—
157C3×#125—58no—
158C2×C3×C4⋊C4—52no—
159C2×C3×C4⋊C4—51no—
160C2×C3×C4⋊C4—52no—
161C2×C3×C4⋊C4—51no—
162C2×C3×C42C2×D511yescharacteristic
163C2×(C2×D3)⋊C4—52no—
164C2×(C2×D3)⋊C4—52no—
165C2×(C2×D3)⋊C4—52no—
166C2×(C2×D3)⋊C4—52no—
167C2×(C2×D3)⋊C4—52no—
168C2×C4⋊Dic3—52no—
169C2×C4⋊Dic3C2×D511yescharacteristic
170C2×C4⋊Dic3C2×D511yescharacteristic
171C2×Dic3⋊C4—52no—
172C2×Dic3⋊C4C2×D512yes—
173C2×Dic3⋊C4C2×D512yes—
174C2×C4×Dic3—52no—
175#898—516no—
176#898—516no—
177#897—516no—
178#896—516no—
179#894—516no—
180#888—516no—
181#875—58no—
182#870C2×D518yescomplemented
183C23×D5—32no—
184C22×C4×C5—32no—
185C22×C4×C5—32no—
186C22×C4×C5C22×D311yescharacteristic
187C22×C4×C5C22×D311yescharacteristic
188C22×C4×C5C22×D311yescharacteristic
189C22×Dic5—32no—
190C22×Dic5C22×D312yes—
191C22×Dic5C22×D312yes—
192C2×D20—34no—
193C2×D20—122no—
194C2×C4×D5—38no—
195C2×C4×D5—124no—
196C2×Dic10—34no—
197C2×Dic10—38no—
198C2×Dic10—124no—
199C2×Dic10—122no—
200C5×C4⋊C4—38no—
201C5×C4⋊C4—38no—
202C5×C4⋊C4—34no—
203C5×C4⋊C4—34no—
204C42×C5C22×D314yes—
205#685—38no—
206#685—38no—
207#685—38no—
208#685—38no—
209#685—38no—
210C4⋊Dic5—38no—
211C4⋊Dic5C22×D314yes—
212C4⋊Dic5C22×D314yes—
213#683—38no—
214#683C22×D318yes—
215#683C22×D318yes—
216C4×Dic5—38no—
217#563—1516no—
218C2×C4⋊Q8—152no—
219C2×#126—152no—
220C2×#126—152no—
221C2×#125—31no—
222C2×#125—51no—
223C2×#124—151no—
224C2×#123—152no—
225C2×C22⋊Q8—152no—
226C2×C22⋊Q8—152no—
227C22×C3×C5H(5,𝔽2)12yes—
228C22×C3×C52−1+412yes—
229C22×C3×C52−1+412yes—
230C22×C3×C5C2511yescharacteristic
231C2×D15—24no—
232C2×D15—24no—
233C2×D15—24no—
234C2×D15—48no—
235C3×C4×C5—22no—
236C3×C4×C5—22no—
237C3×C4×C5—22no—
238Dic15—24no—
239C3×Dic5—24no—
240C3×Dic5—24no—
241C5×Dic3—24no—
242C5×Dic3—24no—
243C23×D3—52no—
244C22×C3×C4—52no—
245C22×C3×C4—52no—
246C22×C3×C4C22×D511yescharacteristic
247C22×C3×C4C22×D511yescharacteristic
248C22×C3×C4C22×D511yescharacteristic
249C22×Dic3—52no—
250C22×Dic3C22×D512yes—
251C22×Dic3C22×D512yes—
252C2×D12—54no—
253C2×D12—202no—
254C2×C4×D3—58no—
255C2×C4×D3—204no—
256C2×Dic6—54no—
257C2×Dic6—58no—
258C2×Dic6—204no—
259C2×Dic6—202no—
260C3×C4⋊C4—58no—
261C3×C4⋊C4—58no—
262C3×C4⋊C4—54no—
263C3×C4⋊C4—54no—
264C3×C42C22×D514yes—
265(C2×D3)⋊C4—58no—
266(C2×D3)⋊C4—58no—
267(C2×D3)⋊C4—58no—
268(C2×D3)⋊C4—58no—
269(C2×D3)⋊C4—58no—
270C4⋊Dic3—58no—
271C4⋊Dic3C22×D514yes—
272C4⋊Dic3C22×D514yes—
273Dic3⋊C4—58no—
274Dic3⋊C4C22×D518yes—
275Dic3⋊C4C22×D518yes—
276C4×Dic3—58no—
277C23×C5C23×D311yescharacteristic
278C22×D5—34no—
279C22×D5—64no—
280C22×D5—64no—
281C22×D5—64no—
282C2×C4×C5—22no—
283C2×C4×C5—22no—
284C2×C4×C5—22no—
285C2×C4×C5—34no—
286C2×C4×C5—34no—
287C2×C4×C5—64no—
288C2×C4×C5—64no—
289C2×C4×C5C23×D312yes—
290C2×C4×C5C23×D312yes—
291C2×C4×C5C23×D312yes—
292C2×Dic5—24no—
293C2×Dic5—24no—
294C2×Dic5—34no—
295C2×Dic5—64no—
296C2×Dic5C23×D314yes—
297C2×Dic5C23×D314yes—
298D20—124no—
299C4×D5—128no—
300Dic10—128no—
301Dic10—124no—
302C22×Q8—152no—
303C22×Q8—151no—
304C22×D4—151no—
305C23×C4—152no—
306C4⋊Q8—1516no—
307#126—1516no—
308#126—1516no—
309#125—38no—
310#125—58no—
311#124—158no—
312#123—1516no—
313C22⋊Q8—1516no—
314C22⋊Q8—1516no—
315C2×C4⋊C4—31no—
316C2×C4⋊C4—32no—
317C2×C4⋊C4—32no—
318C2×C4⋊C4—31no—
319C2×C4⋊C4—51no—
320C2×C4⋊C4—52no—
321C2×C4⋊C4—52no—
322C2×C4⋊C4—51no—
323C2×C4⋊C4—152no—
324C2×C4⋊C4—152no—
325C2×C22⋊C4—152no—
326C2×C22⋊C4—152no—
327C2×C22⋊C4—152no—
328C2×C22⋊C4—152no—
329C2×C22⋊C4—152no—
330C2×C42—152no—
331C2×C42D3×D511yescharacteristic
332C2×C3×C5#56314yescomplemented
333C2×C3×C5C2×H(5,𝔽2)11yescharacteristic
334C2×C3×C5C2×2−1+411yescharacteristic
335C2×C3×C5C2×2−1+411yescharacteristic
336D15—44no—
337C23×C3C23×D511yescharacteristic
338C22×D3—54no—
339C22×D3—104no—
340C22×D3—104no—
341C22×D3—104no—
342C2×C3×C4—22no—
343C2×C3×C4—22no—
344C2×C3×C4—22no—
345C2×C3×C4—54no—
346C2×C3×C4—54no—
347C2×C3×C4—104no—
348C2×C3×C4—104no—
349C2×C3×C4C23×D512yes—
350C2×C3×C4C23×D512yes—
351C2×C3×C4C23×D512yes—
352C2×Dic3—24no—
353C2×Dic3—24no—
354C2×Dic3—54no—
355C2×Dic3—104no—
356C2×Dic3C23×D514yes—
357C2×Dic3C23×D514yes—
358D12—204no—
359C4×D3—208no—
360Dic6—208no—
361Dic6—204no—
362C22×C5#100712yes—
363C22×C5#100912yes—
364C22×C5#101012yes—
365C22×C5C24×D311yescharacteristic
366C2×D5—64no—
367C2×D5—64no—
368C2×D5—64no—
369C2×D5—128no—
370C4×C5—22no—
371C4×C5—22no—
372C4×C5—22no—
373C4×C5—64no—
374C4×C5—64no—
375Dic5—24no—
376Dic5—24no—
377Dic5—64no—
378C24—152no—
379C2×Q8—158no—
380C2×Q8—154no—
381C2×Q8—602no—
382C2×Q8—604no—
383C2×D4—154no—
384C2×D4—602no—
385C22×C4—32no—
386C22×C4—32no—
387C22×C4—52no—
388C22×C4—52no—
389C22×C4—158no—
390C22×C4—152no—
391C22×C4—604no—
392C22×C4C2×D3×D511yescharacteristic
393C22×C4C2×D3×D511yescharacteristic
394C22×C4C2×D3×D511yescharacteristic
395C4⋊C4—34no—
396C4⋊C4—38no—
397C4⋊C4—38no—
398C4⋊C4—34no—
399C4⋊C4—54no—
400C4⋊C4—58no—
401C4⋊C4—58no—
402C4⋊C4—54no—
403C4⋊C4—158no—
404C4⋊C4—158no—
405C22⋊C4—158no—
406C22⋊C4—158no—
407C22⋊C4—158no—
408C22⋊C4—158no—
409C22⋊C4—158no—
410C42—158no—
411C42C2×D3×D514yes—
412C3×C5C2×#56311yescharacteristic, complemented, Hall
413C22×C3#414712yes—
414C22×C3#414912yes—
415C22×C3#415012yes—
416C22×C3C24×D511yescharacteristic
417C2×D3—104no—
418C2×D3—104no—
419C2×D3—104no—
420C2×D3—208no—
421C3×C4—22no—
422C3×C4—22no—
423C3×C4—22no—
424C3×C4—104no—
425C3×C4—104no—
426Dic3—24no—
427Dic3—24no—
428Dic3—104no—
429C2×C5#569914yescomplemented
430C2×C5C2×#100711yescharacteristic
431C2×C5C2×#100911yescharacteristic
432C2×C5C2×#101011yescharacteristic
433D5—124no—
434C23—154no—
435C23—304no—
436C23—304no—
437C23—304no—
438C23C22×D3×D511yescharacteristic
439Q8—604no—
440Q8—608no—
441D4—604no—
442C2×C4—22no—
443C2×C4—22no—
444C2×C4—22no—
445C2×C4—34no—
446C2×C4—34no—
447C2×C4—54no—
448C2×C4—54no—
449C2×C4—64no—
450C2×C4—64no—
451C2×C4—104no—
452C2×C4—104no—
453C2×C4—154no—
454C2×C4—304no—
455C2×C4—608no—
456C2×C4C22×D3×D512yes—
457C2×C4C22×D3×D512yes—
458C2×C4C22×D3×D512yes—
459C2×C3#6603314yescomplemented
460C2×C3C2×#414711yescharacteristic
461C2×C3C2×#414911yescharacteristic
462C2×C3C2×#415011yescharacteristic
463D3—204no—
464C5C2×#569911yescharacteristic, complemented, Sylow
465C22—304no—
466C22—304no—
467C22—304no—
468C22—608no—
469C22#9200012yes—
470C22#9200412yes—
471C22#9201912yes—
472C22C23×D3×D511yescharacteristic
473C4—22no—
474C4—22no—
475C4—22no—
476C4—64no—
477C4—64no—
478C4—104no—
479C4—104no—
480C4—304no—
481C3C2×#6603311yescharacteristic, complemented, Sylow
482C2—604no—
483C2#1175247414yesdirect product factor
484C2C2×#9200011yescharacteristic
485C2C2×#9200411yescharacteristic
486C2C2×#9201911yescharacteristic
subgroupG/H
centerC23C22×D3×D5
commutator, cosocleC22×C3×C5C25
FrattiniC22C23×D3×D5
FittingC2×C3×C42×C5C22
socleC23×C3×C5C24
prime radicalC22×D60C22
hypercenterC2×C42D3×D5
hypocenterC3×C5C2×#563
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C42

Complements

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

NH ≅ G/Ntype
2. C2×C5×#870C2semidirect
3. C2×C3×#4016C2semidirect
5. C2×#91429C2semidirect
6. C2×#91419C2semidirect
7. C2×#91345C2semidirect
8. C2×#91337C2semidirect
9. C2×#91328C2semidirect
10. #11752474C2direct (N × H)
24. C5×#870C22semidirect
29. C3×#4016C22semidirect
37. #91429C22semidirect
38. #91419C22semidirect
39. #91345C22semidirect
40. #91337C22semidirect
41. #91328C22semidirect
52. C2×#4016D3semidirect
96. C2×#870D5semidirect
126. #4016C2×D3semidirect
182. #870C2×D5semidirect
332. C2×C3×C5#563semidirect
412. C3×C5C2×#563semidirect
429. C2×C5#5699semidirect
459. C2×C3#66033semidirect
464. C5C2×#5699semidirect
481. C3C2×#66033semidirect
483. C2#11752474direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#917642
C2×C5×#8702
C2×C3×#40162
C2×#914332
C2×#914292
C2×#914192
C2×#913452
C2×#913372
C2×#913282
#117524742
C2×#660333
C2×#56995