Finite Group Databasebeta

← C2×#11753375 #14481414

Subgroups — C2×#11753375 #14481414

Order 1920.

2264 conjugacy classes of subgroups.

The subgroup lattice has 592 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#91782C211yesmaximal, characteristic, complemented
2C2×C5×#888C211yesmaximal, characteristic, complemented
3C2×C3×#4043C211yesmaximal, characteristic, complemented
4C2×#91455C212yesmaximal, complemented
5C2×#91450C211yesmaximal, characteristic
6C2×#91383C212yesmaximal, complemented
7C2×#91374C211yesmaximal, characteristic, complemented
8C2×#91357C211yesmaximal, characteristic, complemented
9C2×#91354C212yesmaximal, complemented
10C2×#91345C212yesmaximal, complemented
11C2×#91331C211yesmaximal, characteristic, complemented
12#11753375C2116yesmaximal, direct product factor
13C2×#66067—31nomaximal, Hall
14C22×D60C2211yescharacteristic
15C22×C4×D15C2212yes—
16C22×C5×Dic6C2212yes—
17C22×C3×C4×D5C2211yescharacteristic
18C22×C3×Dic10C2211yescharacteristic
19C22×#1202C2212yes—
20C2×C3×C5×C4⋊C4C2211yescharacteristic
21C2×#6714C2212yes—
22C2×C4×Dic15C2211yescharacteristic
23#91782C2218yescomplemented
24C2×C5×C4⋊Dic3C2211yescharacteristic
25C2×C5×Dic3⋊C4C2212yes—
26C2×C4×C5×Dic3C2211yescharacteristic
27C5×#888C2218yescomplemented
28C2×C3×#685C2212yes—
29C2×C3×C4⋊Dic5C2212yes—
30C3×#4043C2218yescomplemented
31C2×#6668C2211yescharacteristic
32C2×#6668C2212yes—
33C2×#6668C2211yescharacteristic
34C2×#6667C2212yes—
35C2×#6667C2212yes—
36C2×#6667C2212yes—
37C2×#6665C2211yescharacteristic
38C2×#6665C2211yescharacteristic
39C2×Dic3×Dic5C2212yes—
40#91455C22116yescomplemented
41#91450C2218yes—
42#91383C22116yescomplemented
43#91374C2218yescomplemented
44#91357C2218yescomplemented
45#91354C22116yescomplemented
46#91345C22116yescomplemented
47#91331C2218yescomplemented
48C2×#5643—51nomaximal, Hall
49C2×C5×C4⋊Q8—31no—
50C2×#4093—32no—
51C2×#4092—32no—
52#66067—316no—
53C2×#4044—32no—
54C2×#4043D311yescharacteristic, complemented
55C2×#4041—31no—
56C2×#4040—32no—
57C2×#4039—31no—
58C2×#4033—31no—
59C2×#4021—31no—
60C2×#4018—31no—
61C23×D15C2312yes—
62C23×C3×D5C2311yescharacteristic
63C22×C3×C4×C5C2312yes—
64C22×C3×C4×C5C2311yescharacteristic
65C22×Dic15C2311yescharacteristic
66C22×Dic15C2311yescharacteristic
67C2×D60—24no—
68C2×D60C2314yes—
69C2×C4×D15—44no—
70C2×C4×D15C2318yes—
71C22×C5×Dic3C2311yescharacteristic
72C22×C5×Dic3C2311yescharacteristic
73C22×C5×Dic3C2312yes—
74C2×C5×Dic6—44no—
75C2×C5×Dic6C2318yes—
76C22×C3×Dic5C2312yes—
77C22×C3×Dic5C2311yescharacteristic
78C2×C3×C4×D5—24no—
79C2×C3×C4×D5C2314yes—
80C2×C3×Dic10—24no—
81C2×C3×Dic10C2314yes—
82C2×#1202—44no—
83C2×#1202C2318yes—
84C3×C5×C4⋊C4C2314yes—
85#6714C2318yes—
86C4×Dic15C2314yes—
87C5×C4⋊Dic3C2314yes—
88C5×Dic3⋊C4C2318yes—
89C4×C5×Dic3C2314yes—
90C3×#685C2318yes—
91C3×C4⋊Dic5C2318yes—
92#6668C2318yes—
93#6668C2314yes—
94#6668C2314yes—
95#6667C2318yes—
96#6667C2318yes—
97#6667C2318yes—
98#6665C2314yes—
99#6665C2314yes—
100Dic3×Dic5C2318yes—
101C2×C3×C22⋊Q8—51no—
102C2×#947—51no—
103C2×Q8×Dic3—51no—
104C2×#930—51no—
105C2×#926—51no—
106#5643—516no—
107C2×#898—52no—
108C2×#897—52no—
109C2×#893—51no—
110C2×#888D511yescharacteristic, complemented
111C2×#885—52no—
112C2×#883—52no—
113C22×C5×Q8—32no—
114C22×D20—31no—
115C22×D20—32no—
116C22×C4×D5—32no—
117C22×C4×D5C2×D311yescharacteristic
118C22×Dic10C2×D311yescharacteristic
119C5×C4⋊Q8—38no—
120C2×C5×C4⋊C4—31no—
121C2×C5×C4⋊C4—32no—
122C2×C5×C4⋊C4C2×D311yescharacteristic
123C2×C42×C5—31no—
124#4093—316no—
125#4092—316no—
126C2×#685—31no—
127C2×#685—31no—
128C2×#685—32no—
129C2×#685—32no—
130C2×#685—32no—
131C2×#685—32no—
132C2×#685C2×D312yes—
133C2×C4⋊Dic5C2×D312yes—
134C2×#683—31no—
135C2×#683—31no—
136C2×#683—32no—
137C2×C4×Dic5—31no—
138C2×C4×Dic5—32no—
139#4044—316no—
140#4043C2×D318yescomplemented
141#4041—38no—
142#4040—316no—
143#4039—38no—
144#4033—38no—
145#4021—38no—
146#4018—38no—
147C2×#542—151noSylow
148C23×C3×C5C2411yescharacteristic
149C22×D15—24no—
150C22×D15—24no—
151C22×D15—24no—
152C22×D15C2414yes—
153C22×C3×D5—22no—
154C22×C3×D5—22no—
155C22×C3×D5—22no—
156C22×C3×D5C2412yes—
157C2×C3×C4×C5—24no—
158C2×C3×C4×C5#4114yes—
159C2×C3×C4×C5C2414yes—
160C2×C3×C4×C5C2412yes—
161C2×Dic15—22no—
162C2×Dic15—22no—
163C2×Dic15C2412yes—
164C2×Dic15C2412yes—
165D60—28no—
166C4×D15—48no—
167C2×C5×Dic3—24no—
168C2×C5×Dic3—22no—
169C2×C5×Dic3—22no—
170C2×C5×Dic3C2414yes—
171C2×C5×Dic3C2412yes—
172C2×C5×Dic3C2412yes—
173C5×Dic6—48no—
174C2×C3×Dic5—24no—
175C2×C3×Dic5—22no—
176C2×C3×Dic5C2414yes—
177C2×C3×Dic5C2412yes—
178C3×C4×D5—28no—
179C3×Dic10—28no—
180#1202—48no—
181C22×C3×Q8—51no—
182C23×C3×C4—51no—
183C22×#67—52no—
184C22×D12—51no—
185C22×C4×D3—52no—
186C22×Dic6C2×D512yes—
187C3×C22⋊Q8—58no—
188C2×C3×C4⋊C4—52no—
189C2×C3×C4⋊C4C2×D511yescharacteristic
190C2×C3×C22⋊C4—52no—
191C2×#217—51no—
192C2×#217—51no—
193#947—58no—
194Q8×Dic3—58no—
195#930—58no—
196C2×(C2×D3)⋊C4—52no—
197C2×(C2×D3)⋊C4—52no—
198C2×(C2×D3)⋊C4—52no—
199C2×(C2×D3)⋊C4—52no—
200#926—58no—
201C2×C4⋊Dic3—52no—
202C2×C4⋊Dic3—51no—
203C2×C4⋊Dic3—51no—
204C2×C4⋊Dic3C2×D511yescharacteristic
205C2×Dic3⋊C4C2×D512yes—
206C2×C4×Dic3—51no—
207C2×C4×Dic3—52no—
208C2×C4×Dic3C2×D511yescharacteristic
209#898—516no—
210#897—516no—
211#893—58no—
212#888C2×D518yescomplemented
213#885—516no—
214#883—516no—
215C23×D5—32no—
216C23×D5C22×D311yescharacteristic
217C2×C5×Q8—38no—
218C2×C5×Q8—124no—
219C22×C4×C5—31no—
220C22×C4×C5—31no—
221C22×C4×C5—32no—
222C22×C4×C5C22×D312yes—
223C22×C4×C5C22×D311yescharacteristic
224C22×Dic5—31no—
225C22×Dic5—31no—
226C22×Dic5C22×D312yes—
227C22×Dic5C22×D311yescharacteristic
228C2×D20—34no—
229C2×D20—38no—
230C2×D20—64no—
231C2×D20—124no—
232C2×C4×D5—24no—
233C2×C4×D5—38no—
234C2×C4×D5—124no—
235C2×C4×D5C22×D314yes—
236C2×Dic10—24no—
237C2×Dic10C22×D314yes—
238C5×C4⋊C4—38no—
239C5×C4⋊C4—34no—
240C5×C4⋊C4C22×D314yes—
241C42×C5—34no—
242#685—34no—
243#685—34no—
244#685—38no—
245#685—38no—
246#685—38no—
247#685—38no—
248#685C22×D318yes—
249C4⋊Dic5C22×D318yes—
250#683—34no—
251#683—34no—
252#683—38no—
253C4×Dic5—34no—
254C4×Dic5—38no—
255#542—1516no—
256C2×C4⋊Q8—31no—
257C2×#126—152no—
258C2×#124—151no—
259C2×#124—152no—
260C2×#123—152no—
261C2×C22⋊Q8—51no—
262C2×C22⋊Q8—152no—
263C2×#121—151no—
264C2×C4×Q8—151no—
265C2×C4×D4—151no—
266C2×#117—151no—
267C22×C3×C5C2×#4112yes—
268C22×C3×C5H(5,𝔽2)12yes—
269C22×C3×C52−1+412yes—
270C22×C3×C5C2511yescharacteristic
271C2×D15—24no—
272C2×D15—24no—
273C2×D15—24no—
274C2×D15—48no—
275C2×C3×D5—22no—
276C2×C3×D5—22no—
277C2×C3×D5—22no—
278C2×C3×D5—44no—
279C3×C4×C5—24no—
280C3×C4×C5C2×#4114yes—
281Dic15—22no—
282Dic15—22no—
283C3×Dic5—24no—
284C3×Dic5—22no—
285C5×Dic3—24no—
286C5×Dic3—22no—
287C5×Dic3—22no—
288C24×C3—51no—
289C23×D3—52no—
290C2×C3×Q8—54no—
291C2×C3×Q8—104no—
292C22×C3×C4—52no—
293C22×C3×C4—51no—
294C22×C3×C4—54no—
295C22×C3×C4—104no—
296C22×C3×C4C22×D512yes—
297C22×C3×C4C22×D511yescharacteristic
298C2×#67—58no—
299C2×#67—204no—
300C22×Dic3—51no—
301C22×Dic3—51no—
302C22×Dic3C22×D512yes—
303C22×Dic3C22×D511yescharacteristic
304C22×Dic3C22×D511yescharacteristic
305C2×D12—54no—
306C2×D12—104no—
307C2×C4×D3—58no—
308C2×C4×D3—204no—
309C2×Dic6—44no—
310C2×Dic6C22×D518yes—
311C3×C4⋊C4—58no—
312C3×C4⋊C4C22×D514yes—
313C3×C22⋊C4—58no—
314#217—54no—
315#217—54no—
316(C2×D3)⋊C4—58no—
317(C2×D3)⋊C4—58no—
318(C2×D3)⋊C4—58no—
319(C2×D3)⋊C4—58no—
320C4⋊Dic3—58no—
321C4⋊Dic3—54no—
322C4⋊Dic3—54no—
323C4⋊Dic3C22×D514yes—
324Dic3⋊C4C22×D518yes—
325C4×Dic3—54no—
326C4×Dic3—58no—
327C4×Dic3C22×D514yes—
328C23×C5C23×D311yescharacteristic
329C22×D5—22no—
330C22×D5—22no—
331C22×D5—22no—
332C22×D5—34no—
333C22×D5—64no—
334C22×D5—64no—
335C22×D5—64no—
336C22×D5C23×D312yes—
337C5×Q8—128no—
338C2×C4×C5—24no—
339C2×C4×C5—34no—
340C2×C4×C5—32no—
341C2×C4×C5—32no—
342C2×C4×C5—62no—
343C2×C4×C5—64no—
344C2×C4×C5—62no—
345C2×C4×C5#23714yes—
346C2×C4×C5C23×D314yes—
347C2×C4×C5C23×D312yes—
348C2×Dic5—24no—
349C2×Dic5—22no—
350C2×Dic5—32no—
351C2×Dic5—32no—
352C2×Dic5—62no—
353C2×Dic5—62no—
354C2×Dic5C23×D314yes—
355C2×Dic5C23×D312yes—
356D20—68no—
357D20—128no—
358C4×D5—28no—
359C4×D5—128no—
360Dic10—28no—
361C22×Q8—32no—
362C22×Q8—51no—
363C22×D4—152no—
364C22×D4—151no—
365C23×C4—51no—
366C23×C4—152no—
367C4⋊Q8—38no—
368#126—1516no—
369#124—158no—
370#124—1516no—
371#123—1516no—
372C22⋊Q8—58no—
373C22⋊Q8—1516no—
374#121—158no—
375C4×Q8—158no—
376C4×D4—158no—
377#117—158no—
378C2×C4⋊C4—31no—
379C2×C4⋊C4—32no—
380C2×C4⋊C4—52no—
381C2×C4⋊C4—151no—
382C2×C4⋊C4—151no—
383C2×C4⋊C4—152no—
384C2×C4⋊C4D3×D511yescharacteristic
385C2×C22⋊C4—52no—
386C2×C22⋊C4—152no—
387C2×C22⋊C4—152no—
388C2×C22⋊C4—152no—
389C2×C22⋊C4—151no—
390C2×C22⋊C4—151no—
391C2×C22⋊C4—152no—
392C2×C42—31no—
393C2×C42—152no—
394C2×C42—151no—
395C2×C3×C5#54214yescomplemented
396C2×C3×C5C22×#4111yescharacteristic
397C2×C3×C5C2×H(5,𝔽2)11yescharacteristic
398C2×C3×C5C2×2−1+411yescharacteristic
399D15—44no—
400C3×D5—42no—
401C23×C3—52no—
402C23×C3—102no—
403C23×C3—102no—
404C23×C3—102no—
405C23×C3C23×D511yescharacteristic
406C22×D3—54no—
407C22×D3—104no—
408C22×D3—104no—
409C22×D3—104no—
410C3×Q8—108no—
411C2×C3×C4—24no—
412C2×C3×C4—54no—
413C2×C3×C4—52no—
414C2×C3×C4—102no—
415C2×C3×C4—104no—
416C2×C3×C4—108no—
417C2×C3×C4#71314yes—
418C2×C3×C4C23×D514yes—
419C2×C3×C4C23×D512yes—
420#67—208no—
421C2×Dic3—22no—
422C2×Dic3—24no—
423C2×Dic3—22no—
424C2×Dic3—52no—
425C2×Dic3—52no—
426C2×Dic3—102no—
427C2×Dic3—102no—
428C2×Dic3C23×D514yes—
429C2×Dic3C23×D512yes—
430C2×Dic3C23×D512yes—
431D12—108no—
432C4×D3—208no—
433Dic6—48no—
434C22×C5C2×#23712yes—
435C22×C5#100712yes—
436C22×C5#100912yes—
437C22×C5C24×D311yescharacteristic
438C2×D5—22no—
439C2×D5—22no—
440C2×D5—22no—
441C2×D5—44no—
442C2×D5—64no—
443C2×D5—64no—
444C2×D5—64no—
445C2×D5—128no—
446C4×C5—24no—
447C4×C5—62no—
448C4×C5—64no—
449C4×C5—62no—
450C4×C5C2×#23714yes—
451Dic5—24no—
452Dic5—22no—
453Dic5—62no—
454Dic5—62no—
455C24—51no—
456C24—152no—
457C2×Q8—38no—
458C2×Q8—54no—
459C2×Q8—104no—
460C2×Q8—124no—
461C2×D4—158no—
462C2×D4—154no—
463C2×D4—304no—
464C2×D4—604no—
465C22×C4—31no—
466C22×C4—31no—
467C22×C4—32no—
468C22×C4—52no—
469C22×C4—51no—
470C22×C4—54no—
471C22×C4—104no—
472C22×C4—151no—
473C22×C4—151no—
474C22×C4—158no—
475C22×C4—604no—
476C22×C4C2×D3×D512yes—
477C22×C4C2×D3×D511yescharacteristic
478C4⋊C4—34no—
479C4⋊C4—38no—
480C4⋊C4—58no—
481C4⋊C4—154no—
482C4⋊C4—154no—
483C4⋊C4—158no—
484C4⋊C4C2×D3×D514yes—
485C22⋊C4—58no—
486C22⋊C4—158no—
487C22⋊C4—158no—
488C22⋊C4—158no—
489C22⋊C4—154no—
490C22⋊C4—154no—
491C22⋊C4—158no—
492C42—34no—
493C42—158no—
494C42—154no—
495C3×C5C2×#54211yescharacteristic, complemented, Hall
496C22×C3—102no—
497C22×C3—102no—
498C22×C3—102no—
499C22×C3—204no—
500C22×C3#414412yes—
501C22×C3C2×#71312yes—
502C22×C3#414712yes—
503C22×C3C24×D511yescharacteristic
504C2×D3—104no—
505C2×D3—104no—
506C2×D3—104no—
507C2×D3—208no—
508C3×C4—24no—
509C3×C4—102no—
510C3×C4—104no—
511C3×C4C2×#71314yes—
512Dic3—22no—
513Dic3—24no—
514Dic3—22no—
515Dic3—102no—
516Dic3—102no—
517C2×C5#564314yescomplemented
518C2×C5C22×#23711yescharacteristic
519C2×C5C2×#100711yescharacteristic
520C2×C5C2×#100911yescharacteristic
521D5—42no—
522D5—124no—
523C23—52no—
524C23—102no—
525C23—102no—
526C23—102no—
527C23—154no—
528C23—304no—
529C23—304no—
530C23—304no—
531C23C22×D3×D511yescharacteristic
532Q8—108no—
533Q8—128no—
534D4—308no—
535D4—608no—
536C2×C4—24no—
537C2×C4—32no—
538C2×C4—32no—
539C2×C4—34no—
540C2×C4—54no—
541C2×C4—52no—
542C2×C4—64no—
543C2×C4—62no—
544C2×C4—62no—
545C2×C4—102no—
546C2×C4—104no—
547C2×C4—108no—
548C2×C4—152no—
549C2×C4—152no—
550C2×C4—302no—
551C2×C4—302no—
552C2×C4—608no—
553C2×C4#676614yes—
554C2×C4C22×D3×D514yes—
555C2×C4C22×D3×D512yes—
556C2×C3—202no—
557C2×C3#6606714yescomplemented
558C2×C3C2×#414411yescharacteristic
559C2×C3C22×#71311yescharacteristic
560C2×C3C2×#414711yescharacteristic
561D3—204no—
562C5C2×#564311yescharacteristic, complemented, Sylow
563C22—102no—
564C22—102no—
565C22—102no—
566C22—204no—
567C22—304no—
568C22—304no—
569C22—304no—
570C22—608no—
571C22C2×#676612yes—
572C22#9202712yes—
573C22#9202912yes—
574C22C23×D3×D511yescharacteristic
575C4—24no—
576C4—64no—
577C4—62no—
578C4—62no—
579C4—102no—
580C4—104no—
581C4—302no—
582C4—302no—
583C4C2×#676614yes—
584C3C2×#6606711yescharacteristic, complemented, Sylow
585C2—202no—
586C2—604no—
587C2#1175337514yesdirect product factor
588C2C22×#676611yescharacteristic
589C2C2×#9202711yescharacteristic
590C2C2×#9202911yescharacteristic
subgroupG/H
centerC23C22×D3×D5
commutator, cosocleC22×C3×C5C25
FrattiniC22C23×D3×D5
FittingC2×C3×C5×C4⋊C4C22
socleC23×C3×C5C24
prime radicalC2×#91450C2
hypercenterC2×C4⋊C4D3×D5
hypocenterC3×C5C2×#542
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
1. C2×#91782C2semidirect
2. C2×C5×#888C2semidirect
3. C2×C3×#4043C2semidirect
4. C2×#91455C2semidirect
6. C2×#91383C2semidirect
7. C2×#91374C2semidirect
8. C2×#91357C2semidirect
9. C2×#91354C2semidirect
10. C2×#91345C2semidirect
11. C2×#91331C2semidirect
12. #11753375C2direct (N × H)
23. #91782C22semidirect
27. C5×#888C22semidirect
30. C3×#4043C22semidirect
40. #91455C22semidirect
42. #91383C22semidirect
43. #91374C22semidirect
44. #91357C22semidirect
45. #91354C22semidirect
46. #91345C22semidirect
47. #91331C22semidirect
54. C2×#4043D3semidirect
110. C2×#888D5semidirect
140. #4043C2×D3semidirect
212. #888C2×D5semidirect
395. C2×C3×C5#542semidirect
495. C3×C5C2×#542semidirect
517. C2×C5#5643semidirect
557. C2×C3#66067semidirect
562. C5C2×#5643semidirect
584. C3C2×#66067semidirect
587. C2#11753375direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#917822
C2×C5×#8882
C2×C3×#40432
C2×#914552
C2×#914502
C2×#913832
C2×#913742
C2×#913572
C2×#913542
C2×#913452
C2×#913312
#117533752
C2×#660673
C2×#56435