Finite Group Databasebeta

← C2×#11754608 #14483710

Subgroups — C2×#11754608 #14483710

Order 1920.

2504 conjugacy classes of subgroups.

The subgroup lattice has 632 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×C4⋊Dic15C211yesmaximal, characteristic, complemented
2C2×C5×#930C211yesmaximal, characteristic, complemented
3C2×C3×C4×C22⋊D5C211yesmaximal, characteristic, complemented
4C2×#91571C212yesmaximal, complemented
5C2×#67⋊Dic5C212yesmaximal, complemented
6#11754608C2116yesmaximal, direct product factor
7C2×#91457C212yesmaximal, complemented
8C2×#91449C211yesmaximal, characteristic, complemented
9C2×Dic5×D12C211yesmaximal, characteristic, complemented
10C2×#91388C211yesmaximal, characteristic, complemented
11C2×#91358C211yesmaximal, characteristic, complemented
12C2×#91342C212yesmaximal, complemented
13C2×#65952—31nomaximal, Hall
14C23×C3×C4×C5C2211yescharacteristic
15C23×Dic15C2212yes—
16C22×C5×#67C2212yes—
17C22×C5×D12C2211yescharacteristic
18C22×C3×C22⋊D5C2211yescharacteristic
19C22×C3×C4×D5C2211yescharacteristic
20C22×#1201C2212yes—
21C22×D3×Dic5C2212yes—
22C2×C4⋊Dic15C2211yescharacteristic
23C2×C4⋊Dic15C2212yes—
24C2×C4⋊Dic15C2211yescharacteristic
25C2×C4⋊Dic15C2218yescomplemented
26C5×#930C2218yescomplemented
27C2×C5×(C2×D3)⋊C4C2212yes—
28C2×C5×C4⋊Dic3C2211yescharacteristic
29C2×C3×C22⋊Dic5C2211yescharacteristic
30C3×C4×C22⋊D5C2218yescomplemented
31C2×C3×#685C2211yescharacteristic
32C2×C3×#683C2211yescharacteristic
33C2×C3×C4×Dic5C2211yescharacteristic
34#91571C22116yescomplemented
35#67⋊Dic5C22116yescomplemented
36C2×Dic3⋊Dic5C2212yes—
37C2×#6669C2212yes—
38C2×(C2×D3)⋊Dic5C2212yes—
39C2×(C2×D3)⋊Dic5C2212yes—
40C2×#6665C2212yes—
41C2×Dic3×Dic5C2212yes—
42#91457C22116yescomplemented
43#91449C2218yescomplemented
44Dic5×D12C2218yescomplemented
45#91388C2218yescomplemented
46#91358C2218yescomplemented
47#91342C22116yescomplemented
48C2×#5565—51nomaximal, Hall
49C2×C5×#121—31no—
50C2×#4085—32no—
51C2×#4084—31no—
52C2×D4×Dic5—31no—
53C2×D4×Dic5—32no—
54C2×C4×C22⋊D5D311yescharacteristic, complemented
55C22×C4⋊Dic5—31no—
56#65952—316no—
57C2×#4038—31no—
58C2×#4036—31no—
59C2×#4032—32no—
60C2×#4025—32no—
61C24×C3×C5C2311yescharacteristic
62C23×C5×D3C2312yes—
63C23×C3×D5C2311yescharacteristic
64C22×C3×C4×C5—24no—
65C22×C3×C4×C5C2311yescharacteristic
66C22×C3×C4×C5C2314yes—
67C22×C3×C4×C5C2311yescharacteristic
68C22×Dic15—44no—
69C22×Dic15C2312yes—
70C22×Dic15C2312yes—
71C22×Dic15C2318yes—
72C2×C5×#67—28no—
73C2×C5×#67C2318yes—
74C22×C5×Dic3C2312yes—
75C2×C5×D12—24no—
76C2×C5×D12C2314yes—
77C2×C3×C22⋊D5—24no—
78C2×C3×C22⋊D5C2314yes—
79C22×C3×Dic5C2311yescharacteristic
80C22×C3×Dic5C2311yescharacteristic
81C22×C3×Dic5C2311yescharacteristic
82C2×C3×C4×D5—24no—
83C2×C3×C4×D5C2314yes—
84C2×#1201—44no—
85C2×#1201C2318yes—
86C2×D3×Dic5—44no—
87C2×D3×Dic5C2318yes—
88C4⋊Dic15C2314yes—
89C4⋊Dic15C2314yes—
90C4⋊Dic15C2318yes—
91C5×(C2×D3)⋊C4C2318yes—
92C5×C4⋊Dic3C2314yes—
93C3×C22⋊Dic5C2314yes—
94C3×#685C2314yes—
95C3×#683C2314yes—
96C3×C4×Dic5C2314yes—
97Dic3⋊Dic5C2318yes—
98#6669C2318yes—
99(C2×D3)⋊Dic5C2318yes—
100(C2×D3)⋊Dic5C2318yes—
101#6665C2318yes—
102Dic3×Dic5C2318yes—
103C2×C3×C4×D4—51no—
104C2×#939—52no—
105C2×#930—51no—
106C2×#930D511yescharacteristic, complemented
107C2×#926—51no—
108C22×C4⋊Dic3—51no—
109#5565—516no—
110C2×#898—52no—
111C2×#883—52no—
112C2×#67⋊C4—52no—
113C2×C4×D12—51no—
114C2×#870—51no—
115C22×C5×D4—32no—
116C22×C5×D4—31no—
117C23×C4×C5C2×D311yescharacteristic
118C22×C22⋊D5—32no—
119C22×C22⋊D5C2×D311yescharacteristic
120C23×Dic5—32no—
121C23×Dic5—32no—
122C22×C4×D5C2×D311yescharacteristic
123C5×#121—38no—
124C2×C5×C4⋊C4—31no—
125C2×C5×C22⋊C4—32no—
126C2×C22⋊Dic5—32no—
127C2×C22⋊Dic5—32no—
128C2×C22⋊Dic5C2×D311yescharacteristic
129#4085—316no—
130#4084—38no—
131D4×Dic5—316no—
132D4×Dic5—38no—
133C4×C22⋊D5C2×D318yescomplemented
134C2×#685—32no—
135C2×#685C2×D311yescharacteristic
136C2×C4⋊Dic5—32no—
137C2×C4⋊Dic5—31no—
138C2×C4⋊Dic5—31no—
139C2×C4⋊Dic5—38no—
140C2×C4⋊Dic5—32no—
141C2×#683—32no—
142C2×#683C2×D311yescharacteristic
143C2×C4×Dic5—32no—
144C2×C4×Dic5C2×D311yescharacteristic
145#4038—38no—
146#4036—38no—
147#4032—316no—
148#4025—316no—
149C2×#553—151noSylow
150C23×C3×C5—22no—
151C23×C3×C5—22no—
152C23×C3×C5#4114yes—
153C23×C3×C5C2412yes—
154C23×C3×C5C2411yescharacteristic
155C22×C5×D3—24no—
156C22×C5×D3—24no—
157C22×C5×D3—24no—
158C22×C5×D3C2414yes—
159C22×C3×D5—22no—
160C22×C3×D5—22no—
161C22×C3×D5—22no—
162C22×C3×D5C2412yes—
163C2×C3×C4×C5—22no—
164C2×C3×C4×C5—28no—
165C2×C3×C4×C5#4114yes—
166C2×C3×C4×C5C2412yes—
167C2×C3×C4×C5C2412yes—
168C2×Dic15—24no—
169C2×Dic15—24no—
170C2×Dic15—48no—
171C2×Dic15C2414yes—
172C2×Dic15C2414yes—
173C5×#67—216no—
174C2×C5×Dic3—24no—
175C2×C5×Dic3C2414yes—
176C5×D12—28no—
177C3×C22⋊D5—28no—
178C2×C3×Dic5—22no—
179C2×C3×Dic5—22no—
180C2×C3×Dic5—22no—
181C2×C3×Dic5C2412yes—
182C2×C3×Dic5C2412yes—
183C2×C3×Dic5C2412yes—
184C3×C4×D5—28no—
185#1201—48no—
186D3×Dic5—48no—
187C22×C3×D4—51no—
188C23×C3×C4—51no—
189C23×C3×C4C2×D511yescharacteristic
190C22×#67—52no—
191C22×#67C2×D512yes—
192C23×Dic3—52no—
193C22×D12C2×D511yescharacteristic
194C22×C4×D3—52no—
195C3×C4×D4—58no—
196C2×C3×C4⋊C4—51no—
197C2×C3×C22⋊C4—51no—
198C2×C3×C22⋊C4—51no—
199C2×C3×C42—51no—
200C2×#217—52no—
201#939—516no—
202#930—58no—
203#930C2×D518yescomplemented
204C2×(C2×D3)⋊C4—52no—
205C2×(C2×D3)⋊C4—52no—
206C2×(C2×D3)⋊C4C2×D512yes—
207#926—58no—
208C2×C4⋊Dic3—51no—
209C2×C4⋊Dic3—51no—
210C2×C4⋊Dic3—52no—
211C2×C4⋊Dic3—58no—
212C2×C4⋊Dic3C2×D511yescharacteristic
213C2×Dic3⋊C4—52no—
214C2×Dic3⋊C4—52no—
215C2×C4×Dic3—52no—
216#898—516no—
217#883—516no—
218#67⋊C4—516no—
219C4×D12—58no—
220#870—58no—
221C24×C5—32no—
222C24×C5C22×D311yescharacteristic
223C23×D5C22×D311yescharacteristic
224C2×C5×D4—38no—
225C2×C5×D4—34no—
226C2×C5×D4—68no—
227C2×C5×D4—64no—
228C22×C4×C5—24no—
229C22×C4×C5—32no—
230C22×C4×C5C22×D311yescharacteristic
231C22×C4×C5C22×D314yes—
232C22×C4×C5C22×D311yescharacteristic
233C2×C22⋊D5—24no—
234C2×C22⋊D5—38no—
235C2×C22⋊D5—124no—
236C2×C22⋊D5C22×D314yes—
237C22×Dic5—32no—
238C22×Dic5—32no—
239C22×Dic5—38no—
240C22×Dic5—38no—
241C22×Dic5—124no—
242C22×Dic5—124no—
243C22×Dic5C22×D311yescharacteristic
244C22×Dic5C22×D311yescharacteristic
245C22×Dic5C22×D311yescharacteristic
246C2×C4×D5—24no—
247C2×C4×D5C22×D314yes—
248C5×C4⋊C4—34no—
249C5×C22⋊C4—38no—
250C22⋊Dic5—38no—
251C22⋊Dic5—38no—
252C22⋊Dic5C22×D314yes—
253#685—38no—
254#685C22×D314yes—
255C4⋊Dic5—38no—
256C4⋊Dic5—34no—
257C4⋊Dic5—34no—
258C4⋊Dic5—38no—
259#683—38no—
260#683C22×D314yes—
261C4×Dic5—38no—
262C4×Dic5C22×D314yes—
263#553—1516no—
264C2×#126—152no—
265C2×#125—151no—
266C2×#123—152no—
267C2×#121—31no—
268C2×#121—151no—
269C2×#121—152no—
270C2×C4×D4—51no—
271C2×C4×D4—151no—
272C2×C4×D4—152no—
273C2×#117—151no—
274C22×C4⋊C4—151no—
275C22×C3×C5—28no—
276C22×C3×C5—22no—
277C22×C3×C5—22no—
278C22×C3×C5C2×#4114yes—
279C22×C3×C5C2×#4112yes—
280C22×C3×C5C2×#4112yes—
281C22×C3×C5H(5,𝔽2)12yes—
282C22×C3×C5C2511yescharacteristic
283C2×C5×D3—24no—
284C2×C5×D3—24no—
285C2×C5×D3—24no—
286C2×C5×D3—48no—
287C2×C3×D5—22no—
288C2×C3×D5—22no—
289C2×C3×D5—22no—
290C2×C3×D5—44no—
291C3×C4×C5—22no—
292C3×C4×C5C2×#4114yes—
293Dic15—24no—
294Dic15—24no—
295C3×Dic5—22no—
296C3×Dic5—22no—
297C3×Dic5—22no—
298C5×Dic3—24no—
299C24×C3—51no—
300C24×C3C22×D511yescharacteristic
301C23×D3C22×D512yes—
302C2×C3×D4—54no—
303C2×C3×D4—104no—
304C22×C3×C4—24no—
305C22×C3×C4—51no—
306C22×C3×C4—51no—
307C22×C3×C4—51no—
308C22×C3×C4—54no—
309C22×C3×C4—104no—
310C22×C3×C4C22×D511yescharacteristic
311C22×C3×C4C22×D514yes—
312C22×C3×C4C22×D511yescharacteristic
313C2×#67—28no—
314C2×#67—58no—
315C2×#67—204no—
316C2×#67C22×D518yes—
317C22×Dic3—52no—
318C22×Dic3—52no—
319C22×Dic3—58no—
320C22×Dic3—204no—
321C22×Dic3C22×D512yes—
322C2×D12—24no—
323C2×D12C22×D514yes—
324C2×C4×D3—58no—
325C2×C4×D3—204no—
326C3×C4⋊C4—54no—
327C3×C22⋊C4—54no—
328C3×C22⋊C4—54no—
329C3×C42—54no—
330#217—58no—
331(C2×D3)⋊C4—58no—
332(C2×D3)⋊C4—58no—
333(C2×D3)⋊C4C22×D518yes—
334C4⋊Dic3—54no—
335C4⋊Dic3—54no—
336C4⋊Dic3—58no—
337C4⋊Dic3C22×D514yes—
338Dic3⋊C4—58no—
339Dic3⋊C4—58no—
340C4×Dic3—58no—
341C23×C5—22no—
342C23×C5—22no—
343C23×C5—34no—
344C23×C5—64no—
345C23×C5—64no—
346C23×C5—64no—
347C23×C5#23514yes—
348C23×C5C23×D312yes—
349C23×C5C23×D311yescharacteristic
350C22×D5—22no—
351C22×D5—22no—
352C22×D5—22no—
353C22×D5C23×D312yes—
354C5×D4—616no—
355C5×D4—68no—
356C2×C4×C5—22no—
357C2×C4×C5—28no—
358C2×C4×C5—34no—
359C2×C4×C5—64no—
360C2×C4×C5#23714yes—
361C2×C4×C5C23×D312yes—
362C2×C4×C5C23×D312yes—
363C22⋊D5—28no—
364C22⋊D5—128no—
365C2×Dic5—22no—
366C2×Dic5—22no—
367C2×Dic5—22no—
368C2×Dic5—34no—
369C2×Dic5—34no—
370C2×Dic5—64no—
371C2×Dic5—64no—
372C2×Dic5—128no—
373C2×Dic5—128no—
374C2×Dic5C23×D312yes—
375C2×Dic5C23×D312yes—
376C2×Dic5C23×D312yes—
377C4×D5—28no—
378C22×D4—32no—
379C22×D4—31no—
380C22×D4—51no—
381C22×D4—152no—
382C23×C4—51no—
383C23×C4—152no—
384C23×C4—152no—
385C23×C4D3×D511yescharacteristic
386#126—1516no—
387#125—158no—
388#123—1516no—
389#121—38no—
390#121—158no—
391#121—1516no—
392C4×D4—58no—
393C4×D4—158no—
394C4×D4—1516no—
395#117—158no—
396C2×C4⋊C4—31no—
397C2×C4⋊C4—51no—
398C2×C4⋊C4—152no—
399C2×C4⋊C4—152no—
400C2×C4⋊C4—151no—
401C2×C4⋊C4—151no—
402C2×C4⋊C4—152no—
403C2×C4⋊C4—158no—
404C2×C22⋊C4—32no—
405C2×C22⋊C4—51no—
406C2×C22⋊C4—51no—
407C2×C22⋊C4—152no—
408C2×C22⋊C4—152no—
409C2×C22⋊C4—152no—
410C2×C42—51no—
411C2×C42—152no—
412C2×C3×C5—24no—
413C2×C3×C5#55314yescomplemented
414C2×C3×C5C22×#4111yescharacteristic
415C2×C3×C5C22×#4111yescharacteristic
416C2×C3×C5C2×H(5,𝔽2)11yescharacteristic
417C3×D5—42no—
418C5×D3—44no—
419C23×C3—22no—
420C23×C3—22no—
421C23×C3—52no—
422C23×C3—102no—
423C23×C3—102no—
424C23×C3—102no—
425C23×C3#71114yes—
426C23×C3C23×D512yes—
427C23×C3C23×D511yescharacteristic
428C22×D3—24no—
429C22×D3—24no—
430C22×D3—24no—
431C22×D3C23×D514yes—
432C3×D4—108no—
433C2×C3×C4—22no—
434C2×C3×C4—28no—
435C2×C3×C4—52no—
436C2×C3×C4—52no—
437C2×C3×C4—52no—
438C2×C3×C4—102no—
439C2×C3×C4—102no—
440C2×C3×C4—102no—
441C2×C3×C4—108no—
442C2×C3×C4#71114yes—
443C2×C3×C4C23×D512yes—
444C2×C3×C4C23×D512yes—
445#67—216no—
446#67—208no—
447C2×Dic3—24no—
448C2×Dic3—54no—
449C2×Dic3—54no—
450C2×Dic3—104no—
451C2×Dic3—104no—
452C2×Dic3—208no—
453C2×Dic3C23×D514yes—
454D12—28no—
455C4×D3—208no—
456C22×C5—22no—
457C22×C5—22no—
458C22×C5—28no—
459C22×C5—64no—
460C22×C5—64no—
461C22×C5—64no—
462C22×C5—128no—
463C22×C5C2×#23514yes—
464C22×C5C2×#23512yes—
465C22×C5C2×#23712yes—
466C22×C5#100912yes—
467C22×C5C24×D311yescharacteristic
468C2×D5—22no—
469C2×D5—22no—
470C2×D5—22no—
471C2×D5—44no—
472C4×C5—22no—
473C4×C5—64no—
474C4×C5C2×#23714yes—
475Dic5—22no—
476Dic5—22no—
477Dic5—22no—
478Dic5—64no—
479Dic5—64no—
480C24—32no—
481C24—51no—
482C24C2×D3×D511yescharacteristic
483C2×D4—38no—
484C2×D4—34no—
485C2×D4—54no—
486C2×D4—68no—
487C2×D4—64no—
488C2×D4—104no—
489C2×D4—158no—
490C2×D4—604no—
491C22×C4—24no—
492C22×C4—32no—
493C22×C4—51no—
494C22×C4—51no—
495C22×C4—51no—
496C22×C4—54no—
497C22×C4—104no—
498C22×C4—158no—
499C22×C4—152no—
500C22×C4—152no—
501C22×C4—158no—
502C22×C4—604no—
503C22×C4—604no—
504C22×C4C2×D3×D514yes—
505C22×C4C2×D3×D511yescharacteristic
506C22×C4C2×D3×D511yescharacteristic
507C4⋊C4—34no—
508C4⋊C4—54no—
509C4⋊C4—158no—
510C4⋊C4—158no—
511C4⋊C4—154no—
512C4⋊C4—154no—
513C4⋊C4—158no—
514C22⋊C4—38no—
515C22⋊C4—54no—
516C22⋊C4—54no—
517C22⋊C4—158no—
518C22⋊C4—158no—
519C22⋊C4—158no—
520C42—54no—
521C42—158no—
522C3×C5C2×#55311yescharacteristic, complemented, Hall
523C22×C3—28no—
524C22×C3—22no—
525C22×C3—22no—
526C22×C3—102no—
527C22×C3—102no—
528C22×C3—102no—
529C22×C3—204no—
530C22×C3C2×#71114yes—
531C22×C3C2×#71112yes—
532C22×C3C2×#71112yes—
533C22×C3#414912yes—
534C22×C3C24×D511yescharacteristic
535C2×D3—24no—
536C2×D3—24no—
537C2×D3—24no—
538C2×D3—48no—
539C3×C4—22no—
540C3×C4—102no—
541C3×C4—102no—
542C3×C4—102no—
543C3×C4C2×#71114yes—
544Dic3—24no—
545Dic3—104no—
546Dic3—104no—
547C2×C5—24no—
548C2×C5—124no—
549C2×C5#556514yescomplemented
550C2×C5C22×#23511yescharacteristic
551C2×C5C22×#23711yescharacteristic
552C2×C5C2×#100911yescharacteristic
553D5—42no—
554C23—22no—
555C23—22no—
556C23—34no—
557C23—52no—
558C23—64no—
559C23—64no—
560C23—64no—
561C23—102no—
562C23—102no—
563C23—102no—
564C23#677214yes—
565C23C22×D3×D512yes—
566C23C22×D3×D511yescharacteristic
567D4—616no—
568D4—68no—
569D4—108no—
570D4—608no—
571C2×C4—22no—
572C2×C4—28no—
573C2×C4—34no—
574C2×C4—52no—
575C2×C4—52no—
576C2×C4—52no—
577C2×C4—64no—
578C2×C4—102no—
579C2×C4—102no—
580C2×C4—102no—
581C2×C4—108no—
582C2×C4—154no—
583C2×C4—154no—
584C2×C4—304no—
585C2×C4—304no—
586C2×C4—608no—
587C2×C4—608no—
588C2×C4#678014yes—
589C2×C4C22×D3×D512yes—
590C2×C4C22×D3×D512yes—
591C2×C3—24no—
592C2×C3—202no—
593C2×C3#6595214yescomplemented
594C2×C3C22×#71111yescharacteristic
595C2×C3C22×#71111yescharacteristic
596C2×C3C2×#414911yescharacteristic
597D3—44no—
598C5C2×#556511yescharacteristic, complemented, Sylow
599C22—22no—
600C22—28no—
601C22—22no—
602C22—64no—
603C22—64no—
604C22—64no—
605C22—102no—
606C22—102no—
607C22—102no—
608C22—128no—
609C22—204no—
610C22C2×#677214yes—
611C22C2×#677212yes—
612C22#9201912yes—
613C22C2×#678012yes—
614C22C23×D3×D511yescharacteristic
615C4—22no—
616C4—64no—
617C4—102no—
618C4—102no—
619C4—102no—
620C4—304no—
621C4—304no—
622C4C2×#678014yes—
623C3C2×#6595211yescharacteristic, complemented, Sylow
624C2—24no—
625C2—124no—
626C2—202no—
627C2#1175460814yesdirect product factor
628C2C22×#677211yescharacteristic
629C2C2×#9201911yescharacteristic
630C2C22×#678011yescharacteristic
subgroupG/H
centerC23C22×D3×D5
commutator, cosocleC22×C3×C5C25
FrattiniC22C23×D3×D5
FittingC23×C3×C4×C5C22
socleC23×C3×C5C24
hypercenterC23×C4D3×D5
hypocenterC3×C5C2×#553
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C23×C4

Complements

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

NH ≅ G/Ntype
1. C22×C4⋊Dic15C2semidirect
2. C2×C5×#930C2semidirect
3. C2×C3×C4×C22⋊D5C2semidirect
4. C2×#91571C2semidirect
5. C2×#67⋊Dic5C2semidirect
6. #11754608C2direct (N × H)
7. C2×#91457C2semidirect
8. C2×#91449C2semidirect
9. C2×Dic5×D12C2semidirect
10. C2×#91388C2semidirect
11. C2×#91358C2semidirect
12. C2×#91342C2semidirect
25. C2×C4⋊Dic15C22semidirect
26. C5×#930C22semidirect
30. C3×C4×C22⋊D5C22semidirect
34. #91571C22semidirect
35. #67⋊Dic5C22semidirect
42. #91457C22semidirect
43. #91449C22semidirect
44. Dic5×D12C22semidirect
45. #91388C22semidirect
46. #91358C22semidirect
47. #91342C22semidirect
54. C2×C4×C22⋊D5D3semidirect
106. C2×#930D5semidirect
133. C4×C22⋊D5C2×D3semidirect
203. #930C2×D5semidirect
413. C2×C3×C5#553semidirect
522. C3×C5C2×#553semidirect
549. C2×C5#5565semidirect
593. C2×C3#65952semidirect
598. C5C2×#5565semidirect
623. C3C2×#65952semidirect
627. C2#11754608direct (N × H)

Maximal subgroups

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

subgroupindex
C22×C4⋊Dic152
C2×C5×#9302
C2×C3×C4×C22⋊D52
C2×#915712
C2×#67⋊Dic52
#117546082
C2×#914572
C2×#914492
C2×Dic5×D122
C2×#913882
C2×#913582
C2×#913422
C2×#659523
C2×#55655