Finite Group Databasebeta

← C2×#11756324 #14483603

Subgroups — C2×#11756324 #14483603

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
1C2×C3×C4×C5×D4C211yesmaximal, characteristic, complemented
2C2×#91828C212yesmaximal, complemented
3C2×#91819C211yesmaximal, characteristic, complemented
4C2×#91819C211yesmaximal, characteristic, complemented
5C2×#91815C211yesmaximal, characteristic, complemented
6C22×C4⋊Dic15C211yesmaximal, characteristic, complemented
7#11756324C2116yesmaximal, direct product factor
8C2×#91787C212yesmaximal, complemented
9C2×#91772C212yesmaximal, complemented
10C2×#91770C212yesmaximal, complemented
11C2×C4×D60C211yesmaximal, characteristic, complemented
12C2×#91759C211yesmaximal, characteristic, complemented
13C2×#65899—31nomaximal, Hall
14C22×C3×C5×D4C2211yescharacteristic
15C23×C3×C4×C5C2211yescharacteristic
16C23×C3×C4×C5C2211yescharacteristic
17C22×#1220C2212yes—
18C22×#1220C2212yes—
19C23×Dic15C2212yes—
20C22×D60C2211yescharacteristic
21C22×C4×D15C2212yes—
22C3×C4×C5×D4C2218yescomplemented
23C2×C3×C5×C4⋊C4C2211yescharacteristic
24C2×C3×C5×C22⋊C4C2211yescharacteristic
25C2×C3×C5×C22⋊C4C2211yescharacteristic
26C2×C3×C42×C5C2211yescharacteristic
27C2×#6719C2212yes—
28#91828C22116yescomplemented
29#91819C2218yescomplemented
30#91819C2218yescomplemented
31C2×#6714C2212yes—
32C2×#6714C2212yes—
33C2×#6714C2212yes—
34#91815C2218yescomplemented
35C2×C4⋊Dic15C2211yescharacteristic
36C2×C4⋊Dic15C2211yescharacteristic
37C2×C4⋊Dic15C2212yes—
38C2×C4⋊Dic15C2211yescharacteristic
39C2×C4⋊Dic15C2218yescomplemented
40C2×#6712C2212yes—
41C2×#6712C2212yes—
42C2×C4×Dic15C2212yes—
43#91787C22116yescomplemented
44#91772C22116yescomplemented
45#91770C22116yescomplemented
46C4×D60C2218yescomplemented
47#91759C2218yescomplemented
48C2×#5565—51nomaximal, Hall
49C2×C4×C5×D4D311yescharacteristic, complemented
50C2×#4085—32no—
51C2×#4076—31no—
52C2×#4076—31no—
53C2×#4072—31no—
54C22×C4⋊Dic5—31no—
55#65899—316no—
56C2×#4044—32no—
57C2×#4029—32no—
58C2×#4027—32no—
59C2×C4×D20—31no—
60C2×#4016—31no—
61C24×C3×C5C2311yescharacteristic
62C24×C3×C5C2311yescharacteristic
63C23×D15C2312yes—
64C2×C3×C5×D4—24no—
65C2×C3×C5×D4C2314yes—
66C22×C3×C4×C5—24no—
67C22×C3×C4×C5—24no—
68C22×C3×C4×C5C2311yescharacteristic
69C22×C3×C4×C5C2311yescharacteristic
70C22×C3×C4×C5C2311yescharacteristic
71C22×C3×C4×C5C2311yescharacteristic
72C22×C3×C4×C5C2314yes—
73C22×C3×C4×C5C2314yes—
74C22×C3×C4×C5C2311yescharacteristic
75C2×#1220—28no—
76C2×#1220—44no—
77C2×#1220C2318yes—
78C2×#1220C2318yes—
79C22×Dic15—44no—
80C22×Dic15C2312yes—
81C22×Dic15C2312yes—
82C22×Dic15C2312yes—
83C22×Dic15C2318yes—
84C2×D60—24no—
85C2×D60C2314yes—
86C2×C4×D15—44no—
87C2×C4×D15C2318yes—
88C3×C5×C4⋊C4C2314yes—
89C3×C5×C22⋊C4C2314yes—
90C3×C5×C22⋊C4C2314yes—
91C3×C42×C5C2314yes—
92#6719C2318yes—
93#6714C2318yes—
94#6714C2318yes—
95#6714C2318yes—
96C4⋊Dic15C2314yes—
97C4⋊Dic15C2314yes—
98C4⋊Dic15C2314yes—
99C4⋊Dic15C2318yes—
100#6712C2318yes—
101#6712C2318yes—
102C4×Dic15C2318yes—
103C2×C3×C4×D4D511yescharacteristic, complemented
104C2×#939—52no—
105C2×#930—51no—
106C2×#930—51no—
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×D4C2×D311yescharacteristic
116C23×C4×C5C2×D311yescharacteristic
117C23×C4×C5C2×D311yescharacteristic
118C22×C22⋊D5—32no—
119C22×C22⋊D5—32no—
120C23×Dic5—32no—
121C22×D20—31no—
122C22×C4×D5—32no—
123C4×C5×D4C2×D318yescomplemented
124C2×C5×C4⋊C4C2×D311yescharacteristic
125C2×C5×C22⋊C4C2×D311yescharacteristic
126C2×C5×C22⋊C4C2×D311yescharacteristic
127C2×C42×C5C2×D311yescharacteristic
128C2×C22⋊Dic5—32no—
129#4085—316no—
130#4076—38no—
131#4076—38no—
132C2×#685—32no—
133C2×#685—32no—
134C2×#685—32no—
135#4072—38no—
136C2×C4⋊Dic5—31no—
137C2×C4⋊Dic5—31no—
138C2×C4⋊Dic5—31no—
139C2×C4⋊Dic5—32no—
140C2×C4⋊Dic5—38no—
141C2×#683—32no—
142C2×#683—32no—
143C2×C4×Dic5—32no—
144#4044—316no—
145#4029—316no—
146#4027—316no—
147C4×D20—38no—
148#4016—38no—
149C2×#553—151noSylow
150C23×C3×C5—22no—
151C23×C3×C5—22no—
152C23×C3×C5—22no—
153C23×C3×C5—22no—
154C23×C3×C5—22no—
155C23×C3×C5#4114yes—
156C23×C3×C5C2412yes—
157C23×C3×C5C2412yes—
158C23×C3×C5C2411yescharacteristic
159C22×D15—24no—
160C22×D15—24no—
161C22×D15—24no—
162C22×D15C2414yes—
163C3×C5×D4—28no—
164C2×C3×C4×C5—22no—
165C2×C3×C4×C5—22no—
166C2×C3×C4×C5—22no—
167C2×C3×C4×C5—22no—
168C2×C3×C4×C5—28no—
169C2×C3×C4×C5—28no—
170C2×C3×C4×C5#4114yes—
171C2×C3×C4×C5C2412yes—
172C2×C3×C4×C5C2412yes—
173C2×C3×C4×C5C2412yes—
174C2×C3×C4×C5C2412yes—
175C2×C3×C4×C5C2412yes—
176#1220—216no—
177#1220—48no—
178C2×Dic15—24no—
179C2×Dic15—24no—
180C2×Dic15—24no—
181C2×Dic15—48no—
182C2×Dic15C2414yes—
183C2×Dic15C2414yes—
184C2×Dic15C2414yes—
185D60—28no—
186C4×D15—48no—
187C22×C3×D4C2×D511yescharacteristic
188C23×C3×C4C2×D511yescharacteristic
189C23×C3×C4C2×D511yescharacteristic
190C22×#67—52no—
191C22×#67—52no—
192C23×Dic3—52no—
193C22×D12—51no—
194C22×C4×D3—52no—
195C3×C4×D4C2×D518yescomplemented
196C2×C3×C4⋊C4C2×D511yescharacteristic
197C2×C3×C22⋊C4C2×D511yescharacteristic
198C2×C3×C22⋊C4C2×D511yescharacteristic
199C2×C3×C42C2×D511yescharacteristic
200C2×#217—52no—
201#939—516no—
202#930—58no—
203#930—58no—
204C2×(C2×D3)⋊C4—52no—
205C2×(C2×D3)⋊C4—52no—
206C2×(C2×D3)⋊C4—52no—
207#926—58no—
208C2×C4⋊Dic3—51no—
209C2×C4⋊Dic3—51no—
210C2×C4⋊Dic3—51no—
211C2×C4⋊Dic3—52no—
212C2×C4⋊Dic3—58no—
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×C5C22×D311yescharacteristic
222C24×C5C22×D311yescharacteristic
223C23×D5—32no—
224C2×C5×D4—24no—
225C2×C5×D4C22×D314yes—
226C22×C4×C5—24no—
227C22×C4×C5—24no—
228C22×C4×C5C22×D311yescharacteristic
229C22×C4×C5C22×D311yescharacteristic
230C22×C4×C5C22×D311yescharacteristic
231C22×C4×C5C22×D311yescharacteristic
232C22×C4×C5C22×D314yes—
233C22×C4×C5C22×D314yes—
234C22×C4×C5C22×D311yescharacteristic
235C2×C22⋊D5—38no—
236C2×C22⋊D5—38no—
237C2×C22⋊D5—68no—
238C2×C22⋊D5—124no—
239C22×Dic5—32no—
240C22×Dic5—32no—
241C22×Dic5—32no—
242C22×Dic5—38no—
243C22×Dic5—124no—
244C2×D20—34no—
245C2×D20—64no—
246C2×C4×D5—38no—
247C2×C4×D5—124no—
248C5×C4⋊C4C22×D314yes—
249C5×C22⋊C4C22×D314yes—
250C5×C22⋊C4C22×D314yes—
251C42×C5C22×D314yes—
252C22⋊Dic5—38no—
253#685—38no—
254#685—38no—
255#685—38no—
256C4⋊Dic5—34no—
257C4⋊Dic5—34no—
258C4⋊Dic5—34no—
259C4⋊Dic5—38no—
260#683—38no—
261#683—38no—
262C4×Dic5—38no—
263#553—1516no—
264C2×#126—152no—
265C2×#125—151no—
266C2×#123—152no—
267C2×#121—152no—
268C2×#121—151no—
269C2×#121—151no—
270C2×C4×D4—152no—
271C2×C4×D4—151no—
272C2×C4×D4D1511yescharacteristic, complemented
273C2×#117—151no—
274C22×C4⋊C4—151no—
275C22×C3×C5—28no—
276C22×C3×C5—22no—
277C22×C3×C5—22no—
278C22×C3×C5—22no—
279C22×C3×C5—22no—
280C22×C3×C5—22no—
281C22×C3×C5—44no—
282C22×C3×C5C2×#4114yes—
283C22×C3×C5C2×#4112yes—
284C22×C3×C5C2×#4112yes—
285C22×C3×C5H(5,𝔽2)12yes—
286C22×C3×C5C2511yescharacteristic
287C2×D15—24no—
288C2×D15—24no—
289C2×D15—24no—
290C2×D15—48no—
291C3×C4×C5—22no—
292C3×C4×C5—22no—
293C3×C4×C5—22no—
294C3×C4×C5—22no—
295C3×C4×C5C2×#4114yes—
296Dic15—24no—
297Dic15—24no—
298Dic15—24no—
299C24×C3C22×D511yescharacteristic
300C24×C3C22×D511yescharacteristic
301C23×D3—52no—
302C2×C3×D4—24no—
303C2×C3×D4C22×D514yes—
304C22×C3×C4—24no—
305C22×C3×C4—24no—
306C22×C3×C4C22×D511yescharacteristic
307C22×C3×C4C22×D511yescharacteristic
308C22×C3×C4C22×D511yescharacteristic
309C22×C3×C4C22×D511yescharacteristic
310C22×C3×C4C22×D514yes—
311C22×C3×C4C22×D514yes—
312C22×C3×C4C22×D511yescharacteristic
313C2×#67—58no—
314C2×#67—58no—
315C2×#67—108no—
316C2×#67—204no—
317C22×Dic3—52no—
318C22×Dic3—52no—
319C22×Dic3—52no—
320C22×Dic3—58no—
321C22×Dic3—204no—
322C2×D12—54no—
323C2×D12—104no—
324C2×C4×D3—58no—
325C2×C4×D3—204no—
326C3×C4⋊C4C22×D514yes—
327C3×C22⋊C4C22×D514yes—
328C3×C22⋊C4C22×D514yes—
329C3×C42C22×D514yes—
330#217—58no—
331(C2×D3)⋊C4—58no—
332(C2×D3)⋊C4—58no—
333(C2×D3)⋊C4—58no—
334C4⋊Dic3—54no—
335C4⋊Dic3—54no—
336C4⋊Dic3—54no—
337C4⋊Dic3—58no—
338Dic3⋊C4—58no—
339Dic3⋊C4—58no—
340C4×Dic3—58no—
341C23×C5—22no—
342C23×C5—22no—
343C23×C5—22no—
344C23×C5—22no—
345C23×C5—22no—
346C23×C5#23514yes—
347C23×C5C23×D312yes—
348C23×C5C23×D312yes—
349C23×C5C23×D311yescharacteristic
350C22×D5—34no—
351C22×D5—64no—
352C22×D5—64no—
353C22×D5—64no—
354C5×D4—28no—
355C2×C4×C5—22no—
356C2×C4×C5—22no—
357C2×C4×C5—22no—
358C2×C4×C5—22no—
359C2×C4×C5—28no—
360C2×C4×C5—28no—
361C2×C4×C5#23714yes—
362C2×C4×C5C23×D312yes—
363C2×C4×C5C23×D312yes—
364C2×C4×C5C23×D312yes—
365C2×C4×C5C23×D312yes—
366C2×C4×C5C23×D312yes—
367C22⋊D5—616no—
368C22⋊D5—128no—
369C2×Dic5—34no—
370C2×Dic5—34no—
371C2×Dic5—34no—
372C2×Dic5—64no—
373C2×Dic5—64no—
374C2×Dic5—64no—
375C2×Dic5—128no—
376D20—68no—
377C4×D5—128no—
378C22×D4—152no—
379C22×D4—152no—
380C22×D4—151no—
381C22×D4C2×D1511yescharacteristic
382C23×C4—152no—
383C23×C4—152no—
384C23×C4C2×D1511yescharacteristic
385C23×C4C2×D1511yescharacteristic
386#126—1516no—
387#125—158no—
388#123—1516no—
389#121—1516no—
390#121—158no—
391#121—158no—
392C4×D4—1516no—
393C4×D4—158no—
394C4×D4C2×D1518yescomplemented
395#117—158no—
396C2×C4⋊C4—151no—
397C2×C4⋊C4—151no—
398C2×C4⋊C4—152no—
399C2×C4⋊C4—158no—
400C2×C4⋊C4—151no—
401C2×C4⋊C4—152no—
402C2×C4⋊C4—152no—
403C2×C4⋊C4C2×D1511yescharacteristic
404C2×C22⋊C4—152no—
405C2×C22⋊C4—152no—
406C2×C22⋊C4—152no—
407C2×C22⋊C4—152no—
408C2×C22⋊C4C2×D1511yescharacteristic
409C2×C22⋊C4C2×D1511yescharacteristic
410C2×C42—152no—
411C2×C42C2×D1511yescharacteristic
412C2×C3×C5—24no—
413C2×C3×C5—42no—
414C2×C3×C5#55314yescomplemented
415C2×C3×C5C22×#4111yescharacteristic
416C2×C3×C5C22×#4111yescharacteristic
417C2×C3×C5C2×H(5,𝔽2)11yescharacteristic
418D15—44no—
419C23×C3—22no—
420C23×C3—22no—
421C23×C3—22no—
422C23×C3—22no—
423C23×C3—22no—
424C23×C3#70914yes—
425C23×C3C23×D512yes—
426C23×C3C23×D512yes—
427C23×C3C23×D511yescharacteristic
428C22×D3—54no—
429C22×D3—104no—
430C22×D3—104no—
431C22×D3—104no—
432C3×D4—28no—
433C2×C3×C4—22no—
434C2×C3×C4—22no—
435C2×C3×C4—22no—
436C2×C3×C4—22no—
437C2×C3×C4—28no—
438C2×C3×C4—28no—
439C2×C3×C4#71114yes—
440C2×C3×C4C23×D512yes—
441C2×C3×C4C23×D512yes—
442C2×C3×C4C23×D512yes—
443C2×C3×C4C23×D512yes—
444C2×C3×C4C23×D512yes—
445#67—1016no—
446#67—208no—
447C2×Dic3—54no—
448C2×Dic3—54no—
449C2×Dic3—54no—
450C2×Dic3—104no—
451C2×Dic3—104no—
452C2×Dic3—104no—
453C2×Dic3—208no—
454D12—108no—
455C4×D3—208no—
456C22×C5—22no—
457C22×C5—22no—
458C22×C5—28no—
459C22×C5—22no—
460C22×C5—22no—
461C22×C5—22no—
462C22×C5—44no—
463C22×C5C2×#23514yes—
464C22×C5C2×#23512yes—
465C22×C5C2×#23712yes—
466C22×C5#100912yes—
467C22×C5C24×D311yescharacteristic
468C2×D5—64no—
469C2×D5—64no—
470C2×D5—64no—
471C2×D5—128no—
472C4×C5—22no—
473C4×C5—22no—
474C4×C5—22no—
475C4×C5—22no—
476C4×C5C2×#23714yes—
477Dic5—64no—
478Dic5—64no—
479Dic5—64no—
480C24—152no—
481C24C22×D1511yescharacteristic
482C24C22×D1511yescharacteristic
483C2×D4—24no—
484C2×D4—158no—
485C2×D4—158no—
486C2×D4—154no—
487C2×D4—308no—
488C2×D4—304no—
489C2×D4—604no—
490C2×D4C22×D1514yes—
491C22×C4—24no—
492C22×C4—24no—
493C22×C4—152no—
494C22×C4—152no—
495C22×C4—152no—
496C22×C4—158no—
497C22×C4—158no—
498C22×C4—604no—
499C22×C4—604no—
500C22×C4C22×D1514yes—
501C22×C4C22×D1514yes—
502C22×C4C22×D1511yescharacteristic
503C22×C4C22×D1511yescharacteristic
504C22×C4C22×D1511yescharacteristic
505C22×C4C22×D1511yescharacteristic
506C22×C4C22×D1511yescharacteristic
507C4⋊C4—154no—
508C4⋊C4—154no—
509C4⋊C4—158no—
510C4⋊C4—154no—
511C4⋊C4—158no—
512C4⋊C4—158no—
513C4⋊C4C22×D1514yes—
514C22⋊C4—158no—
515C22⋊C4—158no—
516C22⋊C4—158no—
517C22⋊C4—158no—
518C22⋊C4C22×D1514yes—
519C22⋊C4C22×D1514yes—
520C42—158no—
521C42C22×D1514yes—
522C3×C5C2×#55311yescharacteristic, complemented, Hall
523C22×C3—28no—
524C22×C3—22no—
525C22×C3—22no—
526C22×C3—22no—
527C22×C3—22no—
528C22×C3—22no—
529C22×C3—44no—
530C22×C3C2×#70914yes—
531C22×C3C2×#70912yes—
532C22×C3C2×#71112yes—
533C22×C3#414912yes—
534C22×C3C24×D511yescharacteristic
535C2×D3—104no—
536C2×D3—104no—
537C2×D3—104no—
538C2×D3—208no—
539C3×C4—22no—
540C3×C4—22no—
541C3×C4—22no—
542C3×C4—22no—
543C3×C4C2×#71114yes—
544Dic3—104no—
545Dic3—104no—
546Dic3—104no—
547C2×C5—24no—
548C2×C5—42no—
549C2×C5#556514yescomplemented
550C2×C5C22×#23511yescharacteristic
551C2×C5C22×#23711yescharacteristic
552C2×C5C2×#100911yescharacteristic
553D5—124no—
554C23—22no—
555C23—22no—
556C23—22no—
557C23—22no—
558C23—22no—
559C23—154no—
560C23—304no—
561C23—304no—
562C23—304no—
563C23#681714yes—
564C23C23×D1512yes—
565C23C23×D1512yes—
566C23C23×D1511yescharacteristic
567D4—28no—
568D4—3016no—
569D4—308no—
570D4—608no—
571C2×C4—22no—
572C2×C4—22no—
573C2×C4—22no—
574C2×C4—22no—
575C2×C4—28no—
576C2×C4—28no—
577C2×C4—154no—
578C2×C4—154no—
579C2×C4—154no—
580C2×C4—304no—
581C2×C4—304no—
582C2×C4—304no—
583C2×C4—608no—
584C2×C4—608no—
585C2×C4#681914yes—
586C2×C4C23×D1512yes—
587C2×C4C23×D1512yes—
588C2×C4C23×D1512yes—
589C2×C4C23×D1512yes—
590C2×C4C23×D1512yes—
591C2×C3—24no—
592C2×C3—42no—
593C2×C3#6589914yescomplemented
594C2×C3C22×#70911yescharacteristic
595C2×C3C22×#71111yescharacteristic
596C2×C3C2×#414911yescharacteristic
597D3—204no—
598C5C2×#556511yescharacteristic, complemented, Sylow
599C22—28no—
600C22—22no—
601C22—22no—
602C22—22no—
603C22—22no—
604C22—22no—
605C22—44no—
606C22—304no—
607C22—304no—
608C22—304no—
609C22—608no—
610C22C2×#681714yes—
611C22C2×#681712yes—
612C22C2×#681912yes—
613C22#9210012yes—
614C22C24×D1511yescharacteristic
615C4—22no—
616C4—22no—
617C4—22no—
618C4—22no—
619C4—304no—
620C4—304no—
621C4—304no—
622C4C2×#681914yes—
623C3C2×#6589911yescharacteristic, complemented, Sylow
624C2—24no—
625C2—42no—
626C2—604no—
627C2#1175632414yesdirect product factor
628C2C22×#681711yescharacteristic
629C2C22×#681911yescharacteristic
630C2C2×#9210011yescharacteristic
subgroupG/H
centerC23C23×D15
commutator, cosocleC22×C3×C5C25
FrattiniC22C24×D15
FittingC2×C3×C4×C5×D4C2
socleC23×C3×C5C24
hypercenterC2×C4×D4D15
hypocenterC3×C5C2×#553
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×C4×D4

Complements

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

NH ≅ G/Ntype
1. C2×C3×C4×C5×D4C2semidirect
2. C2×#91828C2semidirect
3. C2×#91819C2semidirect
4. C2×#91819C2semidirect
5. C2×#91815C2semidirect
6. C22×C4⋊Dic15C2semidirect
7. #11756324C2direct (N × H)
8. C2×#91787C2semidirect
9. C2×#91772C2semidirect
10. C2×#91770C2semidirect
11. C2×C4×D60C2semidirect
12. C2×#91759C2semidirect
22. C3×C4×C5×D4C22semidirect
28. #91828C22semidirect
29. #91819C22semidirect
30. #91819C22semidirect
34. #91815C22semidirect
39. C2×C4⋊Dic15C22semidirect
43. #91787C22semidirect
44. #91772C22semidirect
45. #91770C22semidirect
46. C4×D60C22semidirect
47. #91759C22semidirect
49. C2×C4×C5×D4D3semidirect
103. C2×C3×C4×D4D5semidirect
123. C4×C5×D4C2×D3semidirect
195. C3×C4×D4C2×D5semidirect
272. C2×C4×D4D15semidirect
394. C4×D4C2×D15semidirect
414. C2×C3×C5#553semidirect
522. C3×C5C2×#553semidirect
549. C2×C5#5565semidirect
593. C2×C3#65899semidirect
598. C5C2×#5565semidirect
623. C3C2×#65899semidirect
627. C2#11756324direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C4×C5×D42
C2×#918282
C2×#918192
C2×#918152
C22×C4⋊Dic152
#117563242
C2×#917872
C2×#917722
C2×#917702
C2×C4×D602
C2×#917592
C2×#658993
C2×#55655