Finite Group Databasebeta

← C2×#11756373 #14483606

Subgroups — C2×#11756373 #14483606

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×C5×#121C211yesmaximal, characteristic, complemented
2C2×#91827C211yesmaximal, characteristic, complemented
3C2×#91827C212yesmaximal, complemented
4C2×#91824C212yesmaximal, complemented
5C2×D4×Dic15C211yesmaximal, characteristic, complemented
6C2×C4×#1220C211yesmaximal, characteristic, complemented
7C2×#91815C211yesmaximal, characteristic, complemented
8#11756373C2116yesmaximal, direct product factor
9C2×C4⋊C4×D15C211yesmaximal, characteristic, complemented
10C2×#91779C211yesmaximal, characteristic, complemented
11C2×#91770C212yesmaximal, complemented
12C2×#91768C212yesmaximal, complemented
13C2×#65948—31nomaximal, Hall
14C22×C3×C5×D4C2211yescharacteristic
15C22×C3×C5×D4C2212yes—
16C23×C3×C4×C5C2211yescharacteristic
17C22×#1220C2211yescharacteristic
18C22×#1220C2212yes—
19C23×Dic15C2212yes—
20C22×C4×D15C2211yescharacteristic
21C22×C4×D15C2212yes—
22C3×C5×#121C2218yescomplemented
23C2×C3×C5×C4⋊C4C2211yescharacteristic
24C2×C3×C5×C22⋊C4C2212yes—
25C2×#6719C2212yes—
26C2×#6719C2211yescharacteristic
27C2×#6719C2212yes—
28C2×#6719C2212yes—
29#91827C22116yescomplemented
30#91827C2218yescomplemented
31#91824C22116yescomplemented
32D4×Dic15C2218yescomplemented
33C4×#1220C2218yescomplemented
34C2×#6714C2211yescharacteristic
35#91815C2218yescomplemented
36C2×C4⋊Dic15C2211yescharacteristic
37C2×C4⋊Dic15C2211yescharacteristic
38C2×C4⋊Dic15C2212yes—
39C2×#6712C2211yescharacteristic
40C2×#6712C2212yes—
41C2×#6712C2212yes—
42C2×C4×Dic15C2212yes—
43C2×C4×Dic15C2211yescharacteristic
44C4⋊C4×D15C2218yescomplemented
45#91779C2218yescomplemented
46#91770C22116yescomplemented
47#91768C22116yescomplemented
48C2×#5614—51nomaximal, Hall
49C2×C5×#121D311yescharacteristic, complemented
50C2×#4084—32no—
51C2×#4084—31no—
52C2×#4081—32no—
53C2×D4×Dic5—31no—
54C2×C4×C22⋊D5—31no—
55C2×#4072—31no—
56#65948—316no—
57C2×D5×C4⋊C4—31no—
58C2×#4036—31no—
59C2×#4027—32no—
60C2×#4025—32no—
61C24×C3×C5C2311yescharacteristic
62C24×C3×C5C2312yes—
63C23×D15C2311yescharacteristic
64C2×C3×C5×D4—24no—
65C2×C3×C5×D4—44no—
66C2×C3×C5×D4C2314yes—
67C2×C3×C5×D4C2318yes—
68C22×C3×C4×C5—24no—
69C22×C3×C4×C5C2311yescharacteristic
70C22×C3×C4×C5C2312yes—
71C22×C3×C4×C5C2314yes—
72C22×C3×C4×C5C2311yescharacteristic
73C2×#1220—24no—
74C2×#1220—28no—
75C2×#1220C2318yes—
76C2×#1220C2314yes—
77C22×Dic15—44no—
78C22×Dic15C2311yescharacteristic
79C22×Dic15C2312yes—
80C22×Dic15C2311yescharacteristic
81C22×Dic15C2312yes—
82C22×Dic15C2311yescharacteristic
83C22×Dic15C2318yes—
84C2×C4×D15—24no—
85C2×C4×D15—44no—
86C2×C4×D15C2314yes—
87C2×C4×D15C2318yes—
88C3×C5×C4⋊C4C2314yes—
89C3×C5×C22⋊C4C2318yes—
90#6719C2318yes—
91#6719C2318yes—
92#6719C2318yes—
93#6719C2314yes—
94#6714C2314yes—
95C4⋊Dic15C2318yes—
96C4⋊Dic15C2314yes—
97C4⋊Dic15C2314yes—
98#6712C2314yes—
99#6712C2318yes—
100#6712C2318yes—
101C4×Dic15C2318yes—
102C4×Dic15C2314yes—
103C2×C3×#121D511yescharacteristic, complemented
104C2×#938—52no—
105C2×#938—51no—
106C2×#935—52no—
107C2×D4×Dic3—51no—
108C2×C4×#67—51no—
109C2×#926—51no—
110#5614—516no—
111C2×D3×C4⋊C4—51no—
112C2×#890—51no—
113C2×#67⋊C4—52no—
114C2×#879—52no—
115C22×C5×D4C2×D311yescharacteristic
116C22×C5×D4C2×D312yes—
117C23×C4×C5C2×D311yescharacteristic
118C22×C22⋊D5—32no—
119C22×C22⋊D5—31no—
120C23×Dic5—32no—
121C22×C4×D5—31no—
122C22×C4×D5—32no—
123C5×#121C2×D318yescomplemented
124C2×C5×C4⋊C4C2×D311yescharacteristic
125C2×C5×C22⋊C4C2×D312yes—
126C2×C22⋊Dic5—32no—
127C2×C22⋊Dic5—32no—
128C2×C22⋊Dic5—32no—
129C2×C22⋊Dic5—31no—
130#4084—316no—
131#4084—38no—
132#4081—316no—
133D4×Dic5—38no—
134C4×C22⋊D5—38no—
135C2×#685—31no—
136#4072—38no—
137C2×C4⋊Dic5—32no—
138C2×C4⋊Dic5—31no—
139C2×C4⋊Dic5—31no—
140C2×#683—31no—
141C2×#683—32no—
142C2×#683—32no—
143C2×C4×Dic5—32no—
144C2×C4×Dic5—31no—
145D5×C4⋊C4—38no—
146#4036—38no—
147#4027—316no—
148#4025—316no—
149C2×#553—151noSylow
150C23×C3×C5—22no—
151C23×C3×C5—24no—
152C23×C3×C5—22no—
153C23×C3×C5—24no—
154C23×C3×C5—24no—
155C23×C3×C5—22no—
156C23×C3×C5C2412yes—
157C23×C3×C5C2414yes—
158C23×C3×C5C2411yescharacteristic
159C22×D15—22no—
160C22×D15—22no—
161C22×D15#4114yes—
162C22×D15C2412yes—
163C3×C5×D4—28no—
164C3×C5×D4—48no—
165C2×C3×C4×C5—22no—
166C2×C3×C4×C5—24no—
167C2×C3×C4×C5—28no—
168C2×C3×C4×C5#4114yes—
169C2×C3×C4×C5C2412yes—
170C2×C3×C4×C5C2414yes—
171C2×C3×C4×C5C2412yes—
172#1220—216no—
173#1220—28no—
174C2×Dic15—22no—
175C2×Dic15—24no—
176C2×Dic15—22no—
177C2×Dic15—24no—
178C2×Dic15—22no—
179C2×Dic15—48no—
180C2×Dic15C2412yes—
181C2×Dic15C2414yes—
182C2×Dic15C2412yes—
183C2×Dic15C2414yes—
184C2×Dic15C2412yes—
185C4×D15—28no—
186C4×D15—48no—
187C22×C3×D4C2×D511yescharacteristic
188C22×C3×D4C2×D512yes—
189C23×C3×C4C2×D511yescharacteristic
190C22×#67—52no—
191C22×#67—51no—
192C23×Dic3—52no—
193C22×C4×D3—51no—
194C22×C4×D3—52no—
195C3×#121C2×D518yescomplemented
196C2×C3×C4⋊C4C2×D511yescharacteristic
197C2×C3×C22⋊C4C2×D512yes—
198C2×#217—52no—
199C2×#217—52no—
200C2×#217—52no—
201C2×#217—51no—
202#938—516no—
203#938—58no—
204#935—516no—
205D4×Dic3—58no—
206C4×#67—58no—
207C2×(C2×D3)⋊C4—51no—
208#926—58no—
209C2×C4⋊Dic3—52no—
210C2×C4⋊Dic3—51no—
211C2×C4⋊Dic3—51no—
212C2×Dic3⋊C4—51no—
213C2×Dic3⋊C4—52no—
214C2×Dic3⋊C4—52no—
215C2×C4×Dic3—52no—
216C2×C4×Dic3—51no—
217D3×C4⋊C4—58no—
218#890—58no—
219#67⋊C4—516no—
220#879—516no—
221C24×C5C22×D311yescharacteristic
222C24×C5C22×D312yes—
223C23×D5—31no—
224C2×C5×D4—24no—
225C2×C5×D4—44no—
226C2×C5×D4C22×D314yes—
227C2×C5×D4C22×D318yes—
228C22×C4×C5—24no—
229C22×C4×C5C22×D311yescharacteristic
230C22×C4×C5C22×D312yes—
231C22×C4×C5C22×D314yes—
232C22×C4×C5C22×D311yescharacteristic
233C2×C22⋊D5—38no—
234C2×C22⋊D5—34no—
235C2×C22⋊D5—68no—
236C2×C22⋊D5—64no—
237C22×Dic5—31no—
238C22×Dic5—32no—
239C22×Dic5—31no—
240C22×Dic5—32no—
241C22×Dic5—31no—
242C22×Dic5—38no—
243C22×Dic5—124no—
244C2×C4×D5—34no—
245C2×C4×D5—38no—
246C2×C4×D5—64no—
247C2×C4×D5—124no—
248C5×C4⋊C4C22×D314yes—
249C5×C22⋊C4C22×D318yes—
250C22⋊Dic5—38no—
251C22⋊Dic5—38no—
252C22⋊Dic5—38no—
253C22⋊Dic5—34no—
254#685—34no—
255C4⋊Dic5—38no—
256C4⋊Dic5—34no—
257C4⋊Dic5—34no—
258#683—34no—
259#683—38no—
260#683—38no—
261C4×Dic5—38no—
262C4×Dic5—34no—
263#553—1516no—
264C2×#126—152no—
265C2×#125—151no—
266C2×#123—152no—
267C2×#121—152no—
268C2×#121—151no—
269C2×#121D1511yescharacteristic, complemented
270C2×C4×D4—151no—
271C2×C4×D4—151no—
272C2×C4×D4—152no—
273C2×#117—151no—
274C22×C4⋊C4—151no—
275C22×C3×C5—22no—
276C22×C3×C5—24no—
277C22×C3×C5—22no—
278C22×C3×C5—24no—
279C22×C3×C5—24no—
280C22×C3×C5—22no—
281C22×C3×C5—44no—
282C22×C3×C5—48no—
283C22×C3×C5C2×#4112yes—
284C22×C3×C5C2×#4112yes—
285C22×C3×C5H(5,𝔽2)12yes—
286C22×C3×C5C2511yescharacteristic
287C2×D15—28no—
288C2×D15—22no—
289C2×D15—22no—
290C2×D15C2×#4114yes—
291C3×C4×C5—22no—
292C3×C4×C5—24no—
293C3×C4×C5C2×#4114yes—
294Dic15—22no—
295Dic15—24no—
296Dic15—22no—
297Dic15—24no—
298Dic15—22no—
299C24×C3C22×D511yescharacteristic
300C24×C3C22×D512yes—
301C23×D3—51no—
302C2×C3×D4—24no—
303C2×C3×D4—44no—
304C2×C3×D4C22×D514yes—
305C2×C3×D4C22×D518yes—
306C22×C3×C4—24no—
307C22×C3×C4C22×D511yescharacteristic
308C22×C3×C4C22×D512yes—
309C22×C3×C4C22×D514yes—
310C22×C3×C4C22×D511yescharacteristic
311C2×#67—58no—
312C2×#67—54no—
313C2×#67—108no—
314C2×#67—104no—
315C22×Dic3—51no—
316C22×Dic3—52no—
317C22×Dic3—51no—
318C22×Dic3—52no—
319C22×Dic3—51no—
320C22×Dic3—58no—
321C22×Dic3—204no—
322C2×C4×D3—54no—
323C2×C4×D3—58no—
324C2×C4×D3—104no—
325C2×C4×D3—204no—
326C3×C4⋊C4C22×D514yes—
327C3×C22⋊C4C22×D518yes—
328#217—58no—
329#217—58no—
330#217—58no—
331#217—54no—
332(C2×D3)⋊C4—54no—
333C4⋊Dic3—58no—
334C4⋊Dic3—54no—
335C4⋊Dic3—54no—
336Dic3⋊C4—54no—
337Dic3⋊C4—58no—
338Dic3⋊C4—58no—
339C4×Dic3—58no—
340C4×Dic3—54no—
341C23×C5—22no—
342C23×C5—24no—
343C23×C5—24no—
344C23×C5—22no—
345C23×C5—22no—
346C23×C5—24no—
347C23×C5C23×D312yes—
348C23×C5C23×D314yes—
349C23×C5C23×D311yescharacteristic
350C22×D5—34no—
351C22×D5—32no—
352C22×D5—62no—
353C22×D5—62no—
354C5×D4—28no—
355C5×D4—48no—
356C2×C4×C5—22no—
357C2×C4×C5—24no—
358C2×C4×C5—28no—
359C2×C4×C5#23714yes—
360C2×C4×C5C23×D312yes—
361C2×C4×C5C23×D314yes—
362C2×C4×C5C23×D312yes—
363C22⋊D5—616no—
364C22⋊D5—68no—
365C2×Dic5—32no—
366C2×Dic5—34no—
367C2×Dic5—32no—
368C2×Dic5—34no—
369C2×Dic5—32no—
370C2×Dic5—62no—
371C2×Dic5—64no—
372C2×Dic5—62no—
373C2×Dic5—62no—
374C2×Dic5—64no—
375C2×Dic5—128no—
376C4×D5—68no—
377C4×D5—128no—
378C22×D4—151no—
379C22×D4—152no—
380C22×D4C2×D1511yescharacteristic
381C22×D4C2×D1512yes—
382C23×C4—152no—
383C23×C4—151no—
384C23×C4—152no—
385C23×C4C2×D1511yescharacteristic
386#126—1516no—
387#125—158no—
388#123—1516no—
389#121—1516no—
390#121—158no—
391#121C2×D1518yescomplemented
392C4×D4—158no—
393C4×D4—158no—
394C4×D4—1516no—
395#117—158no—
396C2×C4⋊C4—151no—
397C2×C4⋊C4—152no—
398C2×C4⋊C4—151no—
399C2×C4⋊C4—152no—
400C2×C4⋊C4—151no—
401C2×C4⋊C4—152no—
402C2×C4⋊C4—158no—
403C2×C4⋊C4C2×D1511yescharacteristic
404C2×C22⋊C4—152no—
405C2×C22⋊C4—152no—
406C2×C22⋊C4—151no—
407C2×C22⋊C4—152no—
408C2×C22⋊C4—151no—
409C2×C22⋊C4C2×D1512yes—
410C2×C42—152no—
411C2×C42—151no—
412C2×C3×C5—42no—
413C2×C3×C5—44no—
414C2×C3×C5#55314yescomplemented
415C2×C3×C5C22×#4111yescharacteristic
416C2×C3×C5C22×#4111yescharacteristic
417C2×C3×C5C2×H(5,𝔽2)11yescharacteristic
418D15—24no—
419C23×C3—22no—
420C23×C3—24no—
421C23×C3—24no—
422C23×C3—22no—
423C23×C3—22no—
424C23×C3—24no—
425C23×C3C23×D512yes—
426C23×C3C23×D514yes—
427C23×C3C23×D511yescharacteristic
428C22×D3—54no—
429C22×D3—52no—
430C22×D3—102no—
431C22×D3—102no—
432C3×D4—28no—
433C3×D4—48no—
434C2×C3×C4—22no—
435C2×C3×C4—24no—
436C2×C3×C4—28no—
437C2×C3×C4#71114yes—
438C2×C3×C4C23×D512yes—
439C2×C3×C4C23×D514yes—
440C2×C3×C4C23×D512yes—
441#67—1016no—
442#67—108no—
443C2×Dic3—52no—
444C2×Dic3—54no—
445C2×Dic3—52no—
446C2×Dic3—54no—
447C2×Dic3—52no—
448C2×Dic3—102no—
449C2×Dic3—104no—
450C2×Dic3—102no—
451C2×Dic3—104no—
452C2×Dic3—102no—
453C2×Dic3—208no—
454C4×D3—108no—
455C4×D3—208no—
456C22×C5—22no—
457C22×C5—24no—
458C22×C5—24no—
459C22×C5—22no—
460C22×C5—22no—
461C22×C5—24no—
462C22×C5—44no—
463C22×C5—48no—
464C22×C5C2×#23712yes—
465C22×C5#100412yes—
466C22×C5D3×#4112yes—
467C22×C5C24×D311yescharacteristic
468C2×D5—34no—
469C2×D5—62no—
470C2×D5—62no—
471C2×D5—68no—
472C4×C5—22no—
473C4×C5—24no—
474C4×C5C2×#23714yes—
475Dic5—62no—
476Dic5—64no—
477Dic5—62no—
478Dic5—64no—
479Dic5—62no—
480C24—151no—
481C24C22×D1511yescharacteristic
482C24C22×D1512yes—
483C2×D4—24no—
484C2×D4—44no—
485C2×D4—154no—
486C2×D4—158no—
487C2×D4—304no—
488C2×D4—308no—
489C2×D4C22×D1514yes—
490C2×D4C22×D1518yes—
491C22×C4—24no—
492C22×C4—152no—
493C22×C4—151no—
494C22×C4—152no—
495C22×C4—151no—
496C22×C4—151no—
497C22×C4—158no—
498C22×C4—154no—
499C22×C4—158no—
500C22×C4—304no—
501C22×C4—604no—
502C22×C4—604no—
503C22×C4C22×D1514yes—
504C22×C4C22×D1511yescharacteristic
505C22×C4C22×D1512yes—
506C22×C4C22×D1511yescharacteristic
507C4⋊C4—154no—
508C4⋊C4—158no—
509C4⋊C4—154no—
510C4⋊C4—158no—
511C4⋊C4—154no—
512C4⋊C4—158no—
513C4⋊C4C22×D1514yes—
514C22⋊C4—158no—
515C22⋊C4—158no—
516C22⋊C4—154no—
517C22⋊C4—158no—
518C22⋊C4—154no—
519C22⋊C4C22×D1518yes—
520C42—158no—
521C42—154no—
522C3×C5C2×#55311yescharacteristic, complemented, Hall
523C22×C3—22no—
524C22×C3—24no—
525C22×C3—24no—
526C22×C3—22no—
527C22×C3—22no—
528C22×C3—24no—
529C22×C3—44no—
530C22×C3—48no—
531C22×C3C2×#71112yes—
532C22×C3#414412yes—
533C22×C3D5×#4112yes—
534C22×C3C24×D511yescharacteristic
535C2×D3—54no—
536C2×D3—108no—
537C2×D3—102no—
538C2×D3—102no—
539C3×C4—22no—
540C3×C4—24no—
541C3×C4C2×#71114yes—
542Dic3—102no—
543Dic3—104no—
544Dic3—102no—
545Dic3—104no—
546Dic3—102no—
547C2×C5—42no—
548C2×C5—44no—
549C2×C5#561414yescomplemented
550C2×C5C22×#23711yescharacteristic
551C2×C5C2×#100411yescharacteristic
552C2×C5C2×D3×#4111yescharacteristic
553D5—64no—
554C23—22no—
555C23—24no—
556C23—24no—
557C23—22no—
558C23—22no—
559C23—24no—
560C23—154no—
561C23—152no—
562C23—302no—
563C23—302no—
564C23C23×D1512yes—
565C23C23×D1514yes—
566C23C23×D1511yescharacteristic
567D4—28no—
568D4—48no—
569D4—308no—
570D4—3016no—
571C2×C4—22no—
572C2×C4—24no—
573C2×C4—28no—
574C2×C4—154no—
575C2×C4—152no—
576C2×C4—154no—
577C2×C4—152no—
578C2×C4—152no—
579C2×C4—302no—
580C2×C4—304no—
581C2×C4—302no—
582C2×C4—304no—
583C2×C4—302no—
584C2×C4—308no—
585C2×C4—608no—
586C2×C4—608no—
587C2×C4#681914yes—
588C2×C4C23×D1512yes—
589C2×C4C23×D1514yes—
590C2×C4C23×D1512yes—
591C2×C3—42no—
592C2×C3—44no—
593C2×C3#6594814yescomplemented
594C2×C3C22×#71111yescharacteristic
595C2×C3C2×#414411yescharacteristic
596C2×C3C2×D5×#4111yescharacteristic
597D3—104no—
598C5C2×#561411yescharacteristic, complemented, Sylow
599C22—22no—
600C22—24no—
601C22—24no—
602C22—22no—
603C22—22no—
604C22—24no—
605C22—44no—
606C22—48no—
607C22—154no—
608C22—302no—
609C22—302no—
610C22—308no—
611C22C2×#681912yes—
612C22#9209512yes—
613C22#41×D1512yes—
614C22C24×D1511yescharacteristic
615C4—22no—
616C4—24no—
617C4—302no—
618C4—304no—
619C4—302no—
620C4—304no—
621C4—302no—
622C4C2×#681914yes—
623C3C2×#6594811yescharacteristic, complemented, Sylow
624C2—42no—
625C2—44no—
626C2—304no—
627C2#1175637314yesdirect product factor
628C2C22×#681911yescharacteristic
629C2C2×#9209511yescharacteristic
630C2C2×#41×D1511yescharacteristic
subgroupG/H
centerC23C23×D15
commutator, cosocleC22×C3×C5C25
FrattiniC22C24×D15
FittingC2×C3×C5×#121C2
socleC23×C3×C5C24
hypercenterC2×#121D15
hypocenterC3×C5C2×#553
LCSG ▷ C22×C3×C5 ▷ C3×C5
UCSC1 ◁ C23 ◁ C2×#121

Complements

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

NH ≅ G/Ntype
1. C2×C3×C5×#121C2semidirect
2. C2×#91827C2semidirect
3. C2×#91827C2semidirect
4. C2×#91824C2semidirect
5. C2×D4×Dic15C2semidirect
6. C2×C4×#1220C2semidirect
7. C2×#91815C2semidirect
8. #11756373C2direct (N × H)
9. C2×C4⋊C4×D15C2semidirect
10. C2×#91779C2semidirect
11. C2×#91770C2semidirect
12. C2×#91768C2semidirect
22. C3×C5×#121C22semidirect
29. #91827C22semidirect
30. #91827C22semidirect
31. #91824C22semidirect
32. D4×Dic15C22semidirect
33. C4×#1220C22semidirect
35. #91815C22semidirect
44. C4⋊C4×D15C22semidirect
45. #91779C22semidirect
46. #91770C22semidirect
47. #91768C22semidirect
49. C2×C5×#121D3semidirect
103. C2×C3×#121D5semidirect
123. C5×#121C2×D3semidirect
195. C3×#121C2×D5semidirect
269. C2×#121D15semidirect
391. #121C2×D15semidirect
414. C2×C3×C5#553semidirect
522. C3×C5C2×#553semidirect
549. C2×C5#5614semidirect
593. C2×C3#65948semidirect
598. C5C2×#5614semidirect
623. C3C2×#65948semidirect
627. C2#11756373direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C3×C5×#1212
C2×#918272
C2×#918242
C2×D4×Dic152
C2×C4×#12202
C2×#918152
#117563732
C2×C4⋊C4×D152
C2×#917792
C2×#917702
C2×#917682
C2×#659483
C2×#56145