Finite Group Databasebeta

← C2×#11753775 #14483726

Subgroups — C2×#11753775 #14483726

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×#91779C211yesmaximal, characteristic, complemented
2C2×C5×D3×C4⋊C4C211yesmaximal, characteristic, complemented
3C2×C3×#4041C211yesmaximal, characteristic, complemented
4C2×#91456C212yesmaximal, complemented
5C2×#91456C211yesmaximal, characteristic, complemented
6C2×#91441C212yesmaximal, complemented
7C2×C4×#1201C211yesmaximal, characteristic, complemented
8C2×#91434C211yesmaximal, characteristic, complemented
9C2×#91407C212yesmaximal, complemented
10C2×#91362C212yesmaximal, complemented
11C2×#91358C211yesmaximal, characteristic, complemented
12#11753775C2116yesmaximal, direct product factor
13C2×#65848—31nomaximal, Hall
14C22×C4×C5×D3C2211yescharacteristic
15C22×C4×C5×D3C2212yes—
16C22×C3×D20C2211yescharacteristic
17C22×C3×D20C2212yes—
18C22×C3×C4×D5C2211yescharacteristic
19C22×#1201C2211yescharacteristic
20C22×#1201C2212yes—
21C22×D5×Dic3C2212yes—
22C2×C3×C5×C4⋊C4C2211yescharacteristic
23C2×C4⋊Dic15C2212yes—
24C2×C4⋊Dic15C2211yescharacteristic
25C2×#6712C2212yes—
26C2×C4×Dic15C2211yescharacteristic
27#91779C2218yescomplemented
28C2×C5×C4⋊Dic3C2211yescharacteristic
29C2×C5×Dic3⋊C4C2212yes—
30C5×D3×C4⋊C4C2218yescomplemented
31C2×C3×#685C2212yes—
32C3×#4041C2218yescomplemented
33C2×#6669C2211yescharacteristic
34C2×(C2×D3)⋊Dic5C2211yescharacteristic
35C2×#6665C2212yes—
36C2×#6665C2212yes—
37C2×#6665C2211yescharacteristic
38C2×#6665C2212yes—
39C2×Dic3×Dic5C2212yes—
40#91456C22116yescomplemented
41#91456C2218yescomplemented
42#91441C22116yescomplemented
43C4×#1201C2218yescomplemented
44#91434C2218yescomplemented
45#91407C22116yescomplemented
46#91362C22116yescomplemented
47#91358C2218yescomplemented
48C2×#5614—51nomaximal, Hall
49C22×C5×C4⋊C4—31no—
50C2×#4076—31no—
51C2×#4076—32no—
52C2×C4×C22⋊D5—32no—
53C2×C4×C22⋊D5—31no—
54#65848—316no—
55C2×#4044—32no—
56C2×#4041D311yescharacteristic, complemented
57C2×#4040—32no—
58C2×#4039—31no—
59C2×#4038—31no—
60C2×#4036—31no—
61C23×C5×D3C2311yescharacteristic
62C23×C3×D5C2312yes—
63C23×C3×D5C2311yescharacteristic
64C22×C3×C4×C5C2312yes—
65C22×C3×C4×C5C2311yescharacteristic
66C22×Dic15C2311yescharacteristic
67C22×Dic15C2312yes—
68C22×Dic15C2311yescharacteristic
69C22×C5×Dic3C2311yescharacteristic
70C22×C5×Dic3C2312yes—
71C2×C4×C5×D3—24no—
72C2×C4×C5×D3—44no—
73C2×C4×C5×D3C2314yes—
74C2×C4×C5×D3C2318yes—
75C22×C3×Dic5C2311yescharacteristic
76C2×C3×D20—24no—
77C2×C3×D20—44no—
78C2×C3×D20C2314yes—
79C2×C3×D20C2318yes—
80C2×C3×C4×D5—24no—
81C2×C3×C4×D5C2314yes—
82C2×#1201—24no—
83C2×#1201—28no—
84C2×#1201C2314yes—
85C2×#1201C2318yes—
86C2×D5×Dic3—44no—
87C2×D5×Dic3C2318yes—
88C3×C5×C4⋊C4C2314yes—
89C4⋊Dic15C2314yes—
90C4⋊Dic15C2318yes—
91#6712C2318yes—
92C4×Dic15C2314yes—
93C5×C4⋊Dic3C2314yes—
94C5×Dic3⋊C4C2318yes—
95C3×#685C2318yes—
96#6669C2314yes—
97(C2×D3)⋊Dic5C2314yes—
98#6665C2314yes—
99#6665C2318yes—
100#6665C2318yes—
101#6665C2318yes—
102Dic3×Dic5C2318yes—
103C2×C3×#121—51no—
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⋊C4D511yescharacteristic, complemented
112C2×#890—51no—
113C2×#67⋊C4—52no—
114C2×#879—52no—
115C23×C4×C5—31no—
116C23×C4×C5—32no—
117C22×C22⋊D5—32no—
118C22×C22⋊D5—31no—
119C22×D20C2×D311yescharacteristic
120C22×D20C2×D312yes—
121C22×C4×D5—32no—
122C22×C4×D5C2×D311yescharacteristic
123C2×C5×C4⋊C4—31no—
124C2×C5×C4⋊C4—32no—
125C2×C5×C4⋊C4—38no—
126C2×C5×C4⋊C4C2×D311yescharacteristic
127C2×C22⋊Dic5—31no—
128#4076—38no—
129#4076—316no—
130C4×C22⋊D5—38no—
131C4×C22⋊D5—316no—
132C2×#685—32no—
133C2×#685—32no—
134C2×#685—31no—
135C2×#685—32no—
136C2×#685C2×D312yes—
137C2×C4⋊Dic5—31no—
138C2×C4⋊Dic5—32no—
139C2×#683—32no—
140C2×#683—31no—
141C2×C4×Dic5—31no—
142C2×C4×Dic5—32no—
143#4044—316no—
144#4041C2×D318yescomplemented
145#4040—316no—
146#4039—38no—
147#4038—38no—
148#4036—38no—
149C2×#553—151noSylow
150C23×C3×C5C2411yescharacteristic
151C22×C5×D3—22no—
152C22×C5×D3—22no—
153C22×C5×D3#4114yes—
154C22×C5×D3C2412yes—
155C22×C3×D5—24no—
156C22×C3×D5—22no—
157C22×C3×D5—22no—
158C22×C3×D5—24no—
159C22×C3×D5—24no—
160C22×C3×D5—22no—
161C22×C3×D5C2412yes—
162C22×C3×D5C2414yes—
163C2×C3×C4×C5—24no—
164C2×C3×C4×C5#4114yes—
165C2×C3×C4×C5C2414yes—
166C2×C3×C4×C5C2412yes—
167C2×Dic15—24no—
168C2×Dic15—22no—
169C2×Dic15—22no—
170C2×Dic15C2414yes—
171C2×Dic15C2412yes—
172C2×Dic15C2412yes—
173C2×C5×Dic3—22no—
174C2×C5×Dic3—24no—
175C2×C5×Dic3C2414yes—
176C2×C5×Dic3C2412yes—
177C4×C5×D3—28no—
178C4×C5×D3—48no—
179C2×C3×Dic5—22no—
180C2×C3×Dic5C2412yes—
181C3×D20—28no—
182C3×D20—48no—
183C3×C4×D5—28no—
184#1201—28no—
185#1201—216no—
186D5×Dic3—48no—
187C22×C3×D4—51no—
188C22×C3×D4—52no—
189C23×C3×C4—51no—
190C22×#67—52no—
191C22×#67—51no—
192C23×Dic3—52no—
193C22×C4×D3C2×D511yescharacteristic
194C22×C4×D3C2×D512yes—
195C3×#121—58no—
196C2×C3×C4⋊C4C2×D511yescharacteristic
197C2×C3×C22⋊C4—52no—
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⋊Dic3C2×D511yescharacteristic
212C2×Dic3⋊C4—52no—
213C2×Dic3⋊C4—51no—
214C2×Dic3⋊C4C2×D512yes—
215C2×C4×Dic3—51no—
216C2×C4×Dic3—52no—
217D3×C4⋊C4C2×D518yescomplemented
218#890—58no—
219#67⋊C4—516no—
220#879—516no—
221C24×C5—31no—
222C23×D5C22×D312yes—
223C23×D5C22×D311yescharacteristic
224C22×C4×C5—32no—
225C22×C4×C5—31no—
226C22×C4×C5—34no—
227C22×C4×C5—38no—
228C22×C4×C5—64no—
229C22×C4×C5—124no—
230C22×C4×C5C22×D312yes—
231C22×C4×C5C22×D311yescharacteristic
232C2×C22⋊D5—34no—
233C2×C22⋊D5—38no—
234C2×C22⋊D5—68no—
235C2×C22⋊D5—64no—
236C22×Dic5—31no—
237C22×Dic5—32no—
238C22×Dic5—31no—
239C22×Dic5C22×D311yescharacteristic
240C2×D20—24no—
241C2×D20—44no—
242C2×D20C22×D314yes—
243C2×D20C22×D318yes—
244C2×C4×D5—24no—
245C2×C4×D5—38no—
246C2×C4×D5—124no—
247C2×C4×D5C22×D314yes—
248C5×C4⋊C4—38no—
249C5×C4⋊C4—34no—
250C5×C4⋊C4C22×D314yes—
251C22⋊Dic5—34no—
252#685—38no—
253#685—38no—
254#685—34no—
255#685—38no—
256#685C22×D318yes—
257C4⋊Dic5—34no—
258C4⋊Dic5—38no—
259#683—38no—
260#683—34no—
261C4×Dic5—34no—
262C4×Dic5—38no—
263#553—1516no—
264C2×#126—152no—
265C2×#125—151no—
266C2×#123—152no—
267C2×#121—51no—
268C2×#121—152no—
269C2×#121—151no—
270C2×C4×D4—152no—
271C2×C4×D4—151no—
272C2×C4×D4—151no—
273C2×#117—151no—
274C22×C4⋊C4—31no—
275C22×C3×C5C2×#4112yes—
276C22×C3×C5C2×#4112yes—
277C22×C3×C5H(5,𝔽2)12yes—
278C22×C3×C5C2511yescharacteristic
279C2×C5×D3—28no—
280C2×C5×D3—22no—
281C2×C5×D3—22no—
282C2×C5×D3C2×#4114yes—
283C2×C3×D5—22no—
284C2×C3×D5—24no—
285C2×C3×D5—22no—
286C2×C3×D5—24no—
287C2×C3×D5—24no—
288C2×C3×D5—22no—
289C2×C3×D5—44no—
290C2×C3×D5—48no—
291C3×C4×C5—24no—
292C3×C4×C5C2×#4114yes—
293Dic15—24no—
294Dic15—22no—
295Dic15—22no—
296C3×Dic5—22no—
297C5×Dic3—22no—
298C5×Dic3—24no—
299C24×C3—51no—
300C24×C3—52no—
301C23×D3C22×D511yescharacteristic
302C2×C3×D4—54no—
303C2×C3×D4—58no—
304C2×C3×D4—104no—
305C2×C3×D4—204no—
306C22×C3×C4—51no—
307C22×C3×C4—54no—
308C22×C3×C4—104no—
309C22×C3×C4C22×D512yes—
310C22×C3×C4C22×D511yescharacteristic
311C2×#67—58no—
312C2×#67—54no—
313C2×#67—108no—
314C2×#67—104no—
315C22×Dic3—51no—
316C22×Dic3—51no—
317C22×Dic3—52no—
318C22×Dic3—58no—
319C22×Dic3—204no—
320C22×Dic3C22×D512yes—
321C22×Dic3C22×D511yescharacteristic
322C2×C4×D3—24no—
323C2×C4×D3—44no—
324C2×C4×D3C22×D514yes—
325C2×C4×D3C22×D518yes—
326C3×C4⋊C4C22×D514yes—
327C3×C22⋊C4—58no—
328#217—58no—
329#217—58no—
330#217—58no—
331#217—54no—
332(C2×D3)⋊C4—54no—
333C4⋊Dic3—58no—
334C4⋊Dic3—54no—
335C4⋊Dic3C22×D514yes—
336Dic3⋊C4—58no—
337Dic3⋊C4—54no—
338Dic3⋊C4C22×D518yes—
339C4×Dic3—54no—
340C4×Dic3—58no—
341C23×C5—34no—
342C23×C5—32no—
343C23×C5—62no—
344C23×C5—62no—
345C23×C5C23×D311yescharacteristic
346C22×D5—22no—
347C22×D5—24no—
348C22×D5—24no—
349C22×D5—22no—
350C22×D5—24no—
351C22×D5—22no—
352C22×D5C23×D312yes—
353C22×D5C23×D314yes—
354C2×C4×C5—24no—
355C2×C4×C5—34no—
356C2×C4×C5—32no—
357C2×C4×C5—62no—
358C2×C4×C5—64no—
359C2×C4×C5—68no—
360C2×C4×C5—128no—
361C2×C4×C5#23714yes—
362C2×C4×C5C23×D314yes—
363C2×C4×C5C23×D312yes—
364C22⋊D5—68no—
365C22⋊D5—616no—
366C2×Dic5—22no—
367C2×Dic5—32no—
368C2×Dic5—32no—
369C2×Dic5—34no—
370C2×Dic5—62no—
371C2×Dic5—64no—
372C2×Dic5—62no—
373C2×Dic5C23×D312yes—
374D20—28no—
375D20—48no—
376C4×D5—28no—
377C4×D5—128no—
378C22×D4—51no—
379C22×D4—52no—
380C22×D4—152no—
381C22×D4—151no—
382C23×C4—31no—
383C23×C4—32no—
384C23×C4—51no—
385C23×C4—152no—
386#126—1516no—
387#125—158no—
388#123—1516no—
389#121—58no—
390#121—1516no—
391#121—158no—
392C4×D4—1516no—
393C4×D4—158no—
394C4×D4—158no—
395#117—158no—
396C2×C4⋊C4—31no—
397C2×C4⋊C4—32no—
398C2×C4⋊C4—38no—
399C2×C4⋊C4—151no—
400C2×C4⋊C4—151no—
401C2×C4⋊C4—152no—
402C2×C4⋊C4—152no—
403C2×C4⋊C4D3×D511yescharacteristic
404C2×C22⋊C4—52no—
405C2×C22⋊C4—152no—
406C2×C22⋊C4—152no—
407C2×C22⋊C4—151no—
408C2×C22⋊C4—152no—
409C2×C22⋊C4—151no—
410C2×C42—152no—
411C2×C42—151no—
412C2×C3×C5#55314yescomplemented
413C2×C3×C5C22×#4111yescharacteristic
414C2×C3×C5C22×#4111yescharacteristic
415C2×C3×C5C2×H(5,𝔽2)11yescharacteristic
416C3×D5—42no—
417C3×D5—44no—
418C5×D3—24no—
419C23×C3—52no—
420C23×C3—54no—
421C23×C3—102no—
422C23×C3—104no—
423C23×C3—104no—
424C23×C3—102no—
425C23×C3—102no—
426C23×C3—104no—
427C23×C3C23×D511yescharacteristic
428C22×D3—22no—
429C22×D3—22no—
430C22×D3#71314yes—
431C22×D3C23×D512yes—
432C3×D4—108no—
433C3×D4—208no—
434C2×C3×C4—24no—
435C2×C3×C4—52no—
436C2×C3×C4—102no—
437C2×C3×C4—108no—
438C2×C3×C4#70914yes—
439C2×C3×C4C23×D514yes—
440C2×C3×C4C23×D512yes—
441#67—1016no—
442#67—108no—
443C2×Dic3—22no—
444C2×Dic3—24no—
445C2×Dic3—52no—
446C2×Dic3—54no—
447C2×Dic3—52no—
448C2×Dic3—102no—
449C2×Dic3—104no—
450C2×Dic3—102no—
451C2×Dic3—208no—
452C2×Dic3C23×D514yes—
453C2×Dic3C23×D512yes—
454C4×D3—28no—
455C4×D3—48no—
456C22×C5—34no—
457C22×C5—62no—
458C22×C5—62no—
459C22×C5—68no—
460C22×C5C2×#23712yes—
461C22×C5#100412yes—
462C22×C5D3×#4112yes—
463C22×C5C24×D311yescharacteristic
464C2×D5—22no—
465C2×D5—24no—
466C2×D5—24no—
467C2×D5—22no—
468C2×D5—22no—
469C2×D5—24no—
470C2×D5—44no—
471C2×D5—48no—
472C4×C5—24no—
473C4×C5—62no—
474C4×C5—64no—
475C4×C5C2×#23714yes—
476Dic5—22no—
477Dic5—62no—
478Dic5—64no—
479Dic5—62no—
480C24—31no—
481C24—52no—
482C24—51no—
483C2×D4—54no—
484C2×D4—58no—
485C2×D4—104no—
486C2×D4—158no—
487C2×D4—154no—
488C2×D4—204no—
489C2×D4—308no—
490C2×D4—304no—
491C22×C4—32no—
492C22×C4—31no—
493C22×C4—34no—
494C22×C4—38no—
495C22×C4—51no—
496C22×C4—54no—
497C22×C4—64no—
498C22×C4—104no—
499C22×C4—124no—
500C22×C4—158no—
501C22×C4—152no—
502C22×C4—151no—
503C22×C4—151no—
504C22×C4—604no—
505C22×C4C2×D3×D512yes—
506C22×C4C2×D3×D511yescharacteristic
507C4⋊C4—34no—
508C4⋊C4—38no—
509C4⋊C4—154no—
510C4⋊C4—154no—
511C4⋊C4—158no—
512C4⋊C4—158no—
513C4⋊C4C2×D3×D514yes—
514C22⋊C4—58no—
515C22⋊C4—158no—
516C22⋊C4—158no—
517C22⋊C4—154no—
518C22⋊C4—158no—
519C22⋊C4—154no—
520C42—158no—
521C42—154no—
522C3×C5C2×#55311yescharacteristic, complemented, Hall
523C22×C3—102no—
524C22×C3—104no—
525C22×C3—104no—
526C22×C3—102no—
527C22×C3—102no—
528C22×C3—104no—
529C22×C3—204no—
530C22×C3—208no—
531C22×C3C2×#70912yes—
532C22×C3#414412yes—
533C22×C3C2×#71312yes—
534C22×C3C24×D511yescharacteristic
535C2×D3—28no—
536C2×D3—22no—
537C2×D3—22no—
538C2×D3C2×#71314yes—
539C3×C4—24no—
540C3×C4—102no—
541C3×C4C2×#70914yes—
542Dic3—22no—
543Dic3—24no—
544Dic3—102no—
545Dic3—104no—
546Dic3—102no—
547C2×C5—64no—
548C2×C5#561414yescomplemented
549C2×C5C22×#23711yescharacteristic
550C2×C5C2×#100411yescharacteristic
551C2×C5C2×D3×#4111yescharacteristic
552D5—42no—
553D5—44no—
554C23—34no—
555C23—32no—
556C23—54no—
557C23—52no—
558C23—62no—
559C23—62no—
560C23—102no—
561C23—104no—
562C23—104no—
563C23—102no—
564C23—104no—
565C23—102no—
566C23C22×D3×D511yescharacteristic
567D4—108no—
568D4—208no—
569D4—3016no—
570D4—308no—
571C2×C4—24no—
572C2×C4—34no—
573C2×C4—32no—
574C2×C4—52no—
575C2×C4—62no—
576C2×C4—64no—
577C2×C4—68no—
578C2×C4—102no—
579C2×C4—108no—
580C2×C4—128no—
581C2×C4—154no—
582C2×C4—152no—
583C2×C4—152no—
584C2×C4—302no—
585C2×C4—304no—
586C2×C4—302no—
587C2×C4—608no—
588C2×C4#676514yes—
589C2×C4C22×D3×D514yes—
590C2×C4C22×D3×D512yes—
591C2×C3—202no—
592C2×C3—204no—
593C2×C3#6584814yescomplemented
594C2×C3C22×#70911yescharacteristic
595C2×C3C2×#414411yescharacteristic
596C2×C3C22×#71311yescharacteristic
597D3—24no—
598C5C2×#561411yescharacteristic, complemented, Sylow
599C22—34no—
600C22—68no—
601C22—62no—
602C22—62no—
603C22—102no—
604C22—104no—
605C22—104no—
606C22—102no—
607C22—104no—
608C22—102no—
609C22—204no—
610C22—208no—
611C22C2×#676512yes—
612C22#9202512yes—
613C22D3×#71312yes—
614C22C23×D3×D511yescharacteristic
615C4—24no—
616C4—62no—
617C4—64no—
618C4—102no—
619C4—302no—
620C4—304no—
621C4—302no—
622C4C2×#676514yes—
623C3C2×#6584811yescharacteristic, complemented, Sylow
624C2—64no—
625C2—202no—
626C2—204no—
627C2#1175377514yesdirect product factor
628C2C22×#676511yescharacteristic
629C2C2×#9202511yescharacteristic
630C2C2×D3×#71311yescharacteristic
subgroupG/H
centerC23C22×D3×D5
commutator, cosocleC22×C3×C5C25
FrattiniC22C23×D3×D5
FittingC2×C3×C5×C4⋊C4C22
socleC23×C3×C5C24
hypercenterC2×C4⋊C4D3×D5
hypocenterC3×C5C2×#553
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×#91779C2semidirect
2. C2×C5×D3×C4⋊C4C2semidirect
3. C2×C3×#4041C2semidirect
4. C2×#91456C2semidirect
5. C2×#91456C2semidirect
6. C2×#91441C2semidirect
7. C2×C4×#1201C2semidirect
8. C2×#91434C2semidirect
9. C2×#91407C2semidirect
10. C2×#91362C2semidirect
11. C2×#91358C2semidirect
12. #11753775C2direct (N × H)
27. #91779C22semidirect
30. C5×D3×C4⋊C4C22semidirect
32. C3×#4041C22semidirect
40. #91456C22semidirect
41. #91456C22semidirect
42. #91441C22semidirect
43. C4×#1201C22semidirect
44. #91434C22semidirect
45. #91407C22semidirect
46. #91362C22semidirect
47. #91358C22semidirect
56. C2×#4041D3semidirect
111. C2×D3×C4⋊C4D5semidirect
144. #4041C2×D3semidirect
217. D3×C4⋊C4C2×D5semidirect
412. C2×C3×C5#553semidirect
522. C3×C5C2×#553semidirect
548. C2×C5#5614semidirect
593. C2×C3#65848semidirect
598. C5C2×#5614semidirect
623. C3C2×#65848semidirect
627. C2#11753775direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#917792
C2×C5×D3×C4⋊C42
C2×C3×#40412
C2×#914562
C2×#914412
C2×C4×#12012
C2×#914342
C2×#914072
C2×#913622
C2×#913582
#117537752
C2×#658483
C2×#56145