Finite Group Databasebeta

← C2×#11749505 #14332018

Subgroups — C2×#11749505 #14332018

Order 1920.

1816 conjugacy classes of subgroups.

The subgroup lattice has 608 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#6719C211yesmaximal, characteristic
2C22×C5×#217C211yesmaximal, characteristic
3C22×C3×C22⋊Dic5C211yesmaximal, characteristic
4C22×Dic3⋊Dic5C211yesmaximal, characteristic, complemented
5C22×#6669C211yesmaximal, characteristic, complemented
6C22×#6668C211yesmaximal, characteristic, complemented
7#11749505C218yesmaximal, direct product factor
8C2×#90994C211yesmaximal, characteristic, complemented
9C2×#65238—31nomaximal, Hall
10C25×C3×C5C2211yescharacteristic
11C23×Dic15C2211yescharacteristic
12C23×Dic15C2211yescharacteristic
13C23×C5×Dic3C2211yescharacteristic
14C23×C5×Dic3C2211yescharacteristic
15C23×C3×Dic5C2211yescharacteristic
16C23×C3×Dic5C2211yescharacteristic
17C2×#6719—24no—
18C2×#6719C2214yes—
19C2×C5×#217—24no—
20C2×C5×#217C2214yes—
21C2×C3×C22⋊Dic5—24no—
22C2×C3×C22⋊Dic5C2214yes—
23C2×Dic3⋊Dic5—24no—
24C2×Dic3⋊Dic5C2214yescomplemented
25C2×#6669—24no—
26C2×#6669C2214yescomplemented
27C2×#6668—24no—
28C2×#6668C2214yescomplemented
29#90994C2214yescomplemented
30C2×#4949—51nomaximal, Hall
31C22×C5×C22⋊C4—31no—
32C22×C22⋊Dic5—31no—
33C22×C22⋊Dic5D311yescharacteristic
34C22×C4⋊Dic5—31no—
35C22×#683—31no—
36C22×#683—31no—
37C2×#3963—31no—
38#65238—38no—
39C24×C3×C5—22no—
40C24×C3×C5—22no—
41C24×C3×C5—22no—
42C24×C3×C5—22no—
43C24×C3×C5—22no—
44C24×C3×C5—22no—
45C24×C3×C5Q814yes—
46C24×C3×C5C2312yes—
47C24×C3×C5C2311yescharacteristic
48C22×Dic15—22no—
49C22×Dic15—22no—
50C22×Dic15—22no—
51C22×Dic15—22no—
52C22×Dic15—22no—
53C22×Dic15D414yes—
54C22×Dic15C2312yes—
55C22×Dic15C2312yes—
56C22×C5×Dic3—22no—
57C22×C5×Dic3—22no—
58C22×C5×Dic3—22no—
59C22×C5×Dic3—22no—
60C22×C5×Dic3—22no—
61C22×C5×Dic3D414yes—
62C22×C5×Dic3C2312yes—
63C22×C5×Dic3C2312yes—
64C22×C3×Dic5—22no—
65C22×C3×Dic5—22no—
66C22×C3×Dic5—22no—
67C22×C3×Dic5—22no—
68C22×C3×Dic5—22no—
69C22×C3×Dic5D414yes—
70C22×C3×Dic5C2312yes—
71C22×C3×Dic5C2312yes—
72#6719—28no—
73C5×#217—28no—
74C3×C22⋊Dic5—28no—
75Dic3⋊Dic5—28no—
76#6669—28no—
77#6668—28no—
78C22×C3×C22⋊C4—51no—
79C22×#217—51no—
80C22×#217D511yescharacteristic
81C22×C4⋊Dic3—51no—
82C22×Dic3⋊C4—51no—
83C22×Dic3⋊C4—51no—
84C2×C3⋊#95—51no—
85#4949—58no—
86C25×C5C2×D311yescharacteristic
87C23×C4×C5—31no—
88C23×C4×C5—31no—
89C23×Dic5—31no—
90C23×Dic5—31no—
91C23×Dic5C2×D311yescharacteristic
92C23×Dic5C2×D311yescharacteristic
93C2×C5×C22⋊C4—34no—
94C2×C5×C22⋊C4—64no—
95C2×C22⋊Dic5—24no—
96C2×C22⋊Dic5—34no—
97C2×C22⋊Dic5—64no—
98C2×C22⋊Dic5C2×D314yes—
99C2×C4⋊Dic5—34no—
100C2×C4⋊Dic5—64no—
101C2×#683—34no—
102C2×#683—34no—
103C2×#683—64no—
104C2×#683—64no—
105#3963—34no—
106C2×#399—151noSylow
107C23×C3×C5—28no—
108C23×C3×C5—28no—
109C23×C3×C5—28no—
110C23×C3×C5—22no—
111C23×C3×C5—22no—
112C23×C3×C5—22no—
113C23×C3×C5—22no—
114C23×C3×C5—22no—
115C23×C3×C5—22no—
116C23×C3×C5—44no—
117C23×C3×C5—44no—
118C23×C3×C5—44no—
119C23×C3×C5—44no—
120C23×C3×C5C2×D412yes—
121C23×C3×C5C2×D412yes—
122C23×C3×C5C2×D412yes—
123C23×C3×C5C2×Q814yes—
124C23×C3×C5C2×Q812yes—
125C23×C3×C5#4112yes—
126C23×C3×C5#4112yes—
127C23×C3×C5#4112yes—
128C23×C3×C5C2411yescharacteristic
129C2×Dic15—28no—
130C2×Dic15—22no—
131C2×Dic15—22no—
132C2×Dic15—22no—
133C2×Dic15—22no—
134C2×Dic15—22no—
135C2×Dic15—44no—
136C2×Dic15C2×D414yes—
137C2×C5×Dic3—28no—
138C2×C5×Dic3—22no—
139C2×C5×Dic3—22no—
140C2×C5×Dic3—22no—
141C2×C5×Dic3—22no—
142C2×C5×Dic3—22no—
143C2×C5×Dic3—44no—
144C2×C5×Dic3C2×D414yes—
145C2×C3×Dic5—28no—
146C2×C3×Dic5—22no—
147C2×C3×Dic5—22no—
148C2×C3×Dic5—22no—
149C2×C3×Dic5—22no—
150C2×C3×Dic5—22no—
151C2×C3×Dic5—44no—
152C2×C3×Dic5C2×D414yes—
153C25×C3C2×D511yescharacteristic
154C23×C3×C4—51no—
155C23×C3×C4—51no—
156C23×Dic3—51no—
157C23×Dic3—51no—
158C23×Dic3C2×D511yescharacteristic
159C23×Dic3C2×D511yescharacteristic
160C2×C3×C22⋊C4—54no—
161C2×C3×C22⋊C4—104no—
162C2×#217—24no—
163C2×#217—54no—
164C2×#217—104no—
165C2×#217C2×D514yes—
166C2×C4⋊Dic3—54no—
167C2×C4⋊Dic3—104no—
168C2×Dic3⋊C4—54no—
169C2×Dic3⋊C4—54no—
170C2×Dic3⋊C4—104no—
171C2×Dic3⋊C4—104no—
172C3⋊#95—54no—
173C24×C5—22no—
174C24×C5—22no—
175C24×C5—22no—
176C24×C5—22no—
177C24×C5—22no—
178C24×C5—22no—
179C24×C5Dic614yes—
180C24×C5C22×D312yes—
181C24×C5C22×D311yescharacteristic
182C22×C4×C5—34no—
183C22×C4×C5—32no—
184C22×C4×C5—32no—
185C22×C4×C5—62no—
186C22×C4×C5—62no—
187C22×C4×C5—62no—
188C22×C4×C5—62no—
189C22×C4×C5—62no—
190C22×Dic5—22no—
191C22×Dic5—22no—
192C22×Dic5—22no—
193C22×Dic5—22no—
194C22×Dic5—22no—
195C22×Dic5—34no—
196C22×Dic5—32no—
197C22×Dic5—32no—
198C22×Dic5—62no—
199C22×Dic5—62no—
200C22×Dic5—62no—
201C22×Dic5—62no—
202C22×Dic5—62no—
203C22×Dic5#6714yes—
204C22×Dic5C22×D312yes—
205C22×Dic5C22×D312yes—
206C5×C22⋊C4—68no—
207C22⋊Dic5—28no—
208C22⋊Dic5—68no—
209C4⋊Dic5—68no—
210#683—68no—
211#683—68no—
212C22×C4⋊C4—151no—
213C22×C4⋊C4—151no—
214C22×C4⋊C4—151no—
215C22×C22⋊C4—31no—
216C22×C22⋊C4—51no—
217C22×C22⋊C4—151no—
218#399—158no—
219C2×#95—151no—
220C22×C3×C5—24no—
221C22×C3×C5—24no—
222C22×C3×C5—24no—
223C22×C3×C5—42no—
224C22×C3×C5—42no—
225C22×C3×C5—42no—
226C22×C3×C5—42no—
227C22×C3×C5—416no—
228C22×C3×C5C22⋊Q814yes—
229C22×C3×C5C22⋊Q814yes—
230C22×C3×C5C22⋊Q814yes—
231C22×C3×C5#12314yes—
232C22×C3×C5#12314yes—
233C22×C3×C5#12314yes—
234C22×C3×C5#12714yes—
235C22×C3×C5C22×D411yescharacteristic
236C22×C3×C5C22×D411yescharacteristic
237C22×C3×C5C22×D411yescharacteristic
238C22×C3×C5C22×Q811yescharacteristic
239C22×C3×C5C2×#4111yescharacteristic
240C22×C3×C5C2×#4111yescharacteristic
241C22×C3×C5C2×#4111yescharacteristic
242Dic15—24no—
243Dic15—42no—
244C3×Dic5—24no—
245C3×Dic5—42no—
246C5×Dic3—24no—
247C5×Dic3—42no—
248C24×C3—22no—
249C24×C3—22no—
250C24×C3—22no—
251C24×C3—22no—
252C24×C3—22no—
253C24×C3—22no—
254C24×C3Dic1014yes—
255C24×C3C22×D512yes—
256C24×C3C22×D511yescharacteristic
257C22×C3×C4—54no—
258C22×C3×C4—52no—
259C22×C3×C4—52no—
260C22×C3×C4—102no—
261C22×C3×C4—102no—
262C22×C3×C4—102no—
263C22×C3×C4—102no—
264C22×C3×C4—102no—
265C22×Dic3—22no—
266C22×Dic3—22no—
267C22×Dic3—22no—
268C22×Dic3—22no—
269C22×Dic3—22no—
270C22×Dic3—54no—
271C22×Dic3—52no—
272C22×Dic3—52no—
273C22×Dic3—102no—
274C22×Dic3—102no—
275C22×Dic3—102no—
276C22×Dic3—102no—
277C22×Dic3—102no—
278C22×Dic3C22⋊D514yes—
279C22×Dic3C22×D512yes—
280C22×Dic3C22×D512yes—
281C3×C22⋊C4—108no—
282#217—28no—
283#217—108no—
284C4⋊Dic3—108no—
285Dic3⋊C4—108no—
286Dic3⋊C4—108no—
287C23×C5—28no—
288C23×C5—28no—
289C23×C5—28no—
290C23×C5—22no—
291C23×C5—22no—
292C23×C5—22no—
293C23×C5—22no—
294C23×C5—22no—
295C23×C5—22no—
296C23×C5—44no—
297C23×C5—44no—
298C23×C5—44no—
299C23×C5—44no—
300C23×C5C2×Dic614yes—
301C23×C5C2×Dic612yes—
302C23×C5#23512yes—
303C23×C5D3×D412yes—
304C23×C5D3×D412yes—
305C23×C5#23712yes—
306C23×C5#23712yes—
307C23×C5C2×#6712yes—
308C23×C5C23×D311yescharacteristic
309C2×C4×C5—34no—
310C2×C4×C5—68no—
311C2×C4×C5—62no—
312C2×C4×C5—62no—
313C2×C4×C5—62no—
314C2×C4×C5—62no—
315C2×C4×C5—62no—
316C2×C4×C5—124no—
317C2×Dic5—28no—
318C2×Dic5—22no—
319C2×Dic5—22no—
320C2×Dic5—22no—
321C2×Dic5—22no—
322C2×Dic5—22no—
323C2×Dic5—34no—
324C2×Dic5—44no—
325C2×Dic5—68no—
326C2×Dic5—62no—
327C2×Dic5—62no—
328C2×Dic5—62no—
329C2×Dic5—62no—
330C2×Dic5—62no—
331C2×Dic5—124no—
332C2×Dic5C2×#6714yes—
333C25D3×D511yescharacteristic
334C23×C4—31no—
335C23×C4—31no—
336C23×C4—51no—
337C23×C4—51no—
338C23×C4—151no—
339C23×C4—151no—
340C2×C4⋊C4—154no—
341C2×C4⋊C4—154no—
342C2×C4⋊C4—154no—
343C2×C4⋊C4—304no—
344C2×C4⋊C4—304no—
345C2×C4⋊C4—304no—
346C2×C22⋊C4—34no—
347C2×C22⋊C4—54no—
348C2×C22⋊C4—64no—
349C2×C22⋊C4—104no—
350C2×C22⋊C4—154no—
351C2×C22⋊C4—304no—
352#95—154no—
353C2×C3×C5—44no—
354C2×C3×C5#39918yescomplemented
355C2×C3×C5C2×C22⋊Q811yescharacteristic
356C2×C3×C5C2×C22⋊Q811yescharacteristic
357C2×C3×C5C2×C22⋊Q811yescharacteristic
358C2×C3×C5C2×#12311yescharacteristic
359C2×C3×C5C2×#12311yescharacteristic
360C2×C3×C5C2×#12311yescharacteristic
361C2×C3×C5C2×#12711yescharacteristic
362C23×C3—22no—
363C23×C3—22no—
364C23×C3—22no—
365C23×C3—22no—
366C23×C3—22no—
367C23×C3—22no—
368C23×C3—28no—
369C23×C3—28no—
370C23×C3—28no—
371C23×C3—44no—
372C23×C3—44no—
373C23×C3—44no—
374C23×C3—44no—
375C23×C3C2×Dic1014yes—
376C23×C3C2×Dic1012yes—
377C23×C3#70912yes—
378C23×C3D4×D512yes—
379C23×C3D4×D512yes—
380C23×C3#71112yes—
381C23×C3#71112yes—
382C23×C3C2×C22⋊D512yes—
383C23×C3C23×D511yescharacteristic
384C2×C3×C4—54no—
385C2×C3×C4—102no—
386C2×C3×C4—102no—
387C2×C3×C4—108no—
388C2×C3×C4—102no—
389C2×C3×C4—102no—
390C2×C3×C4—102no—
391C2×C3×C4—204no—
392C2×Dic3—22no—
393C2×Dic3—22no—
394C2×Dic3—28no—
395C2×Dic3—22no—
396C2×Dic3—22no—
397C2×Dic3—22no—
398C2×Dic3—44no—
399C2×Dic3—54no—
400C2×Dic3—102no—
401C2×Dic3—102no—
402C2×Dic3—108no—
403C2×Dic3—102no—
404C2×Dic3—102no—
405C2×Dic3—102no—
406C2×Dic3—204no—
407C2×Dic3C2×C22⋊D514yes—
408C22×C5—24no—
409C22×C5—24no—
410C22×C5—24no—
411C22×C5—416no—
412C22×C5—42no—
413C22×C5—42no—
414C22×C5—42no—
415C22×C5—42no—
416C22×C5#87814yes—
417C22×C5#87814yes—
418C22×C5#88314yes—
419C22×C5#88314yes—
420C22×C5#92414yes—
421C22×C5#93514yes—
422C22×C5#94014yes—
423C22×C5C22×Dic611yescharacteristic
424C22×C5C2×#23511yescharacteristic
425C22×C5C2×D3×D411yescharacteristic
426C22×C5C2×D3×D411yescharacteristic
427C22×C5C2×#23711yescharacteristic
428C22×C5C2×#23711yescharacteristic
429C22×C5C22×#6711yescharacteristic
430C4×C5—64no—
431C4×C5—122no—
432Dic5—24no—
433Dic5—42no—
434Dic5—64no—
435Dic5—122no—
436C24—22no—
437C24—22no—
438C24—22no—
439C24—22no—
440C24—22no—
441C24—22no—
442C24#120414yes—
443C24C2×D3×D512yes—
444C24C2×D3×D511yescharacteristic
445C22×C4—34no—
446C22×C4—32no—
447C22×C4—32no—
448C22×C4—54no—
449C22×C4—52no—
450C22×C4—52no—
451C22×C4—62no—
452C22×C4—62no—
453C22×C4—62no—
454C22×C4—62no—
455C22×C4—62no—
456C22×C4—102no—
457C22×C4—102no—
458C22×C4—102no—
459C22×C4—102no—
460C22×C4—102no—
461C22×C4—154no—
462C22×C4—152no—
463C22×C4—152no—
464C22×C4—302no—
465C22×C4—302no—
466C22×C4—302no—
467C22×C4—302no—
468C22×C4—302no—
469C4⋊C4—308no—
470C4⋊C4—308no—
471C4⋊C4—308no—
472C22⋊C4—68no—
473C22⋊C4—108no—
474C22⋊C4—308no—
475C3×C5C2×#39911yescharacteristic, complemented, Hall
476C22×C3—24no—
477C22×C3—24no—
478C22×C3—24no—
479C22×C3—42no—
480C22×C3—42no—
481C22×C3—42no—
482C22×C3—42no—
483C22×C3—416no—
484C22×C3#402414yes—
485C22×C3#402414yes—
486C22×C3#402914yes—
487C22×C3#402914yes—
488C22×C3#407014yes—
489C22×C3#408114yes—
490C22×C3#408614yes—
491C22×C3C22×Dic1011yescharacteristic
492C22×C3C2×#70911yescharacteristic
493C22×C3C2×D4×D511yescharacteristic
494C22×C3C2×D4×D511yescharacteristic
495C22×C3C2×#71111yescharacteristic
496C22×C3C2×#71111yescharacteristic
497C22×C3C22×C22⋊D511yescharacteristic
498C3×C4—104no—
499C3×C4—202no—
500Dic3—24no—
501Dic3—42no—
502Dic3—104no—
503Dic3—202no—
504C2×C5—44no—
505C2×C5#494918yescomplemented
506C2×C5C2×#87811yescharacteristic
507C2×C5C2×#87811yescharacteristic
508C2×C5C2×#88311yescharacteristic
509C2×C5C2×#88311yescharacteristic
510C2×C5C2×#92411yescharacteristic
511C2×C5C2×#93511yescharacteristic
512C2×C5C2×#94011yescharacteristic
513C23—22no—
514C23—22no—
515C23—22no—
516C23—22no—
517C23—22no—
518C23—22no—
519C23—28no—
520C23—28no—
521C23—28no—
522C23—44no—
523C23—44no—
524C23—44no—
525C23—44no—
526C23#677912yes—
527C23#678012yes—
528C23#678212yes—
529C23C2×#120414yes—
530C23C2×#120412yes—
531C23D5×#6712yes—
532C23D3×C22⋊D512yes—
533C23#679012yes—
534C23C22×D3×D511yescharacteristic
535C2×C4—34no—
536C2×C4—54no—
537C2×C4—62no—
538C2×C4—62no—
539C2×C4—68no—
540C2×C4—62no—
541C2×C4—62no—
542C2×C4—62no—
543C2×C4—102no—
544C2×C4—102no—
545C2×C4—108no—
546C2×C4—102no—
547C2×C4—102no—
548C2×C4—102no—
549C2×C4—124no—
550C2×C4—154no—
551C2×C4—204no—
552C2×C4—302no—
553C2×C4—302no—
554C2×C4—308no—
555C2×C4—302no—
556C2×C4—302no—
557C2×C4—302no—
558C2×C4—604no—
559C2×C3—44no—
560C2×C3#6523818yescomplemented
561C2×C3C2×#402411yescharacteristic
562C2×C3C2×#402411yescharacteristic
563C2×C3C2×#402911yescharacteristic
564C2×C3C2×#402911yescharacteristic
565C2×C3C2×#407011yescharacteristic
566C2×C3C2×#408111yescharacteristic
567C2×C3C2×#408611yescharacteristic
568C5C2×#494911yescharacteristic, complemented, Sylow
569C22—24no—
570C22—24no—
571C22—24no—
572C22—42no—
573C22—42no—
574C22—416no—
575C22—42no—
576C22—42no—
577C22#9154814yes—
578C22#9155514yes—
579C22#9156214yes—
580C22#9156714yes—
581C22#9157414yes—
582C22#9157514yes—
583C22#9157614yes—
584C22C2×#677911yescharacteristic
585C22C2×#678011yescharacteristic
586C22C2×#678211yescharacteristic
587C22C22×#120411yescharacteristic
588C22C2×D5×#6711yescharacteristic
589C22C2×D3×C22⋊D511yescharacteristic
590C22C2×#679011yescharacteristic
591C4—64no—
592C4—104no—
593C4—122no—
594C4—202no—
595C4—304no—
596C4—602no—
597C3C2×#6523811yescharacteristic, complemented, Sylow
598C2—44no—
599C2#1174950518yesdirect product factor
600C2C2×#9154811yescharacteristic
601C2C2×#9155511yescharacteristic
602C2C2×#9156211yescharacteristic
603C2C2×#9156711yescharacteristic
604C2C2×#9157411yescharacteristic
605C2C2×#9157511yescharacteristic
606C2C2×#9157611yescharacteristic
subgroupG/H
centerC24C2×D3×D5
commutator, cosocleC23×C3×C5C24
FrattiniC23C22×D3×D5
Fitting, prime radicalC25×C3×C5C22
socleC24×C3×C5C23
hypercenterC25D3×D5
hypocenterC3×C5C2×#399
LCSG ▷ C23×C3×C5 ▷ C3×C5
UCSC1 ◁ C24 ◁ C25

Complements

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

NH ≅ G/Ntype
4. C22×Dic3⋊Dic5C2semidirect
5. C22×#6669C2semidirect
6. C22×#6668C2semidirect
7. #11749505C2direct (N × H)
8. C2×#90994C2semidirect
24. C2×Dic3⋊Dic5C22semidirect
26. C2×#6669C22semidirect
28. C2×#6668C22semidirect
29. #90994C22semidirect
354. C2×C3×C5#399semidirect
475. C3×C5C2×#399semidirect
505. C2×C5#4949semidirect
560. C2×C3#65238semidirect
568. C5C2×#4949semidirect
597. C3C2×#65238semidirect
599. C2#11749505direct (N × H)

Maximal subgroups

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

subgroupindex
C22×#67192
C22×C5×#2172
C22×C3×C22⋊Dic52
C22×Dic3⋊Dic52
C22×#66692
C22×#66682
#117495052
C2×#909942
C2×#652383
C2×#49495