Finite Group Databasebeta

← C2×#21455753 #30547803

Subgroups — C2×#21455753 #30547803

Order 15616.

2346 conjugacy classes of subgroups.

The subgroup lattice has 674 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#16840274C211yesmaximal, characteristic
2C22×#14519735C211yesmaximal, characteristic, complemented
3#21455753C2116yesmaximal, direct product factor
4C2×#16840106C212yesmaximal, complemented
5C2×#16840105C212yesmaximal, complemented
6C2×#16840089C212yesmaximal, complemented
7C2×#16840088C212yesmaximal, complemented
8C2×#16839805C211yesmaximal, characteristic, complemented
9C2×#16839764C211yesmaximal, characteristic, complemented
10C2×#16839761C211yesmaximal, characteristic, complemented
11C2×#16839725C211yesmaximal, characteristic, complemented
12C2×C61×#428C211yesmaximal, characteristic, complemented
13C22×#11758222C2211yescharacteristic
14C23×C4×D61C2211yescharacteristic
15C22×#11758194C2211yescharacteristic
16#16840274—24no—
17#16840274C2218yes—
18C2×#14519761—21no—
19C2×#14519757C2211yescharacteristic
20C2×#14519757C2211yescharacteristic
21C2×#14519752C2211yescharacteristic
22C2×#14519752C2211yescharacteristic
23C2×#14519747—21no—
24C2×C4×D244—21no—
25C2×C4×D244—21no—
26C2×#14519735—21no—
27C2×#14519735—21no—
28C2×#14519735—24no—
29C2×#14519735C2212yes—
30C2×#14519735C2212yes—
31C2×#14519735C2218yescomplemented
32C22×C4⋊Dic61C2211yescharacteristic
33C22×#11758198—21no—
34C22×C4×Dic61—21no—
35#16840106C22116yescomplemented
36#16840105C22116yescomplemented
37#16840089C22116yescomplemented
38#16840088C22116yescomplemented
39#16839805C2218yescomplemented
40C2×#14519710C2212yescomplemented
41C2×#14519708C2212yescomplemented
42#16839764C2218yescomplemented
43#16839761C2218yescomplemented
44C2×#14519702C2212yescomplemented
45C2×#14519700C2212yescomplemented
46C2×#14519699C2212yescomplemented
47C2×#14519697C2212yescomplemented
48#16839725C2218yescomplemented
49C2×#14519689C2212yes—
50C2×#14519689C2212yes—
51C22×M16×C61C2211yescharacteristic
52C2×#117×C61C2211yescharacteristic
53C22×C4⋊C4×C61C2211yescharacteristic
54C61×#428C2218yescomplemented
55C2×#107×C61C2212yescomplemented
56C2×#106×C61C2212yescomplemented
57C24×D61D411yescharacteristic
58C2×#11758222—22no—
59C2×#11758222—24no—
60C2×#11758222—24no—
61C2×#11758222C2314yes—
62C22×Dic122C2311yescharacteristic
63C22×D244—21no—
64C22×D244—21no—
65C22×D244C2311yescharacteristic
66C22×C22⋊D61—21no—
67C23×Dic61—21no—
68C23×Dic61D411yescharacteristic
69C22×C4×D61—24no—
70C22×C4×D61—24no—
71C22×C4×D61—21no—
72C22×C4×D61D411yescharacteristic
73C22×C4×D61D411yescharacteristic
74C22×C4×D61D414yes—
75C22×C4×D61C2311yescharacteristic
76C22×C4×D61C2311yescharacteristic
77C22×C4×D61C2314yes—
78C2×#11758194—24no—
79C2×#11758194C2314yes—
80C22×#92418C2312yes—
81#14519761—28no—
82#14519757—22no—
83#14519757—22no—
84#14519757C2314yes—
85#14519757C2314yes—
86#14519752—22no—
87#14519752—22no—
88#14519752C2314yes—
89#14519752C2314yes—
90#14519747—28no—
91C4×D244—28no—
92C4×D244—28no—
93#14519735—24no—
94#14519735—24no—
95#14519735—28no—
96#14519735—28no—
97#14519735C2318yes—
98#14519735C2318yes—
99C2×C4⋊Dic61—21no—
100C2×C4⋊Dic61—22no—
101C2×C4⋊Dic61—21no—
102C2×C4⋊Dic61—21no—
103C2×C4⋊Dic61C2312yes—
104C2×C4⋊Dic61C2314yes—
105C2×#11758198—22no—
106C2×#11758198—22no—
107C2×#11758198—28no—
108C2×#11758198—21no—
109C2×#11758198—21no—
110C2×#11758198—21no—
111C2×#11758198—21no—
112C2×C4×Dic61—22no—
113C2×C4×Dic61—22no—
114C2×C4×Dic61—28no—
115#14519710C2318yescomplemented
116#14519708C2318yescomplemented
117#14519702C2318yescomplemented
118#14519700C2318yescomplemented
119#14519699C2318yescomplemented
120#14519697C2318yescomplemented
121#14519689C2318yes—
122#14519689C2318yes—
123C23×C4×C61—21no—
124C23×C4×C61C2311yescharacteristic
125C2×M16×C61—24no—
126C2×M16×C61C2314yes—
127C22×C8×C61C2312yes—
128#117×C61—22no—
129#117×C61C2314yes—
130C2×C4⋊C4×C61—21no—
131C2×C4⋊C4×C61—22no—
132C2×C4⋊C4×C61C2312yes—
133C2×C4⋊C4×C61C2314yes—
134C2×C4⋊C4×C61C2311yescharacteristic
135C2×C4⋊C4×C61C2311yescharacteristic
136C2×C22⋊C4×C61—21no—
137C2×C42×C61—21no—
138#107×C61C2318yescomplemented
139#106×C61C2318yescomplemented
140C24×C61C2×D411yescharacteristic
141C23×D61—24no—
142C23×D61—21no—
143C23×D61—42no—
144C23×D61—42no—
145C23×D61C2×D411yescharacteristic
146C23×D61C2×D411yescharacteristic
147C23×D61C2×D414yes—
148C22×C4×C61—21no—
149C22×C4×C61—21no—
150C22×C4×C61—24no—
151C22×C4×C61—22no—
152C22×C4×C61—44no—
153C22×C4×C61C2×D411yescharacteristic
154C22×C4×C61C2×D412yes—
155C22×C4×C61#4114yes—
156C22×C4×C61#4114yes—
157C22×C4×C61C2411yescharacteristic
158C22×C4×C61C2412yes—
159C2×C22⋊D61—24no—
160C2×C22⋊D61—44no—
161C22×Dic61—24no—
162C22×Dic61—21no—
163C22×Dic61—24no—
164C22×Dic61—22no—
165C22×Dic61—44no—
166C22×Dic61C2×D411yescharacteristic
167C22×Dic61C2×D411yescharacteristic
168C22×Dic61C2×D414yes—
169#11758222—28no—
170#11758222—28no—
171#11758222—48no—
172C2×D244—22no—
173C2×D244—22no—
174C2×D244—24no—
175C2×D244—24no—
176C2×D244—21no—
177C2×D244—44no—
178C2×D244—44no—
179C2×D244C2412yes—
180C2×C4×D61—24no—
181C2×C4×D61—24no—
182C2×C4×D61—216no—
183C2×C4×D61—22no—
184C2×C4×D61—22no—
185C2×C4×D61—28no—
186C2×C4×D61—22no—
187C2×C4×D61—22no—
188C2×C4×D61—28no—
189C2×C4×D61—24no—
190C2×C4×D61—44no—
191C2×C4×D61C2×D414yes—
192C2×C4×D61C2×D414yes—
193C2×C4×D61C2×D412yes—
194C2×C4×D61C2×D412yes—
195C2×C4×D61C2412yes—
196C2×C4×D61C2412yes—
197C2×Dic122—22no—
198C2×Dic122—22no—
199C2×Dic122—21no—
200C2×Dic122C2412yes—
201M16×C61—28no—
202C2×C8×C61—24no—
203C2×C8×C61C2414yes—
204C4⋊C4×C61—24no—
205C4⋊C4×C61—22no—
206C4⋊C4×C61—21no—
207C4⋊C4×C61—21no—
208C4⋊C4×C61C2412yes—
209C4⋊C4×C61C2414yes—
210C4⋊C4×C61C2412yes—
211C22⋊C4×C61—24no—
212C42×C61—24no—
213#11758198—28no—
214#11758198—28no—
215#11758198—24no—
216#11758198—24no—
217#11758198—24no—
218#11758198—24no—
219C4⋊Dic61—24no—
220C4⋊Dic61—22no—
221C4⋊Dic61—24no—
222C4⋊Dic61—24no—
223C4⋊Dic61C2414yes—
224C4×Dic61—28no—
225C4×Dic61—28no—
226#11758194—28no—
227C2×#92418—24no—
228C2×#92418C2414yes—
229C23×C61—22no—
230C23×C61—22no—
231C23×C61#12314yes—
232C23×C61C22×D412yes—
233C23×C61C22×D411yescharacteristic
234C22×D61—28no—
235C22×D61—22no—
236C22×D61—22no—
237C22×D61—22no—
238C22×D61—22no—
239C22×D61—22no—
240C22×D61—22no—
241C22×D61—22no—
242C22×D61—44no—
243C22×D61—44no—
244C22×D61—416no—
245C22×D61—42no—
246C22×D61—42no—
247C22×D61—42no—
248C22×D61C22×D412yes—
249C22×D61C22×D412yes—
250C2×C4×C61—24no—
251C2×C4×C61—22no—
252C2×C4×C61—22no—
253C2×C4×C61—48no—
254C2×C4×C61—42no—
255C2×C4×C61—42no—
256C2×C4×C61—44no—
257C2×C4×C61#12314yes—
258C2×C4×C61#12318yes—
259C2×C4×C61C22×D412yes—
260C2×C4×C61C22×D411yescharacteristic
261C2×C4×C61C2×#4112yes—
262C2×C4×C61C2×#4114yes—
263C2×C4×C61C2×#4112yes—
264C2×C4×C61C2×#4114yes—
265C2×C4×C61C2511yescharacteristic
266C22⋊D61—48no—
267C2×Dic61—28no—
268C2×Dic61—22no—
269C2×Dic61—22no—
270C2×Dic61—24no—
271C2×Dic61—22no—
272C2×Dic61—48no—
273C2×Dic61—42no—
274C2×Dic61—44no—
275C2×Dic61C22×D412yes—
276C2×Dic61C22×D412yes—
277D244—22no—
278D244—22no—
279D244—48no—
280D244—48no—
281D244—42no—
282C4×D61—24no—
283C4×D61—24no—
284C4×D61—28no—
285C4×D61—28no—
286C4×D61—22no—
287C4×D61—22no—
288C4×D61—22no—
289C4×D61—22no—
290C4×D61—48no—
291Dic122—22no—
292Dic122—22no—
293Dic122—42no—
294C8×C61—24no—
295#92418—24no—
296C2×#3093—611nomaximal, Sylow
297C22×C61—28no—
298C22×C61—22no—
299C22×C61—22no—
300C22×C61C2×#12314yes—
301C22×C61C2×#12312yes—
302C22×C61#57512yes—
303C22×C61#57512yes—
304C22×C61C23×D411yescharacteristic
305C2×D61—22no—
306C2×D61—22no—
307C2×D61—22no—
308C2×D61—22no—
309C2×D61—22no—
310C2×D61—22no—
311C2×D61—416no—
312C2×D61—44no—
313C2×D61—44no—
314C2×D61—42no—
315C2×D61—42no—
316C2×D61—42no—
317C2×D61—84no—
318C4×C61—42no—
319C4×C61—42no—
320C4×C61—44no—
321C4×C61C2×#12312yes—
322C4×C61C2×#12314yes—
323C4×C61C22×#4111yescharacteristic
324C4×C61C22×#4111yescharacteristic
325Dic61—22no—
326Dic61—22no—
327Dic61—42no—
328Dic61—44no—
329C2×#525—611no—
330C22×#117—611no—
331#3093—6116no—
332C2×#485—612no—
333C2×#485—612no—
334C2×#482—612no—
335C2×#482—612no—
336C2×#428—611no—
337C2×#428D6111yescharacteristic, complemented
338C2×#417—611no—
339C2×#417—611no—
340C2×#407—611no—
341C2×C61—24no—
342C2×C61#309314yescomplemented
343C2×C61C22×#12311yescharacteristic
344C2×C61C2×#57511yescharacteristic
345C2×C61C2×#57511yescharacteristic
346D61—42no—
347D61—42no—
348D61—82no—
349C22×#41—611no—
350C24×C4—611no—
351C22×M16—611no—
352C22×M16C2×D6111yescharacteristic
353#525—618no—
354#525—1224no—
355C2×#123—1221no—
356C2×C22⋊Q8—611no—
357C2×C22⋊Q8—611no—
358C2×#121—611no—
359C2×#121—611no—
360C2×#120—1221no—
361C2×C4×D4—1221no—
362C2×C4×D4—1221no—
363C2×#117—612no—
364C2×#117—612no—
365C2×#117—618no—
366C2×#117—1221no—
367C2×#117—1221no—
368C2×#117—1224no—
369C2×#117C2×D6111yescharacteristic
370C22×C4⋊C4—611no—
371C22×C4⋊C4C2×D6111yescharacteristic
372C22×C22⋊C4—1221no—
373C22×C42—1221no—
374#485—6116no—
375#485—6116no—
376#482—6116no—
377#482—6116no—
378#428—618no—
379#428C2×D6118yescomplemented
380C2×#107—612no—
381C2×#107C2×D6112yescomplemented
382C2×#106—612no—
383C2×#106C2×D6112yescomplemented
384#417—618no—
385#417—618no—
386C2×#103—612no—
387C2×#103—612no—
388C2×#102—612no—
389C2×#102—612no—
390#407—618no—
391C2×C22⋊C8—612no—
392C2×C22⋊C8—612no—
393C61C2×#309311yescharacteristic, complemented, Sylow
394C25—611no—
395C2×#41—614no—
396C2×#41—1222no—
397C2×#41—1224no—
398C2×#41—1224no—
399C22×Q8—611no—
400C22×D4—611no—
401C22×D4—1221no—
402C22×D4—1221no—
403C22×D4—1221no—
404C23×C4—21no—
405C23×C4—611no—
406C23×C4—611no—
407C23×C4—611no—
408C23×C4—614no—
409C23×C4—611no—
410C23×C4—611no—
411C23×C4—614no—
412C23×C4—1224no—
413C23×C4—1224no—
414C23×C4—1221no—
415C23×C4—1221no—
416C23×C4C22×D6111yescharacteristic
417C2×M16—24no—
418C2×M16—614no—
419C2×M16—1224no—
420C2×M16C22×D6114yes—
421C22×C8—612no—
422C22×C8C22×D6112yes—
423#123—1228no—
424C22⋊Q8—614no—
425C22⋊Q8—614no—
426C22⋊Q8—1222no—
427C22⋊Q8—1222no—
428#121—614no—
429#121—614no—
430#121—1222no—
431#121—1222no—
432#120—1228no—
433C4×D4—1228no—
434C4×D4—1228no—
435#117—22no—
436#117—618no—
437#117—618no—
438#117—1228no—
439#117—1228no—
440#117—1224no—
441#117—1224no—
442#117C22×D6114yes—
443C2×C4⋊C4—22no—
444C2×C4⋊C4—21no—
445C2×C4⋊C4—612no—
446C2×C4⋊C4—614no—
447C2×C4⋊C4—1221no—
448C2×C4⋊C4—1221no—
449C2×C4⋊C4—1222no—
450C2×C4⋊C4—1221no—
451C2×C4⋊C4C22×D6112yes—
452C2×C4⋊C4C22×D6114yes—
453C2×C4⋊C4C22×D6111yescharacteristic
454C2×C4⋊C4C22×D6111yescharacteristic
455C2×C22⋊C4—21no—
456C2×C22⋊C4—1221no—
457C2×C22⋊C4—1221no—
458C2×C22⋊C4—1228no—
459C2×C22⋊C4—1222no—
460C2×C22⋊C4—1222no—
461C2×C22⋊C4—1221no—
462C2×C22⋊C4—1221no—
463C2×C42—21no—
464C2×C42—1228no—
465C2×C42—1222no—
466C2×C42—1222no—
467#107—618no—
468#107C22×D6118yescomplemented
469#106—618no—
470#106C22×D6118yescomplemented
471#103—618no—
472#103—618no—
473#102—618no—
474#102—618no—
475C22⋊C8—618no—
476C22⋊C8—618no—
477C24—611no—
478C24—611no—
479C24—614no—
480C24—1224no—
481C24—1221no—
482C24—2442no—
483C24—2442no—
484C24D4×D6111yescharacteristic
485#41—1228no—
486#41—1228no—
487#41—2448no—
488C2×Q8—612no—
489C2×Q8—1221no—
490C2×Q8—1222no—
491C2×Q8—1222no—
492C2×D4—612no—
493C2×D4—1224no—
494C2×D4—1224no—
495C2×D4—1224no—
496C2×D4—1221no—
497C2×D4—1222no—
498C2×D4—1222no—
499C2×D4—2444no—
500C2×D4—2444no—
501C2×D4—2444no—
502C22×C4—24no—
503C22×C4—21no—
504C22×C4—21no—
505C22×C4—22no—
506C22×C4—44no—
507C22×C4—614no—
508C22×C4—611no—
509C22×C4—611no—
510C22×C4—614no—
511C22×C4—614no—
512C22×C4—612no—
513C22×C4—612no—
514C22×C4—612no—
515C22×C4—612no—
516C22×C4—1224no—
517C22×C4—1222no—
518C22×C4—1222no—
519C22×C4—1228no—
520C22×C4—1222no—
521C22×C4—1222no—
522C22×C4—1228no—
523C22×C4—1224no—
524C22×C4—1224no—
525C22×C4—12216no—
526C22×C4—1224no—
527C22×C4—1222no—
528C22×C4—1221no—
529C22×C4—1224no—
530C22×C4—2444no—
531C22×C4—2444no—
532C22×C4#1175822214yes—
533C22×C4D4×D6112yes—
534C22×C4D4×D6111yescharacteristic
535C22×C4#1175822614yes—
536C22×C4C23×D6112yes—
537C22×C4C23×D6111yescharacteristic
538M16—28no—
539M16—1228no—
540C2×C8—24no—
541C2×C8—614no—
542C2×C8—1224no—
543C2×C8C23×D6114yes—
544C4⋊C4—24no—
545C4⋊C4—22no—
546C4⋊C4—21no—
547C4⋊C4—21no—
548C4⋊C4—614no—
549C4⋊C4—1224no—
550C4⋊C4—1224no—
551C4⋊C4—1224no—
552C4⋊C4—1222no—
553C4⋊C4C23×D6114yes—
554C4⋊C4C23×D6112yes—
555C4⋊C4C23×D6112yes—
556C22⋊C4—24no—
557C22⋊C4—1224no—
558C22⋊C4—1224no—
559C22⋊C4—1228no—
560C22⋊C4—1228no—
561C22⋊C4—1224no—
562C22⋊C4—1224no—
563C42—24no—
564C42—1228no—
565C42—1228no—
566C23—22no—
567C23—22no—
568C23—612no—
569C23—612no—
570C23—1228no—
571C23—1222no—
572C23—1222no—
573C23—1222no—
574C23—1222no—
575C23—1222no—
576C23—1222no—
577C23—1222no—
578C23—2444no—
579C23—2444no—
580C23—24416no—
581C23—2442no—
582C23—2442no—
583C23—2442no—
584C23#1451976214yes—
585C23C2×D4×D6112yes—
586C23C2×D4×D6111yescharacteristic
587Q8—1222no—
588Q8—1222no—
589Q8—2442no—
590D4—1222no—
591D4—1222no—
592D4—2448no—
593D4—2448no—
594D4—2448no—
595D4—2442no—
596C2×C4—22no—
597C2×C4—22no—
598C2×C4—24no—
599C2×C4—48no—
600C2×C4—44no—
601C2×C4—42no—
602C2×C4—42no—
603C2×C4—612no—
604C2×C4—612no—
605C2×C4—1228no—
606C2×C4—1222no—
607C2×C4—1222no—
608C2×C4—1228no—
609C2×C4—1228no—
610C2×C4—1224no—
611C2×C4—1224no—
612C2×C4—1222no—
613C2×C4—1222no—
614C2×C4—1222no—
615C2×C4—1222no—
616C2×C4—1224no—
617C2×C4—1222no—
618C2×C4—2448no—
619C2×C4—2442no—
620C2×C4—2444no—
621C2×C4—2448no—
622C2×C4#1451976214yes—
623C2×C4#1451976218yes—
624C2×C4C2×D4×D6112yes—
625C2×C4C2×D4×D6111yescharacteristic
626C2×C4C2×#1175822212yes—
627C2×C4C2×#1175822214yes—
628C2×C4C2×#1175822612yes—
629C2×C4C2×#1175822614yes—
630C2×C4C24×D6111yescharacteristic
631C8—24no—
632C8—1224no—
633C22—22no—
634C22—22no—
635C22—28no—
636C22—1222no—
637C22—1222no—
638C22—1222no—
639C22—1222no—
640C22—1222no—
641C22—1222no—
642C22—4884no—
643C22—24416no—
644C22—2442no—
645C22—2442no—
646C22—2442no—
647C22—2444no—
648C22—2444no—
649C22C2×#1451976214yes—
650C22C2×#1451976212yes—
651C22#1684055812yes—
652C22#1684056112yes—
653C22C22×D4×D6111yescharacteristic
654C4—44no—
655C4—42no—
656C4—42no—
657C4—1222no—
658C4—1222no—
659C4—2444no—
660C4—2442no—
661C4C2×#1451976214yes—
662C4C2×#1451976212yes—
663C4C22×#1175822211yescharacteristic
664C4C22×#1175822611yescharacteristic
665C2—24no—
666C2—4882no—
667C2—2442no—
668C2—2442no—
669C2#2145575314yesdirect product factor
670C2C22×#1451976211yescharacteristic
671C2C2×#1684055811yescharacteristic
672C2C2×#1684056111yescharacteristic
subgroupG/H
centerC23C2×D4×D61
commutator, cosocleC2×C4×C61C25
FrattiniC2×C4C24×D61
FittingC2×C61×#428C2
socleC23×C61C22×D4
prime radicalC2×#16840274C2
hypercenterC2×#428D61
hypocenterC61C2×#3093
LCSG ▷ C2×C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
2. C22×#14519735C2semidirect
3. #21455753C2direct (N × H)
4. C2×#16840106C2semidirect
5. C2×#16840105C2semidirect
6. C2×#16840089C2semidirect
7. C2×#16840088C2semidirect
8. C2×#16839805C2semidirect
9. C2×#16839764C2semidirect
10. C2×#16839761C2semidirect
11. C2×#16839725C2semidirect
12. C2×C61×#428C2semidirect
31. C2×#14519735C22semidirect
35. #16840106C22semidirect
36. #16840105C22semidirect
37. #16840089C22semidirect
38. #16840088C22semidirect
39. #16839805C22semidirect
40. C2×#14519710C22semidirect
41. C2×#14519708C22semidirect
42. #16839764C22semidirect
43. #16839761C22semidirect
44. C2×#14519702C22semidirect
45. C2×#14519700C22semidirect
46. C2×#14519699C22semidirect
47. C2×#14519697C22semidirect
48. #16839725C22semidirect
54. C61×#428C22semidirect
55. C2×#107×C61C22semidirect
56. C2×#106×C61C22semidirect
115. #14519710C23semidirect
116. #14519708C23semidirect
117. #14519702C23semidirect
118. #14519700C23semidirect
119. #14519699C23semidirect
120. #14519697C23semidirect
138. #107×C61C23semidirect
139. #106×C61C23semidirect
337. C2×#428D61semidirect
342. C2×C61#3093semidirect
379. #428C2×D61semidirect
381. C2×#107C2×D61semidirect
383. C2×#106C2×D61semidirect
393. C61C2×#3093semidirect
468. #107C22×D61semidirect
470. #106C22×D61semidirect
669. C2#21455753direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168402742
C22×#145197352
#214557532
C2×#168401062
C2×#168401052
C2×#168400892
C2×#168400882
C2×#168398052
C2×#168397642
C2×#168397612
C2×#168397252
C2×C61×#4282
C2×#309361