Finite Group Databasebeta

← C2×#23773343 #35156070

Subgroups — C2×#23773343 #35156070

Order 20224.

2326 conjugacy classes of subgroups.

The subgroup lattice has 622 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C22⋊Q8×D79C211yesmaximal, characteristic
2C22×#14579119C211yesmaximal, characteristic
3#23773343C2116yesmaximal, direct product factor
4C2×#19098899C211yesmaximal, characteristic, complemented
5C2×#19098896C211yesmaximal, characteristic, complemented
6C2×#19098907C212yesmaximal, complemented
7C2×#19098903C212yesmaximal, complemented
8C2×#19098893C211yesmaximal, characteristic, complemented
9C2×#19098602C212yesmaximal, complemented
10C2×#19098570C212yesmaximal, complemented
11C2×#19098525C211yesmaximal, characteristic
12C2×#483×C79C211yesmaximal, characteristic, complemented
13C22×Q8×D79C2211yescharacteristic
14C23×C4×D79C2211yescharacteristic
15C22×#11929607C2212yes—
16C2×#14579141C2211yescharacteristic
17C2×#14579140C2211yescharacteristic
18C2×#14579139—21no—
19C2×#14579138—21no—
20C2×#14579137C2211yescharacteristic
21C22⋊Q8×D79—24no—
22C22⋊Q8×D79C2218yes—
23C2×#14579119—21no—
24C2×#14579119—24no—
25C2×#14579119—21no—
26C2×#14579119C2212yes—
27C2×#14579119C2212yes—
28C2×#14579119C2218yes—
29C22×C4⋊Dic79C2211yescharacteristic
30C2×C4⋊C4×D79—21no—
31C2×C4⋊C4×D79C2211yescharacteristic
32C22×(C2×D79)⋊C4—21no—
33C2×C22⋊C4×D79—21no—
34C22×C4×Dic79—21no—
35#19098907C22116yescomplemented
36#19098903C22116yescomplemented
37#19098899C2218yescomplemented
38#19098896C2218yescomplemented
39#19098893C2218yescomplemented
40#19098602C22116yescomplemented
41C2×#14579095C2212yescomplemented
42C2×#14579094C2212yescomplemented
43C2×#14579092C2212yescomplemented
44#19098570C22116yescomplemented
45C2×#14579086C2212yescomplemented
46C2×#14579085C2212yescomplemented
47C2×#14579084C2212yescomplemented
48#19098525C2218yes—
49C2×C22⋊(C79⋊C8)C2211yescharacteristic
50C2×(C2×D79)⋊C8C2212yes—
51C2×C22⋊Q8×C79C2211yescharacteristic
52C22×C4⋊C4×C79C2211yescharacteristic
53#483×C79C2218yescomplemented
54C2×#106×C79C2212yescomplemented
55C2×#103×C79C2212yescomplemented
56C2×C22⋊C8×C79C2211yescharacteristic
57C24×D79D411yescharacteristic
58C22×Dic158—21no—
59C22×Dic158C2311yescharacteristic
60C2×Q8×D79—22no—
61C2×Q8×D79—28no—
62C2×Q8×D79C2314yes—
63C23×Dic79—21no—
64C23×Dic79D411yescharacteristic
65C22×C4×D79—28no—
66C22×C4×D79—21no—
67C22×C4×D79—21no—
68C22×C4×D79D411yescharacteristic
69C22×C4×D79D414yes—
70C22×C4×D79D411yescharacteristic
71C22×C4×D79C2311yescharacteristic
72C22×C4×D79C2314yes—
73C22×C4×D79C2311yescharacteristic
74C2×#11929607—28no—
75C2×#11929607C2318yes—
76C22×C79⋊C8C2312yes—
77#14579141—22no—
78#14579141C2314yes—
79#14579140—22no—
80#14579140C2314yes—
81#14579139—28no—
82#14579138—28no—
83#14579137—22no—
84#14579137C2314yes—
85#14579119—28no—
86#14579119—24no—
87#14579119—28no—
88#14579119—24no—
89#14579119C2318yes—
90#14579119C2318yes—
91C2×C4⋊Dic79—21no—
92C2×C4⋊Dic79—22no—
93C2×C4⋊Dic79—21no—
94C2×C4⋊Dic79C2314yes—
95C2×C4⋊Dic79C2311yescharacteristic
96C2×C4⋊Dic79C2312yes—
97C2×Dic79⋊C4—21no—
98C2×Dic79⋊C4—21no—
99C2×Dic79⋊C4—21no—
100C4⋊C4×D79—28no—
101C4⋊C4×D79—22no—
102C4⋊C4×D79C2314yes—
103C2×C22⋊Dic79—21no—
104C2×(C2×D79)⋊C4—21no—
105C2×(C2×D79)⋊C4—21no—
106C2×(C2×D79)⋊C4—22no—
107C2×(C2×D79)⋊C4—28no—
108C2×(C2×D79)⋊C4—22no—
109C22⋊C4×D79—28no—
110C2×C4×Dic79—22no—
111C2×C4×Dic79—28no—
112C2×C4×Dic79—22no—
113#14579095C2318yescomplemented
114#14579094C2318yescomplemented
115#14579092C2318yescomplemented
116#14579086C2318yescomplemented
117#14579085C2318yescomplemented
118#14579084C2318yescomplemented
119C22⋊(C79⋊C8)C2314yes—
120(C2×D79)⋊C8C2318yes—
121C22×Q8×C79C2311yescharacteristic
122C23×C4×C79—21no—
123C23×C4×C79C2311yescharacteristic
124C22×C8×C79C2312yes—
125C22⋊Q8×C79—22no—
126C22⋊Q8×C79C2314yes—
127C2×C4⋊C4×C79—21no—
128C2×C4⋊C4×C79—22no—
129C2×C4⋊C4×C79—21no—
130C2×C4⋊C4×C79C2311yescharacteristic
131C2×C4⋊C4×C79C2312yes—
132C2×C4⋊C4×C79C2314yes—
133C2×C22⋊C4×C79—21no—
134#106×C79C2318yescomplemented
135#103×C79C2318yescomplemented
136C22⋊C8×C79C2314yes—
137C23×D79—24no—
138C23×D79—24no—
139C23×D79—42no—
140C23×D79C2×D411yescharacteristic
141C23×D79C2×D414yes—
142C23×D79C2×D411yescharacteristic
143C2×Dic158—24no—
144C2×Dic158—21no—
145C2×Dic158—24no—
146C2×Dic158—44no—
147C2×Dic158C2412yes—
148Q8×D79—216no—
149Q8×D79—48no—
150C22×Dic79—24no—
151C22×Dic79—21no—
152C22×Dic79—24no—
153C22×Dic79—22no—
154C22×Dic79—21no—
155C22×Dic79—44no—
156C22×Dic79C2×D411yescharacteristic
157C22×Dic79C2×D414yes—
158C22×Dic79C2×D411yescharacteristic
159C2×C4×D79—24no—
160C2×C4×D79—216no—
161C2×C4×D79—216no—
162C2×C4×D79—24no—
163C2×C4×D79—24no—
164C2×C4×D79—24no—
165C2×C4×D79—24no—
166C2×C4×D79—44no—
167C2×C4×D79—44no—
168C2×C4×D79C2×D414yes—
169C2×C4×D79C2×D412yes—
170C2×C4×D79C2×D414yes—
171C2×C4×D79C2×D412yes—
172C2×C4×D79#4118yes—
173C2×C4×D79C2412yes—
174C2×C4×D79C2412yes—
175#11929607—216no—
176C2×C79⋊C8—24no—
177C2×C79⋊C8C2414yes—
178C4⋊Dic79—24no—
179C4⋊Dic79—21no—
180C4⋊Dic79—24no—
181C4⋊Dic79—22no—
182C4⋊Dic79C2414yes—
183C4⋊Dic79C2412yes—
184Dic79⋊C4—24no—
185Dic79⋊C4—24no—
186Dic79⋊C4—24no—
187C22⋊Dic79—24no—
188(C2×D79)⋊C4—24no—
189(C2×D79)⋊C4—24no—
190(C2×D79)⋊C4—28no—
191(C2×D79)⋊C4—28no—
192C4×Dic79—28no—
193C4×Dic79—28no—
194C24×C79C2×D411yescharacteristic
195C2×Q8×C79—21no—
196C2×Q8×C79—24no—
197C2×Q8×C79C2412yes—
198C22×C4×C79—24no—
199C22×C4×C79—21no—
200C22×C4×C79—21no—
201C22×C4×C79—24no—
202C22×C4×C79—22no—
203C22×C4×C79—44no—
204C22×C4×C79C2×D411yescharacteristic
205C22×C4×C79C2×D412yes—
206C22×C4×C79C2412yes—
207C22×C4×C79C2411yescharacteristic
208C2×C8×C79—24no—
209C2×C8×C79C2414yes—
210C4⋊C4×C79—24no—
211C4⋊C4×C79—22no—
212C4⋊C4×C79—24no—
213C4⋊C4×C79—21no—
214C4⋊C4×C79C2414yes—
215C4⋊C4×C79C2412yes—
216C22⋊C4×C79—24no—
217C23×C79—22no—
218C23×C79—22no—
219C23×C79#13714yes—
220C23×C79C22×D412yes—
221C23×C79C22×D411yescharacteristic
222C22×D79—22no—
223C22×D79—22no—
224C22×D79—22no—
225C22×D79—28no—
226C22×D79—22no—
227C22×D79—22no—
228C22×D79—28no—
229C22×D79—44no—
230C22×D79—416no—
231C22×D79#12314yes—
232C22×D79C22×D412yes—
233C22×D79C22×D412yes—
234Q8×C79—24no—
235Q8×C79—42no—
236C2×C4×C79—22no—
237C2×C4×C79—24no—
238C2×C4×C79—24no—
239C2×C4×C79—24no—
240C2×C4×C79—22no—
241C2×C4×C79—22no—
242C2×C4×C79—42no—
243C2×C4×C79—48no—
244C2×C4×C79—44no—
245C2×C4×C79—42no—
246C2×C4×C79C22×D412yes—
247C2×C4×C79C22×D411yescharacteristic
248C2×C4×C79C2×#4114yes—
249C2×C4×C79C2511yescharacteristic
250C2×Dic79—22no—
251C2×Dic79—28no—
252C2×Dic79—24no—
253C2×Dic79—22no—
254C2×Dic79—22no—
255C2×Dic79—48no—
256C2×Dic79—42no—
257C2×Dic79—42no—
258C2×Dic79—44no—
259C2×Dic79#12314yes—
260C2×Dic79C22×D412yes—
261C2×Dic79C22×D412yes—
262C4×D79—28no—
263C4×D79—24no—
264C4×D79—24no—
265C4×D79—28no—
266C4×D79—48no—
267C4×D79—48no—
268C4×D79#12318yes—
269C4×D79C2×#4118yes—
270Dic158—24no—
271Dic158—42no—
272Dic158—48no—
273C8×C79—24no—
274C79⋊C8—24no—
275C22×C79—28no—
276C22×C79—22no—
277C22×C79—22no—
278C22×C79C2×#12312yes—
279C22×C79C2×#13714yes—
280C22×C79C2×#13712yes—
281C22×C79#57512yes—
282C22×C79C23×D411yescharacteristic
283C2×D79—28no—
284C2×D79—22no—
285C2×D79—22no—
286C2×D79—22no—
287C2×D79—22no—
288C2×D79—22no—
289C2×D79—416no—
290C2×D79—44no—
291C2×D79C2×#12314yes—
292C4×C79—22no—
293C4×C79—42no—
294C4×C79—44no—
295C4×C79—42no—
296C4×C79C2×#12312yes—
297C4×C79C22×#4112yes—
298Dic79—22no—
299Dic79—42no—
300Dic79—42no—
301Dic79—44no—
302Dic79C2×#12314yes—
303C2×#3095—791nomaximal, Sylow
304C2×C79—24no—
305C2×C79#309514yescomplemented
306C2×C79C22×#12311yescharacteristic
307C2×C79C22×#13711yescharacteristic
308C2×C79C2×#57511yescharacteristic
309D79—24no—
310D79—42no—
311C22×C22⋊Q8—791no—
312C22×#117—791no—
313#3095—7916no—
314C2×#484—791no—
315C2×#484—791no—
316C2×#485—792no—
317C2×#485—792no—
318C2×#483—791no—
319C2×#483D7911yescharacteristic, complemented
320C2×#428—792no—
321C2×#419—792no—
322C2×#407—791no—
323C79C2×#309511yescharacteristic, complemented, Sylow
324C23×Q8—791no—
325C24×C4—791no—
326C22×M16—792no—
327C2×C22⋊Q8—791no—
328C2×C22⋊Q8—791no—
329C2×C22⋊Q8—798no—
330C2×C22⋊Q8—791no—
331C2×C22⋊Q8—1581no—
332C2×C22⋊Q8—1581no—
333C2×C22⋊Q8—1584no—
334C2×C22⋊Q8C2×D7911yescharacteristic
335C2×#117—792no—
336C2×#117—798no—
337C2×#117—792no—
338C2×#117—1581no—
339C2×#117—1584no—
340C2×#117—1581no—
341C22×C4⋊C4—791no—
342C22×C4⋊C4—791no—
343C22×C4⋊C4—1581no—
344C22×C4⋊C4C2×D7911yescharacteristic
345C22×C22⋊C4—1581no—
346C22×C22⋊C4—1581no—
347C22×C42—1581no—
348#485—7916no—
349#485—7916no—
350#484—798no—
351#484—798no—
352#483—798no—
353#483C2×D7918yescomplemented
354#428—7916no—
355C2×#107—792no—
356C2×#107—792no—
357C2×#106—792no—
358C2×#106C2×D7912yescomplemented
359#419—7916no—
360C2×#103—792no—
361C2×#103—792no—
362C2×#103—792no—
363C2×#103C2×D7912yescomplemented
364#407—798no—
365C2×C22⋊C8—791no—
366C2×C22⋊C8—792no—
367C2×C22⋊C8C2×D7911yescharacteristic
368C25—791no—
369C22×Q8—794no—
370C22×Q8—791no—
371C22×Q8—1581no—
372C22×Q8—1588no—
373C22×Q8—1582no—
374C22×Q8C22×D7911yescharacteristic
375C23×C4—21no—
376C23×C4—791no—
377C23×C4—791no—
378C23×C4—791no—
379C23×C4—791no—
380C23×C4—794no—
381C23×C4—791no—
382C23×C4—794no—
383C23×C4—1581no—
384C23×C4—1581no—
385C23×C4—1588no—
386C23×C4—1581no—
387C23×C4C22×D7911yescharacteristic
388C2×M16—798no—
389C2×M16—1588no—
390C22×C8—792no—
391C22×C8C22×D7912yes—
392C22⋊Q8—22no—
393C22⋊Q8—794no—
394C22⋊Q8—794no—
395C22⋊Q8—794no—
396C22⋊Q8—1588no—
397C22⋊Q8—1588no—
398C22⋊Q8—1582no—
399C22⋊Q8—1582no—
400C22⋊Q8—1582no—
401C22⋊Q8C22×D7914yes—
402#117—798no—
403#117—798no—
404#117—1584no—
405#117—1588no—
406#117—1584no—
407#117—1588no—
408C2×C4⋊C4—21no—
409C2×C4⋊C4—22no—
410C2×C4⋊C4—21no—
411C2×C4⋊C4—792no—
412C2×C4⋊C4—794no—
413C2×C4⋊C4—791no—
414C2×C4⋊C4—794no—
415C2×C4⋊C4—1582no—
416C2×C4⋊C4—1588no—
417C2×C4⋊C4—1581no—
418C2×C4⋊C4—1581no—
419C2×C4⋊C4—1581no—
420C2×C4⋊C4—1582no—
421C2×C4⋊C4—1581no—
422C2×C4⋊C4—1581no—
423C2×C4⋊C4C22×D7911yescharacteristic
424C2×C4⋊C4C22×D7912yes—
425C2×C4⋊C4C22×D7914yes—
426C2×C22⋊C4—21no—
427C2×C22⋊C4—1588no—
428C2×C22⋊C4—1588no—
429C2×C22⋊C4—1582no—
430C2×C22⋊C4—1582no—
431C2×C22⋊C4—1581no—
432C2×C22⋊C4—1581no—
433C2×C22⋊C4—1581no—
434C2×C42—1588no—
435C2×C42—1582no—
436C2×C42—1582no—
437#107—798no—
438#107—798no—
439#106—798no—
440#106C22×D7918yescomplemented
441#103—798no—
442#103—798no—
443#103—798no—
444#103C22×D7918yescomplemented
445C22⋊C8—794no—
446C22⋊C8—798no—
447C22⋊C8C22×D7914yes—
448C24—3162no—
449C24—794no—
450C24—791no—
451C24—791no—
452C24—1584no—
453C24—1584no—
454C24D4×D7911yescharacteristic
455C2×Q8—21no—
456C2×Q8—24no—
457C2×Q8—3164no—
458C2×Q8—3168no—
459C2×Q8—792no—
460C2×Q8—1584no—
461C2×Q8—15816no—
462C2×Q8—1584no—
463C2×Q8—1581no—
464C2×Q8C23×D7912yes—
465C22×C4—24no—
466C22×C4—24no—
467C22×C4—22no—
468C22×C4—21no—
469C22×C4—21no—
470C22×C4—44no—
471C22×C4—3164no—
472C22×C4—3164no—
473C22×C4—3164no—
474C22×C4—791no—
475C22×C4—798no—
476C22×C4—792no—
477C22×C4—792no—
478C22×C4—794no—
479C22×C4—791no—
480C22×C4—792no—
481C22×C4—794no—
482C22×C4—794no—
483C22×C4—792no—
484C22×C4—1584no—
485C22×C4—1584no—
486C22×C4—1584no—
487C22×C4—15816no—
488C22×C4—15816no—
489C22×C4—1584no—
490C22×C4—1584no—
491C22×C4—1582no—
492C22×C4—1581no—
493C22×C4—1581no—
494C22×C4—1584no—
495C22×C4—1584no—
496C22×C4D4×D7912yes—
497C22×C4D4×D7911yescharacteristic
498C22×C4C23×D7912yes—
499C22×C4C23×D7911yescharacteristic
500M16—15816no—
501C2×C8—24no—
502C2×C8—794no—
503C2×C8—1584no—
504C2×C8C23×D7914yes—
505C4⋊C4—24no—
506C4⋊C4—22no—
507C4⋊C4—24no—
508C4⋊C4—21no—
509C4⋊C4—794no—
510C4⋊C4—792no—
511C4⋊C4—1584no—
512C4⋊C4—1584no—
513C4⋊C4—1584no—
514C4⋊C4—1584no—
515C4⋊C4—1584no—
516C4⋊C4—1582no—
517C4⋊C4—1581no—
518C4⋊C4C23×D7914yes—
519C4⋊C4C23×D7912yes—
520C22⋊C4—24no—
521C22⋊C4—1588no—
522C22⋊C4—1588no—
523C22⋊C4—1584no—
524C22⋊C4—1584no—
525C22⋊C4—1584no—
526C42—1588no—
527C42—1588no—
528C23—22no—
529C23—22no—
530C23—31616no—
531C23—3164no—
532C23—794no—
533C23—792no—
534C23—792no—
535C23—1588no—
536C23—1582no—
537C23—1588no—
538C23—1582no—
539C23—1582no—
540C23—1582no—
541C23—1582no—
542C23#1457919014yes—
543C23C2×D4×D7912yes—
544C23C2×D4×D7911yescharacteristic
545Q8—24no—
546Q8—42no—
547Q8—3168no—
548Q8—3162no—
549Q8—1584no—
550C2×C4—22no—
551C2×C4—24no—
552C2×C4—24no—
553C2×C4—24no—
554C2×C4—22no—
555C2×C4—22no—
556C2×C4—42no—
557C2×C4—48no—
558C2×C4—44no—
559C2×C4—42no—
560C2×C4—3168no—
561C2×C4—3168no—
562C2×C4—3168no—
563C2×C4—3162no—
564C2×C4—3162no—
565C2×C4—3164no—
566C2×C4—794no—
567C2×C4—792no—
568C2×C4—798no—
569C2×C4—798no—
570C2×C4—792no—
571C2×C4—1588no—
572C2×C4—1588no—
573C2×C4—1584no—
574C2×C4—1582no—
575C2×C4—1582no—
576C2×C4—1588no—
577C2×C4—1582no—
578C2×C4—1584no—
579C2×C4—1584no—
580C2×C4C2×D4×D7912yes—
581C2×C4C2×D4×D7911yescharacteristic
582C2×C4#41×D7914yes—
583C2×C4C24×D7911yescharacteristic
584C8—24no—
585C8—1584no—
586C22—22no—
587C22—22no—
588C22—28no—
589C22—31616no—
590C22—3164no—
591C22—794no—
592C22—1582no—
593C22—1582no—
594C22—1582no—
595C22—1582no—
596C22—1582no—
597C22—1588no—
598C22#123×D7912yes—
599C22C2×#1457919014yes—
600C22C2×#1457919012yes—
601C22#1909936212yes—
602C22C22×D4×D7911yescharacteristic
603C4—22no—
604C4—44no—
605C4—42no—
606C4—42no—
607C4—3162no—
608C4—3164no—
609C4—3162no—
610C4—794no—
611C4—1582no—
612C4#123×D7912yes—
613C4C2×#41×D7912yes—
614C2—24no—
615C2—3162no—
616C2—1584no—
617C2#2377334314yesdirect product factor
618C2C2×#123×D7911yescharacteristic
619C2C22×#1457919011yescharacteristic
620C2C2×#1909936211yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutator, cosocleC2×C4×C79C25
FrattiniC2×C4C24×D79
FittingC2×#483×C79C2
socleC23×C79C22×D4
prime radicalC24×D79D4
hypercenterC2×#483D79
hypocenterC79C2×#3095
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#483

Complements

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

NH ≅ G/Ntype
3. #23773343C2direct (N × H)
4. C2×#19098899C2semidirect
5. C2×#19098896C2semidirect
6. C2×#19098907C2semidirect
7. C2×#19098903C2semidirect
8. C2×#19098893C2semidirect
9. C2×#19098602C2semidirect
10. C2×#19098570C2semidirect
12. C2×#483×C79C2semidirect
35. #19098907C22semidirect
36. #19098903C22semidirect
37. #19098899C22semidirect
38. #19098896C22semidirect
39. #19098893C22semidirect
40. #19098602C22semidirect
41. C2×#14579095C22semidirect
42. C2×#14579094C22semidirect
43. C2×#14579092C22semidirect
44. #19098570C22semidirect
45. C2×#14579086C22semidirect
46. C2×#14579085C22semidirect
47. C2×#14579084C22semidirect
53. #483×C79C22semidirect
54. C2×#106×C79C22semidirect
55. C2×#103×C79C22semidirect
113. #14579095C23semidirect
114. #14579094C23semidirect
115. #14579092C23semidirect
116. #14579086C23semidirect
117. #14579085C23semidirect
118. #14579084C23semidirect
134. #106×C79C23semidirect
135. #103×C79C23semidirect
305. C2×C79#3095semidirect
319. C2×#483D79semidirect
323. C79C2×#3095semidirect
353. #483C2×D79semidirect
358. C2×#106C2×D79semidirect
363. C2×#103C2×D79semidirect
440. #106C22×D79semidirect
444. #103C22×D79semidirect
617. C2#23773343direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C22⋊Q8×D792
C22×#145791192
#237733432
C2×#190988992
C2×#190988962
C2×#190989072
C2×#190989032
C2×#190988932
C2×#190986022
C2×#190985702
C2×#190985252
C2×#483×C792
C2×#309579