Finite Group Databasebeta

← C2×#23773316 #35156039

Subgroups — C2×#23773316 #35156039

Order 20224.

2378 conjugacy classes of subgroups.

The subgroup lattice has 638 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19099076C211yesmaximal, characteristic
2C22×#14579119C211yesmaximal, characteristic, complemented
3#23773316C2116yesmaximal, direct product factor
4C2×#19098881C211yesmaximal, characteristic, complemented
5C2×#19098903C212yesmaximal, complemented
6C2×#19098890C212yesmaximal, complemented
7C2×#19098891C211yesmaximal, characteristic, complemented
8C2×#19098877C211yesmaximal, characteristic, complemented
9C2×#19098600C212yesmaximal, complemented
10C2×#19098558C212yesmaximal, complemented
11C2×C22⋊C8×D79C211yesmaximal, characteristic, complemented
12C2×#484×C79C211yesmaximal, characteristic, complemented
13C22×#11929625C2211yescharacteristic
14C23×C4×D79C2211yescharacteristic
15C22×C8×D79C2212yes—
16#19099076—24no—
17#19099076C2218yes—
18C2×#14579146—21no—
19C2×#14579137C2211yescharacteristic
20C2×#14579136C2211yescharacteristic
21C2×#14579135C2211yescharacteristic
22C2×#14579131—21no—
23C2×#14579123—21no—
24C2×(C22⋊D79)⋊C4—21no—
25C2×#14579119—21no—
26C2×#14579119—21no—
27C2×#14579119—24no—
28C2×#14579119C2212yes—
29C2×#14579119C2212yes—
30C2×#14579119C2211yescharacteristic
31C2×#14579119C2218yescomplemented
32C22×C4⋊Dic79C2211yescharacteristic
33C22×(C2×D79)⋊C4—21no—
34C22×C4×Dic79—21no—
35#19098903C22116yescomplemented
36#19098891C2218yescomplemented
37#19098890C22116yescomplemented
38#19098881C2218yescomplemented
39#19098877C2218yescomplemented
40#19098600C22116yescomplemented
41C2×#14579095C2212yescomplemented
42C2×#14579092C2212yescomplemented
43C2×#14579092C2212yescomplemented
44#19098558C22116yescomplemented
45C2×#14579085C2212yescomplemented
46C2×#14579083C2212yescomplemented
47C2×#14579081C2212yescomplemented
48C2×C22⋊(C79⋊C8)C2211yescharacteristic
49C2×(C2×D79)⋊C8C2212yes—
50C22⋊C8×D79C2218yescomplemented
51C2×C22⋊Q8×C79C2211yescharacteristic
52C22×C4⋊C4×C79C2211yescharacteristic
53#484×C79C2218yescomplemented
54C2×#107×C79C2212yescomplemented
55C2×#103×C79C2212yescomplemented
56C2×C22⋊C8×C79C2211yescharacteristic
57C24×D79D411yescharacteristic
58C2×#11929625—22no—
59C2×#11929625—28no—
60C2×#11929625C2314yes—
61C22×D316—21no—
62C22×D316—21no—
63C22×D316C2311yescharacteristic
64C22×C22⋊D79—21no—
65C23×Dic79—21no—
66C23×Dic79D411yescharacteristic
67C22×C4×D79—28no—
68C22×C4×D79—21no—
69C22×C4×D79D411yescharacteristic
70C22×C4×D79D411yescharacteristic
71C22×C4×D79D414yes—
72C22×C4×D79C2311yescharacteristic
73C22×C4×D79C2314yes—
74C22×C4×D79C2311yescharacteristic
75C22×C79⋊C8C2312yes—
76C2×C8×D79—28no—
77C2×C8×D79C2318yes—
78#14579146—28no—
79#14579137—22no—
80#14579137C2314yes—
81#14579136—22no—
82#14579136C2314yes—
83#14579135—22no—
84#14579135C2314yes—
85#14579131—28no—
86#14579123—28no—
87(C22⋊D79)⋊C4—28no—
88#14579119—24no—
89#14579119—24no—
90#14579119—28no—
91#14579119—28no—
92#14579119—22no—
93#14579119C2318yes—
94#14579119C2318yes—
95#14579119C2314yes—
96C2×C4⋊Dic79—21no—
97C2×C4⋊Dic79—22no—
98C2×C4⋊Dic79C2314yes—
99C2×C4⋊Dic79C2312yes—
100C2×C4⋊Dic79C2311yescharacteristic
101C2×Dic79⋊C4—21no—
102C2×(C2×D79)⋊C4—21no—
103C2×(C2×D79)⋊C4—22no—
104C2×(C2×D79)⋊C4—22no—
105C2×(C2×D79)⋊C4—28no—
106C2×(C2×D79)⋊C4—21no—
107C2×(C2×D79)⋊C4—21no—
108C2×(C2×D79)⋊C4—21no—
109C2×C4×Dic79—22no—
110C2×C4×Dic79—22no—
111C2×C4×Dic79—28no—
112C2×C4×Dic79—21no—
113#14579095C2318yescomplemented
114#14579092C2318yescomplemented
115#14579092C2318yescomplemented
116#14579085C2318yescomplemented
117#14579083C2318yescomplemented
118#14579081C2318yescomplemented
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×C79C2314yes—
131C2×C4⋊C4×C79C2311yescharacteristic
132C2×C4⋊C4×C79C2312yes—
133C2×C22⋊C4×C79—21no—
134#107×C79C2318yescomplemented
135#103×C79C2318yescomplemented
136C22⋊C8×C79C2314yes—
137C23×D79—24no—
138C23×D79—24no—
139C23×D79—21no—
140C23×D79—42no—
141C23×D79C2×D411yescharacteristic
142C23×D79C2×D414yes—
143C23×D79C2×D411yescharacteristic
144#11929625—216no—
145#11929625—48no—
146C2×D316—24no—
147C2×D316—24no—
148C2×D316—21no—
149C2×D316—24no—
150C2×D316—44no—
151C2×D316—44no—
152C2×D316C2412yes—
153C2×C22⋊D79—24no—
154C2×C22⋊D79—44no—
155C22×Dic79—24no—
156C22×Dic79—24no—
157C22×Dic79—22no—
158C22×Dic79—21no—
159C22×Dic79—44no—
160C22×Dic79C2×D411yescharacteristic
161C22×Dic79C2×D414yes—
162C22×Dic79C2×D411yescharacteristic
163C2×C4×D79—24no—
164C2×C4×D79—216no—
165C2×C4×D79—216no—
166C2×C4×D79—24no—
167C2×C4×D79—24no—
168C2×C4×D79—24no—
169C2×C4×D79—44no—
170C2×C4×D79C2×D414yes—
171C2×C4×D79C2×D412yes—
172C2×C4×D79C2×D414yes—
173C2×C4×D79C2×D412yes—
174C2×C4×D79#4118yes—
175C2×C4×D79C2412yes—
176C2×C4×D79C2412yes—
177C2×C79⋊C8—24no—
178C2×C79⋊C8C2414yes—
179C8×D79—216no—
180C4⋊Dic79—24no—
181C4⋊Dic79—22no—
182C4⋊Dic79—21no—
183C4⋊Dic79C2414yes—
184C4⋊Dic79C2412yes—
185Dic79⋊C4—24no—
186(C2×D79)⋊C4—24no—
187(C2×D79)⋊C4—28no—
188(C2×D79)⋊C4—28no—
189(C2×D79)⋊C4—24no—
190(C2×D79)⋊C4—24no—
191(C2×D79)⋊C4—24no—
192C4×Dic79—28no—
193C4×Dic79—28no—
194C4×Dic79—24no—
195C24×C79C2×D411yescharacteristic
196C2×Q8×C79—21no—
197C2×Q8×C79—24no—
198C2×Q8×C79C2412yes—
199C22×C4×C79—24no—
200C22×C4×C79—21no—
201C22×C4×C79—21no—
202C22×C4×C79—24no—
203C22×C4×C79—22no—
204C22×C4×C79—44no—
205C22×C4×C79C2×D411yescharacteristic
206C22×C4×C79C2×D412yes—
207C22×C4×C79C2412yes—
208C22×C4×C79C2411yescharacteristic
209C2×C8×C79—24no—
210C2×C8×C79C2414yes—
211C4⋊C4×C79—24no—
212C4⋊C4×C79—22no—
213C4⋊C4×C79—24no—
214C4⋊C4×C79—21no—
215C4⋊C4×C79C2414yes—
216C4⋊C4×C79C2412yes—
217C22⋊C4×C79—24no—
218C23×C79—22no—
219C23×C79—22no—
220C23×C79#13514yes—
221C23×C79C22×D412yes—
222C23×C79C22×D411yescharacteristic
223C22×D79—22no—
224C22×D79—22no—
225C22×D79—22no—
226C22×D79—28no—
227C22×D79—22no—
228C22×D79—22no—
229C22×D79—28no—
230C22×D79—22no—
231C22×D79—44no—
232C22×D79—416no—
233C22×D79—42no—
234C22×D79—42no—
235C22×D79—42no—
236C22×D79#12314yes—
237C22×D79C22×D412yes—
238C22×D79C22×D412yes—
239Q8×C79—24no—
240Q8×C79—42no—
241C2×C4×C79—22no—
242C2×C4×C79—24no—
243C2×C4×C79—24no—
244C2×C4×C79—24no—
245C2×C4×C79—22no—
246C2×C4×C79—22no—
247C2×C4×C79—48no—
248C2×C4×C79—42no—
249C2×C4×C79—42no—
250C2×C4×C79—44no—
251C2×C4×C79C22×D412yes—
252C2×C4×C79C22×D411yescharacteristic
253C2×C4×C79C2×#4114yes—
254C2×C4×C79C2511yescharacteristic
255C22⋊D79—48no—
256C2×Dic79—22no—
257C2×Dic79—28no—
258C2×Dic79—24no—
259C2×Dic79—22no—
260C2×Dic79—48no—
261C2×Dic79—44no—
262C2×Dic79—42no—
263C2×Dic79#12314yes—
264C2×Dic79C22×D412yes—
265C2×Dic79C22×D412yes—
266D316—24no—
267D316—48no—
268D316—48no—
269D316—42no—
270C4×D79—28no—
271C4×D79—24no—
272C4×D79—24no—
273C4×D79—28no—
274C4×D79—48no—
275C4×D79#12318yes—
276C4×D79C2×#4118yes—
277C8×C79—24no—
278C79⋊C8—24no—
279C22×C79—28no—
280C22×C79—22no—
281C22×C79—22no—
282C22×C79C2×#12312yes—
283C22×C79C2×#13514yes—
284C22×C79C2×#13512yes—
285C22×C79#57512yes—
286C22×C79C23×D411yescharacteristic
287C2×D79—28no—
288C2×D79—22no—
289C2×D79—22no—
290C2×D79—22no—
291C2×D79—22no—
292C2×D79—22no—
293C2×D79—416no—
294C2×D79—44no—
295C2×D79—42no—
296C2×D79—42no—
297C2×D79—42no—
298C2×D79—84no—
299C2×D79C2×#12314yes—
300C4×C79—22no—
301C4×C79—42no—
302C4×C79—44no—
303C4×C79—42no—
304C4×C79C2×#12312yes—
305C4×C79C22×#4112yes—
306Dic79—22no—
307Dic79—42no—
308Dic79—44no—
309Dic79C2×#12314yes—
310C2×#3091—791nomaximal, Sylow
311C2×C79—24no—
312C2×C79#309114yescomplemented
313C2×C79C22×#12311yescharacteristic
314C2×C79C22×#13511yescharacteristic
315C2×C79C2×#57511yescharacteristic
316D79—24no—
317D79—42no—
318D79—82no—
319C2×#525—791no—
320C22×#117—791no—
321#3091—7916no—
322C2×#484D7911yescharacteristic, complemented
323C2×#481—791no—
324C2×#485—792no—
325C2×#482—792no—
326C2×#483—791no—
327C2×#480—791no—
328C2×#427—792no—
329C2×#416—792no—
330C22×C22⋊C8—791no—
331C79C2×#309111yescharacteristic, complemented, Sylow
332C22×#41—791no—
333C24×C4—791no—
334C23×C8—792no—
335#525—798no—
336#525—1584no—
337C2×#123—1581no—
338C2×C22⋊Q8—791no—
339C2×C22⋊Q8C2×D7911yescharacteristic
340C2×#121—791no—
341C2×#121—791no—
342C2×#120—1581no—
343C2×C4×D4—1581no—
344C2×C4×D4—1581no—
345C2×#117—792no—
346C2×#117—798no—
347C2×#117—792no—
348C2×#117—791no—
349C2×#117—1581no—
350C2×#117—1581no—
351C2×#117—1584no—
352C22×C4⋊C4—791no—
353C22×C4⋊C4C2×D7911yescharacteristic
354C22×C22⋊C4—1581no—
355C22×C42—1581no—
356#485—7916no—
357#484C2×D7918yescomplemented
358#483—798no—
359#482—7916no—
360#481—798no—
361#480—798no—
362#427—7916no—
363C2×#107—792no—
364C2×#107C2×D7912yescomplemented
365C2×#106—792no—
366C2×#106—792no—
367#416—7916no—
368C2×#103—792no—
369C2×#103C2×D7912yescomplemented
370C2×#102—792no—
371C2×#102—792no—
372C2×C22⋊C8—791no—
373C2×C22⋊C8—792no—
374C2×C22⋊C8—798no—
375C2×C22⋊C8C2×D7911yescharacteristic
376C25—791no—
377C2×#41—794no—
378C2×#41—1588no—
379C2×#41—1582no—
380C22×Q8C22×D7911yescharacteristic
381C22×D4—791no—
382C22×D4—1581no—
383C22×D4—1581no—
384C22×D4—1581no—
385C23×C4—21no—
386C23×C4—791no—
387C23×C4—791no—
388C23×C4—791no—
389C23×C4—791no—
390C23×C4—794no—
391C23×C4—791no—
392C23×C4—794no—
393C23×C4—1581no—
394C23×C4—1588no—
395C23×C4—1581no—
396C23×C4C22×D7911yescharacteristic
397C22×C8—792no—
398C22×C8—798no—
399C22×C8—1588no—
400C22×C8C22×D7912yes—
401#123—1588no—
402C22⋊Q8—22no—
403C22⋊Q8—794no—
404C22⋊Q8—1582no—
405C22⋊Q8C22×D7914yes—
406#121—794no—
407#121—794no—
408#121—1582no—
409#121—1582no—
410#120—1588no—
411C4×D4—1588no—
412C4×D4—1588no—
413#117—798no—
414#117—798no—
415#117—794no—
416#117—1584no—
417#117—1588no—
418#117—1584no—
419#117—1588no—
420#117—1582no—
421C2×C4⋊C4—22no—
422C2×C4⋊C4—21no—
423C2×C4⋊C4—21no—
424C2×C4⋊C4—792no—
425C2×C4⋊C4—794no—
426C2×C4⋊C4—791no—
427C2×C4⋊C4—1581no—
428C2×C4⋊C4—1582no—
429C2×C4⋊C4—1581no—
430C2×C4⋊C4C22×D7912yes—
431C2×C4⋊C4C22×D7914yes—
432C2×C4⋊C4C22×D7911yescharacteristic
433C2×C22⋊C4—21no—
434C2×C22⋊C4—1581no—
435C2×C22⋊C4—1588no—
436C2×C22⋊C4—1582no—
437C2×C22⋊C4—1582no—
438C2×C22⋊C4—1581no—
439C2×C22⋊C4—1581no—
440C2×C22⋊C4—1581no—
441C2×C42—1588no—
442C2×C42—1582no—
443C2×C42—1582no—
444C2×C42—1581no—
445#107—798no—
446#107C22×D7918yescomplemented
447#106—798no—
448#106—798no—
449#103—798no—
450#103C22×D7918yescomplemented
451#102—798no—
452#102—798no—
453C22⋊C8—794no—
454C22⋊C8—798no—
455C22⋊C8C22×D7914yes—
456C24—3162no—
457C24—794no—
458C24—791no—
459C24—791no—
460C24—1584no—
461C24—1581no—
462C24—1584no—
463C24D4×D7911yescharacteristic
464#41—3168no—
465#41—15816no—
466C2×Q8—21no—
467C2×Q8—24no—
468C2×Q8C23×D7912yes—
469C2×D4—3164no—
470C2×D4—3164no—
471C2×D4—3164no—
472C2×D4—792no—
473C2×D4—1584no—
474C2×D4—1584no—
475C2×D4—1584no—
476C2×D4—1584no—
477C2×D4—1581no—
478C22×C4—24no—
479C22×C4—24no—
480C22×C4—22no—
481C22×C4—21no—
482C22×C4—21no—
483C22×C4—44no—
484C22×C4—3164no—
485C22×C4—3164no—
486C22×C4—791no—
487C22×C4—798no—
488C22×C4—792no—
489C22×C4—792no—
490C22×C4—794no—
491C22×C4—791no—
492C22×C4—792no—
493C22×C4—794no—
494C22×C4—792no—
495C22×C4—794no—
496C22×C4—1584no—
497C22×C4—1584no—
498C22×C4—15816no—
499C22×C4—15816no—
500C22×C4—1584no—
501C22×C4—1584no—
502C22×C4—1582no—
503C22×C4—1581no—
504C22×C4—1584no—
505C22×C4—1584no—
506C22×C4D4×D7912yes—
507C22×C4D4×D7911yescharacteristic
508C22×C4C23×D7912yes—
509C22×C4C23×D7911yescharacteristic
510C2×C8—24no—
511C2×C8—794no—
512C2×C8—15816no—
513C2×C8—1584no—
514C2×C8C23×D7914yes—
515C4⋊C4—24no—
516C4⋊C4—22no—
517C4⋊C4—24no—
518C4⋊C4—21no—
519C4⋊C4—794no—
520C4⋊C4—792no—
521C4⋊C4—1584no—
522C4⋊C4—1584no—
523C4⋊C4—1582no—
524C4⋊C4—1581no—
525C4⋊C4C23×D7914yes—
526C4⋊C4C23×D7912yes—
527C22⋊C4—24no—
528C22⋊C4—1584no—
529C22⋊C4—1588no—
530C22⋊C4—1588no—
531C22⋊C4—1584no—
532C22⋊C4—1584no—
533C22⋊C4—1584no—
534C42—1588no—
535C42—1588no—
536C42—1584no—
537C23—22no—
538C23—22no—
539C23—31616no—
540C23—3164no—
541C23—3162no—
542C23—3162no—
543C23—3162no—
544C23—794no—
545C23—792no—
546C23—792no—
547C23—1588no—
548C23—1582no—
549C23—1582no—
550C23—1588no—
551C23—1582no—
552C23—1582no—
553C23—1582no—
554C23—1582no—
555C23#1457918414yes—
556C23C2×D4×D7912yes—
557C23C2×D4×D7911yescharacteristic
558Q8—24no—
559Q8—42no—
560D4—3168no—
561D4—3168no—
562D4—3168no—
563D4—3162no—
564D4—1584no—
565C2×C4—22no—
566C2×C4—24no—
567C2×C4—24no—
568C2×C4—24no—
569C2×C4—22no—
570C2×C4—22no—
571C2×C4—48no—
572C2×C4—42no—
573C2×C4—44no—
574C2×C4—42no—
575C2×C4—3168no—
576C2×C4—3168no—
577C2×C4—3164no—
578C2×C4—3162no—
579C2×C4—794no—
580C2×C4—792no—
581C2×C4—798no—
582C2×C4—798no—
583C2×C4—792no—
584C2×C4—1588no—
585C2×C4—1588no—
586C2×C4—1584no—
587C2×C4—1582no—
588C2×C4—1588no—
589C2×C4—1582no—
590C2×C4—1584no—
591C2×C4—1584no—
592C2×C4C2×D4×D7912yes—
593C2×C4C2×D4×D7911yescharacteristic
594C2×C4#41×D7914yes—
595C2×C4C24×D7911yescharacteristic
596C8—24no—
597C8—1584no—
598C22—22no—
599C22—22no—
600C22—28no—
601C22—31616no—
602C22—3164no—
603C22—3162no—
604C22—3162no—
605C22—3162no—
606C22—794no—
607C22—6324no—
608C22—1582no—
609C22—1582no—
610C22—1582no—
611C22—1582no—
612C22—1582no—
613C22—1588no—
614C22#123×D7912yes—
615C22C2×#1457918414yes—
616C22C2×#1457918412yes—
617C22#1909935912yes—
618C22C22×D4×D7911yescharacteristic
619C4—22no—
620C4—44no—
621C4—42no—
622C4—42no—
623C4—3164no—
624C4—3162no—
625C4—794no—
626C4—1582no—
627C4#123×D7912yes—
628C4C2×#41×D7912yes—
629C2—24no—
630C2—3162no—
631C2—6322no—
632C2—1584no—
633C2#2377331614yesdirect product factor
634C2C2×#123×D7911yescharacteristic
635C2C22×#1457918411yescharacteristic
636C2C2×#1909935911yescharacteristic
subgroupG/H
centerC23C2×D4×D79
commutator, cosocleC2×C4×C79C25
FrattiniC2×C4C24×D79
FittingC2×#484×C79C2
socleC23×C79C22×D4
prime radicalC2×#19099076C2
hypercenterC2×#484D79
hypocenterC79C2×#3091
LCSG ▷ C2×C4×C79 ▷ C2×C79 ▷ C79
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#484

Complements

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

NH ≅ G/Ntype
2. C22×#14579119C2semidirect
3. #23773316C2direct (N × H)
4. C2×#19098881C2semidirect
5. C2×#19098903C2semidirect
6. C2×#19098890C2semidirect
7. C2×#19098891C2semidirect
8. C2×#19098877C2semidirect
9. C2×#19098600C2semidirect
10. C2×#19098558C2semidirect
11. C2×C22⋊C8×D79C2semidirect
12. C2×#484×C79C2semidirect
31. C2×#14579119C22semidirect
35. #19098903C22semidirect
36. #19098891C22semidirect
37. #19098890C22semidirect
38. #19098881C22semidirect
39. #19098877C22semidirect
40. #19098600C22semidirect
41. C2×#14579095C22semidirect
42. C2×#14579092C22semidirect
43. C2×#14579092C22semidirect
44. #19098558C22semidirect
45. C2×#14579085C22semidirect
46. C2×#14579083C22semidirect
47. C2×#14579081C22semidirect
50. C22⋊C8×D79C22semidirect
53. #484×C79C22semidirect
54. C2×#107×C79C22semidirect
55. C2×#103×C79C22semidirect
113. #14579095C23semidirect
114. #14579092C23semidirect
115. #14579092C23semidirect
116. #14579085C23semidirect
117. #14579083C23semidirect
118. #14579081C23semidirect
134. #107×C79C23semidirect
135. #103×C79C23semidirect
312. C2×C79#3091semidirect
322. C2×#484D79semidirect
331. C79C2×#3091semidirect
357. #484C2×D79semidirect
364. C2×#107C2×D79semidirect
369. C2×#103C2×D79semidirect
446. #107C22×D79semidirect
450. #103C22×D79semidirect
633. C2#23773316direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#190990762
C22×#145791192
#237733162
C2×#190988812
C2×#190989032
C2×#190988902
C2×#190988912
C2×#190988772
C2×#190986002
C2×#190985582
C2×C22⋊C8×D792
C2×#484×C792
C2×#309179