Finite Group Databasebeta

← C2×#22610162 #33086844

Subgroups — C2×#22610162 #33086844

Order 18688.

1478 conjugacy classes of subgroups.

The subgroup lattice has 594 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989070C211yesmaximal, characteristic
2C2×#17988627C211yesmaximal, characteristic, complemented
3C22×#14556518C211yesmaximal, characteristic
4C2×#17988452C211yesmaximal, characteristic, complemented
5#22610162C218yesmaximal, direct product factor
6C2×#17988355C211yesmaximal, characteristic, complemented
7C2×#17988348C211yesmaximal, characteristic, complemented
8C2×#417×C73C211yesmaximal, characteristic
9C22×#11924012C2211yescharacteristic
10#17989070—22no—
11#17989070—22no—
12#17989070—22no—
13#17989070C2214yes—
14C2×Q8×Dic73—21no—
15C2×D4×Dic73—22no—
16C2×D4×Dic73—21no—
17C2×#14556550—21no—
18C2×#14556550C2211yescharacteristic
19C22×C4⋊Dic73—21no—
20C22×C4⋊Dic73C2211yescharacteristic
21C22×C4×Dic73C2211yescharacteristic
22#17988627—22no—
23#17988627C2214yescomplemented
24C2×#14556524—21no—
25C2×#14556522—21no—
26C2×#14556518C418yescomplemented
27C2×#14556518C2214yes—
28#17988452—22no—
29#17988452C2214yescomplemented
30C2×#14556498—22no—
31#17988355C2214yescomplemented
32#17988348C2214yescomplemented
33C22×#41×C73C2211yescharacteristic
34C22×M16×C73C2211yescharacteristic
35C22×C4⋊C4×C73C2211yescharacteristic
36#417×C73—22no—
37#417×C73C2214yes—
38C2×#103×C73—21no—
39C2×#102×C73—21no—
40C23×Dic73—22no—
41C23×Dic73—21no—
42C23×Dic73D412yes—
43C2×#11924012—21no—
44C2×#11924012—22no—
45C2×#11924012C2×C414yescomplemented
46C2×#11924012C2312yes—
47C22×#10596831—21no—
48Q8×Dic73—24no—
49Q8×Dic73—24no—
50D4×Dic73—28no—
51D4×Dic73—24no—
52D4×Dic73—28no—
53D4×Dic73—24no—
54#14556550—24no—
55#14556550—24no—
56#14556550—21no—
57#14556550—21no—
58#14556550D412yes—
59#14556550C2312yes—
60C2×C4⋊Dic73—22no—
61C2×C4⋊Dic73—24no—
62C2×C4⋊Dic73—24no—
63C2×C4⋊Dic73—21no—
64C2×C4⋊Dic73—21no—
65C2×C4⋊Dic73—21no—
66C2×C4⋊Dic73—21no—
67C2×C4⋊Dic73D412yes—
68C2×C4⋊Dic73D411yescharacteristic
69C2×C4⋊Dic73D411yescharacteristic
70C2×C4⋊Dic73C2312yes—
71C2×C22⋊Dic73—22no—
72C2×C22⋊Dic73—21no—
73C2×C4×Dic73—22no—
74C2×C4×Dic73—21no—
75C2×C4×Dic73—21no—
76C2×C4×Dic73—22no—
77C2×C4×Dic73C2×C414yescomplemented
78C2×C4×Dic73C2312yes—
79#14556524—24no—
80#14556522—24no—
81#14556518C2×C4116yescomplemented
82#14556498—28no—
83C2×#41×C73—21no—
84C2×#41×C73—21no—
85C2×#41×C73—22no—
86C2×#41×C73—22no—
87C2×#41×C73C2×C414yes—
88C2×#41×C73D412yes—
89C2×#41×C73Q814yes—
90C2×#41×C73C2312yes—
91C22×Q8×C73D411yescharacteristic
92C22×D4×C73—21no—
93C22×D4×C73D411yescharacteristic
94C23×C4×C73—21no—
95C23×C4×C73—21no—
96C23×C4×C73C2311yescharacteristic
97C2×M16×C73—21no—
98C2×M16×C73—22no—
99C2×M16×C73—22no—
100C2×M16×C73C2312yes—
101C22×C8×C73—21no—
102C2×C4⋊C4×C73—21no—
103C2×C4⋊C4×C73—21no—
104C2×C4⋊C4×C73—21no—
105C2×C4⋊C4×C73—21no—
106C2×C4⋊C4×C73—21no—
107C2×C4⋊C4×C73C2312yes—
108#103×C73—24no—
109#102×C73—24no—
110C22×Dic73—24no—
111C22×Dic73—21no—
112C22×Dic73—22no—
113C22×Dic73—21no—
114C22×Dic73—22no—
115C22×Dic73—28no—
116C22×Dic73—22no—
117C22×Dic73—42no—
118C22×Dic73—42no—
119C22×Dic73—44no—
120C22×Dic73—84no—
121C22×Dic73C2×D414yes—
122#11924012—24no—
123#11924012—22no—
124#11924012C22×C414yescomplemented
125C2×#10596831—24no—
126C2×#10596831—22no—
127C4⋊Dic73—28no—
128C4⋊Dic73—22no—
129C4⋊Dic73—21no—
130C4⋊Dic73—22no—
131C4⋊Dic73—21no—
132C4⋊Dic73—22no—
133C4⋊Dic73—22no—
134C4⋊Dic73C2×D412yes—
135C4⋊Dic73C2×D412yes—
136C22⋊Dic73—28no—
137C22⋊Dic73—22no—
138C22⋊Dic73—22no—
139C4×Dic73—28no—
140C4×Dic73—24no—
141C4×Dic73—22no—
142C4×Dic73—22no—
143C4×Dic73—22no—
144C4×Dic73C22×C414yescomplemented
145C24×C73—21no—
146C24×C73C2×D411yescharacteristic
147#41×C73—24no—
148#41×C73—24no—
149#41×C73—24no—
150#41×C73—24no—
151#41×C73—24no—
152#41×C73—22no—
153#41×C73—22no—
154#41×C73C4⋊C418yes—
155#41×C73C22×C414yes—
156#41×C73C2×Q814yes—
157C2×Q8×C73—21no—
158C2×Q8×C73—22no—
159C2×Q8×C73C4⋊C414yes—
160C2×Q8×C73C2×D412yes—
161C2×D4×C73—24no—
162C2×D4×C73—22no—
163C2×D4×C73—24no—
164C2×D4×C73—22no—
165C2×D4×C73—21no—
166C2×D4×C73—22no—
167C2×D4×C73C4⋊C414yes—
168C2×D4×C73C2×D412yes—
169C22×C4×C73—21no—
170C22×C4×C73—24no—
171C22×C4×C73—22no—
172C22×C4×C73—24no—
173C22×C4×C73—22no—
174C22×C4×C73—22no—
175C22×C4×C73—44no—
176C22×C4×C73—41no—
177C22×C4×C73—41no—
178C22×C4×C73C22×C412yes—
179C22×C4×C73C2×D411yescharacteristic
180C22×C4×C73C2×D411yescharacteristic
181C22×C4×C73C2×D411yescharacteristic
182C22×C4×C73C2×D411yescharacteristic
183C22×C4×C73C2×D411yescharacteristic
184C22×C4×C73C2×Q812yes—
185C22×C4×C73#4112yes—
186C22×C4×C73#4112yes—
187C22×C4×C73C2411yescharacteristic
188M16×C73—22no—
189M16×C73—22no—
190M16×C73—42no—
191C2×C8×C73—22no—
192C2×C8×C73—42no—
193C4⋊C4×C73—22no—
194C4⋊C4×C73—22no—
195C4⋊C4×C73—41no—
196C4⋊C4×C73—41no—
197C23×C73—22no—
198C23×C73—22no—
199C23×C73—22no—
200C23×C73—24no—
201C23×C73—42no—
202C23×C73—42no—
203C23×C73C4×D412yes—
204C23×C73#12011yescharacteristic
205C23×C73#12011yescharacteristic
206C23×C73C22⋊Q812yes—
207C23×C73C22×D411yescharacteristic
208Q8×C73—22no—
209Q8×C73—24no—
210Q8×C73C2×C4⋊C414yes—
211D4×C73—24no—
212D4×C73—24no—
213D4×C73—28no—
214D4×C73—22no—
215D4×C73—24no—
216D4×C73C2×C4⋊C414yes—
217C2×C4×C73—22no—
218C2×C4×C73—24no—
219C2×C4×C73—24no—
220C2×C4×C73—24no—
221C2×C4×C73—28no—
222C2×C4×C73—44no—
223C2×C4×C73—42no—
224C2×C4×C73—82no—
225C2×C4×C73—82no—
226C2×C4×C73C2×C4⋊C412yes—
227C2×C4×C73C2×C4⋊C412yes—
228C2×C4×C73C4×D412yes—
229C2×C4×C73C4×D412yes—
230C2×C4×C73C4×D412yes—
231C2×C4×C73#12011yescharacteristic
232C2×C4×C73#12011yescharacteristic
233C2×C4×C73C22⋊Q812yes—
234C2×C4×C73C22⋊Q812yes—
235C2×C4×C73C22⋊Q812yes—
236C2×C4×C73#12312yes—
237C2×C4×C73#12312yes—
238C2×C4×C73C23×C411yescharacteristic
239C2×C4×C73C22×D411yescharacteristic
240C2×C4×C73C22×D411yescharacteristic
241C2×C4×C73C22×Q811yescharacteristic
242C2×C4×C73C2×#4111yescharacteristic
243C2×C4×C73C2×#4111yescharacteristic
244C2×Dic73—24no—
245C2×Dic73—22no—
246C2×Dic73—24no—
247C2×Dic73—22no—
248C2×Dic73—48no—
249C2×Dic73—42no—
250C2×Dic73—42no—
251C2×Dic73—42no—
252C2×Dic73—44no—
253C2×Dic73—42no—
254C2×Dic73—44no—
255C2×Dic73—82no—
256C2×Dic73—88no—
257C8×C73—42no—
258#10596831—24no—
259C22×C73—24no—
260C22×C73—24no—
261C22×C73—21no—
262C22×C73—21no—
263C22×C73—21no—
264C22×C73—21no—
265C22×C73—24no—
266C22×C73—48no—
267C22×C73—42no—
268C22×C73—42no—
269C22×C73#38512yes—
270C22×C73#38512yes—
271C22×C73#45412yes—
272C22×C73#45512yes—
273C22×C73C2×C4×D411yescharacteristic
274C22×C73C2×#12011yescharacteristic
275C22×C73C2×C22⋊Q811yescharacteristic
276C4×C73—24no—
277C4×C73—82no—
278C4×C73#38512yes—
279C4×C73#38512yes—
280C4×C73C22×C4⋊C411yescharacteristic
281C4×C73C2×C4×D411yescharacteristic
282C4×C73C2×C22⋊Q811yescharacteristic
283C4×C73C2×#12311yescharacteristic
284Dic73—44no—
285Dic73—44no—
286Dic73—42no—
287Dic73—42no—
288C2×#1860—731nomaximal, Sylow
289C2×C73—21no—
290C2×C73—21no—
291C2×C73—22no—
292C2×C73—44no—
293C2×C73#186014yescomplemented
294C2×C73C2×#38511yescharacteristic
295C2×C73C2×#45511yescharacteristic
296C2×C73C2×#45411yescharacteristic
297C2×#520—731no—
298C2×#428—731no—
299C22×#104—731no—
300C2×#417D7311yescharacteristic
301C2×#382—731no—
302#1860—738no—
303C2×#343—731no—
304C2×#339—731no—
305C73C2×#186011yescharacteristic, complemented, Sylow
306C22×#41C2×D7311yescharacteristic
307C22×M16—731no—
308C22×M16C2×D7311yescharacteristic
309#520—734no—
310#520—1462no—
311#520—1462no—
312#520—1462no—
313C2×C4×Q8—1461no—
314C2×C4×D4—1462no—
315C2×C4×D4—1461no—
316C2×#117—731no—
317C2×#117—1461no—
318C22×C4⋊C4—731no—
319C22×C4⋊C4—1461no—
320C22×C4⋊C4C2×D7311yescharacteristic
321C22×C42—731no—
322#428—734no—
323#428—1462no—
324C2×#107—1461no—
325C2×#106—1461no—
326C2×#104—738no—
327C2×#104—734no—
328#417—22no—
329#417C2×D7314yes—
330C2×#103—21no—
331C2×#102—21no—
332#382—734no—
333#382—1462no—
334C2×#95—1462no—
335#343—734no—
336#339—734no—
337C2×#41—21no—
338C2×#41—21no—
339C2×#41—22no—
340C2×#41—22no—
341C2×#41Dic14614yes—
342C2×#41C4×D7314yes—
343C2×#41C22⋊D7312yes—
344C2×#41C22×D7312yes—
345C22×Q8C22⋊D7311yescharacteristic
346C22×D4—21no—
347C22×D4C22⋊D7311yescharacteristic
348C23×C4—21no—
349C23×C4—21no—
350C23×C4—732no—
351C23×C4—1461no—
352C23×C4—1462no—
353C23×C4C22×D7311yescharacteristic
354C2×M16—21no—
355C2×M16—22no—
356C2×M16—22no—
357C2×M16—734no—
358C2×M16—732no—
359C2×M16—1461no—
360C2×M16—1462no—
361C2×M16C22×D7312yes—
362C22×C8—21no—
363C22×C8—1461no—
364C4×Q8—1464no—
365C4×Q8—1464no—
366C4×D4—1468no—
367C4×D4—1468no—
368C4×D4—1464no—
369C4×D4—1464no—
370#117—732no—
371#117—732no—
372#117—1464no—
373#117—1464no—
374#117—1461no—
375#117—1461no—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—731no—
382C2×C4⋊C4—731no—
383C2×C4⋊C4—732no—
384C2×C4⋊C4—732no—
385C2×C4⋊C4—1464no—
386C2×C4⋊C4—1462no—
387C2×C4⋊C4—1464no—
388C2×C4⋊C4—1461no—
389C2×C4⋊C4—1461no—
390C2×C4⋊C4—1461no—
391C2×C4⋊C4—1461no—
392C2×C4⋊C4C22×D7312yes—
393C2×C22⋊C4—1462no—
394C2×C22⋊C4—1461no—
395C2×C42—734no—
396C2×C42—732no—
397C2×C42—1462no—
398C2×C42—1461no—
399C2×C42—1461no—
400C2×C42—1462no—
401#107—1464no—
402#106—1464no—
403#104—7316no—
404#103—24no—
405#102—24no—
406#95—1468no—
407C24—21no—
408C24D4×D7311yescharacteristic
409#41—24no—
410#41—24no—
411#41—24no—
412#41—24no—
413#41—24no—
414#41—22no—
415#41—22no—
416#41#1192400718yes—
417#41C2×C4×D7314yes—
418#41C2×Dic14614yes—
419C2×Q8—21no—
420C2×Q8—22no—
421C2×Q8#1192400714yes—
422C2×Q8C2×C22⋊D7312yes—
423C2×D4—24no—
424C2×D4—22no—
425C2×D4—24no—
426C2×D4—22no—
427C2×D4—21no—
428C2×D4—22no—
429C2×D4#1192400714yes—
430C2×D4C2×C22⋊D7312yes—
431C22×C4—21no—
432C22×C4—24no—
433C22×C4—22no—
434C22×C4—24no—
435C22×C4—22no—
436C22×C4—22no—
437C22×C4—44no—
438C22×C4—41no—
439C22×C4—41no—
440C22×C4—2922no—
441C22×C4—2922no—
442C22×C4—2924no—
443C22×C4—5844no—
444C22×C4—734no—
445C22×C4—1464no—
446C22×C4—1461no—
447C22×C4—1461no—
448C22×C4—1462no—
449C22×C4—1462no—
450C22×C4—1468no—
451C22×C4—1462no—
452C22×C4C2×C4×D7312yes—
453C22×C4D4×D7311yescharacteristic
454C22×C4D4×D7311yescharacteristic
455C22×C4D4×D7311yescharacteristic
456C22×C4C2×C22⋊D7311yescharacteristic
457C22×C4C2×C22⋊D7311yescharacteristic
458C22×C4C2×Dic14612yes—
459C22×C4#1192402712yes—
460C22×C4#1192402712yes—
461C22×C4C23×D7311yescharacteristic
462M16—22no—
463M16—22no—
464M16—42no—
465M16—734no—
466M16—1464no—
467M16—1462no—
468C2×C8—22no—
469C2×C8—42no—
470C2×C8—1464no—
471C2×C8—1462no—
472C4⋊C4—22no—
473C4⋊C4—22no—
474C4⋊C4—41no—
475C4⋊C4—41no—
476C4⋊C4—732no—
477C4⋊C4—732no—
478C4⋊C4—1468no—
479C4⋊C4—1461no—
480C4⋊C4—1462no—
481C4⋊C4—1462no—
482C4⋊C4—1461no—
483C4⋊C4—1462no—
484C4⋊C4—1462no—
485C22⋊C4—1468no—
486C22⋊C4—1462no—
487C22⋊C4—1462no—
488C42—734no—
489C42—1468no—
490C42—1464no—
491C42—1462no—
492C42—1462no—
493C42—1462no—
494C23—22no—
495C23—22no—
496C23—24no—
497C23—22no—
498C23—42no—
499C23—42no—
500C23#1455655112yes—
501C23#1455655911yescharacteristic
502C23#1455655911yescharacteristic
503C23#1455656812yes—
504C23C2×D4×D7311yescharacteristic
505Q8—24no—
506Q8—22no—
507Q8C2×#1192400714yes—
508D4—28no—
509D4—24no—
510D4—24no—
511D4—24no—
512D4—22no—
513D4C2×#1192400714yes—
514C2×C4—24no—
515C2×C4—22no—
516C2×C4—28no—
517C2×C4—24no—
518C2×C4—24no—
519C2×C4—44no—
520C2×C4—42no—
521C2×C4—82no—
522C2×C4—82no—
523C2×C4—2922no—
524C2×C4—2922no—
525C2×C4—2928no—
526C2×C4—2922no—
527C2×C4—2924no—
528C2×C4—2922no—
529C2×C4—2924no—
530C2×C4—5842no—
531C2×C4—5848no—
532C2×C4—1464no—
533C2×C4—1462no—
534C2×C4—1462no—
535C2×C4—1464no—
536C2×C4C2×#1192400712yes—
537C2×C4C2×#1192400712yes—
538C2×C4#1455655112yes—
539C2×C4#1455655112yes—
540C2×C4#1455655112yes—
541C2×C4#1455655911yescharacteristic
542C2×C4#1455655911yescharacteristic
543C2×C4#1455656812yes—
544C2×C4#1455656812yes—
545C2×C4#1455656812yes—
546C2×C4#1455657212yes—
547C2×C4#1455657212yes—
548C2×C4C22×C4×D7311yescharacteristic
549C2×C4C2×D4×D7311yescharacteristic
550C2×C4C22×C22⋊D7311yescharacteristic
551C2×C4C22×Dic14611yescharacteristic
552C2×C4C2×#1192402711yescharacteristic
553C2×C4C2×#1192402711yescharacteristic
554C8—42no—
555C8—1464no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—24no—
561C22—24no—
562C22—24no—
563C22—42no—
564C22—42no—
565C22—48no—
566C22#1798845912yes—
567C22#1798845912yes—
568C22#1798875812yes—
569C22#1798876312yes—
570C22C2×#1455655111yescharacteristic
571C22C2×#1455655911yescharacteristic
572C22C2×#1455656811yescharacteristic
573C4—24no—
574C4—82no—
575C4—2924no—
576C4—2924no—
577C4—2922no—
578C4—2922no—
579C4#1798845912yes—
580C4#1798845912yes—
581C4C22×#1192400711yescharacteristic
582C4C2×#1455655111yescharacteristic
583C4C2×#1455656811yescharacteristic
584C4C2×#1455657211yescharacteristic
585C2—21no—
586C2—22no—
587C2—21no—
588C2—44no—
589C2#2261016214yesdirect product factor
590C2C2×#1798845911yescharacteristic
591C2C2×#1798876311yescharacteristic
592C2C2×#1798875811yescharacteristic
subgroupG/H
centerC23#14556559
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC2×#417×C73C2
socleC23×C73#120
prime radicalC22×#41×C73C22
cosocleC22×C4×C73C24
hypercenterC2×#417D73
hypocenterC73C2×#1860
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#417

Complements

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

NH ≅ G/Ntype
2. C2×#17988627C2semidirect
4. C2×#17988452C2semidirect
5. #22610162C2direct (N × H)
6. C2×#17988355C2semidirect
7. C2×#17988348C2semidirect
23. #17988627C22semidirect
26. C2×#14556518C4semidirect
29. #17988452C22semidirect
31. #17988355C22semidirect
32. #17988348C22semidirect
45. C2×#11924012C2×C4semidirect
77. C2×C4×Dic73C2×C4semidirect
81. #14556518C2×C4semidirect
124. #11924012C22×C4semidirect
144. C4×Dic73C22×C4semidirect
293. C2×C73#1860semidirect
305. C73C2×#1860semidirect
589. C2#22610162direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179890702
C2×#179886272
C22×#145565182
C2×#179884522
#226101622
C2×#179883552
C2×#179883482
C2×#417×C732
C2×#186073