Finite Group Databasebeta

← C2×#22610161 #33086843

Subgroups — C2×#22610161 #33086843

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×#17989067C211yesmaximal, characteristic
2C22×#14556517C211yesmaximal, characteristic
3C2×#17988586C211yesmaximal, characteristic
4C2×#17988452C211yesmaximal, characteristic, complemented
5#22610161C218yesmaximal, direct product factor
6C2×#17988355C211yesmaximal, characteristic, complemented
7C2×#17988346C211yesmaximal, characteristic, complemented
8C2×#428×C73C211yesmaximal, characteristic, complemented
9C22×#11924028C2211yescharacteristic
10C22×#11924012C2211yescharacteristic
11#17989067—22no—
12#17989067—22no—
13#17989067—22no—
14#17989067C2214yes—
15C2×C4×Dic146—21no—
16C2×C4×D292—21no—
17C2×#14556551—22no—
18C2×#14556549—21no—
19C22×C4⋊Dic73C2211yescharacteristic
20C2×C4⋊C4×D73—21no—
21C22×C4×Dic73C2211yescharacteristic
22C2×#14556517C418yescomplemented
23C2×#14556517C2214yes—
24#17988586—22no—
25#17988586C2214yes—
26C2×#14556514—21no—
27C2×#14556511—21no—
28#17988452—22no—
29#17988452C2214yescomplemented
30C2×#14556498—22no—
31#17988355C2214yescomplemented
32#17988346C2214yescomplemented
33C22×M16×C73C2211yescharacteristic
34C2×#117×C73C2211yescharacteristic
35C22×C4⋊C4×C73C2211yescharacteristic
36#428×C73—22no—
37#428×C73C2214yescomplemented
38C2×#107×C73—21no—
39C2×#106×C73—21no—
40C2×#11924028—21no—
41C2×#11924028—21no—
42C2×#11924028—22no—
43C2×#11924028—22no—
44C2×#11924028C2×C414yes—
45C2×#11924028D412yes—
46C2×#11924028Q814yes—
47C2×#11924028C2312yes—
48C22×Dic146D411yescharacteristic
49C22×D292D411yescharacteristic
50C22×C22⋊D73—21no—
51C23×Dic73—21no—
52C23×Dic73D412yes—
53C22×C4×D73—22no—
54C22×C4×D73—21no—
55C2×#11924012—21no—
56C2×#11924012—22no—
57C2×#11924012—22no—
58C2×#11924012C2312yes—
59C22×#10596831—21no—
60C4×Dic146—24no—
61C4×Dic146—24no—
62C4×D292—24no—
63C4×D292—24no—
64#14556551—28no—
65#14556551—28no—
66#14556549—24no—
67#14556549—24no—
68C2×C4⋊Dic73—21no—
69C2×C4⋊Dic73—21no—
70C2×C4⋊Dic73—21no—
71C2×C4⋊Dic73—21no—
72C2×C4⋊Dic73D412yes—
73C2×C4⋊Dic73C2312yes—
74C2×#11924007—22no—
75C4⋊C4×D73—24no—
76C4⋊C4×D73—24no—
77C2×#11924005—22no—
78C2×C4×Dic73—22no—
79C2×C4×Dic73—21no—
80C2×C4×Dic73—22no—
81C2×C4×Dic73C2×C414yescomplemented
82C2×C4×Dic73C2312yes—
83#14556517C2×C4116yescomplemented
84#14556514—24no—
85#14556511—24no—
86#14556498—28no—
87C23×C4×C73—21no—
88C23×C4×C73C2311yescharacteristic
89C2×M16×C73—21no—
90C2×M16×C73—22no—
91C2×M16×C73C2×C414yescomplemented
92C2×M16×C73C2312yes—
93C22×C8×C73—21no—
94#117×C73—21no—
95#117×C73—21no—
96#117×C73D412yes—
97#117×C73C2312yes—
98C2×C4⋊C4×C73—21no—
99C2×C4⋊C4×C73—21no—
100C2×C4⋊C4×C73—21no—
101C2×C4⋊C4×C73—21no—
102C2×C4⋊C4×C73—21no—
103C2×C4⋊C4×C73D411yescharacteristic
104C2×C4⋊C4×C73D411yescharacteristic
105C2×C4⋊C4×C73C2312yes—
106C2×C22⋊C4×C73—21no—
107C2×C42×C73—21no—
108#107×C73—24no—
109#106×C73—24no—
110C23×D73—21no—
111#11924028—24no—
112#11924028—24no—
113#11924028—24no—
114#11924028—24no—
115#11924028—24no—
116#11924028—22no—
117#11924028—22no—
118#11924028C4⋊C418yes—
119#11924028C22×C414yes—
120#11924028C2×Q814yes—
121C2×Dic146—21no—
122C2×Dic146—22no—
123C2×Dic146C4⋊C414yes—
124C2×Dic146C2×D412yes—
125C2×D292—21no—
126C2×D292—22no—
127C2×D292C4⋊C414yes—
128C2×D292C2×D412yes—
129C2×C22⋊D73—24no—
130C2×C22⋊D73—22no—
131C2×C22⋊D73—24no—
132C2×C22⋊D73—22no—
133C22×Dic73—21no—
134C22×Dic73—24no—
135C22×Dic73—22no—
136C22×Dic73—22no—
137C22×Dic73—22no—
138C22×Dic73—42no—
139C22×Dic73—42no—
140C22×Dic73—44no—
141C22×Dic73C2×D414yes—
142C2×C4×D73—24no—
143C2×C4×D73—22no—
144C2×C4×D73—24no—
145C2×C4×D73—22no—
146C2×C4×D73—28no—
147C2×C4×D73—84no—
148#11924012—22no—
149#11924012—22no—
150#11924012—42no—
151C2×#10596831—22no—
152C2×#10596831—42no—
153C4⋊Dic73—22no—
154C4⋊Dic73—22no—
155C4⋊Dic73—22no—
156C4⋊Dic73—22no—
157#11924007—28no—
158#11924005—28no—
159C4×Dic73—28no—
160C4×Dic73—24no—
161C4×Dic73—22no—
162C4×Dic73C22×C414yescomplemented
163C24×C73C2×D411yescharacteristic
164C22×C4×C73—21no—
165C22×C4×C73—21no—
166C22×C4×C73—22no—
167C22×C4×C73—44no—
168C22×C4×C73—41no—
169C22×C4×C73—41no—
170C22×C4×C73C22×C412yes—
171C22×C4×C73C2×D411yescharacteristic
172C22×C4×C73C2×D411yescharacteristic
173C22×C4×C73C2×D411yescharacteristic
174C22×C4×C73C2×D411yescharacteristic
175C22×C4×C73C2×D411yescharacteristic
176C22×C4×C73C2×Q812yes—
177C22×C4×C73#4112yes—
178C22×C4×C73#4112yes—
179C22×C4×C73C2411yescharacteristic
180M16×C73—24no—
181M16×C73—22no—
182M16×C73C22×C414yescomplemented
183C2×C8×C73—24no—
184C2×C8×C73—22no—
185C4⋊C4×C73—21no—
186C4⋊C4×C73—21no—
187C4⋊C4×C73—22no—
188C4⋊C4×C73—22no—
189C4⋊C4×C73—41no—
190C4⋊C4×C73—41no—
191C4⋊C4×C73C2×D412yes—
192C4⋊C4×C73C2×D412yes—
193C22⋊C4×C73—22no—
194C22⋊C4×C73—22no—
195C42×C73—22no—
196C42×C73—22no—
197C23×C73—22no—
198C23×C73—22no—
199C23×C73C4×D412yes—
200C23×C73#12011yescharacteristic
201C23×C73#12011yescharacteristic
202C23×C73C22⋊Q812yes—
203C23×C73C22×D411yescharacteristic
204C22×D73—22no—
205C22×D73—24no—
206C22×D73—42no—
207C22×D73—42no—
208C2×C4×C73—22no—
209C2×C4×C73—22no—
210C2×C4×C73—42no—
211C2×C4×C73—42no—
212C2×C4×C73—44no—
213C2×C4×C73—42no—
214C2×C4×C73—82no—
215C2×C4×C73—82no—
216C2×C4×C73C2×C4⋊C412yes—
217C2×C4×C73C2×C4⋊C412yes—
218C2×C4×C73C4×D412yes—
219C2×C4×C73C4×D412yes—
220C2×C4×C73C4×D412yes—
221C2×C4×C73#12011yescharacteristic
222C2×C4×C73#12011yescharacteristic
223C2×C4×C73C22⋊Q812yes—
224C2×C4×C73C22⋊Q812yes—
225C2×C4×C73C22⋊Q812yes—
226C2×C4×C73#12312yes—
227C2×C4×C73#12312yes—
228C2×C4×C73C23×C411yescharacteristic
229C2×C4×C73C22×D411yescharacteristic
230C2×C4×C73C22×D411yescharacteristic
231C2×C4×C73C22×Q811yescharacteristic
232C2×C4×C73C2×#4111yescharacteristic
233C2×C4×C73C2×#4111yescharacteristic
234C22⋊D73—24no—
235C22⋊D73—24no—
236C22⋊D73—28no—
237C2×Dic73—22no—
238C2×Dic73—24no—
239C2×Dic73—24no—
240C2×Dic73—24no—
241C2×Dic73—48no—
242C2×Dic73—42no—
243C2×Dic73—42no—
244C2×Dic73—44no—
245C2×Dic73—44no—
246C2×Dic73—82no—
247D292—22no—
248D292—24no—
249D292C2×C4⋊C414yes—
250C4×D73—24no—
251C4×D73—24no—
252C4×D73—28no—
253C4×D73—88no—
254Dic146—22no—
255Dic146—24no—
256Dic146C2×C4⋊C414yes—
257C8×C73—24no—
258#10596831—42no—
259C22×C73—24no—
260C22×C73—24no—
261C22×C73—21no—
262C22×C73—21no—
263C22×C73—21no—
264C22×C73—21no—
265C22×C73#38512yes—
266C22×C73#38512yes—
267C22×C73#45412yes—
268C22×C73#45512yes—
269C22×C73C2×C4×D411yescharacteristic
270C22×C73C2×#12011yescharacteristic
271C22×C73C2×C22⋊Q811yescharacteristic
272C2×D73—24no—
273C2×D73—48no—
274C2×D73—42no—
275C2×D73—42no—
276C4×C73—42no—
277C4×C73—42no—
278C4×C73—82no—
279C4×C73#38512yes—
280C4×C73#38512yes—
281C4×C73C22×C4⋊C411yescharacteristic
282C4×C73C2×C4×D411yescharacteristic
283C4×C73C2×C22⋊Q811yescharacteristic
284C4×C73C2×#12311yescharacteristic
285Dic73—24no—
286Dic73—44no—
287Dic73—44no—
288C2×#1860—731nomaximal, Sylow
289C2×C73—21no—
290C2×C73—21no—
291C2×C73—22no—
292C2×C73#186014yescomplemented
293C2×C73C2×#38511yescharacteristic
294C2×C73C2×#45511yescharacteristic
295C2×C73C2×#45411yescharacteristic
296D73—44no—
297C2×#520—731no—
298C2×#428D7311yescharacteristic, complemented
299C22×#104—731no—
300C2×#417—731no—
301C2×#382—731no—
302#1860—738no—
303C2×#343—731no—
304C2×#339—731no—
305C73C2×#186011yescharacteristic, complemented, Sylow
306C22×#41—731no—
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—1461no—
315C2×C4×D4—1462no—
316C2×#117—1461no—
317C2×#117C2×D7311yescharacteristic
318C22×C4⋊C4—731no—
319C22×C4⋊C4—1461no—
320C22×C4⋊C4C2×D7311yescharacteristic
321C22×C42—731no—
322#428—22no—
323#428C2×D7314yescomplemented
324C2×#107—21no—
325C2×#106—21no—
326C2×#104—738no—
327C2×#104—734no—
328#417—734no—
329#417—1462no—
330C2×#103—1461no—
331C2×#102—1461no—
332#382—734no—
333#382—1462no—
334C2×#95—1462no—
335#343—734no—
336#339—734no—
337C2×#41—734no—
338C2×#41—732no—
339C2×#41—734no—
340C2×#41—732no—
341C2×#41—1461no—
342C2×#41—1461no—
343C2×#41—1462no—
344C2×#41—1462no—
345C22×Q8—731no—
346C22×D4—731no—
347C22×D4—1461no—
348C23×C4—21no—
349C23×C4—732no—
350C23×C4—1462no—
351C23×C4—1461no—
352C23×C4—1461no—
353C23×C4C22×D7311yescharacteristic
354C2×M16—21no—
355C2×M16—22no—
356C2×M16—732no—
357C2×M16—1462no—
358C2×M16—1462no—
359C2×M16—1461no—
360C2×M16C4×D7314yescomplemented
361C2×M16C22×D7312yes—
362C22×C8—21no—
363C22×C8—1461no—
364C4×Q8—1464no—
365C4×Q8—1464no—
366C4×D4—1464no—
367C4×D4—1464no—
368C4×D4—1468no—
369C4×D4—1468no—
370#117—21no—
371#117—21no—
372#117—1464no—
373#117—1464no—
374#117C22⋊D7312yes—
375#117C22×D7312yes—
376C2×C4⋊C4—21no—
377C2×C4⋊C4—21no—
378C2×C4⋊C4—21no—
379C2×C4⋊C4—21no—
380C2×C4⋊C4—21no—
381C2×C4⋊C4—732no—
382C2×C4⋊C4—732no—
383C2×C4⋊C4—1464no—
384C2×C4⋊C4—1462no—
385C2×C4⋊C4—1461no—
386C2×C4⋊C4—1461no—
387C2×C4⋊C4—1461no—
388C2×C4⋊C4—1461no—
389C2×C4⋊C4—1464no—
390C2×C4⋊C4C22⋊D7311yescharacteristic
391C2×C4⋊C4C22⋊D7311yescharacteristic
392C2×C4⋊C4C22×D7312yes—
393C2×C22⋊C4—21no—
394C2×C22⋊C4—1462no—
395C2×C42—21no—
396C2×C42—734no—
397C2×C42—732no—
398C2×C42—1462no—
399C2×C42—1461no—
400C2×C42—1462no—
401#107—24no—
402#106—24no—
403#104—7316no—
404#103—1464no—
405#102—1464no—
406#95—1468no—
407C24—1461no—
408C24D4×D7311yescharacteristic
409#41—738no—
410#41—734no—
411#41—734no—
412#41—1464no—
413#41—1464no—
414#41—1464no—
415#41—1464no—
416#41—1464no—
417#41—1462no—
418#41—1462no—
419C2×Q8—734no—
420C2×Q8—732no—
421C2×Q8—1461no—
422C2×Q8—1462no—
423C2×D4—734no—
424C2×D4—732no—
425C2×D4—1464no—
426C2×D4—1462no—
427C2×D4—1464no—
428C2×D4—1462no—
429C2×D4—1461no—
430C2×D4—1462no—
431C22×C4—21no—
432C22×C4—21no—
433C22×C4—22no—
434C22×C4—44no—
435C22×C4—41no—
436C22×C4—41no—
437C22×C4—2922no—
438C22×C4—2922no—
439C22×C4—2924no—
440C22×C4—5844no—
441C22×C4—734no—
442C22×C4—1468no—
443C22×C4—1464no—
444C22×C4—1462no—
445C22×C4—1462no—
446C22×C4—1462no—
447C22×C4—1461no—
448C22×C4—1464no—
449C22×C4—1462no—
450C22×C4—1464no—
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×C4Q8×D7312yes—
459C22×C4#1192402812yes—
460C22×C4#1192402912yes—
461C22×C4C23×D7311yescharacteristic
462M16—24no—
463M16—22no—
464M16—2922no—
465M16—1462no—
466M16—1462no—
467M16C2×C4×D7314yescomplemented
468C2×C8—24no—
469C2×C8—22no—
470C2×C8—2922no—
471C2×C8—1462no—
472C4⋊C4—21no—
473C4⋊C4—21no—
474C4⋊C4—22no—
475C4⋊C4—22no—
476C4⋊C4—41no—
477C4⋊C4—41no—
478C4⋊C4—1468no—
479C4⋊C4—1462no—
480C4⋊C4—1462no—
481C4⋊C4—1462no—
482C4⋊C4—1462no—
483C4⋊C4C2×C22⋊D7312yes—
484C4⋊C4C2×C22⋊D7312yes—
485C22⋊C4—22no—
486C22⋊C4—22no—
487C22⋊C4—1468no—
488C42—22no—
489C42—22no—
490C42—734no—
491C42—1468no—
492C42—1464no—
493C42—1462no—
494C23—22no—
495C23—22no—
496C23—2922no—
497C23—2922no—
498C23—1464no—
499C23—1462no—
500C23#1455655312yes—
501C23#1455655911yescharacteristic
502C23#1455655911yescharacteristic
503C23#1455656912yes—
504C23C2×D4×D7311yescharacteristic
505Q8—734no—
506Q8—1464no—
507Q8—1462no—
508D4—734no—
509D4—1468no—
510D4—1464no—
511D4—1464no—
512D4—1464no—
513D4—1462no—
514C2×C4—22no—
515C2×C4—22no—
516C2×C4—42no—
517C2×C4—42no—
518C2×C4—44no—
519C2×C4—42no—
520C2×C4—82no—
521C2×C4—82no—
522C2×C4—2922no—
523C2×C4—2922no—
524C2×C4—2928no—
525C2×C4—2924no—
526C2×C4—2924no—
527C2×C4—5848no—
528C2×C4—5842no—
529C2×C4—1464no—
530C2×C4—1464no—
531C2×C4—1464no—
532C2×C4—1462no—
533C2×C4—1468no—
534C2×C4—1464no—
535C2×C4—1464no—
536C2×C4C4⋊C4×D7312yes—
537C2×C4C4⋊C4×D7312yes—
538C2×C4C4×C22⋊D7312yes—
539C2×C4C4×C22⋊D7312yes—
540C2×C4#1455655312yes—
541C2×C4#1455655911yescharacteristic
542C2×C4#1455655911yescharacteristic
543C2×C4#1455656712yes—
544C2×C4#1455656712yes—
545C2×C4#1455656912yes—
546C2×C4#1455657612yes—
547C2×C4#1455657612yes—
548C2×C4C22×C4×D7311yescharacteristic
549C2×C4C2×D4×D7311yescharacteristic
550C2×C4C22×C22⋊D7311yescharacteristic
551C2×C4C2×Q8×D7311yescharacteristic
552C2×C4C2×#1192402811yescharacteristic
553C2×C4C2×#1192402911yescharacteristic
554C8—24no—
555C8—2922no—
556C22—21no—
557C22—21no—
558C22—21no—
559C22—21no—
560C22—24no—
561C22—24no—
562C22—2922no—
563C22—2922no—
564C22—2928no—
565C22—1464no—
566C22#1798845812yes—
567C22#1798845812yes—
568C22#1798875812yes—
569C22#1798876312yes—
570C22C2×#1455655311yescharacteristic
571C22C2×#1455655911yescharacteristic
572C22C2×#1455656911yescharacteristic
573C4—42no—
574C4—42no—
575C4—82no—
576C4—2924no—
577C4—2924no—
578C4—1464no—
579C4#1798845812yes—
580C4#1798845812yes—
581C4C2×C4⋊C4×D7311yescharacteristic
582C4C2×C4×C22⋊D7311yescharacteristic
583C4C2×#1455656711yescharacteristic
584C4C2×#1455657611yescharacteristic
585C2—21no—
586C2—22no—
587C2—21no—
588C2—2924no—
589C2#2261016114yesdirect product factor
590C2C2×#1798845811yescharacteristic
591C2C2×#1798876311yescharacteristic
592C2C2×#1798875811yescharacteristic
subgroupG/H
centerC23#14556559
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC2×#428×C73C2
socleC23×C73#120
prime radicalC22×#11924028C22
cosocleC22×C4×C73C24
hypercenterC2×#428D73
hypocenterC73C2×#1860
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#428

Complements

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

NH ≅ G/Ntype
4. C2×#17988452C2semidirect
5. #22610161C2direct (N × H)
6. C2×#17988355C2semidirect
7. C2×#17988346C2semidirect
8. C2×#428×C73C2semidirect
22. C2×#14556517C4semidirect
29. #17988452C22semidirect
31. #17988355C22semidirect
32. #17988346C22semidirect
37. #428×C73C22semidirect
81. C2×C4×Dic73C2×C4semidirect
83. #14556517C2×C4semidirect
91. C2×M16×C73C2×C4semidirect
162. C4×Dic73C22×C4semidirect
182. M16×C73C22×C4semidirect
292. C2×C73#1860semidirect
298. C2×#428D73semidirect
305. C73C2×#1860semidirect
323. #428C2×D73semidirect
360. C2×M16C4×D73semidirect
467. M16C2×C4×D73semidirect
589. C2#22610161direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#179890672
C22×#145565172
C2×#179885862
C2×#179884522
#226101612
C2×#179883552
C2×#179883462
C2×#428×C732
C2×#186073